design of wide beam flooring systems...abstract 3 abstract the thesis addresses the design of...

Design of wide beam flooring systems

A thesis submitted to Imperial College London for the degree of Doctor of

Philosophy (PhD)

Mohammed Tagelsir Mustafa Abdelsalam

B.Sc. (Eng.), D.I.C., M.Sc.

Department of Civil and Environmental Engineering

Imperial College of Science, Technology and Science

London, SW7 2AZ, United Kingdom

July 2017

Declaration

I hereby confirm that this thesis is the result of my own work carried out in the Structures

Section of the Department of Civil and Environmental Engineering at Imperial College

London, and that I give appropriate references and citations whenever I referred to, described,

or quoted any work from others, whether published or unpublished.

The copyright of this thesis rests with the author and is made available under a Creative

Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to copy,

distribute or transmit the thesis on the condition that they attribute it, that they do not use it for

commercial purposes and that they do not alter, transform or build upon it. For any reuse or

redistribution, researchers must make clear to others the licence terms of this work.

Mohammed Abdelsalam

July 2017

Abstract

3

Abstract

The thesis addresses the design of reinforced concrete wide beam solid slabs for which international

design standards like BS8110, EC2 and ACI318 provide no guidance. Currently, the only commonly

available UK guidance on the design of wide beam slabs is provided by The Concrete Centre (TCC).

The TCC design method assumes that support moments, about an axis parallel to the direction of wide

beam span, are uniformly distributed along beams. The TCC approach is questionable since elastic

finite element analysis shows the transverse bending moment distribution to be far from uniform and

sharply peaked near columns. The research was motivated by concern that crack widths could be

excessive in wide beam slabs designed for uniformly distributed transverse moments.

Nonlinear finite element analysis (NLFEA) is used to investigate the influence on structural response

of varying the distribution of transverse flexural reinforcement along wide beams. Uniform and banded

transverse reinforcement arrangements are considered. The banded arrangement is based on the elastic

moment field. The influence of compressive membrane action (CMA) on flexural resistance is

investigated. The thesis also considers the influence of transverse reinforcement distribution, one-way

loading from the beam and slab continuity on punching resistance at internal and edge columns.

Punching resistance is investigated with NLFEA using solid elements as well as EC2 and the critical

shear crack theory as implemented in fib MC2010.

A design method is developed for wide beam slabs on the basis of parametric studies. It is proposed

that transverse reinforcement should be provided in two bands depicted “column band” and “span

band”. The width of the column band is shown to vary linearly with wide beam span. The proposed

procedure for banding transverse reinforcement is shown to effectively reduce steel strains and, hence,

crack widths as well as enhance punching shear resistance.

Acknowledgments

4

Acknowledgements

I would like to express my sincere gratitude to my supervisor Dr Robert Vollum for his expert guidance

and continuous encouragement throughout the research period. I am deeply thankful for his patience

and kindness. It has been a brilliant experience to work closely with him, which I learnt a lot from it

both personally and academically.

My sincere acknowledgement is to the late Professor Shawki Saad for his generous bequest and The

University of Khartoum for granting me this scholarship. This research would not have been possible

without the financial support of the scholarship. Special thanks are due to Ms Fionnuala Donovan, the

General and Postgraduate Administrator, for her assistance since I was in Sudan and continued during

my stay at Imperial. Her efforts to facilitate the life for my family in London are highly appreciated. I

would also like to thank Ms Tina Mikellides for her support.

During my study at Imperial, I had to the opportunity to share the office with wonderful and smart guys.

I take this chance to offer my gratitude to Vasileios, Luis Fernando, Yuan, Oluwole Kunle and Wenru.

Thanks also due to my colleagues in the structures group, Luis, Jean-Paul, Mariana and Abobakr. I am

also indebted to my Sudanese community at Imperial; Khalid Elhaj, Khalid Nur, Omer Hassan, Elsmani

and Abobakr. Special thanks to my friends outside the college, Suhaib, Hytham, Abdalla, Ashraf, Ali,

Samih, Ibrahim, Nabil, Alghazouli and Mustafa and their respective families for their support and love.

Most importantly, I would like to express my deepest love, respect and gratitude to my mother Zainab,

and my father Tagelsir, to whom I owe absolutely everything. Their love, guidance and prayers have

been always with me in whatever I pursue. My thanks are also due to my sisters; Weam, Shyma and

Roua and to my brothers; Musab, Abubakr and Muaz for their constant encouragement. I would like to

pay my tributes to my grandmother, Saida, my grandfather, Mohammed and my uncle, Osman, who

passed away during my study. Their unconditional love and kindnesses will always be remembered.

Last but not the least I wish to thank my wife, Hiba; for her loving and support. I want to thank her

specially for taking the major part of our household responsibilities throughout my study. My thanks

extend also to my adorable sons, Omer, Ahmed and Yahya, may Allah bless them, who are a true

unending inspiration.

Table of Contents

5

Table of Contents

Declaration ........................................................................................................................................ 3

Abstract ............................................................................................................................................. 3

Acknowledgements ........................................................................................................................... 4

Table of Contents .............................................................................................................................. 5

List of Figures ................................................................................................................................. 10

List of Tables .................................................................................................................................. 25

Symbols .......................................................................................................................................... 29

Introduction.................................................................................................................. 35

1.1 Background ...................................................................................................................... 35

1.2 Aims and objectives ......................................................................................................... 36

1.3 Thesis Organization ......................................................................................................... 37

Literature Review – Background on Structural Design methods for RC Slabs in Codes of

Practice 40

2.1 Introduction ..................................................................................................................... 40

2.2 Flexural design for RC slabs spanning in one direction ..................................................... 41

UK Practice (CP 110, CP 114, BS 8110 and EC2) .................................................... 41

USA Practice (ACI 318) ........................................................................................... 42

2.3 Flexural design for RC flat slabs....................................................................................... 43

Direct Design Method .............................................................................................. 44

Equivalent Frame Method (EFM) ............................................................................. 46

Yield Line Method ................................................................................................... 51

Flexural design in accordance with a predetermined field of moments ...................... 54

Table of Contents

6

2.4 Punching shear design ...................................................................................................... 57

Introduction .............................................................................................................. 57

Review of Punching shear design in EC2 .................................................................. 58

Review of Punching shear design in fib MC2010 ...................................................... 62

2.5 Calculation of Deflection in EC2 ...................................................................................... 67

2.6 Flexural Cracking in EC2 ................................................................................................. 70

Background .............................................................................................................. 70

EC2 design procedure ............................................................................................... 71

2.7 Conclusions ..................................................................................................................... 72

Literature Review – Previous Research into Wide Beam Slabs ..................................... 74

3.1 Introduction ..................................................................................................................... 74

3.2 Transverse Distribution of Bending Moments in Wide Beams .......................................... 75

General..................................................................................................................... 75

Research by Paultre and Moisan ............................................................................... 75

Research by Tay ....................................................................................................... 78

Research by Shuraim and Al-Negheimish, 2011 ....................................................... 82

Conclusion ............................................................................................................... 88

3.3 One-Way Shear in Wide Beams ....................................................................................... 88

The Influence of support width ................................................................................. 88

The Influence of Transverse stirrup distribution and configurations........................... 89

Conclusion ............................................................................................................... 90

3.4 Shear Failure Modes and Crack Patterns in Wide Beams .................................................. 90

3.5 Current design procedure recommended by The Concrete Centre for wide beam slabs ...... 93

3.6 Conclusions ..................................................................................................................... 93

Methodology – Nonlinear Finite Element Analyses (NLFEA) ...................................... 95

4.1 Introduction ..................................................................................................................... 95

4.2 General Background ......................................................................................................... 96

4.3 Constitutive Models for Materials .................................................................................... 96

Constitutive Models for concrete .............................................................................. 96

Table of Contents

7

Steel Reinforcement Modelling .............................................................................. 106

4.4 Nonlinear Analysis ......................................................................................................... 108

Solution Methods ................................................................................................... 108

Convergence Criteria .............................................................................................. 110

4.5 Other Aspects................................................................................................................. 110

Load Application .................................................................................................... 110

Eccentric connections ............................................................................................. 112

4.6 Conclusions ................................................................................................................... 113

Validation Studies ...................................................................................................... 115

5.1 Introduction ................................................................................................................... 115

5.2 Validation Studies for DIANA Model ............................................................................ 117

Clark and Speirs (135) ............................................................................................ 117

CMA tests of Lahlouh & Waldron (137) ................................................................. 120

Guandalini et al (136) ............................................................................................. 124

Sagaseta et al.(140) ................................................................................................. 127

5.3 Validation Studies for ATENA Model ............................................................................ 130

Beam Shear Tests by Fang ...................................................................................... 130

Punching Shear Tests on slabs ................................................................................ 139

5.4 Punching Shear Calculation using MC2010 LoA IV ....................................................... 149

Symmetrical punching tests by Regan (78) ............................................................. 149

5.5 Conclusions ................................................................................................................... 155

Flexural Design for wide beam slabs .......................................................................... 157

6.1 Introduction ................................................................................................................... 157

6.2 Case Study ..................................................................................................................... 158

General................................................................................................................... 158

Load arrangements of actions ................................................................................. 161

Design of steel reinforcement ................................................................................. 161

6.3 Analysis of Transverse Moment Distribution .................................................................. 167

Elastic FE modelling .............................................................................................. 170

Table of Contents

8

Results of FE Elastic analysis ................................................................................. 171

Transverse reinforcement distribution ..................................................................... 173

NLFEA Modelling ................................................................................................. 181

Results and Discussions .......................................................................................... 186

6.4 Conclusions ................................................................................................................... 212

Chapter 7 Punching Shear Resistance of Wide Beams Slabs ....................................................... 214

7.1 Introduction ................................................................................................................... 214

7.2 Internal column Connection ........................................................................................... 215

7.2.1 Effect of asymmetrical load introduction on the punching shear resistance .............. 215

7.2.2 ATENA Results vs. MC2010 predictions with rotations according to LoAs II and IV

223

7.2.3 Effect of continuity according to MC2010 LoA IV ................................................. 235

7.3 Edge Column Connection ............................................................................................... 239


7.3.2 ATENA Results vs. MC2010 predictions with rotations according to LoAs II and IV

245

7.3.3 Effect of continuity according to MC2010 LoA IV ................................................. 253

7.4 Conclusions ................................................................................................................... 257

Chapter 8 Parametric Studies and Design Recommendations ...................................................... 259

8.1 Introduction ................................................................................................................... 259

8.2 Parametric Studies for Flexure........................................................................................ 260

8.2.1 General................................................................................................................... 260

8.2.2 Numerical Model .................................................................................................... 260

8.2.3 Results and Discussion ........................................................................................... 263

8.2.4 Development of design procedure for transverse distribution of support moment. ... 268

8.2.5 Uniform Steel Distribution versus Proposed Band Steel Distribution....................... 277

8.3 Parametric Studies on Punching shear strength of internal wide beam connection without

shear reinforcement ................................................................................................................... 289

8.3.1 General................................................................................................................... 289

8.3.2 Analytical Model .................................................................................................... 289

Table of Contents

9

8.3.3 Results of Load-deflection curves and Discussions ................................................. 293

8.3.4 Reinforcement Strains and Crack Patterns............................................................... 295


8.3.6 Predictions of punching shear resistance according to ATENA analyses, MC2010 with

rotations according to LoA IV and EC2. ................................................................................ 299

8.3.7 Investigation on the coefficient of eccentricity ........................................................ 301

8.4 Conclusions ................................................................................................................... 302

Chapter 9 Conclusions................................................................................................................ 303

9.1 Introduction ................................................................................................................... 303

9.2 Summary of the thesis .................................................................................................... 304

9.2.1 Literature review .................................................................................................... 304

9.2.2 Methodology .......................................................................................................... 304

9.2.3 Flexural design of wide beam slabs ......................................................................... 305

9.2.4 Punching shear in wide beams ................................................................................ 306

9.2.5 Parametric studies................................................................................................... 307

9.3 Recommendations for Future Work ................................................................................ 308

References..................................................................................................................................... 309

List of Figures

10

List of Figures

Figure 1-1: Types of wide beam slab: (a) solid slab with band beams, (b) ribbed slab with wide beams

(1) ................................................................................................................................................... 36

Figure 2-1: The slab section proposed by Nichols, 1914. .................................................................. 45

Figure 2-2: Equivalent column section as proposed by Corley & Jirsa, (18) ..................................... 48

Figure 2-3: Cross section of transverse beam considered for moment of inertia calculations: (a)

internal beam, (b) edge beam ........................................................................................................... 48

Figure 2-4: Sections used in calculating torsional constants for: (a) beamless slab, (b) slab with beam.

........................................................................................................................................................ 49

Figure 2-5: column and middle strips in an interior panel ................................................................. 50

Figure 2-6: Stepped method yield criterion by Johansen ................................................................... 52

Figure 2-7: a slab element with orthogonal reinforcement ................................................................ 55

Figure 2-8: Kinnunen & Nylander Model for punching .................................................................... 58

Figure 2-9: Verification model for punching shear in EC2 and the control perimeters ...................... 60

Figure 2-10: EC2 shear distribution due to unbalanced moment at internal connection ..................... 61

Figure 2-11: Correlation between opening of critical shear crack and rotation according to the CSCT

........................................................................................................................................................ 63

Figure 2-12: Calculation of eccentricity between the position of resultant shear force and the centroid

of basic control perimeter (10) ......................................................................................................... 65

Figure 2-13: Effective tension area according to EC2 ....................................................................... 72

Figure 3-1:- Sketch showing a typical slab panel with continuous drop panel considered by Paultre &

Moisan (102) ................................................................................................................................... 76

Figure 3-2: FE model for slab with 4x4 panels as considered by Paultre & Moisan (102) ................. 76

Figure 3-3: Transverse moment distribution in slab with continuous drop panels (102) .................... 77

List of Figures

11

Figure 3-4: Sagging moment taken by column strip as a function of inertia ratio and panel aspect ratio

(after P. Paultre & C. Moisan, 2002) ................................................................................................ 78

Figure 3-5: A typical normalised moments versus distance across interior panel of wide beam slab as

presented by Tay, 2006: Section 1-1: passing through the face of columns supporting the internal wide

beam, Section 2-2: passing through the wide beam face and Section 3-3: passing through the Slab

mid-span. ........................................................................................................................................ 79

Figure 3-6: Plan view showing the wide beam floor and the critical sections considered by Tay ....... 79

Figure 3-7: Definition of column and middle strips as proposed by Tay (6) ...................................... 80

Figure 3-8: Transverse distribution of support and span moments for wide beam slab panel as

proposed by Tay .............................................................................................................................. 81

Figure 3-9: Floor plan layout showing member designations and critical sections as presented by A.

Shuraim & A. Al-Negheimish (7). ................................................................................................... 82

Figure 3-10: Joists’ End Moment Variation from average moment at section INCF at the face of

columns supporting the internal wide beam...................................................................................... 83

Figure 3-11: Moment profiles at two critical sections from WSB and FDB models (7) ..................... 83

Figure 3-12: Typical moment diagram in beam –girder grid system showing variation in negative

moments (7) .................................................................................................................................... 84

Figure 3-13: Typical Equivalent Frame as defined by A. Shuraim & A. Al-Negheimish. (7) ............ 84

Figure 3-14: Critical sections for torsional member with variable width as Shuraim & Al-Negheimish

(7) ................................................................................................................................................... 85

Figure 3-15: Plan of interior frame and the low and high rigidity zones as suggested by Shuraim & Al-

Negheimish (7) ................................................................................................................................ 87

Figure 3-16: Widths of rigidity zones in wide-shallow girders B3 & B4 as suggested by Shuraim &

Al-Negheimish (7) ........................................................................................................................... 87

Figure 3-17: wide beam failure mechanism as described by Lau & Clark (8).................................... 91

Figure 3-18: Sketch showing the assumed punching failure with losses: (a) plan view, (b) section. (8)

........................................................................................................................................................ 92

Figure 3-19: Side view showing the cracking patterns after failure for three different specimens (109)

........................................................................................................................................................ 92

Figure 4-1: Compressive hardening/softening and compressive characteristic length (ATENA v. 5.1.1)

........................................................................................................................................................ 98

List of Figures

12

Figure 4-2: Reduction factor due to lateral cracking (after Vecchio & Collins) ................................. 99

Figure 4-3: Predefined compression behaviour for Total Strain model in DIANA .......................... 100

Figure 4-4: Thorenfeldt Compression Curve .................................................................................. 100

Figure 4-5: Tension softening laws in DIANA ............................................................................... 102

Figure 4-6: Equivalent stress-strain relationship for tensioned concrete (After Damjanic & Owen,

1984) ............................................................................................................................................. 103

Figure 4-7: linear tension softening for concrete as proposed by Tay, 2006 .................................... 103

Figure 4-8: Fixed Crack Model (ATENA Manual) ......................................................................... 104

Figure 4-9: Shear retention factor (ATENA) .................................................................................. 105

Figure 4-10: Constant shear retention curve (DIANA Manual) ....................................................... 105

Figure 4-11: Rotating Crack Model (ATENA Manual) .................................................................. 106

Figure 4-12: Bilinear Law for reinforcement (ATENA Manual) ..................................................... 107

Figure 4-13: multi-linear Law for reinforcement (ATENA Manual) ............................................... 107

Figure 4-14: Newton-Raphson Method (DIANA Manual) .............................................................. 109

Figure 4-15: Quasi-Newton Method (DIANA Manual) .................................................................. 109

Figure 4-16: Arc length solution for: (a) Snap-through; (b) Snap-back phenomena (DIANA Manual)

...................................................................................................................................................... 110

Figure 4-17: Load Application: (a) Load control; (b) Displacement control (DIANA Manual) ....... 111

Figure 4-18: Solid Element types: (a) CHX60-DIANA Models; (b) 8 nodes CCIsoBrick-ATENA

Models .......................................................................................................................................... 112

Figure 4-19: 8-node CQ40S curved shell element (DIANA Manual) .............................................. 112

Figure 4-20: Eccentric Connection (DIANA Manual) .................................................................... 113

Figure 5-1: tension rebar arrangements for slabs 1. 4 &7. (135) ..................................................... 117

Figure 5-2: Mesh Discretization and boundary conditions for slabs 1, 4 & 7 .................................. 118

Figure 5-3: Comparisons of moment-curvature curves between the results of NLFEA and test data for

Clark & Speirs slabs 1, 4 & 7......................................................................................................... 119

Figure 5-4: Comparison between steel strains obtained from tests and NLFEA for slabs 1, 4 & 7. .. 119

Figure 5-5: Sketch showing the CMA in axially restrained RC slabs. (146) .................................... 120

Figure 5-6: Sketch showing the geometry and loading of tested specimens by Lahlouh & Waldron.121

List of Figures

13

Figure 5-7: Mesh Discretization, load introduction and boundary conditions for slab strip tests by

Lahlouh & Waldron....................................................................................................................... 122

Figure 5-8: Comparison between the load-deflection curves from test and NLFEA results ............. 123

Figure 5-9: Measured and predicted loads against midspan deflections according to Lahlouh &

Waldron (137) 124

Figure 5-10: Geometry for tested slabs. ......................................................................................... 124

Figure 5-11: Mesh Discretization, boundary conditions and load application for slabs PG8 & PG9 125

Figure 5-12: L-d curves for slab PG8 for mesh sizes: 25mm, 50mm and 100mm. ........................... 126

Figure 5-13: Load-deflection curves obtained from the test and the NLFEA for PG8 and PG9. ...... 126

Figure 5-14: NLFEA sensitivity to concrete elastic modulus (Ec) 127

Figure 5-15: NLFEA sensitivity to concrete tensile strength (ft) 127

Figure 5-16: Definition of test specimens for PT-series slabs: (a) general geometry; (b) type of loading

(12) ............................................................................................................................................... 128

Figure 5-17: Load-rotation curves of PT slabs given by test and NLFEA results from DIANA model

...................................................................................................................................................... 129

Figure 5-18: Investigation on the flexural response of slab PT23 using: (a) Tay’s model with 0.5 fct ,

0.5εs (b) Tay’s model with 0.25 fct , 0.5εs (c) Tay’s model with 0.5 fct , 0.25εs (d) β=0.1, (e) β=0.05 (f)

β=0.01 ........................................................................................................................................... 130

Figure 5-19: Cross section and reinforcement detailing for A- & S- series beams (138) .................. 131

Figure 5-20: A- series beams geometries, loading arrangement and bearing plates sizes. (138) ....... 132

Figure 5-21: S- series beams geometries, loading arrangement, bearing plate sizes and stirrups

distribution (138). (All dimensions are in mm). .............................................................................. 133

Figure 5-22: Mesh sensitivity study for beam A-1 with mesh sizes: 25mm, 50mm, 75mm & 100mm.

...................................................................................................................................................... 135

Figure 5-23: Example of FE beam model showing the mesh elements used for the analysis ........... 135

Figure 5-24: Comparison between the results given by NLFEA using ATENA with fixed and rotated

crack models and test results for beam specimens A-1 & S1-1 ....................................................... 136

Figure 5-25: Load-displacement curves given by the tests and the NLFEA for A & S beam series . 136

Figure 5-26: Comparisons between the crack patterns obtained from tests and NLFEA.(138). ........ 139

List of Figures

14

Figure 5-27: Definition of test specimens for PL-series slabs: (a) general geometry; (b) Placing shear

studs (139) .................................................................................................................................... 141

Figure 5-28: Definition of test specimens for Gomes & Regan slabs: (a) test set-up and general

geometry, b) placing of shear offcuts of steel I-section beams (c) details of shear reinforcement (15).

(All dimensions are in mm). .......................................................................................................... 141

Figure 5-29: Comparison of mesh size of slab PT23 in terms of L-R curves in the direction of: (a) x-

axis (b) y-axis ................................................................................................................................ 142

Figure 5-30: Typical FE slab model showing the mesh discretization. ............................................ 143

Figure 5-31: Load-rotation curves obtained from the test data and NLFEA results for PT series ..... 144

Figure 5-32: : Sensitivity to elastic modulus for concrete, Ec 144

Figure 5-33: Sensitivity to tensile strength for concrete, ft 145

Figure 5-34: : Sensitivity to plastic displacement, wd 145

Figure 5-35: Sensitivity to limited crack concrete compressive strength reduction factor, rc 145

Figure 5-36: Comparison of crack patterns along the transverse section of specimen PT33 from the

test (137) and NLFEA ................................................................................................................... 146

Figure 5-37: Load-rotation curves obtained from the test data and NLFEA results for PL series ..... 147

Figure 5-38: Comparison of the crack patterns in specimens PL6, PL7, PL10 & PL12 between the test

results (149) and predictions of the ATENA model. (Crack widths shown in the FE model ≥ 0.3 mm).

...................................................................................................................................................... 147

Figure 5-39: Load-rotation curves obtained from the test data and NLFEA results for Gomes & Regan

slabs .............................................................................................................................................. 148

Figure 5-40: Reinforcement details for symmetrical punching test slabs (I1-I6) (140) .................... 151

Figure 5-41: Mesh discretization of FE model for slabs I1-I6 ......................................................... 152

Figure 5-42: Comparisons of load-deflection curves from the symmetrical punching tests and NLFEA

...................................................................................................................................................... 152

Figure 5-43: Mesh discretization for a quarter of Regan slab used for symmetric punching tests..... 153

Figure 5-44: Load-deflection curves of Regan slabs predicted by DIANA model and from the test. 154

Figure 5-45: Load-rotation curves and corresponding failure criteria for slabs I1-I6 ....................... 154

Figure 6-1: Case study: three-storey wide beam floor building ....................................................... 159

List of Figures

15

Figure 6-2: Physical Model: (a) Plan (b) Elevation ......................................................................... 160

Figure 6-3: Elevation showing the sub-frame of wide beam with the load combination for ULS. .... 161

Figure 6-4: Plan view showing bottom flexural reinforcement for slab, edge and internal wide beams

used in uniform design. ................................................................................................................. 164

Figure 6-5: Plan view showing top flexural reinforcement for slab, edge and internal wide beams used

in uniform design. ......................................................................................................................... 165

Figure 6-6: Shear reinforcement details in internal and edge wide beams used for both uniform and

band designs .................................................................................................................................. 166

Figure 6-7: Typical column section: dimensions and reinforcement details .................................... 167

Figure 6-8: Plan showing the critical sections under study in the wide beam floor. ......................... 169

Figure 6-9: Eccentric connection (11) ............................................................................................ 170

Figure 6-10: Mesh Discretization for the FE model used to simulate wide beam floor. ................... 171

Figure 6-11: Elastic transverse hogging moment distribution of slab across the internal wide beam at

column faces, centre and wide beam-slab interfaces. ...................................................................... 172

Figure 6-12: Elastic transverse hogging moment distribution of slab across the edge wide beam at

column face, centre and wide beam-slab interface. ......................................................................... 172

Figure 6-13: Elastic transverse sagging moment distribution of slab across the wide beam at end and

internal panel slab midspan. ........................................................................................................... 173

Figure 6-14: Elastic twisting moment distribution about the longitudinal axes of the edge and internal

wide beams. .................................................................................................................................. 173

Figure 6-15: Required reinforcement areas to resist Wood-Armer moments at internal support and end

panel slab sections. ........................................................................................................................ 174

Figure 6-16: Required reinforcement areas to resist Wood-Armer moments at edge support. .......... 175

Figure 6-17: Transverse moments about axis parallel to direction of wide beam span along sections A-

A to D-D. ...................................................................................................................................... 175

Figure 6-18: Transverse reinforcement across the edge and internal wide beams according to the

proposed band distribution. ............................................................................................................ 177

Figure 6-19: Plan showing the uniform distribution of top flexural reinforcement along the edge and

internal wide beams ....................................................................................................................... 178

Figure 6-20: Plan showing band distribution of top flexural reinforcement along the edge and internal

wide beams. .................................................................................................................................. 179

List of Figures

16

Figure 6-21: Plan showing distribution of slab bottom flexural reinforcement used with uniform and

banded rebar designs ..................................................................................................................... 180

Figure 6-22: Comparison of load-deflection curves between FE models with three mesh sizes; 50mm,

100mm & 200mm. ........................................................................................................................ 183

Figure 6-23: Comparison of nonlinear hogging moments along section 6-6 between FE models with

three mesh sizes; 50mm, 100mm & 200mm. .................................................................................. 184

Figure 6-24: Comparison of nonlinear hogging moment along section 2-2 between FE models with

three mesh sizes; 50mm,100mm & 200mm. ................................................................................... 184

Figure 6-25: Comparison of nonlinear sagging moment along section 4-4 between FE models with

three mesh sizes; 50mm,100mm & 200mm. ................................................................................... 184

Figure 6-26: L-D diagrams for the shell and solid models. ............................................................. 186

Figure 6-27: Load-deflection curves for the model with transverse uniform steel distribution and

model with steel placed in bands. ................................................................................................... 187

Figure 6-28: Contour plot showing deflection at ultimate load in plan design ultimate load

(Vu=3408kN) form the NLFEA for model with: (a) uniformly distributed steel, (b) steel placed in

bands. ............................................................................................................................................ 188

Figure 6-29: Transverse moment distribution at section (6-6) through the internal column faces along

the internal wide beam resulting from the uniform and band designs at quasi-permanent load

(Vs=1772kN). ................................................................................................................................ 189


the edge wide beam resulting from the uniform and band designs at quasi-permanent load

(Vs=1772kN). ................................................................................................................................ 189

Figure 6-31: Transverse moment distribution at section (5-5) through the beam face along the internal

wide beam resulting from uniform and band distributions at quasi-permanent load (Vs=1772kN). .. 190

Figure 6-32: Transverse moment distribution at section (3-3) through the beam face along the edge

wide beam resulting from uniform and band distributions at quasi-permanent load (Vs=1772kN). .. 190

Figure 6-33: Transverse moment distribution at section (4-4) through the slab at midspan along the

internal wide beam resulting from uniform and band distributions at quasi-permanent load

(Vs=1772kN). ................................................................................................................................ 190


the internal wide beam resulting from the uniform and band designs at design ultimate load

(Vu=3408kN). ................................................................................................................................ 191

List of Figures

17


the edge wide beam resulting from the uniform and band designs at design ultimate load

(Vu=3408kN). ................................................................................................................................ 191

Figure 6-36: Transverse moment distribution at section (5-5) through the beam face along the internal

wide beam resulting from uniform and band distributions at design ultimate load (Vu=3408kN)..... 191

Figure 6-37: Transverse moment distribution at section (3-3) through the beam face along the edge

wide beam resulting from uniform and band distributions at design ultimate load (Vu=3408kN)..... 192

Figure 6-38: Transverse moment distribution at section (4-4) through the slab midspan along the wide

beam resulting from uniform and band distributions at design ultimate load (Vu=3408kN). ............ 192

Figure 6-39: The in-plane forces along section (6-6) passing through the faces of internal column at

design service load (Vs=1772kN) and design ultimate load (Vu=3408kN) for uniform and banded

distributions. ................................................................................................................................. 193

Figure 6-40: The in-plane forces along section (4-4) passing through the midspan parallel to the wide

beam at ULS (Vu=3408kN) for uniform and banded distributions. ................................................. 193

Figure 6-41: Comparison between the bending moments in the internal wide beam at the column face

section given by the NLFEA and calculated from section analysis under the CMA at ULS

(Vu=3408kN) for uniform distribution. .......................................................................................... 194

Figure 6-42: Comparison between the bending moments in the internal wide beam at column face

section given by the NLFEA and calculated from section analysis under the CMA at ULS

(Vu=3408kN) for the band distribution ........................................................................................... 194

Figure 6-43: Plan view showing the arrangement of C1, C2, C3 and C4 in the FE model and

horizontal reactions on the slab level at design ultimate load (Vu=3408kN) for model with transverse

steel uniformly distributed. ............................................................................................................ 196

Figure 6-44: Plan view showing the arrangement of C1, C2, C3 and C4 in the FE model and

horizontal reactions on the slab level at design ultimate load (Vu=3408kN) for model with transverse

steel distributed in bands................................................................................................................ 196

Figure 6-45: Internal column moments in the x- and y- directions at slab level for elastic FEA and

NLFEA models at design ultimate load (Vu=3408kN) with the transverse steel distributed uniformly

and placed in bands. ...................................................................................................................... 197

Figure 6-46: NLFEA steel strains along the critical sections in the internal wide beam and end bay

slab at quasi-permanent load (1772 kN) for the uniform steel and banded steel models. ................. 199

Figure 6-47: NLFEA steel strains along the critical sections in the internal wide beam and end bay

slab at design ULS (3408 kN) for the uniform steel model and the band model. ............................. 199

List of Figures

18

Figure 6-48: NLFEA steel strains along the column and beam faces in the edge wide beam at quasi-

permanent load (1772 kN) for the uniform steel and banded steel models....................................... 199

Figure 6-49: NLFEA steel strains along the column and beam faces in the edge wide beam at design

ULS (3408 kN) for the uniform steel and banded steel models. ...................................................... 200

Figure 6-50: Contour plot showing the slab moment distribution across the end bay at ULS (3408 kN)

for: (a) uniform rebar distribution, (b) banded rebar distribution..................................................... 200

Figure 6-51: EC2 steel strains and NLFEA strains along the critical sections in the internal wide beam

and end bay slab at quasi-permanent load (1772 kN) for the uniform rebar design. ......................... 202

Figure 6-52: EC2 steel strains and NLFEA strains along the critical sections in the internal wide beam

and end bay slab at quasi-permanent load (1772 kN) for the band rebar design .............................. 202

Figure 6-53: EC2 steel strains and NLFEA strains along the column and beam faces in the edge wide

beam at quasi-permanent load (1772 kN) for the uniform rebar design. .......................................... 202

Figure 6-54: EC2 steel strains and NLFEA strains along the column and beam faces in the edge wide

beam at quasi-permanent load (1772 kN) for the band rebar design. ............................................... 203

Figure 6-55: EC2 steel strains without 0.6 εs limit and NLFEA strains along the critical sections in the

internal wide beam and end bay slab at quasi-permanent load (1772 kN) for the uniform rebar design.

...................................................................................................................................................... 203

Figure 6-56: EC2 steel strains without 0.6 εs limit and NLFEA strains along the critical sections in the

internal wide beam and end bay slab at quasi-permanent load (1772 kN) for the band rebar design. 203

Figure 6-57: EC2 steel strains without 0.6 εs limit and NLFEA strains along the column and beam

faces in the edge wide beam at quasi-permanent load (1772 kN) for the uniform rebar design. ....... 204

Figure 6-58: EC2 steel strains without 0.6 εs limit and NLFEA strains along the column and beam

faces in the edge wide beam at quasi-permanent load (1772 kN) for the band rebar design. ............ 204

Figure 6-59: Crack width based on NLFEA steel strains at critical sections along the internal wide

beam face and end bay slab at quasi-permanent load (1772 kN) for the lateral uniform and banded

rebar distributions. ......................................................................................................................... 207

Figure 6-60: Crack width based on NLFEA steel strains along the column and beam faces in the edge

wide beam at quasi-permanent load (1772 kN) for the lateral uniform and banded rebar distributions.

...................................................................................................................................................... 207

Figure 6-61: Crack width based on NLFEA steel strains at critical sections along the internal wide

beam face and end bay slab at design ultimate load (3408 kN) for the lateral uniform and banded rebar

distributions. ................................................................................................................................. 207

List of Figures

19

Figure 6-62: Crack width based on NLFEA steel strains along the column and beam faces in the edge

wide beam at design ultimate load (3408 kN) for the lateral uniform and banded rebar distributions.

...................................................................................................................................................... 208

Figure 6-63: Comparison of crack width based on steel strains given by EC2 and NLFEA at critical

sections along the internal wide beam face and end bay slab at the quasi-permanent load (1772 kN)

for the uniform rebar distribution. .................................................................................................. 208

Figure 6-64: Comparison of crack width based on steel strains given by EC2 and NLFEA at critical

sections along the internal wide beam face and end bay slab at the quasi-permanent load (1772 kN)

for the banded rebar distribution. ................................................................................................... 208

Figure 6-65: Comparison of crack width based on steel strains given by EC2 and NLFEA along the

column and beam faces in the edge wide beam at the quasi-permanent load (1772 kN) for the uniform

rebar distribution. .......................................................................................................................... 209

Figure 6-66: Comparison of crack width based on steel strains given by EC2 and NLFEA along the

column and beam faces in the edge wide beam at the quasi-permanent load (1772 kN) for the banded

rebar distribution. .......................................................................................................................... 209

Figure 6-67: Crack strain in the model with uniformly distributed steel at design service load (1772

kN). ............................................................................................................................................... 210

Figure 6-68: Crack strain in the model with lateral steel placed in bands at design service load (1772

kN). ............................................................................................................................................... 210

Figure 6-69: Comparison of steel strains obtained from the NLFEA and those based on cracked

section analysis (EC2) for different reinforcement ratios. ............................................................... 212

Figure 7-1: Transverse uniform steel distribution for the internal column assembly ........................ 216

Figure 7-2: Transverse band steel distribution for the internal column assembly ............................. 216

Figure 7-3: Plan view showing the shear reinforcement around the internal column ....................... 217

Figure 7-4: Load distribution subjected to the internal connection. ................................................. 218

Figure 7-5: ATENA mesh discretization for internal connection assembly used for punching analysis

...................................................................................................................................................... 218

Figure 7-6: Load-deflection curves for solid assembly for internal connection for uniform and band

reinforcement designs (deflections are given at points A, B, C & D) .............................................. 219

Figure 7-7: Plan view showing steel strains at failure for internal column sub-assemblages with

transverse reinforcement placed (a) uniformly (1201 kN), (b) in bands (1285 kN). ......................... 220

List of Figures

20

Figure 7-8: Plan view of cracking patterns at failure (w>0.3 mm) in the internal column sub-

assemblages with transverse reinforcement placed (a) uniformly, (b) in bands. .............................. 220

Figure 7-9: Deflected shape of the internal column sub-assemblages at failure with transverse

reinforcement placed (a) uniformly, (b) in bands. ........................................................................... 221

Figure 7-10: Comparison of load-rotation responses of internal connection solid assembly under

different load distributions including symmetrical load and unbalanced moments .......................... 222

Figure 7-11: Calculation of beam rotation in solid assembly. ......................................................... 223

Figure 7-12: DIANA mesh discretization for internal connection assembly used for punching analysis.

...................................................................................................................................................... 224

Figure 7-13: Punching shear strength of internal column connection with rotations according to LoA

II ................................................................................................................................................... 225

Figure 7-14: Rotations of wide beam along the longitudinal and lateral axes of the internal column at

DIANA’s failure load (1390 kN) for the uniform and band sub-assemblies. ................................... 226

Figure 7-15: Deflected shape of the internal column in shell sub-assembly: (a) longitudinal direction

(x-x), (b) transverse direction (y-y). ............................................................................................... 227

Figure 7-16: Comparison of L-R responses between the shell and solid assemblages for each side of

control perimeter around the internal column for the uniform and band steel distributions. ............. 228

Figure 7-17: Load-Rotation curves of the wide beam’s internal connection for shell and solid sub-

assemblies for uniform and band steel designs. .............................................................................. 229

Figure 7-18: Division of control perimeter into segments in x & y directions as proposed by Sagaseta

et al. (137) ..................................................................................................................................... 231

Figure 7-19: Comparison of shear force distribution along the control perimeter at 0.5d from the

internal column face for elastic FEA and NLFEA with uniform and band reinforcement distributions.

...................................................................................................................................................... 232

Figure 7-20: dimensions of EC2's control perimeter (10). .............................................................. 234

Figure 7-21: Rotations of wide beam along the longitudinal and lateral axes of the internal column at

ultimate loads of 1711kN & 1749 kN for the uniform and band full-scale models, respectively. ..... 236

Figure 7-22: Comparison of L-R responses between the full scale model and shell assemblages for

each side of control perimeter around the internal column for the uniform and band steel distributions.

...................................................................................................................................................... 237

Figure 7-23: Load-Rotation curves of the wide beam’s internal column connection for full-scale shell

model and shell sub-assemblies for uniform and band steel designs. .............................................. 238

List of Figures

21

Figure 7-24: Transverse uniform reinforcement distribution for the edge column assembly ............ 240

Figure 7-25: Transverse band reinforcement distribution for the edge column assembly ................. 240

Figure 7-26: Plan view showing the shear reinforcement around the edge column .......................... 240

Figure 7-27: Load distribution subjected to the edge connection. ................................................... 241

Figure 7-28: ATENA mesh discretization for edge connection assembly. ....................................... 242

Figure 7-29: Load-deflection curves for solid assembly for edge connection for uniform and band

reinforcement designs. ................................................................................................................... 242

Figure 7-30: steel strains at failure in the edge column sub-assemblages for transverse uniform and

band reinforcement distributions .................................................................................................... 243

Figure 7-31: the crack patterns at failure (w ≥ 0.3 mm) in edge connection sub-assemblages with

uniform reinforcement distribution. ............................................................................................... 244

Figure 7-32: the crack patterns at failure (w ≥ 0.3 mm) in edge connection sub-assemblages with

banded reinforcement distribution. ................................................................................................. 244

Figure 7-33: DIANA mesh discretization for edge connection assembly used for punching analysis.

...................................................................................................................................................... 245

Figure 7-34: Punching shear strength of edge column connection with rotations according to LoA II

...................................................................................................................................................... 247

Figure 7-35: Rotations of wide beam along the longitudinal and lateral axes of the edge column for

the uniform and band sub-assemblies at uniform assembly ultimate load (641 kN). ........................ 248


control perimeter around the edge column for the uniform and band steel distributions. ................. 248

Figure 7-37: Load-Rotation curves of the wide beam’s internal connection with shear reinforcement

for shell and solid sub-assemblies for uniform and band steel designs ............................................ 249

Figure 7-38: Comparison of shear force distribution along the control perimeter at 0.5d from the edge

column face for elastic FEA and NLFEA with uniform and band reinforcement distributions......... 250

Figure 7-39: EC2’s basic control perimeter for edge column .......................................................... 252

Figure 7-40: Rotations of wide beam along the longitudinal and lateral axes of the wide beam at

ultimate load of 701 kN for the uniform and band full-scale models. .............................................. 253

Figure 7-41: comparison of bending moments across the transverse and longitudinal axes of edge

column at ultimate load between the uniform and band full-scale models. ...................................... 254

List of Figures

22


control perimeter around the edge column for the uniform and band steel distributions. ................. 255

Figure 7-43: Load-Rotation curves of the wide beam’s edge column connection for full-scale shell

model and shell sub-assemblies for uniform and band steel designs ............................................... 256

Figure 8-1: Geometry of the model used for the parametric study .................................................. 261

Figure 8-2: Required transverse reinforcement areas to resist Wood-Armer moments along sections

(1-1), (2-2) and the minimum steel area. ........................................................................................ 262

Figure 8-3: Influence of varying wide beam span on transverse distribution of elastic support moment

at the first internal column at section passing through column and beam faces along the internal wide

beam. ............................................................................................................................................ 264


at the first internal column at section passing through column and beam faces along the edge wide

beam. ............................................................................................................................................ 264


at the end column at section passing through column and beam faces along the internal wide beam.

...................................................................................................................................................... 264


at the end column at section passing through column and beam faces along the edge wide beam. ... 265

Figure 8-7: Influence of varying slab span on transverse distribution of elastic support moment at the

first internal column at section passing through column and beam faces along the internal wide beam.

...................................................................................................................................................... 265


first internal column at section passing through column and beam faces along the edge wide beam. 266


end column at section passing through column and beam faces along the internal wide beam. ........ 266


end column at section passing through column and beam faces along the edge wide beam. ............ 266

Figure 8-11: Influence of varying width of wide beam on transverse distribution of elastic support

moment at the first internal column at section passing through column and beam faces along the

internal wide beam. ....................................................................................................................... 267


moment at the first internal column at section passing through column and beam faces along the edge

wide beam. .................................................................................................................................... 267

List of Figures

23


moment at the end column at section passing through column and beam faces along the internal wide

beam. ............................................................................................................................................ 268


moment at the end column at section passing through column and beam faces along the edge wide

beam. ............................................................................................................................................ 268

Figure 8-15: Proposed division for wide beam into column and span band widths .......................... 269

Figure 8-16: Influence of a) beam span, b) slab span and c) beam width on the width of column band

in the internal beam ....................................................................................................................... 270

Figure 8-17: Influence of a) beam span, b) slab span and c) beam width on the width of column band

in the edge beam............................................................................................................................ 270

Figure 8-18: Relationship between the wide beam span and the width of column band over the

internal beam. ................................................................................................................................ 271

Figure 8-19: Relationship between the wide beam span and the width of column band over the edge

beam ............................................................................................................................................. 271

Figure 8-20: Influence of a) beam span, b) slab span and c) beam width on the moment across the

column band width in the internal beam ......................................................................................... 272

Figure 8-21: Influence of a) beam span, b) slab span and c) beam width on the moment across the

column band width in the edge beam ............................................................................................. 272

Figure 8-22: Relationship between the slab aspect ratio and the moment across column band width in

the internal wide beam. .................................................................................................................. 273

Figure 8-23: Relationship between the slab aspect ratio and the moment across column band width in

the edge wide beam. ...................................................................................................................... 273

Figure 8-24: Comparison of transverse flexural reinforcement area between the uniform and banded

rebar designs along the internal and edge wide beams for models A, B, C, D, E, F & G. ............... 275

Figure 8-25: Transverse uniform and band rebar distribution along the internal and edge beams

without considering the minimum steel rule for models A-G.. ........................................................ 276

Figure 8-26: Comparison of Load-deflection curve for models A, B, C, D, E, F & G between the

uniform and band rebar distributions. ............................................................................................. 277

Figure 8-27: Comparison between elastic, uniform and band moment distribution at design ultimate

load at the column face section (1-1) along the internal wide beam for models A, B, C, D, E, F & G.

...................................................................................................................................................... 279

List of Figures

24

Figure 8-28: Comparison between elastic, uniform and band moment distribution at design ultimate

load at the column face section (4-4) along the edge wide beam for models A, B, C, D, E, F & G. . 280

Figure 8-29: The in-plane forces at design ultimate loads at sections (1-1 & 4-4) along the internal and

edge wide beams for the uniform and band rebar designs for models A, B, C, D, E, F & G ............ 281

Figure 8-30: Comparison of steel stains at sections (1-1 & 2-2) passing through the column and beam

faces along internal wide beam for uniform and band rebar designs for models A, B, C, D, E, F & G)

at SLS load (1.0 D.L+1.0 I.L) ........................................................................................................ 283

Figure 8-31: Comparison of steel stains at sections (4-4 & 3-3) passing through the column and beam

faces along edge wide beam for uniform and band rebar designs for models A, B, C, D, E, F & G) at

SLS load (1.0 D.L+1.0 I.L). ........................................................................................................... 284

Figure 8-32: Comparison of crack width along the internal wide beam at section 1-1 for transverse

uniform and proposed band distributions at quasi-permanent load for models A, B, C, D, E, F & G.

...................................................................................................................................................... 287

Figure 8-33: Comparison of crack width along the edge wide beam at section 4-4 for transverse

uniform and proposed band distributions at quasi-permanent load for models A, B, C, D, E, F & G.

...................................................................................................................................................... 288

Figure 8-34: sample of the solid assembly used in the parametric study showing its geometry. ....... 291

Figure 8-35: Plan view showing the shear reinforcement distribution around the internal column in

models A-G. (All dimensions are in mm) ....................................................................................... 292

Figure 8-36: Sketch showing the points at which the deflection is extracted. .................................. 293

Figure 8-37: Load-deflection curves for uniform and banded rebar distributions for models A,B,C,

D,E, F & G. (deflections are given at points a and b). ..................................................................... 294

Figure 8-38: Plan view showing steel strains at ultimate load for models A, B, C, D, E, F & G with

transverse reinforcement placed (I) uniformly (II) in bands. ........................................................... 296

Figure 8-39: Plan view of cracking patterns at ultimate load (w>0.3 mm for models A, B, C, D, E, F

& G with transverse reinforcement placed uniformly and in bands. ................................................ 297

Figure 8-40: Comparison of load-deflection responses of models A-G subjected to symmetrical and

asymmetrical loadings. .................................................................................................................. 298

Figure 8-41: Load-Rotation curves of the wide beam’s internal connection models A – G for uniform

and band steel designs. .................................................................................................................. 300

List of Tables

25

List of Tables

Table 2-1: Approximate values of bending moments in uniformly loaded beams and slabs continuous

over three or more spans as in CP114 (13) ....................................................................................... 41

Table 2-2: Ultimate bending moments in continuous beams and one way slabs according to CP110

(15) ................................................................................................................................................. 41

Table 2-3: Design of ultimate bending moments in uniformly-loaded continuous beams and one-way

spans as in BS8110 (2) .................................................................................................................... 42

Table 2-4: the approximate design values for bending moments in the ACI318-56 (18) up to date.... 43

Table 2-5: the moment coefficients for an end span according to ACI 318-11 (2) ............................. 46

Table 2-6: Distribution of an interior panel moment between column and middle strips according to

BS 8110, ACI 318 and EC2 ............................................................................................................. 50

Table 2-7: EC2 basic span-effective depth ratios for RC members without axial compression .......... 68

Table 3-1: Tay’s proposed distribution of transverse moments between column and middle strips

across wide beam slab panel ............................................................................................................ 81

Table 5-1: Material properties for slab specimens 1, 4 and 7 .......................................................... 118

Table 5-2: Steel details for slab specimens 1, 4 & 7. ...................................................................... 118

Table 5-3: Comparison between the strengths of slabs 1, 4 and 7 from tests and NLFEA ............... 119

Table 5-4: Material properties and steel details. ............................................................................. 121

Table 5-5: Comparison between the specimen strengths from the test and NLFEA ......................... 123

Table 5-6: Geometry and material properties for specimens PG8 and PG9 ..................................... 125

Table 5-7: Main characteristic of PT-series slab series. .................................................................. 128

Table 5-8: Main characteristics of Fang beams .............................................................................. 131

Table 5-9: Default values for material parameters used in ATENA models for beams and slabs ..... 134

List of Tables

26

Table 5-10: Comparison between the results given by the beam tests and the NLFEA in terms of

ultimate failure load and deflection. ............................................................................................... 136

Table 5-11: Main characteristic of PL-series slab series ................................................................. 140

Table 5-12: Main characteristic of Gomes & Regan slab series ...................................................... 140

Table 5-13: Comparison of the estimated punching loads using ACI, EC2, CSCT and FEA (ATENA)

for PT slabs ................................................................................................................................... 143

Table 5-14: Comparison of the estimated punching loads using ACI, EC2, CSCT and NLFEA for PL

slabs .............................................................................................................................................. 146

Table 5-15: Estimated punching loads using FEA (ATENA) for Gomes & Regan slabs ................. 148

Table 5-16: Material properties for symmetrical punching test slabs (I1-I6) ................................... 150

Table 5-17: Failure load results obtained from the symmetrical punching tests and ATENA .......... 152

Table 5-18: Estimation of rotations according to CSCT levels I, II and III using DIANA 9.6 ......... 155

Table 5-19: Comparisons between punching strengths given by CSCT LoAs I, II, III & IV and from

the test ........................................................................................................................................... 155

Table 6-1: Material properties for the model used as case study ..................................................... 159

Table 6-2: Longitudinal and shear reinforcement details for wide beams and columns ................... 163

Table 6-3: Material properties for wide beam floor under study ..................................................... 182

Table 6-4: Load cases as applied in NLFE Model .......................................................................... 182

Table 6-5: Comparison in terms of computational time and size between the FE models with mesh

sizes; 50 mm, 100 mm & 200 mm. ................................................................................................ 185

Table 6-6: Comparison between the column moments and vertical reactions at design ultimate load

(3408 kN) for TCC and band steel designs ..................................................................................... 197

Table 6-7: Maximum crack spacing for internal wide beam and end bay slab for uniform and band

steel distributions .......................................................................................................................... 206

Table 6-8: Maximum crack spacing for edge wide beam for uniform and band steel distributions .. 206

Table 6-9: Comparison of steel strains calculated with EC2 method and from NLFEA for different

rebar ratios at quasi permanent load ............................................................................................... 212

Table 7-1: Flexural reinforcement details for uniform and band assemblies. ................................... 216

Table 7-2: Estimated punching shear resistance for internal connection using MC2010 LoA II for

uniform and band steel designs (ke=0.9) ......................................................................................... 225

List of Tables

27

Table 7-3: Punching resistances for models with shear reinforcement using EC2, MC2010 level II &

IV and ATENA analysis for lateral uniform and banded steel distribution (ke=0.9). ....................... 230

Table 7-4: Calculation of ke using linear elastic FEA and NLFEA for uniform and band reinforcement

arrangements at ultimate flexural load 1390 kN ............................................................................. 232

Table 7-5: Calculation of coefficient of eccentricity based on the fib MC2010 ............................... 233

Table 7-6: Calculation of coefficient of eccentricity based on EC2 ................................................. 234

Table 7-7: Comparison of Punching shear resistances for full-scale models and shell assemblages

estimated using EC2, MC2010 level IV and NLFEA for lateral uniform and banded steel distribution

(ke=0.9)......................................................................................................................................... 238

Table 7-8: Estimated punching shear resistance for edge column connection using MC2010 LoA II for

uniform and band steel designs (ke=0.7) ......................................................................................... 247

Table 7-9:Punching resistances around edge column for models with shear reinforcement using EC2,

MC2010 level II & IV and NLFEA for lateral uniform and banded steel distribution (ke=0.7). ....... 249

Table 7-10: Calculation of ke using linear elastic FEA and NLFEA for uniform and band

reinforcement arrangements ........................................................................................................... 251

Table 7-11: Calculation of coefficient of eccentricity for edge column based on the fib MC2010 ... 251

Table 7-12: Calculation of coefficient of eccentricity based on EC2 ............................................... 253

Table 7-13: Comparison of Punching shear resistances for full-scale models and shell assemblages

estimated using EC2, MC2010 level IV and NLFEA for lateral uniform and banded steel distribution

(ke=0.7). ........................................................................................................................................ 256

Table 8-1: members dimensions of the models used in the parametric studies ................................ 262

Table 8-2: Division of models used in the parametric studies according to the relevant investigated

parameter ...................................................................................................................................... 263

Table 8-3: Maximum crack spacing and rebar spacing for internal wide beam for uniform and band

steel designs for models A, B, C, D, E, F & G................................................................................ 285

Table 8-4: Maximum crack spacing and rebar spacing for edge wide beam for uniform and band steel

designs for models A, B, C, D, E, F & G ....................................................................................... 286

Table 8-5: Geometry details for the assemblies used for the parametric study................................. 290

Table 8-6: Loads extracted from the elastic FEA subjected to the assemblies used in parametric study

...................................................................................................................................................... 290

Table 8-7: Reinforcement details for the assemblies used for parametric study ............................... 291

List of Tables

28

Table 8-8: NLFEA’s ultimate loads for models A to G with transverse uniform and band rebar

designs. ......................................................................................................................................... 295

Table 8-9: Punching resistances for models A-G using EC2, MC2010 with rotations according to LoA

IV and ATENA analysis for lateral uniform and banded steel distribution (ke=0.9). ....................... 301

Table 8-10: The effect of ρy/ρx on the variability of punching strength prediction with MC2010, LoA

IV and EC2 ................................................................................................................................... 301

Table 8-11: Calculation of the coefficient of eccentricities, ke and β using eccentricity-based formulae

in fib MC2010 and EC2 ................................................................................................................. 302

Symbols

29

Symbols

The symbols listed below are used in this thesis. Where there is more than one parameter assigned to

the same symbol, the valid definition is explained in the relevant text.

b Width of the banded beam

b0 Control perimeter according to MC2010

b0x Basic control perimeter in x directions

b0y Basic control perimeter in y directions

b1 Basic control perimeter according to MC2010

bs Width of support strip across

bu Diameter of a circle having the same area as that enclosed by the basic control perimeter

c Concrete cover

c1 Size of support in the span direction

c2 Size of support normal to the span direction

d Effective depth

dagg Mean aggregate size

dg Maximum aggregate size

dg0 Reference aggregate size

dv Effective depth of slab accounting for penetration of the supported area in the slab

e Eccentricity

epar Eccentricity parallel to the slab edge resulting from a moment about an axis perpendicular to

the slab edge

eu Eccentricity of shear force with respect to the basic control perimeter

eui Eccentricity in the direction i considered

fbd Bond strength

fck Concrete cylinder strength

fcu Concrete cube strength

f`c0 Initial compressive strength for concrete

fp Peak compression stress in concrete

Symbols

30

fy Yield strength for main rebar

fyk Yield stress of shear links

fywd,ef Effective design strength of punching shear reinforcement

h Crack bandwidth

heq Equivalent length of crack

hi Horizontal displacements of column in the longitudinal direction

k Factor accounts for the size effect

ke Coefficient of eccentricity

kex Coefficients of eccentricity in the x direction

key Coefficients of eccentricity in the y direction

kt Factor accounts for the duration of loading

k1 Coefficient accounting for the bond properties of the bonded reinforcement

k2 Coefficient accounting for the distribution of strain

l1 Slab span between support centres

l2 Width of slab panel measured between the support centres

mEd Average moment per unit length in support strip

mhog slab col Design moment in the slab in the column strip at the face of the internal beam

mhog slab middle Design moment in the slab in the middle strip at the face of the internal beam face

mmax Maximum span moment

mRd Average flexural strength moments per unit length in support strip

mux Wood & Armer design bending moment of the infinitesimal element in the x-axis

muy Wood & Armer design bending moment of the infinitesimal element in the y-axis

mx Applied bending moment field per unit width in the x-axis

mxy Applied twisting moment field per unit width.

my Applied bending moment field per unit width in the y-axis

m* Wood-Armer moment at any point

m*av Average Wood & Armer bending moment

m*column Wood & Armer’s moment across the column band width

m*span Wood & Armer’s moment across the beam band width

rg Shear retention factor according to ATENA

rs Distance from column axis to line of contraflexure

s Depth of equivalent rectangular stress block

si Horizontal displacements of column in the transverse direction

sr Spacing of shear links in the radial direction

sr,max Maximum crack spacing

st Spacing of shear links in the tangential direction

Symbols

31

u1 Basic control perimeter

u1* Reduced basic control perimeter

νperp,d,av Average shear force per unit length perpendicular to the basic control perimeter

νperp,d,max Maximum shear force per unit length perpendicular to the basic control perimeter

vRd,c Shear resistance provided by concrete

vRd,cs Punching shear resistance provided by shear reinforcement

w Opening of the critical shear crack

wd Plastic displacement

wk Design crack width

wlim Crack width controlled by crack height

w0 Crack width over bar

x Shorter dimension of a rectangular area of a cross section for torsional constant calculation

y Longer dimension of a rectangular area of a cross section for torsional constant calculation

Ac,eff Effective tension area of concrete

Act Area of concrete in tension

As,min Minimum flexural steel area

As,prov Provided steel area at the section

As,req Required steel area at the section

Ast,av Average transverse flexural reinforcement per unit length

Ast,i Average transverse reinforcement over the band considered

Asw Area of shear reinforcement per perimeter

C Torsional constant

Ec Elastic modulus of concrete

Ec,eff Effective modulus of elasticity of concrete

Es Elastic modulus of steel

Esh Hardening modulus for steel

F Total design ultimate load

G the shear modulus of elasticity

GC Initial shear modulus for concrete

GF Fracture energy

Gk Characteristic dead load

GR Reduced shear modulus

I Second moment of area of the section

Ic Second moment of inertia for column

Icm Modified second moment of inertia for column (as suggested by Shuraim & Al-Negheimish)

Symbols

32

Isd Moment of inertia of the slab-beam from the column centreline to the face of the column,

bracket or capital

Isc Moment of inertia of the slab-beam at the face of the column

J2 Second invariant of the stress deviator sensor

Kc Column rotational stiffness

Kca Axial stiffness of the column

Kcc Stiffness of the equivalent column

Kec Equivalent column stiffness

Kg Vertical rigidity of the wide beam

Kt Torsional stiffness of torsional member

Lb Span of the wide beam

Lc Column strip width according to Tay’s proposed design method

Ls Span of the slab

M Average midspan design moments

Mcr Cracking moment

MEd,x Unbalanced moment about the x- axis

MEd,y Unbalanced moment about the y- axis

MEQF1 Moment given by the equivalent frame analysis at a distance 0.5b - dslab from the column

centreline

MEQF2 Moment given by the equivalent frame analysis at the face of the beam

M0 Total design moments

M’ Average support design moments

Mfi Frame moment (as suggested by Shuraim & Al-Negheimish)

Mhi Moment in the high rigidity zone (as suggested by Shuraim & Al-Negheimish)

Mli Moment in the low rigidity zone

N Axial compressive force

Qk Characteristic imposed load

S First moment of area of reinforcement about the centroid of the section

VATENA Resistance predicted by ATENA analysis

VEd Applied punching load

Vu Design ultimate load

VII Punching shear strength given by MC2010 with rotations according to LoA II

VIV Punching shear strength given by MC2010 with rotations according to LoA IV

W Total uniformly distributed load

Wd Total dead load per span

Ws Total imposed load per span

Symbols

33

Zcolumn Width of column band

Zi Sum of band widths with the same transverse reinforcement area

Zspan Width of span band

α Angle between the plane of the slab and shear reinforcement

αD Ratio of the flexural stiffness of the wide beam to the stiffness of slab bounded laterally by the

centrelines of neighbouring slabs either side

αe Effective modular ratio

αlat Average lateral damage variable

αp Peak compression strain in concrete

α1 Deformation parameter calculated for uncracked section

α11 Deformation parameter calculated for fully-cracked section

β Distribution factor accounting for the effect of eccentricity in EC2

βεcr Reduction factors due to the lateral cracking for the peak strain

βσcr Reduction factors due to the lateral cracking for the peak stress

γc Partial safety factor for concrete

γs Partial safety factor for steel reinforcement

εcm Mean strain the concrete between cracks

εcs Free shrinkage strain

εn Crack normal strain

εsm Mean strain in the reinforcement taking into account the effects of tension stiffening and the

effect of imposed deformations

εu Ultimate crack strain

ɛcr Crack strain

εcp Plastic strain at the maximum compressive strength

ε0 Initial strain

ζ Distribution factor accounts for the tension stiffening in the section

ζh Zone intensity factor

µc Poisson’s ratio for concrete

µs Poisson’s ratio for steel

ρp,eff Effective reinforcement ratio

ρx Flexural reinforcement ratio in x direction

ρy Flexural reinforcement ratio in y direction

ρ0 Reference reinforcement index

ρ1 Mean of the reinforcement ratios in orthogonal directions

ρ' Compression reinforcement ratio

σs Stress in the tension steel

Symbols

34

σsr stress in the tension steel calculated on the basis of a cracked section under loading conditions

causing first cracking

σswd Shear reinforcement stress

φ Creep coefficient

φu Curvature at ultimate load

φy Curvature at yield load

ψ Rotation of slab outside the column region

ϕw Diameter of the shear reinforcement

Introduction Chapter 1

35

Introduction

1.1 Background

Reinforced concrete wide beam slabs are widely used in buildings and bridge decks. They are

characterised by relatively shallow wide beams of substantial width that support one-way spanning

slabs. Wide beams are typically much wider than the supporting columns. The use of such a structural

system is advantageous as it is simple and fast to construct. For typical domestic and office live loads,

wide floors are a cost-effective solution for slab spans of up to 9 m between the centreline of supports

and beam spans of up to 16 m (1). Additionally, large and small holes can be easily accommodated. The

slab can be either solid or ribbed as illustrated in Figure 1-1. In the case of ribbed slabs, the wide beam

depth is usually similar to the overall depth of the ribbed slab, while for solid slabs the wide beam depth

is greater than the slab thickness. Beams are classified as wide in this research if their width is greater

than three times their depth. This research addresses the behaviour and design of solid slab wide beam

floor systems.


36

(a) (b)

International design codes for concrete structures such as BS8110 (2), EC2 (3) and ACI318 (4) give

comprehensive guidance on the design of flat slabs as well as one- and two-way spanning slabs

supported on beams of similar width to columns. These design codes, however, give no guidance on

the design of wide beam slabs. In the UK, design guidance on wide beam slabs is provided by The

Concrete Centre (TCC) (1,5). The TCC design procedure for wide beam slabs is similar to that used for

conventional one-way slabs but with additional checks for punching shear around columns (1,5).

Flexural design methods based on either elastic analysis or yield line theory are also allowed.

The flexural design method recommended by TCC assumes a uniform distribution for transverse

hogging moments over wide beams. However, numerical and experimental studies (6,7) have shown

that the transverse moment distribution at columns is far from uniform and the flexural behaviour of a

wide beam slab is more similar to that of flat slabs. This suggests that crack widths near columns may

exceed permissible limits at the serviceability limit state.

Further, experimental evidence (8,9) shows that punching shear failure could occur in wide beam slabs.

TCC (5) procedure for calculating punching shear resistance for wide beam slabs is similar to that given

by EC2 (3) for flat slabs. This overlooks the fact that loads are introduced into columns less uniformly

in wide beam slabs than in flat slabs.

1.2 Aims and objectives

The overall aim of the research is to develop an improved design method for wide beam slabs which is

consistent with their elastic response and satisfies both the serviceability and ultimate limit states. This

has been achieved by:

Figure 1-1: Types of wide beam slab: (a) solid slab with band beams, (b) ribbed slab with wide beams (1)


37

Developing nonlinear finite element (NLFEA) procedures for modelling the behaviour of wide

beam, slabs in flexure and shear.

Designing a typical wide beam slab for use as a case study.

Investigating the influence on structural response of varying the transverse (perpendicular to

direction of beam span) hogging flexural reinforcement arrangement along the length of wide

beams. The analysis investigates the effect of i) uniformly distributing the transverse hogging

reinforcement along the length of the beam as suggested by TCC and ii) placing the transverse

reinforcement in bands based on the elastic moment field. The comparisons are in terms of the

steel strain, crack width and deflection at quasi-permanent load. In addition, comparisons

include failure loads, failure modes and bending moments in slabs, beams and columns.

Developing improved guidance for distributing transverse hogging flexural reinforcement

along wide beams based on elastic FEA.

Using FEA to investigate the shear force distribution around the punching shear control

perimeter at internal and edge columns of wide beam slabs.

Predicting the punching shear resistance of wide beam slabs at internal and edge columns with

fib Model Code 2010 (MC2010) (10), EC2 and NLFEA. This includes studying the influence

of varying the transverse rebar distribution on punching shear resistance at internal and edge

columns.

Determining the beneficial effect of flexural continuity and compressive membrane action

(CMM) on the punching resistance of wide beam floors.

Carrying out parametric studies to systematically investigate the effect of varying the

reinforcement distribution on reinforcement strains and crack widths at the quasi-permanent

load and ultimate limit states.

Finally, design recommendations are made for wide beam slabs.

It is noteworthy that this research is based entirely on numerical analyses (elastic FEA and NLFEA).

Two software have been implemented namely: DIANA v 9.6 (11) and ATENA v 5.1.1 (12).

1.3 Thesis Organization

Chapter 1 describes the research problem and scope of work. It also underlines the objectives and the

organization of the thesis.

Chapter 2 describes the background to the flexural design methods for one way solid slabs and flat slabs

in BS8110, EC2 and ACI318. The Equivalent Frame Method (EQFM) and the yield line theory are also

reviewed. The principles of designing reinforcement for slabs according to their moment fields are

outlined as well. In addition, Chapter 2 reviews punching shear design provisions in EC2 and MC2010

as well as discussing the basis of the Critical Shear Crack Theory (CSCT) on which the MC2010 design


38

recommendations are based. Finally, the control of deflection with span-depth ratios and crack width

calculations according to EC2 are reviewed.

Chapter 3 reviews previous experimental and numerical studies into wide beam slabs subjected to static

uniformly distributed loading. The main focus is on the transverse hogging bending moment distribution

along wide beams and design for shear in wide beams. Both one-way shear and punching shear are

reviewed with emphasis on the influence on shear strength of wide beam width and lateral spacing of

stirrups.

Chapter 4 defines the constitutive models used in the NLFEA for concrete in compression, tension and

shear. Similarly, the adopted constitutive models for reinforcing steel are described. Brief descriptions

are given of the solution methods adopted in nonlinear analyses and the convergence criteria applied.

Other issues which influence the nonlinear analysis results such as mesh size, mesh element type and

boundary conditions are highlighted.

Chapter 5 describes the studies carried out to validate the DIANA and ATENA modelling procedures

used in the research. In the modelling of wide beam slabs, DIANA is used with shell elements to

simulate the behaviour of wide beam floors in flexure, while ATENA is used with solid elements to

model shear failure at edge and internal column connections. The test data used in the validation were

carefully selected to be representative of the issues involved in the modelling of wide beam slabs. These

issues include the FE model’s ability to capture the flexural behaviour of slabs at all loading stages.

Also examined is the capability of the DIANA analysis to predict enhancement in strength and stiffness

due to the effects of CMA. It is shown that punching shear resistance can be accurately evaluated using

MC2010 LoA IV with rotations from NLFEA using DIANA. ATENA software is used to estimate

linear shear and punching shear resistance of sub-assemblies comprising solid elements. The ATENA

sub-assembly analyses are validated against tests of beams without and with transverse reinforcement

as well as slabs failing in punching without and with shear reinforcement. The analysis of punching

includes slabs with both axis-symmetry as well as non-axis-symmetry conditions. Moreover, the sub-

assembly is used to simulate tests with different types and arrangements of shear reinforcement.

Chapter 6 presents a case study of the type of wide beam floor considered in this research. The results

of the case study are used to inform the development of a rational design method for wide beam slabs.

The case study focusses on the design of hogging reinforcement transverse to the direction of span of

the wide beam. The influence of transverse reinforcement arrangement on transverse bending moment

distribution is studied for uniform and banded arrangements of transverse reinforcement. The banded

reinforcement arrangement is based on the elastic bending moment field. Behaviour is studied at both

the serviceability limit state (SLS) and ultimate limit state (ULS). The main variables considered in

these analyses are steel strain, crack width and deflection. In addition, comparisons are made for the


39

two reinforcement arrangements of bending moment distribution in slabs, beams and columns as well

as failure load and failure mode. The effect of compressive membrane action is examined as well.

Chapter 7 investigates the punching resistance of wide beam slabs using EC2, MC2010 and NLFEA

with emphasis on the influence of slab geometry on the shear force distribution around the control

perimeter of internal and edge columns. It also studies the influence on punching shear resistance of

banding transverse reinforcement over internal and edge columns. The beneficial effects of flexural

continuity and CMM on punching resistance for uniform and banded transverse reinforcement

arrangements are also investigated. Finally, the distribution of the shear forces along the critical

perimeters around columns are investigated and the values of ke and β accounting for the influence of

eccentricity in MC2010 and EC2 respectively are reviewed.

Chapter 8 describes the parametric studies carried out to establish simplified rules for determining

banded transverse steel distributions that satisfy the design SLS and ULS conditions. Rules are

developed for determining the width and amount of reinforcement required in each band. The chapter

also investigates the influence of transverse and longitudinal flexural reinforcement ratios on punching

shear resistance calculated according to MC2010 LoA IV and EC2. The modelling of eccentric shear

in the punching provisions of fib MC2010 and EC2 is also reviewed for wide beam slabs and design

recommendations are made.

Chapter 9 summarizes the conclusions reached throughout this research. It also highlights the

limitations of this work. Furthermore, suggestions for future work in order to tackle these shortcomings

are proposed.

Literature review-Background on Structural Design Methods for RC Slabs in Codes of Practice Chapter 2

40

Literature Review – Background on Structural

Design methods for RC Slabs in Codes of Practice

2.1 Introduction

The design provisions of EC2 (3) and ACI 318 (4) and the superseded BS 8110 (2) for slab design are

the outcome of practical experience as well as extensive theoretical and experimental research. This

chapter reviews the background to the flexural design methods in these design codes for RC solid slabs

spanning in one direction and two-way flat slabs. In addition, it highlights the fundamentals of

alternative code permitted design methods such as yield line theory and design based on elastic analysis.

The background to the EC2 and fib MC2010 (10) design provisions for punching shear are reviewed in

detail. This includes the Critical Shear Crack Theory (CSCT), which is the basis of the MC2010 design

method for punching. Finally, the EC2 design procedures for deflection and crack width calculations

are outlined.


41

2.2 Flexural design for RC slabs spanning in one direction

In general, building design codes, such as EC2 and ACI 318, allow the use of any design method that

satisfies equilibrium and geometrical compatibility, provided that the design strength at any section is

not less than the required strength and serviceability conditions are met. Codes of practice usually treat

solid slabs spanning in one direction in a similar manner to beams of unit width. Approximate values

for span and support bending moments are tabulated in ACI 318 and BS8110 but not EC2. For flat

slabs, several design methods are recommended in codes of practice. This chapter explains in detail the

direct design method and the Equivalent Frame Method (EQFM) of ACI 318 and EC2. Use of Yield

Line Theory (YLT) and the Finite Element Method (FEM) for slab design are also discussed. The

assumptions and limitations associated with each method are underlined.

UK Practice (CP 110, CP 114, BS 8110 and EC2)

Bending moment coefficients for one way slabs were provided in early codes of practice in both the

UK and USA as an alternate to frame analysis provided that certain conditions were fulfilled. The

coefficients were based on the results of elastic analysis for slabs with different spans, with limited

moment redistribution. In the UK, for instance, CP 114: 1957 (13) provided approximate coefficients

for calculating bending moments in slabs spanning in one direction. Table (15) in CP 114: 1957 is

represented in Table 2-1. This approach requires slabs to be continuous over three or more equal spans.

Two spans are considered approximately equal, according to CP 114 when they do not differ by more

than 15% of the longest. Another condition is that slabs should be subjected to uniform loading.

Table 2-1: Approximate values of bending moments in uniformly loaded beams and slabs continuous over three or more spans as in CP114 (13)

Near middle of end span

At support next to end support

At middle of interior spans

At other interior supports

Moment due to dead load + 12 − 10 + 24 − 12

Moment due to imposed load + 10 − 9 + 12 − 9 Note: = total dead load per span, = total superimposed load per span and is the effective span.

Table 2-2: Ultimate bending moments in continuous beams and one way slabs according to CP110 (15)

Note: F= total design ultimate load (1.4Gk + 1.6Qk).

These factors remained the same in the following CP 114: 1969 (14). However, changes were made in

1972 when CP 110 was released. The total dead and superimposed loads were combined together and

At outer support Near middle of end span

At first interior support

At middle of interior spans

At interior supports

0 +

11 −

9 +

14 −

10


42

replaced by the total design ultimate load. The additional condition that the characteristic imposed load

should not exceed the characteristic dead load was added. Consequently, the factors were changed as

shown in Table 2-2 which is extracted from Table (4) in CP 110: 1972 (15).

BS 8110: 1985 (16) adopted the same bending moment coefficients as CP110 apart from a minor change

to the coefficient at the interior support from -0.1 to -0.08. In 1993, the bending moment coefficients in

BS 8110: 1985 were modified as shown in Table 2-3. These modifications reflect the effect of end

support/slab connection conditions on the longitudinal distribution of the total moment in one way slabs.

No further modifications were made in BS 8110: 1997 (2) and its subsequent amendments prior to its

withdrawal and replacement with EC2 (17) in 2010. In the current EC2, no values are specified for the

bending moments in slabs spanning in one direction as this is considered to be text book information.

Table 2-3: Design of ultimate bending moments in uniformly-loaded continuous beams and one-way spans as in BS8110 (2)

End support/slab connection At first interior support

Middle of interior spans

Interior supports

Simple Continuous

At outer support

Near middle of end spans

At outer support

Near middle of end spans

0 0.086FL -0.04FL 0.075FL -0.086FL 0.063FL -0.063FL

Note: F= total design ultimate load (1.4Gk + 1.6Qk), and L is the effective span.

USA Practice (ACI 318)

The design rules for determination of bending moments for one way slabs in ACI 318 are applicable if

the longer of adjacent spans does not exceed the shorter by more than 20%. Additionally, the slab should

be uniformly loaded and the live load should not exceed three times the dead load. When compared

with UK codes of practice, it can be seen that the ACI codes provisions for calculating bending moments

in one way slabs provide more details in terms of the relative beam/slab stiffness to column stiffness.

ACI 318-56 (18), for example, gives estimations for bending moments in slabs with two or more spans.

It also considers cases where end spans are unrestrained as well as built integrally with the support.

Moments are also given at the first interior support of slabs with two or more spans compared with three

in UK codes of practice. Moreover, the design moment at the interior face of a support depends on the

type of support. Types of support taken into consideration are spandrel beams, girders and columns.

Table 2-4 shows the approximate design values for bending moments in the ACI 318-56 (18). The


43

bending moment coefficients for one way spanning slabs in Table 2-4 from ACI 318-56 have remained

unchanged in later revisions of the code.

In summary, the superseded UK codes of practice and ACI 318 provide coefficients for the calculation

of bending moments in one-way spanning slabs based on elastic analysis with limited moment

redistribution. It is tacitly assumed that the bending moments are distributed uniformly across the width

of slab. This assumption is strictly accurate for slabs which conform to the geometrical and loading

limitations specified in the codes. In practice, however, slabs of irregular shape with non-uniform

column grids are increasingly used. Thus, alternative methods are needed since simplified tabular

design methods are not always applicable.

Table 2-4: the approximate design values for bending moments in the ACI318-56 (18) up to date

Sagging Moment

(Positive)

End Spans

Discontinuous end

Unrestrained 111

Integral with support 114

Interior Spans 116

Hogging Moment (Negative)

At exterior face of first interior support

Two spans 19

More than two spans 110

At other faces of interior supports 111

At face of all supports for Slabs with spans ≤ 10 ft. (3.048 m) 112

Beams and girders where column stiffness to beam stiffness > 8.0

112

At interior faces of exterior supports for members built integrally with their supports

support is a spandrel beam or girder 124

support is a column 116

Note: = uniformly distributed load per unit area of slab.

2.3 Flexural design for RC flat slabs

Flat slabs are solid slabs that are supported directly by columns. They may be thickened with drop

panels in the column area to increase punching resistance. The column cross section can also be locally

increased with a capital to increase punching resistance. Flat slabs are typically designed using one of

the direct design method of ACI 318, the equivalent frame method, Johansen’s yield line method or the


44

finite element method. Explicit provisions are available for the design of flat slabs using the equivalent

frame method in EC2 and ACI 318. This section reviews these methods with emphasis placed on the

determination of the total static moment and the longitudinal and transverse distribution of the total

moment in a slab panel.

Direct Design Method

The direct design method of ACI 318 and the empirical design method of BS 8110 provide simple

coefficients for calculating bending moments in slabs. To apply the direct design method, certain

conditions need to be satisfied. The slab system should consist of a minimum of three spans in each

direction with adjacent spans differing by no more than one-third of the longer span. This prevents the

development of hogging moments beyond the point where the top rebar is curtailed (19). In addition,

panels should be rectangular with aspect ratio between 0.5 and 2.0. Further, columns must be located

near the corners of panels with a maximum offset of 10% of the span from the column centreline in

each direction. In the case of pattern loading, the imposed load should not exceed twice the dead load.

The direct design method involves three basic steps. First, the determination of the total static bending

moment of the panel under consideration. Then, the division of total factored bending moment into

midspan (positive) and support (negative) moments. Finally, the distribution of factored midspan and

support moments to column and middle strips and to the beams if any.

The total static moment is defined as the sum of the average support moment and the midspan moment

in a slab panel. In 1914, Nichols (20) studied the equilibrium of a slab free body of a typical interior

panel bounded by the centrelines of adjacent panels supported on circular columns as shown in Figure

2-1. He suggested that the total static bending moment in a slab panel is as follows:

= + = (2.1)

Where defines the total uniformly distributed load on the panel, is the slab span between support

centres and , are the average support, midspan and total design moments respectively.

Nichols’s work was verified later by Westergaard and Slater (21) who analysed several interior slab

panels using elastic analysis. Nevertheless, ACI 318-20 (22) adopted smaller value for the total design

moment than that suggested by Nichols. The total design moment given by ACI 318-20:

= 0.09 1 − (2.2)

Where denotes the total uniformly distributed load on the panel, is the slab span between support

centres and is the size of support in the span direction.


45

ACI 318-56 introduced a correction factor, F to the ACI 318-20 (22) expression for the total design

moment which depends on the ratio between the column diameter and slab span:

= 0.09 (2.3)

= 1.15 − ≥ 1.0 (2.4)

Equations (2.3) & (2.4) were used in ACI 318-63 (23) for calculating the total design moment without

major modifications (only the 0.09 was substituted by 0.1). Conversely, ACI 318-95 (24) follows

Nichols’s expression but defines the maximum support moment as that along the faces of supports

perpendicular to the span.

= − (2.5)

The superseded UK code CP 114 defined the total design moment for flat slabs in a similar manner to

that of the ACI code as follows:

= − (2.6)

where is total load per unit area on the panel and is the diameter of the column head. However,

equation (2.6) was omitted in CP 110: 1972 (15) and replaced by:

= − (2.7)

Figure 2-1: The slab section proposed by Nichols, 1914.


46

In ACI 318 for an interior panel, the total static bending moment is divided into 65% for support

moments, located at the support face, and 35% for the midspan moments. For an end panel, the moments

are calculated using the equivalent column stiffness as defined by Corley, et al., (25) among others

(26,27). This yields the moment coefficients for an end span which have long been used in ACI 318

codes. Table 2-5 lists the moment coefficients for an end span as shown in ACI 318-11 (4).

The direct design method is a useful tool for designing flat slabs within its constraints. Nevertheless, it

is clear that a more general method is needed.

Table 2-5: the moment coefficients for an end span according to ACI 318-11 (2)

Exterior edge unrestrained

Slab with beams

between all supports

Slabs without beams between interior supports

Exterior edge fully

restrained

without edge beam

with edge beam

Interior negative factored moment

0.75 0.70 0.70 0.70 0.65

Positive factored moment

0.63 0.57 0.52 0.50 0.35

Exterior negative factored moment

0 0.16 0.26 0.30 0.65

Equivalent Frame Method (EFM)

The equivalent frame analysis was initially introduced into ACI 318 as a unified design method for two

way slabs, with or without beams, to tackle the shortcomings and limited applicability associated with

the direct design method and other empirical methods. It splits the structure into a series of parallel

plane frames in both longitudinal and transverse directions. Each frame is bounded laterally by the

centrelines of adjacent panels and is subjected predominantly to gravity loads. The equivalent frame

method for slab systems was first codified in the California Uniform Building Code in 1933, USA.

Since then, several modifications have been applied to obtain closer results to those from the empirical

method, such as those proposed by Peabody (28) and Dewell (29). These modifications were mainly

incorporated in ACI 318-41 (30) and ACI 318-63 (23). Despite these amendments, the equivalent frame

method was criticized because it did not account for the discontinuity between the slab and columns or

provide a unified design procedure for other types of slabs such as two way spanning slabs supported

on beams. In response to these issues, a large research programme was launched in 1956 by the


47

University of Illinois at Urbana-Champaign, Department of Civil Engineering (31). The programme

involved testing five nine-panel slabs with different rebar designs. Tests were performed for the five

prototypes with scale of ¼. In addition, the Portland Cement Association carried out another test of a

¾-scale nine-panel flat slab (32). Based on the results of these tests and numerical analyses of a large

number of slabs, new provisions were included in the ACI318-71 (33). The significant changes in ACI

318-71 proposed by Corley (27) were associated with the definition of stiffness of the frame members

and the computation of the fixed-end moments in order to provide design solutions for two-way slabs

as well as flat slabs.

2.3.2.1 The determination of stiffness of frame members

According to the ACI318 EFM, bending moment is transferred from the slab to the column by an

assemblage consisting of the column and transverse beam as shown in Figure 2-2. The column is

assumed to have infinite stiffness throughout the depth of the joint. The moment of inertia Isc of the

slab-beams from the column centreline to the face of the column, bracket or capital is defined as:

= 1 − (2.8)

where Isd is the moment of inertia of the slab-beam at the face of the column, bracket or capital. c2 and

L2 are the column size and span between the centrelines of the supports both measured perpendicular to

the direction of span being considered. Use of the increased stiffness Isc within the support width gives

good comparisons with test results (31,32). Moreover, it gives a valid solution for slabs supported on

walls since the moment of inertia of over walls becomes infinite.

Another important aspect is the determination of flexural stiffness of columns. The ACI318 EFM

accounts for the discontinuity in width between the slab and columns and the torsional stiffness of the

transverse member framing into the column. Previous EFMs were based on the simple frame solution

proposed by Dwell (29) in which the stiffness of column and slab correspond to the moments of inertia

of their respective sections. This procedure, however, underestimates midspan moments especially in

exterior panels and overestimates column moments (34). Corley (27) resolved these issues by

introducing a hammerhead-like equivalent column which consists of the column above and below the

slab and the transverse beam or slab as shown in Figure 2-2. The geometry of transverse beam section

considered for calculating the moment of inertia is illustrated in Figure 2-3


48

In this assemblage the transverse beam is allowed to rotate irrespective of the column stiffness. The

induced torque is assumed to vary linearly along the beam from zero at midspan of the beam to a

maximum at the column. The stiffness of assemblage can be written in terms of flexibilities using the

Cross distribution procedure (35) as follows:

= + (2.9)

where Kc is the flexural stiffness of the columns above and below the slab and Kt is the torsional stiffness

of torsional member, whether it is a beam or part of the slab, per unit rotation. More details are given

in the building codes regarding the calculation of effective width of the torsional member. The torsional

Figure 2-3: Cross section of transverse beam considered for moment of inertia calculations: (a) internal beam, (b) edge beam

Figure 2-2: Equivalent column section as proposed by Corley & Jirsa, (18)


49

member is assumed to be effective to either side of the column width c2. The average flexibility of the

transverse member is defined as (27):

= 1 − (2.10)

= ∑ 1 − 0.63 (2.11)

Where G is the shear modulus of elasticity, and are the shorter and longer dimensions of a

rectangular area of a cross section. The rectangular areas should be divided to minimize the common

lengths. Figure 2-4 illustrates the beam and slab sections used for torsion constants calculation. x is the

shorter side of a rectangular area and y is the longer side of the same area.

The design procedure proposed by Corley (27) has been the basis of the equivalent frame analysis for

slab design in the ACI 318 code since 1971. Nevertheless, it has been criticised in several aspects by

researchers including (36,37). For instance, the physical model which represents the equivalent column

is criticised for being artificial and unrealistic. Also, the approximations used to relate the frame

considered in the analysis to a slab system are complicated and ambiguous and in some cases deviate

from the results of linear elastic analysis. As a result, researchers (38-42) have proposed modifications

to the equivalent frame method. With regards to the equivalent column, Fraser (38) suggested modifying

the column moments and retaining the column as it is. Long (43) suggested, based on analysis and test

results, a direct method for determining the equivalent column stiffness.

BS 8110 recommends dividing the flat slab panels into column and middle strips as for the simplified

method. Thus, the structure comprises of a series of intersecting frames in the longitudinal and

transverse directions. The calculation of stiffness is based on uncracked sections. Unlike ACI, BS 8110

does not account for the relative discontinuity in the connection between the slab and column by

Figure 2-4: Sections used in calculating torsional constants for: (a) beamless slab, (b) slab with beam.


50

modifying the stiffness of column. Instead, the design moment transferred to the column by the column

strip width is limited to 0.15 where fcu is the concrete cube strength. In EC2 the limit is given

as 0.17 where fck is the concrete cylinder strength.

2.3.2.2 Division of moments between column and middle strips.

The equivalent frame method involves dividing the flat slab panels into column and middle strips. For

slabs without drop panels, the width of column strip is taken as one half of the shorter panel width, with

the remainder of the panel width being the middle strip. Figure 2-5 shows the column and middle strips

for an interior panel. After calculating the total design bending moment in a slab panel, this moment is

distributed across the column and middle strips. In general, building design codes, including BS 8110,

ACI 318 and EC2 assign larger portion of the design sagging and hogging moments to column strip.

Table 2-6 gives comparison of the sagging (positive) and hogging (negative) moments in percentage in

an interior panel assigned to column and middle strips in BS8110, ACI 318 and EC2.

Table 2-6: Distribution of an interior panel moment between column and middle strips according to BS 8110, ACI 318 and EC2

strip BS 8110 ACI 318 EC2 Negative Moment

Positive Moment

Negative Moment

Positive Moment

Negative Moment

Positive Moment

Column strip 75 55 75 60 60-80 50-70 Middle strip 25 45 25 40 40-20 50-30

Figure 2-5: column and middle strips in an interior panel


51

Yield Line Method

2.3.3.1 Background

The yield-line method is an upper bound method for designing reinforced concrete slabs. The basis of

yield line method was first introduced by Ingerslev (44), but it has been significantly developed and

become well-established method due to the pioneering work of Johansen (45,46). The early popularity

of yield-line method among designers pertained to its simplicity, versatility and economy for slabs with

and without beams as well as regular and irregular shapes. Moreover, good agreement between the

actual collapse loads of concrete slab bridges of short and medium spans and predicted ones using yield-

line theory has been reported by Clark (47). Thus, it is a valid approach for both design of new slabs

and assessing existing ones. More recently, elastic design methods have become prevalent for slabs due

to the availability of user friendly finite element packages.

In yield-line design, the ultimate load is evaluated by postulating a collapse mechanism that is consistent

with the boundary conditions. The term ‘yield-line’ refers to a band of cracking across which the steel

reinforcement has yielded and along which plastic deformations occur. At failure, yield-lines divide the

slab into rigid plane regions that rotate about the yield lines and pivot about lines of support. The

maximum bending moment at any point in the slab is not greater than its ultimate moment of resistance.

In yield-line analysis the key objective is to identify the yield line pattern with the least failure load.

Thus, it is crucial to examine all the possible basic collapse mechanisms. Some design references, such

as practical yield line design (48) and other text books (19,49,50), give sets of rules to facilitate the

determination of the potential yield line patterns.

A number of assumptions are made in the analysis of a reinforced concrete slab using yield-line theory.

First, the slab sections must have sufficient ductility to develop the collapse mechanism since the slab

may undergo considerable moment redistribution. The ductility of a reinforced concrete section is

measured by the ratio of curvature at ultimate load and yield load, (φu/φy), and is known as the curvature

or ductility factor. Design codes, such as EC2 provide constraints on the neutral axis depth and

reinforcement ductility to ensure failure is characterised by significant reinforcement yielding rather

than concrete crushing. For example, EC2 does not allow the ratio of xu/d to exceed 0.25 for concrete

class less than C50/60 and ductility class B or C should be used for reinforcement. Previously, BS 8110-

1985 limited the ratio between support and span moments to 0.5-1.5. In the 1997 version, BS 8110

restricted the ratio to be similar to that from elastic theory.

Another assumption is that large deformations are concentrated in the yield lines while the elastic

deformations at any point in the rigid portions between yield lines are ignored. Further, the slab is

assumed to be governed by a flexural collapse mode. This implies that the shear and punching shear

strengths of slab can resist the applied shear forces adequately.


52

2.3.3.2 Flexural reinforcement distribution

The steel reinforcement in a slab system designed according to the yield line method is usually uniformly

distributed and the steel ratio does not vary across the slab. However, different reinforcement ratios can

be used for the top and bottom reinforcement in each direction. Reinforcement can also be placed in

directions other than 90° where appropriate as in skew slabs.

2.3.3.3 Johansen’s yield criterion

Johansen (45) introduced a yield criterion for a slab element reinforced in the x- and y-axes and

subjected to ultimate loads which gives applied moment field per unit width, mx, my and mxy in the x-

and y-axes. The bending moments of resistance of the element in the x- and y- axes, mux and muy,

correspond to the field moments mx, my and mxy. In cases where the yield line lies at an angle other than

90⁰ to the x- and y- axes, the moment of resistance about the yield line is assumed to be related to the

moment of resistance in the x- and y- directions. The actual yield line is replaced with an equivalent

stepped yield line that consists of a number of steps at right angles to the reinforcement in order to

eliminate the torsional effect as shown in Figure 2-6. It is assumed that the ultimate bending moments,

mux, and muy are principal moments, i.e., twisting moments equal zero. Equation (2.12) considers the

equilibrium of moments about the yield line:

= + (2.12)

Similarly, the twisting moment, munt is found as follows:

= − (2.13)

Figure 2-6: Stepped method yield criterion by Johansen (45)


53

For isotropic reinforcement, the moment of resistances are similar and the twisting moment equals zero

in all directions. According to Johansen, the yield strength of a slab element is attained when the normal

component of moment field, mx, my and mxy, equals the ultimate normal moment of resistance.

+ + 2 ≤ + (2.14)

Equation (2.12) has been criticised for various reasons. First, twisting moments are set to zero when

calculating the moments of resistance, mux, and muy. Nevertheless, slab tests carried out by several

researchers (51-54) show that twisting moment has little effect on mux, and muy. Second, calculating mun

as the algebraic sum of the ultimate moments of resistance mux, and muy, rather than directly from the

forces acting on the reinforcement bar in the n-direction as in Figure 2-6, results in small errors in the

internal lever arm and hence mun. However, Jain & Kennedy (55) have reported less than 2% difference

between the mun values found by the two approaches.

Another assumption of Johansen’s yield criterion is that bars remain straight when crossing the crack

along the yield line. In reality, kinking can occur causing the forces in the steel bars to change direction.

Wood (50) investigated the effect of kinking on the ultimate moment of resistance, mun and found it to

increase the ultimate moment of resistance along the yield line. However, experimental evidence

reported by Lenschow (56) and others (55,57) shows the influence of kinking on the ultimate moment

of resistance to be insignificant.

Lastly, it should also be noted that the ultimate normal moment of resistance suggested in Johansen’s

yield criterion is sufficiently accurate for designing slab systems where membrane forces are not

present.

2.3.3.4 Determination of the ultimate load

There are two approaches for determining the ultimate load from the yield line patterns; virtual work

and the equations of equilibrium. The principle of virtual work is that at failure the work done by the

external loads must balance the internal energy dissipated. The segments of slab bounded by yield lines

are assumed to be rigid bodies since the plastic deformations are concentrated only at yield lines. The

sum of expenditure of external energy induced by external loads on all rigid bodies is equated with the

total energy dissipated due rotations at yield lines. Another method may be used for finding the ultimate

load is the equations of equilibrium in which the equilibrium is examined for each segment of the yield

line pattern under the applied loads and resulting actions. Although the equilibrium equations are

applied, the yield line theory is still an upper bound solution since the equilibrium is only studied at the

yield lines rather than throughout the slab.


54

2.3.3.5 Conclusion

The yield line design is concerned with slab analysis at ultimate limit state and gives a theoretical upper

bound solution. This is because yield line theory does not study the complete field of bending moments

throughout the slab. Thus, the conditions of serviceability, such as deflection and crack width, need to

be checked, particularly if the chosen distribution of moments deviates largely from the elastic one.

Flexural design in accordance with a predetermined field of moments

2.3.4.1 General

Designs based on elastic theory are considered as lower bound solutions. This means that both the

equilibrium and boundary conditions are satisfied throughout the slab. In addition, the yield criterion

must not be violated at any section in the slab. The design procedure involves the determination of

design elastic moment fields. It has become increasingly common to use finite elements programmes

based on elastic theory for slab analysis. As a result, the determination of a complete field of moments

mx, my and mxy can easily be obtained. However, designing the steel reinforcement in order to follow

the principal moment trajectories is not practical. Therefore, it is important to arrange the steel

reinforcement in a convenient way, usually in two orthogonal directions. The main challenge would be

to determine the additional amount of reinforcement needed to account for twisting moments.

The strip method, which was pioneered by Hillerborg (58), tackles this problem by eliminating the

twisting moments. Hillerborg postulates an imaginary field of moments without any twist moments,

i.e., mxy = 0, in which the general equation of equilibrium for a slab is replaced by strip actions in the x

and y directions as shown below:

The equilibrium equation for a slab is:

+ − = − (2.15)

The strip-action in the x and y directions is divided as follows:

+ 0 − 0 = − (2.16)

0 + − 0 = −(1 − ) (2.17)

where may have any reasonable value between 0 – 1 and is the external load.

Nevertheless, the strip method failed to deal with flat slabs, slabs with openings and re-entrant corners.

Consequently, Hillerborg (59) introduced radial stress elements in the advanced strip method to design


55

such slabs. More efficiently, Wood and Armer (60) developed a simple strip method to solve all cases

of practical slabs. They suggested that reinforcement be heavily distributed within a band of the slab

with considerable width, (i.e., sufficient to accommodate the reinforcement), to act as a beam. This band

is usually referred to as a strong band.

2.3.4.2 Arrangement of reinforcement at right angles

It should be noted that the strip method, as a lower bound approach, must satisfy the normal moment

yield criterion. This means, the normal moment resulting from the design field of moments must always

be less than the ultimate moment of resistance provided by steel reinforcement. This condition must be

met in all directions where yield line might occur.

For instance, consider P as an arbitrary point at line on slab with normal and transverse directions n and

t respectively as shown in Figure 2-7. If the normal moment mn, corresponds to the principal moments

of the stress field m1 and m2 in directions 1 and 2, then, to satisfy the normal moment yield criterion, it

should not exceed the moment of resistance mun provided by the reinforcement in that direction.

Hence, the transformation of moments given by Hillerborg as follows:

= + − 2 (2.18)

= + + 2 (2.19)

= ( − ) + ( − ) (2.20)

If mn and mt are principal moments, then the axis of mn is orientated at an angle ϕ which is given by

2∅ = (2.21)

Figure 2-7: a slab element with orthogonal reinforcement

ϴ

Positive Sign convention

ϕ


56

Wood (61) compared the lower-bound stress field of Hillerborg with Johansen’s stepped yield criterion,

which satisfies the following:

= − (2.22)

= − (2.23)

= ( − ) (2.24)

According to Wood (61), Hillerborg (59) adopted Equation (2.22) without referring to the work of

Johansen (45). That is probably because the moment-arm of reinforcement was calculated slightly

differently for mux. Wood concluded that mn from Equation (2.22) must always be greater than mn from

Equation (2.18)

The design moments of resistance in the orthogonal directions, x and y according to Hillerborg can be

calculated as follows:

For positive moment fields

= + K (2.25)

= + (2.26)

For negative moment fields,

= − K (2.27)

= − (2.28)

Hillerborg (62) suggested values for K and 1/K close to unity in order to obtain economical steel design,

where K is tan∅.

Hence, the most effective arrangement of reinforcement for positive moment fields is obtained when

= + (2.29)

= + (2.30)

The most effective arrangement of reinforcement for negative moment fields is obtained when

= − (2.31)


57

= − (2.32)

For mixed moment fields, difficult cases occur where one of the principal moments is positive in the

given moment field and the other is negative. Wood assumes that muy = 0 when evaluating mux and vice

versa. This yields:

= (2.33)

Thus, it is concluded that:

= + with = 0 (2.34)

= + with = 0 (2.35)

2.4 Punching shear design

Introduction

Punching failure occurs when a slab fails locally in shear around a column or concentrated load. The

failure is characterized by the separation of a truncated cone of concrete around the column and can be

accompanied by yielding of top steel reinforcement over the support. At early loading stages, a

tangential crack forms around the column perimeter because of the negative moments in the radial

direction. As the load increases, wider surface cracks develop radially due to the negative moments in

the tangential direction. These radial cracks divide the slabs into portions which rotate as rigid bodies

about axes near the column. The diagonal tension cracks start to develop at approximately two-third of

the punching failure load and tend to initiate at about half of the slab thickness from the periphery of

the column (34,63).

Existing design and assessment methods for punching shear at slab-column connections can be

categorized into four approaches. First, there are strut-and-tie models like that of Alexander and

Simmonds (64,65). Second, there are beam analogies in which the slab adjacent to the column is

modelled as a series of orthogonal beams, framing into column, subject to combined moment, shear and

torsion. Many such beam analogies have been developed by researchers (66,67). Third, there are

empirical nominal shear stress based models in which the design shear strength is defined on a basic

control perimeter. This approach is adopted in many codes of practice including ACI 318, BS8110 and

EC2. Fourth, there are mechanically based models of which the models of Kinnunen and Nylander (68),

Broms (69) and Muttoni (70) are representative. Kinnunen and Nylander (68) assume that punching

shear is resisted by the inclined concrete strut between the crack tip and the bottom of the slab as shown


58

in Figure 2-8. In their model, the failure occurs when the maximum shear strain at the bottom surface

of slab below the root of shear crack exceeds εc= -1.96% (63). They show that the flexural reinforcement

over the column affects significantly the punching resistance of the slab since it governs crack widths.

This conclusion has been confirmed by several researchers such as Yitzhaki (71), Muttoni (72) and

Dilger (73). Other parameters that influence punching shear resistance are concrete strength and the

size effect. The latter refers to the reduction in shear stress at failure with increasing slab depth. This

phenomenon has been observed by many researchers including Regan (34), Bažant (74), Muttoni (75)

and Birkle & Dilger (76) among others. Both EC2 and MC2010, but not ACI 318, consider the size

effect on punching shear calculation (see Sections 2.4.2 & 2.4.3).

The design rules for calculation of punching shear resistance including the definition of nominal shear

stress and control perimeters given by the fib MC 2010 and EC2 will be highlighted. Punching resulting

from both concentric and eccentric loading are discussed. In addition, some background is given about

the key parameters which influence the punching shear resistance of slab such as the flexural

reinforcement ratio, slab effective depth and column dimensions.

Review of Punching shear design in EC2

The EC2 design rules were derived from experimental observations and test results including those

carried out by Kinnunen and Nylander (68), Regan (77-79), Marzouk and Hussien (80), Hallgren (81),

Ramadane (82) among others. In order to calculate punching resistance of a slab without shear

reinforcement, two variables need to be defined: the control perimeter across which the potential

Figure 2-9: Kinnunen & Nylander Model for punching


59

punching failure may occur, and the shear strength of the slab (vRd,c). The basic control parameter u1 in

EC2 is constructed at distance 2d from the column faces with rounded corners. A further perimeter uout,ef

should be examined where shear reinforcement is no longer required. Figure 2-9 illustrates the model

adopted for punching shear in EC2 and the corresponding control parameters.

For concentric loading, the maximum shear stress is defined as:

=

(2.36)

where denotes the applied punching load, is the length of the control perimeter under

consideration and d is the algebraic mean of the effective depths of slab in the y and z directions.

The failure criterion implies that the maximum shear stress vEd must not exceed the shear resistance of

the slab vRd,c, which is given by the following expression:

, = , (100 ) (2.37)

= 1 + ≤ 2, = , , = .

in which fck is the concrete cylindrical strength, k is a factor accounts for the size effect and is the

mean of the reinforcement ratios in orthogonal directions. Equation (2.37) is similar to Equation (2.38)

below which was used in the superseded UK code BS8110. The differences are in the size effect, which

equals k in Equation (2.37) and (400/d) (1/4) in Equation (2.38), and the value of , . Additionally,

equations in BS 8110 are expressed in terms of the cube strength rather that the cylinder strength in

EC2.

, = . (100 ) (2.38)


60

When the applied load is eccentric with respect to the basic control perimeter, the unbalanced moment

is resisted by a combination of bending, uneven shear and torsion. The assumed distribution of shear

along the control perimeter in EC2 is justified by reference to the work of Mast (83). The maximum

shear stress due to the combination of applied load and moment transferred by uneven shear can be

written as follows:

= + (2.39)

= ∫ | |

where KMEd denotes the part of the moment transferred by uneven shear, K is a coefficient dependent

on the ratio between the column dimensions c1 and c2 along and transverse to the axis about which MED

acts, e is the eccentricity, W1 is a function of the basic control perimeter u1 and corresponds to a shear

distribution. Figure 2-10 shows the shear distribution due to unbalanced moment at an internal

connection according to EC2.

EC2 represents Equation (2.39) in a general form:

=

(2.40)

= 1 + ∙ (2.41)

Figure 2-10: Verification model for punching shear in EC2 and the control perimeters


61

Good agreement between the Equations (2.37) & (2.39) and test results have been reported by

Stamenkovic & Chapman (84) and Regan (34) among others. For braced structures and where the

adjacent spans do not differ by more than 25%, instead of applying Equation (2.41), EC2 allows the

design punching force to be increased by 15%, 40% and 50% for internal, edge and corner columns

respectively.

Slabs with shear reinforcement

Where shear reinforcement is required, it should be calculated using the following expression:

, = 0.75 , + 1.5 ∙ , (2.42)

, = 250 + 0.25 ≤ [ ]

where vRd,cs is the punching shear resistance of the reinforced slab, Asw is the area of shear reinforcement

per perimeter, fywd,ef is the effective design strength of punching shear reinforcement and α is the angle

between the plane of the slab and shear reinforcement, (for vertical links α = 90 ͦ). This implies that

EC2 limits the contribution of unreinforced concrete in resisting punching to 75% of the resistance

without shear reinforcement. Shear reinforcement should be placed between not more than 0.5d from

the column face or the loaded area and 1.5d inside the control perimeter at which shear reinforcement

is no longer required. EC2 requires at least two perimeters of link legs not more than 0.75d apart, with

minimum leg area for vertical links:

, ≥ 0.053 ∙ . . (2.43)

Figure 2-11: EC2 shear distribution due to unbalanced moment at internal connection


62

Where sr and st are the spacing of shear links in the radial and tangential directions respectively and fck

is the concrete strength and fyk is the yield stress of the links. For the first perimeter, the spacing of the

legs around a perimeter should not be greater than 1.5d and should not be more than 2d for perimeters

outside the control perimeter.

Review of Punching shear design in fib MC2010

2.4.3.1 General

Recently, Muttoni (75) presented the Critical Shear Crack Theory (CSCT) as a rational method for

punching shear in flat slabs. The CSCT has become the basis for the design of punching shear in the fib

MC 2010 (10). It relates the punching resistance of a slab to the width of critical shear crack, and hence,

to the rotation of slab relative to the column. In his CSCT, Muttoni has modified the model initially

developed by Kinnunen and Nylander (68) in which the failure criterion for punching shear of a slab is

defined as a function of its rotation at failure. However, their mechanical model involves rigorous

calculations, which makes it difficult to implement in the design codes of practice.

2.4.3.2 Failure Criterion according to the CSCT

The failure criterion describes the relationship between the punching shear resistance of a slab and its

rotation at failure with the resistance reducing as the slab rotation increases. Increasing rotation is

associated with the widening of the critical shear crack which reduces significantly the punching

resistance of the slab as observed from experiments (68,72,85). Muttoni & Schwartz (72) suggested that

the opening of the critical shear crack, w is proportional to the product of rotation and effective depth

of the slab (ψd). Across the critical shear crack, the shear is transferred by the action of aggregate

interlock which is largely influenced by the roughness of the crack. Walraven (86) and Vecchio &

Collins (87) relate the shear force carried by the critical crack roughness to the term (ψd / (dg0+dg)),

where dg0 is the maximum aggregate size and dg is a reference size taken as 16 mm. Based on that,

Muttoni developed the failure criterion for punching in slabs without shear reinforcement as follows:

= . (2.44)

where VR is the punching resistance and b0 is the control perimeter, dv is the effective depth of slab

accounting for penetration of the supported area in the slab and fc is the concrete strength. The fib MC

2010 recommends the use of the failure criteria described by the CSCT as shown in Figure 2-11 with a

slight modification in the right hand term of Equation (2.44) and the introduction of the partial safety

factor for concrete, , which results in the following:

= (2.45)


63

=. .

≤ 0.6 (2.46)

If the maximum aggregate size dg ≤ 16 mm, then kdg is given by:

= ≥ 0.75 (2.47)

For slabs with high strength or lightweight concrete, dg is set to zero because of the potential reduction

in the contribution of aggregate interlock.

The failure criterion described by the CSCT agrees very well with experimental data in literature,

however, some issues have been raised recently. In 2016, Broms (69), criticised the CSCT failure

criterion in many aspects. According to Broms, the assumption of the critical shear crack being

propagating down through the inclined compressive zone is inconsistent with the strain variation in

concrete prior to failure since failure is observed to start from the bottom of the slab at the column

perimeter. Moreover, the CSCT relates the width of the critical shear crack to the rotation of the slab,

which implies that punching shear resistance decreases in the long term due to the creep and shrinkage

of concrete due to increase in rotation with time. Nevertheless, it has been experimentally shown by

Moe (88) that the punching resistance increases slightly with time while the rotation increases as well.

The increase in resistance could be due to concrete strength increasing with time.

2.4.3.3 Control perimeters

The fib MC 2010 places the basic control perimeter b1, at 0.5 dv from the periphery of column. In

addition, it is necessary to define the shear-resisting control perimeter b0, along which the shear force

is assumed to be uniformly distributed. The length of the shear-resisting control perimeter b0 is

calculated by introducing a coefficient of eccentricity ke to account for moment transfer between the

slab and the column.

ψ

Figure 2-13: Correlation between opening of critical shear crack and rotation according to the CSCT


64

2.4.3.4 Coefficient of Eccentricity, ke

The fib MC2010 gives three methods to evaluate the coefficient of eccentricity which accounts for the

non-uniform distribution of the shear forces along the basic control perimeter. A non-uniform

distribution can occur for many reasons. For example, supports with relatively large dimensions may

lead to concentrations of shear force around corners. Additionally, the presence of large unbalanced

moments at columns results in uneven shear force distributions. In order to calculate ke, three methods

are suggested in MC2010:

(1) The shear field around the column is given by the elastic analysis, which accounts for shear

concentration along the control perimeter. Thus, can be calculated as follows:

= =, ,

× , , = , ,

, , (2.48)

where νperp,d,av and νperp,d,max are the average and maximum shear force per unit length perpendicular to

the basic control perimeter respectively.

(2) = (2.49)

where eu denotes the eccentricity of shear force with respect to the basic control perimeter, and bu the

diameter of a circle having the same area as that enclosed by the basic control perimeter b1. eu is defined

by the fib MC2010 as shown in Figure 2-12, and determined as follows:

= , + , (2.50)

In which VEd is the resultant of shear forces, MEd,x and MEd,y are the unbalanced moment about the x-

and y- axes respectively and c is the column size.

= + × × + × (2.51)

(3) Provided that the structure is braced by means other than frame action of slabs and columns

and the adjacent slab spans do not differ by more than 25%, the fib MC 2010 permits the

coefficient of eccentricity to be taken as 0.9, 0.70, and 0.65 for internal, edge and corner

columns respectively.


65

2.4.3.5 Punching shear strength

Fernandez Ruiz & Muttoni (89) extended the CSCT to the calculation of punching resistance for slabs

with shear reinforcement. Unlike EC2 which assumes a constant concrete contribution to punching

resistance, Fernandez and Muttoni assume that the concrete contribution varies with slab rotation. Thus,

the concrete contribution is significantly influenced by the amount of shear reinforcement which is not

the case for EC2. For slabs where shear reinforcement is required, the fib MC 2010 gives the following

expressions to estimate the shear reinforcement contribution to punching resistance:

, = ∑ (2.52)

= 1 + ∙∅

(2.53)

where ∑ is the sum of the cross-sectional area of shear reinforcement within the zone bounded by

0.35 dv and dv from the column face, is shear reinforcement stress, ∅ is the diameter of the shear

reinforcement, fywd and fbd are the yield strength and bond strength respectively. According to fib

MC2010, the slab should possess adequate deformation capacity, therefore the shear reinforcement

should resist at least half of the design punching load.

2.4.3.6 Calculation of load-rotation behaviour using the levels-of-approximation approach

According to fib MC2010, it is necessary to calculate the rotation along the two main directions of the

reinforcement. The larger rotation is used to estimate the punching shear resistance since it produces

the least strength. Based on the work of Muttoni & Ruiz (90), the fib MC2010 introduces four different

levels of approximation (LoAs) for establishing the load-rotation response. The degree of accuracy of

each level depends on the intended purpose of use whether it is for designing a new structure (LoAs I,

II & III) or assessing an existing structure (LoA IV). The main aim is to simplify the design process and

to avoid unnecessary laborious work especially at the preliminary design stages. At the same time, it

Figure 2-14: Calculation of eccentricity between the position of resultant shear force and the centroid of basic control perimeter (10)


66

ensures better results in terms of the strength and stiffness of members where more accuracy is needed,

as for the detailed design stages. The general expression of the load-rotation relationship specified in

the fib MC2010 is:

= 1.5 ∙ ∙.

(2.54)

where rs denotes the distance from column axis to line of contraflexure, i.e., zero moment, fyd the yield

strength of flexural steel, Es the elastic modulus of steel, mEd and mRd the average moment and average

flexural strength moments per unit length in support strip respectively. The width of support strip across

which mEd acts may be determined as follows:

= 1.5 ∙ , ∙ , ≤ (2.55)

where rs,x & rs,y corresponds to the distance from the support axis in x and y directions respectively to

the position where the radial moment is zero, and can be approximated as 0.22 Lx or 0.22 Ly for the x

and y directions respectively. Lmin refers to the slab span for the considered direction.

The application of the LoAs approach to estimating the punching shear resistance of a slab can be

summarized in the following steps:

LoA I: It is intended for preliminary design and suitable for regular flat slabs with aspect ratios between

0.5 and 2 and designed according to an elastic analysis method with limited moment redistribution. The

slab is assumed to be governed by its flexural capacity, and hence, all the top reinforcement over the

support strip yield and the slab fails in a very ductile manner. This leads to msd = mRd which yields:

= 1.5 ∙ (2.56)

LoA II: in this level the average moment acting in the support strip is defined in terms of the unbalanced

moment and the corresponding shear force, VEd. Depending on the support position, the mED can be

approximated for internal columns as:

= ∙ + ,

∙ (2.57)

For edge columns where the tension reinforcement is parallel to the edge:

= ∙ + ,

∙≥ (2.58)

For edge columns where the tension reinforcement is perpendicular to the edge:


67

= ∙ + , (2.59)

For corner columns where tension reinforcement in each direction:

= ∙ + , ≥ (2.60)

Where eu,i denotes the eccentricity of applied shear load with respect to the centroid of basic control

perimeter in the direction considered (see Figure 2-12). These simple approximations prove to produce

accurate predictions of rotations as reported by (75,91).

LoA III: Better estimation for the values of mEd and rs can be attained if the flexural reinforcement

design of slab is based on a linear elastic analysis, which implies the use of uncracked model. Also, the

influence of torsion is directly incorporated in the average moment mEd. In such case, the coefficient

1.5 in Equation (2.54) may be reduced to 1.2.

LoA IV: This level is primarily intended for an accurate assessment of the structure. The load-rotation

relationship is usually established directly from the output of nonlinear analysis for the structure. This

approximation level appears to be ideal for slabs with low flexural reinforcement ratios over supports,

which are influenced significantly by tension stiffening of concrete, and slabs with potentially large

moment redistribution. Nevertheless, the nonlinear analysis might be lengthy and time-consuming. In

addition, the accuracy of the results is not always guaranteed since it is largely dependent on several

factors including the choice of material modelling, mesh types and sizes and solution methods.

When comparing the shear reinforcement required for flat slabs required by MC 2010 and BS 8110,

Soares and Vollum (92) pointed out that fib MC 2010 can require almost double the shear reinforcement

required by BS 8110. Furthermore, there is no evidence that slabs designed to BS 8110 are unsafe in

practice. Soares and Vollum (92) made use of MC 2010 LoA IV to explain the satisfactory performance

of slabs designed with BS8110. They showed that flexural continuity in continuous slabs can

significantly reduce slab rotations below the values calculated in accordance with MC 2010 LoAs I to

III. At the same time Einpaul (93) carried out related studies into the influence of slab continuity on

punching resistance. On the basis of this work Einpaul (94) developed a simplified design method for

including the effect of slab continuity into the CSCT.

2.5 Calculation of Deflection in EC2

EC2 considers that the function and appearance of a RC beam or slab subjected to quasi-permanent load

could be impaired when the calculated sag of the element relative to the supports exceeds a maximum

of span/250. In addition, a limit of span/500 is specified for the deflection occurs after fit-out of finishes.


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These conditions are deemed to be satisfied if the structural element is dimensioned according to the

EC2 span/depth ratio rules. However, more rigorous checks should be carried out for deflection critical

structures. The EC2 limiting span/ effective depth ratios are as follows:

= 11 + 1.5 + 3.2 − 1.

≤ (2.61)

= 11 + 1.5`+ ` > (2.62)

where l/d is the limit span-depth ratio, K accounts for the structural system, ρ0 is the reference

reinforcement index = 10-3× and fck in MPa units. ρ & ρ` are the required tension and compression

reinforcement ratios at midspan respectively.

According to EC2, Equations (2.61) & (2.62) are calibrated for a mid-span cracked section steel stress

of 310 MPa under the design service load. The stress of 310 MPa corresponds to a reinforcement yield

strength of 500 MPa. For other stress levels, a correction factor should be applied, 310/σs which is given

by the following expression:

= ,

, (2.63)

As,prov & As,req denote the provided and required steel area at the section respectively. Table 2-7 lists

typical values for basic span/effective depth for RC members with rectangular cross sections and

without axial compression.

Table 2-7 can also be used for flanged sections but a correction factor of 0.8 should be applied if the

flange width is three times greater than the width of the web. For flat slabs, the long span is considered

in the calculation while for slabs spanning two-way the short span is considered.

Table 2-7: EC2 basic span-effective depth ratios for RC members without axial compression Structural system K Concrete highly stressed

ρ=1.5 % Concrete lightly stressed

ρ=0.5 % S.S beam, one or two spanning S.S. slab 1.0 14 20 End span of continuous beam or one-way continuous slab or two-way slab continuous over one long side

1.3 18 26

Interior span of continuous beam or one-way or two-way spanning slab

1.5 20 30

Flat slab based on longer span 1.2 17 24 Cantilever 0.4 6 8

The values in the table are based for rectangular cross sections class C30/35 and grade 500 steel.


69

Vollum (95) states that the EC2 span-depth ratio rules were derived from parametric studies in which

the total characteristic load (qtot = g + q) was assumed to equal 0.71 qu, where g is the total unfactored

load, q is the unfactored imposed load and the qu is the design ultimate load. The short-term construction

load was taken as the permanent load of g + 0.3q and is half the ultimate load. Vollum pointed out that

the choice of the construction load in the derivation of the EC2 span to depth rules is unconservative.

He suggested modifying the EC2 L/d ratios to account for the reduction in stiffness arising from the

severest cracking which could arise during construction.

EC2 offers rigorous calculation for deformation in cases where the conditions for applying the basic

span/depth ratio rules are not met. The behaviour of flexural members is an intermediate state of the

uncracked and fully cracked conditions. The predicted behaviour is formulated as follows:

= + (1 − ) (2.64)

where α denotes the deformation parameter, which can be strain, curvature or rotation. It may be taken

as deflection as well. α1, α11 refer to the parameters calculated for uncracked and fully-cracked sections

respectively. is a distribution factor accounts for the tension stiffening in the section and may be found

from the following expression:

= 1 − (2.65)

where is 0 for uncracked sections, β is a factor accounts for the duration and type of loading; β = 1.0

for a single short-term loading and β = 0.5 for sustained or repeated loads. σs is the stress in the tension

steel calculated on the basis of a cracked section and σsr is the stress in the tension steel calculated on

the basis of a cracked section under loading conditions causing first cracking. For flexure, the term

can be replaced by . Vollum (95) suggests that when calculating deflection under the quasi

permanent load should be calculated for the load case giving rise to severest cracking which could

arise during construction. In addition, EC2 recommends the use of tensile strength, fctm, and the effective

modulus of elasticity of concrete, Ec,eff which is given as:

, =( , )

(2.66)

= 22.

/ (2.67)

where φ (∞, t0) denotes the creep coefficient relevant for the load and time interval (the ratio of creep

strain to initial elastic strain). The effect of shrinkage can be evaluated as follows:


70

= (2.68)

where 1/rs denotes the curvature due to shrinkage, εcs denotes the free shrinkage strain, S is the first

moment of area of reinforcement about the centroid of the section, I is the second moment of area of

the section and αe is the effective modular ratio = Es/Ec,eff. Note that EC2 requires that both S & I to be

calculated for uncracked and fully cracked section and the final curvature is obtained from Equation

(2.64).

2.6 Flexural Cracking in EC2

Background

Generally, crack width needs to be controlled in RC structures to prevent the corrosion of steel

reinforcement, leakage and adverse appearance of the structure. When a slab is subjected to bending

moment greater than the cracking moment, flexural cracks will form in the tension side of the slab.

Flexural cracks form perpendicular to the axis of the slab if shear forces are insignificant, while diagonal

cracks indicate the presence of significant shear force. The accurate calculation of crack widths in RC

members is not generally possible due to uncertainties in loading, concrete material properties and

inaccuracies in modelling. The EC2 design crack width has a 20% chance of exceedance. This has been

adopted from the superseded CP110 as suggested by Beeby (96). The work of Beeby (97) has led to

better understanding of cracking behaviour in one-way spanning slabs. He showed that at any point in

the tension zone cracking is a combination of two patterns: (1) a pattern controlled by the initial height

of the cracks; (2) a pattern controlled by the proximity of reinforcement. The crack width in the former

pattern is a function of the crack height and is given by the following expression:

= ℎ (2.69)

In the latter the crack width is influenced by the concrete cover, bar diameter, the quality of bond

between the concrete and bars, the effective steel ratio and the ratio between the tension steel area and

the concrete area immediately surrounding the reinforcement bars. The crack width according to this

pattern is evaluated as follows:

= + ∅ (2.70)

where wlim is the crack width controlled by crack height, w0 is the crack width over bar. c is the cover,

ϕ is the bar diameter, k1 & k2 are constants. ρ is the effective steel ratio.


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EC2 design procedure

EC2 gives the following expression for calculating the design crack width, wk:

= , ( − ) (2.71)

where sr,max, is the maximum crack spacing, εsm denotes the mean strain in the reinforcement taking into

account the effects of tension stiffening and the effect of imposed deformations, εcm denotes the mean

strain in the concrete between cracks. The term (εsm - εcm) is given by the expression:

− =,

,( ,

≥ 0.6 (2.72)

where σs is the stress in tension steel calculated using the cracked section. kt is a factor accounts for the

duration of loading: 0.6 for short-term loading and 0.4 for long-term loading.

For cases where bonded reinforcement is fixed at reasonably close centres within the tension zone; i.e.

spacing = 5(c + ϕ/2), the maximum crack spacing is given by:

, = + ∅/ , (2.73)

where ϕ is the bar diameter, or an average bar diameter. If a mixture of bar diameters is used in a section,

then ϕeq should be used and is given for a section with n1 bars of diameter ϕ1 and n2 bars of diameter ϕ2

as:

∅ = ∅ ∅∅ ∅

(2.74)

c denotes the cover to the longitudinal reinforcement, k1 is a coefficient accounting for the bond

properties of the bonded reinforcement: 0.8 for high bond, 1.6 for plain bars, k2 is a coefficient

accounting for the distribution of strain: 0.5 for bending and 1.0 for pure tension. ρp,eff is the effective

reinforcement ratio As/Ac,eff , where As is the area of reinforcement within an effective tension area of

concrete, Ac,eff. Figure 2-13 defines the effective tension areas for typical cases according to EC2. The

recommended values for k3 & k4 are 3.4 and 0.425 respectively.


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EC2 specifies an upper bound to the crack width for cases where the spacing of the bonded

reinforcement exceeds 5(c + ϕ/2) or where there is no bonded reinforcement within the tension zone by

assuming a maximum crack spacing:

, = 1.3 (ℎ − ) (2.75)

It is noteworthy that the EC2 design procedure for crack width has been criticized by many researchers.

For example, after comparing the predictions of the theory which leads to the derivation of the parameter

ϕ/ρp,eff with experiment results, Beeby (98) concluded that the effect of ϕ/ρp,eff is minimal. Instead, he

suggested that the cover has the dominant influence on cracking behaviour. This conclusion is

consistent with the findings of the experimental work of Kong et al. (99). Furthermore, Forth et al.

(100) also questioned the EC2 method of crack width calculation in the presence of greater cover than

needed for durability. They recommend that the factor k3 should be reduced from 3.4 to 2.1 for flexural

elements.

2.7 Conclusions

The chapter reviewed the development of flexural design methods for RC solid slabs spanning in one

direction and two-way flat slabs in ACI 318, BS 8110 and EC2. These methods include the direct design

method, the EQFM and the yield line method. In addition, the design of flexural reinforcement for slabs

with predetermined moment fields was also discussed. The design rules for placing rebar at right angles

across slabs as proposed by Wood & Armer (61) were outlined. They have been followed to obtain the

design bending moments for wide beam slabs in Chapter 6. The chapter also discussed the current

design methods for punching in the EC2 and the fib MC2010. The design procedures for calculating

deflection and flexural cracking according to EC2 were described. Material from this chapter is used

elsewhere in the thesis. For example, in Chapter 6, the EC2 span-effective depth method is used to

Figure 2-15: Effective tension area according to EC2


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determine the required member depth for deflection control. For crack width calculation, Equation

(2.71) is used in Chapter 6, Section 6.3.5.4 to calculate crack widths along critical sections in the wide

beam floor under investigation. The term ( − ) was obtained directly from the FEA results using

DIANA (11). The FEA strains are compared with results from Equation (2.72) in which the bending

moment at SLS was calculated using the quasi-permanent load.

Previous Research into Wide Beam Slabs Chapter 3

74

Literature Review – Previous Research into Wide

Beam Slabs

3.1 Introduction

Wide beam flooring systems are used to minimise structural depth in long span reinforced concrete

floor plates. Unlike flat slabs, neither EC2 nor ACI318 give any guidance on the design of wide beam

slabs on which little published research is available. Design guidance published by The Concrete Centre

(1) suggests that wide beam slabs can be designed in the same manner as conventional one-way solid

slab and beam floor systems. This approach is questionable since recent studies by Tay (6) and Shuraim

& Al-Negheimish (101) show the flexural behaviour of wide beam slabs to be more similar to that of

flat slabs. This chapter reviews previous experimental and numerical studies into the bending moment

distribution in wide beam floor plates subject to static uniformly distributed loading.

Unlike conventional beams in one way spanning slab systems, wide beams can fail in punching shear

(two way shear) as well as in beam shear (one way shear) as shown experimentally by Lau and Clark

(8). Furthermore, design standards typically neglect the influence of the ratio of support to beam width

on shear resistance which can result in shear resistance being overestimated. Research also shows that


75

the shear resistance of wide beams depends on the distribution and configuration of transverse stirrups

across the beam width. The experimental and numerical programmes investigating the effect of these

parameters on the shear capacity of wide beams are briefly discussed and their design recommendations

are outlined. Finally, the current procedure for designing wide beam slabs according to TCC are

highlighted.

3.2 Transverse Distribution of Bending Moments in Wide Beams

General

Simplified design methods for wide beam slabs (1) typically assume that the transverse distribution of

bending moments is uniform along the length of the wide beam. This assumption is incorrect as shown

numerically by Paultre and Moisan (102) and Tay (6) as well as experimentally by Shuraim and Al-

Negheimish (101). This section briefly reviews these studies and their associated design

recommendations.

Research by Paultre and Moisan

Paultre and Moisan carried out a linear elastic finite element study to establish transverse moment

distribution factors across the width of wide beam slabs. The main variables were the panel aspect ratio,

slab thickness, and the width of drop panel. In their study, they used the term slabs with continuous

drop panels to refer to wide beam slabs. Figure 3-1 shows the slab system they considered in their study.

CSA-A23.3-94 (103) states that the proportion of hogging moment taken by the column strip should be

between 0.6-1.0 of the total hogging moment. They questioned using a factor as low as 0.60 which may

lead to cracking problems. Thus, the objective was to determine the proportion of hogging moment

assigned to the column strip that would be acceptable when continuous drop panels are used.

Their study considered a floor with four slab spans ranged between 4.0 m and 11.0 m, supported on

square columns above and below the slab fixed at their far ends, with different panel aspect ratios. The

wide beam width was not greater than column strip width. In addition, the overall depth of drop panels

was less than or equal to twice the slab thickness. Figure 3-2 shows the FE model used in the analyses.

The results of linear elastic analysis showed that, in the direction of span of the banded beam, the mid-

span moment equalled 33% of the total static moment. The support moments at the first internal column

and central column were 72% and 62% of the total static moment, respectively. These results were

consistent with those given in CSA-A23.3-94 (103). The transverse moment distribution, along the band

slab was discussed in detail for a model with a 200 mm thick slab spanning 8.0 m × 8.0 m with a

continuous drop panel 2.44 m wide and 0.4 m thick. Figure 3-3 illustrates the transverse bending

moment distribution in the internal bay for this model. The transverse hogging and sagging moments

assigned to the column strip were 94% and 84% of the total moments respectively.


76

Figure 3-1:- Sketch showing a typical slab panel with continuous drop panel considered by Paultre & Moisan (102)

Figure 3-2: FE model for slab with 4x4 panels as considered by Paultre & Moisan (102)


77

The continuous drop panel resisted 90% and 80% of the total hogging and sagging moments

respectively, while the remaining part of column strip resisted only 4% of the total hogging and sagging

moments. Paultre and Moisan observed significant variation in the transverse moment distribution along

the length of the drop panel. This opposes the common design practice of assuming transverse moments

are uniformly distributed.

Paultre and Moisan investigated the influences of the drop panel flexural stiffness relative to the slab

stiffness and panel aspect ratio on the transverse distribution of bending moment along the length of

continuous drop panels. To measure the relative rigidity of the wide beam, a parameter αD was

introduced. It is defined as the ratio of the flexural stiffness of the wide beam to the stiffness of slab

bounded laterally by the centrelines of neighbouring slabs either side. The wide beam stiffness is

calculated for a T section with flange outstands, to either side of the beam, of width equal to that of the

down-stand of the wide beam below the slab. Figure 3-4 shows the distribution of the transverse moment

across the drop panel at the interior column and mid span as a function of the parameter αD and panel

aspect ratios (l2 /l1 ) between 0.67 - 1.5, where l1 and l2 are the slab span and the length of continuous

drop panel respectively. Each point represents a slab with different inertia ratio αD and panel aspect

ratio. It was reported that for square slabs (i.e., panel aspect ratio =1) the distribution of transverse

moments seems to be independent of the inertia ratio αD at both interior column and mid span strips.

Increasing the panel aspect ratio increases the bending moments in the column strips, while the variance

in moment distribution at mid span is insignificant for different span ratios. The coefficient for moment

distribution in the column strip at midspan is always 0.55. In the case of relatively rigid wide beams,

the reinforcement design of the slab is controlled by minimum reinforcement requirements.

Figure 3-3: Transverse moment distribution in slab with continuous drop panels (102)


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On the basis of their studies, Paultre and Moisan suggested the CSA-A23.3-94 recommendation for the

minimum proportion of the hogging moment to be resisted by the column strip should be increased

from 0.6 to 0.8.

Research by Tay

Tay (6) carried out elastic linear and nonlinear finite element analysis to investigate the distribution of

bending moments in RC solid slab wide beam floors. The main aim was to investigate the transverse

distribution of moment along the length of the wide beam. The dimensions of wide beam floor

considered were Lslab = 9.0 m, Lbeam = 15.0 m, Wbeam = 2.4 m, dbeam =0.6 m, tslab = 0.2 m. It was assumed

that the conventional beam design methods for flexure, shear and punching shear are adequate for

designing wide beams.

Figure 3-5 shows the transverse distribution of moments obtained with elastic FEA in an internal slab

panel along sections passing through 1) the faces of columns supporting an internal wide beam, 2) the

face of an internal wide beam and 3) midspan of the slab. These sections are illustrated in Figure 3-6.

Figure 3-5 shows that the distribution of hogging moment is not uniform along the wide beam and peaks

very sharply around columns. Along the face of the wide beam, the support moment distribution tends

to increase slightly near columns. On the other hand, the span moments remain fairly uniform along the

length of the wide beam. However, transverse bending moments are commonly assumed to be

uniformly distributed along the length of the wide beam. Moreover, Tay reported that the Economic

Concrete Frame Element (ECFE) design method (5) overestimates the span moments in slabs and does

not consider adequately the magnitude of the hogging moments in the slab at the junction with the wide

beam or their lateral distribution along the length of the beam. The assumption of a uniform transverse

Figure 3-4: Hogging and sagging moment taken by column strip as a function of inertia ratio and panel aspect ratio (102)


79

hogging bending moment distribution along the length of the wide beam leads to a uniform distribution

of flexural hogging reinforcement. This gives rise to potential serviceability problems due to high steel

strains and excessive crack widths around columns. Tay also stated that wide beam slabs tend to behave

more like flat slabs rather than conventional beam and slab solutions as commonly presumed.

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

m* /m

*av

Distance along the wide beam: m

Section 1-1

Section 2-2

Section 3-3

Figure 3-5: A typical normalised moments versus distance across interior panel of wide beam slab as presented by Tay, 2006: Section 1-1: passing through the face of columns supporting the internal wide beam, Section 2-2: passing through the wide beam face and Section 3-3: passing through the Slab mid-span.

Figure 3-6: Plan view showing the wide beam floor and the critical sections considered by Tay


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Tay proposed a design method for wide beam slabs based on the equivalent frame method used for the

design of flat slabs. Design moments for reinforcement orientated perpendicular to the wide beam are

calculated using an equivalent frame analysis. The span moment is assumed to be uniformly distributed

along the length of the wide beam. For the support moments, the slab is divided into a column strip of

width Lc=0.5Lx, and middle strip of width (Lb-Lc) where Lx is the slab span and Lb is the span of the

banded beam. The definition of column and middle strips for wide beam slab is illustrated in Figure

3-7. The design moment in the slab in the middle strip at the face of the internal beam face is given by

the following expression:

= 0.5 [ − − 0.5 ] − ( . / ) (3.1)

where Lx is slab span between column centrelines, b is the width of internal beam, wslab is the ultimate

design load (kN/m2), mmax is the maximum span moment given by EQFM, xmax defines the position of

mmax measured from the centreline of the edge column.

The design moment in the slab in the column strip at the face of the internal beam is given by:

= − (3.2)

where, is the moment given by the equivalent frame analysis at a distance 0.5b - dslab from the

centreline of the column, where dslab is the effective slab depth, = Lb-Lc and b is the width of the

banded beam, but not less than 1.2 where is the moment given by the equivalent analysis

at the face of the beam.

Figure 3-7: Definition of column and middle strips as proposed by Tay (6)


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When detailing the hogging reinforcement, within the column strip and normal to the beam, Tay

suggested that 75% of the required area of reinforcement should be provided within the central half of

the column strip and 25% in the edges of the column strip. The proposed transverse distribution of

support and span moments for wide beam slabs is illustrated in Figure 3-8. Table 3-1 summarizes the

distribution of transverse moments between column and middle strips across wide beam slab panel as

proposed by Tay.

Table 3-1: Tay’s proposed distribution of transverse moments between column and middle strips across wide beam slab panel

Column strip Middle strip Hogging moments at

the centre of the

beam

75% Mcolumn strip total (width of column strip centre =0.5 Lc)

0.25% Mcolumn strip total (width of column strip edge =0.5 Lc)

≥ Mhog middle strip

Mhog middle strip total is divided

evenly across the middle

strip of width Lb-Lc

Hogging moment at the beam face

Mcolumn strip total is evenly spaced over the column strip width Lc

Sagging moment Mspan total calculated from EQFM and evenly distributed across slab panel width

Tay carried out a parametric study to investigate the effect of wide beam width, beam span and slab

span on the width of the column strip. He concluded that the width of wide beam and beam span has

Middle strip Middle stripColumn strip

Lb

Sagging moment

Hogging moment

0.5(Lb-Lc) Lc 0.5(Lb-Lc)

Mcolumn strip total/Lx

2Mcolumn strip total/Lx

3Mcolumn strip total/Lx

Mhog middle strip total /(Lb-0.5Lx)

Mspan total/Lb

Figure 3-8: Transverse distribution of support and span moments for wide beam slab panel as proposed by Tay


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little influence on determining the width of column strip and the beam span has no effect at all. On the

other hand, the column strip width appeared to be proportional to the slab span.

The proposed design procedure yielded low strains in the transverse rebar over columns. This was

partially due to banding the rebar over supports but also due to the relatively large depth adopted for

the wide beam (0.6 m). In addition, the rebar design for slab support moment may well be governed by

the minimum reinforcement, which was not considered by Tay.

Research by Shuraim and Al-Negheimish, 2011

Shuraim and Al-Negheimish (7) carried out a full-scale test on a RC joist floor system in order to

develop a design procedure for computing the longitudinal and transverse distribution of slab moments

with sufficient accuracy to satisfy the strength and serviceability criteria. The tested floor consisted of

a one-way spanning reinforced concrete joist floor with wide shallow beams supported on narrow-width

columns. The slab was uniformly loaded. Figure 3-9 shows the layout of floor plan and the arrangement

of structural members. A portion of the tested floor was constructed as flat slab and two drop beams

were also built for comparison purposes. The distribution of moments in the floor was obtained

numerically using 3D nonlinear finite element analysis.

The experiment results revealed that the end moments in the joists, which were supported on wide

shallow beams, varied significantly along the length of the beam. For instance, the end moment of a

joist near the wide beam support was much higher than the end moment of a joist at the mid-span of the

wide beam as shown in Figure 3-10. The end moments in the joists were greatest around the wide beam

Figure 3-9: Floor plan layout showing member designations and critical sections as presented by A. Shuraim & A. Al-Negheimish (7).


83

supports and reached up to 200% higher than the average end moment. Shuraim & Al-Negheimish

noted that similar deviations from average moments were reported in flat slabs by Jofriet (104).

Figure 3-11 shows the transverse bending moment distributions at section INCF through the face of

columns and at mid span section JJ for Wide Shallow Beam slab (WSB) and Floor with Drop Beam

(FDB) models. The FDB floor had two stiff narrow beams. These moments were calculated with 3D-

NLFEA. The distributions of span moments in both slab systems are fairly uniform and almost identical.

Nevertheless, the behaviours of the WSB and FDB are quite different as illustrated in Figure 3-11:

Moment profiles at two critical sections from WSB and FDB models (7). For the FDB model, the

moment distribution is fairly uniform along section INCF. However, the moments vary significantly in

the WSB model as the moments around columns peaked remarkably and decrease considerably near

mid-span of the wide beam.

Figure 3-10: Joists’ End Moment Variation from average moment at section INCF at the face of columns supporting the internal wide beam.

Figure 3-11: Moment profiles at two critical sections from WSB and FDB models (7)


84

The variation in the interior negative moments was largely attributed to the vertical rigidity of the

supporting girder. The researchers illustrated the role of vertical rigidity of the girder in terms of the

spring and grid analogies. The authors analysed a set of joists supported on a simply supported girder

with the same depth at equal intervals as shown in Figure 3-12. The negative moments vary nonlinearly

along the girder having high values at the girder support and decreasing gradually towards the midspan

of the girder.

Following these observations, Shuraim & Al-Negheimish introduced the so-called rigidity-based

equivalent frame (RB-EQF) design procedure for wide beam slabs. The RB-EQB consists of three steps;

1) representing the floor by parallel frames, 2) the stiffness of the structural members of each frame is

modified, as described below, to compute the longitudinal moments along the span, 3) the longitudinal

moments are distributed laterally along the wide beam taking the wide beam rigidity into account.

Figure 3-13: Typical Equivalent Frame as defined by A. Shuraim & A. Al-Negheimish. (7)

Figure 3-12: Typical moment diagram in beam –girder grid system showing variation in negative moments (7)


85

The equivalent frame proposed by Shuraim & Al-Negheimish is shown in Figure 3-13 and Figure 3-15.

The following assumptions were made to account for the torsional rigidity of the wide beam and the

effect of narrow-width column. First, the slab-beam horizontal members should be transformed to

equivalent T-sections or rectangular sections as appropriate. Second, the moment of inertia of the

column Ic is modified to Icm as follows:

= (3.3)

where Kc is the column rotational stiffness. For the high rigidity at column ends, the expression

suggested by Meyer (105) for Kc is adopted:

=.

(3.4)

The equivalent column stiffness Kec is defined as:

= + (3.5)

= ∑ (3.6)

= ∑ 1 − 0.63 (3.7)

where Kt is the torsional stiffness, l2 and are the length of the span and width of the column transverse

to the frame direction respectively, r1 & r2 the short and long dimensions of the torsional member cross

section, respectively. These dimensions are shown in Figure 3-14.

Figure 3-14: Critical sections for torsional member with variable width as Shuraim & Al-Negheimish (7)


86

In addition, the column cross sectional area was modified by applying a reduction factor to account for

the vertical rigidity of the supporting wide beam. This factor was defined as the ratio of the column

width transverse to the frame direction to the width of frame, having a minimum value of 0.1.

Furthermore, the frame width was divided into high and low rigidity zones as shown in Figure 3-16.

The moment in the high rigidity zone, Mhi, at any position i is given by:

= (3.8)

where α1 is the ratio between the width of high rigidity zone to the width of the frame, Mfi is the frame

moment. is the zone intensity factor. The value of depends mainly on two parameters; the ratio

between the width of the high rigidity zone and the frame width and the intensity zone factor. The

researcher’s recommended values of 0.3 and 0.45 for the exterior and interior frames respectively. For

the intensity zone factor, ℎ, they suggested three different expressions depending on the moment type

and location. For sagging moment the factor is always 1.0, for exterior hogging moment the factor is

the ratio between the widths of the frame and high rigidity zone, while the third expression assigns a

value between these two limits as illustrated in Equations (3.9) & (3.10).

For positive moment, = 1 (3.9a)

For exterior negative moment, = (3.9b)

For interior negative moment, = ≥ 1 (3.9c)

= (3.10)

In which Kca is the axial stiffness of the column (AcE/lc), is the vertical rigidity of the wide beam, E

is the concrete modulus of elasticity, b is width of wide beam, h & l are the depth and length of the wide

beam respectively. The coefficients and are 0.265 and -0.562 respectively, obtained from the

nonlinear FE analysis. The variable reflects the flexibility coefficients of the elastic girder subjected

to concentrated unit load. For interior frames the recommended value is 22.475 and for exterior frames

is 7.111.

The moment in the low rigidity zone is computed as follows:

= − (3.11)


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The width of the high rigidity zone at a discontinuous end of a girder is 0.15 l, and for the low rigidity

zone its width extends from the discontinuous end (0.15 l) to the midspan (0.5 l). In case of a continuous

support the width is taken as 0.225 l. Figure 3-15 & Figure 3-16 show the widths of the rigidity zone

considered in this study. The proposed procedure and coefficients need more investigations since they

are case-dependent.

Figure 3-15: Plan of interior frame and the low and high rigidity zones as suggested by Shuraim & Al-Negheimish (7)

Figure 3-16: Widths of rigidity zones in wide-shallow girders B3 & B4 as suggested by Shuraim & Al-Negheimish (7)


88

Conclusion

These studies show that the transverse distribution of support moments in wide beam floor systems is

not uniform as commonly assumed in practical design. This includes slab systems in which wide beams

support solid slabs or secondary beams. As a consequence, serviceability conditions may not be met if

transverse reinforcement is uniformly distributed as commonly done [ECFE], which could lead to

excessive cracking around columns. To tackle this problem, Tay (6) and Shuraim and Al-Negheimish

(7) proposed the use of the Equivalent Frame Method for flat slabs with some modifications. For

computing the longitudinal moments Tay noted that the Equivalent Frame Method gives reasonable

results, while Shuraim and Al-Negheimish introduced a modifier to account for the vertical rigidity of

the wide beam. In order to distribute the hogging moments along the wide beam, the length of wide

beam should be divided into middle and column strips as proposed by Tay, or low and high rigidity

zones as suggested by Shuraim & Al-Negheimish. The width of the column strip, as in the EQFM,

depends on the shorter span of slab, while the width of the high rigidity zone is a function of the length

of wide beam. It is concluded that more research is needed to gain a better understanding of the flexural

behaviour of wide beam flooring systems at the serviceability and ultimate limit states.

3.3 One-Way Shear in Wide Beams

EC2 and ACI318 do not include any special requirements for calculating the shear resistance of wide

beams which can result in shear resistance being overestimated. For example, current procedures

involve the use of the full width of the member in estimating the shear strength regardless the width of

the support. Furthermore, most existing design provisions do not account for the influence of the

distribution and configuration of transverse stirrups across the width of wide beams. Various

experimental programmes, Serna-Ros, et al. (106), Sherwood, et al. (107), Lubell, et al. (108) and

Shuraim (109) have investigated the effect of these parameters on the shear capacity of wide beams.

The results of these experiments show that support width, transverse stirrup distribution and stirrup

configuration affect the magnitude of shear capacity in the tested beams as discussed below in more

detail.

The Influence of support width

Leonhardt & Walther (110) investigated the effect of support width on shear strength and concluded

that there is little effect. Regan & Rezai-Jorabi (79) tested 29 one-way spanning slabs with widths

ranging between 0.4 m -1.0 m and 0.1 m thickness, subjected to concentrated loads and came to the

same conclusion.

Subsequently, Serna-Ros, et al. (106) conducted 18 tests of wide beams measuring 0.75 m in width and

0.25 m deep. The beams were subjected to two point loads and failed in shear. They concluded that the


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use of concentrated supports clearly reduces the shear strength of wide beams. In addition, they

proposed a coefficient to improve the prediction of shear strength of wide beams. However, this

coefficient is applicable for members with shear reinforcement only.

More recently, Lubell et al. (108) studied one-way shear in wide beams supported by narrow supports.

They tested eight large-scale wide beams. The primary variable was the ratio of the support width to

the beam width. The research indicated that one-way shear strength reduces moderately as the support

width to beam width ratio decreases. This observation was evident in members without and with shear

reinforcement. The researchers also questioned the suitability of ACI 318 using the average shear stress

as a measure for evaluating the structural performance. This is because ACI 318 takes no account of

the three-dimensional force flow necessary to achieve an even shear stress distribution across the beam

width. On this basis, Lubell et al. introduced a reduction factor to reduce the shear strength of wide

beams according to ACI 318. The reduction factor is a function of the minimum of a) the ratio of the

support width to member width, b) the loaded width to member width.

The Influence of Transverse stirrup distribution and configurations

In 1985, Hsiumg & Frantz (111) tested five large beams with width/depth ratios between 0.33-1. Each

specimen was 2.896 m long, 0.457 m deep and 0.152 m, 0.304 m or 0.447 m wide. All beams had similar

flexural reinforcement ratios (1.82 % for tension and 0.22% for compression). The primary purpose of

the tests was to examine the influence of varying the transverse stirrups spacing and web width on shear

strength. They concluded that the measured ultimate shear capacity is proportional to the width of beam.

In addition, the influence of transverse distribution of stirrup legs across the width is insignificant. They

also reported that with uniform distribution of transverse stirrups the interior legs carry higher shear

loads than the exterior ones do.

Anderson & Ramirez (112) showed experimentally that the shear strength of wide beams depends on

the transverse spacing of stirrups with the effect becoming greater at high shear stresses. Moreover,

they attributed the conclusions of Hsiung & Frantz to the low levels of the shear stress developed in

their beams. Hence, they recommended maximum transverse spacing of 200 mm for high shear stress

and 400 mm otherwise. Limitations for transverse stirrup spacing set out EC2 were found to be

conservative even with larger spacing as pointed out by Serna-Ros, et al (106). The EC2 limits the

transverse stirrups spacing to lesser of 0.75 d and 600 mm, where d is the effective depth of the beam:

More recently, experimental work by Shuraim (109) includes testing of 16 two span continuous wide

beams supported on a narrow interior column. The beams measured 0.7 m wide by 0.18 m thickness by

3.2 m long. The main aim of the research was to investigate the influence of the transverse distribution

and configuration of stirrups on the shear strength of wide beams. The widths of loading plates and


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supports were the same as the beam width. The research showed that four-leg closed stirrups were

significantly more effective than two-leg closed stirrups. It also showed that small areas of shear

reinforcement can significantly improve shear strength. On the other hand, two-legged stirrups

performed poorly. It was also concluded from the experiment that the closer the transverse stirrups, the

higher the efficiency.

Neither ACI 318 nor EC2 provide any criterion for selecting the transverse stirrups configuration in

wide beams. EC2 limits the transverse spacing of stirrup legs to the lesser of 0.75 d and 600 mm, and

hence it is independent of the shear stress level. Conversely, ACI 318 does not give transverse spacing

limits for stirrups. It assumes that the transverse spacing of stirrup legs across the web width does not

affect the shear capacity of a RC beam.

Conclusion

EC2 and ACI 318 do not provide specific design provisions for calculating the shear resistance of wide

beams. Consequently, the shear capacity could be overestimated when the transverse spacing of the

vertical legs of stirrups is excessive. Numerical and experimental studies show that the key parameters

affecting shear strength include support width, transverse stirrup distribution and stirrup configuration

influence the magnitude of shear capacity of a wide beam. However, there is no consensus about design

rules in order to account for the effect of these parameters. To conclude, although the one-way shear in

wide beam slabs needs more investigations, it is out of the scope of this research. Hence, it is not

discussed any further.

3.4 Shear Failure Modes and Crack Patterns in Wide Beams

Lau & Clark (8) carried out 20 tests on micro-concrete wide beam ribbed slabs at internal column

locations. They observed that punching shear failures of wide beam ribbed slabs are very similar to

punching failures in flat slabs but the shear capacity could be reduced due to the potential reduction in

the shear failure surface. The punching shear failure surface could form either within the wide beam or

it can pass including part of the slab as shown in Figure 3-17 & Figure 3-18. They concluded that BS

8110 generally underestimates the failure loads due to punching. In some cases, it fails to predict the

correct failure mode as punching failure occurred when wide beam shear failure was predicted.


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Lau & Clark (9) also investigated punching shear failures at the edge column connections of a wide

ribbed slab. The observed failure mode was similar to that of a flat slab with the slab rotating downwards

at the inner column face in the presence of moment orthogonal to the slab edge. The applied moment

increased with increasing column eccentricity inducing torsion on the sides of the edge columns which

led to reductions in punching shear capacities.

Shuraim (109) tested 16 continuous wide beams subjected to two point loads and supports on interior

column and two roller supports at the ends extending across the full width of the beam. There were two

main characteristic cracking patterns formed in the test beams; side cracking and top surface cracking.

The side cracking pattern consisted of vertical flexural cracks initiated at maximum positive and

negative bending moment sections near mid span and the internal column respectively. Diagonal cracks

due to shear and flexure were also monitored close to the point loads and they continued until the end

supports.

Figure 3-17: wide beam failure mechanism as described by Lau & Clark (8)


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All the tested specimens were reported to exhibit top surface cracking pattern in which diagonal and

tangential cracks were observed. This form of cracking was attributed to the narrow width of the

supporting columns. Shuraim stated that the top surface cracking pattern is qualitatively similar to that

usually observed in punching shear failures of flat slabs. Figure 3-19 shows the major cracking patterns

after failure for three different beams.

Figure 3-18: Sketch showing the assumed punching failure with losses: (a) plan view, (b) section. (8)

Figure 3-19: Side view showing the cracking patterns after failure for three different specimens (109)


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3.5 Current design procedure recommended by The Concrete Centre

for wide beam slabs

The Concrete Centre (TCC) is a professional body which promotes best practice in the design and use

of concrete in the UK. TCC publishes also a wide range of authoritative design guidance, which include

structural design, sustainable construction and Eurocodes.

With respect to designing RC wide beam floors, TCC follows the design procedures recommended by

EC2 for one way slab supported on beams, which implies uniform distribution of the span and support

reinforcements across wide beams. According to TCC (1,5), different analysis methods for wide beam

slabs may be used. One possible way is to treat the wide beam slab as a continuous beam supported on

rigid knife edge supports. An example of a wide beam ribbed slab design illustrating this method can

be found in Worked Examples to Eurocode 2: Volume 1 (5). This method, however, does not account

for the relative stiffness of the wide beam and slab. TCC permits also the use of moment coefficients

which is based on the effective span of the slab. Relevant design charts and tabulated data for wide

beam slab design are published in Economic Concrete Frame Elements (1). For an internal span, the

effective span is equal to the clear span of the slab plus the slab thickness, while for the end span is

equal to the clear span plus half of the thickness of slab. Hence, the slab span is reduced considerably

as the beam width is relatively wide. Consequently, the total moment in a slab panel is considerably

reduced since it is a function of the effective span. Although the moment coefficients are no longer

included in the EC2, the approximate moment coefficients method has been long practiced in the UK

and remain in BS 8110 which was superseded by EC2 in 2010. TCC also allows the option of rigid

frame analysis.

All design methods specified in TCC imply uniform distribution for transverse moments across the

wide beams. For wide beams, they are designed in the same manner as the ordinary beams. TCC

requires additional checks for punching shear at critical perimeters around columns supporting wide

beams. These include definition of critical perimeters around internal, edge and corner columns as

specified in the EC2.

3.6 Conclusions

Studies on flexural design of wide beam flooring system are few and they are mostly based on the

numerical analyses. Conversely, several experimental investigations have been performed on beam

shear to determine the key factors influencing the wide beam capacity including as the ratio of beam

width to support width and spacing and configuration of transverse stirrups. Additionally, few tests on

micro-specimens studying the behaviour of wide beam ribbed slabs under punching shear and beam


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shear are available in the literature. This chapter describes these studies and the following conclusions

can be drawn:

The transverse distribution of support moments in wide beam floor systems is not uniform as

commonly assumed in practical design. This includes slab systems in which wide beams

support solid slabs or secondary beams. As a consequence, serviceability conditions may not

be met if transverse reinforcement is uniformly distributed as commonly done according to

TCC, which could lead to excessive cracking around columns.

Both Tay (6) and Shuraim and Al-Negheimish (7) proposed placing the transverse rebar in

bands approximately following the elastic analysis. In the process they proposed coefficients

for dividing the total design transverse moments across the bands. Tay adopted a design

procedure similar to those in flat slabs. For validating such a procedure, Tay used a relatively

‘deep’ wide beam which seems to reduce the strains in transverse rebar over band beam

significantly. Shuraim and Al-Negheimish (7) did not propose a generalised design procedure

for transverse rebar distribution along wide beam slabs.

EC2 and ACI318 do not provide specific design provisions for calculating the shear resistance

of wide beams.

Numerical and experimental studies show that the shear capacity of wide beams is influenced

by the support width, transverse stirrup distribution and stirrup configuration. However, there

is no consensus about design rules in order to account for the effect of these parameters. For

this research the longitudinal and transverse stirrups of wide beam is designed in accordance to

TCC and EC2 guidelines for ordinary beams.

The works of Lau & Clark (8,9,113) clearly highlights the lack of current understanding about

wide beam shear and punching shear behaviour in wide beam slabs. This is also true, but to a

lesser extent, for flat slabs where the punching shear provisions of EC2, BS 8110 and ACI318

are based on empirical equations. The merits of the fib MC2010 method, which based on CSCT

discussed in Chapter 2, are considered in this research for the evaluation of punching shear

resistance around internal and edge columns.

Finally, the design practice of wide beam flooring systems is outlined according to TCC, which reflects

the UK practice.

Methodology – Nonlinear Finite Element Analyses Chapter 4

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Methodology – Nonlinear Finite Element Analyses

(NLFEA)

4.1 Introduction

The main objective of this research is to develop a rational design method for wide beam flooring

systems which is consistent with their elastic structural response and meets both the SLS and ULS

conditions. The research was entirely carried out using numerical techniques; elastic linear (EFA) and

nonlinear finite element analyses (NLFEA). Two commercial software packages are used for analysis,

namely ATENA v.5.3.3 (114) DIANA v.9.6 (11). The former was primarily used for elastic linear

analysis and to simulate the nonlinear flexural behaviour of wide beam slabs with shell elements. This

is due to the wide range of element types, especially shell elements, available in DIANA’s library. For

analyses where shear and punching shear investigations are of prime concern, the ATENA was adopted

since with the adopted constitutive models it was found to give better results than DIANA.

This chapter is not intended to give exhaustive description for the finite element method, which can be

found in text books such as Maekawa, et al. (115), Rombach (116) and De Borst (117) among others.

Instead, it focuses on describing the constitutive models for concrete and implemented in the research


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as well as reinforcement modelling. Also, iterative solution methods and convergence criteria are briefly

discussed. Subsequently, the discretisation strategy, including element types and integration schemes,

followed in the research is highlighted.

4.2 General Background

The Finite Element Method (FEM) can be defined as a numerical method which provides approximate

solutions to complex physical problems. When running a finite element analysis, it is essential to obtain

a solution with minimum error in order for the FE model to represent adequately the physical problem.

However, this can be a very challenging task because the accuracy of the NLFEA depends on several

factors. Material modelling, for instance, especially concrete is challenging. The behaviour of concrete

is complex because it is a brittle material susceptible to cracking under tension. Another challenging

issue is the discretisation strategy used for the FE model. This involves the selection of mesh type and

size. To tackle this problem and to establish a robust model, sensitivity studies, that involve examining

meshes with different types and sizes, should be undertaken. Additionally, the selection of iterative

solver, convergence criteria and boundary conditions should be chosen carefully as they influence the

accuracy of NLFEA results.

4.3 Constitutive Models for Materials

Constitutive Models for concrete

The constitutive models for concrete implemented in the research are the fracture-plasticity model in

ATENA (114) and the total strain model in DIANA (11). Both models use the smeared crack approach.

Cracks can be modelled as either fixed or rotating. Perfect bond between concrete and reinforcement is

assumed.

The behaviour of concrete in the elastic state (uncracked concrete) is modelled as an isotropic material.

Hence, the stiffness matrix of the uncracked concrete is given as follows:

=1 0

1 00 0

(4.1)

For linear elastic analysis, only the Young’s modulus, E and the Poisson’s ratio, ν need to be defined.

The behaviour of cracked concrete in compression, tension and shear are discussed below.

4.3.1.1 Compressive Behaviour

The behaviour of concrete in compression is described in terms of plasticity in ATENA. The

compressive failure for concrete is based on the three-parameter failure criterion proposed by Menetrey


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& Willam (118). The failure criterion describes the triaxial strength of concrete in terms of three

independent stress invariants. It is represented geometrically in principal stress space by a convex and

smooth shape. It is defined as:

( , , ) = √1.5 +√

( , ) +√

− = 0 (4.2)

= 3 × (4.3)

( , ) = ( )

( ) ( )[ ( ) ] (4.4)

where , are the hydrostatic and deviatoric stress invariants, respectively, (Heigh-Vestergaard

coordinates); is the deviatoric polar angle; m & c are measures for cohesive and frictional strengths.

, designate the uniaxial compressive strength and tension strength, respectively. The parameter, e

defines the roundness of the failure surface. For perfectly circular failure surface around the hydrostatic

axis, e = 1.0. If the surface has sharp corners, e = 0.5. The strain hardening is a function of the equivalent

plastic strain, and is given by:

Δ = min(Δ ) (4.5)

The hardening/softening law for the failure surface of Menétrey & Willam is based on the uniaxial

compression and depends on the parameter c, which is calculated as follows:

=

(4.6)

where the term denotes the hardening/softening law. It is noteworthy that the hardening part

of the curve is nonlinear, while the softening law is linear as illustrated in Figure 4-1.

The hardening curve is described by the following relationship:

= + ( − ) 1 − (4.7)

where is the initial compressive strength, is plastic strain at the maximum compressive strength

and Lc is the length scale parameter, which converts the equivalent plastic strain to the plastic

displacement, wd. Mier (119), based on his experimental investigations, recommended a value of 0.5mm

for the plastic displacement, wd for normal concrete, which is used as a default value in the definition

of the compression softening in ATENA. The element dimension Lc is measured in the direction of the


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principal axis with minimum stress. Figure 4-1 shows that the hardening is a function of compressive

strain, whilst the displacement, wd, controls the softening part of the curve. This enables the effect of

mesh sensitivity in compression failure to be considered in the FE solution.

In the total strain model adopted in DIANA, the compressive behaviour is defined according to the

Modified Compression Field Theory (MCFT) presented by Vecchio & Collins (87) and extended later

by Selby & Vecchio (120) for 3D. Concrete subjected to uniaxial compression exhibits a softening

relationship after the ultimate strength. If lateral confinement is provided, the strength and ductility of

concrete show an increase with increasing isotropic stress. This is modelled in DIANA using the four-

parameter failure surface Hsieh, et al. (121), which is defined by the following:

= 2.0108 + 0.9174 + 9.1412 + 0.2312 − 1 = 0 (4.8)

J2 & I1 are invariants defined in terms of the concrete stress, σci:

= (( − ) + ( − ) + ( − ) ) (4.9)

= + + (4.10)

Conversely, lateral cracking in concrete can occur and leads to a reduction in compressive strength. As

a result, the compressive strength fp becomes a function of the internal variables governing tensile

damage in the lateral directions, αl,1 and αl,2. In DIANA, these effects are taken into account by

modifying the compressive stress-strain relationship. The peak stress, fp and the corresponding peak

strain αp are given by:

= . (4.11)

Figure 4-1: Compressive hardening/softening and compressive characteristic length (ATENA v. 5.1.1)


99

= . (4.12)

where fcf and εp are the parameters for compressive stress-strain function determined from a failure

criterion considering the lateral confinement effect. & are reduction factors due to the lateral

cracking for the peak stress and strain, respectively and are functions of the average lateral damage

variable, = , + , . Vecchio & Collins (87) suggested the following expression to account

for the reduction in strength due to lateral cracking:

= ≤ 1 (4.13)

= 0.27 − − 0.37 (4.14)

where ε0 is the initial strain; the reduction factor = 1. Figure 4-2 shows the variation of reduction

factor due to lateral cracking as suggested by Vecchio and Collins. DIANA offers several predefined

hardening-softening curves for concrete in compression as shown in Figure 4-3.

Figure 4-2: Reduction factor due to lateral cracking (after Vecchio & Collins)


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The model proposed by Thorenfeldt (122) has been implemented in the research. Hence, some

theoretical background for this model is given. The compressive stress-strain diagram of Thorenfeldt is

presented in Figure 4-4 and defined as:

= − (4.15)

= 0.8 + ; = 1 < < 0

0.67 + ≤ (4.16)

Figure 4-4: Thorenfeldt Compression Curve

Figure 4-3: Predefined compression behaviour for Total Strain model in DIANA


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The peak strain, αp corresponds to the peak stress, fp is given by the following expression:

= . (4.17)

Although the Thorenfeldt model is not fracture energy based, it can be scaled to element size according

to DIANA (11). This is implemented by replacing the post-peak strain α by the term [αp + (α- αp).

h/l] where h is the crack bandwidth and l is a scaling parameter for the post-peak part of Thorenfeldt

curve. It is important to note that the parameters of Thorenfeldt compression curves are unit-free. Also,

the compressive strength of concrete should be specified (in MPa).

4.3.1.2 Tensile Behaviour

The tensile behaviour of concrete is modelled in ATENA using fracture energy-based models, where

the concrete failure in tension is determined by the Rankine failure criterion. In each model, the crack-

opening law describes the tensile softening behaviour of concrete due to crack propagation. Other

models based on stress-strain relationships are also available, but they are not recommended for normal

cases of crack propagation in concrete. Thus, the discussion will be restricted to fracture energy-based

models. ATENA offers a number of softening models for reinforced concrete which include;

exponential crack opening law (Hordijk), linear crack opening law and linear softening based on local

strain. The exponential curve by Hordijk was implemented in the NLFEA performed by ATENA in this

research.

DIANA offers four softening functions based on fracture energy and crack bandwidth to model the

tensile behaviour of reinforced concrete and to be implemented in the Total Strain Crack models. These

include a linear softening curve, an exponential softening curve, the Reinhardt (123) nonlinear softening

curve and Hordijk (124) exponential softening curve. Tension softening might also be described in the

Total Strain Crack model using functions related to the stress-strain relationship, such as a constant

tensile behaviour, a multi-linear behaviour, and a brittle behaviour. The area under the stress-strain

curve represents the fracture energy, GF divided by the equivalent length, heq. Figure 4-5 illustrates the

tension softening functions as presented in DIANA.


102

In the current research, the nonlinear softening curve according to Hordijk was used in the NLFEA

throughout this study for members modelled with brick elements. Hordijk proposed an exponential

expression to describe the softening behaviour of concrete in which a zero tensile stress was assumed

at the ultimate strain. Hence, the tensile stress is given as follows:

= 1 + exp − − (1 + ) exp (− ) (4.18)

This applies when 0 ≤ ≤ . If > , then = 0.

where ft denotes the tensile strength of concrete, ɛu the ultimate crack strain and ɛcr the crack strain. c1

and c2 are parameters having values of 3.0 and 6.93, respectively. The ultimate strain may be calculated

as follows:

= 5.136 (4.19)

In analyses using shell elements where plane sections remain plane, instead of using fracture energy

based approaches, it is more appropriate to model tension stiffening by modifying the tensile stress-

strain curve diagram for concrete. Researchers including Gilbert & Warner (125), Damjanic & Owen

(126) and Carreira & Chu (127) among others (128-130) have developed models based on this approach

for implementation in FE analyses. However, significant differences exist between these models in

terms of shapes and bounds. Figure 4-6 shows the tension softening model for concrete proposed by

Damjanic & Owen (126). As shown in Figure 4-6, the softening function is defined by α1 & α2, ft and

Figure 4-5: Tension softening laws in DIANA


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the ultimate strain at zero stress, ɛu. Different values have been suggested by researchers for α1 & α2.

Torres, et al. (130) and Tay (6) pointed out that these parameters depend significantly on the

reinforcement index and the mode of failure.

For this research, the model developed by Tay (6) was employed to describe the tensile behaviour of

cracked concrete for members modelled with shell elements. The linear tension softening stress-strain

relationship proposed by Tay is shown in Figure 4-7. A value of 0.5ft is recommended for the maximum

tensile strength, . The residual tensile stress is assumed to be zero at strain c3 equal to 0.5ɛs, where

ɛs is the yield strain of steel reinforcement. In addition, a minimum of nine integration points should be

used through the element thickness. It is important to note that the application of this model is limited

to flexural members with reinforcement ratio greater than 0.15%. Moreover, the predicted steel strains

are average values since the model is based on the smeared crack approach.

4.3.1.3 Crack Modelling

There are two ways of modelling cracking in reinforced concrete in NLFEA; namely the discrete and

smeared approaches. In the discrete crack approach, the crack is presented as a geometrical

discontinuity. The process of crack propagation happens as the force at the node representing the tip of

Figure 4-6: Equivalent stress-strain relationship for tensioned concrete (After Damjanic & Owen, 1984)

Figure 4-7: linear tension softening for concrete as proposed by Tay

0.5εy

0.5ft

εcr


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crack becomes greater than the concrete tensile resistance. Then, the tip of the crack propagates to a

new node. Several limitations are associated with the discrete crack approach. A mesh bias is introduced

as the crack is constrained to follow a predefined path along the element boundaries. In addition, the

finite element mesh needs to be modified at each crack increment. Moreover, the implementation of the

discrete crack models in three-dimensional problems is complicated. (131,132).

In the smeared crack approach, the cracked element is considered as a continuum and the crack is

assumed to be uniformly distributed within the material volume. This concept is widely adopted,

because the original topology of FE mesh is preserved. Also, crack propagation can occur in any

direction. However, the smeared crack approach can exhibit strong mesh sensitivity as reported by

Bažant & Oh (133). Important improvements have been introduced to the smeared crack model,

including the work of Cervenka, et al. (131). It has been shown, that the smeared crack model, based

on the refined crack band theory, can successfully describe discrete crack propagation in plain, as well

as reinforced concrete. With respect to the research, all the NLFEA models used, either in DIANA or

ATENA, were based on the smeared crack approach. Two crack models are available within the

smeared crack approach; namely the fixed and rotating crack models. For both models the crack forms

when the principal tensile stress exceeds the concrete tensile strength. In the fixed crack model, the

principal stress direction is fixed and defined by the initial crack direction. In general, the principal

strain directions do not coincide with orthotropic axes m1, m2 as shown in Figure 4-8. Consequently, a

shear stress develops at the crack face, and the resulting stresses, normal and parallel to the crack plane

are not the principal stresses as illustrated in Figure 4-8.

It is known that the shear stiffness of cracked concrete contributes significantly to the shear strength of

RC members without shear reinforcement. Thus, it is essential in fixed crack models to use a shear

retention factor as the shear stiffness is reduces after cracking. The retention shear factor can be fixed

or variable. In ATENA, a variable reduction in the shear modulus was adopted. The strain normal to

Figure 4-9: Fixed Crack Model (ATENA Manual)


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the crack increases as the shear modulus decreases as shown in Figure 4-9. Equation (4.20) gives the

expression for the retention shear factor as follows:

= , = (4.20)

= 7 + 333( − 0.005), = 10 − 167( − 0.005), 0 ≤ ≤ 0.02

where G is the reduced shear modulus, GC is the initial shear modulus for concrete, rg is the shear

retention factor, is the transformed reinforcing ratio and c1 is the scaling factor defined by the user.

The default value of c1 in ATENA is 1.

DIANA provides the option of using both fixed and variable shear retention factors for Total Strain

Fixed Crack models. The use of a constant shear retention factor is not generally recommended and

thorough post-analysis is suggested due to the possibility of spurious principal tensile stress (134).

Figure 4-10 shows the constant shear retention curve as presented in DIANA.

Figure 4-10: Shear retention factor (ATENA)

Figure 4-11: Constant shear retention curve (DIANA Manual)


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DIANA offers two multi-linear shear retention curves based respectively on shear stress and shear strain

and shear strain and shear retention. In addition, DIANA offers damage-based shear retention and

aggregate size-based shear retention. The latter was used in the DIANA models presented in this work.

In the aggregate size-based model the shear retention is related to the mean aggregate size, dagg, and the

crack width. It is assumed that the shear stiffness of a crack reduces to zero when the crack opening

exceeds half of the mean aggregate size. DIANA describes a linear softening for the shear stiffness and

gives the shear retention factor as:

= 1 − ℎ, (0 ≤ ≤ 1) (4.21)

where β is the shear retention factor, dagg denotes the mean aggregate size, εn the crack normal strain

and h is the crack bandwidth.

In the rotating crack model, the principal strain and stress axes coincide. Hence, shear stress does not

develop in the plane of the crack. Figure 4-11 shows the principal axes of stress and strain in the rotating

crack model.

Steel Reinforcement Modelling

Steel reinforcement can be modelled in NLFEA as either discrete or embedded reinforcement. The latter

was used for all the models in this research. Embedded steel reinforcement can have two forms; discrete

bars or reinforcement grid. Both forms were adopted in the presented NLFEA models. The stiffness of

embedded reinforcement contributes to the total stiffness of surrounding concrete. In DIANA, the

stiffness of the reinforcement grid is a function of its equivalent thickness, which is calculated by

dividing the total area of steel bars by the corresponding member width. Moreover, the mother material

in which the reinforcement is embedded is not affected by the reduction of embedded reinforcement

Figure 4-12: Rotating Crack Model (ATENA Manual)


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volume in terms of weight and stiffness. Perfect bond between the embedded steel reinforcement and

concrete is assumed. As a result, the steel strains correspond to the strains in concrete, and they are

computed from the displacement field of the surrounding concrete. Further, the state of uniaxial stress

is assumed for embedded reinforcements, in a form of bars or grid. ATENA provides two formulations

for the stress-strain law of embedded reinforcement of embedded reinforcement. First, the bilinear law,

in which an elastic- plastic curve is considered, as shown in Figure 4-12.

The slope Es represents the elasticity part of the curve and the second line represents the plasticity part.

In addition the strain hardening can be considered by specifying a value for the hardening modulus, Esh.

In order to model the steel ductility, a suitable value for limit strain parameter, εL should be selected.

The second law is a multi-linear curve which consists of four regions. These regions reflect the

behaviour of reinforcement during the loading stages from the start of load application up to rupture as

presented in Figure 4-13. These stages are the elasticity, yielding plateau, hardening and fracture.

Figure 4-13: Bilinear Law for reinforcement (ATENA Manual)

Figure 4-14: multi-linear Law for reinforcement (ATENA Manual)


108

On the other hand, DIANA provides two material models that can be used for reinforcement; the elastic

linear and Von Mises plasticity models. For the NLFEA performed using DIANA, the reinforcements

were modelled using Von Mises type elasto-plastic material model without strain hardening. The Von

Mises yield function is based only on the uniaxial yield stress, σy and the formulation can be written as:

= − 13 = 0 (4.22)

where J2 is the second invariant of the stress deviator sensor.

4.4 Nonlinear Analysis

Solution Methods

There are two types of solution methods: direct and iterative solution methods. The direct methods are

only suitable for small-scale problems and are not recommended for large-size models (e.g. 3D models).

On the other hand, iterative methods can provide robust solutions for large NLFEMs. Iterative solution

techniques usually involve an incremental process for defining the loading history. This is important

because relatively large steps would usually yield inaccurate solutions and lead to premature

divergence. In this research two iterative solvers are used; the Quasi-Newton method for DIANA

models and the Arc-length method for ATENA models. In the subsequent subsections, brief

descriptions for the two methods are presented. Also, the Newton-Raphson method is discussed.

4.4.1.1 Newton-Raphson

In the Newton–Raphson method the displacement iterates until equilibrium is found, while the load

increment is fixed. The stiffness matrix is computed at each iteration which may lead to predictions that

don’t satisfy equilibrium. The main advantage of the Newton-Raphson method is that it needs only a

few iterations to converge due to its quadratic convergence characteristic. However, the quadratic

convergence is only guaranteed if a correct stiffness matrix is used and if the prediction is already close

to the final solution. Moreover, each iteration is relatively time consuming. Figure 4-14 illustrates the

solution procedure of Newton-Raphson method for the first two iterations.


109

4.4.1.2 Quasi-Newton (Secant Method)

The Quasi- Newton method differs from the standard Newton–Raphson method in that it does not set

up a completely new stiffness matrix every iteration. Instead, it uses positions on the equilibrium path

that are already known in order to evaluate the stiffness of the structure. Figure 4-15 demonstrates the

solution process for the first and second iterations using the Secant method.

4.4.1.3 Arc-Length

In the Arc-length method both the displacement and force are incremental during the iteration, while

the solution path is kept constant. This solves the instability problems associated with the Newton-

Raphson method such as the snap back and snap through phenomena shown in Figure 4-16. The

instability problem arises when the load-displacement curve becomes almost horizontal since using

Figure 4-15: Newton-Raphson Method (DIANA Manual)

Figure 4-16: Quasi-Newton Method (DIANA Manual)


110

fixed load increment yields very large displacements. The Arc-length method limits the displacement

increment to a prescribed value at the end of each step. This is done by simultaneously adapting the

increment size.

Convergence Criteria

It is known that NLFEA produces approximate solutions. Therefore, convergence criteria are required

to stop the analysis when an acceptable solution is obtained or divergence occurs. In DIANA four

convergence norms can be used. These are the force norm, displacement norm, energy norm and

residual norm. ATENA, however, does not use the force norm and the absolute residual norm is

employed instead. With regard to ATENA models, all convergence criteria were applied, while in

DIANA, the energy norm was employed.

4.5 Other Aspects

Load Application

External loads are applied incrementally at the start of each step either by load control or displacement

control. In load control, the external force is directly increased. In the latter, a prescribed displacement

is applied at the start of the increment instead of the external load. Figure 4-17 illustrates the two

methods.

Figure 4-17: Arc length solution for: (a) Snap-through; (b) Snap-back phenomena (DIANA Manual)


111

4.5.1.1 Element Types

This subsection gives some details on the characteristics of elements which are implemented in FEA

and NLFEA. For each element type, the stress state, degrees of freedom and geometry are described. In

addition, the corresponding integration scheme adopted in the analysis is determined. The elements are

solid and curved shell elements. Brief discussion on the eccentric connection used in the model is given

as well. For the models performed by ATENA, solid elements were used for all members. Thus, the

discussion is limited to the solid element.

4.5.1.2 Solid Elements

The CHX60 type was used for solid elements in DIANA models. It is a twenty-node isoparametric

solid brick element. Its degrees of freedom are the translations ux, uy and uz in the local element

directions. The strain εxx and stress σxx vary linearly in x direction and quadratically in y and z directions.

Similarly, εyy and σyy vary linearly in y direction and quadratically in x and z directions. εzz and σzz vary

linearly in z direction and quadratically in the other directions. This element is based on Gauss

integration. The 3×3×3 default integration scheme is adopted in the analysis. In ATENA, a lower order

brick element of eight nodes was adopted. The element has a linear interpolation function for the

displacement field. The geometry of both brick elements is depicted in Figure 4-18.

Figure 4-18: Load Application: (a) Load control; (b) Displacement control (DIANA Manual)


112

(a) (b)

4.5.1.3 Curved Shell Elements

Curved shell elements in DIANA are based on two assumptions. First, normal planes remain straight,

but not necessarily normal to the reference surface. Second, the normal stress, σzz equal zero. This

implies that the in-plane strains, εxx, εyy and γxy vary linearly in the thickness direction, while the

transverse shear strains γxz and γyz are assumed to be constant in the thickness direction. At each node

there are five degrees of freedom; three translations and two rotations. The eight-node CQ40S, shown

in Figure 4-19, was chosen for this work. It is based on quadratic interpolation and the default 2 × 2

Gauss integration over the η-ξ plane. In the thickness direction, using 9 integration points were

implemented in this work for nonlinear analyses as recommended from previous studies (6).

Eccentric connections

The 3D problem modelled with DIANA involves simulating the wide beam slab using curved shell

elements for both slab and wide beam members. This results in introduction of eccentricity between the

Figure 4-19: Solid Element types: (a) CHX60-DIANA Models; (b) 8 nodes CCIsoBrick-ATENA Models

Figure 4-20: 8-node CQ40S curved shell element (DIANA Manual)


113

slab and supporting wide beams. Curved shell elements may be connected eccentrically to their nodes

as shown in Figure 4-20. The eccentricity must be defined in the nodal element xyz directions.

4.6 Conclusions

This chapter outlines the main features of the NLFEA implemented in this thesis. Two finite element

programmes, namely DIANA and ATENA are described with discussion limited to the modelling

procedures used in the thesis. DIANA is used for analysis with curved shell elements and ATENA for

analysis with solid elements since it was found to give better strength predictions with the adopted

constitutive models. A Total Strain crack model is used in DIANA while a Fracture-Plasticity model is

used in ATENA for punching shear analysis. The features of these two concrete constitutive models are

described for both compression and tension. With regard to cracking, the smeared crack concept is

adopted throughout the thesis. The differences between the fixed and rotating crack models are clearly

highlighted. The modelling of steel reinforcement is addressed as well. Generally, reinforcement can

be modelled with either discrete or embedded elements. The latter is adopted throughout this thesis.

Embedded reinforcement can be in the form of either discrete bars or grid. Both forms are utilized in

the NLFEA. Moreover, Von Mises plasticity material without hardening was assigned for all types of

embedded reinforcement. The chapter also gives an overview on the iterative solution methods used in

the NLFEA. This includes the Quasi-Newton which implemented in DIANA models and the Arc length

method which is used in ATENA models. In addition, the main characteristics of Newton-Raphson

method are noted. The convergence criteria associated with each software are listed and preferences

were made based on user’s manual recommendations and previous studies. The methods of applying

the incremental loads in the finite element models, load and displacement control, are briefly described.

Other aspects are reviewed including element types, solid and shell elements and integration schemes.

Figure 4-21: Eccentric Connection (DIANA Manual)


114

The findings of this chapter are implemented in the FE models that form the basis of the thesis. The

next chapter describes the validation of the NLFEA in detail.

Validation Studies Chapter 5

115

Validation Studies

5.1 Introduction

The overall objective of the research is to develop numerically an improved design method for wide

beam slabs. The NLFEA of wide beam slabs considers the influence of reinforcement arrangement on

the structural response of wide beam floors at both the serviceability limit state (SLS) and ultimate limit

state (ULS). At the SLS, emphasis is on the influence of reinforcement arrangement on crack width. At

the ULS, the limit states of flexure and shear are of prime concern. The NLFEA directly accounts for

the influence of compressive membrane action (CMA) on deformation and flexural capacity of slab.

To achieve these objectives, two finite element software packages were used, namely: TNO DIANA

v9.6 (11) and ATENA v5 (12). The former was employed to simulate a wide beam floor with shell

elements, while the latter was used to simulate the edge and internal column connections with solid

elements. The main outcomes from the DIANA analysis are the load-deflection and load-rotation

curves, steel strains and crack widths. In addition, the flexural failure load and the effect of CMA on

the deformation and flexural capacity of slab are obtained. Further, the punching resistance of wide

beams is computed using the Critical Shear Crack Theory (CSCT) of Muttoni (75). According to the

CSCT, the punching resistance is related to the width of the critical shear crack which is proportional

to the slab rotation relative to the column. The rotations are easily obtained from DIANA since shell

elements were used for modelling the wide beam slabs.


116

Besides the load-deflection response and ultimate failure load, the ATENA analysis also gives insight

into the failure mechanism and crack pattern.

This chapter describes the studies carried out to validate the DIANA and ATENA analyses developed

in the research. The test data used in the validation have been carefully selected to be representative of

the issues involved in the modelling of wide beam slabs. In order to validate the DIANA model, slabs

tested by Clark & Spiers (135) and Guandalini et al (136) have been chosen to assess the FE model’s

ability to capture the flexural behaviour of slabs at all loading stages. Furthermore, the capability of the

DIANA model to predict the enhancement in strength and stiffness due to the effects of CMA was

examined by simulating the one-way slab specimens tested by Lahlouh & Waldron (137). It is

noteworthy that the DIANA model adopted in this research is similar to that used by Tay (6) in his

assessments of slab deflection. As part of his research, he validated the DIANA model, in which curved

shell element type CQ40S was used for modelling slabs, against test data. These tests consisted of one-

way slabs and two-way slabs failing primarily in flexure. The one way spanning slab tests included tests

by Clark & Spiers (135), Jain & Kennedy (55), McNeice (138) and Lambotte & Taerwe (139).

The DIANA model has also been validated against slabs failing in punching (78,140). The main

variables were the arrangement of the flexural reinforcement over support, the ratio between the flexural

steel ratios in the orthogonal directions and loading arrangement. The punching shear resistance was

evaluated according to the MC2010 Level of Approximation (LoA) IV in which rotations are obtained

directly from NLFEA.

On the other hand, ATENA software is used to estimate linear shear and punching shear resistance of

sub-assemblies comprising solid elements. Sub-assemblies are modelled due to the high computational

cost associated with solid element modelling of concrete structures. The ATENA modelling procedure

is validated against tests including beams without and with transverse reinforcement (141), slabs failing

in punching without and with shear reinforcement for both axis-symmetry as well as non-axis-symmetry

conditions (140,142). In addition, the FE model is used to simulate tests with different types and

arrangements of shear reinforcement (143). Lastly, it is noteworthy that all tests considered for

validating the FE models were based on short-term loading.


117

5.2 Validation Studies for DIANA Model

All tests modelled with DIANA use the same constitutive models for concrete and steel reinforcement.

Chapter 4 gives details of the adopted constitutive models with only key details repeated here.

Thorenfeldt’s softening model (122) is adopted for concrete in compression. Tay’s linear softening

relationship is used for concrete in tension. In this model, the maximum tensile strength of concrete is

limited to 0.5fct and, after cracking, the stress is assumed to reduce linearly to zero at a strain of 0.5εy,

where εy is the yield strain for reinforcement (6). Tay validated his model for slabs failing in flexure

tested by Jain & Kennedy (55), McNeice (138) and Lambotte & Taerwe (139). A variable retention

shear factor based on aggregate interlock is used with an aggregate size of 10 mm. Steel reinforcement

was modelled with embedded discrete bars using a bi-linear stress-strain relationship. Eight-noded

curved shell element type CQ40S are used for modelling slabs, with a minimum of nine integration

points through the slab thickness. Columns and plates are modelled with CHX60-brick elements with

20 nodes with the default integration scheme of 3×3×3. The mesh size for each study was selected on

the basis of a mesh sensitivity study some of which are presented in the relevant sections.

Clark and Speirs (135)

5.2.1.1 Experimental Models

Clark & Speirs (135) performed nine tests on one way spanning slabs to investigate the influence on

deflection of the tension stiffening effect provided by concrete between cracks. The main variables were

the flexural reinforcement ratio and the arrangement of steel bars. Three slab specimens were selected

to validate the DIANA model; namely slab 1, slab 4 and slab 7. Each slab was 3500 mm long, 900 mm

wide and 200 mm deep. The concrete cover was 35 mm. The slabs were subjected to two point loads

which yielded a constant moment zone of length 1200 mm and two shear spans of 1000 mm each. The

strains in steel were measured at Demec points at bottom of slab. Figure 5-1 shows the arrangement of

main steel in cross section. The material properties for the slabs and reinforcement details are given in

Table 5-1 and Table 5-2.

Figure 5-1: tension rebar arrangements for slabs 1. 4 &7. (135)


118

Table 5-1: Material properties for slab specimens 1, 4 and 7 Specimen Concrete (measured) Main steel (measured)

fcu: (MPa) ft: (MPa) Ec: (GPa) Area (mm2) fy: (MPa) Slab 1 35.4 2.65 27.8 1885 450 Slab 4 33.6 3.26 28.9 1206 450 Slab 7 25.2 2.04 25.9 679 450

Table 5-2: Steel details for slab specimens 1, 4 & 7. Specimen Width

(mm) Depth (mm)

Bottom steel Top steel deff (mm) No. of bars Dia.(mm) No. of bars Dia.(mm)

Slab 1 902 204 169 6 20 4 8 Slab 4 901 204 169 6 16 4 8 Slab 7 900 204 169 6 12 4 8

5.2.1.2 FE Modelling

The slabs were modelled using 50 mm × 50 mm 8 noded curved shell elements of type CQ40S with

2×2×9 integration scheme. CHX60-brick elements with 20 nodes, and default integration scheme, were

adopted for the loading plate. One half of the slab was modelled due to symmetry. The slab was

restrained vertically and horizontally at its end across the width of the slab. In order to model the

symmetry conditions, the slab was restrained at mid span in the longitudinal direction and against the

rotation about the transverse axis. Figure 5-2 illustrates the mesh discretization and boundary conditions

imposed on the model. Load control was adopted by applying a point load through a steel plate.

5.2.1.3 Experimental versus FEA results

Figure 5-3 shows the moment-curvature responses for slabs 1, 4 and 7 obtained from tests and NLFEA.

In general, the predicted responses are in good agreement with test data. The FE models predicted the

strengths of all specimens with reasonable accuracy as shown in Table 5-3. In addition, Figure 5-4

Figure 5-2: Mesh Discretization and boundary conditions for slabs 1, 4 & 7

Z = 0

P Line of symmetry

x=0, Ryy=0

z

x y


119

presents the steel strains obtained from NLFEA and those from test data. It can be seen that the model

gives good predictions of the steel strains. It can also be observed that the experimental and NLFEA

plots tend to converge at high strains.

Table 5-3: Comparison between the strengths of slabs 1, 4 and 7 from tests and NLFEA Specimen Mtest: (kN.m) MFE:(kN.m) Mtest/MFE Slab 1 115 125 1.09 Slab 4 74.3 81.9 1.10 Slab 7 44.8 47.3 1.06 Mean

1.08

COV* 0.02 *COV = the Coefficient of variance is defined as the ratio between the standard deviation and the mean.

5.2.1.4 Conclusion

Three slabs, which were part of tests carried out by Clark & Speirs, were simulated to examine the

ability of DIANA’s model to capture the flexural behaviour as well as the tension stiffening

phenomenon. The model predicted accurately flexural response and accounted well for the

enhancement in stiffness due to tension stiffening.

0

20

40

60

80

100

120

140

0.0E+00 4.0E-04 8.0E-04 1.2E-03 1.6E-03 2.0E-03

Mom

ent:

kN.m

steel strain

Slab1-Test

Slab4-Test

Slab7-Test

Slab1-FE

Slab4-FE

Slab7-FE

Figure 5-4: Comparison between mean surface strains obtained from tests and NLFEA for slabs 1, 4 & 7

Figure 5-3: Comparisons of moment-curvature curves between the results of NLFEA and test data for Clark & Speirs slabs 1, 4 & 7

0

25

50

75

100

125

150

0.0E+00 1.0E-05 2.0E-05 3.0E-05 4.0E-05

Mom

ent:

kN.m

Curvature: mm-1

Slab1-FE

Slab1-Test

0

20

40

60

80

100

0.0E+00 1.0E-05 2.0E-05 3.0E-05M

omen

t: kN

.m

Curvature: mm-1

Slab4-FE

Slab4-Test

0

10

20

30

40

50

0.0E+00 5.0E-06 1.0E-05 1.5E-05 2.0E-05 2.5E-05

Mom

ent:

kN.m

Curvature: mm-1

Slab7-FE

Slab7-Test


120

CMA tests of Lahlouh & Waldron (137)

5.2.2.1 Introduction

It has been experimentally proven that compressive membrane action can significantly increase the

load-carrying capacity of reinforced concrete slabs. Compressive membrane forces develop as a result

of axial restraint of extension which would otherwise occur due to flexural cracking. Figure 5-5

illustrates the development of CMA in a reinforced concrete slab under uniformly distributed load.


In order to study the effect of CMA on the strength of reinforced concrete slabs, Lahlouh and Waldron

(6) tested experimentally three H-shaped subassemblies each of which represents a reinforced concrete

slab supported by walls at each end. The dimensions of a typical slab specimen were 2500 mm long,

300 mm wide and 150 mm thickness. All side walls were 1800 mm high and 300 mm wide. The main

variable was the degree of restraint, which was controlled by varying the thickness of side walls. The

wall thicknesses for specimens H-100, H-200 and H-300 were 100 mm, 200 mm and 300 mm

respectively. Table 5-4 gives the material properties and the reinforcement details for three

subassemblies.

Vertical loads were applied to the slab at four points spaced at equal intervals of 625 mm. The upper

and lower ends of the side walls were restrained laterally during testing. The lower ends were also

restrained vertically. Load increment of 5 kN was applied at the beginning of the test. As the specimen

approached its design failure load, the load increment was reduced to 2.5 kN until failure. Figure 5-6

shows the geometry and loading for the tested specimens.

Figure 5-5: Sketch showing the CMA in axially restrained RC slabs. (137)


121

Table 5-4: Material properties and steel details. Subassembly Cube strength:

(MPa) Effective depth of

slab: (mm) Steel ratio of slab: % Effective

depth of wall: (mm)

Steel ratio of wall: %

H-100 H-200 H-300

71.8 78.7 64.4

121 121 121

0.54 0.54 0.54

71 171 271

1.34 0.38 0.24


The FE element implemented in the NLFEA was an 8 nodes curved shell element (CQ40S) with

integration scheme of 2 × 2 × 9. The wall was simulated using a 20 nodes brick element (CHX60) with

3 × 3 × 3 integration scheme. Curved shell elements were connected to solid elements through the

automatic tying function. The slab strip was meshed with 39 mm × 50 mm element size while 50 mm

was used for the wall. The nodes of the steel loading plate coincided with slab mesh nodes. Concrete

was modelled in tension with Tay’s linear softening model and in compression with Thorenfeldt’s

model as previously described. The rebar was modelled with discrete embedded bars. By taking

advantage of symmetry, only half of the structure was modelled. Thus, the slab was restrained in the

longitudinal direction at mid span. The side wall was restrained vertically at the mid node of the bottom

end. In addition, the side wall was restrained horizontally at its top and bottom ends. Figure 5-7 shows

the FE model used DIANA analysis.

Figure 5-6: Sketch showing the geometry and loading of tested specimens by Lahlouh & Waldron.


122


Figure 5-8 shows the results of the load-deflection diagrams from the tests and NLFEA. The measured

and predicted structural responses of subassemblies H-100 and H-200 agree fairly well throughout the

loading regime. Predictions are less good for H-300 as also found by Lahlouh and Waldron as shown

in Figure 5-9. However, Lahlouh and Waldron did not report any reasons that caused the FE results

varies significantly from test results for H-300 specimen. The figure also shows the results of load-

deflection diagrams without considering CMA. This was done by releasing the longitudinal

displacements at midspan. With reference to Table 5-5, which shows a comparison between the ultimate

loads from the tests and NLFEA as well as the increase in strength due to CMA from the test and

NLFEA, it can be concluded that the FE model captured the increase in strength due to CMA

adequately.

P P

Plane of symmetry

Rxx =0, x=0

x=0, y=0

Pinned Support

y

x

Figure 5-7: Mesh Discretization, load introduction and boundary conditions for slab strip tests by Lahlouh & Waldron


123

Table 5-5: Comparison between the specimen strengths from the test and NLFEA

Subassembly Plastic load (No

CMA): kN

VTest : kN

VFE : kN

VFE without

CMA: kN VFE (Lahlouh &

Waldron): kN increase in

strength due CMA %

from FEA

increase in strength

due CMA % from test

VFE/VTest

H-100 H-200 H-300

81.2 81.7 80.5

84.7 109.7 143.1

84.0 104.4 117.2

78.4 84.4 75.6

94.2 108.5 127.5

7.1 23.7 55.0

4.3 34.3 77.8

0.99 0.95 0.82

Mean 0.92 COV 0.10

Figure 5-8: Comparison between the load-deflection curves from test and NLFEA results

0102030405060708090

0 20 40 60 80

Load

: kN

Central deflection: mm

H-100-FE

H-100-Test

H-100-FE(without CMA)

0

20

40

60

80

100

120

0 20 40 60 80

Load

: kN


H-200-FE

H-200-Test

H-200 -FE(without CMA)

0

30

60

90

120

150

0 10 20 30 40 50

Load

: kN


H-300-FE

H-300-Test

H-300-FE(without CMA)

Figure 5-9: Measured and predicted loads against mid-span deflections according to Lahlouh & Waldron (137)


124

5.2.2.5 Conclusion

Three laterally restrained slabs were modelled using TNO DIANA to examine whether the FE model

can simulate with reasonable accuracy the CMA effects on the load-carrying capacity and deformation

in slabs. The CMA in the tests is governed by the thickness of the supporting walls. The results show

that the NLFEA captured the influence of CMA fairly well, but the effect was underestimated for H-

300 supported by the thickest walls as for the analysis of Lahlouh and Waldron.

Guandalini et al (136)


Guandalini et al. (136) carried out tests on eleven square RC slabs representing internal slab-column

connections without transverse reinforcement. The main objective of the tests was to investigate the

punching resistance of slabs with low reinforcement ratios. For all specimens, the columns were square

with sides were slightly longer than the thickness of slab. The main variable was the flexural

reinforcement ratio. The test programme also studied the size effect and influence of reinforcement

stress-strain curve; i.e., hot rolled and cold formed. All slabs failed in punching with different load-

rotation responses due to different levels of stress in rebar. Some slabs failed with yielding of

reinforcement limited to the area over the column while the other parts were still elastic. Other slabs

exhibited large deformations at failure and reached their plastic plateau. The FE model was validated

for slabs PG8 and PG9, which were selected because they reached their full flexural capacity. The slab

geometry and material properties are shown in Figure 5-10 and Table 5-6. Figure 5-10 also illustrates

the loading arrangement which consisted of eight symmetrically positioned concentrated loads around

the slab perimeter.

Figure 5-10: Geometry for tested slabs. (136)

Section A-A


125

Table 5-6: Geometry and material properties for specimens PG8 and PG9 Specimen Geometry Concrete Steel Reinforcement

B (m) h (m) c (m) deff (m) fc (MPa) Layout (mm) fy (MPa) ρ % PG8 1.5 0.125 0.130 0.117 34.7 Φ8@155 525 0.28 PG9 1.5 0.125 0.130 0.117 34.7 Φ8@196 525 0.22


The slab was simulated using square 8-node curved shell elements (CQ40S) with an integration scheme

of 2×2×13, while the column and loading plates was simulated using a 20 nodes brick element (CHX60)

with 3×3×3 integration scheme. The steel reinforcement was modelled using embedded grids. Only one

quarter of the slab was modelled due to symmetry which was enforced by restraining the slab at mid

span against both rotation and translation. The column was vertically restrained at the bottom. Load

control was implemented during the analysis, while the loading was applied through steel plates. The

loading and mesh discretization of slab is shown in Figure 5-11.

A mesh sensitivity study was carried out to determine the mesh element size. Three element length sizes

were investigated: 25 mm, 50mm and 100 mm. The load-deflection curve was plotted for each size and

shown in Figure 5-12. As can be seen, 25 mm and 50 mm elements give very similar results with results

from the 100 mm mesh not very different. However, 25 mm was chosen for this analysis.

Lines of symmetry

Rxx=0, y=0 Ryy=0, x=0

Pinned support

P P

Figure 5-5: Mesh Discretization, boundary conditions and load application for slabs PG8 & PG9


126


Figure 5-13 compares the measured and predicted responses of slabs PG8 and PG9. The vertical axis

has been normalised by dividing the punching shear load by the quantity . , whereas in the

horizontal axis the deflection has been divided by the slab effective depth.

The FE model captured the structural behaviour of slabs generally well. The FE model appears,

however, to overestimate the initial stiffness of both specimens. As the value of elastic modulus was

not reported in the test, values were estimated according to EC2. One possible source for the

overestimate in stiffness could be differences between the assumed values of elastic modulus used in

the NLFEA and the actual ones. There could also have been some bedding-in of the test specimen that

is not simulated in the NLFEA. As the load increases, the predicted and actual responses become very

similar.

5.2.3.4 Sensitivity study to Ec & ft

A sensitivity study was carried out to investigate the influence of varying the concrete elastic modulus

and tensile strength. Slab PG8 was selected for this sensitivity study. The baseline values of Ec and ft

were calculated according to EC2. In addition, values of 0.75 Ec & 1.25 Ec were used. Similarly, 0.75 ft

and 1.25 ft were used to investigate the influence of concrete tensile strength.

0.00

0.05

0.10

0.15

0.20

0.25

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

V/b 0

.d.fc

0.5 :

MPa

0.5

w/d

PG8-Test

PG8-NLFEA

0.00

0.05

0.10

0.15

0.20

0.25

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

V/b 0

.d.fc

0.5 :

MPa

0.5

w/d

PG9-Test

PG9-NLFEA

Figure 5-8: Load-deflection curves obtained from the test and the NLFEA for PG8 and PG9.

Figure 5-7: L-d curves for slab PG8 for mesh sizes: 25mm, 50mm and 100mm.

020406080

100120140160

0 4 8 12 16 20Lo

ad: k

NDeflection: mm

25 mm

50 mm

100 mm


127

Figure 5-14 & Figure 5-15 present the load-deflection results from the FEA with varying Ec and ft

respectively. The load-deflection response from the test is also plotted for comparison. It can be seen

that varying Ec from 0.75 Ec to 1.25 Ec affects the stiffness and strength at early loading stages, while

the effect becomes less significant close to the ultimate load. It is also noted that analysis with the

reduced elastic modulus results in a more ductile response. Figure 5-15 shows that varying the concrete

tensile strength affects the slab behaviour most significantly around first cracking. The load-deflection

curves for the three models converge close to the yield load.

Sagaseta et al.(140)

5.2.4.1 Experimental Model

Sagaseta et al., (140) investigated the load-rotation responses and punching shear resistance for RC

slabs without transverse shear reinforcement under non-axis symmetrical conditions. Such

asymmetrical conditions include varying the flexural rebar ratios and loading arrangements in the x-

and y axes. The main aim of this study was to examine the capability of the DIANA model to capture

correctly the load-rotation relationship in the weak and strong reinforced directions. This is particularly

important since the model will be used to simulate the wide beam connection where the flexural rebar

Figure 5-9: NLFEA sensitivity to concrete elastic modulus (Ec)

Figure 5-10: NLFEA sensitivity to concrete tensile strength (ft)

0

20

40

60

80

100

0 1 2 3 4 5

Load

: kN


(a) Early load stage

0.75Ec=25.72 GPa

Ec=34.29 GPa

1.25Ec=42.86 GPa

Test0

20

40

60

80

100

120

140

160

180

0 10 20 30 40 50

Load

: kN


(b) Overall response

0.75Ec=25.72 GPa

Ec=34.29 GPa

1.25Ec=42.86 GPa

Test

0

25

50

75

100

0 1 2 3 4 5

Load

: kN


(a) Early load stage

0.75ft=1.065 MPa

ft=1.42 MPa

1.25ft=1.78 MPa

Test 0

25

50

75

100

125

150

175

0 10 20 30 40

Load

: kN



0.75ft=1.065 MPa

ft=1.42 MPa

1.25ft=1.78 MPa

Test


128

ratios vary significantly in the orthogonal directions. The geometry and load arrangement are illustrated

in Figure 5-16 and the material properties are summarized in Table 5-7. All slabs were 250 mm thick.

(a) (b)

Table 5-7: Main characteristic of PT-series slab series. Slab Loading Reinforcement dav:

mm fc : MPa ρx/ρy Bar dia.: x – y

direction: mm fyx - fyy : MPa

PT21 Two-way Asymmetric 192 67.5 1.64/0.84 20 - 16 597 - 552 PT22 Two-way Symmetric 196 67.0 0.82/0.82 16 - 16 552 - 552 PT23 Two-way Asymmetric 189 66.0 0.85/0.36 16 - 10 552 - 568 PT31 Two-way Symmetric 212 66.3 1.48/1.48 20 - 20 540 - 540 PT32 Two-way Asymmetric 215 40.0 1.46/0.75 20 - 16 540 - 558 PT33 Two-way Asymmetric 212 40.2 0.76/0.32 16 - 10 558 - 533 PT34 One-way Symmetric 216 47.0 0.74/0.74 16 - 16 558 - 558


The PT slabs were modelled similarly to PG slabs in terms of the constitutive model for concrete and

steel, mesh type and size, boundary conditions and load application method. The details are described

in Section 5.2.3.2.


Figure 5-17 compares the measured and predicted load-rotation responses of the PT slabs as well as

showing the failure criterion for punching according to the CSCT which was calculated as follows:

( ) . = ./

(5.1)

Figure 5-11: Definition of test specimens for PT-series slabs: (a) general geometry; (b) type of loading (12)


129

where VR denotes the punching shear strength, b0 the control perimeter length, d the average effective

depth, fc the concrete cylinder strength, ψ the rotation of slab outside the column region and dg is the

maximum aggregate size. According to the CSCT, the punching resistance is given by the intersection

of the load rotation and resistance curves.

The DIANA analyses simulate the measured load-rotation curve well for slabs PT22, PT31 & PT32

with symmetric reinforcement and two-way loading. Good agreement was also obtained for slab PT21

with asymmetric reinforcement and two-way loading. The measured response is also reasonably

captured for slab PT34 with one-way loading and symmetrical reinforcement but the predicted response

is overly stiff in the y direction. In the case of slabs, PT23 & PT33 with asymmetric reinforcement, the

model overestimates significantly the flexural failure load. This is due to the very low reinforcement

ratios of 0.36% & 0.32% for PT23 & PT33 respectively in the weaker direction.

0

500

1000

1500

2000

0 5 10 15 20 25 30

Load

: kN

Rotation: mRad

PT21 Test-X

FE-X

Test-Y

FE-Y

Failure criterion 0250500750

1000125015001750

0 5 10 15 20 25 30 35 40

Load

: kN

Rotation: mRad

PT22 Test-X

FE-X

Test-Y

FE-Y

Failure criterion

0200400600800

1000120014001600

0 20 40 60

Load

: kN

Rotation: mRad

PT23Test-X

FE-X

Test-Y

FE-Y

Failure criterion0

500

1000

1500

2000

0 5 10 15 20 25

Load

: kN

Rotation: mRad

PT31 Test-X

FE-X

Test-Y

FE-Y

Failure criterion

0

250

500

750

1000

1250

1500

0 5 10 15 20 25

Load

: kN

Rotation: mRad

PT32Test-X

FE-X

Test-Y

FE-Y

Failure criterion0

500

1000

1500

0 5 10 15 20 25 30 35 40

Load

: kN

Rotation: mRad

PT33Test-X

FE-X

Test-Y

FE-Y

Failure criterion

0

250

500

750

1000

1250

1500

0 5 10 15 20 25 30 35 40

Load

: kN

Rotation: mRad

PT34Test-X

FE-X

Test-Y

FE-Y

Failure criterion

Figure 5-17: Load-rotation curves of PT slabs given by test and NLFEA results from DIANA model


130

In an attempt to improve the calculated response, the following six alternative tension

stiffening/softening models were examined: (a) Tay’s tension stiffening model with the concrete tensile

strength 0.5 fct. (b) Tay’s tension stiffening model with the concrete tensile strength reduced to 0.25 fct

instead of 0.5 fct (c) Tay’s tension stiffening model with the concrete strain corresponding to zero tensile

strength in concrete reduced to 0.25 εs instead of 0.5εs. (d) Constant shear retention factor with β equal

0.1. (e) Constant retention shear factor with β equal 0.05. (f) Constant shear retention factor with β equal

0.01. Tay’s tension stiffening model with the concrete tensile strength 0.5 fct and zero tensile strength

corresponds to strain equal 0.5εs was adopted for the analyses (d), (e) & (f). The results are presented

in Figure 5-18

Figure 5-18 shows that the initial load-rotation response is best predicted by Tay’s model without

applying any modifications. However, Tay’s model overestimates the flexural failure load and no

improvement has been gained from modifying its parameters. Using a constant shear retention factor,

gives improved predictions of the flexural failure load with best results obtained with β = 0.01. MC

2010 LoA IV gives the best estimates of punching resistance with β = 0.01 despite the initial load-

rotation response being overly soft. The Tay tension stiffening model with variable shear retention

factor also gives reasonable predictions of punching resistance using MC2010 LoA IV. Therefore, a

variable retention shear factor based on aggregate interlock will be implemented in the DIANA model

used for punching investigations with MC2010 LoA IV in Chapter 7.

5.3 Validation Studies for ATENA Model

Beam Shear Tests by Fang

These tests were modelled to develop a satisfactory procedure for using ATENA to simulate the

structural behaviour of RC beams failing in shear. Besides predicting ultimate loads, correctly

0

200

400

600

800

1000

1200

1400

1600

0 10 20 30 40

Load

: kN

Rotation: mRad

PT23-X directionTest-X

0.5fct-X

0.25fct-X

0.25εct-X

B=0.1-X

B=0.05-X

B=0.01-X

MC2010 failurecriterion 0

200

400

600

800

1000

1200

1400

1600

0 20 40 60 80

Load

: kN

Rotation: mRad

PT23-Y directionTest-Y

0.5fct-Y

0.25fct-Y

0.25εct-Y

B=0.1-Y

B=0.05-Y

B=0.01-Y

MC2010 failurecriterion

Figure 5-18: Investigation on the flexural response of slab PT23 using: (a) Tay’s model with 0.5 fct , 0.5εs (b) Tay’s model with 0.25 fct , 0.5εs (c) Tay’s model with 0.5 fct , 0.25εs (d) β=0.1, (e) β=0.05 (f) β=0.01


131

simulating the mode of failure and associated crack pattern were of prime concern. This is particularly

relevant to the analysis of the wide beam sub-assemblies undertaken in Chapter 7 with ATENA.


ATENA was used to model six short shear span RC beams tested to failure in shear by Fang (141). The

beams were grouped into two series with two beams without shear reinforcement in series A and four

beams with shear reinforcement in series S. The objective of the tests was to investigate the influence

on shear resistance of shear reinforcement, (typical ratios; ρ% = 0 - 0.609), loading arrangement, (two

and four-point loading), and bearing plate size (typical sizes were 100 mm ×100 mm, or 200 mm×200

mm). All beams were 3.0 m length, 0.5 m deep and 0.165 m wide.

Both A- and S- series beams had the same longitudinal reinforcement (4H25) at tension face and two

bars (2H16) at the compression face.

The main characteristics and material properties of the beams are summarised in Table 5-8. The cross

sections of beams and reinforcement details are shown in Figure 5-19 and the beam geometries are

shown in Figure 5-20 and Figure 5-21 for A- and S- series respectively.

Table 5-8: Main characteristics of Fang beams Beam

Designation fc:

(MPa) fct:

(MPa) fy:

(MPa) Shears stirrups ratio %

d: (mm)

No. of loading points

Size of Bearing Plates (mm)

Loading Plate

width: (mm)

H25 H16 T8 Left Right

A-1 33.1 3.2 560 540 540 0 437 4 200 100 100 A-2 34.6 3.2 560 540 540 0 437 2 200 100 100 S1-1 33.7 3.2 560 540 540 0.305 437 4 200 100 100 S1-2 36.0 3.2 560 540 540 0.305 437 2 200 100 100 S2-1 35.2 3.2 560 540 540 0.609 437 4 200 100 100 S2-2 36.7 3.2 560 540 540 0.609 437 2 200 100 100

Figure 5-19: Cross section and reinforcement detailing for A- & S- series beams (141)


132

(1) Beam A-1

(2) Beam A-2

(a) Beam S1-1

Figure 5-20: A- series beams geometries, loading arrangement and bearing plates sizes. (141)

(All dimensions are in mm).


133

(b) Beam S1-2

(c) Beam S2-1

(d) Beam S2-2

5.3.1.2 Finite Element Modelling

Material Modelling

The material selected to simulate concrete was CC3DNonlinear Cementitious2, which consists of a

combined fracture –plastic model (See Chapter 4, Section 4.3.1). Further details about the constitutive

models can be found elsewhere (15). CC3DNonlinear Cementitious2 gives the option of using either

fixed or rotating crack models. The latter was adopted since it gives better predictions as shown in

Figure 5-12: S- series beams geometries, loading arrangement, bearing plate sizes and stirrups distribution (141). (All dimensions are in mm).


134

Figure 5-23. The steel reinforcements were modelled as embedded discrete bars and perfect bond was

assumed between reinforcement and concrete. The stress-strain relationship was a bilinear. The

numerical values for the basic material parameters used in ATENA model are listed in Table 5-8. For

any missing data ATENA automatically generates concrete parameters based on the default formulas

and values shown in Table 5-9.

Table 5-9: Default values for material parameters used in ATENA models for beams and slabs Parameter Formula

Cylinder strength fc = -0.8 fcu Tensile strength ft = 0.24 (fcu)0.6667 Initial elastic modulus Ec = 2150 ( fcm /10 )0.333 Softening compression wd = -0. 5 mm Reduction in compressive strength due to cracks rc = 0.2 Maximum aggregate size 20 mm Tension stiffening stress σst = 0 Fracture energy GF = 73 fcm 0.18 Shear Factor SF=20 Poisson’s ratio 0.2

Load Application

There are two load application methods offered by ATENA: displacement control and load control. The

latter was adopted for all beams with load applied through steel plates in increments of 5kN.

Solution Method

The standard Arc length method was selected as it provides robust solutions and can capture snap back

and snap through phenomena. Tolerances of 0.001, 0.001 and 0.0001 were specified for displacement,

force, and energy criteria respectively.

Finite Element Mesh

ATENA 3D 5.1.1 software offers three types of elements: brick, tetra and mixed (brick and tetra). The

beams were simulated with 50 mm cubic brick-linear elements. The element size was based on a mesh

sensitivity study with cubic elements of side length: 25 mm, 50 mm, 75 mm and 100 mm. Beam A-1

was employed for the mesh sensitivity study. Figure 5-22 shows that a mesh size of 25 mm produces

almost identical results to the test. A mesh size of 50 mm was adopted for subsequent analyses since it

gives reasonably accurate results and is less costly in terms of solution time.

Thus, brick elements with 50 mm size was adopted for this analysis, which yielded ten elements through

the beam depth. The finite element mesh is shown in Figure 5-23. The whole beam was modelled


135

because of differences in the bearing plate sizes in each shear span and at the supports. Default

integration scheme (2×2×2) was adopted.


The results of ultimate shear resistance for all beams obtained from NLFEA are presented in Table

5-10. Comparisons between the load-deflection curves obtained from the test for the A and S beam

series and those from the NLFEA are shown in Figure 5-25. In addition, a comparison between

deflection at failure obtained from the NLFEA and test are shown as well in Table 5-10. It can be

observed that the FE model predicted the failure load in beams with four point-loading very well. On

the other hand, it appears to overestimate the failure loads in beams with two-point loading as in beams

A-2 and S1-2. The structural response was well captured and it was almost identical up to the formation

of initial crack in all beams. Then, slight increase in stiffness was predicted as load increased until

failure.

Figure 5-14: Example of FE beam model showing the mesh elements used for the analysis

0

200

400

600

800

1000

1200

0 2 4 6 8 10 12

Load

: kN


25 mm 50 mm 75 mm100 mm Test

Figure 5-13: Mesh sensitivity study for beam A-1 with mesh sizes: 25mm, 50mm, 75mm & 100mm.


136

Table 5-10: Comparison between the results given by the beam tests and the NLFEA in terms of ultimate failure load and deflection.

Slab fc: MPa

Pu-test: kN Pu-NLFEA: kN Pu-test / Pu-NLFEA dtest: mm dNLFEA: mm dNLFEA / dtest

A-1 33.1 823.0 821.6 1.00 8.6 6.6 0.77 A-2 34.6 349.0 465.0 1.33 4.0 4.9 1.23 S1-1 33.7 1000.0 1067.2 1.07 8.4 7.2 0.86 S1-2 36.0 601.0 745.2 1.25 5.8 6.8 1.17 S2-1 35.2 1179.0 1198 1.02 11.0 8.1 0.74 S2-2 36.7 820.0 888.0 1.08 8.6 7.3 0.85

Average 1.13 0.93 COV 0.12 0.23

0

200

400

600

800

1000

0 2 4 6 8 10 12

Load

: kN

Central displacement: mm

A1 Test

NLFEA

0

200

400

600

0 1 2 3 4 5 6

Load

: kN


A2 Test

NLFEA

0

300

600

900

1200

0 3 6 9 12

Load

: kN


S1-1 Test

NLFEA

0

200

400

600

800

0 1 2 3 4 5 6 7

Load

: kN


S1-2 Test

NLFEA

0

300

600

900

1200

1500

0 2 4 6 8 10 12 14

Load

: kN


S2-1 Test

NLFEA

0

200

400

600

800

1000

0 2 4 6 8 10 12

Load

: kN


S2-2 Test

NLFEA

Figure 5-16: Load-displacement curves given by the tests and the NLFEA for A & S beam series

0

200

400

600

800

1000

0 3 6 9 12

Load

: kN


A1 Test

Rotated crack

Fixed crack

0

300

600

900

1200

0 3 6 9 12

Load

: kN


S1-1 TestRotated crackFixed crack

Figure 5-15: Comparison between the results given by NLFEA using ATENA with fixed and rotated crack models and test results for beam specimens A-1 & S1-1


137

5.3.1.4 Crack patterns

Figure 5-26 compares the FEA and test crack patterns for A & S beam-series. The crack patterns

developed in the NLFEA agree well with those observed experimentally. First, small bending cracks

(˂0.1 mm) initiated at the bottom of the beam. Diagonal cracks subsequently developed and increased

in width as the loading was increased. At failure, for A-series beams all the crack widths larger than 0.1

mm were due to shear, while for beams S2-1, S2-2, there were bending cracks with widths larger than

0.1 mm, initiating from the bottom going upwards, as well as shear cracks. For beams S1-1 & S1-2 both

types of cracks developed, but flexural cracks with widths larger than 0.1 mm only developed at the

bottom of the beam.

(1) Beam A-1

(2) Beam A-2


138

(3) Beam S1-1

(4) Beam S1-2

(5) Beam S2-1


139

(6) Beam S2-2

5.3.1.5 Conclusion

Six tests of reinforced concrete beams with and without shear reinforcement were simulated using

ATENA 3D v5.1.1. The beams had different loading arrangements and bearing plate sizes. The

objective was to examine the capability of the FE model in capturing the behaviour of the beams failing

in shear. This includes estimating the ultimate shear strength, deflection and predicting the crack pattern

during the loading process until failure. In order to achieve that, brick-linear elements with mesh size

of 0.05 m was adopted for the NLFEA. It can be concluded that the results obtained from the FE model

using ATENA 3D v5.1.1 agreed reasonably well with the test results in terms of failure loads for beams

either with or without vertical shear reinforcement. The predicted deflections, however, were less

accurate. It is also can be noted that the FE model captured correctly the failure modes of all beams.

Punching Shear Tests on slabs


Twenty punching shear tests of reinforced concrete slabs were simulated with ATENA. The specimens

consisted of seven slabs tested by Sagaseta et al. (PT series) (140), seven slabs tested by Lips et al. (PL

series) (142) and six slabs tested by Gomes & Regan (143). Both the PT slabs series and PL slabs series

were part of different investigation works carried out at EPFL.

The PT-series slabs tested by Sagaseta et al., (140) were unreinforced in shear. The tested variables

included the flexural reinforcement, which was independently varied in the x and y directions, and the

Figure 5-17: Comparisons between the crack patterns obtained from tests and NLFEA.(141).

Crack widths shown are not less than 0.1 mm.


140

loading arrangement which was either one- or two-way. The objective was to investigate the effect of

non-axis symmetrical conditions on punching shear resistance and load-rotation response. This series

is of particular relevance to wide beam slabs in which the slab is reinforced strongly in the longitudinal

axis and weakly in the transverse orthogonal axis.

The PL-series slabs tests, which were carried out by Lips et al., (142), aimed to study the influence of

various geometrical and mechanical parameters on the punching resistance of slabs with transverse

shear reinforcement (studs). The investigated parameters included the column size (130 mm – 440 mm),

slab thickness (250 mm – 400 mm) and the amount of transverse shear reinforcement.

Gomes & Regan (143) carried out tests on slabs with transverse shear reinforcement. The shear

reinforcement was provided using short offcuts of steel I-beams. The main variables were the area,

number and distribution of the shear steel elements.

The main characteristics of the slabs are summarised in Table 5-7, Table 5-11and Table 5-12. The slab

geometries and loading arrangements are shown in Figure 5-16, Figure 5-27 & Figure 5-28 for PT-

series, PL-series and Gomes & Regan slabs respectively.

Table 5-11: Main characteristic of PL-series slab series Slab h:mm C:mm Stud Dia:

mm Stud length :mm Rows No. No. in a row ρ,% fc: MPa

PL6 250 130 14 215 12 6 1.59 36.6 PL7 250 260 14 215 16 7 1.59 35.9 PL8 250 520 14 215 24 7 1.57 36.0 PL9 320 340 18 285 16 6 1.59 32.1 PL10 400 440 22 365 16 5 1.55 33.0 PL11 250 260 10 215 8 7 1.56 34.2 PL12 250 260 10 215 16 7 1.56 34.6

Table 5-12: Main characteristic of Gomes & Regan slab series Slab fcu (MPa) d (mm) Asv.fy per layer (kN) Method of distributing Shear steel

4 40.1 159 172.9 CDL* 5 43.4 159 270.2 CDL* 6 46.7 159 270.2 Radial 7 42.3 159 389.1 Radial 8 42.6 159 389.1 Radial 9 50 159 404.2 Radial

*CDL refers to cross double line pattern.


141

(a) (b)

Figure 5-27: Definition of test specimens for PL-series slabs: (a) general geometry; (b) Placing shear studs (142)

(a)

(b) (c) Figure 5-28: Definition of test specimens for Gomes & Regan slabs: (a) test set-up and general geometry, b) placing of shear offcuts of steel I-section beams (c) details of shear reinforcement (143).

(All dimensions are in mm).


142

5.3.2.2 Finite Element Modelling

Material Modelling

The material models for concrete and steel reinforcement were the same as used in Section 5.3.1.2.

Furthermore, the same solver and convergence criteria were applied.

FE mesh

All the slabs were modelled using brick-linear elements. Only a quarter of the slab was modelled taking

advantage of symmetry to accelerate the analysis. In order to determine the optimum mesh size, a

sensitivity study was carried out using the slab PT23 which was 250 mm thick. First, three cubic element

sizes were examined: 50 mm, 75 mm and 100 mm without considering refinement. Then, a quarter of

the FE model close to column was discretized with a finer mesh 25 mm for the models with mesh sizes

of 50 mm and 100 mm. This yielded 10 brick elements through the slab thickness compared to 5

elements in the other parts. The results are presented in terms of load-rotation curves about x- & y- axis

and shown in Figure 5-29.

Reasonable load rotation predictions were obtained about both axes with cubic elements with a mesh

size of 50 mm but even better results were obtained by refining the 50 mm – mesh, in the punching

failure zone around the column with a mesh size 25 mm. This discretization of using 25 mm cubic

elements around the column and 50 mm elsewhere was adopted in the ATENA analyses of the PT, PL-

series and Gomes & Regan slabs. The mesh discretization chosen for ATENA model for slabs is shown

in Figure 5-30 .

Load Application

The load was applied as point loads at the slab edge through steel plates. The slab was vertically

restrained at its centre.

(a) (b)

0100200300400500600700

0 5 10 15 20 25 30 35

Load

: kN

Rotation: mRad

Test-X 50 mm75 mm 100 mm100mm+refined 25mm 50mm+refined 25mm

0100200300400500600700

0 10 20 30 40 50

Load

: kN

Rotation: mRad

Test-Y 50 mm75 mm 100 mm100mm+refined 25mm 50mm+refined 25mm

Figure 5-29: Comparison of mesh size of slab PT23 in terms of L-R curves in the direction of: (a) x-axis (b) y-axis


143


PT slabs

The results obtained for the rotations and comparisons to EC2 (17), CSCT (75) and ACI 318 (4) are

presented in Table 5-13. Figure 5-31 compares the measured and predicted slab rotations for the PT

slabs. Rotations were calculated about the x and y axes from displacements at 200 mm centres adjacent

to the inclinometers used in the tests.

The NLFEA correctly predicted that all the tested slabs failed in punching shear within the shear

reinforcement area as observed. The average value of the ratio between the experimental and the

numerical failure loads was 0.93 with a coefficient of variation (COV) of 0.09.

Table 5-13: Comparison of the estimated punching loads using ACI, EC2, CSCT and FEA (ATENA) for PT slabs

Slab Vu-test (kN)

Vu-EC2 (kN)

Vu-ACI (kN)

Vu-NLFEA (kN)

V u-test/V u-EC2 V u-test/V u-ACI V u-test/V u-CSCT V u-test/V u-NLFEA

PT21 959 891.9 1064.5 1210.4 0.93 1.11 0.96 0.79 PT22 989 1038.5 1107.7 1058.2 1.05 1.12 1.07 0.86 PT23 591 673.7 673.7 573.3 1.14 1.14 0.97 1.01 PT31 1433 1590.6 2092.2 1676.6 1.11 1.46 1.17 0.86 PT32 1157 1353.7 1723.9 1388.4 1.17 1.49 1.20 0.96 PT33 602 674.2 674.2 590.0 1.12 1.12 0.98 1.01 PT34 879 896.6 896.6 879.0 1.02 1.02 1.00 1.03

Average 1.08 1.21 1.05 0.93 COV 0.08 0.15 0.08 0.09

Figure 5-19: Typical FE slab model showing the mesh discretization.


144

5.3.2.4 Sensitivity Study to Ec, ft, wd, rc

A sensitivity study was carried out on slab PT23 to investigate the influence of varying input parameters

from the ATENA default values used to describe concrete behaviour (NonLinCementitious2). The

varied parameters include the concrete elastic modulus and tensile strength, plastic displacement wd and

the limiting cracked concrete compressive strength reduction factor rc.

0

300

600

900

1200

1500

0 5 10 15 20 25 30

V: k

N

ψ: mRad

PT21

X-test

Y-test

X-FE

Y-FE0

300

600

900

1200

1500

0 5 10 15 20 25 30

V: k

N

ψ: mRad

PT22

X-testY-testX-FEY-FE

0

150

300

450

600

750

0 5 10 15 20 25 30

V: k

N

ψ: mRad

PT23


0

300

600

900

1200

1500

1800

0 5 10 15 20

V: k

N

ψ: mRad

PT31

X-test

Y-test

X-FE

Y-FE0

300

600

900

1200

1500

0 5 10 15 20 25 30

V: k

N

ψ: mRad

PT32


0

150

300

450

600

750

0 5 10 15 20 25 30 35

V: k

N

ψ: mRad

PT33

X-test

Y-test

X-FE

Y-FE

0

150

300

450

600

750

900

0 5 10 15 20 25 30 35

V: k

N

ψ: mRad

PT34

X-test

Y-test

X-FE

Y-FE

Figure 5-20: Load-rotation curves obtained from the test data and NLFEA results for PT series

Figure 5-21: Sensitivity to elastic modulus for concrete, Ec: - (a) Early loading stage, (b) overall response

0

100

200

300

400

500

0 1 2 3 4 5 6 7 8 9 10

Load

: kN


(a) Early loading stage

0.75Ec=31.7 GPaEc=42.27 GPa1.25Ec=52.84 GPaTest

0

100

200

300

400

500

600

700

0 25 50 75 100 125 150 175

Load

: kN


(b) Overall resonse

0.75Ec=31.7 GPaEc=42.27 GPa1.25Ec=52.84 GPaTest


145

Figure 5-32 shows the load-deflection curves from NLFEA carried out to study the sensitivity of the

FE model to the concrete elastic modulus. The deflection was monitored in the weak direction. The

elastic modulus was taken as the default value Ec, calculated in terms of the concrete cube strength,

0.75Ec and 1.25Ec. The load-deflection curve from the test is plotted as well. At early loading stage, the

slab response varies with Ec since the uncracked deflection inversely proportional to Ec. However, at

high loading levels, the difference diminishes and the ultimate load is virtually the same for three

models. A similar conclusion is obtained for the influence of concrete tensile strength as shown in

Figure 5-33. Figure 5-34 and Figure 5-35 indicate that the plastic displacement wd and the limited

reduction factor rc have no effect on the strength and stiffness.

Figure 5-22: Sensitivity to tensile strength for concrete, ft: (a) Early loading stage, (b) overall response

Figure 5-23: Sensitivity to plastic displacement, wd

Figure 5-24: Sensitivity to limited crack concrete compressive strength reduction factor, rc

0

100

200

300

400

500

0 1 2 3 4 5 6 7 8 9 10

Load

: kN


(a) Early stage loading

0.75ft=3.276 MPa

ft=4.368 MPa

1.25ft=5.46 MPa

Test

0

100

200

300

400

500

600

700

0 25 50 75 100 125 150 175

Load

: kN



0.75ft=3.276 MPa

ft=4.368 MPa

1.25ft=5.46 MPa

Test

0

100

200

300

400

500

600

700

0 25 50 75 100 125 150 175

Load

: kN


wd=0.25 mmwd=0.5 mm (default)wd=1.0 mmTest

0

100

200

300

400

500

600

0 25 50 75 100 125 150 175

Load

: kN


rc=0.7rc=0.5rc=0.2 (default)Test


146

Failure modes and crack patterns

All the tested slabs failed in asymmetrical punching except specimens PT22 & PT31 both of which had

symmetrical flexural rebar ratios in the orthogonal directions as well as symmetrical loading. In the

tests, the slope of the shear failure cone is reported to have been steepest (≈ 45o) on the side of the

maximum rotation. The ATENA model captured this characteristic reasonably well as illustrated for

slab specimen PT33 in Figure 5-36. The ATENA model is concluded to give good predictions of the

behaviour and the strength of PT slabs.

PL slabs

Table 5-14 summarizes the measured and predicted strengths at failure for the PL slabs. It also compares

the measured punching strengths with those calculated according to ACI 318, EC2, the CSCT and

NLFEA. Figure 5-37 compares the measured and predicted load-deflection curves which agree

reasonably well. The average value of the measured to the predicted strength using NLFEA is 1.00 and

the COV is 0.13.

Table 5-14: Comparison of the estimated punching loads using ACI, EC2, CSCT and NLFEA for PL slabs Slab Vu-test (kN) V u-test/V u-EC2 V u-test/V u-ACI V u-test/V u-CSCT V u-test/V u-NLFEA PL6 1363 1.41 1.30 1.02 0.95 PL7 1773 0.94 1.23 1.09 0.98 PL9 3132 1.03 1.29 1.06 1.16 PL10 5193 1.00 1.26 1.05 1.15 PL11 1176 1.03 1.08 1.04 0.80 PL12 1633 1.05 1.28 1.05 0.95 Average 1.08 1.24 1.05 1.00 COV 0.14 0.06 0.02 0.13

N W

Figure 5-25: Comparison of crack patterns along the transverse section of specimen PT33 from the test (140) and NLFEA


147

Failure modes and crack patterns

Figure 5-38 compares the crack patterns at failure predicted by the ATENA model and from the tests

for slab specimens; PL6, PL7, PL10 & PL12. The test crack patterns are from saw-cuts along the weak

axis after the tests, while the cracks shown in the FE model have widths not less than 0.3 mm. It is

concluded that the FE model predicts the observed crack patterns with adequate accuracy.

0

500

1000

1500

2000

0 5 10 15 20 25

V: k

N

ψ: mRad

PL6

Test

FE

0250500750

10001250150017502000

0 5 10 15 20 25 30 35

V: k

N

ψ: mRad

PL7

Test

FE

0

1000

2000

3000

4000

0 5 10 15 20 25 30

V: k

N

ψ: mRad

PL9

Test

FE

0

1000

2000

3000

4000

5000

6000

0 5 10 15 20

V: k

N

ψ: mRad

PL10

Test

FE

0

500

1000

1500

2000

0 10 20 30

V: k

N

ψ: mRad

PL11

Test

FE

0

500

1000

1500

2000

0 10 20 30

V: k

N

ψ: mRad

PL12

Test

FE

Figure 5-37: Load-rotation curves obtained from the test data and NLFEA results for PL series

PL6

PL7

PL10

PL12

Figure 5-38: Comparison of the crack patterns in specimens PL6, PL7, PL10 & PL12 between the test results (154) and predictions of the ATENA model. (Crack widths shown in the FE model ≥ 0.3 mm).


148

Gomes & Regan slabs

Table 5-15 presents the measured and predicted failure loads for the slabs of Gomes and Regan. In

addition, Figure 5-39 shows measured and calculated load-displacement curves. The measured and

calculated displacement was measured at a distance of 185 mm from the column centre. All the slabs

failed in punching outside the shear reinforced regions. The NLFEA failed to predict the failure mode

correctly. This could be because of the improved anchorage of short offcuts of steel I-beams which were

used as shear reinforcement instead of studs for which the model proved to give good agreements with

other test results (PL slabs series). The average punching load for this slabs group is 1.11 with COV of

0.10.

Table 5-15: Estimated punching loads using FEA (ATENA) for Gomes & Regan slabs Slab Vu-test (kN) Vu-NLFEA (kN) Vtest/VNLFEA

4 853 843.7 1.01 5 853 882.6 0.97 6 1040 981.4 1.06 7 1120 947.4 1.18 8 1200 953.9 1.26 9 1227 1024.0 1.20

Average 1.11 COV 0.10

0

200

400

600

800

1000

0 5 10 15 20 25

V: k

N

Displacement: mm

Slab 4

Test

NLFEA

0

200

400

600

800

1000

0 5 10 15 20 25

V: k

N

Displacement: mm

Slab 5

Test

NLFEA

0

250

500

750

1000

1250

0 5 10 15 20 25 30 35

V: k

N

Displacement: mm

Slab 6

Test

NLFEA

0

250

500

750

1000

1250

0 5 10 15 20 25 30 35

V: k

N

Displacement: mm

Slab 7

Test

NLFEA

0

250

500

750

1000

1250

0 5 10 15 20 25 30

V: k

N

Displacement: mm

Slab 8

Test

NLFEA

0

250

500

750

1000

1250

0 5 10 15 20 25 30 35 40

V: k

N

Displacement: mm

Slab 9

Test

NLFEA

Figure 5-39: Load-rotation curves obtained from the test data and NLFEA results for Gomes & Regan slabs


149

5.3.2.5 Conclusion

Twenty experimental reinforced concrete slabs with (using studs or short offcuts of steel I-beams) and

without shear reinforcement have been simulated using ATENA 3D 5.1.1. The objective is to examine

the capability of the FEA at estimating punching resistance, slab rotation and deflection as well as

correct failure mode and crack patterns. The mesh has been refined around the column to better capture

punching failure. A sensitivity study was carried out to investigate the effect of varying the concrete

material properties from the baseline values adopted elsewhere in the thesis. It is shown that varying Ec

& ft affects the initial response but their influence on the ultimate load is insignificant. On the other

hand, varying wd & rc in the analyses with ATENA had no effect of the slab response. Thus, the default

values for such parameters are adopted in the FE models presented in subsequent chapters. In general,

the FE model gives results in good agreement with experimental results in terms of punching strength

and deformation for all three slab series. In addition, the model predicts correctly the modes of failure

for PL and PT slabs series. However, it fails to capture the observed punching failure outside the

punching reinforcement zone in the tests of Gomes & Regan.

5.4 Punching Shear Calculation using MC2010 LoA IV

Symmetrical punching tests by Regan (78)

These punching tests of Regan (11) were selected for analysis because they explored the influence on

punching resistance of varying the reinforcement distribution across the slab width from uniform to

follow the elastic distribution of bending moments. This is pertinent to Chapter 7 which investigates

the influence on punching resistance of adopting banded and uniform reinforcement arrangements in

wide beam slabs. The slabs were analysed using both ATENA with solid elements and DIANA with

shell elements.

The shell elements in DIANA fail to capture punching shear failure because only the reduction in shear

stiffness due to cracking is modelled. However, the punching shear resistance of slabs may be evaluated

by virtue of the Critical Shear Crack Theory (CSCT), which forms the basis of the punching shear

design recommendations in the fib MC2010. The CSCT relates the punching resistance to the width of

the critical shear crack at failure, which is defined as a function of slab rotation. Thus, by extracting the

rotations from the FE model, the punching shear resistance can be calculated. The fib MC2010 offers

four levels of approximation (LoA) for calculating rotations, three of which are intended for the design

of new structures, while LoA IV is intended for the evaluation of existing structures. The punching

resistance is calculated below for all four LoA. The results are compared with strengths evaluated with

EC2 and ACI 318 as well as ATENA.

The following equations give expression for rotations for each level of approximation (10):


150

I = 1.5 . . (5.2)

II = 1.5 . . ..

(5.3)

III = 1.2 . . ..

(5.4)

where denotes the position where radial moments is zero, d is the shear-resisting depth, and are

the yield stress and elastic modulus for reinforcement respectively. is the average bending moment

per unit length and is the average moment of resistance per unit length in the support strip.

For level II, may be estimated for inner columns as follows:

=18

+ ,

2 (5.5)

where eu,i refers to the eccentricity of the shear force resultant with respect to the centroid of the basic

control perimeter and bs is the width of the support strip. For level III, mEd and rs are calculated using

an uncracked model analysis. For level IV, the rotations are obtained from the NLFEA.


A group of six RC slabs was part of a large experimental scheme carried out to study the effects of the

flexural reinforcement arrangement and ratio among other parameters. Typical dimensions of slab

specimens in this group were 2.0 m square and 100 mm thick. The specimens were simply supported

along four sides with spans of 1.83 m. The load was applied at the centre of slab through a 200 mm

monolithic column which projected above and below the slab. The slabs had only tension reinforcement

and were divided into three pairs. Each pair had the same amount of reinforcement. For a specimen in

a pair, the arrangement of reinforcement was based on the elastic moment distribution, while in the

other was uniformly spaced. The material properties are listed in Table 5-15 and the reinforcement

details are shown in Figure 5-40.

Table 5-16: Material properties for symmetrical punching test slabs (I1-I6) Specimen fcu: (MPa) d : mm ρ: % fy: (MPa) Bar dia.: mm Rebar dist. details

I1 32.2 77 1.2 500 10 Elastic I2 29.3 77 1.2 500 10 Uniform I3 34.3 77 0.92 500 10 Elastic I4 40.4 77 0.92 500 10 Uniform I5 35.2 79 0.75 480 8 Elastic I6 27.4 79 0.75 480 8 Uniform


151

5.4.1.2 Prediction of Punching Shear Resistance using ATENA

Finite Element Modelling

The slabs were modelled in ATENA using cubic brick-linear elements with mesh size of 12.5 mm close

to the column and a coarser mesh with 25 mm at the remainder of the slab as shown in Figure 5-41. The

fine mesh was considered to increase the number of elements to 8 elements through the thickness of

slab at the critical zone to capture the punching behaviour adequately. This was based on the results of

mesh sensitivity study performed earlier (see Section 5.3.2.2). Reinforcement was modelled as discrete

bars. In order to reduce the time of the analysis, only quarter of the specimen was modelled taking the

advantage of symmetry. The parameters of the constitutive model for concrete were obtained from the

concrete strength using Table 5-9.

Figure 5-26: Reinforcement details for symmetrical punching test slabs (I1-I6) (143)


152

NLFEA Results

Load-deflection responses obtained from the tests and NLFEA using ATENA are plotted and presented

in Figure 5-42. In all specimens, the structural response was initially well predicted up to the crack

loading. Subsequently, the NLFEA showed more flexible behaviour than occurred in the tests. The

NLFEA correctly predicted all slabs to fail in punching. Moreover, the failure loads obtained from the

analyses compared fairly well with those from the tests, except for I1, as illustrated in Table 5-17.

Table 5-17: Failure load results obtained from the symmetrical punching tests and ATENA Specimen Vu-Test: kN Vu- ATENA: kN Vu-ATENA/ Vu-Test

I1 194 249.8 1.29 I2 176 198.8 1.13 I3 194 218.8 1.13 I4 194 193.8 1.00 I5 165 153.1 0.93 I6 165 158.4 0.96

Average 1.07 COV 0.13

0

50

100

150

200

250

300

0 5 10 15 20 25 30 35

Load

: kN


I1-ATENAI2-ATENAI1-TestI2-Test 0

50

100

150

200

250

0 5 10 15 20 25 30 35

Load

: kN


I3-ATENAI4-ATENAI3-TestI4-Test

0

50

100

150

200

0 5 10 15 20 25 30 35

Load

: kN


I5-ATENAI6-ATENAI5-TestI6-Test

Figure 5-28: Comparisons of load-deflection curves from the symmetrical punching tests and NLFEA

Figure 5-27: Mesh discretization of FE model for slabs I1-I6


153

5.4.1.3 Prediction of Punching Shear Resistance according to MC2010 LoA IV with rotations

obtained from DIANA shell elements.

Finite Element Modelling

The Regan slabs were also modelled with DIANA. Curved shell elements CQ40S-quadrilateral, 8 nodes

type, with integration scheme of 2×2×9 were used to model the slab. Brick elements CHX60-brick, 20

nodes, with default integration scheme of 3×3×3 were adopted for the column. DIANA automatically

generates tying to connect the solid and shell elements together. The reinforcement was modelled using

embedded discrete bars. The mesh size is 25 mm throughout the slab. Only quarter of the model was

considered. The slab was restrained vertically at the edges along line of 1.83 m, while the corners were

free to rotate. The symmetry conditions were imposed by restraining the corresponding translation and

rotation. Tay’s (6) linear softening and Thorenfeldt (122) nonlinear functions were assigned for the

constitutive models for concrete in tension and compression respectively. Figure 5-43 shows the mesh

discretization adopted in DIANA.

Results given by DIANA Analyses

The load-deflection curves obtained from DIANA model are plotted in Figure 5-44 compared to those

from the test. The deflection was considered at the centre of slab specimen. Figure 5-45 shows the

NLFEA load-rotation curves for slabs I1 to I6. Additionally, the failure criterion for punching specified

by the CSCT is also plotted for each slab. The CSCT failure load is given by the intersection of the

rotation and resistance curves. The estimated rotations for all slabs according to MC2010 LoAs I, II and

III are listed in Table 5-18. These rotations correspond to the larger rotations in the x- and y- axes and

calculated at the predicted failure load. It is found that in all slabs the maximum rotation developed

along the x-axis since it is the weaker reinforced direction. Table 5-19 presents the punching resistances

Figure 5-29: Mesh discretization for a quarter of Regan slab used for symmetric punching tests

Lines of symmetry

Ryy=0, x=0 Rxx=0, y=0

Z=0 Z=0


154

corresponding to the rotations in Table 5-18. The MC2010 analyses gave conservative estimates for

punching shear strength. As expected, the degree of accuracy depends on the LoA considered. For

instance, the average punching shear strength ratio (Vu/Vtest) obtained with LoA I & II are 0.48 and 0.59

respectively. Better predictions have been achieved with LoAs III & IV (Vu/Vtest = 0.71, 0.83

respectively).

0

50

100

150

200

250

300

0 0.005 0.01 0.015 0.02 0.025

Load

: kN

Rotation: Rad

Slab-I3

Failure Criterion

Load-Rotation Curve0

50

100

150

200

250

0 0.005 0.01 0.015 0.02 0.025

Load

: kN

Rotation: Rad

Slab-I2

Failure Criterion


50

100

150

200

250

0 0.005 0.01 0.015 0.02 0.025

Load

: kN

Rotation: Rad

Slab-I1

Failure Criterion

Load-Rotation Curve

0

50

100

150

200

250

300

0 0.005 0.01 0.015 0.02 0.025

Load

: kN

Rotation: Rad

Slab-I4

Failure Criterion


50

100

150

200

250

300

0 0.005 0.01 0.015 0.02 0.025

Load

: kN

Rotation: Rad

Slab-I5

Failure Criterion


50

100

150

200

250

0 0.005 0.01 0.015 0.02 0.025

Load

: kN

Rotation: Rad

Slab-I6

Failure Criterion

Load-RotationCurve

Figure 5-32: Load-rotation curves and corresponding failure criteria for slabs I1-I6

0

50

100

150

200

250

300

0 5 10 15 20 25 30 35

Load

: kN


I1-DIANAI2-DIANAI1-TestI2-Test 0

50

100

150

200

250

0 5 10 15 20 25 30 35

Load

: kN


I3-DIANAI4-DIANAI3-TestI4-Test 0

50

100

150

200

250

0 5 10 15 20 25 30 35

Load

: kN


I5-DIANAI6-DIANAI5-TestI6-Test

Figure 5-31: Load-deflection curves of Regan slabs predicted by DIANA model and from the test.


155

Table 5-18: Estimation of rotations according to CSCT levels I, II and III using DIANA 9.6

Note: Es is assumed 200 GPa, Ec is calculated according to the fib MC90 (144).

Table 5-19: Comparisons between punching strengths given by CSCT LoAs I, II, III & IV and from the test Slab Vu-test :kN Vu-CSCT : kN Vu-CSCT /Vu-test

Level I Level II Level III Level IV Level I Level II Level III Level IV I1 194 83.9 121.7 133.8 160 0.43 0.63 0.69 0.82 I2 176 80.1 114.9 121.6 150 0.45 0.65 0.69 0.85 I3 194 86.6 108.6 143.1 156 0.45 0.56 0.74 0.80 I4 194 94.0 103.2 140.0 158 0.48 0.53 0.72 0.81 I5 165 93.1 110.5 135.9 148 0.56 0.67 0.82 0.90 I6 165 82.1 86. 9 117.3 131 0.50 0.53 0.74 0.79 Mean

0.48 0.59 0.71 0.83

COV 0.10 0.11 0.07 0.04

The test results show that the measured deflections are significantly less for the elastic rebar distribution

than for the uniform distribution. Consequently, the rotations would be less as well. The analyses using

LoA I fail to capture this behaviour. For example, LoA I predicts similar rotations for elastic and

uniform rebar distributions. In all cases LoAs II & III predict higher rotations at predicted failure loads

for the slabs with uniform distributions. The FE analyses with LoA IV capture this but overestimate the

deflections for both reinforcement distributions.

The test results also show that the punching resistances are virtually the same for the uniform and elastic

rebar distributions. Hence, it is debatable whether it is beneficial to distribute the rebar for elastic

moments. In view of this, it can be concluded that the CSCT incorrectly predicts that the elastic rebar

distribution increases punching resistance which is not the case. Interestingly, the ATENA analysis also

predicts the elastic rebar distribution to increase punching resistance.

5.5 Conclusions

The chapter summarises the studies performed to validate the FE models against test data using DIANA

and ATENA. In DIANA, slab elements have been modelled with 8 nodes curved shell elements, while

supporting columns and walls have been modelled using 20 nodes brick elements. The focus is mainly

on examining the capability of DIANA model to simulate the flexural behaviour of one-way slabs in

terms of load-deflection response, steel strains and flexural capacity. Besides that, predicting the

Slab dx: mm

dy: mm

fc: MPa

Level I Level II Level III ψ: Rad ψ: Rad ψ: Rad

I1 77 87 27.4 4.87 x 10-2 2.61 x 10-2 2.15 x 10-2 I2 77 87 24.9 4.87 x 10-2 2.66 x 10-2 2.38 x 10-2 I3 77 87 29.2 4.87 x 10-2 3.39 x 10-2 1.99 x 10-2 I4 77 87 34.3 4.87 x 10-2 4.22 x 10-2 2.47 x 10-2 I5 79 87 29.9 4.56 x 10-2 3.46 x 10-2 2.37 x 10-2 I6 79 87 23.3 4.56 x 10-2 4.18 x 10-2 2.48 x 10-2


156

enhancement in stiffness and strength of slabs due to the CMA and the effect of the rebar distribution

is of prime concern.

The results of validation studies show that predicted responses by the DIANA model agreed well with

those of the tests. Additionally, the effect of CMA was simulated fairly well. Moreover, the rotations

obtained from the DIANA model were applied in the CSCT to calculate the punching shear resistances

of slabs. The test results and those given by the CSCT show good agreement in terms of load-rotation

curves. Thus, Tay’s tension stiffening model in conjunction with a variable shear retention factor

(aggregate interlock-based) will be implemented in all nonlinear analyses with DIANA.

The ATENA analysis was implemented primarily to simulate the shear failure of beams and slabs

without and with shear reinforcement. The focus was on the load-deflection response, failure load,

failure mode and crack patterns. Therefore, the validation studies were limited to beams failing in shear,

and slabs, without and with shear reinforcement, failing in punching. In the ATENA analyses, only

brick elements were adopted to model the structural members. Because the variation of shear stress

through the thickness of slab or slab at critical sections near supports and concentrated forces is

nonlinear, several elements are needed through the thickness to model the shear behaviour properly. It

was found from the parametric studies five elements through the thickness are adequate to capture the

shear behaviour.

It can be concluded that the ATENA model gives close estimations for shear strength to test results of

beams with and without transverse reinforcement. Similarly, the ATENA model predicted adequately

the punching shear resistance in slabs without and with shear reinforcement. Furthermore, the failure

modes and crack patterns were captured accurately in all beams and slabs except for Gomes & Regan

slabs in which the FE model predicted incorrectly the failure within the shear-reinforced area, while it

was outside the reinforced zone. This is probably due to the increased efficiency of the shear

reinforcement in the tests resulting from the use of short offcuts of steel I-beams as transverse shear

reinforcement.

Accordingly, the DIANA models have been implemented in the elastic and nonlinear finite element

analyses to study the flexural behaviour of wide beam slabs. The details of these investigations are

presented in Chapter 6. The NLFE analyses performed by ATENA models are concerned with punching

shear resistance of wide beams around the internal and edge columns as detailed in Chapter 7.

Flexural Design for wide beam slabs Chapter 6

157

Flexural Design for wide beam slabs

6.1 Introduction

Design codes of practice such as EC2 (3) and ACI-318 (4) do not give guidance on designing wide

beam slabs. Such systems are commonly assumed to behave structurally in the same way as

conventional one-way spanning slabs on knife edge supports (5). This assumption implies that the

hogging bending moment distribution in the slab is uniform along the length of the supporting beams.

This is not the case for wide beam slabs which Tay (6) showed numerically to behave similarly to flat

slabs. More recently, Shuraim et al., (7) showed experimentally that support moments in the slab of

wide-shallow beam floors peak sharply near supports and are distributed fairly uniformly within the

central region of the supporting beams. In the following discussions, the terms “transverse” and

“longitudinal” refer to the bending moments resisted by reinforcement perpendicular and parallel to the

wide beam.

The main aim of this chapter is to develop a rational design method for wide beam slabs which satisfies

equilibrium and the SLS requirements of cracking and deflection. This is achieved by:

Determining the distribution of transverse bending moment in wide beam slabs using elastic

finite element analysis.


158

Proposing a method for distributing transverse reinforcement in wide beam slabs based on

elastic finite element analysis.

Investigating the structural performance of wide beam slabs with uniformly distributed

transverse reinforcement as adopted by Worked Examples to EC2 Volume 1 (5), as well as

banded reinforcement. Assessed performance criteria include strength, transverse moment

distribution, deflections, crack widths and reinforcement strains. In addition, the effects of

compressive membrane action on load-carrying capacity, steel strains and crack width are

studied.

To accomplish these objectives a systematic numerical programme was carried out. The study was

performed on a typical floor of the multi-storey office building shown in Figure 6-1. The wide beam

floor (see Figure 6-2) was dimensioned in accordance with the general recommendations in ECFE (1)

which are described in Chapter 3. Two designs for transverse reinforcement were studied: the uniform

distribution and a banded distribution based on the elastic analysis. The design moments for slabs were

found from elastic finite element analysis, while sub-frame analysis was used to obtain design moments

for wide beams in their direction of span. The same longitudinal steel reinforcement was provided in

the wide beams and columns for both the uniform and banded steel NLFEA models.

6.2 Case Study

General

The investigated floor consisted of three equal slab spans of 8.0 m between beam centrelines supported

on 400 mm thick wide beams. The slab was 200 mm thick. Wide beams were continuous over three

equal spans of 10 m. The internal beam was 2400 mm wide while the edge beam width was 1400 mm.

The beams were supported on 400 mm square columns. The storey height was 4200 mm. Figure 6-2

shows the plan and elevation for the physical model. The floor was designed to carry gravity loading

consisting of the self-weight of the structural elements, a superimposed dead load of 1.5 kN/m2 for

finishes and an imposed load of 5.0 kN/m2. The concrete covers for slab and beam were 31 mm and 45

mm respectively. The structure was designed as braced structure. The lateral resistance was considered

to be provided to the structure by means of shear walls not shown in Figure 6-1. Although in practice it

is common to have transverse beams around the external perimeter of the floor, they are not included

in the direction of span of the slab in order to exclude their effect on slab bending moments. No cladding

load was considered since the cladding was assumed to be supported from ground level. Figure 6-1

shows a 3 dimensional view of the first two floors of the building model. The material properties (see

Table 6-1) were derived with the following formulae from BS EN 1992-1 (3) :

= 0.3 ( )

(6.1)


159

= 22[ + 8)/10] . , ( ) (6.2)

The fracture energy is calculated using the expression developed by VOS (145), which has been adopted

in the validation studies in Chapter 5:

= 0.000025 (6.3)

where denotes the characteristic strength of concrete, , is the characteristic cubic strength of

concrete, is the mean tensile of strength of concrete, Ec and Es are the elastic moduli for concrete

and steel respectively, μc and μs are the Poisson’s ratio for concrete and steel respectively and is the

fracture energy.

Table 6-1: Material properties for the model used as case study

Member

Concrete Steel

Ec:GPa μc fcd: MPa fct: MPa GF: kN/mm Mean aggregate size: mm

Es: GPa μs fy: MPa

Beam & slab 34.1 0.2 35 2.862 0.0642 16

200 0.3 500 Column 36.2 0.2 40 3.128 0.0702 16

Figure 6-1: Case study: three-storey wide beam floor building

Floor used in the analyses


160

(a) Plan

(b) Elevation Figure 6-3: Physical Model: (a) Plan (b) Elevation


161

Load arrangements of actions

The load combination used in the FEA for reinforcement design at the ULS was all slab spans fully

with a load of 1.35 Gk plus 1.5 Qk where Gk and Qk are the characteristic dead and live loads. No pattern

loading was considered since the aim was to examine the structural performance of a structure with area

of reinforcement exactly equal to that required for the load case used for assessment. For SLS

assessment, characteristic loads were used with partial safety factors equal to 1.0.

Figure 6-3 shows a sub-assembly consisting of three spans of the wide beam and its supporting columns

which are assumed to be rigidly fixed at their ends. The column height is measured to the centroidal

axis of the beams immediately above and below.

Design of steel reinforcement

6.2.3.1 Design of wide beam rebar in longitudinal direction

Flexural Reinforcement:

Bending moments and shear forces were obtained from analysing the sub-frame shown in Figure 6-3.

The support moments were redistributed downwards by 15% and the span moments increased

accordingly to maintain equilibrium. Figure 6-4 & Figure 6-5 show the flexural rebar detailing. The

cover of slab and wide beams were 31 mm and 45 mm, respectively, which yielded effective depths of

169 mm for slab and effective depth of 345 mm for wide beams. The flexural reinforcement was

designed using partial material factors of 1.0 for steel and concrete. The main reinforcement was

checked against the minimum flexural steel as specified by EC2 with the following expression: -

, = , ≥ 0.0013 (6.5)

Figure 6-4: Elevation showing the sub-frame of wide beam with the load combination for ULS.


162

where Act is the area of concrete that is in tension just before the formation of the first crack, fyk is the

yield strength for steel reinforcement, fct,eff is the tensile strength of concrete at time of cracking, kc the

stress distribution factor (1.0 for pure tension and 0.4 for flexure), k is the non-linear stress distribution

factor ( for web depth or flange width < 300 mm, then k =1.0, for web depth or flange width > 800 mm,

k = 0.65 (Interpolation is permitted for intermediate values). For the present design, the minimum steel

criteria governs the area of top rebar in the edge beam perpendicular to its direction of span.

Beam shear Reinforcement:

The beam shear reinforcement was designed in accordance with EC2. The minimum shear

reinforcement requirements were found to control the design of stirrups in the longitudinal and

transverse directions of the beam. This is mainly due to the considerable width of the beam. Thus,

stirrups were provided longitudinally and transversely at a spacing of 0.7d (250 mm), which is slightly

less than the maximum value permitted by EC2 of 0.75d, where d is the effective depth. Figure 6-6

shows the provided shear reinforcement in elevation and cross section.

Punching Shear Reinforcement:

Additional checks were carried out for punching shear failure in the wide beam around the columns.

The checks were made according to EC2. It was found that punching shear reinforcement was required

around the internal and edge columns. Thus, punching shear reinforcement was provided around the

internal and edge columns at distances 0.4 d (150 mm), 1.1 d (400 mm) and 1.8 d (650 mm) from the

column face where d is the effective depth. The partial material factors used for beam shear and

punching shear designs were 1.15 for steel and 1.5 for concrete. Further investigations on punching

shear in wide beams are presented in Chapter 7.

All the columns had the same longitudinal reinforcements with nominal stirrups. Figure 6-7 illustrates

the geometry and reinforcement for a typical column.

Table 6-2 summarizes the beam longitudinal rebar and column reinforcement details used in the

assessment of the uniform and band steel designs.


163

Table 6-2: Longitudinal and shear reinforcement details for wide beams and columns

End Panel Internal Panel Effective depth

Support

(mm2)

Span

(mm2)

Support

(mm2)

Span

(mm2)

longitudinal

direction: mm

transverse

direction: mm

T-Beam 1639 6695 6760 3592 355 369 L-beam 1334 3276 3492 1962 355 369

Flexural reinforcements links

Column

All columns : 8bars T22 H10 @ 200 mm c/c

Shear reinforcement

T-Beam H10 @ 250 mm c/c both ways L-Beam H10 @ 250 mm c/c both ways

6.2.3.2 Design of transverse steel reinforcement

TCC Method: Transverse uniformly spaced steel distribution

Economic Concrete Frame Elements (ECFE) provides design charts for wide beam slab design.

Tabulated data are given for end and internal spans. The slab span is assumed to be L - 1.2 m +h/2 for

end spans and L -2.4 m + h for internal spans where L is the slab span between the centreline of supports,

h is the slab depth and 2.4 m is the assumed wide beam width. Worked Examples to Eurocode 2: Volume

1 (5) includes an example for the design of a wide beam ribbed slab. The methodology of this example

is defined in this thesis as the TCC method. In Worked Examples, the slab is designed as a one way

spanning member supported on knife edge supports at the column centrelines. Flexural reinforcement

is designed for the slab at its intersection with wide beam and at the centreline of the wide beam

assuming that the hogging bending moment distribution is uniform along the length of the wide beam.

The numerical values for the transverse reinforcement provided according to the TCC method is shown

later in Section 6.3.3.1


164

Figure 6-5: Plan view showing bottom flexural reinforcement for slab, edge and internal wide beams used in uniform design.


165

Figure 6-6: Plan view showing top flexural reinforcement for slab, edge and internal wide beams used in uniform design.


166

Figure 6-7: Shear reinforcement details in internal and edge wide beams used for both uniform and band designs


167

6.3 Analysis of Transverse Moment Distribution

The objective of this section is to investigate the transverse bending moment distribution in the wide

beam slab using FE software, TNO DIANA 9.6 (11). Initially, the elastic bending moment distribution

was investigated. Subsequently, the influence of transverse reinforcement distribution on the bending

moment distribution was investigated using NLFEA. The bending moment distribution was examined

at different sections along the wide beam length. These included sections along the column faces, slab

midspan and at slab-beam interfaces as shown in Figure 6-8. Details of the sections taken into account

are listed below:

1-1: Section passing through the column centrelines of the edge beam.

2-2: Section passing through the column faces of the edge beam.

3-3: Section passing along the edge beam-slab interface.

4-4: Section passing through the midspan of the end panel.

5-5: Section passing through the internal beam-slab interface (left side).

6-6: Section at the internal beam through the left column faces.

7-7: Section passing through the internal beam-slab interface (right side).

8-8: Section at the internal wide beam through the column centrelines.

9-9: Section at the internal beam through the right column faces.

Figure 6-8: Typical column section: dimensions and reinforcement details


168

10-10: Section passing through the midspan of the internal panel.

A-A: Section normal to the wide beam axes passing through the edge columns faces

B-B: Section normal to the wide beam axes passing at midspan of end slab panel.

C-C: Section normal to the wide beam axes passing through the outer faces of internal columns.

D-D: Section normal to the wide beam axes passing at midspan of internal slab panel.


169

Figure 6-9: Plan showing the critical sections under study in the wide beam floor.


170

Elastic FE modelling

6.3.1.1 Material Modelling

In the linear elastic analysis concrete is modelled as an isotropic material. In this case, the elastic

modulus and poison’s ratio are the only required material inputs. The relevant values for structural

members used in the EFA are given in Table 6-1.

6.3.1.2 Mesh discretization

Only one quarter of the floor was modelled due to symmetry. The curved shell element type CQ40S-

quadrilateral, 8 nodes was adopted for slabs and beams. Integration scheme of 2 x 2 x 9 was used for

the FEA and NLFEA, which yielded 4 integration points on plan and 9 through the slab thickness.

DIANA allows curved shell elements to be connected eccentrically to their nodes as shown in Figure

6-9. This feature was utilized to connect the slab elements with the wide beam elements with an offset

of 100 mm between the mid surfaces of slab and wide beams. The mesh size for both slabs and wide

beams was 100 mm × 100 mm. Element type CHX60-brick, 20 nodes was chosen for columns with

element size of 100 mm × 100 mm × 100 mm. The default integration scheme of brick element 3 x 3 x

3 was used for both FEA and NLFEA. Figure 6-10 shows the mesh discretization the FE model and its

boundary conditions.

Figure 6-10: Eccentric connection (11)


171

6.3.1.3 Load Application

The floor was subjected to a uniformly distributed load consisting of its self-weight and that of the

finishes which was assumed to be 1.5 kN/m2. In addition, a characteristic design imposed load of 5

kN/m2 was applied. The concrete density was taken as 25 kN/m3. The partial safety factors used for

characteristic dead and imposed loads were 1.35 and 1.5 respectively. This yielded a total factored load

of 16.275 kN/m2 applied to the slab and 23.025 kN/m2 applied to wide beams.

6.3.1.4 Boundary Conditions

The slab was restrained against axial translation and rotation at the lines of symmetry whilst the other

two edges were free. All columns were 4.2 m high between floors and were modelled as pinned at mid-

height. The columns were restrained against vertical and horizontal translations at the centre of their

bottom end and against horizontal translation at their top end.

Results of FE Elastic analysis

Figure 6-11 and Figure 6-12 show the distribution of slab bending moments, at the internal and edge

wide beams respectively, along longitudinal sections passing through the column centreline, column

faces and wide beam face. At column face sections, hogging moments are distributed uniformly away

from columns along the wide beam span. However, they peak sharply near the columns. For example,

in the internal wide beam, the peak moments at the right and left internal column faces are 223 kN.m/m

Free edge

Line of symmetry

y=0, Rxx=0

Pinned support at top and

bottom column centres

Figure 6-11: Mesh Discretization for the FE model used to simulate wide beam floor.

Free edge

Line of symmetry

x=0, Ryy=0


172

and 356 kN.m/m respectively compared with around 65 kN.m/m at midspan. The transverse bending

moments are similarly distributed along the edge beams. On the other hand, hogging moments at the

wide beam-slab interfaces are significantly less than at the column face section and more uniformly

distributed. For instance, the hogging moments at the interface between the end bay slab and the internal

wide beam vary from 35 kN.m/m at midspan to 70 kN.m/m near supports. The distribution of transverse

sagging moments at midspan of the end and internal bays is plotted in Figure 6-13. The distribution is

fairly uniform but greatest around columns. The moment varies between 35 – 49 kN.m/m in the end

panel and 11.6 – 26.0 kN.m/m for the internal panel. Figure 6-14 shows the distribution of twisting

moments about the longitudinal axes of the edge and internal wide beams. It reveals that the twisting

moments in the edge beam are significantly higher than in the internal wide beam. For example, the

twisting moment developed close to the internal column are approximately 31 kN.m/m and 64 kN.m/m

for the internal and edge beams respectively. However, the twisting moment near the end of the edge

wide beam reaches 228 kN.m/m, compared to 37 kN.m/m for the internal beam.

Figure 6-14: Elastic transverse hogging moment distribution of slab across the edge wide beam at column face, centre and wide beam-slab interface.

0

100

200

300

400

0 2 4 6 8 10 12 14 16

Tran

sver

se M

omen

t: kN

.m/m


Column face (left)- section 6-6Column face (Right)- section 9-9Column centre - section 8-8Beam face (left)-section 5-5Beam face (Right)-section 7-7

Figure 6-13: Elastic transverse hogging moment distribution of slab across the internal wide beam at column faces, centre and wide beam-slab interfaces.

0

200

400

600

800

0 2 4 6 8 10 12 14 16

Tran

sver

se M

omen

t: kN

.m/m


Column centres- Section 1-1 Column face (Right)-Section 2-2

Beam face (Right)-Section 3-3


173

Transverse reinforcement distribution

6.3.3.1 Uniform Distribution

Figure 6-15 shows the areas of transverse reinforcement required for the FEA elastic hogging and

sagging bending moments. Reinforcement areas are shown for sections at the column and beam face of

the internal beam as well as midspan of the slab end bay. Figure 6-16 shows the areas of transverse

rebar corresponding to the elastic hogging moments for sections at column centre, column face and

beam face of the edge beam. The flexural reinforcement is designed using partial material factors of 1.0

for steel and concrete in order to minimise the differences between the design ultimate and flexural

failure loads. The steel is uniformly arranged across the wide beam length. Design moments are

calculated according to the following expressions of Wood & Armer (61) for combined normal and

twisting moments.

For bottom reinforcement:

-100

-50

0

50

100

150

200

250

0 2 4 6 8 10 12 14 16

Tran

sver

se M

omen

t: kN

.m/m


Edge beam long. axis- Section 1-1

Internal beam-long. axis-Section 8-8

Figure 6-16: Elastic twisting moment distribution about the longitudinal axes of the edge and internal wide beams.

-60

-50

-40

-30

-20

-10

00 2 4 6 8 10 12 14 16

Tran

sver

se M

omen

t: kN

.m/m


End panel midspan moment-Section 4-4

Int. Panel midspan moment-section 10-10

Figure 6-15: Elastic transverse sagging moment distribution of slab across the wide beam at end and internal panel slab midspan.


174

= + (6.6a)

= + (6.6b)

If either or is found to be negative, the negative moment is put to zero and the other moment

is given as either:

= + ℎ = 0 (6.7a)

or

= + ℎ = 0 (6.7b)

For top reinforcement:

= − (6.8a)

= − (6.8b)

If either or is found to be negative, the negative moment is put to zero and the other moment

is given as either:

= − ℎ = 0 (6.9a)

or

= − ℎ = 0 (6.9b)

0

300

600

900

1200

1500

1800

2100

2400

0 2 4 6 8 10 12 14 16Stee

l are

a pe

r uni

t wid

th: m

m2/

m


Column Face (sec. 6-6)-FEA Beam Face (sect. 5-5)-FEAMidspan (sec. 4-4)-FEA Column Face (sec. 6-6)-TCCBeam Face (sec. 5-5)-TCC Midspan (sec. 4-4)-TCC

Figure 6-17: Required reinforcement areas to resist Wood-Armer moments at internal support and end panel slab sections.


175

The transverse bending moments in the floor system are plotted in Figure 6-17 for sections normal to

the wide beam longitudinal axis passing through the edge and internal column face lines (sections A-A

& C-C respectively) and the end and internal slab panel midspan sections (section B-B & D-D

respectively). It can be seen that the span moments are fairly uniform along the length of the wide beam.

The support moments, however, are sharply peaked at the columns and decrease significantly within

the span. Away from the columns, the edge beam acts as a simple support without moment restraint.

It is concluded that the distribution of transverse hogging bending moment is far from uniform

particularly along the sections at the column face. This is in contrast to the uniform distribution assumed

by Goodchild in his TCC publication (5) which models wide beam slabs as one–way spanning slabs

supported on knife edge supports.

-100

0

100

200

300

400

500

600

0 2 4 6 8 10 12 14

Long

itudi

nal

mom

ent:

kN.m

/m

Distance along the slab span: m

Section A-A

Section B-B

Section C-C

Section D-D

Figure 6-19: Transverse moments about axis parallel to direction of wide beam span along sections A-A to D-D.

0

400

800

1200

1600

2000

2400

2800

3200

3600

0 2 4 6 8 10 12 14 16

Stee

l are

a pe

r uni

t wid

th: m

m2 /m


Column centre (sec. 1-1)-FEAColumn Face (sec. 2-2)-FEA

Beam Face (sec. 3-3)-FEAColumn face (sec. 2-2)-TCC

Figure 6-18: Required reinforcement areas to resist Wood-Armer moments at edge support.


176

The extreme difference between the elastic and uniform bending moment distributions suggests that the

structural performance of wide beam slabs designed with uniform rebar distribution could be

suboptimum. In particular, steel strains and crack widths might exceed the allowable limits at SLS. In

addition, it is necessary to determine whether additional transverse reinforcement steel is required over

the columns to avoid local flexural or punching failure. This raises the question of whether designing

the reinforcement for the elastic moments would improve structural performance, particularly at the

serviceability limit state. In response to these questions, nonlinear analyses have been carried out to

study the deflections, crack widths and steel strains in the wide beam slab at the serviceability limit

state. The failure load and mode of failure have been studied as well.

6.3.3.2 Proposed Method: Band distribution

FEA shows that adopting a uniform distribution of transverse reinforcement across the wide beam as

done in the TCC method (5) is inconsistent with the elastic bending moment distribution at sections

along the wide beam at column faces. This could lead to excessive steel strains and crack widths at the

SLS around the columns in particular. The influence of transverse reinforcement distribution is

investigated by comparing the response of slabs reinforced with uniform and banded arrangements of

transverse steel across the wide beam. The banded distribution is based on elastic FEA. In this

distribution the transverse reinforcement along the wide beam is divided into three bands, each of which

corresponds to the average moment across the relevant width. The bands are located over columns,

midspan and between the column area and midspan band as shown in Figure 6-18. The reinforcement

in each band is designed to resist the average moment within the band. The width of each band and

design moment is determined from the elastic moment field as follows:

Calculate the average elastic moment using Wood-Armer’s expression, mav0 at section passing

through column faces along the wide beam length.

Compare the moment in each finite element to the average moment, mav0. Then, calculate the

average moment, m1 for the element moments less than mav0 and sum up the widths of

corresponding elements, z1.

Calculate the average for element moments, mav1 that are greater than the average moment, mav0.

After that, find the average for the element moments which are less than mav1 but greater than

mav0 and denote it as m2. The sum of element widths related to m2 is z2. Similarly, find the

average for the element moments which are greater than mav1 and denote it as m3. Calculate the

width, z3, which corresponds to m3.

The elastic moments are calculated at the column face section since it governs the transverse steel design

along the wide beam. However, the difference of resulting average flexural reinforcement areas based

on the beam section at the column and the slab section at the beam face of internal beam is small for


177

the chosen wide beam geometry, i.e. 694 mm2/m for column face and 634 mm2/m for beam face. Similar

procedure has been used for the edge beam.

Assuming a constant flexural lever arm, the sum of the transverse steel areas in all bands equals that

provided along the beam for the uniform distribution:

, = ∑ , (6.10)

∑ = (6.11)

where Ast,av denotes the average transverse flexural reinforcement per unit length across the wide beam

given by the transverse uniform distribution, Lb is the length of the wide beam, Ast,i denotes the average

transverse reinforcement over the band considered, i, Zi is the sum of band widths with the same

transverse reinforcement area per unit length.

The provided steel reinforcement in each band is taken as the greatest of the calculated area and the

minimum flexural steel area specified in EC2, which is given in Equation (6.5). For the section passing

through the column faces the effective depth of wide beam is used in the calculation, while for the

effective thickness of slab is considered for the section across the slab/beam interface.

Figure 6-18 presents the banded reinforcement distribution along the internal and edge wide beams.

The presented areas of reinforcement equal the areas calculated from the relevant bending moments.

Figure 6-18 also shows the minimum area of transverse reinforcement required in the beam by EC2

which governs within the central zone (z1) of the internal and edge beams.

0

500

1000

1500

2000

2500

0 2 4 6 8 10 12 14 16

Rein

forc

emen

t are

a: m

m2 /m


Internal column face section (6-6) Edge Column face section (1-1)

Minmum steel area

Column positions

Z3Z3

Z2

Z2 Z2

Z1 Z1

Figure 6-20: Transverse reinforcement across the edge and internal wide beams according to the proposed band distribution.


178

6.3.3.3 Adopted reinforcement distributions

The adopted arrangement of uniformly distributed transverse reinforcement is shown in Figure 6-19

while Figure 6-20 shows adopted the banded reinforcement distribution. The total area of transverse

rebar across the internal wide beam increased by about 15% due to the proposed distribution. The same

span reinforcement was adopted in the slab for uniform and banded transverse rebar distributions as

shown in Figure 6-21. Chapter 8 presents generalised expressions for the band widths and

corresponding moments derived from parametric studies.

Figure 6-21: Plan showing the uniform distribution of top flexural reinforcement along the edge and internal wide beams


179

Figure 6-22: Plan showing band distribution of top flexural reinforcement along the edge and internal wide beams.


180

Figure 6-23: Plan showing distribution of slab bottom flexural reinforcement used with uniform and banded rebar designs


181

NLFEA Modelling

The same element types and sizes used in the elastic FEA are adopted for the NLFEA as described in

Section 6.3.1.2 with a few modifications as described below. The integration scheme for the curved

shell element type CQ40S-quadrilateral, 8 nodes, adopted for slabs and beams, is increased to 2 × 2 ×

9 which yields four integration points on the surface and nine integration points through the thickness

as recommended by other researchers (141,146). This model is similar to the one used in the calibration

studies. Brief details of the adopted material models are presented below. Further details can be found

in Chapter 4.

6.3.4.1 Material modelling

Concrete: Total strain crack fixed model was adopted for concrete. The models for tension, compression

and shear are described below. More details for these model can be found in Chapter 4.

Tension model for concrete: The linear softening model, which proposed by Tay (6), was applied to

shell elements (slabs and beams). In this model, the concrete tensile strength has a peak value of 0.5fct

(1.291 MPa) and reduces to zero at a strain equal to 0.5 of the yield strain of the reinforcement (1.25 x

10-3). The tension softening model of Hordijk (124) was used for solid elements (columns). The tensile

strength of the concrete and the fracture energy were 3.128 MPa and 7.821 × 10-2 MN/m according to

CEB-fib MC90 (144).

Compression model for concrete: The Thorenfeldt (122) model was considered in the NLFEA analyses.

It is defined by the peak compressive strength fp, which was taken as 35 MPa for the slab and beams

and 40 MPa for columns, and the corresponding strain αp. In this analysis the ultimate strain-based

model was used.

Shear Behaviour: Aggregate size based shear retention was chosen to describe the decay in shear

stiffness. The model assumes the contact, and hence shear stiffness, is lost when the crack width is

wider than half the mean aggregate size, which was taken as 10 mm for this NLFEA.

Tension-Compression Interaction: It was assumed that lateral cracking reduces the compressive

strength of concrete. The model presented by Vecchio & Collins (147) was selected.

Compression-Compression Interaction: No confinement effect was assumed.

Reinforcement: Reinforcement was modelled as embedded elements and perfect bond between the steel

and concrete was assumed. The constitutive behaviour of the reinforcement was considered as

elastoplastic. The yield criterion of Von Mises was adopted and an idealised elasto-plastic stress-strain

curve was assumed. A yield strength of 500 MPa was specified for the reinforcement. Flexural


182

reinforcements in slab and beams were modelled as embedded grid reinforcements since the steel

reinforcement in these members is consisted of a series of bars placed at fixed spacing. Embedded

discrete bars were used for main column steel, column links, beam shear reinforcement and punching

reinforcement.

All columns were modelled up to mid height. The 300 mm-distance at top and bottom ends of column

height were modelled elastically to avoid any local failure, while the rest of the height was modelled

nonlinearly. All columns were restrained against vertical and horizontal translations at the bottom

surface centres and against horizontal translation at the top as illustrated in Figure 6-10. Table 6-3

summarizes the material properties used in the analysis.

Table 6-3: Material properties for wide beam floor under study Parameter Wide beam Column Ec: GPa 34.41 36.16 fck: MPa 35 40 fct: MPa 1.431 3.128 GF: N/mm 7.155 × 10-2 7.821 × 10-2

6.3.4.2 Load Application

The floor has been designed to carry the self-weight of the structural elements, other dead loads

including finishes and imposed load. In order to obtain close results to the actual response of the slab

the SLS loading was applied in five stages to simulate the staged loading that occurs during construction

and service. Table 6-4 shows the loading stages, partial safety factors, the load values and load steps

for the slab and beams. Subsequently, additional load stage was applied to increase loads to failure. It

consisted of half the total factored load of the floor applied as UDL, divided into 20 steps.

Table 6-4: Load cases as applied in NLFE Model Load Partial safety

factor Load in Slab (kN/m2)

Load in Beam (kN/m2)

Load steps

Self -weight 1.0 5.0 10.0 1-5 Permanent load & Finishes 1.0 1.5 1.5 6-9 Imposed Load 0.3 5.0 5.0 10-13 self-weight + finishes 0.35 6.5 6.5 14-18 Additional imposed load 1.2 5.0 5.0 19-28

6.3.4.3 Solution Method

The Quasi-Newton method was chosen as the iterative solution method. Energy norm was used as

convergence criterion with convergence tolerance 0.001. The maximum number of iterations

considered was 300.


183

6.3.4.4 Mesh Sensitivity Study

Mesh Sizes (50 mm, 100 mm & 200 mm)

It is known that reducing the FE element size improves the accuracy of FE results. However, this

increases the computational cost, especially for relatively large models. Thus, the mesh sensitivity study

aims to determine the optimum size of the element in terms of the accuracy of results and computational

time. To achieve that, three mesh sizes were investigated; 50 mm, 100 mm and 200 mm. Non-linear

analyses were carried out using the NLFE model described earlier. The steel reinforcement adopted in

the analysis was similar to that detailed in Table 6-2.

Figure 6-22 shows the comparison between the NLFEA results of models with the three mesh sizes in

terms of load-deflection curves. The plotted load equals the total vertical load while deflections are at

the centre of the edge panel. The plot indicates that all models behaved similarly until the cracking of

concrete. Then, the 200 mm-model exhibited a softer response than the other models. On the other hand,

the 50 mm and 100 mm element size models had almost identical load deflection responses. All three

models failed at approximately the same load but the 200 mm- model failed in a more ductile manner.

Comparisons are also presented of the nonlinear transverse support moments along the length of the

wide beam. The sections investigated include section 6-6 and section 2-2 passing through the column

faces as shown in Figure 6-23 & Figure 6-24. Additionally, the slab moment distribution along section

4-4 at midspan is plotted in Figure 6-25.

0

500

1000

1500

2000

2500

3000

3500

4000

0 100 200 300 400

Tota

l Ver

tical

Loa

d: k

N


50mm

100mm

200mm

Figure 6-24: Comparison of load-deflection curves between FE models with three mesh sizes; 50mm, 100mm & 200mm.


184

It can be concluded from the figures above that there are no significant differences in the results obtained

using mesh sizes 50 mm and 100 mm. However, the 200 mm-model yields relatively different results

from those obtained with the finer meshes. For instance, the distribution of sagging moments from

distance zero (i.e. slab end) to the midspan (lb = 5 m) varies from those given by the other models.

0

100

200

300

400

500

600

700

0 2 4 6 8 10 12 14 16

Mom

ent:

kN.m

/m


Mesh size 50 mmMesh size 100 mmMesh size 200 mm

Figure 6-25: Comparison of nonlinear hogging moments along section 6-6 between FE models with three mesh sizes; 50mm, 100mm & 200mm.

-100

0

100

200

300

400

500

600

0 2 4 6 8 10 12 14 16

Mom

ent:

kN.m

/m


Mesh size 50 mm

Mesh size 100 mm

Mesh size 200 mm

Figure 6-26: Comparison of nonlinear hogging moment along section 2-2 between FE models with three mesh sizes; 50mm,100mm & 200mm.

-100

-80

-60

-40

-20

00 4 8 12 16

Mom

ent:

kN.m

/m


Mesh size 50 mmMesh size 100 mmMesh size 200 mm

Figure 6-27: Comparison of nonlinear sagging moment along section 4-4 between FE models with three mesh sizes; 50mm,100mm & 200mm.


185

Table 6-5 shows the number of elements generated by each FE model. In addition, it compares the time

required to complete the NLFEA. The size of output files are also presented for the three model. Based

on these findings, 100 mm seems to be the optimum mesh size as it gives close results to those of 50

mm but at less computational cost. Therefore, a mesh size of 100 mm was adopted in all elastic and

nonlinear finite element analyses.

Table 6-5: Comparison in terms of computational time and size between the FE models with mesh sizes; 50 mm, 100 mm & 200 mm.

Element Size : mm No. of elements Computational time : days Output file size: GB 200 7324 1 8.3 100 21232 4-5 29.7 50 70480 9 48.9

Solid Model versus Shell Model

Solid elements can be used to model any structural member. However, they normally yield larger

systems of equations than shell elements. Yet, it is useful to compare the results predicted by the shell

model with that of solid model.

The shell model described in Section 6.3.4.1 was used for this comparison with mesh size of 100 mm.

For the solid model, element type CHX60-brick, 20 nodes with the default integration scheme 3 x 3 x 3

was used with element size of 100 mm x 100 mm x 100 mm. The modelling of materials was similar to

that shown in Section 6.3.4.1, except for tension model of concrete where Hordijk model was adopted

instead of the linear softening model by Tay. The tensile strength of the concrete and the fracture energy

were 2.862 MPa and 7.116 x 10-2 MN/m respectively.

The boundary conditions were similar for both models regarding the translation restraints, while the

rotation restraints were applied only to the shell model as illustrated in Figure 6-10. The load application

and the solution method adopted were discussed in Sections 6.3.4.2 & 6.3.4.3 respectively.

Figure 6-26 compares the load-deflection curves between the solid and shell models. The deflection

was monitored at the centre of the end slab panel. It can be seen that the responses of both models were

similar at the linear elastic stage and early stage of cracking. However, as load increased, the solid

model had slightly stiffer response than the shell one. The difference in stiffness can be attributed to the

different models for tension stiffening/softening adopted in the two models This response continued

until failure which characterised by considerable ductility in both models. It can be concluded that the

responses of shell and solid model agreed fairly well.


186

6.3.4.5 Final Mesh Selection

Shell elements were adopted for the slab in subsequent analyses. The mesh size for both slabs and wide

beams was 100 mm x 100 mm based on the mesh sensitivity study discussed in Section 6.5.2.4. The slab

was restrained against translation and rotation at the lines of symmetry whilst the other two edges were

free. Element type CHX60-brick, 20 nodes with the default integration scheme 3 x 3 x 3 was used for

columns with element size of 100 mm x 100 mm x 100 mm.

Results and Discussions

In order to evaluate the performance of the slab with the transverse steel placed in bands, comparisons

were carried out between the behaviour of wide beam slabs with uniform and banded transverse

reinforcement. The response of each model was assessed in terms of load-deflection curves, steel strains

and crack width at the SLS and ULS. In addition, the slab moments, beam moments and column

moments developed from the two design methods were compared as well. Furthermore, shear force

distributions along the critical perimeters around the internal column were plotted and investigated for

the two design methods. The rotations given by NLFEA using TNO DIANA software were measured

and then the punching shear resistance was evaluated for the two designs using the fib MC 2010 Levels

III & IV (10) and compared with the punching resistance from the EC2 approach. These are discussed

in Chapter 7.

6.3.5.1 Load-Deflection Curves

Figure 6-27 compares load-deflection curves for the slabs reinforced with banded and uniform

transverse reinforcement. The deflection was monitored at the centre of end panel. In addition, Figure

6-28 shows the variation of deflection on plan between the two models in contour plots at load close to

the ultimate design load (3408 kN). The longitudinal reinforcement in wide beams was the same in both

0500

100015002000250030003500400045005000

0 50 100 150 200 250

Tota

l Ver

tical

Loa

d: k

N


Shell

Solid

Figure 6-28: L-D diagrams for the shell and solid models.


187

slabs. Both slabs were uniformly loaded to flexural failure which occurred just above the design ultimate

load. It is evident that the structural response of both models is identical until first cracking.

Subsequently, the uniformly reinforced slab is slightly stiffer than the one with banded reinforcement.

The two models follow almost parallel load paths until they reach their yield limits. The ultimate load

is 3663 kN for the model with uniformly distributed steel and 3578 kN for the model with bands. These

are comparable to the design ultimate load (3408 kN) with load factors of 1.07 and 1.05 for the uniform

model and the band model respectively. As the load increases beyond the yield limit, both models

undergo considerable yielding. It is important to note that only flexural failure mode can be captured

since shell analysis in DIANA software does not capture shear failure which is investigated in Chapter

7 using 3-D solid element analysis with ATENA.

0

500

1000

1500

2000

2500

3000

3500

4000

0 50 100 150 200 250 300 350

Tota

l ver

tical

load

: kN


Transverse steel distributed uniformly

Transverse steel placed in bands

Design ultimate load

Figure 6-29: Load-deflection curves for the model with transverse uniform steel distribution and model with steel placed in bands.


188

(a) (b)

6.3.5.2 Bending Moments

Transverse Moments

Figure 6-29 to Figure 6-33 show distributions of transverse bending moments (mxx) from elastic FEA

and NLFEA with uniform and banded reinforcement at SLS which corresponds to the quasi-permanent

load (1772 kN). Figure 6-34 to Figure 6-38 show the transverse moment distributions at ULS (3408

kN). Moments from the NLFEA analyses with uniform and banded reinforcement are compared with

the corresponding design moments of resistance provided by the reinforcement. Moments are shown at

longitudinal sections passing through the column face (section 6-6), beam face (section 5-5) along the

internal wide beam, the column face (section 2-2), beam face (section 3-3) along the edge wide beam

and the slab at midspan (section 4-4).

It is evident from Figure 6-29 & Figure 6-30 the distribution of quasi-permanent bending moments

resulting from elastic analysis and banded reinforcement are virtually the same near columns. For the

uniform rebar distribution the corresponding moments are significantly less in internal and edge wide

beams. For instance, the peak moment at the internal column face along the internal wide beam is 170.8

kN.m/m for the uniform model while it is 220.5 kN.m/m for the band model as shown in Figure 6-29.

This indicates that banding reinforcement succeeds in controlling the cracks over columns. Thorough

investigations are in Section 6.3.5.4. Figure 6-34 shows that the ULS peak bending moments (section

Figure 6-30: Contour plot showing deflection in plan at ultimate load design ultimate load (Vu=3408kN) from the NLFEA for model with: (a) uniformly distributed steel, (b) steel placed in bands.


189

6-6) in the internal wide beam from the NLFEA are significantly greater than the flexural resistance of

the provided reinforcement for both the uniform and banded reinforcement arrangements. However, in

the edge beam the ULS peak moments (section 2-2) seem to agree well with those provided by the

flexural reinforcement as illustrated in Figure 6-35. Reasons for this are explored in the next section

where the effect of localised compressive membrane action is shown to be significant.

Figure 6-36 shows that the ULS NLFEA bending moments along the face of internal wide beam, at the

intersection with the slab (section 5-5), for both arrangements are less than the flexural resistance for

uniformly distributed rebar (i.e. TCC design method). This also holds true for the beam face along the

edge beam (section 3-3). In fact, the moments are sagging moments except close to columns as shown

in Figure 6-37. It can be seen that from Figure 6-33 & Figure 6-38 that sagging bending moments in

the slab at the NLFEA (section 4-4) are less uniform than given by elastic FEA at SLS and ULS. This

is possibly due to the influence of cracking. In Figure 6-38, the span moment at distance ≈ 4.0 m for

uniform and band rebar arrangements is significantly greater than elastic moment. However, the average

span moments along the wide beam for elastic, uniform and band distributions are comparable (mElastic

= 42.5 kN.m/m, mTCC = 43.3 kN.m/m & mband = 45.5 kN.m/m).

0

50

100

150

200

250

0 2 4 6 8 10 12 14 16Tran

sver

se M

omen

t: kN

.m/m


Column face-SLS (TCC)

Column face-SLS (Band)

Column Face-Elastic

-50

0

50

100

150

200

250

0 2 4 6 8 10 12 14 16

Tran

sver

se M

omen

t: kN

.m/m



Column face-SLS (Band)

Column Face-Elastic

Figure 6-31: Transverse moment distribution at section (6-6) through the internal column faces along the internal wide beam resulting from the uniform and band designs at quasi-permanent load (Vs=1772kN).

Figure 6-32: Transverse moment distribution at section (2-2) through the internal column faces along the edge wide beam resulting from the uniform and band designs at quasi-permanent load (Vs=1772kN).


190

0

510

1520

25

3035

4045

0 2 4 6 8 10 12 14 16

Tran

sver

se M

omen

t: kN

.m/m


Beam face-SLS (TCC)

Beam face-SLS (Band)

Beam Face-Elastic

Figure 6-33: Transverse moment distribution at section (5-5) through the beam face along the internal wide beam resulting from uniform and band distributions at quasi-permanent load (Vs=1772kN).

-20

-15

-10

-5

0

5

10

15

20

0 2 4 6 8 10 12 14 16

Tran

sver

se M

omen

t: kN

.m/m


Beam face-SLS (TCC)

Beam face-SLS (Band)

Beam Face-Elastic

Figure 6-34: Transverse moment distribution at section (3-3) through the beam face along the edgewide beam resulting from uniform and band distributions at quasi-permanent load (Vs=1772kN).

-35-30-25-20-15-10

-50

0 2 4 6 8 10 12 14 16

Tran

sver

se M

omen

t: kN

.m/m


Midspan-SLS (TCC) Midspan-SLS (Band) Midspan-Elastic

Figure 6-35: Transverse moment distribution at section (4-4) through the slab at midspan along the internal wide beam resulting from uniform and band distributions at quasi-permanent load (Vs=1772kN).


191

Figure 6-38: Transverse moment distribution at section (5-5) through the beam face along the internal wide beam resulting from uniform and band distributions at design ultimate load (Vu=3408kN).

-100

0

100

200

300

400

500

600

700

0 2 4 6 8 10 12 14 16Tran

sver

se M

omen

t: kN

.m/m


Column Face-ULS (TCC)

Column face-ULS (band)

MR with provided steel(TCC)MR with provided steel(Band)Column Face-Elastic

Figure 6-37: Transverse moment distribution at section (2-2) through the internal column faces along the edge wide beam resulting from the uniform and band designs at design ultimate load (Vu=3408kN).

0

100

200

300

400

500

600

0 2 4 6 8 10 12 14 16

Tran

sver

se M

omen

t: kN

.m/m


Column Face-ULS (TCC)

Column face-ULS (band)

MR with provided steel(TCC)MR with provided steel(Band)Column Face-Elastic

Figure 6-36: Transverse moment distribution at section (6-6) through the internal column faces along theinternal wide beam resulting from the uniform and band designs at design ultimate load (Vu=3408kN).

0

50

100

150

200

250

300

0 2 4 6 8 10 12 14 16Tran

sver

se M

omen

t: kN

.m/m


Beam Face-ULS (TCC) Beam face-ULS (band)MR with provided steel (TCC) MR with provided steel (Band)Beam Face-Elastic


192

The effect of Compressive Membrane Action (CMA) on flexural capacity

The peak moments from both NLFEA analyses are greater than the moment of resistance provided by

the reinforcement, with partial material factors of 1.0, in the absence of axial compression. Therefore,

the influence of compressive membrane action was investigated as a possible explanation for the

enhanced moment capacity (137,148,149). The in-plane forces are plotted in Figure 6-39 along section

(6-6) passing through the column faces at the design SLS load (1772 kN) and ULS load (3408 kN). The

sign convention followed is positive for tension and negative for compression. It can be seen that

significant compressive membrane forces develop locally near columns. The compressive membrane

force reduces gradually with distance from the column becoming tensile near midspan of the wide beam.

It is also noticeable that the distribution of internal forces is very similar for both reinforcement

distributions. Figure 6-40 shows the in-plane forces that developed perpendicular to section 4-4 at

midspan for uniform and banded transverse support steel. Although compressive forces developed in

the slab to either side of the column centreline, they are significantly less than at section 6-6.

Figure 6-40: Transverse moment distribution at section (4-4) through the slab midspan along the wide beam resulting from uniform and band distributions at design ultimate load (Vu=3408kN).

Figure 6-39: Transverse moment distribution at section (3-3) through the beam face along the edgewide beam resulting from uniform and band distributions at design ultimate load (Vu=3408kN).

-100-50

050

100150200250300350400

0 2 4 6 8 10 12 14 16

Tran

sver

se M

omen

t: kN

.m/m


Beam Face-ULS (TCC) Beam face-ULS (band)MR with provided steel (TCC) MR with provided steel (Band)Beam Face-Elastic

-80

-60

-40

-20

00 2 4 6 8 10 12 14 16

Tran

sver

se M

omen

t: kN

.m/m


Midspan-ULS (TCC) Midspan-ULS (band)

MR with provided steel Midspan-Elastic


193

Figure 6-41 and Figure 6-42 compare moments of resistance calculated with section analysis including

axial compression with those obtained from the NLFEA at ULS for uniform and banded transverse

reinforcement. Axially compressed sections are considered in the analysis (i.e. near columns), while

those are subjected to tensile forces are ignored. The principles of equilibrium, compatibility of strains

and strain-stress relationships are applied as follows:

= 0.85 − (6.12)

Hence, =.

(6.13)

= 0.85 − + − (6.14)

where N is the axial compressive force, fck is the concrete strength, b is the breadth of the section which

is taken as 1.0 m for this analysis, s is the depth of equivalent rectangular stress block which is calculated

-400

-300

-200

-100

0

100

200

0 2 4 6 8 10 12 14 16

Axi

al fo

rce:

kN/

m

Distance parallel to the wide beam length: m

ULS-Unifrom

ULS-Band

Columns

Figure 6-42: The in-plane forces along section (4-4) passing through the midspan parallel to the wide beam at ULS (Vu=3408kN) for uniform and banded distributions.

-1600

-1200

-800

-400

0

400

800

0 2 4 6 8 10 12 14 16A

xial

forc

e: k

N/m



Column Face-SLS (Band)

Column face-ULS-(TCC)

Column face-ULS (Band)

+ Tension- Compression

Columns

Figure 6-41: The in-plane forces along section (6-6) passing through the faces of internal column at design service load (Vs=1772kN) and design ultimate load (Vu=3408kN) for uniform and banded distributions.


194

from Equation (6.13), As is the steel reinforcement area in the tension zone and fs is the corresponding

tensile stress. Both the axial compressive force, N, and reinforcement tensile stress, fs, have been

extracted from the NLFEA. The partial safety factor for concrete is taken 1.0.

Axial compression significantly increases the moment of resistance for both steel designs. However,

there are some differences between the moment from the NLFEA and those given by section analysis,

especially across the internal columns. This might be due to the fact that the moments were extracted

at nodes rather than Gauss points resulting in errors due to extrapolation from Gauss points to nodes.

Additionally, localised convergence problems close to failure could occur. Nevertheless, it can be

concluded that the compressive forces developed in wide beams could reasonably justify the increase

in the flexural strength predicted by the NLFEA.

0

100

200

300

400

500

9.2 9.6 10 10.4 10.8

Tran

sver

se M

omen

t: kN

.m/m


Internal column NLFEA-uniformSection Analysis-uniformMR for pure flexure

Column0

100

200

300

400

500

0 0.2 0.4 0.6 0.8Tran

sver

se M

omen

t: kN

.m/m


Edge Column NLFEA-uniform

Section Analysis-uniformMR for pure flexure

Column

0

150

300

450

600

9.2 9.6 10 10.4 10.8

Tran

sver

se M

omen

t: kN

.m/m


Internal ColumnNLFEA-Band

Section Analysis-Band

MR for pure flexure

Column0

100

200

300

400

500

600

0 0.2 0.4 0.6 0.8

Tran

sver

se M

omen

t: kN

.m/m


Edge Column NLFEA-BandSection Analysis-BandMR for pure flexure

Column

Figure 6-43: Comparison between the bending moments in the internal wide beam at the column face section given by the NLFEA and calculated from section analysis under the CMA at ULS (Vu=3408kN) for uniform distribution.

Figure 6-44: Comparison between the bending moments in the internal wide beam at column face section given by the NLFEA and calculated from section analysis under the CMA at ULS (Vu=3408kN) for the band distribution


195

Column Moments

Figure 6-43 & Figure 6-44 present the resultant in-plane forces acting at the slab level for uniform and

band steel distributions respectively. The in-plane forces developed as a result of slab expansion at its

centreline due to cracking. This expansion is restrained by columns which are modelled with zero

displacement at their mid heights. The resultant in plane forces are insignificant in terms of the overall

moment of resistance of the slab. The results shown in Figure 6-43 & Figure 6-44 are an upper bound

to the restraint forces that would develop in reality since the columns are not fully laterally restrained

at their mid heights as assumed.

Figure 6-45, which should be read in conjunction with Figure 6-43 & Figure 6-44, shows column

moments from the FEA and NLFEA transferred to the slab. Moments act about the x-axis denote as Mx

and those act about y-axis as My. The column moments are plotted against the vertical reaction in the

column. In addition, Table 6-6 compares the column moments and vertical reactions at design ultimate

load (3408 kN) for the TCC and band steel designs. The presented column moments are the algebraic

sum of the upper and lower column moments. The sign convention adopted in Figure 6-45 and Table

6-6 is as follows: moments acting anti-clockwise are considered positive and vertical reactions acting

upward are positive and vice versa. Figure 6-45 shows column moments were greater in the NLFEA

with banded transverse reinforcement. The increase in moment varies according to the column location

and the moment axis considered. For instance, for the corner column, C1 at a vertical reaction 150 kN,

the corresponding bending moment about the x-axis is 117 kN.m for uniform reinforcement and 137

kN.m for banded reinforcement. The corresponding moments about the y-axis are 96 kN.m and 155

kN.m for uniform and banded models respectively. Figure 6-45 shows that the column moment about

the x-axis (Mx) reduces as load increases due to the yielding of column rebar. This is significant in the

corner column C1 and edge column C4 which are supporting the edge wide beam.

In conclusion, the effect of transverse reinforcement distribution across wide beams on the column

moments appears to be minimal. However, when considering band distribution, the design of columns,

especially corner columns, need to be checked and additional bars may be required.


196

Figure 6-46: Plan view showing the arrangement of C1, C2, C3 and C4 in the FE model and horizontal reactions on the slab level at design ultimate load (Vu=3408kN) for model with transverse steel distributed in bands.

Figure 6-45: Plan view showing the arrangement of C1, C2, C3 and C4 in the FE model and horizontal reactions on the slab level at design ultimate load (Vu=3408kN) for model with transverse steel uniformly distributed.


197

Table 6-6: Comparison between the column moments and vertical reactions at design ultimate load (3408 kN) for TCC and band steel designs

Column TCC design Band design Vertical reaction: kN

Mx: kN.m My: kN.m Vertical reaction: kN

Mx: kN.m My: kN.m

C1 300.3 101.3 325.7 291.7 122.0 386.8 C2 694.1 356.8 -153.8 667.0 353.5 -166.5 C3 1710.7 -111.8 -210.6 1748.8 -111.7 -246.2 C4 701.9 -28.1 387.0 697.3 -12.4 400.9

050

100150200250300350

0 50 100 150 200 250 300 350 400

Col

umn

Mom

ent:

kN.m

Vertical Reaction: kN

Mx@ C1 - Elastic Mx@ C1- TCC

Mx@ C1- Band

0

100

200

300

400

500

0 50 100 150 200 250 300 350 400C

olum

n M

omen

t: kN

.mVertical Reaction: kN

My@ C1 - ElasticMy@ C1- TCCMy@ C1- Band

0

100

200

300

400

500

600

0 100 200 300 400 500 600 700 800

Col

umn

Mom

ent:

kN.m


Mx@ C2 - Elastic Mx@ C2- TCCMx@ C2- Band

-200

-150

-100

-50

00 100 200 300 400 500 600 700 800

Col

umn

Mom

ent:

kN.m


My@ C2 - Elastic My@ C2- TCCMy@ C2- Band

-200

-150

-100

-50

0

50

100

0 400 800 1200 1600 2000

Col

umn

Mom

ent:

kN.m


Mx@ C3 - ElasticMx@ C3- TCCMx@ C3- Band

-400

-300

-200

-100

00 250 500 750 1000 1250 1500 1750 2000

Col

umn

Mom

ent:

kN.m



-75

-50

-25

0

25

0 100 200 300 400 500 600 700 800

Col

umn

Mom

ent:

kN.m


Mx@ C4 - ElasticMx@ C4- TCCMx@ C4- Band

0

100

200

300

400

500

600

0 150 300 450 600 750 900

Col

umn

Mom

ent:

kN.m



Figure 6-47: Internal column moments in the x- and y- directions at slab level for elastic FEA and NLFEA models at design ultimate load (Vu=3408kN) with the transverse steel distributed uniformly and placed in bands.


198

6.3.5.3 Transverse Steel Strains

Figure 6-46 to Figure 6-49 compare reinforcement strains from NLFEA with uniform and banded

reinforcement at the quasi-permanent load (1772 kN) and ultimate loads (3408 kN). It can be seen that

placing transverse steel in bands reduces the steel strains around the columns significantly at both SLS

and ULS. For instance, Figure 6-46 shows that the maximum SLS steel strains at the column faces along

the internal beam decreases from approximately 2.09 x 10-3 for uniform reinforcement to 1.56 x 10-3 for

banded reinforcement. The corresponding maximum strains at the design ultimate load are 1.3 x 10-2

and 8.5 x 10-3 as shown in Figure 6-47. Away from the columns, the strains along the section at the

column face are similar for both reinforcement arrangements at SLS and ULS. The strains in the

transverse support rebar at the edge beam face are nearly zero for both reinforcement arrangements at

SLS and ULS as illustrated in Figure 6-48 & Figure 6-49.

Figure 6-46 & Figure 6-47 show that although the strains in the transverse support steel at the internal

beam face are similar for both reinforcement arrangements around the columns, placing transverse steel

in bands has an adverse effect on strains within the beam span at both SLS and ULS. This is possibly

due to the difference in the reinforcement area between the TCC and band models in the span region

(694 mm2/m for TCC model and 616 mm2/m for band model). Similarly, the strains from uniform

distribution are significantly less than those for banded rebar distribution along the midspan section.

The flexural reinforcement area is the same for both rebar arrangements (514 mm2/m). The difference

in strain between the uniform and banded reinforcement designs is probably because the steel strains

are plotted at the same section for uniform and band rebar arrangements, while the position of maximum

span moment within the span is different for uniform and banded steel. Figure 6-50 compares the

contour diagrams of moments in end bay slab for uniform and banded rebar arrangements.

It should be noted that the steel strains obtained from the NLFEA are average strains due to the tension

stiffening provided by cracked concrete. The adopted tension stiffening model of Tay assumes that the

concrete tensile stress reduces from a peak value of 0.5fct to zero at a strain equal to half the

reinforcement yield strain (1.25 x 10-3). This implies that the effect of tension stiffening is negligible

once the reinforcement yields.


199

0.0E+00

2.0E-03

4.0E-03

6.0E-03

8.0E-03

1.0E-02

1.2E-02

1.4E-02

0 4 8 12 16

Stee

l stra

in


Column Face-TCC Beam Face- TCCMidspan - TCC Column Face- BandBeam Face - Band Midspan- BandYield Strain

Figure 6-49: NLFEA steel strains along the critical sections in the internal wide beam and end bay slab at design ULS (3408 kN) for the uniform steel model and the band model.

-5.0E-04

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-03

3.5E-03

0 2 4 6 8 10 12 14 16

Stee

l stra

in


Column Face-TCC

Beam Face- TCC

Column Face- Band

Beam Face - Band

Yield Strain

Figure 6-50: NLFEA steel strains along the column and beam faces in the edge wide beam at quasi-permanent load (1772 kN) for the uniform steel and banded steel models.

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-03

3.5E-03

0 2 4 6 8 10 12 14 16

Stee

l stra

in


Column Face-TCC

Beam Face- TCC

Midspan - TCC

Column Face- Band

Beam Face - Band

Midspan- Band

Yield Strain

Figure 6-48: NLFEA steel strains along the critical sections in the internal wide beam and end bay slab at quasi-permanent load (1772 kN) for the uniform steel and banded steel models.


200

(a) (b)

-2.5E-030.0E+002.5E-035.0E-037.5E-031.0E-021.3E-021.5E-021.8E-022.0E-022.3E-022.5E-02

0 2 4 6 8 10 12 14 16

Stee

l stra

in


Column Face-TCC

Beam Face- TCC

Column Face- Band

Beam Face - Band

Yield Strain

Figure 6-51: NLFEA steel strains along the column and beam faces in the edge wide beam at design ULS (3408 kN) for the uniform steel and banded steel models.

Figure 6-52: Contour plot showing the slab moment distribution across the end bay at ULS (3408 kN) for: (a) uniform rebar distribution, (b) banded rebar distribution


201

It is instructive to compare strains from the NLFEA at the SLS with strains calculated for a fully cracked

section assuming linear elastic concrete behaviour. The moments applied in this analysis are obtained

from the NLFEA.

The depth to the neutral axis x of a singly reinforced rectangular section is determined as follows:

= ( ) (6.15)

The strain in steel before yielding is given by:

=

(6.16)

where b denotes the breadth of the cracked section, denotes the effective modular ratio, As denotes

the tension reinforcement and d denotes the effective depth.

( − ) =,

,,

≥ 0.6 (6.17)

where kt is a factor to account for the duration of loading (0.4 for long term load and 0.6 for short term

load). In current calculation kt is taken 0.4. ρp,eff denotes the effective reinforcement ratio, As/Ac,eff where

As is the area of reinforcement within an effective tension area of concrete, Ac,eff.

Figure 6-51 & Figure 6-52 compare steel strains obtained with NLFEA and cracked section analysis

along the critical sections in the internal wide beam and midspan at quasi-permanent load (1772 kN) for

both models. Figure 6-53 & Figure 6-54 compare the steel strains along the column and beam faces

along the edge wide beam. It can be seen that the cracked section analysis gives close estimations for

NLFEA steel strains near columns for both arrangements. Away from the column zones the EC2 method

gives higher steel strains than those predicted from the NLFEA. The calculated steel strains away from

column zones are governed by term 0.6 σs/Es. This holds true for strains along beam face and end bay

midspan sections. For purpose of comparison, Figure 6-51 to Figure 6-54 are redrawn with EC2 strains

evaluated without considering the limit of 0.6 σs/Es, and presented in Figure 6-55 to Figure 6-58.


202

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-03

0 2 4 6 8 10 12 14 16

Stee

l stra

in


Column Face-TCC (NLFEA) Beam Face-TCC (NLFEA)Midspan-TCC (NLFEA) Column Face- TCC (EC2)Beam Face- TCC (EC2) Midspan- TCC (EC2)Yield limit

Figure 6-53: EC2 steel strains and NLFEA strains along the critical sections in the internal wide beam and end bay slab at quasi-permanent load (1772 kN) for the uniform rebar design.

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-03

0 2 4 6 8 10 12 14 16

Stee

l stra

in


Column Face-Band (NLFEA) Beam Face-Band (NLFEA)Midspan-Band (NLFEA) Column Face- Band (EC2)Beam Face-Band (EC2) Midspan- Band (EC2)Yield limit

Figure 6-54: EC2 steel strains and NLFEA strains along the critical sections in the internal wide beam and end bay slab at quasi-permanent load (1772 kN) for the band rebar design

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-03

3.5E-03

4.0E-03

0 2 4 6 8 10 12 14 16

Stee

l stra

in


Column Face-TCC (NLFEA) Beam Face-TCC (NLFEA)

Column Face- TCC (EC2) Beam Face-TCC (EC2)

yield limit

Figure 6-55: EC2 steel strains and NLFEA strains along the column and beam faces in the edge wide beam at quasi-permanent load (1772 kN) for the uniform rebar design.


203

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-03

0 2 4 6 8 10 12 14 16

Stee

l stra

in


Column Face-TCC (NLFEA) Beam Face-TCC (NLFEA)Midspan-TCC (NLFEA) Column Face- TCC (EC2)Beam Face- TCC (EC2) Midspan- TCC (EC2)Yield limit

Figure 6-57: EC2 steel strains without 0.6 εs limit and NLFEA strains along the critical sections in the internal wide beam and end bay slab at quasi-permanent load (1772 kN) for the uniform rebar design.

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-03

0 2 4 6 8 10 12 14 16

Stee

l stra

in


Column Face-Band (NLFEA) Beam Face-Band (NLFEA)Midspan-Band (NLFEA) Column Face- Band (EC2)Beam Face-Band (EC2) Midspan- Band (EC2)Yield limit

Figure 6-58: EC2 steel strains without 0.6 εs limit and NLFEA strains along the critical sections in the internal wide beam and end bay slab at quasi-permanent load (1772 kN) for the band rebar design.

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-03

0 2 4 6 8 10 12 14 16

Stee

l stra

in


Column Face-Band (NLFEA) Beam Face-Band (NLFEA)

Column Face- Band (EC2) Beam Face-Band (EC2)

yield limit

Figure 6-56: EC2 steel strains and NLFEA strains along the column and beam faces in the edge wide beam at quasi-permanent load (1772 kN) for the band rebar design.


204

To conclude, placing the transverse reinforcement in bands reduces the peak steel strains near the

columns. In addition, it keeps the steel strains at beam face and end bay midspan sections at SLS below

the yield strain by considerable margin. Although the NLFEA yields different estimates for strains in

span regions calculated with the EC2 method based on crack width model, the results from NLFEA and

EC2 method agree reasonably well in terms of maximum steel strain, which occur near columns.

6.3.5.4 Crack width

Calculated crack widths depend on the crack spacing and steel strain. The crack width calculations

presented in this section are essentially relative comparisons, since they have not been validated with

test data.

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-03

3.5E-03

4.0E-03

0 2 4 6 8 10 12 14 16

Stee

l stra

in




yield limit

Figure 6-59: EC2 steel strains without 0.6 εs limit and NLFEA strains along the column and beam faces in the edge wide beam at quasi-permanent load (1772 kN) for the uniform rebar design.

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-03

0 2 4 6 8 10 12 14 16

Stee

l stra

in




yield limit

Figure 6-60: EC2 steel strains without 0.6 εs limit and NLFEA strains along the column and beam faces in the edge wide beam at quasi-permanent load (1772 kN) for the band rebar design.


205

Two approaches have been used to estimate the crack width. In the first approach the steel strains are

extracted from the NLFEA, while in the second the steel strains were obtained from the full cracked

section analysis. The EC2 (3) approach for crack width calculation is summarised below:

= , ( − ) (6.18)

where wk is the crack width, sr,max is the maximum crack spacing, εsm is the mean strain in the

reinforcement allowing for the effects of concrete tension stiffening and shrinkage and εcm is the mean

strain in the concrete between cracks. ( − ) is calculated as in Equation (6.17). With respect to

the NLFEA, steel strains correspond directly to ( − ).

, = 2.5 (ℎ − ) ≤ (ℎ − )/3 ≤ ℎ/2 (6.19)

= ,

(6.20)

, =( , )

(6.21)

= 22 ( ) ./ (6.22)

where denotes the effective modular ratio, Es and Ecm denotes the elastic modulus of reinforcement

and concrete respectively. is the creep coefficient equal to the ratio of creep strain to initial elastic

strain, which is assumed 2.63 in this work.

If bar spacing ≯ 5 + ∅ , , = 3.4 + . ∅

, (6.23)

Otherwise, , = 1.3(ℎ − ) (6.24)

where c is the cover, k1 is a factor accounting for bond properties of the steel, which is considered 0.8

in this analysis, k2 is a coefficient considering the type of the strain distribution, which is flexure in this

case, (k2 = 0.5), and ∅ is the bar size. In the internal beam, the bar diameter is taken as 10 mm for the

slab and band 1 in the wide beam (band 1 corresponds to the weak reinforced band). For the intermediate

and strong reinforced bands, band 2 & 3, the bar sizes are 12 mm & 16 mm respectively. For the edge

beam the bar diameters are 10 mm, 12 mm and 20 mm for bands 1,2 & 3 respectively. For uniformly

spaced reinforcement the rebar size is 10 mm. The maximum crack spacing used in the crack width

calculation for each reinforcement arrangement in internal wide beam and end bay slab is shown in

Table 6-7. The maximum crack spacing used in the edge wide beam is presented in Table 6-8.


206

Table 6-7: Maximum crack spacing for internal wide beam and end bay slab for uniform and band steel distributions

Table 6-8: Maximum crack spacing for edge wide beam for uniform and band steel distributions Crack/steel spacing or bar diameter: mm

TCC steel design Band steel design Edge wide beam:

mm Wide beam

(band 1) Wide beam

(band 2) Wide beam

(band 3) Bar diameter 10 10 12 20 Maximum crack spacing, Sr,max 308.7 308.7 305.9 224.1 Rebar spacing 150 150 150 100 5(c+∅/2). 150 150 155 175 1.3(h-x) 404.3 404.3 394.2 321.9

The asterisk mark (*) indicates that the upper bound limit Sr,max = 1.3(h –x ) governs, since the

reinforcement spacing exceeds 5 + ∅ .

Figure 6-59 and Figure 6-60 compare crack widths calculated employing steel strains extracted from

the NLFEA (first approach) at critical sections along the internal and edge beams and end bay slab for

both reinforcement arrangements at the quasi-permanent load. Figure 6-61 & Figure 6-62 compare the

crack widths at ULS. Figure 6-63 & Figure 6-64 show crack widths obtained with EC2 at the SLS for

uniformly and banded rebar distributions respectively. Similarly, comparison of crack width along the

column and beam faces in the edge beam are presented in Figure 6-65 & Figure 6-66 for uniform and

band rebar arrangements respectively. A crack width of 0.3 mm is set as a reference crack width in

accordance with EC2 guidelines. Figure 6-59 and Figure 6-60, with crack widths obtained using strains

from the NLFEA, show that the proposed band distribution yields better crack width control. For

instance, in the internal wide beam, under quasi-permanent loading the maximum NLFEA crack widths

at the internal and edge columns are 0.62 mm and 0.47 mm respectively for uniform rebar distribution,

while for banded steel the corresponding crack widths reduce to 0.36 mm and 0.29 mm respectively.

Figure 6-63 to Figure 6-66 show that the NLFEA predictions of maximum crack width close to columns

at the column face sections are comparable to those from the EC2 calculations for uniformly and banded

rebar distributed. Away from columns crack widths EC2 estimates are significantly higher than those

given by the NLFEA. This is because the crack-inducing strain is governed by 0.6 σs/Es, which results

in relatively higher strains. Similarly, the crack widths obtained from the EC2 method and NLFEA vary

significantly along the beam face and end bay midspan sections. However, they are less than 0.3 mm.

Crack/steel spacing or bar diameter: mm

TCC steel design Band steel design Internal wide

beam: mm End bay

slab Wide beam

(band 1) Wide beam

(band 2) Wide beam

(band 3) End bay

Slab Bar diameter 10 10 10 12 16 10 Maximum crack spacing, Sr,max 295.2 361.7 319.0 281.4 232.2 361.7 Rebar spacing 125 175 125 150 125 175 5(c+∅/2). 150 150 150 155 165 150 1.3(h-x) 398.0 192.0* 407.8 388.4 350.9 192.0*


207

It is also observed that good agreement is achieved for sections with relatively high flexural steel ratio.

This is investigated in more details in the next section.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 2 4 6 8 10 12 14 16

Cra

ck w

idth

: mm


Column Face-TCC (NLFEA)

Beam Face-TCC (NLFEA)

Midspan- TCC (NLFEA)

Column Face- Band(NLFEA)Beam Face-Band (NLFEA)

Midspan- Band (NLFEA)

Reference crack width

Figure 6-61: Crack width based on NLFEA steel strains at critical sections along the internal wide beam face and end bay slab at quasi-permanent load (1772 kN) for the lateral uniform and banded rebar distributions.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 2 4 6 8 10 12 14 16

Cra

ck w

idth

: mm


Column Face-TCC (NLFEA)Beam Face-TCC (NLFEA)Midspan- TCC (NLFEA)Column Face- Band (NLFEA)Beam Face-Band (NLFEA)Midspan- Band (NLFEA)Reference crack width

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12 14 16

Cra

ck w

idth

: mm




Column Face- Band (NLFEA)

Beam Face-Band (NLFEA)


Figure 6-62: Crack width based on NLFEA steel strains along the column and beam faces in the edge wide beam at quasi-permanent load (1772 kN) for the lateral uniform and banded rebar distributions.

Figure 6-63: Crack width based on NLFEA steel strains at critical sections along the internal wide beam face and end bay slab at design ultimate load (3408 kN) for the lateral uniform and banded rebar distributions.


208

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 2 4 6 8 10 12 14 16

Cra

ck w

idth

: mm


Column Face-TCC (NLFEA)Beam Face-TCC (NLFEA)Midspan- TCC (NLFEA)Column Face- TCC (EC2)Beam Face-TCC (EC2)Midspan- TCC (EC2)Reference crack width

0.00

0.10

0.20

0.30

0.40

0 2 4 6 8 10 12 14 16

Cra

ck w

idth

: mm


Column Face-Band (NLFEA) Beam Face-Band (NLFEA)Midspan-Band (NLFEA) Column Face- Band (EC2)Beam Face-Band (EC2) Midspan- Band (EC2)Reference crack width

Figure 6-66: Comparison of crack width based on steel strains given by EC2 and NLFEA at critical sections along the internal wide beam face and end bay slab at the quasi-permanent load (1772 kN) for the banded rebar distribution.

Figure 6-65: Comparison of crack width based on steel strains given by EC2 and NLFEA at critical sections along the internal wide beam face and end bay slab at the quasi-permanent load (1772 kN) for the uniform rebar distribution.

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

0 2 4 6 8 10 12 14 16

Cra

ck w

idth

: mm




Column Face- Band (NLFEA)

Beam Face-Band (NLFEA)


Figure 6-64: Crack width based on NLFEA steel strains along the column and beam faces in the edge wide beam at design ultimate load (3408 kN) for the lateral uniform and banded rebar distributions.


209

The crack patterns for the uniformly distributed and banded rebar models are depicted in Figure 6-67

& Figure 6-68 respectively. The plots are extracted from DIANA at quasi-permanent load 1772 kN.

DIANA presents the crack pattern in terms of the normal crack strains in the integration points. The

result monitor indicates that the maximum crack strain is approximately equal to 1.03×10-2 in the

uniform model and 8.3×10-3 in the band model. Note that the green areas remain uncracked. In both

models the crack strains are high close to columns. It is interesting to note that the edge beams in both

models do not crack except in the column zones. Although placing reinforcement in bands decreases

crack widths over column zones, it increases the crack width at beam face and midspan sections in

comparison with uniform distribution. The crack widths, however, do not exceed the allowable limit.

It can be concluded that placing the transverse reinforcement in bands improves significantly the

structural performance of the wide beam slab at serviceability in terms of steel strains, crack width and

crack width control.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 2 4 6 8 10 12 14 16

Cra

ck w

idth

: mm





Figure 6-67: Comparison of crack width based on steel strains given by EC2 and NLFEA along the column and beam faces in the edge wide beam at the quasi-permanent load (1772 kN) for the uniform rebar distribution.

0.0

0.1

0.1

0.2

0.2

0.3

0.3

0.4

0.4

0.5

0 2 4 6 8 10 12 14 16

Cra

ck w

idth

: mm





Figure 6-68: Comparison of crack width based on steel strains given by EC2 and NLFEA along the column and beam faces in the edge wide beam at the quasi-permanent load (1772 kN) for the banded rebar distribution.


210

Figure 6-69: Crack strain in the model with uniformly distributed steel at design service load (1772 kN).

Figure 6-70: Crack strain in the model with lateral steel placed in bands at design service load (1772 kN).


211

6.3.5.5 Further investigation on steel strains

The previous section shows that reinforcement strains calculated with NLFEA can be significantly less

than given by the EC2 crack width calculation method. To assess this further, a one element wide simply

supported beam was analysed with the steel and concrete models used in the NLFEA. The same material

properties were adopted as for the wide beam slab analysis. The dimensions of the S.S beam was 2400

mm length, 400 mm depth and one FE element width (100 mm). The effective depth was 369 mm. Four

flexural reinforcement ratios were provided in the tension region. They represent the transverse steel

rebar across the internal wide beam for uniform and band rebar designs. Figure 6-69 compares the steel

strains obtained from the NLFEA with those calculated with EC2 method based on crack width model

for different flexural steel ratios. The black dashed line is related to steel strains calculated with

expression (6.17), the red solid line is related to the same expression without the limit 0.6 σs/Es. The

black dotted line represents the strains based on fully cracked section analysis. The moment at quasi-

permanent load is determined approximately by 0.52 Mu, where 0.52 is the ratio between the quasi-

permanent load and the design ultimate load in the full-scale model, and marked with thick blue

horizontal line in Figure 6-69. Table 6-9 presents the comparison of steel strains calculated with EC2

method and the NLFEA predictions for different rebar ratios at the quasi permanent load.

It is clear that the NLFEA strains are significantly less than given by EC2 at quasi-permanent moments.

This is because the EC2 strains are governed by 0.6 σs/Es at low moments. However, they are

comparable to the EC2 strains without considering such limit. The NLFEA predictions improve as steel

ratio increases. For instance, in Figure 6-69(b), the NLFEA and EC2 strains at quasi-permanent load

agree well with ρ=0.45%. For uniform rebar distribution, the strains vary significantly as NLFEA strain

is 1.02×10-4 while EC2 strain is 8.72×10-4. Therefore, it is concluded that disregarding the limit of 0.6

σs/Es reduces the differences between the NLFEA predictions and EC2 strains at low moments.


212

Table 6-9: Comparison of steel strains calculated with EC2 method and from NLFEA for different rebar ratios at quasi permanent load

Method Uniform rebar (ρ=0.19%) Band 1 (ρ=0.17%) Band 2 (ρ=0.25%) Band 3 (ρ=0.45%) NLFEA 1.02×10-4 7.58×10-5 2.55×10-4 9.66×10-4 EC2 method(without the limit 0.6 εs)

3.25×10-4 2.1×10-4 5.58×10-4 9.16×10-4

0.6 εs 8.72×10-4 8.84×10-4 8.65×10-4 8.52×10-4

6.4 Conclusions

Wide beam slabs are commonly designed in a similar manner to one-way spanning slabs in which the

transverse reinforcement is distributed uniformly along the length of the wide beam. Elastic FEA shows

that the transverse distribution of slab span moment is fairly uniform. However, the transverse support

moment peaks sharply near columns which, for uniformly distributed transverse reinforcement, raises

concern about steel strains and crack width control at SLS. A banded lateral distribution of transverse

reinforcement across the wide beam based on the elastic analysis is proposed. Each reinforcement band

corresponds to the average transverse elastic bending moment across its width. Comparisons were made

between a model with the conventional uniform distribution and the proposed distribution in terms of

0

2

4

6

8

10

12

14

0.0E+00 5.0E-04 1.0E-03 1.5E-03 2.0E-03 2.5E-03 3.0E-03

Mom

ent:

kN.m

steel strain

(a) Uniform: ρ=0.19%

Mean strain from NLFEA

Mean strain from crackedsection Analysis, (EC2)Mean strain from crack width calculation (εs-εcm)εs-εcm

quasi-permanent moment

0

5

10

15

20

25

30

35

0.0E+00 5.0E-04 1.0E-03 1.5E-03 2.0E-03 2.5E-03 3.0E-03

Mom

ent:

kN.m

steel strain

(b) Band 3: ρ=0.45%




0

2

4

6

8

10

12

14

16

18

0.0E+00 5.0E-04 1.0E-03 1.5E-03 2.0E-03 2.5E-03 3.0E-03

Mom

ent:

kN.m

steel strain

(c) Band 2: ρ=0.25%




0

2

4

6

8

10

12

14

0.0E+00 5.0E-04 1.0E-03 1.5E-03 2.0E-03 2.5E-03 3.0E-03

Mom

ent:

kN.m

steel strain

(d) Band1: ρ=0.17%




Figure 6-71: Comparison of steel strains obtained from the NLFEA and those based on cracked section analysis (EC2) for different reinforcement ratios.


213

bending moments in slab, wide beams and columns, load-deflection curves, steel strains, crack width

at SLS and ULS. The following conclusions may be drawn:

Varying the transverse steel distribution does not significantly influence the flexural load

capacity of the slab. The calculated effect on the bending moment distribution in the slab, wide

beams and columns is also small due to the effect of localised CMA around columns.

Banding the transverse reinforcement as proposed reduces significantly reinforcement strains.

Consequently, crack widths reduce significantly adjacent to columns at both SLS and ULS.

This is significant since the NLFEA suggests that uniformly distributing transverse hogging

reinforcement in wide beams can result in excessive steel strains and hence crack widths over

supports at the SLS.

The NLFEA analyses show that the CMA developed locally around columns. The CMA leads

to differences between the NLFEA strains and those calculated by EC2 method, particularly

along the beam face and end bay midspan sections as well as the parts far from column zones

along the column face section.

Reinforcement strains calculated with NLFEA and EC2 method agree reasonably well at high

reinforcement strains but the NLFEA strains are significantly less than EC2 strains at the

cracking moment. This is probably due to the assumed tension stiffening.

Parametric studies have been carried out to determine the suitable widths of transverse reinforcement

bands and define the key parameters. The details of the models implemented and the discussion of

results are presented in Chapter 8.

Punching Shear Resistance of Wide Beam slabs Chapter 7

214

Chapter 7

Punching Shear Resistance of Wide Beams Slabs

7.1 Introduction

Wide beams are relatively shallow and, therefore, susceptible to punching failure as shown

experimentally (8,9). This chapter investigates the punching resistance of wide beam slabs with

emphasis on the influence of slab geometry on the shear force distribution around the control perimeter

of internal and edge columns. The influence of shear force distribution is overlooked in the TCC (5)

procedure for calculating punching shear resistance at columns of wide beam slabs which is the same

as given in EC2 (3) for flat slabs (see Chapter 3). The influence on punching resistance of banding

transverse reinforcement over internal and edge columns is studied using NLFEA and fib MC2010 (10).

MC2010 LoA IV is also used to investigate the beneficial effect of flexural continuity and CMA on

punching resistance for uniform and banded transverse reinforcement arrangements.

Solid element modelling with ATENA is used to obtain best estimates of punching resistance, load-

deflection response, failure mode and crack patterns are obtained for internal and edge column sub-

assemblages. Modelling is carried out using material parameters and modelling procedures validated in

Chapter 5 with relevant test data. The ATENA results are compared with the predictions of fib MC2010

LoAs II and IV. LoA IV rotations are obtained from NLFEA with shell elements using DIANA.

Recommendations are made for the choice of the coefficient ke which accounts for the effect of

eccentricity on punching resistance in MC2010. Lastly, the effect of slab continuity is assessed by


215

comparing punching resistances calculated with MC2010 LoA IV for sub-assemblies and continuous

slabs modelled with nonlinear shell elements.

7.2 Internal column Connection

7.2.1 Effect of asymmetrical load introduction on the punching shear resistance

This section investigates the effect of wide beam slab geometry on the shear force distribution around

the basic punching shear control perimeter and the coefficient ke which accounts for eccentric shear.

Also studied is the influence of lateral reinforcement distribution on punching resistance, failure mode

and crack pattern at internal wide beam column connections. This is achieved by performing NLFEA

with ATENA using solid elements. The modelling procedure and material parameters are based on the

validation studies presented in Chapter 5. The analysis in Chapter 5 of the PT-series of slabs tested by

Sagaseta et al. (140) is particularly pertinent owing to some slabs being strongly reinforced in one

direction and weakly reinforced in the other orthogonal direction. Furthermore, some of the slabs tested

by Sagaseta et al. were loaded non uniformly analogously to wide beam slabs.

7.2.1.1 Physical Model

The sub-assembly used in the ATENA analysis consists of a wide beam of depth 400 mm and length

equal to the distance between the lines of contraflexure in the beam to either side of the internal column

(400 mm × 400 mm). The distances from the column centre to the lines of contraflexure in the adjacent

end and internal spans of the wide beam were determined from the elastic FEA of the full-scale slab to

be 2.0 m and 2.2 m respectively. In the full-scale model, as recommended by Worked Examples to EC2

(5), the top flexural reinforcement in the beam is distributed over a width of 3.6 m which includes widths

of 0.6 m in the slab to either side of the wide beam. Just the rectangular cross section of the wide beam

is modelled in the sub-assembly. Therefore, the flexural reinforcement ratio has been adjusted in the

sub-assembly to compensate for the area of flexural reinforcement placed in the flange to either side of

the beam in the full-scale shell model. The reinforcement ratio is consequently increased from 0.53%

in the full scale model to 0.79% in the sub-assemblage. The total area of top longitudinal reinforcement

over the internal column is the same in the sub-assembly and full scale shell model. In the sub-assembly

with uniform transverse reinforcement, steel bars with diameter 13.3 mm are placed at 200 mm across

the wide beam as shown in Figure 7-1. The arrangement of banded reinforcement is illustrated in Figure

7-2. The flexural reinforcement in both sub-assemblies is summarised in Table 7-1. Note that the chosen

bar sizes, which are non-standard, are adopted in order to eliminate effects arising from provision of

additional reinforcement over that calculated for strength. The beam shear and punching shear

reinforcement was designed according to EC2 provisions including the design load (1.15 VEd) and

partial safety factors (γc=1.5, γs=1.15). Minimum reinforcement calculated according to EC2 was found

to be sufficient for beam shear. This shear reinforcement consisting of vertical H10 stirrups spaced at


216

250 mm centres along and perpendicular to the beam axis is also adequate for punching shear according

to EC2. Figure 7-3 illustrates the shear reinforcement distribution in plan around the internal column.

It also shows the control perimeters as defined by EC2 and MC2010. The internal column is reinforced

by 8 H22 bars and H10 links @ 150 mm c/c. Detailing of column reinforcement is shown in Figure 6-

7. The material properties used throughout this analyses are 35 MPa for concrete strength, 500 MPa for

characteristic steel yield strength and 34.41 GPa & 200 GPa for elastic moduli for concrete and steel

respectively.

Table 7-1: Flexural reinforcement details for uniform and band assemblies. Transverse reinforcement design

d – x-axis: mm d – y-axis: mm ρx% ρy%

Uniform 355 369 0.79 0.19 Band 355 369 0.79 a 0.45 b 0.24 c 0.17

a corresponds to the band over the column with a width of 1.3 m. b corresponds to two bands with a width of 0.95 m at either side of the strong band. c corresponds to two bands adjacent to the moderately reinforced bands with widths of 0.3 m & 0.7 m

Figure 7-2: Transverse uniform steel distribution for the internal column assembly

Figure 7-3: Transverse band steel distribution for the internal column assembly

Internal beam

End beam

Internal beam span

End beam span


217

7.2.1.2 FE Modelling of sub-assembly with ATENA

Material Modelling: Concrete was modelled in ATENA using CC3DNonlinear Cementitious2 (12),

which as described in Chapter 4, consists of a combined fracture –plastic model. Cracking was modelled

with the rotated crack option. The Hordijk (124) curve was used to describe the softening behaviour of

concrete in tension. In addition, a bilinear relationship was adopted for reinforcement with no hardening

assumed. Both flexural and transverse reinforcements were modelled as embedded discrete bars. Perfect

bond was assumed between concrete and reinforcement.

Load Application: A design ultimate surface load of 23.025 kN/m2 was applied to the wide beam as in

the full scale shell analyses. In addition, equivalent line loads were applied to the edges of the wide

beam as shown in Figure 7-4. The line loads were extracted at the design ultimate load from the elastic

FEA performed with shell elements using DIANA (refer to Section 6.3.1). Load control was used in the

NLFEA.

Figure 7-5: Plan view showing the shear reinforcement around the internal column


218

Mesh sizes & Types: The wide beam was meshed using linear cubic brick elements with a maximum

element size of 80 mm, which yielded five elements through its depth. A finer mesh with cubic elements

of size 40 mm was used around the column over a rectangular zone measuring 1200 mm x 800 mm. A

cubic mesh size of 100 mm was assigned to the column apart from end regions of 300 mm depth which

were modelled with finer mesh size of 50 mm. Figure 7-5 shows the mesh of the internal wide beam

assembly. The mesh discretization adopted herein has been validated against tests of flat slabs with

symmetric and orthotropic flexural reinforcements, and also with and without shear reinforcements.

The details of relevant validation studies are found in Chapter 5.

Boundary Conditions: The column was modelled up to its mid height above and below the wide beam.

Vertical restraint was applied to the central node of the bottom surface of the column. The top and

bottom surfaces of the column were restrained against horizontal displacement. No restraints were

imposed on the wide beam.

Solution Method & Convergence Criteria: The arc-length solver was adopted for the NLFEA. For

convergence, energy and displacement error tolerances of 0.0001, 0.01 were applied, respectively.

Figure 7-6: Load distribution subjected to the internal connection.

Surface translation

restraints

Vertical support Surface translation

restraints

Figure 7-7: ATENA mesh discretization for internal connection assembly used for punching analysis

Face A

Face B Face C

Face D


219

7.2.1.3 Load deflection curves and discussion

Figure 7-6 presents load-deflection curves for the solid assembly with uniform and banded

reinforcement arrangements. The total applied load is plotted against deflections along the longitudinal

and transverse axes at points A, B, C & D in Figure 7-6. The ultimate loads for the sub-assemblies with

uniform and banded reinforcement are 1201 kN (VFE/VEd = 0.73) and 1285 kN (VFE/VEd = 0.78)

respectively, where the design ultimate load is 1646.5 kN. Flexural failure occurred since support

moments were redistributed downwards by 15% in the design of flexural reinforcement and partial

material factors of 1.0 were assumed in the design of flexural reinforcement. Both sub-assemblies

exhibited large plastic deformations prior to failure. Large plastic displacements developed in the

transverse direction along face D in the uniformly reinforced sub-assembly and in the longitudinal

direction along face C in the sub-assembly with banded reinforcement. In both models, the deflection

is least at point B in the transverse direction. Prior to yield deflections along the longitudinal axis at

points A and C are similar for both transverse reinforcement arrangements but final deflections are

greater for banded reinforcement. In the transverse direction, concentrating transverse steel over the

column leads to a significant reduction in deflection at point D. This suggests that the transverse steel

design influences cracking and deformation without greatly influencing the load resistance.

7.2.1.4 Reinforcement Strains and Crack Patterns

Figure 7-7 depicts the steel strains at failure for assemblies with banded and uniform reinforcement.

The scale monitor indicates that all transverse bars of the uniform assembly yielded significantly (εs ≈

1.9×10-1=76 εy). The longitudinal top flexural bars at column faces underwent approximately similar

yielding, with remaining top bars away from the column yielding considerably (εs ≈2.0×10-2= 8 εy) but

to a lesser extent than the transverse reinforcement. In the assembly with lateral banded reinforcement,

strains were greatest in the longitudinal reinforcement with strains reaching up to 1.53×10-1, εs ≈ 60 εy,

at the column face. Limited yielding occurred in the transverse flexural reinforcement (εs ≈ 3 εy). The

0

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400

600

800

1000

1200

1400

0 10 20 30 40 50 60 70 80 90 100

Load

: kN

Deflection: mm

Face A-Uniform

Face A-Band

Face B-Uniform

Face B-Band

Face C-Uniform

Face C-Band

Face D-Uniform

Face D-Band

Face D

Face CFace A

Face B

Assembly

Long. axisTrans. axis

Figure 7-9: Load-deflection curves for solid assembly for internal connection for uniform and band reinforcement designs (deflections are given at points A, B, C & D)


220

shear reinforcements placed along the two most loaded faces, C & D, yielded considerably. Other shear

reinforcement, however, remained at low strain levels.

(a) (b)

Figure 7-8 shows plan views of the crack patterns at failure for the uniform and banded reinforcement

arrangements. The figure presents cracks with widths of 0.3 mm and greater Figure 7-8 compares the

deflected shapes at failure for both lateral steel distributions. It is clear that the crack pattern is non-

symmetric for both transverse steel arrangements. Figure 7-8–a shows that wide cracks form along the

connection due to the significant yielding of transverse reinforcement. However, these cracks appear to

be limited effectively by placing the transverse reinforcement in bands as shown in Figure 7-8-b.

Banding the reinforcement also reduces the width of diagonal surface cracks. Moreover, banding the

transverse reinforcement distributes the surface diagonal cracks more uniformly than with unifrom

transverse reinforcement.

(a) (b) Figure 7-12: Plan view of cracking patterns at failure (w>0.3 mm) in the internal column sub-assemblages with transverse reinforcement placed (a) uniformly, (b) in bands.

Face D

Face

Face C Face A Face A

Face C

Face D

Face

Figure 7-10: Plan view showing steel strains at failure for internal column sub-assemblages with transverse reinforcement placed (a) uniformly (1201 kN), (b) in bands (1285 kN).


221

(a) (b)

7.2.1.5 Comparison of punching shear resistance between symmetrical and asymmetrical

load distributions

In order to study the influence of load arrangement on the punching resistance of the subassembly, three

cases are considered:

(i) LC1: All loads are applied to the wide beam edges transverse to the direction of span.

(ii) LC2: Equal line loads applied to opposite edges with load ratios between long and short

edges equal to that in the elastic FEA.

(iii) LC3: Actual load arrangement obtained from the elastic FEA.

For LC1 & LC2, the column was shifted so that its centroid coincided with the centroid of the assembly

on plan. Load cases i) to iii) were applied to the solid sub-assembly using ATENA software. The

material modelling, boundary conditions, mesh choice, solution method and convergance criteria were

similar to those described in Section 7.2.1.2. Uniform transverse flexural reinforcement was provided

in the model.

Figure 7-10 compares the solid element sub-assembly load-rotation (L-R) responses for the three load

cases. The rotations are beam rotations measured relative to the column. Beam rotations were calculated

as the slope over a length of 200 mm in the direction considered (see Figure 7-11). The column rotation

was calculated from its deformed shape (see section 7.2.2.3).

Figure 7-13: Deflected shape of the internal column sub-assemblages at failure with transverse reinforcement placed (a) uniformly, (b) in bands.


222

The loading arrangement affects both the deformation and strength of the wide beam. For LC1, where

load is only applied to the sides of wide beam transverse to the direction of span, the ultimate load is

922.7 kN. The ultimate load increases significantly to 1245 kN for LC2 when line loads are applied to

all four edges of the wide beam. This is comparable to LC3 (1201 kN) with loading taken from the

elastic FEA with shell elements. Moreover, under LC1, large rotations develop about the transverse

direction at centres of face A & C. Although the model with LC2 loading undergoes considerable

rotation, it is stiffer than under LC1. The failure loads for LC2 and LC3 are similar but the behaviour

is different. For instance, in the case of LC3, the L-R curves corresponding to faces A and C in the

longitudinal direction differ significantly. Similarly, for faces B and D in the transverse direction.

Consequently, punching shear resistances calculated with MC2010, on the basis of the maximum

rotation, represent a lower bound since shear resistance would be increased by redistribution of shear

force around the control perimeter as reported by Sagaseta et al. (140)

0

200

400

600

800

1000

1200

1400

0.0E+00 2.5E-02 5.0E-02 7.5E-02 1.0E-01

Load

: kN

Rotation: Rad

Face A-LC1

Face A-LC2

Face A-LC3

Face A

0

200

400

600

800

1000

1200

1400

0.0E+00 2.0E-02 4.0E-02 6.0E-02 8.0E-02

Load

: kN

Rotation: Rad

Face B-LC1Face B-LC2Face B-LC3

Face B

0

200

400

600

800

1000

1200

1400

0.0E+00 2.5E-02 5.0E-02 7.5E-02 1.0E-01

Load

: kN

Rotation: Rad

Face C-LC1

Face C-LC2

Face C-LC3

Face C

0

200

400

600

800

1000

1200

1400

0.0E+00 2.0E-02 4.0E-02 6.0E-02 8.0E-02

Load

: kN

Rotation: Rad

Face D-LC1Face D-LC2Face C-LC3

Face D

Figure 7-14: Comparison of load-rotation responses of internal connection solid assembly under different load distributions including symmetrical load and unbalanced moments


223

7.2.2 ATENA Results vs. MC2010 predictions with rotations according to LoAs

II and IV

This section compares the results from ATENA with the predictions of the fib MC2010 according to

LoA II and IV. The LoA IV rotations were obtained from a shell element sub-assembly modelled with

DIANA as well as from the ATENA solid element sub-assembly. The same physical model for the solid

sub-assembly described in Section 7.2.1.1 is used.


The material modelling in DIANA is similar to that described in Chapter 6, Section 6.3.4.1. Loading

and boundary conditions were introduced in a similar manner to those described for the solid assembly

(see Section 7.2.1.2).

Mesh sizes & types: The curved shell element type CQ40S-quadrilateral, 8 nodes, was adopted for the

wide beam with a 2 x 2 x 13 integration scheme. The mesh size for the wide beam is 50 mm x 50 mm.

Element type CHX60-brick, 20 nodes with the default integration scheme 3 x 3 x 3 was used for columns

with element size of 50 mm x 50 mm x 50 mm. A 300 mm length at the top and bottom ends of the

column was modelled elastically with a mesh size of 25 mm. Figure 7-12 depicts the mesh discretization

of the internal connection using DIANA as well as the loading, which correspond to the elastic FEA,

and boundary conditions.

Figure 7-15: Calculation of beam rotation in solid assembly.


224

Solution Method: The NLFEA was performed using the Quasi-Newton iterative solver. The

convergence criterion is the energy norm with convergence tolerance 0.001. The maximum number of

iterations was limited to 300.

7.2.2.2 Calculation of rotations and punching resistance according to MC2010 LoA II

The rotation calculation according to LoA II is given in Chapter 2. Recall Equations 2.54 and 2.57

which give the rotation at an internal column as:

= 1.5 ∙ ∙.

(2.54)

= ∙ + ,

∙ (2.57)

mRd is the average flexural strength per unit length in support strip, which is calculated from the

following expression:

= . . . 1 − 0.5.

(7.1)

where is the tension reinforcement ratio in the considered direction. fy and fc are the yield strength for

steel and compression strength for concrete, respectively.

The rotations were calculated in the directions of the longitudinal and transverse axes of the internal

wide beam. Uniform and banded lateral reinforcement distributions were considered. For the uniform

Translation restraints applied

to bottom and top surfaces

Vertical support

23.025kN/m2

Figure 7-16: DIANA mesh discretization for internal connection assembly used for punching analysis.

Face A

Face B

Face C

Face D


225

steel arrangement, ρuniform is 0.19%. The banded reinforcement distribution consists of different

flexural reinforcement ratios as shown in Table 7-1 and Figure 7-2. This can be averaged as follows:

= × × (7.2)

where Wband1, Wband2 are the width of bands 1 & 2 respectively. Thus, the average flexural reinforcement

ratio across the support strip is 0.36 %. The reinforcement ratio in the longitudinal direction is 0.79%

for both cases.

Figure 7-13 shows the punching shear resistance calculated with LoA II with the provided transverse

rebar arranagements. The calculated punching resistance is given by the intersection of the load-rotation

and resistance curves. Table 7-2 presents the rotations calculated according to LoA II at failure loads in

punching and the corresponding punching shear resistances for transverse uniform and banded

reinforcement distributions. It is clear that the maximum LoA II rotations occur in the transverse

direction of the wide beam for both reinforcement distributions. The corresponding punching shear

resistance for uniformly spaced reinforcement in the transverse direction is 1079 kN (γc= γs= 1, ke=0.9).

The increase in the punching shear resistance due to banding reinforcement is 17.3%. The predicted

punching shear strengths with LoA II rotations are less than predicted with ATENA (VII/VATENA = 0.90

& 0.99 for the uniform and banded reinforcement distributions respectively).

Table 7-2: Estimated punching shear resistance for internal connection using MC2010 LoA II for uniform and band steel designs (ke=0.9)

Steel arrangement Rotation: Rad VMC2010-II: kN γc=1.5, γs=1.15 γc= γs= 1.0 γc=1.5, γs=1.15 γc= γs= 1.0

Lateral Uniform 1.48 x 10-2 2.01 x 10-2 878 1079 Band 9.72 x 10-3 1.40 x 10-2 993 1266

Longitudinal 4.81 x 10-3 6.87 x 10-3 1265 1605

Figure 7-18: Punching shear strength of internal column connection with rotations according to LoA II

0

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800

1200

1600

2000

2400

2800

3200

0.0E+00 7.5E-03 1.5E-02 2.3E-02 3.0E-02 3.8E-02

Load

: kN

Rotation: Rad

Transverse direction-UniformTransverse direction-BandLongitudinal directionMC2010 Failure criterion (γc=γs=1.0)MC2010 Failure criterion (γc=1.5, γs=1.15)


226

7.2.2.3 Derivation of rotations for calculation of punching resistance with MC2010 LoA IV

In the LoA IV assessment, rotations are obtained directly from the NLFEA results of shell sub-

assemblies made in DIANA software which allow for the effect of tension stiffening/softening, cracks

and yielding of steel reinforcement. Thus, it provides more accurate estimation of rotations, and hence

punching resistance, than MC2010 LoA II. It is also instructive to compare the rotations obtained from

the shell element and solid sub-assemblies.

Figure 7-14 shows the maximum rotations along the transverse and longitudinal axes of the beam

plotted at the design ultimate column reaction of 1390 kN. The load-rotation curves show nodal beam

rotations relative to the column. Rotations about axes perpendicular and parallel to the beam are defined

as longitudinal and transverse respectively. Column rotations were obtained from the relevant deflected

shape as illustrated in Figure 7-15. For instance, the relative rotation of beam in the longitudinal

direction across the faces A & C is found as follows:

= + , = − (7.3)

Similarly, in the transverse direction:

= + , = − (7.4)

where h1 & h2 are the horizontal displacements in mm of column in the longitudinal direction, and s1 &

s2 are the horizontal displacements in mm of column in the transverse direction as illustrated in Figure

7-15.

Figure 7-19: Rotations of wide beam along the longitudinal and lateral axes of the internal column at DIANA’s failure load (1390 kN) for the uniform and band sub-assemblies.

-2.5E-02-2.0E-02-1.5E-02-1.0E-02

-5.0E-030.0E+005.0E-031.0E-021.5E-022.0E-022.5E-02

-2.4 -2.0 -1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 1.6 2.0 2.4Rot

atio

n: R

ad

Distance across the longitudinal/transverse axis of column: m

Transvers-Uniform Transverse-Band

Longitudinal-Uniform Longitudina-Band

Face A

Face BFace C

Face D

Maximum rotation


227

(a) Longitudinal direction (b) Transverse direction

Figure 7-14 shows that the DIANA rotations beyond the effective depth, d from the column face are

almost constant within rs as observed in punching tests of isolated specimens (70). Based on Figure

7-14 it can be concluded that the position of maximum rotation is at distance approximately 0.8 m from

the column face (i.e. ≈ 2.2 d) in the longitudinal direction for both lateral steel designs. Longitudinal

rotations are greatest for both reinforcement arrangements. This clearly contradicts the result from fib

MC2010 LoA II which predicts transverse rotations to be greatest for both reinforcement arrangements.

Furthermore, it seems that the banded transverse reinforcement arrangement reduces rotations of the

beam particularly in the transverse direction. Similar conclusions were reached by Regan (78) who

carried out internal column punching tests with uniform and banded reinforcement. He reported that the

deflection is significantly less for elastic reinforcement distribution, but the punching shear strength is

virtually the same. Detailed discussion on Regan’s symmetrical punching tests can be found in Chapter

5, Section 5.4.1.

Figure 7-16 compares the load-rotation responses obtained with the shell and solid element assemblages

for each face of control perimeter for the uniform and band steel distributions. A, B, C and D denotes

the control perimeter faces, where A & C are the faces transverse to the direction of wide beam span

and B & D are the faces parallel to the direction of span. Face C has the maximum rotation in the

longitudinal direction and face D in the transverse direction. This can also be seen in Figure 7-12.

In general, the results obtained from solid and shell sub-assemblies show similar trends for all faces.

For example, both assemblies predict similar load-rotation responses across face A for lateral uniform

and banded reinforcement designs. In addition, they show that the load-rotation curves differ across

face D. Nevertheless, the rotations predicted by the shell sub-assemblies are less than given by the solid-

assemblies for both lateral steel designs. This might be attributed to the different tension softening

models adopted in the solid and shell element sub-assemblies. For the shell assemblage Tay’s linear

tension softening model was adopted, with variable shear retention based on aggregate interlock, while

Figure 7-20: Deflected shape of the internal column in shell sub-assembly: (a) longitudinal direction (x-x), (b) transverse direction (y-y).


228

the nonlinear model developed by Hordijk was used in the solid element assemblage. Another potential

source of difference could be the tendency of shell elements to overestimate torsional stiffness which

is significant for out-of-balance loading.

Concentrating the transverse reinforcement across the width of support strip has little influence on the

load-rotation response in the longitudinal direction. A greater increase in stiffness is observed in the

solid assemblage especially in the transverse direction across faces B & D.

7.2.2.4 Calculation of Punching Shear Resistance with MC2010 LoA IV

Figure 7-17 shows the load–rotation relationships for the shell and solid sub-assemblies for both

transverse steel arrangements. The rotations shown are maximum rotations, which correspond to face

C in the longitudinal direction. The coefficient of eccentricity, ke is taken as 0.9. The MC2010 failure

criterion is plotted without and with partial safety factors ( c =1.5 and s = 1.15). In addition, the

punching shear resistances evaluated with EC2 approach with and without partial safety factors are also

shown. The punching shear resistance is the sum of the resistances provided by concrete and shear

reinforcement (12H10 per perimeter). The background and equations for punching shear calculations

according to the EC2 and MC2010 are detailed in Chapter 2, Sections 2.4.2 & 2.4.3.

0

250

500

750

1000

1250

1500

0.0E+00 3.0E-03 6.0E-03 9.0E-03 1.2E-02 1.5E-02

Ver

tical

Rea

ctio

n: k

N

Rotation: Rad

Face A

Assembly shell-Unifrom

Assembly shell-Band

Assembly solid-Unifrom

Assembly solid-Band0

250

500

750

1000

1250

1500

0.0E+00 1.5E-03 3.0E-03 4.5E-03 6.0E-03

Ver

tical

Rea

ctio

n: k

N

Rotation: Rad

Face B


Assembly shell-Band


Assembly solid-Band

0

250

500

750

1000

1250

1500

0.0E+00 6.0E-03 1.2E-02 1.8E-02 2.4E-02 3.0E-02

Ver

tical

Rea

ctio

n: k

N

Rotation: Rad

Face C

Assembly shell-Uniform

Assembly shell-Band


Assembly solid-Band

0

250

500

750

1000

1250

1500

0.0E+00 3.0E-03 6.0E-03 9.0E-03 1.2E-02 1.5E-02 1.8E-02

Ver

tical

rea

ctio

n: k

N

Rotation: Rad

Face D

Assembly shell-Uniform

Assembly shell-Band


Assembly solid-Band

Figure 7-21: Comparison of L-R responses between the shell and solid assemblages for each side of control perimeter around the internal column for the uniform and band steel distributions.


229

The design ultimate shear forces calculated in the frame design are 1646.4 kN and 1893.6 kN for

MC2010 and EC2 respectively. The difference arises from the fact that the design punching load

according to EC2 is increased by 15% to account for moment transfer from slabs into internal columns.

Instead, the fib MC2010 reduces the punching shear strength by 10% for internal connections. Table

7-3 gives the punching shear resistances predicted using NLFEA with solid elements, MC2010 with

rotations calculated according to the LoAs II, IV, as well as EC2. Results are shown for uniform and

banded transverse reinforcement arrangements.

It is concluded that the fib MC2010 LoA IV predictions for punching resistance agree well with the

ATENA results for both lateral steel distributions. This holds true for results obtained from shell and

solid assemblies. For shell assemblies the ratio VIV/VATENA is 1.08 and 1.04 for uniform and band

reinforcement distributions respectively. For solid assemblies VIV/VATENA is 0.99 and 0.98 for both

uniform and band reinforcement distributions respectively. Flexural failure is critical according to EC2

which appears to overestimate the punching shear strength with both uniform and banded reinforcement

designs by approximately 40% assuming that actual failure is due to combined shear and flexure.

Banding the transverse reinforcement increases the punching shear resistance by 7% according to the

ATENA analysis and 8.5% according to EC2. MC2010 LoA IV gives an increase of 6.5% for banded

transverse reinforcement when rotations are extracted from the ATENA analysis with solid elements

but only 3% for rotations from the shell assembly.

Figure 7-22: Load-Rotation curves of the wide beam’s internal connection for shell and solid sub-assemblies for uniform and band steel designs.

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

0.0E+00 2.0E-03 4.0E-03 6.0E-03 8.0E-03 1.0E-02 1.2E-02 1.4E-02 1.6E-02 1.8E-02

Ver

tical

col

umn

reac

tion:

kN

Rotation: Rad

VRd,cs -EC2 (Uniform design) γc=γs=1.0 VRd,cs - EC2 (Band design) γc=γs=1.0Failure criterion γc=γs=1.0 Failure criterion γc=1.5, γs=1.15VRd,cs - EC2 (Uniform design) γc=1.5, γs=1.15 VRd,cs -EC2 (Band design) γc=1.5, γc=1.15L-R response (Uniform-shell assemblage) L-R response (Band-shell assemblage)L-R response (Uniform-solid assemblage) L-R response (Band-solid assemblage)Design Ultimate Load


230

Table 7-3: Punching resistances for models with shear reinforcement using EC2, MC2010 level II & IV and ATENA analysis for lateral uniform and banded steel distribution (ke=0.9).

Lateral reinforceme

nt design

EC2*: kN MC2010-LoA IV: kN

MC2010-LoA II: kN

ATENA’s results: kN

(γc=1.5, γs=1.15)

(γc= γs=1.0)

(γc=1.5, γs=1.15) (γc= γs=1.0) (γc=1.5, γs=1.15)

(γc= γs=1.0) shell solid shell solid

Uniform 1329.0 1690.6 1088 1021 1299 1184 878 1079 1201 Band 1390.7 1783.2 1100 1047 1338 1261 993 1266 1285

*The punching strength calculated with EC2 is divided by 1.15 to present it in a similar from to that of MC2010.

7.2.2.5 Investigation on the coefficient of eccentricity

The fib MC2010 method for ke calculation

Chapter 2, Section 2.4.3.4 describes the methods specified in fib MC2010 for calculating ke. All these

methods are considered in this investigation of ke. For convenience these expressions are rewritten

below.

(i)

= = , ,

, , (2.48)

where b0 is the basic control perimeter, b1 is the shear-resisting perimeter,νperp,d,av and νperp,d,max are

the average and maximum shear force per unit length perpendicular to the basic control perimeter

respectively.

(ii) = (2.49)

= , + , (2.50)

= + × × + × (2.51)

where VEd is the resultant of shear forces, MEd,x and MEd,y are the unbalanced moment about the x- and

y- axes respectively and c is the column size. In the current case the x-axis corresponds to the

longitudinal axis of the wide beam, and the y-axis to the transverse axis.

(iii) MC2010 permits ke = 0.90 for internal columns where the frame action does not resist lateral loads

and the adjacent spans do not differ by more than 25%.

(iv) Alternatively, the coefficient of eccentricity may be calculated for each direction separately since

the flexural behaviour of the assemblage is biaxial. The first and second methods stated above have


231

been used to calculate the coefficient of eccentricity in each orthogonal direction. Equation (2.48) can

be modified as:

= = , , ,

, , ,, = = , , ,

, , ,, (7.5)

where kex, key are the coefficients of eccentricity in the x & y directions respectively. vperp,d,av,x, vperp,d,av,y

are average shear force per unit length perpendicular to the basic control length in the x and y directions

respectively as shown in Figure 7-18. vperp,d,max,x & vperp,d,max,y, are the maximum shear force per unit

length perpendicular to the basic control length in the x and y directions respectively. To calculate the

basic control perimeter in x & y directions b0x, b0y, the perimeter can be divided in a similar manner to

that proposed by Sagaseta et al. Figure 7-18 illustrates the division of the control perimeter into

segments in the x & y directions as proposed by Sagaseta et al.(140). Having obtained kex & key, the

punching shear resistance can be calculated for each rotation in the orthogonal directions in conjunction

with the appropriate ke.

In order for account for the eccentricity in each direction, Equation (2.50) has been modified as:

= , (7.6)

where i is the direction under consideration.

Figure 7-23: Division of control perimeter into segments in x & y directions as proposed by Sagaseta et al. (140)


232

Results and Discussion

Figure 7-19 compares the shear force distributions along the control perimeter obtained with elastic

FEA and NLFEA for uniform and banded steel arrangements. The perimeter has been approximated as

square instead of having rounded corners at distance 0.5 d from the column face. Table 7-4 presents the

coefficient of eccentricity and the average and maximum shear force per unit length obtained from the

elastic FEA and NLFEA for both steel designs using the shell sub-assembly.

Table 7-4: Calculation of ke using linear elastic FEA and NLFEA for uniform and band reinforcement arrangements at ultimate flexural load 1390 kN

νperp,d,av: N/mm νperp,d,max: N/mm ke Elastic 507.5 742.5 0.68 NLFEA-Uniform 507.5 1085.5 0.47 NLFEA-Band 507.5 861.0 0.59

Table 7-5 compares the values of ke obtained with methods i) to iv). The first column gives ke calculated

with method (i) with linear FEA and nonlinear FEA with banded and uniform reinforcement

distributions. The second column presents ke calculated with method (ii) using Equation (7.4) as well

as kex and key obtained using method (iv). Note that kex, key are the coefficients of eccentricity in the

longitudinal & transverse directions respectively. The default value of ke = 0.9 given by method (iii) is

in the third column.

1.09 KN/mm

0.99 kN/mm 0.74 kN/mm

0.76 kN/mm

0.67 kN/mm

0.86 kN/mm 0.66

Elastic Uniform Band

Face A

Face BFace C

Face D

Longitudinal direction

Transverse direction

Figure 7-24: Comparison of shear force distribution along the control perimeter at 0.5d from the internal column face for elastic FEA and NLFEA with uniform and band reinforcement distributions.


233

Method (i) is seen to give very conservative estimates for ke, particularly with shear forces from the

NLFEA. Method (ii) yields close results for ke when considering eccentricity with Equation (2.50) and

eccentricities along the longitudinal and transverse directions separately. These values are comparable

to the approximate value of 0.9 specified for internal columns by fib MC2010. In view of this, 0.9 seems

a reasonable value for ke for the current study, and hence will be adopted.

Table 7-5: Calculation of coefficient of eccentricity based on the fib MC2010 = , ,

, , =

11 +

MC2010 approximation

for internal column

Method (i) ke - Elastic 0.68 ke

0.94

0.90

ke -Uniform 0.46 ke -Band 0.49

Method (iv) about the x-axis

kex - Elastic 0.73 kex

0.95 kex-Uniform 0.47

kex -Band 0.51 Method (iv) about

the y-axis key - Elastic 0.73

key

0.96 key -Uniform 0.49 key -Band 0.53

EC2 calculation method for β

EC2 accounts for eccentricity differently from MC2010. Instead of reducing the basic control perimeter,

it increases the design ultimate load by a ratio . The expression of EC2 for calculating β is given in

Chapter 2:

= 1 + ∙ (2.41)

= + + 4 + 16 + 2 (7.7)

where K is a coefficient dependent on the ratio between the column dimensions c1 & c2 and its value is

a function of the proportions of the unbalanced moment transmitted by uneven shear and by bending

and torsion. For square columns K = 0.6. u1 is the length of the basic control perimeter (see Figure

7-20).

EC2 also gives expression to evaluate β for internal rectangular columns with eccentricities in both

directions:

= 1 + 1.8 + (7.8)


234

where ez, ey denotes the eccentricities MEd/VEd along z and y axes respectively. bz, by are the dimensions

of the control perimeter as shown in Figure 7-20.

In the current analysis the x- and y- axes correspond to the longitudinal and transverse axes. The column

is square (400 mm × 400 mm) and the corresponding K is 0.6, the effective depth is 362 mm. VEd is

1679.3 kN, MEd,x (51.3 kN.m) and MEd,y (70.6 kN.m).

Equation (2.41) gives βx = 1.03, and βy = 1.04 when MEd is substituted as MEd,x (51.3 kN.m) and MEd,y

(70.6 kN.m). The coefficient of eccentricity, β is 1.05 according to Equation (7.8). These results are

summarized in Table 7-6.

In conclusion, calculating β gives values below 1.15 which is the EC2 recommended value for internal

columns of braced frames provided specified geometrical requirements are satisfied (see Chapter 2,

Section 2.4.2). In addition, the calculated values indicate that the shear force distribution along the

control perimeter is not uniform and varies slightly more in the transverse direction. For the current

analysis, β is taken as 1.15 since lower values would yield higher estimates for punching shear strength

which appears to be overestimated by EC2. However, it should be noted that the maximum shear force

that can be applied to the analysed sub-assembly is limited by flexural failure.

Table 7-6: Calculation of coefficient of eccentricity based on EC2 Axis considered

= 1 + ∙ = 1 + 1.8 +

EC2 approximation for internal column

Biaxial - 1.05 1.15 x-axis 1.04 -

y-axis 1.03 -

Figure 7-25: dimensions of EC2's control perimeter (10).


235

7.2.3 Effect of continuity according to MC2010 LoA IV

This section investigates the effect of flexural continuity on punching shear resistance. This is done by

comparing punching resistances calculated with MC2010 LoA IV using rotations extracted from the

shell sub-assembly and full-scale model. Both uniform and banded reinforcement distributions are

considered.

The full-scale model has been fully described in Chapter 6, Section 6.3.4. The same material modelling,

load application, mesh choice and solution method were implemented in this analysis. The transverse

flexural reinforcement ratios are 0.19 % and 0.45% for the models with uniform and banded

distribution, respectively. The longitudinal flexural reinforcement ratio (0.53%) is the same for both

models, which yielded mean flexural reinforcement ratios 0.32 % and 0.49% for the uniform and band

models, respectively.

7.2.3.1 Calculation of Rotations in full-scale model: Level IV

The rotations of the wide beam about the internal column centrelines in the longitudinal and transverse

directions are plotted in Figure 7-21. The rotations plotted are not corrected for the column rotation. It

is clear that longitudinal rotations are greatest for both reinforcement arrangements. The position of

maximum rotation is at distance about 1.1 m (i.e. ≈ 3.0 d) and 1.3 m (i.e. ≈ 3.6 d) from the column face

in the longitudinal direction for the lateral uniform and banded reinforcement distributions respectively.

Furthermore, it seems that the lateral arrangement of reinforcement does not affect rotations of the beam

in either direction. As a result, MC2010 LoA IV gives similar punching shear resistances for both

transverse steel designs. This raises the question of whether the punching resistance really depends on

maximum rotation as considered in MC2010 or the flexural reinforcement ratio as suggested by tests

of sub-assemblies. In defence of MC2010, it can be argued that isolated punching shear tests are not

representative of punching failure in continuous flat slabs where moment redistribution occurs causing

a shift in the line of radial moment contraflexure. NLFEA by Soares and Vollum (150) of continuous

sub-assemblies using ATENA, with solid elements, supports this view and suggests that punching

resistance is indeed dependent on rotation as assumed in the CSCT and MC2010.


236

7.2.3.2 Comparison of Load - Rotation diagrams

Figure 7-22 compares the load-rotation responses of the full-scale model and shell sub-assemblage for

each face of the control perimeter for uniform and banded steel distributions. A, B, C and D denote the

control perimeter faces, where A & C are the faces perpendicular to the direction of wide beam span

and B & D are the faces parallel to the direction of span. In the subassembly analysis, C & D are the

most heavily loaded edges in the longitudinal and the transverse directions respectively.

The lateral reinforcement distribution is seen to have little effect on rotations for all models. There are

some noticeable differences between the behaviour of full-scale models and sub-assemblies with the

full-scale models exhibiting a stiffer response than the sub-assemblies in both the longitudinal and

transverse directions. Moreover, the full-scale models have larger strengths than the sub-assemblies.

This is mainly due to the differences in the boundary conditions of the models. Also, the points of

contraflexure were fixed in the subassemblies, but varied with loading in the full scale models. Further,

the analyses from the previous chapter have shown that significant in-plane forces developed locally

around columns due to CMA.

-1.5E-02

-1.0E-02

-5.0E-03

0.0E+00

5.0E-03

1.0E-02

1.5E-02

-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Rot

atio

n: R

ad

Distance along the beam transverse/longitudinal centreline: m

Transverse-Uniform Transverse-Band

Longitudinal-Uniform Longitudinal-Band

Maximum Rotation

Face C

Face D

Face B

Face A

Figure 7-26: Rotations of wide beam along the longitudinal and lateral axes of the internal column at ultimate loads of 1711kN & 1749 kN for the uniform and band full-scale models, respectively.


237

Recently, Einpaul et al. (93) conducted numerical studies to explore the influence of moment

redistribution and compressive membrane action on punching strength of slabs. They concluded that

the stiffness and strength of continuous slabs can be increased significantly by compressive membrane

action arising from the restraint provided by the surrounding structural elements and by in-plane

stiffness of the sagging moment area. Soares & Vollum (92) came to similar conclusions from numerical

investigations of continuous flat slabs.

7.2.3.3 Comparison of Punching shear resistance

Figure 7-23 shows the load–rotation relationship for the full-scale models and shell sub-assemblies for

both transverse steel arrangements. The rotations considered are the maximum rotations. In addition,

the failure criterion according to fib MC2010 is plotted without and with partial safety factors (γc=1.5

& γs=1.15). The coefficient of eccentricity, ke is taken as 0.9.

Punching resistances evaluated with EC2 for both transverse reinforcement arrangements are also

shown. As previously described, EC2 accounts for uneven shear at internal slab column connections by

increasing the design load by 15%, while the fib MC2010 allows the shear strength to be reduced by

10%. In order to establish a valid comparison of the punching shear strength calculated with EC2 and

the fib MC2010 in the same figure, the punching shear strength given by EC2 is divided by 1.15.

0200400600800

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Rotation: Rad

Face D

Full scale-UniformFull scale-BandAssembly shell-UniformAssembly shell-Band

0200400600800

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Rotation: Rad

Face C

Full scale-UniformFull scale-BandAssembly shell-UniformAssembly shell-Band

0

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Rotation: Rad

Face A

Full scale-UnifromFull scale-BandAssembly shell-UnifromAssembly shell-Band

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Ver

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N

Rotation: Rad

Face B

Full scale-Uniform

Full scale-Band


Assembly shell-Band

Figure 7-27: Comparison of L-R responses between the full scale model and shell assemblages for each side of control perimeter around the internal column for the uniform and band steel distributions.


238

It is noted that rotations are similar for uniform and banded transverse reinforcement throughout the

loading stages. This also holds true for the shell assemblages. However, the full-scale models are stiffer

than shell assemblages. According to MC2010 LoA IV, the full scale models also have greater punching

shear strengths due to flexural continuity and CMA. Figure 7-23 indicates that the punching shear

design with EC2 is unsafe and additional shear reinforcement is required.

Table 7-7 lists the punching shear resistances calculated for the full-scale and shell sub-assemblages

using MC2010 LoA IV as well as EC2 and ATENA sub-assemblages. In this case, the increase in

punching shear resistance due to continuity and CMA is relatively low for both lateral reinforcement

arrangements according to MC2010 LoA IV. However, it is slightly greater for the lateral uniform

reinforcement distribution.

Table 7-7: Comparison of Punching shear resistances for full-scale models and shell assemblages estimated using EC2, MC2010 level IV and NLFEA for lateral uniform and banded steel distribution (ke=0.9).

Lateral steel dist.

EC2*: kN MC2010-level IV (γc=1.5, γs=1.15)

MC2010 -level IV – (γc= γs =1.0) NLFEA : kN

γc=1.5 γs=1.15

γc= γs=1.0

Full scale:

kN

shell assemblage:

kN

Increase due to

CMA %

Full scale:

kN

Shell assemblage:

kN

Increase due to

CMA% Uniform 1329.0 1690.6 1191 1088 9.5 1438 1299 10.8 1201

Band 1390.7 1783.2 1167 1100 6.1 1421 1338 6.3 1285 *The punching strength calculated with EC2 is divided by 1.15 to present it in a similar from to that of MC2010

Figure 7-28: Load-Rotation curves of the wide beam’s internal column connection for full-scale shell model and shell sub-assemblies for uniform and band steel designs.

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

0.0E+00 2.0E-03 4.0E-03 6.0E-03 8.0E-03 1.0E-02 1.2E-02 1.4E-02 1.6E-02 1.8E-02

Ver

tical

col

umn

reac

tion:

kN

Rotation: Rad

VRd,cs -EC2 (Uniform design) γc=γs=1.0 VRd,cs - EC2 (Band design) γc=γs=1.0Failure criterion γc=γs=1.0 Failure criterion γc=1.5, γs=1.15VRd,cs - EC2 (Uniform design) γc=1.5, γs=1.15 VRd,cs -EC2 (Band design) γc=1.5, γc=1.15L-R response (Uniform-shell assemblage) L-R response (Band-shell assemblage)L-R response (Uniform-full scale model) L-R response (Band-full scale model)Design Ultimate Load


239

7.3 Edge Column Connection


This section investigates punching failure at connections of internal wide beams to edge columns. It

studies the influence of eccentricity, lateral reinforcement distribution at edge columns on punching

resistance, failure mode and crack patterns. This is done in a similar manner to the investigations into

internal wide beam column connections. First, NLFEA is carried out on sub-assemblies using ATENA

with solid elements to investigate the influence of lateral reinforcement distribution on punching

resistance, failure mode and crack patterns. Then, similar analyses are performed with shell sub-

assemblies in DIANA to study the influence of banding reinforcement on rotations, shear force

distribution along the control perimeter and punching shear resistance evaluated with EC2 and fib

MC2010 with rotations calculated according to LoAs II & IV. Lastly, the effect of beam continuity has

been studied through comparing the results of NLFEA in terms of rotation and punching shear strength

obtained from the shell sub-assembly and the full-scale model.

7.3.1.1 Physical Model

The length of the sub-assembly is taken from the edge of wide beam to the elastic point of contraflexure

along the longitudinal axis of the wide beam, (1.2 m). The distance to the line of contraflexure was

determined from the elastic full-scale FEA model in DIANA (Chapter 6). The wide beam width is 2.4

m. The steel reinforcement details are similar to those in the full scale model (see Figure 6-5). The wide

beam longitudinal top reinforcement area is 1639 mm2 distributed over a width of 800 mm within the

middle third of the beam width. A minimum longitudinal steel ratio of 0.17% is provided in the beam

to either side of the middle third. Figure 7-24 and Figure 7-25 illustrate the uniform and banded

transverse reinforcement distributions respectively. For the transverse reinforcement with uniform

spacing, the arrangement is similar to that at the internal column connection; i.e., reinforcement

diameter of 13.3 mm at 200 mm centre to centre. For banded reinforcement two bar sizes are used 16

mm and 13 mm. The edge column is reinforced similarly to the internal column. H10 @ 250 mm c/c

both ways vertical shear stirrups were provided around the edge column. Figure 7-26 shows the

distribution of stirrups in the wide beam around the column and the punching shear control perimeters

according to EC2 and MC2010.


240

Figure 7-29: Transverse uniform reinforcement distribution for the edge column assembly

Figure 7-30: Transverse band reinforcement distribution for the edge column assembly

assembly

Figure 7-31: Plan view showing the shear reinforcement around the edge column


241


The material modelling, boundary conditions, solution methods and convergence criteria are similar to

those used in solid sub-assemblies for internal column. (See section 7.2.2.1).

Load Application: The design ultimate surface load is 23.025 kN/m2. Additionally, two line loads of

87.7 kN/m and 68.6 kN/m have been applied at the edges perpendicular to the width of the beam. A line

load of 184.7 kN/m has been imposed along the internal edge of assembly parallel to the beam width.

Figure 7-27 illustrates the loads subjected to the sub-assembly. These equivalent loads were extracted

from the elastic FEA carried out with DIANA software (refer to Section 6.3.1). Load control was used

in the NLFEA.

Mesh sizes & Types: The wide beam was meshed with linear cubic brick elements with element size of

50 mm. This yields eight elements through the beam depth. A mesh size of 50 mm has been assigned

for the column with its 200 mm-thick ends modelled elastically. Figure 7-28 depicts the mesh of the

edge connection assembly and its boundary conditions.

Figure 7-32: Load distribution subjected to the edge connection.


242

7.3.1.3 Results and Discussion

Figure 7-29 compares the NLFEA results in terms of load-deflection curves for the edge column

assembly with lateral uniform and band steel arrangements. Deflections are plotted at the intersections

of the edges of sub-assembly and the longitudinal and transverse axes of column as shown in Figure

7-29. The ultimate loads for the uniform and band assembly are 460.4 kN and 530.3 kN respectively.

Thus, banding reinforcement results in about 15% increase in punching strength of edge column.

Despite that, it does not seem that lateral reinforcement distribution significantly influenced the stiffness

of edge column connections in the longitudinal direction.

Figure 7-33: ATENA mesh discretization for edge connection assembly.

0

100

200

300

400

500

600

0 5 10 15 20 25 30

Load

: kN

Deflection: mm

Face G-UniformFace G-BandFace E-UniformFace E-BandFace F-UniformFace F-Band

Face G Face F

Face E

Deflection points

Load-deflection curves for solid assembly for edge connection for uniform and band reinforcement designs.

Figure 7-35: Load-deflection curves for solid assembly for edge connection for uniform and band reinforcement designs.


243

7.3.1.4 Failure Modes and Crack Patterns

Figure 7-30 compares the steel strains in the edge column assembly at failure for transverse uniform

and band rebar distributions. Prior to failure, both sub-assemblies exhibit ductile behaviour in the

longitudinal direction (normal to the slab edge). All the top flexural reinforcement across the effective

transfer width underwent significant yielding at the column face. The maximum steel strain reached 6.8

x 10-2 and 7.6 x 10-2 in the uniform and band assemblies respectively, which corresponds to 27 ɛs & 30

ɛs. Considerable yielding occurred in the transverse bars, parallel to wide beam width, at the column

faces in the uniform sub-assembly, but the strains in transverse bars were relatively low in the band

model.

Figure 7-31 & Figure 7-32 depict crack patterns at failure for the uniform and band sub-assemblies

respectively. Cracks are shown if their width is greater than 0.3 mm. The failures are characterised by

localised yield lines around the column with flexural cracks spreading along the column faces. Diagonal

cracks are also observed accompanied by torsional cracks across the beam sides adjacent to the column.

Placing transverse reinforcement in bands reduces these crack widths significantly. On the other hand,

it seems that there is no significant change in crack widths transverse to the direction of span of the

wide beam where cracks extend across the full beam width. The maximum crack width occurred along

the column face parallel to beam width for both lateral reinforcement distributions was around 5.5 mm

at failure.

Uniform distribution Band distribution

Figure 7-36: steel strains at failure in the edge column sub-assemblages for transverse uniform and band reinforcementdistributions


244

Figure 7-37: the crack patterns at failure (w ≥ 0.3 mm) in edge connection sub-assemblages with uniform reinforcementdistribution.

Figure 7-38: the crack patterns at failure (w ≥ 0.3 mm) in edge connection sub-assemblages with banded reinforcement distribution.


245

7.3.2 ATENA Results vs. MC2010 predictions with rotations according to LoAs

II and IV

Similar procedures to those for internal column connection, described in Section 7.2.2, were followed

for edge connections. The material modelling, mesh sizes and types, solution method and convergence

criteria are the same as given in Section 7.2.2.1. The load application for edge connection is illustrated

in Section 7.3.1.2. Figure 7-33 shows the mesh discretization of the DIANA shell sub-assembly for

edge connections.

7.3.2.1 Calculation rotations according to LoA II

The method of calculating the rotations with LoA II for edge columns is described in Chapter 2, Section

2.4.3.6 as follows:

= 1.5 ∙ ∙.

(2.54)

where tension reinforcement is parallel to the edge:

= ∙ + ,

∙≥ (2.58)

where tension reinforcement is perpendicular to the slab edge:

= ∙ + , (2.59)

Figure 7-39: DIANA mesh discretization for edge connection assembly used for punching analysis.

Translation restraints applied

at bottom and top surfaces

Vertical support

Face G

Face F

Face E

23.025 kN/m2


246

The average flexural strength per unit length in support strip, mRd is given by Equation (7.1).

The width of support strip parallel to the edge of the slab is calculated according to the fib

MC2010, in which the width parallel to the slab edge is reduced to:

= + 2 (7.9)

where c1 & c2 are the column size parallel and perpendicular to the edge of the slab. The width

of the support strip normal to the edge of the beam is found as follows:

= + (7.10)

where

= 1.5 , × , ≤ , ≤ (7.11)

Rotations have been calculated in the directions of the longitudinal and transverse axes of the internal

wide beam. Uniform and band lateral reinforcement distributions have been considered. The flexural

reinforcement ratio of the support strip with respect to the uniform lateral reinforcement design, ρuniform

is 0.19%. Across the support width of the edge connection, the flexural reinforcement ratio in the banded

arrangement isρBand(1) = 0.46% over a width of 0.6 m from the outside edge of the beam and ρBand(2) =

0.25% across the remaining support strip width of 0.6 m. The average flexural reinforcement ratio is

calculated using Equation (7.2), which yields ρ = 0.35%. The flexural reinforcement ratio in the

longitudinal direction (i.e., in the direction of span of the wide beam) is 0.44% for both the uniform and

banded transverse steel designs.

Figure 7-34 shows the load-rotation behaviour of slab as a function of rotation in the longitudinal and

transverse directions for the provided transverse rebar arranagements. Table 7-8 presents the punching

shear resistances calculated with MC2010 LoA II and the corresponding rotations. The total shear acting

on the control perimeter can be obtained from a linear analysis of the structure (column reaction minus

forces acting within control perimeter) as 658.4 kN. The minimum punching shear strength corresponds

to the maximum LoA II rotations which occur in the longitudinal direction of the wide beam (i.e., due

to the deflection of face E) for both lateral reinforcement designs. The longitudinal rotations are the

same for both uniform and banded transverse reinforcement arrangements since the longitudinal

reinforcement in the beam is the same in each case. Therefore, banding the transverse reinforcement

does not increase the punching shear strength according to the CSCT since the rotation in the

longitudinal direction governs the punching design.


247

Table 7-8: Estimated punching shear resistance for edge column connection using MC2010 LoA II for uniform and band steel designs (ke=0.7)

Steel arrangement Rotation: Rad VMC2010-II: kN γc=1.5, γs=1.15 γc= γs =1.0 γc=1.5, γs=1.15 γc= γs =1.0

Lateral Uniform 1.03 x 10-2 1.46 x 10-2 463 585 Band 5.70 x 10-3 8.00 x 10-3 582 734

Longitudinal 1.55 x 10-2 2.15 x 10-2 414 494

7.3.2.2 Calculation of rotations according to LoA IV

Figure 7-35 plots the internal beam rotations at the edge column from the DIANA sub-assembly

analyses. Rotations are shown along and transverse to the direction of span of the beam for each

transverse reinforcement distribution. Rotations are shown at 641 kN which is the failure load of the

shell sub-assembly with uniform transverse reinforcement. The failure load of the banded reinforcement

assembly is 697 kN. The column reaction at the design ultimate load from the elastic analysis is 701 kN.

It is evident that concentrating the transverse reinforcement at the edge column decreases the rotations

in both directions significantly. Moreover, the rotation in the banded reinforcement sub-assembly tends

to be constant in both directions. The maximum rotations occur in the longitudinal directions for both

reinforcement distributions as found for LoA II.

0

200

400

600

800

1000

1200

1400

1600

0.0E+00 5.0E-03 1.0E-02 1.5E-02 2.0E-02 2.5E-02 3.0E-02 3.5E-02 4.0E-02

Load

: kN

Rotation: Rad

Transverse direction-UniformTransverse direction-BandLongitudinal directionMC2010 failure criterion (ke=0.7, γc= γs=1.0)MC2010 failure criterion (ke=0.7& γc=1.5,γs=1.15)

Figure 7-41: Punching shear strength of edge column connection with rotations according to LoA II


248

Figure 7-36 compares the load-rotation relationships for the solid and shell sub-assemblies for both

transverse reinforcement distributions. The load-rotation curves are plotted for the three faces of the

control perimeter. Face E corresponds to the rotation in the longitudinal direction, while faces G & F

correspond to rotations in the transverse directions with face G having the highest line load as shown

in Figure 7-33. The depicted rotations are rotations of the wide beam relative to the edge column.

Figure 7-42: Rotations of wide beam along the longitudinal and lateral axes of the edge column for the uniform and band sub-assemblies at uniform assembly ultimate load (641 kN).

0

150

300

450

600

750

0.0E+00 5.0E-03 1.0E-02 1.5E-02 2.0E-02 2.5E-02 3.0E-02

Load

: kN

Rotation: Rad

Longitudinal axis-Face E

Shell-UniformSolid-UniformShell-BandSolid-Band

0

150

300

450

600

750

0.0E+00 2.0E-03 4.0E-03 6.0E-03 8.0E-03 1.0E-02

Load

: kN

Rotation: Rad

Transverse axis-Face G

Shell-Uniform

Solid-Uniform

Shell-Band

Solid-Band

0

150

300

450

600

750

0.0E+00 6.0E-04 1.2E-03 1.8E-03 2.4E-03 3.0E-03

Load

: kN

Rotation: Rad

Transverse axis-Face F

Shell-Uniform

Solid-Uniform

Shell-Band

Solid-Band

Figure 7-43: Comparison of L-R responses between the shell and solid assemblages for each side of control perimeter around the edge column for the uniform and band steel distributions.

-0.04

-0.02

0

0.02

0.04

0.06

0.08

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

Rot

atio

n: R

ad

Distance along the transverse/longitudinal axis of column: m

Longitudinal direction-Uniform Longitudinal direction-Band

Transverse drection-Uniform Transverse direction-Band

Max. Rotation


249

In general, the shell sub-assemblages are stiffer than the solid assemblies. This is probably because the

shell elements overestimate the torsional stiffness when subjected to out-of-balance loads. Additionally,

it could be due to the use of different tension stiffening models in the shell and solid assemblies as

described in Section 7.2.2.3. It is also noted that the load-rotation responses for uniform and banded

steel designs are comparable for shell and solid assemblages.

7.3.2.3 Calculation of Punching Shear Resistance

Figure 7-37 shows the load-rotation curves for the shell and solid sub-assemblies for uniform and

banded reinforcement. The rotations shown are the maximum rotations which occur in the longitudinal

direction as concluded previously. The failure criterion according to the fib MC2010 is also plotted for

partial safety factors 1.0 & 1.5 with ke =0.7. EC2 estimates for punching shear strength are shown in

the figure for both steel distributions with partial safety factors for concrete 1.0 & 1.5 and 1.0& 1.15

for steel with β=1.4. The design ultimate load on the sub-assembly is 658.4 kN. Table 7-9 presents

punching shear resistances calculated with EC2 and the fib MC2010 with rotations according to LoAs

II & IV and ATENA analyses.

Table 7-9:Punching resistances around edge column for models with shear reinforcement using EC2, MC2010 level II & IV and NLFEA for lateral uniform and banded steel distribution (ke=0.7).

Lateral reinforceme

nt design

EC2*: kN MC2010-LoA IV: kN

MC2010-LoA II: kN

ATENA’s ultimate load:

kN γc=1.5, γs=1.15

γc= γs= 1.0 γc=1.5, γs=1.15 γc= γs= 1.0 γc=1.5, γs=1.15

γc= γs= 1.0 shell solid shell solid

Uniform 546.4 695.1 481 424 542 457 241.7 312.3 460.4 Band 578.3 743.1 494 443 563 513 241.7 312.3 530.3

*The punching strength calculated with EC2 is divided by 1.4 to present it in a similar from to that of MC2010.

0

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600

800

1000

1200

1400

1600

0.0E+00 3.0E-03 6.0E-03 9.0E-03 1.2E-02 1.5E-02 1.8E-02 2.1E-02 2.4E-02 2.7E-02

Ver

tical

col

umn

reac

tion:

kN

Rotation: Rad

VRd,cs -EC2 (Uniform design) γc=γs=1.0 VRd,cs - EC2 (Band design) γc=γs=1.0Failure criterion γc=γs=1.0- ke=0.7 Failure criterion γc=1.5,γs=1.15-ke=0.7VRd,cs - EC2 (Uniform design) γc=1.5,γs=1.15 VRd,cs -EC2 (Band design) γc=1.5,γs=1.15L-R response (Uniform-shell assembly) L-R response (Band-shell assembly)L-R response (Uniform-solid assembly) L-R response (Band-solid assembly)Design Ultimate Load

Figure 7-44: Load-Rotation curves of the wide beam’s internal connection with shear reinforcement for shell and solid sub-assemblies for uniform and band steel designs


250

The fib MC2010, LoA IV predictions of punching shear resistance agree well with the ATENA results

for solid and shell sub-assemblies with band reinforcement distribution (Vshell/VATENA = 1.06, Vsolid/VATENA

= 0.97). However, the shell assembly with lateral uniform distribution overestimates the punching shear

resistance (Vshell/VATENA = 1.18). The reason is that the critical load-rotation responses for both lateral

reinforcement distributions is in the longitudinal direction for which the flexural reinforcement is the

same. Thus, similar predictions of punching shear strength are obtained for both steel designs. The

results from the fib MC2010 with rotations calculated with LoA II confirm this conclusion.

EC2 yields high estimates for the punching shear strength for lateral uniform and banded reinforcement

distributions, Vuniform/VATENA = 1.51, Vband/VATENA = 1.4 and hence the flexural strength governs. This

could be because EC2 assumes the punching shear resistance provided by concrete is constant

throughout the loading stages whereas in reality it reduces as assumed in the CSCT. ATENA analyses

show that only the punching shear reinforcements placed in front of the column in the longitudinal

direction have yielded prior to failure. This is due to the non-uniform shear stress distribution along the

control perimeter, as illustrated in Figure 7-38. However, both EC2 and MC2010 require punching shear

reinforcement to be uniformly distributed.

7.3.2.4 Investigation on the coefficient of eccentricity

The fib MC2010 method for ke calculation

Figure 7-38 compares shear force distributions obtained with elastic FEA and NLFEA along the

punching control perimeter for both uniform and banded steel arrangements. The perimeter is

approximated as rectangular with sides at 0.5d from the column face. The shear distribution along the

control perimeter is not uniform for either reinforcement distribution. Figure 7-38 suggests that banding

transverse reinforcement leads to more uniform distribution for shear forces in orthogonal directions.

Elastic Uniform Band

Face E

Face FFace

2.41 kN/mm 2.39 kN/mm

2.32 kN/mm1.92 kN/mm

1.43 kN/mm

1.12 kN/mm

Column

Edge of the wide beam

Figure 7-45: Comparison of shear force distribution along the control perimeter at 0.5d from the edge column face for elastic FEA and NLFEA with uniform and band reinforcement distributions.


251

Table 7-10 presents the average and maximum shear force per unit length obtained from the elastic FEA

and NLFEA for both steel designs. In addition, it shows the coefficient of eccentricity, ke, which

calculated with Equation (2.48) (see Section 7.2.2.5).

Table 7-10: Calculation of ke using linear elastic FEA and NLFEA for uniform and band reinforcement arrangements

νperp,d,av: N/mm νperp,d,max: N/mm ke Elastic 557.8 890.5 0.63 NLFEA-Uniform 557.8 2407.5 0.23 NLFEA-Band 557.8 1925 0.29

Table 7-11 compares the results of ke obtained from methods (i), (ii) and (iv) described in Section

7.2.2.5. It can be concluded that method (i) generally gives overly conservative estimates for ke if shear

forces are derived with NLFEA. Method (ii) yields a similar result for ke to that recommended by

MC2010 for edge columns of braced flat slabs with regular layout. However, results are not sensible

when calculated separately for eccentricities in the longitudinal and transverse directions. In such cases,

the average and maximum shear force per unit length are obtained for each direction separately and

applied according to Equation (7.5) to calculate ke in the transverse and longitudinal directions

separately. Then, they are applied to calculate the punching shear resistance for each rotation.

ke(trans) & ke(long) in Table 7-11 denotes the ke in the transverse and longitudinal directions respectively.

Similarly, ke was calculated in each direction using Equation (7.6). In conclusion, this investigation

shows that the effect of eccentricity is adequately considered with ke =0.7.

Table 7-11: Calculation of coefficient of eccentricity for edge column based on the fib MC2010 = , ,

, , =

11 +

MC2010 approximation for internal column

Method (i) ke - Elastic 0.63 ke

0.69

0.70

ke -Uniform 0.24 ke -Band 0.43

Method (iv) – transverse direction

ke(trans)- Elastic 0.73 ke(trans)

0.96 ke(trans)-Uniform 0.30

ke(trans) -Band 0.40 Method (iv) - longitudinal direction

ke(long)- Elastic 0.67 ke(long)

0.47 ke(long)-Uniform 0.26

ke(long) -Band 0.32

EC2 calculation method for β

The shear enhancement factor β is calculated in a similar manner to that described in Section 7.2.2.5.

Only the modifications related to edge columns are mentioned herein.


252

= 1 + ∙ (2.41)

At edge columns for bending about an axis perpendicular to the slab edge:

= + + 4 + 8 + (7.12)

For edge square columns K = 0.45. u1 is the length of the basic control perimeter.

EC2 gives expression for β for edge rectangular columns with eccentricities in both directions:

= ∗ + (7.13)

where epar denotes the eccentricity parallel to the slab edge resulting from a moment about an axis

perpendicular to the slab edge. u1* is the reduced basic control perimeter as shown in Figure 7-39.

Table 7-12 presents values of β calculated with respect to transverse and longitudinal axes of the edge

column. The x- and y- axes correspond to the longitudinal and transverse axes. The column dimensions

are 400 mm × 400 mm and the corresponding K is 0.45 (3). The results reveal that the coefficient of

eccentricity is governed by the eccentricity in the longitudinal direction. It is also concluded from the

punching shear calculation presented in Section 7.3.2.3 that using β = 1.4 overestimates significantly

the punching shear strength at the edge column. Therefore, the coefficient β = 1.56 is considered in the

calculation of punching shear strength in Section 7.3.3.

Figure 7-46: EC2’s basic control perimeter for edge column


253

Table 7-12: Calculation of coefficient of eccentricity based on EC2 Axis considered MEd: kN.m

= 1 + ∙ = ∗ + EC2 approximation for internal column

Biaxial - - 1.16

1.4 Longitudinal axis 544.8 1.56

Transverse axis 27.5 1.03

7.3.3 Effect of continuity according to MC2010 LoA IV

7.3.3.1 Calculation of Rotations in full-scale model: Level IV

Figure 7-40 shows rotations obtained from the NLFEA along the longitudinal and transverse axes of

the wide beam at the design ultimate load (701 kN). The rotations are taken along the column centrelines

in the orthogonal directions without correction for column rotation. It is concluded that the maximum

rotation may be extracted at distance d (≈ 400 mm) from the column face in the longitudinal direction

and 2d (≈ 800 mm) in the transverse direction. In addition, the rotations in the longitudinal direction are

higher than those in the transverse direction for both transverse reinforcement arrangements.

Moreover, rotations in the band model are higher than in uniform model which is surprising since this

would result in lower punching shear strength for band model according to the CSCT although it has a

higher flexural reinforcement ratio than the uniform model. To explore the problem more closely, the

bending moments are plotted across the transverse and longitudinal axes of the wide beam plotted at

ultimate load for both steel designs as shown in Figure 7-41.

0

0.003

0.006

0.009

0.012

0.015

0.018

0.021

0.024

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

Rot

atio

n: R

ad

Distance along the longitudinal / transverse axis: m

Longitudinal axis-UniformTransverse axis-UniformLongitudinal axis-BandTransverse axis-Band

Position of max. rotation

Position of max. rotation

Figure 7-47: Rotations of wide beam along the longitudinal and lateral axes of the wide beam at ultimate load of 701 kN for the uniform and band full-scale models.


254

The bending moments in the band model are relatively higher than in the uniform model across the

entire width of the wide beam. In both cases, the longitudinal reinforcement yielded across the column.

This appears to have resulted in higher rotations for banded than uniform transverse reinforcement.

7.3.3.2 Comparison of Load - Rotation diagrams

Figure 7-42 compares the load-rotation responses for the shell assembly and full-scale model for

uniform and banded lateral reinforcement distributions. The load-rotation curves are plotted for the

three faces of the control perimeter in which face E corresponds to rotation in the longitudinal direction,

while faces G & F relate to rotations in the transverse directions. Note that the rotations herein refer to

the rotations of the wide beam relative to the edge column.

The predicted behaviour of the shell full-scale model with banded transverse steel is slightly less stiff

in all directions than the model with uniform reinforcement. This is attributed to the higher moments

taken by the longitudinal reinforcement in the banded model as discussed in the previous section. In the

model with uniform lateral reinforcement, slab continuity reduces rotations significantly in both

orthogonal directions. Towards flexural failure, the opposite is the case for banded transverse

reinforcement where the sub-assembly with banded reinforcement has a stiffer response than the full-

scale model. This is most noticeable in the transverse direction. This could be a result of the shell model

underestimating rotations due to overestimate of torsional stiffness as suggested by Figure 7-36.

0

100

200

300

400

500

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

Ben

ding

Mom

ent:

kN.m

/m

Distance along the transverse axis: m

UniformBand

Position of maximum rotation

69.4 47.9

-150

-100

-50

0

50

100

150

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

Ben

ding

Mom

ent:

kN.m

/m

Distance across the longitudinal axis: m

UniformBand

8.9

51.8

Figure 7-48: comparison of bending moments across the transverse and longitudinal axes of edge column at ultimate load between the uniform and band full-scale models.

Position of maximum

125.5 73.6


255

7.3.3.3 Comparison of Punching shear resistance

Figure 7-43 shows the load-rotation curves of the edge connection obtained from the NLFEA for full-

scale models with both uniform and banded transverse reinforcement. The rotations are wide beam

rotations relative to the edge column. Figure 7-40 shows that the wide beam maximum rotation occurs

in the longitudinal direction at distance slightly greater than the effective depth d from the column face.

The column rotation has been calculated in a similar manner to that of the internal column (see Figure

7-15). In addition, the failure criterion according to fib MC2010 is plotted with partial safety factors

(γc=1.5, γs=1.15) and (γc =γs=1.0). The coefficient of eccentricity, ke was taken as 0.7. The EC2

predictions for punching shear strength, calculated with β = 1.56 as calculated in Section 7.3.2.4, are

also shown for both transverse steel distributions. Table 7-13 presents the numerical values of the

punching shear resistances calculated with NLFEA, fib MC2010 level IV, and EC2.

0

100

200

300

400

500

600

700

0.0E+00 6.0E-03 1.2E-02 1.8E-02 2.4E-02 3.0E-02

Load

: kN

Rotation: Rad

Longitudinal axis-Face E

Shell assembly-Uniform

Full-scale model-Uniform

Shell assembly-Band

Full-scale model-Band0

100

200

300

400

500

600

700

0.0E+00 2.0E-03 4.0E-03 6.0E-03 8.0E-03 1.0E-02

Load

: kN

Rotation: Rad

Transverse axis-Face G

Shell assembly-Uniform

Full-scale model-Uniform

Shell assembly-Band

Full-scale model-Band

0

100

200

300

400

500

600

700

0.0E+00 6.0E-04 1.2E-03 1.8E-03 2.4E-03 3.0E-03

Load

: kN

Rotation: Rad

Transverse axis-Face F

Shell assembly-UniformFull-scale model-UniformShell assembly-BandFull-scale model-Band

Figure 7-50: Comparison of L-R responses between the shell and solid assemblages for each side of control perimeter around the edge column for the uniform and band steel distributions.


256

Table 7-13: Comparison of Punching shear resistances for full-scale models and shell assemblages estimated using EC2, MC2010 level IV and NLFEA for lateral uniform and banded steel distribution (ke=0.7).

Lateral steel dist.

EC2*: kN MC2010-level IV (γc=1.5, γs=1.15)

MC2010 -level IV – (γc= γs=1.0) NLFEA : kN

γc=1.5, γs=1.15

γc= γs=1.0

Full scale:

kN

shell assemblage:

kN

Increase due to

CMA %

Full scale:

kN

Shell assemblage:

kN

Increase due to

CMA% Uniform 490.3 623.8 537 481 11.6 628 542 15.9 460.4

Band 519.0 666.9 510 494 3.2 604 563 7.3 530.3 *The punching strength calculated with EC2 is divided by 1.56 to present it in a similar from to that of MC2010.

In general, the load-rotation responses of the shell assemblages are comparable to those of full-scale

models. Continuity of slab results in a slightly stiffer response, and hence, slightly greater punching

shear resistances as evaluated with MC2010 LoA IV. It is interesting to note that, with γc =γs=1.0, the

increase in punching shear strength due to slab continuity in the model with lateral uniform

reinforcement design is almost double that with banded reinforcement design.

Another interesting point is that using β=1.56 seems to improve the EC2 prediction for punching shear

strength. The results are close to the MC2010 estimates for the full scale slab with partial safety factors

γc=1.5, γs=1.15 (VEC2/VMC2010 = 1.02, 1.05 for uniform and band distributions respectively). With γc

=γs=1.0 the differences between the EC2 and MC2010 estimates increase (15% & 18% for uniform and

band reinforcement distributions respectively).

EC2 gives a high estimate for the punching shear resistance of the sub-assembly with uniform transverse

steel, (VEC2/VATENA =1.35), with partial safety factors γc =γs=1.0. Closer agreement is obtained with the

banded sub-assembly (VEC2/VATENA =1.26). It can be seen that, although still high, the EC2 prediction for

band reinforcement distribution is relatively better than for the uniform distribution. This could be

0

200

400

600

800

1000

1200

1400

1600

0.0E+00 2.0E-03 4.0E-03 6.0E-03 8.0E-03 1.0E-02 1.2E-02 1.4E-02 1.6E-02 1.8E-02 2.0E-02 2.2E-02

Ver

tical

col

umn

reac

tion:

kN

Rotation: Rad

VRd,cs -EC2 (Uniform design) γc=γs=1.0 VRd,cs - EC2 (Band design) γc=γs=1.0Failure criterion γc=γs=1.0- ke=0.7 Failure criterion γc=1.5,γs=1.15-ke=0.7VRd,cs - EC2 (Uniform design) γc=1.5,γs=1.15 VRd,cs -EC2 (Band design) γc=1.5,γs=1.15L-R response (Uniform-shell assembly) L-R response (Band-shell assembly)L-R response (Uniform-full scale model) L-R response (Band-full scale model)Design Ultimate Load

Figure 7-51: Load-Rotation curves of the wide beam’s edge column connection for full-scale shell model and shell sub-assemblies for uniform and band steel designs


257

explained since banding reinforcement leads to more uniform shear distribution around the control

perimeter than in the uniform model.

7.4 Conclusions

This chapter investigates the influence of wide beam transverse steel distribution on punching resistance

and shear stress distribution at internal and edge columns. It also evaluates the suitability of EC2 and

the fib MC2010 design methods for calculating the punching shear resistance of wide beams in which

both loading and reinforcement arrangements are greatest in the direction of span.

These matters have been investigated systematically for the banded slab investigated in Chapter 6. First,

the punching shear resistance was evaluated by analysing sub-assemblies with ATENA using solid

elements. The modelling procedure was validated against relevant test data in Chapter 5. The resulting

punching resistances are compared with the predictions of MC2010 with rotations calculated according

to LoAs II and IV. The rotations according to LoA IV were obtained from DIANA analyses using shell

elements as well as from the ATENA sub-assemblies. Investigations were also carried out to determine

the best choice of the coefficient of eccentricity, ke. Similar investigations have also been performed for

the shear enhancement multiplier β used in EC2. After that, comparison is made between the results of

shell sub-assembly and full scale-model in terms of rotations and punching shear strength to study the

effect of continuity according to MC2010 LoA IV.

In summary, the following conclusions are drawn from this work:

Banding the transverse steel has little influence on shear force distributions along the fib MC2010

control perimeter for punching shear at internal columns. The effect is more pronounced around the

edge columns where banding the reinforcement results in a more uniform distribution of shear force.

ATENA analyses shows that banding transverse steel improves the punching shear resistance due

to the increase of average flexural reinforcement ratio, but the enhancement in stiffness is

insignificant.

The transverse steel distribution does not greatly affect the beam rotations according to DIANA

analyses. As a result, MC2010 LoA IV gives virtually similar estimates of punching resistance for

both uniform and banded transverse steel designs at internal and edge column connections.

Based on comparisons with ATENA analyses, the EC2 approach overestimates the punching

strength of internal and edge column connections for both transverse steel arrangements. This is

probably because the wide beam was close to flexural failure at the design ultimate load. Hence,

EC2 overestimates the contribution of concrete to punching shear resistance.

The shear reinforcement is not fully utilized due to the non-symmetrical distribution of shear force

around the critical section. This suggests that it could be more efficient to concentrate punching


258

shear reinforcement where shear stresses are greatest. Nevertheless, this would require experimental

investigation and could be at the expense of construction speed.

It is evident that the flexural continuity and CMA influences the punching resistance of wide beam

slabs. This has been found by comparing punching resistances obtained with MC2010 LoA IV in

full-scale models and sub-assemblies. Yet, further experimental research is required to accurately

quantify the effect of CMA and flexural continuity on punching resistance.

Parametric studies will be carried out in the Chapter 8 to investigate the effect of varying the ratio

between the flexural reinforcement ratios in the transverse and longitudinal directions, ρy /ρx on

punching shear strength evaluated with the fib MC2010 with rotations according to LoA IV and EC2.

In addition, the coefficient of eccentricities, ke, and β which account for eccentricity of internal column

reaction relative to the control perimeter according to fib MC2010 and EC2 respectively, for internal

column in internal wide beam floors will be investigated as well.

Parametric Studies and Design Recommendations Chapter 8

259

Chapter 8

Parametric Studies and Design Recommendations

8.1 Introduction

The main aim of the thesis is to develop a rational design method for wide beam floors that satisfies the

design serviceability limit state of cracking over columns as well as the design ultimate limit state.

Chapter 6 shows that crack widths are reduced by concentrating the transverse hogging reinforcement

in bands over internal columns, in accordance with the elastic moment distribution, rather than

uniformly distributing the reinforcement along the beam as commonly done. Chapter 7 investigates the

influence of transverse reinforcement distribution on punching resistance at internal and edge columns

using NLFEA with solid elements as well as with MC2010 LoA IV using rotations from NLFEA.

Comparisons are also made with the strength predictions of EC2 and MC2010 LoA II. For the cases

considered, banding transverse reinforcement marginally increased punching shear resistance according

to EC2 and NLFEA with solid elements using ATENA but had no significant influence according to

MC2010 LoA IV.

Design moments for wide beam slabs can either be calculated using equivalent frames as done by TCC

(1) or with FEA. In either case, the transverse reinforcement should be banded for adequate crack

control. This chapter describes a series of parametric studies which were carried out to establish

simplified rules for determining banded transverse steel distributions that satisfy the design SLS and

ULS conditions. In the proposed method, the transverse hogging reinforcement is placed across the

wide beam in bands of width related to the elastic bending moment distribution. Making use of


260

parametric studies, rules are developed for determining the width and amount of reinforcement in each

band. The chapter also investigates the influence of transverse and longitudinal flexural reinforcement

ratios on the punching shear resistance according to MC2010 LoA IV and EC2. The modelling of

eccentric shear in the punching provisions of fib MC2010 and EC2 is also reviewed for wide beam slabs

and design recommendations are made.

8.2 Parametric Studies for Flexure

8.2.1 General

The NLFEA analyses in Chapter 6 show that the flexural failure load of wide beam slabs is not

significantly affected by the transverse reinforcement distribution over wide beams. However,

reinforcement strains, and hence crack widths, depend significantly on the transverse reinforcement

distribution. In particular, steel strains and crack widths reduce significantly when transverse

reinforcement is distributed in accordance with the elastic design moments rather than being uniformly

distributed. Uniformly distributing the transverse hogging reinforcement along the length of wide

beams can violate the allowable crack width limits specified by EC2. For design it is helpful to

determine simple rules for determining the widths of the bands across which the reinforcement is

distributed. Possible factors affecting the transverse hogging moment distribution are the spans of the

wide beam (Lb) and slab (Ls) as well as the width of wide beam (Wb). This chapter presents the results

of a parametric study carried out to systematically investigate the influence of these parameters.

8.2.2 Numerical Model

Linear elastic FEA with shell elements was carried out to determine the distribution of elastic bending

moment along critical sections. Then, nonlinear analysis using shell elements was performed in order

to assess the structural performance. Full description of the procedures adopted for the linear elastic and

nonlinear analyses are found in Sections 6.3.1 & 6.3.4 respectively. The numerical values of Lb, Ls and

Wb are chosen to be representative of the practical range for reinforced concrete wide beam slabs. In the

author’s opinion post tensioned concrete should be considered for larger spans to minimise structural

depth. Figure 8-1 shows the geometry of the model used in the parametric study, whilst Table 8-1 shows

the parameters considered in the study. The member sizes given in Table 8-1 are chosen to comply with

ECFE (1). The design of slab thickness and wide beam depth are governed by deflection. The deflection

calculations are based on the EC2 span to effective depth rules. Reinforcement was designed for a single

load case of (1.35 D.L. +1.5 I.L) in accordance with EC2 to satisfy both the ULS and SLS requirements.

No moment redistribution was considered.


261

Figure 8-1: Geometry of the model used for the parametric study


262

Table 8-1: members dimensions of the models used in the parametric studies

Wbi & Wbe refer to widths of internal and edge wide beams respectively.

Although the span of RC wide beams can be up to 14 m (1), it is limited to 10 m in these parametric

studies because greater spans only require minimum transverse flexural reinforcement along the wide

beam. For instance, Figure 8-2 compares the average transverse steel required along sections 1-1 & 2-

2 (see Figure 8-1) with the minimum reinforcement area required by EC2 for wide beam and slab spans

of 12 m & 8 m respectively. For the adopted design procedure, minimum reinforcement governs since

the average moment in the column band strip is less than the moment of resistance provided by

minimum reinforcement.

Model designation

Slab span Ls (m)

Beam span Lb (m)

Slab thickness ts (m)

Beam depth Db (m)

Beam width Wbi - Wbe (m)

Column size C1 (mm) x C2 (mm)

A 7 10 0.18 0.38 2.4-1.4 0.4 × 0.4 B 8 10 0.20 0.40 2.4-1.4 0.4 × 0.4 C 9 10 0.22 0.42 2.4-1.4 0.4 × 0.4 D 7 8 0.18 0.26 2.4-1.4 0.4 × 0.4 E 7 9 0.18 0.32 2.4-1.4 0.4 × 0.4 F 7 9 0.18 0.33 2.0-1.2 0.4 × 0.4 G 7 9 0.19 0.34 1.8-1.1 0.4 × 0.4

Figure 8-4: Required transverse reinforcement areas to resist Wood-Armer moments along sections (1-1), (2-2) and the minimum steel area.

0

200

400

600

800

1000

1200

1400

1600

1800

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Tran

sver

se f

lexu

ral s

teel

are

a: m

m2/

m

Distance along the wide beam span: m

Section 1-1 Section 2-2Minimum steel area Average steel area- Section 1-1Average steel area-Section 2-2


263

Loading:

The floor was designed to carry the self-weight of the structural elements, other dead loads including a

superimposed dead load of 1.5 kN/m2 for finishes and an imposed load of 5.0 kN/m2. The material

properties for concrete and steel reinforcement are the same as used in Chapter 6 (refer to Table 6-1).

Expected Output:

The main outputs from this parametric study are the width of the reinforcement bands within which

greater than minimum steel reinforcement is required to control cracking as well as rules for distributing

reinforcement between bands. Additionally, the parameters influencing the transverse moment

distribution and the band widths are identified.

The models are divided into three groups. Each group is used to study the influence of one parameter,

as shown in Table 8-2.

Table 8-2: Division of models used in the parametric studies according to the relevant investigated parameter Group Models in the group Investigated parameter

I A, D & E Span of wide beam II A, B & C Span of slab III E, F, G Width of wide beam

8.2.3 Results and Discussion

8.2.3.1 The influence of wide beam span

Figure 8-3 & Figure 8-4 show the transverse distribution of elastic hogging moments along the wide

beam to either side of the first internal column for internal and edge beams respectively. Moments are

shown along sections at the column and faces from the column centreline to mid span. Results are

shown for models A, B and C. Figure 8-5 & Figure 8-6 show the same results at the end columns of the

internal and edge beams. The elastic moments are plotted along the wide beam sections (1-1, 2-2, 3-3

& 4-4) passing through column and beam faces. Moment distributions are shown for three values of

wide beam span: 8 m, 9 m & 10 m. The elastic moments are presented in normalised form (m*/m*av),

where m* is the Wood-Armer moment (61) at any point at the section and m*av is the average Wood-

Amer moment. The normalised moment is plotted against the distance along the wide beam divided by

the wide beam span.


264

The figures show that the normalised moments are very similar for each span considered. Furthermore,

the width over which m*/m*av > 1 is almost proportional to the wide beam span. The peak normalised

moments in each figure are similar but increase slightly with wide beam span.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

m* /m

* av

Distance along wide beam/beam span

Ls=7.0m, Wb=2.4m Column face, Lb=8mColumn face, Lb=9mColumn face, Lb=10mBeam face, Lb=8mBeam face, Lb=9mBeam face, Lb=10m

Figure 8-5: Influence of varying wide beam span on transverse distribution of elastic support moment at thefirst internal column at section passing through column and beam faces along the internal wide beam.

Figure 8-6: Influence of varying wide beam span on transverse distribution of elastic support moment at the first internal column at section passing through column and beam faces along the edge wide beam.

0

1

2

3

4

5

6

7

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

m* /m

* av


Ls=7.0m, Wb=1.4m Column face, Lb=8mColumn face, Lb=9mColumn face, Lb=10mBeam face, Lb=8mBeam face, Lb=9mBeam face, Lb=10m


265

8.2.3.2 The influence of slab span

Figure 8-7 and Figure 8-8 show the transverse elastic moment distribution along the wide beam length

at the first interior column for internal and edge wide beams respectively. Results are shown for slab

spans of 7 m, 8 m and 9 m. Figure 8-9 and Figure 8-10 show the same distribution at the end columns

for internal and edge wide beams. For both the internal and edge beams the slab span seems to have

little effect on the normalised elastic support moment and the width of steel bands. This finding

contradicts that of Tay (6) who noted that the column strip width is influenced largely by slab span. Tay

computed the column strip width as a function of slab span. However, Shuraim & Al-Negheimish (7)

suggested that the width of high rigidity zones, which is conceptually similar to the width of column

strips, should be calculated as percentage of the wide beam span rather than the slab span. This is

consistent with results of the current parametric study.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 0.1 0.2 0.3 0.4 0.5

m* /m

* av


Ls=7.0m, Wb=2.4m

Column face, Lb=8mColumn face, Lb=9mColumn face, Lb=10mBeam face, Lb=8mBeam face, Lb=9mBeam face, Lb=10m

Figure 8-7: Influence of varying wide beam span on transverse distribution of elastic support moment atthe end column at section passing through column and beam faces along the internal wide beam.

0

1

2

3

4

5

6

7

8

9

10

0 0.1 0.2 0.3 0.4 0.5

m* /m

* av


Ls=7.0m, Wb=1.4m

Column face, Lb=8mColumn face, Lb=9mColumn face, Lb=10mBeam face, Lb=8mBeam face, Lb=9mBeam face, Lb=10m

Figure 8-8: Influence of varying wide beam span on transverse distribution of elastic support moment atthe end column at section passing through column and beam faces along the edge wide beam.


266

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

m*/

m*a

v


Lb=10m, Wb=2.4mColumn face, Ls=7mColumn face, Ls=8mColumn face, Ls=9mBeam face, Ls=7mBeam face, Ls=8mBeam face, Ls=9m

Figure 8-9: Influence of varying slab span on transverse distribution of elastic support moment at the first internal column at section passing through column and beam faces along the internal wide beam.

0

1

2

3

4

5

6

7

8

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

m* /m

* av


Lb=10m, Wb=1.4m

Column face, Ls=7mColumn face, Ls=8mColumn face, Ls=9mBeam face, Ls=7mBeam face, Ls=8mBeam face, Ls=9m

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 0.1 0.2 0.3 0.4 0.5

m* /m

* av


Lb=10m, Wb=2.4mColumn face, Ls=7mColumn face, Ls=8mColumn face, Ls=9mBeam face, Ls=7mBeam face, Ls=8mBeam face, Ls=9m

Figure 8-12: Influence of varying slab span on transverse distribution of elastic support moment at the first internal column at section passing through column and beam faces along the edge wide beam.

Figure 8-15: Influence of varying slab span on transverse distribution of elastic support moment at the end column at section passing through column and beam faces along the internal wide beam.


267

8.2.3.3 The influence of wide beam width

Figure 8-11 and Figure 8-12 compare transverse elastic moment distributions at the first internal

column, for internal and edge wide beams respectively, for beam widths of 1.8 m, 2.0 m & 2.4 m. Figure

8-13 and Figure 8-14 present the same results at the end columns. It is clear that the distributions of

transverse elastic moment across the wide beams are nearly identical. Similarly, the width

corresponding to the high intensity moment doesn’t seem to be influenced by the width of wide beam.

The same conclusion was reported by Tay (6).

Figure 8-16: Influence of varying slab span on transverse distribution of elastic support moment at the end column at section passing through column and beam faces along the edge wide beam.

0

1

2

3

4

5

6

7

8

9

10

0 0.1 0.2 0.3 0.4 0.5

m* /m

* av


Column face, Ls=7mColumn face, Ls=8mColumn face, Ls=9mBeam face, Ls=7mBeam face, Ls=8mBeam face, Ls=9m

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

m* /m

* av


Lb=9.0m, Ls=7.0m Column face, Wb=1.8mColumn face, Wb=2.0mColumn face, Wb=2.4mBeam face, Wb=1.8mBeam face, Wb=2.0mBeam face, Wb=2.4m

Figure 8-17: Influence of varying width of wide beam on transverse distribution of elastic support moment at the first internal column at section passing through column and beam faces along the internal wide beam.


268

Figure 8-19: Influence of varying width of wide beam on transverse distribution of elastic support moment at the end column at section passing through column and beam faces along the edge wide beam.

0

1

2

3

4

5

6

7

8

9

0 0.1 0.2 0.3 0.4 0.5

m* /m

* av


Lb=9m, Ls=7m

Column face, Wb=1.1mColumn face, Wb=1.2mColumn face, Wb=1.4mBeam face, Wb=1.1mBeam face, Wb=1.2mBeam face, Wb=1.4m

0

1

2

3

4

5

6

7

8

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

m* /m

* av


Column face, Wb=1.1mColumn face, Wb=1.2mColumn face, Wb=1.4m

Beam face, Wb=1.1mBeam face, Wb=1.2mBeam face, Wb=1.4m

Figure 8-18: Influence of varying width of wide beam on transverse distribution of elastic support moment at the first internal column at section passing through column and beam faces along the edge wide beam.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 0.1 0.2 0.3 0.4 0.5 0.6

m* /m

* av


Lb=9 m, Ls=7 m

Column face, Wb=1.8mColumn face, Wb=2.0mColumn face, Wb=2.4mBeam face, Wb=1.8mBeam face, Wb=2.0mBeam face, Wb=2.4m

Figure 8-20: Influence of varying width of wide beam on transverse distribution of elastic support moment at the end column at section passing through column and beam faces along the internal wide beam.


269

8.2.4 Development of design procedure for transverse distribution of support

moment.

Chapter 6 shows that placing the transverse flexural reinforcement uniformly across the wide beam can

result in excessive crack widths. Consequently, a banded distribution is proposed for transverse flexural

reinforcement with resistance following approximately the elastic moment distribution. This involves

dividing the wide beam length into three zones with three different reinforcement bands as discussed in

Chapter 6 (see Section 6.3.3.2). In order to avoid providing large areas of reinforcement in narrow

bands, which could impede constructability, the number of zones is reduced to two; namely column

band and slab band as illustrated in Figure 8-15. Within each band, the reinforcement is uniformly

distributed. In order to develop a generalised design approach for laterally distributing the reinforcement

across the wide beam, two quantities need to be determined. These are the widths of each band and the

design bending moment for each band. In order to establish relationships between the column band

width and beam span, slab span and beam width, Figure 8-3 to Figure 8-14 are presented in different

form.

8.2.4.1 Band width

Column band width:

The column band width is defined as the width across which the acting moment is greater than the

average moment along the entire length of the wide beam, i.e., (m* >m*av). The average moment is

chosen because it represents the design moment according to TCC method. Therefore, at any section if

m* >m*av there is a possibility of exceeding the allowable limit of crack width at SLS. The band width

Span band width Span band width Span band width

First internalcolumn band width

First internalcolumn band width End column

band widthEnd column band width

Wide beam

Figure 8-21: Proposed division for wide beam into column and span band widths


270

is determined for each model and plotted against the wide beam span, slab span and wide beam width

for the first internal and end columns as shown in Figure 8-16.

(a) (b) (c)

Figure 8-17 shows the influence of the wide beam span, slab span and wide beam width on the width

of column band in the edge wide beam for the first internal and end columns.

(a) (b) (c)

Figure 8-16 & Figure 8-17 show clearly that the width of the column band varies linearly with beam

span at end and internal columns. They also show that the column band width is almost independent of

slab span. Additionally, the effect of beam width appears to be insignificant. Based on this, a

relationship has been developed between the wide beam span and the width of the column band for

internal and edge beams as illustrated in Figure 8-18 and Figure 8-19 respectively.

Figure 8-22: Influence of a) beam span, b) slab span and c) beam width on the width of column band in the internal beam

Figure 8-23: Influence of a) beam span, b) slab span and c) beam width on the width of column band in the edge beam

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8

Ban

d w

idth

: m (f

or m

* >m

* av)

Beam width: m

First internal column

End column

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 1 2 3 4 5 6 7 8 9 10

Ban

d w

idth

: m(fo

r m

* >m

* av)

Slab span: m


End column

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 1 2 3 4 5 6 7 8 9 10 11

Ban

d w

idth

: m(fo

r m

* >m

* av)

Beam span: m


End column

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 1 2 3 4 5 6 7 8 9 10 11

Ban

d w

idth

: m (f

or m

* >m

*av)

Beam span: m


End column

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 2 4 6 8 10

Ban

d w

idth

: m (f

or m

* >m

* av)

Slab span: m


End column

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 0.4 0.8 1.2 1.6 2 2.4Ban

d w

idth

: m (f

or m

* >m

* av)

Beam width: m

First internal columnEnd column


271

Span band width:

The span band width is defined as the width across which the acting moment is less than the average

moment along the entire length of wide beam, i.e., (m* <m*av). The width of span band is computed

after the width of column band is determined using the principles described in Chapter 6, Section

6.3.3.2. Thus, the width of span band for internal and edge wide beams is defined as follows:

= − ∑ (8.1)

where Zspan & Zcolumn denote the width of span and column bands respectively. Lb is the span of wide

beam.

Figure 8-24: Relationship between the wide beam span and the width of column band over the internal beam.

Figure 8-25: Relationship between the wide beam span and the width of column band over the edge beam

y = 0.1167x + 1.9667

y = 0.0729x + 0.7729

1.0

1.5

2.0

2.5

3.0

3.5

7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5Ban

d w

idth

: m(fo

r m* >

m* av

)

Wide beam span: m

First internal columnEnd columnLinear (First internal column)Linear (End column)

y = 0.1042x + 1.2042

y = 0.0979x + 0.2479

0.50.70.91.11.31.51.71.92.12.32.5

7 8 9 10 11

Ban

d w

idth

: m (f

or m

* >m

* av)

Wide beam span: m



272

8.2.4.2 Moments across the band widths.

Moments across the column band width

The moments acting across the column band width need to be quantified. To achieve this, the key

parameters are plotted against the ratio of the average Wood & Armer’s moment across the column

band width, m*column, to the average Wood & Armer’s moment across the entire wide beam length, m*

av.

Results are shown in Figure 8-20 and Figure 8-21 for internal and edge wide beams at first internal and

end columns.

(a) (b) (c)

(a) (b) (c)

It can be concluded from Figure 8-20 & Figure 8-21 that the normalised average moment across the

column band width increases linearly with wide beam span and reduces linearly with slab span. The

beam width, however, does not seem to affect the normalised average moment across the column band

width.

0.0

0.5

1.0

1.5

2.0

2.5

0 2 4 6 8 10

m* co

lum

n/m* av

Slab span: m

Interior panelEdge panel

0.0

0.5

1.0

1.5

2.0

0 0.4 0.8 1.2 1.6 2 2.4 2.8

m* co

lum

n/m* av

Beam width: m


0.0

0.5

1.0

1.5

2.0

2.5

0 2 4 6 8 10 12

m* co

lum

n/m* av

Beam span: m

Interior panel

Edge panel

Figure 8-26: Influence of a) beam span, b) slab span and c) beam width on the moment across the column band width in the internal beam.

0.0

0.4

0.8

1.2

1.6

2.0

0 2 4 6 8 10 12

m* co

lum

n/m* av

Beam span: m

Interior panel

Edge panel

0.0

0.4

0.8

1.2

1.6

2.0

0 2 4 6 8 10

m* co

lum

n/m* av

Slab span: m


0.0

0.5

1.0

1.5

2.0

2.5

0 0.4 0.8 1.2 1.6

m* co

lum

n/m* av

Beam width: m


Figure 8-27: Influence of a) beam span, b) slab span and c) beam width on the moment across the column band width in the edge beam.


273

Having acknowledged this, it is instructive to plot the normalised moment, m*column/m*

av, against the

span aspect ratio, Lb/Ls. Figure 8-22 & Figure 8-23 present these plots for internal and edge beams

respectively at first internal and end columns.

Moments across the span band width

The moment within the span band width is quantified using the procedure described in Section 6.3.3.2.

The transverse average bending moment across the span band is given by the following expression:

∗ =∗ . ∑ ∗ . (8.2)

where m*span is the average Wood & Armer’s moment across the span band width, Zspan. The flexural

reinforcement corresponding to the moments across the span band width should satisfy the minimum

reinforcement criterion.

Figure 8-29: Relationship between the slab aspect ratio and the moment across column band width in the edge wide beam.

Figure 8-28: Relationship between the slab aspect ratio and the moment across column band width in the internal wide beam.

y = 0.1673x + 3.2104R² = 0.0157

y = 1.1528x + 2.8021R² = 0.2962

0

1

2

3

4

5

0.0 0.3 0.6 0.9 1.2 1.5

m* C

olum

n/m

* av

Lb/Ls


y = 0.5737x + 1.1255R² = 0.2495

y = 0.4084x + 1.2134R² = 0.2063

0.0

0.4

0.8

1.2

1.6

2.0

2.4

0.0 0.3 0.6 0.9 1.2 1.5

m* C

olum

n/m* av

Lb/Ls



274

8.2.4.3 Results from the Regression Analysis

In order to obtain numerical expressions for the band width and the corresponding band moment,

regression analyses have been carried out on the data presented in Figures 8-16, 8-17, 8-20 and 8-21.

For simplicity, the band widths and design moments are assumed to be proportional to and

respectively. Consequently, the resulting equations for band widths and design moments are only

applicable within the range of parameters considered in the parametric study. The correlation with the

data from the parametric studies is obtained with relationships of the form = + . For the internal

wide beam, the band width and the corresponding band moment are defined as follows:

For first internal column:

= 0.12 + 2 (8.3)

∗

∗ = 0.57 + 1.13 (8.4)

For end column:

= 0.07 + 0.77 (8.5)

∗

∗ = 0.41 +1.2 (8.6)

For an edge wide beam, the band width and the corresponding band moment can be defined as follows:

For first internal column:

= 0.10 + 1.2 (8.7)

∗

∗ = 1.15 + 2.8 (8.8)

For edge column:

= 0.1 + 0.25 (8.9)

∗

∗ = 0.17 + 3.21 (8.10)

It should be noted that in Equations (8.3), (8.5), (8.7) and (8.9) the beam span is in metres.

Figure 8-24 compares the transverse flexural reinforcement distribution for the uniform and banded

rebar designs along the internal and edge wide beams. The total banded steel area is significantly greater


275

than the uniform area. This is because the minimum steel area governs for the uniform arrangement and

span band.

0

500

1000

1500

2000

0 2 4 6 8 10 12 14 16

As:

mm

2 /m


Model C Band-Internal beam Uniform-Internal beam

Band-Edge beam Uniform-Edge beam

0

300

600

900

1200

1500

1800

0 2 4 6 8 10 12 14

As:

mm

2 /m


Model E Band-Internal beam Uniform-Internal beam


0

300

600

900

1200

1500

1800

0 2 4 6 8 10 12 14

As:

mm

2 /m


Model F Band-Internal beam Uniform-Internal beam


0

500

1000

1500

2000

0 2 4 6 8 10 12 14

As:

mm

2 /m


Model G Band-Internal beam Uniform-Internal beam


Figure 8-30: Comparison of transverse flexural reinforcement area between the uniform and banded rebar designs along the internal and edge wide beams for models A, B, C, D, E, F & G.

0

500

1000

1500

2000

0 2 4 6 8 10 12 14 16

As:

mm

2/m


Model B Band-Internal beam Uniform-Internal beam


0

500

1000

1500

2000

0 2 4 6 8 10 12 14 16

As:

mm

2/m


Model A Band-Internal beam Unifrom-Internal beam


0

300

600

900

1200

1500

1800

0 2 4 6 8 10 12 14

As:

mm

2/m


Model D Band-Internal beam Uniform-Internal beam



276

Figure 8-25 shows the required transverse flexural reinforcement distribution along the internal and

edge wide beams, for the uniform and banded rebar designs, without and with minimum reinforcement.

Minimum reinforcement is seen to govern within the span of all models except across the internal wide

beam span in Model D.

-500

0

500

1000

1500

2000

0 2 4 6 8 10 12 14 16

As:

mm

2/m


Model A Band-Internal beam Unifrom-Internal beam


Minimum steel

0

500

1000

1500

2000

0 2 4 6 8 10 12 14 16

As:

mm

2/m


Model B Band-Internal beam Uniform-Internal beamBand-Edge beam Uniform-Edge beamMinimum steel

0

500

1000

1500

2000

0 2 4 6 8 10 12 14 16

As:

mm

2/m


Model C Band-Internal beam Uniform-Internal beam


Minimum steel

-300

0

300

600

900

1200

1500

1800

0 2 4 6 8 10 12 14

As:

mm

2/m


Model D Band-Internal beam Uniform-Internal beamBand-Edge beam Uniform-Edge beamMinimum steel

0

300

600

900

1200

1500

1800

0 2 4 6 8 10 12 14

As:

mm

2/m


Model E Band-Internal beam Uniform-Internal beamBand-Edge beam Uniform-Edge beamMinimum steel

0

300

600

900

1200

1500

1800

0 2 4 6 8 10 12 14

As:

mm

2/m


Model F Band-Internal beam Uniform-Internal beamBand-Edge beam Uniform-Edge beam

Minimum steel

0

500

1000

1500

2000

0 2 4 6 8 10 12 14

As:

mm

2/m


Model G Band-Internal beam Uniform-Internal beamBand-Edge beam Uniform-Edge beamMinimum steel

Figure 8-31: Transverse uniform and band rebar distribution along the internal and edge beams without considering the minimum steel rule for models A-G.


277

8.2.5 Uniform Steel Distribution versus Proposed Band Steel Distribution

The design of flexural reinforcement for models A-G was carried out similarly to the case study in

Chapter 6. The provided reinforcement areas are equal to the greatest of the calculated and the minimum

reinforcement areas. Material modelling, meshing details, solution method and convergence criteria are

similar that described in Chapter 6, Section 6.3.4.

8.2.5.1 The influence of banding transverse rebar on Load-deflection response

Figure 8-26 compares the load-deflection curves for models A to G with uniform and banded rebar

arrangements. The deflection is normalised by dividing it by the slab thickness, while the load is divided

by the total design ultimate load. It is concluded that the transverse rebar distribution does not affect

deflection.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.1 0.2 0.3 0.4 0.5 0.6

P/Pu

Deflection/slab thickness

Model A

Uniform

Band

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.1 0.2 0.3 0.4 0.5 0.6

P/Pu


Model B

Uniform

Band

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

P/Pu


Model C

Uniform

Band

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

P/Pu


Model D

Uniform

Band

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.1 0.2 0.3 0.4 0.5 0.6

P/Pu


Model E

Uniform

Band

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.1 0.2 0.3 0.4 0.5 0.6

P/Pu


Model F

Uniform

Band

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.1 0.2 0.3 0.4

P/Pu


Model G

Uniform

Band

Figure 8-32: Comparison of Load-deflection curve for models A, B, C, D, E, F & G between the uniform and band rebar distributions.


278

8.2.5.2 Bending Moment

Figure 8-27 & Figure 8-28 compare the design ultimate transverse bending moments from NLFEA for

uniform and banded rebar distributions for models, A – G. Moments are shown along sections (1-1 &

4-4) passing through column faces along the internal and edge wide beams respectively. The elastic

transverse moments are also plotted for comparison. In general, design ultimate moments calculated

with NLFEA are similar for uniformly spaced and banded lateral reinforcement. Thus, it seems that

concentrating the reinforcement around the supports has little influence on the bending moment

distribution at the ULS. In internal beams, the NLFEA moments are larger than the elastic moments,

for both transverse steel distributions, especially near supports. Furthermore, the peak moments

calculated with NLFEA clearly exceed the flexural capacity of the section calculated with section

analysis neglecting CMA. However, in edge beams, the elastic moments are greater than the NLFEA

moments for both rebar designs.

It was shown in Chapter 6 that the development of CMA around columns supporting the internal wide

beam also led to NLFEA moments exceeding the flexural capacity calculated with section analysis

neglecting CMA. In order to investigate this, the in-plane forces along the column face sections in the

internal and edge wide beams (sections 1-1 & 4-4) are plotted in Figure 8-29. It is apparent that

significant compressive in-plane forces develop in the slab to either side of columns with equilibrating

tensile membrane forces developing over the central region of the span. This is consistent with the

behaviour seen earlier in Chapter 6. It is also noted that the in-plane forces around internal columns

supporting the internal wide beam are significantly larger than those at the edge beam. This is reasonable

since the edge beam, unlike the internal wide beam, has a discontinuous side which prevents

development of significant compressive forces. Thus, it can be concluded that CMA influences the

flexural behaviour of the internal wide beam along the longitudinal section at the column faces.


279

Figure 8-33: Comparison between elastic, uniform and band moment distribution at design ultimate load at the column face section (1-1) along the internal wide beam for models A, B, C, D, E, F & G.

0

100

200

300

400

500

0 2 4 6 8 10 12 14 16

Mom

ent:

kN.m

/m


Model AVu=2927.1 kN

Column Face-ElasticColumn Face-UniformColumn Face-Band

0

100

200

300

400

500

0 2 4 6 8 10 12 14 16

Mom

ent:

kN.m

/m


Model BVu=3407.9 kN

Column Face-Elatsic

Column Face-Uniform

Column Face-Band

0

100

200

300

400

500

600

700

0 2 4 6 8 10 12 14 16

Mom

ent:

kN.m

/m


Model CVu=3919.6 kN

Column Face-Elastic

Column Face-Uniform

Column Face-Band

0

50

100

150

200

250

300

350

0 2 4 6 8 10 12 14

Mom

ent:

kN.m

/m


Model DVu=2161.6 kN

Column Face-Elastic

Column Face-Uniform

Column Face-Band

0

50

100

150

200

250

300

350

0 2 4 6 8 10 12 14

Mom

ent:

kN.m

/m


Model EVu=2532.8 kN


0

50

100

150

200

250

300

350

0 2 4 6 8 10 12 14

Mom

ent:

kN.m

/m


Model FVu=2508.7 kN

Column Face-Elastic

Column Face-Uniform

Column Face-Band

0

50

100

150

200

250

300

0 2 4 6 8 10 12 14

Mom

ent:

kN.m

/m


Model GVu=2537.4 kN



280

Figure 8-34: Comparison between elastic, uniform and band moment distribution at design ultimate load at the column face section (4-4) along the edge wide beam for models A, B, C, D, E, F & G.

0

100

200

300

400

0 2 4 6 8 10 12 14 16

Mom

ent:

kN.m

/m


Model AVu=2927.1 kNColumn Face-Elastic

Column Face-Uniform

Column Face-Band

0

100

200

300

400

500

600

0 2 4 6 8 10 12 14 16

Mom

ent:

kN.m

/m


Model BVu=3407.9 kN

Column Face-Elatsic

Column Face-Uniform

Column Face-Band

0

100

200

300

400

500

600

0 2 4 6 8 10 12 14 16

Mom

ent:

kN.m

/m


Model CVu=3919.6 kN

Column Face-Elastic

Column Face-Uniform

Column Face-Band

0

50

100

150

200

250

300

0 2 4 6 8 10 12 14

Mom

ent:

kN.m

/m


Model DVu=2161.6 kN

Column Face-Elastic

Column Face-Uniform

Column Face-Band

0

50

100

150

200

250

300

350

0 2 4 6 8 10 12 14

Mom

ent:

kN.m

/m


Model EVu=2532.8 kN

Column Face-Elastic

Column Face-Uniform

Column Face-Band

0

50

100

150

200

250

300

350

0 2 4 6 8 10 12 14

Mom

ent:

kN.m

/m


Model FVu=2508.7 kN


0

50

100

150

200

250

300

0 2 4 6 8 10 12 14

Mom

ent:

kN.m

/m


Model GVu=2537.4 kN



281

-2000

-1500

-1000

-500

0

500

1000

1500

0 2 4 6 8 10 12 14 16

In-p

lane

for

ce: N

/mm

Distance along the beam: m

Model A

Internal beam-Uniform Internal beam-BandEdge beam-Uniform Edge beam-Band

Compression (-ve)

Tension (+ve)

Column

-2000

-1500

-1000

-500

0

500

1000

1500

0 2 4 6 8 10 12 14 16

In-p

lane

for

ce: N

/mm


Model B


Compression (-ve)

Tension (+ve)

Column

-2500

-2000

-1500

-1000

-500

0

500

1000

1500

0 2 4 6 8 10 12 14 16

In-p

lane

for

ce: N

/mm


Model C


Compression (-ve)

Tension (+ve)

Column

-2000

-1500

-1000

-500

0

500

1000

1500

0 2 4 6 8 10 12 14

In-p

lane

for

ce: N

/mm


Model D


Compression (-ve)

Tension (+ve)

Column

-2000

-1500

-1000

-500

0

500

1000

1500

0 2 4 6 8 10 12 14

In-p

lane

for

ce: N

/mm


Model E


Compression (-ve)

Tension (+ve)

Column

-1500

-1000

-500

0

500

1000

1500

0 2 4 6 8 10 12 14

In-p

lane

for

ce: N

/mm


Model F


Compression (-ve)

Tension (+ve)

Column

-2000

-1500

-1000

-500

0

500

1000

1500

0 2 4 6 8 10 12 14

In-p

lane

for

ce: N

/mm


Model G


Compression (-ve)

Tension (+ve)

Column

Figure 8-35: The in-plane forces at design ultimate loads at sections (1-1 & 4-4) along the internal and edge wide beams for the uniform and band rebar designs for models A, B, C, D, E, F & G.


282

8.2.5.3 Steel Strains

An important output of the parametric study is the steel strain at the service load. For this study, the

service load is taken as the sum of the unfactored dead and imposed loads. Figure 8-30 compares steel

strains from the NLFEA in the transverse reinforcement along the internal wide beam for the uniform

and banded reinforcement distributions. Strains are shown for the internal beam along sections 1-1 &

2-2 passing through the column and beam faces respectively. Figure 8-31 shows strains in the transverse

reinforcement of the edge along sections 3-3 & 4-4.

Figure 8-30 and Figure 8-31 show that banding the transverse reinforcement as proposed significantly

reduces the steel strains around columns in all models. Moreover, away from supports, the stains in the

transverse reinforcement across the span of wide beam are very small. For sections along the interface

between the internal wide beam and slab the steel strains are fairly uniform with relatively small values

for both steel distributions. In the case of edge beams, the strains are hardly developed along the beam

face.


283

0.E+00

1.E-03

2.E-03

3.E-03

4.E-03

0 2 4 6 8 10 12 14 16

Stee

l stra

in


Model BColumn Face-Uniform Beam/slab Face-UniformColumn Face-Band Beam/Slab Face-Band

0.E+00

1.E-03

2.E-03

3.E-03

4.E-03

0 2 4 6 8 10 12 14 16

Stee

l stra

in


Model CColumn Face-Uniform Beam/slab Face-UniformColumn Face-Band Beam/Slab Face-Band

0.E+00

1.E-03

2.E-03

3.E-03

4.E-03

0 2 4 6 8 10 12 14

Stee

l stra

in


Model DColumn Face-Uniform Beam/slab Face-UniformColumn Face-Band Beam/Slab Face-Band

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-03

0 2 4 6 8 10 12 14

Stee

l stra

in


Model EColumn Face-Uniform Beam/slab Face-UniformColumn Face-Band Beam/Slab Face-Band

0.E+00

1.E-03

2.E-03

3.E-03

4.E-03

0 2 4 6 8 10 12 14

Stee

l stra

in


Model FColumn Face-Uniform Beam/slab Face-UniformColumn Face-Band Beam/Slab Face-Band

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-03

0 2 4 6 8 10 12 14

Stee

l stra

in


Model GColumn Face-Uniform Beam/slab Face-UniformColumn Face-Band Beam/Slab Face-Band

Figure 8-36: Comparison of steel stains at sections (1-1 & 2-2) passing through the column and beam faces along internal wide beam for uniform and band rebar designs for models A, B, C, D, E, F & G) at SLS load (1.0 D.L+1.0 I.L)

0.E+00

1.E-03

2.E-03

3.E-03

4.E-03

0 2 4 6 8 10 12 14 16

Stee

l stra

in


Model AColumn Face-Uniform Beam/slab Face-UniformColumn Face-Band Beam/Slab Face-Band


284

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-03

3.5E-03

4.0E-03

0 2 4 6 8 10 12 14 16

Stee

l stra

in


Model A Column Face-UniformBeam/slab Face-UniformColumn Face-BandBeam/Slab Face-Band

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-03

3.5E-03

0 2 4 6 8 10 12 14 16

Stee

l stra

in


Model B Column Face-UniformBeam/slab Face-UniformColumn Face-BandBeam/Slab Face-Band

0.0E+00

1.0E-03

2.0E-03

3.0E-03

4.0E-03

5.0E-03

0 2 4 6 8 10 12 14 16

Stee

l stra

in


Model CColumn Face-UniformBeam/slab Face-UniformColumn Face-BandBeam/Slab Face-Band

0.0E+00

1.0E-03

2.0E-03

3.0E-03

4.0E-03

5.0E-03

6.0E-03

0 2 4 6 8 10 12 14

Stee

l stra

in


Model D Column Face-UniformBeam/slab Face-UniformColumn Face-BandBeam/Slab Face-Band

0.0E+00

1.0E-03

2.0E-03

3.0E-03

4.0E-03

0 2 4 6 8 10 12 14

Stee

l str

ain


Model E Column Face-Uniform

Beam/slab Face-Uniform

Column Face-Band

Beam/Slab Face-Band

0.0E+00

1.0E-03

2.0E-03

3.0E-03

0 2 4 6 8 10 12 14

Stee

l stra

in


Model F Column Face-UniformBeam/slab Face-UniformColumn Face-BandBeam/Slab Face-Band

0.0E+00

1.0E-03

2.0E-03

3.0E-03

0 2 4 6 8 10 12 14

Stee

l stra

in


Model G Column Face-Uniform

Beam/slab Face-Uniform

Column Face-Band

Beam/Slab Face-Band

Figure 8-37: Comparison of steel stains at sections (4-4 & 3-3) passing through the column and beam faces along edge wide beam for uniform and band rebar designs for models A, B, C, D, E, F & G) at SLS load (1.0 D.L+1.0 I.L).


285

8.2.5.4 Crack Width Control

Crack widths have been computed in accordance with EC2 as outlined in Chapter 6, Section 6.3.4.5.

The steel strains used in the calculations are average steel strains extracted from the NLFEA. The

service load at which the crack widths are calculated corresponds to the quasi-permanent load, which

is defined as 1.0 D.L. plus 0.3 I.L. It has been shown from Chapter 6 that the NLFEA strains are

significantly less than EC2 strains at low moments. Therefore, the crack width calculations are

essentially relative comparisons. The calculated crack spacing depends on the assumed bar diameter.

Table 8-3 and Table 8-4 present the rebar diameter and spacing and the maximum crack spacing

calculated according to EC2 using the NLFEA strains at quasi-permanent load for internal and edge

beams respectively for both steel designs.

Table 8-3: Maximum crack spacing and rebar spacing for internal wide beam for uniform and band steel designs for models A, B, C, D, E, F & G

*Sr,max is limited by the upper bound 1.3(h-x) because rebar spacing exceeds 5 (c + ϕ/2).

Model Panel Rebar spacing: mm Rebar diameter: mm Maximum crack spacing,

Sr,max: mm Uniform

Column

band Span band

Uniform

Column band

Span band Uniform

Column band

Span band

Model A Internal

150 100

150 10 12

10 484.7 339

494.6 Edge 100 12 352

Model B Internal

125 100

150 10 12

10 425.0 322.6

463.1 Edge 100 12 334.8

Model C Internal

125 150

125 10 16

10 406.2 355.2

458.2 Edge 160 16 369.4

Model D Internal

100 100

275 12 12

10 441.5 321.3

272.7* Edge 100 12 333.3

Model E Internal

125 175

200 10 16

10 332.5* 287.8*

332.5* Edge 200 16 291.2*

Model F Internal

150 100

200 10 12

10 470.0 350.2

342.5* Edge 100 12 364.3

Model G Internal

150 100

175 10 12

10 469.8 350.1

352.4* Edge 100 12 364.4


286

Table 8-4: Maximum crack spacing and rebar spacing for edge wide beam for uniform and band steel designs for models A, B, C, D, E, F & G

*Sr,max is limited by the upper bound 1.3(h-x) because rebar spacing exceeds 5 (c + ϕ/2).

Figure 8-32 shows the crack width at section (1-1) passing through the column faces along the internal

wide beam for the uniform and band rebar designs at the quasi-permanent load. Similarly, Figure 8-33

presents the same comparison along section (4-4) for the edge wide beam. It can be seen that a

considerable reduction in crack width is obtained with the proposed reinforcement distribution in all

models. In some models, banding the reinforcement reduces crack widths by a multiple of 0.5 or less.

Moreover, it is evident that the minimum reinforcement provides adequate crack width control along

the wide beam away from supports.

Model Panel Rebar spacing: mm Rebar diameter: mm Maximum crack spacing, Sr,max: mm

Uniform Column band

Span band

Uniform Column band

Span band

Uniform Column band

Span band

Model A Internal

150 175

150 10 16

10 494.6 349.4*

494.6 Edge 125 16 340.2

Model B Internal

150 175

150 10 16

10 463.1 368.6*

463.1 Edge 125 16 335.0

Model C Internal

125 150

125 10 16

10 458.2 364.6

458.2 Edge 100 16 320.9

Model D Internal

200 100

275 12 12

10 266.8* 322.2

272.7* Edge 100 12 287.4

Model E Internal

200 175

200 10 16

10 332.5* 288.8*

332.5* Edge 125 16 344.5

Model F Internal

200 100

200 10 12

10 342.5* 341.3

342.5* Edge 150 16 352.3

Model G Internal

175 100

175 10 12

10 352.4* 346.7

352.4* Edge 150 16 358.0


287

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 2 4 6 8 10 12 14 16

Cra

ck w

idth

: mm

Distane along the wide beam span: m

Model B Uniform

Band

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10 12 14

Cra

ck w

idth

: mm


Model D Uniform

Band

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10 12 14

Cra

ck w

idth

: mm


Model F Uniform

Band

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0 2 4 6 8 10 12 14

Cra

ck w

idth

: mm


Model G Uniform

Band

Figure 8-38: Comparison of crack width along the internal wide beam at section 1-1 for transverse uniform and proposed band distributions at quasi-permanent load for models A, B, C, D, E, F & G.

0.00.10.20.30.40.50.60.70.80.91.0

0 2 4 6 8 10 12 14

Cra

ck w

idth

: mm


Model E UniformBand

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 2 4 6 8 10 12 14 16

Cra

ck w

idth

: mm


Model C Uniform

Band

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 2 4 6 8 10 12 14 16

Cra

ck w

idth

: mm


Model A Uniform

Band


288

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 10 12 14

Cra

ck w

idth

: mm


Model D Uniform

Band

Figure 8-39: Comparison of crack width along the edge wide beam at section 4-4 for transverse uniform and proposed band distributions at quasi-permanent load for models A, B, C, D, E, F & G.

0.00.10.20.30.40.50.60.70.80.91.0

0 2 4 6 8 10 12 14 16

Cra

ck w

idth

: mm


Model A Uniform

Band

0.00.10.20.30.40.50.60.70.80.91.0

0 2 4 6 8 10 12 14 16

Cra

ck w

idth

: mm


Model B Uniform

Band

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 2 4 6 8 10 12 14 16

Cra

ck w

idth

: mm


Model C Uniform

Band

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 2 4 6 8 10 12 14

Cra

ck w

idth

: mm


Model E Uniform

Band

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10 12 14

Cra

ck w

idth

: mm


Model F Uniform

Band

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0 2 4 6 8 10 12 14

Cra

ck w

idth

: mm


Model G Uniform

Band


289

8.3 Parametric Studies on Punching shear strength of internal wide

beam connection without shear reinforcement

8.3.1 General

It is commonly thought that the flexural reinforcement ratio influences punching shear resistance

(73,151-153). EC2 considers the effect of the flexural reinforcement in both directions through the

expression:

= × (8.11)

where , are the flexural ratio in longitudinal and transverse directions of the wide beam,

respectively.

Punching shear resistance according to fib MC2010, which adopts the CSCT (ψmax), is inversely

proportional to the product of rotation and the flexural effective depth, (ψd). This term depends on the

flexural reinforcement in the direction considered.

In wide beam slabs the flexural reinforcement ratios over supports can vary significantly in the

orthogonal directions. Neither, EC2 nor the fib MC2010 specify minimum limits for

in their

punching shear design provisions. A parametric study has been carried out to study the effect of varying

on punching resistance with the adopted values of / corresponding to the transverse uniform

and banded rebar arrangements considered for models A – G.

Also investigated are the influences on punching resistance of i) moment transfer from the wide beam

into the internal column and ii) non-uniform load introduction from the wide beam into the column.

These are investigated with MC2010 LoA IV and EC2.

8.3.2 Analytical Model

The model adopted for the parametric studies is similar to the solid assemblage used for the NLFEA of

internal connections in Chapter 7, Section 7.2.1.2 in terms of material modelling, mesh types, boundary

conditions, solution method and convergence criteria. The element size is modified for each model to

obtain sufficient number of elements through the wide beam depth (at least 8 elements around the

column). Figure 8-34 shows the geometry of the sub-assemblages used in the parametric study. The

length of each sub-assemblage is defined as the distance between the points of contraflexure in the wide

beam to either side of the internal column as found by elastic FEA. The dimensions of all the

assemblages are presented in Table 8-5. The loads applied to each assemblage were extracted from the


290

elastic FEA of the full-scale shell model. The material properties used throughout these analyses are 35

MPa for concrete strength, 500 MPa for characteristic steel yield strength and 34.41 GPa & 200 GPa

for elastic moduli of concrete and steel respectively. Table 8-6 presents the loading applied to each

model. The beam shear and punching shear reinforcement was designed according to EC2 assuming a

design shear force of 1.15 VEd and partial safety factors of γc=1.5 and γs=1.15. Table 8-7 shows the beam

flexural and shear reinforcement details for each model. Note that the bar diameters used in this analysis

do not necessarily follow the standard sizes in order to eliminate any effect of surplus steel area on

punching shear strength. The shear reinforcement distribution in plan around the internal column is

illustrated in Figure 8-35 for each model. The internal column in all models is reinforced typically with

8 H22 bars and H10 links @ 150 mm c/c. Detailing of typical column reinforcement is shown in Chapter

6, Figure 6-7.

Table 8-5: Geometry details for the assemblies used for the parametric study

Table 8-6: Loads extracted from the elastic FEA subjected to the assemblies used in parametric study

Model span L :(m)

L1 L2 :m Width, W: m

Depth, D: m

Effective depth (x axis-

y-axis): m

Main flexural rebar ratio: %

Transverse flexural rebar ratio: %

ρy / ρx

Uniform Band Uniform Band

A 4.2 2.0-2.2 2.4 0.38 0.335 – 0.349 0.80 0.16 0.35 0.21 0.44 B 4.2 2.0-2.2 2.4 0.40 0.355 - 0.369 0.79 0.19 0.37 0.24 0.44 C 4.3 2.0-2.3 2.4 0.42 0.375 – 0.389 0.87 0.23 0.39 0.27 0.42 D 3.2 1.6-1.6 2.4 0.26 0.215 – 0.229 1.07 0.35 0.60 0.32 0.56 E 3.7 1.8-1.9 2.4 0.32 0.275 – 0.289 0.90 0.23 0.45 0.26 0.50 F 4.1 2.0-2.1 2.0 0.33 0.285 – 0.299 1.00 0.20 0.39 0.20 0.39 G 3.7 1.8-1.9 1.8 0.34 0.295 – 0.309 1.06 0.19 0.38 0.18 0.36

Assembly Edge A: kN/m

Edge B: kN/m

Edge C: kN/m

Edge D: kN/m

w: kN/m2

A 131.85 51.53 168.42 62.72 22.35 B 142.9 71.2 186.4 85.2 23.03 C 142.25 84.54 217.39 102.21 23.70 D 89.18 59.28 102.26 69.42 18.30 E 109.49 55.12 133.51 65.91 20.33 F 114.59 64.45 142.33 74.97 20.70 G 131.90 68.32 161.60 79.65 21.00


291

Table 8-7: Reinforcement details for the assemblies used for parametric study

Model Top Rein. Bottom Rein.

Transverse rebar Uniform design

Transverse rebar Column band

Transverse rebar Span band

Beam stirrups

A 13T25 13T20 H12@200 mm c/c H18@200 mm c/c H11.4@200 mm c/c H10@250 mm c/c B 12T26.8 12T20 H13.3@200 mm c/c H19.5@200 mm c/c H12@200 mm c/c H10@250 mm c/c C 13T27.8 13T20.8 H12@125 mm c/c H15.6@125 mm c/c H12@200 mm c/c H8@125 mm c/c D 13T23.3 13T18 H10@100 mm c/c H13.3@100 mm c/c H8@200 mm c/c H8@150 mm c/c E 13T24 13T18.5 H13@200 mm c/c H18.7@200 mm c/c H10@200 mm c/c H8@200 mm c/c F 10T25.7 10T20 H12.4@200 mm c/c H18@200 mm c/c H10.2@200 mm c/c H8@200 mm c/c G 9T28 9T21.3 H12.4@200 mm c/c H18@200 mm c/c H10.2@200 mm c/c H8@200 mm c/c

Column size 0.4m x 0.4m

D W

Figure 8-40: Sample of the solid assembly used in the parametric study showing its geometry.

Edge A

Edge B Edge C

Edge D


292

Figure 8-42: Plan view showing the shear reinforcement distribution around the internal column in models A-G. (All dimensions are in mm)


293

8.3.3 Results of Load-deflection curves and Discussions

NLFEA was carried out with ATENA for models A – G. Deflections were extracted at points a and b

shown in Figure 8-36. The load-deflection curves for models A - G are shown in Figure 8-37 for

transverse uniform and banded rebar arrangements. The NLFEA failure loads of each model are listed

in Table 8-8 for uniform and banded transverse steel arrangements. Also shown are design punching

shear forces calculated with EC2.

Figure 8-43: Sketch showing the points at which the deflection is extracted.

Face D- End bay

1-1 Longitudinal axis, 2-2 Lateral axis

Face C

1 1

2

2

Face B-Internal bay

Face A

End

beam

span

Internal beam span

b

a


294

0

200

400

600

800

1000

1200

0 20 40 60 80 100 120

Load

: kN

Deflection: mm

Model A

Uniform-Long. direction

Uniform-Lateral direction

Band-Long. direction

Band-Lateral direction

0

200

400

600

800

1000

1200

1400

1600

0 20 40 60 80 100 120

Load

: kN

Deflection: mm

Model C





0

200

400

600

800

1000

0 8 16 24 32 40 48 56 64 72 80

Load

: kN

Deflection: mm

Model D





0

200

400

600

800

1000

1200

0 10 20 30 40 50 60 70 80 90 100

Load

: kN

Deflection: mm

Model E





0

200

400

600

800

1000

0 10 20 30 40 50 60 70 80 90

Load

: kN

Deflection: mm

Model F





0

200

400

600

800

1000

1200

0 10 20 30 40 50 60 70 80

Load

: kN

Deflection: mm

Model G

Uniform-Long. directionUniform-Lateral directionBand-Long. directionBand-Lateral direction

Figure 8-44: Load-deflection curves for uniform and banded rebar distributions for models A,B,C, D,E, F & G. (deflections are given at points a and b).

0

200

400

600

800

1000

1200

1400

1600

0 20 40 60 80 100 120 140

Load

: kN

Deflection: mm

Model B






295

Table 8-8: NLFEA’s ultimate loads for models A to G with transverse uniform and band rebar designs. Model VEd, EC2: kN Vuniform, FE: kN Vband, FE: kN Increase in V due to

banding transverse rebar % A 1355.9 1001 1116.0 11.5 B 1643.3 1201 1338.0 11.4 C 1826.3 1463 1510.0 3.2 D 981.5 689.0 857.0 24.4 E 1164.1 801.4 982.7 22.6 F 1204.3 828.9 838.0 1.1 G 1161.4 938.6 1004.0 7.0

8.3.4 Reinforcement Strains and Crack Patterns

Figure 8-38 compares steel strains at failure for models A-G with uniform and banded rebar

arrangements. For uniformly distributed transverse reinforcement, the scale monitor indicates that all

transverse bars yielded significantly except in Model F where only transverse steel over the column

yielded excessively. Banding the transverse rebar increased the flexural capacity in the transverse

direction. Consequently, the flexural capacity for each is model is governed by its flexural strength in

the longitudinal direction. This is evident since the top longitudinal reinforcement yielded significantly

in all models (εlong. ≈ 10 εtrans).

Figure 8-39 compares the crack patterns at failure for uniform and band rebar distributions for models

A-G. Crack width of 0.3 mm or more are shown in plan. It can be seen that for models with uniform

rebar design, yield lines formed along the length of the sub-assemblies with cracks concentrated near

the column (face D). This holds true but to lesser extent for models F & G, in which cracks distributed

more uniformly around the column. On the other hand, in models with banded rebar yield lines formed

near the column face (face C) across the beam width. Banding the transverse rebar led to formation of

more scattered cracks with smaller widths.


296

Figure 8-45: Plan view showing steel strains at ultimate load for models A, B, C, D, E, F & G with transverse reinforcement placed (I) uniformly (II) in bands.

Model A (VFE-uniform=1001 kN) Model A (VFE-Band=1116 kN)

Model B (VFE-uniform=1205 kN) Model B (VFE-Band=1338 kN)

Model C (VFE-uniform=1463 kN) Model C (VFE-Band=1510 kN)

Model D (VFE-uniform=689 kN) Model D (VFE-Band=857 kN)

Model E (VFE-uniform=801.4 kN) Model E (VFE-uniform=982.7 kN)

Model F (VFE-uniform=828.9 kN) Model F (VFE-uniform=838 kN)

Model G (VFE-uniform=938.6 kN) Model G (VFE-Band=1004 kN)


297

Figure 8-47: Plan view of cracking patterns at ultimate load (w>0.3 mm for models A, B, C, D, E, F & G with transverse reinforcement placed uniformly and in bands.

Model A-Uniform Model A-Band

Model B-Uniform Model B-Band

Model C-Uniform

Model D-Uniform

Model E-Uniform

Model F-Uniform

Model G-Uniform

Model C-Band

Model D-Band

Model E-Band

Model F-Band

Model G-Band


298


The effect of load introduction on punching strength was investigated. Subsequently, ke was evaluated

using MC2010’s eccentricity-based formula. All models were subjected to symmetrical line loads along

opposite edges. The ratio of the total line load in the long and short edges was kept the same as the ratio

of the loading from the elastic FEA (i.e., asymmetrical loading). The results are compared with those

subjected to loads given by the elastic FEA (asymmetrical loading). Figure 8-40 shows the NLFEA

load-deflection curves for models A-G. Deflections were monitored at points a & b along the

longitudinal and transverse axes of the column respectively as shown in Figure 8-36. It can be concluded

that the influence of load introduction on the strength and stiffness for these models is little.

0

200

400

600

800

1000

1200

0 10 20 30 40 50 60

Load

: kN

Deflection: mm

Model A

Asymmetrical-point aAsymmetrical-point bSymmetrical-point aSymmetrical-point b

0200400600800

100012001400

0 20 40 60 80 100 120

Load

: kN

Deflection: mm

Model B

Asymmetrical-point aAsymmetrical-point bsymmetrical-point aSymmetrical-point b

0200400600800

1000120014001600

0 10 20 30 40 50 60 70 80

Load

: kN

Deflection: mm

Model C


0

200

400

600

800

0 8 16 24 32 40 48 56 64 72 80

Load

: kN

Deflection: mm

Model D


0

200

400

600

800

1000

0 10 20 30 40 50 60 70 80 90 100

Load

: kN

Deflection: mm

Model E

Asymmetrical-point aAsymmetrical-point bSymmetrical-ponit aSymmetrical-point b

0

200

400

600

800

1000

0 10 20 30 40 50 60 70 80 90

Load

: kN

Deflection: mm

Model F

Asymmetrical-point aAsymmetrical-point bSymmetrical-pointaSymmetrical-point b

0

200

400

600

800

1000

1200

0 10 20 30 40 50 60 70 80

Load

: kN

Deflection: mm

Model G


Figure 8-48: Comparison of load-deflection responses of models A-G subjected to symmetrical and asymmetrical loadings.


299

8.3.6 Predictions of punching shear resistance according to ATENA analyses,

MC2010 with rotations according to LoA IV and EC2.

Figure 8-41 shows load–rotation relationships at points a and b (see Figure 8-36) for all models with

both banded and uniform transverse steel arrangements. The method of calculating rotations is

described in Chapter 7 (Section 7.2.2.3 & Figure 7-11). In addition, the failure criterion according to

fib MC2010 is plotted without and with partial safety factors (γc=1.5 & γs=1.15). The coefficient of

eccentricity, ke is taken as 0.9. The EC2 punching shear resistances for both transverse reinforcement

arrangements are also shown. The EC2 punching shear strengths shown in Figure 8-41 are divided by

1.15 for direct comparison with those of MC2010.

It is clear that maximum rotations about the transverse axis are similar for uniform and banded

transverse reinforcement for models C, F & G throughout loading stages. For other models with banded

rebar design, however, the rotations become less as they reach the ultimate load. Banding the transverse

rebar results in a stiffer response about the longitudinal axis of the beam in all models except for model

G, with the least beam width (1.8 m).

Table 8-9 presents the punching shear resistances predicted for models A-G using MC2010 LoA IV,

EC2 and ATENA. The MC2010 rotations were derived from displacements calculated in the ATENA

analyses. It can be concluded that the predictions of fib MC2010 with rotations according to LoA IV

for punching shear resistance agree well with the ATENA results for both transverse rebar distributions

for all models. EC2, however, overestimates the punching shear strength for uniform and banded

reinforcement designs in all models with average VEC2/VATENA of 1.49 and 1.44 for uniform and band

rebar arrangements respectively as presented in Table 8-10. The plastic plateau of the load deflection

and rotation responses implies that EC2 predicts flexural failure for all models. It can be observed form

Table 8-10 that fib MC2010 yields close results to ATENA analyses with different values for ρy/ρx.

Similarly, the accuracy of EC2 predictions is not influenced by varying ρy/ρx.


300

0

400

800

1200

1600

2000

2400

2800

3200

0.0E+00 1.0E-02 2.0E-02 3.0E-02 4.0E-02 5.0E-02 6.0E-02

Load

: kN

Rotation: Rad

Model AUniform-Trans. directionUniform-Long. directionBand-Trans. directionBand-Long directionMC2010 failure criterion (γc=γs=1.0)MC2010 failure criterion (γc=1.5,γs=1.15)Vcs,EC2 (γc=γs=1.0)-UniformVcs,EC2 (γc=1.5,γs=1.15)-UniformVcs,EC2 (γc=γs=1.0)-BandVcs,EC2 (γc=1.5,γs=1.15)-Band

0400800

1200160020002400280032003600

0.0E+00 5.0E-03 1.0E-02 1.5E-02 2.0E-02 2.5E-02 3.0E-02

Load

: kN

Rotation: Rad

Model CUniform-Trans. directionUniform-Long. directionBand-Trans. directionBand-Long. directionMC2010 failure criterion (γc=γs=1.0)MC2010 failure criterion (γc=1.5,γs=1.15)Vcs,EC2 (γc=γs=1.0)-UniformVcs,EC2 (γc=1.5,γs=1.15)-UniformVcs,EC2 (γc=γs=1.0)-BandVcs,EC2 (γc=1.5,γs=1.15)-Band

0200400600800

10001200140016001800

0.0E+00 1.0E-02 2.0E-02 3.0E-02 4.0E-02 5.0E-02 6.0E-02

Load

: kN

Rotation: Rad

Model DUniform-Trans. directionUniform-Long. directionBand-Trans. directionBand-Long. directionMC2010 failure criterion (γc=γs=1.0)MC2010 failure criterion (γc=1.5,γs=1.15)Vcs,EC2 (γc=γs=1.0)-UniformVcs,EC2 (γc=1.5,γs=1.15)-UniformVcs,EC2 (γc=γs=1.0)-BandVcs,EC2 (γc=1.5,γs=1.15)-Band

0200400600800

10001200140016001800200022002400

0.0E+00 1.0E-02 2.0E-02 3.0E-02 4.0E-02 5.0E-02 6.0E-02

Load

: kN

Rotation: Rad

Model EUniform-Trans. directionUniform-Long. directionBand-Trans. directionBand-Long. directionMC2010 failure criterion (γc=γs=1.0)MC2010 failure criterion (γc=1.5,γs=1.15)Vcs,EC2 (γc=γs=1.0)-UniformVcs,EC2 (γc=1.5,γs=1.15)-UniformVcs,EC2 (γc=γs=1.0)-BandVcs,EC2 (γc=1.5,γs=1.15)-Band

0

300

600

900

1200

1500

1800

2100

2400

0.0E+00 1.0E-02 2.0E-02 3.0E-02 4.0E-02 5.0E-02 6.0E-02

Load

: kN

Rotation: Rad

Model FUniform-Trans. directionUniform-Long. directionBand-Trans. directionBand-Long. directionMC2010 failure criterion (γc=γs=1.0)MC2010 failure criterion (γc=1.5,γs=1.15)Vcs,EC2 (γc=γs=1.0)-UniformVcs,EC2 (γc=1.5,γs=1.15)-UniformVcs,EC2 (γc=γs=1.0)-BandVcs,EC2 (γc=1.5,γs=1.15)-Band

0200400600800

100012001400160018002000220024002600

0.0E+00 1.0E-02 2.0E-02 3.0E-02 4.0E-02 5.0E-02

Load

: kN

Rotation: Rad

Model GUniform-Trans. directionUniform-Long. directionBand-Trans. directionBand-Long. directionMC2010 failure criterion (γc=γs=1.0)MC2010 failure criterion (γc=1.5,γs=1.15)Vcs,EC2 (γc=γs=1.0)-UniformVcs,EC2 (γc=1.5,γs=1.15)-UniformVcs,EC2 (γc=γs=1.0)-BandVcs,EC2 (γc=1.5,γs=1.15)-Band

0

400

800

1200

1600

2000

2400

2800

3200

0.0E+00 1.0E-02 2.0E-02 3.0E-02 4.0E-02 5.0E-02 6.0E-02

Load

: kN

Rotation: Rad

Model BUniform-Trans. directionUniform-Long. directionBand-Trans. directionBand-Long. directionMC2010 failure criterion (γc=γs=1.0)MC2010 failure criterion (γc=1.5,γs=1.15)Vcs,EC2 (γc=γs=1.0)-UniformVcs,EC2 (γc=1.5,γs=1.15)-UniformVcs,EC2 (γc=γs=1.0)-BandVcs,EC2 (γc=1.5,γs=1.15)-Band

Figure 8-49: Load-Rotation curves of the wide beam’s internal connection models A – G for uniform and band steel designs.


301

Table 8-9: Punching resistances for models A-G using EC2, MC2010 with rotations according to LoA IV and ATENA analysis for lateral uniform and banded steel distribution (ke=0.9).

Model EC2*: kN MC2010-LoA IV: kN ATENA’s results: kN

(γc=1.5, γs=1.15) (γc= γs=1.0) (γc=1.5, γs=1.15) (γc= γs=1.0) Uniform

Band Uniform Band Uniform Band Uniform Band Uniform Band

A 1213.9 1301.8 1539.0 1670.8 922 1002 990 1113 1001.0 1116.0 B 1340.6 1421.8 1708.0 1829.7 1020 1082 1145 1308 1201.0 1338.0 C 1917.7 1989.2 2318.4 2425.7 1150 1163 1377 1441 1463.0 1510.0 D 1005.5 1044.3 1211.6 1269.8 686 787 688 846 689.0 857.0 E 882.2 943.1 1145.5 1236.8 721 757 790 915 801.4 982.7 F 922.6 986.4 1198.3 1294.0 811 824 828 838 828.9 838.0 G 968.6 1035.9 1259.4 1360.3 920 935 983 1002 983.6 1004.0

*The punching strength calculated with EC2 is divided by 1.15 to present it in a similar form to that of MC2010.

Table 8-10: The effect of ρy/ρx on the variability of punching strength prediction with MC2010, LoA IV and EC2

8.3.7 Investigation on the coefficient of eccentricity

Chapter 2, Section 2.4.3.4 describes the methods specified in fib MC2010 for calculating ke. These

methods have been applied in Chapter 7, Section 7.2.2.5. The coefficient of eccentricity ke is calculated

herein for models A-G using the eccentricity-based formula (Equations 2.49-2.51 & 7.6). Similarly,

EC2 provides eccentricity-based formula to evaluate the coefficient of eccentricity, β. According to

EC2, for internal rectangular columns where the loading is eccentric to one axis β is evaluated using

expressions (2.41) & (7.7). Additionally, for internal rectangular columns with eccentricities in both

directions β can be calculated with Equation (7.8) as shown in Chapter 7, Section 7.2.2.5.

Table 8-11 presents the values of ke and β calculated according to the eccentricity based formulae in fib

MC2010 and EC2. Mx refers to moments requiring reinforcement placed parallel to the x-axis (parallel

to direction of wide beam span) and My refers to moments requiring reinforcement placed parallel to

the y-axis (normal to direction of wide beam span). The results of ke are in good agreement with the

approximate value of 0.9 specified for internal columns of braced frames by fib MC2010. It can be seen

that the approximate value of β for internal columns, 1.15, is an upper bound for all models. Thus,

higher punching shear strengths would be predicted. This, however, does not seem sensible since EC2

predicts flexural failure in all models.

Model ρt / ρl ρt / ρl VEC2/VATENA VMC2010/VATENA Uniform Band Uniform Band Uniform Band

A 0.21 0.44 1.54 1.50 0.99 1.0 B 0.24 0.44 1.42 1.37 0.95 0.98 C 0.27 0.42 1.58 1.61 0.94 0.95 D 0.32 0.56 1.76 1.48 1.00 0.99 E 0.26 0.50 1.43 1.26 0.99 0.93 F 0.20 0.39 1.45 1.54 1.00 1.00 G 0.18 0.36 1.28 1.35 1.00 1.00

Mean 1.49 1.44 0.98 0.98 COV 0.10 0.08 0.03 0.03


302

Table 8-11: Calculation of the coefficient of eccentricities, ke and β using eccentricity-based formulae in fib MC2010 and EC2

Model Column reaction:

kN

Unbalanced Moment: kN.m

ke=1/(1+eu/bu) β= 1+K . β=1+1.8 +

x-axis y-axis x-axis y-axis Both x-axis y-axis Both A 1425.8 41.7 56.4 0.97 0.95 0.94 1.03 1.04 1.05 B 1679.3 51.3 70.6 0.96 0.95 0.94 1.03 1.04 1.05 C 1910.8 101.1 91.2 0.94 0.95 0.92 1.05 1.04 1.07 D 1011.8 50.2 38.9 0.93 0.95 0.92 1.07 1.05 1.09 E 1211.5 46.1 47.9 0.95 0.95 0.93 1.04 1.05 1.06 F 1255.2 51.0 43.1 0.95 0.96 0.94 1.05 1.04 1.06 G 1215.7 38.1 37.7 0.96 0.96 0.95 1.03 1.03 1.05

8.4 Conclusions

This chapter presents parametric studies conducted to determine the key parameters influencing the

geometry of the proposed design transverse bending moment distribution over wide beams. Based on

the parametric studies, a simple method is proposed for banding the transverse reinforcement along the

length of the wide beams. In the proposed method, transverse hogging reinforcement over wide beams

is placed in two bands referred to as column and span bands. It is shown that there is a linear relationship

between the width of the column band and the wide beam span.

Subsequently, reinforcement was designed for the models used in the parametric study and the models

were analysed with both uniform and banded transverse reinforcement using NLFEA with shell

elements. The results show that the proposed procedure for banding transverse reinforcement

effectively reduces steel strains and, hence, crack widths.

NLFEA parametric studies were conducted with solid element sub-assemblies to investigate the

influence on punching resistance of varying the ratio between the transverse flexural steel ratio and the

longitudinal steel ratio,

on the punching shear strength. The punching resistance was determined

with MC2010 LoA IV and EC2 in addition to NLFEA. It was concluded that varying

does not affect

the relative punching shear resistances given by ATENA, EC2 and MC2010 LoA IV.

The coefficient of eccentricity ke was also calculated using the eccentricity based equation given in fib

MC2010. It was found that ke values calculated for these models agree well with the fib MC2010’s

value (0.9) specified for internal columns of braced frames. In addition, the results of β for all models

calculated using the eccentricity-based formula in EC2 are less than default value of 1.15 for internal

columns.

It should be noted that these conclusions are restricted to wide beam slabs with dimensions within the

range used in this study.

Conclusions Chapter 9

303

Chapter 9

Conclusions

9.1 Introduction

The main aim of the thesis is to develop a rational design method for wide beam floors that satisfies the

serviceability limit state of cracking as well as the ultimate limit state. Emphasis is placed on

determining a suitable distribution of reinforcement to resist hogging bending moments transverse to

the direction of span of the wide beams. These bending moments are referred to as transverse throughout

the thesis. The objectives are achieved by studying a representative nine-panel wide beam floor

subjected to uniformly distributed loading. Adopted structural arrangements are based on guidance

given in ECFE (1). The EC2 span-effective depth method is used to determine the required member

depths for deflection control. Uniform and banded transverse hogging reinforcement arrangements are

investigated numerically with NLFEA. Uniform reinforcement spacing is investigated because it is

recommended by TCC (1,5) while the banded arrangement is based on the elastic bending moment

distribution albeit with some lateral moment redistribution. Performances of the two reinforcement

arrangements are compared in terms of deflection, reinforcement strain, crack width at quasi-permanent

load and failure load. The influence of CMA on flexural capacity is also investigated. The results of

these studies are presented in Chapter 6.

The influence of transverse reinforcement arrangement on the punching shear resistance of wide beam

slabs is investigated at internal and edge columns using NLFEA with ATENA as well as the critical

shear crack theory of Muttoni (70) as implemented in fib MC2010. The ATENA analyses are carried


304

out on sub-assemblies consisting of the wide beam, between adjacent lines of contraflexure, and the

upper and lower lifts of the supporting column. The punching shear resistances obtained with ATENA

are compared with resistances given by fib MC2010, with rotations according to LoAs II & IV, as well

as EC2. Also investigated is the influence of banding transverse reinforcement on the shear force

distribution along the control perimeter for punching at internal and edge columns. The influence of

eccentric shear on punching resistance is also considered. fib MC2010 accounts for eccentric shear by

reducing the basic control perimeter by a multiple ke, while EC2 increases the design shear force by a

multiple β. Recommendations are made for the choice of these parameters at internal and edge columns

of wide beam slabs. The beneficial effect of flexural continuity is investigated as well. Finally, design

recommendations for wide beam slabs are made in Chapter 8.

9.2 Summary of the thesis

The main findings of the research are outlined below.

9.2.1 Literature review

BS8110, EC2 and ACI318 give no guidance on designing wide beam flooring systems. Little research

has been carried out into the design of wide beams slabs for either flexure or punching shear. Elastic

FEA shows that the transverse distribution of support moments in wide beam floors is not uniform as

commonly assumed in design. Instead, transverse moments peak sharply over columns. As a

consequence, serviceability conditions may not be met, due to excessive cracking around columns, if

transverse reinforcement is uniformly distributed as recommended by TCC (5). Beam shear strength is

influenced by the support width, transverse stirrup distribution and stirrup configuration. However,

there is no consensus on how to account for the effect of these effects. In the current research, the design

rules of EC2 for beam shear are assumed adequate for wide beam shear.

9.2.2 Methodology

The thesis is based solely on Finite Element Analysis (FEA) which is carried out using DIANA v9.6

and ATENA v5.1.1. Descriptions of the main features of the NLFEA and the adopted constitutive

models for concrete and steel reinforcement are given in Chapter 4, which also outlines the modelling

procedures used in the thesis.

Chapter 5 describes the validation of the NLFEA procedures used in the research. With regards to

DIANA, the validation focusses on simulating the short-term flexural behaviour of slabs in terms of

load-deflection response, steel strains and flexural capacity. In DIANA, slab elements are modelled

with 8 node curved shell elements and supporting columns and walls using 20 node brick elements. A

total strain fixed crack model is used in all the DIANA analyses. For concrete, Thorenfeldt’s softening


305

model (122), is adopted for concrete in compression. Tay’s linear softening relationship (6) is used for

concrete in tension. The input parameters of Tay’s model are the effective concrete tensile strength and

the strain at zero tensile stress which are respectively taken as 0.5fct and 0.5εy, where εy is the

reinforcement yield strain. Use of the aggregate interlock based variable shear retention factor and the

Quasi-Newton solver are shown to give good predictions of the considered test results. The adopted

modelling procedure is shown to predict the experimentally observed enhancement in stiffness and

strength of slabs due to CMA. The analysis is also shown to capture the influence of rebar distribution

on the deflection of slabs tested by Regan (78). Making use of shell elements used for modelling slabs,

rotations are extracted to predict the punching shear resistance with MC2010 LoA IV.

The validation studies focus on the use of ATENA to simulate shear failures of beams and slabs without

and with shear reinforcement. The focus is on the load-deflection response, failure load, failure mode

and crack patterns. In ATENA, linear cubic brick elements are adopted. To accurately capture the

punching behaviour, at least five elements are needed through the thickness. The analyses with ATENA

use Thorenfeldt’s softening model for compression and Hordijk’s softening curve for tension. Best

results in terms of failure load, load-deflection response and crack patterns were obtained with the

rotated crack model and standard arc-length solver.

The steel reinforcement is modelled as embedded elements. Embedded reinforcement can be in the form

of either discrete bars or grid. Both forms of reinforcement were utilized in DIANA, while

reinforcement was modelled as discrete bars in ATENA. Moreover, Von Mises plasticity material

without hardening was assigned for all types of embedded reinforcement in both types of analyses.

9.2.3 Flexural design of wide beam slabs

Chapter 6 presents a case study in which a typical wide beam floor is designed and assessed using FEA.

Elastic FEA is carried out to determine the transverse bending moment distribution along critical

sections passing through column faces, beam faces and slab at midspan. It is concluded that the

transverse moment distribution at slab midspan section and at the beam/slab face are fairly uniform.

However, transverse bending moment distribution varies significantly along a section passing through

the column face and tends to peak sharply over columns. Subsequently, the wide beam floor was

analysed nonlinearly, using DIANA, with two transverse rebar designs. First, the top transverse rebar

was spaced equally over the wide beams as recommended by TCC. Second, the rebar was placed in

three bands along the wide beam with band widths and design moments based on elastic analysis. Each

reinforcement band corresponds to the average transverse elastic bending moment across its width,

which was determined from the elastic FEA. Comparisons are made between the uniform and banded

reinforcement arrangements in terms of bending moments in the slab, wide beams and columns, load-

deflection curves, steel strains and crack widths. The following conclusions are drawn:


306

Varying the transverse steel distribution, across the wide beams, does not significantly

influence the flexural load capacity of the slab. The effect of transverse reinforcement

arrangement on the bending moment distribution in the slab, wide beams and columns is also

small.

Banding the transverse reinforcement as proposed reduces significantly reinforcement strains.

Consequently, crack widths reduce significantly adjacent to columns at both SLS and ULS.

This is significant since NLFEA suggests that uniformly distributing transverse hogging

reinforcement in wide beams can result in excessive steel strains and hence crack widths over

supports at the SLS.

The NLFEA analyses show that the CMA developed locally around columns.

Reinforcement strains calculated with NLFEA and EC2 method agree reasonably well at

relatively high reinforcement strains but the NLFEA strains are significantly less than EC2

strains at low strains due to differences in the adopted tension stiffening models.

9.2.4 Punching shear in wide beams

The influence of wide beam transverse steel distribution on punching resistance and shear stress

distribution at internal and edge columns is investigated. The suitability of the EC2 and fib MC2010

design methods for punching shear are evaluated for wide beam slabs in which both loading and flexural

reinforcement are greatest in the direction of wide beam span. Investigations are also carried out to

determine the best choice of the coefficient ke which allows for eccentric shear in MC2010. Similar

investigations are performed to determine the best choice of shear enhancement multiplier β to be used

in EC2. After that, comparisons are made between slab rotations obtained with shell element sub-

assemblies and full scale-models. The slab rotations are used to determine the influence of flexural

continuity on punching shear strength according to MC 2010 LoA IV. The following conclusions are

drawn:

Banding the transverse steel has little influence on shear force distributions along the fib

MC2010 control perimeter for punching shear at internal columns. The effect is more

pronounced at edge columns where banding the reinforcement results in a more uniform

distribution of shear force.

Banding transverse steel improves the punching shear resistance due to the increase of average

flexural reinforcement ratio, but the enhancement in stiffness is marginal.

Based on comparisons with ATENA analyses, EC2 overestimates the punching strength of

wide beam internal and edge column connections for both transverse steel arrangements. This

could be because the punching resistance of the analysed slabs is close to the flexural capacity

as will typically be the case if surplus flexural reinforcement is not provided.


307

The transverse steel distribution does not greatly affect beam rotations according to NLFEA.

As a result, MC2010 LoA IV gives virtually the same estimates of punching resistance for

uniform and banded transverse steel designs at both internal and edge column connections.

The punching shear strength provided by shear reinforcement is not fully utilized due to the

non-symmetrical distribution of shear force around the critical sections. This suggests that it

could be more efficient to concentrate punching shear reinforcement where shear stresses are

greatest. Nevertheless, this would require experimental investigation and could be at the

expense of construction speed.

It appears that the flexural continuity and CMA enhance the punching resistance of wide beam

slabs. This was established by comparing punching resistances obtained with MC2010 LoA IV

for full-scale models and sub-assemblies. However, further experimental research is required

to confirm the predicted increase in strength.

9.2.5 Parametric studies

Parametric studies were carried out to determine the band widths for transverse reinforcement and the

magnitudes of bending moments acting over them. The key parameters which influence the width of

the elastic transverse bending moment distribution were determined. The main conclusions may be

summarized as follows:

The width of a panel; either first internal or end panel, is divided into two bands namely; column

band and slab band.

The width of the column band is linearly proportional to the wide beam span. A linear

relationship is found between the transverse bending moment corresponding to the column band

and the ratio between the wide beam span and slab span.

A simple method is proposed for banding the transverse reinforcement along the length of wide

beams.

NLFEA shows that the proposed procedure for banding transverse reinforcement effectively

reduces steel strains and, hence, crack widths.

Due to lack of experimental data for validation, the comparison of crack widths is essentially

relative.

Parametric studies are conducted with solid element sub-assemblies to investigate the influence of

varying the ratio between the transverse flexural steel ratio and the longitudinal steel ratio,

on the

punching shear strength. Additionally, recommendations are made for the choice of ke and β which are

used to account for eccentric punching shear in MC2010 and EC2 respectively. Following this,

punching resistances are determined with MC2010 LoA IV, EC2 and ATENA. The main conclusions

are:


308

Varying

does not affect the relative punching shear resistances given by ATENA, EC2 and

MC2010 LoA IV.

Calculated ke values at internal columns agree reasonably well with the approximate value of

ke = 0.9 specified in MC2010 for internal columns of braced frames.

The values of β calculated using the eccentricity-based formula in EC2 are less than the default

value of 1.15 for given in EC2 for internal columns. Lower values than 1.15 would yield higher

estimates for punching shear strength which appears to be overestimated by EC2. However, it

should be noted that the maximum shear force that can be applied to the analysed sub-

assemblies is limited by flexural failure.

9.3 Recommendations for Future Work

The proposed design procedure for banding transverse rebar along wide beams is based solely on

numerical analyses. Therefore, experimental evidence is required to validate the procedure. Full-scale

testing would be very costly due to the large sizes of wide beam floors. Thus, it is recommended to

conduct 1/4 scale test. Alternately, a ½ scale wide beam sub-assemblage bounded by slab centrelines at

either side could be tested. The experimental investigations should include detailed measurement of

crack widths with digital image correlation as well as reinforcement strain. Experimental evidence is

also needed to confirm the findings about suitable values for ke and β for wide beam assemblages at

internal and edge columns.

The research should be extended in the future to study the influence of torsion on the performance of

edge beams as well as the performance of connections between edge beams and edge columns. The key

factors influencing the performance of connections should be investigated more fully. These include

the ratio of column size to beam width and the effect of eccentricity between the column and edge beam.

Additionally, the influence of reinforcement detailing at connections between wide beams and columns

should be investigated experimentally.

References

309

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