analysis and design of building structures aci 318r-11 with seismic considerations ibc 2012

ONE MAN'S MAGIC IS ANOTHER

MAN'S ENGINEERING.

SCIENTISTS INVESTIGATE THAT

WHICH ALREADY IS; ENGINEERS CREATE

THAT WHICH HAS NEVER BEEN.'

Compiled by:

IMRAN AKBER “10CE03” (Group Leader)

AQEEL JOKHIO “10CE131” (AGL)

ASHFAQUE CHANNA “10CE143”

MUHAMMAD SHOAIB “10CE147”

Supervised By:

PROF. PERVAIZ SHEIKH

DEPARTMENT OF CIVIL ENGINEERING MEHRAN UNIVERSITY OF ENGINEERING & TECHNOLOGY

JAMSHORO, SINDH

Submitted in partial fulfilment of the requirement for the degree of Bachelor of Civil Engineering

December 2013

DEPARTMENT OF CIVIL ENGINEERING

CERTIFICATE

This is to certify that

MR. IMRAN AKBER (GROUP LEADER) “10CE03”

MR. AQEEL JOKHIO “10CE131”

MR. ASHFAQUE CHANNA “10CE143”

MR. MUHAMMAD SHOAIB “10CE147”

Have completed the Thesis work entitled “ANALYSIS AND DESIGN OF

BUILDING STRUCTURES ACI 318R-11 WITH SEISMIC CONSIDERATIONS

IBC 2012” as a partial requirement for the Degree of Bachelor of Engineering.

(PERVAIZ SHEKH) Professor

Department of Civil Engineering Mehran University Of Engineering &

Technology, Jamshoro

(Dr. G. B. Khaskheli) Chairman

Department of Civil Engineering Mehran University Of Engineering &

Technology, Jamshoro

(AGL)

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ACKNOWLEDGEMENT

Foremost, we would like to express our sincere gratitude to our supervisor Prof.

Pervaiz Sheikh for the continuous support of our thesis, for his patience, motivation,

enthusiasm, and immense knowledge. His guidance helped us in all time of research

and writing of this thesis.

Besides our supervisor, we would like to thank our professional structural

engineers for their good advice and support on both academic and on personal level,

for which we are extremely grateful; Sir Zahoor Ahmed Memon for compilation of our

Section of Manual methods of Design and Sir Fahad Samo for compilation of our

Automated Section of Analysis and Design using ETABS and SAFE.

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ABSTRACT

We have worked under the title “Analysis & Design of Building Structures ACI

318R-11 with seismic considerations IBC 2012”.

We have collected Methods and procedures of Analysis and Design of Building

Structures along with Earthquake effects. We did it to introduce modern methods for

seismic loads calculation despite of IBC has replaced UBC in the year of 2000, still our

practices are being carried out as per UBC 1997.

In this achievement we have got remarkable data from US Geological Survey

which provided the basic parameters. Now, we learned to calculate Earthquake effects

using modern procedures of IBC 2012. As observing present situation of Building Code

of Pakistan we have achieved a new trend to do practice on.

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CONTENTS AT A GLANCE

STRUCTURAL ANALYSIS

1. Introduction to Structural Analysis

2. Structural Analysis: Basic Methods

3. Structural Analysis: Matrix Methods

STRUCTURAL DESIGN

4. Introduction to Structural Design

5. Design of Structural Members

EARTHQUAKE RESISTANT DESIGN

6. Earthquakes and Fundamentals of Ground Motion

7. Structural Response

8. Seismic Loading UBC 1997 & BCP, SP 2007

9. International Building Code IBC 2012

COMPUTERS & STRUCTURES

10. Seismic Analysis and Design of Multistoried RC Building using ETABS

C o

n t

e n

t s a

t a

G l

a n

c e

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TABLE OF CONTENTS

CHAPTER 1 ............................................................................................................................ 1

INTRODUCTION TO STRUCTURAL ANALYSIS .......................................................................... 1

1.1 INTRODUCTION ......................................................................................................... 1

1.2 ROLE OF STRUCTURAL ANALYSIS IN STRUCTURAL ENGINEERING PROJECTS ............... 1

1.3 CLASSIFICATION OF STRUCTURES .............................................................................. 3

1.3.1 Tension Structures ................................................................................................ 3

1.3.2 Compressive Structures......................................................................................... 4

1.3.3 Trusses.................................................................................................................. 4

1.3.4 Shear Structures ................................................................................................... 5

1.3.5 Bending Structures ............................................................................................... 6

1.4 ANALYSIS ................................................................................................................... 6

1.4.1 Free Body Diagrams .............................................................................................. 7

1.4.2 Sign Convention .................................................................................................... 8

1.5 STRUCTURAL RESPONSE ................................................................................................... 9

CHAPTER 2 .......................................................................................................................... 11

STRUCTURAL ANALYSIS: BASIC APPROACH .......................................................................... 11

2.1 DOUBLE INTEGRATION METHOD ............................................................................. 11

2.1.1 Elastic Curve ....................................................................................................... 11

2.1.2 Load or Moment Function .................................................................................. 12

2.1.3 Slope and Elastic Curve ....................................................................................... 12

2.2 MOMENT AREA THEOREM .............................................................................................. 13

2.2.1 M/EI Diagram ..................................................................................................... 13

2.2.2 Elastic Curve ....................................................................................................... 14

2.2.3 Moment‐Area Theorems ..................................................................................... 15

2.3 CONJUGATE BEAM METHOD ............................................................................................ 16

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2.3.1 Conjugate Beam ................................................................................................. 16

2.3.2 Equilibrium ......................................................................................................... 17

2.4 UNIT LOAD METHOD OR METHOD OF VIRTUAL WORK ............................................................ 18

2.4.1 Virtual Moments m or mu ................................................................................... 18

2.4.2 Real Moments .................................................................................................... 19

2.4.3 Virtual‐Work Equation ........................................................................................ 19

2.5 THREE MOMENT EQUATIONS ........................................................................................... 20

2.6 SLOPE DEFLECTION METHOD ........................................................................................... 23

2.6.1 Procedure ........................................................................................................... 24

CHAPTER 3 .......................................................................................................................... 26

STRUCTURAL ANALYSIS: MATRIX APPROACH ...................................................................... 26

3.1 INTRODUCTION ............................................................................................................ 26

3.1.1 Force Analysis: .................................................................................................... 26

3.1.2 Deformation Analysis: ........................................................................................ 26

3.1.3 Requirement ....................................................................................................... 26

3.2 FLEXIBILITY AND STIFFNESS ............................................................................................. 27

3.3 FORCE OR FLEXIBILITY METHOD ........................................................................................ 27

3.3.1 Basic concepts of Force or Flexibility method ...................................................... 27

3.3.2 Flexibility Coefficient ........................................................................................... 28

3.3.3 Generation of Flexibility Matrices [3.3] ............................................................... 28

3.3.4 Procedure to Apply Force Method ...................................................................... 32

3.4 DISPLACEMENT OR STIFFNESS METHOD .............................................................................. 32

3.4.1 Basic concepts of Displacement or Stiffness method: .......................................... 32

3.4.2 Stiffness, Stiffness Coefficient and Stiffness Matrix: ............................................ 33

3.4.3 Principles of Stiffness Method for Beams and Plane Frames ............................... 34

3.4.4 Generation of Stiffness Matrices [3.4] ................................................................. 34

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3.4.5 Direct Stiffness Method ...................................................................................... 54

3.4.6 Procedure to Apply Displacement Method .......................................................... 56

3.5 COMPARISON OF BOTH METHODS .................................................................................... 56

CHAPTER 4 .......................................................................................................................... 58

INTRODUCTION TO STRUCTURAL DESIGN ........................................................................... 58

4.1 REINFORCED CONCRETE ................................................................................................. 58

4.2 ADVANTAGES AND DISADVANTAGES OF REINFORCED CONCRETE ........................... 60

4.3 STRUCTURAL DESIGN ..................................................................................................... 61

4.3.1 Objectives ........................................................................................................... 61

4.4 DESIGN PHILOSOPHY AND CONCEPTS...................................................................... 61

4.4.1 Working Stress Method (WSM)........................................................................... 62

4.4.2 Unified Design Method (UDM) ............................................................................ 62

4.4.3 Ultimate Load Method (ULM) ............................................................................. 63

4.4.4 Limit States Method (LSM).................................................................................. 64

4.4.5 Summary of Design Methods .............................................................................. 67

4.5 CODES OF PRACTICE ...................................................................................................... 68

4.5.1 Purpose of Codes ................................................................................................ 68

4.5.2 Basic Codes for Design ........................................................................................ 68

4.6 LOADS ....................................................................................................................... 69

4.7 SAFETY PROVISIONS [4.2] .............................................................................................. 71

CHAPTER 5 .......................................................................................................................... 73

DESIGN OF STRUCTURAL MEMBERS .................................................................................... 73

5.1 INTRODUCTION ............................................................................................................ 73

5.2 RECTANGULAR BEAMS ................................................................................................... 74

5.2.1 Types of Flexural Failure and Strain Limits [5.1] .................................................. 74

5.2.2 Equivalent Compressive Stress Distribution ........................................................ 78

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5.2.3 Beam Design ....................................................................................................... 81

5.3 ANALYSIS AND DESIGN OF T SECTIONS ........................................................................ 92

5.3.1 Description ......................................................................................................... 92

5.3.2 Effective Width ................................................................................................... 93

5.3.3 T ‐Sections Behaving as Rectangular Sections ..................................................... 94

5.3.4 Analysis of a T‐Section ........................................................................................ 96

5.3.5 Design of T‐Sections ............................................................................................ 99

5.4 COLUMNS ................................................................................................................. 102

5.4.1 Types of Columns .............................................................................................. 102

5.4.2 Behavior of Axially Loaded Columns ................................................................. 105

5.4.3 ACI Code Limitations ......................................................................................... 107

5.4.4 Spiral Reinforcement ........................................................................................ 111

5.4.5 DESIGN EQUATIONS ......................................................................................... 113

5.5 ONE WAY SLABS ......................................................................................................... 115

5.5.1 One‐Way Beam–Slab Systems .......................................................................... 115

5.5.2 Temperature and Shrinkage Reinforcement ..................................................... 118

5.5.3 Design of One‐Way Solid Slabs ......................................................................... 119

5.6 TWO‐WAY SLABS ....................................................................................................... 126

5.6.1 Types of Two‐Way Slabs ................................................................................... 126

5.6.2 Economical Choice of Concrete Floor Systems ................................................... 130

5.6.3 Design Concepts ............................................................................................... 130

5.6.4 Column and Middle Strips ................................................................................. 131

5.6.5 Minimum Slab Thickness to Control Deflection ................................................. 133

5.6.6 Analysis of Two‐Way Slabs by The Direct Design Method ................................. 136

5.6.7 Summary of the Direct Design Method (DDM) .................................................. 145

5.7 FOUNDATIONS ........................................................................................................... 148

5.7.1 Types of Foundations ........................................................................................ 150

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5.7.2 Distribution of Soil Pressure .............................................................................. 153

5.7.3 Design Considerations [5.1] .............................................................................. 154

5.7.4 Combined Footings ........................................................................................... 162

5.7.5 Footings under Eccentric Column Loads ............................................................ 165

CHAPTER 6 ......................................................................................................................... 168

EARTHQUAKES AND FUNDAMENTALS OF GROUND MOTION ............................................. 168

6.1 EARTHQUAKES ........................................................................................................... 168

6.2 FUNDAMENTALS OF EARTHQUAKE GROUND MOTION ........................................................ 172

6.2.1 Introduction ...................................................................................................... 172

6.2.2 Recorded Ground Motion ................................................................................. 173

6.2.3 Characteristics of Earthquake Ground Motion .................................................. 174

6.2.4 Factors Influencing Ground Motion .................................................................. 174

CHAPTER 7 ......................................................................................................................... 175

STRUCTURAL RESPONSE .................................................................................................... 175

7.1 GENERAL ............................................................................................................... 175

7.2 STRUCTURAL CONSIDERATION ....................................................................................... 175

7.3 MEMBER CONSIDERATIONS .......................................................................................... 180

CHAPTER 8 ......................................................................................................................... 182

SEISMIC LOADING UBC 1997 & BCP, SP 2007 ...................................................................... 182

8.1 BUILDING CODES ........................................................................................................ 182

8.2 UNIFORM BUILDING CODE, UBC 1997 ........................................................................... 183

8.2.1 Division IV—Earthquake Design [8.3] ............................................................... 183

8.3 BUILDING CODE OF PAKISTAN, SEISMIC PROVISION, BCP SP, 2007 ......................................... 195

8.3.1 Static lateral force procedure ........................................................................... 196

8.3.2 Dynamic lateral force procedure ....................................................................... 196

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CHAPTER 9 ......................................................................................................................... 208

INTERNATIONAL BUILDING CODE 2012 .............................................................................. 208

9.1 SECTION 1613 ‐ EARTHQUAKE LOADS ............................................................................ 208

9.2 IBC DESIGN CRITERIA ................................................................................................... 209

9.2.1 Mapped Acceleration Parameters .................................................................... 209

9.2.2 Site Class........................................................................................................... 210

9.2.3 Site Coefficients and Adjusted Maximum Considered Earthquake Spectral

Response Acceleration Parameters .......................................................................................... 210

9.2.4 Design Spectral Acceleration Parameters [9.4] ................................................. 210

9.2.5 Design Response Spectrum ............................................................................... 210

9.2.6 Importance Factor and Occupancy Category [9.4] ............................................ 211

9.2.7 Seismic Design Category ................................................................................... 212

9.3 DESIGN REQUIREMENTS FOR SEISMIC DESIGN CATEGORY A ................................................. 212

9.4 DESIGN REQUIREMENTS FOR SEISMIC DESIGN CATEGORIES B, C, D, E, AND F [9.4] ................. 213

9.4.1 Structural System Selection ............................................................................... 214

9.4.2 Structural Irregularities ..................................................................................... 218

9.4.3 Analysis Procedure Selection............................................................................. 218

9.4.4 Equivalent Lateral Force Procedure .................................................................. 219

9.4.5 P‐∆ Effect .......................................................................................................... 239

9.4.6 Diaphragm ....................................................................................................... 240

9.4.7 Building Separation .......................................................................................... 241

9.4.8 Anchorage of Concrete or Masonry Walls ......................................................... 241

CHAPTER 10 ....................................................................................................................... 242

SEISMIC ANALYSIS AND DESIGN OF MULTISTORIED RC BUILDING USING ETABS ................. 242

10.1 ETABS (EXTENDED 3D ANALYSIS OF BUILDING SYSTEM) ..................................................... 242

10.1.1 Features and Benefits of ETABS .................................................................... 242

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10.2 PROJECT ................................................................................................................... 243

10.2.1 General Description ..................................................................................... 243

10.2.2 Drawings ...................................................................................................... 243

10.3 ETABS 2013 ............................................................................................................. 244

10.3.1 Material Properties ...................................................................................... 244

10.3.2 Load Cases ................................................................................................... 244

10.3.3 Analysis ........................................................................................................ 245

CONCLUSION ..................................................................................................................... 261

APPENDIX – A .................................................................................................................... 262

FIXED‐END MOMENTS ....................................................................................................... 262

APPENDIX‐B....................................................................................................................... 263

SHEAR FORCE AND BENDING MOMENT DIAGRAMS FOR SELECTED LOADING CASES ........... 263

APPENDIX‐C ....................................................................................................................... 266

VALUES FOR RU MAX, , ........................................................................................... 266

APPENDIX‐D ...................................................................................................................... 266

REBAR SIZE / SPACING CHART ............................................................................................ 266

APPENDIX‐E ....................................................................................................................... 268

REINFORCEMENT DESIGN AIDS .......................................................................................... 268

APPENDIX‐F ....................................................................................................................... 269

MINIMUM THICKNESS OF BEAMS & ONE‐WAY SOLID SLABS .............................................. 269

APPENDIX‐G ...................................................................................................................... 270

MINIMUM BEAM WIDTH (IN.) USING STIRRUPS ................................................................. 270

APPENDIX‐H ...................................................................................................................... 271

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RECTANGULAR SECTIONS WITH COMPRESSION STEEL MINIMUM STEEL PERCENTAGE FOR

COMPRESSION STEEL TO YIELD .................................................................................................... 271

APPENDIX‐I ........................................................................................................................ 272

MODULUS OF ELASTICITY OF CONCRETE (KSI) .................................................................... 272

APPENDIX‐J ....................................................................................................................... 273

AREAS OF GROUPS OF STANDARD U.S. BARS IN SQUARE INCHES ....................................... 273

.......................................................................................................................................... 273

APPENDIX‐K ....................................................................................................................... 274

AREAS OF BARS IN SLABS (SQUARE INCHES PER FOOT) ....................................................... 274

APPENDIX‐L ....................................................................................................................... 275

CENTROIDS OF AREAS ........................................................................................................ 275

REFERENCES ...................................................................................................................... 276

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LIST OF FIGURES

FIGURE 1.2‐1: A FLOWCHART SHOWING THE VARIOUS PHASES OF A TYPICAL STRUCTURAL ENGINEERING PROJECT ............ 2

FIGURE 1.3‐1: TENSION STRUCTURES ............................................................................................................ 3

FIGURE 1.3‐2: COMPRESSION STRUCTURES ..................................................................................................... 4

FIGURE 1.3‐3: TRUSSES .............................................................................................................................. 5

FIGURE 1.3‐4: SHEAR STRUCTURES ............................................................................................................... 5

FIGURE 1.3‐5: BENDING STRUCTURES ............................................................................................................ 6

FIGURE 1.4‐1: ANALYSIS ............................................................................................................................. 7

FIGURE 1.4‐2: FREE BODY DIAGRAM ............................................................................................................. 8

FIGURE 1.4‐3: SIGN CONVENTION ................................................................................................................ 9

FIGURE 2.1‐1: DOUBLE INTEGRATION METHOD ............................................................................................. 11

FIGURE 2.1‐2: ELASTIC CURVE ................................................................................................................... 12

FIGURE 2.2‐1: MOMENT AREA METHOD ...................................................................................................... 13

FIGURE 2.2‐2: MOMENT DIAGRAM ............................................................................................................. 14

FIGURE 2.2‐3: ELASTIC CURVE (A) ............................................................................................................... 15

FIGURE 2.2‐4: ELASTIC CURVE (B) ............................................................................................................... 15

FIGURE 2.3‐1: CONJUGATE BEAM METHOD .................................................................................................. 17

FIGURE 2.4‐1: VIRTUAL LOAD .................................................................................................................... 18

FIGURE 2.4‐2: VIRTUAL MOMENT .............................................................................................................. 18

FIGURE 2.4‐3: REAL LOAD ......................................................................................................................... 19

FIGURE 2.4‐4: REAL MOMENT ................................................................................................................... 19

FIGURE 2.4‐5: UNIT LOAD EQUATION .......................................................................................................... 19

FIGURE 2.5‐1: THREE MOMENTS METHOD ................................................................................................... 21

FIGURE 2.6‐1: SLOPE DEFLECTION METHOD ................................................................................................. 24

FIGURE 3.3‐1: STRUCTURE WITH SINGLE FLEXIBILITY COORDINATE ...................................................................... 28

FIGURE 3.3‐2: STRUCTURE WITH TWO FLEXIBILITY COORDINATES ....................................................................... 29

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FIGURE 3.3‐3: STRUCTURE WITH THREE FLEXIBILITY COORDINATES ...................................................................... 30

FIGURE 3.4‐1: MEMBERS SUBJECTED TO VARYING AXIAL LOAD ........................................................................... 33

FIGURE 3.4‐2: GRAPH OF LOAD VERSES DISPLACEMENT .................................................................................... 33

FIGURE 3.4‐3: SINGLE COORDINATE ............................................................................................................ 34

FIGURE 3.4‐4: MOMENT DIAGRAM ............................................................................................................. 35

FIGURE 3.4‐5: TWO COORDINATES ............................................................................................................. 35

FIGURE 3.4‐6 ......................................................................................................................................... 35

FIGURE 3.4‐7 ......................................................................................................................................... 36

FIGURE 3.4‐8 ......................................................................................................................................... 37

FIGURE 3.4‐9: FRAME .............................................................................................................................. 38

FIGURE 3.4‐10: FRAME (A) ....................................................................................................................... 39

FIGURE 3.4‐11 ....................................................................................................................................... 39

FIGURE 3.4‐12 ....................................................................................................................................... 40

FIGURE 3.4‐13: THREE COORDINATES STIFFNESS COEFFICIENTS ......................................................................... 43

FIGURE 3.4‐14: MEMBER OR ELEMENT STIFFNESS MATRIX .............................................................................. 44

FIGURE 3.4‐15: STIFFNESS COEFFICIENTS ..................................................................................................... 46

FIGURE 3.4‐16: STIFFNESS COEFFICIENTS ..................................................................................................... 48

FIGURE 3.4‐17 ....................................................................................................................................... 50

FIGURE 3.4‐18 ....................................................................................................................................... 52

FIGURE 3.4‐19: DIRECT STIFFNESS MATRIX METHOD ...................................................................................... 55

FIGURE 3.5‐1: FLEXIBILITY AND STIFFNESS COMPARISON .................................................................................. 57

FIGURE 4.1‐1: REINFORCED CONCRETE ........................................................................................................ 59

FIGURE 5.2‐1: REINFORCED CONCRETE BEAM ............................................................................................... 74

FIGURE 5.2‐2: STRESS AND STRAIN DIAGRAMS .............................................................................................. 75

FIGURE 5.2‐3:STRAIN LIMIT DISTRIBUTION ................................................................................................... 77

FIGURE 5.2‐4: BALANCED STRAIN CONDITION ............................................................................................... 77

FIGURE 5.2‐5: ULTIMATE FORCES IN A RECTANGULAR SECTION .......................................................................... 79

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FIGURE 5.2‐6: VALUES OF BETA FOR DIFFERENT FC'. ....................................................................................... 80

FIGURE 5.2‐7: INTERNAL EQUILIBRIUM OF RCC SECTION ................................................................................. 80

FIGURE 5.2‐8: LOAD FACTOR COMBINATIONS ............................................................................................... 82

FIGURE 5.2‐9: STRESS IN BEAM .................................................................................................................. 83

FIGURE 5.2‐10: CRITICAL SHEAR................................................................................................................. 86

FIGURE 5.2‐11: DOUBLY REINFORCED RECTANGULAR BEAM. ............................................................................. 88

FIGURE 5.3‐1: (A) T‐SECTION AND (B) I‐SECTION, WITH (C) ILLUSTRATION OF EFFECTIVE FLANGE WIDTH B_E ............ 93

FIGURE 5.3‐2: EFFECTIVE FLANGE WIDTH OF T‐BEAMS ..................................................................................... 94

FIGURE 5.3‐3: RECTANGULAR SECTION BEHAVIOR (A) WHEN THE NEUTRAL AXIS LIES WITHIN THE FLANGE AND (B) WHEN

THE STRESS DISTRIBUTION DEPTH EQUALS THE SLAB THICKNESS. ................................................................. 95

FIGURE 5.3‐4: T‐SECTION BEHAVIOR. ........................................................................................................... 96

FIGURE 5.3‐5:T ‐SECTION ANALYSIS. ............................................................................................................ 97

FIGURE 5.3‐6: ACI CODE, SECTION 8.12 ..................................................................................................... 99

FIGURE 5.4‐1 TYPES OF COLUMNS ............................................................................................................. 103

FIGURE 5.4‐2: FAILURE OF SHORT COLUMN ................................................................................................ 104

FIGURE 5.4‐3: FAILURE OF LONG COLUMN ................................................................................................. 104

FIGURE 5.4‐4: FAILURE OF TIED COLUMN ................................................................................................... 104

FIGURE 5.4‐5: FAILURE OF SPIRAL COLUMN ................................................................................................ 105

FIGURE 5.4‐6: BEHAVIOR OF TIED AND SPIRAL COLUMNS ................................................................................ 106

FIGURE 5.4‐7: ACI CODE, SECTION 10.9.1 ................................................................................................ 108


FIGURE 5.4‐9: ARRANGEMENT OF BARS AND TIES IN COLUMNS ........................................................................ 109

FIGURE 5.4‐10: ACI CODE, SECTION 7 .10.4 .............................................................................................. 110

FIGURE 5.4‐11: ACI CODE, SECTION 7 .10.5 .............................................................................................. 110

FIGURE 5.4‐12: DIMENSIONS OF A COLUMN SPIRAL ...................................................................................... 112

FIGURE 5.4‐13: ACI CODE, SECTION 10.3.6 .............................................................................................. 114

FIGURE 5.5‐1: ONE WAY SLAB .................................................................................................................. 115

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FIGURE 5.5‐2: MOMENTS AND REINFORCEMENT LOCATIONS IN CONTINUOUS BEAMS ........................................... 117

FIGURE 5.5‐3: CRITICAL SHEAR (A) ............................................................................................................ 118

FIGURE 5.5‐4: CRITICAL SHEAR (B) ............................................................................................................ 118

FIGURE 5.5‐5: ONE‐WAY SLAB BAR BENDING AND PLACING DETAIL ................................................................... 118


FIGURE 5.5‐7: RECTANGULAR STRIP IN ONE WAY SLAB ................................................................................... 120

FIGURE 5.5‐8: ACL CODE, SECTION 8.3 ..................................................................................................... 121

FIGURE 5.5‐9: MOMENT COEFFICIENTS FOR CONTINUOUS BEAMS AND SLABS ..................................................... 122

FIGURE 5.5‐10: COVER IN SLABS ............................................................................................................... 123


FIGURE 5.6‐1: TWO‐WAY SLABS ON BEAMS ............................................................................................... 126

FIGURE 5.6‐2: FLAT SLABS ...................................................................................................................... 127

FIGURE 5.6‐3: FLAT SLABS WITH DROP PANELS AND COLUMN CAPITALS ............................................................. 127

FIGURE 5.6‐4: FLAT SLABS WITH DROP PANELS ONLY ..................................................................................... 128

FIGURE 5.6‐5: FLAT SLABS WITH COLUMN CAPITALS ONLY .............................................................................. 128

FIGURE 5.6‐6: FLAT‐PLATE FLOORS ........................................................................................................... 129

FIGURE 5.6‐7: WAFFLE SLAB ................................................................................................................... 129

FIGURE 5.6‐8: COLUMN AND MIDDLE STRIPS; X = 0.25/1 OR 0.25H WHICHEVER IS SMALLER. ................................ 132

FIGURE 5.6‐9 BENDING MOMENT IN A FIXED – END BEAM. .............................................................................. 137

FIGURE 5.6‐10 CRITICAL SECTIONS FOR NEGATIVE DESIGN MOMENTS. A‐A, SECTION FOR NEGATIVE MOMENT AT EXTERIOR

SUPPORT WITH BRACKET. ................................................................................................................ 138

FIGURE 5.6‐11: DISTRIBUTION OF MOMENTS IN AN INTERIOR PANEL. ................................................................ 139

FIGURE 5.6‐12 EXTERIOR PANEL. .............................................................................................................. 140

FIGURE 5.6‐13 DISTRIBUTION OF TOTAL STATIC MOMENT INTO NEGATIVE AND POSITIVE SPAN MOMENTS. ................ 142

FIGURE 5.6‐14 WIDTH OF THE EQUIVALENT RIGID FRAME (EQUAL SPANS IN THIS FIGURE) AND DISTRIBUTION OF MOMENTS

IN FLAT PLATES, FLAT SLABS, AND WAFFLE SLABS WITH NO BEAMS. ........................................................... 143

FIGURE 5.7‐15.7‐8: DISTRIBUTION OF SOIL PRESSURE ASSUMING UNIFORM PRESSURE ......................................... 153

xvi

FIGURE 5.7‐25.7‐9: SOIL PRESSURE DISTRIBUTION IN COHESION LESS SOIL (SAND) ............................................... 154

FIGURE 6.1‐1: EARTH'S TECTONIC PLATES ................................................................................................... 168

FIGURE 6.1‐2: DIGITAL TECTONIC ACTIVITY MAP OF THE EARTH ...................................................................... 169

FIGURE 6.1‐3: EARTHQUAKE WAVES ......................................................................................................... 170

FIGURE 6.1‐4: EPICENTER ....................................................................................................................... 171

FIGURE 6.1‐5: EFFECT OF INERTIA IN A BUILDING WHEN SHAKEN AT ITS BASE ....................................................... 171

FIGURE 6.1‐6: INERTIA FORCE AND RELATIVE MOTION WITHIN A BUILDING ......................................................... 171

FIGURE 6.1‐7: ARRIVAL OF SEISMIC WAVES AT A SITE ..................................................................................... 172

FIGURE 6.2‐1: EARTHQUAKE RECORD ........................................................................................................ 173

FIGURE 7.2‐1: MODAL SHAPES FOR A THREE STOREY BUILDING (A) FIRST MODE; (B) SECOND MODE; (C) THIRD MODE .. 176

FIGURE 7.2‐2: UPPER STOREYS OF OPEN GROUND STOREY MOVE TOGETHER AS SINGLE BLOCK ................................ 177

FIGURE 7.2‐3: GROUND STOREY OF REINFORCED CONCRETE BUILDING LEFT OPEN TO FACILITATE ............................. 177

FIGURE 7.2‐4: SIMPLE PLAN SHAPE BUILDINGS DO WELL DURING EARTHQUAKE.................................................... 178

FIGURE 7.2‐5: BUILDINGS WITH ONE OF THEIR OVERALL SIZES MUCH LARGER OR MUCH SMALLER THAN OTHER TWO .... 179

FIGURE 7.2‐6: BUILDINGS WITH SETBACKS .................................................................................................. 179

FIGURE 7.2‐7: HAMMERING OR POUNDING ................................................................................................ 180

FIGURE 7.3‐1: FRAME SUBJECTED TO LATERAL LOADING (A) DEFLECTED SHAPE; (B) MOMENTS ACTING ON BEAM‐COLUMN

JOINT ......................................................................................................................................... 181

FIGURE 8.2‐1: SECTION 1612.2 LOAD COMBINTIONS ................................................................................... 187

FIGURE 8.3‐1: SESIMIC HAZARD ZONES OF PAKISTAN [8.4] ............................................................................. 198

FIGURE 8.3‐2: SEISMIC ZONING OF PAKISTAN [8.4] ...................................................................................... 199

FIGURE 8.3‐3: SEISMIC ZONING SINDH [8.4] .............................................................................................. 200

FIGURE 8.3‐4: SEISMIC ZONING PUNJAB [8.4] ............................................................................................ 201

FIGURE 8.3‐5: SEISMIC ZONING BALUCHISTAN [8.4] ..................................................................................... 202

FIGURE 8.3‐6: SEISMIC ZONING KPK, JK, NORTHERN AREAS [8.4] .................................................................. 203

FIGURE 9.2‐1: DESIGN RESPONSE SPECTRUM ............................................................................................... 211

FIGURE 9.4‐1: BEARING WALL SYSTEM ....................................................................................................... 215

xvii

FIGURE 9.4‐2: BUILDING FRAME SYSTEM .................................................................................................... 216

FIGURE 9.4‐3: MOMENT‐RESISTING FRAME SYSTEM FBD ............................................................................... 216

FIGURE 9.4‐4: MOMENT‐RESISTING FRAME SYSTEM ...................................................................................... 217

FIGURE 9.4‐5: SHEAR WALL‐FRAME INTERACTIVE SYSTEM ............................................................................... 218

FIGURE 9.4‐6: SCHEMATIC CENTER OF MASS IN SHEAR WALL ........................................................................... 236

FIGURE 9.4‐7: CANTILEVER SHEAR‐WALL DEFLECTION .................................................................................... 236

FIGURE 9.4‐8: SCHEMATIC CENTER OF RIGIDITY IN A SHEAR WALL ..................................................................... 236

FIGURE 9.4‐9: SHEAR DISTRIBUTION FORMULATION ...................................................................................... 238

xviii

LIST OF TABLES

TABLE 2.3‐1: CONJUGATE BEAM METHOD ................................................................................................... 16

TABLE 4.4‐1: SUMMARY OF DESIGN METHODS .............................................................................................. 67

TABLE 4.6‐1: TYPICAL UNIFORMLY DISTRIBUTED DESIGN LOADS ........................................................................ 70

TABLE 4.6‐2: DENSITY AND SPECIFIC GRAVITY OF VARIOUS MATERIALS ............................................................... 71

TABLE 5.2‐1: BENDING MOMENTS .............................................................................................................. 82

TABLE 5.2‐2: BAR DIAMETER CHART ........................................................................................................... 84

TABLE 5.4‐1: MAXIMUM SPACINGS OF TIES ................................................................................................ 111

TABLE 5.4‐2: SPIRALS FOR CIRCULAR COLUMNS (FY = 60 KSI) ......................................................................... 112

TABLE 5.5‐1: MINIMUM THICKNESS H OF NON‐PRESTRESSED ONE‐WAY SLABS .................................................... 116

TABLE 5.5‐2:ACI CODE, TABLE 9.5A ......................................................................................................... 123

TABLE 5.5‐3: ACI CODE, TABLE 9.5B ........................................................................................................ 124

TABLE 5.6‐1: MINIMUM SLAB THICKNESS ................................................................................................... 134

TABLE 5.6‐2 DISTRIBUTION OF MOMENTS IN AN END PANEL .......................................................................... 141

TABLE 5.6‐3 PERCENTAGE OF LONGITUDINAL MOMENT IN COLUMN STRIPS, INTERIOR PANELS (ACI CODE, SECTION

13.6.4) ..................................................................................................................................... 142

TABLE 5.6‐4 PERCENTAGE OF MOMENTS IN TWO‐WAY INTERIOR SLABS WITHOUT BEAMS (Α1 = 0) ........................ 143

TABLE 5.6‐5 PERCENTAGE OF LONGITUDINAL MOMENT IN COLUMN STRIPS, EXTERIOR PANELS (ACI CODE, SECTION

13.6.4) ..................................................................................................................................... 144

TABLE 5.6‐6 PERCENTAGE OF LONGITUDINAL MOMENT IN COLUMN AND MIDDLE STRIPS, EXTERIOR PANELS (FOR ALL

RATIOS OF L2/L1 ), GIVEN Α1 = Β1= 0 ................................................................................................ 144

TABLE 8.2‐1: MAXIMUM ALLOWABLE DEFLECTION FOR STRUCTURAL MEMBERS ............................... 190

TABLE 8.2‐2: SEISMIC ZONE FACTOR Z ................................................................................................. 190

TABLE 8.2‐3: SOIL PROFILE TYPES ........................................................................................................ 190

TABLE 8.2‐4: OCCUPANCY CATEGORY ................................................................................................. 191

TABLE 8.2‐5: STRUCTURAL SYSTEMS ................................................................................................... 192

xix

TABLE 8.2‐6: SEISMIC COEFFICIENT CA ................................................................................................. 194

TABLE 8.2‐7: SEISMIC COEFFICIENT CV ................................................................................................. 194

TABLE 8.3‐1: SEISMIC ZONES OF TEHSILS OF PAKISTAN (A) [8.4] ..................................................................... 204

TABLE 8.3‐2: SEISMIC ZONES OF TEHSILS OF PAKISTAN (B) [8.4] ..................................................................... 205

TABLE 8.3‐3: SEISMIC ZONES OF TEHSILS OF PAKISTAN (C) [8.4] ..................................................................... 206

TABLE 8.3‐4: SEISMIC ZONES OF TEHSILS OF PAKISTAN (D) [8.4] ..................................................................... 207

TABLE 9.4‐1: PERMITTED ANALYTICAL PROCEDURES ..................................................................................... 219

TABLE 9.4‐2: VALUE OF CT AND X ............................................................................................................. 223

TABLE 9.4‐3: COEFFICIENT FOR UPPER LIMIT ON CALCULATED PERIOD .............................................................. 224

TABLE 9.4‐4: ALLOWABLE STORY DRIFT (ΔA) .............................................................................................. 224

TABLE 9.4‐5: GROUND MOTION SPECTRAL RESPONSE ACCELERATION FOR SOME CITIES OF SINDH [9.3] .................. 225

TABLE 9.4‐6: SITE CLASSIFICATION: ........................................................................................................... 226

TABLE 9.4‐7: VALUES OF SITE COEFFICIENT FA ..................................................................................... 227

TABLE 9.4‐8: VALUES OF SITE COEFFICIENT FV ..................................................................................... 227

TABLE 9.4‐9: SEISMIC IMPORTANCE .......................................................................................................... 228

TABLE 9.4‐10: SEISMIC DESIGN CATEGORY BASED ON SHORT PERIOD RESPONSE ACCELERATION PARAMETER ........... 229

TABLE 9.4‐11: SEISMIC DESIGN CATEGORY BASED ON 1‐SECOND PERIOD RESPONSE ACCELERATION PARAMETER ...... 229

TABLE 9.4‐12: DESIGN COEFFICIENTS AND FACTORS FOR SEISMIC FORCE‐RESISTING SYSTEMS ............................... 230

TABLE 9.4‐13: ALLOWABLE STORY DRIFT (ΔA) ............................................................................................ 239

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• Structural Analysis

Section - I

Chapter 1 • Introduction to Structural Analysis

Chapter 2 • Structural Analysis: Basic Methods

Chapter 3 • Structural Analysis: Matrix Methods

1

CHAPTER 1

INTRODUCTION TO STRUCTURAL ANALYSIS

1.1 INTRODUCTION

Structural analysis is the prediction of the performance of a given structure under

prescribed loads and or other external effects, such as support moments and temperature

changes. The performance characteristics commonly of interest in the design of

structures are (1) stresses or stress resultants, such as axial force, shear forces, and

bending moments; (2) deflection; and (3) support reactions. Thus, the analysis of a

structure usually involves determination of these quantities as caused by a given loading

condition. [1.1].

1.2 ROLE OF STRUCTURAL ANALYSIS IN STRUCTURAL

ENGINEERING PROJECTS

Structural engineering is the science and art of planning, designing, and

constructing safe and economical structures that will serve their intended purposes.

Structural analysis is an integral part of any structural engineering project, its function

being the prediction of the performance of the proposed structure. A flowchart showing

the various phases of a typical structural engineering project is presented in Fig. 1.2-1.

1. Planning Phase The planning phase usually involves the establishment of the

functional requirements of the proposed structure, the general layout and

dimensions of the structure, consideration of the possible types of materials to

be used (e.g., structural steel or reinforced concrete ).

2. Preliminary Structural Design In the preliminary structural design phase,

the size of the various members of the structural system selected in the

2

planning phase are estimated based on the approximate analysis, past

experience, and code requirements. The member sizes thus selected are used

in the next phase to estimate the weight of the structure.

Figure 1.2-1: A flowchart showing the various phases of a typical structural engineering project

3. Estimation of loads Estimation of loads involves determination of all the

loads that can be expected to act on the structure.

4. Structural Analysis In structural analysis, values of the loads are used to carry

out the analysis of the structure in order to determine the stresses or stress

resultant in the members and the deflections at various points of the structure.

3

5. Safety and Serviceability Checks The results of the analysis are used to

determine whether or not the structure satisfies the safety and serviceability

requirements of the design codes.

6. Revised structural design If the code requirements are not satisfied, then the

member size are revised, and phases 3 through 5 are repeated until all the

safety and serviceability requirements are satisfied. [1.1].

1.3 CLASSIFICATION OF STRUCTURES

Commonly used structures can be classified into five basic categories, depending

on the type of primary stresses that may develop in their members under major design

loads.

1.3.1 Tension Structures

The members of tension structures are subjected to pure tension under the action

of external loads. Tension structures composed of flexible steel cable are frequently

employed to support bridges and long-span roofs. Because of their flexibility, cables

have negligible bending stiffness and can develop only tension. Examples of tension

structures include vertical rods used as hangers (for example, to support balconies or

tanks) and membrane structures such as tents.

Figure 1.3-1: Tension Structures

4

1.3.2 Compressive Structures

Compression structures develop mainly compressive stresses under the action

of external loads. Two common examples of such structures are columns and arches.

Columns are straight members subjected to axially compressive loads. When a straight

member is subjected to lateral loads and/or moments in addition to axial loads, it is

called a beam-column.

Figure 1.3-2: Compression Structures

An arch is curved structure, with a shape similar to that of an inverted cable.

Such structures are frequently used to support bridges and long-span roofs.

1.3.3 Trusses

Trusses are composed of straight members connected at their ends by hinged

connections to form a stable configuration. When the loads are applied to a truss only

at the joint, its members either elongated or shorten. Thus, the members of an ideal truss

are always either in uniform tension or in uniform compression.

Trusses, because of their light weight and high strength, are among the most

commonly used types of structures. Such structures are used in a variety of applications,

ranging from supporting roofs of buildings to serving as support structures in space

stations.

5

Figure 1.3-3: Trusses

1.3.4 Shear Structures

Shear structures, such as reinforced concrete shear walls, are used in multistory

buildings to reduce lateral movements due to wind loads and earthquake excitations.

Shear structures develop mainly in-plane shear, with relatively small bending stress

under the action of external loads.

Figure 1.3-4: Shear Structures

6

1.3.5 Bending Structures

Bending structures develop mainly bending stresses under the action of external

loads. Some of the most commonly used structures, such as beams, rigid frames, slabs,

and plates, can be classified as bending structures.

Figure 1.3-5: Bending Structures

From statics and mechanics of materials, the bending (normal) stress varies

linearly over the depth of a beam from the maximum compressive stress at the fiber

farthest from the neutral axis on the concave side of the bent beam to the maximum

tensile stress at the outermost fiber on the convex side. For example, in the case of a

horizontal beam subjected to a vertically downward load, the bending stress varies from

the maximum compressive stress at the top edge to the maximum tensile stress at the

bottom edge of the beam. To utilize the material of a beam cross section most efficiently

under this varying stress distribution, the cross sections of beams are often I-shaped,

with most of the material in the top and bottom flanges. The I-shaped cross sections are

most effective in resisting bending moments. [1.1].

1.4 ANALYSIS

In analysis, we require the entire details of the structure, loading sectional

properties. To proportion a structure, we must first know how it will behave under

loading. Therefore, the process of analysis and design forms an integral part in design.

7

Figure 1.4-1: Analysis

In practice, the properties of members are so chosen as to obtain a specified

structure, and then analysis is carried out. Often the designer may have to readjust his

initial dimensions in order to get desired response from the structure. Therefore, the

intended purpose of any analysis is to know how the structure responds to a given

loading and thereby evaluate the stresses and deformations.

Analysis helps the designer to choose right type of sections consistent with

economy and safety of structure. The purpose of structural analysis is to determine the

reactions, internal forces, such as axial, shear, bending and torsional, and deformations

at any point of a given structure caused by the applied loads and forces.

1.4.1 Free Body Diagrams

The analysis of all the structures is based on the fact that the structure is in

equilibrium under the action of external loads and reactions. The magnitudes of the

reactions are such that the loads are exactly counteracted according the reactions to

8

Newton’s third law. Further, any part of structure is in equilibrium along with the

structure as a whole. This fact is used to determine the internal forces in a structure by

drawing what are known as free-body diagrams for parts of a structure. Free-body

diagrams are so useful in studying structural analysis that their importance cannot be

over-emphasized.

Figure 1.4-2: Free Body Diagram

1.4.2 Sign Convention

An essential part of structural analysis is the adoption is an appropriate sign

convention for representing forces and displacements. It will become clear with the

development of different methods of analysis that there are advantages in not following

the same sign convention.

Axial Forces: An axial force is considered positive when it produces tension in the

member. A compressive forces is, therefore, negative.

9

Shear Forces: Shear force which tends to shear the member is considered positive.

Notice that the positive shear force forms a clockwise couple on a segment.

Bending Moment: There are two conventions used for bending moment:

(1) The beam convention based on the nature of stress the moment produces,

(2) The static sign convention based on the direction the moment tends to rotate the

joint or end of a member.

In the beam convention, the moment which produces compressive stresses in the

top fibers or tensile stresses in the bottom fibers is positive. In the joint convention, the

moment that tends to rotate the joint clockwise or the member end anti-clockwise is

denoted positive. [1.2].

Figure 1.4-3: Sign Convention

1.5 STRUCTURAL RESPONSE

Structural systems subjected to static loading exhibit their response in the form of

induced internal stresses and consequent displacements. In general, the members of

elements of structures are subjected to four types of internal forces, viz., an axial load,

a shear force, bending moment and twisting moments. The resulting internal stresses

give rise to linear displacements. Thus the structural system as a whole undergoes a set

of displacement. The external forces acting on the structure undergo these

displacements and consequently lose their potential energy. In accordance with the law

10

of conservation of energy, the loss of potential energy of the external forces is

compensated by an equal amount of energy stored in the structure in the form of strain

energy. The main object of structural analysis is to evaluate the response of the structure

exhibited by way of induced internal stresses and resulting displacement because these

are directly related to the safety and serviceability of the structural system. [1.3].

11

CHAPTER 2

STRUCTURAL ANALYSIS: BASIC APPROACH

2.1 DOUBLE INTEGRATION METHOD

The following procedure provides a method for determining the slope and

deflection of a beam (or shaft) using the method of double integration. It should be

realized that this method is suitable only for elastic deflections for which the beam’s

slope is very small. Furthermore, the method considers only deflections due to bending.

Additional deflection due to shear generally represents only a few percent of the

bending deflection, and so it is usually neglected in engineering practice.

Figure 2.1-1: Double Integration Method

2.1.1 Elastic Curve

• Draw an exaggerated view of the beam’s elastic curve. Recall that points of zero

slope and zero displacement occur at a fixed support, and zero displacement occurs at

pin and roller supports.

• Establish the x and v coordinate axes. The x axis must be parallel to the un-

deflected beam and its origin at the left side of the beam, with a positive direction to

the right.

12

• If several discontinuous loads are present, establish x coordinates that are valid

for each region of the beam between the discontinuities.

• In all cases, the associated positive v axis should be directed upward.

2.1.2 Load or Moment Function

• For each region in which there is an x coordinate, express the internal moment

M as a function of x.

• Always assume that M acts in the positive direction when applying the equation

of moment equilibrium to determine

2.1.3 Slope and Elastic Curve

• Provided EI is constant, apply the moment equation which requires two

integrations. For each integration it is important to include a constant of integration.

The constants are determined using the boundary conditions for the supports and the

continuity conditions that apply to slope and displacement at points where two

functions meet.

Figure 2.1-2: Elastic Curve

• Once the integration constants are determined and substituted back into the slope

and deflection equations, the slope and displacement at specific points on the elastic

13

curve can be determined. The numerical values obtained can be checked graphically by

comparing them with the sketch of the elastic curve.

• Positive values for slope are counterclockwise and positive displacement is

upward. [2.1].

2.2 MOMENT AREA THEOREM

Theorem 1: The change in slope between any two points on the elastic curve

equals the area of the M/EI diagram between these two points.

Theorem 2: The vertical deviation of the tangent at a point (A) on the elastic

curve with respect to the tangent extended from another point (B) equals the “moment”

of the area under the M/EI diagram between the two points (A and B). This moment is

computed about point A (the point on the elastic curve), where the deviation tA/B is to

be determined.

Figure 2.2-1: Moment Area Method

The following procedure provides a method that may be used to determine the

displacement and slope at a point on the elastic curve of a beam using the moment-area

theorems.

2.2.1 M/EI Diagram

• Determine the support reactions and draw the beam’s M/EI diagram.

14

• If the beam is loaded with concentrated forces, the M/EI diagram will consist of

a series of straight line segments, and the areas and their moments required for the

moment-area theorems will be relatively easy to compute.

Figure 2.2-2: Moment Diagram

• If the loading consists of a series of concentrated forces and distributed loads, it

may be simpler to compute the required M/EI areas and their moments by drawing the

M/EI diagram in parts, using the method of superposition.

In any case, the M/EI diagram will consist of parabolic or perhaps higher-order

curves, and it is suggested that the table on the inside back cover be used to locate the

area and centroid under each curve.

2.2.2 Elastic Curve

• Draw an exaggerated view of the beam’s elastic curve. Recall that points of zero

slope occur at fixed supports and zero displacement occurs at all fixed, pin, and roller

supports.

• If it becomes difficult to draw the general shape of the elastic curve, use the

moment (or M/EI) diagram. Realize that when the beam is subjected to a positive

moment the beam bends concave up, whereas negative moment bends the beam

concave down. Furthermore, an inflection point or change in curvature occurs where

the moment in the beam (or M/EI) is zero.

15

Figure 2.2-3: Elastic Curve (a)

• The displacement and slope to be determined should be indicated on the curve.

Since the moment-area theorems apply only between two tangents, attention should be

given as to which tangents should be constructed so that the angles or deviations

between them will lead to the solution of the problem. In this regard, the tangents at the

points of unknown slope and displacement and at the supports should be considered,

since the beam usually has zero displacement and/or zero slope at the supports.

Figure 2.2-4: Elastic Curve (b)

2.2.3 Moment-Area Theorems

• Apply Theorem 1 to determine the angle between two tangents, and Theorem 2

to determine vertical deviations between these tangents.

• Realize that Theorem 2 in general will not yield the displacement of a point on

the elastic curve. When applied properly, it will only give the vertical distance or

deviation of a tangent at point A on the elastic curve from the tangent at B.

16

• After applying either Theorem 1 or Theorem 2, the algebraic sign of the answer

can be verified from the angle or deviation as indicated on the elastic curve. [2.1].

2.3 CONJUGATE BEAM METHOD

The following procedure provides a method that may be used to determine the

displacement and slope at a point on the elastic curve of a beam using the conjugate-

beam method.

2.3.1 Conjugate Beam

Table 2.3-1: Conjugate Beam Method

17

• Draw the conjugate beam for the real beam. This beam has the same length as

the real beam and has corresponding supports as listed in Table 2.3-1.

• In general, if the real support allows a slope, the conjugate support must develop

a shear; and if the real support allows a displacement, the conjugate support must

develop a moment.

• The conjugate beam is loaded with the real beam’s M/EI diagram. This loading

is assumed to be distributed over the conjugate beam and is directed upward when M/EI

is positive and downward when M/EI is negative. In other words, the loading always

acts away from the beam.

Figure 2.3-1: Conjugate Beam Method

2.3.2 Equilibrium

• Using the equations of equilibrium, determine the reactions at the conjugate

beam’s supports.

• Section the conjugate beam at the point where the slope and displacement of the

real beam are to be determined. At the section show the unknown shear and moment

acting in their positive sense.

18

• Determine the shear and moment using the equations of equilibrium. And equal

and, respectively, for the real beam. In particular, if these values are positive, the slope

is counterclockwise and the displacement is upward. [2.1].

2.4 UNIT LOAD METHOD OR METHOD OF VIRTUAL WORK

The following procedure may be used to determine the displacement and/or the

slope at a point on the elastic curve of a beam or frame using the method of virtual

work.

2.4.1 Virtual Moments m or mu

• Place a unit load on the beam or frame

at the point and in the direction of the desired

displacement.

• If the slope is to be determined, place

a unit couple moment at the point.

• Establish appropriate x coordinates that

are valid within regions of the beam or frame

where there is no discontinuity of real or virtual

load.

• With the virtual load in place, and all

the real loads removed from the beam or frame, calculate the internal moment m or as

a function of each x coordinate.

• Assume m or acts in the conventional positive direction.

Figure 2.4-1: Virtual Load

Figure 2.4-2: Virtual Moment

19

2.4.2 Real Moments

• Using the same x coordinates as those

established for m or determine the internal

moments M caused only by the real loads.

• Since m or was assumed to act in the

conventional positive direction, it is important that

positive M acts in this same direction. This is

necessary since positive or negative internal work

depends upon the directional sense of load and

displacement.

2.4.3 Virtual-Work Equation

• Apply the equation of virtual work to determine the desired displacement or

rotation. It is important to retain the algebraic sign of each integral calculated within its

specified region.

Figure 2.4-5: Unit Load Equation

Figure 2.4-4: Real Moment

Figure 2.4-3: Real Load

20

• If the algebraic sum of all the integrals for the entire beam or frame is positive,

or is in the same direction as the virtual unit load or unit couple moment, respectively.

If a negative value results, the direction of or is opposite to that of the unit load or unit

couple moment. [2.1].

2.5 THREE MOMENT EQUATIONS

The three-moment equation represents, in a general form, the compatibility

condition that the slope of the elastic curve be continuous at an interior support of the

continuous beam. Since the equation involves three moments—the bending moments

at the support under consideration and at the two adjacent supports—it commonly is

referred to as the three-moment equation. When using this method, the bending

moments at the interior (and any fixed) supports of the continuous beam are treated as

the redundants. The three-moment equation is then applied at the location of each

redundant to obtain a set of compatibility equations which can be solved for the

unknown redundant moments.

The following step-by-step procedure can be used for analyzing continuous

beams by the three-moment equation.

1. Select the unknown bending moments at all interior supports of the beam as the

redundants.

2. By treating each interior support successively as the intermediate support c,

write a three-moment equation. When writing these equations, it should be realized that

bending moments at the simple end supports are known. For such a support with a

cantilever overhang, the bending moment equals that due to the external loads acting

on the cantilever portion about the end support. The total number of three-moment

21

equations thus obtained must be equal to the number of redundant support bending

moments, which must be the only unknowns in these equations.

Figure 2.5-1: Three Moments Method

22

3. Solve the system of three-moment equations for the unknown support bending

moments.

4. Compute the span end shears. For each span of the beam, (a) draw a free-body

diagram showing the external loads and end moments and (b) apply the equations of

equilibrium to calculate the shear forces at the ends of the span.

5. Determine support reactions by considering the equilibrium of the support

joints of the beam.

6. If so desired, draw shear and bending moment diagrams of the beam by using

the beam sign convention. [2.2].

If E is constant, the Eq. becomes

If E and I are constant, then

For the application of three-moment equation to continuous beam, points 1, 2, and 3

are usually unsettling supports, thus h1 and h3 are zero. With E and I constants, the

equation will reduce to [2.3].

Equation 2.5-1: Three Moments

23

2.6 SLOPE DEFLECTION METHOD

The slope-deflection method for the analysis of indeterminate beams and frames

takes into account only the bending deformations of structures. Although the slope-

deflection method is itself considered to be a useful tool for analyzing indeterminate

beams and frames, an understanding of the fundamentals of this method provides a

valuable introduction to the matrix stiffness method, which forms the basis of most

computer software currently used for structural analysis.

24

Figure 2.6-1: Slope Deflection Method

When a continuous beam or a frame is subjected to external loads, internal

moments generally develop at the ends of its individual members.

The slope-deflection equations relate the moments at the ends of a member to the

rotations and displacements of its ends and the external loads applied to the member.

[2.2].

Equation 2.6-1: Slope Deflection Equation

2.6.1 Procedure

The procedure is as follows:

(1) Determine the fixed end moments at the end of each span due to applied loads acting

on span by considering each span as fixed ended. Assign ± Signs w.r.t. above sign

convention.

(2) Express all end moments in terms of fixed end moments and the joint rotations by

using slope – deflection equations.

(3) Establish simultaneous equations with the joint rotations as the unknowns by

applying the condition that sum of the end moments acting on the ends of the two

members meeting at a joint should be equal to zero.

(4) Solve for unknown joint rotations.

25

(5) Substitute back the end rotations in slope – deflection equations and compute the

end moments.

(6) Determine all reactions and draw S.F. and B.M. diagrams and also sketch the elastic

curve. [2.4].

26

CHAPTER 3

STRUCTURAL ANALYSIS: MATRIX APPROACH

3.1 INTRODUCTION

The Structures are basically built to withstand the applied loads. The application

of these loads develops internal Forces and deformations within the structural

components. Structural analysis is related with the determination of forces and

deformations experienced by the structure.

It is divided into two types.

1. Force Analysis.

2. Deformation Analysis.

3.1.1 Force Analysis:

Force Analysis involves the calculation of reactions of the supports and the

determination of variation of internal actions (Normal forces, Shear forces, Bending

moments etc.) within the structure.

3.1.2 Deformation Analysis:

Deformation Analysis involves the evaluation of deformation (displacements

and strains) of the elements of the structure as well as of the whole structure.

3.1.3 Requirement

Following main requirements are to be satisfied by any method of analysis for

any structure. These are:

I. Equilibrium of forces. Equilibrium between the internal forces and external loads.

27

2. Compatibility of displacement. Displacement of a structure at a particular point must

be compatible with the strains developed within the structure at that point.

3. Force displacement relationship. Specified by geometric and elastic properties of

the elements. [3.1].

3.2 FLEXIBILITY AND STIFFNESS

Flexibility and its converse, known as stiffness, are important properties which

characterize the response of a structure by means of the force-displacement

relationship. In a general sense the flexibility of a structure is defined as the

displacement caused by a unit force and the stiffness is defined as the force required for

a unit displacement. [3.2].

The various methods of analyzing indeterminate structure generally fall in two

classes. [3.1].

1. Force method (Flexibility method)

2. Displacement method (Stiffness method)

3.3 FORCE OR FLEXIBILITY METHOD

3.3.1 Basic concepts of Force or Flexibility method

In this method redundant constraints are removed and corresponding redundant

forces (or moments) are placed. An equation of compatibility of deformation is written

in terms of these redundants and the corresponding displacements. The redundants are

determined from these simultaneous equations. Equations of statics are then used for

the calculation of desired internal action. In this method forces are treated as the basic

unknowns. [3.2].

28

3.3.2 Flexibility Coefficient

The flexibility coefficients characterize the behavior of the structure by specifying

the displacement response to the applied forces at the coordinates. [3.3].

3.3.3 Generation of Flexibility Matrices [3.3]

Single Coordinate

Consider a simple example of a cantilever beam in Fig. 3.3-1a with a single

coordinate indicated for force displacement measurements. The deformation of the

structure may be expressed as

D = f.P (3.1)

Where.

D = deformation at coordinate point 1.

f = flexibility coefficient which is defined as the displacement at

coordinate 1 caused by a unit force at 1.

P = load applied at coordinate 1.

Using the moment area method, we find for the beam of Fig. 3.3-1a

3 (3.2)

Figure 3.3-1: Structure with single flexibility coordinate

29

Two Coordinates

Fig. 3.5-2a shows a cantilever beam with two coordinates. Let us relate the forces

and the corresponding displacements through flexibility coefficients.

To do this, we apply the superposition of forces as follows: First we apply a unit

force at coordinate 1 only (Fig. 3.3-2b) and designate the displacements at 1 and 2 as

f11 and f21 respectively. Next, we apply a unit force at 2 only (Fig. 3.3-2c) and designate

the displacements at I and 2 as f12 and f22 respectively.

Figure 3.3-2: Structure with two flexibility coordinates

The displacements D1 and D2 due to forces P1 and P2 acting simultaneously are

D1 = f11 PI + f12 P2

D2 = f21 P1 + f22 P2 (3.3)

This can be written in the form of a matrix as

(3.4)

Or simply,

D = f. P (3.5)

The matrix f is the flexibility matrix for the structure of Fig. 3.3-2. It may be noted

that the elements of the first column of this matrix are generated by applying a unit

force at I only and the elements of the second column by applying a unit force at 2 only.

The elements of the flexibility matrix for the structure are

30

3 2

2 1 (3.6)

Three Coordinates (Frame)

To generate the elements in the first column of the flexibility matrix f we apply a

unit force at coordinate I only and compute the displacements at the coordinates. The

flexibility coefficients are indicated in Fig. 3.3-3b. The displacements correspond to the

translation at coordinate 1 and rotations at coordinates 2 and 3. Displacements only due

to bending are considered. Any method such as the moment area or virtual work method

can be used in the computation of displacements.

The corresponding displacements or flexibility coefficients are:

3,

2,

2 (3.7)

Figure 3.3-3: Structure with three flexibility coordinates

31

To arrive at the second column of the matrix f we again apply a unit force (in this

case a unit couple) at coordinate 2 only as indicated in Fig. 3.3-3c. The resulting

displacements at the coordinates give:

2, , (3.8)

Lastly, a unit couple is applied at only coordinate 3 (Fig. 3.3-3d) and the elements

in the third column of matrix f are determined. The flexibility coefficients are:

2, ,

2 (3.9)

The complete flexibility matrix f is:

3 2 2

21 1

21 2

(3.10)

It is seen that flexibility matrix f is a square matrix and is symmetric, that is, fij =

fji.

Flexibility and Stiffness Matrices in n Coordinates

Consider a linear elastic structure with n coordinates. To generate the elements of

column I of the flexibility matrix, we apply a unit force at coordinate 1 only and

compute displacements at all the coordinates fi1 (i = I, 2, .... n). This will give the

elements in column 1. To generate, again say, column n of matrix f, we apply a unit

force at coordinate n only and compute displacements fin (i = I , 2, .... n). The values of

these displacements form the elements in the nth column of matrix f. In general, to

generate the elements in the jth column, apply a unit force at coordinate j only and

compute the displacements fi1 (i = 1, 2 ... , n). The values of these displacements form

32

the elements of the jth column of the matrix f. Thus, it is seen that the complete

flexibility matrix f will have n rows and n columns forming a square matrix n x n.

3.3.4 Procedure to Apply Force Method

1. Determine Static Indeterminacy of the structure.

2. Choose redundants and remove them.

3. Compute deformations for removed redundants. Rotations and deflections.

This will give matrix D.

4. Generate Flexibility matrix for redundants in unloaded structure. This will

give matrix f.

5. Compute unknowns using relation P = - f-1 . D.

3.4 DISPLACEMENT OR STIFFNESS METHOD

In this method the rotations or the nodal displacements are treated as unknowns.

These are then related to corresponding forces. [3.1].

3.4.1 Basic concepts of Displacement or Stiffness method:

Displacement or stiffness method allows one to use the same method to analyze

both statically determinate and indeterminate structures. It is generally easier to

formulate the necessary matrices for the computer operations using the displacement

method. In this method nodal displacements are the basic unknown. Equilibrium

equations in terms of unknown nodal displacements and known stiffness coefficients

(force due a unit displacement) are written. These equations are solved tor nodal

displacements and when the nodal displacements are known the forces in the members

of the structure can be calculated from force displacement relationship.

33

3.4.2 Stiffness, Stiffness Coefficient and Stiffness Matrix:

The stiffness of a member is defined as the force which is to be applied at some

point to produce a unit displacement when all other displacement are restrained to be

zero. If a member which behaves elastically is subjected to varying axial tensile load

(W) as shown in fig. 3.4-1 and a graph is drawn of load (W) versus displacement (∆)

the result will be a straight line as shown in fig. 3.4-2, the slope of this line is called

stiffness.

Figure 3.4-1: Members subjected to varying axial load

Figure 3.4-2: Graph of load verses displacement

Mathematically it can be expressed as

K=W/∆

In other words Stiffness 'K' is the force required at a certain point to cause a unit

displacement at that point. Equation 3.1 can be written in the following form.

W = K∆

where,

W = Force at a particular point

34

K = Stiffness

∆ = Unit displacement of the particular point.

The above equation relates the force and displacement at a single point. This can

be extended for the development of a relationship between load and displacement for

more than one point on a structure.

The term "force" and the symbol "W" refers to the moments as well as forces and

the term "deformation" and symbol "∆" refer to the both rotations and deflection. [3.1].

3.4.3 Principles of Stiffness Method for Beams and Plane Frames

As a frame element is subjected not only to axial forces but also to shear forces

and bending moments, therefore three degrees of freedom per joint of a frame element

are present. A degree of freedom is an independent deformation of a joint or a node.

These are:

i) Axial deformation.

ii) End rotations.

iii) Normal translations.

Out of these three, axial deformation is normally neglected, so element or member

stiffness matrix for an element subjected to shear force and bending moment will only

be developed. [3.1].

3.4.4 Generation of Stiffness Matrices [3.4]

Single Coordinate

In fig 3.4-3, k is the stiffness coefficient,

the force required to produce unit displacement.

Therefore, we will induce unit displacement and

compute the required force ‘k’. Figure 3.4-3: Single Coordinate

35

Applying Moment Theorems.

∆12

..

23

2. .2.3

3

Putting ∆ 1

∴3

1

3 (3.11)

Two Coordinates (Beams)

In this case, we will first induce displacement at 1 i.e: ∆ and compute the force

required at coordinates 1 and 2, as shown in fig 3.4-5.

Indicates force required at

coordinate 1 due to displacement at

coordinate 1 and indicates force

required at coordinate 2 due to

displacement at coordinate 1.

Applying Moment Area Theorems,

using moment diagram by parts to avoid

unknowns to group together, as shown in

fig below.

Figure 3.4-4: Moment Diagram

Figure 3.4-6

Figure 3.4-5: Two Coordinates

36

Theorem 2, for deflection. At Fig (3.4-6)

∆12

. . .23

. 2

26

2

3

2

∆ 1

3

21 (3.12)

Theorem 1, for rotation at 2, Fig (3.4-7)

12

. .

2.

0.

2

2 (3.13)

Putting value of in Eq. (3.12)

2 213

321

2

21

2 213

321

2

21

4 3 6

2162 (3.14)

Putting value of 21in Eq (3.13)

Figure 3.4-7

37

112 6

2

1112

3 (3.15)

Secondly, we induce

displacement at 2 i.e; ,and compute

the force required at coordinates 1

and 2 , as shown in fig 3.4-8.

21 indicates force required at

coordinates 1 due to rotation at

coordinate 2 and 22indicates force required at coordinate 2 due to rotation at

coordinate 2.

Apply moment Area Theorems.

Theorem 2 for deflection at 1.

∆12

23

..2

26

2

∆ 0

222

22 12

3

6

224 12

3

6 2

32

(3.16)

Theorem 1 for rotation at 2

2.

Figure 3.4-8

38

2

1

21 (3.17)

Putting value of from Eq, 3.16 in Eq (3.17)

32

2

1

1 3

4 (3.18)

12 6

6 4 (3.19)

Three Coordinates (Frames)

Figure 3.4-9: Frame

39

STEP 1: FIRST COLUMN OF STIFFNESS MATRIX

∆ 1, 0

In this case, we will first induce displacement at 1 i.e: ∆ and compute the force

required at coordinates 1 and 2, as shown in fig 3.4-10.

Indicates force required at

coordinate 1 due to displacement at

coordinate 1 and indicates force

required at coordinate 2 due to

displacement at coordinate 1.

Applying Moment Area Theorems,

using moment diagram by parts to avoid

unknowns to group together.

Theorem 2, for deflection. At Fig

(3.4-11)

∆12

. . .23

.

2 (3.20)

26

2

3

2

∆ 1

3

21 (3.21)

Figure 3.4-11

Figure 3.4-10: Frame (a)

40

Theorem 1, for rotation at 2, Fig

(3.4-12)

12

. .

2.

0.

2

2 (3.22)

Putting value of in Eq. (3.21)

2 213

321

2

21

2 213

321

2

21

4 3 6

2162 (3.23)

Putting value of 21in Eq (3.22)

112 6

2

1112

3 (3.24)

and k31=0.

Figure 3.4-12

41

STEP 2: SECOND COLUMN OF STIFFNESS MATRIX

∆ 0, 1 0

We induce displacement at 2 i.e;

,and compute the force required at

coordinates 1 and 2, as shown in fig.

21 indicates force required at

coordinates 1 due to rotation at coordinate 2 and 22indicates force required at

coordinate 2 due to rotation at coordinate 2.

Apply moment Area Theorems.

Theorem 2 for deflection at 1.

∆12

23

. .2

26

2

∆ 0

222

22 12

3

6

224 12

3

6 2

32

(3.25)

Theorem 1 for rotation at 2

3.

12

2

42

Here, 1

21 (3.26)

Putting value of from Eq. 3.25 in Eq (3.26)

32

2

1

1 3

4 (3.27)

Here, add EI/L for the effect of rotation at C in Eq. 3.27

5 (3.28)

For Apply Moment Theorem 1

1

(3.29)

STEP 3: THIRD COLUMN OF STIFFNESS MATRIX

∆ 0, 1

0 (3.30)

For and .

Apply moment theorem 1

1

1

43

(3.31)

Applying Moment Theorem 1

1

1 (3.33)

12 60

65 1

0 1 1

(3.34)

Figure 3.4-13: Three Coordinates Stiffness Coefficients

44

Four Coordinates

There are two forces (shear force w3 and a moment w1) acting at near end of the

joint and correspondingly there are two deformations (vertical translations 3 and

rotation 1). Similarly there are two forces (shear force w4 and a moment w2) acting at

the far end of the joint and correspondingly two deformations (vertical translation 4

and rotation 2).

As there are four forces and four corresponding deformations then the stiffness

equation can be expanded in the following form:

Figure 3.4-14: Member or Element Stiffness Matrix

(3.35)

Where each element of the stiffness matrix is called stiffness coefficient. It

represents the place occupied by it with respect to row and columns. Any stiffness

coefficient may be represented by kij; where i and j are number of rows and columns.

45

The above mentioned element stiffness matrix “[k]” is formed by applying a unit value

of each end deformation in turn and the corresponding column of the matrix of equation

3.35 gives the various end forces developed at the member ends while other

deformations are restrained. This procedure is as follows:

Apply unit positive deformation (clockwise rotation) 1 = 1 and equating all

other deformations to zero ( 2 = 3 = 4 = 0). The element would be deformed as

shown in figure 3.4-14. From the definition of stiffness, the forces induced at both ends

due to unit clockwise rotation of near end are as under.

w1 = k11 =Moment produced at '1' due to unit clockwise rotation at 1.

w2 = k21 = Moment produced at '2' due to unit clockwise rotation at 1.

w3 = k31 = Vertical reaction produced at '3' due to unit clockwise rotation at 1.


The values of k11. k21, k31 and k41 can be obtained by using the moment area

theorems.

As according to moment area theorem no. l change in slope between two points

on an elastic curve is equal to area of the M/EI diagram between these two points.

Looking at figure 3.4-15 change in slope between two ends is equal to unity. Adding

the areas of figures.

2 21 (3.36)

According to theorem no. 2 of moment area method tangential deviation of a

certain point with respect to the tangent at another point is equal to the moment of M/EI

diagram between the two points calculated about the point where the deviation is to be

46

determined. From the above definition the moment of M/El diagram figures 3.4-15 (b)

and (c)) about the left of the member is equal to zero.

2 3 223

0 (3.37)

Following values of k11 and k21 are obtained by solving equations 3.36 and 3.37.

4,

2 (3.38)

Figure 3.4-15: Stiffness Coefficients

Reaction k41 and k31 can be obtained using equation of equilibrium. Summation

of moments about the right end is equal to zero see figure 3.4-15 (d).

MB = 0.

47

. 0

4 2

6 (3.39)

Applying force equation of equilibrium to figure 3.4-15 (d).

0

6 (3.40)

4,

2,

6,

6 (3.41)

This gives the first column of the element stiffness matrix. As this matrix is

symmetric so it also provides the first row. To obtain 2nd column of stiffness matrix

deformation (rotation) 2 =1 is imposed on the far end equating all other deformations

to zero 1 = 3 = 4 = 0. The element would be deformed as shown in figure 3.4-16 (a).

From the definition of stiffness, the forces induced at both ends due to unit

rotation at far end can be defined as:

w1 = k12 = Moment produced at ‘1’ due to unit clockwise rotation at 2.

w2 = k22 = Moment produced at '2' due to unit clockwise rotation at 2.


W4 = k42 = Vertical reaction produced at '4' due to unit clockwise rotation at 2.

The values of k12, k22, k32 and k42 can be obtained by using the moment area theorems.

48

Applying moment area theorem no. 1 and using bending moment diagram of figure 3.4-

16 (b,c). For this case change in slope between both ends is equal to unity so

2 21 (3.42)

However according to moment area theorem no. 2 the moment of M/EI diagram

about the right end of the member is equal to zero.

Figure 3.4-16: Stiffness Coefficients

2 3 223

0 (3.43)

Solving equations 3.8 and 3.9.

49

4,

2 (3.44)

Reaction k42 and k32 can be obtained using equation of equilibrium. Summation of

moment about the left end see figure 3.4-16 (d)

∑ =0

= +

=

6

(3.45)

Applying force equation of equilibrium to figure 3.4-16 (d)

∑ 0

- = 0

(3.46)

6

(3.47)

These are the forces and moments as shown in figure 3.4-16 a, b, c. d. On

comparison with figure 3.4-14 the correct sign are obtained and these are defined by

the following equations

2

,4

,6

,6

(3.48)

Continuing this process of applying unit vertical translation ∝ 1 and

solving for forces and moments shown in figure 3.4-14 (a, b c & d) and unit vertical

translation ∝ 1 and solving for forces and moments shown in figure 3.4-15 (a, b, c

50

& d ) third row , third column fourth row and fourth column can be obtained . Following

is the summary of these calculations.

For ∝ 1 change in slope between both ends is zero. Therefore

.2

.2

0

(3.49)

Moments of bending moment diagrams in figures 3.4-17 b and c about left end

is equal to unity, therefore

.2

23

.

2 31 (3.50)

Figure 3.4-17

51

Substituting the values of 13from Eq. 3.49 in Eq. 3.50

.2

.23

.

2 .3

1

22

31

.6

1

6

(3.51)

Taking moment about right end see figure 3.4-17d

∑ 0

0

12 (3.52)

Applying the equation of equilibrium according to figure 3.4-17d

∑ = 0

(3.53)

(3.54)

Correct signs can be obtained by comparing with figure 3.4-14. These are defined as

under:

6,

6,

12,

12 (3.55)

For the case when 1 see the figure 3.4-18 (a) change the slope between

both ends is zero, therefore

52

.2

.2

(3.56)

Moment of bending moment diagrams figure 3.4-18 a, b.

About right end is equal to unity, therefore

Figure 3.4-18

53

.2

23

.2

.3

1

.3 6

1 (3.57)

Substituting the value of from Eq. 3.56 in Eq. 3.57

61

Applying equation of moment equilibrium about left see figure 3.4-18 (d)

∑ =0

=

12 (3.60)

Applying equation of force equilibrium to figure 34-18 (d)

∑ = 0

12

(3.61)

Correct sign can be obtained by comparing these values with figure 3.5-1.

These defined as

k6EIL

,6

,12

,12

(3.62)

On combining the calculation in equation (3.41), (3.48), (3.55), (3.62) fallowing

element stiffness matrix is obtained.

6 (3.58)

6 (3.59)

54

4EIL

2EIL

2EIL

4EIL

6EIL

6EIL

6EIL

6EIL

6EIL

6EIL

6EIL

6EIL

12EIL

6EIL

12EIL

12EIL

(3.63)

And the force, stiffness and deformation relationship is as under.

4EIL

2EIL

2EIL

4EIL

6EIL

6EIL

6EIL

6EIL

6EIL

6EIL

6EIL

6EIL

12EIL

6EIL

12EIL

12EIL

(3.64)

3.4.5 Direct Stiffness Method

In the analysis of structures by the stiffness method, the formation of stiffness

matrix K is a major step in the process. The numerical work involved in manual

computations tends to become voluminous even for a simple structure. For this reason

the transformation procedure may not be the best way of assembling structure matrix

K. This matrix can be deduced more easily by noting the fact that any stiffness element

Kij is the nodal force corresponding to degree of freedom i caused by the imposition of

a unit displacement corresponding to degree of freedom j. The same result is, therefore

more simply obtained if the forces caused by the displacements as they are imposed,

one at a time, on the restrained structure are computed and assembled. [3.3].

For the continuous beam, the joint or nodal stiffness values, Kij are easily

computed from a knowledge of the member stiffness values. Therefore, [3.3]

55

Figure 3.4-19: Direct Stiffness Matrix Method

56

3.4.6 Procedure to Apply Displacement Method

1. Determine kinematic indeterminacy of given structure.

2. Compute Fixed End Moments.

3. Compute the forces required to hold the restrained structure. [P’].

4. Generate Stiffness Matrix.

5. Use Stiffness Relation, i.e., P + k. D = 0.

6. Using Slope Deflection Equations, compute end moments.

22

22

(3.65)

7. Compute unknowns from separated structure using basic equations of

equilibrium.

3.5 COMPARISON OF BOTH METHODS

The flexibility and stiffness methods of structural analysis are quite similar in

many respects, especially in the formulation of the problem. For this reason the choice

of one method or the other is primarily a matter of computational convenience.

In the flexibility method there are several alternatives as to redundants, and the

choice of redundants has a significant effect on the nature and amount of computational

effort required. In the stiffness method, on the other hand, there is no choice of

unknowns since the structure can be restrained in a definite manner; thus, the method

of analysis follows a rather set procedure. However, there are both advantages and

disadvantages in both approaches and when carrying out the analysis by hand

computations, the method that produces fewer unknowns generally involves the least

amount of computations. For example, the inversion of a flexibility or stiffness matrix

57

depends upon the number of unknowns involved. For a structure that has numerous

redundants but very few joint displacements as shown in Fig 3.5-1a, the stiffness

method will be preferred. The flexibility method needs an inversion of a 7 x 7 matrix,

whereas the stiffness method needs an inversion of a 2 x 2 matrix. When there are fewer

redundants in a structure than the number of joint displacements, as in Fig. 3.5-1b, the

flexibility method is preferred. Since the structure is redundant to the second degree,

the flexibility method requires an inversion of a 2 x 2 matrix. On the other hand, the

stiffness method requires the inversion of a 9 x 9 matrix in order to compute

displacements. [3.3].

Figure 3.5-1: Flexibility and Stiffness Comparison

• Structural Design

Section - II

Chapter 4 • Introduction to Structural Design

Chapter 5 • Design of Structural Members

58

CHAPTER 4

INTRODUCTION TO STRUCTURAL DESIGN

4.1 REINFORCED CONCRETE

Concrete may be remarkably strong in compression, but it is equally remarkably

weak in tension. [Fig. 4.1-1(a)]. Its tensile ‘strength’ is approximately one-tenth of its

compressive ‘strength’. Hence, the use of plain concrete as a structural material is

limited to situations where significant tensile stresses and strains do not develop, as in

hollow (or solid) block wall construction, small pedestals and ‘mass concrete’

applications (in dams, etc.).

Concrete would not have gained its present status as a principal building material,

but for the invention of reinforced concrete, which is concrete with steel bars embedded

in it. The steel bars embedded in the tension zone of the concrete compensate for the

concrete’s incapacity for tensile resistance, effectively taking up all the tension, without

separating from the concrete [Fig. 4.1-1(b)]. The bond between steel and the

surrounding concrete ensures strain compatibility, i.e., the strain at any point in the steel

is equal to that in the adjoining concrete. Moreover, the reinforcing steel imparts

ductility to a material that is otherwise brittle. In practical terms, this implies that if a

properly reinforced beam were to fail in tension, then such a failure would, fortunately,

be preceded by large deflections caused by the yielding of steel, thereby giving ample

warning of the impending collapse [Fig.4.1-1(c)].

Tensile stresses occur either directly, as in direct tension or flexural tension, or

indirectly, as in shear, which causes tension along diagonal planes ‘diagonal tension’.

Temperature and shrinkage effects may also induce tensile stresses. In all such cases,

59

reinforcing steel is essential, and should be appropriately located, in a direction that

cuts across the principal tensile planes (i.e., across potential tensile cracks). If

insufficient steel is provided, cracks would develop and propagate, and could possibly

lead to failure.

Figure 4.1-1: Reinforced Concrete

60

Reinforcing steel can also supplement concrete in bearing compressive forces, as

in columns provided with longitudinal bars. These bars need to be confined by

transverse steel ties [Fig. 4.1-1(d)], in order to maintain their positions and to prevent

their lateral buckling. The lateral ties also serve to confine the concrete, thereby

enhancing its compression load-bearing capacity. [4.1].

4.2 ADVANTAGES AND DISADVANTAGES OF REINFORCED

CONCRETE

Reinforced concrete, as a structural material, is widely used in many types of

structures. It is competitive with steel if economically designed and executed.

The advantages of reinforced concrete can be summarized as follows:

1. It has a relatively high compressive strength.

2. lt has better resistance to fire than steel.

3. It has a long service life with low maintenance cost.

4. ln some types of structures, such as dams, piers, and footings, it is the most

economical structural material.

5. It can be cast to take the shape required, making it widely used in precast structural

components. It yields rigid members with minimum apparent deflection.

The disadvantages of reinforced concrete can be summarized as follows:

1. It has a low tensile strength of about one-tenth of its compressive strength.

2. It needs mixing, casting, and curing, all of which affect the final strength of concrete.

3. The cost of the forms used to cast concrete is relatively high.

4. It has a low compressive strength as compared to steel (the ratio is about 1:10,

depending on materials), which leads to large sections in columns of multistory

buildings.

61

5. Cracks develop in concrete due to shrinkage and the application of live loads. [4.2].

4.3 STRUCTURAL DESIGN

The design of different structures is achieved by performing, in general, two main

steps: (I) determining the different forces acting on the structure using proper methods

of structural analysis, and (2) proportioning all structural members economically,

considering the safety, stability, serviceability, and functionality of the structure. [4.2].

4.3.1 Objectives

The design of a structure must satisfy three basic requirements:

1) Stability to prevent overturning, sliding or buckling of the structure, or parts of

it, under the action of loads;

2) Strength to resist safely the stresses induced by the loads in the various

structural members; and

3) Serviceability to ensure satisfactory performance under service load

conditions.

There are two other considerations for a sensible designer; economy and

aesthetics. One can always design a massive structure, which has more-than-adequate

stability, strength and serviceability, but the ensuing cost of the structure may be

exorbitant, and the end product, far from aesthetic.

It is indeed a challenge, and a responsibility, for the structural designer to design

a structure that is not only appropriate for the architecture, but also strikes the right

balance between safety and economy. [4.1].

4.4 DESIGN PHILOSOPHY AND CONCEPTS

The design of a structure is the process of selecting the proper materials and

proportioning the different elements of the structure according to engineering principles

62

and, the structure must meet the conditions of safety, serviceability, economy, and

functionality. This can be achieved using design approach-based strain limits in

concrete and steel reinforcement.

4.4.1 Working Stress Method (WSM)

The basis of WSM assumes that the structural material behaves in a linear elastic

manner, and the adequate safety can be ensured by suitably restricting the stresses in

the material induced by the expected ‘working loads’ (service loads) on the structure.

As the specified permissible (‘allowable’) stresses are kept well below the material

strength (i.e., in the initial phase of the stress-strain curve), the assumption of linear

elastic behaviour is considered justifiable. The ratio of the strength of the material to

the permissible stress is often referred to as the factor of safety.

The stresses under the applied loads are analyzed using simple bending theory.

In order to apply such theory to a composite material like reinforced concrete, strain

compatibility (due to bond) is assumed, whereby the strain in the reinforcing steel is

assumed to be equal to that in the adjoining concrete to which it is bonded.

The design usually results in relatively large sections of structural members

(compared to ULM and LSM), thereby resulting in better serviceability performance

(less deflections, crack-widths, etc.) under the usual working loads. The method is also

notable for its essential simplicity — in concept, as well as application. [4.1].

4.4.2 Unified Design Method (UDM)

The unified design method (UDM) is based on the strength of structural members

assuming a failure condition, whether due to the crushing of the concrete or to the yield

of the reinforcing steel bars. Although there is some additional strength in the bars after

yielding (due to strain hardening), this additional strength is not considered in the

63

analysis of reinforced concrete members. In this approach, the actual loads, or working

loads, are multiplied by load factors to obtain the factored design loads. The load factors

represent a high percentage of the factor for safety required in the design. [4.2].

4.4.3 Ultimate Load Method (ULM)

Due to shortcomings of WSM in reinforced concrete design, and with improved

understanding of the behaviour of reinforced concrete at ultimate loads, the ultimate

load method of design (ULM) evolved in the 1950’s and became an alternative to

WSM. This method is sometimes also referred to as the load factor method or the

ultimate strength method.

In this method, the stress condition at the state of impending collapse of the

structure is analyzed, and the non-linear stress−strain curves of concrete and steel are

made use of. The concept of ‘modular ratio’ and its associated problems are avoided

entirely in this method. The safety measure in the design is introduced by an appropriate

choice of the load factor, defined as the ratio of the ultimate load (design load) to the

working load. The ultimate load method makes it possible for different types of loads

to be assigned different load factors under combined loading conditions, thereby

overcoming the related shortcoming of WSM.

This method generally results in more slender sections, and often more

economical designs of beams and columns (compared to WSM), particularly when high

strength reinforcing steel and concrete are used.

However, the satisfactory ‘strength’ performance at ultimate loads does not

guarantee satisfactory ‘serviceability’ performance at the normal service loads. The

designs sometimes result in excessive deflections and crack-widths under service loads,

64

owing to the slender sections resulting from the use of high strength reinforcing steel

and concrete. [4.1].

4.4.4 Limit States Method (LSM)

The philosophy of the limit states method of design (LSM) represents a definite

advancement over the traditional design philosophies. Unlike WSM, which based

calculations on service load conditions alone, and unlike ULM, which based

calculations on ultimate load conditions alone, LSM aims for a comprehensive and

rational solution to the design problem, by considering safety at ultimate loads and

serviceability at working loads.

The LSM philosophy uses a multiple safety factor format which attempts to

provide adequate safety at ultimate loads as well as adequate serviceability at service

loads, by considering all possible ‘limit states’. [4.1].

Limit States

A limit state is a state of impending failure, beyond which a structure ceases to

perform its intended function satisfactorily, in terms of either safety or serviceability;

i.e., it either collapses or becomes unserviceable.

There are two types of limit states:

1. Ultimate limit states (or ‘limit states of collapse’), which deal with strength,

overturning, sliding, buckling, fatigue fracture, etc.

2. Serviceability limit states, which deal with discomfort to occupancy and/or

malfunction, caused by excessive deflection, crack-width, vibration, leakage, etc., and

also loss of durability, etc.

65

Multiple Safety Factor Formats

The objective of limit states design is to ensure that the probability of any limit

state being reached is acceptably low. This is made possible by specifying appropriate

multiple safety factors for each limit state (Level I reliability). Of course, in order to be

meaningful, the specified values of the safety factors should result (more-or-less) in a

‘target reliability’.

Load and Resistance Factor Design Format

Of the many multiple safety factor formats in vogue, perhaps the simplest to

understand is the Load and Resistance Factor Design (LRFD) format, which is adopted

by the ACI Code. Applying the LRFD concept to the classical reliability model,

adequate safety requires the following condition to be satisfied:

∅ (4.1)

where Rn and Sn denote the nominal or characteristic values of resistance R and

load effect S respectively; φ and γ denote the resistance factor and load factor

respectively. The resistance factor φ accounts for ‘under-strength’, i.e., possible

shortfall in the computed ‘nominal’ resistance, owing to uncertainties related to

material strengths, dimensions, theoretical assumptions, etc., and accordingly, it is less

than unity. On the contrary, the load factor γ, which accounts for ‘overloading’ and the

uncertainties associated with Sn, is generally greater than unity.

Eq. 4.1 may be rearranged as:

∅ (4.2)

66

which is representative of the safety concept underlying WSM, γ/φ here

denoting the ‘factor of safety’ applied to the material strength, in order to arrive at the

permissible stress for design.

Alternatively, Eq. 4.2 may be rearranged as:

/∅ (4.3)

which is representative of the safety concept underlying ULM, γ/φ here

denoting the so-called ‘load factor’ (ULM terminology) applied to the load in order to

arrive at the ultimate load for design. [4.1].

67

4.4.5 Summary of Design Methods

Tabl

e 4.

4-1:

Sum

mar

y of

Des

ign

Met

hods

68

4.5 CODES OF PRACTICE

4.5.1 Purpose of Codes

National building codes have been formulated in different countries to lay down

guidelines for the design and construction of structures. The codes have evolved from

the collective wisdom of expert structural engineers, gained over the years. These codes

are periodically revised to bring them in line with current research, and often, current

trends.

The codes serve at least four distinct functions. Firstly, they ensure adequate

structural safety, by specifying certain essential minimum requirements for design.

Secondly, they render the task of the designer relatively simple; often, the results of

sophisticated analyses are made available in the form of a simple formula or chart.

Thirdly, the codes ensure a measure of consistency among different designers. Finally,

they have some legal validity, in that they protect the structural designer from any

liability due to structural failures that are caused by inadequate supervision and/or

faulty material and construction. [4.1].

4.5.2 Basic Codes for Design

Most codes specify design loads, allowable stresses, material quality, construction

types, and other requirements for building construction. The most significant American

code for structural concrete design is the Building Code Requirements for Structural

Concrete, ACI 318, or the ACI Code. Other American codes of practice and material

specifications include the International Building Code, the Uniform Building Code,

Standard Building Code, National Building Code, Basic Building Code, South Florida

Building Code, American Association of State Highway and Transportation Officials

(AASHTO) specifications, and specifications issued by the American Society for

69

Testing and Materials (ASTM), American Railway Engineering Association (AREA),

and Bureau of Reclamation, Department of the Interior. Different codes other than those

of the United States include the British Standard (BS) Code of Practice for Reinforced

Concrete, CP 110 and BS 8110; the National Building Code of Canada; the German

Code of Practice for Reinforced Concrete, DIN 1045; Specifications for Steel

Reinforcement (U.S.S.R.); and Technical Specifications for the Theory and Design of

Reinforced Concrete Structures, CC-BA (France), and the CEB Code (Comite

European Du Beton). [4.2].

4.6 LOADS

Structural members must be designed to support specific loads. In general, loads

may be classified as dead or live.

Dead loads include the weight of the structure (its self-weight) and any permanent

material placed on the structure, such as tiles, roofing materials, and walls. Dead loads

can be determined with a high degree of accuracy from the dimensions of the elements

and the unit weight of materials.

Live loads are all other loads that are not dead loads. They may be steady or

unsteady or movable or moving; they may be applied slowly, suddenly, vertically, or

laterally, and their magnitudes may fluctuate with time. In general, live loads include

the following:

o Occupancy loads caused by the weight of the people, furniture, and goods

o Forces resulting from wind action and temperature changes

o The weight of snow if accumulation is probable

o The pressure of liquids or earth on retaining structures

o The weight of traffic on a bridge

70

o Dynamic forces resulting from moving loads (impact), earthquakes, or blast

loading.

The ACI Code does not specify loads on structures; however, occupancy loads on

different types of buildings are prescribed by the American National Standards Institute

(ANSI). Some typical values are shown in Table 4.6-1. Table 4.6-2 shows weights and

specific gravity of various materials.

Table 4.6-1: Typical Uniformly Distributed Design Loads

Snow loads on structures may vary between 10 and 40 lb/ft2 (0.5 and 2 kN/m2),

depending on the local climate.

Wind loads may vary between 15 and 30 lb/ft2, depending on the velocity of wind.

The wind pressure of a structure, F, can be estimated from the following equation:

F = 0.00256 Cs V2 (4.1)

where

V = velocity of air (mi/h)

71

Cs = shape factor of the structure

F = the dynamic wind pressure (lb/ft2)

The shape factor, Cs varies with the horizontal angle of incidence of the wind. On

vertical surfaces of rectangular buildings, Cs may vary between 1.2 and 1.3. [4.2].

Table 4.6-2: Density and Specific Gravity of Various Materials

4.7 SAFETY PROVISIONS [4.2]

Structural members must always be proportioned to resist loads greater than the

service or actual load in order to provide proper safety against failure. In the strength

design method, the member is designed to resist factored loads, which are obtained by

multiplying the service loads by load factors. Different factors are used for different

loadings. Because dead loads can be estimated quite accurately, live loads have a high

degree of uncertainty. Several load combinations must be considered in the design. The

72

ACI Code presents specific values of load factors to be used in the design of concrete

structures.

A safe design is achieved when the structure's strength, obtained by multiplying

the nominal strength by the reduction factor∅, exceeds or equals the strength needed

to withstand the factored loadings (service loads times their load factors). For example,

∅ ∅ (4.2)

Where,

Mu and Vu = external factored moment and shear forces,

Mn and Vn = nominal flexural strength and shear strength of the member,

respectively.

73

CHAPTER 5

DESIGN OF STRUCTURAL MEMBERS

5.1 INTRODUCTION

The concrete building may contain some or all of the following main structural

elements, which are explained in detail in subsequent sections:

• Beams are long, horizontal or inclined members with limited width and depth.

Their main function is to support loads from slabs.

• Columns are critical members that support loads from beams or slabs. They

may be subjected to axial loads or axial loads and moments.

• Frames are structural members that consist of a combination of beams and

columns or slabs, beams, and columns. They may be statically determinate or statically

indeterminate frames.

• Slabs are horizontal plate elements in building floors and roofs. They may carry

gravity loads as well as lateral loads. The depth of the slab is usually very small relative

to its length or width.

• Walls are vertical plate elements resisting gravity as well as lateral loads as in

the case of basement walls.

• Footings are pads or strips that support columns and spread their loads directly

to the soil. [5.1].

In this text, preceding members are designed using Ultimate Strength Design

Method.

74

5.2 RECTANGULAR BEAMS

Figure 5.2-1: Reinforced Concrete Beam

5.2.1 Types of Flexural Failure and Strain Limits [5.1]

Flexural Failure

Three types of flexural failure of a structural member can be expected depending

on the percentage of steel used in the section.

1. Steel may reach its yield strength before the concrete reaches its maximum

strength, Fig. 5.2-2a. In this case, the failure is due to the yielding of steel reaching a

high strain equal to or greater than 0.005. The section contains a relatively small amount

of steel and is called a tension-controlled section.

2. Steel may reach its yield strength at the same time as concrete reaches its

ultimate strength, Fig. 5.2-2b. The section is called a balanced section.

75

Figure 5.2-2: Stress and Strain Diagrams

76

3. Concrete may fail before the yield of steel, Fig. 5.2-2c, due to the presence of

a high percentage of steel in the section. In this case, the concrete strength and its

maximum strain of 0.003 are reached, but the steel stress is less than the yield strength,

that is, fs is less than fy. The strain in the steel is equal to or less than 0.002. This section

is called a compression-controlled section.

It can be assumed that concrete fails in compression when the concrete strain

reaches 0.003. A range of 0.0025 to 0.004 has been obtained from tests and the ACI

Code assumes a strain of 0.003.

In beams, designed as tension-controlled sections, steel yields before the crushing

of concrete. Cracks widen extensively, giving warning before the concrete crushes and

the structure collapses. The ACI Code adopts this type of design. In beams, designed

as balanced or compression-controlled sections, the concrete fails suddenly, and the

beam collapses immediately without warning. The ACI Code does not allow this type

of design.

Strain Limits for Tension and Tension-Controlled Sections

The design provisions for both reinforced and prestressed concrete members are

based on the concept of tension or compression-controlled sections, ACI Code, Section

10.3. Both are defined in terms of net tensile strain (NTS), , in the extreme tension

steel at nominal strength. Moreover, two other conditions may develop: (1) the balanced

strain condition and (2) the transition region condition. These four conditions are

defined as follows:

1. Compression-controlled sections are those sections in which the net tensile strain,

NTS, in the extreme tension steel at nominal strength is equal to or less than the

compression-controlled strain limit at the time when concrete in compression reaches

77

its assumed strain limit of 0.003, ( = 0.003). For grade 60 steel, (fy = 60 ksi), the

compression-controlled strain limit may be taken as a net strain of 0.002, Fig. 5.2-3a.

This case occurs mainly in columns subjected to axial forces and moments.

Figure 5.2-3:Strain Limit Distribution

2. Sections in which the NTS in the extreme tension steel lies between the compression-

controlled strain limit (0.002 for fy = 60 ksi) and the tension-controlled strain limit of

0.005 constitute the transition region, Fig. 5.2-3b.

3. Tension-controlled sections are those sections in which the NTS, is equal to or

greater than 0.005 just as the concrete in the compression reaches its assumed strain

limit of 0.003, Fig. 5.2-3c.

Figure 5.2-4: Balanced Strain Condition

78

4. The balanced strain condition develops in the section when the tension steel, with the

first yield, reaches a strain corresponding to its yield strength, fy or = fy/Es, just as

the maximum strain in concrete at the extreme compression fibers reaches 0.003, Fig.

5.2-4.

5.2.2 Equivalent Compressive Stress Distribution

The distribution of compressive concrete stresses at failure may be assumed to be

a rectangle, trapezoid, parabola, or any other shape that is in good agreement with test

results. When a beam is about to fail, the steel will yield first if the section is under-

reinforced, and in this case the steel is equal to the yield stress. If the section is over-

reinforced, concrete crushes first and the strain is assumed to be equal to 0.003, which

agrees with many tests of beams and columns. A compressive force, C, develops in the

compression zone and a tension force, T, develops in the tension zone at the level of

the steel bars. The position of force T is known, because its line of application coincides

with the center of gravity of the steel bars. The position of compressive force C is not

known unless the compressive volume is known and its center of gravity is located. If

that is done, the moment arm, which is the vertical distance between C and T, will

consequently be known.

In Fig. 5.2-5, if concrete fails, = 0.003, and if steel yields, as in the case of a

balanced section, fs = fy.

The compression force, C, is represented by the volume of the stress block, which

has the non-uniform shape of stress over the rectangular hatched area. This volume may

be considered equal to C = ba( ), where is an assumed average stress of the

non-uniform stress block.

79

The position of compression force C is at a distance z from the top fibers, which

can be considered as a fraction of the distance c (the distance from the top fibers to the

neutral axis), and z can be assumed to be equal to , where < 1. The values of

and have been estimated from many tests.

Figure 5.2-5: Ultimate forces in a rectangular section

To derive a simple rational approach for calculations of the internal forces of a

section, the ACI Code adopted an equivalent rectangular concrete stress distribution,

which was first proposed by C. S. Whitney and checked by Mattock and others. A

concrete stress of 0.85fc’ is assumed to be uniformly distributed over an equivalent

compression zone bounded by the edges of the cross-section and a line parallel to the

neutral axis at a distance from the fiber of maximum compressive strain, where

80

c is the distance between the top of the compressive section and the neutral axis (Fig.

5.2-5). The fraction is 0.85 for concrete strengths fc’ ≤ 4000 psi (27.6 MPa) and is

reduced linearly at a rate of 0.05 for each 1000 psi (6.9 MPa) of stress greater than

4000psi (Fig. 5.2-6), with a minimum value of 0.65. [5.1].

Figure 5.2-6: Values of Beta for different fc'.

Figure 5.2-7: Internal Equilibrium of RCC Section

C = compression in concrete = stress x area = 0.85 fc’.b.a

T = tension in steel = stress x area = Asfy

C = T and Mn = T(d-a/2)

where

fc’ = concrete compression strength

a = height of stress block

81

β1 = factor based on fc’

x = location to the neutral axis

b = width of stress block

fy = steel yield strength

As = area of steel reinforcement

d = effective depth of section = depth to n.a. of reinforcement

With C = T, As fy = 0.85 fc’ b.a so ‘a’ can be determined as: [5.2]

0.85 (5.1)

5.2.3 Beam Design

Reinforcement Ratio

The amount of steel reinforcement is limited. Too much reinforcement, or over-

reinforcing will not allow the steel to yield before the concrete crushes and there is a

sudden failure. A beam with the proper amount of steel to allow it to yield at failure is

said to be under reinforced. [5.2].

82

Design of Singly Reinforced Beams

Design for Flexure:

Step-1: Compute dead and five loads and multiply them with their respective design

factors combinations.

Figure 5.2-8: Load Factor Combinations

Step-2: Compute Bending Moment (Factored) from factored loads using appropriate

formula.

Table 5.2-1: Bending Moments

83

Step-3: Note out Maximum positive and negative bending Moments, and Assume

suitable dimension b and d for Beam. Assume d as 1 in. for every foot of span length.

Step-4: Compute Reinforcement Ratios.

a). Minimum Reinforcement Ratio:

200 (5.1)

b). Balanced section Reinforcement ratio:

0.85 . 8787 (5.2)

Here,

∈

∈ ∈

∈ strain in concrete = 0.003

Figure 5.2-9: Stress in Beam

84

c). Maximum Reinforcement ratio:

∈ ⁄∈ ∈

(5.3)

∈ = strain in concrete = 0.003

∈ = strain in Steel = 0.005

d). Required Reinforcement Ratio:

0.851 1

41.7∅

(5.4)

e). Actual provided reinforcement Ratio:

emin < e < emax (5.5)

Step-5: Compute Area of steel, Bar dia, and bar Nos.

As = bd (5.6)

Table 5.2-2: Bar Diameter Chart

85

Bar Nos:

(5.7)

Step-6: Check moment capacity of designed section.

a). Revise provided area of steel.

As = Ab x No.s

b). Compute depth of stress block ‘a’

0.85 (5.8)

c). Moment capacity

M T or C da2

(5.9)

∅M ∅ T or C da2

(5.10)

d). Check

iff Mn > Mu

then section is safe for bending

Design for Shear:

Step-7: Compute maximum sheer force from factored loads, we get factored sheer force

Vu.

Step-8: Compute Vu at d from support. Fig: 5.2.10.

Step-9: Compute shear strength provided by concrete.

∅ V ∅ (5.11)

Here, = 0.75 for compression and shear members.

86

Figure 5.2-10: Critical Shear

Step-10: From figure 5.2-10, the point at which web reinforcement theoretically is no

longer required is [5.3], say x.

2∅

(5.12)

From support face. However, according to the ACI code, 11.5.5 at least a

minimum amount of web reinforcement is required whenever the shear force exceeds

is Vc/2. As shown in figure 5.2-10, this applies to a distance [5.3] say x’.

2∅ /2

(5.13)

From the support face. At least the minimum amount web reinforcement within

the distance of “x’ ” from supports, and with “x” the web steel must provide for the

shear force corresponding to the shaded area [5.3].

Step-11: Web reinforcement

a). Spacing

(5.14)

A= cross-sectional areas of standard stirrup; twice the area of bar.

87

b). Minimum Web Reinforcement

0.75.

50.

(5.16)

It is undesirable to space vertical stirrups closer their about 4 in. the size of the

stirrups when vertical stirrups are required over a comparatively short distance, it is a

good practice to space them uniformly over the entire distance, the spacing being

calculated for the point of greater shear (minimum spacing).

Where web reinforcement is required, the code requires it to be spaced so that

every 450 line, representing a potential diagonal crack and extending from the mid-

depth d/2 of the crossed by atleast one line of reinforcement, the code specifies a

maximum spacing of 24 in. when Vs exceeds 4. bd, these spacing’s are shattered.

[5.3]. For usual case of stirrups, with Vs < 4. bd the maximum spacing of stirrups

is the smallest of.

0.75 50 (5.16)

2 (5.17)

Smax = 24 in (5.18)

Design of Doubly Reinforced Beams

If a beam cross section is limited because of architectural or other considerations,

it may happen that the concrete cannot develop the compression force required to resist

the given bending moment. In this case, reinforcement is added in the compression

zone, resulting in a so-called doubly reinforced beam, i.e., one with compression as

well as tension reinforcement (see Fig. 5.2-11). The use of compression reinforcement

has decreased markedly with the use of strength design methods, which account for the

88

full strength potential of the concrete on the compressive side of the neutral axis.

However, there are situations in which compressive reinforcement is used for reasons

other than strength. It has been found that the inclusion of some compression steel will

reduce the long term deflections of members In addition, in some cases, bars will be

placed in the compression zone for minimum-moment loading or as stirrup-support bars

continuous throughout the beam span.

Figure 5.2-11: Doubly reinforced rectangular beam.

If, in a doubly reinforced beam, the tensile reinforcement ratio is less than or

equal to b, the strength of the beam may be approximated within acceptable limits by

disregarding the compression bars. The strength of such a beam will be controlled by

tensile yielding, and the lever arm of the resisting moment will ordinarily be but little

affected by the presence of the compression bars. If the tensile reinforcement ratio is

larger than b, a somewhat more elaborate analysis is required. In Fig. 5.2-11a, a

rectangular beam cross section is shown with compression steel As’ placed a distance

d' from the compression face and with tensile steel As, at effective depth d. It is assumed

initially that both As’ and As are stressed to fy at failure. The total resisting moment can

be thought of as the sum of two parts. The first part, Mn1 is provided by the couple

consisting of the force in the compression steel As’ and the force in an equal area of

89

tension steel as shown in Fig. 5.2-11d. The second part, Mn2 is the contribution of the

remaining tension steel As – As’ acting with the compression concrete. [5.3].

Compression steel will provide compressive forced in addition to the compressive

force in the concreting area [5.1].

Assuming one row of tension bars:

The procedure for designing a rectangular section with compression steel when

Mu, fc’, b, d and d are given can be summarized as follows:

Step 1: Calculate the balanced and maximum steel ratio, max, using Eqs. (5.2) and

(5.3).

0.8587

87 (5.19)

∈ ⁄

∈ ∈ (5.20)

Calculate Asmax = maxbd. (maximum steel over singly reinforced section).

Step 2: Calculate Rumax using max, = 0.9.

∅ 11.7

(5.21)

Step 3: Calculate the moment strength of the section, Mu1, as singly reinforced, using

max and Rumax bd2.

(5.22)

If Mu1 < Mu (the applied moment), the compression steel is needed. Go the next

step.

If Mu1 > Mu, the compression steel is not needed. Design the section as singly

reinforced section as explained above.

90

Step 4: The moment to be resisted by compression steel:

Mu2 = Mu – Mu1 (5.23)

Step 5: Calculate As2.

M ∅ A (5.24)

Here,

d = effective depth for tension steel to top fibre.

d’ = effective depth for compression steel to top fibre.

Then, total steel will be:

As = As1 + As2 (5.25)

Step 6: Calculate the stress in the compression steel follows:

a. Calculate

87′

, cannot exceed f (5.26)

here,

0.85 (5.27)

b. or ∈ can be calculated from the strain diagram, and ∈

. If ∈ >, ∈ then

compression steel yields and .

c. Calculate from Mu2 = ∅A f (d - d’). if f = fy then A = As2 . if f < fy, then

A > As2, and A = As2 (fy/f ).

Choose bars for As and As’ to fit within the section width, b. in most cases As bars

will be placed in two rows, whereas bars are placed in one row.

91

Step 7: Calculate h = d + 2.5 in. for one row of tension bars and h = d+ 3.5 in. for two

rows of tension steel. Round h to the next higher inch. Now check that [-p’( / )] <

max using the new d, or check that Asmax = bd [-p’ ( / )] > As (used).

(5.29)

′′

(5.30)

The check may not be needed if max is used in the basic section.

Assuming two rows of Tension Bars:

In the case of two rows of bars, it can be assumed that d = h – 3.5 in. and dt = h –

2.5 in. = d+1.0 in.

Two approaches may be used to design the section:

One approach is to assume a strain at the level of the centroid of the tension steel

equal to 0.005 or s = 0.005 ( at d level). In this case, the strain in the lower row of bars

is greater than 0.005 t = (d= – c/c) 0.003 > 0.005, which still meets the ACI code

limitation. For this case, follow the above steps.

A second approach is to assume a strain t = 0.005 at the level of the lower row

of bars, dt in this case, the stain at the level of the centroid of bars is less than 0.005: s

= [(dt –c)/c] 0.003 < 0.005, which is still acceptable. The solution can be summarized

as follows. [5.1].

a. Calculate dt = h – 2.5 and then form the strain diagram and calculate c,

the depth of the neutral axis.

0.0030.003

(5.31)

For t = 0.005,

92

38.

(5.32)

b. Calculate the compression force in the concrete.

C1 = 0.85 ab = T1 = As1 fy (5.33)

Determine As1. Calculate Mu1 = Asi. fy (d - a/2). 1 = As1/bd, = 0.9.

c. Calculate Mu2 = Mu – Mu1; assume d’ = 2.5 in.

d. Calculate As2: Mu2 = As2, fy (d –d’) = fy, =0.9 Total As = As1 +

As2.

e. Check if compression steel yields similar to step 6 above.

5.3 ANALYSIS AND DESIGN OF T SECTIONS

5.3.1 Description

It is normal to cast concrete slabs and beams together, producing a monolithic

structure. Slabs have smaller thicknesses than beams. Under bending stresses, those

parts of the slab on either side of the beam will be subjected to compressive stresses.

The part of the slab acting with the beam is called the flange, and it is indicated in Fig.

5.3-1a by area . The rest of the section confining the area (h - t) is called the stem,

or web.

In an I-section there are two flanges, a compression flange, which is actually

effective, and a tension flange, which is ineffective, because it lies below the neutral

axis and is thus neglected completely. Therefore, the analysis and design of an I-beam

is similar to that of a T-beam. [5.1].

93

5.3.2 Effective Width

In a T-section, if the flange is very wide, the compressive stresses are at a

maximum value at points adjacent to the beam and decrease approximately in a

parabolic form to almost 0 at a distance x from, the face of the beam. Stresses also vary

vertically from a maximum at the top fibers of the flange to a minimum at the lower

fibers of the flange. This variation depends on the position of the neutral axis and the

change from elastic to inelastic deformation of the flange along its vertical axis.

An equivalent stress area can be assumed to represent the stress distribution on

the width b of the flange, producing an equivalent flange width, , of uniform stress

(Fig. 5.3-1c).

Figure 5.3-1: (a) T-section and (b) I-section, with (c) illustration of effective flange width b_e

Other variables that affect the effective width are (Fig. 5.3-2).

• Spacing of beams.

94

• Width of stem (web) of beam .

• Relative thickness of slab with respect to the total beam depth.

• End conditions of the beam (simply supported or continuous)

• The way in which the load is applied (distributed load or point load)

• The ratio of the length of the beam between points of zero moment to the width of the

web and the distance between webs. [5.1].

Figure 5.3-2: Effective flange width of T-beams

5.3.3 T -Sections Behaving as Rectangular Sections

In this case, the depth of the equivalent stress block ‘a’ lies within the flange,

with extreme position at the level of the bottom fibers of the compression flange (a ≤

t). When the neutral axis lies within the flange (Fig. 5.3-3a), the depth of the equivalent

compressive distribution stress lies within the flange producing a compressed area equal

to a. The concrete below the neutral axis is assumed ineffective, and the section is

considered singly reinforced with ‘be’ replaced by b. Therefore,

95

0.85 ′ (5.34)

Figure 5.3-3: Rectangular section behavior (a) when the neutral axis lies within the flange and (b) when the stress distribution depth equals the slab thickness.

And

∅ ∅2

(5.35)

If the depth ‘a’ is increased such that a = t, then the factored moment capacity

is that of a singly reinforced concrete section:

∅ ∅2

(5.36)

In this case

0.85 ′ (5.37)

96

In this analysis, the limit of the steel area in the section should apply: ,

and 0.005. [5.1].

5.3.4 Analysis of a T-Section

In this case the depth of the equivalent compressive distribution stress lies below

the flange. Consequently, the neutral axis also lies in the web. This is due to an amount

of tension steel more than that calculated by Eq. 5.37. Part of the concrete in the

web will now be effective in resisting the external moment. In Fig. 5.3-4, the

compressive force C is equal to the compression area of the flange and web multiplied

by the uniform stress of 0.85 ′

C = 0.85 ′ [ (a - t )] (5.38)

The position of C is at the centroid of the T-shaped compressive area at a

distance z from top fibers.

Figure 5.3-4: T-section behavior.

The analysis of a T-section is similar to that of a doubly reinforced concrete

section, considering an area of concrete as equivalent to the compression

steel area ′ . The analysis is divided into two parts, as shown in Fig. 5.3-5:

97

1. A singly reinforced rectangular basic section, d, and steel reinforcement · The

compressive force, , is equal to 0.85 ′ , the tensile force, , is equal to

, and the moment arm is equal to (d - a/2).

Figure 5.3-5:T -section analysis.

2. A section that consists of the concrete overhanging flange sides 2 x /2

developing the additional compressive force (when multiplied by 0.85 ′ ) and a

moment arm equal to (d - t/2). If is the area of tension steel that will develop a

force equal to the compressive strength of the overhanging flanges, then

=0.85 ′

0.85 ′

(5.39)

The total steel used in the T-section is equal to + , or

- (5.40)

98

The T-section is in equilibrium, so = , = , and = .

Considering equation = for the basic section, then = 0.85 ′ or (

- )/y = 0.85 ′ ; therefore,

0.85 ′ (5.41)

Note that is used to calculate a. The factored moment capacity of the section is the

sum of the two moments and .

∅

∅2

∅ 2

where

-

and

0.85 ′

∅2

(5.42)

∅ =∅ (5.43)

Considering the web section d, the net tensile strain (NTS), , can be

calculated from a, c, and as follows:

If c = (from Eq. 5.41) and = h - 2.5 in., then = 0.003 (c - )/c. For

tension-controlled section in the web, 0.005. [5.1].

99

5.3.5 Design of T-Sections

In slab-beam-girder construction, the slab dimensions as well as the spacing and

position of beams are established first. The next step is to design the supporting beams,

namely, the dimensions of the web and the steel reinforcement.

In many cases web dimensions can be known based on the flexural design of the

section at the support in a continuous beam. The section at the support is subjected to a

negative moment, the slab being under tension and considered not effective, and the

beam width is that of the web.

In the design of a T-section for a given factored moment, Mu, the flange thickness,

t, and width, b, would have been already established from the design of the slab and the

ACI Code limitations for the effective flange width, b, The web thickness, , can be

assumed to very between 8 in. and 20 in., with a practical width of 12 to 16 in. [5.1].

Step 1: Calculate effective width of flange′ . Fig: 5.3-6. Smallest of all will be used.

Figure 5.3-6: ACI Code, Section 8.12

Step 2: Effective depth ‘d‘. Depth of the web is commonly known from flexural design

of beam plus the thickness of slab, subtracting from it the cover, mathematically.

d = + t – t’ (5.45)

100

where,

= depth of beam .

t = slab thickness .

t’ = cover. i.e., distance from bottom fiber to centroid of tension steel .

Step 3: Shape compression block

a. Check if section act as a rectangular or T-section by assuming a = t and

calculating the moment strength of whole flange :

∅ ∅ 0.85 /2 (5.46)

If > ∅ then a > t, and the section behaves as a T-section

If < ∅ then a < t, and the section behave as a rectangular section.

Here, is the factored maximum positive bending moment.

Step 4: Area of steel.

If a < t, then calculate using equation 5.47. Also check > .

0.85 ′1 1

41.7∅

(5.47)

bd

Or else, If a > t, then switch to step 6.

Step 5: Moment capacity of section equation 5.34 and 5.35.

Depth of compression block ‘a’:

a0.85

∅ =∅ /2

∅ >

Section safe for bending.

101

Step 6: If a > t determine for overhanging portion of flange eq: 5.39.

0.85 t/

Step 7: Moment resisted by web eq: 5.42

∅2

Step 8: Calculate , using and d in equation 5.47 and determine

Total

Then check that .Also check that

If a = t, then ∅ 0.85 /

Step 9: Moment capacity of T-section.

∅ =∅ (5.49)

0.85

∅ (5.50)

Section is safe for bending.

102

5.4 COLUMNS

Columns are members used primarily to support axial compressive loads and have

a ratio of height to the least lateral dimension of 3 or greater. In reinforced concrete

buildings, beams, floors and columns are cast monolithically, causing some moments

in the columns due to end restraint. Moreover, perfect vertical alignment of columns in

a multistory building is not possible, causing loads to be eccentric relative to the center

of columns. The eccentric loads will cause moments in columns. Therefore, a column

subjected to pure axial loads does not exist in concrete buildings. However, it can be

assumed that axially loaded columns are those with relatively small eccentricity, e, of

about 0.1h or less, where ‘h’ is the total depth of the column and e is the eccentric

distance from the center of the column. Because concrete has a high compressive

strength and is an inexpensive material, it can be used in the design of compression

members economically. [5.1].

5.4.1 Types of Columns

Columns may be classified based on the following different categories (Fig 5.4-

1):

1. Based on loading, columns may be classified as follows:

a. Axially loaded columns, where loads are assumed acting at the center of the

column section.

b. Eccentrically loaded columns, where loads are acting at a distance e from the

center of the column section. The distance e could be along the x- or y-axis, causing

moments either about the x- or y-axis.

c. Biaxially loaded columns, where the load is applied at any point on the column

section, causing moments about both the x- and y-axes simultaneously.

103

Figure 5.4-1 types of columns

2. Based on length, columns may be classified as follows:

a. Short columns, where the column's failure is due to the crushing of concrete or

the yielding of the steel bars under the full load capacity of the column. Figure 5.4-2.

b. Long columns, where buckling effect and slenderness ratio must be taken into

consideration in the design, thus reducing the load capacity of the column relative to

that of a short column. Figure 5.4-3.

3. Based on the shape of the cross-section:

a. Column sections may be square, rectangular, round, L-shaped, octagonal, or

any desired shape with an adequate side width or dimensions.

104

Figure 5.4-2: Failure of Short Column

Figure 5.4-3: Failure of Long Column

4. Based on column ties, columns may be classified as follows:

a. Tied columns containing steel ties to confine the main longitudinal bars in the

columns. Ties are normally spaced uniformly along the height of the column. Figure

5.4-4.

Figure 5.4-4: Failure of tied Column

105

Figure 5.4-5: Failure of spiral Column

b. Spiral columns containing spirals (spring-type reinforcement) to hold the main

longitudinal reinforcement and to help increase the column ductility before failure. In

general, ties and spirals prevent the slender, highly stressed longitudinal bars from

buckling and bursting the concrete cover. Figure 5.4-5. [5.1].

5.4.2 Behavior of Axially Loaded Columns

When an axial load is applied to a reinforced concrete short column, the concrete

can be considered to behave elastically up to a low stress of about (1/3 fc’). If the load

on the column is increased to reach its ultimate strength, the concrete will reach the

maximum strength and the steel will reach its yield strength fy. The nominal load

capacity of the column can be written as follows:

0.85 (5.51)

Where,

An and Ast = the net concrete and total steel compressive areas,

respectively.

106

An = Ag - Ast

Ag = gross concrete area

Two different types of failure occur in columns, depending on whether ties or

spirals are used. For a tied column, the concrete fails by crushing and shearing outward,

the longitudinal steel bars fail by buckling outward between ties, and the column failure

occurs suddenly, much like the failure of a concrete cylinder.

A spiral column undergoes a marked yielding, followed by considerable

deformation before complete failure. The concrete in the outer shell fails and spalls off.

The concrete inside the spiral is confined and provides little strength before the

initiation of column failure. A hoop tension develops in the spiral, and for a closely

spaced spiral, the steel may yield. A sudden failure is not expected. Figure 5.4-6 shows

typical load deformation curves for tied and spiral columns. Up to point a, both columns

behave similarly. At point a, the longitudinal steel bars of the column yield, and the

spiral column shell spalls off. After the factored load is reached, a tied column fails

suddenly (curve b), whereas a spiral column deforms appreciably before failure (curve

c). [5.1].

Figure 5.4-6: Behavior of tied and spiral columns

107

5.4.3 ACI Code Limitations

The ACI Code presents the following limitations for the design of compression

members:

1. For axially as well as eccentrically loaded columns, the ACI Code sets the

strength-reduction factors as ∅ = 0.65 for tied columns and ∅ = 0.75 for spirally

reinforced columns. The ACI Code Limitations difference of 0.05 between the two

values shows the additional ductility of spirally reinforced columns.

The strength-reduction factor for columns is much lower than those for flexure

(∅ = 0.9) and shear (∅ = 0.75). This is because in axially loaded columns, the strength

depends mainly on the concrete compression strength, whereas the strength of members

in bending is less affected by the variation of concrete strength, especially in the case

of an under-reinforced section. Furthermore, the concrete in columns is subjected to

more segregation than in the case of beams. Columns are cast vertically in long, narrow

forms, but the concrete in beams is cast in shallow, horizontal forms. Also, the failure

of a column in a structure is more critical than that of a floor beam.

2. The minimum longitudinal steel percentage is 1%, and the maximum

percentage is 8% of the gross area of the section (ACI Code, Section 10.9.1), Figure

5.4-7. Minimum reinforcement is necessary to provide resistance to bending, which

may exist, and to reduce the effects of Creep and Shrinkage of the concrete under

sustained compressive stresses. Practically, it is very difficult to fit more than 8% of

steel reinforcement into a column and maintain sufficient space for concrete to flow

between bars.

108

Figure 5.4-7: ACI Code, Section 10.9.1

3. At least four bars are required for tied circular and rectangular members and

six bars are needed for circular members enclosed by spirals (ACI Code, Section

10.9.2), Figure 5.4-8. For other Shapes, one bar should be provided at each comer, and

proper lateral reinforcement must be provided. For tied triangular columns, at least

three bars are required. Bars shall not be Located at a distance greater than 6 in. clear

on either side from a laterally supported bar.


Figure 5.4-9 shows the arrangement of longitudinal bars in tied columns and the

distribution of ties. Ties shown in dotted lines are required when the clear distance on

either side from laterally supported bars exceeds 6 in. The minimum concrete cover in

columns is 1.5 in.

4. The minimum ratio of spiral reinforcement, according to the ACI Code

Section 10.9.3, is limited to:

0.45 1 (5.52)

109

Figure 5.4-9: Arrangement of bars and ties in columns

where

Ag = gross area of section

Ach = area of core of spirally reinforced column measured to the outside diameter

of spiral

fyt = yield strength of spiral reinforcement (60 ksi; ACI Code, Section 10.9.3)

5. The minimum diameter of spirals is 3/8 in., and their clear spacing should not

be more than 3 in. nor less than 1 in., according to the ACI Code, Section 7 .10.4, Figure

5.4-10.

6. Ties for columns must have a minimum diameter of 3/8 in. to enclose

longitudinal bars of No. 10 size or smaller and a minimum diameter of 1/2 in. for larger

bar diameters (ACI Code, Section 7 .10.5), Figure 5.4-11.

110

Figure 5.4-10: ACI Code, Section 7 .10.4

7. Spacing of ties shall not exceed the smallest of 48 times the tie diameter, 16

times the Longitudinal bar diameter, or the least dimension of the column. Table 10.1

gives spacing’s For no. 3 and no. 4 ties. The Code does not give restrictions on the size

of columns to allow wider utilization of reinforced concrete columns in smaller sizes.

[5.1].

Figure 5.4-11: ACI Code, Section 7 .10.5

111

Table 5.4-1: Maximum Spacings of Ties

5.4.4 Spiral Reinforcement

Spiral reinforcement in compression members prevents a sudden crushing of

concrete and buckling of longitudinal steel bars. It has the advantage of producing a

tough column that undergoes Gradual and ductile failure. The minimum spiral ratio

required by the ACI Code is meant to provide an additional compressive capacity to

compensate for the spelling of the column shell.

The strength contribution of the shell is:

0.85 (5.53)

Where,

Ag is the gross concrete area and Ach is the core area (Fig. 5.4-12).

In spirally reinforced columns, spiral steel is at least twice as effective as

longitudinal bars; therefore, the strength contribution of spiral equals 2 where

is the ratio of volume of spiral reinforcement to total volume of core.

112

Figure 5.4-12: Dimensions of a column spiral

Table 5.4-2: Spirals for Circular Columns (fy = 60 ksi)

If the strength of the column shell is equated to the spiral strength contribution,

then:

0.85 2 (5.54)

0.425 1

113

The ACI Code adopted a minimum ratio of according to the following

equation:

0.45 1 (5.52)

The design relationship of spirals may be obtained as follows (Fig. 5.4-12):

4

4 (5.55)

Where

as = area of spiral reinforcement

Dch = diameter of the core measured to the outside diameter of

spiral

D = diameter of the column

ds = diameter of the spiral

S = spacing of the spiral

Table 5.4-2 gives spiral spacing’s for no. 3 and no. 4 spirals with fy = 60 ksi.

5.4.5 DESIGN EQUATIONS

The nominal load strength of an axially loaded column was given in Eq. 5.50.

Because a perfect axially loaded column does not exist, some eccentricity occurs on the

column section, thus reducing its load capacity, P0 • To take that into consideration, the

ACI Code specifies that the Maximum nominal load, P0 , should be multiplied by a

factor equal to 0.8 for tied columns and 0.85 for spirally reinforced columns.

Introducing the strength reduction factor, the axial load Strength of columns according

to the ACI Code, Section 10.3.6, figure 5.4-13, are as follows:

114


∅ ∅ 0.80 0.85 (5.56)

For tied columns and

∅ ∅ 0.85 0.85 (5.57)

For spiral columns, where,

Ag = gross concrete area

Ast = total steel compressive area

∅ = 0.65 for tied columns and 0.70 for spirally reinforced

columns

Equations 5.56 and 5.57 may be written as follows:

∅ ∅ 0.85 (5.58)

Where ∅ = 0.65 and K = 0.8 for tied columns and ∅ = 0.75 and K = 0.85 for spiral

columns.

If the gross steel ratio is = Ast / Ag, or Ast = g Ag, then Eq. 5.58 may be written

as follows:

∅ ∅ 0.85 0.85 (5.59)

Equation 5.58 can be used to calculate the axial load strength of the column,

whereas Eq. 5.59 is used when the external factored load is given and it is required to

115

calculate the size of the column section, Ag, based on an assumed steel ratio, g between

a minimum of 1% and a maximum of 8%.

It is a common practice to use grade 60 reinforcing steel bars in columns with a

concrete compressive strength of 4 ksi or greater to produce relatively small concrete

column sections. [5.1].

5.5 ONE WAY SLABS

Structural concrete slabs are constructed to provide flat surfaces, usually

horizontal, in building floors, roofs, bridges, and other types of structures. The slab may

be supported by walls, by reinforced concrete beams usually cast monolithically with

the slab, by structural steel beams, by columns, or by the ground. The depth of a slab is

usually very small compared to its span. See Fig. 5.5-1. [5.1].

Figure 5.5-1: One way slab

5.5.1 One-Way Beam–Slab Systems

The selection of a beam–slab structural system is most frequently driven by the

geometry of a given column bay. Rectangular bays, with an aspect ratio exceeding 2:1,

will function to distribute nearly 100% of the shear and moments in the short direction.

Under standard loading conditions, slabs can be kept thin (see Table 5.5-1), with

reinforcing steel provided primarily in one direction only. Nominal transverse

temperature and shrinkage reinforcement must always be provided to prevent cracking.

It is more effective to use smaller diameter bars at closer spacing than larger diameter

116

bars at larger spacing. The former is essential to controlling cracking development in

the slabs.

The distribution pattern of the

primary reinforcing steel closely follows

the pattern of the bending moment

diagram (see Figure 5.5-2). Where

negative moments are greatest (over the

beam supports), top reinforcing steel is

provided. The cutoff point for the top

steel occurs where the concrete no longer requires steel to resist tension stresses. The

ACI Code requires that reinforcement must extend beyond this point a distance equal

to the greater of the effective depth of the slab or 12db, the diameter of the bar. Also, at

least 1/3 of the total tension reinforcing provided for negative moment must be extended

beyond the point of inflection not less than the effective depth of the slab, 12db, or 1/16

the clear span, whichever is greater.

Where extremely heavy loads (exceeding 250 psf) are experienced, the slab shear

capacity should be checked. In accordance with Chapter 11 of the ACI 318 Code,

figures 5.5-3 and 5.5-4, the critical shear plane is located at a dimension ‘d’ away from

the face of the support for one-way slabs.

No special detailing at columns is required, as the beam element is intended to

carry 100% of the load from the slab to the column. One-way slab systems have the

following advantages:

Table 5.5-1: Minimum Thickness h of non-prestressed one-way slabs

117

• Long-span capability of the beam elements permits wide column Spacings and

frame elements for lateral resistance.

• Predictable slab thicknesses, reinforcing requirements, and deflection

performance allow the designer to concentrate design efforts elsewhere.

• Reinforcing detailing and placement are prioritized in one direction only,

reducing complication at the construction site.

Figure 5.5-2: Moments and reinforcement locations in continuous beams

One-way concrete-slab systems are frequently used in parking structures, where

the predictable traffic patterns require long open-column bays in one direction but

permit shorter bays in the other direction. [5.6].

118

Figure 5.5-3: Critical Shear (a)

Figure 5.5-4: Critical Shear (b)

Figure 5.5-5: One-way slab bar bending and placing detail

5.5.2 Temperature and Shrinkage Reinforcement

Concrete shrinks as the cement paste hardens, and a certain amount of shrinkage

is usually anticipated. If a slab is left to move freely on its supports, it can contract to

accommodate the shrinkage. However, slabs and other members are joined rigidly to

other parts of the structure, causing a certain degree of restraint at the ends. This results

in tension stresses known as shrinkage stresses. A decrease in temperature and

shrinkage stresses is likely to cause hairline cracks. Reinforcement is placed in the slab

119

to counteract contraction and distribute the cracks uniformly. As the concrete shrinks,

the steel bars are subjected to compression.

Reinforcement for shrinkage and temperature stresses normal to the principal

reinforcement should be provided in a structural slab in which the principal

reinforcement extends in one direction only. The ACI Code, Section 7.12.2, specifies

the following minimum steel ratios, figure 5.5-6.


For temperature and shrinkage reinforcement, the whole concrete depth ‘h’

exposed to shrinkage shall be used to calculate the steel area. [5.1].

5.5.3 Design of One-Way Solid Slabs

If the concrete slab is cast in one uniform thickness without any type of voids, it

can be referred to as a solid slab. In a one-way slab, the ratio of the length of the slab

120

to its width is greater than 2. Nearly all the loading is transferred in the short direction,

and the slab may be treated as a beam. A unit strip of slab, usually 1ft (or 1m) at right

angles to the supporting girders, is considered a rectangular beam. The beam bas a unit

width with a depth equal to the thickness of the slab and a span length equal to the

distance between the supports. A one-way slab thus consists of a series of rectangular

beams placed side by side (Fig. 5.5-7).

Figure 5.5-7: Rectangular strip in one way slab

The ACl Code, Section 8.3, figure 5.5-8, permits the use of moment and shear

coefficients in the case of two or more approximately equal spans (Fig. 5.5-9). When

these conditions are not satisfied, structural analysis is required.

121

Figure 5.5-8: ACl Code, Section 8.3

122

Figure 5.5-9: Moment coefficients for continuous beams and slabs

Design Limitations according to the ACI Code

The following limitations are specified by the ACI Code.

1. A typical imaginary strip 1 ft (or 1 m) wide is assumed.

2. The minimum thickness of one-way slabs using grade 60 steel according to the

ACI Code, Table 9.5a, see table 5.5-2:

3. Deflection is to be checked when the slab supports are attached to construction

likely to be damaged by large deflections. Deflection limits are set by the ACI Code,

Table 9.5b, see table 5.5-3.

4. It is preferable to choose slab depth to the nearest 1/2 in. (or 10 mm).

5. Shear should be checked, although it does not usually control.

123

Table 5.5-2:ACI Code, Table 9.5a

6. Concrete cover in slabs shall not be less than 3/4 in. (20 mm) at surfaces not

exposed to weather or ground. In this case, d = h- (3/4 in.) - (half-bar diameter). Fig.

5.5-10.

Figure 5.5-10: Cover in slabs

7. In structural slabs of uniform thickness, the minimum amount of reinforcement

in the direction of the span shall not be less than that required for shrinkage and

temperature reinforcement (ACI Code, Section 7.12), figure 5.5-6.

8. The principal reinforcement shall be spaced not farther apart than three times

the slab thickness nor more than 18 in. (ACI Code, Section 7.6.5), figure 5.5-11. [5.1].

124

Table 5.5-3: ACI Code, Table 9.5b

125


Steps for Design of One-way Slab

Step 1: Calculate Factored Dead and Live loads.

Step 2: Determine Minimum thickness requirement as per ACI Code.

Step 3: Calculate Factored Positive Bending Moments at centre of spans and Negative

Bending Moments at faces of supports.

Step 4: Now consider one foot strip and design it as rectangular beam with one foot

width and depth as of slab.

Step 5: After getting Area of Steel, compute number of bars per one foot strip as:

(5.60)

Step 6: Calculate spacing:

12 (5.61)

126

Step 7: Calculate Nominal Steel as per ACI Code:

0.2% (5.62)

Step 8: Calculate Nos and spacing for nominal steel using Eqs. 5.60 And 5.61.

5.6 TWO-WAY SLABS

When the slab is supported on all four sides and the length, L, is less than twice

the width, S, the slab will deflect in two directions, and the loads on the slab are

transferred to all four supports. This slab is referred to as a two-way slab. The bending

moments and deflections in such slabs are less than those in one-way slabs; thus, the

same slab can carry more load when supported on four sides. [5.1].

5.6.1 Types of Two-Way Slabs

1. Two-Way Slabs on Beams: This case occurs when the two-way slab is

supported by beams on all four sides. The loads from the slab are transferred to all four

supporting beams, which, in turn, transfer the loads to the columns. Fig: 5.6-1.

Figure 5.6-1: Two-Way Slabs on Beams

2. Flat Slabs: A flat slab is a two-way slab reinforced in two directions that usually

does not have beams or girders, and the loads are transferred directly to the supporting

columns. The column tends to punch through the slab, which can be treated by three

methods. Fig: 5.6-2.

127

a. Using a drop panel and a column capital. Fig: 5.6-3.

b. Using a drop panel without a column capital. The concrete panel around the

column capital should be thick enough to withstand the diagonal tensile stresses arising

from the punching shear. Fig: 5.6-4.

c. Using a column capital without drop panel, which is not common. Fig: 5.6-5.

Figure 5.6-2: Flat Slabs

Figure 5.6-3: Flat Slabs with drop panels and column capitals

128

Figure 5.6-4: Flat Slabs with drop panels only

Figure 5.6-5: Flat Slabs with column capitals only

3. Flat-Plate Floors: A flat-plate floor is a two-way slab system consisting of a

uniform slab that rests directly on columns and does not have beams or column capitals

(Fig: 5.6-6). In this case the column tends to punch through the slab, producing diagonal

tensile stresses. Therefore, a general increase in the slab thickness is required or special

reinforcement is used.

129

Figure 5.6-6: Flat-Plate Floors

Two-Way Ribbed Slabs and the Waffle Slab System: This type of slab consists

of a floor slab with a length-to-width ratio less than 2. The thickness of the slab is

usually 2 to 4 in. and is supported by ribs (or joists) in two directions. The ribs are

arranged in each direction at spacing’s of about 20 to 30 in., producing square or

rectangular shapes. Fig: 5.6-7.

Figure 5.6-7: Waffle Slab

130

5.6.2 Economical Choice of Concrete Floor Systems

A general guide for the economical use of floor systems can be summarized as

follows:

1. Flat Plates: Flat plates are most suitable for spans of 20 to 25 ft and live loads

between 60 and 100 psf. Flat plates have low shear capacity and relatively low Stiffness,

which may cause noticeable deflection

2. Flat Slabs: Flat slabs are most suitable for spans of 20 to 30 ft and for live loads of

80 to 150 psf.

3. Waffle Slabs: Waffle slabs are suitable for spans of 30 to 48 ft and live loads of 80

to 150 psf. They carry heavier loads than flat plates.

4. Slabs on Beams: Slabs on beams are suitable for spans between 20 and 30 ft and live

loads of 60 to 120 psf. The beams increase the stiffness of the slabs, producing relatively

low deflection. [5.1]

5.6.3 Design Concepts

An exact analysis of forces and displacements in a two-way slab is complex, due

to its highly indeterminate nature; Numerical methods such as finite elements can be

used, but simplified methods such as those presented by the ACI Code are more suitable

for practical design. Factored load capacity of two-way slabs with restrained boundaries

is about twice that calculated by theoretical analysis, because a great deal of moment

redistribution occurs in the slab before failure. At high loads, large deformations and

deflections are expected; thus, a minimum slab thickness is required to maintain

adequate deflection and cracking conditions under service loads.

131

The ACI Code specifies two methods for the design of two-way slabs:

l. The direct design method, DDM (ACI Code, Section 13.6), is an approximate

procedure for the analysis and design of two-way slabs. It is limited to slab systems

subjected to uniformly distributed loads and supported on equally or nearly equality

spaced columns. The method uses a set of coefficients to determine the design moments

at critical sections. Two-way slab systems that do not meet the limitations of the ACI

Code, Section 13.6.1, must be analyzed by more accurate procedures.

2. The equivalent frame method, EFM (ACI Code, Section 13.7), is one in which

a three-dimensional building is divided into a series of two-dimensional equivalent

frames by cutting the building along lines midway between columns. The resulting

frames are considered separately in the longitudinal and transverse directions of the

building [5.1]

5.6.4 Column and Middle Strips

Figure 5.6-8 shows an interior panel of a two-way slab supported on columns A,

B, C, and D. If the panel is loaded uniformly, the slab will deflect in both directions,

with maximum deflection at the center, O. The highest points will be at the columns A,

B, C, and D; thus, the part of the slab around the columns will have a convex shape.

Sections at O, E, F, G, and H will have positive bending moments, whereas the

periphery of the columns will have maximum negative bending moments. Considering

a strip along AF8, the strip bends like a continuous beam (Fig. 5.6-8b), having negative

moments at A and B and positive bending moment at F. This strip extends between the

two columns A and B and continues on both sides of the panel, forming a column strip.

Similarly, a strip along EOG will have negative bending moments at E and G and a

positive moment at O, forming a middle strip.

132

Figure 5.6-8: Column and middle strips; x = 0.25/1 or 0.25h whichever is smaller.

The panel can be divided into three strips, one in the middle along EOG, referred

to as the middle strip, and one on each side, along AFB and DHC, referred to as column

strips (Fig. 5.6-8a). Each of the three strips behaves as a continuous beam. In a similar

way, the panel is divided into three strips in the other direction, one middle strip along

FOH and two column strips along AED and BGC, respectively (Fig. 5.6-8e). The

column strips carry more load than the middle strips. The positive bending moment in

each column strip (at E, F, G, and H) is greater than the positive bending moment at O

in the middle strip. Also, the negative moments at the columns A, B, C, and D in the

133

column strips are greater than the negative moments at E. F, G, and H in the middle

strips. The extent of each of the column and middle strips in a panel is defined by the

ACI Code, Section 13.2. The column strip is defined by a slab width on each side of

the column centerline, x in Fig. 5.6-8, equal to one-fourth the smaller of the panel

dimensions L2 and L2, including beams if they are present, where:

L1 = span length, center to center of supports, in the direction moments are being

determined

L2 = span length, center to center of supports, in the direction perpendicular to L1.

The portion of the panel between two column strips defines the middle strip. [5.1]

5.6.5 Minimum Slab Thickness to Control Deflection

The ACI Code, Section 9.5.3, specifies a minimum slab thickness in two-way

slabs to control deflection. By increasing the slab thickness, the flexural stiffness of the

slab is increased, and consequently the slab deflection is reduced. The ACI Code limits

the thickness of these slabs by adopting the following three empirical limitations, if

these limitations are not met, it will be necessary to compute deflections.

1. For 0.2 2

.,

.

.

.

(5.63)

But not less than 5 in.

134

2. For 2.0,

.,

.

(5.64)

But not less than 3.5 in.

3. For 0.2,

h = min slab thickness without interior beams (Table 5.6-1) (5.65)

Where:

ln = Clear span in the long direction measured face to face of columns (or face

to face of beams for slabs with beams).

The ratio of the long to the short clear spans.

Table 5.6-1: Minimum slab thickness

= the average value of a for all beams on the sides of a panel

= the ratio of flexural stiffness of a beam section Ecblb to the flexural stiffness

of the slab , bounded laterally by the centerlines of the panels on each side of the

beam

135

(5.66)

Where Ecb and Ecs are the moduli of elasticity of concrete in the beam and the

slab, respectively, and

the gross moment of inertia of the beam section about the centroid axis (the

beam section includes a slab length on each side of the beam equal to the projection of

the beam above or below the slab, whichever is greater, but not more than four times

the slab thickness) the moment of inertia of the gross section of the slab.

However, the thickness of any slab shall not be less than the following:

1. For slabs with a 0.2 then thickness ≥ 5.0 in. (125 mm)

2. For slabs with a 0.2 then thickness ≥ 3.5 in. (90 mm)

If no beams are used, as in the case of flat plates, then = 0 and = 0. The

ACI Code equations for calculating slab thickness, h, take into account the effect of the

span length, the panel shape, the steel reinforcement yield stress, fy, and the flexural

stiffness of beams. For flat plates and flat slabs, when no interior beams are used, the

minimum slab thickness may be determined directly from Table 9.5c of the ACI Code,

which is shown here as Table 5.6-1.

Other ACI Code limitations are summarized as follows:

1. For panels with discontinuous edges, end beams with a minimum α equal to 0.8

must be used; otherwise, the minimum slab thickness calculated by Eqs. 5.63 and 5.64

must be increased by at least 10% (ACI Code, Section 9.5.3).

2. When drop panels are used without beams, the minimum slab thickness may

be reduced by 10%. The drop panels should extend in each direction from the centerline

of support a distance not less than one-sixth of the span length in that direction between

136

center to center of supports and also project below the slab at least h/4. This reduction

is included in Table 5.6-1.

3. Regardless of the values obtained by Eqs. 5.63 and 5.64, the thickness of two-

way slabs shall not be less than the following: (1) for slabs without beams or drop

panels, 5 in. (125 mm); (2) for slabs without beams but with drop panels, 4 in. (100

mm); (3) for slabs with beams on all four sides with ≥ 2.0, 3(1/2)in. (90 mm), and

for < 2.0, 5 in. (125 mm) (ACI Code, Section 9.5.3.). [5.1].

5.6.6 Analysis of Two-Way Slabs by The Direct Design Method

The direct design method is an approximate method established by the ACI Code

to determine the design moments in uniformly loaded two-way slabs. To use this

method, some limitations must be met, as indicated by the ACI Code, Section 13.6.1.

Limitations

1. There must be a minimum of three continuous spans in each direction.

2. The panels must be square or rectangular; the ratio of the longer to the shorter

span within a panel must not exceed 2.0.

3. Adjacent spans in each direction must not differ by more than one-third of the

longer span.

4. Columns must not be offset by a maximum of 10% of the span length, in the

direction of offset, from either axis between centerlines of successive columns.

5. All loads must be uniform, and the ratio of the unfactored live to unfactored

dead load must not exceed 2.0.

6. If beams are present along all sides, the ratio of the relative stiffness of beams

in two perpendicular directions, must not be less than 0.2 nor greater than 5.0.

137

Total Factored Static Moment

If a simply supported beam carries a uniformly distributed load w K/ft, then the

maximum positive bending moment occurs at midspan and equals , where

is the span length. If the beam is fixed at both ends or continuous with equal negative

moments at both ends, then the total moment (positive moment at midspan)

+ Mn (negative moment at support) = (Fig.5.6-9). Now if the beam AB carries the

load W from a slab that has a width perpendicular to , then , and the total

moment is , where Wu = load intensity in k/ft2 . In this expression, the

actual moment occurs when l1 equals the clear span between supports A and B. If the

clear span is denoted by ln, then

(ACI Code, Eq. 13.4) (5.67)

Figure 5.6-9 bending moment in a fixed – end beam.

138

Figure 5.6-10 Critical sections for negative design moments. A-A, section for negative moment at exterior support with bracket.

The face of the support where the negative moments should be calculated is

illustrated in Fig.5.6-10. The length l2 is measured in a direction perpendicular to ln and

equals the direction between center to center of supports (width of slab). The total

moment M0 calculated in the long direction will be referred to here as Mol and that in

the short direction, as Mos·

Once the total moment, Mo, is calculated in one direction, it is divided into a

positive moment, Mp, and a negative moment, Mn, such that M0 = Mp + Mn. Then each

moment, MP and Mn, is distributed across the width of the slab between the column and

middle strips, as is explained shortly.

139

Longitudinal Distribution of Moments in Slabs

In a typical interior panel, the total static moment, M0, is divided into two

moments, the positive moment, Mp at midspan, equal to 0.35M0, and the negative

moment, Mn, at each support, equal to 0.65M0, as shown in Fig. 5.6-10. These values

of moment are based on the assumption that the interior panel is continuous in both

directions, with approximately equal spans and loads, so that the interior joints have no

significant rotation. Moreover, the moment values are approximately the same as those

in a fixed-end beam subjected to uniform loading, where the negative moment at the

support is twice the positive moment at midspan. In Fig. 5.6-11, if L1 > L2, then the

distribution of moments in the long and short directions is as follows:

Figure 5.6-11: Distribution of moments in an interior panel.

140

If the magnitudes of the negative moments on opposite sides of an interior support

are different because of unequal span lengths, the ACI Code specifies that the larger

moment should be considered to calculate the required reinforcement.

In an exterior panel, the slab load is applied to the exterior column from one side

only, causing an unbalanced moment and a rotation at the exterior joint. Consequently,

there will be an increase in the positive moment at midspan and in the negative moment

at the first interior support. The magnitude of the rotation of the exterior joint

determines the increase in the moments at midspan and at the interior support. For

example, if the exterior edge is a simple support, as in the case of a slab resting on a

wall (Fig. 5.6-12), the slab moment at the face of the wall there is 0, the positive moment

at midspan can be taken as Mp = 0.63M0, and the negative moment at the interior

support is Mn = 0.75 M0. These values satisfy the static equilibrium equation

8, 0.35 0.65 (5.68)

8, 0.35 0.65 (5.69)

Figure 5.6-12 exterior panel.

According to Section 13.6.3 of the ACI Code, the total static moment M0 in an

end span is distributed in different ratios according to Table 5.6-2 and fig 5.6-13.

141

Transverse Distribution of Moments

The transverse distribution of the longitudinal moments to the middle and

column strips is a function of the ratios L2/L1.

beam stiffnessslab stiffness

(5.70)

2 (5.71)

(5.72)

Table 5.6-2 Distribution of Moments in an End Panel

Where

10.63

3 (5.73)

Where x and y are the shorter and longer dimension of each rectangular

component of the section. The percentages of each design moment to be distributed to

column and middle strips for interior and exterior panels are given in Tables 5.6-3

through Table 5.6-6 In a typical interior panel, the portion of the design moment that is

not assigned to the column strip (Table 5.6-2) must be resisted by the corresponding

half-middle strips. When no beams are used 0.

142

Figure 5.6-13 Distribution of total static moment into negative and positive span Moments.

Table 5.6-3 Percentage of Longitudinal Moment in Column Strips, Interior Panels (ACI Code, Section 13.6.4)

For exterior panels, the portion of the design moment that is not assigned to the

column strip (Table 5.6-5) must be resisted by the corresponding half-middle strips.

143

From Table 5.6-5 it can be seen that when no edge beam is used at the exterior

end of the slab, β1 = 0 and 100% of the design moment is resisted by the column strip.

The middle strip will not resist any moment; therefore, minimum steel reinforcement

must be provided.

Figure 5.6-14 Width of the equivalent rigid frame (equal spans in this figure) and distribution of moments in flat plates, flat slabs, and waffle slabs with no beams.

Table 5.6-4 Percentage of Moments in Two-Way Interior Slabs Without Beams (α1 = 0)

144

Table 5.6-5 Percentage of Longitudinal Moment in Column Strips, Exterior Panels (ACI Code, Section 13.6.4)

Table 5.6-6 Percentage of Longitudinal Moment in Column and Middle Strips, Exterior Panels (For All Ratios of l2/l1 ), Given α1 = β1= 0

Reinforcement Details

After all the percentages of the static moments in the column and middle strips

are determined, the steel reinforcement can be calculated for the negative and positive

moments in each strip:

∅2

(5.74)

Calculate Ru and determine the steel ratio p using the tables in Appendix or use

the following equation:

∅ 11.7

(5.75)

where ∅ = 0.9. The steel area is As = bd.

145

The spacing of bars in the slabs must not exceed the ACI limits of maximum

spacing: 18 in. (450 mm) or twice the slab thickness, whichever is smaller.

5.6.7 Summary of the Direct Design Method (DDM)

Case 1. Slabs without beams (flat slabs and flat plates).

1. Check the limitation requirements. If limitations are not met, DDM cannot be

used.

2. Determine the minimum slab thickness (hmin) to control deflection using

values in Table 5.6-1. Exterior panels without edge beams give the highest hmin (ln/30

for fy = 60 ksi). It is a common practice to use the same slab depth for all exterior and

interior panels.

3. Calculate the factored loads, Wu = 1.2Wd + 1.6WL.

4. Check the slab thickness, h, as required by one-way and two-way shear. If the

slab thickness, h, is not adequate, either increase h or provide shear reinforcement. If

no shear reinforcement is provided the shearing force at a distance d from the face of

the beam Vud, must be equal to:

∅ ∅ (5.76)

Where

2. (5.77)

Whenn slab is supported without beams, the shear strength of concrete is smaller

of Eqs 5.78 and 5.79.

∅ 24

4∅ (5.78)

Where,

bo = perimeter of critical section

146

b = ratio of the long side of column to short side

∅ ∅°

2 (5.79)

Where, is 40 for interior and 30 for edge and 20 for corner column Fig 5.6-15.

5. Calculate the total static moment, M0 , in both directions (Eq.5.67).

6. Determine the distribution factors for the positive and negative moments in the

longitudinal and transverse directions for each column and middle strip in both interior

and exterior panels as follows:

a. For interior panels, use the moment factors given in Table 5.6-4 or Fig. 5.6-14.

b. For exterior panels without edge beams, the panel moment factors are given in

Table 5.6-2 or Fig. 5.6-13 (Case 5). For the distribution of moments in the transverse

direction, use

Table 5.6-6 or Fig. 5.6-14 for column-strip ratios. The middle strip will resist the

portion of the moment that is not assigned to the column strip.

c. For exterior panels with edge beams, the panel moment factors are given in

Table 5.6-2 or Fig. 5.6-13 (Case 4 ). For the distribution of moments in the transverse

direction, use Table 5.6-5 for the column strip. The middle strip will resist the balance

of the panel moment.

7. Determine the steel reinforcement for all critical sections of the column and

middle strips and extend the bars throughout the slab.

8. Compute the unbalanced moment and check if transfer of unbalanced moment

by flexure is adequate. If not, determine the additional reinforcement required in the

critical width.

147

9. Check if transfer of the unbalanced moment by shear is adequate. If not,

increase h or provide shear reinforcement.

Case 2. Slabs with interior and exterior beams.

1. Check the limitation requirements as explained.

2. Determine the minimum slab thickness (hmin) to control deflection using Eqs.

5.63 through 5.65 In most cases, Eq. 5.64 controls. Eq. 5.63 should be calculated first.

3. Calculate the factored load, Wu.

4. Check the slab thickness, h, according to one-way and two-way shear

requirements. In general, shear is not critical for slabs supported on beams.

5. Calculate the total static moment, M0 in both directions (Eq. 5.67).

6. Determine the distribution factors for the positive and negative moments in the

longitudinal and transverse directions for each column and middle strips in both interior

and exterior panels as follows:

a. For interior panels, use moment factors in Fig. 5.6-13 (Case 3) or Fig. 5.6-11.

For the distribution of moments in the transverse direction, use Table 5.6-3 for column

strips.

The middle strips will resist the portion of the moments not assigned to the column

strips. Calculate a1 from Eq. 5.71

b. For exterior panels, use moment factors in Table 5.6-2 or Fig. 5.6-13 (Case 3).

For the distribution of moments in the transverse direction, use Table 5.6-5 for the

column strip. The middle strip will resist the balance of the panel moment.

148

c. In both cases (a) and (b), the beams must resist 85% of the moment in the

column strip when 1.0 1.0, whereas the ratio varies between 85% and 0%

when varies between 1.0 and 0.

7. Determine the steel reinforcement for all critical sections in the column strip,

beam, and middle strip; then extend the bars throughout the slab.

8. Compute the unbalanced moment and then check the transfer of moment by

flexure and shear.

5.7 FOUNDATIONS

Reinforced concrete footings are structural members used to support columns and

walls and to transmit and distribute their loads to the soil. The design is based on the

assumption that the footing is rigid, so that the variation of the soil pressure under the

footing is linear. Uniform soil pressure is achieved when the column load coincides

with the centroid of the footing. The proper design of footings requires that

1. The load capacity of the soil is not exceeded.

2. Excessive settlement, differential settlement, or rotations arc avoided.

3. Adequate safety against sliding and/or overturning is maintained.

The most common types of footings used in buildings are the single footings and wall

footings (Figs. 5.7-1 and 5.7-2). When a column load is transmitted to the soil by the

footing, the soil becomes compressed. If different footings of the same structure have

different settlements, new stresses develop m the structure. Excessive differential

settlement may lead to the damage of nonstructural members in the buildings or even

failure of the affected parts.

149

Figure 5.7-1 Wall footing

Figure 5.7-2 Single footing

150

5.7.1 Types of Foundations

The most common types are as follows:

1. Wall footings are used to support structural walls that carry loads from other

floors or to support nonstructural walls. They have a limited width and a

continuous length under the wall (Fig. 5.7-1).

2. Isolated, or single, footings are used to support single columns (Fig. 5.7-.2).

3. Combined footings (Fig.5.7-3) usually support two columns or three columns

not in a row. The shape of the footing in plan may be rectangular or trapezoidal,

depending on column loads. Combined footings are used when two columns

are so close that single footings cannot be used or when one column is located

at or near a property line.

4. Cantilever, or strap, footings (Fig. 5.7-4) consist of two single footings

connected with a beam or a strap and support two single columns.

5. Continuous footings (Fig. 5.7-5) support a row of three or more columns. They

have limited width and continue under all columns.

6. Raft, or mat, foundations (Fig. 5.7-6) consist of one footing, usually placed

under the entire building area, and support the columns of the building. They

are used when

a. The soil-bearing capacity is low.

b. Column loads are heavy.

c. Single footings cannot be used.

d. Piles are not used.

e. Differential settlement must be reduced through the entire footing system

151

7. Pile caps (Fig. 5.7-7) are thick slabs used to tie a group of piles together and to

support and transmit column loads to the piles. [5.1]

Figure 5.7-3: Combined footing.

Figure 5.7-4: Strap footing.

152

Figure 5.7-5: Continuous footing.

Figure 5.7-6: Raft, mat foundation.

153

Figure 5.7-7: Pile cap footing.

5.7.2 Distribution of Soil Pressure

Fig. 5.7-8 shows a footing supporting a single column. When the column load, P,

is applied on the centroid of the footing, a uniform pressure is assumed to develop on

the soil surface below the footing area. However, the actual distribution of soil pressure

is not uniform. The pressure is maximum under the center of the footing and decreases

toward the ends of the footing. If the footing is resting on a cohesive soil such as clay,

the pressure under the edges is greater than at the center of the footing (Fig.5.7-10). The

clay near the edges has a strong cohesion with the adjacent clay surrounding the footing,

causing the non-uniform pressure distribution.

Figure 5.7-15.7-8: Distribution of soil pressure assuming uniform pressure

154

Figure 5.7-25.7-9: Soil pressure distribution in cohesion less soil (sand)

Figure 5.7-10: Soil pressure distribution in cohesive soil (clay).

The allowable bearing soil pressure, , is usually determined from soil tests. ,

for sedimentary rock is 30 ksf, for compacted gravel is 8 ksf, for well-graded compacted

sand is 6 ksf, and for silty-gravel soils is 3 ksf. [5.1]

5.7.3 Design Considerations [5.1]

Footings must be designed to carry the column loads and transmit them to the soil

safely. The design procedure must take the following strength requirements into

consideration:

1. The area of the footing based on the allowable bearing soil capacity.

2. On-way shear.

3. Two -way shear, or punching shear.

4. Bending moment and steel reinforcement required.

5. Bearing capacity of columns at their base and dowel requirements.

155

Size of Footings

The area of the footings can be determined from the actual external loads

(unfactored forces and moments) such that the allowable soil pressure is not exceeded.

In general, for vertical loads

Area of footing =

, (5.80)

or

Area =

(5.80)

where the total service load is the unfactored design load. Once the area is determined,

a factored soil pressure is obtained by dividing the factored load, = 1.2D 1.6L, by the

area of the footing. This is required to design the footing by the strength design method.

(5.82)

The allowable soil pressure, q0 is obtained from soil test and is based on service

load conditions.

One-Way Shear beam (Beam Shear) ( )

For footings with bending action in one direction, the critical section is located at a

distance d from the face of the column. The diagonal tension at section m-m in Fig. 5.7-

11 can be checked as was done before in beams. The allowable shear in this case is

equal to

∅ 2∅ ′ (∅ 0.75) (5.83)

Where b = width of section m-m. The factored shearing force at section m-m can be

calculated as follows:

156

2 2 (5.84)

If no shear reinforcement is to be used, then d can be determined, assuming ∅ .

d 1

2∅ ′

(5.85)

Two-Way Shear (Punching Shear) ( )

Two-way shear is a measure of the diagonal tension caused by the effect of the

column load on the footing. Inclined cracks may occur in the footing at a distance d/2

from the face of the column on all sides. The footing will fail as the column tries to

punch out part of the footing (Fig. 5.7-12).

The ACI Code, Section 11.11.2 allows a shear strength, , in footings without

shear reinforcement for two-way shear action, the smallest of

4 ′ (5.86)

2 ′ (5.87)

2 ′

(5.88)

Where

= Ratio of long side to short side of the rectangular column

= perimeter of the critical section taken at d/2 from the loaded area (column section)

(see Fig. 5.7-12)

d = effective depth of footing

= is a modification factor for type of concrete (ACI 8.6.1)

=1.0 Normal-weight concrete

157

= 0.85 sand-lightweight concrete

= 0.75 for all-lightweight concrete

Figure 5.7-12: Punching shear (two-way).

For Eq. 5.88, is assumed to be 40 for interior columns, 30 for edge columns,

and 20 for comer columns. Based on the preceding three values of , the effective

depth, d, required for two-way shear is the largest obtained from the following formulas

(∅ = 0.75):

158

2

4∅ ′0

( 2) (5.89)

2

∅ ′

( 2) (5.90)

2

∅0

2

(5.91)

The two-way shearing force, , and the effective depth, d, required (if shear

reinforcement is not provided) can be calculated as follows (refer to Fig. 13.12):

1. Assume d.

2. Determine : = 4(c +d) for square columns, where one side = c. = 2 ( +d) +

2( + d) for rectangular columns of sides and .

3. The shearing force acts at a section that has a length = 4 (c + d) or [2( +d)

+ 2( + d)] and a depth d; the section is subjected to a vertical downward load, , and

a vertical upward pressure, . Therefore,

= - (c + d)2 for square columns (5.92a)

= - ( + d) ( + d) for rectangular column (5.92b)

4. Determine the largest d (of and ). If d is not close to the assumed d, revise your

assumption and repeat.

Flexural Strength and Footing Reinforcement

The critical sections for moment occur at the face of the column (section n-n, Fig.

5.7-13). The bending moment in each direction of the footing must be checked and the

appropriate reinforcement must be provided. In square footings and square columns,

the bending moments in both directions are equal. To determine the reinforcement

159

required, the depth of the footing in each direction may be used. Because the bars in

one direction rest on top of the bars in the other direction, the effective depth, d, varies

with the diameter of the bars used. An average value of d may be adopted. A practical

value of d may be assumed to be (h - 4.5) in.

Figure 5.7-13: Critical section of bending moment.

The depth of the footing is often controlled by shear, which requires a depth greater

than that required by the bending moment. The steel reinforcement in each direction

can be calculated in the case of flexural members as follows:

∅1.7 ′

(5.93)

160

Also, the steel ratio, , can be determined as follows:

0.851 1

2

∅ 0.85 (5.94)

Where = / b . The ACI Code, Section 10.5, indicates that for structural slabs

of uniform thickness, the minimum area and maximum spacing of steel bars in the

direction of bending shall be as required for shrinkage and temperature reinforcement.

The reinforcement in one-way footings and two-way footings must be distributed across

the entire width of the footing. In the case of two-way rectangular footings, the ACI

Code, Section 15.4.4, specifies that in the long direction, a portion of the total

reinforcement distributed uniformly along the width of the footing. In the short

direction, a certain ratio of the total reinforcement in this direction must be placed

uniformly within a bandwidth equal to the length of the short side of the footing

according to

21

(5.95)

Where

= (5.96)

The bandwidth must be centered on the centerline of the column (Fig. 5.7-14). The

remaining reinforcement in the short direction must be uniformly distributed outside

the bandwidth. This remaining reinforcement percentage shall not be less than that

required for shrinkage and temperature.

161

Bearing Capacity of Column at Base

The loads from the column act on the footing at the base of the column, on an area equal

to the area of the column cross-section. Compressive forces are transferred to the

footing directly by bearing on the concrete.

Forces acting on the concrete at the base of the column must not exceed the bearing

strength of concrete as specified by the ACI Code, Section 10.14:

Bearing strength = ∅ 0.85 (5.97)

Where ∅ = 0.65 and = the bearing area of the column .When the supporting surface

is wider on all sides than the loaded area. Here is the area of the part of the supporting

footing that, the factor2

1

is greater than unity, indicating that the allowable bearing

strength is increased because of the lateral support from the footing area surrounding

the column base.

Figure 5.7-14: Bandwidth for reinforcement distribution.

162

The modified bearing strength is

= ∅ 0.85 ′1

2

12∅ 0.85 ′

1 (5.98)

If the factored force, , is greater than either or reinforcement must be provided

to transfer the excess force. This is achieved by providing dowels or extending the

column bars into the footing. The excess force is = - and the area of the dowel

bars is = ( / ) 0.005 , where , is the area of the column section. At least

four bars should be used at the four comers of the column. If the factored force is less

than either or , then minimum reinforcement must be provided. The ACI Code,

Section 15.8.2, indicates that the minimum area of the dowel reinforcement is at least

0.005 (and not less than four bars), where is the gross area of the column section.

The development length of the dowels must be checked to determine proper transfer of

the compression force into the footing.

5.7.4 Combined Footings

When a column is located near a property line, a better design can be achieved by

combining the footing with the nearest internal column footing, forming a combined

footing. The center of gravity of the combined footing coincides with the resultant of

the loads on the two columns.

Figure 5.7-20: Combined footing.

163

Another case where combined footings become necessary is when the soil is poor

and the footing of one column overlaps the adjacent footing. The shape of the combined

footing may be rectangular or trapezoidal (Fig. 5.7-20).

The length and width of the combined footing are chosen to the nearest 3 in. For

a uniform upward pressure, the footing will deflect, as shown in Fig. 5.7-21. A simple

method of analysis is to treat the footing as a beam in the longitudinal direction, loaded

with uniform upward pressure, qu. For the transverse direction, it is assumed that the

column load is spread over a width under the column equal to the column width plus

‘d’ on each side, whenever that is available. In other words, the column load acts on a

beam under the column within the footing, which has a maximum width of (c + 2d) and

a length equal to the short side of the footing (Fig. 5.7-22). A smaller width, down to (c

+d), may be used.

Figure 5.7-21: Upward deflection of a combined footing in two directions.

164

Figure 5.7-22: Analysis of combined footing in transverse direction.

In the design of combined footing, first we calculate point at which resultant of

two column load act, by taking moments on one column.

ΙI distance to colunm I

2ccdistance

1 2

(5.99)

The projections and should have such value that the C.G of footing coincide with

the C.G of the column loads. To meet this requirement [5.7]:

(5.100)

Also

2

2 (5.101)

(5.102)

165

Where

, = projections of footing.

= distance from C.L of column to mid of both columns.

L = length of combined footing.

l = distance b/w both columns.

5.7.5 Footings under Eccentric Column Loads

When a column transmits axial loads only, the footing can be designed such that

the load acts at the centroid of the footing, producing uniform pressure under the

footing. However, in some cases, the column transmits an axial load and a bending

moment, as in the case of the footings of fixed-end frames. The pressure q that develops

on the soil will not be uniform and can be evaluated from the following equation:

0 (5.103)

where A and I are the area and moment of inertia of the footing, respectively. Different

soil conditions exist, depending on the magnitudes of P and M, and allowable soil

pressure. The different design conditions are shown in Fig. 5.7-25 and are summarized

as follows:

1. When e = M I P < L/6, the soil pressure is trapezoidal

62 (5.104)

62 (5.105)

166

2. When e = M I P = L/6, the soil pressure is triangular.

62

2 (5.106)

06 6

(5.107)

Figure 5.7-25: Single footing subjected to eccentric loading: L = length of footing, B width, and h = height

3. When e > L/6, the soil pressure is triangular.

x = 3 = 2 – e

167

P = (3

2) (5.108)

23

43 2

4. When the footing is moved a distance e from the axis of the column to produce

uniform soil pressure under the footing. Maximum moment occurs at section n-n.

M M’ – Hh and e (5.109)

• Earthquake Resistant Design

Section - III

Chapter 6 • Earthquakes and Fundamentals of Ground Motion

Chapter 7 • Structural Respone

Chapter 8 • Seismic Loading UBC 1997 & BCP, SP 2007

Chapter 9 • International Building Code IBC 2012

168

CHAPTER 6

EARTHQUAKES AND FUNDAMENTALS OF GROUND

MOTION

6.1 EARTHQUAKES

Earthquake results from the sudden movement of the tectonic plates in the earth’s

crust, figure 6.1-2. The movement takes place at

the fault lines, and the energy released is

transmitted through the earth in the form of

waves, figure 6.1-3, that cause ground motion

many miles from the epicentre, figure 6.1-4.

Regions adjacent to active fault lines are the most

prone to experience earthquakes. As experienced

by structures, earthquakes consist of random

horizontal and vertical movements of the earth’s surface. As the ground moves, inertia

tends to keep structures in place, figures 6.1-5, resulting in the imposition of

displacements and forces that can have catastrophic results, figure 6.1-6. The purpose

of seismic design is to proportion structures so that they can withstand the

displacements and the forces induced by the ground motion.

The horizontal components of an earthquake usually exceed the vertical

component. Experience has shown that the horizontal components are the most

destructive. For structural design, the intensity of an earthquake is usually described in

terms of the ground acceleration as a fraction of the acceleration of gravity, i.e., 0.1,

0.2, or 0.3g. Although peak acceleration is an important design parameter, the

frequency characteristics and duration of an earthquake are also important; the

Figure 6.1-1: Earth's Tectonic plates

169

Figure 6.1-2: Digital Tectonic Activity Map of the Earth

170

closer the frequency of the earthquake motion is to the natural frequency of a structure

and the longer the duration of the earthquake, the greater the potential for damage.

Figure 6.1-3: Earthquake Waves

171

Figure 6.1-4: Epicenter

Figure 6.1-5: Effect of inertia in a building when shaken at its base

Figure 6.1-6: Inertia force and relative motion within a building

172

Figure 6.1-7: Arrival of seismic waves at a site

Designers of structures that may be subjected to earthquakes, therefore, are faced

with a choice: (a) providing adequate stiffness and strength to limit the response of

structures to the elastic range or (b) providing lower-strength structures, with

presumably lower initial costs, that have the ability to withstand large inelastic

deformations while maintaining their load-carrying capability. [6.1].

6.2 FUNDAMENTALS OF EARTHQUAKE GROUND MOTION

6.2.1 Introduction

When transmitted through a structure, ground acceleration, velocity, and

displacements (referred to as ground motion) are in most cases amplified. The amplified

motion can produce forces and displacements that may exceed those the structure can

sustain. Many factors influence ground motion and its amplification. Earthquake

ground motion is usually measured by a strongmotion accelerograph that records the

acceleration of the ground at a particular location. The maximum values of the ground

motion (peak ground acceleration, peak ground velocity, and peak ground

displacement) are of interest for seismic analysis and design. These parameters,

173

however, do not by themselves describe the intensity of shaking that structures or

equipment experience. Other factors, such as the earthquake magnitude, distance from

fault or epicenter, duration of strong shaking, soil condition of the site, and frequency

content of the motion, also influence the response of a structure. Some of these effects,

such as the amplitude of motion, duration of strong shaking, frequency content, and

local soil conditions, are best represented through the response spectrum. [6.2].

Figure 6.2-1: Earthquake Record

6.2.2 Recorded Ground Motion

Ground motion during an earthquake is measured by a strong-motion

accelerograph that records the acceleration of the ground at a particular location. Three

orthogonal components of the motion, two in the horizontal direction and one in the

vertical, are recorded by the instrument. The instruments may be located in a free field

or mounted in structures. [6.2].

174

6.2.3 Characteristics of Earthquake Ground Motion

Characteristics of earthquake ground motion that are important in earthquake

engineering applications include:

• Peak ground motion (peak ground acceleration, peak ground velocity, and peak

ground displacement)

• Duration of strong motion

• Frequency content

Each of these parameters influences the response of a structure. Peak ground

motion primarily influences the vibration amplitudes. Duration of strong motion has a

pronounced effect on the severity of shaking. A ground motion with a moderate peak

acceleration and long duration may cause more damage than a ground motion with a

larger acceleration and a shorter duration. In a structure, ground motion is most

amplified when the frequency content of the motion and the vibration frequencies of

the structure are close to each other. [6.2].

6.2.4 Factors Influencing Ground Motion

Earthquake ground motion and its duration at a particular location are influenced

by a number of factors, the most important being: (1) earthquake magnitude, (2)

distance of the source of energy release (epicentral distance or distance from causative

fault), (3) local soil conditions, (4) variation in geology and propagation of velocity

along the travel path, and (5) earthquake-source conditions and mechanism (fault type,

stress conditions, stress drop). [6.2].

175

CHAPTER 7

STRUCTURAL RESPONSE

7.1 GENERAL

For many years, the goal of earthquake design has been to construct buildings

that will withstand moderate earthquakes without damage and severe earthquakes

without collapse. Building codes have undergone regular modification as major

earthquakes have exposed weaknesses in existing design criteria.

Design for earthquakes differs from design for gravity and wind loads in

the relatively greater sensitivity of earthquake-induced forces to the geometry of the

structure. Without careful design, forces and displacements can be concentrated in

portions of a structure that are not capable of providing adequate strength or ductility.

Steps to strengthen a member for one type of loading may actually increase the forces

in the member and change the mode of failure from ductile to brittle.

7.2 STRUCTURAL CONSIDERATION

The closer the frequency of the ground motion is to one of the natural frequencies

of a structure, the greater the likelihood of the structure experiencing resonance,

resulting in an increase in both displacement and damage. Therefore, earthquake

response depends strongly on the geometric properties of a structure, especially height.

Tall buildings respond more strongly to long-period (low frequency) ground motion,

while short buildings respond more strongly to short period (high frequency) ground

motion. Figure 7.2-1 shows the shapes for the principal modes of vibration of a three

storey frame structure. The relative contribution of each mode to the lateral

displacement of the structure depends on the frequency characteristics of the ground

motion.

176

Figure 7.2-1: Modal shapes for a three storey building (a) first mode; (b) second mode; (c) third mode

The first mode, figure 7.2-1a, usually provides the greatest contribution to lateral

displacement. The taller a structure, the more susceptible it is to the effects of higher

modes of vibration, which are generally additive to the effects of the lower modes and

tend to have the greatest influence on the upper stories. Under any circumstances, the

longer the duration of an earthquake, the greater the potential of damage.

The configuration of a structure also has a major effect on its response to an

earthquake. Structures with a discontinuity in stiffness or geometry can be

subjected to undesirably high displacements or forces. For example, the

discontinuance of shear walls, infill walls or even cladding at a particular story level,

will have the result of concentrating the displacement in the open, or “soft,” story,

figures 7.2-2 and 7.2-3. The high displacement will, in turn, require a large amount of

ductility if the structure is not to fail. Such a design is not recommended, and the

stiffening members should be continued to the foundation.

177

Figure 7.2-2: Upper storeys of open ground storey move together as single block

Figure 7.2-3: Ground storey of reinforced concrete building left open to facilitate

Similarly, any kind of horizontal or vertical mass or stiffness irregularity in

structures places them in undesirable position against earthquake forces. Buildings

with simple geometry in plan, figure 7.2-4, perform well during strong earthquakes.

178

Buildings with re-entrant corners, like those U, V, H and + shaped in plan, figure 7.2-

4b, have sustained significant damage in past earthquakes. Many times, the bad effects

of these interior corners in the plan of buildings are avoided by making the buildings in

two parts. For example, an L-shaped plan can be broken up into two rectangular plan

shapes using a separation joint at the junction, figure 7.2-4c.

Figure 7.2-4: Simple plan shape buildings do well during earthquake

Figure 7.2-5 shows buildings with one of their dimensions much larger or much

smaller than the other two. Such shapes do not perform well during the earthquakes.

Buildings with vertical setbacks (like the hotel buildings with a few storeys wider than

the rest) cause a sudden jump in earthquake forces at the level of discontinuity,

figure 7.2-6. Within a structure, stiffer members tend to pick up a greater portion of the

load. When a frame is combined with a shear wall, this can have the positive effect of

reducing the displacements of the structure and decreasing both structural and non-

179

structural damage. However, when the effects of higher stiffness members, such as

masonry infill walls, are not considered in the design, unexpected and often undesirable

results can occur.

Figure 7.2-5: Buildings with one of their overall sizes much larger or much smaller than other two

Figure 7.2-6: Buildings with setbacks

Finally, any discussion of structural considerations would be incomplete without

emphasizing the need to provide adequate separation between structures. Lateral

displacements can result in structures coming in contact during an earthquake, resulting

in major damage due to hammering, figure 7.2-7. Spacing requirements to ensure that

adjacent structures do not come into contact as a result of earthquake induced motion

are specified in relevant codes.

180

Figure 7.2-7: Hammering or Pounding

7.3 MEMBER CONSIDERATIONS

Members designed for seismic loading must perform in a ductile fashion and

dissipate energy in a manner that does not compromise the strength of the structure.

Both the overall design and the structural details must be considered to meet this goal.

The principal method of ensuring ductility in members subject to shear and

bending is to provide confinement for the concrete. This is accomplished through the

use of closed hoops or spiral reinforcement, which enclose the core of the beams and

columns. When confinement is provided, beams and columns can undergo nonlinear

cyclic bending while maintaining their flexural strength and without deteriorating due

to diagonal tension cracking. The formation of ductile hinges allows reinforced

concrete frames to dissipate energy.

Successful seismic design of frames requires that the structures be proportioned

so that hinges occur at locations that least compromise strength. For a frame undergoing

lateral displacement, such as shown in figure 7.3-1a, the flexural capacity of the

members at a joint, figure 7.3-1b, should be such that the columns are stronger than the

181

beams. In this way, hinges will form in the beams rather than the columns, minimizing

the portion of the structure affected by nonlinear behaviour and maintaining the overall

vertical load capacity. For these reasons, the “weak beam-strong column” approach

is used to design reinforced concrete frames subject to seismic loading.

Figure 7.3-1: Frame subjected to lateral loading (a) deflected shape; (b) Moments acting on beam-column joint

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CHAPTER 8

SEISMIC LOADING UBC 1997 & BCP, SP 2007

8.1 BUILDING CODES

The 1997 UBC, the 75th and last UBC issued, was replaced by the IBC in 2000.

The UBC underwent several modifications, some changes undoubtedly influenced by

significant seismic events such as the 1933 Long Beach earthquake. The 1960 UBC

provided the following equation in addressing the total lateral force acting at the base

of the structure:

V = KCW for base shear (8.1)

This approach was soon modified. Subsequent editions of the UBC included zone

factor Z, which depends on the expected severity of earthquakes in various regions;

coefficient C, which represents the vibration characteristics based on the fundamental

period T of the structure; horizontal force factor K, which measures the strength of the

structure against earthquake impact; and the total dead-load weight W of the structure:

V = ZKCW expression for the base shear. (8.2)

The coefficients ‘I’ and ‘S’ were added to the formula in the 1970s, where ‘I’

represents the importance of the structure (such as hospitals and fire and police

stations) and ‘S’ the soil structure, a ‘‘site structure’’ resonance factor in UBC 1982:

V = ZIKCSW (8.3)

The 1988 UBC provided a modified expression,

(8.4)

where ‘C’ incorporated the soil structure response factor ‘S’ into the expression

for the fundamental period of the structure and ‘RW’ assumed the role of the former

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‘K’ factor for basic braced frame, special moment-resisting space frame (SMRSF), and

other types of structures. [8.1].

8.2 UNIFORM BUILDING CODE, UBC 1997

The 1997 UBC was drafted after the 1994 Northridge earthquake. It resulted in

significant change in structural configurations, member sizes, and types of beam-to-

column connections.

Among the new features introduced by the 1997 UBC is the modification of

internal design forces such as column loads and forces in braces and connections in an

attempt to increase design values. Two design examples; one using load and resistance

factor design (LRFD) and the other based on allowable stress design (ASD) analysis.

The design seismic forces in the 1997 UBC are based on earthquake zones with due

consideration given to existing faults. [8.1].

8.2.1 Division IV—Earthquake Design [8.3]

Section 1626 — General

1626.1 Purpose. The purpose of the earthquake provisions herein is primarily to

safeguard against major structural failures and loss of life, not to limit damage or

maintain function.

1626.2 Minimum Seismic Design. Structures and portions thereof shall, as a

minimum, be designed and constructed to resist the effects of seismic ground motions

as provided in this division.

Section 1627 — Definitions

For the purposes of this division, certain terms are defined as follows:

BASE is the level at which the earthquake motions are considered to be imparted

to the structure or the level at which the structure as a dynamic vibrator is supported.

184

BASE SHEAR, V, is the total design lateral force or shear at the base of a

structure.

BEARING WALL SYSTEM is a structural system without a complete vertical

load-carrying space frame.

BOUNDARY ELEMENT is an element at edges of openings or at perimeters of

shear walls or diaphragms.

BRACED FRAME is an essentially vertical truss system of the concentric or

eccentric type that is provided to resist lateral forces.

BUILDING FRAME SYSTEM is an essentially complete space frame that

provides support for gravity loads.

COLLECTOR is a member or element provided to transfer lateral forces from a

portion of a structure to vertical elements of the lateral-force-resisting system.

CONCENTRICALLY BRACED FRAME is a braced frame in which the

members are subjected primarily to axial forces.

ECCENTRICALLY BRACED FRAME (EBF) is a steel-braced frame

designed in conformance with Section 2213.10.

DESIGN BASIS GROUND MOTION is that ground motion that has a 10

percent chance of being exceeded in 50 years as determined by a site-specific hazard

analysis or may be determined from a hazard map. A suite of ground motion time

histories with dynamic properties representative of the site characteristics shall be used

to represent this ground motion. The dynamic effects of the Design Basis Ground

Motion may be represented by the Design Response Spectrum. See Section 1631.2.

DESIGN SEISMIC FORCE is the minimum total strength design base shear,

factored and distributed in accordance with Section 1630.

185

DIAPHRAGM is a horizontal or nearly horizontal system acting to transmit

lateral forces to the vertical-resisting elements. The term “diaphragm” includes

horizontal bracing systems.

DUAL SYSTEM is a combination of moment-resisting frames and shear walls

or braced frames designed in accordance with the criteria of Section 1629.6.5.

INTERMEDIATE MOMENT-RESISTING FRAME (IMRF) is a concrete

frame designed in accordance with Section 1921.8.

LATERAL-FORCE-RESISTING SYSTEM is that part of the structural

system designed to resist the Design Seismic Forces.

MOMENT-RESISTING FRAME is a frame in which members and joints are

capable of resisting forces primarily by flexure.

MOMENT-RESISTING WALL FRAME (MRWF) is a masonry wall frame

especially detailed to provide ductile behavior and designed in conformance with

Section 2108.2.5.

ORDINARY BRACED FRAME (OBF) is a steel-braced frame designed in

accordance with the provisions of Section 2213.8 or 2214.6, or concrete-braced frame

designed in accordance with Section 1921.

ORDINARY MOMENT-RESISTING FRAME (OMRF) is a moment-

resisting frame not meeting special detailing requirements for ductile behavior.

SPECIAL MOMENT-RESISTING FRAME (SMRF) is a moment-resisting

frame specially detailed to provide ductile behavior and comply with the requirements

given in Chapter 19 or 22.

186

P∆ EFFECT is the secondary effect on shears, axial forces and moments of

frame members induced by the vertical loads acting on the laterally displaced building

system.

SHEAR WALL is a wall designed to resist lateral forces parallel to the plane of

the wall (sometimes referred to as vertical diaphragm or structural wall).

SOFT STORY is one in which the lateral stiffness is less than 70 percent of the

stiffness of the story above. See Table 16-L.

STORY is the space between levels. Story x is the story below Level x.

STORY DRIFT is the lateral displacement of one level relative to the level above

or below.

STORY DRIFT RATIO is the story drift divided by the story height.

STORY SHEAR, Vx, is the summation of design lateral forces above the story

under consideration.

WEAK STORY is one in which the story strength is less than 80 percent of the

story above. See Table 16-L.

Section 1629 — Criteria Selection

1629.1 Basis for Design. The procedures and the limitations for the design of

structures shall be determined considering seismic zoning, site characteristics,

occupancy, configuration, structural system and height in accordance with this section.

Structures shall be designed with adequate strength to withstand the lateral

displacements induced by the Design Basis Ground Motion, considering the inelastic

response of the structure and the inherent redundancy, over-strength and ductility of the

lateral-force-resisting system. The minimum design strength shall be based on the

Design Seismic Forces determined in accordance with the static lateral force procedure

187

of Section 1630, except as modified by Section 1631.5.4. Where strength design is used,

the load combinations of Section 1612.2 shall apply, Fig: 8.2-1. Where Allowable

Stress Design is used, the load combinations of Section 1612.3 shall apply. One- and

two-family dwellings in Seismic Zone 1 need not conform to the provisions of this

section.

Figure 8.2-1: Section 1612.2 Load Combintions

1629.2 Occupancy Categories. For purposes of earthquake resistant design, each

structure shall be placed in one of the occupancy categories listed in Table 16-K. Table

16-K assigns importance factors, I and Ip, and structural observation requirements for

each category.

1629.3 Site Geology and Soil Characteristics. Each site shall be assigned a soil

profile type based on properly substantiated geotechnical data using the site

categorization procedure set forth in Division V, Section 1636 and Table 16-J.

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1629.3.1 Soil profile type. Soil Profile Types SA, SB, SC, SD and SE are defined in Table

16-J and Soil Profile Type SF is defined as soils requiring site-specific evaluation as

follows:

1. Soils vulnerable to potential failure or collapse under seismic loading, such as

liquefiable soils, quick and highly sensitive clays, and collapsible weakly cemented

soils.

2. Peats and/or highly organic clays, where the thickness of peat or highly organic clay

exceeds 10 feet (3048 mm).

3. Very high plasticity clays with a plasticity index, PI > 75, where the depth of clay

exceeds 25 feet (7620 mm).

4. Very thick soft/medium stiff clays, where the depth of clay exceeds 120 feet (36 576

mm).

1629.4 Site Seismic Hazard Characteristics. Seismic hazard characteristics

for the site shall be established based on the seismic zone and proximity of the site to

active seismic sources, site soil profile characteristics and the structure’s importance

factor.

1629.4.1 Seismic zone. Each site shall be assigned a seismic zone in accordance with

Figure 16-2. Each structure shall be assigned a seismic zone factor Z, in accordance

with Table 16-I.

1629.4.3 Seismic response coefficients. Each structure shall be assigned a seismic

coefficient, Ca, in accordance with Table 16-Q and a seismic coefficient, Cv, in

accordance with Table 16-R.

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1630.2 Static Force Procedure

1630.2.1 Design base shear. The total design base shear in a given direction shall be

determined from the following formula:

(8.1)

The total design base shear need not exceed the following:

2.5 (8.2)

The total design base shear shall not be less than the following:

0.11 (8.3)

In addition, for Seismic Zone 4, the total base shear shall also not be less than the

following:

0.8 (8.4)

1630.2.2 Structure period. For all buildings, the value T may be approximated from

the following formula:

/ (8.5)

Where:

Ct = 0.035 (0.0853) for steel moment-resisting frames.

Ct = 0.030 (0.0731) for reinforced concrete moment-resisting frames and

eccentrically braced frames.

Ct = 0.020 (0.0488) for all other buildings.

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Table 8.2-1: MAXIMUM ALLOWABLE DEFLECTION FOR STRUCTURAL MEMBERS

Table 8.2-2: SEISMIC ZONE FACTOR Z

Table 8.2-3: SOIL PROFILE TYPES

191

Table 8.2-4: OCCUPANCY CATEGORY

192

Table 8.2-5: STRUCTURAL SYSTEMS

194

Table 8.2-6: SEISMIC COEFFICIENT Ca

Table 8.2-7: SEISMIC COEFFICIENT Cv

195

8.3 BUILDING CODE OF PAKISTAN, SEISMIC PROVISION,

BCP SP, 2007

In Pakistan, the design criteria for earthquake loading are based on design

procedures presented in chapter 5, division II of Building Code of Pakistan, seismic

provision 2007 (BCP, SP 2007), which have been adopted from chapter 16, division II

of UBC-97 (Uniform Building Code), volume 2.

The design seismic force can be determined based on the UBC-97 static lateral

force procedure [sec. 1630.2, UBC-97 or Sec. 5.30.2, BCP 2007] and/or the dynamic

lateral force procedure [sec. 1631, UBC-97 or sec. 5.31, BCP-2007]. The static lateral

force procedures (section 1630 of the UBC-97) may be used for the following

structures:

1. All structures, regular or irregular, in Seismic Zone 1 and in Occupancy

Categories 4 and 5 in Seismic Zone 2.

2. Regular structures under 240 feet (73152 mm) in height with lateral force

resistance provided by systems listed in Table 16-N, except where section 1629.8.4,

Item 4, applies.

3. Irregular structures not more than five stories or 65 feet (19812 mm) in height.

4. Structures having a flexible upper portion supported on a rigid lower portion

where both portions of the structure considered separately can be classified as being

regular, the average story stiffness of the lower portion is at least 10 times the average

story stiffness of the upper portion and the period of the entire structure is not greater

than 1.1 times the period of the upper portion considered as a separate structure fixed

at the base.

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The dynamic lateral force procedure of section 1631 shall be used for all other

structures including the following:

1. Structures 240 feet (73152 mm) or more in height, except as permitted by

Section 1629.8.3, Item 1.

2. Structures having a stiffness, weight or geometric vertical irregularity of Type

1, 2 or 3, as defined in Table 16-L, or structures having irregular features not described

in Table 16-L or 16-M, except as permitted by Section 1630.4.2.

3. Structures over five stories or 65 feet (19812 mm) in height in Seismic Zones

3 and 4 not having the same structural system throughout their height except as

permitted by Section 1630.4.2.

4. Structures, regular or irregular, located on Soil Profile Type SF, which have a

period greater than 0.7 second. The analysis shall include the effects of the soils at the

site and shall conform to Section 1631.2, Item 4. [8.2].

8.3.1 Static lateral force procedure

Refer to Section 8.2 above.

8.3.2 Dynamic lateral force procedure

UBC-97 section 1631 include information on dynamic lateral force procedures

that involve the use of (a) response spectra, or (b) time history analyses of the structural

response based on a series of ground motion acceleration histories that are

representative of ground motion expected at the site. The details of these methods are

presented in sections 1631.5 and 1631.6 of the UBC-97. [8.2].

198

Figure 8.3-1: Sesimic hazard zones of Pakistan [8.4]

199

Figure 8.3-2: Seismic zoning of Pakistan [8.4]

200

Figure 8.3-3: Seismic Zoning Sindh [8.4]

201

Figure 8.3-4: Seismic Zoning Punjab [8.4]

202

Figure 8.3-5: Seismic Zoning Baluchistan [8.4]

203

Figure 8.3-6: Seismic Zoning KPK, JK, Northern Areas [8.4]

204

Table 8.3-1: Seismic Zones of Tehsils of Pakistan (a) [8.4]

205

Table 8.3-2: Seismic Zones of Tehsils of Pakistan (b) [8.4]

206

Table 8.3-3: Seismic Zones of Tehsils of Pakistan (c) [8.4]

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Table 8.3-4: Seismic Zones of Tehsils of Pakistan (d) [8.4]

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CHAPTER 9

INTERNATIONAL BUILDING CODE 2012

9.1 SECTION 1613 - EARTHQUAKE LOADS

1613.1 Scope. Every structure, and portion thereof, including nonstructural

components that are permanently attached to structures and their supports and

attachments, shall be designed and constructed to resist the effects of earthquake

motions in accordance with ASCE 7, excluding Chapter 14 and Appendix 11A. The

seismic design category for a structure is permitted to be determined in accordance with

Section 1613 or ASCE 7.

Exceptions:

1. Detached one- and two-family dwellings, assigned to Seismic Design Category

A, B or C, or located where the mapped short-period spectral response acceleration, SS,

is less than 0.4 g.

2. The seismic force-resisting system of wood-frame buildings that conform to

the provisions of Section 2308 are not required to be analyzed as specified in this

section.

3. Agricultural storage structures intended only for incidental human occupancy.

4. Structures that require special consideration of their response characteristics

and environment that are not addressed by this code or ASCE 7 and for which other

regulations provide seismic criteria, such as vehicular bridges, electrical transmission

towers, hydraulic structures, buried utility lines and their appurtenances and nuclear

reactors. [9.1].

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9.2 IBC DESIGN CRITERIA

The International Building Code (IBC, 2012) references the provisions of ASCE

7-10 (ASCE, 2010) for lateral seismic loads. The basic guideline of the IBC provisions

is that a minor seismic event should cause little or no damage, and a major seismic

event should not result in the collapse of the structure. Accordingly, the building is

expected to behave elastically when subjected to frequently occurring earthquakes and

exhibit inelastic behavior when influenced only by infrequent strong earthquakes. Most

low-rise concrete buildings fall into the regular structure type of the International

Building Code and accordingly are designed for a loading condition resulting from

equivalent static lateral force. This static load depends on the site geology and soil

characteristics, the building occupancy, the building configuration and height, and the

structural system being used to support the lateral load. [9.4].

9.2.1 Mapped Acceleration Parameters

Chapter 22 of ASCE 7-10 contains the mapped maximum considered earthquake

(MCE) ground motion parameters Ss and S1. Both Ss and S1 are spectral response

acceleration parameters with 5% damping; SS is determined from a 0.2-second response

acceleration, and S1 is determined from a 1-second response acceleration. Chapter 22

provides detailed maps developed by the U.S. Geological Survey (USGS), but it is not

practical to determine site parameters from the maps provided. The recommended

method of obtaining these parameters is to use the Ground Motion Parameter

Calculator, provided on the USGS website. [9.4]. Table 9.4-5 shows some common

cities of Sindh [9.3].

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9.2.2 Site Class

The soil type on which a structure is founded has a great impact on the ground

motion at the site, different soil types are assigned values, depending on their stiffness,

from Site Class A for rocklike material to Site Class F for a soil profile that requires a

site-specific analysis. Table 9.4-6 lists the site coefficients for different soil types. [9.4].

9.2.3 Site Coefficients and Adjusted Maximum Considered

Earthquake Spectral Response Acceleration Parameters

The mapped acceleration parameters are based on Site Class B. The acceleration

parameters must be modified based on the actual site class; for example, SS must be

multiplied by an adjustment factor (Fa) to determine SMS, the maximum considered

earthquake (MCE) spectral response acceleration at short periods adjusted for site class

effects. The same holds true for the MCE spectral acceleration at a period of 1 second;

S1 is multiplied by Fv to determine SM1. [9.4]. Refer to Table 9.4-7 and Table 9.4-8 for

the site coefficients Fa and Fv, respectively.

9.2.4 Design Spectral Acceleration Parameters [9.4]

The design earthquake spectral response acceleration parameters are determined

by multiplying the modified MCE spectral response acceleration parameters by 2/3:

SMS = FaSS (9.1)

Where, Fa = Site coefficient based upon Site Class

SM1 = FvS1 (9.2)

Where, Fv = Site coefficient based upon Site Class

9.2.5 Design Response Spectrum

When required by ASCE 7-10 and the IBC, the design response spectrum shall be

developed as shown in Figure 9.2-1. [9.4].

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Figure 9.2-1: Design response spectrum

9.2.6 Importance Factor and Occupancy Category [9.4]

The importance factor (I) depends on the occupancy category of a given structure.

Table 9.4-9. The International Building Code and ASCE 7-10 distinguish between four

different categories:

• Buildings and other structures that represent a low hazard to human life in the

event of failure

• All other structures besides those listed in Occupancy Categories I, III, and IV

• Buildings and structures that represent a substantial hazard to human life in the

event of failure

• Essential facilities and structures containing highly toxic substances

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9.2.7 Seismic Design Category

Structures shall be assigned the more severe seismic design category as

determined from Table 9.4-10 or Table 9.4-11 regardless of the period of vibration of

the structure. In addition, when the mapped spectral response acceleration parameter at

the 1-second period (S1) is greater than or equal to 0.75, then the structures in

Occupancy Categories I, II, or III shall be assigned to Seismic Design Category E, and

structures in Occupancy Category IV shall be assigned to Seismic Design Category F.

There are exceptions that allow for the seismic design category to be determined from

Table 9.2-10 alone. To use this exception, S1 must be less than 0.75, and all of the

following requirements must be met:

• The approximate fundamental period of vibration (Ta), as determined by

Equation 9.10 in each of the two orthogonal directions, is less than 0.8TS, where TS =

SD1/SDS.

• The fundamental period of the structure that is used to calculate the story drift

in the two orthogonal directions is less than TS.

• The seismic response coefficient (CS) is determined by Equation 9.6.

• The diaphragms are rigid or, where diaphragms are considered flexible, the

spacing between vertical elements of the lateral-force-resisting system does not exceed

40 feet. [9.4].

9.3 DESIGN REQUIREMENTS FOR SEISMIC DESIGN

CATEGORY A

Structures designated to Seismic Design Category A are designed for minimal

seismic forces. A lateral force of 1% of the dead load of each level shall be applied in

each of two orthogonal directions independently. Load path connections must be

213

designed to transfer the lateral forces induced by the elements being connected. Smaller

portions of a structure are required to be tied to the remainder of the structure.

Connections to supporting elements require a positive connection to resist a horizontal

force acting parallel to the member being connected. This positive connection may be

obtained by connecting an element to slabs designed to act as diaphragms. Concrete

and masonry walls are required to be anchored to all floors and the roof, as well as

members that provide lateral support for the wall or that are supported by the wall. [9.4].

9.4 DESIGN REQUIREMENTS FOR SEISMIC DESIGN

CATEGORIES B, C, D, E, AND F [9.4]

Similar to Seismic Design Category A, Seismic Design Categories B through F

have some minimum requirements with respect to member design, connection design,

and load path. These requirements may be found at the beginning of Chapter 12 of

ASCE 7-10. The design of building structures assigned to Seismic Design Categories

B through F requires the following steps:

• Determine the structural system or systems (may include a combination of systems in

different directions, in the same direction, or vertical combinations).

• Determine if any structural irregularities exist.

• Determine redundancy factor ρ if the structure is assigned to Seismic Design

Categories D through F.

• Determine the appropriate analytical procedure (i.e., equivalent lateral force analysis,

modal response spectrum analysis, or seismic response history procedure).

• Apply the appropriate seismic load combinations to determine member design forces.

• Determine the diaphragm, chord, and collector design requirements.

214

• Check allowable story drift and determine building separation requirements, if

necessary.

• Check deformation compatibility of members not included in the seismic-force-

resisting system.

• Address foundation design requirements.

• Address material specific seismic design and detailing requirements of the structural

system.

9.4.1 Structural System Selection

The maximum elastic response acceleration of a structure during a severe

earthquake can be several times the magnitude of the maximum ground acceleration

and depends on the mass and stiffness of the structure and the amplitude of the damping.

Because it is unnecessary to design a structure to respond in the elastic range to the

maximum seismic inertia forces, it is of the utmost importance that a well-designed

structure be able to dissipate seismic energy by inelastic deformations in certain

localized regions of the lateral-force-resisting system. This translates into

accomplishing flexural yielding of the members and avoiding all forms of brittle failure.

The code-specified design seismic force recognizes such inelastic behavior and

damping and scales down the inertia forces corresponding to a fully elastic response

based on the structural system used. This is accounted for in the response modification

factor (R). The structural systems for buildings and the corresponding R values are

listed in Table 9.4-12 and are defined as follows:

• Bearing wall system—A structural system without a complete vertical-load-

carrying frame. Bearing walls provide support for all or most gravity loads. Resistance

to lateral load is provided by shear walls or light-frame walls with flat-strap bracing. In

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Seismic Design Categories D, E, and F, concrete and masonry shear walls must be

specially detailed to satisfy IBC requirements. These special reinforced-concrete or

masonry shear walls are limited to a height of 160 feet in Seismic Design Categories D

and E, while they are limited to a height of 100 feet in Seismic Design Category F.

Figure 9.4-1: Bearing wall system

• Building frame system—A structural system with an essentially complete

frame providing support for gravity loads. Resistance to lateral load is provided by

shear walls or braced frames. Shear walls in Seismic Design Categories D, E, and F are

required to be specially designed and detailed to satisfy the IBC requirements. In

addition, other structural elements not designated part of the lateral-load-resisting

system must be able to sustain their gravity load-carrying capacity at a lateral

displacement equal to a multiple times the computed elastic displacement of the lateral-

force-resisting system under code-specified design seismic forces. The IBC restricts the

216

building frame system to a maximum height of 160 feet for Seismic Design Categories

D and E but lowers the limit to 100 feet for Seismic Design Category F.

Figure 9.4-2: Building frame system

• Moment-resisting frame system—A structural system with an essentially

complete frame providing support for gravity

loads. Moment-resisting frames provide

resistance to lateral loads primarily by

flexural action of members. In Seismic

Design Category B, the moment-resisting

frames can be ordinary moment-resisting

frames (OMRFs) proportioned to satisfy the

IBC requirements. In Seismic Design

Category C, reinforced-concrete frames resisting forces induced by earthquake motions

at minimum must be intermediate moment-resisting frames (IMRFs). In Seismic

Design Categories D, E, and F, reinforced-concrete frames resisting forces induced by

earthquake motions must be special moment-resisting frames (SMRFs).

Figure 9.4-3: Moment-resisting frame system FBD

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Figure 9.4-4: Moment-resisting frame system

• Dual system—Structural system with the following features: (1) an essentially

complete frame that provides support for gravity loads; (2) resistance to lateral load

provided by shear walls or braced frames and moment-resisting frames (SMRFs and

IMRFs), with the moment-resisting frames designed to independently resist at least

25% of the design base shear; and (3) designed to resist total design base shear in

proportion to relative rigidities considering the interaction of the dual system at all

levels.

• Shear wall-frame interactive system—A structural system with ordinary

reinforced-concrete moment frames and ordinary reinforced-concrete shear walls. This

system is permitted only in Seismic Design Category B.

• Cantilevered column system—A structural system consisting of cantilevered

column elements detailed to conform to the requirements of various moment frame

systems.

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Figure 9.4-5: Shear wall-frame interactive system

9.4.2 Structural Irregularities

The IBC and ASCE 7-10 require that a structure be designated as regular or

irregular. Regular structures have no significant physical discontinuities in plan or

vertical configurations or in their lateral-force-resisting systems. Irregular structures are

those with irregular features. Vertical irregularities are defined as a distribution of mass,

stiffness, or strength that results in lateral forces or deformations, over the height of the

structure, that are significantly different from the linearly varying distribution obtained

from an equivalent lateral force analysis. Plan irregularities are encountered where

diaphragm characteristics create significant diaphragm deformations or stress

concentrations.

9.4.3 Analysis Procedure Selection

The IBC and ASCE 7-10 recognize three different analysis procedures for the

determination of seismic effects on structures. Refer to Table 9.4-1 for the applicability

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of each of the three analysis options. The equivalent lateral force analysis procedure is

allowed when certain criteria of building period, occupancy, and regularity are met; the

modal response spectrum analysis (dynamic analysis) procedure is always permissible

for design. The third analysis option is the seismic response history procedure, which

is beyond the scope of this thesis.

Table 9.4-1: Permitted Analytical Procedures

9.4.4 Equivalent Lateral Force Procedure

The International Building Code and ASCE 7-10 require that structures be

designed for seismic forces in each of the two orthogonal directions. Even though not

all structures are permitted to be designed using the equivalent lateral force analysis

procedure, the procedure is often used to help determine those structures that require

220

more exhaustive analysis. When a dynamic analysis is to be performed, an equivalent

lateral force analysis generally has to be the first step of the process.

Seismic Base Shear [9.2]

Step 1: Determine Ground Motion Spectral Response Acceleration:

SS = Ground acceleration at short (0.2 second) period = see Table 9.4-5.

S1 = Ground acceleration at longer (1 second) period = see Table 9.4-5.

Step 2: Determine “Site Class”:

Site class is based on seismic shear wave velocity, Vs, traveling through the top

100 feet of ground. Site class is determined from Table 9.4-6.

Step 3: Determine “Maximum Considered Earthquake” Spectral Response:

SMS = FaSS (9.1)

Where, Fa = Site coefficient based upon Site Class = From Table 9.4-7.

SM1 = FvS1 (9.2)

Where, Fv = Site coefficient based upon Site Class = From Table 9.4-8

Step 4: Determine Design Spectral Response Acceleration:

SDS = 2/3 (SMS) (9.3)

SD1 = 2/3 (SMI) (9.4)

Step 6: Determine the Effective Seismic Weight of Structure “W”:

W = Effective seismic weight of structure = Total dead load of structure +

1. In areas used for storage, a minimum of 25% of the reduced floor live load (floor

live load in public garages and open parking structures need not be included)

2. Where an allowance for partition load is included in the floor load design, the actual

partition weight or a minimum weight of 10 PSF of floor area, whichever is greater

3. Total operating weight of permanent equipment

221

4. 20% of uniform flat roof snow load where the flat roof snow load “Pf” exceeds

30PSF.

Step 7: Determine Seismic Importance Factor “IE”.

Step 8: Determine Seismic Base Shear “V”:

V = CS W (9.5)

Use largest of Cs :

(9.6)

(9.7)

0.5, 0.6

(9.9)

0.01 (9.8)

where:

CS = seismic response coefficient.

W = total seismic dead load of the system.

I = importance factor in Table 9.4-9.

R = response modification coefficient per Table 9.4-12.

T = fundamental period of vibration of the building.

V = total design lateral force or shear at the base.

SDS = design, 5% damped, spectral response acceleration at short periods of vibration.

SD1 = design, 5% damped, spectral response acceleration at a period of 1 second.

S1 = mapped MCE, 5% damped, spectral response acceleration at a period of 1 second.

The fundamental period of the structure (T) shall be determined using the

structural properties and deformation characteristics of the resisting elements. As an

alternative to performing an analysis to determine the fundamental period (T), it is

222

permitted to use the approximate building period (Ta). For all buildings, the value Ta

may be approximated from the following formula:

(9.10)

where hn is the height in feet above the base to the highest level of the structure,

and the numerical coefficients Ct and x are determined from Table 9.4-2.

The fundamental period (T) calculated by methods other than the approximate

method shall not exceed an upper limit as determined from:

(9.11)

where Cu is determined from Table 9.4-3.

Alternatively, the approximate period may be determined from the following

equations when the structure does not exceed 12 stories in height and consists entirely

of concrete or steel moment resisting frames with a story height of at least 10 feet:

0.1 (9.12)

where N = number of stories

For concrete or masonry shear-wall structures, the approximate period (Ta) may

be determined from the following formula:

0.0019 (9.13)

where CW is calculated from the following formula:

100

1 0.83

(9.14)

where:

AB = area of base of structure (ft2).

Ai = web area of shear wall i (ft2).

223

Di = length of shear wall i (ft).

hi = height of shear wall i (ft).

x = number of shear walls in the building effective in resisting lateral forces in the

direction under consideration.

Table 9.4-2: Value of Ct and x

Step 9: Determine Vertical Distribution of Seismic Shears:

(9.10)

Where,

∑ (9.11)

Where,

h = height above base (ft).

x = portion of weight at that level.

wi, wx = the portion of the total effective seismic weight (W) located or assigned to level i or x.

hi, hx = the height (ft or m) from the base of the structure to level i or x.

k = an exponent related to the structure period as follows: (1) when T < 0.5 seconds, k = 1, and (2) when T ≥ 2.5 seconds, k = 2; for structures having a period between 0.5 and 2.5 sec, k shall be taken equal to 2 or shall be determined by linear interpolation between 1 and 2.

224

Table 9.4-3: Coefficient for Upper Limit on Calculated Period

Table 9.4-4: Allowable Story Drift (Δa)

225

Table 9.4-5: Ground Motion Spectral Response Acceleration for some cities of Sindh [9.3]

City SS S1

Hyderabad 0.87 0.35

Karachi 0.74 0.30

Thatta 1.19 0.48

Dadu 0.67 0.27

Badin 1.09 0.44

Sukkur 1.05 0.42

Haala 0.74 0.30

Larkana 0.82 0.33

Kotri 0.90 0.36

Jamshoro 0.88 0.35

Umerkot 0.69 0.28

Tharparkar 0.71 0.28

Sehwan 0.69 0.28

Matiari 0.78 0.31

Nawabshah 0.67 0.27

226

Table 9.4-6: Site Classification:

227

Table 9.4-7: VALUES OF SITE COEFFICIENT Fa

Table 9.4-8: VALUES OF SITE COEFFICIENT Fv

228

Table 9.4-9: Seismic Importance

229

Table 9.4-10: Seismic Design Category Based on Short Period Response Acceleration Parameter

Table 9.4-11: Seismic Design Category Based on 1-Second Period Response Acceleration Parameter

230

Table 9.4-12: Design Coefficients and Factors for Seismic Force-Resisting Systems

235

Horizontal Distribution of Shear and Horizontal Torsional

Moments [9.2]

The seismic design story shear (Vx, the sum of the forces Fi above that story) in

any story shall be distributed to the various elements of the vertical lateral-force-

resisting system in proportion to their rigidities, considering the rigidity of the

diaphragm. Furthermore, the IBC and ASCE 7-10 require that provisions be made for

the increased shears resulting from horizontal torsion where diaphragms are not

flexible. To account for the uncertainties in load locations, the IBC and ASCE 7-10

further require that the mass at each level be assumed to be displaced from the

calculated center of mass in each direction a distance equal to 5% of the building

dimension at that level perpendicular to the direction of the force under consideration.

This is often referred to as the accidental torsion. The torsional design moment at a

given story is the moment resulting from the combination of this accidental torsional

moment and the inherent torsional moment between the applied design lateral forces

and the center of rigidity of the vertical lateral-force-resisting elements in that story (see

Figure 9.4-7). When seismic forces are applied concurrently in two orthogonal

directions, the required 5% displacement of the center of mass need not be applied in

both orthogonal directions at the same time but shall be applied in the direction that

produces the greater effect.

The following procedure addresses the distribution of shear and torsional

moments on the basis that the lateral-force-resisting system consists of shear walls. The

same discussion also applies to buildings with moment frames or combination of shear

walls and moment frames. The center of mass of the floor is first calculated as follows:

236

Figure 9.4-6: Schematic center of mass in shear wall

Figure 9.4-7: Cantilever shear-wall deflection

Figure 9.4-8: Schematic center of rigidity in a shear wall

237

∑∑

(9.12)

∑∑

(9.13)

Figure 9.4-6 serves Eqs. 9.11 and 9.12. Next, the rigidity of each wall is

calculated. The deflection at the top of the wall is calculated based on the following

formula:

∆ ∆ ∆3

1.2 (9.14)

where r is the rigidity or stiffness of the wall panel equal to 1/deflection (see Figure 9.4-

6, 9.4-7, and 9.4-8). The center of rigidity of the floor is then calculated as follows:

∑

(9.15)

∑ (9.16)

Refer to Figure 9.4-9 for the shear distribution formulation.

Story Drift Determination and Limitation [9.2]

The IBC and ASCE 7-10 define story drift as the relative displacement between

adjacent stories (above or below) due to the design lateral forces. The design story drift

is computed as the difference of the deflections at the center of mass at the top and

bottom of the story under consideration. For structures assigned to Seismic Design

Categories C, D, E, or F having horizontal irregularities of Type 1a or Type 1b, the

design story drift (∆) shall be computed as the largest difference of the deflections along

any of the edges of the structure at the top and bottom of the story under consideration.

The design story drift (∆) shall not exceed the allowable story drift (∆a) determined

from Table 9.4-13.

238

Figure 9.4-9: Shear distribution formulation

The deflections used to determine the design story drift shall be determined from

the following equation:

(9.17)

where:

Cd = the deflection amplification factor in Table 9.4-12.

239

δxe = the deflection determined by elastic analysis without the upper limit on the

fundamental period (CuTa).

I = the importance factor determined from Table 9.4-9.

A limitation has been put into place on the allowable story drift requiring that

the design story drift not exceed ∆a/ρ for structures assigned to Seismic Design

Categories D, E, or F.

9.4.5 P-∆ Effect

In structural engineering, the P-Δ or P-Delta effect refers to the abrupt changes in

ground shear, overturning moment, and/or the axial force distribution at the base of a

sufficiently tall structure or structural component when it is subject to a critical lateral

displacement.

Table 9.4-13: Allowable Story Drift (Δa)

240

The P-Delta effect is a destabilizing moment equal to the force of gravity

multiplied by the horizontal displacement a structure undergoes as a result of a lateral

displacement. For example: In a perfectly rigid body subject only to small

displacements, the effect of a gravitational or concentrated vertical load at the top of

the structure is usually neglected in the computation of ground reactions. However,

structures in real life are flexible and can exhibit large lateral displacements in unusual

circumstances. The lateral displacements can be caused by wind or seismically induced

inertial forces. Given the side displacement, the vertical loads present in the structure

can adversely perturb the ground reactions. This is known as the P-Δ effect. [9.5].

9.4.6 Diaphragm

In structural engineering, a diaphragm is a structural system used to transfer

lateral loads to shear walls or frames primarily through in-plane shear stress. These

lateral loads are usually wind and earthquake loads, but other lateral loads such as

lateral earth pressure or hydrostatic pressure can also be resisted by diaphragm action.

The diaphragm of a structure often does double duty as the floor system or roof

system in a building, or the deck of a bridge, which simultaneously supports gravity

loads.

The two primary types of diaphragm are flexible and rigid. Flexible diaphragms

resist lateral forces depending on the tributary area, irrespective of the flexibility of the

members that they are transferring force to. On the other hand, rigid diaphragms transfer

load to frames or shear walls depending on their flexibility and their location in the

structure. The flexibility of a diaphragm affects the distribution of lateral forces to the

vertical components of the lateral force resisting elements in a structure. [9.6].

241

9.4.7 Building Separation

All structures have to be separated from adjoining structures a distance sufficient

to avoid damaging contact under total deflection δx. The separation is to be based on

the square root sum of the squares of the estimated maximum seismic displacement of

the two structures. The separation was calculated as:

(9.18)

where δx1 and δx2 are the displacement of adjacent buildings and δS is the

calculated required building separation. [9.2].

9.4.8 Anchorage of Concrete or Masonry Walls

Concrete or masonry walls shall be provided with a positive direct connection to

all floors and roofs that provide them lateral support. Such connections shall be capable

of resisting the horizontal forces induced by the seismic excitement. ASCE 7-10,

Section 12.11 provides provisions for 9.4.8 Anchorage of Walls [9.2].

• Computers & Structures

Section - IV

Chapter 10•Seismic Analysis and Design of Multistoried RC Building using ETABS 2013 and SAFE 12

242

CHAPTER 10

SEISMIC ANALYSIS AND DESIGN OF MULTISTORIED RC

BUILDING USING ETABS

10.1 ETABS (EXTENDED 3D ANALYSIS OF BUILDING

SYSTEM)

ETABS is a program for linear, nonlinear, static and dynamic analysis, and the

design of building systems. From an analytical standpoint, multistorey buildings

constitute a very special class of structures and therefore deserve special treatment. The

concept of special programs for building type structures was introduced over 30 years

ago and resulted in the development of the TABS series of computer programs.

10.1.1 Features and Benefits of ETABS

The input, output and numerical solution techniques of ETABS are

specifically designed to take advantage of the unique physical and

numerical characteristics associated with building type structures. As a

result, this analysis and design tool expedites data preparation, output

interpretation and execution throughput.

The need for special purpose programmes has never been more evident as

Structural Engineers put non-linear dynamic analysis into practice and use

the greater computer power available today to create larger analytical

models.

Over the past two decades, ETABS has numerous mega-projects to its

credit and has established itself as the standard of the industry. ETABS

software is clearly recognised as the most practical and efficient tool for

243

the static and dynamic analysis of multistorey frame and shear wall

buildings.

10.2 PROJECT

10.2.1 General Description

The building is situated and is under construction at London Town, near Sehrish

Nagar, Qasimabad, Hyderabad. It is G+6 storied Residential Project with Flat System.

Basement is provided for parking purpose. Building Frame is modeled using ETABS

and Slabs and Foundation is designed using SAFE. The purpose is to show

effectiveness of these tools and to show our previously researched modern methods of

seismic load calculations, by comparing them.

10.2.2 Drawings

See Plan of Project on next Page.

244

10.3 ETABS 2013

10.3.1 Material Properties

Concrete Type-I: fc’ = 2500 psi

Concrete Type-II: fc’ = 3000 psi

Concrete Type-I is used for casting beams and slabs, and Concrete Type-II is used

for casting columns. This difference is provided to establish the concept of “weaker

beams stronger columns”.

10.3.2 Load Cases

Gravity Loads

Dead Load = 150 pcf.

Live Load = 40 psf on slabs.

Superimposed Dead Load = 500 psf on beams, 25 psf on slabs.

Seismic Loads UBC 97

Earthquake Load in X-Direction

Earthquake Load in X-Direction

Soil Profile type is SE.

Seimic Zone factor = 0.15

Overstrength factor, R = 5.5

Importance factor, I = 1

Ct = 0.03

Seismic Loads IBC 2012

0.2 Seconds Spectral Acceleration Table 9.4-5 = SS = 0.87

0.1 Seconds Spectral Acceleration Table 9.4-5 = SD = 0.35

245

Long-Period Transition Period Eq. 9.10 = T = 0.8165

Site Class Table 9.4-6 = C

Seismic Design Category Table 9.4-10 & Table 9.4-11 = D

10.3.3 Analysis

Results using UBC 1997

A U T O S E I S M I C U B C 9 7

Case: EQX

AUTO SEISMIC INPUT DATA

Direction: X

Typical Eccentricity = 5%

Eccentricity Overrides: No

Period Calculation: Program Calculated

Ct = 0.03 (in feet units)

Top Story: ROOF

Bottom Story: BASE

R = 5.5

I = 1

hn = 86.000 (Building Height)

Soil Profile Type = SE

Z = 0.15

Ca = 0.3000

Cv = 0.5000

246

AUTO SEISMIC CALCULATION FORMULAS

Ta = Ct (hn^(3/4))

If Z >= 0.35 (Zone 4) then: If Tetabs <= 1.30 Ta then T = Tetabs, else T

= 1.30 Ta

If Z < 0.35 (Zone 1, 2 or 3) then: If Tetabs <= 1.40 Ta then T = Tetabs, else T

= 1.40 Ta

V = (Cv I W) / (R T) (Eqn. 1)

V <= 2.5 Ca I W / R (Eqn. 2)

V >= 0.11 Ca I W (Eqn. 3)

If T <= 0.7 sec, then Ft = 0

If T > 0.7 sec, then Ft = 0.07 T V <= 0.25 V

AUTO SEISMIC CALCULATION RESULTS

Ta = 0.8472 sec

T Used = 1.1861 sec

W Used = 30666.17

V (Eqn 1) = 0.0766W

V (Eqn 2) = 0.1364W

V (Eqn 3) = 0.0330W

V (Eqn 4) = 0.0349W

V Used = 0.0766W = 2350.41

Ft Used = 195.15

247

AUTO SEISMIC STORY FORCES

STORY FX FY FZ MX MY MZ

ROOF (Forces reported at X = 65.9286, Y = 53.8677, Z = 79.0000)

483.45 0.00 0.00 0.000 -57.272 -1.364

6TH (Forces reported at X = 66.0000, Y = 54.0095, Z = 69.0000)

441.97 0.00 0.00 0.000 -30.182 0.052


385.69 0.00 0.00 0.000 -26.211 0.037


330.41 0.00 0.00 0.000 -22.240 0.077

3RD (Forces reported at X = 65.9986, Y = 53.9236, Z = 39.0000)

273.77 0.00 0.00 0.000 -18.268 -0.251

2ND (Forces reported at X = 65.9967, Y = 53.8598, Z = 29.0000)

215.46 0.00 0.00 0.000 -14.297 -0.361

1ST (Forces reported at X = 65.9981, Y = 53.8688, Z = 19.0000)

162.05 0.00 0.00 0.000 -24.651 -0.499

248

GL (Forces reported at X = 65.9738, Y = 49.9083, Z = 3.0000)

57.62 0.00 0.00 0.000 -32.016 -25.712

A U T O S E I S M I C U B C 9 7

Case: EQY


Direction: Y





Top Story: ROOF

Bottom Story: BASE

R = 5.5

I = 1


Soil Profile Type = SE

Z = 0.15

Ca = 0.3000

Cv = 0.5000

249

AUTO SEISMIC CALCULATION FORMULAS

Ta = Ct (hn^(3/4))

If Z >= 0.35 (Zone 4) then: If Tetabs <= 1.30 Ta then T = Tetabs, else T =

1.30 Ta

If Z < 0.35 (Zone 1, 2 or 3) then: If Tetabs <= 1.40 Ta then T = Tetabs, else T

= 1.40 Ta

V = (Cv I W) / (R T) (Eqn. 1)

V <= 2.5 Ca I W / R (Eqn. 2)

V >= 0.11 Ca I W (Eqn. 3)

If T <= 0.7 sec, then Ft = 0

If T > 0.7 sec, then Ft = 0.07 T V <= 0.25 V


Ta = 0.8472 sec

T Used = 1.1861 sec

W Used = 30666.17

V (Eqn 1) = 0.0766W

V (Eqn 2) = 0.1364W

V (Eqn 3) = 0.0330W

V (Eqn 4) = 0.0349W

V Used = 0.0766W = 2350.41

Ft Used = 195.15

250




0.00 483.45 0.00 57.272 0.000 0.736


0.00 441.97 0.00 30.182 0.000 0.000


0.00 385.69 0.00 26.211 0.000 0.000


0.00 330.41 0.00 22.240 0.000 0.000


0.00 273.77 0.00 18.268 0.000 0.005


0.00 215.46 0.00 14.297 0.000 0.009


0.00 162.05 0.00 24.651 0.000 0.007

251


0.00 57.62 0.00 32.016 0.000 0.164

Results using IBC 2012

A U T O S E I S M I C I B C 2 0 1 2

Case: EQX


Direction: X





x = 0.9

Top Story: ROOF

Bottom Story: BASE

R = 5.5

I = 1

Ss = 0.87g

S1 = 0.35g

TL = 0.88

Site Class = C

Fa = 1.052

Fv = 1.45

252



Sds = 0.6102g

Sd1 = 0.3383g

T Used = 1.2339 sec

W Used = 30666.17

V Used = 0.0356W = 1090.26

K Used = 1.3670




170.12 0.00 0.00 0.000 -20.153 -0.480


249.23 0.00 0.00 0.000 -17.020 0.029


206.52 0.00 0.00 0.000 -14.035 0.020

253


166.56 0.00 0.00 0.000 -11.211 0.039


128.40 0.00 0.00 0.000 -8.568 -0.118


92.36 0.00 0.00 0.000 -6.129 -0.155


61.64 0.00 0.00 0.000 -9.377 -0.190


15.44 0.00 0.00 0.000 -8.577 -6.888

A U T O S E I S M I C I B C 2 0 1 2

Case: EQY


Direction: Y





254

x = 0.9

Top Story: ROOF

Bottom Story: BASE

R = 5.5

I = 1

Ss = 0.87g

S1 = 0.35g

TL = 0.88

Site Class = C

Fa = 1.052

Fv = 1.45



Sds = 0.6102g

Sd1 = 0.3383g

T Used = 1.2339 sec

W Used = 30666.17

V Used = 0.0356W = 1090.26

K Used = 1.3670

255




0.00 170.12 0.00 20.153 0.000 0.259


0.00 249.23 0.00 17.020 0.000 0.000


0.00 206.52 0.00 14.035 0.000 0.000


0.00 166.56 0.00 11.211 0.000 0.000


0.00 128.40 0.00 8.568 0.000 0.002


0.00 92.36 0.00 6.129 0.000 0.004


0.00 61.64 0.00 9.377 0.000 0.003

256


0.00 15.44 0.00 8.577 0.000 0.044

257

ACI 318-11 Concrete Frame Design Report

Prepared by

Imran

Model Name: Bond Street_IBC.edb

13 October 2013

258

Design Preferences

Consider Minimum Eccentricity = YesNumber of Interaction Curves = 24 Number of Interaction Points = 11

Pattern Live Load Factor = 0.750 Utilization Factor Limit = 0.950

Phi (Tension Controlled) = 0.900 Phi (Comp. Controlled Tied) = 0.650

Phi (Comp. Controlled Spiral) = 0.700 Phi (Shear and/or Torsion) = 0.750

Phi (Shear Seismic) = 0.600 Phi (Shear Joint) = 0.850

259

Load Combinations Load Combinations

Combination Combination

Name Definition

UDCON1 1.400*DEAD + 1.400*SD

UDCON2 1.200*DEAD + 1.600*LIVE + 1.200*SD

UDCON3 1.200*DEAD + 0.500*LIVE + 1.200*SD + 1.000*EQX

UDCON4 1.200*DEAD + 0.500*LIVE + 1.200*SD – 1.000*EQX











UDCON15 0.900*DEAD + 0.900*SD + 1.000*EQX

UDCON16 0.900*DEAD + 0.900*SD – 1.000*EQX

UDCON17 0.900*DEAD + 0.900*SD

UDCON18 0.900*DEAD + 0.900*SD

UDCON19 0.900*DEAD + 0.900*SD

UDCON20 0.900*DEAD + 0.900*SD

UDCON21 0.900*DEAD + 0.900*SD

UDCON22 0.900*DEAD + 0.900*SD

UDCON23 0.900*DEAD + 0.900*SD

UDCON24 0.900*DEAD + 0.900*SD

UDCON25 0.900*DEAD + 0.900*SD

UDCON26 0.900*DEAD + 0.900*SD

Material Property Data Material Property Data

Name Mass per

Weight per

Modulus of

Poisson's Thermal Shear

Unit Volume

Unit Volume

Elasticity

Ratio Coefficient Modulus

C2500 4.6573E-

03 1.4999E-

01410400.

0000.2000 5.5000E-06 171000.

000

C3000 4.6573E-

03 1.5001E-

01449568.

0000.2000 5.5000E-06 187320.

000

Material Property Data - Concrete Design Material Property Data - Concrete Design

Name Lightweight Concrete Rebar Rebar Lightweight

Concrete fc fy fys Reduc. Factor

C2500 No 360.0008640.00

08640.00

0 N/A

C3000 No 432.0008640.00

08640.00

0 N/A

260

Frame Section Property Data - Concrete Columns Frame Section Property Data - Concrete Columns

Frame Section Material Column Column Rebar Concrete Bar Corner

Name Name Depth Width Pattern Cover SizeBar

Size

C8X24 C3000 2.000 0.667 RR-2-6 0.179 #6 #6

C15X27 C3000 2.250 1.250 RR-3-7 0.188 #8 #8

C8X30 C3000 2.500 0.667 RR-3-8 0.188 #5 #5

C15X24 C3000 2.000 1.250 RR-3-6 0.188 #6 #6

C12X24 C3000 2.000 1.000 RR-3-6 0.188 #5 #5

C8X27 C3000 2.250 0.667 RR-2-5 0.188 #6 #6

C12X27 C3000 2.250 1.000 RR-3-6 0.188 #6 #6

C15X30 C3000 2.500 1.250 RR-3-5 0.188 #8 #8

C12X30 C3000 2.500 1.000 RR-3-5 0.188 #8 #8

C18X33 C3000 2.750 1.500 RR-4-6 0.188 #8 #8

C15X33 C3000 2.750 1.250 RR-3-6 0.188 #8 #8

Frame Section Property Data - Concrete Beams Part 1 of 2 Frame Section Property Data - Concrete Beams Part 1 of 2

Frame Section Material Beam Beam Top Bottom

Name Name Depth Width Cover Cover

B12X24 C2500 2.000 1.000 0.208 0.208

B8X33 C2500 2.750 0.667 0.208 0.208

B12X33 C2500 2.750 1.000 0.208 0.208

Frame Section Property Data - Concrete Beams Part 2 of 2 Frame Section Property Data - Concrete Beams Part 2 of 2

Frame Section Rebar Rebar Rebar Rebar

Name AT-1 AT-2 AB-1 AB-2

B12X24 C2500 2.000 1.000 0.208

B8X33 C2500 2.750 0.667 0.208

B12X33 C2500 2.750 1.000 0.208

261

CONCLUSION

We have worked under the title “Analysis & Design of Building Structures ACI

318R-11 with seismic considerations IBC 2012”.

We have collected Methods and procedures of Analysis and Design of Building

Structures along with Earthquake effects. We did it to introduce modern methods for

seismic loads calculation despite of IBC has replaced UBC in the year of 2000, still our

practices are being carried out as per UBC 1997.

In this achievement we have got remarkable data from US Geological Survey

which provided the basic parameters. Now, we learned to calculate Earthquake effects

using modern procedures of IBC 2012. As observing present situation of Building Code

of Pakistan we have achieved a new trend to do practice on.

262

APPENDIX – A

FIXED-END MOMENTS

263

APPENDIX-B

SHEAR FORCE AND BENDING MOMENT DIAGRAMS FOR

SELECTED LOADING CASES

266

APPENDIX-C

VALUES FOR RU MAX, ,

APPENDIX-D

REBAR SIZE / SPACING CHART

268

APPENDIX-E

REINFORCEMENT DESIGN AIDS

269

APPENDIX-F

MINIMUM THICKNESS OF BEAMS & ONE-WAY SOLID SLABS

270

APPENDIX-G

MINIMUM BEAM WIDTH (IN.) USING STIRRUPS

271

APPENDIX-H

RECTANGULAR SECTIONS WITH COMPRESSION STEEL MINIMUM STEEL PERCENTAGE FOR

COMPRESSION STEEL TO YIELD

272

APPENDIX-I

MODULUS OF ELASTICITY OF CONCRETE (KSI)

273

APPENDIX-J

AREAS OF GROUPS OF STANDARD U.S. BARS IN SQUARE INCHES

274

APPENDIX-K

AREAS OF BARS IN SLABS (SQUARE INCHES PER FOOT)

275

APPENDIX-L

CENTROIDS OF AREAS

DESIGN AID J.1-1 Areas of Reinforcing Bars

Total Areas of Bars (in.2)

Bar Size

Number of Bars 1 2 3 4 5 6 7 8 9 10

No. 3 0.11 0.22 0.33 0.44 0.55 0.66 0.77 0.88 0.99 1.10 No. 4 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 No. 5 0.31 0.62 0.93 1.24 1.55 1.86 2.17 2.48 2.79 3.10 No. 6 0.44 0.88 1.32 1.76 2.20 2.64 3.08 3.52 3.96 4.40 No. 7 0.60 1.20 1.80 2.40 3.00 3.60 4.20 4.80 5.40 6.00 No. 8 0.79 1.58 2.37 3.16 3.95 4.74 5.53 6.32 7.11 7.90 No. 9 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 No. 10 1.27 2.54 3.81 5.08 6.35 7.62 8.89 10.16 11.43 12.70 No. 11 1.56 3.12 4.68 6.24 7.80 9.36 10.92 12.48 14.04 15.60

Areas of Bars per Foot Width of Slab (in.2/ft)

Bar Size

Bar Spacing (in.) 6 7 8 9 10 11 12 13 14 15 16 17 18

No. 3 0.22 0.19 0.17 0.15 0.13 0.12 0.11 0.10 0.09 0.09 0.08 0.08 0.07 No. 4 0.40 0.34 0.30 0.27 0.24 0.22 0.20 0.18 0.17 0.16 0.15 0.14 0.13 No. 5 0.62 0.53 0.46 0.41 0.37 0.34 0.31 0.29 0.27 0.25 0.23 0.22 0.21 No. 6 0.88 0.75 0.66 0.59 0.53 0.48 0.44 0.41 0.38 0.35 0.33 0.31 0.29 No. 7 1.20 1.03 0.90 0.80 0.72 0.65 0.60 0.55 0.51 0.48 0.45 0.42 0.40 No. 8 1.58 1.35 1.18 1.05 0.95 0.86 0.79 0.73 0.68 0.63 0.59 0.56 0.53 No. 9 2.00 1.71 1.50 1.33 1.20 1.09 1.00 0.92 0.86 0.80 0.75 0.71 0.67 No. 10 2.54 2.18 1.91 1.69 1.52 1.39 1.27 1.17 1.09 1.02 0.95 0.90 0.85 No. 11 3.12 2.67 2.34 2.08 1.87 1.70 1.56 1.44 1.34 1.25 1.17 1.10 1.04

mrs

Typewritten Text

mrs

Typewritten Text

mrs

Typewritten Text

mrs

Typewritten Text

mrs

Typewritten Text

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314 Design Aid

PositiveMoment

NegativeMoment

Shear

n n n n

Prismatic members

n

nuw nuw nuw

nuw avgnuw avgnuw nuw nuwSpandrelSupport

ColumnSupport

nuw

nuw nuwnuw nuwnuw nuw

nuw

nnavgn

Note A nuw avgnuw avgnuw nuw nuw

Two or more spans

Uniformly distributed load wu (L/D 3)

nuw

DESIGN AID J.1-2

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314 Design Aid

DESIGN AID J.1-4

Simplified Calculation of sA Assuming Tension-Controlled Section and Grade 60 Reinforcement

cf ′ (psi) sA (in.2)

3,000 d

M u

84.3

4,000 d

M u

00.4

5,000 d

M u

10.4

uM is in ft-kips and d is in inches

In all cases, d

MA us 4

= can be used.

Notes:

• d

ff

f

MA

c

yy

us

×

−

=

'85.05.0

1ρ

φ

• For all values of ρ < 0.0125, the simplified As equation is slightly conservative. • It is recommended to avoid ρ > 0.0125 when using the simplified As equation.

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314 Design Aid

DESIGN AID J.1-51

Assumptions:

yf

cc

sf

Bar Size

Beam Width (in.)

Minimum number of bars, nmim:

1)5.0(2

sdcb

n bcwmin

where

s

cs

f

cf

s

000,4012

5.2000,4015

1 Alsamsam, I.M. and Kamara, M. E. (2004). Simplified Design Reinforced Concrete Buildings of Moderate Size and Heights, EB104, Portland Cement Association, Skokie, IL.

db

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314 Design Aid

DESIGN AID J.1-61

Assumptions:

yf

sc

Bar Size

Beam Width (in.)

Maximum number of bars, nmax:

1space)(Clear

)(2

b

sswdrdcb

nmax

1 Alsamsam, I.M. and Kamara, M. E. (2004). Simplified Design Reinforced Concrete Buildings of Moderate Size and Heights, EB104, Portland Cement Association, Skokie, IL.

db

pfs

314 Design Aid

1 2 3

5.18/1h 21/2h 8/3h

1 2 3

24/1h 28/2h 10/3hSolid One-way Slabs

Applicable to one-way construction not supporting or attached to partitions or other construction likely to be damaged by large deflections.

Values shown are applicable to members with normal weight concrete ( 145cw lbs/ft3) and Grade 60 reinforcement. For other conditions, modify the values as follows:

For structural lightweight having cw in the range 90-120 lbs/ft3, multiply the values by .09.1005.065.1 cw

For yf other than 60,000 psi, multiply the values by .000,100/4.0 yf

For simply-supported members, minimum slabsway -one ribbedor beamsfor 16/

slabsway -one solidfor 20/h

Beams or Ribbed One-way Slabs

DESIGN AID J.1-7h

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314 Design Aid

DESIGN AID J.1-8

Reinforcement Ratio tρ for Tension-Controlled Sections Assuming Grade 60 Reinforcement

cf ′ (psi) tρ when εt = 0.005 tρ when εt = 0.004

3,000 0.01355 0.01548

4,000 0.01806 0.02064

5,000 0.02125 0.02429 Notes:

1. ( )bcfC c 1'85.0 β=

ys fAT = ( ) ysc fAbcfTC =⇒= 1'85.0 β

a. When εt = 0.005, c/dt = 3/8.

( ) ystc fAbdf =83'85.0 1β

y

c

t

st f

f

bdA )8

3(85.0 1 ′==

βρ

b. When εt = 0.004, c/dt = 3/7.

( ) ystc fAbdf =73'85.0 1β

y

c

t

st f

f

bdA )7

3(85.0 1 ′==

βρ

2. β1 is determined according to 10.2.7.3.

mrs

Typewritten Text

mrs

Typewritten Text

mrs

Typewritten Text

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314 Design Aid

DESIGN AID J.1-9

Simplified Calculation of wb Assuming Grade 60 Reinforcement and maxρ=ρ 5.0

cf ′ (psi) wb (in.)*

3,000 26.31

d

M u

4,000 27.23

d

M u

5,000 20.20

d

M u

* uM is in ft-kips and d is in inches

In general:

( ) 211 2143.01

600,36

df

Mb

c

uw

βρ−′βρ=

where maxρρ=ρ / , cf ′ is in psi, d is in inches and uM is in ft-kips and

003.0004.0003.085.0 1+

′β=ρ

y

cmax f

f (10.3.5)

pfs

314 Design Aid

1s

fhh =

+−

+

+

≤

2443

612lengthSpan

121

1

1

1sbb

hb

b

b

ww

w

w

e

1eb

2s 1wb 2wb

2eb

++

+−

+≤

242

164lengthSpan

21312

22ssbbb

hbb

www

we

3wb

2w

fbhh ≥=

wb

we bb 4≤

Isolated T-beam

DESIGN AID J.1-10 T-beam Construction

8.12

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314 Design Aid

DESIGN AID J.1-11

Values of cus VVV φ−=φ (kips) as a Function of the Spacing, s*

s No. 3 U-stirrups No. 4 U-stirrups No. 5 U-stirrups d/2 19.8 36.0 55.8

d/3 29.7 54.0 83.7

d/4 39.6 72.0 111.6 * Valid for Grade 60 ( 60=ytf ksi) stirrups with 2 legs (double the tabulated values for

4 legs, etc.).

In general:

sdfA

V ytvs

φ=φ (11.4.7.2)

where ytf used in design is limited to 60,000 psi, except for welded deformed wire reinforcement, which is limited to 80,000 psi (11.4.2).

pfs

314 Design Aid

DESIGN AID J.1-12

Minimum Shear Reinforcement */, sA minv

cf ′ (psi) s

A minv,

in.in.2

500,4≤ wb00083.0

5,000 wb00088.0

* Valid for Grade 60 ( 60=ytf ksi) shear reinforcement.

In general:

yt

w

yt

wc

minv

fb

fb

fs

A 5075.0, ≥′= Eq. (11-13)

where ytf used in design is limited to 60,000 psi, except for welded deformed wire reinforcement, which is limited to 80,000 psi (11.4.2).

pfs

314 Design Aid

DESIGN AID J.1-15

Approximate Equation to Determine Immediate Deflection, i∆ , for Members Subjected to Uniformly Distributed Loads

ec

ai IE

KM48

5 2=∆

where =aM net midspan moment or cantilever moment

= span length (8.9)

=cE modulus of elasticity of concrete (8.5.1)

= cc fw ′335.1 for values of cw between 90 and 155 pcf

=cw unit weight of concrete

=eI effective moment of inertia (see Flowchart A.1-5.1)

=K constant that depends on the span condition

Span Condition K

Cantilever* 2.0

Simple 1.0

Continuous **)/(2.02.1 ao MM−

* Deflection due to rotation at supports not included

** 8/2wM o = (simple span moment at midspan)

pfs

314 Design Aid

DESIGN AID J.2-1

f

f

scs

bcbf IE

IE

cE

cc fw cw

sb II

Page 1 of 11

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314 Design Aid

DESIGN AID J.2-2

½-Middle strip

½-Middle strip

1

Column strip

Minimum of 1/4 or ( 2)A/4

Minimum of 1/4 or ( 2)B/4( 2)A

( 2)B

Page 2 of 11

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314 Design Aid

DESIGN AID J.2-3

n

n

Page 3 of 11

pfs

314 Design Aid

DESIGN AID J.2-4

Flat Plate or Flat Slab

Flat Plate or Flat Slab with Spandrel Beams

t

Page 4 of 11

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314 Design Aid

DESIGN AID J.2-4

Flat Plate or Flat Slab with End Span Integral with Wall

Flat Plate or Flat Slab with End Span Simply Supported on Wall

Page 5 of 11

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314 Design Aid

DESIGN AID J.2-4

Two-Way Beam-Supported Slab

f t

Page 6 of 11

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314 Design Aid

DESIGN AID J.2-5

f

CC2

ha

b

beff = b + 2(a – h) b + 8h

Beam, Ib

Slab, Is

2

ha

b

Beam, Ib

Slab, Is

CL

beff = b + (a – h) b + 4h

Interior Beam Edge Beam

scs

bcbf IE

IE

cE

cc fw cw

hIs

beffeffbb yhahbhbhayhabhabI

habhb

habhahby

eff

eff

b

Page 7 of 11

pfs

314 Design Aid

DESIGN AID J.2-6

t

CC2

ha

b

beff = b + 2(a – h) b + 8h

Interior Beam

Case A

yxyx

yxyxCA

Case B

yxyx

yxyxCB

C AC BC

scs

cbt IE

CE

hIs cc fwE cw

x2x1

y1

y2y2

x2

x1

y1

y2

Page 8 of 11

pfs

314 Design Aid

2

ha

b

CL

beff = b + (a – h) b + 4h

DESIGN AID J.2-6

t

Edge Beam

Case A

yxyx

yxyxCA

Case B

yxyx

yxyxCB

C AC BC

scs

cbt IE

CE

hIs cc fwE cw

x2x1

y1

y2

x2

x1

y1

y2

Page 9 of 11

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314 Design Aid

DESIGN AID J.2-7

Nc NcNFk NFC NFm

sbcsNFsb IEkK

uFN qmFEM

FN cc FN cc uqPCA Notes on ACI 318-11

Nc

Nc

Fc

Fc

Page 10 of 11

pfs

314 Design Aid

DESIGN AID J.2-8

cHABk ABC

ccccBABAc

ccccABABc

IEkK

IEkK

PCA Notes on ACI 318-11

H c

Page 11 of 11

pfs

314 Design Aid

276

REFERENCES

CHAPTER 1

[1.1] Structural Analysis, Fourth Edition, Aslam Kassimali

[1.2] Basic Structural Analysis, Third Edition, C. S. Reddy

[1.3] Structural Analysis A Matrix Approach, 26th Reprint, Pundit &

Gupta

CHAPTER 2

[2.1] Structural Analysis, Eighth Edition, R. C. Hibbeler

[2.2] Structural Analysis, Fourth Edition, Aslam Kassimali

[2.3] http://www.mathalino.com/reviewer/strength-materials/three-

moment-equation

[2.4] syedalirizwan.com/downloads/4.pdf

CHAPTER 3

[3.1] Analysis of Structures Stiffness Method, Dr. Saeed Ahmed

[3.2] Structural Analysis A Matrix Approach, 26th Reprint, Pundit &

Gupta

[3.3] Basic Structural Analysis, Third Edition, C. S. Reddy

[3.4] Extract from different books

CHAPTER 4

[4.1] Reinforced Concrete Design, Third Edition, Pillai & Menon

[4.2] Structural Concrete Theory & Design, Fourth Edition, Nadim

Hassoun and Akthem Al-Manaseer

277

CHAPTER 5

[5.1] Structural Concrete Theory & Design, Fourth Edition, Nadim

Hassoun and Akthem Al-Manaseer

[5.2] http://faculty.arch.tamu.edu/anichols/index_files/courses/

arch331/NS22-1cncrtdesign.pdf

[5.3] Design of Concrete Structures, Thirteenth Edition, Arthur H.

Nelson, David Darwin, Charles W. Dolan

[5.4] Building Code Requirements for Structural Concrete (ACI 318-

011) and Commentary

[5.5] Google Images

[5.6] Concrete Construction Engineering Handbook, Second Edition

Edward G. Nawy

[5.7] Treasure of RCC Designs, Sushil Kumar

CHAPTER 6

[6.1] Earthquake Resistant Design of Reinforced Concrete Structures

By Professor Dr. Qaisar Ali


Edward G. Nawy

[6.3] Earthquake Engineering Application to Design, 2007, Charles K.

Erdey

278

CHAPTER 7



CHAPTER 8

[8.1] Earthquake Engineering Application to Design, 2007, Charles K.

Erdey



[8.3] Uniform Building Code, Volume 2, 1997 (UBC)

[8.4] Building Code of Pakistan, Seismic Provisions 2007, BCP-SP

CHAPTER 9

[9.1] International Building Code, 2012 (IBC)

[9.2] Lecture 27 – Seismic Loads as per IBC,

faculty.delhi.edu/hultendc/AECT360-Lecture%2027.pdf

[9.3] USGS, United States Geological Survey, Geologic Hazards

Science Centre, Worldwide Seismic "Design Maps" Web

Application, http://www.usgs.gov/


Edward G. Nawy

[9.5] Lindeburg, M.R., Baradar, M. Seismic Design of Building

Structures : A Professional’s Introduction to Earthquake Forces

and Design Details (8th ed.). Professional Publications, Inc.

Belmont, CA (2001).

279

[9.6] An Investigation of the Influence of Diaphragm Flexibility on

Building Design through a Comparison of Forced Vibration

Testing and Computational Analysis

analysis and design of building structures aci 318r-11 with seismic considerations ibc 2012

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