Optimal Structural Design under Stability Constraints

The first optimal design problem for an elastic column subject to buckling was formulated by Lagrange over 200 years ago. However, rapid development of structural optimization under stability constraints occurred only in the last twenty years. In numerous optimal structural design problems the stability phenomenon becomes one of the most important factors, particularly for slender and thin-walled elements of aerospace structures, ships, precision machines, tall buildings etc. In engineering practice stability constraints appear more often than it might be expected; even when designing a simple beam of constant width and variable depth, the width - if regarded as a design variable - is finally determined by a stability constraint (lateral stability). Mathematically, optimal structural design under stability constraints usually leads to optimization with respect to eigenvalues, but some cases fall even beyond this type of problems. A total of over 70 books has been devoted to structural optimization as yet, but none of them has treated stability constraints in a sufficiently broad and comprehensive manner. The purpose of the present book is to fill this gap. The contents include a discussion of the basic structural stability and structural optimization problems and the pertinent solution methods, followed by a systematic review of solutions obtained for columns, arches, bar systems, plates, shells and thin-walled bars. A unified approach based on Pontryagin's maximum principle is employed inasmuch as possible, at least to problems of columns, arches and plates. Parametric optimization is discussed as well.

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.