# Unified Theory of Concrete Structures

# Unified Theory of Concrete Structures

ISBN: 978-0-470-68874-8

Apr 2010

518 pages

## Description

*Unified Theory of Concrete Structures*develops an integrated theory that encompasses the various stress states experienced by both RC & PC structures under the various loading conditions of bending, axial load, shear and torsion. Upon synthesis, the new rational theories replace the many empirical formulas currently in use for shear, torsion and membrane stress.

The unified theory is divided into six model components: a) the struts-and-ties model, b) the equilibrium (plasticity) truss model, c) the Bernoulli compatibility truss model, d) the Mohr compatibility truss model, e) the softened truss model, and f) the softened membrane model. Hsu presents the six models as rational tools for the solution of the four basic types of stress, focusing on the significance of their intrinsic consistencies and their inter-relationships. Because of its inherent rationality, this unified theory of reinforced concrete can serve as the basis for the formulation of a universal and international design code.

- Includes an appendix and accompanying website hosting the authors’ finite element program SCS along with instructions and examples
- Offers comprehensive coverage of content ranging from fundamentals of flexure, shear and torsion all the way to non-linear finite element analysis and design of wall-type structures under earthquake loading.
- Authored by world-leading experts on torsion and shear

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**This item: **Unified Theory of Concrete Structures

Dynamic Behavior of Concrete and Seismic Engineering *(Hardcover £158.00)*

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

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

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

**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

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 toWall-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