Advanced Methods of Structural Analysis (eBook)

Preface 6
Acknowledgments 10
Contents 12
Introduction 19
Part I Statically Determinate Structures 23
1 Kinematical Analysis of Structures 24
1.1 Classification of Structures by KinematicalViewpoint 24
1.2 Generation of Geometrically UnchangeableStructures 26
1.3 Analytical Criteria of the InstantaneouslyChangeable Structures 28
1.4 Degrees of Freedom 32
Problems 34
2 General Theory of Influence Lines 36
2.1 Analytical Method for Construction of InfluenceLines 36
2.1.1 Influence Lines for Reactions 37
2.1.1.1 Simply Supported Beam (Fig.2.1) 37
2.1.1.2 Simply Supported Beam with Overhang (Fig.2.2) 39
2.1.1.3 Cantilevered Beam (Fig.2.3) 40
2.1.2 Influence Lines for Internal Forces 41
2.1.2.1 Bending Moment Mk 41
2.1.2.2 Influence Line Qk 43
2.1.2.3 Discussion 45
2.2 Application of Influence Lines for Fixed andMoving Loads 48
2.2.1 Fixed Loads 48
2.2.2 Moving Loads 51
2.3 Indirect Load Application 54
2.4 Combining of Fixed and Moving Load Approaches 56
2.5 Properties of Influence Lines 57
Problems 58
3 Multispan Beams and Trusses 60
3.1 Multispan Statically Determinate Beams 60
3.1.1 Generation of Multispan Statically Determinate Hinged Beams 60
3.1.2 Interaction Schemes and Load Path 61
3.1.3 Influence Lines for Multispan HingedBeams 63
3.1.4 Summary 66
3.2 The Generation of Statically Determinate Trusses 68
3.2.1 Simple Trusses 68
3.2.2 Compound Trusses 69
3.2.3 Complex Trusses 70
3.3 Simple Trusses 70
3.4 Trusses with Subdivided Panels 75
3.4.1 Main and Auxiliary Trusses and Load Path 76
3.4.2 Baltimore and Subdivided Warren Trusses 78
3.5 Special Types of Trusses 82
3.5.1 Three-Hinged Trusses 82
3.5.2 Trusses with a Hinged Chain 85
3.5.3 Complex Trusses 89
3.5.4 Summary 91
Problems 93
4 Three-Hinged Arches 98
4.1 Preliminary Remarks 98
4.1.1 Design Diagram of Three-Hinged Arch 98
4.1.2 Peculiarities of the Arches 99
4.1.3 Geometric Parameters of Circularand Parabolic Arches 100
4.2 Internal Forces 101
4.3 Influence Lines for Reactions and Internal Forces 107
4.3.1 Influence Lines for Reactions 109
4.3.2 Influence Lines for Internal Forces 109
4.3.2.1 Bending Moment 109
4.3.2.2 Shear Force 110
4.3.2.3 Axial Force 112
4.3.2.4 Properties of the Influence Lines for Internal Forces 113
4.3.3 Application of Influence Lines 113
4.4 Nil Point Method for Construction of InfluenceLines 115
4.4.1 Bending Moment 115
4.4.2 Shear Force 116
4.4.3 Axial Force 117
4.5 Special Types of Arches 118
4.5.1 Askew Arch 118
4.5.1.1 Reactions and Bending Moment at Section k 119
4.5.1.2 Influence Lines for Thrust and Bending Moment Mk. 120
4.5.2 Parabolic Arch with Complex Tie 121
4.5.2.1 Reactions and Bending Moment at Section k 122
4.5.2.2 Influence Lines for Thrust and Bending Moment at the Section k 123
Problems 124
5 Cables 129
5.1 Preliminary Remarks 129
5.1.1 Direct and Inverse Problems 130
5.1.2 Fundamental Relationships 131
5.2 Cable with Neglected Self-Weight 133
5.2.1 Cables Subjected to Concentrated Load 133
5.2.1.1 Direct Problem 134
5.2.1.2 Inverse Problem 135
5.2.2 Cable Subjected to Uniformly DistributedLoad 136
5.2.2.1 Direct Problem 136
5.2.2.2 Inverse Problem 137
5.3 Effect of Arbitrary Load on the Thrust and Sag 142
5.4 Cable with Self-Weight 145
5.4.1 Fundamental Relationships 145
5.4.2 Cable with Supports Located at the SameLevel 147
5.4.3 Cable with Supports Locatedon the Different Elevations 150
5.4.3.1 Saddle Point Within the Span 150
5.4.3.2 Saddle Point Outside of the Span 152
5.5 Comparison of Parabolic and Catenary Cables 155
5.6 Effect of Axial Stiffness 157
5.6.1 Elastic Cable with Concentrated Load 157
5.6.2 Elastic Cable with Uniformly DistributedLoad 159
Problems 160
6 Deflections of Elastic Structures 164
6.1 Introduction 164
6.2 Initial Parameters Method 166
6.3 Maxwell–Mohr Method 178
6.3.1 Deflections Due to Fixed Loads 178
6.3.2 Deflections Due to Change of Temperature 184
6.3.3 Summary 189
6.4 Displacement Due to Settlement of Supports and Errors of Fabrication 189
6.5 Graph Multiplication Method 195
6.6 Elastic Loads Method 204
6.7 Reciprocal Theorems 208
6.7.1 Theorem of Reciprocal Works(Betti Theorem) 208
6.7.2 Theorem of Reciprocal Unit Displacements (Maxwell Theorem) 209
6.7.3 Theorem of Reciprocal Unit Reactions (Rayleigh First Theorem) 211
6.7.4 Theorem of Reciprocal Unit Displacements and Reactions (Rayleigh Second Theorem) 212
6.7.5 Summary 212
Problems 214
Part II Statically Indeterminate Structures 228
7 The Force Method 229
7.1 Fundamental Idea of the Force Method 229
7.1.1 Degree of Redundancy, Primary Unknowns and Primary System 229
7.1.2 Compatibility Equation in Simplest Case 232
7.2 Canonical Equations of Force Method 235
7.2.1 The Concept of Unit Displacements 235
7.2.2 Calculation of Coefficients and Free Termsof Canonical Equations 237
7.3 Analysis of Statically Indeterminate Structures 240
7.3.1 Continuous Beams 240
7.3.2 Analysis of Statically IndeterminateFrames 242
7.3.3 Analysis of Statically IndeterminateTrusses 251
7.3.4 Analysis of Statically Indeterminate Arches 255
7.4 Computation of Deflections of Redundant Structures 261
7.5 Settlements of Supports 264
7.6 Temperature Changes 269
Problems 277
8 The Displacement Method 288
8.1 Fundamental Idea of the Displacement Method 288
8.1.1 Kinematical Indeterminacy 289
8.1.2 Primary System and Primary Unknowns 291
8.1.3 Compatibility Equation. Concept of UnitReaction 292
8.2 Canonical Equations of Displacement Method 293
8.2.1 Compatibility Equations in General Case 293
8.2.2 Calculation of Unit Reactions 294
8.2.3 Properties of Unit Reactions 296
8.2.4 Procedure for Analysis 297
8.3 Comparison of the Force and Displacement Methods 308
8.3.1 Properties of Canonical Equations 309
8.4 Sidesway Frames with Absolutely Rigid Crossbars 311
8.5 Special Types of Exposures 313
8.5.1 Settlements of Supports 313
8.5.2 Errors of Fabrication 317
8.6 Analysis of Symmetrical Structures 319
8.6.1 Symmetrical and Antisymmetrical Loading 319
8.6.2 Concept of Half-Structure 320
Problems 322
9 Mixed Method 330
9.1 Fundamental Idea of the Mixed Method 330
9.1.1 Mixed Indeterminacy and PrimaryUnknowns 330
9.1.2 Primary System 331
9.2 Canonical Equations of the Mixed Method 333
9.2.1 The Matter of Unit Coefficientsand Canonical Equations 333
9.2.2 Calculation of Coefficients and Free Terms 334
9.2.3 Computation of Internal Forces 335
Problems 336
10 Influence Lines Method 339
10.1 Construction of Influence Lines by the ForceMethod 339
10.1.1 Continuous Beams 341
10.1.2 Hingeless Nonuniform Arches 347
10.1.2.1 Unit Coefficients 350
10.1.2.2 Free Terms 350
10.1.2.3 Reactions of Support A 351
10.1.2.4 Bending Moment at Crown C 352
10.1.3 Statically Indeterminate Trusses 355
10.1.3.1 Influence Line for Primary Unknown X1 358
10.2 Construction of Influence Linesby the Displacement Method 360
10.2.1 Continuous Beams 362
10.2.1.1 Influence Line for Primary Unknown Z1 362
10.2.1.2 Influence Line for Bending Moment Mk 364
10.2.1.3 Influence Line for Shear Force Qk 365
10.2.2 Redundant Frames 369
10.3 Comparison of the Force and DisplacementsMethods 371
10.4 Kinematical Method for Construction of InfluenceLines 374
Problems 380
11 Matrix Stiffness Method 385
11.1 Basic Idea and Concepts 385
11.1.1 Finite Elements 386
11.1.2 Global and Local Coordinate Systems 386
11.1.3 Displacements of Joints and Degreesof Freedom 387
11.2 Ancillary Diagrams 388
11.2.1 Joint-Load (J-L) Diagram 388
11.2.2 Displacement-Load (Z-P) Diagram 392
11.2.3 Internal Forces-Deformation (S-e)Diagram 393
11.3 Initial Matrices 395
11.3.1 Vector of External Joint Loads 395
11.3.2 Vector of Internal Unknown Forces 396
11.4 Resolving Equations 397
11.4.1 Static Equations and Static Matrix 397
11.4.2 Geometrical Equations and DeformationMatrix 402
11.4.3 Physical Equations and Stiffness Matrix in Local Coordinates 403
11.5 Set of Formulas and Procedure for Analysis 406
11.5.1 Stiffness Matrix in Global Coordinates 406
11.5.2 Unknown Displacements and InternalForces 407
11.5.3 Matrix Procedures 408
11.6 Analysis of Continuous Beams 409
11.7 Analysis of Redundant Frames 420
11.8 Analysis of Statically Indeterminate Trusses 426
11.9 Summary 430
Problems 431
Part III Special Topics 437
12 Plastic Behavior of Structures 438
12.1 Idealized Stress–Strain Diagrams 438
12.2 Direct Method of Plastic Analysis 442
12.3 Fundamental Methods of Plastic Analysis 445
12.3.1 Kinematical Method 445
12.3.2 Static Method 445
12.4 Limit Plastic Analysis of Continuous Beams 447
12.4.1 Static Method 448
12.4.2 Kinematical Method 450
12.5 Limit Plastic Analysis of Frames 456
12.5.1 Beam Failure 457
12.5.2 Sidesway Failure 459
12.5.3 Combined Failure 459
12.5.4 Limit Combination Diagram 459
Problems 460
13 Stability of Elastic Systems 464
13.1 Fundamental Concepts 464
13.2 Stability of Structures with Finite Number Degreesof Freedom 468
13.2.1 Structures with One Degree of Freedom 468
13.2.2 Structures with Two or More Degreesof Freedom 473
13.3 Stability of Columns with Rigid and ElasticSupports 476
13.3.1 The Double Integration Method 476
13.3.1.1 Uniform Clamped-Free Column 476
13.3.1.2 Uniform Columns with Elastic Supports 478
13.3.2 Initial Parameters Method 481
13.4 Stability of Continuous Beams and Frames 486
13.4.1 Unit Reactions of the Beam-Columns 486
13.4.2 Displacement Method 488
13.4.3 Modified Approach of the DisplacementMethod 496
13.5 Stability of Arches 498
13.5.1 Circular Arches Under Hydrostatic Load 499
13.5.2 Complex Arched Structure:Arch with Elastic Supports 505
13.6 Compressed Rods with Lateral Loading 506
13.6.1 Double Integration Method 507
13.6.2 Initial Parameters Method 509
13.6.3 P-Delta Analysis of the Frames 514
13.6.4 Graph Multiplication Methodfor Beam-Columns 517
13.6.4.1 Simply Supported Beam-Column 518
13.6.4.2 Fixed-Free Beam-Column 518
Problems 519
14 Dynamics of Elastic Systems 527
14.1 Fundamental Concepts 527
14.1.1 Kinematics of Vibrating Processes 527
14.1.2 Forces Which Arise at Vibrations 527
14.1.3 Degrees of Freedom 529
14.1.4 Purpose of Structural Dynamics 533
14.1.5 Assumptions 533
14.2 Free Vibrations of Systems with Finite Number Degrees of Freedom: Force Method 534
14.2.1 Differential Equations of Free Vibration in Displacements 534
14.2.2 Frequency Equation 535
14.2.3 Mode Shapes Vibration and Modal Matrix 536
14.3 Free Vibrations of Systems with Finite Number Degrees of Freedom: Displacement Method 544
14.3.1 Differential Equations of Free Vibrationin Reactions 544
14.3.2 Frequency Equation 546
14.3.3 Mode Shape Vibrations and Modal Matrix 546
14.3.4 Comparison of the Force and DisplacementMethods 552
14.4 Free Vibrations of One-Span Beams with Uniformly Distributed Mass 552
14.4.1 Differential Equation of Transversal Vibration of the Beam 554
14.4.2 Fourier Method 555
14.4.3 Krylov–Duncan Method 557
Problems 560
Bibliography 600
Index 602