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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 6, No. I. 1970
SURVEY PAPER
Optimization of Structural Design I.~
W. PRAGER 3
Abstract. Typical problems of optimal structural design are discussed to indicate mathematical techniques used in this field. An introductory example(Section 2) concerns the design of a beam for prescribed maximal deflection and shows how suitable discretization may lead to a problem of nonlinear programming, in this case, convex programming. The problem of optimal layout of a truss (Section 3) is discussed at some length. A new method of establishing optimality criteria (Section 4) is illustrated by the optimal design of a statically indeterminate beam of segmentwise constant or continuously varying cross section for given deflection under a single concentrated load. Other applications of this method (Section 5) are briefly discussed, and a simple example of multipurpose design (Section 6) concludes the paper.
1. Introduction
The most general problem of structural optimization may be stated as follows: from all structural designs that satisfy certain constraints, select one of minimal cost. Note that this statement does not necessarily define a unique design; there may be several optimal designs of the same minimal cost.
Typical design constraints that will be considered in the following specify upper bounds for deformations or stresses, or lower bounds for load-carrying capacity, buckling load, or fundamental natural frequency. Both singlepurpose and multipurpose structures will be considered, that is, structures that are respectively subject to a single design constraint or a multiplicity of constraints.
The term cost in the statement of the design objective may refer to the manufacturing cost or to the total cost of manufacture and operation over the expected lifetime of the structure. In aerospace structures, the cost of the fuel needed to carry a greater weight frequently overshadows the cost of manufacture to such an extent that minimal weight becomes the sole design objective. This point of view will be adopted in the following.
In the first part of this paper, typical problems of optimal design will be discussed to illustrate mathematical techniques that have been used in this field. The second part will be concerned with a promising technique of wide applicability that has been developed recently. Throughout the paper, it will be emphasized that the class of structures within which an optimum is sought must be carefully defined if meaningless solutions are to be avoided. The fact will also be stressed that certain intuitive optimality criteria of great appeal to engineers do not necessarily furnish true optima. For greater clarity in the presentation of design principles, the majority of examples will be concerned with single-prupose structures even though multipurpose structures are of far greater practical importance.
2. Discretization
To explore the mathematical character of a problem of structural optimization, it is frequently useful to replace the continuous structure by a discrete analog. Consider, for instance, the simply-supported elastic beam in Fig. 1. The maximum deflection produced by the given load 6P is not to exceed a given value To discretize the problem, replace the beam by a sequence of rigid rods that are connected by elastic hinges. In Fig. 1, only
Fig. 1. Discrete analog of elastic beam.
three hinges have been introduced; but, to furnish realistic results, the discretization would have to use a much greater number of hinges. The bending moment transmitted across the ith hinge is supposed to be related to the angle of flexure by
= (1)
where is the elastic stiffness of the hinge. Since the beam is statically determinate, the bending moments at the hinges are independent of the stiffnesses ; thus,
=5Ph=, =3Ph=, =Ph=. (2)
In the following, the angles of flexure , will be treated as small. In a design space with the rectangular Cartesian coordinates, i = 1, 2, 3, the nonnegative character of the angles of flexure and the constraints on the deflections at the hinges define the convex feasible domain
，，0,
5+3+-6/h0,
3+9-3-6/h0, （3）
+3+5-6/h0，
As will be shown in connection with a later example, the cost (in terms of weight) of providing a certain stiffness may be assumed to be proportional to this stiffness. The design objective thus is ++=Min or, by (2),
5/+3/+1/=Min (4)
Note that, for the convex program (3)-(4), a local optimum is necessarily a global optimum. This remark is important because a design that can only be stated to be lighter than all neighboring designs satisfying the constraints is of little practical interest. Note also that the optimum will not, in general, correspond to a point of design space that lies on an edge or coincides with a vertex of the feasible domain. This remark shows that the intuitively appealing concept of competing constraints is not necessarily valid. Suppose, for instance, that a design,, has been found for which<<=. If denotes a sufficiently small change of stiffness, the design +,-, , which has the same weight, might then be expected to have deflection ，， satisfying <,<<=, and all three stiffnesses could be decreased in proportion until the deflection at the first hinge has again the value. If this argument were correct, this process of reducing the structural weight could be repeated until the deflections at the hinges 1 and 2 had both the value &. In subsequent design changes, and would be increased by the same small amount while would be decreased by twice this amount to keep the weight constant. In this way, it might be argued that the optimal design must correspond to a point on an edge or at a vertex of the feasible domain, that is, that, for the optimal design, two or three of the constraining inequalities must be fulfilled as equations. This concept of competing constraints, to which appeal is frequently made in the engineering literature, is obviously not applicable to the problem on hand.
Minimum-weight design of beams with inequality constraints on deflection has recently been discussed by Haug and Kirmser (Ref. 1). Earlier investigations (see, for instance, Refs. 2-4) involved inequality constraints on the deflection at a specific point, for instance, at the point of application of a concentrated load. In special cases, where the location of the point of maximum deflection is known a priori, for instance, from symmetry considerations, a constraint on the maximum deflection can be formulated in this way. As Barnett (Ref. 3) has pointed out, however, constraining a specific rather than the maximum deflection may lead to paradoxical results. For example, when some loads acting on a horizontal beam are directed downward while others are directed upward, it may be possible to find a design for which the deflection at the specified point is zero. Since it will remain zero as all stiffnesses are decreased in proportion, the design constraint is compatible with designs of arbitrarily small weight.
3. Optimal
In the preceding example, the type and layout of the structure (simply supported, straight beam) were given and only certain local parameters (stiffness values) were at the choice of the designer. A much more challenging problem arises when type and/or layout must also be chosen optimally.
Figure 2a shows the given points of application of loads P and Q that are to be transmitted to the indicated supports by a truss, that is, a structure consisting of pin-connected bars, the layout of which is to be determined to minimize the structural weight. To simplify the analysis, Dorn, Gomory, and Greenberg (Ref. 5) discretized the problem by restricting the admissible locations of the joints of the truss to the points of a rectangular grid with horizontal spacing l and vertical spacing h (Fig. 2a). Optimization is then found to require the solution of a linear program. The optimal layout depends
Fig. 2. Optimal layout of truss according to Dorn, Gomory, and Greenberg (Ref. 5).
on the values of the ratios h/l and P/Q. Figures 2b through 2d show optimal layouts for h/l = 1 and P/Q = O, 0.5, and 2.0.
For h/l = 1 and a given value of P/Q, the optimal layout is unique except for certain critical values of P/Q, at which the optimal layout changes, for instance, from the form in Fig. 2c to that in Fig. 2d. The next example, however, admits an infinity of optimal layouts that are all associated with the same structural weight.
Three forces of the same intensity P, with concurrent lines of action that form angles of 120 ° with each other, have given points of application that form an equilateral triangle (Fig. 3@ A truss that connects these points is to be designed for minimal weight, when an upper bound is prescribed for the magnitude of the axial stress in any bar.
Figures 3b and 3c show feasible layouts. After the forces in the bars of these statically determinate trusses have been found from equilibrium considerations, the cross-sectional areas are determined to furnish an axial stress of magnitude in each bar.
The following argument, which is due to Maxwell (Ref. 6, pp. 175-177), shows that the two designs have the same weight.
Imagine that the planes of the trusses are subjected to the same virtual, uniform, planar dilatation that produces the constant unit extension e for all line elements. By the principle of virtual work, the virtual external work of the loads P on the virtual displacements of their points of application
Fig. 3. Alternative optimal designs.
equals the virtual internal work =Fof the bar forces F on the virtual elongations ~ of the bars. If cross-sectional area and length of the typical bar are denoted by A and L, then F=A and =L. Thus,
=AL=V （5）
where V is the total volume of material used for the bars of the truss. Now, depends only on the loads and the virtual displacements of their points of application but is independent of the layout of the bars; therefore, it has the same value for both trusses. If follows from=and (5) that the two trusses use the same amount of material.
If all cross-sectional areas of the two trusses are halved, each of the new trusses will be able to carry loads of the common intensity P/2 without violating the design constraint. Superposition of these trusses in the manner shown in Fig. 3d then results in an alternative truss for the full load intensity P that has the same weight as the trusses in Figs. 3b and 3c.
Fig. 4. Alternative solution to problem in Fig. 3a.
Figure 4 shows another solution to the problem. The center lines of the heavy edge members are circular arcs. The axial force in each of these members has constant magnitude corresponding to the tensile axial stress . The other bars are comparatively light. They are also under the tensile axial stress and are prismatic, except for the bars AO, BO, and CO, which are tapered.
The bars that are normal to the curved edge members must be densely packed. If only a finite number is used, as in Fig. 4, and the edge members are made polygonal rather than circular, a slightly higher weight results. This statement, however, ceases to be valid when the weight of the connections between bars (gusset plates and rivets or welds) is taken into account.
The interior bars in Fig. 4 may also be replaced by a web of uniform thickness under balanced biaxiat tension. While fully competitive as to weight, this design has, however, been excluded by the unnecessarily narrow formulation of the problem, which called for the design of a truss. In this case, the excluded design does not happen to be lighter than the others. However, unless the class of structures within which an optimum is sought is defined with sufficient breadth, it may only furnish a sequence of designs of decreasing weight that converges toward an optimum that is not itself a member of the considered class.
Figure 5 illustrates this remark. The discrete radial loads at the periphery are to be transmitted to the central ring by a structure of minimal weight.
If the word structure in this statement were to be replaced by the expression
Fig. 5. Optimal structure for transmitting peripheral loads to central ring is truss rather than disk
disk of continuously varying thickness, the optimal structure of Fig. 5 would be excluded. Note that Fig. 5 shows only the heavy members. Between these, there are densely packed light members along the logarithmic spirals that intersect the radii at
The problem indicated in Fig. 3a has an infinity of solutions, each of which contains only tension members. Figure 6 illustrates a problem that requires the use of compression as well as tension members and has a unique solution. The horizontal load P at the top of the figure is to be transmitted to the curved, rigid foundation at the bottom by a trusslike structure of
Fig. 6. Unique optimal structure for transmission of load P to curved, rigid wall.
minimal weight, the stresses in the bars of which are to be bounded by- and . The optimal truss has heavy edge members; the space between them
is filled with densely packed, light members, only a few of which are shownin Fig. 6. Note that the displacements of the densely packed joints of thestructure define a displacement field that leaves the points of the foundation fixed. A displacement field satisfying this condition wilt be called kinematically admissible.
There is a kinematically admissible displacement field that everywhere has the principal strains =/ E and =-/E, where E is Young's modulus. Indeed, if u and v are the (infinitesimal) displacement components with respect to rectangular axes x and y, the fact that the invariant + vanishes furnishes the relation
+=0， （6）where the subscripts x and y indicate differentiation with respect to the coordinates. Similarly, the fact that the maximum principal strain has the constant value e1 yields the relation
4*-(+)( +)=-4 （7）In view of (6), there exists a function such that
=,=- （8）Substitution of (8) into (7) finally furnishes
4 +=4 (9)Along the foundation are, u = v = O, which is equivalent to
=0， =0 （10）where is the derivative of T along the normal to the foundation are.
The partial differential equation (9) is hyperbolic, and its characteristics are the lines of principal strain. The Cauchy conditions (10) on the foundation arc uniquely determine the function , and hence the displacements (8), in a neighborhood of this arc.
These displacements will now be used as virtual displacements in the application of the principle of virtual work to an arbitrary trusslike structure that transmits the load P to the foundation are (Fig. 6) and in which each bar is under an axial stress of magnitude %. With the notations used above in the presentation of Maxwell's argmnent, ==. Here, ｜｜=A
and ｜｜, because no line element experiences a unit extension or contraction of a magnitude in excess of /E. Accordingly,
=∑｜F｜｜｜ (/E)V, (11) where V is again the total volume of material used in the structure.
Next, imagine a second trusslike structure whose members follow the lines of principal strain of the considered virtual displacement field and undergo the corresponding strains. Quantities referring to this structure will be marked by an asterisk. Applying the principle of virtual work as before, one has =, but *=and = with correspondence of signs. Accordingly,
== (12)In view of =, comparison of (11) and (12) reveals that the second structure cannot use more material than the first.
The argument just presented is due to Michell (Ref. 7), who, however, considered purely static boundary conditions and, consequently, failed to arrive at a unique optimal structure. The importance of kinematic boundary conditions for the uniqueness of optimal design was pointed out by the present author (Ref. 8).
Figure 7 illustrates an important geometric property of the orthogonal curves of principal strain in a field that has constant principal strains of equal magnitudes and opposite signs. Let ABC and DEF be two fixed curves of one family. The angle c~ formed by the tangents of these curves at their points of intersection with a curve of the other family does not depend on the choice of the latter curve. In the theory of plane plastic flow, orthogonal families of
Fig. 7. Geometry of optimal layout.
curves that have this geometric property indicate the directions of the maximum shearing stresses (slip lines). In this context, they are usually named after Hencky (Ref. 9) and Prandtl (Ref. 10); their properties have been studied extensively (see, for instance, Refs. 11-13).
Figure 8 shows the optimal layout where the space available for the structure is bounded by the verticals through d and B. Because the foundation arc is a straight-line segment, there are no bars inside the triangle dBC. Here again, the edge members are heavy, and the other members, of which only a few are shown, are comparatively light. The layout of these bars strongly resembles the trajectoriat system of the human femur (see, for instance, ReL 14, p. 12, Fig. 6). For further examples of Michell structures, see Refs. 15-16.
4. New Method of Establishing Optimality Criteria
The beam in Fig. 9 is built in at A and simply supported by B and C.
Its deflection at the point of application of the given load P is to have the given value. The beam is to have sandwich section of constant core breadth B and constant core height H. The face sheets are to have the common breadth B,
and their constant thicknesses 《H and 《H in the spans and are to be determined to minimize the structural weight of the beam. Since the
Fig. 8. Optimal layout when available space is bounded by verticals through A and B.
dimensions of the core are prescribed, minimizing the weight of the beam means minimizing the weight of the face sheets. Moreover, since the elastic bending stiffness s i of the cross section with face sheet thickness , i = 1, 2, is, where E is Young's modulus,
(13)
may be regarded as the quantity that is to be minimized.
Fig. 9. Beam with spanwise constant cross section.
Let be the distance of the typical cross section in the span from the Left end of this span, and denote curvature and bending moment at this cross section by and . The prescribed quantity may then be written as
==(14)where the integration is extended over the span
Within the framework of the problem, a beam design is determined by the values of , i = t, 2. If s i and si are two designs satisfying the design constraint (given value of ), and and are the curvatures that they assume under the given load, it follows from (14) that
= （15）Moreover, since the curvature is kinematically admissible (i.e., derived from a deflection satisfying the constraints at the support) for the design, it follows from the principle of minimum potential energy for the designthat
(16) Suppressing the terms in (16) and using (15), one obtains the inequality
（17） where
（18） is the mean-square curvature in the span . If
(19) it follows from (17) and (13) that the design s~ that satisfies (19) in addition to the design constraint cannot be heavier than an arbitrary design that satisfies only the design constraint. The condition (19) thus is sufficient for optimality; that it is also necessary may be shown as follows.
With the definition
（20） the condition that the design s i should not be heavier than the design takes the form
. (21)
On the other hand, the inequality (17), which followed from the principle of minimum potential energy, becomes
. (22)
The quantities , and, will be regarded as the components of vectors and with respect to the same