Analysis and Design of Elastic Beams: Computational Methods

ANALYSIS AND DESIGN OF ELASTIC BEAMS......Page 3
CONTENTS......Page 9
PREFACE......Page 15
1.1.1 Kinematical Strain–Displacement Equations......Page 19
1.1.2 Material Law......Page 22
1.1.3 Equations of Equilibrium......Page 25
1.1.4 Surface Forces and Boundary Conditions......Page 26
1.1.5 Other Forms of the Governing Differential Equations......Page 29
1.2 Bending Stresses in a Beam in Pure Bending......Page 30
1.3 Principal Bending Axes......Page 42
1.4 Axial Loads......Page 49
1.5 Elasticity Solution for Pure Bending......Page 50
References......Page 56
2 BEAM ELEMENTS......Page 58
2.1.1 Geometry of Deformation......Page 59
2.1.2 Force–Deformation Relations......Page 61
2.1.3 Equations of Equilibrium......Page 62
2.1.4 Boundary Conditions......Page 64
2.1.5 Displacement Form of the Governing Differential Equations......Page 65
2.1.6 Mixed Form of the Governing Differential Equations......Page 77
2.1.7 Principle of Virtual Work: Integral Form of the Governing Equations......Page 79
2.2.1 First-Order Form of the Governing Equations......Page 83
2.2.2 Sign Conventions for Beams......Page 90
2.2.3 Definition of Stiffness Matrices......Page 94
2.2.4 Determination of Stiffness Matrices......Page 95
2.2.5 Development of an Element by Mapping from a Reference Element......Page 116
2.3 Mass Matrices for Dynamic Problems......Page 120
2.3.1 Consistent Mass Matrices......Page 121
2.3.2 Lumped Mass Matrices......Page 123
2.3.3 Exact Mass and Dynamic Stiffness Matrices......Page 124
2.4 Geometric Stiffness Matrices for Beams with Axial Loading......Page 127
References......Page 128
3 BEAM SYSTEMS......Page 130
3.1.2 Transformation of Forces and Displacements......Page 131
3.2 Displacement Method of Analysis......Page 135
3.2.1 Direct Stiffness Method......Page 136
3.2.2 Characteristics of the Displacement Method......Page 153
3.3 Transfer Matrix Method of Analysis......Page 159
3.4.1 Free Vibration Analysis......Page 162
3.4.2 Forced Response......Page 164
3.5 Stability Analysis......Page 168
3.6 Analyses Using Exact Stiffness Matrices......Page 169
References......Page 170
4.1 Shape Functions......Page 171
4.2 Transformation of Derivatives and Integrals......Page 175
4.3 Integrals......Page 176
4.4 Cross-Sectional Properties......Page 179
References......Page 184
5.1 Fundamentals of Saint-Venant Torsion......Page 185
5.1.1 Force Formulation......Page 196
5.1.2 Membrane Analogy......Page 203
5.2 Classical Formulas for Thin-Walled Cross Sections......Page 204
5.2.1 Open Sections......Page 205
5.2.2 Closed Sections, Hollow Shafts......Page 208
5.3 Composite Cross Sections......Page 217
5.4.1 Principle of Virtual Work......Page 220
5.4.2 Weighted Residual Methods......Page 224
5.4.3 Isoparametric Elements......Page 226
5.5 Assembly of System Matrices......Page 228
5.6 Calculation of the Torsional Constant and Stresses......Page 233
5.7 Alternative Computational Methods......Page 240
5.7.1 Boundary Integral Equations......Page 241
5.7.2 Boundary Element Method......Page 244
References......Page 246
6.1.1 Approximate Shear Stress Formulas Based on Engineering Beam Theory......Page 248
6.1.2 Theory of Elasticity Solution......Page 253
6.1.3 Composite Cross Section......Page 259
6.1.4 Finite Element Solution Formulation......Page 261
6.2.1 y Coordinate of the Shear Center......Page 266
6.2.2 Axis of Symmetry......Page 267
6.2.3 Location of Shear Centers for Common Cross Sections......Page 269
6.2.5 Finite Element Solution Formulation......Page 270
6.2.6 Trefftz’s Definition of the Shear Center......Page 272
6.3 Shear Deformation Coefficients......Page 275
6.3.1 Derivation......Page 277
6.3.2 Principal Shear Axes......Page 278
6.3.3 Finite Element Solution Formulation......Page 279
6.3.4 Traditional Analytical Formulas......Page 287
6.4.1 Governing Equations......Page 290
6.4.2 Transfer Matrix......Page 293
6.4.3 Stiffness Matrix......Page 294
6.4.4 Exact Geometric Stiffness Matrix for Beams with Axial Loading......Page 299
6.4.5 Shape Function–Based Geometric Stiffness and Mass Matrices......Page 309
6.4.6 Loading Vectors......Page 327
References......Page 328
7.1 Restrained Warping......Page 330
7.2 Thin-Walled Beams......Page 335
7.2.1 Saint-Venant Torsion......Page 337
7.2.2 Restrained Warping......Page 340
7.3.1 Governing Equations......Page 343
7.3.2 Boundary Conditions......Page 344
7.3.3 Response Expressions......Page 345
7.3.4 First-Order Governing Equations and General Solution......Page 347
7.4 Warping Constant......Page 350
7.5 Normal Stress due to Restrained Warping......Page 351
7.6 Shear Stress in Open-Section Beams due to Restrained Warping......Page 352
7.7 Beams Formed of Multiple Parallel Members Attached at the Boundaries......Page 373
7.7.1 Calculation of Open-Section Properties......Page 378
7.7.2 Warping and Torsional Constants of an Open Section......Page 381
7.7.3 Calculation of the Effective Torsional Constant......Page 383
7.8 More Precise Theories......Page 384
References......Page 386
8.1.1 State of Stress......Page 387
8.1.2 Principal Stresses......Page 388
8.1.3 Invariants of the Stress Matrix......Page 390
8.1.4 Extreme Values of Shear Stress......Page 391
8.1.5 Beam Stresses......Page 393
8.2 Yielding and Failure Criteria......Page 397
8.2.3 Von Mises Criterion......Page 398
References......Page 400
9.1 Concept of a NURBS Curve......Page 401
9.2 Definition of B-Spline Basis Functions......Page 403
9.3 B-Spline and Rational B-Spline Curves......Page 409
9.4 Use of Rational B-Spline Curves in Thin-Walled Beam Analysis......Page 414
References......Page 416
10.1 Design Velocity Field......Page 417
10.2 Design Sensitivity Analysis......Page 421
10.2.1 Derivatives of Geometric Quantities......Page 423
10.2.3 Derivatives of the Torsional Constant and the Shear Stresses......Page 424
10.3 Design Sensitivity of the Shear Deformation Coefficients......Page 428
10.4 Design Sensitivity Analysis for Warping Properties......Page 435
10.5 Design Sensitivity Analysis for Effective Torsional Constant......Page 437
10.6 Optimization......Page 438
Reference......Page 439
A.1 Overview of the Programs......Page 440
A.2 Input Data File for Cross-Section Analysis......Page 441
A.3 Output Files......Page 449
B.1 Closed Elliptical Tube......Page 452
B.2 Symmetric Channel Section......Page 455
B.3 L Section without Symmetry......Page 459
B.4 Open Circular Cross Section......Page 462
B.5 Welded Hat Section......Page 463
B.6 Open Curved Section......Page 467
B.7 Circular Arc......Page 469
References......Page 472
INDEX......Page 473