Table Of ContentINTERNATIONAL CENTRE FOR MECHANICAL SCIENCES
COURSES AND LECfURES -No. 299
MATHEMATICAL PROGRAMMING METHODS
IN STRUCTURAL PLASTICITY
EDITED BY
D. LLOYD SMITH
IMPERIAL COLLEGE, LONDON
SPRINGER-VERLAG WIEN GMBH
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© 1990 by Springer-Verlag Wien
Originally published by Springer Verlag Wien-New York in 1990
In order to make this volume available as economically and as
rapidly as possible the authors' typescripts have been
reproduced in their original forms. This method unfortunately
has its typographical limitations but it is hoped that they in no
way distract the reader.
ISBN 978-3-211-82191-6 ISBN 978-3-7091-2618-9 (eBook)
DOI 10.1007/978-3-7091-2618-9
PREFACE
Civil engineering structures tend to be fabricated from materials that respond elastically at
norma/levels of loading. Most such materials, however, would exhibit a marked and ductile inelasticity
if the structure were overloaded by accident or by some improbable but naturally occurring
phenomenon. Indeed, the very presence of such ductility is an important safety provision for large
scale constructions where human life is at risk. In the mathematical theory ofp lasticity there is found a
material mode/which is both fairly simple and generally representative of the essential characteristics of
observed behaviour. It must be considered unrealistic, therefore, not to include the effects ofp lasticity
in the comprehensive evaluation of safety in a structural design.
Mathematical Programming (MP) is concerned with seeking to optimise a function whose
variables are also required to satisfy additional conditions or constraints. Such constraints may be
represented by systems of equations, or of inequalities, connecting the variables, and they would
include the possibility that some of the variables might be constrained in sign. It is clear that the search
for a solution to a system of inequalities, or of equations, without the optimising of a function is a
special case of the above problem.
During the second world war, the need to address complex problems ofp lanning and resource
allocation for economic as well as military purposes coincided with the increasing availability of large
scale computing facilities. This provided the climate for the rapid developlment ofM P, and, since the
mathematical description ofp lasticity necessarily involves inequality conditions, it was not long before
the link between these two subjects was established.
In the ensuing twenty-five years there has been a thorough investigation of the role ofM P in all
the major problem types in structural plasticity. The appeal of MP in this context has been twofold:
computational and theoretical. MP can provide a ready fund of general algorithms. More importantly, it
invests applications with a refined mathematical formalism, rich in fundamental theorems, which often
gives additional insight into known results and occasionally leads to new ones.
The conspectus offered by the lectures at the C.I.S.M. Advanced School in Udine, and
contained in this book, is hardly a complete one. It can be said, however, that these lectures address a
selection of the most interesting and potentially most useful applications of MP in structural plasticity:
the ultimate strength and elastoplastic deformability of sections ,framed and continuum structures; the
ability ofa structure to shake down or to adapt itself to respond elastically to a complex programme of
loading; the assessment ofp ractical upper bounds on specific measures of deformation for a structure
responding either quasi-statically or dynamically to such a complex loading programme; the evolutive
dynamic behaviour of structures with rigid-plastic and elastic-plastic constitutive laws; the static
response and instability of an elastoplastic structure in a regime of large displacements; the use of
stochastic and fuzzy programming methods for representing the role of uncertainty in the assessment of
ultimate strength.
In the best traditions of applied mechanics, the following lectures successfully implement
applications from a foundation off undamental mechanics. Their success, if I may be permitted to say,
is achieved through a systematic and consistent mathematical modelling of both structure and material.
In this regard, particular attention has been given to the mesh and nodal network modelling of structural
systems, to the virtues of representing general non linear material behaviour through asymptotic
expansions, and to the achievement of a consistent two-field approximation of stresses and strain rates
in the finite-element modelling of continuum plasticity.
Beyond the interest that the subject generates as a fertile field for the nurturing of research projects is
the conviction that its educational value thoroughly befits a university discipline. This genuine
pedagogic concern is amply reflected in the lectures of my colleagues, the distinguished lecturers at the
C.I.S.M. Advanced School, through their many years of experience of giving graduate classes in
structural plasticity. While the lectures begin with a brief survey of the most important results of MP,
ofp articular interest is the unifying pedagogic role that complementarity theory brings to a broad class
ofp roblems in structural mechanics.
In the long term, the enthusiasm with which a subject becomes popularly embraced is
determined, it seems to me, by the scope it affords for computational and theoretical development.
Practical espousement is largely dependent upon the availability of efficient commercial software. Some
notable achievements for plastic limit analysis and design are described in one of the following lectures,
but the consensus must be that not nearly enough has yet been done on software and algorithm
development, particularly for large-scale structural systems. On the theoretical front, however, the
subject is vibrantly healthy. The ability of MP and complementarity theory to encompass a class of
problems much wider than those of structural plasticity, has already been mentioned. Furthermore, the
exploration of links between finite-dimensional complementarity problems and the variational
inequalities of continua seems to be providing an ideal mathematical basis for mechanical problems in
which the essential characteristics (material or structural) are non-smooth.
As I write these prefatory remarks, I am constantly reminded of one whose duty it should have
been. The C.I.SM. Advanced School in MP Methods in Structural Plasticity was conceived by the late
John Munro and was to be coordinated by him. Sadly, Professor Munro died on 27 February 1985.
The subject of his Chair -Civil Engineering Systems -was the vehicle through which he sought to
broaden perspectives in civil engineering education. His finest original researches resulted from a
special ability to distil engineering wisdom by reducing a problem to its simplest terms: if that required
the questioning of established axioms, he was prepared to do it. The passing of this refined and
sensitive man has removed a singular spirit and, for many, a most respected friend.
A centra/figure in MP Methods in Structural Plasicity is Giulio Maier. While unable to take an
active part in the Advanced School, he remained a constant source of encouragement and advice to this
writer: to Professor Maier I am much in debt. In my closing address at the Advanced School/
conveyed the sincere thanks to the lecturers to the secreterial staff of C.I.SM., especially Miss Elsa
Venir, and to the participants who, through informal and interactive discussions, made a valuable
contribution to the success and enjoyment of the meeting. Time has not changed those sentiments.
David Lloyd Smith
CONTENTS
Page
Preface
Chapter 1
Mathematical programming
by D. Lloyd Smith . ................................................................................... 1
Chapter2
Linear programming
by D. Lloyd Smith ................................................................................... 2 3
Chapter 3
Quadratic programs and complementarity
by D. Lloyd Smith ................................................................................... 3 7
Chapter4
Statics and kinematics
by D. Lloyd Smith .................................................................................. 4 7
Chapter 5
Plastic limit analysis
by D. Lloyd Smith .................................................................................. 6 1
Chapter6
Piecewise-linear elastic-plastic stress-strain relations
by J. A. Teixeira de Freitas ......................................................................... 8 3
Chapter?
Complementarity problems and unilateral constraints
by A. Borkowski ................................................................................... 1 1 5
Chapter 8
Elastic-plastic analysis of structural cross-sections
by J. A. Teixeira de Freitas ....................................................................... 1 3 5
Chapter9
Elastoplastic analysis of skeletal structures
by J. A. Teixeira de Freitas ....................................................................... 1 5 3
Chapter 10
A gradient method for elastic-plastic analysis of structures
by J. A. Teixeira de Freitas ....................................................................... 1 7 1
Chapter 11
Plastic shakedown analysis
by Nguyen Dang Hung and P. Morelle ........................................................ 1 8 1
Chapter 12
Optimal plastic design and the development of practical software
by Nguyen Dang Hung and P. Morelle ........................................................ 2 0 7
Chapter 13
Variational statements and mathematical programming formulations in
elastic-plastic analysis
by L. Corradi ....................................................................................... 2 3 1
Chapter 14
Finite element modelling of the elastic-plastic problem
by L. Corradi ....................................................................................... 2 5 5
Chapter 15
Rigid plastic dynamics
by D. Lloyd Smith and C. L. Sahlit ............................................................. 2 9 3
Chapter 16
Bounding techniques and their application to simplified plastic analysis of structures
by T. Panzeca, C. Polizzotto and S. Rizzo ..................................................... 3 1 5
Chapter 17
Mathematical programming methods for the evaluation of dynamic
plastic deformations
by G. Borino, S. Caddemi and C. Polizzotto ................................................... 3 4 9
Chapter 18
Structural analysis for nonlinear material behaviour
by J. A. Teixeira de Freitas ....................................................................... 3 7 3
Chapter 19
Large displacement elastoplastic analysis of structures
by J. A. Teixeira de Freitas ....................................................................... 3 8 7
Chapter20
Ultimate load analysis by stochastic programming
by K. A. Sikorski and A. Borkowski ......................................................... .4 0 3
Chapter21
Fuzzy linear programming in plastic limit design
by D. Lloyd Smith, P-H. Chuang and J. Munro ............................................. .4 2 5
CHAPTER 1
MATHEMATICAL PROGRAMMING
D. Lloyd Smith
Imperial College, London, U.K.
ABSTRACT
A brief review is made of those basic aspects of the theory of mathematical
programming which are of most relevance in the mathematical description of the theory
of structures with plastic constitutive laws. Firstly, the classical problem of optimisation
is used to introduce the method of Lagrange multipliers. The same method is then
applied to a mathematical program with inequality constraints, and the necessary
optimality criteria of Karush, Kuhn and Tucker are obtained. With the imposition of
convexity, the concept of duality in mathematical programming is described, and the
solutions of the consequent dual mathematical programs are related to the saddlepoint
property of the associated Lagrange function.
2 D. Lloyd Smith
INTRODUCTION
Mathematical programming provides an appealing formalism for encoding the behaviour
of discrete structures formed from materials in which plasticity is a constitutive
component. It can be associated with any basis for the discretisation of a structure,
especially that of the finite element; it offers a refined mathematical foundation, rich in
theorems which present interesting and useful structural interpretations; it has a small
collection of well-constructed algorithms through which numerical solutions may be
obtained. The integration of such features sets practical problem-solving in structural
plasticity within a strong scientific discipline: one could hardly ask for more.
Optimal design provides an obvious field of application for mathematical programming.
In the absence of viable alternative analytical methods, it has proved really quite
successful; and yet one might feel a sense of surprise that optimisation has not so far
had a stronger impact upon engineering design. In engineering plastic analysis, a
considerable body of literature [1 ,2] attests to the vigour with which mathematical
programming is being used to explore an ever-widening range of problem classes, to
unify their theoretical foundation and to obtain general results of relevance in practical
design.
For those problems of plastic analysis most often addressed in relation to practical
design, other, quite different, methods have achieved popular currency. The nonlinear
boundary value problem of incremental elastoplastic analysis, for instance, is usually
solved by an ad hoc numerical scheme in which a linear elastic solver is employed
iteratively within each increment of the control parameter. There is no clear evidence
that such procedures are more efficient than that of a properly formulated mathematical
programming approach. What is clear, however, is that a considerable body of
research effort has been expended on the development of reliable and professionally
written computer software for the analysis of linear elastic systems. To extend the
range of application of the procedures to elastoplastic systems is an entirely sensible
capitalisation on the previous investment of effort; and the availability of good software
ensures its own reward. The lesson is transparent: for whatever appealing features it
may have, mathematical programming will only receive widespread professional utilisation
through the development of high quality applications software.
Structural mechanics is a consistently logical and intellectually challenging scientific
discipline. It uses appropriate mathematical apparatus to describe, as closely as
possible, all the relevant physical laws that must be obeyed by a structural system
which is subjected to a specific disturbance in its environment, and to deduce general
results or theorems pertaining to its response. It demands frequent development to
accord with innovation in structural form, construction materials and in mathematics.
In the longer term, the progressive use of particular mathematical apparatus in
structural mechanics may be determined by its scope for encompassing a broader class
of relevant problems. We may observe that the mathematical descriptions of structural
plasticity, of the contact between elastic bodies and of the movement of bodies against
frictional forces have certain features in common. The setting of such disparate
physical problems within a generalising mathematical context often conveys greater
physical insight and certainly widens the circle of interaction between those who will
provide the impetus for advancement. The developing and generalising field of
non-smooth mechanics[3] displays an elegant, if rather sophisticated, mathematical
