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Conditions for structural optimality
Contpurcrr & Yrurrurrs. Vol. 2, pp. 833440. PergamonPress1972.Printedin Great Britain
CONDITIONS
FOR STRUCTURAL WILLIAM
Division of Engineermg,
OPTIMALITYt
PRACER
Brown University, Providence,
R.I., U.S.A.
Abstmct-Typical difficulties encountered in the formulation of problems of optimal structural design and the inadequacy of widely used cmp&ical optimality criteria are illustrated by examples. For the optimal design of a statically determinate or indeterminate truss of given layout, a method is presented by which necessary and sufficient conditions for global optimality may be derived when an upper bound is prescribed for the compliance of the truss under one or several sets of loads and a lower bound is prescribed for the cross sectional area of each bar. The extension of the method to other structures and constraints is briefly discussed with reference to the literature, and the general form of the resulting optimality conditions is given.
PROBLEM
FORMULATION
STRUCTURAL optimization is a field in which apparently reasonable problems often turn out to be ill formulated. Special care must therefore be taken to avoid problems that are not well-posed. A few examples will illustrate this remark. A rod of the given length I, which is fixed at one end (root) and free at the other end (tip), is to be made of a given amount of a specified elastic material. It is to have constant cross sectional area in each of the two segments between a specified cross-section and the ends of the rod, and the fundamental natural frequency of its longitudinal vibrations is to be made as great as possible. Jt the formulation of the problem is taken at face value, the optimal design uses all material for the root segment, but this design may not be acceptable because it may be essential for the rod to have the length 1. To rule out the unacceptable design, a minimum cross sectional area may be prescribed for both segments. The optimal design then uses this area for the tip segment and places the remaining material in the root segment. Figure 1 shows a beam that is built in at one end, simply supported at the other end, and subjected to a load P and a couple C. The beam is to be designed for minimum weight subject to the constraint that its elastic deflection at a specified point A between the points of application of P and C should not exceed a given absolute value.
x FIG. 1. Minimum-weightdesign for given upper bound on deflection at A. t Research supported by the U.S. Army Research O&e, Durham. presented as an Invited Paper at the National Symposiumon ComputerizedStructural Analysis and Design, at the School of Engineering and Applied Science, George Washington D.C., 27-29 March (1972). 833
University,
Washington,
834
WILLIAM PRAGER
(4
As Barnett [1] has pointed out, the weight of a design satisfying the constraint can be made arbitrarily small. Indeed, in view of the opposing influences of P and C, there exists a design for which the deflection at .4 vanishes. This beam obviously satisfies the constraint. If all its cross-sectional dimensions are reduced in the same proportion, the resulting lighter beam still satisfies the constraint. Since the limit of this process of reducing weight, that is the beam of zero weight, will not be acceptable, there is no minimum-weight design for this problem. The difficulty can, of course, be avoided by the imposition of an upper bound on the absolute value of any deflection rather than only the deflection at A. The next example concerns the plastic minimum-weight design of an annular circular sandwich plate, the outer and inner radii of which have the ratio 2. The plate is simply supported at the outer edge and free at the inner edge; its core is to have a given constant thickness. The continuously varying common thickness f of the cover sheets is to be determined in such a manner that plastic collapse occurs at a given value of the uniformly distributed load while the plate is as light as possible. Figure 2, which is based on Ref. [2], shows the optimal variation of the thickness t. when this is required to be constant over 4 or 8 concentric annuli of equal width. The figure suggests that the continuously varying thickness will become extremely great or infinite at the inner edge (see Ref. [3]). The resulting design is unacceptable because it violates the basic assumption of the analysis that the core is much thicker than the cover sheets. This design suggests, however, that the formulation of the problem was too narrow because it overlooked the possibility that the optimal design might have a reinforcing circular beam at the inner edge.
FIG. 2. Plastic
minimum-weight
EMPIRICAL
design
of stepped
OPTIMALITY
annular
sandwich
plate.
CRITERIA
Traditionally, near-optimization of structural design has been sought by trial and error methods. It is only natural that, having chosen a design in this way, the designer should look for characteristic features that would have steered him more directly to the final design. As a consequence of this, the field of structural optimization abounds with empirical optimality criteria of uncertain validity. Consider, for instance, the plane truss in Fig. 3, which is to be designed for minimum weight under the alternative loadings shown in Figs. 3a and b, when an upper bound 6, is prescribed for the absolute value of the stress
835
Conditions for Structural Optimality
in any bar. It is widely believed that the minimum-weight design of a truss for alternative loadings is statically indeterminate andjXy stressed. For the present example, this would mean that the optimal truss contains all three bars with cross sections that will let the absolute value of the stress in any bar reach the prescribed upper bound in at least one state of loading.
,v
,v,‘,z’
la (b)
(a)
FIG. 3a, b. Plane truss subject to alternative loadings. Figure 4, which is based on Ref. [4], shows that this belief is not justified. Only if the load point with coordinates P, Q falls into one of the shaded regions is the statically indeterminate design lighter than a statically determinate one with only two bars, and only if the load point lies in one of the shaded regions marked F is the minimum-weight design fully stressed.
OMIT
OMIT
EAR
BAR
E
C
Only for combinations P, Q eorreaponding to points in shaded regions is statically indeterminate design lighter than determinate design. Only for combinations P, Q corresponding to points in regions marked F is indeterminate truss of minimum weight fully stressed.
FIG. 4.
As this example shows, empirical optimality criteria may fail to have the broad validity that is attributed to them. On the other hand, the designer would obviously prefer using a general optimality criterion of wide applicability to deriving appropriate optimality
WILLIAM PRAGER
836
criteria from first principles for each new combination of structural objectives and constraints. A general criterion of this kind is discussed in the following. For additional examples, the reader should consult the references. SIMPLE
EXAMPLE
As a simple example of structural optimization, consider a statically determinate or indeterminate truss that is carrying a single set of loads. The joints of the truss are to have given positions and to be connected in a given manner by elastic bars. For each bar. the structural material and its positive unit cost Ci, (I’= 1, 2, . . . ), are specified as well as a lower bound Ui on the cross-sectional area. The truss is to have a given elastic compliance CO, which is defined as the virtual work of the given loads on the joint displacements caused by them. Designs satisfying these constraints will be called admissible. An admissible design is to be determined, for which the total cost of the bars is as small as possible. Note that this formulation includes minimum-weight design for which the “cost” of a bar must be identified with its weight. The bars of the truss may be divided into two groups. Bar i belongs to groups 1 or 2, according to whether its cross-sectional area A, is governed by the constraint on compliance or cross-sectional area. The following feature of the optimal design will be established. When the optimal truss is subjected to the given loads, the numerical value of the elastic strain energy oj’ each bar is equal to or smaller than u.fiued multiple qf the numerical value of the cost of this bar depending on whether the bur belongs to group 1 or to group 2. The following proof of this
theorem involves two designs of the compliance C,,, which will be referred to as designs vi and 0:. Here, ui is the volume of the ith bar of the Iirst design, and I$, the volume of the ith bar of the second design. The common modulus of elasticity of the two bars will be denoted by Ei, and their axial strains under the given loads by Ei and a*. The fact that both trusses have the compliance COis expressed by the relation C,= EEi&;ui= z:E$:%* )
(1)
in which the summations include all bars of the considered truss. Since the work of the loads has the fixed value C,,, the principle of minimum potential energy becomes a principle of minimum strain energy. Applied to the design $, this principle states that
where the axial strains gi must be kinematically admissible, that is, must derive from joint displacements that satisfy the kinematic constraints at the supports. Since the axial strains of the design ui are kinematically admissible for the design VT,it follows from (1) and (2) that
This inequality shows that an arbitrary design $ with the prescribed compliance C, cannot cost less than the design ui that, in addition to having this compliance, experiences axial strains E, under the given loads satisfying EiEf
= k2ci for all bars in group 1, ~.k2ci for all bars in group 2.
(4)
837
Conditions for Structural Optimality
Here, kZ denotes a positive constant that does not depend on i. Indeed, the inequality (3) may be written in the form ZE,$(o;
-U,)~~k2Ci(U~-Ui)-C(U~-UiXli2C~-_,E~)~0.
(5)
According to (4), the factor k2c,- E,E~’ in the last term of (5) vanishes for the bars in group 1 and is nonnegative for the bars in group 2, for which OF> u1because their crosssectional areas are at the lower bound. The last term of (5) may therefore be dropped without invalidating this inequality. Accordingly, EC*Ui2ZCiUiy
(6)
which shows that the design OF cannot cost less than the design ui. When both sides of the optimality condition (4) are multiplied by ui, it is seen that the strain energy Eisfu, in bar i is equal to or smaller than k2 times the cost ciui of this bar, depending on whether the bar belongs to group 1 or to group 2. The principle of minimum strain energy, from which the optimality condition (4) was derived, is a global minimum principle. Accordingly, the condition (4) assures globul optimality. This remark is important, because the designer needs to be certain that there exist, not only no neighboring admissible designs, but no admissible designs at all that cost less than the chosen design. While the preceding discussion only shows the sufficiency of the optimality condition (4) its necessity is readily established along familiar lines of the calculus of variations. An important consequence of the optimality condition (4) deserves mention. To simplify the following discussion, assume that Young’s modulus E,, the specific cost pi, and the lower bound a, on cross-sectional area have the same values for all bars so that the subscript i may be dropped from these quantities. Because the form of the optimality condition (4) does not depend on the value a of the minimal cross-sectional area, this condition remains valid for u+O. This remark becomes important when the given layout of the truss is such that the truss would be statically indeterminate if all bars had nonvanishing cross-sections. With a = 0, the designer would feel free to omit bars from the original statically indeterminate truss. The first part of the optimality condition (4) would require the remaining axial bars to have axial strains of the same absolute value E,. In general, this condition can only be fulfilled if the remaining bars form a statically determinate truss. The second part of (4) indicates that this statically determinate truss cannot be optimal if its joint displacements under the given load imply an axial strain of an absolute value in excess of E, in any one of the omitted bars. The case of two (or more) compliance constraints may be handled in a similar manner. Consider, for instance, a truss of given layout subject to two alternative sets of loads for each of which an upper bound on elastic compliance is prescribed. When the compliance under a set of loads reaches its upper bound, this compliance constraint is said to be relevant. Only the case that both compliance constraints are relevant needs to be considered here, because the problem otherwise reduces to the preceding one. If E; and e; denote the axial strains of bar i of the optimal truss in the two states of loading, it can be shown that the condition Ei(lE;2 +(l -n)eF’}
= k2c, d k2c,
for all bars in group 1, for all bars in group 2.
is necessary and sufficient for global optimality.
(OG16 1)
(7)
WILLIAM PRAGER
838
EXTENSION
TO OTHER KINDS OF CONSTRAINT
As was remarked by Prager and Taylor [5], the procedure by which the optimality condition (4) has been derived may be used whenever the single constraint concerns a quantity, such as compliance, that is characterized by a minimum principle, such as the principle of minimum strain energy used above. Depending on whether this principle has global or local character, the resulting optimality condition is s@icient for global or locul optimality. As a rule, the necessity of this condition is readily established by calculus of variations. If the value presented by the behavioral constraint may only be characterized as a stationary rather than a minimum value, the resulting condition is sufic.ient for the cost to be stationary in the neighborhood of the considered design. Optimal design for a prescribed value of one of the following quantities has been treated in this manner: (a) static elastic compliance [5-91; (b) static elastic deflection at a specified point [IO. 1I]; in presence of body forces [ 12, 131; in presence of thermal strains [14, 151; (c) rate of compliance in steady creep, [9, 161; (d) dynamic elastic compliance under harmonically varying loads [17-191; (e) fundamental natural frequency [5, 9, 20-221; (f) elastic buckling load [5, 23, 241; (g) load factor for plastic collapse [2, 3, 25-271. For the constraints (a), (c), (e), (f) and (g), known global minimum principles of structural theory were used to derive sufficient conditions for global optimality, and for the constraint (d) a suitable minimum principle could be established [17, 181. For the constraint (b), a principle of stationary mutual potential energy was established in [ll]. For statically determinate structures, this yields a sufficient condition for global optimality, but for indeterminate structures, it only furnishes a sufficient condition for the cost of the structure to be stationary in the neighborhood of the considered design. Note that (g) was the first constraint for which a global minimum principle was used to derive a sufficient condition for global optimality [25]. Multiple constraints lead to optimality conditions that resemble condition (7) [28-341. Consider, for instance, an elastic truss of given layout that carries a point mass m, at the typical joint c1, (c1= 1, 2, . . . ), and let upper bounds be prescribed on the fundamental natural frequency and on the static compliance under the weights gm,. For brevity, bounds on cross-sectional area will not be considered here, and it will be assumed that, for the determination of the fundamental frequency o, the mass of bar i may bc accounted for by adding +ppI to the point masses at the tetminal joints of this bar? iji being the density of the bar, Subject to the frequency and compliance constraints, the cost of the truss is to be minimized. If only the compliance constraint is relevant, the optimality condition is J&E;~= k2ci,
(8)
where E: is the axial strain of bar i under the weights gm,. lf only the frequency constraint is relevant, Rayleigh’s principle may be used in a similar way as the principle of minimum strain energy was used above to derive the optimality condition
839
Conditions for Structural Optimality
Here, UCis the displacement vector of joint a and E; the axial strain of bar i in the fundamental mode, and aeiF{i}if
joint a{:: ..,lan
endpoint of bar i.
Note that the optimality condition (9) states that, in the optimal truss, the contribution of a bar to the difference between the amplitudes of the strain and kinetic energies of the fundamental mode is proportional to this bar’s contribution to the cost of the truss. When both constraints are relevant, the optimality condition assumes the form AE,e;’ + (1 -A) E,E;' -
(O<& ~p.,P,luq=k2*,
CONCLUDING
1).
(11)
REMARKS
It is worth recognizing that the optimality conditions derived from minimum principles have a common form, The structure that is to be optimized will, in general, have parts with prescribed dimensions (fixed parts) and parts with dimensions that are at the choice of the designer (free parts). In the following, use of the term free part will imply that the considered part can be designed independently of all other free parts. To avoid degenerate structures, a lower bound may be set for a critical dimension of a free part. A free part will be said to be unilaterally or bilaterally free according to whether its critical dimension is or is not at the lower bound. Cbnsider first optimal design for a single constraint that specifies the value of a scalar quantity Q (usually an energy) describing a global feature of structural behavior. If the value of Q for a given structure is characterieed by a minimum principle, the optimality condition states that the contribution of each free part of the optimal structure to the value of Q is equal to or less than a tied multiple of the cost of this part according to whether the part is bilaterally or only unilaterally free. If there are several constraints that specify the values Q,, Q,, . . . of scalar quantities describing global structural behavior, and each of these quantities is characterized by a minimum principle, the words “contribution of each free part of the optimal structure to the value of Q” in the optimality condition above must be replaced by the words “a bveighted average of the contributions of each free part of the optimal structure to the values of Q,, Q,, . . . “. (The weights used in averaging the contributions to Q,, Q,, . . . must be the same for all free parts.)
REFERJZNCES [l] R. L. BARNETT. Minimum-weightdesignof beamsfor deflection. Proc. ASCE87, EMl, 75-109(1961). [2] C. Y. SHEUand W. PRAYER, Optimal plastic design of circular and annular sandwich plates with piecewiseconstant cross-section. J. Mech. Phys. Solidr 17. 11-16 (1969). [3] 0. J. MEOAREFS, Minimal design of sandwich axisymme& plat&. &oc. ASCE 93, EM6, 245-269 1967)and 94, EMl. 177-198(1968). [4] J.-M. CHURN and W. %AcaER. Optimal design of trusses for alternative loads. Ztzgeni&w-Archiv (to appear).
840
WULIAM PRACER
151 W. PRAGERand J. E. TAYLOR, Problems of optimal structural design. J. appl. Mech. 35, 102-106 (1968). 161W. DZIENISZEWSKI,Optimum design of plates of variable thickness for minimum potential energy. Bull. Acad. Polon., Ser. Sci. Tech. 13,45-52 (1965). On the equivalence of the design principles: minimum potential-constant volume Vl Z. WASIIJTYSJSKI, and minimum volume-constant potential. Bull. Acad. Polon., Ser. Sci. Tech. 14, 537-539 (1966). PI C. Y. SHEU and W. PRAGER, Minimum-weight design with piecewise constant specific stiffness. J. Optimization Theory Appl. 2, 179-186 (1968). [91 G. A. HEGEMIERand W. PRAGER,On Michell trusses. Int. J. mech. Sci. 11, 209-215 (1969). WI R. T. SHIELDand W. PRAG~R, Optimal structural design for given deflection. J. uppl. Math. Phys. (ZAMP) 21, 513-523 (1970). 1111 W. PRAGER.Optimal design of statically determinate beams for given deflection. Int. J. mech. Sci. 13, 893 (1971). WI J.-M. CHERN and W. PIUGER, Optimal design of rotating disk for given radial displacement of edge. J. Optim. Theory Appl. 6,161-170 (1970). v31 J.-M. CHERN, Optimal structural design for given deflection in presence of body forces. Inf. J. Solids Struct. 7, 373-382 (1971). B41 W. ~RAGER,Optimal thermoelastic design for given deflection. ht. J. mech. Sci. 12, 705-709 (1970). J.-M. CHERN, Optimal thermoelastic design for given deformation. J. uppl. Mech. 38,538-540 (1971). t::; W. PRAGER, Optimal structural design for-given stiffness in stationary creep. J. appl. Math. Phys. (ZAMP) 19,252-256 (1968). 1171 L. J. ICERMAN,Optimal structural design for given dynamic deflection. In/. J. Solids Struct. 5,473-490 (1969). WI Z. MR~z, Optimal design of elastic structures subjected to dynamic, harmonically varying loads. Z. ungew. Math. Mech. 50, 303-309 (1970). v91 R. H. PLAUT, Optimal structural design for given deflection under periodic loading. Q. appl. Math. 29, 315-318 (1971). [20] J. E. TAYLOR,Minimum-mass bar for axial vibration at specified natural frequency. AIAA Journal 5, 1911-1913 (1967). [21] J. E. TAYLOR,Optimum design of a vibrating bar with specified minimum cross-section. AZAA Jourrud 6, 1379-1381 (1968). [22] C. Y. SHEU, Elastic minimum-weight design for specified fundamental frequency. Int. J. Solids Struct. 4,953-958 (1968). [23] J. E. TAYLOR,The strongest column: an energy approach. J. appl. Mech. 34,486-487 (1967). [24] J. E. TAYLORand C. Y. LIU, Optimal design of columns. AIAA Journal 6, 1497-1502 (1968). [25] D. C. DRUCKERand R. T. SHIELD,Design for minimum weight. Proc. 9th Internatl. Congress Appl. Mech., Vol. 5, pp. 212-222. Brussels (1957). [26] R. T. SHIELD,Plasticity (edited by E. H. LEE and P. S. SYMONDS),pp. 580-591. Pergamon Press, Oxford (1960). [271 R. T. SHIELD,Optimum design methods for multiple loading. J. appl. Math. Phys. (ZAMP) 14, 38-45 (1963). [28] R. MAYEDAand W. PRAGER, Minimum-weight design fo beams for multiple loading. Int. J. Solids Struct. 3, 1~001-1011(1967). [29] W. PRAGERand R. T. SHIELD, Optimal design of multi-purpose structures. ht. J. So/ids Struct. 4, 469-475 (1969). [30] C. Y. SHEU and W. PIUGER, Optimal design of sandwich beams for elastic deflection and load factor at plastic collapse. J. appl. Math. Phys. 20,289-297 (1969). [31] J.-M, CHERN and W. PRAGER, Optimal design for prescribed compliance under alternative loads. J. Ootimization Theory A&. 5.424-431 (1970). [321 J. B: MARTIN, Optimal &sign~ of elastic‘structures for multipurpose loading. J. Optimization Theory Appl. 6, 22-40 (1970). 1331 J.-M. CHERN and W. PUG~R. Minimum-weight design of statically determinate trusses subject to multiple constraints. Znt. J. Solids Struct. (to appear). -7, 931-940 (1971). [341 W. PRAGER,Foulkes mechanism in optimal plastic design for alternative loads. Int. J. mech. Sci. 13, 971-973 (1971). (Received 28 January 1972)
CONDITIONS
FOR STRUCTURAL WILLIAM
Division of Engineermg,
OPTIMALITYt
PRACER
Brown University, Providence,
R.I., U.S.A.
Abstmct-Typical difficulties encountered in the formulation of problems of optimal structural design and the inadequacy of widely used cmp&ical optimality criteria are illustrated by examples. For the optimal design of a statically determinate or indeterminate truss of given layout, a method is presented by which necessary and sufficient conditions for global optimality may be derived when an upper bound is prescribed for the compliance of the truss under one or several sets of loads and a lower bound is prescribed for the cross sectional area of each bar. The extension of the method to other structures and constraints is briefly discussed with reference to the literature, and the general form of the resulting optimality conditions is given.
PROBLEM
FORMULATION
STRUCTURAL optimization is a field in which apparently reasonable problems often turn out to be ill formulated. Special care must therefore be taken to avoid problems that are not well-posed. A few examples will illustrate this remark. A rod of the given length I, which is fixed at one end (root) and free at the other end (tip), is to be made of a given amount of a specified elastic material. It is to have constant cross sectional area in each of the two segments between a specified cross-section and the ends of the rod, and the fundamental natural frequency of its longitudinal vibrations is to be made as great as possible. Jt the formulation of the problem is taken at face value, the optimal design uses all material for the root segment, but this design may not be acceptable because it may be essential for the rod to have the length 1. To rule out the unacceptable design, a minimum cross sectional area may be prescribed for both segments. The optimal design then uses this area for the tip segment and places the remaining material in the root segment. Figure 1 shows a beam that is built in at one end, simply supported at the other end, and subjected to a load P and a couple C. The beam is to be designed for minimum weight subject to the constraint that its elastic deflection at a specified point A between the points of application of P and C should not exceed a given absolute value.
x FIG. 1. Minimum-weightdesign for given upper bound on deflection at A. t Research supported by the U.S. Army Research O&e, Durham. presented as an Invited Paper at the National Symposiumon ComputerizedStructural Analysis and Design, at the School of Engineering and Applied Science, George Washington D.C., 27-29 March (1972). 833
University,
Washington,
834
WILLIAM PRAGER
(4
As Barnett [1] has pointed out, the weight of a design satisfying the constraint can be made arbitrarily small. Indeed, in view of the opposing influences of P and C, there exists a design for which the deflection at .4 vanishes. This beam obviously satisfies the constraint. If all its cross-sectional dimensions are reduced in the same proportion, the resulting lighter beam still satisfies the constraint. Since the limit of this process of reducing weight, that is the beam of zero weight, will not be acceptable, there is no minimum-weight design for this problem. The difficulty can, of course, be avoided by the imposition of an upper bound on the absolute value of any deflection rather than only the deflection at A. The next example concerns the plastic minimum-weight design of an annular circular sandwich plate, the outer and inner radii of which have the ratio 2. The plate is simply supported at the outer edge and free at the inner edge; its core is to have a given constant thickness. The continuously varying common thickness f of the cover sheets is to be determined in such a manner that plastic collapse occurs at a given value of the uniformly distributed load while the plate is as light as possible. Figure 2, which is based on Ref. [2], shows the optimal variation of the thickness t. when this is required to be constant over 4 or 8 concentric annuli of equal width. The figure suggests that the continuously varying thickness will become extremely great or infinite at the inner edge (see Ref. [3]). The resulting design is unacceptable because it violates the basic assumption of the analysis that the core is much thicker than the cover sheets. This design suggests, however, that the formulation of the problem was too narrow because it overlooked the possibility that the optimal design might have a reinforcing circular beam at the inner edge.
FIG. 2. Plastic
minimum-weight
EMPIRICAL
design
of stepped
OPTIMALITY
annular
sandwich
plate.
CRITERIA
Traditionally, near-optimization of structural design has been sought by trial and error methods. It is only natural that, having chosen a design in this way, the designer should look for characteristic features that would have steered him more directly to the final design. As a consequence of this, the field of structural optimization abounds with empirical optimality criteria of uncertain validity. Consider, for instance, the plane truss in Fig. 3, which is to be designed for minimum weight under the alternative loadings shown in Figs. 3a and b, when an upper bound 6, is prescribed for the absolute value of the stress
835
Conditions for Structural Optimality
in any bar. It is widely believed that the minimum-weight design of a truss for alternative loadings is statically indeterminate andjXy stressed. For the present example, this would mean that the optimal truss contains all three bars with cross sections that will let the absolute value of the stress in any bar reach the prescribed upper bound in at least one state of loading.
,v
,v,‘,z’
la (b)
(a)
FIG. 3a, b. Plane truss subject to alternative loadings. Figure 4, which is based on Ref. [4], shows that this belief is not justified. Only if the load point with coordinates P, Q falls into one of the shaded regions is the statically indeterminate design lighter than a statically determinate one with only two bars, and only if the load point lies in one of the shaded regions marked F is the minimum-weight design fully stressed.
OMIT
OMIT
EAR
BAR
E
C
Only for combinations P, Q eorreaponding to points in shaded regions is statically indeterminate design lighter than determinate design. Only for combinations P, Q corresponding to points in regions marked F is indeterminate truss of minimum weight fully stressed.
FIG. 4.
As this example shows, empirical optimality criteria may fail to have the broad validity that is attributed to them. On the other hand, the designer would obviously prefer using a general optimality criterion of wide applicability to deriving appropriate optimality
WILLIAM PRAGER
836
criteria from first principles for each new combination of structural objectives and constraints. A general criterion of this kind is discussed in the following. For additional examples, the reader should consult the references. SIMPLE
EXAMPLE
As a simple example of structural optimization, consider a statically determinate or indeterminate truss that is carrying a single set of loads. The joints of the truss are to have given positions and to be connected in a given manner by elastic bars. For each bar. the structural material and its positive unit cost Ci, (I’= 1, 2, . . . ), are specified as well as a lower bound Ui on the cross-sectional area. The truss is to have a given elastic compliance CO, which is defined as the virtual work of the given loads on the joint displacements caused by them. Designs satisfying these constraints will be called admissible. An admissible design is to be determined, for which the total cost of the bars is as small as possible. Note that this formulation includes minimum-weight design for which the “cost” of a bar must be identified with its weight. The bars of the truss may be divided into two groups. Bar i belongs to groups 1 or 2, according to whether its cross-sectional area A, is governed by the constraint on compliance or cross-sectional area. The following feature of the optimal design will be established. When the optimal truss is subjected to the given loads, the numerical value of the elastic strain energy oj’ each bar is equal to or smaller than u.fiued multiple qf the numerical value of the cost of this bar depending on whether the bur belongs to group 1 or to group 2. The following proof of this
theorem involves two designs of the compliance C,,, which will be referred to as designs vi and 0:. Here, ui is the volume of the ith bar of the Iirst design, and I$, the volume of the ith bar of the second design. The common modulus of elasticity of the two bars will be denoted by Ei, and their axial strains under the given loads by Ei and a*. The fact that both trusses have the compliance COis expressed by the relation C,= EEi&;ui= z:E$:%* )
(1)
in which the summations include all bars of the considered truss. Since the work of the loads has the fixed value C,,, the principle of minimum potential energy becomes a principle of minimum strain energy. Applied to the design $, this principle states that
where the axial strains gi must be kinematically admissible, that is, must derive from joint displacements that satisfy the kinematic constraints at the supports. Since the axial strains of the design ui are kinematically admissible for the design VT,it follows from (1) and (2) that
This inequality shows that an arbitrary design $ with the prescribed compliance C, cannot cost less than the design ui that, in addition to having this compliance, experiences axial strains E, under the given loads satisfying EiEf
= k2ci for all bars in group 1, ~.k2ci for all bars in group 2.
(4)
837
Conditions for Structural Optimality
Here, kZ denotes a positive constant that does not depend on i. Indeed, the inequality (3) may be written in the form ZE,$(o;
-U,)~~k2Ci(U~-Ui)-C(U~-UiXli2C~-_,E~)~0.
(5)
According to (4), the factor k2c,- E,E~’ in the last term of (5) vanishes for the bars in group 1 and is nonnegative for the bars in group 2, for which OF> u1because their crosssectional areas are at the lower bound. The last term of (5) may therefore be dropped without invalidating this inequality. Accordingly, EC*Ui2ZCiUiy
(6)
which shows that the design OF cannot cost less than the design ui. When both sides of the optimality condition (4) are multiplied by ui, it is seen that the strain energy Eisfu, in bar i is equal to or smaller than k2 times the cost ciui of this bar, depending on whether the bar belongs to group 1 or to group 2. The principle of minimum strain energy, from which the optimality condition (4) was derived, is a global minimum principle. Accordingly, the condition (4) assures globul optimality. This remark is important, because the designer needs to be certain that there exist, not only no neighboring admissible designs, but no admissible designs at all that cost less than the chosen design. While the preceding discussion only shows the sufficiency of the optimality condition (4) its necessity is readily established along familiar lines of the calculus of variations. An important consequence of the optimality condition (4) deserves mention. To simplify the following discussion, assume that Young’s modulus E,, the specific cost pi, and the lower bound a, on cross-sectional area have the same values for all bars so that the subscript i may be dropped from these quantities. Because the form of the optimality condition (4) does not depend on the value a of the minimal cross-sectional area, this condition remains valid for u+O. This remark becomes important when the given layout of the truss is such that the truss would be statically indeterminate if all bars had nonvanishing cross-sections. With a = 0, the designer would feel free to omit bars from the original statically indeterminate truss. The first part of the optimality condition (4) would require the remaining axial bars to have axial strains of the same absolute value E,. In general, this condition can only be fulfilled if the remaining bars form a statically determinate truss. The second part of (4) indicates that this statically determinate truss cannot be optimal if its joint displacements under the given load imply an axial strain of an absolute value in excess of E, in any one of the omitted bars. The case of two (or more) compliance constraints may be handled in a similar manner. Consider, for instance, a truss of given layout subject to two alternative sets of loads for each of which an upper bound on elastic compliance is prescribed. When the compliance under a set of loads reaches its upper bound, this compliance constraint is said to be relevant. Only the case that both compliance constraints are relevant needs to be considered here, because the problem otherwise reduces to the preceding one. If E; and e; denote the axial strains of bar i of the optimal truss in the two states of loading, it can be shown that the condition Ei(lE;2 +(l -n)eF’}
= k2c, d k2c,
for all bars in group 1, for all bars in group 2.
is necessary and sufficient for global optimality.
(OG16 1)
(7)
WILLIAM PRAGER
838
EXTENSION
TO OTHER KINDS OF CONSTRAINT
As was remarked by Prager and Taylor [5], the procedure by which the optimality condition (4) has been derived may be used whenever the single constraint concerns a quantity, such as compliance, that is characterized by a minimum principle, such as the principle of minimum strain energy used above. Depending on whether this principle has global or local character, the resulting optimality condition is s@icient for global or locul optimality. As a rule, the necessity of this condition is readily established by calculus of variations. If the value presented by the behavioral constraint may only be characterized as a stationary rather than a minimum value, the resulting condition is sufic.ient for the cost to be stationary in the neighborhood of the considered design. Optimal design for a prescribed value of one of the following quantities has been treated in this manner: (a) static elastic compliance [5-91; (b) static elastic deflection at a specified point [IO. 1I]; in presence of body forces [ 12, 131; in presence of thermal strains [14, 151; (c) rate of compliance in steady creep, [9, 161; (d) dynamic elastic compliance under harmonically varying loads [17-191; (e) fundamental natural frequency [5, 9, 20-221; (f) elastic buckling load [5, 23, 241; (g) load factor for plastic collapse [2, 3, 25-271. For the constraints (a), (c), (e), (f) and (g), known global minimum principles of structural theory were used to derive sufficient conditions for global optimality, and for the constraint (d) a suitable minimum principle could be established [17, 181. For the constraint (b), a principle of stationary mutual potential energy was established in [ll]. For statically determinate structures, this yields a sufficient condition for global optimality, but for indeterminate structures, it only furnishes a sufficient condition for the cost of the structure to be stationary in the neighborhood of the considered design. Note that (g) was the first constraint for which a global minimum principle was used to derive a sufficient condition for global optimality [25]. Multiple constraints lead to optimality conditions that resemble condition (7) [28-341. Consider, for instance, an elastic truss of given layout that carries a point mass m, at the typical joint c1, (c1= 1, 2, . . . ), and let upper bounds be prescribed on the fundamental natural frequency and on the static compliance under the weights gm,. For brevity, bounds on cross-sectional area will not be considered here, and it will be assumed that, for the determination of the fundamental frequency o, the mass of bar i may bc accounted for by adding +ppI to the point masses at the tetminal joints of this bar? iji being the density of the bar, Subject to the frequency and compliance constraints, the cost of the truss is to be minimized. If only the compliance constraint is relevant, the optimality condition is J&E;~= k2ci,
(8)
where E: is the axial strain of bar i under the weights gm,. lf only the frequency constraint is relevant, Rayleigh’s principle may be used in a similar way as the principle of minimum strain energy was used above to derive the optimality condition
839
Conditions for Structural Optimality
Here, UCis the displacement vector of joint a and E; the axial strain of bar i in the fundamental mode, and aeiF{i}if
joint a{:: ..,lan
endpoint of bar i.
Note that the optimality condition (9) states that, in the optimal truss, the contribution of a bar to the difference between the amplitudes of the strain and kinetic energies of the fundamental mode is proportional to this bar’s contribution to the cost of the truss. When both constraints are relevant, the optimality condition assumes the form AE,e;’ + (1 -A) E,E;' -
(O<& ~p.,P,luq=k2*,
CONCLUDING
1).
(11)
REMARKS
It is worth recognizing that the optimality conditions derived from minimum principles have a common form, The structure that is to be optimized will, in general, have parts with prescribed dimensions (fixed parts) and parts with dimensions that are at the choice of the designer (free parts). In the following, use of the term free part will imply that the considered part can be designed independently of all other free parts. To avoid degenerate structures, a lower bound may be set for a critical dimension of a free part. A free part will be said to be unilaterally or bilaterally free according to whether its critical dimension is or is not at the lower bound. Cbnsider first optimal design for a single constraint that specifies the value of a scalar quantity Q (usually an energy) describing a global feature of structural behavior. If the value of Q for a given structure is characterieed by a minimum principle, the optimality condition states that the contribution of each free part of the optimal structure to the value of Q is equal to or less than a tied multiple of the cost of this part according to whether the part is bilaterally or only unilaterally free. If there are several constraints that specify the values Q,, Q,, . . . of scalar quantities describing global structural behavior, and each of these quantities is characterized by a minimum principle, the words “contribution of each free part of the optimal structure to the value of Q” in the optimality condition above must be replaced by the words “a bveighted average of the contributions of each free part of the optimal structure to the values of Q,, Q,, . . . “. (The weights used in averaging the contributions to Q,, Q,, . . . must be the same for all free parts.)
REFERJZNCES [l] R. L. BARNETT. Minimum-weightdesignof beamsfor deflection. Proc. ASCE87, EMl, 75-109(1961). [2] C. Y. SHEUand W. PRAYER, Optimal plastic design of circular and annular sandwich plates with piecewiseconstant cross-section. J. Mech. Phys. Solidr 17. 11-16 (1969). [3] 0. J. MEOAREFS, Minimal design of sandwich axisymme& plat&. &oc. ASCE 93, EM6, 245-269 1967)and 94, EMl. 177-198(1968). [4] J.-M. CHURN and W. %AcaER. Optimal design of trusses for alternative loads. Ztzgeni&w-Archiv (to appear).
840
WULIAM PRACER
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