Optimization of stay cables in cable-stayed bridges using finite element,genetic algorithm,and B-spline combinedtechniqueM.M.Hassan ⇑AMEC Americas Limited,Calgary,Alberta,CanadaDepartment of Civil Engineering,Faculty of Engineering,Port Said University,Port Said,Egypta r t i c l e i n f o Article history:Received 22August 2011Revised 5November 2012Accepted 29November 2012Available online 21January 2013Keywords:Cable-stayed bridge Stay cables Finite element Genetic algorithm B-spline curvesStructural optimizationa b s t r a c tTraditionally,a trial-and-error procedure is carried out to design cross-sectional areas of stay cables in cable-stayed bridges.This design process is monotonous,expensive,time-consuming,and incapable of finding the optimum design solution.The aim of this study is to develop a robust design optimization technique in order to achieve the minimum cross-sectional areas of stay cables.The developed optimiza-tion technique integrates finite element method,B-spline curves,and genetic algorithm.The capability and efficiency of the proposed optimization technique is tested and assessed by applying it to a practical sized cable-stayed bridge.Ó2012Elsevier Ltd.All rights reserved.1.IntroductionBecause of their aesthetic appeal,ease of erection,efficient uti-lization of structural materials,and other notable advantages,cable-stayed bridges have found wide applications all over the world in the last few decades [1].Bridges of this type have recently entered a new era with main spans exceeding a value of 1000m.In modern long-span cable-stayed bridges,such as the Sutong Bridge in China (2088m),a large number of stay cables would be required in order to achieve reasonable distribution of bending moments along the bridge deck.The unit cost of stay cables is relatively high compared to other construction materials;therefore,there is a need for the development of an optimization technique to deter-mine the minimum cost of stay cables in cable-stayed bridges.In the current practice,the design process of stay cables is per-formed in two stages.The first stage involves the determination of initial post-tensioning cable forces,which are evaluated corre-sponding to zero vertical deflections of the deck and zero horizontal deflections of the pylons’tops under only self-weight of the bridge.These forces are required to determine the initial configuration of the bridge.In the second stage,the cross-sectional areas of stay cables are determined under the combined effect of self-weight,ini-tial post-tensioning cable forces,and live load cases.To date,this de-sign stage is based on a trial-and-error procedure,which depends on the designer’s experience and skills [2,3].A set of cross-sectional areas of stay cables is first assumed.Structural analysis for the bridge is then carried out in order to obtain the bridge deflections and stres-ses.If the deflections and stresses satisfy the requirements imposed by design codes,the assumed cross-sectional areas of stay cables are adopted.Otherwise,the cross-sectional areas are modified and the structural process is repeated until all the design criteria are met.The previous iterative design procedure is expensive,tedious,and time consuming.Moreover,it does not guarantee that the final solu-tion will be the best of all the possible design solutions that satisfy the requirements of design codesThere have been many studies concerning the determination of the optimum post-tensioning cable forces under self-weight [4–15];however,there have been only a few attempts to deter-mine the optimum cross-sectional areas of stay cables under self-weight,initial post-tensioning cable forces,and live load cases.One of the first attempts was conducted by [16].In their study,a convex scalar function was used to minimize the cost of a box-girder deck cable-stayed bridge.The proposed function combines dimensions of the cross-sections of the bridge and post-tensioning cable forces.This method is very sensitive to the constraints,which should be imposed very cautiously to obtain a practical output [8].In the research done by [3],the optimization module implemented in MATLB (fmincon ),together,with the commercial finite element software,ABAQUS,are employed to evaluate the minimum cost of stay cables for cable stayed bridges.It should be noted that the two previous studies are based on direct search optimization techniques.The drawback of these0141-0296/$-see front matter Ó2012Elsevier Ltd.All rights reserved./10.1016/j.engstruct.2012.11.036⇑Corresponding author.Address:Department of Civil Engineering,Faculty ofEngineering,Port Said University,Port Said,Egypt.Tel.:+15872271140.E-mail address:mhassa7@alumni.uwo.cadirect techniques is that they begin the search procedure with a guess solution,which is often chosen randomly in the search space. If the guess solution is not chosen close enough to the global min-imum solution,the optimization technique will be trapped in local minima.As a result,thefinal solutions of these previous studies may not be the global minimum[3].On the other hand,the cross-sectional areas of stay cables are considered as discrete de-sign variables in both studies.With the increase in the number of stay cables,the number of design variables becomes quite large leading to potential numerical problems in the optimization tech-nique.In addition,the increase in the number of stay cables makes thefinal distribution of the cross-sectional areas of stay cables non-smooth.Hence,the resulting values from these methods may be impractical in such cases.The objective of the current study is to present a powerful opti-mization design technique in order to achieve the optimum cross-sectional areas of stay cables,which is directly proportional to the cost of the material.The proposed study focuses on the second de-sign stage,where self-weight,initial post-tensioning cable forces, and live load cases are applied to the bridge.The proposed optimi-zation technique involves interaction between three numerical schemes:finite element method(FEM),B-spline curves,and Real Coded Genetic Algorithm(RCGA).The novelty of this combined technique lies in the adoption of the B-spline curves to represent the distribution of cross-sectional areas of stay cables along the bridge length,which significantly reduces the number of design variables.In addition,RCGA,which is a global optimization meth-od,is capable offinding the global optimal solution.The remainder of the paper is organized as follows.In the next section,the geometry,finite element modeling,and design loads of the bridge chosen for the study are described.In Section3,a description of the design variables,design constraints,objective function,and optimization technique is presented.A detailed description of the optimization design technique that involves a combination between the FEM,B-spline curves,and RCGA is pre-sented in Section4.In Section5,detailed presentation and discus-sion of the numerical optimization results are given.Finally, Section6presents the main conclusions drawn from the study. 2.Description,finite element modeling,and design loads of the bridge2.1.Description of the bridgeThe geometry of the bridge chosen for this study is similar to the Quincy Bayview Bridge,located in Illinois,USA[17].The length of the main span(M)is285.6m,with two side spans(S)of128.1m. Therefore,the total length of the bridge(L)is541.8m,as shown in Fig.1a.The deck superstructure is supported by double planes of stay cables in a semi-harp type arrangement,where forty cables are an-chored into each transverse H frame shaped pylon.As such,eighty stay cables support the whole bridge,where forty cables support the main span and twenty cables support each side span.The bridge has two lanes of traffic having a width of12.2m measured between centers of the cables.The typical cross-section of the bridge deck(Fig.1b)consists of a precast concrete deck hav-ing a thickness of0.23m and a width of14.20m.Two steel main girders are located at the outer edge of the deck.These girders are interconnected by a set of equally spacedfloor steel beams. The distance between each pair offloor steel beams is9.0m.The pylons consist of two concrete legs,interconnected with a pair of struts.The upper strut cross beam connects the upper legs and the lower strut cross beam supports the deck.The lower legs of the pylon are connected by a1.22m thick wall,which is placed as a web between the two legs,as shown in Fig.1c.2.2.Finite element modeling of the bridgeIn general,the elastic stay cables are assumed to be perfectly flexible and to resist a tensile force only.The inclined stay cables of cable-stayed bridges will sag into a catenary shape due to their self-weight[18].The tension stiffness of a cable,which varies depending on the sag,is modeled by using one straight truss ele-ment with an equivalent modulus of elasticity.The concept of an equivalent cable modulus of elasticity wasfirst proposed by[19]. The equivalent cable’s modulus of elasticity used to account for the sag effect is given by:E eq¼E cs1þðw cs L cÞA i E cs12T3ð1Þwhere E eq is the equivalent modulus of elasticity;E cs is the cable material effective modulus;L c is horizontal projected length of a cable;w cs is the weight per unit length of the cable;A i is the cross-sectional area of the cable i;and T is the tension in the cable.On the other hand,three-dimensional frame elements are used to model the deck and the pylon.The deck is modeled using a sin-gle spine passing through its shear center.The axial stiffness of the deck and the moments of inertia about the vertical and transverse axes are obtained by converting the concrete slab to an equivalent steel section,using the ratio of the two elastic moduli.Thefinite element model of this bridge is shown in Fig.1d.NomenclatureA i cross-sectional area of the cable iA max maximum cross-sectional area of the cableC the objective functionE eq equivalent modulus of elasticity of cablesE cs modulus of elasticity of cablesH pylon heightL total length of the bridgeL i length of the cable iL c horizontal projected length of the cableLC total number of live load casesM main span lengthN cables total number of stay cablesN Area total number of cross-sectional areas of stay cables N Pylons number of pylons S side span lengthT tension in the cablew cs weight per unit length of cablesr kitensile stress in stay cable i for the live load case kr Max maximum allowable tensile stress in stay cablesd kivertical deflection of the deck at the cable position i forthe load case kd Max maximum allowable vertical deflection of the deckD kjhorizontal deflection of the tops of the pylon j for theload case kD Max maximum allowable horizontal deflection of the tops ofpylonsc cable unit weight of stay cables644M.M.Hassan/Engineering Structures49(2013)643–6542.3.Design loadsIn this study,the optimum design of cross-sectional areas of stay cables are evaluated under the combined effect of the follow-ing loads: The self-weight of the bridge includes the weight of all structure components and appendagesfixed to the structure,such as wearing surface,and traffic barriers.The initial post-tensioning cable forces.All possible live load cases depicted in Fig.2,as provided by[20]. M.M.Hassan/Engineering Structures49(2013)643–654645In the current study,the magnitude of the design live load is as-sumed to be 9kN/m/lane as specified by [21].3.Formulation of the optimization problem 3.1.Design variablesIn previous optimization techniques conducted to achieve an optimum design of stay cables,the number of design variables de-pends upon the number of the stay cables.As such,increasing the number of stay cables increases the number of design variables,which increases the complexity and computational expense of the optimization problem.On the other hand,the use of a large number of stay cables has become common in modern long-span cable-stayed bridges,making these previous techniques inefficient for obtaining the optimum design solution of this type of bridge.In order to overcome this problem,B-spline curve is utilized in the current study to define the cable area function,which represents the distribution of cross-sectional areas of stay cables along the bridge length.A p th degree B -spline curve S ðu Þ,shown in Fig.3a ,is defined as follows:S ðu Þ¼X n i ¼0N i ;p ðu ÞP i 06u 61ð2Þwhere u is the independent variable,P i is the set of control points approximating the curve,N i,p (u )are the p th degree B -spline basis functions given as:N i ;o ðu Þ¼1if u i 6u 6u i þ10otherwise&ð3-a ÞN i ;p ðu Þ¼u Àu i u i þp Àu iN i ;p À1ðu Þþu i þp þ1Àuu i þp þ1Àu i þ1N i þ1;p À1ðu Þð3-b Þand it is defined on nonperiodic and nonunifrom knot vectorU ¼0;...;0;|fflfflfflfflffl{zfflfflfflfflffl}p þ1u p þ1;...;u m Àp À1;1;...;1|fflfflfflffl{zfflfflfflffl}p þ1&'ð4Þwhere p is the degree of the basic function,(m +1)is the number of knots,(n +1)is the number of control points.The properties and the advantages of the B-spline curves are given by [22].In general,B-spline curves are considered to be the most popular mathematical forms that offer a unified mathematical form for representation of free-form curves and surfaces [23].The B-spline curves can be used to model complex curves with lower degree polynomials.Slight modifications in the control points’locations change the shape of the B-spline curve.Therefore,the control points are used as the de-sign variables in this study.It should be noted that using the control points as the design variables instead of the areas of stay cables sig-nificantly reduces the number of design variables.This reduction decreases the computational time required to get the optimum solution.Moreover,it improves the performance of the optimiza-tion technique by increasing the probability of finding the global optimum solution.3.1.1.Representation of cable area functions by B-spline curvesReferring to Fig.3b ,the following steps are used to locate a point on a cable area function.1.Assign the number of the control points (n +1),the degree of the function (p ),and then calculate the number of knots (m +1)using the formula (m =n +p +1).646M.M.Hassan /Engineering Structures 49(2013)643–6542.Define the x and y-coordinates of the B-spline control points(P iin Fig.3b),which are the design variables.pute the nonzero basis functions,Eq.(3).4.Multiply the values of the nonzero basis function with the cor-responding control points.In Fig.3b,the cross-sectional area of a stay cable is equal to the y-coordinate of the B-spline curve.Cables number i,j,k and l are mapped to their respective cross-sectional area values on the B-spline curves for exemplary purpose.3.2.Design constraintsThe following design constraints related to stresses in stay cables,vertical deflections of the deck,and horizontal deflections of the pylons’tops are considered in the optimization process.(a)Stay cable stressg1¼r k iÀr Max60i¼1;2;...;N Cables k¼1;2;...;LCð5Þwhere r k i=the tensile stress in stay cable i for the load case k; r Max=the maximum allowable tensile stress of the stay cables, which is taken as1600MPa;N Cables=the total number of stay cables.;LC=the total number of live load cases.(b)Vertical deflections of the deckg2¼j d kijÀd Max60i¼1;2;...;ðN Cables=2Þk¼1;2;...;LCð6Þwhere d ki=the vertical deflection of the deck at the cable position i for the load case k;d Max=the maximum allowable vertical deflec-tion of the deck.(c)Horizontal deflections of the pylons’topsg3¼j D kjjÀD Max60j¼1;...;N Pylons k¼1;2;......:;LCð7Þwhere j D kjj=the horizontal deflection of the tops of the pylon j in the longitudinal direction of the bridge deck for the load case k;D Max=the maximum allowable horizontal deflection of the pylons’tops;N Pylons=the total number of pylons of the bridge.In the current study,the maximum allowable vertical deflection of the deck (d Max)and the maximum allowable horizontal deflection of thepylons’tops(D Max)are taken as M550ÀÁand H550ÀÁ,respectively,as pro-vided by[3].3.3.Objective functionThe objective function in this study is set to be the minimum weight of steel in stay cables and can be expressedas:M.M.Hassan/Engineering Structures49(2013)643–654647C¼c cables XN cablesi¼1L i A ið8Þwhere c cable=the unit weight of stay cables,which is equal to 77kN/m3;L i=the length of the cable i;A i=the cross-sectional area of the cable i.3.4.Optimization techniqueMost optimization methods available in literature for solving general nonlinear optimization problems could be classified into two groups known as direct and global optimization methods.Di-rect optimization methods are local search techniques as they be-gin with a starting guess solution and search for a local minimum, while the global optimization methods are global optimizers.In general,cable-stayed bridges are highly statically indeterminate structures,making the solution of the minimum weight of steel in stay cables not singular.Moreover,the search space of the cur-rent objective function is expected to contain several hills and val-leys(local minima)due to the intersection of the design constraints with the objective function.Accordingly,a global search method is needed tofind the global optimum solution and to avoid the possibility of being trapped in a local optimum. Since the control points of the B-spline curve are continuous in nature,the optimization method should be more suited to contin-uous variables.In light of the above,the RCGA is selected.The RCGA is easy-to-implement and efficient infinding the global opti-mal solution[24].A complete description of genetic algorithm techniques can be found in[25–27]among other available refer-ences.Section4describes how the RCGA is adapted tofind the glo-bal optimum solution for the optimization problem in hand.3.4.1.Genetic operatorsThe RCGA is an evolutionary optimization method,which starts first by introducing a number of initial solutions(cable area func-tions).The initial solutions are chosen randomly between the low-er and upper bounds of the design variables.The genetic algorithm improves these initial solutions by applying crossover and muta-tion operators to selected pairs of parent solutions in order to gen-erate new solutions having minimum objective functions values (minimum weight of steel in stay cables).These new solutions re-place the worst ranked ones.The above procedure is repeated until termination condition is reached.In this study,the x-and y-coordinates of the B-spline control points(P i in Fig.3b)are used as the design variables.Positions of these control points determine the shape of the B-spline curves, which represent the distribution of cross-sectional areas of stay cables along the bridge length.In general,the lower and upper bounds for the x-coordinates are zero and the span length,respec-tively,i.e.(06x6span length),as shown in Fig.3b.The lower and upper bounds for the y-coordinates are zero and a preset value for the maximum cross-sectional area of the cable(A max),respectively,i.e.(06y6A max).In this study,the A max is taken as0.20m2.3.4.1.1.Crossovers operators.The RCGA uses these crossover operators as averaging techniques that produce new‘‘cable area functions’’from parent‘‘cable area functions’’having good objec-tive function values(minimum weight of steel in stay cables). The crossover operators used are the arithmetic,uniform and heu-ristic crossovers.Thefirst produce new‘‘cable area functions’’in the functional landscape of the parent solutions,while the last one extrapolates the parent solutions into a promising direction. Details of such operators are given by[28].3.4.1.2.Mutation operators.The RGCA uses mutation operators to alter an existing solution in order to search for solutions in remote areas of the objective function.In the current study,boundary,uni-form,non-uniform mutations are used.Thefirst operator searches the boundaries of the independent variables for optima lying there, the second is a totally random search element,and the third is a random search that decreases its random movements with the progress of the search.Details of such operators are given by[28].In the current investigation,the following operators are applied on each population with a size of100candidates(cable areas func-tions)according to the following configuration:(1)4Instances undergo uniform mutation.(2)4Instances undergo boundary mutation.(3)4Instances undergo non-uniform mutations.(4)2Instances undergo arithmetic crossover.(5)2Instances undergo uniform crossover.(6)8Instances undergo heuristic crossover.It can be noticed that the heuristic crossover operator is in-creased relative to other operators since the weight function of stay cables is a monotonic function and it is expected that the opti-mum solutions will tend to lie on the constraint boundaries.There-fore,a rapid movement of the search points towards the constraint boundaries is needed in this case.3.4.2.Penalized objective functionThe RCGA is suitable only for unconstrained optimum design problem.Therefore,the objective function of the optimization prob-lem in hand(Eq.(8))is penalized by adding a penalty function,con-sistent with constraint violations.The penalized objective function for a problem with m constraints is generally defined as follows: C p¼CþX mi¼1a i d ið9Þwhered i¼g i;for g i>0d i¼0;for g i60:&C p=the penalized objective function. C=the unpenalized objective function defined in Eq.(8).d i=a vio-lation factor for the i th constraint.g i=the design constraint defined in Section3.2.a i=a penalty parameter imposed for violation of constraint i, which depends on the type of the optimization problem.This con-stant ensures that the summation terms in the above equation have the same order of magnitude so that the search is not dominated by one of the constraint functions.The values of these constants are obtained by running a Monte Carlo Simulation of the independent values prior to the optimization step and obtaining the value of the objective and constraint functions corresponding to each simu-lation.In this study,the values of the penalty parameter a i used to estimate the penalized objective function are assumed to be 50Â103and100Â106for the stress constraint and the deflection constraints,respectively,which are defined in Section3.2.4.Finite element,genetic algorithm,and B-spline combined techniqueThe general procedure,which involves interaction between the FEM,B-spline curves,and the RCGA forfinding the optimum weight of steel in stay cables,is given as follows:1.A three dimensionalfinite element model of the cable-stayedbridge is developed based on the assumed geometry and physical properties of the bridge,as described in Section2.2.2.The initial post-tensioning cable forces due to self-weight ofthe bridge are calculated using the optimization method developed by[29].648M.M.Hassan/Engineering Structures49(2013)643–6543.The design variables,which are the x and y -coordinates of the B-spline control points (P i in Fig.3b ),are randomly selected by the RCGA between the lower and upper bounds of each design variable.Based on this random selection,a set of candidate functions for the cross-sectional areas of stay cables (100cable area functions)is created to form the ini-tial population,as described in Section 3.1.1.For each candi-date function,the cross-sectional areas of the stay cables along the bridge length are calculated.4.The finite element analysis of the bridge is conducted con-sidering the calculated cross-sectional areas of stay cables under self-weight of the bridge,initial post-tensioning cable forces,and live loads cases to predict the internal forces and displacements of the bridge elements.The objective function (C ),which is set as the weight of steel in the stay cables,is then calculated (Eq.(8)).5.The design constraints defined in Section 3.2are checked.In case of violating any of these constraints,the result of this specific bridge is excluded by applying the penalty function given by Eq.(9).6.The initial population is sorted in an ascending order based on the value of the objective function,such thatthe first ranked candidate has the minimum weight of steel.7.A new population (100cable area functions)is generated by applying the RCGA crossover and mutation operators,which are defined in Section 3.4.1.These operators are applied on the high ranked functions evaluated in the previous step.These operators direct the search towards the global opti-mum solution.8.The previous population is replaced with the newer one,containing new candidates with better fitness,in addi-tion to the best candidate function found so far (elitist selection).9.Steps 4–9are repeated for a certain number of generations until a global optimum solution is reached.In this study,these steps are repeated for 100generations to reach the optimum solution.10.Deliver the candidate function with the highest fitness (smallest values of the objective function)obtained at step 9as the final solution.All previous steps are summarized in the flow chart shown in Fig.4.M.M.Hassan /Engineering Structures 49(2013)643–6546495.Results and discussion5.1.Optimum number of control pointsReducing the number of design variables greatly increases the performance and efficiency of the optimization procedure.Conse-quently,it is very crucial to use the least number of control points (design variables)to represent the cable area functions without compromising theflexibility of the B-spline curves.Shapes of the cable area functions are not identical,since they are affected by the number of stay cables;the height of the pylon above the deck; and the ratio of the main span length to the total length of the bridge.For the current bridge,the objective function is calculated by repeating the optimization technique using three,four,andfives control points.The objective function values(weight of steel in stay cables)obtained from analyses are1339.2kN,1351.2kN, and1346.3kN,respectively.The distributions of cross-sectional areas of stay cables are plotted in Fig.5a.The results show that the objective function value and the distribution of cross-sectional areas of stay cables remain almost unchanged with increase of the number of control points.Given thefindings of the previous trials, it is concluded that three control points are sufficient to represent the cable area functions of the current bridge,as shown in Fig.5b.By reviewing Fig.5a,it should be noted that the optimization re-sults lead to larger cross-sectional areas for back-stay cables at-tached to both ends of the deck.Those back-stay cables are used to balance large overturning moments on the pylon since they brace the whole bridge in the longitudinal direction.Similarly,the stay cables close to the bridge’s center plane have cross-sectional areas larger than those close to the pylons in order to fulfill the deck deflection constraint.It is noticed also from thisfigure that the stay cables adjacent to the pylon have the smallest cross-sectional areas, since most of the load at this zone is carried by the rigid pylon.650M.M.Hassan/Engineering Structures49(2013)643–654。
