Mechanics of Structural Elements (eBook)

PREFACE 6
CONTENTS 20
BASIC VARIATIONAL PRINCIPLES OF STATICS AND GEOMETRY IN STRUCTURAL MECHANICS 29
Preliminaries 29
Basic integral identity 33
Various types of stress and strain fields 39
The general principle of statics and geometry 41
Final comments to Chapter 1 53
References 55
BASIC VARIATIONAL PRINCIPLES OF STRUCTURAL MECHANICS 57
Energy space 57
Lagrange variational principle principle 78
Castigliano variational principle 82
Sensitivity of the strain energy to modifications of a system 87
Generalized forces and generalized displacements 106
Basic variational principles in problems with initial strains 117
Statically determinate and statically indeterminate systems 121
Final comments to Chapter 2 123
References 124
ADDITIONAL VARIATIONAL PRINCIPLES OF STRUCTURAL MECHANICS 126
Reissner mixed variational principle 126
Principle of stationarity of the boundary conditions functional 134
A variational principle for physical relationships 136
Hu–Washizu mixed variational principle 137
A generalized mixed variational principle 139
Gurtin’s variational principle 147
Geometric interpretation of functionals used in structural mechanic 152
Final comments to Chapter 3 158
References 159
PARTICULAR CLASSES OF PROBLEMS IN STRUCTURAL MECHANICS – part 1 161
Variations of the operator formulations in structural mechanics 161
Spatial elasticity 164
Plane elasticity 172
Lengthwise deformation of a straight bar 177
Bernoulli-type beam on elastic foundation 180
Timoshenko-type beam on elastic foundation 189
Planar curvilinear bar, shear ignored 197
Planar curvilinear bar, shear considered 230
Final comments to Chapter 4 245
References 246
PARTICULAR CLASSES OF PROBLEMS IN STRUCTURAL MECHANICS – part 2 247
Thin plate bending — Kirchhoff–Love theory 247
Static-geometric analogy in the theory of plates 278
Bending of medium-thickness plates – Reissner’s theory 300
Some examples 319
Final comments to Chapter 5 332
References 336
PARTICULAR CLASSES OF PROBLEMS IN STRUCTURAL MECHANICS – part 3 338
Torsion of solid bars – Saint-Venant’s theory 338
Thin-walled open-profile bars – a theory by Vlasov 362
Allowing for shearing in open-profile thin-walled bars 396
A semi-shear theory of open-profile thin-walled bars 411
References 417
PARTICULAR CLASSES OF PROBLEMS IN STRUCTURAL MECHANICS – part 4 419
Closed-profile thin-walled bars – a theory by Umanski 419
Multiple-contour, closed-profile, thin-walled bars 456
References 480
PARTICULAR CLASSES OF PROBLEMS IN STRUCTURAL MECHANICS – part 5 482
Compound-profile thin-walled bars 482
A multiple-contour compound profile 499
Final comments to thin-walled bar theories theories theories theories theories theories theories 507
References 517
THE RITZ METHOD AND ITS MODIFICATIONS 519
The basic theorem of the Ritz method 519
The Ritz method in application to mixed functionals 529
Method of two functionals 538
References 559
VARIATIONAL PRINCIPLES IN SPECTRAL PROBLEMS 561
Basic concepts. Terminology 561
The spectral problem as a variational problem 564
A general spectral problem 591
The Ritz method in the spectral problem 595
Final comments to Chapter 10 10 10 10 10 623
References 625
VARIATIONAL PRINCIPLES IN STABILITY/ BUCKLING ANALYSIS 627
Stability of systems with a finite number of degrees of freedom 628
Variational description of critical loads 656
Geometrically nonlinear problems in elasticity 677
Stability of equilibrium of an elastic body body body body 685
Stability of equilibrium in particular classes of problems 697
Mixed functionals in the stability analysis 716
Final comments to Chapter 11 718
References 719
CONCLUSION 722
APPENDIX 724
AUTHOR INDEX 800
SUBJECT INDEX1 804