ARCHIEF
Recent Deveiopmens ¡n Otmc!
Structural Design
C. Y. SHEU AND W. PRAGER
UNIVERSITY OF CALIFORNIA, SAN DIEGO LA JOLLA, CALIFORNIA
very comprehensive review of the literature on optimal structural design from Galileo to the
end of 1962 appeared in Applied Mechanics
Reviews in May, 1963 [Ref. i]. The present review of this field is therefore restricted to papers published after 1962. The general background of structural optimization is developed in Section 1. Contributions to the general methodology of the field are reviewed in Section 2, so-lutions of specific problems in Section 3, and relevant books and expository articles in Section 4.
1. NTRODUCTION
To avoid overlap with the comprehensive review by Wasiutyr(ski and Brandt [Ref. i] in the May 1963 issue
of Applied Mechanics Reviews, the present review of work on optimal structural design is essentially re-stricted to papers published after 1962. That another review seems worthwhile after so short a period of time indicates the increasing importance of the field and the
vigor of its development. This vigor is also
demon-strated by the fact that the present review, covering a
span of only five years, has more than half the number of references of the former paper.
A well-posed problem of optimal structural design
in-volves specifications of: (1) the purpose of the
struc-ture, (2) the geometric design Constraints, (3) the be-havioral design constraints, and (4) the design objective.
The general purpose of a structure is to support given static or dynamic loads in given environments. Whereas a
struc-ture has in general to carry several alternative systems of
loads, it may happen that the design of the structure is gov-erned by a single one or a few of these. The description of
the environment may specify a single application or re-peated applications of the loads, and a fixed or random sequence of alternative load systems. Further elements of the specification of environment concern the
tempera-ture range in which the structempera-ture has to serve, its
ex-posure to corroding or cavitating fluids, etc. The terms
"single-purpose" or "multipurpose structure" will be
used to indicate whether a structure serves only a sin-gle purpose or several purposes.
Geometric design constraints specify at
least the
space that is available for the structure, but usually goLab. y.
Scheepsbouwktmde
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much further in restricting the choices of structural type and shape that are open to the designer. Consider, for example, a circular plate of radius R under
rotationally-symmetric loading. A minimal geometric constraint
would set an upper bound H on the thickness of the plate but leave the designer free to choose the manner
in which the thickness h varies with the distance r from
the center. Somewhat tighter geometric constraints might divide the planform of the plate into a central
cir-cular area and n - i surrounding concentric rings with
given radii and require that the plate thickness have a constant value in each of these parts. In the first case, a design is specified by a function h (r) H defined in
O r R; in the second case, by n thickness values
h1, h..., h,, none of which can exceed H. Still tighter geometrical constraints would result from the
require-ment that h1, h2, . .., h,, must be selected from a finite
number of available thicknesses. Similar problems con-cerning a finite number of available sections are treated in Refs. 74, 136, and 138.
Behavioral design constraints set bounds on
quanti-ties that characterize the behavior of the structure un-der the conditions of service for which it is being
de-signed. Examples of constraints of this kind are upper
bounds on certain stresses or deflections in the elastic
range or rates of deflection in stationary creep, and lower bounds on natural frequencies or on the ratio in which the given loads would have to be increased
be-fore they cause failure by structural instability or un-restricted plastic flow. The terms ''single-constraint''
or "multiconstraint design" will be used to indicate
whether a design is subject to a single behavioral con-straint or several behavioral concon-straints.
The general design objective is minimization of the
combined cost of the manufacture of the structure and its operation over the expected lifetime. It is typical for aerospace structures that the cost of the fuel that
would be needed to carry additional weight and the
ac-companying reduction in payload are much more
im-portant considerations than the manufacturing cost. In
these circumstances, minimization of structural weight
becomes the sole design objective. A vast majority of
papers on optimal structural design are concerned with minimum-weight design, but the combined costs of
terial and manufacture are considered by some authors [Refs. 2, 3, 7, 136, 1401. Again, a vast majority of
pa-pers view optimal structural design as a deterministic
problem, but attempts have been made at probabilistic treatment in which loads and allowable stresses are re-garded as random variables with given probability
dis-tributions [Refs. 4-7, 137].
2. GENERAL METHODS
Traditionally, near-optimization of structural design
has been achieved by trial and error based on experi-ence with structures of similar purpose. Several de-signs in the neighborhood of an intuitive optimum are deve loped to a point where their relative merits can be
assessed and the best design be chosen. The success
of this method obviously depends on the available ex-perience with similar designs; it works well in a period
of orderly evolution of designs, but cannot cope with
revolutionary changes. It is not surprising, therefore, that interest in more rational methods of optimal design has sharply increased in recent years when the develop-ment of aerospace, ocean, and nuclear engineering has brought revolutionary changes in design concepts and
requirements.
The general mathematical character of problems of
optimal structural design is most readily discussed in
connection with problems involving discrete design
pa-rameters (e.g., cross sections of members of a truss)
rather than continuous design functions (e.g., continu-ously-varying bending stiffness of a beam). Consider,
for instance, an elastic truss, with given layout of its
n bars, that is internally and externally determinate and subject to a single load applied at an unsupported joint. Subject to the behavioral design constraint that the
elas-tic response of the truss to the load, defined as the
virtual work of the load on the elastic displacement of its point of application, have a given value R, the crosssections of the bars are to be determined to minimize
the total volume V of the bars.
If E is Young's modulus of the material used for the bars, and the length, cross-sectional area, and (statically-determinate) axial force of the typical bar are denoted
by ,j, A1, and F1, respectively, the equivalent problem of minimizing the response R for a given structural
vol-ume V may be formulated as follows:
For given values of the constants ',j > 0, F, (i = 1,
n), and V, determine the values of the non-negative variables A1, (i 1, .. ., n), to minimize
R
',j F2/(E A)
(1) subject to'jA1=V.
(2)This is a nonlinear programming problem. In the
do-main A 0, (i =
i...n),
of an o-dimensional de-sign space with the rectangular Cartesian coordinates A1, the nesting surfaces corresponding to constantpositive values of R are convex towards the origin, from
which they recede as the value of R is decreased. On account of their convexity, only one of these surfaces is tangent to the hyperplane (2); the point of contact
represents the unique optimal design. A necessary and sufficient condition for optimality states that R and V,
considered as functions of the design variables A1, have gradients of opposite directions at this point. lt
follows from this condition that (F1/A1)2 is independent
of i; that is, the optimal truss is uniformly stressed.
This enables the designer to take care of the additional
behavioral constraint that the axial stress u
in the typical bar should be in the allowable interval cre cjiAs this simple example shows, naturally discrete or
artificially discretized problems of optimal elastic
de-sign are essentially problems of nonlinear programming.
The fact that the behavioral design constraint can be written as an explicit function of the design variables [ see (1)1 greatly facilitates the solution but is, un-fortunately, not typical for problems of optimal struc-turai design. Accordingly, use of the analytical and
numerical methods of nonlinear programming [see, for
instance, Ref. 8] may be difficult or even impossible.
As a consequence of this, a wide variety of techniques
are advocated, including linearization [Ref. 9], linear
perturbation and Monte Carlo methods [Ref. io], dynamic
programming [Refs. 2, 136], and steepest descent
[Ref. 3] and gradient projection methods [Refs. 11, 146]. Fox [Ref. 12] describes a technique that may be useful in overcoming the handicap resulting from the lack of
an explicit expression for the behavioral design con-straint in terms of the design variables. While some
papers in this area are computer-oriented [Refs. 11, 13-. 15, 19], others primarily explore the general mathemati-cal character of various optimization problems [Ref.
16-22] or general features of optimal designs [Refs. 23, 144].
Because the truss considered above was statically
determinate, the bar forces F1 in (1) could be
deter-mined without reference to the choice of the cross
sec-tions A. In the optimal design of a
statically-indeter-minate truss of given layout of bars, both the behavioral
variables F1 and the design variables A1 must be de-termined to minimize the response R in (1). The bar forces F1 are subject to the equations of equilibrium at
the joints, and the cross sections A to the condition (2) and the inequalities A1 0. Minimization with re-spect to the behavioral variables F is dictated by the
principle of minimum complementary energy, and mini-mization with respect to the design variables A1 by the
desire to achieve the smallest response possible for a
given structural volume. By the use of Lagrangean mul-tipliers and slack variables, the constrained minimiza-tion problem may be transformed into an unconstrained
problem. Alternatively, non-negative penalty functions that vanish when the Constraints are satisfied may be
used, as proposed by Schmit and Fox [Ref. 24].
It is worth noting that minimization with respect to the cross sections A1 again furnishes the condition that F?/Al2 should be independent of i. It follows that all
bars should experience unit extensions of the same
absolute value, a condition that cannot, in general, be
fulfilled in an indeterminate truss. This means that the optimal truss will as a rule be statically determinate.
Since the given layout of the truss may involve many
joints and bars that need not appear in the statically
determinate optimal truss a discretized form of the prob-lem of optimal layout is included in this formulation of
the problem [Ref. 251. Of course, when the admissible
positions of joints are not discretized in this manner, the optimal truss is a Michell truss. Several
contribu-tions to the theory of these trusses appeared in the
re-view period of Refs. 26-30. Whereas the usual behavioral
design constraint in this theory is an upper bound on the allowable absolute value of axial stress, it follows
from the preceding discussion that the elastic response
may be prescribed as alternative design constraint. lt
can, moreover, be shown that the Michell layout is also optimal for several other design constraints, for exam-ple a given fundamental natural frequency of a truss of
massless bars that carries given point masses at t'ne
joints [Ref. 31]. Note, however, that while the layout
of bars is the same, the cross sections of this optimal
truss differ from those of the truss with prescribed elastic response to static loads.
Whereas optimal structurai design in the elastic range involves nonlinear programming, optimal design for a given load factor at plastic collapse may often be for-mulated as a problem in linear programming, as was first
recognized by Foulkes [Ref. 32]. Since the
straight-forward application of the simplex method to optimal
plastic design of, say, a building frame with numerous
stories and bays involves very large numbers of con-straints and variables, special methods have been de-veloped [see, for instance, Ref. 33]. The large
rapid-access memories of recent computers, however, have all but eliminated the need for methods of this kind. Toakley [Ref. 34] states that the simplex method is as efficient as the special methods and estimates that the
capability of handling linear programming problems with
10,000 constraints and 30,000 variables may be
ex-pected in 3 to 5 years.
The linear character of these problems of optimal plastic design is due to the assumption of a linear re-lation between plastic resistance (e.g., limiting value
of bending moment of a beam) and unit weight. If this
assumption is not justified, or if a lower bound is set
on plastic resistance, problems of convex programming may result [Refs. 35-37].
As has already been indicated in the discussion of optimal design of a truss for given elastic response,
extremum principles of structural theory may be used to
establish sufficient conditions of optimality. This is pointed out by Prager and Taylor [Ref. 38] who illus-trate the technique by examples pertaining to optimal
design for given elastic response, buckling load, funda-mental natural frequency, or load factor for plastic col-lapse. Whereas these authors establish the necessity of their optimality conditions by variational calculus, Sheu and Prager [Ref. 39] indicate a simpler way of
proving necessity. While these papers are concerned with optimal design of sandwich beams and plates, much more general problems were treated by MrSz [Ref. 40] and Prager [Ref. 41]: the optimal structure occupies the
volume V with the surface S, and each element of S is
supposed to belong to one and only one of the sets S', S", and S", where nonvanishing surface tractions are
prescribed on S'; vanishing surface tractions on S", and
vanishing displacements on S". The surfaces S' and S" and the boundary conditions thereon are treated as
fixed ingredients of the design problem, so that any al-ternative structure can differ from the optimal structure
only in the shape of the traction-free surfaces. The
considered behavioral design constraints are elastic response or load factor for plastic collapse in [Ref. 40]
and any quantity that can be characterized by an
ex-tremum principle in Ref. 41.
Prager and Shield [Ref. 42] extend the method of
Ref. 38 to multiconstraint design. Optimum plastic
de-sign for multiple loading is discussed by Shield [Ref. 43].
3. SPECIFIC PROBLEMS
(a) Static compliance. Depending on the field of ap-plication, the compliance of a structure under given static loads may be defined as (i) the maximum deflec-tion, (ii) the deflection at a specified point, or (iii) the virtual work of the given loads on the displacements of their points of application.
Constraint (i) is treated by Haug and Kirmser [Ref. 44],
and constraint (ii) by Barnett [Refs. 45, 46]. In special circumstances, where the location of the point of
maxi-mum deflection is known a priori, for example from
sym-metry considerations, constraints (i) and (ii) are
inter-changeable. As Barnett [Ref. 45] points out, however, constraining a specified rather than the maximum
de-flection may yield paradoxical results. For example,
when some loads acting on a horizontal beam are
di-rected downwards and others upwards, there may exist
a design with vanishing deflection at the specified
point. Since this design will continue to have
vanish-ing deflection when all cross-sectional dimensions are
reduced in proportion, the design Constraint is
satis-fied by designs of arbitrarily-small weight.
Tadjbaksh [Ref. 47] discusses the optimal design of a nonlinear cantilever elastica for given tip deflection.
Whereas the usual aim of designs with stiffness
con-straints is to obtain the greatest stiffness that can be achieved with a givqn amount of a specific structural
material, it occasionally becomes necessary to design
for maximum compliance with a given allowable stress. A problem of this kind, concerning the suspension of a satellite during altitude control tests, is treated by Mansfield [Ref. 48].
The design constraint (iii) above is discussed by
Dzieniszewski [Ref. 49] and Wasiutydski [Refs. 50, 51] [see also Refs. 31, 38-42]. In Ref. 51, theinflu-ence
of the structural weight on the
deflections istaken into account. When this becomes necessary,
how-ever, the Constraint (iii) is less readily justified. Con-sider, for instance, a beam on two supports carrying a
given central load in addition to its own weight. When
the influence of the latter can be neglected, the
com-pliance definitious (i), (ii), (iii) are equally acceptable.
When structural weight must be considered, the
com-bined virtual work of central load and structural weight
on the deflections they produce is a less convincing
measure of compliance than maximal deflection, though
it does of course increase as the stiffness of the
struc-ture is reduced.
As has been illustrated by the truss example in
Sec-tion 2, the optimal design for given elastic compliance is often identical with the optimal design for given
al-lowable stress. Moreover, in view of the well-known
analogy between body forces and thermal gradients
subjected to given thermal gradients can be treated in essentially the same manner, provided the temperature field is regarded as given a priori and not as dependent
on the unknown distribution of Structural material.
Hackman and Richardson [Ref. 53] discuss this kind of
problem.
Optimal design for given rate of compliance in
sta-tionary
creep is discussed by Prager [Ref. 54] and
Hegemier and Prager [Ref. 311.
(b) Dynamic compliance. Constraints on dynamic
compliance of a structure may be formulated as
con-straints on the fundamental natural frequency or as any
one of the constraints (i), (ii), (iii), considered under (a) above, on the deflections under impulsive loading, step loads, or harmonically-varying loads.
The first kind of constraint is discussed by Niordson [Ref. 551, Turner [Ref. 561, Taylor [Refs. 57, 58],
Zar-ghamee [Ref. 591, and Sheu [Ref. 60] [see also Ref. 381, and the second by Prager [Ref. 61] and Icerman [Ref. 621.
Optimal design of a wing for torsional divergence is
treated by Mcintosh and Eastep [Ref. 631, and optinal
design of rotors for fundamental critical speed by
French [Ref. 641. Brach [Ref. 65] discusses optimal design of beams for dynamic deflection under impulsive or step loading.
(e) Buckling load. Optimal design for prescribed
buckling load is treated by Tadjbaksh and Keller [Ref. 661,
Keller and Niordson [Ref. 671, Taylor [Ref. 68], Taylor and Lio [Ref. 691, and Huang and Sheu [Ref. 70] [see also Ref. 3S1. Whereas these authors consider only a
single buckling mode, Shanley [Ref. 71] discusses the case in which several buckling modes need to be con-sidered (e.g., column buckling and local buckling of the wall of a tubular column). Optimality is then generally assumed to require simultaneous occurrence of the
di-verse buckling modes and yielding in compression.
Cohen [Ref. 72] shows, however, that this criterion is not generally valid and gives sufficient conditions for
its validity.
Load factor at plastic collapse. Optimal plastic
design of rings is treated by Prager [Ref. 731; and opti-mal plastic design of building frames, by Rubinschmidt and Karagozian [Ref. 142] (who limit the elastic
deflec-tion under service loads), and Hill [Ref. 143], and Bigelow
and Gaylord [Ref. 741, who consider a nonlinear relation between plastic moment and unit weight. Optimal
plas-tic design of shells is discussed by Felton and Dobbs
[Ref. 751, Hoffman [Ref. 761, who studies the influence of the yield condition, and Sharniev [Ref. 771. Optimal plastic design of plates is treated by Lackman and Ault
LRef. 781, Mancai and Prager [Ref. 351, Marçal [Ref. 36]
and Megarefs [Ref. 79]. The last author draws attention
to singularities of the thickness of annular plates that
occur unless an upper bound is set on thickness. Sheu and Prager [Ref. SO] treat this effect for annular plates with ringwise constant thickness and study the transi-tion to the Megarefs singularity as the number of rings is increased indefinitely.
Multipurpose and rnuliiconszraint designs. Elastic
problems in this area are treated by Ghista [Refs. 4, 17], Marks [Ref. sil, Prager and Shield [Ref. 42], and
Wasiutyrski [Ref. 821; and plastic problems, by ibragi-may [Ref. 831, Gross and Prager [Ref. 841, Mayeda and
Prager [Ref. SS], Prager [Ref. 86], Save and Prager
[Ref. 871, Save and Shield [Ref. 881, and Shield [Ref. 431.
It is worth noting that the optimality criteria derived in Reis. 42 and 82 do not agree. Refs. 81, 84, 87, and 88
treat moving loads that may act at any point of the
s tru e ture.
Optimal layout. In the vast majority of problems of optimal structural design treated in the literature, the
"skeleton" of the structure is given, and the designer is free only to put more or less "flesh'' here or there.
A small number of problems have, however, been
dis-cussed in which the layout of the structure is at the
choice of the designer. Foremost among these are truss problems [see Refs. 4, 24-311. Other problems
in this area are investigated by Brown [Ref. 89], Durelli, Parks, and Uribe [Ref. 90], Gerard and Lakshmikanrham [Ref. 911, Mazurkiewicz and Zyczkowski [Ref. 92], and
Schmidler [Ref. 931. The last paper discusses the un-usual features and requirements of orbiting structures
of large size.
( g) Reinforced and prestressed structures. Whereas
the most important development of the general theory of optimal structural design is likely to come in this area, the vast majority of papers published to date deal with specific problems in ways that are not readily general-ized to similar problems. There are only a very few
pa-pers concerned with general methodology.
Reinforcements of a structure may be classified as
external (e.g., stiffeners) or integrated (e.g., reinforcing
steel in concrete). Reis. 94-101 deal with a wide va-riety of problems of optimization of external
reinforce-ments as indicated by the titles of the papers.
Craw-ford and Burns [Ref. 951 introduce loading indices and efficiency factors in their discussion of stiffened Struc-tures. Crawford and Schwartz [Ref. 96] show that a rib-stiffened shell may have considerably higher
effi-ciency than a nibless shell of the same weight. Kloppel
and Molier [Ref. 971 and Rocky and Cook [Ref. 100]
in-vestigate the optimum distribution of ribs. Maksimov [Ref. 981 compares the efficiencies of
cylindrical-pressure vessels reinforced by wire-winding or built up from shrink-fitted tubes.
Less conventional problems in this general area are treated by Cloud [Ref. 1021, and Green and Lancaster
[Ref. 1031. Most papers on integrated reinforcements concern reinforced concrete [Refs. 7, 104-112], but
es-sentially the same problems are encountered with other types of composite structures [Reis. 113-1171. Optimal design of reinforced roncrete structures may be based on a rigid, perfectly-plastic model: at the limit load, the
reinforcement yields in tension; and the concrete, in
compression; the concrete cracks under tension, and
this may be interpreted as yielding under vanishing tensile yield stress. Massonnet and Save [Ref. 108],
Morley [Ref. 1091, Mrdz [Ref. 110] and Kaliszky [Ref. 139],
apply these concepts to the design of reinforced slabs.
Pipkin and Riviin [Ref. us] discuss isotensoid
de-signs, in which each fiber is under the same tensilestress, and relate the shape of the membrane to the
distribution of fibers on it. Schuerch and Burggraf
[Ref. 117] treat an axisymmetric filament-wound vessel
under internal pressure and centrifugal forces caused
by rotation about the axis. Cohn [Ref. 141] investigates
continuous beams under moving loads.
addi-tional degree of freedom: he may control not only the shape of the structure and its cross-sectional
dimen-sionsbut also the level ofprestressing. Brandt [Ref. 118]
uses a criterion of minimum strain energy for a given
volume of material andestablishes an interesting
theo-rem. Gawecki [Ref. 119] compares alternative design constraints for simply-supported beams. Rozvany and
Hampson [Ref. 1201 discuss a "transformed membrane
method" to minimize either the thickness of the plate or
the weight of the prestressing cables.
Rozvany [Ref. 121] develops a "reversed deformation method"and applies it to a beam that is suspended from an arch
by equidistant ties of the same cross section. If it is
assumed that all ties operate at the same tensile stress,
the beam and the arch become statically determinate, and their deformations, which are readily determined, furnish the extensions of the ties, which then are
suit-ably prestressed. Optimum design of prestressed plates and beam grillages is also discussed in Ref. 121.
BOOKS AND REVIEWS
The books by Cox [Ref. 122] and Owen [Ref. 123] are
primarily concerned with optimal structural design,
while the books by Massonnet and Save [Refs. 124, 1251 have extensive sections on this subject. Expository ar-ticles and more or less detailed surveys have been
writ-ten by Barnett [Ref. 126], Gerard [Ref. 127], Johnson [Ref. 128], Prager [Refs. 61, 129, 130], Reitman and
Shapiro [Ref. 131], Save [Ref. 132], Schmit [Ref. 133], and Woods and Sams [Ref. 134]. The proceedings of an ASCE joint specialty conference [Ref. 1351 contain ab-stracts of a number of papers on structural optimization
that were presented at this conference.
CONCLUDING REMARKS
Current work on optimal structural design proceeds on two fronts. On one hand, the numerical methods of
mathematical programming are being successfully
ap-plied to specific, highly realistic problems of optimal
design. All the same, the optimal design of complex
structures remains a formidable task, which often taxes
the capacities of present automatic computers. On the
other hand, considerable progress is being made in the analytical treatment of a variety of problems of optimal design of structural elements and simple structures. By
and large, these problems are not realistic because the structures are too simple; the geometrical constraints, too severe; and the behavioral constraints, too few in number. Any attempt at increasing the degree of
real-ism as a rule leads to problems that can no longer be
treated analytically. It therefore appears certain that
the treatment of realistic problems of optimal structural
design will have to resort to the iterative numerical
techniques of mathematical programming. The
analyti-cal treatment of selected simple problems nevertheless
serves a good purpose. It reveals faults in intuitive
design criteria that have long been taken as self-evident see Section 3(c) above for an example), and it provides a deeper insight into the analytical nature of optimality criteria and in this way greatly improves the efficiency of these numerical techniques by enabling the designer to choose a starting point that is closer to the optimum.
ACKNOWLEDGMENT
This review was prepared in the course of research
conducted in the Institute for Pure and Applied Physi-cal Sciences and supported by the Advanced Research
Projects Agency of the Department of Defense and
monitored by the US Army Research OfficeDurham
un-der Contract DA-31-124-ARO-D-257.
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1104-1112, 1965; AMR 19(1966), Rev. 2279.
25 Dorn, W. S., Gomory, R. E., and Greenberg, H. S., Automatic design ol optimal structures, J. de Mécanique 3,
25-52, 1964.
26 Chan, H. S. Y., Optimum Michcll framework for three parallel forces, Coil. Aeronautics, Cranfield, Rep. 167, 1963;
AMR 17(196.), Rcv. 5S25.
27 Chan, Ii. S. Y. , Tabulation of some layouts and virtual
displacement fields in the theory of Michell optimum struc-tures, Coil. Aeronautics, Cranfield, Note Aero. 161, 1964;
AMR 17(196.1), Rev. 6474.
28 Ghista, D. N., Optimum frameworks under single load system, Proc. ASCE 92, ST 5, 261-286, 1966; AMR 20(1967), Rev. 2505.
29 Ghista, D. N., and Resnikoff, M. M., Development of Michell minimum weight structures, NASA Tech. Note D-4345,
1968; AME 21 (1968), Rev. 2642.
30 Hemp, W. S., Studies in the theory of Michell
struc-tures, Proc. 11th Congr. Appl. Mech. (Munich 1964), Springer, Berlin 1966, pp. 621-62S; AME 20(1967), Rev. 4826.
31 liegemier, G. A., and Prager, W., On Michell trusses, University ai California, San Diego, Report 3 to U.S. Army
Rcscarch Office-Durham, 196S.
32 Foulkes, J. 1)., Minimum-weight design and the theory
of plastic collapse, Quart. Appt. Math. 10, 347-358, 1953; AMR 6(1953), Rev. 2787.
33 llcyman, J., and Prager, W., Automatic minimum weight
design of stccl frames, J. Franklin inst. 266, 339-364, 1958;
AMR 12(1959), Rev. 2406.
34 Toakley, A. R., Sorne computational aspects of
opti-mum rigid-plastic design, Intcrnat'l J. Mech. SCiS. 10,
531-537, 1968.
35 Marçai, P. V., and Prager, W., A method of optimal plastic design, J. de Mccaniquc 3, 509-530, 1964; AMR 18(1965), Rev. 5288.
36 Marca!, P. V., Optimal plastic design of circular
plates, lnt,'rnat'l. J. Solids Structs. 3, 427-443, 1967; AMR 20(1967), Rev. 7862.
37 Prager, W., and Shield, R. T., A general theory of
optimal plastic design, J. Appt. Mech. 34, 184-186, 1967. 38 Prager, ....irid Taylor, J. E., Problems of optimal structural design, J. Appt. Mech. 35, 102-106, 1968; AME 21(1968), Rev. 6597.
39 Shcu, C. Y., and Prager, W., Minimum-weight design
with piecewise constant specific stiffness, J. Optimization
Theory & Applications 2, 179-186, 1968.
10 Mrz, Z., Limit analysis of plastic structures subject to boundary variations, Archiwum Mech. Stos. 15, 63-76,
1963; AME 17(1964), Rev. 6291.
41 Prager, W., Optimatity criteria in structural design, Proc. Nat'!. Arad. Scis. USA, in press.
42 Prager, W., and Shield, R. T., Optimal design of
multi-purpose structures, Internat'!. J. Solids Structures 4, 469-475, 1968.
43 Shield, R. T., Optimum design methods for multiple
loading, J. App!. Math. Phys. (ZAMP) 14, 38-45, 1963; AMR 17(1964), Rev. 5812.
44 Ifaug, Jr., E. J., and Kirmser, P. G., Minimum weight
design of beams with inequality constraints on Stress and de-flection, J. App!. Mech. 34, 999-1004, 1967; AMR 21(1968),
Rev. 4981.
45 Barnett, R. L., Minimum weight design of beams for
deflection, Proc. ASCE 87, EM 1, 75-109, 1961.
46 Barnetr, R. L., Minimum deflection design of a
uni-formly accelerating cantilever beam, J. Appt. Mech. 30, 466-467, 1963; AMP. 17(1964), Rev. 2574.
47 Tadjbaksh, I., An optimum design problem for the non-linear elastica, IBM Research Paper RC-1949, 1967.
48 Mansfield, E. IL, Optimum tapers for eccentrically
loaded ties, J. Roy. Aero. Soc. 71, 647-650, 1967; AMR 21 (1968), Rev. 2639.
49 Dzieniszewski, W., Optimum design of plates of vari-able thickness for minimum potential energy, Bull. Arad. Polonaise Scis. , Shr. Scis. Techn. 13, 45-52, 1965; AMR
20(1967), Rev. 77.
50 Wasiutyiski, Z., On the equivalence of design
prin-ciple s: minimum potential-constant volume and minimum volume-constant potential, Bull. Arad. Polonaise Sczs.,
S6r. Scis. Techn. 14, 537-539, 1966; AME 20(1967), Rev.
7868.
51 Wssiurytski, Z., On the criterion of minimum
deforma-bility design of elastic structures: effect of own weight of
material, Bull. Arad. Polonaise Scis., Svr. Scis. Techn. 14,
529-532, 1966.
52 Boley, B. A., and Weiner, J. H., Theory of thermal stresses, John Wiley & Sons, New York, 1960, p. 77; AMR
14(1961), Rev. 1790.
53 Hackrnan, L. E., and Richardson, J. E., Design opti-mization of aircraft structures with thermal gradients, J.
Air-craft 1, 27-32, 1964; MIR 17(1964), Rev. 5835.
54 Prager, W., Optimal structural design for given stiff-ness in stationary creep, J. Appt. Math. Phys. (ZAMP) 19,
252-256, 1968.
55 Niordson, F. I., On the optimal design of a vibrating
beam, Quart. Appt. Iiiath. 23, 47-53, 1965; AME 18 (1965), Rev. 5343.
56 Turner, M., Design of minimum-mass structures with specified natural frequencies, A1AA J. 5, 406-412, 1967;
AME 20 (1967), Rev. 9278.
57 Taylor, J. E., Minimum-mass bar for axial vibration at specified natural frequency, AIAA J. 5, 1911-1913, 1967; AMR 21 (1968), Rev. 5628.
58 Taylor, J. E., Optimum design of a vibrating bar with specified minimum cross section, AIAA J. 6, 1379-1381,
1968.
59 Zarghamee, M. S., Optimum frequency of structures,
A1AA J. 6, 749-750, 1968; AME 21(1968), Rev. 5655. 60 Sheu, C. Y., Elastic minimum-weight design for
spec-ified fundamental frequency, lnternatl J. Solids Structures
4, in press.
61 Prager, W., Optimization of structural design, Invited
lecture at the Symposium on Optimization, SIAM 1968 National Meeting in Toronto; to be published by SIAM.
62 lcerman, L. J., Optimal structural design for given
dynamic deflection, M.S. Thesis, University of California,
San Diego, 1968.
63 McIntosh, S. C., and Eastep, F. E., Design of
minimum-mass structures with specified stiffness properties, AIRA J.
6, 962-964, 1968; AMR 21(1968), Rev. 6599.
64 French, M. J., The design of rotors to have the mini-mum weight consistent with a required first critical speed,
J. Mech. Engg. Sci. 1, 99-100, 1964; AME 17(1964), Rev. 5688.
65 Brach, R. M., Minimum dynamic response for a class of simply supported beam shapes, lnternat'l. J. i'ijech. Sci. 10, 429-439, 1968.
66 Tad;baksh, I., and Keller, J. B., Strongest columns
and isoperirnetric inequalities for eigenvalues, J. Appt. Mech. 29, 159;164, 1962; AMR 15(1962), Rev. 5774.
67 Keller, J. B., and Niordson, F. I., The tallest column,
J. Math. Mach. 16, 433-446, 1966; AMR 20(1967), Rev. 3975.
68 Taylor, J. E., The strongest column: an energy
ap-proach, J. App!. Mech. 34, 486-487, 1967; AME 21(1968),
Rev. 155.
69 Taylor, J. E., and Liu, C. Y., Optimal design of
col-umns, AIRA J. 6, 1497-1502, 1968.
70 Huang, N. C., and Sheu, C. Y., Optimal design of an elastic column of thin-walled cross section, J. App!. Mech.
35, 285-288, 1968.
71 Shanley, F. R., Optimum design of eccentrically loaded columns, Proc. ASCE 93, ST 4, 201-226, 1967; AMP. 21(1968), Rev. 2496.
72 Cohen, G. A., Optimum design of truss-core sandwich
cylinders under axial compression, AIAA J. 1, 1626-1630, 1963; AME 17(1964), Rev. 141.
Contribu-tions to Mechanics, Edited by D. Abir, Pergamon Press,
1968, pp. 163-169.
74 Bigelow, R. H., arid Gaylord, E. H., Minimum weight
of plastically designed steel frames, Univ. Illinois Eogg.
Exp. Station, Bull. 485, 1966; AMR 20(1967), Rev. 2507.
75 Felton, L. P., and Dobbs, M. W., Optimum design of tubes for bending and torsion, Proc. ASCE 93, ST 4,
185-200, 1967; AMR 21 (1968), Rev. 3245.
76 Hoffman, G. A., Optimal proportions of pressure vessel heads, J. Aerospace Sci. 29, 1471-1475, 1962; AMR 16 (1963), Rev. 5722.
77 Shamiev, F. G., Designing shells of minimum weight (in Russian), Izo. Alead. Nauk Azerb. SSR, Ser. Fiz. -Mat. i
Te/eh. Nauht, no. 5, 37-44, 1963; AMR 18(1965), Rev. 3370.
78 Lackman, L. M., and Ault, R. M., Influence of
pIas-ticity correction factor in minimum weight analysis, AIAA J. 4, 714-715, 1966; AMR 19(1966), Rev. 6914.
79 Megarefs, G. J., Minimal design of sandwich axisyrn-metric plates; I: Proc. ASCE 93, EM 6, 245-269, 1967; 11: ibid. 94, EM 1, 177-198, 1968; AMR 21 (1968), Rev. 7220.
80 Sheu, C. Y., snd Prager, W., Optimal plastic design of circular and annular sandwich plates with piecewise constant cross section, J. Mech. Phys. Solzds, in press.
81 Marks, W., A method to determine the optimum form of
prestressed beams under moving load, (in Polish), Rozprawy
¡nzynierskie 14, 49-68, 1966.
82 Wasiutyriski, Z., On the criterion of optimum design of
elastic structures subjected to n various systems of
solicita-tions, Bull. Aced. Polonaise Scis., Sr. Scis. Techn. 14,
533-535, 1966; AMR 20(1966), 6509.
83 Ibragimov, M. R., Designing a circular plate of mini-mum weight for two independent load systems, (in Russian),
Izo. A/sad. Nauls. Azrb. SSR, Ser. Fiz.-Mat. i Te/eh. Nauls, no. 3, 33-40, 1965; AMR 19(1966), Rev. 7689.
84 Gross, O., and Prager, W., Minimum weight design for
moving loads, Proc. 4th U.S. Nat'l. Congress AppI. Mech., (Berkeley, 1962), ASME, New York, 1962, vol. 2, pp.
1047-1051; £MR 16(1963), Rev. 5042.
85 Mayeda, R., and Prager, W., Minimum weight design of
beams for multiple loading, Internat'!. J. Solids Structures 3,
1001-1011, 1967; AMR 21(1968), Rev. 3230.
86 Prager, W., Optinium plastic design of a portal frame
for alternative loads, J. Appi. Mech. 34, 772-774, 1967; AMR 21(1968), Rev. 2653.
87 Save, M., and Prager, W., Minimum weight design of beams subjected to fixed and moving loads, J. MecS. Phys.
Solids 11, 255-267, 1963.
SS Save, M. A., and Shield, R. T., Minimum-weight de-sign of sandwich shell subjected to fixed and moving loads, Proc. 11th lnternat'l. Congress Appl. Mech. (Munich, 1964),
Springer, Berlin, 1966, pp. 341-349; AMR 20(1967), Rev.
3210.
89 Brown, E. H., The minimum weight design of closed shells of revolution, Quart. J. Mech. AppI. Math. 15, 109-128, 1962; AMR 16(1963), Rev. 745.
90 Durelli, A. J., Parks, V. J., Uribe, S., Optimization of
a slot end configuration in a finite plate subjected to uni-formly distributed load, J. Appl. MecS. 35, 403-406, 1968.
91 Gerard, G., and Lakshmikantharn, C., Optimum
thin-wall pressure vessels of anisotropic materials, J. AppI.
Mech. 33, 623-628, 1966; AMR 20(1967), Rev. 1787.
92 Mazurkiewicz, S., and Zyczkowski, M., Optimum de-sign of cross section of thin-walled bar under combined
tor-sion and bending, Bull. Aced. Polonaise Scis., Sir. Scis.
Techn. 14, 273-281, 1966; AMR 20(1967), Rev. 854. 93 Schmidler, G. M., Natural geometry of orbiting struc-tures, A/AA J. 6, 5 66-567, 1968.
94 Basu, A. K., and Chapman, J. C., Optimum design of plate with symmetrical trapezoidal corrugations subjected to lateral pressure, Quart. Trans. Roy. Insin. Naval Architects
109, 209-221, 1967; AMR 21(1968), Rev. 1676.
95 Crawford, R. F., and Burns, A. B., Minimum weight potentials for stiffened plates and shells, A/AA J. 1,
879-886, 1963; ASIR 17(1964), Rev. 743.
96 Crawford, R. F., and Schwartz, D. B., General insta-bility and optimum design of grid stiffened spherical domes,
A/AA J. 3, 511-515, 1965; AME 18(1965), Rev. 6023.
97 Kloppel, K., and Mollet, K. H., A note on increasing
the buckling load of a longitudinally stiffened rectangular
plate by means of a suitable distribution of stiffeners (in
German), Stahlbau 34, 303-311, 1965; AME 20(1967), Rev.
1639.
98 Maksimov, L. Yu., Design of cylinders to withstand high internal pressure (in Russian), Vesinils Machinostr. 44, 5, 9-12, 1964. (translated in Russian Engg. J. 44, 5, 6-9,
1964); ASIR i 8 ( 1965), Rev. 2186.
99 Mauch, H. R., and Felton, L. P., Optimum design of columns supported by tension ties, Proc. ASCE 93, ST 3,
201-220, 1967; AME 21 (1968), Rev. 3255.
100 Rockey, K. C., and Cook, I. T., Optimum
reinforce-ment by two longitudinal stifferiers of a plate subjected to pure bending, Internat'!. J. Solids Structures 1, 79-92, 1965;
AMR 18(1965), Rev. 7338.
101 Tulchii, V. I., Optimum reinforcement of holes in
plates (in Russian), Pri/eI. hie/eh. 1, 3, 77-53, 1965; ASIR 19(1966), Rev. 1390.
102 Cloud, R. L., Minimum weight design of a radial
noz-Zle in a spherical shell, J. AppI. Mech. 32, 448-449, 1965;
AMR 19(1966), Rev. 6890.
103 Green, A. P., and Lancaster, P. R., Design of a com-posite drawing die with a brittle insert, ¡nternat'l. J. MecS.
Scis. 8, 281-294, 1966.
104 Burru!tt, E. F. P., and Yo, C. W., Reinforced concrete linear structures at ultimate load, Proc. Internat'l. Symposium on Flexural Mechanics of Reinforced Conctete (Miami, 1964), ASCE, New York, 1965, pp. 29-52.
105 Cohn, M. Z. , Optimum limit design for reinforced-concrete continuous beams, Proc. Instn. Czv. Engrs. 30,
675-707, 1965.
106 Cohn, M. Z., Limit design solutions for concrete structures, Proc. ASCE 93, ST I, 37-57, 1967.
107 Korach, M., Economy of reinforced concrete slab
structures (in Hungarian), Magyar Tud. A/sad. Misz. Turi.
Oszt. Kdzl 30, 1/4, 97-119, 1962; ASIR 16(1963), Rev. 5219. 108 Klassonnet, Ch., and Save, M., Optimum design of
beams and frames in reinforced concrete, Progress Appl.
Mech., Macmillan, New York, 1963, pp. 279-294.
109 Morley, C. T., The minimum reinforcement of concrete slabs, I,rternai'l. J. Mech. Scis. 8, 305-319, 1966; AMR 19(1966), Rev. 7849.
110 Mrz, Z., On the optimum design of reinforced slabs, Acta Mechanica 3, 34-55, 1967; AMR 20(1967), Rev. 9372. 111 Petcu, V., Fundamentals of the limit design of
stati-cally indeterminate reinforced concrete structures (in French), Rev. Mécanique Appl. 7, 285-296, 1962; AME 16(1963), Rev. 3929.
112 Roavany, G. I. N., Minimum volume of uncurtailed
orthogonal reinforcement in freely supported slabs, Concrete
Construct'l. Engg. 61, 281-286, 1966; AME 20(1967), Rev.
4100.
113 Aleksandrov, A. Ya., Optimum parameters of
three-layer plates with a phenoplastic filler for compression (in Russian), Teoriya Plast. Obolochek, Akad. Nauk USSR, Kiev,
1962, pp. 463-466; AMR 17(1964), Rev. 5020.
114 Muchtari, Kh. M., The bending theory of
minimum-weight plates of composite materials, Pri/ei. Me/eh. 3, 4, 1-7, 1967.
115 Pipkin, A. C., and Rivlin, R. S., Minimum-weight de-sign for pressure vessels reinforced with inextensible fibers,
J. AppI. Mech. 30, 103-108, 1963; AMR 16(1963), Rev. 5726. 116 Korolev, V. I., Some problems in the choice of an opti-mum structure for glass-reinforced plastics (in Russian),
Inzhenernyi Zhurnal 5, 306-315, 1965 (translated in Soviet
Engg. J. 5, 252-258, 1965).
117 Schuerch, H. U., and Burggraf, O. R., Analytical de-sign for optimum filamentary pressure vessels, A/AA J. 2,
809-820, 1964; AME 17(1964), Rev. 5831.
118 Brandt, A., A theorem on the optimum design of
pre-stressed beams, Proc. Vibration Problems 11, 5 59-566, 1963; ASIR 18(1965), Rev. 2169.
119 Gawecki, A., Optimum design of post-tensioned pre-stressed concrete beams (in Polish), Archiwurn Inzynierzi
Ladowej 11, 503-537, 1965; ASIR 20(1967), Rev. 3350.
120 Rozvany, G. I. N., and Hampson, A. J. K., Optimum
design of prestressed plates, J, Amer. Concrete Inst. 60,
121 Rozvany, G. I. N., Optimum synthesis of prestressed
structures, Proc. ASCE 90, ST 6, 189-211, 1964.
122 Cox, H. L., The design of structures for least weight,
Pergarnon Press, New York, 1965.
123 Owen, J. B. B., The analysis and design of light
structures, E. Arnold, London, 1965.
124 Massonnet, Ch., and Save, M. A., Calcul plastique des
constructions; Vol. 1: Structures d6pcndant d'un parametre, 2nd cd.; ASBL, Brussels, 1967; Vol. 2: Structures spaticies;
ASBL, Brussels, 1963; AMR 17(1964), Rev. 3313.
125 Massor,nct, Ch., and Save, M. A., Plastic analysis and
design, Vol. 1, Blnisdell, New York, 1965 (translation of
[124 , Vol. 1).
126 larnet, R. L., Survey of optimum structural design,
Exp. Mcc6. 6, 12, l9A-26A, 1966; AMR 20(1967), Rev. 3361.
127 Gerard, G., Optimum structurai design concepts for
aerospace vehicles: bibliography and assessment, Allied
Re-search Associates, Inc., ARA Div. TR. no. 272-2, 1965;
AMR 18(1965), Rev. 6147.
128 Johnson, L. G., Practical design problems, Engineer-ing Plasticity, University Press, Cambridge, 1968, pp.
363-383.
129 Prager, W., Lineare Ungleichungen in der Baustatik,
Schweizerische liauzc itung 80, 315-320, 1962.
130 Prager, W., Optimization in structural design,
Mathe-matical Optimization Techniques, edited by R. Bellman,
Univ. California Press, Los Angeles, 1963, pp. 279-289. 131 Reitman, M. I., and Shapiro, G. S., Theory of optimal
design in structural mechanics and the theories of elasticity
arid plasticity (in Russian), Ac. Sci. USSR, Moscow, 1966. 132 Save, M. A., Some aspects of minimum-weight design,
Engineering Plasticity, University Press, Cambridge, 1968,
pp. 611-626.
133 Schmit, L. A., Automated design, Internat'l. Science
and Technology, No. 54, 63-78 and 115-117, 1968.
134 Woods, W. J., and Sams III, J. H., Geometric
optimiza-tion in the theory of structural synthesis, A1AA Paper No. 68-330, presented at AIAA/ASME 9th Structures, Structural Dynamics and Materials Conference, Palm Springs, Cal., April, 1968.
135 Optimization and Nonlinear Problems, Proc. ASCE
Joint Specialty Conference, Chicago, April, 1968.
136 Palmer, A. C., Optimal structural design by dynamic
programming, Proc.ASCE94, ST 8, 1887-1906, 1968.
137 Benjamin, J. R., Probabilistic structural analysis and
design, Proc. ASCE94, ST 7, 1665-1680, 1968.
138 Toakley, A. R., Optimal design using available
sec-tions, Proc. ASCE94, ST 5, 1219-1241, 1968.
139 Kaliszky, S., On the optimum design of reinforced
con-crete structures, Acta Techn. Acad. Sci. Hungarzcae 60,
257-264, 1968.
140 Razani, R,, and GobIe, G. G., Optimum design of
con-stant-depth plate girders, Proc. ASCE 92, ST 2, 253-281,
1966.
141 Cohn, M. Z., Limit design of continuous reinforced
concrete crane girders, Proc. ASCE92, ST 3, 161-177, 1966; AMR 20(1967), Rev. 853.
142 Rubinstein, M. F., and Karagozian, J., Building
de-si8n using linear programming, Proc. ASCE 92, ST 6,
223-245, 1966.
143 Hill, Jr., L. A., Automated optimum cost building
de-sign, Proc. ASCE92, ST 6, 247-263, 1966.
144 Kicher, T. P., Optimum design-minimum weight versus
fully stressed, Proc. ASCE 92, ST 6, 265-279, 1966.
145 Reinschmidt, K. F., Cornell, C. A., and Brotchie, J. F., Iterative design and structural optimization, Proc. ASCE92, ST 6, 281-318, 1966.
146 Brown, D. M., and Ang, A. H. S., Structural optimiza-tion by norilinear programming, Proc. ASCE 92, ST 6,
