THE ELEMENTS OF APPLIED MATHEMATICS: KINETICS, STATICS & HYDROSTATICS
1 BASIC VARIATIONAL PRINCIPLES OF STATIC AND GEOMETRY IN STRUCTURAL MECHANICS 1 11 Preliminaries 1 111 Formally conjugate differential operators 2 12 Basic integral identity 5 13 Various types of stress and strain fields 11 14 The general principle of statics and geometry 13 141 The principle of virtual displacements as an implication of the general principle of statics and geometry 18 142 The principle of virtual stress increments as an implication of the general principle of statics and geometry 20 143 Theorem of field orthogonality 22 144 Integral identity by Papkovich 24 15 Final comments to Chapter 1 25 References 27 2 BASIC VARIATIONAL PRINCIPLES OF STRUCTURAL MECHANICS 29 21 Energy space 29 211 Physically admissible fields 29 212 Betty theorem 30 213 Energy of strain Clapeyron theorem 33 214 Rigid displacements 35 215 Strain compatibility conditions in the form of an integral identity 41 216 Necessary conditions for an equilibrium state of a system to exist 44 217 Theorem of a general form of an arbitrary physically admissible field 45 218 Lagrangian energy space 46 219 Prager-Synge identity 49 22 Lagrang variational principle 50 221 Conservative external forces 50 222 Lagrange functional 51 23 Castigliano variational principle 54 231 Castigliano functional 54 232 Castigliano energy space 56 24 Sensitivity of the strain energy to modifications of a system 59 241 First theorem of the strain energy minimum 59 242 Remarks on the effect of additional constraints (kinematical and force) 61 • Kinematical constraints 61 • Force constraints 63 243 Build-up of a system 68 244 Modification of stiffness properties of a system 70 245 Perturbation of external actions 72 246 Second theorem of the strain energy minimum 74 247 St-Venant principle and its energy-based background 75 25 Generalized forces and generalized displacements 78 251 Force actions Castigliano theorem 78 252 Kinematic actions Lagrange theorem 81 253 Inversion of stiffness and compliance matrices 83 254 Lemma of constraints 84 255 Mohr formula and its reciprocal 86 26 Basic variational principles in problems with initial strains 89 27 Statically determinate and statically inderterminate systems 93 28 Final comments to Chapter 2 95 References 96 3 ADDITIONAL VARIATIONAL PRINCIPLES OF STRUCTURAL MECHANICS 99 31 Reissner mixed variational principle 99 311 Reissner functional 100 312 Principle of minimum for stresses 104 313 Principle of maximum for displacements 106 32 Principle of stationarity of the boundary conditions functional 107 33 A variational principle for physical relationships 109 34 Hu-Washizu mixed variational principle 110 35 A generalized mixed variational principle 112 351 A generalized solution of a problem 113 352 A generalized mixed functional 114 353 A connection between the Lagrange & Reissner functionals and the generalized mixed functional 116 354 Parametrized energy space 118 36 Gurtin’s variational principle 120 361 Gurtin’s functional 121 37 Geometric interpretation of functionals used in structural mechanics 125 371 Generalized mixed functional 125 372 A remark on the Gurtin functional 130 38 Final comments to Chapter 3 131 References 132 4 PARTICULAR CLASSES OF PROBLEMS IN STRUCTURAL MECHANICS – part 1 135 41 Variations of the operator formulations in structural mechanics 135 411 Statement of a problem in displacements 135 412 Statement of a problem in stresses 136 42 Spatial elasticity 138 421 Lagrange functional 144 422 Reissner functional 145 423 Castigliano functional 146 424 Gurtin functional (third form) 146 43 Plane elasticity 146 431 Lagrange functional 149 432 Castigliano functional 150 433 Reissner functional 150 44 Lengthwise deformation of a straight bar 151 45 Bernoulli-type beam on elastic foundation 154 46 Timoshenko-type beam on elastic foundation 163 461 Another remark on kinematic boundary conditions for a Timoshenko beam 170 47 Planar curvilinear bar, shear ignored …………………………… 171 471 Geometric equation 174 472 Equations of equilibrium 181 473 Stresses conjugate to the Vlasov vector of strains 187 474 Variational principles for curvilinear bars 193 • Lagrange functional 195 • Castigliano functional 197 • Reissner functional 198 475 Remark on comparison of solutions in various formulations of curvilinear-bar-related problems 199 48 Planar curvilinear bar, shear considered ………………………… 204 481 Geometry of a curvilinear bar revisited 204 482 Allowing for shear deformations 207 • Bars of medium and big curvature 209 • Bars of small curvature 214 483 Estimation of changes in the strain energy introduced by switching to the refined theory 217 49 Final comments to Chapter 4 219 References 220 5 PARTICULAR CLASSES OF PROBLEMS IN STRUCTURAL MECHANICS – part 2 221 51 Thin plate bending – Kirchhoff-Love theory 221 511 Local basis in points of boundary ? of area ? 230 512 Matrix representation of basic relationship in the Kirchhoff-Love plate theory 233 513 Basic integral identity for thin plate bending ……………… 237 514 Boundary conditions for thin plate bending ……………… 241 515 Important functionals for thin plate bending ……………… 244 • Lagrange functional 244 • Castigliano functional 247 • Reissner functional 249 • Herrmann functional 250 52 Static-geometric analogy in the theory of plates 252 521 A stress function vector in the theory of plates 252 • Physical meaning of the stress function in plane stress 255 • Physical meaning of the stress function in plate bending 257 522 A static-geometric analogy in the theory of plates 261 523 Boundary conditions for deformations in the theory of plates 266 • Boundary conditions for deformations in plane stress 267 • Boundary conditions for deformations in plate bending 272 53 Nending of medium-thickness plates – Reissner’s theory 274 531 A governing system of equations for a Reissner plate with respect to two unknown functions 283 532 Basic integral identity in the Reissner plate theory 286 533 Important functionals for the reissner plate 287 • Lagrange functional 287 • Castigliano functional 288 • A stress function vector for the Reissner plate 290 • A Reissner functional for a Reissner plate 292 54 Some examples 293 541 Example 1 A round plate loaded by a torque on its edge 293 542 Example 2 A square plate loaded by a torques on its edge 297 543 Example 3 A simply supported rectangular plate 299 • Reissner plate 299 • Kirchhoff plate 303 55 Final comments to Chapter 5 306 References 310 6 PARTICULAR CLASSES OF PROBLEMS IN STRUCTURAL MECHANICS – part 3 313 61 Torsion of solid bars – Saint-Venant’s theory 313 611 Saint-Venant torsion function Lagrange functional 316 612 Prandtl stress function Timoshenko functional 319 • Bredt theorem of the tangential stress circulation 324 • Center of twist 326 613 Two-sided estimates of a section’s torsion inertia moment 327 614 A remark on Reissner-type functionals for the bar torsion analysis 331 615 A membrane analogy by Prandtl Torsion of a narrow strip 333 62 Thin-walled open-profile bars – a theory by Vlasov …………… 337 621 Basic assumptions in the theory of thin-walled open-profile bars 340 622 Equations of equilibrium of an open-profile thin-walled bar 351 623 Center of twist, center of bending 356 624 Tangential stresses in the open-profile thin-walled bar theory 358 625 A matrix form of basic relationship in the theory of open-profile thin-walled bars 361 626 Basic integral identity in the theory of open-profile thin-walled bars 363 627 Basic variational principles in the theory of open-profile thin-walled bars 364 628 A remark on non-warped cross-sections in the open-profile thin-walled bars 368 63 Allowing for shearing in open-profile thin-walled bars 371 631 Basic integral identity in the shear theory of open-profile thin-walled bars 379 632 Basic variational principles in the shear theory of open-profile thin-walled bars 380 633 A remark on the shear theory of open-profile thin-walled bars for non-warped cross-sections 381 634 Remark on a matrix of cross-section shape factor, ? 384 64 A semi-shear theory of open-profile thin-walled bars 386 References 392 7 PARTICULAR CLASSES OF PROBLEMS IN STRUCTURAL MECHANICS – part 4 395 71 Closed-profile thin-walled bars – a theory by Umanski 395 711 Pure torsion of a closed-profile thin-walled bar 396 712 A general behavior of a closed-profile thin-walled bar 402 • Tangential stresses 406 • Properties of functions Soz , Soy , So? 409 713 General physical relationships in the theory of closed-profile thin-walled bars 412 714 First (energy-based) version of the theory 415 • Physical relationships in the first (energy-based) version of the semi-shear theory 417 • Governing equations of the first version of the semi-shear theory 419 • A semi-shear theory by Janelidze-Panjvko 420 715 Second (physical) version of the theory 421 • A semi-shear theory by Umanski 427 716 A remark on non-warped cross-sections of closed-profile thin-walled bars 428 • Basic relationships of the theory of closed-profile thin-walled bars for non-warped cross-sections 430 72 Multiple-contour, closed-profile, thin-walled bars 432 721 Pure torsion of a multiple-contour profile 432 • A topologic structure of a multiple-contour profile 437 • A warped function for a multiple-contour profile 442 722 A general behavior of a multiple-contour profile 445 • Tangential stresses in a multiple-contour profile 445 • Governing equations for thin-walled bars having closed multiple-contour profile 455 References 456 8 PARTICULAR CLASSES OF PROBLEMS IN STRUCTURAL MECHANICS – part 5 459 81 Compound-profile thin-walled bars …………………………… 459 811 Pure torsion of a compound-profile thin-walled bars 461 812 A general behavior of a compound-profile thin-walled bar 466 • Average tangential stresses 466 • Properties of functions Soz , Soy , So? and consequences of these properties 469 • Physical relationships of the theory of compound-profile thin-walled bars 474 813 Non-warped compound profiles 475 82 Multiple-contour compound profile …………………………… 476 821 Topology of a multiple-contour compound profile 477 822 Pure torsion of a multiple-contour compound profile 478 • A warp function for a multiple-contour compound profile 479 823 A general behavior of a thin-walled multiple-contour compound profile 482 83 Final comments to thin-walled bar theories …………………… 484 831 A remark on the energy-based comparison between Vlasov’s shear-free theory and semi-shear theory of open-profile thin-walled bars 484 • An example 485 832 A remark on the Luzhin equations for compound-profile thin-walled bars 487 833 Classification criteria for separation between theories of thin-walled bars 491 834 Final remarks 493 References 494 9 THE RITZ METHOD AND ITS MODIFICATIONS 497 91 The basic theorem of the Ritz method 497 92 The Ritz method in application to mixed functionals 507 921 The Ritz method in application to the Reissner functional 507 922 The Ritz method in application to the generalized mixed functional ? 511 • An example 513 93 Method of two functionals … 516 931 Weak and strong solutions with respect to stresses 517 932 A remark on existence of a strong solution with respect to stresses 519 933 A remark on consistence approximations for minimization of functional D 521 934 A remark on the connection between functional D and the Reissner functional 523 • The method of two functionals and the method of conjugate approximations by Oden 524 935 Examples of application of the method of two functionals 526 • Example 1 526 • Example 2 529 • Example 3 533 References 537 10 VARIATIONAL PRINCIPLES IN SPECTRAL PROBLEMS 539 101 Basic concepts Termilogy 539 102 The spectral problem as a variational problem 542 1021 The spectrum of a mechanical system with finite number of degrees of freedom 542 • Metric ideas in the spectral eigenvalue problem 547 1022 Variational description of eigenvalues 551 • A remark on expansion of an arbitrary vector over a basis of eigevectors 553 • A recursive variational definition of eigenvalues and eigenvectors 554 • A remark on the effect of constraints on first eigenvalue 555 • A variational technique to find the eigenvalues independently from one another 558 • A general Routh theorem about the effect of constraints on the eigenvalues of a mechanical system 560 • Kinematic constraints which do not alter the number of dynamic degrees of freedom in a mechanical system 561 1023 A geometric description of eigenvalues and eigenvectors A Rayleigh ellipsoid 565 • A notion of a maximum-rigidity constraints 567 103 A general spectral problem …………………………… 569 1031 An expansion of an arbitrary function over the eigenfunctions 572 104 The Ritz method in the spectral problem 573 1041 The Ritz method in the spectral problem, applied to the Reissner functional 577 1042 The method of two functionals in the spectral problem 582 • Example 1 584 • Example 2 587 1043 A remark on an effect which arrises when the finite element method in its mixed form is applied to the spectral problems 588 1044 A generalized mixed functional in the spectral problem 597 105 Final comments to Chapter 10 601 References 602 11 VARIATIONAL PRINCIPLES IN STABILITY/BUCKLING ANALYSIS 605 111 Stability of systems with a finite number of degrees of freedom 606 1111 A functional of stability – Bolotin’s functional 613 1112 Linear analysis and a linearized formulation of the stability problem 614 1113 Example 1 617 1114 Example 2 – paradoxes in the stability analysis 625 • A remark on a non-invariant critical load to the choise of generalized coordinates 632 112 Variational description of critical loads 634 1121 A Rayleigh ratio and a recursive variational calculation of critical loads 635 1122 A remark on the effect of constraints on the stability of a linearized elasic system 639 1123 Papkovich theorem of convexity of the stability area 642 1124 The geometric stiffness matrix revisited 649 1125 The stability of equilibrium under a non-force-type action 651 113 Geometrically nonlinear problems in elasticity 655 1131 Geometric equations 655 • Expressions of variations of the strain tensor components 657 1132 Equations of equilibrium and static boundary conditions 658 114 Stability of equilibrium of an elastic body 663 1141 A linearized formulation of the equilibrium stability problem for an elastic body …………………………… 665 • A remark on a mechanical interpretation of particular terms in the stability functional 668 • Criteria of a critical state in a system 669 1142 The Ritz method 673 115 Stability of equilibrium in particular classes of problems 675 1151 Stability of equilibrium of a bar in the engineering theory of bars 676 1152 Stability of equilibrium of a Timoshenko bar 679 1153 Stability of equilibrium of a Kirchhoff-Love plate 686 1154 Stability of equilibrium of a Reissner plate 687 1155 Stability of equilibrium of a thin-walled bar 690 116 Mixed functionals in the stability analysis 694 117 Final comments to Chapter 11 696 References 697 CONCLUSION 701