# Ultimate Limit State Analysis and Design of Plated Structures, 2nd Edition

ISBN: 978-1-119-36775-8 March 2018 672 Pages

## Description

Reviews and describes both the fundamental and practical design procedures for the ultimate limit state design of ductile steel plated structures

The new edition of this well-established reference reviews and describes both fundamentals and practical design procedures for steel plated structures. The derivation of the basic mathematical expressions is presented together with a thorough discussion of the assumptions and the validity of the underlying expressions and solution methods.

Furthermore, this book is also an easily accessed design tool, which facilitates learning by applying the concepts of the limit states for practice using a set of computer programs, which can be downloaded.

Ultimate Limit State Design of Steel Plated Structures provides expert guidance on mechanical model test results as well as nonlinear finite element solutions, sophisticated design methodologies useful for practitioners in industries or research institutions, and selected methods for accurate and efficient analyses of nonlinear behavior of steel plated structures both up to and after the ultimate strength is reached.

• Covers recent advances and developments in the field
• Includes new topics on constitutive equations of steels, test database associated with low/elevated temperature, and strain rates
• Includes a new chapter on a semi-analytical method
• Supported by a companion website with illustrative example data sheets
• Provides results for existing mechanical model tests
• Offers a thorough discussion of assumptions and the validity of underlying expressions and solution methods

Designed as both a textbook and a handy reference, Ultimate Limit State Design of Steel Plated Structures, Second Edition is well suited to teachers and university students who are approaching the limit state design technology of steel plated structures for the first time. It also meets the needs of structural designers or researchers who are involved in civil, marine, and mechanical engineering as well as offshore engineering and naval architecture.

Preface xvii

How to Use This Book xxi

1 Principles of Limit State Design 1

1.1 Structural Design Philosophies 1

1.1.1 Reliability-Based Design Format 3

1.1.2 Partial Safety Factor-Based Design Format 5

1.1.3 Failure Probability-Based Design Format 6

1.1.4 Risk-Based Design Format 7

1.2 Allowable Stress Design Versus Limit State Design 7

1.2.1 Serviceability Limit State Design 9

1.2.2 Ultimate Limit State Design 10

1.2.3 Fatigue Limit State Design 11

1.2.4 Accidental Limit State Design 15

1.3 Mechanical Properties of Structural Materials 17

1.3.1 Characterization of Material Properties 17

1.3.1.1 Young’s Modulus,E 19

1.3.1.2 Poisson’s Ratio,v 19

1.3.1.3 Elastic Shear Modulus,G 19

1.3.1.4 Proportional Limit, σP 20

1.3.1.5 Yield Strength, σY, and Yield Strain, εY 20

1.3.1.6 Strain-Hardening Tangent Modulus, Eh, and Strain-Hardening Strain, εh 20

1.3.1.7 Ultimate Tensile Strength, σT 20

1.3.1.8 Necking Tangent Modulus, En 22

1.3.1.9 Fracture Strain, εF, and Fracture Stress, σF 22

1.3.2 Elastic–Perfectly Plastic Material Model 23

1.3.3 Characterization of the Engineering Stress–Engineering Strain Relationship 23

1.3.4 Characterization of the True Stress–True Strain Relationship 25

1.3.5 Effect of Strain Rates 29

1.3.6 Effect of Elevated Temperatures 29

1.3.7 Effect of Cold Temperatures 30

1.3.8 Yield Condition Under Multiple Stress Components 34

1.3.10 Limits of Cold Forming 38

1.3.11 Lamellar Tearing 39

1.4 Strength Member Types for Plated Structures 39

1.6 Basic Types of Structural Failure 42

1.7 Fabrication Related Initial Imperfections 43

1.7.1 Mechanism of Initial Imperfections 44

1.7.2 Initial Distortion Modeling 44

1.7.2.1 Plate Initial Deflection 47

1.7.2.2 Column-Type Initial Deflection of a Stiffener 56

1.7.2.3 Sideways Initial Distortion of a Stiffener 56

1.7.3 Welding Residual Stress Modeling 56

1.7.4 Modeling of Softening Phenomenon 59

1.8 Age Related Structural Degradation 60

1.8.1 Corrosion Damage 60

1.8.2 Fatigue Cracks 69

1.9 Accident Induced Damage 73

References 73

2 Buckling and Ultimate Strength of Plate–Stiffener Combinations: Beams, Columns, and Beam–Columns 79

2.1 Structural Idealizations of Plate–Stiffener Assemblies 79

2.2 Geometric Properties 82

2.3 Material Properties 82

2.4 Modeling of End Conditions 83

2.6 Effective Width Versus Effective Breadth of Attached Plating 85

2.6.1 Shear Lag-Induced Ineffectiveness: Effective Breadth of the Attached Plating 88

2.6.2 Buckling-Induced Ineffectiveness: Effective Width of the Attached Plating 91

2.6.3 Combined Shear Lag-Induced and Buckling-Induced Ineffectiveness 93

2.7 Plastic Cross-Sectional Capacities 93

2.7.1 Axial Capacity 93

2.7.2 Shear Capacity 93

2.7.3 Bending Capacity 94

2.7.3.1 Rectangular Cross Section 94

2.7.3.2 Plate–Stiffener Combination Model Cross Section 95

2.7.4 Capacity Under Combined Bending and Axial Load 96

2.7.4.1 Rectangular Cross Section 97

2.7.4.2 Plate–Stiffener Combination Model Cross Section 98

2.7.5 Capacity Under Combined Bending, Axial Load, and Shearing Force 99

2.8 Ultimate Strength of the Plate–Stiffener Combination Model Under Bending 100

2.8.1 Cantilever Beams 101

2.8.2 Beams Simply Supported at Both Ends 102

2.8.3 Beams Simply Supported at One End and Fixed at the Other End 103

2.8.4 Beams Fixed at Both Ends 106

2.8.5 Beams Partially Rotation Restrained at Both Ends 107

2.8.6 Lateral-Torsional Buckling 110

2.9 Ultimate Strength of the Plate–Stiffener Combination Model Under Axial Compression 110

2.9.1 Large-Deflection Behavior of Straight Columns 110

2.9.2 Elastic Buckling of Straight Columns 112

2.9.3 Effect of End Conditions 113

2.9.4 Effect of Initial Imperfections 115

2.9.5 Collapse Strength of Columns 119

2.9.5.1 The Johnson–Ostenfeld Formulation Method 120

2.9.5.2 The Perry–Robertson Formulation Method 120

2.9.5.3 The Paik–Thayamballi Empirical Formulation Method for a Steel Plate–

Stiffener Combination Model 121

2.9.5.4 The Paik Empirical Formulation Method for an Aluminum Plate–Stiffener Combination Model 122

2.9.6 Local Web or Flange Buckling Under Axial Compression 125

2.9.7 Lateral-Torsional Buckling Under Axial Compression 126

2.10 Ultimate Strength of the Plate–Stiffener Combination Model Under Combined

Axial Compression and Bending 126

2.10.1 The Modified Perry–Robertson Formulation Method 126

2.10.2 Lateral-Torsional Buckling Under Combined Axial Compression and

Bending 129

References 132

3 Elastic and Inelastic Buckling Strength of Plates Under Complex Circumstances 135

3.1 Fundamentals of Plate Buckling 135

3.2 Geometric and Material Properties 136

3.4 Boundary Conditions 137

3.5 Linear Elastic Behavior 138

3.6 Elastic Buckling of Simply Supported Plates Under Single Types of Loads 138

3.7 Elastic Buckling of Simply Supported Plates Under Two Load Components 139

3.7.1 Biaxial Compression or Tension 139

3.7.2 Longitudinal Axial Compression and Longitudinal In-Plane Bending 141

3.7.3 Transverse Axial Compression and Longitudinal In-Plane Bending 142

3.7.4 Longitudinal Axial Compression and Transverse In-Plane Bending 143

3.7.5 Transverse Axial Compression and Transverse In-Plane Bending 144

3.7.6 Biaxial In-Plane Bending 144

3.7.7 Longitudinal Axial Compression and Edge Shear 145

3.7.8 Transverse Axial Compression and Edge Shear 145

3.7.9 Longitudinal In-Plane Bending and Edge Shear 146

3.7.10 Transverse In-Plane Bending and Edge Shear 147

3.8 Elastic Buckling of Simply Supported Plates Under More than Three Load Components 147

3.9 Elastic Buckling of Clamped Plates 149

3.9.1 Single Types of Loads 149

3.10 Elastic Buckling of Partially Rotation Restrained Plates 149

3.10.1 Rotational Restraint Parameters 149

3.10.2 Longitudinal Axial Compression 152

3.10.2.1 Partially Rotation Restrained at Long Edges and Simply Supported at Short Edges 152

3.10.2.2 Partially Rotation Restrained at Short Edges and Simply Supported at Long Edges 153

3.10.2.3 Partially Rotation Restrained at Both Long and Short Edges 154

3.10.3 Transverse Axial Compression 155

3.10.3.1 Partially Rotation Restrained at Long Edges and Simply Supported at Short Edges 155

3.10.3.2 Partially Rotation Restrained at Short Edges and Simply Supported at Long Edges 156

3.10.3.3 Partially Rotation Restrained at Both Long and Short Edges 157

3.11 Effect of Welding-Induced Residual Stresses 158

3.12 Effect of Lateral Pressure Loads 159

3.13 Effect of Opening 163

3.13.1 Longitudinal Axial Compression 163

3.13.2 Transverse Axial Compression 165

3.13.3 Edge Shear 166

3.14 Elastic–Plastic Buckling Strength 168

3.14.1 Single Types of Loads 168

3.14.1.1 Plates Without Opening 168

3.14.1.2 Perforated Plates 170

References 176

4 Large-Deflection and Ultimate Strength Behavior of Plates 179

4.1 Fundamentals of Plate Collapse Behavior 179

4.2 Structural Idealizations of Plates 185

4.2.1 Geometric Properties 185

4.2.2 Material Properties 186

4.2.4 Fabrication Related Initial Imperfections 186

4.2.5 Boundary Conditions 187

4.3 Nonlinear Governing Differential Equations of Plates 189

4.4 Elastic Large-Deflection Behavior of Simply Supported Plates 191

4.4.3 Interaction Effect Between Biaxial Loads and Lateral Pressure 198

4.4.4 Interaction Effect Between Biaxial and Edge Shear Loads 200

4.5 Elastic Large-Deflection Behavior of Clamped Plates 201

4.5.3 Interaction Effect Between Biaxial Loads and Lateral Pressure 206

4.6 Elastic Large-Deflection Behavior of Partially Rotation Restrained Plates 206

4.6.1 Longitudinal Compression 207

4.6.2 Transverse Compression 209

4.6.3 Biaxial Compression 210

4.7 Effect of the Bathtub Deflection Shape 210

4.8 Evaluation of In-Plane Stiffness Reduction Due to Deflection 214

4.8.1 Effective Width 218

4.8.2 Effective Length 224

4.8.3 Effective Shear Modulus 231

4.9 Ultimate Strength 234

4.9.1 Ultimate Strength by Gross Yielding 234

4.9.2 Rigid-Plastic Theory Method 234

4.9.3 Membrane Stress-Based Method 240

4.9.3.1 Ultimate Strength Conditions 240

4.9.3.3 Combined Longitudinal Axial Loads and Lateral Pressure 242

4.9.3.4 Combined Transverse Axial Loads and Lateral Pressure 243

4.9.3.5 Edge Shear 245

4.9.3.6 Combined Edge Shear Loads and Lateral Pressure 247

4.10 Effect of Opening 251

4.10.1 Single Types of Loads 252

4.10.2 Biaxial Compression 253

4.10.3 Combined Longitudinal Compression and Edge Shear 257

4.10.4 Combined Transverse Compression and Edge Shear 257

4.11 Effect of Age-Related Structural Deterioration 257

4.11.1 Corrosion Damage 257

4.11.2 Fatigue Cracking Damage 260

4.12 Effect of Local Denting Damage 260

4.13 Average Stress–Average Strain Relationship of Plates 261

4.13.1 Pre-buckling or Undeflected Regime 262

4.13.2 Post-buckling or Deflected Regime 262

4.13.3 Post-ultimate Strength Regime 265

References 267

5 Elastic and Inelastic Buckling Strength of Stiffened Panels and Grillages 271

5.1 Fundamentals of Stiffened Panel Buckling 271

5.2 Structural Idealizations of Stiffened Panels 272

5.2.1 Geometric Properties 272

5.2.2 Material Properties 273

5.2.4 Boundary Conditions 275

5.2.5 Fabrication Related Initial Imperfections 277

5.3 Overall Buckling Versus Local Buckling 277

5.4 Elastic Overall Buckling Strength 278

5.4.1 Longitudinal Axial Compression 278

5.4.1.1 Longitudinally Stiffened Panels 278

5.4.1.2 Transversely Stiffened Panels 278

5.4.1.3 Cross-Stiffened Panels (Grillages) 278

5.4.2 Transverse Axial Compression 279

5.4.2.1 Longitudinally Stiffened Panels 279

5.4.2.2 Transversely Stiffened Panels 280

5.4.2.3 Cross-Stiffened Panels (Grillages) 280

5.4.3 Edge Shear 280

5.4.4 Combined Biaxial Compression or Tension 281

5.4.5 Combined Uniaxial Compression and Edge Shear 282

5.5 Elastic Local Buckling Strength of Plating Between Stiffeners 283

5.6 Elastic Local Buckling Strength of Stiffener Web 283

5.6.1 Governing Differential Equation 283

5.6.2 Exact Web Buckling Characteristic Equation 284

5.6.3 Closed-Form Expressions of Stiffener Web Buckling Strength 288

5.7 Elastic Local Buckling Strength of Stiffener Flange 289

5.8 Lateral-Torsional Buckling Strength of Stiffeners 291

5.8.1 Fundamentals of Lateral-Torsional Buckling 291

5.8.2 Closed-Form Expressions of Lateral-Torsional Buckling Strength 293

5.8.2.1 Elastic Flexural-Torsional Buckling Strength of Asymmetric Angle Stiffeners 296

5.8.2.2 Elastic Flexural-Torsional Buckling Strength of Symmetric Tee Stiffeners 297

5.8.2.3 Elastic Flexural-Torsional Buckling Strength of Flat-Bar Stiffeners 297

5.8.2.4 Effect of Welding-Induced Residual Stresses 298

5.9 Elastic–Plastic Buckling Strength 299

References 299

6 Large-Deflection and Ultimate Strength Behavior of Stiffened Panels and Grillages 301

6.1 Fundamentals of Stiffened Panel Ultimate Strength Behavior 301

6.2 Classification of Panel Collapse Modes 302

6.3 Structural Idealizations of Stiffened Panels 305

6.3.1 Collapse Modes I and VI 307

6.3.2 Collapse Modes II, III, IV, and V 307

6.4 Nonlinear Governing Differential Equations of Stiffened Panels 307

6.4.1 Large-Deflection Orthotropic Plate Theory 307

6.4.2 Large-Deflection Isotropic Plate Theory 311

6.5 Elastic Large-Deflection Behavior After Overall Grillage Buckling 311

6.5.3 Effect of the Bathtub Deflection Shape 314

6.5.4 Interaction Effect Between Biaxial Loads and Lateral Pressure 315

6.6 Ultimate Strength 315

6.6.1 Mode I: Overall Collapse 315

6.6.1.1 Calculation of σI xu 315

6.6.1.2 Calculation of σIy u 316

6.6.1.3 Calculation of τIu 317

6.6.2 Mode II: Plate Collapse Without Distinct Failure of Stiffener 317

6.6.2.1 Calculation of σII xu 317

6.6.2.2 Calculation of σII yu 318

6.6.2.3 Calculation of τII u 318

6.6.3 Mode III: Beam–Column Collapse 318

6.6.3.1 Calculation of σIII xu 318

6.6.3.2 Calculation of σIII yu 319

6.6.3.3 Calculation of τIII u 319

6.6.4 Mode IV: Collapse by Local Web Buckling of Stiffener 319

6.6.4.1 Calculation of σIV xu 320

6.6.4.2 Calculation of σIV yu 320

6.6.4.3 Calculation of τIV u 320

6.6.5 Mode V: Collapse by Lateral-Torsional Buckling of Stiffener 321

6.6.5.1 Calculation of σV xu 321

6.6.5.2 Calculation of σV yu 322

6.6.5.3 Calculation of τV u 322

6.6.6 Mode VI: Gross Yielding 322

6.6.7 Determination of the Real Ultimate Strength 322

6.7 Effects of Age-Related and Accident Induced Damages 323

6.8 Benchmark Studies 323

References 331

7 Buckling and Ultimate Strength of Plate Assemblies: Corrugated Panels, Plate Girders, Box Columns, and Box Girders 333

7.1 Introduction 333

7.2 Ultimate Strength of Corrugated Panels 334

7.2.1 Ultimate Strength Under Axial Compression 334

7.2.2 Ultimate Strength Under Shearing Force 334

7.2.3 Ultimate Strength Under Lateral Pressure 336

7.3 Ultimate Strength of Plate Girders 337

7.3.1 Ultimate Strength Under Shearing Force 338

7.3.1.1 Simple Post-Critical Buckling Method 339

7.3.1.2 Tension Field Method 339

7.3.2 Ultimate Strength Under Bending Moment 342

7.3.2.1 Mode I 343

7.3.2.2 Mode II 343

7.3.3 Ultimate Strength Under Combined Shearing Force and Bending Moment 345

7.3.4 Ultimate Strength Under Patch Load 346

7.3.5 Ultimate Strength Under Combined Patch Load, Shearing Force, and Bending Moment 347

7.4 Ultimate Strength of Box Columns 347

7.5 Ultimate Strength of Box Girders 349

7.5.1 Simple-Beam Theory Method 349

7.5.1.1 Maximum Bending Stress 351

7.5.1.2 Section Modulus 351

7.5.1.3 First-Yield Bending Moment 353

7.5.1.4 First-Collapse Bending Moment 353

7.5.1.5 Full Plastic Bending Moment 354

7.5.1.6 Exercise for Cross-Sectional Property Calculations 354

7.5.2 The Caldwell Method 358

7.5.3 The Original Paik–Mansour Method 359

7.5.4 The Modified Paik–Mansour Method 361

7.5.5 Interactive Relationship Between Vertical and Horizontal Bending 362

7.5.6 Interactive Relationship Between Combined Vertical or Horizontal Bending and Shearing Force 363

7.5.7 Interactive Relationship Between Combined Vertical Bending, Horizontal Bending, and Shearing Force 363

7.5.8 Effect of Torsional Moment 363

7.6 Effect of Age Related Structural Degradation 365

7.7 Effect of Accident-Induced Structural Damage 365

References 366

8 Ultimate Strength of Ship Hull Structures 369

8.1 Introduction 369

8.2 Characteristics of Ship’s Hull Structures 369

8.3 Lessons Learned from Accidents 377

8.4 Fundamentals of Vessel’s Hull Girder Collapse 380

8.5 Characteristics of Ship Structural Loads 387

8.6 Calculations of Ship’s Hull Girder Loads 388

8.6.2 Long-Term Still-Water and Wave-Induced Loads: IACS Unified Formulas 390

8.6.3 Long-Term Wave-Induced Loads: Direct Calculations 390

8.6.4 Short-Term Wave-Induced Loads: Simplified Direct Calculations Using Parametric Seakeeping Tables 391

8.7 Minimum Section Modulus Requirement 392

8.8 Determination of Ultimate Hull Girder Strength 394

8.9 Safety Assessment of Ships 396

8.10 Effect of Lateral Pressure Loads 398

8.11 Ultimate Strength Interactive Relationships Between Combined Hull Girder Loads 403

8.11.1 Combined Vertical and Horizontal Bending 403

8.11.2 Combined Vertical Bending and Shearing Force 405

8.11.3 Combined Horizontal Bending and Shearing Force 405

8.11.4 Combined Vertical Bending, Horizontal Bending, and Shearing Force 406

8.11.5 Effect of Torsional Moment 407

8.12 Shakedown Limit State Associated with Hull Girder Collapse 408

8.13 Effect of Age Related Structural Degradation 410

8.14 Effect of Accident-Induced Structural Damage 413

References 417

9 Structural Fracture Mechanics 421

9.1 Fundamentals of Structural Fracture Mechanics 421

9.2 Basic Concepts for Structural Fracture Mechanics Analysis 424

9.2.1 Energy-Based Concept 424

9.2.2 Stress Intensity Factor Concept 425

9.3 More on LEFM and the Modes of Crack Extension 427

9.3.1 Useful K Solutions 430

9.3.2 Fracture Toughness Testing 431

9.4 Elastic–Plastic Fracture Mechanics 432

9.4.1 Crack Tip Opening Displacement 433

9.4.1.1 The Irwin Approach 433

9.4.1.2 The Dugdale Approach 435

9.4.1.3 CTOD Design Curve 435

9.4.2 Other EPFM Measures: J-Integral and Crack Growth Resistance Curve 437

9.4.2.1 The J-Integral 437

9.4.2.2 The Crack Growth Resistance Curve 439

9.5 Fatigue Crack Growth Rate and Its Relationship to the Stress Intensity Factor 441

9.6 Buckling Strength of Cracked Plate Panels 443

9.6.1 Fundamentals 443

9.6.2 A Plate with Edge Crack in Uniaxial Compression 446

9.6.3 A Plate with Central Crack in Uniaxial Compression 446

9.6.4 A Plate with Edge or Central Crack in Edge Shear 449

9.6.5 A Plate with Vertical Edge Crack in Biaxial Compression 450

9.7 Ultimate Strength of Cracked Plate Panels 450

9.7.1 Fundamentals 450

9.7.2 A Cracked Plate in Axial Tension 454

9.7.3 A Cracked Stiffened Panel in Axial Tension 456

9.7.4 A Cracked Plate in Axial Compression 458

9.7.5 A Cracked Plate in Edge Shear 464

References 467

10 Structural Impact Mechanics 471

10.1 Fundamentals of Structural Impact Mechanics 471

10.2 Load Effects Due to Impact 473

10.3.1 The Malvern Constitutive Equation 476

10.3.2 Dynamic Yield Strength: The Cowper–Symonds Equation 478

10.3.3 Dynamic Fracture Strain 480

10.3.4 Strain-Hardening Effects 481

10.3.5 Inertial Effects 482

10.3.6 Friction Effects 484

10.4 Ultimate Strength of Beams Under Impact Lateral Loads 485

10.5 Ultimate Strength of Columns Under Impact Axial Compressive Loads 487

10.5.1 Oscillatory Response 487

10.5.2 Dynamic Buckling Response 488

10.6 Ultimate Strength of Plates Under Impact Lateral Pressure Loads 489

10.6.1 Analytical Formulations: Small-Deflection Theory 489

10.6.2 Analytical Formulations: Large-Deflection Theory 492

10.6.3 Empirical Formulations 493

10.7 Ultimate Strength of Stiffened Panels Under Impact Lateral Loads 494

10.8 Crushing Strength of Plate Assemblies 494

10.8.1 Fundamentals of Crushing Behavior 494

10.8.2 A Plate 497

10.8.3 A Stiffened Panel 498

10.8.4 An Inclined Plate 499

10.8.5 L-, T-, and X-Shaped Plate Assemblies 500

10.9 Tearing Strength of Plates and Stiffened Panels 502

10.9.1 Fundamentals of Tearing Behavior 502

10.9.2 Analytical Formulations 503

10.9.3 Empirical Formulations 505

10.9.4 Concertina Tearing 507

10.10 Impact Perforation of Plates 508

10.11 Impact Fracture of Plates and Stiffened Panels at Cold Temperature 510

10.12 Ultimate Strength of Plates Under Impact Axial Compressive Loads 511

10.13 Ultimate Strength of Dented Plates 513

10.13.1 A Dented Plate in Axial Compression 515

10.13.2 A Dented Plate in Edge Shear 518

References 533

11 The Incremental Galerkin Method 539

11.1 Features of the Incremental Galerkin Method 539

11.2 Structural Idealizations of Plates and Stiffened Panels 539

11.3 Analysis of the Elastic–Plastic Large-Deflection Behavior of Plates 542

11.3.2 The Incremental Approach 544

11.3.3 Application to the Plates Simply Supported at Four Edges 547

11.3.4 Treatment of Plasticity 551

11.4 Analysis of the Elastic–Plastic Large-Deflection Behavior of Stiffened Panels 552

11.4.2 The Incremental Approach 554

11.4.3 Application to the Stiffened Panels Simply Supported at Four Edges 557

External Load Increment Vector for the Plate Part 562

Stiffness Matrix Associated with Initial Stress for the Plate Part 563

Bending Stiffness Matrix for the Plate Part 563

Stiffness Matrix Due to Membrane Action for the Plate Part 564

Unknown Coefficient Vector 565

External Load Increment Vector for Stiffeners 565

Stiffness Matrix Associated with Initial Stress for Stiffeners 565

Bending Stiffness Matrix for Stiffeners 566

Stiffness Matrix Due to Membrane Action for Stiffeners 566

11.4.4 Treatment of Plasticity 572

11.5 Applied Examples 572

11.5.1 A Rectangular Plate Under Longitudinal Axial Compression 573

11.5.2 A Rectangular Plate Under Transverse Axial Compression 574

11.5.3 A Rectangular Plate Under Edge Shear 575

11.5.4 A Rectangular Plate Under In-Plane Bending 576

11.5.5 A Rectangular Plate Under Lateral Pressure Loads 576

11.5.6 A Rectangular Plate Under Combined Transverse Axial Compression and Edge Shear 578

11.5.7 A Rectangular Plate Under Other Types of Combined Load Applications 579

11.5.8 A Stiffened Panel with Flat-Bar Stiffeners Under Uniaxial Compression 579

11.5.9 A Stiffened Panel with Three Stiffeners Under Combined Axial Compression and Lateral Pressure Loads 582

11.5.10 A Very Large Crude Oil Carrier’s Deck Structure Under Combined Axial Compression and Lateral Pressure 583

References 586

12 The Nonlinear Finite Element Method 587

12.1 Introduction 587

12.2 Extent of the Analysis 587

12.3 Types of Finite Elements 588

12.4 Mesh Size of Finite Elements 588

12.5 Material Modeling 593

12.6 Boundary Condition Modeling 596

12.7 Initial Imperfection Modeling 597

12.8 Order of Load Component Application 598

References 601

13 The Intelligent Supersize Finite Element Method 603

13.1 Features of the Intelligent Supersize Finite Element Method 603

13.2 Nodal Forces and Nodal Displacements of the Rectangular Plate Element 604

13.3 Strain versus Displacement Relationship 605

13.4 Stress versus Strain Relationship 607

13.5 Tangent Stiffness Equation 608

13.5.1 The Total Lagrangian Approach 608

13.5.2 The Updated Lagrangian Approach 610

13.6 Stiffness Matrix for the Displacement Component, θz 611

13.7 Displacement (Shape) Functions 611

13.8 Local to Global Transformation Matrix 612

13.9 Modeling of Flat Bar Stiffener Web and One-Sided Stiffener Flange 612

13.10 Applied Examples 613

13.10.1 A Rectangular Plate 613

13.10.2 A Box Column 613

13.10.2.1 A Short Box Column with L = 500mm 616

13.10.2.2 A Medium Box Column with L = 8000mm 617

13.10.2.3 A Long Box Column with L = 21 000mm 618

13.10.2.4 Global Buckling of a Box Column 620

13.10.3 A Ship’s Hull Girder: The Dow Test Model 622

13.10.4 A Corroded Steel-Bridge Structure 629

References 632

Appendices 635

A.1 Source Listing of the FORTRAN Computer Program CARDANO 635

A.2 SI Units 636

A.2.1 Conversion Factors 636

A.2.2 SI Unit Prefixes 638

A.3 Density and Viscosity of Water and Air 638

A.4 Scaling Laws for Physical Model Testing 638

A.4.1 Structural Mechanics Model Tests 638

A.4.2 Hydrodynamics Model Tests 639

A.4.2.1 Froude’s Scaling Law 639

A.4.2.2 Reynolds Scaling Law 640

A.4.2.3 Vortex-Shedding Effects 640

A.4.2.4 Surface-Tension Effects 641

A.4.2.5 Compressibility Effects 641

Index 643