Structural Optimization

Adam Borkowski

Institute of Fundamental Technological Problems,
Polish Academy of Sciences,
Swietokrzyska 21, 00-049 Warsaw, Poland
E-mail: abork@ippt. gov. pl

1. Introduction

During the last phase of the Second World War huge quantities of goods were transported across Atlantic ocean from the United States and Canada to Europe. Diminishing the overall cost of this logistic task even by several percents meant sparing millions of dollars. This demand motivated US – authorities to allocate money for research and many mathematicians started to investigate a problem of transportation: how to organize the flow of goods between given locations in order to minimize the total cost of delivery. G. B. Dantzig proposed very efficient method of solving such problems – the simplex algorithm – and named the domain Linear Programming(LP). He did not realize that the second term of this name will soon collide with the vast area of computer programming.

After Dantzig published his first paper on LP (Dantzig, 1948), this approach attracted much interest in the West. It remained unnoticed that similar results were obtained by L. V. Kantorovich in the Soviet Union already before the Second World War (Kantorovich, 1939). At the beginning of the 1950-ties a general theory of Mathematical Programming (MP) was developed, with major contribution given by H. W. Kuhn and A. W. Tucker (Kuhn, Tucker, 1951). The subject of this theory is a Non-Linear Programming Problem (NLP-problem) :

min { f(x)| gt (x) < 0, x є Rn, i = 1,2,…, m } (1)

x>0

Here x є Rn is a column matrix of unknowns, f = f (x) is a cost function and gt = gt (x) are

constraints with i = 1,2,…,m. Usually it is assumed that functions f and gt are convex. In

particular, the cost function may be quadratic and the constraints may be linear. This leads to a particular form of the NLP-problem called Quadratic Programming Problem (QP-problem):

min {-1 xTDx + cTx | Ax > b } (2)

x>0 2

Matrices A є Rmxn, b є Rm, c є Rn and D є Rnxn are given. In order to assure convexity of the cost function, matrix D must be positive definite.

Finally, taking D = 0 in (2), we obtain the simplest form of MP-problems, namely a Linear Programming Problem (LP-problem):

min { cTx | Ax > b } (3)

x>0

It turns out that each minimization problem in LP has its maximization counterpart:

max { bTy | ATy < c } (4)

y >0

Problems (3) and (4) are said to be mutually dual and the entries of y є Rm are called dual variables. For the sake of simplicity, we quote MP-problems in their canonical form: the problems (1) to (4)

639

M. Pandey et al. (eds), Advances in Engineering Structures, Mechanics & Construction, 639-651.

© 2006 Springer. Printed in the Netherlands.

contain only non-negative variables and inequality constraints. In general, free variables and equality constraints can be present as well.

At the beginning Linear Programming was used only in management and economics. The remnants of this period are still present in the terminology (the cost function, the shadow prices, etc.). A typical application of the model (4) would be maximizing the total production of a factory that uses different technological processes and different resources. The unknowns y are then the time slots allocated for each process, the entries of c represent given efficiency of each process, the entries of A tell us how much of each resource is consumed by a particular process and the entries of b describe available amount of each resource.

In parallel to things happening in Mathematical Programming, revolutionary changes occurred in Structural Analysis. Instead of relying on linear elasticity and on admissible stresses, a concept of safety factors against possible ultimate states was introduced. Again the Cold War precluded the exchange of ideas and the pioneering work of A. A. Gvozdev (Gvozdev, 1949) remained unknown in the Western hemisphere.

It was proved soon that the safety factor against plastic collapse does not depend on elastic properties of the structural material and that such factor can be found by maximizing the load multiplier over all statically admissible stress fields. For skeletal structures with a single dominant internal force (e. g. the axial force or the bending moment) a stress state s is statically admissible if each Sj remains less or equal to the yield stress s0J and if s is equilibrated with a load p. Assuming strains q and displacements w to remain small prior to the plastic collapse, we can write the equilibrium equation as CT s = p, where C is the matrix of kinematics: q = C w. Let loading be proportional: p = y p0, where y is an unknown multiplier and p0 is a given reference load. According to the static theorem, the ultimate value yt of the load factor can be found solving the problem:

max { y | s < So, CTs – y po = 0} (5)

У, s

Looking at this model today we see at once that it is a LP-problem. However, the pioneers of the MP-based modeling of structural behavior had to overcome the barrier between economics and mechanics. Having accomplished that, they could enjoy the power of mathematics: the semantics of the production planning problems and the ultimate load problem is completely different but the formal structure of both problems is identical. We believe that the following papers, cited in the alphabetic order after the first author, were important in providing impetus to the MP-oriented approach: Biron & Hodge, 1968; Brown & Ang, 1964; Ceradini & Gavarini, 1965; Hodge, 1966; Koopman & Lance, 1965; Sacchi & Buzzi-Ferraris, 1966; Wolfensberger, 1964.

Special tribute should be given to two persons: Mircea Z. Cohn and Aleksandras Cyras. Already in 1956 Cohn published in Romania his first paper on the plastic structural analysis (Cohn, 1956). In 1972 an inspiring paper on the unified theory of plastic analysis appeared (Cohn et al., 1972). During the NATO Advanced Study Institute that took place in Waterloo in 1977 he was invited to deliver a keynote lecture (Cohn, 1979). The celebration of his 65th birthday in 1991 gathered over 60 contributors from 14 countries. The results of this meeting were published in a book edited by Cohn’s former students D. E. Grierson, A. Franchi and P. Riva (Grierson et al., 1991). Their contribution to the considered domain is substantial (Franchi & Cohn, 1980), (Grierson & Gladwell, 1971), (Grierson, 1972), (Riva & Cohn, 1990).

Since Lithuania was a part of the Soviet Union at the time of his scientific carrier, Cyras was for a long time isolated from the Western scientific community. Most of his early papers were written in Russian (Cyras, 1963) and published in a local Lithuanian journal. His first paper in English, written with the present author, appeared in Poland in 1968 (Cyras & Borkauskas, 1968). Already well known in the Soviet Union and in other countries of the Eastern block, he was invited in 1974 by Waclaw Olszak to present his results at the CISM-course (Cyras, 1974).

In 1969 Cyras published a book that contained many fundamental results on the applications of Linear Programming in the analysis and design of structures made of rigid-perfectly plastic material (Cyras, 1969). Two further books (Cyras, 1971), (Cyras et al., 1974) were also written in Russian language. The first English edition appeared in 1983 – this was the translation of the book (Cyras, 1982). In 2002 the 75th birthday of Aleksandras Cyras was celebrated in Vilnius. This motivated his former students R. Karkauskas and the present author to prepare new edition of the book (Cyras et al., 1974). Substantially updated and rewritten it appeared in 2004 (Cyras et al., 2004).

Being a founder and a long time Rector of the Institute of Civil Engineering in Vilnius (VISI), Cyras inspired many researchers to work on the MP-applications in Structural Analysis and Optimum Design. This group contributed substantially to progress in such areas as the ultimate state under constrained strains (Cyras & Cizas, 1966), the evaluation of displacements prior to collapse (Cyras & Baronas, 1971), the ultimate state of shells (Cyras & Karkauskas, 1971), (Cyras & Kalanta, 1974), the plastic shakedown problem (Cyras & Atkociunas, 1984). The present author took part in the development of general concept of the dual approach (Cyras & Borkauskas, 1969). In 1988 he published a book in Polish that was translated three years later into English (Borkowski, 1988).

The above overview is by no means complete. It reflects personal experience of the author who was involved in this fascinating scientific adventure.