# Stability, Bifurcation and Postcritical Behaviour of Elastic Structures

A comprehensive and systematic analysis of elastic structural stability is presented in this volume. Traditional engineering buckling concepts are discussed in the framework of the Liapunov theory of stability by giving an extensive review of the Koiter approach. The perturbation method for both nonlinear algebraic and differential equations is discussed and adopted as the main tool for postbuckling analysis. The formulation of the buckling problem for the most common engineering structures - rods and frames, plates, shells, and thin-walled beams, is performed and the critical load evaluated for problems of interest. In many cases the postbuckling analysis up to the second order is presented. The use of the Ritz-Galerkin and of the finite element methods is examined as a tool for approximate bifurcation analysis. The volume will provide an up-to-date introduction for non-specialists in elastic stability theory and methods, and is intended for graduate and post-graduate students and researchers interested in nonlinear structural analysis problems. Basic prerequisites are kept to a minimum, a familiarity with elementary algebra and calculus is all that is required of readers to make use of this book.

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Published: December 1991

Imprint: Elsevier

ISBN: 978-0-444-88140-3

## Contents

• 1. The Liapunov Theory of Equilibrium Stability (M. Pignataro). Differential equations. Simple types of equilibrium points. Equilibrium points of non-linear systems. Stability of equilibrium according to Liapunov. Theorems on the stability of equilibrium. Analysis of the stability of equilibrium by linear approximation. Criterion of negative real parts of all the roots of a polynomial. 2. The Stability of Equilibrium and Post-Buckling Behaviour of Discrete Mechanical Systems (M. Pignataro). Lagrange and Hamilton equations of motion. Stability of equilibrium according to Liapunov. Lagrange-Dirichlet theorem. Theorems of Liapunov and Chetayev. Analysis of the stability of linear systems. Criterion of stability of discrete systems. A system of one degree of freedom with: Stable symmetrical post-critical behaviour; Unstable symmetrical post-critical behaviour; Asymmetrical post-critical behaviour; Non-linear pre-critical behaviour. A system of two degrees of freedom. 3. Analysis of Bifurcation for Discrete Systems. Characterisation of the Points of an Equilibrium Path from Examination of Local Properties (N. Rizzi). Local analysis of the properties of points belonging to an equilibrium path. Perturbation method in the asymptotic determination of regular equilibrium paths through a point Q. Search for critical points along a known equilibrium path and local analysis of the bifurcated path in the vicinity of the bifurcation point. Analysis of bifurcation for a system of two degrees of freedom. Outlines of the analysis of bifurcation starting from an approximate equilibrium path. 4. Stability of Equilibrium and Post-Critical Behaviour of Continuous Systems (M. Pignataro). Theorems of stability and instability. Critical condition of equilibrium. Criterion of stability of continuous systems. Construction of equilibrium paths by means of perturbation analysis. 5. Analysis of Beams and Plane Frames (N. Rizzi, M. Pignataro). Beam models. Critical load and post-critical behaviour of beams loaded axially at one end. Particular problems. Simply supported beam axially loaded at mid-span. Stability of beams. Symmetric, simply supported, two bar frame. Hinged symmetrical portal frame. Application of the method of finite elements to problems of bifurcation in plane frames. 6. Thin-Walled Beams with Open Cross-Section (A. Luongo). Hypotheses on the mechanical behaviour of thin-walled beams. Kinematics of the beam. Formulation of the linearised problem of stability. Thin-walled beams subjected to: Uniform compression; Eccentric axial loads; Under pure bending. Flexural-torsional stability of beams subjected to lateral loads. Methods of discretisation. Outline of the post-critical behaviour of thin-walled beams. 7. Analysis of Plates and Shells (A. Luongo, M. Pignataro). Some basic results of the theory of surfaces and shells. Kinematics of shells. Elastic strain energy. Stability of plates. Critical load of: Rectangular plates; Stiffened plates. Infinite cylinder subjected to lateral pressure. Cylinder of finite length subjected to lateral pressure. Cylinders subjected to: Axial pressure; The action of combined loadings. Collapse of cylindrical shells. Sphere subjected to hydrostatic pressure. Post-critical behaviour of plates and shells. Appendix: The Calculus of Variations (M. Pignataro). Stationary values of definite integrals. Basic procedures of the calculus of variations. Commutative properties of the variational process. Stationary value of a definite integral from the calculus of variations. Variations of an integral with auxiliary holonomous conditions. Extremum values of a definite integral. Successive variations of a functional. Stationarity of a function of n variables. Indexes. References are included with each chapter.