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A SEGMENTAL METHOD FOR THE DISCRETE OPTIMUMDESIGN OF STRUCTURESA. B. TEMPLEMAN a & D. F. YATES ba Department of Civil Engineeringb Department of Statistical and Computational Mathematics, The University of Liverpool,P.O.Box 147, Liverpool, U.K.Version of record first published: 27 Feb 2007.

To cite this article: A. B. TEMPLEMAN & D. F. YATES (1983): A SEGMENTAL METHOD FOR THE DISCRETE OPTIMUM DESIGN OFSTRUCTURES, Engineering Optimization, 6:3, 145-155

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Engineering Oprimizotion, 1983, Vol. 6. pp. 0305-215X/83/OM)3-0145$18.50/0

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A SEGMENTAL METHOD FOR THE DISCRETE OPTIMUM DESIGN OF STRUCTURES

A. B. TEMPLEMANt and D. F. YATESf

tDepartment of Civil Engineering, fDeparonent of Slatistical and Computational Mathematics, The Unicersity of Liverpool, P.O. Box 147, Liverpool, U.K.

(October 29, 1982)

A simple method based upon linear programming is described for the design of minimum weight structures under the restrictions that member sizes and/or material properties may be chosen only from discrete sets. The types of structures considered are lhose composed of axial farce bars, membrane plates and shear panels. The method avoids the com- binatorial nature of dixrete optimization by introducing the concept of segmental members. The segmental optimum design is found by linear programming. Its weight is a lower bound to the weight of the discrete optimum design. A aimple mcthod fur achieving a discrete optimum design from a segmenlal optimum design is described. Several examples o f discrete optimum llU51 designs are prcsenled

I INTRODUCTION

Almost all available methods for the optimum design of engineering structures make the assump- tions that member size variables are continuous- valued and that the material and its properties from which each member is made is known in advance, (see, for example, Refs 1 and 2). In practice, however, designers rarely have the free- dom to choose member sizes from a continuous range as this implies a different and expensive fabrication process for each member. Usually designers are restricted to choosing member sizes from a discrete set of commonly available pre- fabricated sizes. Furthermore, although the set of available sizes may sometimes be large, it is com- mon practice to pie-select a small subset of per- haps five or six of these sizes which may actually be used in a design. Typically, a large size. a small size, and three or four other sizes covering this range may form such a subset which, by virtue of standardization of components, reduces many of the indirect costs associated with the design.

Designers may also find difficulty in specifying the material properties of each member before the design process starts. Typically a particular steel section may be produced in several steel grades each with a different set of material properties. Ideally, the design process itself should select the most appropriate material for each member.

Although the discrete nature of member sizes has long been recognised very little research has been devoted to optimum structural design in- corporating discrete size variables. Recently Fleury and B r a i b a d 4 have attempted to add both dis- crete members and material properties to an exist- ing, efficient continuous-valued optimum design method based upon duality. However, the com- plexity of the method is greatly increased, its efficiency is severely reduced and it does not aim to find a global optimum, merely to successively improve an initial design. That rigorous discrete optimum design is significantly more difficult than the continuous problem has been demonstrated by Yates, Templeman and Boffey.' They have shown that minimum weigh1 truss design using discrete member sizes is, in mathematical terms, an NP- hard problem. This result implies that a rigorous and rapid algorithm which guarantees to find a globally optimal discrete design is not achievable within the limits of existing ma~hematical know- ledge, and suggests that research should, instead, concentrate upon developing simple methods for finding discretedesigns which lieascloseas possible to a good lower bound to the discrete optimum.

The continuous optimum design clearly forms a lower bound to the discrete optimum though not necessarily a good one, and much of the work on continuous optimum design contains the impli- cation that the optimum continuous member sizes

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146 A. B. TEMPLEMAN A N D D . F. YATES

can somehow be rounded up or down to discrete sizes, thus yielding a good discrete design of low weight. However, this rounding process turns out to be a combinatorial problem of immense size for all but the simplest of structures. There is no way of determining a priori for each structural member whcther i t should round up or round down in size. I f all members arc rounded up, the discrete design is needlessly heavy; if all are rounded down, some performance constraints must be violated. Ad 11oc rounding rules can be devised for some structures but they ofTer no guarantees of optimality. It is obvious that if the discrete set of sizes is large and the sizes cover the range fairly evenly, the continu- ous optimum design should form a close lower bound to the discrete optimum. The use of non- optimal rounding processes in such cases will not necessarily lead to grossly inefficient discrete designs. However in the more practical situation described earlier of only five or six discrete sizes within a large range it is found that the continuous optimum design does not form a very useful lower bound. Ad iioc rounding rules in this case can often lead to grossly overweight discrete designs. Consequently the use of continuous optimum design with some approximate rounding rules cannot be recommended when member sizes are limited to a small discrete set covering a wide range of sizes. Also. it should be noted that this method does not permit the very desirable inclusion of discrete material properties selection.

This paper describes ;I new mcthod for the mini- mum weight design of structures incorporating both discrete member size variables and discrete m:~terial propcrties. It does not use the contin- uous optimum dcsign in any way, but introduces the somewhat artificial concepts of segmental mcmbcrs and the segmental optimum design which providcs a much closer and more useful lower bound to the optimum discrete design. Section 2 describes the method for the case of minimum weight dcsign of truss structures with discrete mcmbcr sizcs. Section 3 deals with rounding the segmental optimum design t o yield a discrete optimum design. Section 4 descr~bes how the mcthod can be simply extended to include discrete material propcrties and membrane plates, shear panels in addition to axial force bars. Section 5 con- sidcrs how segmcntal members can be incorporated into structural analysis methods which form part of an iterative process for the design of indeter- min:~tc structures. Section 6 gives examples of the use of the new mcthod.

2 SEGMENTAL O P T I M U M DESIGN

Consider an N-member truss composed of axial force bars which must be designed for minimum weight. Ifthe geometry of the trussand the member material properties are specified this becomes a pure bar-sizing problem. The truss may be statically determinate o r indeterminate. singly o r multiply loaded and may have limits upon member maxi- mum stresses, joint displacements and member minimum gauges. I f the choice of member cross- sectional areas can only be made from a discrete set. the optimum design problem can be generally expressed as :

Minimize

Subject to : I

- A, t Ai i= 1, ..., N (Id)

A , e S = ( s , : d = l , . . . , D) i= l ...., N ( le)

In problem (I), ( l a ) is the weight function contain- ing the known member lengths and material densities, L, and p,, and the unknown member areas A,, i = 1 , . . . , N. ( lb) are constraints upon joint displacements. 7;, is the axial force in bar i under load case j, j = I , . . . , J . Tk is the virtual force in bar i associated with virtual unit force at joint k whose displacement must be limited to 6,, under load case j. E, is the elastic modulus of bar I.

For K joint displacement limits in each olJ loading cases there will be a total of J K constraints like (lb). (Ic) are member stress constraints with d, d being known maximum permissible compressive and tensile stresses in bar i. (Id) are minimum gauge constraints in which Xi is the known smallest permissible size for bar i . ( Is) states that all bar sizes must be chosen from set S which contains a total or D known discrete sizes s,, d = I , . . . , D.

For n determinate truss the 7;, and Tk are con- stant for each member and can be uniquely deter- mined by static analysis. For an indeterminate truss 7;; and T,k are implicit functions of the mem- ber sizes A,, i = 1, . . .. N. and will change as the member sizes are changed. Modern structural

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I48 A . 8. TEhlPLEMAN A N D D. F. YATES

isn property oflinear programming problems.Thus in the case of a determinate truss the weight of the segmental optimum design is a global minimum. For an indeterminate truss solved in an iterative fashion each of the sequence of LPs will be solved globally although the sequence of global minima will not necessarily converge to a global minimum weight. This is an inherent feature of the iterative process and applies not just to segmental optimum dcsign but to all methods which solve minimum weight truss design problems iteratively. Second, the weight of the segmental optimum design (problem (3)) is a lower bound to the weight of the discrete optimum design (problem (1)) for the specified values of T j , T,. This can be demonstra- tcd as follows: For the solutions ofproblems ( l )and (3) to be equivalent, the solution of problem (3) must contain only one segment per member. All other segment lengths must be zero. This may be achieved by adding to problem (3) the following constraints:

L, for some d = [I* i = I . . . . . N lid =

0 for d # d* d = l . . . . , D (4)

The eRect of adding constraints to problem (3) will be to increase the value of the minimum weight by some non-negative amount. Therefore the weight obtained by solving problem (3) without extra constraints (4) must be generally lower than the solution weight of problem (I).

I f problem (3) is solved by some LP algorithm based on the simplcx method some further features of the segmental optimum design can be deduced from the structure of the simplex table. The first step in the simplex solution is to add slack variables to convert the J K displacement constraints into equalities. The L P therefore now has (ND + J K ) variables and (N + J K ) constraints. The nature of [he simplex method is such that in the optimum solution each of the constraints will contain one b s i c variable (having a non-zero value). There will therefore be(N + J K ) basicand (ND + JK) - (N + J K ) = N(D - 1 ) non-basic variables in the solution. Suppose that the segmental optimum dcsign has Jx active displacement constraints where 0 s Jx 5 JK . For these active constraints the slack variables are, by definition, zero and will be non-basic. For the remaining ( JK - J x ) dis- pl:icement constraints which are inacti\'e, their slack variables must have some positive value and must therefore be basic. The optimal set of (N + J K ) basic variables must be composed of

( JK - J x ) displacement slack variables from in- active constraints and (N + J K ) - (JK - J x ) = (N + J x ) other variables which can only be segment lengths. The presence of the geometric length equivalence equalities (2) in problem (3) demands that at least one segment length per member must have a non-zero value and will be basic. The solution of problem (3) must therefore have (N + J x ) non-zero segment lengths with at least one for each of the N members. This leads to the conclusion that in the s e ~ e n t a l optimum design there can be at most J K multi-segment members and at least (A' - J X ) members com- posed only of a single segment.

This is a striking deduction. J x , the number of active displacement constraints in the optimum design, is usually a fairly small number; very much smaller than N, the number of members. The seg- mental optimum design, which has already been shown to be globally optimal for specified Ti, T.,and to bea lower bound to thediscrete optimum design, can be found by straightforward linear programming and will have the vast majority of its members composed of a single discrete segment. Only a very few members of the segmental opti- mum design will have more than one segment. It is clear from this that the segmental optimum design forms a very useful lower bound which is a long way along the path towards a discrete optimum design.

3 ACHIEVING A DISCRETE OPTIMUM DESIGN

The discrete optimum design for a truss must have only one segment of discrete size per member. In the segmental optimum design most members will satisfy this requirement but there will be a few multi-segment members which do not. Some sort of rounding operation is necessary. An obvious scheme is simply to round up all the multi-segment members, i t increase the size ofall the smaller size segments in each member until they are of the same discrete cross-sectional area as the largest discrete size within the member. If all multi-segment members are treated in this way the result will be a feasible discrete design which may be the discrete optimum design and will in general be an upper bound to the discrete optimum design. Because only a few members of the truss are concerned in this rounding up operation the percentage weight increase should be very small and the discrete design thus obtained, though perhaps not optimal,

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DISCRETE OPTIMUM DESIGN I49

should have a weight only fractionally larger than the globally optimum discrete design.

This rounding up can be done using the simplex table corresponding t o the segmental optimum design. The rounding u p operation is equivalent to setting the segment length variables of the smaller size segments within a multi-segment member t o zero. T o d o this, the Jx smaller size segment variables a t present in the basic set must leave the basis, to be replaced by other variables. T o avoid creating other multi-segment members the vari- ables which must enter the basis must be the slack variables for the J i active displacement con- straints. The rounding up is therefore accomplished by JX pivoting operations on the simplex table. The ioct that displacement constraint slack vari- ables enter the basis means that in the discrete rounded design, active constraints will in general become slightly slack or inactive. This reflects physical reality in that it is not generally possible to satisfy some exactly defined displacement limit using only discrete sizes.

Having achieved a rounded-up discrete design from the segmental optimum design this is often as Tar as the method will go. Occasionally, however, further refinement of the discrete design may be possible and it is comparatively easy to check this. The check consists of determining whether any complete members in the discrete design can be replaced by complete members of a smaller size without violating any constraints. The simplex table for the rounded up discrete design can be used for this. The slack variables of the displacement constraints will be in the basic set and must remain there with positive or zero values. The other basic variables are segment lengths, one per member with a value equal to the physical length of the member. The objective function coefficients will indicate several candidate segment length variables in the non-bas~c set which, i f they entered the basis, would rcduce the weight of the structure. Each candidate can be examined and pivoted into the basis provided that, I) it pivots a complete segment variable out of the basis, and 2) it does not violate the non-negativity of any other basic variable. These extra checks and pivots will produce a new discrete design even closer to the optimum.

I t should be noted here that only the optimality of the segmental optimum design is guaranteed. The rounding up and refinement proccsscs are not rigorous and have no inherent guarantees of optimality. However, because the segmental opti- mum design is globally optimal and forms an

"almost discrete" design with only a few multi- segment members it is usually a very close lower bound to the discrete optimum design. Because only a few members are involved in the rounding, the rounded up segmental optimum design usually forms a very close upper bound to the discrete optimum design. Any further refinements of this upper bound design will tighten further these already close bounds upon the discrete optimum design.

4 EXTENSIONS O F T H E M E T H O D

4.1 Discrete nturrriul properries

Discrete material properties can be includcd in the segmental approach very easily. They d o not affect the nature of the method but d o increase the size of the segmental LP, problem ( 3 ) . Suppose that the material of each bar of the truss is not known in advance but must be selected from a total of P different discrete materials, p = 1, . . . , P, each material having different density, p, and elastic modulus E. Problem (3) can be modified so that it will selec~ the most appropriate material for each segment lcnglh provided that segments are appro- priately defined. Define I,, to be the unknown length of a segment of member i , i = I . . . . , N, which has o known discrete cross-sectional area sd , d = I , . . . , D, and is made of material p, p = 1 , . . . , P, for which material properties are known. Thus each member will have ;I Iota1 of DP segments defined for it. With this definition, problem (3) can be reformulated as :

Minimize

Subject to: I

I = I . . . . . N I

Problem (5) has the same LP form as problem (3) but with a total of NDP variables instead of h'D.

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I50 A. B. TEMPLEMAN A N D D. 1'. YATES

In order to form problem (3) from problem (1) it was assumed that no segments would be defined in ;I mcmber which would violate stress o r minimum gauge constraints, thus these constraints were omitted. Their omission also from problem (5) tn:~kcs a similar assumption; that no segment will bc defined in a member of such a combination of sizc and material properties as would violate minimum gauge o r stress constraints. This implies that a slightly more comprehensive checking scheme will be necessary to ensure the feasibility with respect to stress and gauge limits of all seg- ments defined in problem (5).

Having set up problem (5), it can be solved in the same way as problem (3). All the results and deduc- tions following problem (3) remain valid and unchanged for problem (5). The number of multi- scgment members in the segmental optimum design will still be, at most, equal to J T , the number of active displacement constraints, as before. The truss members which are composed of only one scgment will have had the optimal material selected lor them. The rounding up process which produces a discrete design from the segmental optimum dcsign throws up an interesting problem. It is possible, in the segmental optimum design, for a multi-segment mcmber to Ii:~vc segments of differ- cnt ni:~tcri:~ls. How is such a member best rounded to yield ;I mcmber of uniform sizc and material? This question can be answered if. as before, the rounding up is done using the simplex table corresponding to the segmental optimum design. The J K slack variables corrcsponding to active displacement constraints at present in the non- basic set, must pivot into ihc basis. This pivoting operation will automaticdly cause J i surplus scgment lengths to leave the basis and the rounding of c;ch multi-segment membcr will occur auto- matically os a result of the pivoting and in an optimal fashion for each mcmber. The method of improving the discrete dcsign by replacing whole members using the simplex table remains un- changed.

All minimum weight truss design methods are c:~p:~ble of being easily extcndcd to include mem- branc plate and shear panel clenicnts in addition to axial force bars. This is becl~usc all these elements sh:~rc tlic property that stiffness is proportional to mass. Figure 2a shows a membrane plate element. The clement shown is rectangular but the shape is

of no consequence. In conventional minimum weight design the plate element would have a known surface area, B, corresponding to the known member length of an axial force bar, and an un- known, plate thickness, r , corrcsponding to the unknown cross sectional area. A , of the bar.

Segmental Plate

FIGURE 2 Conventional and segmental membrane plales.

The key to the segmental approach which sets up problem (3) for bar structures was the definition of segments which change the problem variables. A similar change of variables for plate elements leads to the definition of plate segments which per- mits their inclusion in problem (3). Plate segments are defined as follows:

Assume that the thickness i i of plate element i must be chosen from a discrete set of plate thick- nesses R,, tit = 1, . . . , M. Define hi, to be the unknown surface area of a segment of plate i which has discrete thickness R,, m = I , . . ., M. Figure 2b shows a segmental plate with four segments each having a different discrete thickness. The layout of the individual plate segments within the overall element is immaterial, but the sum of the segment surface areas must be equal to the total plate element surface area. Thus:

11

1 h,, = B, for all plate elements i m = l

(6)

This necessary relationship is the parallel of Eq. (2) which applies to bar members.

With this definition of plate segments and the assumption that n o plate segment will be defined which would violate stress o r minimum gauge

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DISCRETE OPTIMUM DESIGN IS1

constraints. an exact parallel with problem (3) can be set up for plate elements. Combinations of bar and plate elements can be included in a similar for- mulation, and shear panels can be handled in the same way as plates. If discrete material property variables are to be included the definition of seg- ments leads to an exact parallel with problem (5). Define bi,, to be the unknown surldce area of a segment of plate element i which has discrete thickness R , and is made of material p. All aspects of the formulation, solution, round up and refine- ment remain unchanged. In any segmental opti- mum design the total number o i multi-segment bars, plates or panels is determined solely by JX, the number of active displacement constraints.

5 INDETERMINATE STRUCTURES

The methods described in the preceeding sections are applicable to statically determinate structures in which only one optimization is required. The methods can also be applied to each optimization in an iterative sequcnce of analysis/optimization cycles for designing statically indeterminate struc- tures. This latter case is worth closer examination because there are several ways in which the optimiz- ation methods already described can be integrated into an iterative design strategy.

One possible method is t o perform a complete discrete optimum design in each optimization cycle. This will involve finding the segmental optimum design by LP, rounding up and refining to yield a discrete optimum design which is then analyzed for thenext cycle in the iterative sequence. This approach poses n o special difficulties. An alternative approach is to perform only a segmental optimum design in each iterative cycle until con- vergence of the sequence is almost complete. Rounding u p and refinement are only added in at this late stage. This second approach appears to be attractive and to require less computation and hence less computing time. It does, however, raise one small difficulty in that the structural analysis which separates each optimization must be carried out on a structure containing multi-segment mem- bers, and some thought needs to be given to how this is best accomplished.

If a matrix stiffness (displacement) method is used to carry out the analysis. the equations to be solved are of the form

in which Sand P a r e vectors of nodal displacements and externally applied nodal loads, and K is a square stiffness matrix. 6 is found by solving Eqs. (7) for specified K and P. and values for all the Ti. 7;, required in problems (3) and (5) can then be calculated from 6. The elements of the stiffness matrix K are formed from individual member stiffnesses. For a single segment axial force bar the member stiffness is EA/L where E is the elastic modulus of the member material, A is its cross- sectional area and L its length. For a multi-segment member with different lengths. material properties and areas for each segment. the simplest way of accommodating this within K is to calculate an equivalent stiffness value for the member rather than to attempt t o write K in terms of individual segments. The equivalent stiffness k i of a multi- segment axial force bar is simply

P D E , s d ki = 1 1 - for all lidp # 0

p = l d = I I i d P

Using Eq. (8) all elements of K corresponding to a segmental design can easily be calculated and the subsequent determination ofall Ti, T, is straight- forward. Equivalent stiffnesses similar to Eq. (8) can be calculated for plate elements also.

The relative efficiencies and merits of the two strategies for designing indeterminate structures are not yet known. Computational tests o i both methods are needed to determine which is the better strategy.

6 DESIGN EXAMPLES

Figure 3 shows a 38-bar cantilever truss. The truss is statically determinate and carries only one load system asshown. All members arc made of the same material for which p = 7.85 x k g / m m h n d E = 210 kN/mm2. The truss is to be designed for minimum weight with a single constraint, that the vertical displacement of the cantilever tip must not exceed 1Omm. There are no stress o r gauge con- straints.

This example was chosen so that the method and its results could be compared with an alternative method6 which had also been used to solve this example. As this truss is statically determinate with a single displacement constraint it is easy to find an exact continuous optimum design. Table I lists this continuous optimum design which has a

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A . B. T E M P L E M A N A N D D . F. YATES

4 I

F I G U R E 3 311-harcanlileuer truss.

Member wcar (mm' x 10')

Member Continuous Segmcnlul Discrete No. oplimum oplimum optimum

Weight (kg1 8165.7 8456.2 8489.2

weight of 8167.7 kg. Optimum member sizes range between 7 x 103 mm2 and 85 x lo3 mm2. This design is listed only for comparison. For the pur- poses of discrete design it was assumed that only five bar sizes were avnilable and discrete sizes were choscn roughly within the continuous optimum

range. The set of sizes selected was 5.0, 10.0, 20.0, 40.0 and 75.0 x lo3 mm2. Problem (3) for this truss i san LP with 191 variablesand 39'cbnstraints and its solution yields a segmental optimum design as listed in Table I. The segmental optimum design has a weight of 8456.2 kg. As expected. because there is only one displacement constraint which must, therefore, be active, there is only one multi- segment member in this segmental optimum design. All members except member 21 have a single dis- crete-sized segment; member 21 has two segments, one of length 580.41 mm and of size 20 x lo3 mm2 and the other of length 419.59 mm and of size 10 x lo3 mm2.

Rounding up this segmental optimum design t o a fully discrete design is a trivial operation. Clearly, to avoid violating the active displacement con- straint, the smaller size segment of member 21 must be rounded up to have the same size as the other segment. In the LP solution, the constraint slack variable at present in the non-basic set is pivoted into the basis and it is found that the variable corresponding to the smaller size segment or member 21 leaves the basis. This results in a discrete design weighing 8489.2 kg. Further in- spection of the LP solution reveals that no further pivoting of complete members to refine this design is possible. Consequently, this rounded up design represents the minimum weight discrete design found by the methods of this paper. This design is also listed in Table I.

Several comments can be made on this example. The continuous optimum forms a very poor lower bound 10 the discrete optimum. The segmental optimum design, however, is a very close bound to the discrete optimum weight. The continuous optimum design would be very difficult t o round to discrete sizes. In fact 21 o f the 38 members must be rounded up and 17 rounded down to achieve the discrete optimum design from the continuous one,

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DISCRETE OPTIMUM DESIGN I53

but there is no way of knowing which way any particular member should be rounded. The seg- mental optimum design, however, is very simple to convert to a discrete optimum design. The discrete optimum design obtained is only 0.39% heavier than the segmental optimum design. Short of some form of complete enumeration of discrete designs, theglobal optimality ofthe discrete design obtained cannot be guaranteed. However, the discrete design obtained is the same as that obtained in Ref. 6, and its global optimality is strongly suspected.

6.2 3-Bar Truss

This is not the classical indeterminate 3-bar truss but is a very simple determinate one contrived so that some special features of the segmental ap- proach can be demonstrated. Figure 4 shows the truss which is an equilateral triangle of l m side. The member material properties are p = 7.85 x kg/mm3and E = 200 kN/mm2.There is a single load case, 100 k N vertically downwards a t joint B, and a single displacement constraint, that joint B must not displace vertically by more than 10 mm. For purposes of reference only, the continuous minimum weight design has a weight of 2.288 kg. The optimum areas of members AB and BC are 85.355 mm2 and that of AC is 120.71 1 mm2.

Suppose that only three discrete sizes are avail- able for use in a minimum weight design. With only three possible sizes and three members all the 27 possible designs can be enumerated and the globally optimum discrete design found. This may

FIGURE 4 3.bm truss.

be compared with the discrete design found by the methods of this paper. For illustrative purposes two different sets of three sizes have been studied. In case (a) the sizes are 75, 105 and 133 mm2. In case (b) the largest of these sizes is increased to 137 mm2.

Case ( a ) Using the three available sizes, 75, 105 and 133 mm2, the segmental LP, problem (3), has 10 variables and 4 constraints. Its solution yields the segmental optimum design shown in Figure 5a in which the thickness of lines indicates the use of the three discrete sizes. The truss is symmetrically loaded and hasa symmetrical layout. Consequently an alternative "mirror image" segmental optimum design can be found from the LP. In this alterna- tive design the thinnest member and the two- segment member are interchanged. The segmental optimum design weighs 2.327 kg. Rounding u p to a discrete design yields the structure shown in Figure 5b which weighs 2.457 kg. Further refine- ment using the LP table is not possible, so the structure of Figure 5b represents the discrete optimum design found. Enumeration of alterna- tive designs proves that it is in fact the globally optimum design. The interesting feature of this example is that although the structure has a sym- metrical geometry and is symmetrically loaded, minimum weight discrcte dcsign does not have its members symmetrically sized. I f size symmetry is required it must be pre-defined into problem (3),

A

( b )

FIGURE 5 Scgrnenral and discreteoptimum designr.case(a).

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154 A. 8. TEMPLEMAN A N D D. F. YATES

otherwise i t may not be automatically found. A similar cffcct can be found with material properties sclcction in which the optimum design may have non-symmetrical materials. In the case of this ex- :~mplc, symmetry can be defined into problem (3) by having a single set of segment length variables representing both members A B and B C . This results in an LP, problem (3) which has 7 variables and 3 constraints. Its solution yields the segmental optimum design of Figure 6a weighing 2.327 kg. Bccausc one set of segment length variables dc- scribes both A B nnd B C , there are now two multi- scgmcnt members in the segmental optimum dcsign. Rounding up using the LP table yields a discrctc dcsign which, this time, can be refined furthcr by using the LP table as described in Section 3. This dcsign is shown in Figure 6b and is sym- mctric:~l, as expected, weighing 2.473 kg. Enum- cr:~tion shows this to be the globally optimal synimctrical discrctc design. Compared with the structure of Figure 5b in which symmetry waS not prc-defined. i t is seen that the weight penalty nssociatcd with prescribing symmetry is small.

( b )

FIGURE 6 Care(a) with impored symmetry.

Case ( h ) The dcsign problem of Case (a) is now rcpcntcd with n different set of three available discrctc sizes, i.e. 75_ 105 and 137 mm'. The seg- mental optimum design of Figure 7a is found which wcighs 2.331 kg. Rounding up yields the structure of Figure 7b for which further refinement is not possible using the LP t:tble. Figure 7b weighs 2.488 kg and represents the discrete optimum

design. So far everything parallels Case (a) and Figure 5. However, this time,enumeration ofaltern- alive designs shows that the design of Figure 7b is not in Pact the global optimum. The structure shown in Figure 7c is globally optimal for thiscase and weighs 2.473 kg. This shows that the segmental method described in this paper does not always find the globally optimum discrete design. How- ever the discrete design found is only 0.6% heavier than the globally optimum one; a very small differ- ence in practical terms.

(c)

FIGURE 7 Case (bl.

If symmetry of member sizes is required and segment length variables are defined lo control members A B and B C together, results similar to those of Figure 6 are obtained. The segmental optimum design is similar to Figure 6a (but with different segment lengths) and weighs 2.331 kg. Round up and refinement using the LP table yield a structure exactly the same as Figure 6b and weighing 2.473 kg. It can be shown by enumera- tion of alternative designs to be globally optimal.

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DISCRETE O P T I M U M DESIGN 155

C O M M E N T S A N D CONCLUSIONS

Using the segmental approach described in this paper, the discrete minimum weight design of structures composed of elements whose mass is proportional to stiiiness turns out to be a very simple problem. Only standard linear program- ming is nccessary and an ability to carry out simple pivoting operations o n the final table of an L P solution. The entire dcsign process can easily be automated, requiring only minor coding in addi- tion to standard structural analysis and L P routines.

The use of segmental members to develop the method introduced the segmental optimum design which forms a very good and most useful lower bound t o the discrete optimum design: a far better lower bound than that provided by the continuous optimum design. Rounding up and refining the segmental optimum design yields a discrete optimum design which in many cases is globally optimal though this cannot be guaranteed. Even if the discrete design found is not globally optimal its weight will be only slightly heavier.

The examples have demonstrated the valuc of the segmental optimum design and the simplicity with which i t can be convertcd into a discrete optimum design. They also provided a simple demonstration of the fact that discrete minimum weight designs are not necessarily symmetrical even though the structure geometry and loading are symmetrical. Symmetry of mcmber sizes (and hencc grouping of members) can, howevcr, bc imposed if required.

The papcr has shown that discrete member materials selection. membrane plate elements and shear panels can all be easily incorporated into the segmental approach. I I has also discussed possible

iterative schemes for statically indeterminate struc- tures. These are all essentially extensions of the method described here for axial bar trusses and they require computational implementation to prove their value. This constitutes future work. A major requirement for any computational imple- mentation is an efficient L P algorithm tailored to the particular structure of segmental L P problems. This requirement is of major importance because, simple though the method is to use, segmental LPs t m d to be large. This appears to be the only maior . . disadvantageif the method. The advantages must be the simplicity o f the method and the fact that i t does not require or use a continuous optimum design. thc finding of which can sometimes also require very large and complex computer pro- grams.

REFERENCES

I . N. S. Khol. "Algorithms bnxd on optimality crileria lo dcsign minimum weight structurer." L t g . Opr.. 5. (2). 73-90 (1981).

2. A . R a j ~ n m a n and L. A. Schmil. Jr.. "Barir rcductian concepts in large scale slructuml synthesis." Lry. Opr.. 5. (2). 91-1Ml19Rl\ .. .. ~ ~ ~ - ~ , ~

3. C. Fleury and V. Bnibam. "Dimensionnement oplimal cn varvahlcs discr&s." Rrporr A'<,. LTAS SF-104. University or Liege. (1982).

4. C . Flcury and V. Braibant. "Struclural oplimizalion in- volvingdiscrclr design variables." k i m m s c l Colloyrriw,r 164. "Optimimtivn melhodrin rtructuruldrsign."(l982). (Papers lo he published by El-Wirrenschaf~srerlag. Mannheim).

5. D . F. Wales. A . B. Tcmrrleman, and T . B. Bolfw. "Thc complcnily or procedures for delermining minimum urighl Irusses wilh dincrelc memher sizes." 681. J . solid^ Prol-,urm 18. (6) . 487-495 (1982).

6. D . F. Yetes. A . 8 . Tcmplcman. and T . B. Bolfey. " A hcuristc melhod for the dcsign of minimum weigh1 trusses using discrete member sizes." Camp. , M d ~ x . io Appl, Al~d!. ond 0y.. (1983) (10 appitr).

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