Unified theory of concrete structures

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UNIFIED THEORY

OF CONCRETE

STRUCTURES

Thomas T. C. Hsu and Y. L. Mo

University of Houston, USA

A John Wiley and Sons, Ltd., Publication

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UNIFIED THEORY

OF CONCRETE

STRUCTURES

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UNIFIED THEORY

OF CONCRETE

STRUCTURES

Thomas T. C. Hsu and Y. L. Mo

University of Houston, USA

A John Wiley and Sons, Ltd., Publication

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This edition first published 2010

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Library of Congress Cataloging-in-Publication Data

Hsu, Thomas T. C. (Thomas Tseng Chuang), 1933-

Unified theory of concrete structures / Thomas T. C. Hsu and Y. L. Mo.

p. cm.

Includes index.

ISBN 978-0-470-68874-8 (cloth)

1. Reinforced concrete construction. I. Mo, Y. L. II. Title.

TA683.H73 2010

624.1

8341–dc22

2009054418

A catalogue record for this book is available from the British Library.

ISBN 978-0-470-68874-8

Typeset in 10/12pt Times by Aptara Inc., New Delhi, India

Printed in Singapore by Markono

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Contents

About the Authors xi

Preface xv

Instructors’ Guide xvii

1 Introduction 1

1.1 Overview 1

1.2 Structural Engineering 2

1.2.1 Structural Analysis 2

1.2.2 Main Regions vs Local Regions 3

1.2.3 Member and Joint Design 5

1.3 Six Component Models of the Unified Theory 6

1.3.1 Principles and Applications of the Six Models 6

1.3.2 Historical Development of Theories for Reinforced Concrete 7

1.4 Struts-and-ties Model 13

1.4.1 General Description 13

1.4.2 Struts-and-ties Model for Beams 14

1.4.3 Struts-and-ties Model for Knee Joints 15

1.4.4 Comments 20

2 Equilibrium (Plasticity) Truss Model 23

2.1 Basic Equilibrium Equations 23

2.1.1 Equilibrium in Bending 23

2.1.2 Equilibrium in Element Shear 24

2.1.3 Equilibrium in Beam Shear 33

2.1.4 Equilibrium in Torsion 34

2.1.5 Summary of Basic Equilibrium Equations 37

2.2 Interaction Relationships 38

2.2.1 Shear–Bending Interaction 38

2.2.2 Torsion–Bending Interaction 41

2.2.3 Shear–Torsion–Bending Interaction 44

2.2.4 Axial Tension–Shear–Bending Interaction 51

2.3 ACI Shear and Torsion Provisions 51

2.3.1 Torsional Steel Design 52

2.3.2 Shear Steel Design 55

2.3.3 Maximum Shear and Torsional Strengths 56

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2.3.4 Other Design Considerations 58

2.3.5 Design Example 60

2.4 Comments on the Equilibrium (Plasticity) Truss Model 67

3 Bending and Axial Loads 71

3.1 Linear Bending Theory 71

3.1.1 Bernoulli Compatibility Truss Model 71

3.1.2 Transformed Area for Reinforcing Bars 77

3.1.3 Bending Rigidities of Cracked Sections 78

3.1.4 Bending Rigidities of Uncracked Sections 82

3.1.5 Bending Deflections of Reinforced Concrete Members 84

3.2 Nonlinear Bending Theory 88

3.2.1 Bernoulli Compatibility Truss Model 88

3.2.2 Singly Reinforced Rectangular Beams 93

3.2.3 Doubly Reinforced Rectangular Beams 101

3.2.4 Flanged Beams 105

3.2.5 Moment–Curvature (M–φ) Relationships 108

3.3 Combined Bending and Axial Load 112

3.3.1 Plastic Centroid and Eccentric Loading 112

3.3.2 Balanced Condition 115

3.3.3 Tension Failure 116

3.3.4 Compression Failure 118

3.3.5 Bending–Axial Load Interaction 121

3.3.6 Moment–Axial Load–Curvature (M −N − φ) Relationship 122

4 Fundamentals of Shear 125

4.1 Stresses in 2-D Elements 125

4.1.1 Stress Transformation 125

4.1.2 Mohr Stress Circle 127

4.1.3 Principal Stresses 131

4.2 Strains in 2-D Elements 132

4.2.1 Strain Transformation 132

4.2.2 Geometric Relationships 134

4.2.3 Mohr Strain Circle 136

4.2.4 Principle Strains 137

4.3 Reinforced Concrete 2-D Elements 138

4.3.1 Stress Condition and Crack Pattern in RC 2-D Elements 138

4.3.2 Fixed Angle Theory 140

4.3.3 Rotating Angle Theory 142

4.3.4 ‘Contribution of Concrete’ (Vc) 143

4.3.5 Mohr Stress Circles for RC Shear Elements 145

5 Rotating Angle Shear Theories 149

5.1 Stress Equilibrium of RC 2-D Elements 149

5.1.1 Transformation Type of Equilibrium Equations 149

5.1.2 First Type of Equilibrium Equations 150

5.1.3 Second Type of Equilibrium Equations 152

5.1.4 Equilibrium Equations in Terms of Double Angle 153

5.1.5 Example Problem 5.1 Using Equilibrium (Plasticity) Truss Model 154

5.2 Strain Compatibility of RC 2-D Elements 158

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5.2.1 Transformation Type of Compatibility Equations 158

5.2.2 First Type of Compatibility Equations 159

5.2.3 Second Type of Compatibility Equations 160

5.2.4 Crack Control 161

5.3 Mohr Compatibility Truss Model (MCTM) 165

5.3.1 Basic Principles of MCTM 165

5.3.2 Summary of Equations 166

5.3.3 Solution Algorithm 167

5.3.4 Example Problem 5.2 using MCTM 168

5.3.5 Allowable Stress Design of RC 2-D Elements 172

5.4 Rotating Angle Softened Truss Model (RA-STM) 173

5.4.1 Basic Principles of RA-STM 173

5.4.2 Summary of Equations 174

5.4.3 Solution Algorithm 178

5.4.4 Example Problem 5.3 for Sequential Loading 181

5.4.5 2-D Elements under Proportional Loading 188

5.4.6 Example Problem 5.4 for Proportional Loading 194

5.4.7 Failure Modes of RC 2-D Elements 202

5.5 Concluding Remarks 209

6 Fixed Angle Shear Theories 211

6.1 Softened Membrane Model (SMM) 211

6.1.1 Basic Principles of SMM 211

6.1.2 Research in RC 2-D Elements 213

6.1.3 Poisson Effect in Reinforced Concrete 216

6.1.4 Hsu/Zhu Ratios ν12 and ν21 219

6.1.5 Experimental Stress–Strain Curves 225

6.1.6 Softened Stress–Strain Relationship of Concrete in Compression 227

6.1.7 Softening Coefficient ζ 228

6.1.8 Smeared Stress–Strain Relationship of Concrete in Tension 232

6.1.9 Smeared Stress–Strain Relationship of Mild Steel Bars in Concrete 236

6.1.10 Smeared Stress–Strain Relationship of Concrete in Shear 245

6.1.11 Solution Algorithm 246

6.1.12 Example Problem 6.1 248

6.2 Fixed Angle Softened Truss Model (FA-STM) 255

6.2.1 Basic Principles of FA-STM 255

6.2.2 Solution Algorithm 257

6.2.3 Example Problem 6.2 259

6.3 Cyclic Softened Membrane Model (CSMM) 266

6.3.1 Basic Principles of CSMM 266

6.3.2 Cyclic Stress–Strain Curves of Concrete 267

6.3.3 Cyclic Stress–Strain Curves of Mild Steel 272

6.3.4 Hsu/Zhu Ratios υTC and υCT 274

6.3.5 Solution Procedure 274

6.3.6 Hysteretic Loops 276

6.3.7 Mechanism of Pinching and Failure under Cyclic Shear 281

6.3.8 Eight Demonstration Panels 284

6.3.9 Shear Stiffness 287

6.3.10 Shear Ductility 288

6.3.11 Shear Energy Dissipation 289

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viii Contents

7 Torsion 295

7.1 Analysis of Torsion 295

7.1.1 Equilibrium Equations 295

7.1.2 Compatibility Equations 297

7.1.3 Constitutive Relationships of Concrete 302

7.1.4 Governing Equations for Torsion 307

7.1.5 Method of Solution 309

7.1.6 Example Problem 7.1 314

7.2 Design for Torsion 320

7.2.1 Analogy between Torsion and Bending 320

7.2.2 Various Definitions of Lever Arm Area, Ao 322

7.2.3 Thickness td of Shear Flow Zone for Design 323

7.2.4 Simplified Design Formula for td 326

7.2.5 Compatibility Torsion in Spandrel Beams 328

7.2.6 Minimum Longitudinal Torsional Steel 337

7.2.7 Design Examples 7.2 338

8 Beams in Shear 343

8.1 Plasticity Truss Model for Beam Analysis 343

8.1.1 Beams Subjected to Midspan Concentrated Load 343

8.1.2 Beams Subjected to Uniformly Distributed Load 346

8.2 Compatibility Truss Model for Beam Analysis 350

8.2.1 Analysis of Beams Subjected to Uniformly Distributed Load 350

8.2.2 Stirrup Forces and Triangular Shear Diagram 351

8.2.3 Longitudinal Web Steel Forces 354

8.2.4 Steel Stresses along a Diagonal Crack 355

8.3 Shear Design of Prestressed Concrete I-beams 356

8.3.1 Background Information 356

8.3.2 Prestressed Concrete I-Beam Tests at University of Houston 357

8.3.3 UH Shear Strength Equation 364

8.3.4 Maximum Shear Strength 368

8.3.5 Minimum Stirrup Requirement 371

8.3.6 Comparisons of Shear Design Methods with Tests 372

8.3.7 Shear Design Example 375

8.3.8 Three Shear Design Examples 379

9 Finite Element Modeling of Frames and Walls 381

9.1 Overview 381

9.1.1 Finite Element Analysis (FEA) 381

9.1.2 OpenSees–an Object-oriented FEA Framework 383

9.1.3 Material Models 384

9.1.4 FEA Formulations of 1-D and 2-D Models 384

9.2 Material Models for Concrete Structures 385

9.2.1 Material Models in OpenSees 385

9.2.2 Material Models Developed at UH 388

9.3 1-D Fiber Model for Frames 392

9.4 2-D CSMM Model for Walls 393

9.4.1 Coordinate Systems for Concrete Structures 393

9.4.2 Implementation 394

9.4.3 Analysis Procedures 396

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9.5 Equation of Motion for Earthquake Loading 396

9.5.1 Single Degree of Freedom versus Multiple Degrees of Freedom 396

9.5.2 A Three-degrees-of-freedom Building 399

9.5.3 Damping 400

9.6 Nonlinear Analysis Algorithm 402

9.6.1 Load Control Iteration Scheme 402

9.6.2 Displacement Control Iteration Scheme 403

9.6.3 Dynamic Analysis Iteration Scheme 403

9.7 Nonlinear Finite Element Program SCS 406

10 Application of Program SCS to Wall-type Structures 411

10.1 RC Panels Under Static Load 411

10.2 Prestresed Concrete Beams Under Static Load 413

10.3 Framed Shear Walls under Reversed Cyclic Load 414

10.3.1Framed Shear Wall Units at UH 414

10.3.2Low-rise Framed Shear Walls at NCREE 417

10.3.3Mid-rise Framed Shear Walls at NCREE 420

10.4 Post-tensioned Precast Bridge Columns under Reversed Cyclic Load 422

10.5 Framed Shear Walls under Shake Table Excitations 425

10.6 A Seven-story Wall Building under Shake Table Excitations 428

Appendix 433

References 481

Index 489

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About the Authors

Thomas T. C. Hsu is a John and Rebecca Moores Professor at the University of Houston

(UH), Houston, Texas. He received his MS and Ph.D. degrees from Cornell University and

joined the Portland Cement Association, Skokie, Illinois, as a structural engineer in 1962. He

was a professor and then chairman of the Department of Civil Engineering at the University

of Miami, Coral Gables, Florida, 1968–79. After joining UH, he served as the chairman of the

Civil and Environmental Engineering Department, 1980–84, built a strong faculty and became

the founding director of the Structural Research Laboratory, 1982–2003, which later bears his

name. In 2005 he and his wife, Dr. Laura Ling Hsu, established the “Thomas and Laura Hsu

Professorship in Engineering” at UH.

Dr. Hsu is distinguished by his research in construction materials and in structural engi￾neering. The American Concrete Institute (ACI) awarded him its Wason Medal for Materials

Research, 1965; Arthur R. Anderson Research Award, 1990 and Arthur J. Boase Award for

Structural Concrete, 2007. Other national awards include the American Society of Engineer￾ing Education (ASEE)’s Research Award, 1969, and the American Society of Civil Engineers

(ASCE)’s Huber Civil Engineering Research Prize, 1974. In 2009, he was the honoree of the

ACI-ASCE co-sponsored “Thomas T. C. Hsu Symposium on Shear and Torsion in Concrete

Structures” at the ACI fall convention in New Orleans. At UH, Professor Hsu’s many honors

include the Fluor-Daniel Faculty Excellence Award, 1998; Abraham E. Dukler Distinguished

Engineering Faculty Award, 1998; Award for Excellence in Research and Scholarship, 1996;

Senior Faculty Research Award, 1992; Halliburton Outstanding Teacher, 1990; Teaching

Excellence Award, 1989.

Professor Hsu authored numerous research papers on shear and torsion of reinforced con￾crete and published two books: “Unified Theory of Reinforced Concrete” (1993) and “Torsion

of Reinforced Concrete” (1984). In this (his third) book “Unified Theory of Concrete Struc￾tures” (2010), he integrated the action of four major forces (axial load, bending, shear, torsion),

in 1,2,3 – dimensions, which culminated into a set of unified theories to analyze and design

concrete buildings and infrastructure. Significant parts of Dr. Hsu’s work are codified into

the ACI Building Code which guides the building industry in the USA and is freely shared

worldwide.

Intrinsic to Dr. Hsu’s work are two research innovations: (1) the concept that the behavior of

whole structures can be derived from studying and integrating their elemental parts, or panels;

and (2) the design, construction and use of the “Universal Panel Tester” at UH, a unique,

million-dollar test rig (NSF grants) that continues to lead the world in producing rigorous,

research data on the constitutive models of reinforced concrete, relatable to real-life structures.

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xii About the Authors

In his research on construction materials, Dr. Hsu was the first to visually identify micro￾cracks in concrete materials and to correlate this micro-phenomenon to their overt physical

properties. His research on fatigue of concrete and fiber-reinforced concrete materials made it

possible to interpret the behavior of these structural materials by micro-mechanics.

Among his consulting projects, Dr. Hsu is noted for designing the innovative and cost￾saving “double-T aerial guideways” for the Dade County Rapid Transit System in Florida; the

curved cantilever beams for the Mount Sinai Medical Center Parking Structure in Miami Beach,

Florida, and the large transfer girders in the American Hospital Association Buildings, Chicago,

Illinois. He is currently a consultant to the US Nuclear Regulatory Commission (NRC).

Dr. Hsu is a fellow of the American Society of Civil Engineers and of the American Concrete

Institute. He is a member of ACI Committee 215 (Fatigue), ACI-ASCE joint Committees

343 (Concrete Bridge Design) and 445 (Shear and Torsion). He had also served on ACI

Committee 358 (Concrete Guideways), ACI Committee on Publication and ACI Committee

on Nomination.

Y. L. Mo is a professor in the Civil and Environmental Engineering Department, University

of Houston (UH), and Director of the Thomas T.C. Hsu Structural Research Laboratory. Dr.

Mo received his MS degree from National Taiwan University, Taipei, Taiwan and his Ph.D.

degree in 1982 from the University of Hannover, Hannover, Germany. He was a structural

engineer at Sargent and Lundy Engineers in Chicago, 1984–91, specializing in the design of

nuclear power plants. Before joining UH in 2000, Dr. Mo was a professor at the National

Cheng Kung University, Tainan, Taiwan.

Professor Mo has more than 27 years of experience in studies of reinforced and prestressed

concrete structures subjected to static, reversed cyclic or dynamic loading. In addition to

earthquake design of concrete structures, he is an expert in composite and hybrid structures.

His outstanding research achievement is in the synergistic merging of structural engineering,

earthquake engineering and computer application.

Professor Mo is noted for his innovations in the design of nuclear power plants and is cur￾rently a consultant to the US Nuclear Regulatory Commission (NRC). His experience includes

developing a monitoring system for structural integrity using the concept of data mining, as

well as a small-bore piping design expert system using finite element method and artificial

intelligence. Dr. Mo has recently focused on innovative ways to use piezoceramic-based smart

aggregates (SAs) to assess the state of health of concrete structures. He also developed carbon

nanofiber concrete (CNFC) materials for building infrastructures with improved electrical

properties that are required for self health monitoring and damage evaluation.

Professor Mo’s wide-ranging consulting work includes seismic performance of shearwalls,

optimal analysis of steam curing, effect of casting and slump on ductility of RC beams, effect

of welding on ductility of reinforcing bars, early form removal of RC slabs, etc. After the

1999 Taiwan Chi-Chi earthquake, Dr. Mo was selected by Taiwan’s National Science Council

(NSC) to lead a team of twenty professors to study the damages in concrete structures, to

assess causes and to recommend rehabilitation and future research.

Professor Mo is the author of the book “Dynamic Behavior of Concrete Structures” (1994)

and is the editor or co-editor of four books. He has written more than 100 technical papers

published in national and international journals. For his research and teaching, he received the

Alexander von Humboldt Research Fellow Award from Germany in 1995, the Distinguished

Research Award from the National Science Council of Taiwan in 1999, the Teaching Excellent