EFFECTS OF AXIAL PRESTRESS ON THE PUNCHING BEHAVIOUR OF

PLAIN AND FIBRE REINFORCED CONCRETE SLABS

by

Mohamed El Semelawy

A thesis submitted in conformity with the requirements

for the degree of Masters of Applied Science

Graduate Department of Civil Engineering

University of Toronto

© Copyright by Mohamed El Semelawy (2007)

ii

M.A.Sc. Thesis Department of Civil Engineering University of Toronto

EFFECTS OF AXIAL PRESTRESS ON THE PUNCHING BEHAVIOUR OF

PLAIN AND FIBRE REINFORCED CONCRETE SLABS

By Mohamed El Semelawy

ABSTRACT

A slab that is axially prestressed develops compressive membrane action that tends to

increase significantly its strength. The behaviour of axially prestressed slabs was

experimentally examined. Five unreinforced two-way slab specimens were built and

tested to failure under monotonically increasing central load. Three of the specimens

were constructed using plain concrete, while the other two were constructed using fibre

reinforced concrete (FRC). Axial stress was applied using an external post-tensioning

system; the system consisted of Dywidag bars and side steel beams. Parameters, such as

lateral stress level, fibre inclusion, and varying the axial stress in one direction were

investigated.

The level of axial stress was observed to affect all aspects of the behaviour, including

cracking, deflection, stiffness, and failure mode. The higher the stress level, the higher

the ultimate strength and stiffness, and the lower the ductility. A stress level as low as 2.0

MPa was able to provide lateral restraint necessary to prevent premature flexural failure

of Specimen P-3, which failed in a combined flexural-punching mode at a significantly

higher load than expected by flexural failure. Adding steel fibres in an amount equal to

1% by volume fraction resulted in a ductile punching shear failure and improved post-

cracking behaviour and residual load-carrying capacity after reaching maximum load.

iii

ACKNOWLEDGEMENTS

I would like to express my deep gratitude to Professors F.J.Vecchio and P.Gauvreau for

their guidance and support through the course of this project.

The completion of the experimental work would not have been possible without the

assistance of the Structural laboratory staff and fellow graduate students. I would like to

thank everybody who took part in this project either by giving me a hand or a piece of

advice.

iv

TABLE OF CONTENTS

CHAPTER 1: INTRODUCTION 1

1.1 Background 1

1.2 Research Significance 2

1.3 Objective 3

CHAPTER 2: LITERATURE REVIEW 4

2.1 Introduction 4

2.2 Punching Shear Failure 4

2.3 Punching Shear of Restrained Slabs 5

2.4 Testing of Laterally Restrained Slabs 7 2.4.1 Taylor and Hayes 1965: Some Tests on the Effect of Edge Restraint on Punching Shear in

Reinforced Concrete Slabs 7 2.4.2 Aoki and Seki 1974: Shearing Strength and Cracking in Two-Way Slabs Subjected to

Concentrated Load 11

2.5 Modelling of the Behaviour of Restrained Slabs 13 2.5.1. Hewitt and Batchelor 1975: Punching Shear Strength of Restrained Slabs 14

2.6 Example of Laterally Restrained Slabs: Steel-free Slab-on-Girder Bridges 16 2.6.1 Mufti; Jaeger; Bakht; and Wagner 1993: Experimental Investigation of Fibre-Reinforced

Concrete Deck Slabs without Internal Steel Reinforcement 18 2.6.2 Hassan, Kawakami, Niitani, Yoshioka 2002: An Experimental Investigation of Steel-Free

Deck Slabs 21 2.6.3 He 1992: Punching Behaviour of Composite Bridge Decks with Transverse Prestressing 24 2.6.4 Modelling of Slab-on-Girder Bridges 25

2.7 Punching Strength of Fibre Reinforced Concrete (FRC) 27 2.7.1 Introduction 27 2.7.2 Swamy and Ali (1982): Punching Shear Behaviour of Reinforce Slab-Column Connections

Made with Steel Fibre Concrete 28

v

CHAPTER 3: EXPERIMENTAL PROGRAM 33

3.1 Introduction 33

3.2 Test Specimens 33 3.2.1 Geometry and Dimensions 33 3.2.2 Post-Tensioning 36

3.3 Material Properties 41 3.3.1 Concrete 41 3.3.2 Post-Tensioning Steel 46 3.3.3 Discrete Steel Fibres 47

3.4 Test Set-up 48 3.4.1 Loading System 48 3.4.2 Test Instrumentation 51

CHAPTER 4: EXPERIMENTAL RESULTS AND OBSERVATIONS 57

4.1 Introduction 57

4.2 Test Data 57

4.3 Mechanical Behaviour of Slab Specimens 75 4.3.1 Specimens P-1 & P-2 75 4.3.2 Specimen P-3 76 4.3.3 Specimens F-1 & F-2 76

CHAPTER 5: DISCUSSION OF EXPERIMENTAL RESULTS 78

5.1 Behaviour of the Specimens 78

5.2 Failure Modes 80

5.3 Failure loads of Specimen P-1 and Specimen P-2 81

5.4 Parameters Analysis 82 5.4.1 Effect of Lateral Restraint Provided by Post-Tensioning Stresses 82 5.4.2 Effect of Fibres Reinforcement 83 5.4.3 Effect of Varying Lateral Stresses in One Direction 85

5.5 Variation of the top and bottom bars forces during testing 86

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5.6 Performance of testing set-up and collected data 88 5.6.1 Application of pure axial load 88 5.6.2 Centre Displacement 89 5.6.3 Bar strains and Bar forces 89

CHAPTER 6: THEORETICAL PREDICTION 90

6.1 Modified Compression Field Theory prediction 90 6.1.1 Model Description 90 6.1.2 Analytical Models 93 6.1.3 Comparison between Experimental and VecTor3 Model Results 94

6.2 CSA prediction 98

6.3 Prediction using Hewitt and Batchelor (1975) model 99

CHAPTER 7: CONCLUSIONS AND RECOMMENDATIONS 101

7.1 Conclusions 101

7.2 Recommendations 103

CHAPTER 8: REFERENCES 104

APPENDIX A 109

APPENDIX B 139

APPENDIX C 148

APPENDIX D 162

vii

LIST OF FIGURES

FIGURE 2.1: A PHOTOGRAPH OF A SECTION OF A SLAB FAILED IN PUNCHING SHEAR 5 FIGURE 2.2: IDEALIZED RESTRAINED SLABS FORCES AND STRESS DISTRIBUTION 7 FIGURE 2.3 TEST SET-UP (TAYLOR AND HAYES) 8 FIGURE 2.4: LOAD VERSUS CENTRE DEFLECTION RESPONSE (TAYLOR AND HAYES) 10 FIGURE 2.5 CONE OF CONCRETE PUNCHED OUT FROM ONE OF THE UNREINFORCED

SLABS OF SERIES1 (TAYLOR AND HAYES) 10 FIGURE 2.6: TYPICAL GEOMETRY OF SPECIMENS 12 FIGURE 2.7 FACTOR OF SFETY VERSUS f’

c / ρ.fy 13 FIGURE 2.8: MECHANICAL MODEL OF SLAB AT PUNCHING FAILURE 15 FIGURE 2.9 ISOMETRIC VIEW OF SLAB-ON-GIRDER BRIDGE SYSTEM 17 FIGURE 2.10: DETAILS OF THE FIRST MODEL (MUFTI ET AL. 1993) 19 FIGURE 2.11: TEST LOCATIONS ON THE DECK SLAB OF THE THIRD MODEL 20 FIGURE 2.12: LOAD DEFLECTION CURVES (MUFTI ET AL. 1993) 21 FIGURE 2.13: UPPER AND LOWER PLAN OF THE TESTED SPECIMENS 22 FIGURE 2.14:CROSS-SECTION A-A IN THE PRESTRESSED STEEL-FREE SLAB

(HASSAN ET AL. 2002) 23 FIGURE 2.15: LOAD-DEFLECTION CURVE FOR (A) NORMAL-STRENGTH CONCRETE SLAB,

AND (B) HIGH-STRENGTH CONCRETE SLABS (HASSAN ET AL. 2002) 23 FIGURE 2.16: PUNCHING STRENGTH VERSUS. FAILURE LOAD (HE 1992) 25 FIGURE 2.17: RIGID BODY ROTATION OF WEDGES (NEWHOOK 1997) 25 FIGURE 2.18: LOAD VERSUS. DEFLECTION CURVES (NEWHOOK 1997) 26 FIGURE 2.19: ARRANGEMENT OF STEEL REINFORCING BARS FOR SLAB-COLUMN

CONNECTIONS 29 FIGURE 2.20 : TYPICAL LOAD DEFLECTION CHARACTERISTICS OF SERIES 1 AT CENTRE

SPAN OF SLAB COLUMN CONNECTIONS (SWAMY AND ALI 1982) 30 FIGURE 2.21: TYPICAL LOAD-TENSION STEEL STRAIN BEHAVIOUR AT CENTRE OF SPAN

(SERIES 1, 2) 30 FIGURE 3.1: TYPICAL SPECIMEN DIMENSIONS 35 FIGURE 3.2: POST-TENSIONING SYSTEM 38 FIGURE 3.3: BLOW-UP OF SECTION 1-1 39 FIGURE 3.4: SECTION 2-2 39 FIGURE 3.5: AXIAL STRESS DISTRIBUTION IN SPECIMENS 40 FIGURE 3.6: DYWIDAG ELECTRIC POWERED HYDRAULIC JACK 41

viii

FIGURE 3.7: FORMWORK CLAMPED TO THE VIBRATING TABLE 43 FIGURE 3.8: CONCRETE CYLINDER COMPRESSIVE STRENGTH 45 FIGURE 3.9: DIMENSION OF FRACTURE ENERGY TEST SPECIMENS 46 FIGURE 3.10: OBSERVED LOAD-STRAIN RELATIONSHIP OF DYWIDAG THREADBAR® 47 FIGURE 3.11: FIBRE DIMENSION 48 FIGURE 3.12: BALDWIN MACHINE 49 FIGURE 3.13: PODIUM DETAILS 50 FIGURE 3.14: IMAGES OF THE PODIUM 51 FIGURE 3.15: LAYOUT OF HORIZONTAL AND VERTICAL LVDTS 53 FIGURE 3.16: LVDT SET-UP 54 FIGURE 3.17: DYWIDAG BAR STRAIN GAUGES 55 FIGURE 3.18: LOAD CELLS CONFIGURATION 56 FIGURE 4.1: SPECIMEN LOAD-DEFORMATION RESPONSES 60 FIGURE 4.2: REVISED SPECIMEN LOAD-DEFORMATION RESPONSES 61 FIGURE 4.3: LAYOUT OF ADDITIONAL LVDTS FOR SPECIMEN F-1 62 FIGURE 4.4: LOAD-DEFORMATION RESPONSES FROM ADDITIONAL LVDTS OF SPECIMEN

F-1 62 FIGURE 4.5: SKETCHES OF CRACKING PATTERN ON TENSION (BOTTOM) SURFACE 64 FIGURE 4.6: SKETCHES OF CRACKING PATTERN ON COMPRESSION (TOP) SURFACE 65 FIGURE 4.7: SELECTED PHOTOS OF SPECIMENS P-1 66 FIGURE 4.8: SELECTED PHOTOS OF SPECIMENS P-2 66 FIGURE 4.9: SELECTED PHOTOS OF SPECIMENS P-3 67 FIGURE 4.10: SELECTED PHOTOS OF SPECIMENS F-1 67 FIGURE 4.11: SELECTED PHOTOS OF SPECIMENS F-2 68 FIGURE 4.12: AXIAL LOAD VERSUS APPLIED VERTICAL LOAD (SPECIMEN P-1) 69 FIGURE 4.13: AXIAL LOAD VERSUS APPLIED VERTICAL LOAD (SPECIMEN P-2) 69 FIGURE 4.14: AXIAL LOAD VERSUS APPLIED VERTICAL LOAD (SPECIMEN P-3) 70 FIGURE 4.15: AXIAL LOAD VERSUS APPLIED VERTICAL LOAD (SPECIMEN F-1) 70 FIGURE 4.16: AXIAL LOAD VERSUS APPLIED VERTICAL LOAD (SPECIMEN F-2) 71 FIGURE 4.17: AVERAGE SIDE ROTATION VERSUS APPLIED VERTICAL LOAD

(SPECIMEN P-1) 71 FIGURE 4.18: AVERAGE SIDE ROTATION VERSUS APPLIED VERTICAL LOAD

(SPECIMEN P-2) 72 FIGURE 4.19: AVERAGE SIDE ROTATION VERSUS APPLIED VERTICAL LOAD

(SPECIMEN P-3) 72

ix

FIGURE 4.20: AVERAGE SIDE ROTATION VERSUS APPLIED VERTICAL LOAD

(SPECIMEN F-1) 73 FIGURE 4.21: AVERAGE SIDE ROTATION VERSUS APPLIED VERTICAL LOAD

(SPECIMEN F-2) 73 FIGURE 4.22: ESTIMATED END MOMENT VERSUS APPLIED VERTICAL LOAD

(SPECIMEN P-1) 74 FIGURE 4.23: ESTIMATED END MOMENT VERSUS APPLIED VERTICAL LOAD

(SPECIMEN P-2) 74 FIGURE 5.1: GENERAL LOAD-DEFORMATION RESPONSE OF SPECIMENS 79 FIGURE 5.2: GENERAL AXIAL LOAD VERSUS APPLIED VERTICAL LOAD RESPONSE 79 FIGURE 5.3: EFFECT OF POST-TENSIONING STRESSES ON (A) NORMALIZED PUNCHING

LOAD (B) STRAIN ENERGY ABSORBED 83 FIGURE 5.4: EFFECT OF FIBRE REINFORCEMENT ON (A) NORMALIZED PUNCHING LOAD

(B) STRAIN ENERGY ABSORBED 84 FIGURE 5.5: EFFECT OF VARYING POST-TENSIONING STRESS IN ONE DIRECTION ON (A)

NORMALIZED PUNCHING LOAD (B) STRAIN ENERGY ABSORBED 86 FIGURE 5.6: EFFECT OF SIDE ROTATION ON BAR FORCES 87 FIGURE 5.7: VARIATION OF TOP AND BOTTOM BARS FORCES FOR SPECIMEN P-1 AND

SPECIMEN P-2 87 FIGURE 6.1: FINITE ELEMENT MESH 92 FIGURE 6.2: SUPPORT CONDITIONS AND LOAD APPLICATION POINTS 92 FIGURE 6.3: APPLICATION OF END MOMENT TO SPECIMEN P-1 93 FIGURE 6.4: EXPERIMENTAL VERSUS ANALYTICAL LOAD-DEFORMATION RESPONSE

OF SPECIMEN P-1 96 FIGURE 6.5: EXPERIMENTAL VERSUS ANALYTICAL LOAD-DEFORMATION RESPONSE

OF SPECIMEN P-3 96 FIGURE 6.6: EXPERIMENTAL VERSUS ANALYTICAL LOAD-DEFORMATION RESPONSE

OF SPECIMEN F-1 97 FIGURE 6.7: EXPERIMENTAL VERSUS ANALYTICAL LOAD-DEFORMATION RESPONSE

OF SPECIMEN F-2 97 FIGURE 6.8: Z-DISPLACEMENT OF SPECIMEN P-3 AT INTERMEDIATE LOAD STAGE 98

x

LIST OF TABLES

TABLE 2.1: DETAILS OF SLABS AND TEST RESULTS OF SELECTED SPECIMENS. 9 TABLE 2.2: DETAILS OF SLABS AND TEST RESULTS OF SELECTED SPECIMENS 13 TABLE 2.3: DETAILS OF THE TESTED SPECIMENS 22 TABLE 2.4: TEST RESULTS 24 TABLE 2.5: REINFORCEMENT AND STEEL FIBRE DISTRIBUTION OF SERIES 1, 3, AND 4 29 TABLE 2.6: DIAGONAL TENSION CRACKING LOAD, RELATIVE DUCTILITY, AND ENERGY

ABSORPTION OF SLABS 31 TABLE 2.7: SERVICE LOADS BASED ON DEFORMATION CRITERIA 32 TABLE 3.1: NOMINAL STRESS IN CONCRETE AND TOTAL FORCE IN BARS 37 TABLE 3.2: CONCRETE MIX DESIGN (PLAIN CONCRETE) 42 TABLE 3.3: CONCRETE MIX DESIGN (FRC) 42 TABLE 3.4: AVERAGE CONCRETE PROPERTIES 44 TABLE 3.5: CONCRETE CYLINDER COMPRESSIVE STRENGTH 45 TABLE 3.6: POST-TENSIONING STEEL PROPERTIES 46 TABLE 3.7: FIBRE PROPERTIES 47 TABLE 4.1: SLAB SPECIMEN VARIABLES AND TEST RESULTS 63 TABLE 4.2: STRAIN ENERGY ABSORBED 63 TABLE 5.1: EFFECT OF LATERAL RESTRAINT PROVIDED BY POST-TENSIONING STRESSES 83 TABLE 5.2: EFFECT OF FIBRE REINFORCEMENT 84 TABLE 5.3: EFFECT OF LATERAL STRESS IN ONE DIRECTION 85 TABLE 6.1: ANALYTICAL MODELS USED IN THE FE ANALYSIS 93 TABLE 6.2: SUMMARY OF VECTOR3 ANALYSIS 95 TABLE 6.3: CSA PREDICTION OF THE FAILURE LOAD OF SPECIMENS 99 TABLE 6.4: MAXIMUM LOAD PREDICTION USING HEWITT AND BATCHELOR MODEL 100

xi

NOTATION

d effective shear depth of the specimen (= 0.72 * t)

Es elastic modulus of steel

F1average average axial force per unit length of the specimen (Faverage /1.5)

Faverage average axial force applied to the specimen

faverage average axial stresses applied to the specimen

FE-W total bar forces in the East-West direction

fn nominal stress applied to the specimen in one direction ( Ptotal / 1500*t )

fn(N-S) nominal axial stress in concrete in North-South Direction (FN-S/ 1500*t)

fn(E-W) nominal axial stress in concrete in East-West Direction (FE-W/ 1500*t)

FN-S total bar forces in the North-South direction

Pmax applied ultimate load during testing

Ptotal total force applied to the specimen by post-tensioning bars in one direction

t thickness of the specimen

maxPU the area under the revised load-deformation curve up to the ultimate load

80U the area under the revised load-deformation curve up to 80-percent of the ultimate load beyond the peak.

maxUΔ the area under the revised load-deformation curve up to the failure

1

Chapter 1: Introduction

1.1 Background

A conventional slab is designed to fail in flexural. However, if the same slab is laterally

restrained, it will behave differently under loading. A laterally restrained slab fails at

significantly higher loads, and the failure mode may change from flexure to punching

shear. The different behaviour of restrained slabs is attributed to two main mechanisms

that do not develop in unrestrained slabs: first, arching action (also called compressive

membrane action) which results from the restrained lateral expansion of the slab due to

the presence of a stiff boundary element i.e. barrier walls; second, fixed boundary action

which results from the force developed in the reinforcement (Hewitt and Batchelor 1975).

The punching shear resistance of slabs can be enhanced by adding fibre to the concrete

mix (i.e. fibre reinforced concrete FRC). Discrete fibres, when incorporated into the

concrete matrix, improves the overall mechanical behaviour of concrete e.g. toughness,

ductility, energy absorption, cracking resistance, and tensile strength (Shaaban and

Gesund 1994). The fibres in FRC can be made from various materials such as steel,

carbon, aluminum, glass, and plastic. Because these materials have different mechanical

properties, FRC will behave in a slightly different way depending on the type of fibre

chosen. Several studies (e.g. Harajli et al. 1995) have concluded that steel fibres are the

2

most effective in improving mechanical behaviour of concrete, particularly tensile

strength.

1.2 Research Significance

The aforementioned mechanisms, arching action and fixed boundary action, were

recognized by some jurisdictions all over the world. For example, the Canadian Highway

Bridge Design Code states that an FRC deck slab need not to be designed for positive

moments if it satisfies certain conditions concerning longitudinal beams spacing, shear

connector, minimum cross sectional area and spacing of transverse steel straps, and slab

thickness (clause 16.7). These conditions provide the degree of lateral restraint necessary

to transform the expected flexural failure mode into punching.

On the contrary, the design of prestressed concrete bridge deck slabs in transverse

direction is based on flexural failure. No code recognizes this possibility of transforming

the flexural failure to punching failure as a result of the inherited lateral restraint provided

by prestressing. Recognizing the dramatic enhancement of resistance due to lateral

restraint in designing prestressed concrete bridges will allow for the use of thinner slabs,

thereby reducing the self-weight of the deck. Moreover, it is possible to eliminate the

internal flexural rebars that do not contribute to the overall flexural rigidity. Although the

elimination of internal rebars may cause cracking control problems, these problems can

be avoided if FRC is used. In addition, fibres enhance the overall behaviour of concrete.

3

While most of the research done on characterizing the behavior and the modelling of

restrained deck slabs concentrates on its application to FRC composite slab-on-girder

bridges, less attention has been given to its application to the prestressed concrete

bridges.

1.3 Objective

The objective of this research is to study the effect of prestressing and fibre inclusion on

the behaviour and ultimate strength of non-reinforced slab specimens. It is of particular

interest to determine the minimum level of prestressing necessary to provide lateral

restraint for deck slabs, transforming the expected flexural failure into punching failure. It

is obvious that a non-prestressed slab (and non-reinforced) would fail in flexural mode at

the cracking moment.

An experimental program was designed to investigate the above objectives. Five

externally prestressed square slab panels were tested. Two of the slabs contained steel

fibre reinforcements. All specimens were simply supported on all four sides and loaded

centrally through a square loading plate. External post-tensioning was applied by means

of Dywidag bars and side steel beams; post-tensioning force was transmitted to the

specimen at four contact areas on each side. The choice of external unbonded prestressing

over internal prestressing was done to ensure the total elimination of any dowel effect.

4

Chapter 2: Literature Review

2.1 Introduction

The literature review presented in this chapter discusses the punching shear strength of

slabs, with special attention given to punching of restrained slabs. Both experimental and

analytical investigations are presented. An example of the restrained slabs used in the

steel-free slab-on-girder bridge system is discussed. The review also discusses the effect

of steel fibres incorporated into the concrete mix on the punching shear resistance.

2.2 Punching Shear Failure

The ultimate strength of a reinforced concrete slab under a concentrated load is often

determined by the punching shear failure load rather then the flexural load. Over the past

decades many research projects have been dedicated to study the punching shear

phenomenon, both experimentally and analytically.

Moe (1961) defined the punching shear failure as the failure of a concrete slab directly

under a concentrated load that occurs when a concrete plug is pushed out of the slab. The

pushed-out plug takes the shape of a cone with a top area at least equal to the loading

area.

5

The sequence of events occurring in a concrete slab under a monotonically increasing

central load can be summarized as: (1) formation of a roughly circular crack around the

column periphery on the tension side of the slab; (2) formation of new lateral and

diagonal flexural cracks; (3) initiation of shear cracks at mid-depth of the slab at

approximately 50-70 % of the ultimate load. As load increases, the shear crack

propagates towards the top and bottom surfaces of the slab. The crack propagation is

prevented by dowel action of the flexural reinforcement (if any) and by the compression

zone surrounding the loading plate, termed as the critical zone (Theodorakopoulos and

Swamy 2002). Collapse is attained by failure of the concrete in the critical zone, either by

splitting under principal tensile stress or crushing in the radial or tangential direction

(Shehata and Regan 1989). The punching shear strength is governed by: effective depth

of concrete, column size and shape, flexural reinforcement ratio, concrete compressive

strength, and lateral restraint conditions. A photograph of a section through a slab that

failed in punching is shown in Figure 2.1.

Figure 2.1: A photograph of a section of a slab failed in punching shear (Sissakis 2002)

2.3 Punching Shear of Restrained Slabs

To explain the behaviour of laterally restrained slabs, consider the slab shown in Figure

2.2. The edges of the slab are laterally restrained by a stiff boundary element. As the slab

6

is loaded, a flexural crack is developed on the tension side; the edges rotate and translate

in the plane of the slab. The edge movement will be resisted by the stiff boundary

element, inducing compressive forces in the slab. The induced forces will increase the

punching resistance. Two mechanisms explain the forces developed in the slab, namely

compressive membrane action and fixed boundary action.

Compressive membrane action (Figure 2.2 a) results from the net compressive stresses

that develop in the slab in a way similar to the known arching action in beams. Fixed

boundary action is due to moment restraint with no net in-plane forces at the slab

boundary (Figure 2.2 b). Compressive membrane action develops only in cracked

concrete, while fixed boundary action can develop in both cracked and uncracked

concrete. The progressive stages of behaviour of a restrained slab loaded up to failure can

be summarized as: (1) fixed boundary action; (2) cracking; (3) compressive membrane

action with fixed boundary action if the slab is reinforced; and (4) punching failure

(Hewitt and Batchelor 1975).

7

compressive membrane force

in- plane forces

stress distributions

applied load

(a) Compressive membrane action

fixed boundary moment

(b) Fixed boundary action

Figure 2.2: Idealized restrained slabs forces and stress distribution (Hewitt and Batchelor 1975)

2.4 Testing of Laterally Restrained Slabs

Since the late 1950’s and early 1960’s, many tests have been conducted to study the

behaviour of laterally restrained slabs. Compared to unrestrained slabs, tests on restrained

slabs indicated a considerable increase in the load-carrying capacity. For example, Wood

(1961) observed an increase up to 10.9 times those predicted by the yield-line theory.

2.4.1 Taylor and Hayes 1965: Some Tests on the Effect of Edge Restraint on

Punching Shear in Reinforced Concrete Slabs

Taylor and Hayes carried out an experimental program to identify the effects of lateral

edge restraint on the punching shear strength of slabs. The authors pointed out that code

8

requirement for punching shear needed to be revised to take the restraint effect into

consideration.

A total of 22 square slab panels were tested. The slabs were 889 x 889 x 76 mm, simply

supported along all four sides to give a span of 864 mm, and loaded centrally through a

square loading plate. The dimensions of the loading plate varied from 50 x 50 mm to 150

x 150 mm. The specimens were divided into three series depending on the steel

reinforcement ratio: Series 1- no reinforcement; Series 2- 1.57 %; Series 3- 3.14 %. To

study the effect of lateral restraint, pairs of slabs in Series 2 and Series 3 were tested in

restrained and unrestrained conditions. Series 1, having a very low flexural strength, were

tested only under restrained conditions. The test set-up is illustrated in Figure 2.3; a

heavy steel welded frame surrounded the slabs. The inner dimensions of the frame were

927 x 927 mm, creating a gap of 19 mm between the specimen and the frame. For slabs

tested under restrained conditions, this gap was filled with a fairly stiff mortar. Details of

the tested specimens are shown in Table 2.1.

Figure 2.3 Test Set-up (Taylor and Hayes)

9

Table 2.1: Details of slabs and test results of selected specimens.

Designation Reinforcement ratio [%]

Loading plate size

[mm]

Cube strength[MPa]

Failure load in punching

shear [kN]

Increase in failure load

due to lateral restraint

1R2(a) 50 37 83.4 1R2(b) 50 33 87.3

1R4 100 34 147.2 1R6

0.00

150 27 141.3

2S4 85.9 2R4

100 29 136.8

1.59

2S6 96.6 2R6

1.57 150 23

154.4 1.60

3S4 115.3 3R4

3.14 100 28 132.4

1.15

R laterally restrained; S simply supported (unrestrained) Comparing the load deflection curves of the restrained and unrestrained slabs of Series 2

and Series 3 (Figure 2.4), lateral restraint had little effect on the behaviour in early

loading stages. Crack widths and deflections of the restrained slabs were similar to those

of the unrestrained slabs. However, edge restraint evidently affected subsequent stages.

The laterally restrained specimens showed smaller crack widths and deflections when

compared to unrestrained specimens at the same loads.

For Series 1, fewer cracks developed on the underside of the slab compared to Series 2

and Series 3. Cracks started at approximately 20 kN. Though Series 1 had no flexural

reinforcement, specimens maintained their load carrying capacity far beyond their

cracking loads. As the loads increased, the widths of the cracks increased. Failure

occurred when a concrete plug, shown in Figure 2.5, punched out of the slab in a sudden

explosive manner. The failure loads ranged from 4.0 to 7.0 times the cracking load. As

10

shown in Figure 2.4, the load-deflection curve for Series 1 was linear and did not exhibit

any plateau or warning. The failure loads of the tested specimens are listed in Table 2.1.

Figure 2.4: Load versus centre deflection response (Taylor and Hayes)

Figure 2.5 Cone of concrete punched out from one of the unreinforced slabs of Series 1

(Taylor and Hayes) In general, edge restraint increased the punching shear strength. The ratio of the failure

load of the restrained slabs to unrestrained ones ranged from 1.24 to 1.60 for Series 2,

and from 1.00 to 1.16 for Series 3. The extent of increase in Series 3 was less than in

11

Series 2 as the punching shear strength of the unrestrained slabs in Series 3 was enhanced

by using a higher reinforcement ratio.

2.4.2 Aoki and Seki 1974: Shearing Strength and Cracking in Two-Way Slabs

Subjected to Concentrated Load

Additional tests were carried out in 1974 by Aoki and Seki. The authors studied shear

strength, shear stresses, and failure mechanism in the vicinity of the concentrated load.

Special attention was given to the arching action, the mechanism which explains the

tendency of restrained slabs to fail in punching rather than in flexure. A method to

predict the strength of the slabs taking the arch action into consideration was proposed.

The testing program consisted of 14 square slabs with dimensions ranging from 1.2 x 1.2

to 1.6 x 1.6 m. Lateral restraint was provided by concrete beams that were monolithically

cast with the slabs (Figure 2.6). Eight specimens were directly placed on the floor, while

the other six specimens were supported at four corners. Specimens were loaded at the

centre via a 190 mm steel disc in a force-controlled mode. The dimensions and material

properties of the tested panels are presented in Table 2.2.

12

t

b

tf

s

Figure 2.6: Typical geometry of specimens

All of the tested specimens, except FC-1, failed in punching. Failure loads are presented

in Table 2.2. The factor of safety against failure was quite high; the maximum ratio

between the actual failure load and the expected failure load by conventional flexural

design was 2.13. Tests results have indicated that arching action is more efficient in slabs

with higher compressive strength or lower reinforcement ratio. This observation agrees

with the test results obtained by Taylor and Hayes (1965). Figure 2.7 shows the factor of

safety against failure versus (f’c/ρ.fy).

13

Figure 2.7: Factor of safety versus f’

c / ρ.fy Table 2.2: Details of slabs and test results of selected specimens

Specimen dimensions

Boundary frame Designa-

tion d* [mm]

s [mm]

b [mm]

tf [mm]

Tensile reinf. ratio [%]

Compres-sive

strength [MPa]

Support condi-tions**

Punching failure load

[kN]

XC-2 78 1400 400 450 0.91 36.4 C 275.7 XC-3 71 1400 400 600 1.00 34.5 C 275.7 XC-4 62 1400 400 450 2.30 25.0 C 147.2 FC-3 90 1200 450 400 0.00 23.4 E 131.5 FC-4 73 1600 450 400 0.39 23.4 E 186.4 FC-5 70 1600 450 400 0.40 22.9 E 185.4 FC-6 73 1600 450 400 0.77 22.9 E 176.6 FC-7 70 1600 450 400 0.81 20.5 E 168.4

For all specimens t=100 mm * d is the effective depth of the specimen **C supported at four corners, E supported along four edges

2.5 Modelling of the Behaviour of Restrained Slabs

Various approaches have been developed to predict the ultimate punching shear strength

of a concrete slab. These approaches can be divided into two categories: empirical

equations, based on statistical analysis of tests results; and rational models. Rational

14

models idealize the failure mechanism, system geometry, and material properties to build

a mathematical model that seeks to predict the punching shear strength. The following

section reviews the model developed by Hewitt and Bacthelor (1975). The Hewitt and

Batchelor model is an extension of the Kinnunen and Nylander’s model (1960), expanded

to include the restraining effects. The model was further developed by Newhook (1997)

to predict the behaviour of laterally restrained deck slabs of slab-on-girder bridges.

2.5.1. Hewitt and Batchelor 1975: Punching Shear Strength of Restrained Slabs

Hewitt and Batchelor proposed a rational model to predict the behaviour of a restrained

circular slab that fails in punching. The model is based on the analysis of a fractured slab

at failure. The geometrical and material parameters of the system affecting the behaviour

were identified as: depth of the concrete slab, diameter or equivalent diameter of the slab,

diameter or equivalent diameter of the loading area, reinforcement ratio, yield stress of

the steel, and the compressive strength of the concrete. Figure 2.8 shows the idealized

model; the outer wedge of the slab, which is bounded by a shear crack and radial cracks,

is loaded through a compressed conical shell that develops from the loading area to the

end of the shear crack. The thickness of the conical shell is assumed to vary in a way that

the compressive stresses in the radial direction is constant.

The portion of the slab in Figure 2.8 is subjected to the following forces: (1) the external

load, Pβ/2π; (2) the oblique compression force in the compressed conical shell, Тβ/2π; (3)

horizontal forces in the circumferential reinforcement at right angles to the shear crack,

15

with resultant R1; (4) horizontal forces in the radial reinforcement traversing the shear

crack, with resultant R2; (5) horizontal tangential compressive forces in the concrete, with

the resultant R3; and (5) boundary restraint forces Mb and Fb. It should be noted that

dowel action was not explicitly calculated in the model; the enhancement of strength due

to dowel action was estimated as 20% of the failure load of a simply supported similar

slab (i.e. when Mb = Fb = 0).

conical shellshear crack

loadP

reaction P ß2p

M

Fb

b

( a ) Section showing boundary forces

d h

CC

0

c

ya

( B ) Sector element showing slab forces

T ß2p

ß

R3

R1

R1

R3

P ß2p

Mb

Fb

R2

Figure 2.8: Mechanical model of slab at punching failure (Hewitt and Batchelor 1975)

The model adapts the failure criteria used by Kinnunen and Nylander (1960), where

failure is described as the failure of the concrete in the compression zone which takes

place when the tangential strain reaches a characteristic value. Expressions used to

calculate the forces acting on the concrete wedge are given by Hewitt and Batchelor

16

(1975). The failure load can be developed using three equations of equilibrium; the

calculation involves two iterative processes. A computer program was developed to

calculate the theoretical punching strength of the slab using the proposed model. The

model yielded good accuracy when used to analyze previously tested slabs.

It should be noted that a non-circular slab can be modelled as a circular slab having an

equivalent diameter. The equivalent diameter is taken as the diameter of the largest circle

which could be inscribed within the area of the slab. The equivalent diameter of the

loaded area is taken as the diameter of the circle with the same perimeter as the loaded

area.

For slabs where the boundary restraint is not well defined, the authors proposed a

restraint factor, η. The restraint factor ranges from zero for simply supported slabs

unrestrained to unity for fully restrained slabs. Based on the analysis of previously tested

slabs, Hewitt and Batchelor proposed a guide for choosing the value of the restraint factor

for a number of practical cases.

2.6 Example of Laterally Restrained Slabs: Steel-free Slab-on-Girder Bridges

A large number of highway bridges in Canada are designed as a slab-on-girder system

(see Figure 2.9). In this system, concrete deck slabs are constructed integrally with the

supporting beams (i.e. a composite system). The concrete slab is laterally confined in

both directions; longitudinally by the steel beams which are connected to the slab through

17

shear connector, and transversally by steel reinforcement (either internal or external). The

lateral confinement inherent in the system significantly enhances the ultimate resistance

of the slab and causes it to fail in a punching mode rather than flexure.

As part of the development of the Ontario Highway Bridge Design code (OHBDC), a

large research program was conducted to study the behaviour of slab-on-girder bridges.

The program involved the testing of large number of full-half-and quarter-scale models.

The outcome of this program has been included as a design recommendation in OHBDC

and is now included in the Canadian highway bridge design code (CSA 2000). A number

of bridges were built according to the code recommendations and are now in service (for

example, Salmon River Bridge in Nova Scotia, Canada).

Figure 2.9 Isometric view of slab-on-girder bridge system

18

2.6.1 Mufti; Jaeger; Bakht; and Wagner 1993: Experimental Investigation of

Fibre-Reinforced Concrete Deck Slabs without Internal Steel Reinforcement

The main objective of this study was to investigate the possibility of producing a concrete

deck slab that is entirely ferrous free and capable of resisting the same load levels resisted

by conventionally reinforced decks. This objective would be achieved by utilizing the

notion that a deck slab will resist the applied loads through internal arching action

provided that certain confinement conditions are met. External steel straps were used to

provide such confinement. Polypropylene fibres were incorporated into concrete to

control temperature and shrinkage cracks.

An experimental program was conducted to study the appropriate confinement conditions

that will lead to punching failure; four half-scale models were tested. Figure 2.10 shows

the details of the first model. The deck slab was connected to longitudinal steel girders

through pairs of shear connectors spaced at 305 mm. Load was applied using a thick steel

plate and a thin neoprene pad.

19

Figure 2.10: Details of the first model (Mufti et al. 1993)

The first model failed at 173 kN. The mode of failure was flexure, indicating that the

specimen lacked the level of confinement needed to develop a punching shear failure.

The second model tried to achieve lateral constraint by adding end diaphragms to the

steel frame-work. This arrangement resulted in increasing the failure load, but the mode

of failure was still flexural. It was realized that conventional transverse steel was not able

to provide enough lateral confinement for the deck slab to be able to develop arching

action. In the third model, transverse steel straps were attached to the underside of the top

flange of the girder every 457 mm. This time the model was able to develop arching

action and failed in punching shear. Utilizing the localized nature of the failure, the same

model was tested again in several locations. Figure 2.11 shows the test locations. The

20

failure loads were 418 kN, 316 kN, and 209 kN for locations 1, 2, and 3 respectively. The

maximum failure load was achieved at location 1. It can be concluded that the degree of

confinement in the longitudinal direction decreased as the test location moved towards

the transverse free edge of the deck slab.

Figure 2.11: Test locations on the deck slab of the third model (Mufti et al. 1993)

To stiffen the free transverse edge of the bridge, the authors suggested the addition of a

beam with its major flexural rigidity in the horizontal plane. This beam should be

connected to the concrete slab using shear connectors.

The fourth model was devised to test the behaviour of a FRC deck slab under a pair of

equal loads simulating truck axle loads. The deck slab was able to develop arching action

21

for the pair of loads as effectively as a single load. Figure 2.12 shows the load deflection

curves for model 3 and 4.

Figure 2.12: Load deflection curves (Mufti et al. 1993)

2.6.2 Hassan, Kawakami, Niitani, Yoshioka 2002: An Experimental Investigation

of Steel-Free Deck Slabs

In this study, the effect of prestressing on the punching strength of steel-free deck slabs

was investigated. A system of external unbonded prestressing bars was used as the lateral

confinement system rather than the steel straps used by Mufti et al. (1993).

Seven large-scale one-way steel-free deck slabs were built using plain normal- and high-

strength concrete. Details of the tested specimens are presented in Table 2.3, Figure 2.13,

and Figure 2.14. For all specimens, except DS3’, the area of steel was chosen to be 65%

of the minimum area required by the CSA 2000 for normal non-prestressed bars.

22

Specimens were loaded at the centre via a 200 x 400 mm plate. During testing, vertical

deflection along the centre-line of the slab, edge rotation, strain of the top and bottom

fibres of concrete, and strain of steel bars were recorded. Load cells were used to measure

the force in prestressing bars.

Table 2.3: Details of the tested specimens

Slab Name

Concrete comp.

strength [MPa]

Concrete tensile

strength [MPa]

Pre-stressing steel ratio

[%]

Transverse compressive stresses at

lower fibre of concrete [MPa]

Transverse tensile

stresses at upper fibre of concrete

[MPa]

Transverse prestressing

stress at mid-section of concrete [MPa]

DS1 37.8 2.75 0.27 0.00 0.00 0.00 DS2 37.4 2.86 0.27 1.60 0.84 0.38 DS3 38.4 3.77 0.27 2.41 1.43 0.49 DS3’ 36.1 3.03 0.49 2.36 1.41 0.48 DS4 90.7 4.95 0.27 0.00 0.00 0.00 DS5 94.0 5.25 0.27 2.72 1.53 0.59 DS6 88.4 5.64 0.27 4.70 3.00 0.85

Figure 2.13: Upper and lower plan of the tested specimens

(Hassan et al. 2002)

23

Figure 2.14:Cross-section A-A in the prestressed steel-free slab (Hassan et al. 2002)

Generally, the overall behaviour of the specimens was similar to those confined by straps;

the specimens failed in a punching shear mode. For normal strength concrete, the top area

of the punched-out cone exactly matched the loading plate, while in high strength

concrete the top area took the shape of an ellipse.

The load-deflection curves of the specimens are shown in Figure 2.15. It is clear that

prestressing resulted in smoother load-deflection curves. Table 2.4 summarizes the tests

results. Prestressing increased the ultimate strength of specimens (34% comparing DS1

with DS2, and 15% comparing DS5 with DS6). It is also seen that increasing the

prestressing level decreased the maximum deflection at failure (Table 2.4).

Figure 2.15: Load-deflection curve for (a) normal-strength concrete slab, and (b) high-

strength concrete slabs (Hassan et al. 2002)

24

Table 2.4: Test results

Slab Name

Prestressing stress at mid-

section of concrete [MPa]

Cracking load [kN]

Punching failure load [kN]

Maximum deflection at

failure [mm]

Maximum rotation of

edges [○]

DS1 0.00 99 554.6 8.28 0.43 DS2 0.38 124 746.2 7.78 0.30 DS3 0.49 147 730.9 7.44 0.33 DS3’ 0.48 132 696.1 4.53 0.20 DS4 0.00 157 862.9 13.45 0.46 DS5 0.59 225 853.2 10.25 0.31 DS6 0.85 231 980.5 8.77 0.36

2.6.3 He 1992: Punching Behaviour of Composite Bridge Decks with Transverse

Prestressing

The positive effects of prestressing were previously confirmed by He (1992). He tested a

¼-scale slab-on-girder bridge; the model was prestressed using bonded post-tensioned

wires placed at the mid-depth of the slab. Different levels of prestressing were applied to

the model. Prestressing improved the overall behaviour, delayed first cracking of the

model, and increased the punching strength. Based on test results, a linear relation was

found between the level of transverse prestressing and the punching strength (Figure

2.16).

25

Figure 2.16: Punching strength versus. failure load (He 1992)

2.6.4 Modelling of Slab-on-Girder Bridges

Newhook (1997) proposed a rational model to predict the behaviour of ferrous-free slab-

on-girder bridges. The model is based on the analysis of a fractured slab at an

intermediate load level. The geometrical and material parameters of the system that will

affect the behaviour are: depth of the concrete deck, d; clear span between girders, c;

axial stiffness of the transverse strap, K, and its spacing s; dimension of the loading area,

B; and the compressive strength of concrete fc'.

Figure 2.17: Rigid body rotation of wedges (Newhook 1997)

26

Based on the study of equilibrium of the free body shown in Figure 2.17, and the model

proposed by Richart for the behaviour of concrete under confinement, the author

developed an iterative procedure to predict the load-deflection curve for deck slabs under

punching load. It should be noted that the slab strength is greatly influenced by the

horizontal restraining force Fw, which in turn is dependent on the stiffness of the

transverse steel reinforcement. Figure 2.18 shows the predicted and the experimental

load-deflection curve of one specimen.

Figure 2.18: Load versus. deflection curves (Newhook 1997)

27

2.7 Punching Strength of Fibre Reinforced Concrete (FRC)

2.7.1 Introduction

Incorporation of discrete fibres into concrete has proven to be beneficial to the over-all

mechanical behaviour of concrete. For instance, fibre reinforced concrete (FRC) shows

an improvement in toughness, ductility, energy absorption, cracking resistance, and

tensile strength (Shaaban and Gesund 1994).

Improvement in behaviour is mainly attributed to the bridging effects of fibres,

modifying the microcracks and macrocrack mechanics. Depending on length, fibres can

be divided into two main categories; macrofibres and microfibres. Microfibres carry

loads across microcracks, thus affecting precracking behaviour and increasing stiffness

and maximum tensile stresses. Macrofibres carry loads across macrocracks affecting the

postcracking behaviour and increasing toughness.

The extent of improvement in behaviour is affected by: volume fraction, material, length,

aspect ratio, and shape of fibres. Several materials such as steel, carbon, aluminum, and

glass fibres have been used to produce fibres that can be incorporated into the concrete.

Having different mechanical and physical properties, each will behave in a slightly

different way when incorporated within concrete.

28

2.7.2 Swamy and Ali (1982): Punching Shear Behaviour of Reinforce Slab-Column

Connections Made with Steel Fibre Concrete

This research was aimed at studying the effect of fibre reinforcement on the behaviour of

traditionally reinforced slab-column connections with and without shear reinforcement.

A total of 19 full-scale specimens typical of a flat-plate structure were tested. These were

divided into five series to study the following parameters: amount, location and type of

steel fibres, flexural reinforcement reduction, and bent-up bars as shear reinforcement.

The specimens were 1800 x 1800 x 125 mm, with an average effective depth of 100 mm,

simply supported along all four edges and loaded centrally through the stub column.

Details of the specimens are shown in Figure 2.19 and Table 2.5. Ribbed bars of 10 and 8

mm diameter, with characteristic strength of 462 and 480 MPa, respectively, were used as

flexural tension and compression reinforcement. The concrete 28-day compressive cube

(150 mm cube) strength ranged from 44.6 to 50.7 MPa for plain concrete and from 44.4

to 51.6 MPa for fibre concrete. A table vibrator was used for compaction of all

specimens. Three types of steel fibres were used: crimped (50 x 0.5 mm), hooked (50 x

0.5 mm), and plain (50 x 0.6 mm) with ultimate tensile strengths of 2066, 1160, 845

MPa, respectively.

29

125

1690

1810

15

250

1507 - 8 mm Comp.Reinf. EachDirection @ 241mm c/c

Direction @ 141mm c/c12 - 10 mm Comp.Reinf. Each

4 - 10 mm Column Reinforcement

Ties2 - 6 mm Column

Figure 2.19: Arrangement of steel reinforcing bars for slab-column connections

Extensive measurements were taken during the tests, 12 deflection readings, 32

compression and 30 tension face concrete strains, 7 tension and 3 compression steel

strains, and 5 rotations were recorded at all loading stages.

Table 2.5: Reinforcement and steel fibre distribution of series 1, 3, and 4

No. of tension reinf.

10 mm bars

No. of compression

reinf. 10 mm bars

Ser-ies No.

Para-meter

studied

Slab No.

middle outer middle Outer

Steel fibre % by vol.

Steel fibre type*

Remarks

S-1 6 6 3 4 0.0 - Plain conc.

S-2 6 6 3 4 0.6 C.S.F S-3 6 6 3 4 0.9 C.S.F

1 Steel

fibre % by vol.

S-4 6 6 3 4 1.2 C.S.F

Steel fibre distributed

over the whole

S-13 8 4 3 4 0.9 P.S.F S-12 8 4 3 4 0.9 H.S.F 3

Steel fibre type S-11 8 4 3 4 0.9 C.S.F

Steel fibre only 3.5 h

around column

S-8 8 4 3 - 0.9 C.S.F S-16 6 2 3 - 0.9 C.S.F S-10 5 2 3 - 0.9 C.S.F S-9 4 2 3 - 0.9 C.S.F

Steel fibre only 3.5 h

around column

4 Reinf.

reduce-tion

S-19 4 2 3 - 0.0 - Plain conc. * C.F.S. Crimped steel fibres; P.S.F. Plain steel fibres; H.S.F. Hooked steel fibres

30

Compared to the control slab, fibre reinforcement significantly reduced deflection at all

load stages, especially after first cracking. The load-deflection response of Series 1 is

shown in Figure 2.20. Figure 2.21 shows the load-tension steel strain behaviour. The

presence of steel fibres also resulted in a reduction of steel strains at intermediate load

levels. Extensive yielding of tension steel was observed at failure.

Figure 2.20 : Typical load deflection characteristics of series 1 at centre span of slab column connections (Swamy and Ali 1982)

Figure 2.21: Typical load-tension steel strain behaviour at centre of span (Series 1, 2) (Swamy and Ali 1982)

31

Fibre reinforcement changed the shape of the punching failure surface from square

(control specimens) to elliptical. Unlike the failure in the control specimen, the punching

failure in the FRC specimens was gradual and ductile. The presence of fibre enabled

large deformations to be sustained at maximum loads. The concrete in specimens with

fibre reinforcement was able to sustain strains larger than the limiting strain in

compression 0.0035, although these specimens failed in punching shear. Table 2.6 shows

that fibres delayed the formation of the first diagonal crack and increased the maximum

load capacity. Dramatic increases in both ductility and energy absorption were observed.

Specimens with fibres showed a considerably more ductile post-cracking behaviour.

Generally, crimped fibres were more effective than hooked fibres, and the plain fibres

were least effective.

Table 2.6: Diagonal tension cracking load, relative ductility, and energy absorption of slabs

Slab

No.

Steel

fibre %

by vol.

Load at which

diagonal tension

crack developed

[kN]

Maximum

Load

[kN]

Mode of

failure Ductility

Energy

absorption

[kN.mm]

S-1 0.0 78.0 197.7 Punching 33.0 4098 S-2 0.6 121.9 243.6 Punching 53.7 10992 S-3 0.9 140.1 262.9 Punching 62.3 17985 S-4 1.2 162.6 281.0 Punching 70.9 16829 S-11 0.9 214.5 262.0 Punching 58.8 17384 S-19 0.0 55.0 130.7 Flexural 75.3 5191 S-9 0.9 155.8 179.3 Flexural 126.1 22636 S-10 0.9 160.9 203.0 Flexural 74.0 17291 S-16 0.9 165.8 213.0 Flexural 61.2 14378

Based on the observed improved performance of FRC, it should be possible to increase

the service loads in FRC slab-column connections. Table 2.7 shows possible increased

32

service loads based on different deformation criteria of the control specimen S-1 at

service load of 118.1 kN (based on the British code). An increase in service load of 30 to

50 % is expected when 1% fibre volume is used. In other words, a decrease in the

required thickness for the same service load is possible.

Table 2.7: Service loads based on deformation criteria

Service load [kN] Slab

No.

Steel

fibre

% by

vol.

Deflect

-ion

[mm]

Service

load

[kN]

Steel

strain

x106

Service

load

[kN]

Concrete

strain

x103

Service

load

[kN]

Service

load

[kN]

S-1 0.0 6.72 118.1 2369 118.1 159.3 118.1 118.1 S-2 0.6 6.72 133.4 2369 132.5 159.3 153.7 132.9 S-3 0.9 6.72 147.8 2369 161.7 159.3 173.8 157.1 S-4 1.2 6.72 159.2 2369 195.7 159.3 183.2 191.9

33

Chapter 3: Experimental Program

3.1 Introduction

This chapter describes the details of the experimental program including specimen

geometry and dimension, and post-tensioning details. Also discussed are the properties of

the materials used, the testing set-up, and the testing instrumentation.

3.2 Test Specimens

The experimental program consisted of five square two-way slab specimens. Three of the

specimens were plain concrete, while the other two contained discrete steel fibres, i.e.

Fibre Reinforced Concrete (FRC). The fibres added were 1.0 % by volume fraction. The

specimens contained no regular internal rebars, and were externally post-tensioned using

Dywidag bars in the two perpendicular directions.

3.2.1 Geometry and Dimensions

The slab and loading plate dimensions were 1500 x 1500 x 127 mm and 200 x 200 x 50

mm respectively. The specimen geometry and dimensions are depicted in Figure 3.1. To

minimize the effect of local distress due to the applied axial load, the edges of the

34

specimen were reinforced by casting sixteen C 130 x 10 channels, each 204 mm in

length, into the specimen at the bearing areas.

All specimens were identical except Specimen P-1 and P-2. Specimen P-1 was 130 mm

in thickness, with no edge reinforcement. Specimen P-2 was 127 mm in thickness,

reinforced with one continuous channel on each side; each channel was 1500 mm long.

35

127

1500

1350

loadingplate20

0

200

1500

1500

119 204 78 204 290 204 78 204 119

C 130 x 10

Cross-section

Plan Figure 3.1: Typical specimen dimensions

36

3.2.2 Post-Tensioning

3.2.2.1 Post-Tensioning System

The specimens were externally post-tensioned using eight 32-mm diameter Dywidag bars

in each direction. The force in a Dywidag bar was transmitted first to a vertical steel

beam comprised of two C 310 x 45 channels, back-to-back spaced 44 mm apart, then to

the specimen through a 204 x 170 x 38 mm bearing plate bearing against a 32-mm-

diameter round bar welded to the C 130 x 10 channel. Post-tensioning system details are

shown in Figure 3.2 through Figure 3.4. For specimens P-1, the loading plates were

bearing directly against the specimen; while for specimen P-2, they were bearing against

the 1500 mm-long C 130 x 10 channels.

The points of application of axial load were chosen to facilitate the post-tensioning

process and to create a state as close as possible to a uniform axial stress state without

interfering with the applied vertical load. To estimate the axial stress distribution in the

specimen due to the applied axial forces, a linear elastic finite element (FE) model was

constructed using the commercial software SAP 2000. The specimen was modelled as

series of shell elements having 127 mm thickness; meshing of the elements was chosen to

facilitate load application and support conditions. The average element size was 83 x 83

mm. The axial forces were applied as concentrated loads distributed over three nodes on

each of the bearing plates. Figure 3.5 shows the axial stress distribution at successive

cross-sections as a ratio between actual stresses to the nominal stress (Force/Area). It is

37

shown that as the force travels through the specimen the stress distributions becomes

more uniform. Refer to Appendix A for model details.

Different stress levels were applied to each specimen; the nominal stresses in the concrete

and total force in the bars are summarized in Table 3.1.

Table 3.1: Nominal stress in concrete and total force in bars

N-S Direction E-W Direction

Specimen f’c

[MPa]

Total force in

bars [kN]

Nominal stress in concrete

fn [MPa]

Total force in

bars [kN]

Nominal stress in concrete

fn [MPa]

P-1* 65.4 1192 6.1 1162 6.0 P-2 64.1 792 4.2 897 4.7 P-3 68.5 366 1.9 417 2.2 F-1 59.9 1148 6.0 1135 6.0 F-2 54.8 861 4.5 1110 5.8

* 130 mm Thickness

38

1200

LoadingArea20

0

(1)

(2)

(3)

(4)

(8)(7)(6)(5)1

200

LoadingArea20

0 roller support

(1)

(2)

(3)

(4)

Plan

Section 1-1

N

204x170x38 plateback to back Spaced 44 mm2 C 310 x 45 203x127x38

Dywidag anchor plate

C 130 x 10

32-mm-diameterround bar

32-mm Dywidag bar

Figure 3.2: Post-tensioning system

39

127170

305

2 C 310 x 45

32-mm Dywidag bar

C 100 x 11

203 x 127 x 38

nut

specimen

127

32-mm Dywidag bar2

top bar

bottom bar

Dywidag anchor plate

C 130 x 10

44-mm back-to-back

204 mm-length

32-mm-diameter round bar

L 89 x 64 x 7.9

204 x170 x 38 plate

Figure 3.3: Blow-up of Section 1-1

204

44

C 100 x 11

204 x 170 x 38

203 x 127 x 38 Dywidag

anchor plate

plate

Figure 3.4: Section 2-2

40

0.25

0.40

0.55

0.70

0.85

1.00

1.15

1.30

0 300 600 900 1200 1500

Y- Distance

Act

ual S

tress

/ N

omin

al S

tress

At x = 221 At x = 401 At x = 650 At x = 750

X

Figure 3.5: Axial stress distribution in specimens

3.2.2.2 Post-Tensioning Operation

The bars were tensioned individually using the 600 kN Dywidag electric powered jack

shown in Figure 3.6. For each pair of bars, the following post-tensioning procedure was

adopted:

1. Both the top and bottom nuts were hand-tightened.

2. Half of the post-tensioning force was applied to the top bar; as the force was

applied an equal force developed in the bottom bar to satisfy the equilibrium

condition around the roller connection.

3. The top nut was tightened using the jack ratchet handle and then the jack was

released.

41

4. The bottom bar was jacked to full force; the nut was tightened and the jack was

then released.

The previous steps were repeated in the following sequence 2, 3, 4, 1; 6, 7, 8, 5 (see

Figure 3.2). The bars were post-tensioned one or two days before testing. The forces were

monitored overnight up to testing; minor losses were recorded (7-10 % of jacking forces),

mainly occurring immediately after releasing the jack due to grip losses. The maximum

jacking force applied to a bar was 163 kN, which corresponded to 20 % of its ultimate

force.

Figure 3.6: Dywidag electric powered hydraulic jack

3.3 Material Properties

3.3.1 Concrete

The concrete used in the specimens was batched in the concrete laboratory of the

University of Toronto; each specimen was batched on a separate occasion. The targeted

42

28-day compressive strength was 50 to 60 MPa. Several trial batches using plain and

fibre reinforced concrete were performed to design the concrete mix. To balance

workability and strength, different mixes were used for plain and fibre reinforced

concrete. Natural sand, crushed limestone of 10 mm maximum size, and Portland cement

were used in the proportions shown in Table 3.2 and Table 3.3.

To maintain an adequate workability for the fibre-reinforced concrete mix, the

gravel/cement ratio was reduced, and the super plasticizer content was increased. A

higher cement content was needed to reach the intended concrete strength. The

workability was measured using the Slump Cone Test for plain concrete (170 mm), and

the Inverted Slump Cone Test, according to ASTM C 995- 01, for the fibre-reinforced

concrete (17 second).

Table 3.2: Concrete mix design (Plain Concrete)

Mixture proportion Dry weight [kg/m3] Standard Type 10 Portland cement 1.00 375 Sand 2.26 847 Gravel (10 mm) 2.88 1080 Water 0.37 139

Admixtures

mL/ m3 Water reducer 1000 Super-plasticizer 3500

Table 3.3: Concrete mix design (FRC)

Mixture proportion Dry weight [kg/m3] Portland Silica Fume cement 1.00 450 Sand 2.26 1018 Gravel (10 mm) 1.67 750 Water 0.39 176

43

Admixtures

mL/ m3 Water reducer 1000 Super-plasticizer 5000

Four 0.10 m3 batches, from a mixer with a capacity of 0.14 m3, were used to cast each

specimen. Each batch was placed onto the wooden formwork and externally vibrated

using the vibrating table shown in Figure 3.7. The final concrete surface was leveled by a

steel trowel. After the slab had hardened sufficiently, it was covered with wet burlap and

plastic sheets for a period of 4 to 5 days. After curing, the slab was moved and stored

until it was ready for testing.

Figure 3.7: Formwork clamped to the vibrating table

For each specimen, twelve cylinders (150 mm in diameter and 300 mm in height) were

cast. The plain concrete cylinders were consolidated by rodding; the FRC cylinders were

consolidated by an external vibrator, as rodding tends to influence the random fibre

44

distribution and alignment. Three cylinders for plain concrete specimens, and six for FRC

specimens, were moist cured; the others were kept alongside the specimen where their

curing conditions were kept as close as possible to those of the specimen. The cylinders

were tested for axial compressive strength according to ASTM C39 standards, and for

splitting tensile strength according to ASTM C496 standards. The results are presented in

Table 3.4 and Figure 3.8. Relevant strengths are summarized in Table 3.5. Full stress-

strain responses for a number of the tested cylinders are presented in Appendix A.

The fracture energy of the FRC specimens was determined by using notched beams (150

x 150 x 530 mm) tested under 4-point bending according to the Italian standard UNI-

11039 (see Figure 3.9). The fracture energy was defined as the amount of energy

absorbed per unit area of the crack surface up to 25% of the maximum load beyond peak.

All results are summarized in Table 3.4.

Table 3.4: Average concrete properties

Specimen Steel fibre

% by volume

Comp. strength (Day of Testing)

[MPa]

Splitting tensile strength [MPa]

Fracture energy [J/m2]

P-1 0.0 65.4 - - P-2 0.0 64.1 5.53 - P-3 0.0 68.5 5.62 - F-1 1.0 59.9 7.35 13095 F-2 1.0 54.8 8.05 9920

Results are average of three tested samples

45

Table 3.5: Concrete cylinder compressive strength

f ’c [MPa]

Specimen Day 7

(Lab cured)

Day 28

(Lab cured)

Day 28

(Moist cured)

Day (of Test)

(Lab cured)

Age of Specimen

tested [Days]

P-1 53.0 63.6 61.5 65.4 101 P-2 54.3 62.6 - 64.1 35 P-3 55.2 62.6 - 68.5 69 F-1 47.3 - 56.9 59.9 68 F-2 44.3 - 54.4 54.8 62

Results are average of three tested samples

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

0 10 20 30 40 50 60 70 80 90 100

Age [Days]

Com

pres

sive

Str

engt

h [M

Pa]

P-1P-2P-3F-1F-2

Figure 3.8: Concrete cylinder compressive strength

46

150 150 150450530

45

5

150

150

Figure 3.9: Dimension of fracture energy test specimens

3.3.2 Post-Tensioning Steel

A 32 mm-diameter Dywidag threadbar® was used as post-tensioning steel. The bar

ultimate strength was 834 kN; relevant bar properties are summarized in Table 3.6. The

observed load-strain relationship is given in Figure 3.10 (refer to section 3.4.2.3). The

manufacturer data sheet can be found in Appendix A.

Table 3.6: Post-tensioning steel properties

Prestressing Force [kN] Nominal Diameter

[mm]

Ultimate stress [MPa]

Cross section Area

[mm2]

Ultimate Strength

[kN] 0.8 fpu Aps 0.6 fpu Aps

Young’s Modulus

[GPa]

32 1030 806 834 662 500 224.2

47

0

50

100

150

200

250

300

350

0 200 400 600 800 1000 1200 1400 1600 1800

Strain ( x 10 ) -6

Load

[kN

]

DataLinear Regression

Load [kN] = 180716 x Strain x 10-6

Es = 224.2 Gpa

Figure 3.10: Observed load-strain relationship of Dywidag threadbar®

3.3.3 Discrete Steel Fibres

Dramix RC-80/50-BP was the type of fibre used. It is a high carbon wire fibre, with

hooked ends, glued in bundles with a water-soluble glue to ensure better distribution of

fibres in concrete. The fibre properties are summarized in Table 3.7 and the profile of a

typical fibre is depicted in Figure 3.11. The manufacturer’s data sheet can be found in

Appendix A. Considering the density of concrete and steel, one percent of fibre volume

fraction corresponds to 78.8 kg of fibre per cubic meter of concrete.

Table 3.7: Fibre properties

Length (L) [mm]

Diameter (d) [mm] L/d Tensile strength

[MPa] 50 0.60 83 2000

48

L = 50 mm

d = 0.60 mm

Figure 3.11: Fibre dimensions

3.4 Test Set-up

3.4.1 Loading System

The loading system for the specimens consisted of a vertical monotonically increasing

concentric load distributed by means of a loading plate. The loading plate comprising of

two 200 x 200 x 50 mm steel plates held together by four Allen-key bolts. The load was

imposed onto the loading plate by a 5300 kN capacity Baldwin universal testing machine

shown in Figure 3.12. Additional details of the Baldwin machine can be found in

Appendix A.

The slab specimens were simply supported on all four sides by rollers comprised of 44-

mm diameter solid steel rods. The rollers rested on a steel podium placed directly on the

machine base plate. Two of the four rollers were welded to the podium, while the

opposite two rollers remained free to rotate. The rollers were positioned 75 mm from the

edges of the slab specimens, giving a span of 1350 mm for the specimen. Four 152 x 25 x

1500 mm plates were loosely positioned between the slab and the rollers to ensure

appropriate transfer of forces from the slab to the rollers. In order to prevent loose rollers

49

and plates from moving during installation and removal of the specimens, small steel

angles with Allen-key bolts temporarily held the plates and rollers in place. Design

specification and images of the podium are presented in Figure 3.13 and Figure 3.14.

Figure 3.12: Baldwin machine

50

free 44-mm roller75(Typ.)

3

HSS 254x152x13

44-mm-diameter roller 152x25 plate

concrete specimen152x25 plate

1500

1350

HSS 254x152x13

Section 3-3

1196

1500

Plan

Ø60

free 44-mm roller

welded 44-mm roller

welded 44-mm roller

Figure 3.13: Podium details

51

Figure 3.14: Images of the podium

3.4.2 Test Instrumentation

A computer-controlled data acquisition system was used to record all electronic test data

from LVDTs, strain gauges, and load cells.

52

3.4.2.1 Linear Variable Differential Transducers

Linear variable differential transducers (LVDTs) were used to monitor the vertical and

horizontal displacement of the specimen. The vertical displacement was monitored using

six LVDTs; four were placed at the corners, mounted on the top surface of concrete and

aligned with the centreline of the rollers, and two were attached on opposite sides of the

Baldwin machine head. The top and bottom horizontal LVDTs were used to monitor

axial displacements and rotations of the edges. The horizontal LVDTs were placed at the

mid span of each side, 9 mm away from top and bottom surfaces. Aluminum targets were

glued to the concrete surface to create a better surface for reading displacements. The

positions of the horizontal and vertical LVDTs are shown in Figure 3.15 and Figure 3.16.

The vertical displacement of the machine head was used as an approximation of the

vertical displacement of the specimen. It was not possible to mount a LVDT underneath

the specimen due to the limited clearance between the specimen and the loading machine

base plate.

53

LoadingArea

roller support

15001350

Plan

Cross section

LoadingArea

15001350

vertical LVDT

horizontal LVDT

Plan

Cross section

N

machine head

machine head

Figure 3.15: Layout of horizontal and vertical LVDTs

54

Figure 3.16: LVDT set-up

55

3.4.2.2 Strain Gauges

Two electrical strain gauges, with 5-mm gauge lengths, were applied to each Dywidag

bar to monitor the bar strain during post-tensioning and testing. The strains were used to

correlate bar forces in the experiment. The strain gauges were positioned at the middle of

the bar on opposite surfaces to eliminate any bending effects that may occur. To provide

appropriate contact area between the bar and the strain, a minimal surface of the bar was

sanded to a buffed finish with varying grits of emery sanding paper and cleaned with a

solvent to remove any contaminants. See Figure 3.17.

Figure 3.17: Dywidag bar strain gauges

3.4.2.3 Load Cells

At the anchored end of the post-tensioning bar, two load cells were installed to monitor

the force in the bar. As shown in Figure 3.18 the two load cells bear against two 50 mm

thick bearing plate distributing the bar load on the two load cells. A spherical head was

installed to ensure an even distribution of the load. Due to the limited number of available

load cells, the load cells were installed only on two of the sixteen Dywidag bars (top bars

56

of pair 2, 3). Load cells and strain gauges data were used to construct a load-strain curve

for the Dywidag thereadbar® (refer to section 3.3.2).

50 mm thick plate

2 load cells 150 mm spaced

sphirical head38 mm plate

nut

Cross section

Plan

Figure 3.18: Load cells configuration

57

Chapter 4: Experimental Results and Observations

4.1 Introduction

In the following sections, the results obtained from the tests described in the previous

chapter are presented. Five externally post-tensioned slab specimens were tested to

failure under monotonically increasing central load. Although all slabs were

unreinforced, four specimens failed in punching shear, while Specimen P-3 failed in a

combined flexural-punching shear mode. The load, displacements at selected points,

post-tensioning bars strains, and load cells readings were recorded using a computer-

based data acquisition system. This chapter focuses on the presentation of the tests

results. Refer to Chapter 5 for discussion.

4.2 Test Data

The slab specimen variables and test set-up are presented in Table 4.1, Table 4.2, and

Figure 4.1 through Figure 4.23. All relevant plots have been drawn to the same scale, so

that visual comparisons can be made.

Figure 4.1 describes the load-deformation responses recorded for the specimens. The slab

deformation is taken as the difference between the slab specimen deformation at the

supports and at the loading plate. Deflections were partly due to specimen’s deformation

and partly due to the slack of the testing set-up and the loading machine. The load-

58

deformation curves shown in Figure 4.2 were revised to eliminate the part of the

specimen’s deflection that is a consequence of the slack of the testing set-up. The revised

load-deformation curves were obtained by eliminating the initial lower-stiffness branch

of the graph.

To estimate the part of the centre displacement due to the slack of the testing set-up, two

additional LVDTs were installed to measure the relative displacement between the roller

support and the loading plate as shown in Figure 4.3 (only for Specimen F-1). Given in

Figure 4.4 is the average displacement of the two LVDTs.

Table 4.1 summarizes the specimen variables and test results. The strain energy absorbed

by the specimens up to different stages of loading is presented in Table 4.2. The strain

energy absorbed, U, is taken as the area under the load-deformation curve up to a

specified point. For instance, U80 is taken as the area under the revised load-deformation

curve up to 80-percent of the ultimate load beyond the peak. The ductility of the

specimens was quantified in terms of the ratio of the strain energy absorbed up to

maximum displacement to the strain energy absorbed up to maximum load ( max max PU / UΔ ).

Figure 4.5 and Figure 4.6 show sketches of the cracking patterns on the tension and

compression surfaces of the specimens. The tension surface cracks could not be

monitored during testing. After testing, each specimen was picked up and the bottom

surface was inspected. Specimen P-3 was severely damaged and collapsed when picked

up by the crane. Selected photos of the tested specimens are given in Figure 4.7 through

Figure 4.11

59

Figure 4.12 through Figure 4.16 summarize the axial loads applied to the specimen

through the post-tensioning bars. The recorded axial strains were used to correlate to the

bar forces; each bar had two strain gauges to eliminate the effect of any bending stresses

(refer to section 3.3.2). The axial force plotted on the graph is the summation of the

forces of the eight bars in each of the two orthogonal directions. The maximum force

recorded for a single bar was 221 kN which corresponds to 27% of the ultimate strength

(recorded for Specimen P-1).

Figure 4.17 through Figure 4.21 illustrate the average rotation of the North and South,

and the East and West sides. The side rotation is taken as the difference between the side

deformation at top and bottom points divided by the distance between, 110 mm. Refer to

Figure 3.15 for the positions of the LVDTs. The North-South rotation of Specimen P-3

was deemed unreliable and was discarded from plots. A possible source of error is the

excessive rotation of the side beams which may have resulted in movement or twisting of

the LVDT base.

Figure 4.22 and Figure 4.23 show the estimated bending moments transferred to

Specimen P-1 and Specimen P-2. For the other specimens, the presence of the 32-mm-

diameter round bar ensured that only axial load was transferred (refer to section 3.2.2.1).

The bending moment was estimated as the bar force times the distance to the centreline

of the specimen.

60

0

100

200

300

400

500

600

700

0 5 10 15 20 25 30

Displacement [mm]

Load

[kN

]

P-1

P-3

P-2

a) Plain concrete specimens

0

100

200

300

400

500

600

700

0 5 10 15 20 25 30

Displacement [mm]

Load

[kN

]

F-1

F-2

b) FRC specimens

Figure 4.1: Specimen load-deformation responses

61

0

100

200

300

400

500

600

700

0 5 10 15 20 25 30

Displacement [mm]

Load

[kN

]

P-1

P-3

P-2

a) Plain concrete specimens

0

100

200

300

400

500

600

700

0 5 10 15 20 25 30

Displacement [mm]

Load

[kN

]

F-1

F-2

b) FRC specimens

Figure 4.2: Revised specimen load-deformation responses

62

1500135015001350

Machine Head

(1) (2)

Figure 4.3: Layout of additional LVDTs for Specimen F-1

0

100

200

300

400

500

600

700

0 5 10 15 20 25 30

Displacement [mm]

Load

[kN

]

External LVDT (1)

Inner LVDT (2)

Figure 4.4: Load-deformation responses from additional LVDTs of Specimen F-1

63

Table 4.1: Slab specimen variables and test results

Table 4.2: Strain energy absorbed

Speci-men

f’c

[MPa] Vf %

Thick-ness [mm]

fn(N-S) [MPa]

fn(E-W) [MPa]

Failure load Pmax [kN]

Disp. at

max. load [mm]

Failure mode

Nature of failure.

P-1 65.4 0.0 130 6.1 6.0 488 9.78 Punching Brittle

P-2 64.1 0.0 127 4.2 4.7 675 10.98 Punching Brittle

P-3 68.5 0.0 127 1.9 2.2 239 15.18 Punching-Flexural

Moderately-Ductile

F-1 59.9 1.0 127 6.0 6.0 503 12.49 Punching Moderately-Ductile

F-2 54.8 1.0 127 4.5 5.8 457 13.68 Punching Moderately-Ductile

Strain Energy [ J ]

Up to Max. load

Up to 80% of max. load

beyond peak

Up to max. displacement

Specimen

maxPU 80U maxUΔ

Ductility

max

max P

UU

Δ⎡ ⎤⎢ ⎥⎣ ⎦

P-1 2673 - 2673 1.00 P-2 3975 - 3975 1.00 P-3 2493 2903 3166 1.27 F-1 4187 5395 7843 1.87 F-2 4393 5671 7800 1.78

64

P-1

F-1 F-2

P-2 N

Figure 4.5: Sketches of cracking pattern on tension (bottom) surface

65

P-1

P-3

F-1 F-2

P-2 N

Figure 4.6: Sketches of cracking pattern on compression (top) surface

66

Figure 4.7: Selected photos of Specimens P-1

Figure 4.8: Selected photos of Specimens P-2

67

Figure 4.9: Selected photos of Specimens P-3

Figure 4.10: Selected photos of Specimens F-1

68

Figure 4.11: Selected photos of Specimens F-2

69

0

200

400

600

800

1000

1200

1400

1600

0 100 200 300 400 500 600 700

Applied Vertical Load [kN]

Axi

al F

orce

[kN

]

N-SE-W

Figure 4.12: Axial load versus applied vertical load (Specimen P-1)

0

200

400

600

800

1000

1200

1400

1600

0 100 200 300 400 500 600 700

Applied Vertical Load [kN]

Axi

al F

orce

[kN

]

N-SE-W

Figure 4.13: Axial load versus applied vertical load (Specimen P-2)

70

0

200

400

600

800

1000

1200

1400

1600

0 100 200 300 400 500 600 700

Applied Vertical Load [kN]

Axi

al F

orce

[kN

]

N-SE-W

Figure 4.14: Axial load versus applied vertical load (Specimen P-3)

0

200

400

600

800

1000

1200

1400

1600

0 100 200 300 400 500 600 700

Applied Vertical Load [kN]

Axi

al F

orce

[kN

]

N-SE-W

Figure 4.15: Axial load versus applied vertical load (Specimen F-1)

71

0

200

400

600

800

1000

1200

1400

1600

0 100 200 300 400 500 600 700

Applied Vertical Load [kN]

Axi

al F

orce

[kN

]

N-SE-W

Figure 4.16: Axial load versus applied vertical load (Specimen F-2)

-0.01

0

0.01

0.02

0.03

0.04

0.05

0 100 200 300 400 500 600 700

Applied Vertical Load [kN]

Ave

rage

Sid

e R

otat

ion

[rad

]

N-SE-W

+ve+ve

Figure 4.17: Average side rotation versus applied vertical load (Specimen P-1)

72

-0.01

0

0.01

0.02

0.03

0.04

0.05

0 100 200 300 400 500 600 700

Applied Vertical Load [kN]

Ave

rage

Sid

e R

otat

ion

[rad

]

N-SE-W

+ve+ve

Figure 4.18: Average side rotation versus applied vertical load (Specimen P-2)

E-W

-0.01

0

0.01

0.02

0.03

0.04

0.05

0 100 200 300 400 500 600 700

Applied Vertical Load [kN]

Ave

rage

Sid

e R

otat

ion

[rad

]

E-W

+ve+ve

Figure 4.19: Average side rotation versus applied vertical load (Specimen P-3)

73

-0.01

0

0.01

0.02

0.03

0.04

0.05

0 100 200 300 400 500 600 700

Applied Vertical Load [kN]

Ave

rage

Sid

e R

otat

ion

[rad

]

N-SE-W

+ve+ve

Figure 4.20: Average side rotation versus applied vertical load (Specimen F-1)

-0.01

0

0.01

0.02

0.03

0.04

0.05

0 100 200 300 400 500 600 700

Applied Vertical Load [kN]

Ave

rage

Sid

e R

otat

ion

[rad

]

N-SE-W

+ve+ve

Figure 4.21: Average side rotation versus applied vertical load (Specimen F-2)

74

-20

0

20

40

60

80

100

120

0 100 200 300 400 500 600 700

Applied Vertical Load [kN]

Mom

ent [

kN.m

]

N-SE-W

+ve+ve

Figure 4.22: Estimated end moment versus applied vertical load (Specimen P-1)

-20

0

20

40

60

80

100

120

0 100 200 300 400 500 600 700

Applied Vertical Load [kN]

Mom

ent [

kN.m

]

N-SE-W

+ve+ve

Figure 4.23: Estimated end moment versus applied vertical load (Specimen P-2)

75

4.3 Mechanical Behaviour of Slab Specimens

Four of the tested specimens failed in punching shear mode. For two of the specimens,

the failure was very sudden and without any warning. The behaviour of the slabs during

loading is described below.

4.3.1 Specimens P-1 & P-2

Both specimens failed in a punching mode. A punching failure was attained when a cone

of concrete completely punched out of the slab (see photos in Figure 4.7 and Figure 4.8).

The failure was brittle with no visible or audible warnings; the suddenness of the failure

can be gauged from the load-deformation responses recorded.

Specimen P-1 failed at a load of 489 kN with a corresponding vertical displacement of

9.78 mm. The shear fracture on the compressed surface of the specimen occurred

immediately adjacent to the loading plate. On the tension side, the crack was aligned

with the inner face of the 152 x 25 mm plate, with some irregularities at the corners (see

Figure 4.5). The average angle of inclination of the shear failure plane, measured from

the horizontal, was 15°. Specimen P-2 failed at a vertical load of 675 kN with a

corresponding vertical displacement of 10.98 mm. The shear fracture on both the

compression and tension sides were similar to those of Specimen P-1; only, the tension

side crack was pushed further away from the 152 x 25 mm plate. The angle of inclination

of the shear failure plane was 18°.

76

4.3.2 Specimen P-3

The failure mode of Specimen P-3 was interpreted as a combined flexural-punching shear

failure in which both flexural and punching shear cracks were observed to form

simultaneously at failure. On the compressed surface, a few longitudinal flexural cracks

formed approximately at mid-span, while punching shear cracks formed immediately

adjacent to the loading plate. Although the tension surface cracks could not be inspected,

it is believed that the longitudinal cracks on the compression surface initiated on the

tension surface and extended to the full depth of the specimen. Specimen P-3 failed at a

load of 239 kN with a corresponding vertical displacement of 15.18 mm. The formation

of a complete failure plan was gradual; and resulted in a moderately ductile failure.

4.3.3 Specimens F-1 & F-2

Specimens F-1 & F-2 failed in a punching shear mode. The punching failure was

characterized by cracks forming immediately at the loading plate periphery followed by

penetration of the loading plate. On the tension surface, no distinguishable shear fractures

were evident. A few longitudinal and diagonal cracks were observed (Refer to Figure

4.10 and Figure 4.11). Audible signs of distress occurred during the loading and these

were attributed to the fibres debonding and pulling out across the widening cracks. The

formation of a complete failure surface was gradual and occurred over a few minutes

resulting in a moderately ductile failure. Specimen F-1 failed at a load of 502 kN with a

corresponding vertical displacement of 12.49 mm, while Specimen F-2 failed at a load of

77

457 kN with a corresponding vertical displacement of 13.78 mm. At failure, the loading

plate penetrated approximately 15 mm.

78

Chapter 5: Discussion of Experimental Results

5.1 Behaviour of the Specimens

The typical response observed can be divided into four distinct stages as illustrated in

Figure 5.1 and Figure 5.2. Upon loading, the centre of the specimen moved downward

and the edges of the specimen started to rotate and translate in the plane of the slab.

During this stage, specimen remained uncracked and the applied load increased linearly

with the deflection (Stage І); the forces recorded in the post-tensioning bars remained

virtually unchanged. Upon cracking, Stage ІІ, the slab stiffness gradually reduced as the

applied load increased. At the same time, an increase in the bars forces was observed

indicating that the slab was expanding laterally. The amount of force increase was

governed by the axial stiffness of the bars which was constant for all specimens. In Stage

ІІІ, a deformation increase took place without any significant increase in the vertical load.

During this stage, the bar forces increased at a lesser rate, and reached a maximum value

at the end of this stage. The post-peak stage (Stage ІV) indicated further reduction in the

load-carrying capacity. This reduction occurred in several steps, with spreading of cracks

on the tension surface. In this stage, the bars forces decreased considerably, returning

approximately to the values recorded at the beginning of the test. In general, the

behaviour of the specimens was similar to the behaviour of laterally restrained slab-on-

girder bridges.

79

Displacement

Ver

tical

Loa

d

І ІVІІІІІ

Pmax

Pcrack

Figure 5.1: General load-deformation response of specimens

Vertical Load

Axi

al L

oad

І

ІVІІІ

ІІ

PmaxPcrack

Figure 5.2: General axial load versus applied vertical load response

80

5.2 Failure Modes

According to the nature of their failure mode, the specimens can be divided into three

categories: (a) brittle punching failure, (b) moderately ductile flexural-punching failure,

and (c) moderately ductile punching failure. The nature of failure was governed primarily

by the applied level of stress and the fibre reinforcement.

Plain concrete specimens with relatively higher stress levels fall into Category (a); these

specimens reached their maximum load and failed abruptly at the end of Stage ІІ. No

post-peak responses were observed, and the specimens demonstrated poor strain energy-

absorption capacity. Category (a) specimens included Specimen P-1 and Specimen P-2.

The flatness of the shear cracks, 15° and 18° respectively, and the relatively large size of

the punched cone were expected for the high compressive stress applied.

Specimen P-3 fell into Category (b); the maximum load was reached at the end of Stage

ІІІ. Although the specimen was constructed of plain concrete, a short plateau was

observed and the loss of the load-carrying capacity during post peak-response was not as

sudden as for Category (a) specimens. The moderately ductile behaviour was attributed to

the lower level of stress applied, resulting in flexural stresses which led to a combined

flexural-punching failure mode rather than a pure punching failure mode.

The FRC specimens, Category (c), showed satisfactory post-peak responses as seen from

the plateau and descending branch of the load deformation responses. For this category,

the maximum load was reached during Stage ІІІ. The specimens lost their load-carrying

81

capacity gradually. The ductile behaviour was attributed to the debonding, stretching, and

pulling out of the fibre reinforcement bridging the shear cracks.

5.3 Failure loads of Specimen P-1 and Specimen P-2

Although the stress level applied to Specimen P-2 was lower than that on Specimen P-1,

unpredictably, Specimen P-2 achieved a higher load-carrying capacity (6.04 MPa and

489 kN for P-1; 4.43 MPa and 675 kN for P-2). Both specimens were similar in regard to

concrete strength and thickness (65.4 MPa and 130 mm for P-1; 64.1 MPa and 127 mm

for P-2). The discrepancies in the results were attributed to the passive restraint in

Specimen P-2. Specimen P-2 was reinforced with one continuous 1500 mm long C 130 x

10 channel cast on each side, while no such reinforcement was provided for Specimen P-

1 (Refer to section 3.2.1). It is believed that the edge channels, being continuous and tack

welded at the corners, restrained the lateral movement of the specimen, and thus provided

additional lateral restraint which resulted in an increase in the failure load. This

shortcoming was remedied in subsequent tests by casting smaller length channels, 204

mm long, only at bearing areas. Specimen P-2 was excluded from the parametric

analysis.

82

5.4 Parameters Analysis

5.4.1 Effect of Lateral Restraint Provided by Post-Tensioning Stresses

Figure 5.3 and Table 5.1 illustrate the effects of the post-tensioning stress. For the plain

concrete specimens, equal stresses were applied to both orthogonal directions of each

specimen. Direct comparison to Specimen P-2 could not be made due to the additional

passive restraint discussed earlier.

Comparing the behaviour of Specimen P-1 and Specimen P-3, the level of stress was

observed to affect all aspects of the slab behaviour, including cracking, deflection,

stiffness, and failure mode. Increasing the lateral stress level enhanced the ultimate load

capacity and stiffness but had a negative effect on ductility. Increases in the ultimate load

and stiffness can be attributed to the increase in compressive membrane action which, in

turn, was affected by the level of stress.

A stress level as low as 2.0 MPa was able to provide lateral restraint necessary to prevent

premature pure flexural failure of Specimen P-3, which failed in a combined flexural-

punching mode. A linear finite element analysis of Specimen P-3 (using SAP 2000)

showed that it would reach cracking stress (0.33√ f’c = 2.73 MPa) at a load of 61.7 kN;

unreinforced specimens are expected to fail shortly after first cracking. Increasing the

stress level to 6.0 MPa ensured a pure punching failure of Specimen P-1. In conclusion,

a specimen, being predominantly in compression, is more likely to fail in punching mode

rather than flexure.

83

Table 5.1: Effect of lateral restraint provided by post-tensioning stresses

N-S Direction E-W Direction Speci-

men

f’c

[MPa] fn

[MPa] n

c

ff '

fn [MPa]

n

c

ff '

t [mm]

d [mm]

Pmax [kN]

max

c

P. f ' d

maxUΔ [J]

P-1 65.4 6.1 0.76 6.0 0.74 130 93.6 488 644.7 2673 P-3 68.5 1.9 0.23 2.2 0.26 127 91.4 239 315.8 3166

(a) (b)

300

400

500

600

700

800

fn / √ f’c

P max

/ ( d

. √

f’ c )

P-3 P-10.25 0.750.00 1.00 2000

3000

4000

5000

6000

7000

8000

9000

fn / √ f’c

Stra

in E

nerg

y A

bsor

ped

[J]

P-3 P-10.750.25 1.000.00

Figure 5.3: Effect of post-tensioning stresses on (a) Normalized punching load (b) Strain energy absorbed

5.4.2 Effect of Fibres Reinforcement

Table 5.2 and Figure 5.4 show the effect of the presence of fibre on the behaviour of the

specimens. Comparing the behaviour of Specimen P-1 and Specimen F-1, adding steel

fibres in an amount equal to 1 % by volume fraction resulted in a ductile punching shear

failure and improved post-cracking behaviour and residual load-carrying capacity after

reaching maximum load. Improvements in the strain energy absorption capability can be

attributed to the high amount of energy absorbed in debonding, stretching and pulling out

of the fibres after cracking. However, the ultimate punching resistance only increased by

10 %. It is thought that fibre inclusion has less beneficial effects on the ultimate load

84

when combined with axial restraint. Swamy and Ali (1982) reported an increase of 40%

in the ultimate resistance corresponding to 1 % of fibre volume (refer to section 2.7.2).

In addition, the presence of fibres prevented the propagation of the shear crack from the

compression zone (top surface) to the tension zone, impeding the formation of the failure

cone observed in Specimens P-1. Specimen F-1 remained intact after failure.

Therefore, while the use of 1% fibres by volume fraction improved the ductility of the

shear failure of concrete slabs, their ultimate punching resistance was not improved

significantly. It is obvious that the axial stress had greater effect on the ultimate punching

resistance.

Table 5.2: Effect of fibre reinforcement

Speci-men

Fibre % by

volume vf

f’c

[MPa] t

[mm] d

[mm] Pmax [kN]

max

c

P. f ' d

maxUΔ [J]

P-1 0.0 65.4 130 93.6 488 644.7 2673 F-1 1.0 59.9 127 91.4 503 710.1 7843

(a) (b)

300

400

500

600

700

800

Fibre Volume %

P max

/ ( d

. √

f’ c )

P-1 F-10.00 1.00 2000

3000400050006000700080009000

Fibre Volume %

Stra

in E

nerg

y A

bsor

ped

[J]

P-1 F-10.00 1.00

Figure 5.4: Effect of fibre reinforcement on (a) Normalized punching load (b) Strain energy absorbed

85

5.4.3 Effect of Varying Lateral Stresses in One Direction

For Specimen F-1 and Specimen F-2, an effective stress (normalized by √f’c) of

approximately 0.785 was applied in one direction, while the effective stresses in the other

direction were 0.74 and 0.61, respectively. Varying the axial stresses in one direction had

minor effect on the general behaviour of the tested specimens. Both specimens

experienced similar behaviour, strain energy absorption capabilities, and failure mode.

Increasing the effective axial stresses by 27 % corresponded to a 5% increase in failure

loads. The energy absorption capacity remained virtually unchanged. In general, varying

the lateral stress level in one direction had less impact on the behaviour than did the other

variables studied. Figure 5.5 and Table 5.3 summarize the effects of varying the post-

tensioning stresses in one direction.

Table 5.3: Effect of lateral stress in one direction

N-S Direction E-W DirectionSpeci-men

f’c

[MPa] fn

[MPa] n

c

ff '

fn [MPa]

n

c

ff '

d [mm]

Pmax[kN]

max

c

P. f ' d

maxUΔ [J]

F-1 59.9 6.0 0.78 6.0 0.74 503 710.1 7843 F-2 54.8 4.5 0.61 5.8 0.79

91.4 457 675.8 7800

86

(a) (b)

300

400

500

600

700

800

fn / √ f’c

P max

/ ( d

. √

f’ c )

F-2 F-10.770.610.00 1.00

20003000400050006000700080009000

fn / √ f’c

Stra

in E

nerg

y A

bsor

ped

[J]

F-2 F-10.000.61

1.000.77

Figure 5.5: Effect of varying post-tensioning stress in one direction on (a) Normalized punching load (b) Strain energy Absorbed

5.5 Variation of the top and bottom bars forces during testing

Forces in the post-tensioning bars were affected by the following movements of the

specimen during testing:

(1) the specimen expanding laterally, inducing additional strain on the bars which

resulted in an increase in bars forces.

(2) rotation of the sides of the specimen which forced the side beams to rotate (only

for Specimen P-1 and Specimen P-2) resulting in an increase in the bottom bars

forces and a decrease in the top bars forces (see Figure 5.6). For the same side

rotation, the change in bar forces in the N-S direction would be greater than the

change in the E-W direction as longer beams were used in the N-S direction.

Specimen P-3, Specimen F-1, and Specimen F-2 were affected by mechanism (1) only;

the top and bottom bars forces were similar, and followed the behaviour depicted in

Figure 5.2. On the other hand, the observed axial load versus vertical load curves

observed for Specimen P-1 and Specimen P-2 were different due to the effects of side

87

rotation. During the uncracked stage, the bottom bars forces increased while the top bars

forces decreased, resulting in the total force practically remaining constant. In Stage ІІ,

the specimen began to expand laterally resulting in an increase in both the top and bottom

forces. As a result of the combined effect of (1) and (2), the bottom bar forces increased

at a higher rate, while the top bar forces reached a minimum value then started to

increase. The specimens reached their maximum load and failed at the end of Stage ІІ.

Figure 5.7 illustrates the variation of top and bottom bars forces during testing.

Figure 5.6: Effect of side rotation on bar forces

Axial Load

Ver

tical

Loa

d

Top Bars Forces Bottom Bars Forces

ІІІ

PmaxPcrack

Figure 5.7: Variation of top and bottom bars forces for Specimen P-1 and Specimen P-2

88

5.6 Performance of testing set-up and collected data

The overall behaviour of the testing set-up and the data collected were satisfactory.

Concerns, difficulties encountered, and improvement of the testing set-up and testing

procedures are discussed in the following section.

5.6.1 Application of pure axial load

The design of the test set-up was driven by the desire to apply pure axial load on the

specimens. The data obtained from the first two tested specimens (Specimen P-1 and

Specimen P-2) showed that direct contact between the bearing plate and the specimen

could not ensure that. For subsequent tests, a 32-mm round bar was welded to the side

channels, between the bearing plate and the specimen. The round bar, located at the

centre of the specimen, ensured that no bending moment was transferred to the specimen

but created another problem. The side beams were free to rotate; excessive rotations were

observed during testing and during post-tensioning. To minimize beam rotation, two

spacers were positioned between the bearing plate and the specimen on one side of the

specimen. The opposite beam remained free to rotate to ensure that top and bottom bars

forces are equal. The observed behaviour of the specimen tested this way (Specimen F-1)

was satisfactory.

89

5.6.2 Centre Displacement

The centre displacement of the specimen could not be measured directly. The podium,

resting directly on the machine base plate, prevented the possibility of mounting LVDTs

on the bottom surface of the slab. In addition, the machine head and the side beams left

very limited clearance on the top surface. For future tests, it would be more reliable to

develop a method for direct measurement of the centre displacement. By looking at

Figure 4.4, the difference in displacement is due to slack in the testing set-up. The slack

in the testing set-up is caused by bumps and undulations present in both the machine base

plate and the podium, causing uneven contact surfaces. The load-deformation curve

obtained from the inner LVDT was more reliable.

5.6.3 Bar strains and Bar forces

For the bars where the load cells were installed (top bars of pair 2,3 Figure 3.2), a

comparison between the forces obtained from strain readings and from the load cells

readings indicate that the latter were more accurate. They correlated very well in all tests.

The maximum force recorded corresponded to 27% of the ultimate strength; no losses

due to relaxation were observed. In futures tests using the same set-up, if higher stress

levels are to be applied, relaxation losses would be expected and installation of load cells

on each bar may be justified.

90

Chapter 6: Theoretical Prediction

6.1 Modified Compression Field Theory prediction

The specimens were modeled using the nonlinear finite element (FE) analysis software

VecTor3. VecTor3, formerly known as SPARCS, was developed at the University of

Toronto for the analysis reinforced concrete solids. Reinforced concrete is modeled as an

isotropic material before cracking and as an orthotropic material afterwards. Material

models and constitutive relations were derived from the Modified Compression Field

Theory (MCFT) developed by Vecchio and Collins (1986) for 2-D analysis. The program

is based on an iterative total stress and strain formulation in which secant moduli are

defined and progressively refined according to current local stress-strain states. A more

detailed description of the program and the analysis procedures can be found in Vecchio

and Selby 1991. Commercial program GID was used as a pre-and-post processor for

VecTor3.

6.1.1 Model Description

Due to symmetry in both orthogonal directions, only one-quarter of the specimen was

modeled. Concrete was modeled using constant-strain 8-noded hexahedron elements with

orthogonal sides and 24 degrees of freedom. The specimen was divided into 10 elements

across the depth to capture the stress variation in the z-direction. This forced the aspect

ratio of the elements to be 1:1:0.22. The FE mesh is presented in Figure 6.1. The model

91

consisted of 1694 element and 2165 node; the average element size was 58 x 58 x 12.7

mm. For better distribution of the imposed vertical load and to prevent local failure, the

loading plate was modeled as a concrete element with significantly higher compressive

and tensile strengths and stiffnesses. VecTor3 input files of the model are given in

Appendix C.

All nodes across the axes of symmetry were restrained against in-plane displacement.

Nodes corresponding to the location of the roller supports (bottom layer at 75 mm from

the edges) were restrained in the z-direction. The supports conditions are illustrated in

Figure 6.2.

Vertical loading was modeled as an imposed downward displacement at two nodes at the

quarter-points of the loading plate. Axial stresses were applied as concentrated loads in

both orthogonal directions, distributed over all nodes at the edges. The axial forces were

kept constant through the loading stages.

For Specimen P-1, end bending moment was applied as a series of concentrated forces

applied at all nodes (except the top and bottom nodes to avoid stress concentration

problems) as shown in Figure 6.3. Linear variation was assumed to approximate the

variation of the end moments observed during testing. It was reasonable to apply the

vertical load in a force-controlled method, as the specimen failed at peak load; no post-

peak response was expected.

92

Figure 6.1: Finite element mesh

Restraint in Z-DirectionPoint of application of vertical displacement

(0,0)

(750,750)x

y

Figure 6.2: Support conditions and load application points

93

Figure 6.3: Application of end moment to Specimen P-1

6.1.2 Analytical Models

Table 6.1 summarizes the analytical models used in the FE analysis. The effect of

concrete tension softening model was of particular importance as the specimens

contained no internal rebars. For plain concrete specimens, the linear with no-residual

model (Model No. 1) was chosen to represent the brittle behaviour expected for such

specimens. The VecTor3 library contains two models to represent the tension softening

behaviour of FRC (Model No. 6 & 7). Unfortunately, these models were developed for

types and volume percentages of fibres different than those used in the specimens

investigated herein. However, these models were used to gauge, roughly, the

corresponding effect of the behaviour.

Table 6.1: Analytical models used in the FE analysis

Concrete compression base curve Hognestad (Parabola) Concrete compression post-peak Modified Park-Kent Concrete compression softening Vecchio 1992-A

Concrete tension stiffening Modified Bentz Concrete tension softening Linear – no residual * Concrete confinement strength Kupfer / Richart Concrete dilatation Variable – Kupfer Concrete cracking criterion Mohr-Coulomb (stresses)

* other models were used for FRC specimens

94

6.1.3 Comparison between Experimental and VecTor3 Model Results

The analytical responses are summarized in Table 6.2 and Figure 6.4 through Figure 6.7.

For the FRC specimens, the number in parenthesis corresponds to the tension softening

model (Model No. 6 for FRC Dramix 45/30; Model No. 7 for FRC Dramix 80/30). The z-

displacement of Specimen P-3 at intermediate load stage is shown in Figure 6.8.

In general, VecTor3 model was able to predict the ultimate loads of plain concrete

specimens with reasonable accuracy. However, it was not able to replicate the observed

load-deformation response. The FE model demonstrated higher initial stiffness and the

displacements at failure were considerably less than observed. This may be partially due

to the effect of machine and test set-up slack which increased the recorded displacement.

However, the same observation was made for other specimens that failed in punching

shear tested by other researchers (for instance Swamy and Ali 1982).

For Specimen P-3, cracks were initiated on the bottom surface under the point of load

application. As the vertical load increased, cracks propagated in the diagonal and

orthogonal directions. No cracks were observed on the top surface. For Specimen P-1,

cracks were initiated on the top surface as an effect of the applied end moments. As the

vertical load increased, cracks started to form on the bottom surface in a pattern similar to

those of Specimen P-3. The crack pattern and propagation sequence implied that the

damage modes of both models were flexure which is contrary to the test results.

95

There is a discrepancy between the experimental behaviour and the VecTor3 model.

Given that VecTor3 has accurately predicted structural behaviour in a wide range of

cases, it is unlikely that the observed discrepancy is an indicative of a problem with

VecTor3. Further study is required to identify the source of this discrepancy.

For the FRC specimens, there is a need to develop a tension softening model

representative of the fibre type and volume percentage used in the specimens. It is

obvious from the presented load-deformation responses that using the FRC tension

softening models increased the maximum load and the maximum displacements.

Table 6.2: Summary of VecTor3 analysis

Specimen Max. Load

Ptheor. [kN]

Disp. at max. load

[mm]

theor

exp.

PP

P-1 432.0 7.02 0.88 P-3 242.8 0.86 1.02 F-1* 419.6 5.23 0.83 F-2* 365.2 1.20 0.80

* Obtained using tension softening model 1

96

0

100

200

300

400

500

600

-2 0 2 4 6 8 10 12

Displacement [mm]

Loa

d [k

N]

ExperimentalVecTor3

Figure 6.4: Experimental versus Analytical load-deformation response of Specimen P-1

0

50

100

150

200

250

300

0 2 4 6 8 10 12 14 16 18 20

Displacement [mm]

Loa

d [k

N]

ExperimentalVecTor3

Figure 6.5: Experimental versus Analytical load-deformation response of Specimen P-3

97

0

100

200

300

400

500

600

700

0 5 10 15 20 25 30

Displacement [mm]

Loa

d [k

N]

ExperimentalVecTor3(1)VecTor3(6)VecTor3(7)

Figure 6.6: Experimental versus Analytical load-deformation response of Specimen F-1

0

100

200

300

400

500

600

700

0 5 10 15 20 25 30

Displacement [mm]

Loa

d [k

N]

ExperimentalVecTor3(1)VecTor3(6)VecTor3(7)

Figure 6.7: Experimental versus Analytical load-deformation response of Specimen F-2

98

Figure 6.8: z-displacement of Specimen P-3 at intermediate load stage

6.2 CSA prediction

The Canadian code CSA A23.3-04 (Clause 18.12.3.3) determines the punching strength

of prestressed slabs by:

max c oP = v b d (1) where:

c nc p c c

c c

fv = f ' 1 0.33 f '

p

o

Vb d

φβ φφ

+ + (2)

and where fn is the average value of axial stress in the two directions. The second term in

Eq. (2) represents the vertical component of the axial force, which is zero for all

specimens considered here. The resistance factor for concrete, cφ was taken as unity,

p β as 0.33, and d as 0.72 t.

99

Table 6.3: CSA Prediction of the failure load of specimens

Speci-men t f’

c [MPa]

fn(N-S) [MPa]

fn(E-W) [MPa]

faverage [MPa]

d [mm]

bo [mm] Ptheor.

theor

exp.

PP

P-1 130 65.4 6.1 6.0 6.04 93.60 1174.4 529.8 1.09 P-2 127 64.1 4.2 4.7 4.43 91.44 1165.8 460.9 0.68 P-3 127 68.5 1.9 2.2 2.05 91.44 1165.8 385.4 1.61 F-1 127 59.9 6.0 6.0 6.00 91.44 1165.8 498.0 0.99 F-2 127 54.8 4.5 5.8 5.17 91.44 1165.8 459.8 1.01

Comparison reveals that the code equation predicted with reasonable accuracy the

punching failure load of Specimen P-1 and, surprisingly, Specimen F-1 and Specimen F-

2. However, the code equation is not making any allowance for the beneficial influence

of the fibres. Had Specimen F-1 and Specimen F-2 contained plain concrete, their

strengths would likely have been over-predicted. Underestimation of the failure load of

Specimen P-2 was expected due to the additional lateral restraint provided (refer to

section 5.3). Overestimation of the failure load of Specimen P-3 is due to the flexural

stresses observed (longitudinal cracks on the top surface). The failure mode of Specimen

P-3 was combined flexural-punching, while code equation presumes a pure punching

failure mode.

6.3 Prediction using Hewitt and Batchelor (1975) model

The model proposed by Hewitt and Batchelor (1975) was used to predict the punching

shear capacity of the tested specimens. More details of the model, failure criteria, and

input data are given in Section 2.5.1. The program code and output text files are given in

Appendix D. The equivalent diameter of the loaded area is taken as the diameter of the

100

circle with the same perimeter as the loaded area (254.6 mm), and the equivalent

diameter of the slab is taken as the diameter of the largest circle which could be inscribed

within the area of the slab (1350 mm). The Hewitt and Batchelor model was found to

underestimate the ultimate punching load for the tested specimens.

Table 6.4: Maximum load prediction using Hewitt and Batchelor model

Speci-men

f’c

[MPa] d

[mm] F N-S [kN]

F E-W [kN]

Faverage [kN]

F1*

[kN/m]Ptheor. [kN]

Δ calc. [mm]

theor

exp.

PP

P-1 65.4 93.60 1192 1162 1177 785 460.9 4.25 0.94 P-2 64.1 91.44 792 897 845 563 332.1 5.37 0.49 P-3 68.5 91.44 366 417 391 261 154.6 10.71 0.65 F-1 59.9 91.44 1148 1135 1141 761 433.2 4.21 0.86 F-2 54.8 91.44 861 1110 985 657 378.4 4.49 0.83

*Average axial force per unit length applied of the specimen (Faverage /1.5)

101

Chapter 7: Conclusions and Recommendations

7.1 Conclusions

Based on the test results, and the analytical work performed, the following conclusions

can be drawn:

• A prestressed slab specimen, being predominantly in compression, is inclined to

fail in punching shear mode, rather than flexure, and to surpass the ultimate loads

calculated based on flexural strength.

• The level of axial stress applied to the specimen was observed to affect all aspects

of the behaviour, including cracking, deflection, stiffness, and failure mode.

• The higher the restraint provided, in this case the higher the post-tensioning stress,

the higher the ultimate strength and stiffness, and the lower the ductility.

• A stress level as low as 2.0 MPa was able to provide lateral restraint necessary to

prevent premature pure flexural failure of Specimen P-3, which failed in a

combined flexural-punching mode at a significantly higher load than expected by

flexure failure.

• A stress level as high as 6.0 MPa was able to provide the lateral restraint

necessary to ensure pure punching failure of Specimen P-1.

• Adding steel fibres in an amount equal to 1 % by volume fraction resulted in a

ductile punching shear failure and improved post-cracking behaviour and residual

load-carrying capacity after reaching maximum load.

102

• Failure of FRC specimens occurred over several minutes and exhibited audible

and visible signs of distress, contrary to the failure of the plain concrete

specimens.

• The improvement in the ductility of the punching failure due to the use of fibres

may allow reducing the factor of safety used for design.

• Fibre inclusion has less beneficial effect on the ultimate load when combined with

axial restraint.

• Varying the axial stresses in one direction had minor effect on the general

behaviour of the specimens.

• No obvious shear strength reduction was observed when reducing the level of

stress in one direction.

• Studying the effect of the axial stiffness of the bars (EA/L) was not one of the

objectives of this work. However, it certainly affected the force in the post-

tensioning bars at failure which, in turn, affected the punching strength of the

slabs.

• The CSA code formulations predicted the punching strength of the specimens

with reasonable accuracy.

• The MCFT predicted the ultimate punching failure load of the plain concrete

specimens reasonably accurately. However, it was not able to replicate the

observed load-deformation response.

103

7.2 Recommendations

With regard to future research into the behaviour of restrained slabs and testing set-up,

several recommendations and words of advice can be noted:

• The behaviour of the axially restrained slabs is affected by the span-to-depth ratio;

experimental investigation of this parameter is needed.

• There is a need to develop a method for direct measurement of centre

displacement, eliminating the effects of slack in both the loading machine and

testing set-up on the recorded displacement.

• The forces in the post-tensioning bars increased during testing due to specimen

lateral expansion. For future tests, if the axial stresses are needed to be kept

constant, another system should be developed for the application of axial forces.

• Direct contact between the bearing plate and the specimen cannot ensure the

application of pure axial load to the specimen even if the forces in both top and

bottom bars were exactly the same at the beginning of the test. These forces will

vary due to the effect of the rotation of the specimen sides inducing bending

moments on the sides.

• Excessive rotations of the side beams may affect the accuracy of the test results;

the use of a locking mechanism is recommended.

• A thorough investigation is needed to assess the abilities and limitations of

VecTor3 in modeling specimens that failed in punching shear mode.

104

Chapter 8: References

ACI Committee 544, “Design Consideration for Steel Fibre Reinforced Concrete”, ACI

Structural Journal, v 85, n 5, Sep-Oct, 1988, p 563-580.

Aoki, Y., and Seki, H., “Shearing Strength and Cracking in Two-way Slabs Subjected to

Concentrated Load”, Cracking, Deflection, and Ultimate Load of Concrete Slab Systems,

American Concrete Institute Publication SP-30, 1974, p 103-126.

ASTM C39/C39M-94: Standard Test Method for Compression Strength of Cylindrical

Concrete Specimens.

ASTM C496-96: Standard Test Method for Splitting Tensile Strength of Cylindrical

Concrete Specimens.

Canadian Highway Bridge Design Code, 1998, Canadian Standards Association,

Rexdale, Ontario, Canada

CSA. 2000. FRC deck slabs. In Canadian highway bridge design code. Section 16.7.

Canadian Standard Association (CSA), Rexdale, Ont.

105

Harajli, M.H., and Maalouf, D., and Khatib, H. “Effect of Fibers on the Punching Shear

Strength of Slab-Column Connections”, Cement & Concrete Composites, v 17, n 2, 1995,

p 161-170.

Hassan, A.; Kawakami, M.; Niitani, K.; Yoshioka, T. “An experimental investigation of

steel-free deck slabs”, Canadian Journal of Civil Engineering, v 29, n 6, December,

2002, p 831-841.

He, W., “Punching behaviour of composite bridge decks with transverse prestressing”,

PhD Thesis, Queen’s University, Kingston, Canada, 1992.

Hewitt, Brian E., and Batchelor, Barrington deV, “Punching Shear Strength of Restrained

Slabs”, ASCE Journal of Structural Division, v 101, n 9, Sep, 1975, p 1837-1853.

Kinnunen, S., and Nylander, H., “Punching of Concrete Slabs without shear

Reinforcement”, Transaction of the Royal Institute of Technology, Stockholm, Sweden,

No. 158, 1960.

Moe, J., “Shearing Strength of Reinforced Concrete Slabs and Footings under

Concentrated Load”, Bulletin D47, Research and Development Department, Portland

Cement Association, Skokie, Ι11. , 1961.

106

Mufti, A. A., and Newhook John P., “Punching Shear Strength of Restrained Concrete

Bridge Deck Slabs”, ACI Structural Journal, v 95, n 4, Jul-Aug, 1998, p 375-381.

Mufti, A. A.; Jaeger, L. G.; Bakht, B.; and Wagner, L. D., “Experimental Investigation of

Fibre-Reinforced Concrete Deck Slabs Without Internal Steel Reinforcement”, Canadian

Journal of Civil Engineering, V.20, n 3, Jun, 1993, p 398-406.

Newhook John P., “The Behaviour of Steel-Free Concrete Bridge Deck Slabs under

Static Loading Conditions”, PhD Thesis, Department of Civil Engineering, Dalhousie

University, Canada, 1997.

Shaaban, A.M., and Gesund, H., “Punching Shear Strength of Steel Fibre Reinforced

Concrete Flat Plates”, ACI Structural Journals, v 91, n 4, Jul-Aug, 1994, p 406-416

Shehata, Ibrahim A.E.M, and Regan, Paul E. “Punching in R.C. slabs”, Journal of

Structural Engineering, v 115, n 7, Jul, 1989, p 1726-1740.

Sissakis, Ken, “Strengthening concrete slabs for punching shear with CFRP laminates”,

M.A.Sc. Thesis, Department of Civil Engineering, University of Toronto, Canada, 2002

Swamy, R. N., and Ali S. A. R., “Punching Shear Behaviour of Reinforce Slab-Column

Connections Made with Steel Fibre Concrete”, J.ACI Journal, V 79, N 5, Sep-Oct, 1982,

p 392-406.

107

Taylor, R., and Hayes, B., “Some Tests on the Effect of Edge Restraint on Punching

Shear in Reinforced Concrete Slabs”, Magazine of Concrete Research, v 17, n 50, March,

1965, p 39-44.

Theodorakopoulos, D.D., and Swamy, R.N. “Ultimate punching shear strength analysis

of slab-column connections”, Cement and Concrete Composites, v 24, n 6, December,

2002, p 509-521.

UNI-11039, Steel fibre reinforced concrete - Part I: Definitions, classification

specification and conformity - Part II: test method for measuring first crack strength and

ductility indexes, Italian Board for Standardization, 2003.

Vecchio, F.J., and Collins, M.P.” The Modified Compression Field Theory for

Reinforced Concrete Elements Subjected to Shear”, Journal of the American Concrete

Institute, v 83, n 2, March-April, 1986, pp. 219-231

Vecchio, F.J., and Selby, R.G., “Towards Compression Field Analysis of Reinforced

Concrete Solids”, ASCE Journal of Structural Engineering, v 117, n 6, June, 1991,pp.

1740-1758.

108

Wagner, L. D., and Mufti, A. A., “Finite Element Investigation of Fibre-Reinforced

Concrete Deck Slabs without Internal Steel Reinforcement”, Canadian Journal of Civil

Engineering, v 21, n 2, Apr, 1994, p 231-236.

Wood, R. H., Plastic and Elastic Design of Slabs and Plates, London, Thames and

Hudson, 1961. p 253.

109

Appendix A

A.1 Concrete Stress-Strain Curves & Selected Photos

A.2 Fracture Energy Test Results & Selected Photos

A.3 Material Datasheet (from Manufacturer)

A.4 Formwork Details

A.5 Baldwin Universal Testing Machine

A.6 Workshop Drawings of the Podium

A.7 Workshop Drawings of the Side Beams

A.8 SAP model for calculating axial stress distribution

110

A.1 Concrete Stress Strain Curves

0

10

20

30

40

50

60

70

0.0 0.5 1.0 1.5 2.0 2.5

Strain x 10-3 [mm/mm]

Stre

ss [

MPa

]

f’c = 61.5 MPa

εc = 1.81 x 10-3

Concrete Stress Versus Strain (Specimen P-1 28-Days moist cured)

0

10

20

30

40

50

60

70

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Strain x 10-3 [mm/mm]

Stre

ss [

MPa

]

f’c = 63.6 MPa

εc = 2.48 x 10-3

Concrete Stress Versus Strain of 3 Cylenders (Specimen P-1 28-Days lab cured)

111

Selected Photos

Cylinders and beams for one specimen

Cylinders stored at the vibrating table

112

Specimen covered with wet burlap and plastic sheets

Specimens and cylinders stored pending testing

113

A.2 Fracture Energy Test Results

Specimen F-1

Cracking

Load [kN]

Cracking stresses [MPa]

Max. Load [kN]

Absorbed Energy up to 25 % of max.

load [J]

Absorbed Energy per unit area of the crack

[J/m2] Beam-1 24.95 6.79 46.85 260.38 16532 Beam-2 23.32 6.35 46.42 160.99 10222 Beam-3 24.12 6.56 40.42 197.38 12532 Average 24.13 6.57 44.56 205.25 13095

Specimen F-2

Cracking

Load [kN]

Cracking stresses [MPa]

Max. Load [kN]

Absorbed Energy up to 25 % of max.

load [J]

Absorbed Energy per unit area of the crack

[J/m2] Beam-1 21.63 5.89 37.70 94.96 6029 Beam-2 23.30 6.34 44.49 221.80 14083 Beam-3 22.03 5.99 34.23 151.94 9647 Average 22.32 6.07 38.81 156.23 9920

114

0

10

20

30

40

50

0.0 5.0 10.0 15.0 20.0

Displacement [mm]

Load

[ kN

]

B1B2B3

Pmax.(ava.) = 44.6 kN

Beams Load-Deformation ResponsesSpecimen F-1

0

10

20

30

40

50

0.0 5.0 10.0 15.0 20.0

Displacement [mm]

Load

[ kN

]

Beam-1Beam-2Beam-3

Pmax.(ava.) = 38.8 kN

Beams Load-Deformation ResponsesSpecimen F-2

115

0

10

20

30

40

50

0.0 5.0 10.0 15.0 20.0 25.0

Displacement [mm]

Load

[ kN

]

Beam-1Beam-2Beam-3

Pmax.(ava.) = 44.6 kN

Beams Load-CMOD ResponsesSpecimen F-1

0

10

20

30

40

50

0.0 5.0 10.0 15.0 20.0 25.0

Displacement [mm]

Load

[ kN

]

Beam-1Beam-2Beam-3

Pmax.(ava.) = 38.8 kN

Beams Load-CMOD ResponsesSpecimen F-2

Note: Crack mouth open displacement (CMOD) was measured 14 mm from bottom surface (see horizontal LVDT in photo)

116

Selected Photos

Testing Set-up

117

Beam-1

Beam-2

Beam-3

Tested Beams (Specimens F-1)

118

Beam-1

Beam-2

Beam-3

Tested Beams (Specimens F-2)

119

A.3 Material Datasheet (from Manufacturer) A.3.1 Steel Fibre Data

120

A.3.2 Dywidag Bars Post-tensioning System

121

122

123

124

125

A.4 Formwork Details

(All dimensions in mm)(All dimensions in mm)

Formwork Details

126

Formwork Details

127

A.5 Baldwin Universal Testing Machine

Baldwin Testing Machine

128

Tension Crosshead

Movable

Screw

Platform

Specimen

20 hp Motor

Tension CrossheadPositions

Ladder

Sensitive CrossheadConsole

Trench

Main Cylinder

Pump

Piston(flushed totop)

Base Plate

LoadCells

Baldwin Testing Machine (front View)

129

The previous figure illustrates the mechanism of applying a compressive load to the

specimen. Hydraulic oil is pumped to the main cylinder of the moving crosshead below

the base plate. The moving crosshead is pushed down forcing the sensitive crosshead

down by pulling on large screws. The sensitive crosshead may be positioned anywhere

along the screws to adjust to line length of the test specimen. The capacity of the machine

is 5300 kN (1200 kip) and the testing opening is 3000 x 6700 mm (10 x 22 ft.). The

Baldwin machine is force controlled, where the load is monitored through three load cells

located in the sensitive cross head.

130

A.6 Workshop Drawings of the Podium

HS

S 3

05x2

03x1

3 L

= 15

00 m

m2-

1A

LEN

GTH

ma

PC

MK

1A 1B

45 r

od4

1300

DE

SC

RIP

TIO

N

HS

S 2

54x1

52x1

3

HS

S 2

54x1

52x1

3

QTY 2 2

1500

1196

Uni

vers

ity o

f Tor

oto

TITL

E

DW

G. N

O.

Dra

win

g 1

D-1

Pod

ium

Det

ails

HS

S 3

05x2

03x1

3 L

= 11

96 m

m2-

1B

PL

152x

252

pa15

00

PL

152x

252

pb11

96

660

-400

60-4

006

221

282

494

282

(Typ

.)

494

282

6928

2

124.0

56.0

pb

pa

Ø60

(Typ

.)Ø

60

131

Pod

ium

Det

ails

11961

Uni

vers

ity o

f Tor

oto

Pos

ition

ing

of H

SS

Sec

tion

DW

G. N

O.

TITL

E

E1

Sec

tion

1-1

Typ.

152

HSS

203

x152

x13

1500

1B

1A

6(T

yp.)

1A

1B

152x

25 P

late

132

Podi

um D

etai

lsE3

DW

G. N

O.

Uni

vers

ity o

f Tor

oto

Pos

ition

ing

of U

pper

Pla

tes

TITL

E

152

HSS

203

x152

x13

152x

25 P

late

45 m

m R

olle

r

Sec

tion

1-1

Typ.

2

152

HSS

203

x152

x13

152x

25 P

late

45 m

m R

olle

r

Sec

tion

2-2

Typ.

275

(Typ

.)

pd

pcpc

pd

60.00

45°

70.00

45°

pc pd1300

1300

133

1350

Pod

ium

Det

ails

Uni

vers

ity o

f Tor

oto

DW

G. N

O.

TITL

E

E4

Sec

tion

3-3

HSS

254

x152

x13

45 m

m D

iam

eter

Rol

ler

152x

25 P

late

Con

cret

e Sp

ecim

en

152x

25 P

late

1500

(Not

Sen

t)

134

A.7 Workshop Drawings of the Side Beams

204

aa

186

292

DY

WID

AG

Anc

hor P

late

127

494

258

305

Bea

m D

etai

ls

8-2A

2 65

6

127

2 C

310x

45

L=

686

mm

1

6

ma

aa pa

Anc

hor P

late

DY

WID

AG

62A

PC

MK

102

ma6

170

64

ma

61-PL

-pa

Uni

vers

ity o

f Tor

oto

LEN

GTH

281

16C

100x

11

L89x

64x7

.9

(cut

to fi

t Cha

nel P

rofil

e)

PL

170x

38

8 8

175

204

C31

0x45

Cha

nnel

Ass

embl

y

DES

CR

IPTI

ON

Sec

tion

1-1

168

QTY

686

DW

G. N

O.

TITL

E

D-2

Dra

win

g 2

Sec

tion

2-2

89

44

6Ty

p.

44175

6

24-Mar-2006

135

Bea

m D

etai

ls

8-3A

2 C

310x

45

L=

822

mm

Uni

vers

ity o

f Tor

oto

8pa

PL

170x

38

DW

G. N

O.

204

TITL

E

D-3

Dra

win

g 3

254 32

6 360

127

630

305

652

6

Anc

hor P

late

DY

WID

AG

6

ma

1

127

DY

WID

AG

Anc

hor P

late 6

64

170

6 aa

102

1-P

L-pa

6

Sec

tion

1-1

Cha

nnel

Ass

embl

y

(cut

to fi

t Cha

nel P

rofil

e)

aa8

ma

3A

PC

MK

168

QTY 16

L89x

64x7

.9

C10

0x11

C31

0x45

DE

SCR

IPTI

ON

44m

a

6

Sec

tion

2-2

175

281

LEN

GTH

822

Typ.

44

204

175

89

6

136

A.8 SAP 2000 Model for Calculating Axial Stress Distribution

Mesh and applied loads

Support conditions

137

Axial stress distribution due to the applied forces Note: The applied forces were chosen to result in a nominal compressive stress of 1.0 MPa.

138

Deformed Shape

139

Appendix B

Graphs from Experimental Results B.1 Vertical LVDT Readings versus Applied Vertical Load

B.2 Bar Forces versus Applied Vertical Load

140

B.1 Vertical LVDT Readings versus Applied Vertical Load

-12

-8

-4

0

4

8

12

16

20

24

28

32

0 100 200 300 400 500 600 700

Applied Vertical Load [kN]

LV

DT

Dis

plac

emen

t [m

m]

N-EN-WS-ES-WC-EC-W

Specimen P-1

-12

-8

-4

0

4

8

12

16

20

24

28

32

0 100 200 300 400 500 600 700

Applied Vertical Load [kN]

LVD

T D

ispl

acem

ent [

mm

]

N-EN-WS-ES-WC-EC-W

Specimen P-2

141

-12

-8

-4

0

4

8

12

16

20

24

28

32

0 100 200 300 400 500 600 700

Applied Vertical Load [kN]

LVD

T D

ispl

acem

ent [

mm

]

N-EN-WS-ES-WC-EC-W

Specimen P-3

-12

-8

-4

0

4

8

12

16

20

24

28

32

0 100 200 300 400 500 600 700

Applied Vertical Load [kN]

LVD

T D

ispl

acem

ent [

mm

]

N-EN-WS-ES-WC-EC-W

Specimen F-1

142

-12

-8

-4

0

4

8

12

16

20

24

28

32

0 100 200 300 400 500 600 700

Applied Vertical Load [kN]

LVD

T D

ispl

acem

ent [

mm

]N-EN-WS-ES-WC-EC-W

Specimen F-2

143

B.2 Bar Forces versus Applied Vertical Load

0

100

200

300

400

500

600

700

800

900

0 100 200 300 400 500 600 700

Applied Vertical Load [kN]

Axi

al F

orce

[kN

]

Top bars forcesBottom bars forces

Specimen P-1 (North-South Direction)

0

100

200

300

400

500

600

700

800

900

0 100 200 300 400 500 600 700

Applied Vertical Load [kN]

Axi

al F

orce

[kN

]

Top bars forcesBottom bars forces

Specimen P-1 (East-West Direction)

144

0

100

200

300

400

500

600

700

800

900

0 100 200 300 400 500 600 700

Applied Vertical Load [kN]

Axi

al F

orce

[kN

]

Top bars forcesBottom bars forces

Specimen P-2 (North-South Direction)

0

100

200

300

400

500

600

700

800

900

0 100 200 300 400 500 600 700

Applied Vertical Load [kN]

Axi

al F

orce

[kN

]

Top bars forcesBottom bars forces

Specimen P-2 (East-West Direction)

145

0

100

200

300

400

500

600

700

800

900

0 100 200 300 400 500 600 700

Applied Vertical Load [kN]

Axi

al F

orce

[kN

]

Top bars forcesBottom bars forces

Specimen P-3 (North-South Direction)

0

100

200

300

400

500

600

700

800

900

0 100 200 300 400 500 600 700

Applied Vertical Load [kN]

Axi

al F

orce

[kN

]

Top bars forcesBottom bars forces

Specimen P-3 (East-West Direction)

146

0

100

200

300

400

500

600

700

800

900

0 100 200 300 400 500 600 700Applied Vertical Load [kN]

Axi

al F

orce

[kN

]

Top bars forcesBottom bars forces

Specimen F-1 (North-South Direction)

0

100

200

300

400

500

600

700

800

900

0 100 200 300 400 500 600 700

Applied Vertical Load [kN]

Axi

al F

orce

[kN

]

Top bars forcesBottom bars forces

Specimen F-1 (East-West Direction)

147

0

100

200

300

400

500

600

700

0 100 200 300 400 500 600 700 800 900

Applied Vertical Load [kN]

Axi

al F

orce

[kN

]

Top bars forcesBottom bars forces

Specimen F-2 (North-South Direction)

0

100

200

300

400

500

600

700

800

900

0 100 200 300 400 500 600 700

Applied Vertical Load [kN]

Axi

al F

orce

[kN

]

Top bars forcesBottom bars forces

Specimen F-2 (East-West Direction)

148

Appendix C

VecTor3 Input files C.1 vector.job

C.2 vertical.l3r

C.3 px.s3r

C.4 py.s3r

C.5 vector.s3r

C.6 End moment variation of Specimen P-1

149

C.1 Vector.job V e c T o r J O B D A T A Job Title (30 char max) : VecTor3 Job File Name ( 8 char max) : vector Date (30 char max) : Date STRUCTURE DATA -------------- Structure Type : 3 File Name (8 char max) : vector LOADING DATA ------------ No. of Load Stages : 99 Starting Load Stage No. : 1 Load Series ID (5 char max) : v3 Load File Name | Factors | Case (8 char max) Initial Final LS-Inc Type Reps C-Inc 1 vertical 0.010 15.00 0.050 1 1 0.000 2 px 2.200 2.200 0.000 1 1 0.000 3 py 1.900 1.900 0.000 1 1 0.000 4 NULL 0.000 0.000 0.000 1 1 0.000 5 NULL 0.000 0.000 0.000 1 1 0.000

ANALYSIS PARAMETERS ------------------- Analysis Mode (1-2) : 1 Seed File Name (8 char max) : NULL Convergence Limit (>1.0) : 1.001 Averaging Factor (<1.0) : 0.5 Maximum Iterations : 50 Convergence Criteria (1-5) : 2 Results Files (1-4) : 2 Output Format (1-3) : 1

MATERIAL/STRUCTURAL BEHAVIOUR MODELS ------------------------------------ Concrete Compression Base Curve (0-5) : 1 Concrete Compression Post-Peak (0-5) : 1 Concrete Compression Softening (0-9) : 1 Concrete Tension Stiffening (0-6) : 1 Concrete Tension Softening (0-4) : 1 Concrete Tension Splitting (0-2) : 1 Concrete Confined Strength (0-3) : 1 Concrete Dilatation (0-2) : 1 Concrete Cracking Criterion (0-4) : 1 Concrete Crack Slip Check (0-2) : 1 Concrete Crack Width Check (0-5) : 3 Concrete Bond or Adhesion (0-4) : 1 Concrete Creep and Relaxation (0-1) : 1 Concrete Hysteresis (0-5) : 2 Reinforcement Hysteresis (0-3) : 3 Reinforcement Dowel Action (0-1) : 0 Reinforcement Buckling (0-1) : 0 Element Strain Histories (0-1) : 1 Element Slip Distortions (0-10) : 3 Strain Rate Effects (0-1) : 1 Structural Damping (0-1) : 1 Geometric Nonlinearity (0-1) : 0 Crack Allocation Process (0-1) : 1 <<< JOB FILE NOTES>>> [As of June 05, 2002]

150

C.2 vertical.l3r V e c T o r 3 L O A D D A T A LOAD CASE PARAMETERS Structure Title (30 char. max.) : VecTor3 Load Case Title (30 char. max.) : VecTor3 Load Case File Name (8 char. max.) : vector No. of Loaded Joints : 0 No. of Prescribed Support Displacements : 2 No. of Elements with Gravity Forces : 0 No. of Elements with Temperature Change : 0 No. of Elements with Concrete Prestrain : 0 No. of Elements with Ingress Pressure : 0 No. of Element Surfaces w/ Thermal Load : 0 No. of Nodes with Impulse Forces : 0 Ground Acceleration Record (0-1) : 0 JOINT LOADS <NOTE:> UNITS kN <<<<< FORMAT >>>>> NODE Fx Fy Fz [ #NODE d(NODE d(Fx) d(Fy) d(Fz) ] <-- up to 2 directions/ / SUPPORT DISPLACEMENTS <NOTE:> UNITS mm <<<<< FORMAT >>>>> NODE Dx Dy Dz [ #NODE d(NODE d(Dx) d(Dy) d(Dz) ] <-- up to 2 directions / 2152 0.0 0.0 -1/ 2153 0.0 0.0 -1/ / GRAVITY LOADS <NOTE:> UNITS: KG/M3 <<<<< FORMAT >>>>> ELMT DENS GX GY GZ [#ELMT d(ELMT)] <-- up to 3 directions / / TEMPERATURE LOADS <NOTE:> UNITS: C <<<<< FORMAT >>>>> ELMT TEMP [ #ELMT d(ELMT) d(TEMP) ] <-- up to 3 directions / / CONCRETE PRESTRAINS <NOTE:> UNITS: me <<<<< FORMAT >>>>> ELMT STRAIN [ #elmt d(ELMT) d(STRAIN) ] <-- up to 3 directions / / INGRESS PRESSURES <NOTE:> UNITS: MPa <<<<< FORMAT >>>>> ELMT PRESSURE [ #ELMT d(ELMT) d(PRS) ] <-- up to 3 directions / / SURFACE THERMAL LOADS <NOTE:> UNITS: Sec, Degrees C <<<<< FORMAT >>>>>

151

NODE1 NODE2 Tm1 Tp1 Tm2 Tp2 Tm3 Tp3 [#SURF d(NODE)] <-- up to 3 directions / / IMPULSE FORCES <NOTE:> UNITS: Sec, kN <<<<< FORMAT >>>>> NODE DOF T1 F1 T2 F2 T3 F3 T4 F4 [ #NODE d(NODE) ] / / GROUND ACCELERATION <NOTE:> UNITS: Sec, m/s2 <<<<< FORMAT >>>>> TIME ACC-X ACC-Y / <NOTES:>

152

C.3 px.l3r V e c T o r 3 L O A D D A T A LOAD CASE PARAMETERS Structure Title (30 char. max.) : VecTor3 Load Case Title (30 char. max.) : VecTor3 Load Case File Name (8 char. max.) : vector No. of Loaded Joints : 154 No. of Prescribed Support Displacements : 0 No. of Elements with Gravity Forces : 0 No. of Elements with Temperature Change : 0 No. of Elements with Concrete Prestrain : 0 No. of Elements with Ingress Pressure : 0 No. of Element Surfaces w/ Thermal Load : 0 No. of Nodes with Impulse Forces : 0 Ground Acceleration Record (0-1) : 0 JOINT LOADS <NOTE:> UNITS kN <<<<< FORMAT >>>>> NODE Fx Fy Fz [ #NODE d(NODE d(Fx) d(Fy) d(Fz) ] <-- up to 2 directions/ 1 0.432955 0.0 0.0/ 2 0.432955 0.0 0.0/ 3 0.432955 0.0 0.0/ 4 0.432955 0.0 0.0/ 5 0.432955 0.0 0.0/ 6 0.432955 0.0 0.0/ 11 0.432955 0.0 0.0/ 16 0.432955 0.0 0.0/ 21 0.432955 0.0 0.0/ 28 0.432955 0.0 0.0/ 35 0.432955 0.0 0.0/ 7 0.764886 0.0 0.0/ 10 0.764886 0.0 0.0/ 12 0.764886 0.0 0.0/ 15 0.764886 0.0 0.0/ 17 0.764886 0.0 0.0/ 19 0.764886 0.0 0.0/ 25 0.764886 0.0 0.0/ 30 0.764886 0.0 0.0/ 33 0.764886 0.0 0.0/ 44 0.764886 0.0 0.0/ 54 0.764886 0.0 0.0/ 38 0.663864 0.0 0.0/ 40 0.663864 0.0 0.0/ 41 0.663864 0.0 0.0/ 46 0.663864 0.0 0.0/ 48 0.663864 0.0 0.0/ 52 0.663864 0.0 0.0/ 60 0.663864 0.0 0.0/ 67 0.663864 0.0 0.0/ 73 0.663864 0.0 0.0/ 78 0.663864 0.0 0.0/ 84 0.663864 0.0 0.0/ 88 0.663864 0.0 0.0/ 93 0.663864 0.0 0.0/ 96 0.663864 0.0 0.0/ 98 0.663864 0.0 0.0/

153

101 0.663864 0.0 0.0/ 106 0.663864 0.0 0.0/ 113 0.663864 0.0 0.0/ 119 0.663864 0.0 0.0/ 126 0.663864 0.0 0.0/ 132 0.663864 0.0 0.0/ 139 0.663864 0.0 0.0/ 158 0.663864 0.0 0.0/ 161 0.663864 0.0 0.0/ 165 0.663864 0.0 0.0/ 167 0.663864 0.0 0.0/ 169 0.663864 0.0 0.0/ 173 0.663864 0.0 0.0/ 180 0.663864 0.0 0.0/ 187 0.663864 0.0 0.0/ 195 0.663864 0.0 0.0/ 203 0.663864 0.0 0.0/ 211 0.663864 0.0 0.0/ 242 0.663864 0.0 0.0/ 244 0.663864 0.0 0.0/ 245 0.663864 0.0 0.0/ 248 0.663864 0.0 0.0/ 251 0.663864 0.0 0.0/ 254 0.663864 0.0 0.0/ 268 0.663864 0.0 0.0/ 276 0.663864 0.0 0.0/ 286 0.663864 0.0 0.0/ 292 0.663864 0.0 0.0/ 299 0.663864 0.0 0.0/ 347 0.663864 0.0 0.0/ 350 0.663864 0.0 0.0/ 353 0.663864 0.0 0.0/ 357 0.663864 0.0 0.0/ 360 0.663864 0.0 0.0/ 365 0.663864 0.0 0.0/ 373 0.663864 0.0 0.0/ 381 0.663864 0.0 0.0/ 388 0.663864 0.0 0.0/ 395 0.663864 0.0 0.0/ 403 0.663864 0.0 0.0/ 469 0.663864 0.0 0.0/ 471 0.663864 0.0 0.0/ 472 0.663864 0.0 0.0/ 477 0.663864 0.0 0.0/ 479 0.663864 0.0 0.0/ 481 0.663864 0.0 0.0/ 489 0.663864 0.0 0.0/ 497 0.663864 0.0 0.0/ 506 0.663864 0.0 0.0/ 512 0.663864 0.0 0.0/ 519 0.663864 0.0 0.0/ 603 0.663864 0.0 0.0/ 606 0.663864 0.0 0.0/ 612 0.663864 0.0 0.0/ 614 0.663864 0.0 0.0/ 617 0.663864 0.0 0.0/ 620 0.663864 0.0 0.0/ 628 0.663864 0.0 0.0/ 636 0.663864 0.0 0.0/ 650 0.663864 0.0 0.0/ 660 0.663864 0.0 0.0/ 670 0.663864 0.0 0.0/ 761 0.663864 0.0 0.0/

154

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<NOTE:> UNITS: KG/M3 <<<<< FORMAT >>>>> ELMT DENS GX GY GZ [#ELMT d(ELMT)] <-- up to 3 directions / / TEMPERATURE LOADS <NOTE:> UNITS: C <<<<< FORMAT >>>>> ELMT TEMP [ #ELMT d(ELMT) d(TEMP) ] <-- up to 3 directions / / CONCRETE PRESTRAINS <NOTE:> UNITS: me <<<<< FORMAT >>>>> ELMT STRAIN [ #elmt d(ELMT) d(STRAIN) ] <-- up to 3 directions / / INGRESS PRESSURES <NOTE:> UNITS: MPa <<<<< FORMAT >>>>> ELMT PRESSURE [ #ELMT d(ELMT) d(PRS) ] <-- up to 3 directions / / SURFACE THERMAL LOADS <NOTE:> UNITS: Sec, Degrees C <<<<< FORMAT >>>>> NODE1 NODE2 Tm1 Tp1 Tm2 Tp2 Tm3 Tp3 [#SURF d(NODE)] <-- up to 3 directions / / IMPULSE FORCES <NOTE:> UNITS: Sec, kN <<<<< FORMAT >>>>> NODE DOF T1 F1 T2 F2 T3 F3 T4 F4 [ #NODE d(NODE) ] / / GROUND ACCELERATION <NOTE:> UNITS: Sec, m/s2 <<<<< FORMAT >>>>> TIME ACC-X ACC-Y / <NOTES:>

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C.4 py.l3r V e c T o r 3 L O A D D A T A LOAD CASE PARAMETERS Structure Title (30 char. max.) : VecTor3 Load Case Title (30 char. max.) : VecTor3 Load Case File Name (8 char. max.) : vector No. of Loaded Joints : 154 No. of Prescribed Support Displacements : 0 No. of Elements with Gravity Forces : 0 No. of Elements with Temperature Change : 0 No. of Elements with Concrete Prestrain : 0 No. of Elements with Ingress Pressure : 0 No. of Element Surfaces w/ Thermal Load : 0 No. of Nodes with Impulse Forces : 0 Ground Acceleration Record (0-1) : 0 JOINT LOADS <NOTE:> UNITS kN <<<<< FORMAT >>>>> NODE Fx Fy Fz [ #NODE d(NODE d(Fx) d(Fy) d(Fz) ] <-- up to 2 directions/ 1 0.0 0.432955 0.0/ 2 0.0 0.432955 0.0/ 3 0.0 0.432955 0.0/ 4 0.0 0.432955 0.0/ 5 0.0 0.432955 0.0/ 6 0.0 0.432955 0.0/ 11 0.0 0.432955 0.0/ 16 0.0 0.432955 0.0/ 21 0.0 0.432955 0.0/ 28 0.0 0.432955 0.0/ 35 0.0 0.432955 0.0/ 8 0.0 0.764886 0.0/ 9 0.0 0.764886 0.0/ 13 0.0 0.764886 0.0/ 14 0.0 0.764886 0.0/ 18 0.0 0.764886 0.0/ 20 0.0 0.764886 0.0/ 24 0.0 0.764886 0.0/ 29 0.0 0.764886 0.0/ 34 0.0 0.764886 0.0/ 43 0.0 0.764886 0.0/ 53 0.0 0.764886 0.0/ 37 0.0 0.663864 0.0/ 39 0.0 0.663864 0.0/ 42 0.0 0.663864 0.0/ 45 0.0 0.663864 0.0/ 49 0.0 0.663864 0.0/ 51 0.0 0.663864 0.0/ 59 0.0 0.663864 0.0/ 66 0.0 0.663864 0.0/ 74 0.0 0.663864 0.0/ 77 0.0 0.663864 0.0/ 83 0.0 0.663864 0.0/ 89 0.0 0.663864 0.0/ 92 0.0 0.663864 0.0/ 95 0.0 0.663864 0.0/ 97 0.0 0.663864 0.0/

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762 0.0 0.663864 0.0/ 764 0.0 0.663864 0.0/ 766 0.0 0.663864 0.0/ 768 0.0 0.663864 0.0/ 776 0.0 0.663864 0.0/ 786 0.0 0.663864 0.0/ 794 0.0 0.663864 0.0/ 803 0.0 0.663864 0.0/ 808 0.0 0.663864 0.0/ 818 0.0 0.663864 0.0/ 924 0.0 0.663864 0.0/ 926 0.0 0.663864 0.0/ 931 0.0 0.663864 0.0/ 932 0.0 0.663864 0.0/ 939 0.0 0.663864 0.0/ 942 0.0 0.663864 0.0/ 952 0.0 0.663864 0.0/ 961 0.0 0.663864 0.0/ 979 0.0 0.663864 0.0/ 989 0.0 0.663864 0.0/ 999 0.0 0.663864 0.0/ 1129 0.0 0.620568 0.0/ 1133 0.0 0.620568 0.0/ 1135 0.0 0.620568 0.0/ 1140 0.0 0.620568 0.0/ 1142 0.0 0.620568 0.0/ 1147 0.0 0.620568 0.0/ 1155 0.0 0.620568 0.0/ 1164 0.0 0.620568 0.0/ 1172 0.0 0.620568 0.0/ 1178 0.0 0.620568 0.0/ 1183 0.0 0.620568 0.0/ 1317 0.0 0.577273 0.0/ 1318 0.0 0.577273 0.0/ 1323 0.0 0.577273 0.0/ 1324 0.0 0.577273 0.0/ 1328 0.0 0.577273 0.0/ 1333 0.0 0.577273 0.0/ 1342 0.0 0.577273 0.0/ 1350 0.0 0.577273 0.0/ 1361 0.0 0.577273 0.0/ 1364 0.0 0.577273 0.0/ 1370 0.0 0.577273 0.0/ 1520 0.0 0.288636 0.0/ 1523 0.0 0.288636 0.0/ 1524 0.0 0.288636 0.0/ 1527 0.0 0.288636 0.0/ 1533 0.0 0.288636 0.0/ 1535 0.0 0.288636 0.0/ 1544 0.0 0.288636 0.0/ 1552 0.0 0.288636 0.0/ 1561 0.0 0.288636 0.0/ 1567 0.0 0.288636 0.0/ 1575 0.0 0.288636 0.0/ / SUPPORT DISPLACEMENTS <NOTE:> UNITS mm <<<<< FORMAT >>>>> NODE Dx Dy Dz [ #NODE d(NODE d(Dx) d(Dy) d(Dz) ] <-- up to 2 directions / / GRAVITY LOADS

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<NOTE:> UNITS: KG/M3 <<<<< FORMAT >>>>> ELMT DENS GX GY GZ [#ELMT d(ELMT)] <-- up to 3 directions / / TEMPERATURE LOADS <NOTE:> UNITS: C <<<<< FORMAT >>>>> ELMT TEMP [ #ELMT d(ELMT) d(TEMP) ] <-- up to 3 directions / / CONCRETE PRESTRAINS <NOTE:> UNITS: me <<<<< FORMAT >>>>> ELMT STRAIN [ #elmt d(ELMT) d(STRAIN) ] <-- up to 3 directions / / INGRESS PRESSURES <NOTE:> UNITS: MPa <<<<< FORMAT >>>>> ELMT PRESSURE [ #ELMT d(ELMT) d(PRS) ] <-- up to 3 directions / / SURFACE THERMAL LOADS <NOTE:> UNITS: Sec, Degrees C <<<<< FORMAT >>>>> NODE1 NODE2 Tm1 Tp1 Tm2 Tp2 Tm3 Tp3 [#SURF d(NODE)] <-- up to 3 directions / / IMPULSE FORCES <NOTE:> UNITS: Sec, kN <<<<< FORMAT >>>>> NODE DOF T1 F1 T2 F2 T3 F3 T4 F4 [ #NODE d(NODE) ] / / GROUND ACCELERATION <NOTE:> UNITS: Sec, m/s2 <<<<< FORMAT >>>>> TIME ACC-X ACC-Y / <NOTES:>

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C.5 vector.s3r S T R U C T U R E D A T A STRUCTURAL PARAMETERS Structure title (30 char. max.) : VecTor3 Structure file name (8 char. max.) : vector No. of reinforced concrete material types : 2 No. of steel material types : 0 No. of hexahedral elements : 1694 No. of wedge elements : 0 No. of truss elements : 0 No. of nodes : 2165 No. of restraints : 333 MATERIAL SPECIFICATIONS (A) REINFORCED CONCRETE ----------------------- <NOTE:> To be used in 8-node brick,6-node brick,3-node torus,and 4-node torus elements only. CONCRETE -------- MAT NS f'c [f't Ec e0 Mu Cc Agg Dens Kc] [Sx Sy Sz] TYP MPa MPa MPa me /C mm kg/m3 mm2/2 mm mm mm 1 0 68.5 2.66 48285 1.81 0.20 0.0 10 2400 0.0 50 50 50 / 2 0 400 400 200000 20 0.3 0.0 10 2400 0.0 50 50 50 / / REINFORCEMENT COMPONENTS ------------------------ MAT SRF ORIENT. RHO Db Fy Fu Es Esh esh Cs Dep TYP TYP k l m % mm MPa MPa MPa MPa me /C me / (B) STEEL -------- <NOTE:> To be used in truss / ring elements only. MAT REF AREA Db Fy Fu Es Esh esh Cs Dep TYP TYP mm2 mm MPa MPa MPa MPa me /C me / ETC. <NOTES>

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C.5 End moment variation of Specimen P-1

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0

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60

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100

0 50 100 150 200 250 300 350 400 450 500

Vertical Load

Est

imat

ed e

nd m

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ExperimentVecTor3

Bending Moment Applied to the VecTor3 ModelN-S Direction (Specimen P-1)

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Bending Moment Applied to the VecTor3 ModelE-W Direction (Specimen P-1)

162

Appendix D

Hewitt and Batchelor (Program B) D.1 User Interface

D.2 Input Parameters

D. 3 Source Code

D. 4 Output text files

163

D.1 User Interface

164

D.1 Input Parameters

Over all thickness (h)

Effective thickness of the slab (d)

Equivalent diameter of loaded area (b)

b = diameter of the circle having the same perimeter as the loaded area

Equivalent diameter of the slab (C)

C = diameter of the largest circle that could be inscribed within the area of the slab

Cylinder strength of concrete (f’c)

Reinforcement ratio (ρ) = area of steel / area of concrete.

Yield point of reinforcing steel (fy)

Boundary force (Fb) : boundary restraining force per unit length of the slab

Boundary moment (Mb) : boundary restraining moment per unit length of the slab

LoadP

M

Fb

b

d h

C

b

165

D.3 Source Code (Visual Basic 6.0)

General – Declaration Option Explicit Dim fnum As Integer 'TO Writre to file Dim OUTputFile As String Dim NewOutputFile As String Dim VariableString As String Dim OutputString(10000000) As String Dim NoOfOutputString As Long Dim TesT As String ' Test designation Dim T As Double ' Overall thickness Dim H As Double ' Effective thickness Dim B As Double ' Equivalent diameter of loaded area Dim C As Double ' Equivalent diameter of the slab Dim SIGCY As Double ' Cylender strength of concrete Dim Es As Double ' Modulus of Elasticity Dim MU As Double ' Reinforcement ratio Dim SIGSY As Double ' Yield point of steel Dim SIGCU As Double ' Cube strength of concrete Dim IFlag As Integer Dim KflaG As Integer Dim LflaG As Integer Dim MflaG As Integer Dim JflaG As Integer Dim DEFL As Double ' Slab Deflection at punching Dim M As Integer Dim N As Integer Dim I As Integer Dim My As Integer Dim NSY As Integer Dim NSX As Double Dim PDIff As Double Dim X As Double ' 4*pi*BM / P Dim YDelta As Double Dim XDelta As Double Dim XDHolD As Double Dim XDiff As Double Dim XchecK As Double Dim BF As Double ' Boundary Force Dim BM As Double ' Boundary Moment Dim BMPF As Double ' 4 pi* BM Dim YONH As Double ' ratio of Y/H Dim Y As Double ' Depth to root of shear crack (Centre of rotation of the concrete segment

166

Dim HoLD1 As Double Dim HoLD2 As Double Dim HoLD3 As Double Dim HoLD4 As Double Dim HoLD5 As Double Dim HoLD6 As Double Dim Ky As Double Dim KZ As Double Dim TANA As Double Dim FALfa As Double Dim BONH As Double ' B/H ratio Dim F As Double ' Constant = 14.22 Dim SIGT As Double Dim Pu As Double Dim P1 As Double Dim P2 As Double Dim PSI As Double Dim Rs As Double Dim Co As Double ' Radius to shear crack Dim R1 As Double ' Resultant in-plane force at punching shear.... Horizontal force in circumference reinforcement Dim R2 As Double ' Resultant in-plane force at punching shear.... Horizontal force in radial reinforcement Dim R3 As Double ' Resultant in-plane force at punching shear.... Horizontal tangential compressive force in concrete Dim PDHolD As Double Dim PeRR1 As Double ' Error between P1 & P2 Dim PeRR2 As Double ' Error Between X & Xcheck Dim YHolD As Double Dim CDifF As Double Dim RSCNH As Double ' Rs/H ratio Dim COCNH As Double ' Co/H ratio Dim Alfa As Double Dim PHolD As Double Dim PINC As Double Private Sub Command1_Click() OUTputFile = App.Path & "\" & Me.Text1.Text & ".txt" ‘ Output file name If Me.Text1.Text = "" Then OUTputFile = App.Path & "\Hewitt.txt" Call CreateFile(OUTputFile) ‘ Create a new output text file

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NoOfOutputString = 0 'READING DATA '------------------------------------------------------------------------------------------------ 40: TesT = Me.Text1.Text ' Unit conversion to Imperial Units If Me.Combo1.Text = "SI Units" Then T = Val(Me.Text2.Text) / 25.4 ' Overall Thickness H = Val(Me.Text3.Text) / 25.4 ' Effective thickness B = Val(Me.Text4.Text) / 25.4 ' Equivalent Diameter of loaded area C = Val(Me.Text5.Text) / 25.4 ' Equivalent diameter of slab SIGCY = Val(Me.Text6.Text) * 145.03772351661 ' Cylender strength of Concrete MU = Val(Me.Text7.Text) ' Reinforcement ratio SIGSY = Val(Me.Text8.Text) * 145.03772351661 ' Yield point of steel BF = Val(Me.Text9.Text) / 0.17512683523622 ' Boundary Restraining Force BM = Val(Me.Text10.Text) / 0.004448221615 ' Boundary Restraining Moment GoTo 50: End If T = Val(Me.Text2.Text) ' Overall Thickness H = Val(Me.Text3.Text) ' Effective thickness B = Val(Me.Text4.Text) ' Equivalent Diameter of loaded area C = Val(Me.Text5.Text) ' Equivalent diameter of slab SIGCY = Val(Me.Text6.Text) ' Cylender strength of Concrete MU = Val(Me.Text7.Text) ' Reinforcement ratio SIGSY = Val(Me.Text8.Text) ' Yield point of steel BF = Val(Me.Text9.Text) ' Boundary Restraining Force BM = Val(Me.Text10.Text) ' Boundary Restraining Moment 50: Es = 30000000 ' Modulus of elasticity of concrete PSI If B < 0 Then GoTo 390: 'mesh 3aref a2ra Call WriteFile(OUTputFile, "TEST :" & TesT) Call WriteFile(OUTputFile, "") Call WriteFile(OUTputFile, "INPUT :") Call WriteFile(OUTputFile, "********") Call WriteFile(OUTputFile, "")

168

Call WriteFile(OUTputFile, " Overall thickness of slab " & T & " inch = " & T * 25.4 & " mm") Call WriteFile(OUTputFile, " Effective thickness of slab " & H & " inch = " & H * 25.4 & " mm") Call WriteFile(OUTputFile, " Equivalent diameter of loaded area " & B & " inch = " & B * 25.4 & " mm") Call WriteFile(OUTputFile, " Equivalent diameter of slab " & C & " inch = " & C * 25.4 & " mm") Call WriteFile(OUTputFile, " Cylender Strength of concrete " & SIGCY & " PSI = " & SIGCY / 145.03772351661 & " Mpa") Call WriteFile(OUTputFile, " Ration of reinforcement " & MU) Call WriteFile(OUTputFile, " Yield point of steel " & SIGSY & " PSI = " & SIGSY / 145.03772351661 & " Mpa") Call WriteFile(OUTputFile, " Modulus of elasticity of steel " & Es & " PSI = " & Es / 145.03772351661 & " Mpa") Call WriteFile(OUTputFile, "") Call WriteFile(OUTputFile, " Boundary force = " & BF & " LBS/IN' = " & BF * 0.17512683523622 & " kN/m'") Call WriteFile(OUTputFile, " Boundary moment = " & BM & " IN LBS/IN' = " & BM * 0.004448221615 & " kN.m/m'") Call WriteFile(OUTputFile, "") Call WriteFile(OUTputFile, "") Call WriteFile(OUTputFile, "OUTPUT :") Call WriteFile(OUTputFile, "*********") SIGCU = SIGCY / (0.75 + 0.000025 * SIGCY) ' Cube strength of concrete IFlag = 1 X = 0 If MU <> 0 Then GoTo 133: Pu = 0.001 GoTo 240: 133: KflaG = 1 LflaG = 1 MflaG = 1 M = 0 N = 0 I = 0 My = 0 NSY = 0 NSX = 0 PDIff = 0 XDiff = 0

169

YDelta = 0.1 XDelta = 0.01 If IFlag = 1 Then GoTo 136: 'CALCULATE BOUNDARY RESTRAINT '---------------------------------------------------------------------------------------------------------- 135: BMPF = 12.568 * BM / 1000 ' 12.568 = 4 * pi 'SELECT X X = 1 'SELECT Y/H 136: YONH = 0.5 'CALCULATE P1 '------------------------------------------------------------------------------------------------------------ 140: Y = YONH * H If M < 200 And N < 50 And I < 5 Then GoTo 160: If IFlag = 1 Then Call WriteFile(OUTputFile, "") 150: Call WriteFile(OUTputFile, " Solutions have not closed EXIT") If IFlag = 1 Then GoTo 246: GoTo 246: 160: HoLD1 = (1 / 4.7) * (1 + Y / B) * Log(C / (B + 2 * Y)) Ky = (C - B) / (2 * (H - Y / 3)) KZ = Ky - X * C / (4 * (H - Y / 3)) HoLD2 = KZ + HoLD1 HoLD3 = -KZ - 1 HoLD4 = HoLD1 + 1 HoLD5 = (HoLD3 ^ 2 - 4 * HoLD2 * HoLD4) If HoLD5 >= 0 Then GoTo 168: TANA = -HoLD3 / (2 * HoLD2) JflaG = 2

170

GoTo 169: 168: TANA = (-HoLD3 - Sqr(HoLD5)) / (2 * HoLD2) JflaG = 1 169: FALfa = TANA * (1 - TANA) / (1 + TANA ^ 2) BONH = B / H F = 14.22 If BONH > 2 Then GoTo 170: SIGT = 825 * (0.35 + SIGCU / (F * 500)) * (1 - 0.22 * BONH) * F GoTo 180: 170: SIGT = 460 * (0.35 + SIGCU / (F * 500)) * F 180: P1 = 3.142 * BONH * YONH * (B + 2 * Y) * SIGT * FALfa * H ^ 2 / (B + Y) 'CALCULATION OF P2 '-------------------------------------------------------------------------------------------------------- If BONH > 2 Then GoTo 190: PSI = 0.0035 * (1 - 0.22 * BONH) * (1 + 0.5 * B / Y) GoTo 200: 190: PSI = 0.0019 * (1 + 0.5 * B / Y) 200: Rs = H * Es * PSI * (1 - YONH) / SIGSY Co = 0.5 * B + 1.8 * H HoLD6 = MU * SIGSY * H If Rs <= Co Then GoTo 210: If Rs > 0.5 * C Then Rs = 0.5 * C R1 = HoLD6 * ((Rs - Co) + Rs * Log(0.5 * C / Rs)) R2 = HoLD6 * Co GoTo 220: 210: R1 = HoLD6 * Rs * Log(0.5 * C / Co) R2 = HoLD6 * Rs 220: DEFL = PSI * (C - B) / 2

171

If IFlag = 1 Then P2 = 6.284 * (R1 + R2) / KZ Else P2 = 6.284 * (R1 + R2 + BF * C * (H - Y / 3 - DEFL) / (2 * (H - Y / 3))) / KZ End If 'COMPARE P1 & P2 '--------------------------------------------------------------------------------------------------------- PDHolD = PDIff PDIff = Abs(P1 - P2) PeRR1 = 100 * PDIff / P1 If PeRR1 < 1 Then GoTo 230: If My <> 0 And PDIff > PDHolD Then LflaG = 2 'ITERATION WITH Y/H '---------------------------------------------------------------------------------------------------- If KflaG <> 1 Then GoTo 227: If LflaG = 2 Then GoTo 225: If My > 15 Then GoTo 225: YHolD = YONH YONH = YONH * 0.5 * (1 + P2 / P1) If My > 10 Then YONH = (YONH + YHolD) / 2 M = M + 1 My = My + 1 GoTo 140: 'SEARCH FOR Y/H '---------------------------------------------------------------------------------------------------- 225: KflaG = 2 226: YONH = YONH + YDelta M = M + 1 NSY = NSY + 1 GoTo 140: 227: If PDIff < PDHolD Then GoTo 226: YONH = YONH - YDelta PDIff = PDHolD If KflaG = 3 Or NSY > 1 Then GoTo 228:

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KflaG = 3 YDelta = -YDelta GoTo 226: 228: KflaG = 2 YDelta = 0.1 * YDelta NSY = 0 GoTo 226: 230: Pu = (P1 + P2) / 2000 If X = 0 Then GoTo 240: XchecK = BMPF / Pu 'COMPARE X & XCHECK '------------------------------------------------------------------------------------------------------------- XDHolD = XDiff XDiff = Abs(X - XchecK) PeRR2 = Abs(100 * XDiff / X) If PeRR2 < 1 Then GoTo 240: 'ITERATION WITH X '------------------------------------------------------------------------------------------------------------- YDelta = 0.1 My = 0 NSY = 0 KflaG = 1 LflaG = 1 If MflaG <> 1 Then GoTo 237: If N <> 0 And XDiff > XDHolD Then GoTo 235: X = (X + XchecK) / 2 N = N + 1 GoTo 140: 'SEARCH FOR X '------------------------------------------------------------------------------------------------------------- 235: MflaG = 2

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236: X = X + XDelta N = N + 1 NSX = NSX + 1 GoTo 140: 237: If XDiff < XDHolD Then GoTo 236: X = X - XDelta XDiff = XDHolD If MflaG = 3 Or NSX > 1 Then GoTo 238: MflaG = 3 XDelta = -XDelta GoTo 236: 238: MflaG = 2 XDelta = 0.1 * XDelta NSX = 0 GoTo 236: 'PRINT OUTPUT '-------------------------------------------------------------------------------------------------------------- 240: If IFlag = 2 Then GoTo 242: Call WriteFile(OUTputFile, "") 241: Call WriteFile(OUTputFile, "Neglecting Boundary Restraint") Call WriteFile(OUTputFile, "*******************************") Call WriteFile(OUTputFile, "") If MU <= 0 Then GoTo 325: GoTo 246: 242: Call WriteFile(OUTputFile, "") Call WriteFile(OUTputFile, "Considering boundary resraint ") Call WriteFile(OUTputFile, "********************************") Call WriteFile(OUTputFile, "") Call WriteFile(OUTputFile, "") 246: If YONH <= 1 Then GoTo 250:

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Call WriteFile(OUTputFile, "Y/H Calculated to be greater than 1 EXIT") Call WriteFile(OUTputFile, "*********************************************") Call WriteFile(OUTputFile, "") Call WriteFile(OUTputFile, " Total No of Cycles") Call WriteFile(OUTputFile, " For Y/H " & M & " Error " & PeRR1) Call WriteFile(OUTputFile, " For X " & N & " Error " & PeRR2) Call WriteFile(OUTputFile, "") If IFlag = 1 Then GoTo 40: ' after calculation of P for the first restraint factor=0 go solve for the slab where fb , mb is known GoTo 390: 250: Call WriteFile(OUTputFile, " Y/H = " & YONH) If IFlag = 2 Then Call WriteFile(OUTputFile, " X = " & X) RSCNH = Rs / H Call WriteFile(OUTputFile, " Rs/H = " & RSCNH) COCNH = Co / H Call WriteFile(OUTputFile, " Co/H = " & COCNH) Call WriteFile(OUTputFile, " PSI = " & PSI & " Radians") Call WriteFile(OUTputFile, " Defl. = " & DEFL & " inch = " & DEFL * 25.4 & " mm") Alfa = Atn(TANA) Call WriteFile(OUTputFile, " Alpha = " & Alfa & " Radians") If JflaG = 2 Then Call WriteFile(OUTputFile, " (MAXIMIAED)") If IFlag = 2 Then GoTo 350: Call WriteFile(OUTputFile, "") Call WriteFile(OUTputFile, " No. of Cycles " & M & " Error " & PeRR1 & " %") M = 0 325: Call WriteFile(OUTputFile, " ") Call WriteFile(OUTputFile, "**** Ultimate Punching Load Pu = " & Pu & " Kips = " & Pu * 4.448221615 & " kN") Call WriteFile(OUTputFile, " ") Call WriteFile(OUTputFile, " ") PHolD = Pu ' ' ' IFlag = 2 GoTo 133: 350: Call WriteFile(OUTputFile, "")

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Call WriteFile(OUTputFile, " Total No of Cycles") Call WriteFile(OUTputFile, " For Y/H " & M & " Error " & PeRR1 & " %") If X <> 0 Then GoTo 364: GoTo 367: 364: Call WriteFile(OUTputFile, " For X " & N & " Error " & PeRR2 & " %") Call WriteFile(OUTputFile, "") 367: If PHolD <> 0 Then PINC = 100 * (Pu - PHolD) / PHolD Call WriteFile(OUTputFile, " ") Call WriteFile(OUTputFile, "**** Ultimate Punching Load Pu = " & Pu & " Kips = " & Pu * 4.448221615 & " kN") If PHolD > 0.01 Then Call WriteFile(OUTputFile, " % of increase due to Restraint = " & PINC & " %") 390: Me.Label21.Caption = Format(Pu, "#0.000000") & " Kips" Me.Label22.Caption = Format(Pu * 4.448221615, "#0.000000") & " kN" Me.Label24.Caption = Format(DEFL, "#0.000000") & " inch" Me.Label25.Caption = Format(DEFL * 25.4, "#0.000000") & " mm" 'Print to file Call WriteMultiLines(OUTputFile, OutputString()) End Sub

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D.4 Output text files of tested specimens

TEST :Specimen P-1 INPUT : ******** Overall thickness of slab 5.11811023622047 inch = 130 mm Effective thickness of slab 3.68503937007874 inch = 93.6 mm Equivalent diameter of loaded area 10.0255118110236 inch = 254.648 mm Equivalent diameter of slab 53.1496062992126 inch = 1350 mm Cylender Strength of concrete 9485.46711798629 PSI = 65.4 Mpa Ration of reinforcement 0 Yield point of steel 58015.089406644 PSI = 400 Mpa Modulus of elasticity of steel 30000000 PSI = 206842.739065498 Mpa Boundary force = 4482.46551672765 LBS/IN' = 785 kN/m' Boundary moment = 2.24808943112876E-02 IN LBS/IN' = 0.0001 kN.m/m' OUTPUT : ********* Neglecting Boundary Restraint ******************************* **** Ultimate Punching Load Pu = 0.001 Kips = 0.004448221615 kN Considering boundary resraint ******************************** Y/H = 0.441452142833222 X = 2.74178227338951E-06 Rs/H = 2.23978066770606 Co/H = 3.16029914529915 PSI = 7.7546966370595E-03 Radians Defl. = 0.167207135251898 inch = 4.2470612353982 mm Alpha = 0.208956474697714 Radians Total No of Cycles For Y/H 16 Error 0.599861436584726 % For X 26 Error 0.543462577245926 % **** Ultimate Punching Load Pu = 103.612811065353 Kips = 460.892745771815 kN TEST :Specimen P-2 INPUT : ******** Overall thickness of slab 5 inch = 127 mm Effective thickness of slab 3.6 inch = 91.44 mm

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Equivalent diameter of loaded area 10.0255118110236 inch = 254.648 mm Equivalent diameter of slab 53.1496062992126 inch = 1350 mm Cylender Strength of concrete 9296.9180774147 PSI = 64.1 Mpa Ration of reinforcement 0 Yield point of steel 58015.089406644 PSI = 400 Mpa Modulus of elasticity of steel 30000000 PSI = 206842.739065498 Mpa Boundary force = 3214.81284830276 LBS/IN' = 563 kN/m' Boundary moment = 2.24808943112876E-02 IN LBS/IN' = 0.0001 kN.m/m' OUTPUT : ********* Neglecting Boundary Restraint ******************************* **** Ultimate Punching Load Pu = 0.001 Kips = 0.004448221615 kN Considering boundary resraint ******************************** Y/H = 0.334478358379668 X = 3.81429137775175E-06 Rs/H = 3.37596467074688 Co/H = 3.1924321959755 PSI = 9.80969312684321E-03 Radians Defl. = 0.211517066651062 inch = 5.37253349293698 mm Alpha = 0.213714920520966 Radians Total No of Cycles For Y/H 17 Error 0.754775958272554 % For X 25 Error 0.781291411949732 % **** Ultimate Punching Load Pu = 74.6573069070379 Kips = 332.092246301575 kN TEST :Specimen P-3 INPUT : ******** Overall thickness of slab 5 inch = 127 mm Effective thickness of slab 3.6 inch = 91.44 mm Equivalent diameter of loaded area 10.0255118110236 inch = 254.648 mm Equivalent diameter of slab 53.1496062992126 inch = 1350 mm Cylender Strength of concrete 9935.08406088778 PSI = 68.5 Mpa Ration of reinforcement 0 Yield point of steel 58015.089406644 PSI = 400 Mpa Modulus of elasticity of steel 30000000 PSI = 206842.739065498 Mpa Boundary force = 1490.3484074725 LBS/IN' = 261 kN/m' Boundary moment = 2.24808943112876E-02 IN LBS/IN' = 0.0001 kN.m/m'

178

OUTPUT : ********* Neglecting Boundary Restraint ******************************* **** Ultimate Punching Load Pu = 0.001 Kips = 0.004448221615 kN Considering boundary resraint ******************************** Y/H = 0.149893011965051 X = 8.19142948266699E-06 Rs/H = 7.38188976377953 Co/H = 3.1924321959755 PSI = 1.95500634530601E-02 Radians Defl. = 0.421539391799928 inch = 10.7071005517182 mm Alpha = 0.231283466525977 Radians Total No of Cycles For Y/H 23 Error 0.763834296309298 % For X 24 Error 0.727565337624933 % **** Ultimate Punching Load Pu = 34.7449255879332 Kips = 154.553129011811 kN TEST :Specimen F-1 INPUT : ******** Overall thickness of slab 5 inch = 127 mm Effective thickness of slab 3.6 inch = 91.44 mm Equivalent diameter of loaded area 10.0255118110236 inch = 254.648 mm Equivalent diameter of slab 53.1496062992126 inch = 1350 mm Cylender Strength of concrete 8687.75963864494 PSI = 59.9 Mpa Ration of reinforcement 0 Yield point of steel 58015.089406644 PSI = 400 Mpa Modulus of elasticity of steel 30000000 PSI = 206842.739065498 Mpa Boundary force = 4345.42198500604 LBS/IN' = 761 kN/m' Boundary moment = 2.24808943112876E-02 IN LBS/IN' = 0.0001 kN.m/m' OUTPUT : ********* Neglecting Boundary Restraint ******************************* **** Ultimate Punching Load Pu = 0.001 Kips = 0.004448221615 kN

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Considering boundary resraint ******************************** Y/H = 0.457019453373304 X = 2.91624728652131E-06 Rs/H = 2.15886982230915 Co/H = 3.1924321959755 PSI = 7.68885899238178E-03 Radians Defl. = 0.165787540846917 inch = 4.21100353751168 mm Alpha = 0.202327751629418 Radians Total No of Cycles For Y/H 16 Error 0.535058826664847 % For X 26 Error 0.510947977129811 % **** Ultimate Punching Load Pu = 97.3823197046432 Kips = 433.178139429034 kN TEST :Specimen F-2 INPUT : ******** Overall thickness of slab 5 inch = 127 mm Effective thickness of slab 3.6 inch = 91.44 mm Equivalent diameter of loaded area 10.0255118110236 inch = 254.648 mm Equivalent diameter of slab 53.1496062992126 inch = 1350 mm Cylender Strength of concrete 7948.06724871023 PSI = 54.8 Mpa Ration of reinforcement 0 Yield point of steel 58015.089406644 PSI = 400 Mpa Modulus of elasticity of steel 30000000 PSI = 206842.739065498 Mpa Boundary force = 3751.56668087906 LBS/IN' = 657 kN/m' Boundary moment = 2.24808943112876E-02 IN LBS/IN' = 0.0001 kN.m/m' OUTPUT : ********* Neglecting Boundary Restraint ******************************* **** Ultimate Punching Load Pu = 0.001 Kips = 0.004448221615 kN Considering boundary resraint ******************************** Y/H = 0.420476023026906 X = 3.35133879083054E-06 Rs/H = 2.4549343392588 Co/H = 3.1924321959755 PSI = 8.19196678875591E-03 Radians Defl. = 0.176635574921208 inch = 4.48654360299868 mm Alpha = 0.205702317870426 Radians

180

Total No of Cycles For Y/H 16 Error 0.599379329025266 % For X 25 Error 0.889223915655765 % **** Ultimate Punching Load Pu = 85.0629706788997 Kips = 378.378944809993 kN