Table Of ContentINTERNATIONAL CENTRE FOR MECHANICAL SCIENCES
COURSES AND LECTURES -No. 308
STRUCTURAL OPTIMIZATION UNDER
STABILITY AND
VIBRATION CONSTRAINTS
EDITED BY
M. ZYCZKOWSKI
TECHNICAL UNIVERSITY OF CRACOW
Springer-Verlag Wien GmbH
Le spese di stampa di questo volume sono in parte coperte da
contributi del Consiglio N azionale delle Ricerche.
This volume contains 111 illustrations.
This work is subject to copyright.
All rights are reserved,
whether the whole or part of the material is concerned
specifically those of translation, reprinting, re-use of illustrations,
broadcasting, reproduction by photocopying machine
or similar means, and storage in data banks.
e
1989 by Springer-Verlag Wien
Originally published by CISM, Udine in 1989.
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 978-3-211-82173-2 ISBN 978-3-7091-2969-2 (eBook)
DOI 10.1007/978-3-7091-2969-2
PREFACE
Optimal design of structures leads, as a rule, to slender and thin-walled shapes of the
elements, and such elements are subject to the loss of stability. Hence the constraints of
structural optimization usually include stability constraints. Loading parameters corresponding
to the loss of stability are, in most cases, expressed by eingenvalues of certain differential
equations, and hence the problems under consideration reduce to minimization of a certain
functional (volume) under eigenvalues (critical loadings) kept constant. Briefly, wa call such
problems "optimization with respect to eigenvalues", though in many cases the eigenvalue
problems are not visible explicitely.
Optimal design under vibration constraints is related to that under stability constraints
because of at least two reasons. First, the vibration frequencies are also expressed by
eigenvalues of some differential equations, and hence the relevant problems belong also to
optimization with respect to eigenvalues. Second, in nonconservative cases of structural
stability we usually have to apply the kinetic criterion of stability, analyzing the stability of
vibrations in the vicinity of the equilibrium state: hence both problems are directly
interconnected in such cases.
The course on structural optimization under stability and vibration constraints was
given in Udine, 20-24 June 1988, by five researchers particularly active in this field, namely
prof M. Zyczkowski (coordinator) and prof. A. Gajewski from the Technical University of
Cracow, Poland, prof. N. Olhofffrom the University of Aalborg, Denmark, prof. J. Ronda/
from the University of Liege, Belgium, and prof A. P. Seyranian from the Institute of
Problems of Mechanics in Moscow, U.S.S.R.
Part I, by M. Zyczkowski, deals just with optimal structural design under stability
constraints. It gives first a general introduction to structural optimization, discussing typical
objectives, design variables, constraints and equations of state. Then a chapter is devoted to
optimization of shells in the elastic range, but the remaining chapters deal with optimal design
under inelastic stability constraints, mainly· creep buckling constraints (columns, trusses,
arches, plates, shells).
Part 1I, by A. Gajewski, is devoted to optimal design both under stability and vibration
constraints. Basic optimization methods are discussed in detail, and then particular attention is
paid to multimodal optimal design resulting from simultaneous analysis of buckling in two
planes (columns and arches). Further problems discussed in this part are: optimization of a
plate under nonconservative forces, viscoelastic column compressed by a follower force
optimized with respect to its dynamic stability, and a plane bar system in conditions of internal
resonance.
Also part III, by N. Olhoff, deals with optimal structural design under both stability
and vibration constraints, namely with optimal design of conservative mechanical systems
with respect to fundamental and higher order eigenvalues. A unified variational approach for
optimal design of one-dimensional continuum systems with respect to simple fundamental
eigenvalue is presented first, and then extended to multimodal design and higher order
eigenvalues. Examples concern natura/frequencies of axial, torsional and transverse vibration
of rods and beams, critical whirling speeds of rotating shafts, and structural buckling loads.
Part IV, by J. Ronda!, is devoted just to optimal structural design under stability
constraints, namely to optimal design of thin-walled bars and beams. In this case not only
single buckling modes must be taken into account (local plate buckling, flexural column
buckling, torsional buckling, lateral buckling), but also interactive buckling between single
modes. Effects of imperfections are studied in detail, and the results are presented in various
efficiency charts. This part of the present volume is particularly oriented towards engineering
applications.
Part V, by A. P. Seyranian, deals with optimal structural design under both stability
and vibration constraints, and mainly with flutter constraints. Sensitivity analysis is used as a
tool of optimization: it is presented both for discrete cases and for distributed cases.
Gyroscopic systems, nonconservative problems of elastic stability, aeroelastic stability of
panels in supersonic gas flow, and bending - torsional flutter of a wing are considered in
detail.
A reader of the present volume is expected to be familiar with basic problems and
methods of structural optimization, as well as with fundamentals of structural stability and
vibrations.
Finally, the efforts of the CISM Rector, Prof. S. Kaliszky, CISM Secretary General,
Prof. G. Bianchi, the Editor of the Series, Prof. C. Tasso, and all the CISM Staff in Udine
are gratefully acknowledged.
Michal Zyczkowski
Politechnika Krakowska
CONTENTS
Page
Preface
PART I
by M. Zyczkowski
Abstract ........................................................................................... 1
1. General introduction to structural optimization ..........................................' :'2
2. Contours of complete non uniqueness in the case of stability constraints ............ 13
3. Optimal design of cylindrical shells via the concept of uniform stability ............. 20
4. Optimal design of inelastic columns ...................................................... 3 2
5. Multimodal optimization of circular arches against creep buckling ................... 3 7
6. Optimal design of plates and shells under creep buckling constraints ............... .49
7. Recent results on optimal structural design under stability constraints ............... 60
References ...................................................................................... 62
PART II
by A. Gajewski
Abstract ........................................................................................... 69
1. Selected methods of structural optimization ............................................. 7 0
2. The optimal structural design of columns ................................................ 79
3. Multimodal optimal structural design of arches ........................................ 10 3
4. Non-conservative optimization problems of annular plates .......................... 122
5. The parametrical optimization of simple bar systems in certain dynamic
problems ................................................................................... 13 1
References .................................................................................... 13 8
PART III
by N. Olhoff
Abstract .......................................................................................... 145
Introduction .................................................................................... 146
1. Optimal design of one-dimensional, conservative, elastic continuum
systems with respect to a fundamental eigenvalue ..................................... 14 8
2. Optimal design of Euler columns against buckling. Multimodal formulation ...... 161
3. Optimization of transversely vibrating Bernoulli-Euler beams and rotating
shafts with respect to the fundamental natural frequency or critical speed .......... 18 5
4. Optimization of Bernoulli-Euler beams and shafts with respect to higher order
eigenfrequencies ........................................................................... 19 5
References .................................................................................... 207
PARTN
by J. Ronda/
Abstract .......................................................................................... 213
1. Introduction ............................................................................... 214
2. Stability modes of thin-walled bars ..................................................... 216
3. Mode interaction in thin-walled bars .................................................... 235
4. Efficiency charts ........................................................................... 240
5. Optimal design and manufacturability .................................................. 254
6. Optimal ranges of bars .................................................................... 256
7. Optimal design and mathematical programming methods ............................ 262
References .................................................................................... 2 7 0
PARTY
by A. P. Seyranian
Abstract ......................................................................................... 2 7 5
1. Introduction ................................................................................ 2 7 6
2. Sensitivity analysis of vibrational frequencies of mechanical systems .............. 2 7 7
3. Sensitivity analysis for nonconservative problems of elastic stability
(discrete case) .............................................................................. 2 8 3
4. Sensitivity analysis for nonconservative problems of elastic stability
(distributed case) .......................................................................... 290
5. Optimization of critical loads of columns subjected to follower forces ............. 2 99
6. Optimization of aeroelastic stability of panels in supersonic gas flow .............. 3 0 5
7. Bending-torsional flutter. Influence of mass and stiffness distributions on
aeroelastic stability ........................................................................ 3 12
8. Bending-torsional flutter. Optimization of aeroelastic stability ...................... 320
References .................................................................................... 3 2 7
PART I
M. Zyczkowski
Technical University of Cracow, Cracow, Poland
ABSTRACT
Part one consists of seven chapters corresponding to seven
lectures delivered. The first chapter gives general
introduction to structural optimization, discusses typical
objectives, design variables,. constraints and equations of
state. Chapter 2 applies the concept of local shell buckling
to optimization of elastic shells under stability
constraints. The remaining chapters are devoted to
optimization with respect to plastic or creep buckling:
trusses, columns, arches, plates and shells are optimized, in
most cases with rheological properties of the material
allowed for. The last chapter gives a short survey of recent
results, obtained within the years 1984 - 1988.
2 M. Zyczkowski
1. GENERAL INTRODUCTION TO STRUCTURAL OPTIMIZATION
1.1. Formulation of optimization problems
Extensive use of computers resulted in a rapid progress
in structural analysis. However, the next and more advanced
step consists in replacement of analysis by synthesis, namely
by optimal structural design.
Mathematical problems of optimal structural design - as
of most optimization problems consist of four basic
elements: design objective, control variables or decisive
variables called here design variables, constraints, and
equations of state. We look for upper or lower bound <ma:<imum
or mini~AWR) of the design objective specified as a function
or functional of design variables, functions or vectors <sets
of functions or of parameter<5) under certain constraints.
These constraints are usually expressed in terms of some
other variables, called state variables or behavioural
variables; they may also appear in the objective function and
are interrelated and related to design variables by the
equations of state.
A proper choice of the design objective and of
ronstraints constitutes the most important point of the
ciesign philosophy. As a "design objective" or a "constraint"
we understand here a notion or an idea rather than its
mathematical expression. Such a notion may be insufficiently
defined or its definition may be non - unique. Then a certain
criterion is needed to specify the objective function or the
constraint function and to formulate uniquely the relevant
mathematical optimization problem. Sometimes even
substitutive criteria are introduced. Such criteria used in
structural optimization will be discussed in detail in Sec.
1.2. and 1.4. The above remark refers to deterministic
approach; in probabilistic approach such criteria specifying
ubjectives and constraints are even more necessary.
We discussed here design objectives and constraints in
the same manner. Indeed, in some optimization problems the
role of the design objective and of a constraint may be
J.nterchanged. Such an interchange is called sometimes
alternate equivalent, dual or mutual formulation of the
problem <though in mathematical programming the term "dual"
is used also for other interchanges>.
It is convenient to present a structure under
cun•Uderation as a point in an abstract design space, with
t:he design variables serving as coordinates. Any solution
satisfying the constraints and equations of state is called
sn acceptable or feasible solution. In general, optimum is
then not achieved. To find an optimum we have to derive a
General Introduction to Optimization 3
relevant necessary condition, called the optimality
condition. Such conditions in structural optimization were
discussed in detail by W.Prager [1,2J, W.Prager and J.Taylor
[3J, L.Berke and V.B.Venkayya (4J, M.Save [5J, C.Fleury and
M.Geradin [6J, Z.Mr6z and A.Mironov [7J. Sufficient
conditions, important from the theoretical point of view-• are
often much more difficult to be f.ormulated and employed. A
distinction between the local extremum and a global extremum
inside the whole admissible domain is particularly important
here.
The first paper on structural optimization is ascribed
to Galilee Galilei (8J, 1638, who discussed optimal shape of
beams. At present, the number of papers devoted to optimal
structural design, e>:ceeds five thousand. They are discussed
in detail in monographs, te}{tbooks and survey papers; an
e>:tensive list of sources is given in the monograph by
A.Gajewski and M.Zyczkowski [9].
1.2. Design objectives and their criteria
The cost is the most typical objective in many
optimization problems. However, this objective is not
uniquely defined and various criteria must be introduced to
specify the relevant objective function.
In structural optimization the total cost consists, as a
rule, of three elements: material costs, manufacturing costs
and exploitational coasts. If the material is preassigned,
then the material cost is proportional to the volume of the
structure. Similarly, exploitational costs are in many cases
proportional to the mass, and hence to the volume of the
structure (first of all we quote here aircrafts and rockets,
but also all vehicles etc.>. Hence, if the manufacturing
costs are not particularly essential, then volume of the
structure constitutes a reasonable substitutive criterion for
the cost as a design objective.
It should be noted that even if the volume of the
structure is chosen as a design objective, a further
specification is sometimes necessary. For example, if the
structure consists of various materials, then an introduction
of certain weight coefficients is justified: they are
different for material costs (unit prices> and for
exploitational costs <specific weight>. A reasonabl&
criterion to formulate the objective function should then be
introduced.
In the simplest case the volume is a functional
determined as follows
for a bar structure
