Synthesis of structural geometry using approximation concepts

SYNTHESISOF STRUCTURAL GEOMETRY USING AFPROXIMAT~ONCONCEPTS URIKIRSCH? Departmentof Civil Engineering,Technion-IsraelInstitute of Technology Haifa, Israel

Abet-Some approx~matiooconcepts for e&ient synthesis of structural geometryare presents” Using the force methodof anaiysis and negIectingtemporarilythe implicit~mpatibility condemns, an approximateexplicit problem(AD) is introduced.Solvingthe AD, a towerbound of the optimumis e~~ient~yobt~ed. To evaWe the true optimumof the implicitproblem,the ~ompat~~iityconditionsare consideredfor the final g~metry of the AEP. Choosingthe geometric variables as the independent ones, muhifevelsolution proceduresare proposed,To improvethe solutionefficiency,the numberof independentvariablesis reducedby geometricvariablelinking.Also, the number of triai geometriesis reducedby introdu~i~ a coarsegrid in the independentvariablesspace. Several approximationconcepts are proposed for efficient solution of the explicit fixed geometry problem. Linear programmingmodels and approximate treatment of the displacement constraints are presented. The proposedsolution proceduresdo not involve multipleimplicitanalysesof the structure. Numerical examples show that in a variety of structures, where the optimal geometry is not appreciably affectedby the compatibilityconditions, a single exact analysis is sufficientto evaluate the final optimum.The efficiencyof the so&ion process and the quality of the approx~ations used are demonstrated.

While most studies on structural o~timi~t~on assume only cross sections as design variables, it is recognized that considerationof both geometric and cross-sectional design variables in the synthesis process may appreciably improve the optimal design. Synthesis of struct~al geome~y has attracted many researchers in recent years&121. The main di~culties involved in solvingthis problem are: (a) The two types of variables involved are of fundamentally different nature. Changes in the geometric variables (such as coordinates of nodes in framed structures] may be of different magnitude than changes in cross sections. Experience has sbownf2,9] that combining these two types of variablesmay produce a different rate of convergence and ill-~o~ditionjngproblems. (bf Since the behavior ~oustraintsare usually imp~~it functions of the design variables,a complete analysis of the structure must be repeated for any trial design to evaluate the constraint values. The need for m~ti~le repeated anatyses, which usually involve much computational effort, is a major obstacle in applying optimi~tion methods to large structures, (cl The total number of variables~~artic~arlyin large systems) is often large. In many synthesis methods the solution efhciency is highly dependent on the number of variables optimize simultaneously,and it is desired to reduce this numbers fd) While the optimal design is usually improved by ~ons~de~nggeometricvariables,the objective function is often not sensitive to changes in these variables near the optimum. The result might be slow convergence and further computational effort is needed to reach the optimum. In this study some approximation concepts and a multitevelapproach are combinedto overcome the above mentioned di~culties. The concept of optimization in tAssociatePrafessor.

separate design spaces, used by several aut~rs[2, 7, 9, 10, 13-151,is based on choosing the geometric variablesas the independentones and solving the fixed geometry problem by simple algorithms. It combines efficient subo~timi~ationfor member sizes, reduction in the number of design variabies optimized simuI~n~o~sl~,and improvedconvergence.This concert is further developed in this study and viewed as a two-levef approach in which the cross sections are optimizedat the first level for a given geometry, and the geometric variables are optimized at the second level. This two-level approach is justified particularly in cases where the number of georne~i~ variables is relatively small. Alternativepossibilitiesfor mul~lev~lsoIu~o~of the geometric synthesis problem are also discussed. Some approximation concepts are proposed to improve the solution process. To reduce the ~ornp~~tio~~ effort invofved in multiplerepeated analyses, it is common practice now to use approximation concepts in gross-sections optima1 designf15-251. Approximate behavior models used in st~ctural optimization can usually be ctassifiedas follows: (a) Approximations based on information obtained from exact analysis of a limited number of designs. Typical examples for this type of approximations are Taylor series expansion, polynomial fitting, or reduced basis. (b) Approximationsbased on temporary negl~ting the implicitanafysise~ua~ons.This can be done, e.g. by assuming fixed internal forcesfl5,18,221 or neglecting the compatibilitycondit~ons~l~,17,19,24]. In this study the general problem of optimizingthe structural geometry and member cross sections is approximatedby an explicit optimi~tion model. Using the force method of analysis and neglectingtemporarily the ~om~atib~iityconditions~ all the constraints become explicit functions of the variables. Solution of the ap proximate explicit problem, which involves much less computationsthan solution of the implicitproblem, provides a lower bound of the optimum. To evaluate the

URI KIRSCH

306

true optimum of the implicit problem, it is proposed to consider the compatibility conditions only for the final geometry of the explicit problem. In cases, where the difference between the lower bound and the final solution is small, no further calculations may be required. This situation is typical in structures where the optimal geometry is not appreciably affected by the compatibility conditions. Since the explicit problem does not involve multiple implicit analyses, it can be efficiently solved. To further improve the solution process, it is proposed to reduce the number of independent (second level) variables by design variable linking. This is justi~ed particularly in cases where the structural geometry can be represented by a small number of independent design variables. Also, the number of trial geometries is reduced by introducing a coarse grid of points in the space of the geometric variables. This operation may greatly reduce the amount of computations in cases where the objective function is not sensitive to changes in the latter variables near the optimum. Some simplified models for efficient optimization of the cross sectional (first level) variables are presented. These include linear programming formulations and approximate treatment of displacement constraints. 2. G~ERAL FO~ULATION

The general design p~o~~e~ (GDP) discussed in this study can be stated as follows: Find the geometric design variables {Y} and the crosssectional design variables {X} such that 2 = F({X},{Y})-+ min.

(objective function) (stress constraints)

{CrL}~{o}~{uU}

(1)

analyzed, it might be convenient to use the virtual-load method. It is often assumed that the displacements Di can be expressed as

where n is the number of members (or cross-sectional design variables), and Bii represents the contribution of the ith member to Di

(10) Ai is the force in the ith member due to the actual loads (computed by eqn 7); A$ is the force in the ith member due to a virtual load Qi = 1.0 applied in the jth direction; li is the member length; and E is the modulus of elasticity. Equation (9) can be expressed in the matrix form

(11) It should be noted that eqn (11) is based on the assumption that a single force (such as axial force or bending moment) is sufficient to describe the response behavior of each member. However, the present formulation can be extended to the more general case of multiple force members. Also, it is assumed that Xi are the cross sectional areas in trusses or linear functions of the moments of inertia in flexural systems (such as beams or frames).

(2) 3. THE APPROXIMATEEXPLICIT PROBLEM (AEP)

{DL}c (D} s {If’}

{XL} s {X) s {X”} (YL}c{ficfYU}

I

(displacement constraints)

(3)

(side constraints)

(4) (5)

in which L and U are superscripts denoting lower and upper bounds, respectively; and {u} and {@ are vectors of stresses and displacements, respectively. Both {o} and {O}are usually implicit functions of the design variables, given by the analysis equations. Using the force method of analysis the redundant forces {N} can be computed for any given design by solving the set of compatibility equations 11;1IM = I8

(6)

where [1;1 is the flexibility matrix and {S}is the vector of displacements corresponding to redundants due to loads on the primary structure. The internal forces {A} and the displacements {LJ}are then computed by

To reduce the computational effort involved in multiple repeated analyses, it is proposed to introduce explicit approximations of the optimization problem by neglecting the compatibility conditions (eqn 6). Substituting eqns (7) and (11) into eqns (2) and (3), respectively, the following approximate explicit problem (AEP) can be formulated: find {X), {Y} and {N}, such that 2 = F({X}, { Y})-min

(12)

IAL] c L&1+ ]&IIN] s {A”}

(13)

{o”}~[[B]{l/X}~{IDU}

(14)

{XL1g {X) C {X”)

(15)

{YL}S(Y)~{YU}

(16)

where {AL) and {A “} are lower and upper bound vectors for the forces, given by

0) = I&.] -t [&KM

(7)

IALl =

[ClbLl

(17)

ID} = WL] + [Qvl{fv

@I

IA “] = [Clb”}

(18)

in which {AL}, {LX} are the vectors of forces and displacements, respectively, due to loads, and [AN], [IIN] are the matrices of forces and displacements, respectively, due to unit value of redundants. All these quantities are computed for the primary structure. If only a small number of displacements must be

[Cj is a diagonal matrix giving the relationship between the stresses and the forces, its elements being functions of {X). The elements of all the lower and upper bound vectors are usually constant (if buckling is considered, the elements of {&) are functions of both (X) and ( Y}); the elements of [I?] are functions of both {N) and {Y}

Synthesis of structural geometry using approximationconcepts

(for a statically determinate structure {N} = {O});and the elements of {AL} and [AN] are functions of only {Y}. Since all the constraints in this formulation are explicit functions of the variables, the optimization procedure does not involve repeated solution of the analysis eqns (6). Unlike the GDP statement, the redundant forces {N} in the AEP formulation are treated as independent variables. This topic of problem dimensionality will be discussed later in Section 4. The solution of the AEP can be viewed as a lower bound of the optimum, since the compatibility conditions are not considered. (The latter conditions are not needed in a statically determinate structure where both formulations of the GDP and AEP are equivalent.) The proposed procedure for solving the AEP is based on a multilevel approach, as will be presented in Section 4. For each candidate geometry the variables {N} and {X} are optimized. Some simplified models for the fixed geometry optimization problem are discussed in Section 5. The topic of consideration of the compatibility conditions is treated in Section 6. 4. MULTILEVEL SOLUTION

One straight forward approach to solve the AEP is to optimize simultaneously all the variables (Fig. l(a)) by one of the available nonlinear programming methods. However, the problem dimensionality is usually high and experience has shown that combining the two types of design variables (the geometric variables {Y} and the cross-sectional variables {X}) may produce convergence difficulties and ill-conditioning problems. The concept of separated design spaces, used by some authors, combines inproved convergence, reduction in

Optimize

simultaneously

{Yl,{N),{Xj

307

the number of variables optimized simultaneously, and efficient optimization of member sizes. A possible twolevel solution procedure of the AEP is as follows (Fig. l(b)): (1) Assume initial geometry. (2) Optimize the cross-sectional variables and the forces for the given geometry by solving the problem of eqns (12)-(15). (3) Modify the geometric variables subject to eqn (16). (4) Repeat Steps 2 and 3 until the optimal geometry is obtained. It has been shown[l6,17,19] that if displacement and buckling constraints are not considered, the fixed geometry problem of Step 2 (the first level problem) can be stated in a linear programming form and efficiently solved by standard computer programs. Further simplified models for solving this problem will be discussed later in Section 5. The solution efficiency is highly dependent on the number of candidate geometries in Step 3 (the second level problem). The concept of design variable linking, which has been applied to cross sections optimal design[201, can be used also in geometric optimization. The number of design variables is reduced by expressing all the geometric variables in terms of a small number of independent ones. Design variable linking is often necessary due to such considerations as functional requirements, fabrication limitations, etc. Another possibility to reduce the number of candidate geometries at the second level (Step 3) is to use a coarse grid in the space of geometric variables, so that only a small number of {Y} values is considered. This is justified in many

]

Optimize

~N~,~X~ for

I

the given

IY)

t

I

Optimize

Optimal Fig. 1.

{XI

solution

308

URI KIRSCH

cases where the objective function is not sensitive to changes in the geometric variables near the optimum. To optimize the {Y} variables at this step, one of the well known unconstrained minimization techniques [26] can be used, with provisions taken to ensure satisfaction of the side constraints (eqn 16). If the degree of statical indeterminacy is low, it might be desired to employ the following solution procedure (Fig. Ic): (1) Assume initial geometry and force distribution. (2) Optimize the cross-sectional variables. (3) Modify the geometric variables. Each trial geometry involves optimization of the force distribution (Step 4). (4) Modify the force distribution. Each checked force distribution involves cross sections optimization (Step 5). (5) Optimize the cross-sectional variables. If displacement constraints are not considered, Steps 2 and 5 (the first level) are reduced to a direct determination of the cross-sectional variables. The choice of {Y} in Step 3 (the third level) is subject to the side constraints of eqn (16). Also, it is assumed that solution of the second level problem (Step 4) exists for all feasible {Y} values. Available nonlinear programming methods can be used to optimize {N} and {Y}. A direct search approach, while not as efficient as methods based on first or second derivatives, is most convenient since only the objective function values must be calculated. However, this approach is not practical for problems with large numbers of variables. It should be noted that all intermediate values of the variables {Y}(Fig. l(b)) or the variables {Y} and {N} (Fig. l(c)) are feasible. That is, the iteration can be terminated always with a feasible-even though nonoptimal-solution, whatever the number of cycles. While any of the three schemes shown in Fig. 1 may prove useful for a specific application, the two-level procedure of Fig. l(b) is most suitable in many cases. In the next section some simplified models for solving the fixed geometry (first level) problem are discussed.

diagonal. This formulation is often used in optimal plastic design of framed structures[lS]. The amin advantage is that the LPP can be efficiently solved using standard computer programs. The assumption of linear dependence between Z, [Cl and the cross-sectional design variables {X} is usually valid for truss structures. It is approximate for other types of framed structures such as beams and frames, however, it has been noted[27] that the inaccuracy involved in this assumption is minor. A more serious drawback is that the displacement constraints (eqn 14) are not considered in the linear programming solution. Some possible simplifications intended to include the latter constraints in the LPP formulation are presented here. Lagrange multiplier approach

One approach is to solve first the LPP and then to check if the displacement constraints are satisfied. If a certain constraint 4 < Dj” is violated, we may calculate optimal values of X,(i = 1, . . . , n) for which Dj = Di”.

(22)

Substituting eqn (9) into eqn (22), the problem to be solved is: find {X} such that Z = 2 /,X-

min

(23)

i=l

g+=

Dj”.

(24

i=1Ai

Defining the Lagrangian function

(25) the conditions that must be satisfied at the optimum are

$&$-Dj”=O I

5. SOLUTION OF THE FIXED GEOMETRY PROBLEM (FGP) Solving the AEP by the two-level approach of Fig. l(b), the cross-sectional variables and the force distribution in the structure are optimized for any candidate geometry. This operation, which must be repeated many times, usually involves much computational effort and it is essential to employ efficient solution procedures at this step. Several approximation concepts, leading to simplified models and efficient solution of the fixed geometry problem (FGP), are developed in this section. Assuming both the objective function 2 (eqn 12) and the matrix [Cl (eqns 17 and 18) to be linear functions of IX} and considering only the stress and side constraints, we obtain the following linear programming problem (LPP): find {X} and {N} such that Z= {I}‘{X}-min [xl{u?

c {A3 + I&IIN) {X”}~{X}<{X”}

(19) s WIW’J

(20) (21)

where {I} is a vector of constant coefficients (members length, e.g.), and [Xl is a diagonal matrix containing linear functions of the design variables on its principal

!i!LO -$=6-A X,’ h

h=l

, . . . , n.

(27)

Solving for A and X,, yields

X,,=V/aV\l(BjJl,,)

h=l,...,n.

(29)

The required values of X,, may be computed for all displacement constraints and the largest value of each design variable (considering both the displacement constraints and the values obtained by the LLP solution) is selected for the final design. In this procedure the internal forces (and therefore the elements Bji) obtained from the LPP solution are assumed to be constant. Thus, the final design is only an approximation to the optimum of the FGP, its accuracy being dependent on the elements Bii. Linearized displacement constraints

Another possibility is to linearize the displacement functions. A first order Taylor series expansion about the

Synthesis of structural geometry using approximation concepts

eqn (31) is reduced to

solution point of the LPP, {&, {$, yields

D_

_

J--

bjko t bik,k-,Nh-, + bihhNh -%I

b&O$ bik,h_cNh- 1f bj*hNh- Dj”xk C 0.

where 6iio and brik are constant coefficients. Differentiation with respect to X,, and Nrt respectively, gives

X;=C&i=2,3,4.

(34)

The value of the coehicients oi can be selected based on the stress constraints or other considerations. (The ai can be determined, for example, from the LPP solution; the linearized displacement constraint is then added to the LPP and a revised solution is obtained.)

J&g2 Xi

Adding the linearized displacement constraints to the LPP, we may solve an augmented LP problem. The steps of linearization and solution of the modified LP problem may be repeated until convergence. In the present formulation both {X} and {Nf are treated as independent variables. It is interesting to note that eqn (32) is valid also if the compatibility conditions are considered (that is, Nk depend on Xi) since ~~i~~Xi = 0 (eqn 9) as proved by Berke [28].

Statically determinate systems In truss design, uneconomical members are often deleted by the LPP solution if lower bound constraints on the cross-sectional areas ({X”} six)) are not considered. The result might be a statically determinate optimal truss with fixed internal forces. (This is the case, for example, if the truss is subjected to a single loading condition.) Choosing the inverse cross-sectional areas & = I/Xi as design variables, all the constraints can be expressed in linear terms and only the objective function is a nonlinear functjon of the new variables. Experience has shown that solving for the linearized objective function

Selecting the virtuat-load system The internal-force system corresponding to the virtual external load Qj needs only be statically equivalent to Q.. In a statically indeterminate structure, the forces A# (eqn 10) are not unique and various choices exist for the statically equivalent system. Selecting, for example, the system shown in Fig. 2 (with hinges assumed over the intermediate supports) the displacement expression of

Nh-t/1

‘h

th

bt Ah Qj=l.O Cl

(35)

That is, any displacement constraint of a continuous beam can be expressed as a linear function of three variables and added to the LPP. Using a similar approach for the frame shown in Fig. 3(a) and selecting the equivalent system of Fig. 3(b), the vertical displacement constraints (such as D, % .Q” or 4 c 4”) can be expressed in the linear form of eqn (35). To obtain a linear horizontal displacement constraint (I&C 4’), it is necessary to assume a linear relationship between the design variables of the middle column (Fig. 3(b), only flexural deformations are considered)

(31)

~t~-“y

(341

in which h denotes the member on which the virtual load Qj is applied, and Nk_,, Nk are the bending IIIOnWItS over the supports of the hth member. Substituting this equation into the displacement constraint 4 g Q” and rearranging, we obtain the linear form

where m is the number of redundant forces ( = degree of statical indeterminacy). To calculate the displacement derivatives, it can be noted that the internal forces {A) are linear functions of {N} (eqn 7). Also, a staticalty equivalent internal force system corresponding to the virtual external load Qi (eqn 10) may be selected so that the forces AZ are fixed. Thus, the displacement Dj can be expressed as (eqn 9)

r-l

309

+

Fig. 2.

+INh

URIKIRSCH

310

Fig. 3.

the true optimum is,often reached after a small number of modifications of Vi. 6. CONSIDERATIONOFTRECOMPATIBILITYCONDI~ONS Once the complete AEP (eqns 12-16) has been solved, it is necessary to consider the compatibility conditions (eqn 6) for the final geometry. If the resulting design is statically determinate, this step is not necessary and the solution of the explicit problem is the final optimum. AS noted earlier, this would be the case in truss optimal design if members are eliminated by the LPP solution to obtain a statically determinate truss. Several procedures have been proposed to obtain a solution satisfying the compatibility conditions [16,17,19,24]. Denoting the optimal values of the AEP by asterisks *, then the degree of not satisfying the compatibility conditions by this solution is given by (eqn 6) {AS}+%&-{*s}.

(38)

The object is to modify the solution so that {ASI={O]. Farshi and Schmit[l7] proposed to add successively a linearized form of the compatibility conditions and solve a linear pro~amming problem at each iteration. Reinschmidt et aL[l6,19] proposed to employ a fully stressed design procedure to obtain the final design. Another possibility is to solve the complete explicit problem for the given geometry by one of the available nonlinear programming methods. Comparison of the AEP solution with the exact implicit solution indicates the effect of the compatibility conditions on the optimum. In cases of a large difference between the two solutions, it might be desired to solve the implicit problem for modified values of the geometric variables. The following examples show that such cases are not typical. 7.ExAbfPLES

All dimensions are in tons and meters. The allowable stresses are {au}= -{uL}={15,000], the modulus of elasticity is E = 2.1 . fO’, and the objective function 2 represents the volume of material. Example 1. Statically determinate six- bar truss (Fig, 4) The truss is subjected to a single horizontal load and the member areas are chosen as cross sectional design variables, with no side constraints imposed. The single geometric variable Y is subject to the side constraints 3c YGlO

(39)

5.0

5.0

~

Fig. 4,

To illustrate the sensitivity of 2 to changes in Y, two cases were solved. Case A The behavior constraints are related only to stresses. Case B The displacement constraint D, G 0.03

(40)

is considered as an additional requirement (0, is the horizontal displacement at the top of the truss). Variation of minZ with Y for both cases is shown in Fig. 5. While large changes in Y result in considerable variations of min Z, it can be observed that the objective function is relatively not sensitive to changes in the geometric variable near the optimum. That is, a coarse grid of points (say, a minimum step size of 1.Om) could be assumed for Y. Since no redundant forces are involved, compatibility conditions must not be considered and the explicit problem formulation is exact. Example 2. Twenty one bar truss (Fig. 6) The truss is subjected to a single vertical load. The degree of statical indeterminacy is four but due to symmetry of loading and geometry only two redundant forces (N1 and NZ) may be considered. The member areas are chosen as cross sectional design variables and the single geometric variable Y represents the height of

Synthesisof structuralgeometryusing approximationconcepts

311

Fig. 5. the truss. A singe displacement constraint l$GDtU

IO’.min. 2 [ma]

(41)

is considered, where D, is the vertical displacement at the middle of the span. Two cases were solved. Case A DIu =00200 Case B Q U = 0:OlSO. The explicit problem was first solved without considering the displacement constraint. The LPP solution was obtained for various Y values (Fig. 7). It is interesting to note that the three different determinate trusses shown in Fig. 8 represent equivalent optimal solutions for all Y values. (The truss of Fig. 8(a) is unstable, two horizontal members must be added to preserve stability.) The optimal design is Y = 4.0 min Z = 0.0170

I 250 -

200 -

(42)

The actual displacement at the optimum for all three trusses is I), = 0.0180, that is the solution is infeasible for Case B. modifying the cross sections to satisfy only the displacement constraint (eqns (28f and (29)) we obtain the feasible optimal design min Z = 0.0198. In this case the displacement constraint appreciably affects the optimum. Example 3. Continuous beam (Fig. 9) The symmetric continuous beam is subjected to a single loading of four concentrated loads. For simplicity

of presentation, the following relationships have been assumed for the cross-sectional design variables x=Ai=

Wi=J

i-1,2,3,4

(43)

in which Ai = the ith cross-sectional area; Wi = the modulus of section; and 4 = the moment of inertia. The geometric variable Y, representing the location of the two interior supports, is subject to the following side constraints 3s Yc7. A singte displacement constraint is assumed D, co.001.

3.0 _I_ 3.0 _I_ 3.0 Fig. 6. CAS Vol. 15, No. 3-H

_I_ 3.0

W-1

The fixed geometry problem can be stated in a linear programming form. Solving the AEP for various Y values gives the lower bound of the optimum (Fig. 10) Y = 5.0 min Z = 0.00472

(461

URIKIRSCH

312

Fig. 8.

.

Y

4

*

I-

Y

4

10.0

L T

10.0

4

Fig. 9. IO’

optimum (eqn 46) is D, = 0.001, that is the displacement constraint is active.

minZ [ mS]

A 70-

Example 4. Multibay frame (Fig. 11) The symmetric frame is subjected to a single antisymmetric loading of six concentrated loads. The relationships of eqn (43) have been assumed and two geometric variables (Y, and YJ have been considered with the following side constraints

60-

6c Y,<14

z = 2x, Y, t 2X*(20- Y,) t x,Y3.

T I

(48)

The objective function is

AEP

50-

4s Y,<8.

(49)

Assuming steps of 2.0 in the geometric variables space, the (lower bound) solution of the AEP is I I

1

1 3

1

1

1

5

1

7

*y[ml

Y, = 12.0 Y2= 4.0 Z= 602.

(50)

Fig. 10.

Exact solution of the implicit problem for Y = 5.0 yields

The exact solution of the implicit problem for this geometry is Z = 682, which is 13% heavier than the AEP solution. The true optimum is (Fig. 12)

min Z = 0.00515.

Y, = 14.0 Y*= 4.0 2 = 668,

(47)

Since the difference between the lower bound and the exact solution is 9%, it is possible that improved results for modified geometries could be obtained. However, Fig. 10 shows that the solution of eqn (47) is the true optimum. That is, the optimal geometries of the AEP and the implicit problem are identical, despite the large difference in the objective function values. The approximate displacement at the lower bound

1 20-Y, I

1 1

Yl

(51)

corresponding to the lower bound solution, Z = 652. The true optimum is only 2% lighter than the final result obtained from the AEP solution (Z= 682) and 2.5% heavier than the corresponding lower bound solution (Z = 652). Modification of the optimal geometry obtained by the AEP solution (which may be justified if the difference between the lower bound and the exact solution is large) almost did not improve the final design.

1 1

Fig. 11.

Yl

1 *

20-Y,

1

1

Synthesis of structural geometry using approximation concepts IO’

min. Z

313

min.2

min.Z

Fig. 12.

8.

CONCLUSIONS

The combination of multilevel solution and approximation concepts provides a powerful tool for efficient synthesis of structural geometry. Member cross sections and geometric variables are of fundamentally different nature. The multilevel formulation combines simplified sub-optimization for member sizes, reduction in the number of design variables optimized simultaneously, and improved convergence properties of the design problem. Neglecting temporarily the compatibility conditions, the AEP formulation is obtained. it has been observed that the latter conditions do not appreciably affect the optimal geometry in a variety of structures, and therefore can be considered only for the final geometry obtained by the AEP solution. The AEP model provides a lower bound of the optimum and does not involve multiple explicit analyses. In the examples solved, a single implicit analysis was sufficient to obtain a near optimal geometry. The solution efficiency is improved by reducing the number of geometric independent variables and the number of trial geometries. This can be achieved by using linking of variables and a coarse grid in the space of geometric variables. The above approximations are justified in cases where the objective function is not sensitive to changes in the geometric variables near the optimum and if the object is to reach a near optimal solution rather than an exact optimum. The step of cross sections optimization for a fixed geometry (the FGP) consumes most of the computational resources; therefore, it is essentiat to employ efficient methods at this stage. Some linear programming models and approximate treatment of the displacement constraints are proposed to achieve this goal. The concepts presented in this study simplify the geometric optimization process and help to overcome some di~~ulties involved in this problem. Acknowledgemen&The author is indebted to the ‘Fund for the Promotion of Research at the Technion’ for supporting this work. REFERENCES 1. M. W. Dobbs and L. P. Felton, Optimization of truss geome~y. J. S#ruct.Div., ASCE ST10(95),2105-2118(1969). 2. G. N. Vanderplaats and F. Moses, Automated design of

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