Optimal Structural Design under Stability Constraints

1. Elements of the theory of structural stability.- 1.1 Definition of stability.- 1.1.1 Lyapunov's definition of stability.- 1.1.2 Lyapunov's first method.- 1.1.3 Lyapunov's second method.- 1.1.4 The fundamental problem of stability for deformable bodies.- 1.2 Stability of elastic structures.- 1.2.1 Classification of loadings.- 1.2.2 Kinetic analysis.- 1.2.3 Static criterion of the loss of stability.- 1.2.4 Energy approach for conservative systems.- 1.2.5 Energy approach for nonconservative systems.- 1.2.6 Effect of imperfections.- 1.2.7 Coincident critical points and their relation to optimal design.- 1.2.8 Stability under combined loadings.- 1.3 Elastic-plastic stability.- 1.3.1 General remarks.- 1.3.2 Plastically active and passive zones.- 1.3.3 Example of a column.- 1.3.4 Bifurcation and stability.- 1.4 Stability and buckling in creep conditions.- 1.4.1 General remarks.- 1.4.2 Creep stability of perfect structures.- 1.4.3 Creep buckling of imperfect structures.- 1.4.4 Snap-through in creep conditions.- 2. Problems of optimal structural design.- 2.1 Formulation of optimization problems.- 2.2 Design objectives and their criteria.- 2.3 Design variables.- 2.4 Constraints and their criteria.- 2.4.1 Classification of constraints.- 2.4.2 Strength constraints and the shapes of uniform strength.- 2.4.3 Stability constraints.- 2.4.4 Stiffness or compliance constraints.- 2.4.5 Vibration constraints.- 2.4.6 Relaxation constraints.- 2.4.7 Technological constraints.- 2.5 Equation of state.- 2.6 Stability constraints in structural optimization.- 2.6.1 General remarks.- 2.6.2 Eigenvalue as constraints, multimodal optimal design.- 2.6.3 Simultaneous mode design, mode interaction.- 2.6.4 Local stability condition and the shapes of uniform stability.- 2.6.5 Peculiarities of creep buckling constraints.- 2.6.6 Historical notes and surveys.- 3. Methods of structural optimization.- 3.1 Calculus of variations.- 3.1.1 General remarks.- 3.1.2 Classical problems of calculus of variations.- 3.1.3 Equality constraints.- 3.1.4 Functions of functionals.- 3.1.5 Vectorial notation for single integrals.- 3.1.6 Variable ends, transversality conditions, corners.- 3.1.7 Problems of Bolza and Mayer.- 3.1.8 Sufficient conditions.- 3.1.9 Approximate methods of variational calculus.- 3.2 Pontryagin's maximum principle.- 3.2.1 Equations of state and boundary conditions.- 3.2.2 Objective functional.- 3.2.3 Hamiltonian and the maximum principle.- 3.2.4 Inequality constraints.- 3.2.5 Problems of Bolza and Mayer.- 3.2.6 Additional parametric optimization.- 3.2.7 Balakrishnan's e-method in optimal control.- 3.3 Sensitivity analysis.- 3.3.1 General remarks.- 3.3.2 Approach based on differential equations of state.- 3.3.3 Variational approach.- 3.3.4 Eigenvalue problems.- 3.3.5 Optimal structural remodeling and reanalysis.- 3.3.6 Application of perturbation methods.- 3.4. Parametric optimization, mathematical programming.- 3.4.1 Statement of the problem, necessary conditions.- 3.4.2 Methods of transformation linearizing the inequality constraints.- 3.4.3 Finite element discretization.- 3.4.4 Application of sensitivity analysis.- 3.4.5 Numerical methods of parametric optimization.- 3.4.6 Decomposition in parametric structural optimization.- 3.4.7 Multicriterial optimization.- 4. Elastic and inelastic columns.- 4.1 Stability of non-prismatic columns.- 4.1.1 General nonlinear governing equations.- 4.1.2 General precritical state and relevant conditions of loss of stability.- 4.1.3 Momentless precritical state and relevant conditions of loss of stability.- 4.1.4 Inextensible axis and neglecting of shear deformations.- 4.1.5 Examples of loadings independent of state variables.- 4.1.6 Examples of loadings dependent on state variables.- 4.1.7 Effective forms of constitutive equations.- 4.2 Unified approach to optimization of columns.- 4.2.1 General statement of the optimization problems.- 4.2.2 Geometric relations for typical cross-sections.- 4.2.3 Solution by Pontryagin's maximum principle.- 4.2.4 Solution by sensitivity analysis.- 4.2.5 Analytical and numerical methods of evaluation of optimal shapes.- 4.2.6 Multimodal formulation.- 4.2.7 Self-adjoint system of equations of the critical state.- 4.2.8 Non-self-adjoint system of equations of the critical state.- 4.3. Unimodal solutions to linearly elastic problems.- 4.3.1 The optimal condition.- 4.3.2 General solution for affine columns compressed by a concentrated force.- 4.3.3 Plane-affine columns, out-of-taper-plane buckling, ? =1.- 4.3.4 Spatially affine columns, ? =2.- 4.3.5 Plane-affine columns, in-taper-plane buckling, ? =3.- 4.3.6 Some effective elastic solutions for concentrated forces.- 4.3.7 Energy approach to optimization problems.- 4.3.8 Columns with several independent loading parameters.- 4.3.9 Optimization of bars in tension subjected to loss of stability.- 4.3.10 Singularities in optimal solutions.- 4.3.11 Analytical solutions with geometrical constraints.- 4.3.12 Multispan columns.- 4.3.13 Postcritical behaviour of optimal columns.- 4.3.14 Multicritical optimization of columns.- 4.3.15 Optimal elastic non-homogeneity.- 4.3.16 Other problems.- 4.4 Multimodal solutions to conservative problems.- 4.4.1 A clamped-clamped column (the Olhoff-Rasmussen problem).- 4.4.2 Compressed columns in an elastic (Winkler's) medium.- 4.4.3 Multimodal optimization of elastically clamped columns for buckling in two planes.- 4.5 Non-conservative linearly-elastic problems.- 4.5.1 The optimality condition.- 4.5.2 Generalized Hamilton's principle.- 4.5.3 Optimization of Ziegler's model.- 4.5.4 Optimization of real column under anti-tangential force.- 4.5.5 Optimization of real columns under follower force.- 4.5.6 Optimization of real columns under distributed follower forces.- 4.5.7 Optimization in aeroelasticity.- 4.6 Inelastic columns.- 4.6.1 Nonlinearly elastic and elastic-plastic solutions.- 4.6.2 Linearly visco-elastic solutions.- 4.6.3 Optimization of imperfect columns under linear creep buckling constraints.- 4.6.4 Optimization of columns under nonlinear creep buckling constraints.- 5. Arches.- 5.1 Stability of non-prismatic arches.- 5.1.1 Introductory remarks.- 5.1.2 General non-linear governing equations for in-plane motion.- 5.1.3 General precritical state and relevant conditions of in-plane loss of stability.- 5.1.4 Momentless precritical state and relevant conditions of in-plane buckling.- 5.1.5 Momentless precritical state and relevant conditions of out-of-plane buckling.- 5.1.6 Examples of loadings.- 5.2 General statement of the optimization problem.- 5.2.1 Formulation of the problem and historical notes.- 5.2.2 Geometrical characteristics of cross-sections.- 5.2.3 General solution.- 5.3 Funicular arches.- 5.3.1 In-plane buckling.- 5.3.2 Circular arches under hydrostatic loading.- 5.3.3 Simultaneous in-plane and out-of-plane buckling of funicular arches.- 5.4 Extensible arches optimized for in-plane bifurcation and snap-through.- 5.5 Optimal forms of axis of the arch.- 6. Trusses and Frames.- 6.1 Stability of trusses.- 6.1.1 Introductory remarks.- 6.1.2 The Mises classical approach.- 6.1.3 Matrix notation, the Maier-Drucker approach.- 6.1.4 Buckling of individual bars.- 6.2 Optimal design of trusses.- 6.2.1 Optimization of uniform cross-sections in trusses of given geometry and topology.- 6.2.2 Simultaneous optimization of layout and cross-sections.- 6.2.3 Example of optimization in the elastic-plastic range.- 6.2.4 Example of optimization in creep conditions.- 6.2.5 Optimal topology of trusses.- 6.2.6 Inverse problem of structural optimization of trusses.- 6.2.7 Optimal transmission of a force to a given foundation contour.- 6.3 Stability of frames.- 6.4 Optimal design of frames.- 6.4.1 Introductory remarks.- 6.4.2 General formulation of the optimization problem.- 6.4.3 Brief survey of solutions.- 6.4.4 Unimodal and bimodal optimization of a portal frame.- 7. Plates and Panels.- 7.1 Governing equations of stability of plates.- 7.1.1 General remarks.- 7.1.2 Governing equations in Cartesian coordinates.- 7.1.3 Governing equations in polar coordinates.- 7.1.4 Energy approach for conservative loadings.- 7.2 Optimal design of circular and annular plates.- 7.2.1 Optimal control approach.- 7.2.2 Energy approach, Rayleigh quotient.- 7.2.3 Numerical approaches, parametric optimization.- 7.2.4 Stiffened circular plates.- 7.2.5 Optimal prestressing.- 7.3 Optimal design of rectangular plates.- 7.3.1 Solid plates.- 7.3.2 Multilayer plates.- 7.3.3 Stiffened rectangular plates.- 7.3.4 Reinforced rectangular plates.- 7.4 Aeroelastic optimization.- 8. Shells.- 8.1 Stability of shells.- 8.1.1 Introductory remarks.- 8.1.2 General nonlinear equations of shell stability.- 8.1.3 The Lukasiewicz nonlinear equations for variable thickness shells.- 8.1.4 Linear equations for cylindrical shells.- 8.2 Optimal design of cylindrical shells.- 8.2.1 Smooth shells of variable thickness.- 8.2.2 Shells stiffened by ribs.- 8.2.3 Multilayer, composite and reinforced shells.- 8.2.4 Optimization of shells under aeroelastic and dynamic stability constraints.- 8.3 Optimal design of cylindrical shells via the concept of uniform stability.- 8.3.1 The shells of uniform stability.- 8.3.2 Cylindrical shell under overall bending.- 8.3.3 Cylindrical shell under bending with axial force.- 8.3.4 Cylindrical shell under bending with torsion.- 8.4 Optimal design of noncylindrical shells.- 8.4.1 Smooth shells.- 8.4.2 Shells stiffened by ribs.- 8.4.3 Multilayer, composite and reinforced shells.- 8.4.4 Optimization under aeroelastic constraints.- 9. Thin-walled bars.- 9.1 Stability of thin-walled bars.- 9.1.1 Typical buckling modes.- 9.1.2 Buckling mode interaction.- 9.1.3 Overall stability of variable-thickness and variable-profile bars.- 9.2 Optimal design of thin-walled columns.- 9.2.1 Elastic columns with closed cross-sections.- 9.2.2 Elastic-plastic columns with closed cross-sections.- 9.2.3 Columns with open cross-sections.- 9.2.4 Optimization of columns allowing for imperfections and mode interaction.- 9.3 Optimal design of thin-walled beams.- 9.3.1 Beams with closed cross-sections under pure bending.- 9.3.2 Box-beams under pure bending.- 9.3.3 Box-beams under pure torsion.- 9.3.4 Beams with closed cross-sections under combined loadings.- 9.3.5 Beams with open cross-sections.- 9.4 Aeroelastic problems.- 9.5 Optimal design of structures of thin-walled elements.- 9.6 Final remarks.- References.- I. Monographs, textbooks and proceedings of selected symposia.- 1. Optimal structural design.- 2. Structural stability.- 3. Optimization theory and methods.- II. References to individual chapters.- 1. Elements of the theory of structural stability.- 2. Problems of structural design.- 3. Methods of structural optimization.- 4. Elastic and inelastic columns.- 5. Arches.- 6. Trusses and frames.- 7. Plates and panels.- 8. Shells.- 9. Thin-walled bars.- III. References added in proof.- Author Index.