Optimization of cable-stayed bridges: A literature survey

Advances in Engineering Software 149 (2020) 102829 Contents lists available at ScienceDirect Advances in Engineering Software journal homepage: www.elsevier.com/locate/advengsoft Optimization of cable-stayed bridges: A literature survey a,⁎ a Alberto M.B. Martins , Luís M.C. Simões , João H.J.O. Negrão a v a,b T University of Coimbra, ADAI, Department of Civil Engineering, Rua Luís Reis Santos, Pólo II, 3030-788 Coimbra, Portugal University of Coimbra, Department of Civil Engineering, Rua Luís Reis Santos, Pólo II, 3030-788 Coimbra, Portugal A R TICL E INFO A BSTR A CT Keywords: Cable-stayed bridges Cable forces optimization Optimum design Structural health monitoring Structural identification This literature survey concerns the application of optimization techniques to cable-stayed bridges. A search within the Web of Science and Scopus electronic databases was conducted, complemented by a hand search. From a total of 155 articles, 90 were chosen for a detailed review. The search results are summarized, highlighting the procedures, the main conclusions and contributions of each work. The optimization of the cable forces distribution and the optimum design through cost minimization is present in 80% of the publications. In the last decade, optimization algorithms were used for a wide range of applications, such as, the design of hybrid fiber reinforced polymeric deck and cables, structural health monitoring, assessment and identification of existing bridges, design and location of passive and active control devices to improve the dynamic performance. The detailed analysis showed that the optimization of footbridges, long-span bridges and multi-span bridges with innovative cable arrangements like crossing-cables are attracting the interest of researchers. Moreover, the optimization taking into account the wind action, the seismic action and other dynamic effects are areas of major concern in future developments. 1. Introduction The optimization of cable-stayed bridges is a challenging problem for structural engineers. Although the first works on this subject were reported over 40 years ago, it is a research topic of growing interest with more than half of the studies published in the last 10 years. Traditionally, optimization techniques were applied to the cable forces optimization and the optimum design aiming at cost minimization. These two topics represent 80% of the previous research on the optimization of cable-stayed bridges. However, in the last decade, there is an increasing attention in applying optimization algorithms to a wide range of applications, such as, for example, the design of hybrid fiber reinforced polymeric deck and cables, monitoring and assessment of existing bridges, design of passive and active control devices to enhance seismic performance. Cable-stayed bridges are formed by three main structural elements: deck, towers and cable-stays. They feature multiple inclined cable stays that are used to support the deck along its length. This allows the construction of large-span bridges with shallow decks. The deck behaves like a continuous beam elastically supported by the inclined stays which, besides providing vertical support, also provide a natural prestressing in the deck. The cable-stays transfer the deck vertical loads to ⁎ the towers. The towers acting in compression transfer the loads to the foundations. These bridges are highly redundant, with their behavior governed by the stiffness of the load-supporting elements (deck, towers and cable-stays) and the cable forces distribution. They represent an efficient structural solution for medium-to-long spans and are widely used all over the world. The slenderness of the deck and the configuration of the cable suspension system provide these structures with undeniable aesthetical advantages. Fig. 1 shows the load transfer scheme in a symmetrical cable-stayed bridge. The main features and the structural behavior of cable-stayed bridges are thoroughly explained in several references [1–5]. The design of cable-stayed bridges is a challenging task which involves solving some complex problems, such as: definition of the structural system, finding the members cross-sections, calculation of the cable forces distribution, construction stages and geometrical nonlinear effects. For concrete bridges, the time-dependent effects are of major importance and must be considered. Optimization techniques are particularly suited for solving design problems in large and complex structures like cable-stayed bridges. They represent an efficient way to process the large amount of information aiming at reducing the material costs and thus obtaining economical and structurally efficient solutions. Moreover, these techniques are also particularly suited to solve Corresponding author. E-mail addresses: [email protected] (A.M.B. Martins), [email protected] (L.M.C. Simões), [email protected] (J.H.J.O. Negrão). https://doi.org/10.1016/j.advengsoft.2020.102829 Received 9 December 2019; Received in revised form 4 May 2020; Accepted 7 May 2020 0965-9978/ © 2020 Elsevier Ltd. All rights reserved. Advances in Engineering Software 149 (2020) 102829 A.M.B. Martins, et al. Tower Cable-stays Deck Pier Tension Compression Fig. 1. Scheme of load transfer in a cable-stayed bridge. several types of problems in these complex structures due to their capabilities in decision making problems. Optimization algorithms change iteratively the design variables to improve the current design towards the optimum solution. There is a wide range of optimization methods which can be grouped as gradientbased and non-gradient-based. The first approach requires the calculation of the derivatives of the objective function and all the design constraints with respect to the design variables. This information termed sensitivity analysis is used to define the direction, in the design space, according to which the current design variables should be modified seeking an optimum solution. In the second approach, the minimization of the objective function is done with techniques not needing derivatives. These techniques are often called metaheuristic algorithms (evolutionary algorithms, genetic algorithms, particle swarm optimization, simulated annealing), random search or branchand-bound. These strategies are easier to use, however, they are associated with an exponential convergence time related to the number of design variables. They finish with the best solution found so far, not even guaranteeing a local optimum unless the solution satisfies the Karush–Kuhn–Tucker (KKT) optimality conditions. These techniques can also be classified as nonconvex optimization strategies. The former convex optimization strategies converge in polynomial time to a local (not necessarily global) optimum solution. The optimization of cablestayed bridges usually features a large number of design variables and design constraints which, typically, leads to a computational costly problem. With the first works published more than 40 years ago, almost 70 journal articles published in this topic and with an increasing number of articles published on the last 5 to 10 years, it is considered relevant a state-of-the-art survey regarding the optimization of cable-stayed bridges. This work aims to present an overview of previous research works, the current trends and pointing out some future research developments. A search in the Web of Science and Scopus electronic databases was done to identify articles within this topic. This was followed by a hand search to obtain additional articles missed by the database search. From a total of 155 articles from the databases and the hand search, 90 were selected for a detailed review. Some general aspects of the articles analyzed are summarized and a description of each work is presented, highlighting the main characteristics, the principal conclusions and contributions of each. The search results are then analyzed and discussed. In Section 2, the methods used in this survey are described. Section 3 presents the overview of the results of the articles search. Furthermore, a summary of each article is included in the detailed analysis. In Section 4, the main characteristics and major findings of the articles described in Section 3 are examined and the results are discussed. The current trends and possible future developments within this research topic are also pointed out. Finally, Section 5 summarizes the main conclusions of this literature survey. 2. Methods The authors used the search engines in the Web of Science and Scopus electronic databases to find articles concerning the optimization of cable-stayed bridges. The search was limited to articles written in English and published between 1900 and November 2019. The search was conducted using a Boolean search strategy with the operators AND, OR and using the following terms: (“cable-stayed”) AND (“optimization” OR “optimum” OR “optimal” OR “minimum cost” OR “least cost”). The term “cable-stayed” was selected to easily identify the type of structure treated in this survey. The terms “optimization”, “optimum”, “optimal”, “minimum cost” and “least cost” were used to find works that employed optimization algorithms because these are common keywords used in this type of works. These terms were searched within the article's title. This search was also complemented by a hand search to obtain additional articles missed by the electronic database search. This was done to include papers published in conference proceedings which are considered relevant because, to the best of the authors’ knowledge, correspond to the first works on optimization of cable-stayed bridges. In the second step, from the articles obtained by the electronic search, the duplicates were eliminated and the articles published in archival journals were selected. Moreover, conference papers with Digital Object Identifier (DOI) and full-text available were also selected. In the third step, the full-text versions of the articles were obtained and analyzed to exclude the research works that do not use optimization algorithms [6–24] and do not concern cable-stayed-bridges [25–31]. Finally, a detailed assessment of articles’ full-text versions was performed. For each article a summary was done and the main characteristics were listed. Microsoft Excel was used for the statistical analysis considering the main characteristics of each work included in the detailed analysis. All the references from the selected articles were also checked manually to identify relevant studies that might have been missed in the electronic and hand searches and to eliminate duplicates. Fig. 2 depicts the flowchart of the procedure adopted in this literature survey. 3. Results 3.1. Overview From a total of 149 articles from the electronic databases search, 84 were chosen for the detailed review. Another, 6 articles were added from the hand search. The first research works on the optimization of cable-stayed bridges date back to the 1970s. Fig. 3 presents the articles distribution by year of publication, from which it can be stated an increasing interest in this research topic with 65.6% of the articles published in the last decade. The articles can be grouped in three main subjects, namely, “cable forces optimization”, “optimum design” and “other topics”. Fig. 4 presents the distribution of articles within each of the three subjects. The 2 Advances in Engineering Software 149 (2020) 102829 A.M.B. Martins, et al. Electronic search Web of Science database Scopus database articles written in English terms searched within the article’s title: (cable-stayed”) AND (“optimization” OR “optimum” OR “optimal” OR “minimum cost” OR “least cost”) 130 articles Hand search duplicates removed add articles missed by the electronic search 113 articles 149 articles some of the first works about optimization of cable-stayed bridges 69 conference papers 80 journal articles 30 conference papers (with DOI and full-text available) Full-texts analysed 26 articles excluded (don’t use optimization algorithms or don’t concern cable-stayed bridges) 84 articles 6 articles Detailed analysis 90 articles Fig. 2. Flowchart of the literature survey procedure. 2.2% 5.6% 13.3% 20.0% 40.0% 42.2% 13.3% 37.8% 25.6% 1970-1979 1980-1989 1990-1999 2000-2009 2010-2014 2015-2019 Forces Design Other Fig. 3. Articles distribution by year of publication. Fig. 4. Distribution of articles published by subject. “cable forces optimization” represents most of the articles analyzed, followed by the “optimum design” and by “other topics”. Fig. 5 shows the distribution of articles within each topic by year of publication. Besides the growing interest in the optimization of cablestayed bridges, depicted in Fig. 3, there is an increasing interest in all of the main subjects listed. In the last 5 years, the number of published articles about “cable forces optimization”, “optimum design” and “other topics” increased by 16.7%, 200.0% and 42.9%, respectively. This result can be explained by the rapid increase in the computational resources available and on the number of researchers dedicated to this subject. Soft computing not involving the need to understand the nuances of convex optimization algorithms and the need of derivatives for sensitivity analysis represents a big chunk of this increase. Sections 3.2–3.4 present a description of each article analyzed according to the subject. 3.2. Cable forces optimization The calculation of the cable forces is a distinctive aspect of cablestayed bridges design when compared to other types of bridges. Cable tensioning is required to control the geometry, the stress distribution and to correct construction errors or deviations. Bearing this in mind, several researchers focused on applying optimization algorithms to find the cable forces distribution (42.2% of the articles analyzed). The first 3 Advances in Engineering Software 149 (2020) 102829 A.M.B. Martins, et al. 16 the relaxation of prestressing tendons. Sung et al. [40] and Lee et al. [41] addressed the cable forces optimization for asymmetric steel bridges. Sung et al. [40] used an influence matrix of the cable forces and the minimization of the total strain energy expressed as a quadratic function of the post-tensioning cable forces. Lee et al. [41] applied a two-step approach based on the unit load method to obtain the desired deck bending moment distribution for the final stage of the structure under dead load and cable prestressing forces. Sun et al. [42] used the trust region algorithm to determine the optimal cable forces that minimize the bending strain energy of the deck and towers for the bridge under permanent loads. Baldomir and Hernández [43], Baldomir et al. [44] and Hernandez et al. [45] obtained the cable areas for a long span steel bridge by minimizing the cables volume through a gradient-based sequential quadratic programming (SQP) algorithm and the sensitivities computed by finite differences. Zhang and Bai [46] determined the cable forces for a single-pylon double-plane concrete bridge. The minimum bending strain energy was used to express the objective function as a quadratic form of the cable prestressing forces. ANSYS parametric design language (APDL) was used to solve the optimization problem. Yu et al. [47] used the unknown load factor method (ULF) of the software Midas/Civil to optimize the cable forces of an asymmetrical bridge. This method uses the concept of influence matrix and the displacements, reactions and members’ internal forces can be considered as constraints. Hassan et al. [48] computed the cable forces in steel-concrete composite bridges using a real-coded genetic algorithm (RCGA) by minimizing the square root of sum of squares (SRSS) of the deck and towers nodal points’ deflections. Hassan [49] used the same algorithm to obtain the cable areas through the minimization of the cables’ steel weight. In both articles, B-spline interpolation curves were used to describe the cable forces distribution and efficiently handle the large number of variables that arise due to the large number of cables. Sun et al. [50] proposed an optimization method for computing the cable forces considering the construction stages aiming to achieve the desired final state at bridge completion. The forward analysis procedure was used to simulate the construction process. The optimization was formulated as a quadratic programming problem to minimize the difference between the determined final state and the ideal final state of the bridge. Regarding hybrid cable-stayed suspension bridges, Zhang et al. [51] proposed a two-stage approach to obtain the optimal cable tensions for the bridge under dead-load. The finite element method (FEM) considering geometrical nonlinearities and a constraint relaxation quadratic programming method were used. Lonetti and Pascuzzo [52, 53] developed a formulation to obtain the optimum post-tensioning cable forces and cable cross-sections for selfanchored cable-stayed suspension bridges. The FEM was used for structural analysis considering geometrical nonlinearities, dead and live loads. Constrains concerning ultimate and serviceability limit states, maximum allowable stresses and bridge deflections were considered. The sparse nonlinear optimizer (SNOPT) was used to solve a constrained nonlinear programming (NLP) problem. Martins et al. [55] determined the cable forces considering the construction stages, the time-dependent effects of concrete and the geometrical nonlinearities ([54] and [56]). The solution was found by minimizing a constraint aggregation (displacements and stresses) convex scalar function obtained by an entropy-based approach. Regarding long-spans, Asgari et al. [57, 58] presented a multiconstraint optimization strategy based on the application of an inverse problem through the unit load method. Song et al. [59] determined the cable forces, the load and the range of the counterweight in long-span bridges by minimizing the weighted total bending energy of the girder and the tower. 14 12 10 8 6 4 2 0 1970-1979 1980-1989 1990-1999 Forces 2000-2009 Design 2010-2014 2015-2019 Other Fig. 5. Distribution of articles published by year and by topic. articles within this topic date back to the 1970s and 1980s. In this topic were included the articles that employed only the cable forces, the cable areas or both as design variables, independently from the objective function considered. Feder [32] proposed an optimality criteria method to compute the cable prestressing forces in steel bridges. Cable stresses, bending moments in the main girder and at the tower base were considered as the optimality criteria. Influence coefficients of the cable forces were used and the problem was posed as a system of linear equations solved by the least squares method. Furukawa et al. [33] applied a minimum strain energy criterion to obtain the cable forces in steel bridges. Influence coefficients of the cable forces were used to define the objective function. This function includes the constraint that, for the complete bridge, the dead load does not generate bending moment at the tower base. The Broyden–Fletcher–Goldfarb–Shanno (BFGS) variable metric method was used to minimize the objective function. Furukawa et al. [34] also applied the minimization of a strain energy criterion to optimize the cable forces in concrete bridges with prestressed decks considering the concrete creep effect. The tendon forces in the main girder were expressed in terms of linear functions of the cable forces. Osuo et al. [35] developed an optimization method to calculate the shim thicknesses aiming to adjust the cables length and forces in bridges with rocker towers. The problem was solved by minimizing the inner product of shim vector. Qin [36] considered the construction stages by the segmental cantilever construction and addressed the optimum cable-stretching planning through a linear programming (LP) problem. Kasuga et al. [37] studied the cable force adjustments in concrete cable-stayed bridges including the concrete creep effect. An influence matrix of the cable forces was used and the adjustment forces were found through a nonlinear programming (NLP) problem and the minimization of the work due to these forces. Wang et al. [38] studied four methods to optimize the cable forces in order to minimize the deformations and stresses due to dead load of the bridge. The authors considered: minimization of the summation of the squares of the vertical displacements of the cables anchor points at the deck (MMSVD); minimization of the maximum moment along the bridge deck (MMM); continuous beam method (CBM) and the simplysupported beam method (SBM). The latter was selected as the best method by comparing the calculated bending and vertical displacements of the bridge deck. Janjic et al. [39] proposed the unit force method (UFM) to obtain the cable forces aiming to achieve the desired deck bending moment distribution for the complete bridge under dead load. The method includes the construction sequence, the geometrical nonlinearities and the time-dependent effects, namely, concrete creep and shrinkage and 4 Advances in Engineering Software 149 (2020) 102829 A.M.B. Martins, et al. Sun and Xiao [60] proposed a method for the optimization of the dead load state in earth-anchored cable-stayed bridges. The method is based on the rigidly supported continuous beam method and the feasible zone method. The ANSYS APDL language was used for the optimization of self-anchored cable forces through the minimization of the bending strain energy. The earth-anchored cable forces were optimized through the feasible zone of pylons bending moments. Sung et al. [61] addressed the optimum construction planning using the cantilever erection method. The cable forces were computed through the minimization of the deviation between displacements upon completion and the pre-planned construction targets. Constraints on the cables axial forces during each erection stage were considered. Particle swarm optimization (PSO) and simulated annealing (SA) in the mutation operation of a conventional genetic algorithm (GA) were integrated to improve the ability to escape local minimum. Carpentieri et al. [62] developed an algorithm for the optimal design of the cables pre-tensioning sequence. The cables pre-tensioning forces that induce an “optimal” bending moment distribution over the deck were determined through the solution of a linear problem. Fabbrocino et al. [63] calculated the cable prestressing forces in bridges with steel-concrete composite decks to achieve a desired bending moment distribution over the longitudinal beams. An influence matrix of the cable forces was employed and the target bending moment distribution was defined aiming to an optimized use of the materials composing the bridge. Ha et al. [64] presented a three-stage algorithm to optimize the initial cable tensions and total weight of the cables using a nonlinear inelastic analysis. A micro-genetic algorithm (μGA)-based method using a unit load matrix is proposed to reduce the computational effort. The initial cable tensions were obtained through the minimization of the SRSS of the deck vertical and pylon horizontal deflections. The cables areas were obtained by the minimization of the total weight of the cables considering the bridge under dead and live loads and the initial cable tensions previously computed. Constraints on member stresses and deck vertical and pylon horizontal deflections of the bridge were considered. Concerning multi-span bridges, Baldomir et al. [65] minimized the cables volume of the Forth Replacement Crossing Bridge. Two design variables per stay (the cross sectional area and the prestressing force) were considered to accomplish the desired geometry under self-weight. The commercial software Altair Optistruct v11 (2013) was used to solve the optimization problem. Cid et al. [66] determined the cables anchor positions, areas and forces in a multi-span bridge with concrete towers and steel deck, considering the geometrical nonlinearities. The minimization of the cables’ steel volume was done with a gradient-based SQP algorithm and the sensitivities were computed by finite differences. Arellano et al. [67] also focused on multi-span bridges to compute the cable overlap lengths in bridges with criss-cross cables. The minimum cost of the cable system, the minimum of the maximum displacement of the pylon head and the maximum alternate live load on the bridge were considered as objective functions. The multi-objective problem was solved by the non-dominated sorting genetic algorithm II (NSGA II). Wang et al. [68] proposed a two-phase tensioning planning optimization for the system transformation process (STP) of a cable-stayed bridge erected by the incremental launching method. In phase one, a back propagation neural network (BPNN)-assisted global sensitivity analysis is used to identify one-off stretching of stayed cables to minimize construction expense. A BPNN-assisted reliability-based design optimization method is employed in phase two to select two-step stayed-cable stretching lengths such that the optimized scheme satisfies multiple control principles for a successful STP. Guo et al. [69] solved the cable force optimization of a curved bridge with a steel box girder combining SA algorithm and cubic BSpline interpolation curves to represent the cable forces distribution. 3.3. Optimum design The “optimum design” aiming at cost minimization represents 37.8% of the articles analyzed. This topic comprises the articles that considered the cost or volume of the structure in the objective function. This optimization problem features a complex design space given the large number of design variables and design constraints, which are nonlinear and conflicting. The first works concerning the volume or cost minimization of steel bridges were reported in the 1970s and 1980s. Daiguji and Yamada [70] proposed an optimality parameter method for cost minimization of steel bridges considering the cable area and the moment of inertia of stiffening girder as design variables. Bhati et al. [71] presented an optimization technique for the preliminary design of steel bridges through the minimization of the total weight of the structure. The cables areas, the sizes of the deck main girder and of the towers were considered as design variables. Constraints on member stresses, displacements and member sizes were considered. The Linearization method which is based on solving a quadratic programming (QP) problem was used to solve the optimization problem. Torii and Ikeda [72] proposed a non-iterative optimum design method for minimizing the cost of steel bridges considering constraints on member stresses. The objective function was transformed taking the cables redundant forces as design variables and the cross-sectional properties in terms of member forces. Therefore, the solution was obtained through an unconstrained minimization problem solved by the least squares method. Ohkubo and Taniwaki [73] developed a minimum-cost design method for steel bridges. The cables anchor positions on the deck and towers, and the members’ cross-sectional dimensions were considered as design variables. Constraints were imposed on the members’ stresses and the cost minimization problem was solved by using the dual method with mixed direct/reciprocal design variables. Ohkubo et al. [74] also included the pseudo-loads applied to the cables as design variables in a two-stage optimum design process. In the first-stage, the cable arrangement and sizing variables were optimized by using the dual method. In the second-stage, the optimum values of pseudo-loads and of sizing variables were determined to minimize the total cost of the bridge. The sensitivities with respect to the pseudo-loads were used in a modified linear programming algorithm. Simões and Negrão [75] focused on the optimization of steel bridges considering sizing and geometrical design variables, three-dimensional modeling ([76] and [77]), the seismic action through modal/spectral and time-history approaches [80], box-girder decks [78, 79, 81] and reliability [82, 83]. The reliability-based design of glulam footbridges was also reported [84]. An entropy-based approach was used to obtain the solution of the multi-objective (cost, stresses, displacements) optimization problem by minimizing a convex scalar function. Long et al. [85] used an internal penalty function method to minimize the cost of steel-concrete composite bridges taking the cable areas and the cross-sectional dimensions as design variables. Lute et al. [86] presented a GA-based algorithm to minimize the cost of concrete bridges considering sizing design variables. Ferreira and Simões [87] presented an algorithm for the optimum design considering active devices to control the response of steel bridges subjected to earthquakes. Yazdani-Paraei et al. [88] focused on the structural and sensitivity analysis considering the geometrical nonlinearities and construction stages. An optimality criteria (OC) method was used for cost minimization considering sizing, geometrical and mechanical design variables. Hassan et al. [89, 90] developed a RCGA for the optimal design of semi-fan bridges with concrete towers and steel-concrete composite decks. The problem was formulated as the minimization of overall cost considering sizing, geometrical, mechanical and topological design 5 Advances in Engineering Software 149 (2020) 102829 A.M.B. Martins, et al. detection. Azarbayejani et al. [105] solved the optimal sensor network as a multi-objective problem and used an entropy-based objective function to represent monitoring uncertainty and a sensor network cost function for cost limitations. Deshan et al. [106] presented a method for the optimal sensor placement of the SHM system for a long-span railway steel truss bridge. The improved genetic algorithm (IGA), dual structure coding method was used to improve the individual encoding method and thus overcome some disadvantages in other genetic algorithms aiming to the global optimum solution of the sensor placement problem. Hou et al. [107] used the random elimination (RE) and heuristic random elimination (HRE) algorithms to optimize the sensor quantity and the multistage global optimization (MGO) algorithm to determine the sensor location considering the concept of damage identification reliability index (DIRI) under uncertainty. Zhang et al. [110] employed the PSO algorithm for the optimal sensor placement in the SHM system of a long-span bridge. A dual-structure coding was adopted to improve the individual encoding method in the PSO algorithm. The fitness function was established based on the root-meansquare value of the off-diagonal elements of modal assurance criterion matrix. Li et al. [111] solved the optimal sensor placement for longspan bridges using a novel dual-structure coding and mutation particle swarm optimization (DSC-MPSO) algorithm. Casciati et al. [112] applied a bio-inspired optimization algorithm to obtain the optimal deployment of a reduced number of sensors along the deck for the validation of the bridge modal identification. Wu et al. [113] developed a GA-based optimization tool for strain gauges and accelerometers placement application to long-span bridges. The fast messy GA (fmGA) solver, previously used for water distribution optimization, was extended and employed to the sensors placement problem. Optimization algorithms are also being employed for assessment and identification of existing bridges. Nazarian et al. [114] developed a recursive optimization method for monitoring of tension loss in cables. Bao et al. [115] used a sparse l1 optimization-based identification approach for the distribution of moving heavy vehicle loads through cable force measurements. A convex optimization package was employed to solve the optimization problem. Ye et al. [116] presented a method for the assessment of the current state of girder internal forces for an existing single-pylon prestressed concrete bridge. The concrete modulus of elasticity of different girder sections was updated through the minimization of the absolute value of the relative error between calculated and measured girder displacements. To adjust the internal forces, cable forces were optimized by the minimization of the bending strain energy which was formulated as a quadratic programming problem. Zarbaf et al. [117] proposed a GA and PSO-based algorithm for stay cable tension estimation through the minimization of the error between the experimentally measured and the analytical natural frequencies of the cable. The optimization of control devices to enhance the dynamic performance of cable-stayed bridges was also reported. Mohamad et al. [118] proposed a GA-based method for magnetorheological (MR) fluid damper optimization to mitigate the bridge flutter due to seismic and aerodynamic vibration. Feng [108] and Mu et al. [109] applied a multiobjective fuzzy optimization to obtain the optimal dynamic isolation model scheme to enhance the dynamic response of self-anchored cablestayed suspension bridges. Fangfang et al. [119] applied the NSGA-II algorithm to a multi-objective problem to optimize parameters of viscous fluid dampers aiming to control the longitudinal seismic response of multi-span bridges. De et al. [120] used an SQP algorithm for the optimal design and design-under-uncertainty of passive control devices. Yu et al. [121] employed a mixing method called BPNN-NSGA-II combining the back propagation neural network (BPNN) non-dominated sorting genetic algorithm with elitist strategy (NSGA-II) to optimize parameters of longitudinal viscous dampers for a freight railway bridges under braking forces. variables. B-spline interpolation curves were used to describe the cable forces distribution. The loads, design, and constraints were defined based on the Canadian Highway Bridge Design Code CAN/CSA-S6-06. Cai and Aref [91] focused on improving the aerodynamic performance through combining carbon fiber reinforced polymeric (CFRP) materials with steel in the cable system. A GA-based algorithm was developed to optimize the distribution of CFRP and steel maximizing the critical flutter velocity. Cai and Aref [92] also optimized the distribution of glass FRP (GFRP) and concrete in a hybrid deck system, and the distribution of carbon FRP (CFRP) and steel in a hybrid cable system. A GA-based algorithm was used to maximize both static and flutter performances simultaneously. Martins et al. [93] considered the construction stages, the concrete time-dependent effects and the geometrical nonlinearities in the optimum design of concrete bridges with different deck cross-sections and deck prestressing [94]. The multi-objective optimization problem with cost, displacements and stresses was solved by minimizing a convex scalar function obtained through an entropy-based approach. Gao et al. [95] presented a multi-parameter optimization technique for the minimum cost design of concrete bridges with prestress in the girder. The number of prestressing tendons, cable forces, cable areas and girders and towers sizes were considered as design variables. Stress and displacement constraints were imposed to ensure the structural safety and serviceability. Ferreira and Simões [96, 97] addressed the optimization of steel footbridges with active control devices and three-dimensional modeling [99], subjected to a running event and using semi-active and passive mass dampers [98] and curved footbridges [100]. Montoya et al. [101] reported the shape optimization of streamlined decks considering aeroelastic and structural constraints. A surrogatebased optimization was considered. A gradient-based SQP algorithm was used to minimize the volume of the deck and stays considering the cables forces and areas, the deck shape and plate thicknesses as design variables. Martins et al. [102] developed a computational method for the optimization of concrete bridges under seismic action. Three-dimensional modeling, concrete time-dependent effects, geometrical nonlinearities and seismic action through a modal/spectral approach were considered. The problem was solved as a multi-objective optimization with objectives of cost, deflections, natural frequencies and stresses considering both, serviceability and ultimate limit states. Cable-forces, cable areas and the cross-sectional dimensions of deck and towers were considered as design variables. The solution was obtained by a convex optimization technique associated with multiple starting points. Ferreira and Simões [103] proposed an optimization algorithm to solve the simultaneous structural-control design problem of steel bridges under seismic action. Three-dimensional modeling, erection stages and geometrical nonlinearities were considered. Cable-forces, cable areas, properties of control devices, bridge geometry, deck and towers sizes were considered as design variables. A convex optimization algorithm associated with a multi-start strategy was used to solve the multi-objective optimization problem. 3.4. Other topics From 2009, optimization algorithms are also being employed in other problems besides the “cable forces optimization” and the “optimum design”. Therefore, 20% of the overall articles (28.8% and 27.8% of the articles of the last 10 years and 5 years, respectively) reported the application of optimization algorithms in topics different from the previously described. Park et al. [104] employed a sensitivity-based penalty function method for the finite element model updating based on ambient vibration measurements. Regarding structural health monitoring (SHM), several authors focused on the optimization of sensor placement to enhance damage 6 Advances in Engineering Software 149 (2020) 102829 A.M.B. Martins, et al. approach considered a multi-objective problem solved by the minimization of a convex scalar function obtained by an entropy-based approach. This constraint aggregation (cost, displacements and stresses) scalar function creates an inside convex approximation of the original nonconvex domain. The formulation of this problem should include the design constraints (stresses and displacements) already referred for the “cable forces” problem. It is worth referring that the deck and towers stress limits should be checked for the complete bridge under dead and live loading conditions. The traffic live load should be placed in different locations to obtain the most unfavorable design conditions. Bearing this in mind, placing the traffic live load in the central span induces the largest bending deflections and stresses in the towers. The towers deflection is controlled by the backstays and therefore this design situation should be considered for the design of both, the towers and the backstays. Standards and design codes provisions influence the formulation of the optimum design problem. The analysis stage is affected by the loading conditions, the effects that should be included in the design process and the type of structural analysis. In the optimization stage, the service and strength criteria prescribed in the design codes should be expressed in terms of design constraints that should be met to obtain a feasible design. When comparing gradient-based and non-gradientbased optimization methods, there is no difference in terms of constraint evaluation. Gradient-based methods require additional effort in the formulation and implementation of the derivatives required for sensitivity analysis. This facilitates the use of non-gradient-based optimization methods. Nevertheless, the complexity of the analysis stage associated with a large number of design variables can be handled efficiently with convex optimization strategies. Given the number of design variables and constraints present in the sizing of cable-stayed bridges, the optimum design approach is usually limited to engineers familiar with optimization and sensitivity analysis techniques. Although there are some commercial computer programs which contain an optimization module, the domain of many structural design problems, such as, cable-stayed bridges sizing optimization, is nonconvex and in some cases non-connected. At present, the designer must define the initial data, supervise the overall process and interactively modify or adjust some design parameters needed to find an optimum solution. Along with the increasing computational capacity it is foreseeable the development of computer programs that, including the requirements of standards and design codes, will provide optimum solutions for the design problem. These solutions will be checked, chosen or modified by a designer not familiar with optimization and sensitivity analysis techniques. Soft computing strategies are easier to implement and do not require a deep understanding of optimization concepts; hence, they assume some relevance in formulations with a limited number of design variables. Fig. 6 presents the distribution of articles regarding the “optimum design” topic by bridge main structural material. The “optimum design” of steel bridges constitutes the largest number of articles analyzed. Until 4. Discussion The “cable forces optimization” problem involves considering the cable forces, the cable areas or both as design variables. This problem was previously solved considering different objective functions. Some authors focused on minimizing the cables cost, thus, they considered the cables cost or the cables volume as the objective function. Other authors concentrated on controlling the bridge displacements, therefore, the SRSS of the deck vertical displacements and towers horizontal displacements was taken as the objective function. In other works, the cable forces were computed aiming to minimize the total strain energy or to obtain the desired deck bending moment distribution for the complete bridge under dead load. A multi-objective optimization strategy was also considered and the cable forces were found by minimizing a constraint aggregation (displacements and stresses) convex scalar function obtained by an entropy-based approach. Regardless of the objective function considered it is fundamental that the optimization problem includes several structural constraints (displacements, stresses) to ensure the physical meaning of the problem solution. The deck vertical displacements should be controlled in order to achieve the desired deck profile at the end of construction. The towers horizontal displacements should be minimized to control the towers bending deflections and stresses. Cable areas should be obtained minimizing the volume of steel and considering the bridge under permanent and traffic live loads. The cables allowable stress is usually limited to 0.45 to 0.50 of the prestressing steel tensile strength due to the detrimental effect of fatigue. Moreover, for the complete bridge under permanent load the deck and towers stresses should remain within the elastic range. The structural analysis should include the three main sources of geometric nonlinearities, namely, the nonlinear axial force-elongation relationship for the inclined cable stays due to the sag caused by their own weight; the nonlinear axial force and bending moment deformation relationships for the towers and the deck under combined bending and axial forces; and the geometry change caused by large displacements. These bridges are usually erected by the balanced cantilever method. In this method, the geometry of the structure, the stresses and displacements change during the several construction stages. Thus, the corresponding displacement and stress constraints should be included in the formulation of the cable forces optimization problem. In this optimization problem, each cable-stay corresponds to one or two design variables, depending if the cable force, the cable area or both are considered as design variables. Therefore, this problem can easily present a large number of design variables. This aspect may justify that only 18.4% of the works used metaheuristic algorithms. Instead of considering each cable force or cable area as design variables, some authors employed B-spline interpolation curves to describe the cable forces distribution aiming to reduce the number of design variables and thus reducing the computational effort required to solve the problem. In order to compute the structural responses caused by changes in the design variables, the analytical, semi-analytical and finite-differences procedures were used for sensitivity analysis. Some authors employed the concept of influence matrix whose values are obtained by applying unit forces in each cable separately. The “optimum design” problem involves considering sizing (cable areas, cross-sectional dimensions of deck and towers), mechanical (cable prestressing forces), geometrical (tower height, lateral and central span lengths, cable anchor positions) or topological (number of cables) design variables. The large number of design variables and design constraints, which are nonlinear and conflicting, translates into a nonconvex design space and a more computationally costly problem when compared with the “cable forces optimization”. This problem was previously solved by minimizing a single objective function (cost or volume) subjected to a set of constraints. Another 8.8% 5.9% 2.9% 14.7% 67.6% Steel Concrete Hybrid FRP Timber Composite Fig. 6. Distribution of articles within the “optimum design” topic by bridge main structural material. 7 Advances in Engineering Software 149 (2020) 102829 A.M.B. Martins, et al. 1999, only steel bridges were considered. Although concrete bridges and bridges with steel-concrete composite decks are common worldwide, the fewer articles concerning these solutions may be justified by the additional complexity of structural concrete. The instantaneous and time-dependent behavior of concrete should be considered in the structural analysis. Moreover, reinforced concrete adds some complexity when evaluating the constraints related to the strength verification of reinforced concrete members. Concerning the bridge type, it can be referred that 85.3% of the “optimum design” articles considered road bridges and only 14.7% considered footbridges. However, in the last 5 years, the “optimum design” of footbridges represents 25.0% of the “optimum design” articles and 8.3% of the overall articles. Linear programming, nonlinear programming, optimality criteria methods and genetic algorithms were used to minimize the objective function. As in the “cable force optimization” problem, only a small percentage (14.7%) of the works employed metaheuristic algorithms. The large number of design variables and design constraints associated with a complex structural analysis (large structure, several load cases, geometrical nonlinearities, construction stages, dynamic actions) may impair the application of metaheuristic techniques. This literature survey revealed that, besides what can be termed as “classical problems” in the optimization of cable-stayed bridges, in the last decade optimization techniques are becoming popular for solving other problems. The results presented in Fig. 5 highlight the increasing interest in applying optimization algorithms in “other topics”. Fig. 7 shows that, regarding structural health monitoring (SHM), optimization of sensors placement to enhance damage detection represents the largest percentage of articles within “other topics”. Some results seem to indicate that the simultaneous optimization of the bridge and device location is preferable instead of optimally locate devices to control the dynamic response of an existing bridge. The assessment and identification of existing bridges (AIEB), namely, monitoring cable tension loss, cable tension estimation, identification of moving heavy vehicle load distribution and the optimization of control devices (CD) to enhance the dynamic performance of cable-stayed bridges represent 55.5% of the articles in “other topics”. Optimization algorithms were also used for finite element model updating (FEMU) based on ambient vibration measurements. It can be stated that in “other topics” 55.6% of the articles used metaheuristic algorithms. The problems considered in this topic typically consider a small number of design variables when compared to the “cable forces optimization” or “optimum design” problems. The design variables can be either continuous, integer or discrete and can be handled by metaheuristic algorithms. Generally, in these problems, the gradients of the objective function and all the design constraints with respect to the design variables cannot be easily formulated and computed. These algorithms do not require the calculation of these gradients which facilitates the problem solution with standard finite element software, which can be used as a black-box. Therefore, it can be referred that metaheuristic methods are particularly suited for solving the problems classified in “other topics”. In recent years, the optimization of footbridges, curved bridges, long-span bridges and multi-span bridges with innovative cable arrangements like crossing-cables are attracting the interest of some researchers. Regarding the algorithms, there is an increasing use of metaheuristic algorithms, artificial neural networks (ANN) and surrogate models in the optimization field. It can be expected that the use of these techniques will be extended to cable-stayed bridges optimization. Novel contributions for solving the cable force optimization problem are arising. The dynamic actions are particularly relevant in the structural response of these bridges. Therefore, some recent articles focused in the “optimum design” considering dynamic actions, such as, the dynamic effects induced by pedestrians in footbridges, the wind action and the seismic action. The optimization considering the above-mentioned dynamic actions, the dynamic effects due to high-speed railways and the reliability-based design optimization will certainly constitute new challenges in the optimization of cable-stayed bridges. Further developments can be expected concerning the use of optimization algorithms for monitoring, inspection and maintenance aiming to extend the service life of these bridges. Moreover, the application of optimization algorithms for strengthening and retrofitting of existing bridges can also be expected. The structural design should satisfy a set of criteria related to serviceability, safety and economy. Nowadays, the design should also include some sustainability measures (e.g. carbon dioxide emissions, embodied energy). Aspects related to robust and resilient design are also of major concern and should be included. Additionally, the structural response to unforeseen events, in a climate changing world, should also be addressed. Besides ensuring safety, minimizing costs is desirable and may be included in the next generation of design codes. Optimization seems particularly suited for the holistic approach required for the structural design in the XXI century. 5. Conclusions This article presented a literature survey on the application of optimization algorithms to cable-stayed bridges. The following conclusions can be drawn: • The first works date back to the 1970s and 1980s with more than • • • 5.6% half of the works published in the last 10 years. The researchers focused mainly on the cable forces optimization and the optimum design through cost minimization. A significant number of articles employ gradient-based algorithms. A number of recent works use metaheuristics associated with soft computing strategies. In the last decade, optimization algorithms are being used for SHM and assessment of existing bridges, design, location and type of control devices to enhance the dynamic performance. Optimization of footbridges, curved bridges, long-span bridges and multi-span bridges with innovative cable arrangements like crossing-cables are attracting the interest of some researchers. Optimization considering the wind action, the seismic action and other dynamic effects, sustainability, resilience and robustness constitute areas of major interest in future developments. 33.3% 38.9% CRediT authorship contribution statement 22.2% FEMU SHM AIEB Alberto M.B. Martins: Conceptualization, Methodology, Formal analysis, Investigation, Writing - original draft, Writing - review & editing. Luís M.C. Simões: Conceptualization, Methodology, Resources, Supervision, Writing - review & editing. João H.J.O. Negrão: Conceptualization, Resources, Supervision. CD Fig. 7. Distribution by subject of the articles within “other topics”. 8 Advances in Engineering Software 149 (2020) 102829 A.M.B. Martins, et al. Declaration of Competing Interest www.scientific.net/AMM.274.490. [28] Wang XL, Liang T, Qi TD. Study on cable force optimization for extradosed cablestayed bridge under the rational completion stage. 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