Advanced Analysis and Design of Steel Frames

Preface xi

Symbols xiii

Part One Advanced Analysis of Steel Frames 1

Chapter 1 Introduction 3

1.1 Type of Steel Frames 3

1.2 Type of Components for Steel Frames 3

1.3 Type of Beam–Column Connections 7

1.4 Deformation of Joint Panel 7

1.5 Analysis Tasks and Method for Steel Frame Design 8

1.6 Definition of Elements in Steel Frames 9

Chapter 2 Elastic Stiffness Equation of Prismatic Beam Element 11

2.1 General Form of Equation 11

2.1.1 Beam Element in Tension 11

2.1.2 Beam Element in Compression 16

2.1.3 Series Expansion of Stiffness Equations 16

2.1.4 Beam Element with Initial Geometric Imperfection 17

2.2 Special Forms of Elemental Equations 19

2.2.1 Neglecting Effect of Shear Deformation 19

2.2.2 Neglecting Effect of Axial Force 21

2.2.3 Neglecting Effects of Shear Deformation and Axial Force 22

2.3 Examples 22

2.3.1 Bent Frame 22

2.3.2 Simply Supported Beam 24

Chapter 3 Elastic Stiffness Equation of Tapered Beam Element 25

3.1 Tapered Beam Element 25

3.1.1 Differential Equilibrium Equation 25

3.1.2 Stiffness Equation 27

3.2 Numerical Verification 29

3.2.1 Symmetry of Stiffness Matrix 29

3.2.2 Static Deflection 30

3.2.3 Elastic Critical Load 30

3.2.4 Frequency of Free Vibration 30

3.2.5 Effect of Term Number Truncated in Polynomial Series 31

3.2.6 Steel Portal Frame 31

3.3 Appendix 33

3.3.1 Chebyshev Polynomial Approach (Rice, 1992) 33

3.3.2 Expression of Elements in Equation (3.23) 34

Chapter 4 Elastic Stiffness Equation of Composite Beam Element 35

4.1 Characteristics and Classification of Composite Beam 35

4.2 Effects of Composite Action on Elastic Stiffness of Composite Beam 37

4.2.1 Beam without Composite Action 37

4.2.2 Beam with Full Composite Action 38

4.2.3 Beam with Partial Composite Action 39

4.3 Elastic Stiffness Equation of Steel–Concrete Composite Beam Element 40

4.3.1 Basic Assumptions 40

4.3.2 Differential Equilibrium Equation of Partially Composite Beam 41

4.3.3 Stiffness Equation of Composite Beam Element 42

4.3.4 Equivalent Nodal Load Vector 46

4.4 Example 49

4.5 Problems in Present Work 51

Chapter 5 Sectional Yielding and Hysteretic Model of Steel Beam Columns 53

5.1 Yielding of Beam Section Subjected to Uniaxial Bending 53

5.2 Yielding of Column Section Subjected to Uniaxial Bending 53

5.3 Yielding of Column Section Subjected to Biaxial Bending 56

5.3.1 Equation of Initial Yielding Surface 56

5.3.2 Equation of Ultimate Yielding Surface 56

5.3.3 Approximate Expression of Ultimate Yielding Surface 61

5.3.4 Effects of Torsion Moment 62

5.4 Hysteretic Model 64

5.4.1 Cyclic Loading and Hysteretic Behaviour 64

5.4.2 Hysteretic Model of Beam Section 65

5.4.3 Hysteretic Model of Column Section Subjected to Uniaxial Bending 67

5.4.4 Hysteretic Model of Column Section Subjected to Biaxial Bending 67

5.5 Determination of Loading and Deformation States of Beam–Column Sections 68

Chapter 6 Hysteretic Behaviour of Composite Beams 71

6.1 Hysteretic Model of Steel and Concrete Material Under Cyclic Loading 71

6.1.1 Hysteretic Model of Steel Stress–Strain Relationship 71

6.1.2 Hysteretic Model of Concrete Stress–Strain Relationship 71

6.2 Numerical Method for Moment–Curvature Hysteretic Curves 75

6.2.1 Assumptions 75

6.2.2 Sectional Division 75

6.2.3 Calculation Procedure of Moment–Curvature Relationship 76

6.3 Hysteretic Characteristics of Moment–Curvature Relationships 77

6.3.1 Characteristics of Hysteretic Curves 77

6.3.2 Typical Phases 78

6.4 Parametric Studies 79

6.4.1 Height of Concrete Flange hc 79

6.4.2 Width of Concrete Flange Bc 79

6.4.3 Height of Steel Beam hs 80

6.4.4 Strength Ratio g 83

6.4.5 Yielding Strength of Steel fy 84

6.4.6 Compressive Strength of Concrete fck 84

6.4.7 Summary of Parametric Studies 85

6.5 Simplified Hysteretic Model 86

6.5.1 Skeletal Curve 86

6.5.2 Hysteresis Model 89

Chapter 7 Elasto-Plastic Stiffness Equation of Beam Element 93

7.1 Plastic Hinge Theory 93

7.1.1 Hinge Formed at One End of Element 94

7.1.2 Hinge Formed at Both Ends of Element 97

7.2 Clough Model 97

7.3 Generalized Clough Model 98

7.4 Elasto-Plastic Hinge Model 99

7.4.1 Both Ends Yielding 102

7.4.2 Only End 1 Yielding 103

7.4.3 Only End 2 Yielding 103

7.4.4 Summary 104

7.5 Comparison Between Elasto-Plastic Hinge Model and Generalized Clough Model 104

7.5.1 Only End 1 Yielding 104

7.5.2 Both Ends Yielding 105

7.5.3 Numerical Example 106

7.6 Effects of Residual Stresses and Treatment of Tapered Element 107

7.6.1 Effects of Residual Stresses on Plasticity Spread Along Element Section 107

7.6.2 Effects of Residual Stresses on Plasticity Spread Along Element Length 109

7.6.3 Treatment of Tapered Element 110

7.7 Beam Element with Plastic Hinge Between Two Ends 110

7.8 Subdivided Model with Variable Stiffness for Composite Beam Element 113

7.8.1 Subdivided Model 113

7.8.2 Stiffness Equation of Composite Beam Element 114

7.9 Examples 117

7.9.1 A Steel Portal Frame with Prismatic Members 117

7.9.2 A Steel Portal Frame with Tapered Members 118

7.9.3 Vogel Portal Frame 119

7.9.4 Vogel Six-Storey Frame 120

7.9.5 A Single-Storey Frame with Mid-Span Concentrated Load 121

7.9.6 A Single-Storey Frame with Distributed Load 123

7.9.7 A Four-Storey Frame with Mid-Span Concentrated Load 124

7.9.8 A Two-Span Three-Storey Composite Frame 126

Chapter 8 Elastic and Elasto-Plastic Stiffness Equations of Column Element 127

8.1 Force and Deformation of Column Element 127

8.2 Elastic Stiffness Equation of Column Element Subjected to Biaxial Bending 127

8.3 Elasto-Plastic Stiffness Equations of Column Element Subjected to Biaxial Bending 129

8.3.1 Both Ends Yielding 131

8.3.2 Only End 1 Yielding 132

8.3.3 Only End 2 Yielding 133

8.3.4 Summary 133

8.4 Elastic and Elasto-Plastic Stiffness Equations of Column Element Subjected to Uniaxial Bending 134

8.5 Axial Stiffness of Tapered Column Element 135

8.5.1 Elastic Stiffness 135

8.5.2 Elasto-Plastic Stiffness 135

8.6 Experiment Verification 136

8.6.1 Experiment Specimen 136

8.6.2 Set-Up and Instrumentation 139

8.6.3 Horizontal Loading Scheme 140

8.6.4 Theoretical Predictions of Experiments 141

8.6.5 Comparison of Analytical and Tested Results 144

Chapter 9 Effects of Joint Panel and Beam–Column Connection 147

9.1 Behaviour of Joint Panel 147

9.1.1 Elastic Stiffness of Joint Panel 147

9.1.2 Elasto-Plastic Stiffness of Joint Panel 149

9.2 Effect of Shear Deformation of Joint Panel on Beam/Column Stiffness 150

9.2.1 Stiffness Equation of Beam Element with Joint Panel 150

9.2.2 Stiffness Equation of Column Element with Joint Panel Subjected to Uniaxial Bending 153

9.2.3 Stiffness Equation of Column Element with Joint Panel Subjected to Biaxial Bending 154

9.3 Behaviour of Beam–Column Connections 155

9.3.1 Moment–Rotation Relationship 156

9.3.2 Hysteretic Behaviour 161

9.4 Effect of Deformation of Beam–Column Connection on Beam Stiffness 163

9.4.1 Stiffness Equation of Beam Element with Beam–Column Connections 164

9.4.2 Stiffness Equation of Beam Element with Connections and Joint Panels 166

9.5 Examples 166

9.5.1 Effect of Joint Panel 166

9.5.2 Effect of Beam–Column Connection 170

Chapter 10 Brace Element and its Elastic and Elasto-Plastic Stiffness Equations 175

10.1 Hysteretic Behaviour of Braces 175

10.2 Theoretical Analysis of Elastic and Elasto-Plastic Stiffnesses of Brace Element 175

10.3 Hysteretic Model of Ordinary Braces 181

10.4 Hysteretic Characteristics and Model of Buckling-Restrained Brace 183

10.5 Stiffness Equation of Brace Element 185

Chapter 11 Shear Beam and its Elastic and Elasto-Plastic Stiffness Equations 187

11.1 Eccentrically Braced Frame and Shear Beam 187

11.1.1 Eccentrically Braced Frame 187

11.1.2 Condition of Shear Beam 187

11.2 Hysteretic Model of Shear Beam 189

11.3 Stiffness Equation of Shear Beam 190

Chapter 12 Elastic Stability Analysis of Planar Steel Frames 193

12.1 General Analytical Method 193

12.2 Effective Length of Prismatic Frame Column 194

12.2.1 Concept of Effective Length 194

12.2.2 Assumption and Analytical Model 195

12.2.3 Formulations of Effective Length 197

12.2.4 Simplified Formula of Effective Length 202

12.2.5 Modification of Effective Length 203

12.2.6 Effect of Shear Deformation on Effective Length of Column 205

12.2.7 Examples 205

12.3 Effective Length of Tapered Steel Columns 211

12.3.1 Tapered Columns Under Different Boundary Conditions 211

12.3.2 Tapered Column in Steel Portal Frame 213

Chapter 13 Nonlinear Analysis of Planar Steel Frames 219

13.1 General Analysis Method 219

13.1.1 Loading Types 219

13.1.2 Criteria for the Limit State of Ultimate Load-Carrying Capacity 220

13.1.3 Analysis Procedure 221

13.1.4 Basic Elements and Unknown Variables 222

13.1.5 Structural Analysis of the First Loading Type 222

13.1.6 Structural Analysis of the Second Loading Type 223

13.1.7 Numerical Examples 223

13.2 Approximate Analysis Considering P_D Effect 226

13.2.1 Formulation 226

13.2.2 Example 227

13.3 Simplified Analysis Model Considering P_D Effect 228

13.3.1 Development of Simplified Model 228

13.3.2 Example 231

Chapter 14 Seismic Response Analysis of Planar Steel Frames 233

14.1 General Analysis Method 233

14.1.1 Kinetic Differential Equation 233

14.1.2 Solution of Kinetic Differential Equation 235

14.1.3 Determination of Mass, Stiffness and Damping Matrices 238

14.1.4 Numerical Example 240

14.2 Half-Frame Model 241

14.2.1 Assumption and Principle of Half-Frame 241

14.2.2 Stiffness Equation of Beam Element in Half-Frame 244

14.2.3 Numerical Examples 244

14.3 Shear-Bending Storey Model 248

14.3.1 Equivalent Stiffness 248

14.3.2 Inter-Storey Shear Yielding Parameters 251

14.3.3 Examples 252

14.4 Simplified Model for Braced Frame 255

14.4.1 Decomposition and Simplification of Braced Frame 255

14.4.2 Stiffness Matrix of Pure Frame 256

14.4.3 Stiffness Matrix of Pure Bracing System 257

14.4.4 Example 258

Chapter 15 Analysis Model for Space Steel Frames 259

15.1 Space Bar Model 259

15.1.1 Transformation from Local to Global Coordinates 259

15.1.2 Requirement of Rigid Floor 264

15.1.3 Global Stiffness Equation of Frame and Static Condensation 267

15.2 Planar Substructure Model 268

15.2.1 Stiffness Equation of Planar Substructure in Global Coordinates 268

15.2.2 Global Stiffness Equation of Spatial Frame 271

15.2.3 Numerical Example 272

15.3 Component Mode Synthesis Method 274

15.3.1 Principle of Component Mode Synthesis Method 274

15.3.2 Analysis of Generalized Elements 276

15.3.3 Stiffness Equation of Generalized Structure 281

15.3.4 Structural Analysis Procedure 282

15.3.5 Numerical Example 283

Part Two Advanced Design of Steel Frames 287

Chapter 16 Development of Structural Design Approach 289

16.1 Deterministic Design Approach 289

16.1.1 Allowable Stress Design (ASD) (AISC, 1989) 289

16.1.2 Plastic Design (PD) (AISC, 1978) 290

16.2 Reliability Design Approach Based on Limit States of Structural Members 290

16.3 Structural System Reliability Design Approach 292

Chapter 17 Structural System Reliability Calculation 293

17.1 Fundamentals of Structural Reliability Theory 293

17.1.1 Performance Requirements of Structures 293

17.1.2 Performance Function of Structures 293

17.1.3 Limit State of Structures 294

17.1.4 Structural Reliability 294

17.1.5 Reliability Index 296

17.2 The First-Order Second-Moment (FOSM) Methods for Structural Reliability Assessment 297

17.2.1 Central Point Method 298

17.2.2 Design Point Method 299

17.3 Effects of Correlation Among Random Variables 302

17.4 Structural System Reliability and Boundary Theory 302

17.4.1 Basic Concepts 302

17.4.2 Upper–Lower Boundary Method 305

17.5 Semi-Analytical Simulation Method for System Reliability 306

17.5.1 General Principle 306

17.5.2 Random Sampling 307

17.5.3 Exponential Polynomial Method (EPM) 309

17.6 Example 309

17.6.1 A Steel Beam Section 309

17.6.2 A Steel Portal Frame 313

Chapter 18 System Reliability Assessment of Steel Frames 317

18.1 Randomness of Steel Frame Resistance 317

18.2 Randomness of Loads 318

18.3 System Reliability Evaluation of Typical Steel Frames 319

18.3.1 Effect of Correlation Among Random Variables 319

18.3.2 Evaluation of Structural System Reliability Under Vertical Loads 320

18.3.3 Evaluation of Structural System Reliability Under Horizontal and Vertical Loads 323

18.4 Comparison of System Reliability Evaluation 325

Chapter 19 Reliability-Based Advanced Design of Steel Frames 327

19.1 Structural Design Based on System Reliability 327

19.1.1 Target Reliability of Design 327

19.1.2 Load and Load Combination 329

19.1.3 Practical Design Formula 329

19.2 Effect of Correlation on Load and Resistance Factors 335

19.3 Comparison of Different Design Methods 337

19.3.1 For Steel Portal Frames 337

19.3.2 For Multi-Storey Steel Frames 340

References/Bibliography 345

Author Index 363

Subject Index 365