DESIGN OF NONLINEAR FRAMED STRUCTURES USING GENETIC OPTIMIZATION

Transcription

1 DESIGN OF NONLINEAR FRAMED STRUCTURES USING GENETIC OPTIMIZATION By S. Pezeshk, 1 Member, ASCE, C. V. Camp, 2 Assocate Member, ASCE, and D. Chen, 3 Member, ASCE ABSTRACT: In ths paper we present a genetc algorthm (GA)-based optmzaton procedure for the desgn of 2D, geometrcal, nonlnear steel-framed structures. The approach presented uses GAs as a tool to acheve dscrete nonlnear optmal or near-optmal desgns. Frames are desgned n accordance wth the requrements of the AISC-LRFD specfcaton. In ths paper, we employ a group selecton mechansm, dscuss an mproved adaptng crossover operator, and provde recommendatons on the penalty functon selecton. We compare the dfferences between optmzed desgns obtaned by lnear and geometrcally nonlnear analyses. Through two examples, we wll llustrate that the optmal desgns are not affected sgnfcantly by the P- effects. However, n some cases we may acheve a better desgn by performng nonlnear analyss nstead of lnear analyss. INTRODUCTION Many mathematcal programmng methods have been developed durng the last three decades (Gallagher and Zenkewcz 1973; Hller and Leberman 1990). However, no sngle method has been found to be entrely effcent and robust for the entre range of engneerng optmzaton problems (Rajeev and Krshnamoorthy 1992). Most desgn applcatons n cvl engneerng nvolve selectng values for a set of desgn varables that best descrbe the behavor and performance of the partcular problem whle satsfyng the requrements and specfcatons mposed by codes of practce. Mathematcally, these desgn varables are dscrete for most practcal desgn problems. However, most mathematcal optmzaton applcatons are suted and developed for contnuous desgn varables. In dscrete optmzaton problems, searchng for the global or a local optmal soluton becomes a dffcult task. A few mathematcal methods have been reported for solvng problems n dscrete optmzaton. These methods nclude complete enumeraton technques, nteger programmng, branch and bound algorthms, and dynamc programmng. All these methods use mathematcal programmng technques. In ths paper we present a genetc algorthm (GA) approach for optmzed desgn of 2D frames usng dscrete structural elements. GAs are effcent and broadly applcable global search procedures based on a stochastc approach that reles on a survval of the fttest strategy. In recent years, GAs have been used n structural optmzaton by many researchers (Goldberg and Samtan 1986; Rajeev and Krshnamoorthy 1992; Adel and Cheng 1994; Koumouss and Georgou 1994; Rajan 1995; Kocer and Arora 1997; Camp et al. 1996; and Rajeev and Krshnamoorthy 1997). All these studes have shown that the GA can be a powerful desgn tool for dscrete optmzaton. An optmal desgn of 2D steel-framed structures usng a GA approach wll be conducted. Desgns are n complance wth the Amercan Insttute of Steel Constructon Load Resstance Factor Desgn (AISC-LRFD) specfcaton (Manual 1994). To 1 Prof., Dept. of Cv. Engrg., Campus Box , Unv. of Memphs, Memphs, TN Assoc. Prof., Dept. of Cv. Engrg., Campus Box , Unv. of Memphs, Memphs, TN. 3 Programmng Engr., Green Mountan Geophyscs, Boulder, CO Note. Assocate Edtor: Scott A. Burns. Dscusson open untl August 1, To extend the closng date one month, a wrtten request must be fled wth the ASCE Manager of Journals. The manuscrpt for ths paper was submtted for revew and possble publcaton on September 1, Ths paper s part of the Journal of Structural Engneerng, Vol. 126, No. 3, March, ASCE, ISSN /00/ /$8.00 $.50 per page. Paper No learn how the P- effects nfluence optmal desgns, we wll conduct both lnear and geometrcally nonlnear analyss. Through a seres of desgn examples, we wll conclude that the optmal desgn s not sgnfcantly affected by P- effects. But, n some cases we may get a better desgn by consderng P- effects. In addton, we wll demonstrate that our proposed approach can be an effectve optmzaton technque. GENETIC ALGORITHMS A genetc algorthm s a search strategy that models the mechansms of genetc evoluton (Holland 1975; Goldberg 1989). The GA search s based on the prncples of survval of the fttest and adaptaton. In general, a GA s a flexble and farly effcent strategy to search such complex spaces as the soluton space for the desgn of frames. GAs are search algorthms that are based on the concepts of natural selecton and genetc codng. As algorthms, GAs are dfferent from tradtonal optmzaton methods n the followng aspects: They work wth a coded set of the varables and not wth the varables themselves; they search from a populaton of desgn varables rather than mprovng a sngle desgn varable; they use objectve functon nformaton wthout any gradent nformaton; and ther transton scheme s probablstc, whereas tradtonal methods use gradent nformaton (Goldberg 1989). Codng and Decodng An essental characterstc of a GA s the codng of the varables that descrbe the problem. The most common codng method s to transform the varables to a bnary strng of specfc length. In ths paper, we use a smple bnary codng method. Group Selecton The GA used n ths study proceeds by frst randomly generatng a soluton populaton of a specfc sze. From ths populaton, the next generaton of desgns s evolved by performng three dstnct operatons reproducton, crossover, and mutaton. There are several dfferent forms for these operators. There are a number of reproducton schemes commonly used n GAs; these nclude proportonate reproducton, rankng selecton, tournament selecton, steady-state selecton, and greedy over selecton. Comparson of the varous schemes have been performed by Goldberg and Deb (1991). In ths study we used a group selecton scheme for reproducton. In ths scheme ndvduals n the soluton populaton are sorted, accordng to ther ftness values, from best to worst. Then the sorted populaton s dvded nto groups. Each group s as- 382 / JOURNAL OF STRUCTURAL ENGINEERING / MARCH 2000

2 OPTIMIZATION FORMULATION The objectve of our problem s to develop a desgn that mnmzes the total structural weght W whle satsfyng the AISC-LRFD specfcaton. The objectve functon can be expressed as N e mnmze W = AL =1 FIG. 1. Reproducton Scheme Usng Group Selecton sgned a selecton probablty. The assgned selecton probablty s dstrbuted equally among ndvduals n the group. The group selecton scheme s llustrated n Fg. 1. Adaptve Crossover Crossover s one of the most mportant operators n a GA. There are many crossover schemes, such as mult-pont crossover and unform crossover (see Camp et al for detaled dscusson). Although there are some expermental observatons suggestng that two-pont and unform crossover exhbt better behavor than other operators, there s no theoretcal proof as to whch one s best. In ths study we used a modfed form of the adaptve crossover scheme developed by Spears (1994). In an adaptve crossover scheme, two bts are appended to the end of every ndvdual n the populaton. The value coded n these bts represents the crossover operator used to generate the ndvdual. For example, 00 refers to one-pont crossover, 01 to two-pont crossover, 10 to three-pont crossover, and 11 to unform crossover. In ths way, the last two columns of the codng (.e., the last two bts of every ndvdual) are used to record the crossover operator scheme. Because the approach s adaptve, crossover and mutaton are allowed to manpulate ths extra two columns of bts (Chen 1997). The last two bts of each ndvdual are used to select whch crossover operator s used on the matng par. For example, let us say that two ndvduals are chosen for crossover. The last two bts of each ndvdual are examned. If they are the same, say 00 s, one-pont crossover s performed. If they are dfferent, say one s 00 and the other one s 11, the crossover operator s randomly chosen from one-pont crossover and unform crossover. Snce the extra two bts are used to determne whch crossover to apply, ths mechansm should gve greater award to the crossover operator that produces superor offsprng. Note that ths mechansm allows the GA to adjust the relatve mxture of the four crossover operators. For example, the adaptatonal GA can use multpont, unform crossover prmarly, or any combnaton among them. Mutaton Although mutaton s a secondary GA operator, t can play an mportant role n the search. Mutaton can be an exploratve operator by movng the search nto regons of the soluton space t may have never reached. Mutaton s a character-based strng operaton. The procedure for mutaton can be summarzed as follows: For each character of a soluton strng, a randomly generated number s compared aganst a mutaton probablty. If the random number s less than the mutaton probablty, the value of the character at that poston s changed; otherwse, move to the next character poston and repeat the procedure. Typcal values for the mutaton probablty are around 0.1%. where N e = total number of elements, and L, A, and = length, area, and unt weght of materal of th element, respectvely. The AISC-LRFD specfcaton ncludes strength and stablty requrements. These requrements, combned wth dsplacement lmts, consttute the constrants for the optmzaton problem. Here, the dsplacement constrants are the allowable nterstory drft. These constrants are mplct constrants because structural responses lke stresses, strans, and dsplacements are functons of desgn varables. Structural responses are calculated by the fnte-element method. In addton, practcal and constructonal constrants are ncluded. Snce GAs are used to solve unconstraned optmzaton problems, we must transform the constraned problem to an unconstraned problem by usng penalty functons. The selecton of the penalty functon s crtcal. Many researchers beleve that penalty functons should be harsh, so that the GA wll avod the forbdden spaces (Rchardson et al. 1989). However, the foundaton of GA research states that a GA optmzes by combnng partal nformaton from the populaton. Therefore, nfeasble solutons may provde some useful nformaton. If the penalty s too large, the desgn process may converge too quckly, not allowng the GA to explot varous combnatons of strngs. If the penalty s too small, the convergence process may be too slow and the computatonal costs could be hgh. There are several papers devoted to the selecton of penalty functons (Rchardson et al. 1989; Homafar et al. 1994; Camp et al. 1996). In ths study, we use the followng penalty functon F = W[1 C] where F = ftness functon and C = constrant volaton functon. We use the followng expresson for constrant volaton functon: Ne Nn Nc d I =1 =1 =1 C = C C C where C d and I, C, C = constrant volatons for stress, dsplacement, and nteracton formulas of the AISC-LRFD specfcaton. N e, N n, and N c = number of elements, number nodes, and number of beam columns, respectvely. In general, we express the constrant volaton C as 2 0 f 0 C = f0< 1.0 f > 1.0 For stress constrants, we defne as = a 1 (5) a where = stress n the th element and = allowable stress for the th element. For dsplacement constrants we have where d = dsplacement at th node and d = a 1 (6) d = allowable dsa d JOURNAL OF STRUCTURAL ENGINEERING / MARCH 2000 / 383

3 placement for the th node. The penalty terms have been normalzed by ther allowable values. The nteracton formula of AISC-LRFD specfcaton for P u / (P n ) 0.2 s Then becomes Pu 8 Mux Muy < 1.0 (7) P 9 M M n b nx b ny Pu 8 Mux Muy = 1 (8) P 9 M M n b nx b ny where P u = requred axal strength (tenson or compresson); P n = nomnal axal strength (tenson or compresson); = resstance factor (tenson 0.90, compresson 0.85); M ux and M uy = requred flexural strengths n x and y drecton, respectvely; M nx and M ny = nomnal flexural strengths n x and y drecton, respectvely (for 2D structures, M uy s equal to zero); and b = flexural resstance reducton factor ( b = 0.90). Usng the form of penalty functon as explaned above, the penalty quantty wll always be some percentage of the weght of the structure. The larger the volaton s, the heaver the penalty wll be. GA-BASED DESIGN The proposed desgn procedure nvolves a GA, a lnear and geometrcal fnte-element analyses for ftness evaluaton, an enforcement of code provsons, and a calculaton of the penalty functon. Step-by-step operatons of the GA procedure used n ths study can be summarzed as the followng (also see Fg. 2). 1. Select the GA control parameters sutable to the gven problem. These parameters nclude populaton sze, strng length per ndvdual desgn varable, crossover, and mutaton rate. The selecton of these parameters may requre some expermentaton. 2. The ntal populaton s randomly generated. 3. Decode the bnary desgn varables nto decmal values and generate an nput fle for fnte-element analyss (FEA). 4. Perform FEA usng a sutable software package, check the gven constrants, and calculate the value of the penalty functon. 5. Check the convergence crtera. Termnate the desgn process f t s satsfed; otherwse contnue. 6. Calculate the penalzed ftness for every ndvdual of the populaton and generate the next generaton through reproducton, crossover, and mutaton. 7. Repeat steps 3 through 6. DESIGN EXAMPLES To represent the strength and lmtatons of the GA optmzaton, we present three desgn examples. The frst two examples are 2-bay, 3-story frames. The thrd example s a 1- bay, 10-story frame. The cross sectons of all members are assumed to be W shapes. We used 256 avalable cross sectons for each member accordng to the AISC-LRFD. We used a bnary codng length of 8, representng these 256 cross sectons. In addton, we used the same GA control parameters for all examples, whch are populaton of 60, crossover probablty of 0.85, mutaton probablty of 0.01, and rato of random generatng porton of populaton to whole populaton of As mentoned earler, we used a group selecton scheme n our calculatons. The populaton s dvded nto two groups. The frst group occupes 30% of the populaton and a selecton probablty of The second group occupes the rest of the 70% of the populaton and wll have a select probablty of The desgn process may be stopped automatcally f the best feasble desgn s not mproved wthn fve successve generatons. However, n examples presented n ths paper we dd not follow ths convergence crteron. Young s modulus of E = 29,000 ks and a yeld stress of f y = 36 ks are used. For each example, we performed three dfferent analyss and desgn cycles. Case 1. Lnear analyss gnorng the P- effects of the AISC-LRFD specfcaton. Case 2. Lnear analyss consderng the P- effects n accordance wth the AISC-LRFD specfcaton. Case 3. Geometrcally nonlnear analyss n leu of the AISC-LRFD specfcaton s P- effects magnfcaton factors. FIG. 2. Flowchart of GA-Based Optmal Desgn Example 1: Two-Bay, Three-Story Frame Fg. 3 shows the topology of a 2-bay, 3-story frame under a sngle-load case. Ths frame was desgned by Hall et al. (1989) n accordance wth the AISC-LRFD specfcaton. We wll also desgn ths frame usng the proposed GA. The load values ndcated n Fg. 3 are assumed to defne a factored load level that s approprate for drect applcaton of the strength/stablty provsons of the AISC-LRFD specfcaton. Dsplacement constrants are not mposed for the desgn. The member effectve length factors K x are calculated as for a sway-permtted frame,.e., K x 1.0. The coeffcent K x was calculated usng transcendental equatons. Fabrcaton condtons are mposed to group together the relatve szes of the member cross sectons. Specfcally, the sx beam members are requred to have the same W-secton for the desgn. Smlarly, the nne column members are requred to have the same W- secton. Moreover, each column secton s specfed to have a maxmum depth of 10 n. The out-of-plane effectve length factor K y for each column member s specfed to be one, and for each beam member s specfed to be (.e., floor 384 / JOURNAL OF STRUCTURAL ENGINEERING / MARCH 2000

4 FIG. 3. Frame Geometry and Appled Loadng for 2-Bay, 3-Story strngers are assumed at 1/6 ponts of each beam span). The length of the unbraced compresson flange for each column member s drectly calculated durng the desgn process, whle the length of the unbraced compresson flange for each beam member s specfed to be 1/6 of the span length. Optmal Desgn Results To llustrate the effcency of the desgn method, we ran the GA-based desgn procedure 30 tmes and recorded the varaton of desgn varables of each desgn and typcal desgn convergence hstory. In addton, we performed push-over analyss of the best desgn. The followng s the summary of fndngs. Case 1. Lnear Analyss Ignorng P- Effects of AISC-LRFD Specfcaton In ths case, we optmze the 2-bay, 2-story frame wth the GA procedure usng a lnear analyss and gnorng the P- effects of the AISC-LRFD specfcaton. Usng a populaton of 60 wth a crossover probablty of 0.85 and mutaton probablty of 0.01, we determned the best desgn. To see f we arrve at the same best desgn we ran the GA procedure 30 tmes. The results of each of these 30 runs are gven n Table 1. From Table 1, t s apparent that a desgn wth W1060 for columns and W2462 for beams would be the best desgn. The correspondng weght for ths desgn s 18,792 lb. Hall et al. (1989) reported the same desgn. It s also of nterest to menton that an exhaustve search of canddates resulted n the same desgn. The varaton of desgn varables of these 30 runs s shown n Fg. 4(a). Each member of the ntal populaton can take any weght between 10 and 810 lb/ft, as shown n the vertcal axs of Fg. 4(a). Intal populaton can be staggered along the vertcal axs of Fg. 4(a), as llustrated by the shaded areas; however, as t can be seen from Fg. 4(a), the desgn varables for all of the fnal desgns are narrowed down to a relatvely small range. It took 15 mn on average on a sngle-processor SunSPARC workstaton 5 to obtan an optmzed desgn usng the GA approach. The computatonal tme can be reduced to 10 mn f runnng on two workstatons usng the parallel vrtual machne (PVM) technque (Gest et al. 1991). Convergence hstores for three desgns are llustrated n Fg. 4(b). The optmal desgn was obtaned wthn 15 generatons. Ths can be attrbuted to the smplcty of the problem and the effcency of the GA. Case 2. Lnear Analyss Followng P- Consderaton of AISC-LRFD Specfcaton In ths case we used a lnear analyss and followed the P- effects consderatons of the AISC-LRFD specfcaton. The results of 30 runs are summarzed n Table 1. From Table 1, we can observe that the optmal desgn conssts of W1060 Run Case 1 TABLE 1. Optmzed Desgns of 30 Runs for Cases 1 3 Weght (5) Case 2 (6) Weght (7) (8) Case 3 (9) Weght (10) 1 W1060 W ,088 W1060 W ,792 W1068 W ,808 2 W1060 W ,792 W1060 W ,792 W1068 W ,808 3 W1077 W ,618 W1060 W ,088 W1068 W ,512 4 W1060 W ,816 W867 W ,718 W1068 W ,808 5 W1060 W ,088 W1060 W ,088 W1077 W ,322 6 W10112 W ,968 W1060 W ,088 W1077 W ,322 7 W1060 W ,792 W1060 W ,792 W1068 W ,512 8 W1060 W ,816 W1060 W ,792 W10100 W ,392 9 W867 W ,718 W1060 W ,792 W1088 W , W1060 W ,088 W1060 W ,088 W1088 W , W867 W ,422 W1060 W ,792 W1088 W , W1060 W ,792 W1060 W ,088 W1088 W , W1060 W ,088 W1060 W ,088 W1068 W , W1060 W ,792 W1060 W ,088 W1068 W , W1060 W ,816 W1060 W ,088 W1077 W , W1060 W ,088 W1060 W ,032 W1088 W , W1060 W ,088 W1060 W ,792 W1068 W , W1060 W ,088 W867 W ,718 W1077 W , W1060 W ,816 W1060 W ,792 W1068 W , W1060 W ,032 W1060 W ,816 W1088 W , W1060 W ,792 W1060 W ,792 W1068 W , W1060 W ,088 W1060 W ,088 W1077 W , W1060 W ,088 W1060 W ,792 W1077 W , W1060 W ,792 W867 W ,718 W1077 W , W1060 W ,088 W1060 W ,088 W1068 W , W1077 W ,898 W1060 W ,792 W1068 W , W1060 W ,088 W1088 W ,248 W1068 W , W1068 W ,808 W1060 W ,088 W1068 W , W10100 W ,864 W1060 W ,792 W1068 W , W1060 W ,088 W1060 W ,792 W1077 W ,322 JOURNAL OF STRUCTURAL ENGINEERING / MARCH 2000 / 385

5 was taken to be an ellpsod n stress-resultant space, wth prncple axes proportonal to the fully plastc yeld values of each of the stress resultants n absence of the others. The pushover curves are plotted n Fg. 6. The load-carryng capacty of the optmal desgn obtaned usng geometrcally nonlnear analyss s 20% hgher than that of lnear analyss wth about the same postlmt slope as the lnear analyss. Fg. 6 tells us that by provdng an addtonal 4% weght by choosng W1068 for column nstead of W1060, we can acheve a 20% ncrease n strength, and wth the same postlmt loadcarryng capacty. Obvously, the desgn followng nonlnear analyss s a much better but slghtly more expensve desgn. Example 2 To hghlght stablty as an mportant desgn crteron Example 1 s redesgned wth addtonal vertcal loadngs, as llustrated n Fg. 7. Optmal desgns for three cases are gven FIG. 4. Desgn Varables of Typcal Convergence Hstores of Lnear Optmal Desgns wthout P- Effects for columns and W2462 for beams. The correspondng weght of ths desgn s 18,792 lb. We acheved the same desgn n the prevous case where the P- effects were gnored. The optmal desgn was obtaned wthn 27 generatons. Case 3. Geometrcally Nonlnear Analyss n Leu of AISC- LRFD Specfcaton s P- Effects Magnfcaton Factors In ths case, we performed a geometrcally nonlnear analyss n leu of the AISC-LRFD specfcaton. Results of 30 runs are summarzed n Table 1. From Table 1, we can observe that usng W1068 for columns and W2462 for beams results n the best desgn. The correspondng weght of ths desgn s 19,512 lb. Ths s a lttle heaver than the prevous optmal desgn (4%). One typcal convergence hstory s llustrated n Fg. 5. We note that the optmzed desgn s obtaned wthn 18 generatons. However, the desgn process keeps fluctuatng compared wth the prevous lnear desgn. FIG. 5. Varaton of Desgn Varables and Typcal Convergence Hstory of Geometrcally Nonlnear Desgn TABLE 2. Summary of Optmal Desgn Results for Example 1 Analyss procedure Weght Case 1 W2462 W ,792 Case 2 W2462 W ,792 Case 3 W2462 W ,512 Summary and Comparson of Optmal Desgns The optmal desgns for dfferent methods are summarzed n Table 2. We note that the geometrcally nonlnear analyss case resulted n a 4% heaver structure than the other two cases. Surprsngly, the P- effects of the AISC-LRFD specfcaton dd not result n dfferent optmzed desgns for Cases 1 and 2. To thoroughly understand the behavoral dfference between varous cases, we performed push-over analyses of desgns. The push-over analyses of the frames studed here are based upon the geometrcally nonlnear rod model used by Pezeshk (1992 and 1998). The model was extended to account for rate-ndependent elastoplastcty (n stress resultant space) and mplemented n the general purpose fnte-element program FEAP (Zenkewcz 1982, Chapter 24). The yeld surface 386 / JOURNAL OF STRUCTURAL ENGINEERING / MARCH 2000 FIG. 6. Results of Pushover Analyses

6 FIG. 7. Geometry and Appled Loadng for Example 2 TABLE 3. Summary of Optmal Desgn Results for Example 2 Analyss procedure Weght Case 1 W2162 W ,392 Case 2 W2462 W ,392 Case 3 W2462 W ,392 We consdered a ten-story frame as llustrated n Fg. 8 for the thrd example. We optmzed ths frame under a sngleload case, as shown n Fg. 8. Ths frame s desgned followng the AISC-LRFD specfcaton and uses a dsplacement constrant (story drft < story heght/300). The load values presented n Fg. 8 are assumed to defne the servce-load level. The effectve length factors of the members are calculated as K x 1.0 for a sway-permtted frame. Fabrcaton condtons are mposed to group together the relatve szes of the member cross sectons. Specfcally, commencng at the frst story and excludng the roof beam, the beam members are requred to reman the same sze for three consecutve stores. Smlarly, commencng at the foundaton level, the columns are requred to reman the same over two stores. The total number of desgn varables s nne. The out-of-plane effectve length factor for each column member s specfed to be K y = 1.0, whle that for each beam member s specfed to be K y = 0.2 (.e., floor strngers at 1/5 ponts of the span). The length of the unbraced compresson flange for each column member s calculated durng the desgn process, whle that for each beam member s specfed to be 1/5 of the span length. Optmal Desgn Results Smlar to Example 1, we conducted 30 desgns for each of the three desgn procedures consdered. However, we wll report only the best one of the 30 optmal desgns and the varatons of desgn varables for each desgn procedure. The best optmal desgns are lsted n Table 4. Summary of Best Optmal Desgn Results for Ex- TABLE 4. ample 2 Desgn procedure Weght Average CPU tme (mn) (5) Lnear wthout 1-3S (W33118) 1-2S (W14233) 65, P- effect 4-6S (W3090) 3-4S (W14176) 7-9S (W2784) 5-6S (W14159) Roof (W2455) 7-8S (W1499) 9-10S (W1279) Lnear wth 1-3S (W36150) 1-2S (W14233) 69, P- effect 4-6S (W3090) 3-4S (W14211) 7-9S (W2784) 5-6S (W12152) Roof (W1453) 7-8S (W12106) 9-10S (W1068) Geometrcally 1-3S (W36150) 1-2S (W14233) 70, nonlnear 4-6S (W33130) 3-4S (W14176) analyss 7-9S (W2794) 5-6S (W14132) Roof (W1650) 7-8S (W1499) 9-10S (W1265) FIG. 8. Topology of 1-Bay, 10-Story Frame n Table 3. It s nterestng to note that all three cases resulted n smlar desgns havng the same secton for all columns. Although the beams were dfferent, the total weght of all desgns are the same. The push-over analyss for all three cases also resulted n almost the same performance and are not presented here. It appears that the effect of geometrc nonlnear s not mportant for ths example problem, because of the constrants mposed by the AISC-LRFD specfcaton. Example 3: One-Bay, Ten-Story Frame FIG. 9. Capacty Curves of Dfferent Desgns of Example 3 JOURNAL OF STRUCTURAL ENGINEERING / MARCH 2000 / 387

7 nvestgate the result of the P- on desgn. The requrements of the AISC-LRFD specfcaton were followed. We used a group selecton mechansm and presented an adaptng crossover operator. Through several examples, we found that group selecton and adaptng crossover works well for the problems we are consderng. We also presented recommendatons on the penalty functon selecton and mplementaton. Through two desgn examples, we conclude that the optmzed desgns are not affected sgnfcantly by the P- effects. However, n some cases we may acheve a better desgn by performng nonlnear analyss than lnear analyss. Ths concluson needs further research and dscusson. FIG. 10. Dstrbuton of Fnal Desgn Varables and Typcal Convergence Hstores of Example 3 From Table 4 and Fg. 9 we observe that the optmal desgns obtaned from three dfferent desgn procedures are not sgnfcantly dfferent. Because the frame s slender t s the dsplacement constrant, not the strength requrement, that controls the desgn. The desgn obtaned by geometrcally nonlnear analyss s the heavest. From Fgs. 10(a c), we observe that the same desgn varable for dfferent runs vares n a small range of avalable space. In addton, Fgs. 10(a c) show the average,, and average plus and mnus one standard devaton,, of all 30 runs. Ths demonstrates the present optmal desgn method s ablty to converge. From Fgs. 10(d f) we can conclude that the optmal desgn can be acheved wthn 40 generatons. CONCLUSIONS The wrters have presented a desgn procedure usng a genetc algorthm for the desgn of 2D framed structures. Both geometrcally lnear and nonlnear analyss were performed to APPENDIX. REFERENCES Adel, H., and Cheng, N. T. (1994). Concurrent genetc algorthms for optmzaton of large structures. J. Aero. Engrg., 7, Camp, C. V., Pezeshk, S., and Cao, G. (1996). Desgn of 3-D structures usng a genetc algorthm. Proc, 1st U.S.-Japan Sem. on Struct. Optmzaton, Aprl, Chcago. Chen, D. (1997). Least weght desgn of 2-D and 3-D geometrcally nonlnear frame structures usng a genetc algorthm, PhD dssertaton, Unversty of Memphs, Memphs, Tenn. Gallagher, R. H., and Zenkewcz, O. C. (1973). Optmum structural desgn: Theory and Applcatons, Wley, New York. Gest, A., Begueln, A., Dongarra, J., Jang, W., Mancheck, R., and Sunderam, V. (1991). PVM: Parallel vrtual machne: A user s gude and tutoral for networked parallel computng, The MIT Press, Cambrdge, Mass. Goldberg, D. E., and Samtan, M. P. (1986). Engneerng optmzaton va genetc algorthms. Proc., 9th Conf. on Electronc Comput., ASCE, New York, Hall, S. K., Cameron, G. E., and Grerson, D. E. (1989). Least-weght desgn of steel frameworks accountng for P- effects. J. Struct. Engrg., ASCE, 115(6), Hller, F. S., and Leberman, G. J. (1990). Introducton to mathematcal programmng, McGraw-Hll, New York. Homafar, A., Q, C. X., and La, S. H. (1994). Constraned optmzaton va genetc algorthms. Smulaton, Aprl, Manual of steel constructon: Load and resstance factor desgn. (1994). 2nd Ed., Amercan Insttute of Steel Constructon. Pezeshk, S. (1992). Optmal desgn of structures wth knematc nonlnear behavor. J. Engrg. Mech., ASCE, 118, Pezeshk, S. (1998). Desgn of framed structures: An ntegrated nonlnear analyss and optmal mnmum weght desgn. Int. J. Numer. Methods n Engrg., 41, Rajeev, S., and Krshnamoorthy, C. S. (1992). Dscrete optmzaton of structures usng genetc algorthms. J. Struct. Engrg., ASCE, 118(5), Rajan, S. D. (1995). Szng, shape, and topology desgn optmzaton of trusses usng genetc algorthms. J. Struct. Engrg., ASCE, 121(10), Rchardson, J. T., Palmer, M. R., Lepns, G., and Hllard, M. (1989). Some gudelnes for genetc algorthms wth penalty functons. Proc., 3rd Int. Conf. on Genetc Algorthms, Morgan Kaufmann, San Mateo, Calf., Spears, W. M. (1994). Adaptve crossover n genetc algorthms. Artfcal Intellgence Center Internal Rep. #AIC , Naval Research Laboratory, Washngton, D.C. Zenkewcz, O. C. (1982). The fnte element method, 3rd Ed., McGraw- Hll, London. 388 / JOURNAL OF STRUCTURAL ENGINEERING / MARCH 2000