S. Malasri, D.A.Halijan and M.L.Keough Department of Civil Engineering Christian Brothers University Memphis, TN Abstract

Transcription

1 S. Malasr, D.A.Haljan and M.L.Keough Department of Cvl Engneerng Chrstan Brothers Unversty Memphs, TN Abstract Ths paper demonstrates an applcaton of the natural selecton process to the desgn of structural members. Renforced concrete beam desgn s used as the example to show how varous chromosomes representng a desgn soluton can be formulated. Ftter chromosomes (or better solutons) have a better chance of beng selected for cross over; ths n turn creates better generatons. Random mutaton s used to enhance the dversty of the populaton. The evoluton progresses through several generatons, and the best soluton s then used n the desgn. The method gves reasonable results, but sometmes a local (as opposed to the global) optmzed soluton s obtaned. Introducton Structural engneers tradtonally desgn structural elements based on a tral-and-error process. An educated guess s made for a tral sze of the member, then the performance s checked. Adjustments are then made for the next tral. An experenced desgner normally starts wth a reasonable tral sze whch a good desgn s obtaned after a few teratons. For a typcal new desgner, ths process can become tedous. In recent years, genetc algorthms (GA) have been used n varous optmzaton problems (Mchalewcz, 1992). Structural desgn s another form of an optmzaton problem, n whch the desgner looks for the optmal soluton (or a near-optmal soluton) under a set of constrants. Ths paper demonstrates that GA can be appled to structural desgn problems by usng the desgn of a renforced concrete beam as an example. desgn. Chromosome Formulaton. In desgnng a rectangular renforced concrete beam for bendng strength, the desgn soluton conssts of the secton dmensons (wdth and effectve depth) and the steel area, as shown n Fg. l(a) "b" s the secton wdth, "d M s the secton effectve depth (the dstance from the extreme compresson fber to the centrod of the tenson steel), and "A,"s the area of renforcng steel. Materals and Methods The evoluton process starts wth a randomly created frst generaton. Ageneraton conssts of a constant populaton sze, n whch an ndvdual n the populaton s represented by a chromosome. Each chromosome, consstng of genes, represents a desgn soluton. A ftness value s then evaluated for each chromosome. Ftter chromosomes are assgned greater probabltes to be selected as parents for the next generaton. Some of these selected chromosomes exchange genes wth others durng the crossover stage. Some genes are also randomly mutated. The process repeats through several generatons. The fttest chromosome s then used as the desgn soluton. followng sectons wll descrbe the detals of ths rocess n the context of renforced concrete beam the Fg. 1. Renforced Concrete Beam: (a) Dmensons, (b) Stresses, and (c) Forces. Thus, a chromosome must consst of three sets of genes representng these three quanttes. In ths partcular mplementaton, each of these sets s represented by 12 bnary dgts, whch gves the maxmum decmal number of Ths maxmum number s then dvded by 100, so each parameter s n the range of 0 to Ths range covers most of the practcal problems. Fg. 2 shows a chromosome wth ts genes and the parameter range. The Frst Generaton. Populaton n the frst generaton s created usng random numbers. To avod startng the sequence of random numbers at the same locaton 111

2 every tme the program s executed, the current mnute from the computer tme clock s used as the seed value for the random number generator. If"r" s the random number generated for a gene, the value of the j-thgene of the -th chromosome (geney) s determned based on the followng rule: Ifr < 0.5 then gene y = 0, otherwse gene j: 1, 0 <= r <= 1 Fg. 2. Chromosome, genes and parameter range. Ftness Evaluaton. Once the populaton n a generaton s defned, the ftness of each chromosome can be evaluated. For the renforced concrete beam problem, the ftness s determned based on ts bendng strenth M d ), the secton proporton (wdth/depth rato), and the steel rato (A,/(bd)). The bendng strength s gven by the followng equaton whch was derved from engneerng mechancs (Nawy, 1990) based on the stress and force dagrams shown n Fg. 1(b) and (c). M d = (0.9) (A/y ) (d-a/2) There are several combnatons of secton dmensons and steel renforcement that provde suffcent bendng strength. Larger sectons requre less steel, whle smaller sectons requre more steel. There are mnmum and maxmum lmts on the steel renforcement set by the Amercan Concrete Insttute (Amercan Concrete Insttute, 1989) to avod the sudden falure of concrete beams. Steel rato s used n the comparson wth these lmts, as shown below: 200/f y <= As /(bd) <= 0.75(0.85) 8^(87000) / (f y (87000+f y )) B! = (f c -4000)/1000 and 0.65 <= B! <= 0.85 The ftness of a chromosome s then determned from the followng rules: 1. The smaller the dfference ofm d and M u, the hgher the ftness. When M d s less than M u, a penalty s appled. 2. The closer the b/d rato s to 0.5, the hgher the ftness. 3. When the steel rato exceeds the maxmum or mnmum lmts, a penalty s appled. Based on these general rules, the ftness s determned by: Ftness = 106 /( jmd -M u j)/( D.5-b/d D/p^pg = Absolute value Pj = Penalty factor forbendng capacty IfM d M u, then p=l, otherwse pj=2 (form d < MJ p 2 = Penalty factor for steel renforcement Ifthe steel rato s wthn the mnmum and maxmum lmts, P2 = l,otherwse P2 = 1O 106 = Scalng factor to make sure that the ftness value s not too small? V» fy = yeld strength of renforcng steel a 7 = (As f y )/(0.85f c b) and f c = concrete strength at 28 days Ths M d s then compared wth the requred moment (M u ) whch s specfed as part of the nput data. IfM d s greater than or equal to M u, then the secton s acceptable; otherwse, the secton s rejected. There are dfferent secton proportons that provde le desred strength. When "b" s too large compared wth "d", the secton s not economcal. On the other land, when "b" s too small compared wth "d", the seeon s too slender and lateral bucklng can occur. For a ractcal desgn, many desgners keep the wdth/depth ato around 0.5. Populaton Selecton. Once the ftness for each chromosome has been evaluated, they are selected accordng to a probablty weghng scheme as an magnary spnner. The ftter chromosomes occupy larger areas on the spnner. In ths mplementaton, the relatve probablty s used to represent these areas on the spnner. Let ps be the probablty of the -th chromosome. Thus, pj can be computed from the followng equaton: = Ftness/Ftness gen, P Ftnessj = Ftness of the -th chromosome, and Ftness = gen Summaton ofall ftnesses of the generaton. * - < - 112

3 Let n be the number of chromosomes n a generaton (populaton sze). The spnner s spun "n" tmes, durng whch the new populaton s selected. The -th chromosome s selected from a spn fthe random number, r, satsfes the followngcondton: ( Pl +p pu) < r <= (p!+p P) Cross Over. After the spnner s spun and a new pool of chromosomes s selected, a number of chromosomes (based on the probablty of crossover specfed by the user) s selected for cross over. A cross over locaton s randomly determned. The two randomly selected chromosomes exchange ther genes from ths locaton to the rest of the chromosome. The two new chromosomes (offsprng) are tfen used to replace the orgnal two parents. Ifthe two parent chromosomes, each wth 15 genes, are: , and , and the crossover locaton s rght after the 6th gene, the two offsprngs, whch replace the two parents become: , and Mutaton. Mutaton s the process n whch some genes change ther genetc codes. In ths mplementaton, mutaton causes a gene to change ts value from 0 to 1, or vce versa. After several generatons, t s possble that a soluton whch s superor to the others but not really acceptable could take control of the entre populaton by Ipreadng ts genetc codes to others. A better soluton ould then become mpossble. Mutaton njects dversty ) the populaton and often helps to move the evoluton ut from a local optmum stuaton. Results and Dscusson IAs an example, a beam s to be desgned for a bendng oment (M u ) of 2,000,000 lb-n (226 kn-m) usng the concrete strength (f c ) of 4,000 ps (27.6 MPa) and the steel yeld strength (f) of 60,000 ps (414 Mpa), as shown n Fg. 3. Rectangular Beam Desgner - (c) 1994 by S. Malasr Concrete Strength - fc (ps) :? 4000 Steel YeldStrength - fy (ps) :? Requred Moment - Mu(n-lb) :? Press any key to contnue Fg. 3. Input screen. Other nput parameters ncludng the populaton sze, the crossover probablty, the mutaton probablty, and the number of generatons are shown n Fg. 4. After 20 generatons, a 11.96" by 30.29" secton s obtaned wth the moment capacty (M d ) of 2,013,032 n-lb (whch s very close to the requred M u ). The steel rato (Rho) of s also wthn the mnmum steel rato of and the maxmum steel rato of The wdth/depth rato s 0.39 whch s not too far from the desred 0.5. Ths, n fact, s a good desgn. Fg. 4. Screen Showng the Evoluton Process Results. and Ten consecutve runs were made usng dfferent values of populaton sze, crossover and mutaton probabltes, and number of generatons. They are summarzed n Table 1. Most of the runs gve good desgns, except for the followng: 1) Run number 3 has the steel rato of whch s lower than the mnmum of allowed by the Amercan Concrete Insttute Code. The desgn engneer would reject ths desgn. 2) Run number 6 has the steel rato of whch s greater than the maxmum of Ths s not too bad, snce theoretcally, the maxmum steel rato n ths case can go up to However, a conservatve desgner would reject ths desgn. 3) Run numbers 7 and 9 are unnecessarly large, snce they gve the bendng capacty of over 3,000,000 n-lb as compared to the requred moment of 2,000,000 n-lb. Ths soluton s safe but uneconomcal. Out of these 10 runs, sx gve acceptable solutons, two gve safe but uneconomcal solutons, one gves a workng soluton wth less safety margn, and one gves an undesrable soluton. Three of the four runs that have problems (run numbers 3, 7, and 9) use the same populaton sze of 50. Ths populaton sze probably does not provde Proceedngs Arkansas Academy ofscence, Vol.48,

4 114 enough dversty. By ncreasng the mutaton probablty from 0.1 to 0.2 as n run number 10, an acceptable soluton s obtaned. Table 2 shows the evoluton process that took place n run number 10. The soluton starts from a very large secton n the frst generaton that gves almost sx tmes the desred bendng capacty to an acceptable soluton after 9 generatons. After the only mnor changes occur untl the 51st generaton. No better soluton was found from the 51st generaton to the 100th generaton. Table 1. Varous Runs for the Same Desgn Problems. Input*: No. Populaton Crossover Mutaton Number of Sze Probablty Probablty Generatons * Other nput data s shown n Fg. 3. Output: No. Secton b/d M d Steel A s bxd n-lb Rato** n " x 18.02" ,066, " x 19.90" ,071, " x 40.94" ,034, " x 13.93" ,059, *** 11.96" x 30.29" ,013, " x 15.35" ,051, " x 26.18" ,479, " x 23.05" ,046, " x 27.65" ,789, " x 17.93" ,099, ** Mnmum Steel Rato = , Maxmum Steel Rato = *** Also shown n Fg. 4. Table 2. Evolvng from an IntalRandom Soluton to an Acceptable Soluton. Generaton Secton Md (n-lb) Steel " x38.75" 11,473, " x 16.35" 3,372, " x 14.29" 674, " x38.77" 1,948, " x 18.09" 2,307, " x 14.29" 1,609, " x 18.09" 2,114, " x 18.09" 2,128, " x 18.09" 2,123, " x 17.93" 2,092, " x 17.93" 2,087, " x 17.93" 2,099, Concluson Ths paper demonstrates that t s possble to automate the desgn process usng the evoluton process as seen n the renforced concrete beam desgn example. The cumulatve selecton (as opposed to pure random selecton) s a very powerful mechansm n evoluton. As shown n the example, acceptable solutons are obtaned quckly (wthn 20 generatons). In ths problem, the goal s to optmze the bendng capacty wth the three constrants: M d s greater or equal to M u, secton proporton s around 0.5, and steel rato should le wthn the acceptable range. For a more complex problem wth more constrants, more generatons may be needed. To a structural engneer, the desgn of a renforced concrete beam s a smple problem and many desgn ads are avalable. But for other more complex problems desgn ads are not avalable and a resonable tral secton s hard to guess, ths evoluton approach becomes very useful. The current work ncludes the desgn of structural steel columns. Ths problem has more complex constrants. For example, steel sectons come n standard szes, a data base of the avalable standard secton must be checked. Ths puts severe restrctons to the corss over and mutaton mechansms. Lterature Cted Amercan Concrete Insttute Buldng code requrements for renforced concrete and

5 115 commentary. Amercan concrete nsttute, Detrot, 353 pp. Mchalewcz, Z Genetc algorthms + Data structures = evoluton programs. Sprnger-Verlag, New York, 259 pp. Nawy, E.G Renforced concrete a fundamental approach, 2nd edton. Prentce Hall,Englewood Clffs,734 pp. Proceedngs Arkansas Academy of Scence, Vol. 48, 1994