This chapter investigates the buckling mode localization in the periodic multi-span beams and plates. We start our discussion with disorder in two- and three-span elastic plates; then we focus our attention on the multi-span beams and plates with a disorder occurring in an arbitrary single span. The analytical finite difference calculus is employed to derive the transcendental equations from which buckling load is calculated. The underlying treatment is general, and the solution thus obtained is exact within the theory used. Numerical results show that the buckling mode is highly localized in the vicinity of the disordered span of the beam or the plate. In the multi-span, elastic plates are considered with transverse stiffeners, and the discreteness of the stiffeners is accounted for. The torsional rigidity of the stiffener is found to play an important role in the buckling mode pattern. When the torsional rigidity is properly adjusted, the stiffener can be used in passive control; that is, it can serve as an isolator of deformation for the structure at buckling so that the deflection is limited to only a small area.

Localization in Elastic Plates Due to Misplacement in the Stiffener Location

Traditionally, the stability of the stiffened plate has been studied following three different settings.

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.

The purpose of Modern Mathematical Methods for Physicists and Engineers is to help graduate and advanced undergraduate students of the physical sciences and engineering acquire a sufficient mathematical background to make intelligent use of modern computational and analytical methods. This book responds to my students’ repeated requests for a mathematical methods text with a modern point of view and choice of topics.

For the past fifteen years I have taught graduate courses in computational and mathematical physics. Before introducing the course on which this book is based, I found it necessary, in courses ranging from numerical methods to the applications of group theory in physics, to summarize the rudiments of linear algebra and functional analysis before proceeding to the ostensible subjects of the course. The questions of the students who studied early drafts of this work have helped to shape the presentation. Some students working concurrently in nearby telecommunication, semiconductor, or aerospace, industries have contributed significantly to the substance of portions of the book.

The following is an example of the situations that motivated me to take the time to write a mathematical methods text that breaks significantly with the past: Every semester, students come to my office, puzzled over numerical modeis in which minor changes in the data produce drastic changes in the Outputs. Unfortunately most of these students lack the mathematical background needed to conceptualize some of the most common problems of numerical computation. For an engineer, and for the increasingly large fraction of physics graduates who make careers in numerical modeling or electrical engineering, conceptual understanding of analytical and numerical modeis is an absolutely essential ingredient of successful designs. A Computer can be a tool for understanding, and not merely a means for obtaining a numerical answer of unknown reliability and significance, only in the hands of those who understand the foundations and potential shortcomings of numerical methods. Yet the traditional mathematical methods taught to students in engineering and physics for most of the twentieth Century do not provide a sufficient background even for introductory graduate texts on many important contemporary topics, of which numerical computation is only one.

What upper-level undergraduate and first-year graduate students in physics and engineering tend consistently to lack is an understanding of basic mathematical structures - groups, rings, fields, and vector Spaces - and of mappings that preserve these structures.

Index Of Notation