In this chapter, we deal with the analytic, symbolic computation for buckling analysis. Symbolic algebra can significantly reduce the tedium of analytic computation and simultaneously increases its reliability. It has great impact on scientific computation, as more and more analytically minded researchers are using computers, which have been traditionally associated with “number crunching.” Analytic work can now be extended as far as possible, and the numerical side of the analysis can be “delayed.” Calculations with arbitrary arithmetic precision along with the automatic generation of computer codes have opened up more possibilities of computer use. In this chapter, we use a classical problem buckling of non-isotropic plate to illustrate the application of symbolic algebra, a neo-classical analytic-numerical tool.

In this chapter, we treat initial geometric imperfections as spacewise random functions (i.e., random fields). As a result, the buckling load the maximum load the structure can sustain turns out to be a random. We focus our attention on reliability of the shells, namely the probability that a structure will not fail prior to the specified load. We develop efficient techniques of simulation of random fields, based on the knowledge of the mean initial imperfection function and the auto-correlation function. This allows us to conduct, by the Monte Carlo method, an extensive analysis of the buckling of columns on non-linear elastic foundations and of circular cylindrical shells. We consider the cases of both axisymmetric and general, non-symmetric imperfections.

In a traditional probabilistic analysis, the statistical parameters of uncertain quantities initial geometric imperfections or elastic moduli are presumed to be known, which must be inferred from on-site measurements. Because the available data of such measurements are often limited to permit the probabilistic analysis, a new discipline, called convex modeling of uncertainty, is applied to obtain estimates of the upper and lower bounds of the buckling loads. From a structural safety point of view, the least favorable lower buckling load should be used in design. Critical comparison of the probabilistic and convex analyses is performed.

There are at present numerous books available on the theory of stability and its applications to structures. One author even remarked sarcastically that if they were put in a single bookcase, it would buckle under their weight. We do not complain that “…of the making of many books there is no end” (Ecclesiastes 12:12), rather we ask a natural question: Is there a legitimate place for a new book in this field?

The answer to this question is affirmative, if a book has its unique, distinct characteristics. We have chosen to deal with non-classical problems. To the best of our knowledge, none of the subjects, touched upon in this monograph, have been discussed exclusively in the existing books on buckling analysis. Thus we feel that this book will not be just another newbook on buckling. Indeed, most existing books may be classified as belonging to one of the following two categories: textbooks which often look very much alike, maybe not without reason, since the subject is the same and monographs which have an encyclopedic nature, trying to comprise an uncomprisable to cover all or nearly all pertinent topics. This latter task of listing all the results (even only those of major importance) appears to be impossible indeed.

The purpose of this book is to present two competing theories, which incorporate ever-present uncertainty in the stability applications of the real world.

When bodies collide, they come together with some relative velocity at an initial point of contact. If it were not for the contact force that develops between them, the normal component of relative velocity would result in overlap or interference near the contact point and this interference would increase with time. This reaction force deforms the bodies into a compatible configuration in a common contact surface that envelopes the initial point of contact. Ordinarily it is quite difficult and laborious to calculate deformations that are geometrically compatible, that satisfy equations of motion and that give equal but opposite reaction forces on the colliding bodies. To avoid this detail, several different approximations have been developed for analyzing impact: rigid body impact theory, Hertz contact theory, elastic wave theory, etc. This book presents a spectrum of different theories for collision and describes where each is applicable. The question of applicability largely depends on the materials of which the bodies are composed (their hardness in the contact region and whether or not they are rate-dependent), the geometric configuration of the bodies and the incident relative velocity of the collision. These factors affect the relative magnitude of deformations in the contact region in comparison with global deformations.

A collision between hard bodies occurs in a very brief period of time.

When a bat strikes a ball or a hammer hits a nail, the surfaces of two bodies come together with some relative velocity at an initial instant termed incidence. After incidence there would be interference or interpenetration of the bodies were it not for the interface pressure that arises in a small area of contact between the two bodies. At each instant during the contact period, the pressure in the contact area results in local deformation and consequent indentation; this indentation equals the interference that would exist if the bodies were not deformed.

At each instant during impact the interface or contact pressure has a resultant force of action or reaction that acts in opposite directions on the two colliding bodies and thereby resists interpenetration. Initially the force increases with increasing indentation and it reduces the speed at which the bodies are approaching each other.

angle of incidence angle between the direction of the incident relative velocity of the contact points and the common normal direction. Direct or normal collisions have zero angle of incidence, whereas oblique collisions have a nonzero angle of incidence.

angle of rebound angle between the direction of the relative velocity of the contact points at separation and the common normal direction.

attractor steady state solution that is approached asymptotically with increasing time if the system has small dissipation.

coefficient of friction upper limit on ratio of tangential to normal force at contact.

coefficient of stick geometric parameter specifying ratio of tangential to normal force for stick.

collinear (or central) impact configuration colliding bodies oriented so that each center of mass is on common normal line passing through the point of initial contact.

common normal direction normal to common tangent plane that passes through contact point C.

common tangent plane If at least one of the bodies has a topologically smooth surface at the contact point, this is the plane that is tangent to the surface at the point of initial contact. Usually both bodies have smooth surfaces around their respective points of contact, so they have a common tangent plane.

Three-dimensional (3D, or nonplanar) changes in velocity occur in collisions between rough bodies if the configuration is not collinear and the initial direction of sliding is not in-plane with two of the three principal axes of inertia for each body. In collisions between rough bodies, dry friction can be represented by Coulomb's law. If there is a tangential component of relative velocity at the contact point (sliding contact) this law relates the normal and tangential components of contact force by a coefficient of limiting friction. The friction force acts in a direction opposed to sliding. For a collision with planar changes in velocity, sliding is in either one direction or the other on the common tangent plane. In general however, friction results in nonplanar changes in velocity. Nonplanar velocity changes give a direction of sliding that continuously changes, or swerves, during an initial phase of contact in an eccentric impact configuration. This chapter obtains changes in relative velocity during rigid body collisions as a function of the impulse P due to the normal component of the reaction force. The changes in velocity depend on two independent material parameters – the coefficient of friction and an energetic coefficient of restitution.

In this chapter lumped parameter models for compliance of the deforming region are used to examine the influence of factors which previously in this book were assumed to be negligibly small – namely the effects of (a) a viscoelastic or rate-dependent normal compliance relation and (b) tangential compliance. Because these factors depend on the interaction force and not simply the impulse, the analysis of their effects necessarily uses time rather than normal impulse as an independent variable.