A genetic algorithm-based multi-objective optimization for hybrid fiber ...

A genetic algorithm-based multi-objective optimization for hybrid fiber reinforced polymeric deck and cable system of cablestayed bridges Hongwei Cai & Amjad J. Aref

Structural and Multidisciplinary Optimization ISSN 1615-147X Struct Multidisc Optim DOI 10.1007/s00158-015-1266-4

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Author's personal copy Struct Multidisc Optim DOI 10.1007/s00158-015-1266-4

RESEARCH PAPER

A genetic algorithm-based multi-objective optimization for hybrid fiber reinforced polymeric deck and cable system of cable-stayed bridges Hongwei Cai 1,2 & Amjad J. Aref 2

Received: 2 February 2014 / Revised: 8 May 2015 / Accepted: 8 May 2015 # Springer-Verlag Berlin Heidelberg 2015

Abstract As the length of main span of cable-stayed bridge increases, several technical challenges become more prevalent with traditional materials. Such technical challenges include: large axial stresses in main girders, cable sagging effect, and aerodynamic instability, consequently limiting the prospects of extending the span length of future cable-stayed bridges with traditional materials. In order to remedy these issues, we propose fiber reinforced polymeric (FRP) composites for the deck and cable system of cable-stayed bridges in combination with traditional materials. To use FRP composites most effectively, we developed a genetic algorithm (GA)-based optimization procedure to solve for the distribution of Glass FRP and concrete in the hybrid deck system, and the distribution of carbon FRP and steel in the hybrid cable system. This proposed optimization-based procedure aimed at developing two systems: (1) optimized hybrid Glass FRP-concrete deck system (OHDS), and (2) optimized Carbon FRP-steel cable system (OHCS), which can maximize static and aerodynamic performances concurrently. As an example, we utilized an existing long-span composite cable-stayed bridge and implemented these two systems. For a typical long span cablestayed bridge, the results of this benchmark example provide insights about the typical composition of OHDS and OHCS and suggest that these two systems can concurrently improve the static and aerodynamic performances by 33 and 12 %, respectively.

* Hongwei Cai [email protected] 1

Parsons Brinckerhoff, 2777 N. Stemmons Freeway, Dallas, TX, USA

2

Department of Civil, Structural and Environmental Engineering, University at Buffalo-the State University of New York, Buffalo, NY 14260, USA

Keywords Cable-stayed bridge . Glass fiber reinforced polymer-concrete deck system . Carbon fiber reinforced polymer-steel cable system . Genetic algorithm . Critical flutter velocity

1 Introduction Since the concept of cable-stayed bridges was first proposed in the 17th century, modern cable-stayed bridges have entered the era of 1000-m main span length. However, as the span length of cable-stayed bridge increases, there exist several technical challenges pertaining to axial stresses in main girders, cable stiffness, cable strength, and aerodynamic stability. In particular, the following four issues are of key significance: i. As the span length increases, dead loads and cable tensions increase. The axial compressions from stay cables applied on main girders will increase accordingly. ii. As the span length and cable length increase, cables sag more and thus nonlinear cable behavior becomes more evident, which will lead to the deterioration of the cable stiffness. iii. Cables need to hold large tensions to keep taut, reduce sagging effect, and carry the ever-increasing dead loads as the span length increases. However, traditional material can’t provide comparable strength without increasing the cross-sectional area significantly, which will increase the self-weight of cables and result in cable sagging issue in return. iv. For long-span cable-stayed bridges, the critical aerodynamic issue is the flutter instability (Wang 2003). The main factors influencing flutter instability on bridge characteristics are: (i) geometry of the bridge deck; (ii)

Author's personal copy H. Cai, A. J. Aref

frequencies of the bridge; (iii) mass of the bridge, and (iv) mechanical damping of the bridge. As the span length increases, frequencies of the bridge decrease and thus flutter performance deteriorates. These challenges become more prominent for cable-stayed bridges using traditional materials, such as steel and concrete, consequently limiting chances of extending the span length in future construction. Cable-stayed bridges have evolved in concert with the developments and innovations in structural materials. In this paper, new materials, such as FRP composites, are proposed as a remedy to solve these challenges along with the span length extension. Superior mechanical and chemical properties, such as the high strength-to-density ratio, anti-corrosion properties, fatigue-resistance, and tractability for tailoring and replacement make FRP a suitable alternative material to replace traditional materials, and thus potentially increasing the span length. There are two candidate applications of FRP composites in bridge superstructures- decks, and stay cables. The static, dynamic, aerodynamic, and cost issues of bridges with FRP composites or hybrid FRP systems with traditional materials have been studied by many researchers (Wang and Wu 2010a, b; Xiong et al. 2010; Xiong and Xiao 2010; Zhang and Ying 2007; He and Aref 2003; Kitane et al. 2004; Aref et al. 2005; Alnahhal and Aref 2008; Hognestad et al. 1951; Nakamura et al. 2011). However, a study to find the most effective way to use FRP has not been completed by many researchers. He (2002) optimized an FRP web core sandwich deck system for short-span bridges. In He’s study, a unified optimization framework coupling the optimum design of the bridge geometry, sandwich configuration of bridge deck, and material architecture is proposed. In this paper, carbon FRP (CFRP) cable and glass FRP (GFRP) deck systems are implemented to cable-stayed bridges combining with traditional materials in an optimal way to maximize both static and flutter performances concurrently. Figure 1 schematically shows half of the cable-stayed bridge with typical proposed hybrid deck and cable systems. The combination of FRP composites with traditional materials is necessary not only for economical reasons but more importantly to achieve structural advantages. Table 1 shows the properties of materials used in this paper. Compared with the traditional steel cable, the FRP cable is lighter but has a smaller elastic modulus. Therefore, on one hand, the CFRP cable will reduce the self-weight and sagging effect, decreasing the nonlinear cable behavior and increasing the stiffness of the stay cable; on the other hand, CFRP itself, as a composite material, has a lower elastic modulus and thus a lower stiffness level. In order to maximize the stiffness of the cable, it is necessary to combine steel and FRP to form a hybrid CFRPsteel cable. The hybridization can take advantage of the inherent good properties of both materials—namely, the lower

density of CFRP and the higher elastic modulus of steel. The hybrid cable system with maximized stiffness will be generally favorable for improving static and aerodynamic performances. Compared with the traditional concrete deck, a GFRP deck will significantly reduce the dead load, making the chances of reducing axial stresses of main girders more favorable. However, a shortcoming of the reduced dead load is related to the mass reduction, which is potentially unfavorable for the flutter performance. Therefore, in order to improve both static and aerodynamic performances concurrently, it is necessary to combine both GFRP deck and concrete deck to form the hybrid GFRP-concrete deck system. When it comes to the optimization algorithms, the mathematical optimization algorithm cannot outweigh the genetic algorithm (GA) for two inherent drawbacks: (1) the susceptibility to converge to local optima, and (2) complexity to calculate the gradient information. Awad et al. (2012) reviewed optimization techniques used in the design of fiber composite structures for civil engineering applications. Through comparing with other optimization methods, they gave a high overall ranking to GA and stated that GA is particularly suitable to deal with complicated and large variable problems. Therefore, GA is a suitable candidate for optimization, and we utilized it in this paper to address the complex optimization problem at hand. The GA method was first proposed by Holland in the 1960s at the University of Michigan and popularized by one of his students, Goldberg, in 1989. GA is a stochastic search technique that draws upon the mechanism of Darwinian evolution in nature. Accordingly, when a population of individuals is allowed to evolve over generations, individual characteristics that are favorable for survival tend to be passed on to future generations, because individuals carrying them are assigned higher fitness and thus get higher chances to breed (He and Aref 2003; Michalewicz 1994). There are many applications of GAs in the field of structural engineering. The first application to a structural design was the well-known 10bar truss weight minimization problem, developed by Goldberg (1989). Rao et al. (1991) conducted GA optimization to search optimal selection of discrete actuator locations in actively controlled structures. Chen and Chen (1997) presented three structural optimization examples: minimization of the weight of a 3-D 25-bar tower; topology optimization; and maximization of first natural frequency of an in-plane cantilever plate. He (2002) developed a GA optimization algorithm to search for the optimal geometrical parameters, web core sandwich configuration, and material architecture of FRP sandwich bridge deck simultaneously. Dogruel and Dargush (2008) presented a GA-based computational methodology for the optimal life-cycle cost of seismic and wind-excited structures retrofitted with passive energy dissipation devices. In this paper, we construct a two-objective GA-based optimization to search for the optimal FRP distribution in the hybrid deck and cable system for cable-stayed

Author's personal copy A genetic algorithm-based multi-objective optimization for hybrid fiber reinforced polymeric deck and cable...

Fig. 1 Proposed hybrid deck and cable system: a half of typical hybrid cable-stayed bridges, b typical cross section of proposed hybrid cables, c typical cross section of proposed hybrid deck system

model. For more details about how to assign the equivalent mechanical properties to the numerical model, one can refer to the work by Cai (2012). In his work, a detailed ABAQUS finite element model is also constructed to verify the equivalent mechanical properties. Within the numerical model, 3D beam elements are used to model the actual superstructure and pylons and catenary cable elements to model stay cables. The stiffness formulation of 3D beam element and catenary cable element can be found in the references (Kassimali and Abbasnia 1991; Oran 1973; Cai 2012). The analysis engine is capable of conducting geometrical nonlinear static, dynamic modal, and flutter analyses, and can output member forces, displacements, and critical flutter velocity. All geometrical nonlinearities including large displacement effects, large 3D rotation, beamcolumn effect, and cable sagging effect are taken into account in the analysis engine. For the static analysis, initial cable tensions are firstly determined through the “relative stiffness method”. More details about this method can be found in the work by Wang and Zhou (1997) and Cai (2012). Using the Newton–Raphson scheme, the nonlinear static analysis is basically solving the nonlinear forcedisplacement equation (KU=F) and finding the

bridges to concurrently maximize both static and flutter performances.

2 Formulation of the multi-objective GA-based optimization The optimization procedure requires an analysis and an optimization engine. The analysis engine is a numerical program used to evaluate static and flutter performances of cablestayed bridges, and the optimization engine is a GA-based numerical program constructed to evolve initial designs to the final optimal designs. 2.1 Analysis engine The analysis engine is a verified MATLAB computer program for analyzing cable-stayed bridges (Cai 2012; Cai and Aref 2014). In this program, the single-beam model proposed by Wilson and Gravelle (1991) is used to numerically represent the actual cable-stayed bridges. Structural properties of the actual superstructure including mass, bending stiffness, torsional stiffness, warping stiffness, and mass moment of inertia are assigned to the numerical Table 1

Selection of materials

Steel for main girders (A992,Grade 50) Concrete for pylon and deck (normal weight concrete) Steel for cable (Xiong and Xiao 2010) CFRP for cable (Xiong and Xiao 2010) GFRP for deck (Kitane 2003)

E (GPa)

μ

G (GPa)

ρ (kg/m3)

E/ρ

G/ρ

Fy/f′c/Fu (MPa)

199.9 24.86 200 147 18.61

0.3 0.2 NA NA 0.13

76.90 10.36 NA NA 2.9

7849 2403 7850 1600 1565

0.025 0.010 0.025 0.092 0.012

0.010 0.004 NA NA 0.002

Fy =345 f′c =27.58 Fu =1890 Fu =2700 NA


equilibrium state. Nonlinear global stiffness K can be assembled by combining both the beam element stiffness and cable element stiffness and K is updated as the structure deforms. After the static analysis, axial stresses of main girders and maximum deflections of the superstructure and the pylon are evaluated as the measure of the static performance. More details about the algorithm for nonlinear static analysis and the procedure determining the static responses can be found in the work by Cai (2012). The natural vibration frequencies of the bridge structure are determined by solving the equations of motion of a free undamped system, which can be expressed by the :: equation of motion: ½M X g þ ½K fX g ¼ f0g. This equation can be manipulated to be in the form of (−ωn2[M]+ [K]) = {0} with the assumption of harmonic vibration. Stiffness matrix K is based on the deformed geometry after the static analysis. Eigenvalues of the above equation are the natural frequencies of the structure, and the corresponding eigenvectors represent the vibration mode shapes. Due to the interaction between the upcoming wind and bridge structure, self-excited aerodynamic forces are formed and introduced onto the structure. These forces are in the form of displacements or first derivatives of displacements, having influence on the structure’s stiffness and damping. A threedimensional self-excited vibration equation of motion can be expressed based on the dynamic equilibrium condition as fol :: : lows: ½M δ g þ ½C δg þ ½K fδg ¼ f F g, in which {F} is self-excited aerodynamic forces, having the form of : f F g ¼ ½ F s fδl g þ ½ F d δl g. Therefore, the equation of mo :: : tion can then be expressed as: ½M δ g þ ð½C −½ F d Þ δg þð½K −½ F s Þfδg ¼ f0g. More details about the aerodynamic damping matrix [Fd] and aerodynamic stiffness matrix [Fs] can be found in the work by Singh et al. (1995) and Ge and Xiang (2008). The equivalent “aerodynamic damping”, represented by matrix [Fd], decreases the structure’s mechanical damping, which means feeding energy to the system, and the system will be eventually driven to a divergent response, initiating the flutter instability. In this paper, a semi-analytical and semi-experimental approach is adopted to investigate the flutter behavior of cable-stayed bridges. In this approach, the flutter analysis is performed based on the classic self-excited force model, analytical flutter equations of motion, and experimentally obtained flutter derivatives. In order to solve the flutter equations of motion, the pK-F algorithm originally proposed by Namini et al. (1992) is implemented in the analysis engine. Through pK-F method, the critical wind velocity that initiates the flutter instability will be found, which will be the measure of the flutter performance. More details about the pK-F algorithm can be found in the work by Namini et al. (1992), Cheng (1999), and Cai (2012).

2.2 Optimization engine The optimization engine is a validated MATLAB (The MathWorks Inc 2010) computer program based on GA used to solve the multi-objective optimization problems. More details about this program refer to the work by Cai (2012). For multi-objective problems, objectives are generally conflicting or often preventing simultaneous optimization of each objective. A solution that is optimal with respect to one objective requires a compromise in other objectives. Thus, rather than providing a single point solution, we see a set of alternative trade-offs, commonly known as Pareto optimal solutions (Pareto 1896). In this paper, the non-dominated sorting genetic algorithm Π (NSGA-Π) proposed by Deb et al. (2002) is used. NSGA-Π uses a fast non-dominated sorting procedure, an elitist-preserving approach, and a parameterless niching operator, which alleviates inherent difficulties of other evolutionary algorithms (EAs) (Deb 2000). More details about this algorithm can be found in the work by Deb et al. (2002) and Cai (2012). The accuracy and efficiency of the optimization engine is validated through 2 multi-objective optimization examples in Cai’s study (2012).

3 Optimized hybrid CFRP-steel cable system The proposed optimized hybrid CFRP-steel cable system (OHCS) is developed to maximize the stiffness of each cable in the cable system. Since the stiffness plays a pivotal role for flutter performance, we assume the OHCS can also maximize the flutter performance. Cai (2012) and Cai and Aref (2015) did a more comprehensive optimization with GA and the results indicate the assumption is valid, especially for long-span cable-stayed bridges. A common stay cable usually consists of paralleled wires or strands. For the hybrid CFRP-steel cable proposed in this paper, the cable section is illustrated schematically as shown in Fig. 2. FRP wires are arrayed in the center and steel wires are arrayed in the outer layer of the cable. They are wrapped or glued together to have strain compatibility. Detailed composition and the durability issue of the hybrid cable system are beyond the scope of this paper. Following the rule of mixtures, which is often used for composite materials, the equivalent mechanical properties of a CFRP-steel hybrid cable can be calculated based on the mechanical properties of the constituents, i.e., CFRP and steel. When we replace original steel cables with hybrid CFRPsteel cables, two commonly used principles can be adopted to determine the equivalent area of the hybrid cable: the stiffness principle and the strength principle (Wang and Wu 2010b). Basically, the stiffness or strength principle is to keep the stiffness (EA) or strength (SA/SF) of the cable unchanged before and after the replacement, in which E is the elastic


performance ratio,” defined by the ratio of capacity (factored strength) to demand (cable tension), for both the original steel cable system and the hybrid CFRP-steel cable system are kept Sh

Ai

Ss

Ai

the same, namely S FThi h ¼ S FTsi s. h

Fig. 2 Typical hybrid CFRP-steel cable section

modulus, S is the strength, SF is the safety factor, and A is the area. SFCFRP and SFs are employed to be 3.0 and 2.5, respectively to account for the brittleness of carbon fibers (Kao et al. 2006; Zhang and Ying 2007). In this paper, we keep the strength of cables unchanged before and after replacement and optimize the volume ratio of CFRP in hybrid cables to maximize the cable stiffness, and hence the strength principle is adopted. In other words, the original steel cable system, consisting of n steel cables with each cable having area Ais and tension Tsi, is replaced by a hybrid cable system, consisting of n CFRP-steel cables with each cable having equivalent area Aih and tension Tih. We keep cable tensions the same, namely Tsi =Tih and the equivalent hybrid cable area is obtained by strength principle, Aih ¼

S s Ais S F h Sh S F s .

Consequently, “the strength

cable stiffness

s

In order to find the optimal volume ratio of CFRP in each cable of the OHCS, firstly, the stiffness of each cable is calculated with the CFRP volume ratio changing from 0 to 1 with an increment of 0.05. Then the relationship of each cable between the cable stiffness and the CFRP volume ratio can be constructed and the volume ratio of CFRP corresponding to the peak vertical stiffness is the optimal CFRP volume ratio. The three figures in Fig. 3 illustrate typical relationships between cable stiffness and CFRP volume ratio for different types of cables. These figures are not intended to compare stiffness between different types of cables. Details about deriving the cable stiffness and constructing the “CFRP volume ratio-cable stiffness” illustration can be found in the work by Cai (2012).

4 Optimized hybrid GFRP-concrete deck system The proposed optimized hybrid Glass FRP-concrete deck system (OHDS) is developed in this paper to maximize both static and flutter performance at the same time. The multiobjective GA-based optimization is conducted to determine the optimal GFRP and concrete distribution in the hybrid deck system. The formulation of the GA optimization is presented with the following topics. 4.1 Design variables

longer cable/smaller tension

0

cable stiffness

CFRP volume ratio

1

medium cable/medium tension

0

CFRP volume ratio

1

cable stiffness

shorter cable/bigger tension

0

CFRP volume ratio

Figure 4 shows the elevation view of a typical cable-stayed bridge and the schematic plan view of the hybrid deck system. The hybrid deck system is divided into nmax segments along the span; each segment has the length of corresponding cable spacing. For each segment, the cross section is optimized. As shown in Fig. 4, ni is defined as the number of basic FRP element (see Fig. 1c) for segment i, which is in the range of zero and n, where zero means no FRP deck while n is the maximum number of basic FRP element, which can be geometrically included in the cross section. Accordingly, n1,n2, …ni,…nmax, determining the 2-D distribution of FRP deck, are chosen as the design variables. Since FRP deck is constructed by multiplying the basic FRP element, these design variables have to be integers. 4.2 Optimization objective functions

1

Fig. 3 Typical relationships between the cable stiffness and the CFRP volume ratio

Two objective functions are defined. The first objective function is the measure of static performance, which is represented by the maximum axial stress of main girders under the dead

Author's personal copy H. Cai, A. J. Aref Fig. 4 Design variables of the hybrid deck optimization

load and initial cable tensions, σg. The second objective function is the measure of flutter performance, which is represented by the critical flutter velocity, Vflutter. The values of these two objective functions are evaluated by the analysis engine. Clearly, the first objective function is seeking a minimum value, while the second objective function is seeking for a maximum value. The optimization problem can be defined with the following equation: Min σg ðn1 ; n2 ; …ni ; …nmax Þ Max V flutter ðn1 ; n2 ; …ni ; …nmax Þ subject to n1 ; n2 ; …ni ; …nmax ∈½0; nwhere n is an integer ð1Þ

4.3 Selections of major GA operators The selection of major GA operators used in this paper is summarized as follows. More details can be found in the work by Cai (2012). Firstly, regarding the population type, floating point (real number) is adopted; secondly, with respect to the population size, N, we choose N≥20×m, where m is the number of design variables; thirdly, with regard to crossover and mutation operators, the heuristic crossover operator and adaptive feasible mutation operator will be used; fourthly, since the most efficient heuristic rate and crossover rate are problemdependent, parametric tests are conducted to determine them (Cai 2012); lastly, when it comes to the stopping criteria, we set the maximum generations size at 200, which can be always increased for better convergence in the future work, if computational cost is not a big concern.

Mutation and crossover operators are pivotal for an effective and efficient searching. Among commonly used crossover operators, heuristic crossover not only inherits characteristics of both parents but also introduces random new genes based on parents’ performances. Therefore, among these crossover operators, the heuristic crossover will obtain the greatest possible favorable information exchange from parents and will be adopted as the crossover operator in this paper. More details about heuristic crossover operator can be found in the reference. The adaptive feasible mutation is a patented algorithm invented by Kumar (2010) and built in MATLAB (The MathWorks Inc 2011; Cai 2012). It takes into account linear and bound constraints to generate new mutated individuals. In the work by Cai (2012), three mutation operators, namely uniform mutation operator, non-uniform mutation operator, and adaptive feasible mutation operator, are compared and it is shown that the optimization with the adaptive feasible operator will find the global optimum effectively and efficiently. Therefore, the adaptive feasible operator will be used in this paper. 4.4 Procedure to obtain the Optimized Hybrid Deck System (OHDS) The procedure to obtain the OHDS couples the analysis engine and the optimization engine for performance evaluations and GA-based searching. The GA-based optimization procedure is presented in Fig. 5. Through this procedure, the original conventional composite cable-stayed bridge, which consists of steel main girders, concrete deck system, and steel cable system, can be upgraded by


Fig. 5 The GA-based optimization procedure

replacing the conventional deck and cable system with the proposed OHDS and corresponding Optimized Hybrid Cable System (OHCS). The upgraded composite cablestayed bridge has the maximized static and flutter performances due to the implementation of the OHDS and OHCS. To initiate the GA-based optimization, random initial population is firstly generated. Each bridge in the initial population has a different hybrid deck system. The

hybrid cable optimization is then conducted to generate the corresponding OHCS for each bridge. Then each bridge is evaluated by the analysis engine to achieve the static and flutter performances. Then the stopping criterion is checked. If not satisfied, new generation is generated by the optimization engine based on the obtained performances of each bridge in current generation. This process continues until the stopping criterion is satisfied.


5 Numerical example Yangpu Bridge is investigated in this paper as a benchmark example to demonstrate the procedure to obtain the OHDS and OHCD and investigate their merits. Yangpu Bridge is a traditional steel-girder-concrete-deck composite cable-stayed bridge with a main span length of 602 m located in Shanghai, China. The main span length of Yangpu Bridge is only 3 m smaller than the longest composite cable-stayed bridge in the world. The geometry and dimensions of Yangpu Bridge can be found in the work by Cai (2012). The selection of GA operators is listed in Table 2. 5.1 Numerical results Figure 6 shows the Pareto optimal front (Pareto 1896) history and we can see as the generation increases, the Pareto optimal front becomes fitter and fitter and the difference between two adjacent Pareto optimal fronts becomes smaller and smaller. Figure 7 shows the final Pareto optimal solutions at generation 200. The five optimal designs in the front are shown with more details in Fig. 8. In Fig. 8, the color bar legend is used for the OHCS, which is also presented in Fig. 9 for each optimal design. In Fig. 8, the color bar legend is used for the hybrid cable system and values from 0 to 1 stand for the CFRP volume ratio for each cable, which match along with color from blue to red. For example, “0” stands for all-steel cable, “1” means all-CFRP cable, “0.4” represents 40 % CFRP and 60 % steel, and other values have similar meanings. These conventions will apply to all similar figures in this paper. In Fig. 8, the 2D plots demonstrate the optimal architectures of the hybrid deck system, in which the red zones stand for GFRP decks and the white zones represent concrete deck as defined in Fig. 4. In Fig. 9, the x-axis is the cable index, numbered from 1 and starting at the most left and progressing to the right. Table 3 shows the performance indices of the above five optimal designs and the comparison with two extreme

Table 2

Major GA parameters

Item

Value

Population size Crossover Crossover rate Mutation Total generation Stopping criteria

N=1344 Heuristic/R=1.4 c=0.8 Adaptive feasible 200 Total generation criterion

cases, namely an all-concrete deck system with the corresponding OHCS and an all-GFRP deck system with the side corresponding OHCS. In this table: R ¼ WWhmain ¼ ratio of side span weight to half the main span weight; N GFRP ¼ i¼nmax

∑ nif rp l i ¼ index representing the total quantity of the

i¼1

GFRP used in the design, where nifrp = number of basic FRP element used in segment i; li = length of segment i; σg = maximum axial stress of main girder; Dd = vertical displacement at center of main span; Dp = horizontal dis . S icable placement at pylon tip; σc ¼ max , maxiðS ih =S F ih Þ mum specific axial stress of cables, in which Sicable = axial stress of ith hybrid cable; Sih = strength of ith hybrid cable; SFih = safety factor of ith hybrid cable; ωtor = first torsional frequency; ω ver = first vertical bending frequency. As shown in Table 3, all these obtained Pareto optimal designs have much better static and flutter performances than the original “all-concrete” case and “all-GFRP” case. The improvements of flutter performances of these Pareto optimal designs can result from the larger ω tor and ω ver, even tor though ωωver is smaller than the original “all-concrete” case. However, for the “all-GFRP” case, even if it has larger or tor similar ωtor,ωver and ωωver in comparison with the Pareto optimal designs, the critical flutter velocity is still smaller. This phenomenon can be explained by the fact that the “all-GFRP” case has a much smaller mass, whereby the critical flutter velocity is reduced. 5.2 Conclusions and discussions of the numerical example Based on the results of the optimization of the presented example, we can draw some important observations and conclusions as follows: 1) All the Pareto optimal designs are hybrid deck system. None of them are all-FRP or all-concrete deck designs. 2) Multi designs are found in the Pareto optimal front, which proves that the two objectives cannot reach their optimum at the same time. 3) All the designs in the Pareto optimal front are equally optimal, providing a trade-off between static and flutter performances. Higher-level information is needed to choose one of the obtained solutions as the final solution, if necessary. 4) The obtained optimal distributions reconciled two conflicting objectives and can be only achieved through well-defined optimizations. 5) How to use the FRP most effectively to minimize σg can be concluded: the distribution of FRP and concrete

Author's personal copy A genetic algorithm-based multi-objective optimization for hybrid fiber reinforced polymeric deck and cable... Fig. 6 Searching history of Pareto optimal front solutions

103 25th generation 50th generation 75th generation 100th generation 125th generation 150th generation 175th generation 200th generation

101 99 97 95 93 91 89 65

70

in the hybrid deck system plays an important role. Generally speaking, the distributions that make R, namely the ratio of side span weights to half main span weights, close to one are generally favorable for smaller σg. To make R close to one, more FRP deck tends to be used in the main span than the side span. Moreover, as shown in Fig. 8, we can see less FRP deck tendsX to be distributed around the region near pylons in order to have a smaller σ g . This distribution characteristic makes the region where the maximum axial stress of the main girder is most likely to occur have more concrete and larger section accordingly. 6) How to use the FRP most effectively to maximize Vflutter can be also concluded: (1) generally speaking, there is no single factor that dominates the influence of Vflutter; rather,

Fig. 7 Pareto optimal front

75

80

85

90

95

100

105

all the potential influencing factors need to work together in an optimal way to maximize Vflutter; (2) the nonuniform distribution of FRP in the hybrid decks system results in a varied mass and mass moment of inertia of different segments along the span. The distribution pattern of the mass and mass moment of inertia along the span can also influence Vflutter. 7) The maximum possible potential of FRP materials for a typical long-span cable-stayed bridge has also been demonstrated from these results. The GFRP hybrid deck and CFRP hybrid cable together can possibly improve the static and flutter performances by 37 % (design 1) and 12 % (design 5), respectively, while for design 5, both performances can be improved by 33 and 12 % at the same time.

Author's personal copy H. Cai, A. J. Aref Fig. 8 Pareto optimal designs of scenario (iii) (not to scale): a 3D view of design 1, b 2D view of design 1, c 3D view of design 2, d 2D view of design 2, e 3D view of design 3, f 2D view of design 3, g 3D view of design 4, h 2D view of design 4, i 3D view of design 5, j 2D view of design 5

Author's personal copy A genetic algorithm-based multi-objective optimization for hybrid fiber reinforced polymeric deck and cable... 1.2

Fig. 9 The corresponding OHCS of Pareto optimal designs

Design 1 Design 2 Design 3 Design 4 Design 5

1 0.8 0.6 0.4 0.2 0 0

10

20

30

40

50

60

70

0.2

6 Summary

A typical existing long-span cable-stayed bridge is modified as a benchmark example by replacing traditional concrete deck system and steel cable system with the OHDS and OHCS through the proposed optimization procedures. Through the implementation of these two hybrid systems, the static and flutter performances are improved by 33 and 12 % at the same time, which promisingly indicates a 12 % span length extension. Moreover, the obtained optimal distribution patterns of FRP and concrete in the OHDS provide guidelines and insights for future hybrid cable-stayed bridges design. Based on these optimal distribution patterns, a more construction-friendly, regularly (smoothed) shaped, and economical distribution pattern can be obtained in future study, which can also improve both static and flutter performances. Another issue that is not covered in this paper is the fatigue of steel cables. For clear assessment of any degradation due to fatigue or other factors, thorough experimental study must be conducted and that has not been the scope of this research.

This paper presented a procedure that aimed at using FRP materials in the deck and cable systems most effectively through GA-based optimizations. The optimization is conducted with the aid of a built-in and coupled analysis- and optimization-engine. The accuracy and effectiveness of these two engines are validated in previous research work by Cai (2012). Through the optimization, the optimized hybrid GFRP-concrete deck system (OHDS) and corresponding optimized hybrid CFRP-steel cable system (OHCS) are developed, which can maximize both static and flutter performances of cable-stayed bridges concurrently and therefore potentially extend the span length. In the OHCS, each cable has the maximized stiffness with the optimal volume ratio of CFRP and steel; in the OHDS, the optimal 2D distribution of FRP and concrete is found considering both materials’ advantages and two conflicting objectives.

Table 3

Performances and characteristics of the optimal designs

Designs

Design 1 Design 2 Design 3 Design 4 Design 5 All- concrete All- GFRP

Geometry indices

Static performance indices

Flutter performance indices

Relevant modal properties

R

NGFRP

σg (MPa)

Dd (m)

Dp (m)

σc

Vflutter (m/s)

ωflutter (rad/s)

ωtor (rad/s)

ωver (rad/s)

0.919 0.910 0.899 0.892 0.901 0.807 0.807

2158 2144 2161 2156 2004 0 3843

68.38 68.47 68.72 69.01 72.7 109.27 102.83

−0.448 −0.444 −0.443 −0.443 −0.445 −0.607 −0.515

0.204 0.202 0.202 0.201 0.203 0.291 0.251

0.508 0.506 0.506 0.507 0.510 0.972 0.471

98 99 100 101 102 91 92

3.791 3.787 3.782 3.777 3.759 3.388 3.858

3.876 3.868 3.862 3.855 3.833 3.461 3.973

2.045 2.046 2.047 2.046 2.037 1.696 2.043

ωtor ωver

1.8956 1.8905 1.8871 1.8842 1.8812 2.0410 1.9446


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