Planar truss bridge optimization by dynamic programming and linear programming

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1 IABSE-JSCE Jon Conference on Advances n Brdge Engneerng-III, Augus 1-, 015, Dhaka, Bangladesh. ISBN: Amn, Oku, Bhuyan, Ueda (eds.) Planar russ brdge opmzaon by dynamc programmng and lnear programmng P.K.M. Monruzzaman Unversy of Brsh Columba, Vancouver, V6T 1Z4, Canada Tanmoy Bswas Bangladesh Unversy of Engneerng and Technology, Dhaka 1000, Bangladesh Ahmed Shrwa Suleyman Demrel Unversy, Ispara, Turkey Tamanna Tabassum Bangladesh Unversy, Dhaka 107, Bangladesh ABSTRACT: The obecve of he curren sudy s wo-fold. Frs, mnmze he cos of a brdge by deermnng he opmal number and locaon of he pers. Second, mnmze he wegh, sze and cos of he russ for he ndvdual spans deermned n he pror sep. There are hree man caegores nvoke o opmze a russ srucure, namely, szng, shape and opology opmzaon. The am of he curren sudy s o opmze he opology of a planar russ whle mananng he exernal force s balanced n all consdered degree of freedoms and meeng up he Euler bucklng and maeral srengh sasfacory. Two cung-edge opmzaon echnques, namely, dynamc programmng and lnear programmng were employed n hs sudy. In one hand, dynamc programmng s an approach for makng a sequence of decsons n an opmal way for a gven recursve problem. In dynamc programmng, a problem s generally dvded no sages ha gve he bes oucome based on he prevous decson. On he oher hand, lnear programmng s a mehod for opmzng a scenaro ha can be descrbed mahemacally by lnear relaonshps. Resuls showed ha he adoped sraegy can deermne he opmal brdge confguraon boh n small and large scale very well. 1 INTRODUCTION Opmzaon s a echnque used o selec he bes opon from avalable alernaves, subec o ceran condons. There are many dfferen programmng mehods used o opmze a varey of problems. Two ha were used n hs sudy are dynamc programmng (DP) and lnear programmng (LP). DP s an approach for makng a sequence of decsons n an opmal way. A a basc level s akng a small par of a problem, fndng an opmal oucome for ha small par, expandng he problem by a small amoun, and solvng agan, unl he expanded problem encompasses he orgnal problem (Snedovch 010). By hen racng back he opmal decson aken a each sep, he opmal decson for he whole can be found. In dynamc programmng, a problem s generally dvded no sages. Sages can be hough of as a new small problem o be solved ha bulds on he prevous soluon. Each sage hen has a number of saes, decsons and decson updaes. On he oher hand, LP s a mehod for opmzng a scenaro ha can be descrbed mahemacally by lnear relaonshps. Many problems can be formulaed and solved n hs syle of programmng. One example s russ opmzaon, whch ulzes LP, or n some cases nework flow programmng, o opmze wegh or sze or cos of he russ srucure. LP s he mos successful and mos ofen used echnque for solvng russ problem because of s sysem of equaons deal wh member dmensons ha bounds o lnear doman (L e al. 009; Raeev & Krshnamoorhy 199; & Raham e al. 008). The obecve of hs sudy s wo-fold. One s o mnmze he cos of a brdge by deermnng he opmal locaon of he pers. Second s o mnmze he wegh of he russ for ndvdual spans deermned n he prevous sep. 31

2 313 BRIDGE SPAN OPTIMIZATION.1 Dynamc Programmng- Overvew Dynamc Programmng s an approach for opmzng mulsage decson processes. I s based on Bellman s Prncple of Opmaly: an opmal polcy has he propery ha, regardless of he decsons aken o ener a parcular sae n a parcular sage, he remanng decsons mus consue an opmal polcy for leavng ha sae (Snedovch 010). A mulsage decson process s a process ha can be separaed no a number of sequenal seps, or sages, whch may be compleed n one or more ways. The opons for compleng he sages are called decsons. A polcy s a sequence of decsons, one for each sage of he process. The condon of he process a a gven sage s called he sae a ha sage; each decson effecs a ranson from he curren sae o a sae assocaed wh he nex sage. I s o be noed ha a mulsage decson process s fne f here are only a fne number of sages n he process and a fne number of saes assocaed wh each sage (Snedovch 010). Mulsage decson processes have reurns assocaed wh each decson whch vary wh sages and saes. The obecve n analyzng such decson processes s o deermne an opmal polcy, one ha resuls n he bes oal reurn. Thus, DP s a mehod o solve opmzaon problem conanng a specfc obecve.. Conex of he Presen Sudy The conex of he DP par for hs sudy s o desgn a brdge n erms of number of pers and per spacng whch mnmzes he oal cos. The brdge span was consdered as a 150 m lengh and he bedrock profle across he ravne a he brdge se s assumed from a rver bed profle found n GoogleEarh ha was locaed over Narayangan, Bangladesh. The model ncludes boh cos consrans and spaal consrans. The spaal consrans are: he brdge may have no more han 5 pers and no ndvdual span may exceed 500 m. The cos consrans ensure ha a mnmum cos confguraon would be chosen. Smplfed cos esmang formulae are avalable for ndvdual spans of decks and for pers. The cos of a sngle span s assumed proporonal o he square of he span and s gven by: Cos of deck span= DCons 1 + DCons *(span lengh) (1) where DCons 1 and DCons are gven consans as assumed a value of 0000 and, respecvely. The cos of a sngle per s assumed proporonal o s hegh and s gven by: Cos of per= PCons 1 + PCons *(per hegh) () where PCons 1 and PCons are gven consans as assumed a value of and 11000, respecvely..3 Model Assumpons The assumpons made durng he problem formulaon are largely presen n he cos funcons. The cos coeffcens deermne how he model chooses he mos economcal per locaons because of he weghs assgned o per deph and span lengh. Changng hese would have been a sgnfcan mpac on he resul obaned. The choce of lm for he span lengh o 500 m s anoher scope of he sudy, ergo a shorer maxmum lengh would end up wh a dfferen per confguraon resul..4 Defne sages and Sage Numberng A sage was conssed of one deck span and he supporng per a he lef hand (LH) end of hs span. Sage numberng was consdered from lef o rgh wh he lef hand abumen ncluded n sage 1 and he rgh hand (RH) abumen ncluded n sage 7 (Fg. 1a)..5 Defne Saes Sae for a sage was consdered a posve cener lne locaon for a per and was noed from RH abumen. In hs way, an nerval of 50 m was assumed beween dscree sae values. For example, Sae 1 would correspond o he locaon of he RH abumen. Sae 1 (a 1000 m from RH) wll correspond o he locaon of he LH abumen (Fg. 1a)..6 Defne Decson Varable A a parcular sage and sae,.e. for a gven per and cener lne locaon, he decson choce would be he lengh of span o he nex per o he rgh (Fg. 1a).

3 .7 Sae Transformaon Equaon Gven a sae and a decson (.e. span o he nex per o he rgh), sae ransformaon equaon would be he span lengh resulng from he dfference beween sae (secon.5) and decson varable (secon.6)..8 Sage Reurn Funcon One sage cos wll be he sum of he per cos and deck cos. The mnmum cos n a sae would be he decson of ha parcular sage..9 Recurson Equaon The same formulaon s adoped for all of he sages sared from Sage 7 o Sage 1. Per a) (b) Fgure 1. a) Problem defnon and opmal locaons for pers accordng o he DP model, b) DP opmzed per locaon.10 DP Resul The dynamc programmng model yelded a wo-per brdge as he opmal resul, wh he pers locaed a x = 50 m and x = 750 m (Fgure 1b). Gven he profle of he rver bed, here are no obvously shallow locaons o place pers ha mnmze per hegh, so he resul has been domnaed by span lengh and an aemp o have as few pers as possble. Ths mean he opmal spans measured 300 m, 00 m and 500 m, agan from rgh o lef. The decson makes sense, because he deeper secons of he rver are assocaed wh hgher coss because of per hegh. The model has herefore chosen o place one per a he maxmum possble span lengh o avod havng pers n he deepes par of he rver and hen has chosen a balance beween span cos and per cos o place he second per. 3 PLANAR TRUSS OPTIMIZATION There are hree man caegores nvoke o opmze a russ srucure:. Shape opmzaon (varables are nodal coordnaes). Szng opmzaon (varables are cross-seconal areas of he members) and. Topology opmzaon (varables are he locaon of lnks n whch connec nodes). The am of hs sudy s o do opology opmzaon. In hs secon he applcaon of LP for opmzaon of planar russ has been dscussed. A generalzed model whch could be exended o any confguraon has been modelled n a programmng language, namely, AMPL (Appled Mahemacal Programmng Language). The model se up was frs valdaed for a smple russ confguraon. Ths was laer exended o opmze a large scale russ problem. A srucure s called o be a planar russ f s (Hbbeler 1998& Popov1998): exernally (geomerc) sable, and has rmembers [ and r are number of ons and suppor reacons, respecvely]. 314

4 LP Problem Formulaon Then defnon of he srucural analyss problem o solve he russ srucure by LP s descrbed as follows (Ghasem e al. 1997; L e al. 009; Raeev & Krshnamoorhy 199; Raham e al. 008; & Rasmussen & Solpe 008). enson n member {, n doman se A lengh vecor for member {, un vecor for member {, u mnmze subeced o Reformulaed LP formulaon : mnmze ; Compreson - Tenson p poson vecor for on f exernal force vecor for on l u u ; subeced o l, A { l {, A : {, A, : {, A u 0 f l u Bucklng load for member {,, P u ; 1,,..., n f ; crcal, x x y y p p p p 1,,..., n EI K l e ; I Momen of nera and K 1 e (3) (4) (5) (6) (7) (8) (9) (10) (11) (1) 3. Example Problem Workng of he model has been dscussed n hs secon wh he help of a smple confguraon.. A grd sysem was consdered as llusraed n Fgure a. There are 7 nodes sarng from 0 o 6 along x drecon and 5 nodes sarng from 0 o 4 along y drecon. Each node s equally spaced a a dsance of 1m apar.. A load of 5 kn s appled a Pon P (3, 0). The confguraon s arrved on such ha he russ s smple suppored a nodes of (0, 0) and (6, 0).. The maeral properes of he members are consdered as follows: modulus of elascy, E=x10 5 N/mm, densy = kg/m 3 and maxmum allowable sress =50 N/mm. v. Arcs are defned such ha he nodes are conneced n all possble ways (Fg. b). v. No force balance equaon was appled a anchored ons. 3.3 Obecve Funcon The obecve of hs LP formulaon s o mnmze he wegh of he srucure and arrve on an opmal confguraon wh he area of cross-secon for he members beng used (Equaon 10). Snce he force depends on he area of cross-secon, he obecve funcon s defned as mnmzng he oal absolue force.

5 3.4 Consrans The consrans for he opmzaon are. Sasfy he equlbrum equaon ha s sum of forces along x drecon a every node should be zero (Equaon 11). Smlarly he sum of he forces along y drecon should be zero.. An addonal consran s added such ha he crcal member sze does no go beyond 1000 mm due o have maeral s physcal lm.. Forces of he members would be governed by he sably of member relang o Euler bucklng (Equaon 1) and srengh of he maeral up o elasc sage. a) Fgure. a) Node defnon and b) search doman (b) 3.5 LP Resul Fgure 3 shows he srucure ha s obaned afer opmzaon algorhm runs. The resuls obaned are abulaed n Table 1. The frs wo columns shows he sarng node of each member and he hrd and fourh column shows he end node of each member. The ffh column gves he force n each member connecng he wo nodes. Fgure 3. Opmal russ confguraon whn nodes boundary (6 x 4) 3.6 Model Assumpons. Decson varables ha s he cross secon are of he russ members are connuous.. The russ was consdered havng smply suppored boundary condons. 316

6 Pros of he Model. The model s smple and easy o use. The user s requred o specfy he coordnaes for load and suppor condons.. The model s capable o handle complex and large srucural problems whou losng accuracy and/or demandng more compuaonal power. 3.8 Cons of he Model. Snce he cross-secons are consdered as connuous, he model mgh no he precse represenaon for a real case scenaro.. The LP problem solved based on smple suppored russ confguraon. Ths canno be used for srucures ha are no smply suppored.. The russ desgn problem ha we have formulaed presumes ha he russ srucure self s no affeced by s own wegh. Table 1. Bar forces n he opmzed russ srucure. Node1 Node Force n Members x y x y (kn) CONCLUSIONS The sudy was nvesgaed he opmum number of pers and per spacng whch mnmzes he oal cos of a brdge consrucon. The dynamc programmng model was yelded a wo-per brdge as he opmal resul. In DP formulaon he cos funcon for pers consders only he hegh. A more realsc cos funcon would have a erm relang span lengh o per dameer and consequenly would effec on cos behavor. As a follow-up sep, he applcaon of lnear programmng for opmzaon of planar russ sued for he span lengh deermned by DP has been dscussed n hs sudy. A generalzed LP model whch could be exended o any confguraon has been modelled n a programmng language, namely, AMPL. Resuls showed ha he adoped sraegy can deermne he opmal brdge confguraon boh n small and large scale very effcenly n erms of compuaonal cos and accuracy. REFERENCES Ghasem, M; Hnon, E &Wood, R Opmzaon of russes usng genec algorhms for dscree and connuous varables, Journal of Engneerng Compuaons, 1997; 16: Hbbeler, R Srucural Analyss, Fourh edon. Prence Hall publcaon. L, LJ; Huang, ZB & Lu, F A heursc parcle swarm opmzaon mehod for russ srucures wh dscree varables, Journal of Compuers and Srucures, 009;87: Popov, E Engneerng Mechancs of Solds. Prence Hall publcaon. Raham, H; Kaveh, A & Gholpour, Y Szng, geomery and opology opmzaon of russes va force mehod and genec algorhm, Journal of Engneerng Srucures, 008; 30: Raeev, S & Krshnamoorhy, CS Dscree opmzaon of srucures usng genec algorhms, Journal of Srucural Engneerng, 199; 118(5):

7 Rasmussen, MH & Solpe, M Global opmzaon of dscree russ opology desgn problems usng a parallel cu-and-branch mehod, Journal of Compuers and Srucures, 008; 86: Toppng, B Shape opmzaon of skeleal srucures: A revew. Journal of Srucural Engneerng, 109, Snedovch, M Dynamc Programmng: Foundaons and Prncples, Taylor & Francs publcaon. 318