Space truss bridge optimization by dynamic programming and linear programming

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1 306 IABSE-JSCE Join Conference on Advances in Bridge Engineering-III, Ags 1-, 015, Dhaka, Bangladesh. ISBN: Amin, Oki, Bhiyan, Ueda (eds.) Space rss bridge opimizaion by dynamic programming and linear programming P.K.M. Monirzzaman Universiy of Briish Colmbia, Vancover, V6T 1Z4, Canada Tanmoy Biswas Bangladesh Universiy of Engineering and Technology, Dhaka 1000, Bangladesh Ahmed F. Farah Sakarya Universiy, Sakarya, Trkey Faysal M. Omar EELO Universiy, Borama, Somalia ABSTRACT: The objecive of he crren sdy is wo-fold. Firs, minimize he cos of a bridge by deermining he opimal nmber and locaion of he piers. Second, minimize he weigh, size and cos of he rss for he individal spans deermined in he prior sep. There are hree main caegories invoke o opimize a rss srcre, namely, sizing, shape and opology opimizaion. The aim of he crren sdy is o opimize he opology of a space rss while mainaining he exernal force is balanced in all considered degree of freedoms and meeing p he Eler bckling and maerial srengh saisfacory. Two cing-edge opimizaion echniqes, namely, dynamic programming and linear programming were employed in his sdy. In one hand, dynamic programming is an approach for making a seqence of decisions in an opimal way for a given recrsive problem. In dynamic programming, a problem is generally divided ino sages ha give he bes ocome based on he previos decision. On he oher hand, linear programming is a mehod for opimizing a scenario ha can be described mahemaically by linear relaionships. Resls showed ha he adoped sraegy can deermine he opimal bridge configraion boh in small and large scale very well. 1 INTRODUCTION Opimizaion is a echniqe sed o selec he bes opion from available alernaives, sbjec o cerain condiions. There are many differen programming mehods sed o opimize a variey of problems. Two ha were sed in his sdy are dynamic programming (DP) and linear programming (LP). DP is an approach for making a seqence of decisions in an opimal way. A a basic level i is aking a small par of a problem, finding an opimal ocome for ha small par, expanding he problem by a small amon, and solving again, nil he expanded problem encompasses he original problem (Sniedovich 010). By hen racing back he opimal decision aken a each sep, he opimal decision for he whole can be fond. In dynamic programming, a problem is generally divided ino sages. Sages can be hogh of as a new small problem o be solved ha bilds on he previos solion. Each sage hen has a nmber of saes, decisions and decision pdaes. On he oher hand, LP is a mehod for opimizing a scenario ha can be described mahemaically by linear relaionships. Many problems can be formlaed and solved in his syle of programming. One example is rss opimizaion, which ilizes LP, or in some cases nework flow programming, o opimize weigh or size or cos of he rss srcre. LP is he mos sccessfl and mos ofen sed echniqe for solving rss problem becase of is sysem of eqaions deal wih member dimensions hose bond o linear domain (Li e al. 009; Rajeev & Krishnamoorhy 199; & Rahami e al. 008). The objecive of his sdy is wo-fold. One is o minimize he cos of a bridge by deermining he opimal locaion of he piers. Second is o minimize he weigh of he rss for individal spans deermined in he previos sep.

2 BRIDGE SPAN OPTIMIZATION.1 Dynamic Programming- Overview Dynamic Programming is an approach for opimizing mlisage decision processes. I is based on Bellman s Principle of Opimaliy: an opimal policy has he propery ha, regardless of he decisions aken o ener a pariclar sae in a pariclar sage, he remaining decisions ms consie an opimal policy for leaving ha sae (Sniedovich 010). A mlisage decision process is a process ha can be separaed ino a nmber of seqenial seps, or sages, which may be compleed in one or more ways. The opions for compleing he sages are called decisions. A policy is a seqence of decisions, one for each sage of he process. The condiion of he process a a given sage is called he sae a ha sage; each decision effecs a ransiion from he crren sae o a sae associaed wih he nex sage. I is o be noed ha a mlisage decision process is finie if here are only a finie nmber of sages in he process and a finie nmber of saes associaed wih each sage (Sniedovich 010). Mlisage decision processes have rerns associaed wih each decision which vary wih sages and saes. The objecive in analyzing sch decision processes is o deermine an opimal policy, one ha resls in he bes oal rern. Ths, DP is a mehod o solve opimizaion problem conaining a specific objecive.. Conex of he Presen Sdy The conex of he DP par for his sdy is o design a bridge in erms of nmber of piers and pier spacing which minimizes he oal cos. The bridge span was considered as a 150 m lengh and he bedrock profile across he ravine a he bridge sie is assmed from a river bed profile fond in GoogleEarh ha was locaed over Narayanganj, Bangladesh. The model incldes boh cos consrains and spaial consrains. The spaial consrains are: he bridge may have no more han 5 piers and no individal span may exceed 500 m. The cos consrains ensre ha a minimm cos configraion wold be chosen. Simplified cos esimaing formlae are available for individal spans of decks and for piers. The cos of a single span is assmed proporional o he sqare of he span and is given by: Cos of deck span= DCons 1 + DCons *(span lengh) (1) where DCons 1 and DCons are given consans as assmed a vale of 0000 and, respecively. The vales of hese consans are considered based on raional daa available in lierare. The cos of a single pier is assmed proporional o is heigh and is given by: Cos of pier= PCons 1 + PCons *(pier heigh) () where PCons 1 and PCons are given consans as assmed a vale of and 11000, respecively. Akin o earlier case, he vales of hese consans are considered based on raional daa available in lierare..3 Model Assmpions The assmpions made dring he problem formlaion are largely presen in he cos fncions. The cos coefficiens deermine how he model chooses he mos economical pier locaions becase of he weighs assigned o pier deph and span lengh. Changing hese wold have been a significan impac on he resl obained. The choice of limi for he span lengh o 500 m is anoher scope of he sdy, ergo a shorer maximm lengh wold end p wih a differen pier configraion resl..4 Define Sages and Sage Nmbering A sage was consised of one deck span and he spporing pier a he lef hand (LH) end of his span. Sage nmbering was considered from lef o righ wih he lef hand abmen inclded in sage 1 and he righ hand (RH) abmen inclded in sage 7 (Fig. 1a)..5 Define Saes Sae for a sage was considered a posiive cener line locaion for a pier and was noed from RH abmen. In his way, an inerval of 50 m was assmed beween discree sae vales. For example, Sae 1 wold corres- 307

3 308 pond o he locaion of he RH abmen. Sae 1 (a 1000 m from RH) will correspond o he locaion of he LH abmen (Fig. 1a)..6 Define Decision Variable A a pariclar sage and sae, i.e. for a given pier and cener line locaion, he decision choice wold be he lengh of span o he nex pier o he righ (Fig. 1a)..7 Sae Transformaion Eqaion Given a sae and a decision (i.e. span o he nex pier o he righ), sae ransformaion eqaion wold be he span lengh resling from he difference beween sae (secion.5) and decision variable (secion.6)..8 Sage Rern Fncion One sage cos will be he sm of he pier cos and deck cos. The minimm cos in a sae wold be he decision of ha pariclar sage..9 Recrsion Eqaion The same formlaion is adoped for all of he sages sared from Sage 7 o Sage 1. Pier a) (b) Figre 1. a) Problem definiion and opimal locaions for piers according o he DP model, b) DP opimized pier locaion.10 DP Resl The dynamic programming model yielded a wo-pier bridge as he opimal resl, wih he piers locaed a x = 50 m and x = 750 m (Figre 1b). Given he profile of he river bed, here are no obviosly shallow locaions o place piers ha minimize pier heigh, so he resl has been dominaed by span lengh and an aemp o have as few piers as possible. This mean he opimal spans measred 300 m, 00 m and 500 m, again from righ o lef. The decision makes sense, becase he deeper secions of he river are associaed wih higher coss becase of pier heigh. The model has herefore chosen o place one pier a he maximm possible span lengh o avoid having piers in he deepes par of he river and hen has chosen a balance beween span cos and pier cos o place he second pier. 3 SPACE TRUSS OPTIMIZATION There are hree main caegories invoke o opimize a rss srcre: i. Shape opimizaion (variables are nodal coordinaes) ii. Sizing opimizaion (variables are cross-secional areas of he members) and iii. Topology opimizaion (variables are he locaion of links in which connec nodes). The aim of his sdy is o do opology opimizaion. In his secion he applicaion of LP for opimizaion of space rss has been discssed. A generalized model which cold be exended o any configraion has been

4 modelled in a programming langage, namely, AMPL (Applied Mahemaical Programming Langage). The model se p was firs validaed for a simple rss configraion. This was laer exended o opimize a large scale rss problem. A srcre is called o be a space rss if i is (Hibbeler 1998& Popov1998): exernally (geomeric) sable, and has 3 j rmembers [ j and r are nmber of joins and sppor reacions, respecively]. 3.1 LP Problem Formlaion Then definiion of he srcral analysis problem o solve he rss srcre by LP is described as follows (Ghasemi e al. 1997; Li e al. 009; Rajeev & Krishnamoorhy 199; Rahami e al. 008; & Rasmssen & Solpe 008). l ension in member { i, in domain se A lengh vecor for member { i, ( ji ( minimize sbjeced o Reformlaed LP formlaion : p f ji minimize ; sbjeced o ; Compresion - Tension posiion vecor for join i exernal force vecor for join i ni vecor for member { i, l i, A { ( l { i, A ( j: { i, A, j: { i, A 0 Bckling load for member { i,, P l ( f ; ( x x y y z z criical, i i 1,,..., n f p p j j( j( K EI l e p p ; i j i 1,,..., n i ; I Momen of ineria and K j 1 e (3) (4) (5) (6) (7) (8) (9) (10) (11) (1) 3. Example Problem Working of he model has been discssed in his secion wih he help of a simple configraion. i. A grid sysem was considered as illsraed in Figre a. There are 5 nodes saring from 0 o 4 along X direcion and 3 nodes saring from 0 o along Y and Z direcion. Each node is eqally spaced a a disance of 1m apar in X and Y direcions and m apar in Z direcion. ii. A load of 50 kn is applied a poin P (3, 1, 0). The configraion is arrived on sch ha he rss has fixed sppored a nodes of (0, 0, 0), pinned connecion a (0, 0, ) and roller sppor (0,, ). 309

5 310 iii. The maerial properies of he members are considered as follows: modls of elasiciy, E=x10 5 N/mm, densiy = kg/m 3 and maximm allowable sress =50 N/mm. iv. Arcs are defined sch ha he nodes are conneced in all possible ways (Fig. b). v. No force balance eqaion was applied a anchored joins. 3.3 Objecive Fncion The objecive of his LP formlaion is o minimize he weigh of he srcre and arrive on an opimal configraion wih he area of cross-secion for he members being sed (Eqaion 10). Since he force depends on he area of cross-secion, he objecive fncion is defined as minimizing he oal absole force. 3.4 Consrains The consrains for he opimizaion are i. Saisfy he eqilibrim eqaion ha is sm of forces along X direcion a every node shold be zero (Eqaion 11). Similarly he sm of he forces along Y and Z direcion shold be zero. ii. An addiional consrain is added sch ha he criical member size does no go beyond 1000 mm de o have maerial s physical limi. iii. Forces of he members wold be governed by he sabiliy of member relaing o Eler bckling (Eqaion 1) and srengh of he maerial p o elasic sage. Z Y X a) (b) Figre. a) Node definiion and b) search domain 3.5 LP Resl Figre 3 shows he srcre ha is obained afer opimizaion algorihm rns. However, he member force is no showing here de o have space limiaion. Laer on, in order o verify he algorihm resl, he opimized srcre was evalaed by secion c mehod and go he accepable agreemen. 3.6 Model Assmpions i. Decision variables ha is he cross secion are of he rss members are coninos. 3.7 Pros of he Model i. The model is simple and easy o se. The ser is reqired o specify he coordinaes for load and sppor condiions. ii. The model is capable o handle complex and large srcral problems wiho losing accracy and/or demanding more compaional power.

6 P Figre 3. Opimal rss configraion wihin nodes bondary (5 x 3 x 3) 3.8 Cons of he Model i. Since he cross-secions are considered as coninos, he model migh no he precise represenaion for a real case scenario. ii. The rss design problem ha is formlaed in he presen sdy presmes ha he rss srcre iself is no affeced by is own weigh. 4 CONCLUSIONS The sdy was invesigaed he opimm nmber of piers and pier spacing which minimizes he oal cos of a bridge consrcion. The dynamic programming model was yielded a wo-pier bridge as he opimal resl. In DP formlaion he cos fncion for piers considers only he heigh. A more realisic cos fncion wold have a erm relaing span lengh o pier diameer and conseqenly wold effec on cos behavior. As a follow-p sep, he applicaion of linear programming for opimizaion of space rss sied for he span lengh deermined by DP has been discssed in his sdy. A generalized LP model which cold be exended o any configraion has been modelled in a programming langage, namely, AMPL. Resls showed ha he adoped sraegy can deermine he opimal bridge configraion boh in small and large scale very efficienly in erms of compaional cos and accracy. REFERENCES Ghasemi, M; Hinon, E &Wood, R Opimizaion of rsses sing geneic algorihms for discree and coninos variables, Jornal of Engineering Compaions, 1997; 16: Hibbeler, R Srcral Analysis, Forh ediion. Prenice Hall pblicaion. Li, LJ; Hang, ZB & Li, F A herisic paricle swarm opimizaion mehod for rss srcres wih discree variables, Jornal of Compers and Srcres, 009;87: Popov, E Engineering Mechanics of Solids. Prenice Hall pblicaion. Rahami, H; Kaveh, A & Gholipor, Y Sizing, geomery and opology opimizaion of rsses via force mehod and geneic algorihm, Jornal of Engineering Srcres, 008; 30: Rajeev, S & Krishnamoorhy, CS Discree opimizaion of srcres sing geneic algorihms, Jornal of Srcral Engineering, 199; 118(5): Rasmssen, MH & Solpe, M Global opimizaion of discree rss opology design problems sing a parallel c-and-branch mehod, Jornal of Compers and Srcres, 008; 86: Topping, B Shape opimizaion of skeleal srcres: A review. Jornal of Srcral Engineering, 109, Sniedovich, M Dynamic Programming: Fondaions and Principles, Taylor & Francis pblicaion. 311