1. Introduction
Past destructive earthquakes (e.g. the 1994 Northridge earthquake and the 1995 Kobe earthquake) have left a clear signature on the engineering community worldwide, changing thinking of structural engineers [12]. As such, after holding several workshops and conferences, an innovative approach namely PerformanceBased Design (PBD) was presented by modern guidelines [35]. In principle, a structure designed using PBD approach should meet performance objectives in accordance with a set of specified reliabilities over its service life. This is aimed to reach structural design candidates associated with more predictable seismic behavior, quantifying and controlling the risk at an engineered acceptable level.
Both seismic demands and capacity parameters, that are inherently uncertain, are highly influential on the acceptable performance level of a structure. Furthermore, due to the fact that a structure on underlying soil is not rigid, soilstructure interaction (SSI) affects the responses of structures during an earthquake. Obviously, ignoring the SSI effects could lead to unrealistic structural responses and seismic demands. Hence, the effects of SSI should be considered in the seismic responses of structures [6]. Therefore, soil type, material properties of the structure, and ground motion characteristics randomly affect the seismic structural responses.
Deterministic structural optimization without considering the uncertainties in design manufacturing and operating processes may lead to unreliable design resulting in inappropriate balance between cost and safety. A proper design procedure must reasonably account for the inherent uncertain nature of a structural system and associated external load [7]. In structural optimization, nondeterministic performance of structures can be taken into account using robust design optimization (RDO) [8] and reliabilitybased design optimization (RBDO) [9]. RDO aims to minimize variation of the objective function, but RBDO optimizes the structural cost under reliability of the constraints.
A few studies have been implemented in a structural optimization problem, where RBDO is incorporated into PBD concept. Foley
Nonlinear dynamic analysis of structures using finite element method requires much computational effort. This drawback may accentuate when the nonlinear dynamic structural responses are required in RBDO of the structure using the MonteCarlo Simulation (MCS) method, the importance sampling technique and the response surface method. In order to obtain an acceptable confidence within probabilities of the order close to 10^{4}  10^{6}, the MCS method requires a large number of structural analyses. Based on Lagaros
In this chapter, RBDO of RC structures with considering SSI effects under timehistory earthquake loading is presented in accordance with the PBD concept of SEAOC guidelines [3]. In this work, a new discrete gravitational search algorithm (DGSA) and an efficient proposed metamodel were introduced for performing RBDO of RC structures [22]. The objective function is the total cost of the structure while the constraints are treated as deterministic and probabilistic. The annual probability of nonperformance for each performance level is considered as the probabilistic constraint in RBDO procedure. The new DGSA based on the fundamental concept of the standard GSA [23] is introduced for finding the optimal designs in the RBDO procedure. In DGSA, the position of each agent is presented in positive integer numbers. Also, the velocity of each agent is modified based on the particle swarm optimizer with passive congregation (PSOPC) which was proposed by He
Numerical examples show that the wavelet as a kernel function is much better than those of the common kinds as kernel function in WLSSVM. The accuracy and generalization of WWLSSVM is improved using GSA. Furthermore, numerical results demonstrate the efficiency and computational advantages of the proposed DGSA for RBDO of structures.
2. RBDO of RC structures
2.1. Formulation of optimization
Seismic design optimization of RC structures under timehistory earthquake loads is an ongoing research topic and has received great attention among researchers [14, 2733]. As such, RBDO of RC structures with the consideration of SSI effects was investigated in accordance with PBD concept of SEAOC guidelines [3] under seismic loading. This work incorporates the acceptable performance levels and the RBDO theory to compare the achieved annual probability of nonperformance with target values for each performance level. The objective of the RBDO problem is to minimize the total cost whereas the deterministic and probabilistic constraints should not exceed a specified target.
The RBDO problem of RC structures can be formulated in the following form:
where
2.2. Life cycle cost assessment of RC structure
The total cost,
where
where
The repair cost refers to the cost of damage level from earthquake that may occur during the life of a structure. In this study, the overall damage index,
where
where
2.3. Constraint handling approach
A comprehensive overview of the most popular constraint handling approaches used in conjunction with metaheuristic optimization methods was presented in the literature review by Coello Coello [35]. In the present study, the external penalty function method as one of the most common forms of the penalty function in the structural optimization [15, 2729, 3639] is employed to transform constrained RBDO problem into unconstrained one as follows:
where
This formulation allows solutions with violated constraints, and the objective function is always greater than the nonviolated one.
3. Reliability assessment of RC structure
In order to evaluate the system reliability index corresponding to each of the performance levels, RC structures should be assessed in the RBDO procedure [14, 22]. The system reliability index corresponding to each of the performance levels are estimated by MCS method. In the following subsections, the procedure of assessment of RC structures is explained.
3.1. Required database
In PBD approach, many uncertain variables influence the structural seismic responses. In the studies by Khatibinia
3.2. Limit state functions
The operational, life safety and collapse prevention levels have been defined as the performance levels. A performance level depends on some limit state functions. A limit state function,
where
The limit state functions and their probability distribution function (PDF) for the performance levels, according to SEAOC guidelines (2000), are shown in Table 1.




Operational  
Elastic roof displacement  
Lognormal  
Interstorey drift  
Lognormal  
Life safety  
Interstorey drift  
Lognormal  
Max. local damage index  
Beta  
Global damage index  
Beta  
Collapse  
Interstorey drift  
Lognormal  
Max. local damage index  
Beta  
Global damage index  
Beta 
In Table 1,
3.3. Annual probability of nonperformance
The nonperformance probability,
where
For each performance level, the total exceeding probability,
where
4. Seismic responses and SSI system
4.1. Seismic responses of SSI system
There are two main approaches for modeling and analyzing SSI systems, namely the direct method and the substructure method either in time domain or in frequency domain [6]. Considering the discretized dynamic equations of structure and soil simultaneously, the direct method models the soil and structure together, and the responses of soil and structure are determined simultaneously by analyzing SSI system in each time step [22].
In the direct method, the discretization of nonlinear dynamic equations can be expressed in FEM framework as:
where
Over the past two decades, the use of damage and energy concepts for the seismic performance evaluation and design of structures has attracted considerable attention among the researchers [30, 4447]. These concepts can be simultaneously used through a combined damage index namely ParkAng damage index. The index is taken into account as a combined index, defined as the linear combination of the maximum displacement and the hysteretic energy dissipation for a structural element. For this reason, the damage index [44] is one of the indices that have widely been used for damage assessment and damagebased design of RC structures [14, 22, 2830, 4546, 48]. As shown in Table 1, some limit states of the performance levels depend on the damage indices.
An improved version of the index namely modified ParkAng damage index [43] is defined based on the crosssection deformation of structural elements as:
where
where
4.2. Finite element model of SSI system
OpenSEES [49], as an opensource computational software framework, is used for by simulation of SSI system, and performing nonlinear dynamic analyses of SSI system depicted in Fig. 1. Assuming materials of constant properties over its depth, soil encompasses different layers, and the foundation is considered as rigid strip footing. Beams and columns of structure are modeled using forcebased nonlinear beamcolumn element with considering the spread plasticity along the element’s length. The integration along each element is based on GaussLobatto quadrature rule. Also, the infinite boundaries of soil are modeled using the artificial boundaries (Fig. 1). The model of soilstructure system shown in Fig. 1 was successfully used by [14, 22, 31].
The KentScottPark model [50] is utilized for modeling the confined and unconfined concrete of crosssections of structural elements. The constitutive parameters of this model are:
where
In this study, the onedimensional
Soil layers are modeled using isoperimetric fournode quadrilateral finite elements and assuming bilinear displacement interpolation. The plane strain condition is assumed for the soil domain with considering a constant soil thickness corresponding to the interframe distance. The material of the soil is modeled using a modified pressureindependent multiyieldsurface
One of the major problems in SSI system for infinite media has been the modeling of the domain boundaries. Infinite boundaries have to absorb all outgoing waves and reflect no waves back into the computational domain. In this study, the standard viscous boundary proposed by Lysmer and Kuhlemeyer [54] is used for this purpose. This boundary can be described by two series of dashpots oriented normal and tangential to the boundary of a finite element mesh (Fig. 1) as follows:
where
The material damping matrix,
5. Artificial earthquakes
For RBDO of structures, it is then necessary to utilize accelerograms of compatible characteristics with a desired site. It is often difficult or impossible in some cases to choose a proper record for a site, since historically recorded accelerograms for a given site could be limited or scare. Hence, artificial earthquakes that are statistically influenced by desired properties of the given site are very useful for seismic design of structures. In this work, spectral representation method based on time domain procedure is used for the generation of synthetic ground motion records. The nonstationary ground motion is simulated using this method as [13]:
where
where
where





5.12  10.24  20.48 

0.50  1.50  2.00 

4.00  8.00  16.00 

2.0  1.0  0.7 

2.0  2.0  2.0 
NFR  100  200  300 

12  15  15 

0.40  0.40  0.40 
The PGA values are obtained corresponding to hazard curves and produced for a specific region. As shown in Table 3, in this work the hazard curves presented by Möller et al. [13] are used. An artificial earthquake generated based on Eq. (19) is shown in Fig. 4.



50% in 50 years  73  0.27 
10% in 50 years  475  0.6 
5% in 50 years  975  0.8 
6. The new discrete gravitational search algorithm
Based on the work presented by Khatibinia
6.1. Gravitational search algorithm
GSA was introduced by Rashedi
where
To compute the acceleration of an agent, total forces from a set of heavier masses applied to it should be considered based on the law of gravity (Eq. (23)). Afterwards, the next velocity of an agent is calculated as a fraction of its current velocity added to its acceleration (Eq. (24)). Then, its next position could be calculated using Equation (25):
where
6.2. The proposed discrete GSA
The binary GSA (BGSA) for solving discrete problem was developed by Rashedi
Based on Eq. (26), a large computer memory is needed for the position of agents in BGSA. Also, coding and encoding of the position of agents is a time consuming process. In order to overcome the shortcomings of BGSA, a new DGSA based on the fundamental concept of the standard GSA with passive congregation is presented herein. The passive congregation strategy as perturbations operator can transfer information among agents in the optimization procedure [24]. Therefore, the search performance of DGSA can be improved using the passive congregation. To achieve this purpose, Khatibinia
where
In DGSA, the scalar
Therefore, the coding and encoding of the position of agents are omitted; and the position of agents is calculated as the integer value. The current position of agents may violate from the values of the set
where
7. Approximating the structural seismic responses
MCS requires excessive computational cost for RBDO of structures in order to obtain an acceptable accuracy [11]. Because of the drawback, Khatibinia
7.1. Weighted least squares support vector machines
WLSSVM was introduced as excellent machine learning algorithms in largescale problems by Suykens
WLSSVM is described as the following optimization problem in primal weight space [26]:
Subject to the following equality constraints:
where
It is impossible to indirectly compute
Based on the KarushKuhnTucker (KKT) conditions, by eliminating
where
According to Mercer’s condition, a kernel
Consequently, the final WLSSVM model for the prediction of functions becomes:
Weight
where
In WLSSVM, Gaussian radial basis function (RBF) is frequently used as the kernel function, and it is expressed as:
where
Based upon the Suykens
7.2. The new metamodelbased wavelet kernel
Wavelets as kernel function have been introduced and developed in ANNs and SVMs [6063]. It has been shown that wavelet kernel functions are superior to other kernel functions in the training ANN and SVM. Accordingly, the kernel function of WLSSVM is substituted with a specific kind of wavelet functions proposed by Khatibinia et al. [22]. The metamodel based on wavelet kernel function is called WWLSSVM. The cosineGaussian Morlet wavelet is used as the kernel function of WLSSVM. The wavelet function is mathematically written as follows:
where
According to Zhang
where
Therefore, according to Eqs. (40) and (41), the wavelet kernel function of the cosineGaussian Morlet wavelet is given as follows:
The accuracy of WWLSSVM prediction depends on the good selection of its parameters. Selecting appropriate values of these parameters is important for obtaining the excellent predicting performance. Hence, in this study, GSA is used to find the WWLSSVM optimal parameter,
where
The WWLSSVM training stage during GSA is performed according to the
The converged solution is affected by the setting value of parameters in GSA. In this study, the values are selected based on the general recommendations by Rashedi
8. Predicting failure probability of structures
In the RBDO procedure, nonlinear timehistory analysis of SSI system is used and it may be failed regarding a number of random structures [65]. In fact, a number of structures collapse and then lose their stability. Hence, these structures should be identified and eliminated from optimization process. For this purpose, a failure probability is considered as stability criterion. An efficient method is presented to train the failure probability with high performance [65]. This efficient method is consisted of a modified adaptive neuro fuzzy inference system (ANFIS) with a hybrid of fuzzy cmeans (FCM) [66] and fuzzy particle swarm optimization (FPSO) [67]. To train the modified ANFIS, the input–output data are classified by a hybrid algorithm consisting of FCMFPSO clustering. The optimum number of ANFIS fuzzy rules is determined by subtractive algorithm (SA).
8.1. Hybrid of FCM and FPSO for clustering
The FCM algorithm has been extensively studied and is known to converge to a local optimum in nonlinear problems. Moreover, the FPSO algorithm is robust method to increase the probability of achieving the global optimum in comparison with the FCM algorithm. The FCM algorithm is faster than the FPSO algorithm because it requires fewer function evaluations. This shortcoming of FPSO can be dealt with selecting an adequate initial swarm [65].
In this study, a hybrid clustering algorithm called FCMFPSO is presented to use the merits of both FCM and FPSO algorithms and increase the procedure of convergence. In this way, the FCM algorithm finds an adequate initial swarm FPSO algorithm for commencing the FPSO. For this purpose, first, the FCM algorithm is utilized to find a preliminary optimization that shown by
8.2. Modified ANFIS
An ANFIS model depends on the number of ANFIS fuzzy rules and membership functions. In other words, creating an ANFIS model with a minimum number of fuzzy rules can eliminate a wellknown drawback. Therefore, for overcoming of this drawback, Khatibinia
9. Numerical examples
In this work, two RC frame structures shown in Fig. 8 are selected as illustrative examples. Three layers of sand associated with material properties varying over its depth are considered as the soil under the frames. The depth of each soil layer and the entire width of soil domain are considered to be 10 m and 100 m, respectively. The soil is also assumed to have plane strain condition with a constant thickness of 5.0 m in proportion to the interframe distance. Inertia properties of the soil mesh are considered using lumped mass matrices modeling with soil mass density of 17 kN/m^{3} for all soil layers. The values of the dead and live loads are considered to be
For vertical continuity on the dimensions along the height of a column, the section database of columns is divided into three types in the height of RC frame. Hence, a database shown in Table 4 is generated. Similarly, the section database of beams is divided into three types in the height of RC frame. Distribution of beam dimensions along the height of the frame is shown in Table 4. The diameter of longitudinal bars for beams and columns is laid between 12 mm and 32 mm in the databases.
The initial cost is calculated for
The concrete material parameters shown in Table 5 are considered for the cover of column crosssections. The strain corresponding to the peak strength,
where











1  65 
55 
45 
55 
50 
45 
2  60 
50 
40 
55 
50 
45 
3  55 
50 
40 
55 
50 
45 
4  55 
45 
40 
55 
50 
45 
5  55 
45 
35 
50 
45 
40 
6  50 
45 
40 
50 
45 
40 
7  55 
45 
35 
50 
45 
40 
8        50 
45 
35 
The presented DGSA requires the user to specify several internal parameters that can affect convergence behavior at the search space. It is found that a population of 50 agents can be adequate. Higher values are not recommended, as this will increase significantly computation time in RBDO. In addition, different optimization runs are carried out for RBDO model in this study, so optimum designs are found by DGSA about 150 iterations. Due to the effect of decreasing gravity, the actual value of the gravitational constant,
9.1. Example 1: Sixstorey RC frame
Sixstorey RC frame is shown in Fig. 8(a). In the frame, the length of each bay and the height of stories are 5 m and 3 m, respectively. The members of the structure are divided into four groups for the columns C_{1}, C_{2}, C_{3}, C_{4} and four groups B_{1}, B_{2}, B_{3} and B_{4} for the beams. The groups of structural elements are presented in Fig. 8(a).





Concrete  

Lognormal  28  2  

Normal  0.0035  0.00035  
Steel  

Lognormal  340  25  

Lognormal  210000  8000  

Normal  0.015  0.0015  
Soil  

Normal  375  10  

Normal  37.5  1.0  

Normal  300  20  

Normal  37.5  1.0  

Normal  200  10  

Normal  32.5  1.0 





Lognormal  300  100 

Normal    0.15 

Normal  2.5  0.375 
9.1.1. Training and testing the metamodel
To predict the mean,
In order to validate the performance and accuracy of the proposed metamodel, relative rootmeansquared error, i.e.












375.73  2.892  4.274  284.62  1.647  4.048 

386.04  3.804  6.347  314.42  2.104  4.702 

400.48  3.615  5.895  373.67  2.548  3.692 

365.37  4.052  6.329  308.38  1.947  4.082 
The smaller














WWLSSVM  

2.008  2.417  2.337  2.172  2.163  2.075  2.532  3.406  

0.018  0.022  0.020  0.021  0.023  0.024  0.045  0.028  

0.9999  0.9998  0.9998  0.9999  0.9999  0.9998  0.9998  0.9997  
WLSSVM  

5.328  6.302  5.392  5.862  5.386  6.017  6.007  5.737  

0.057  0.074  0.059  0.061  0.079  0.094  0.091  0.064  

0.9988  0.9979  0.9987  0.9987  0.9985  0.9984  0.9988  0.9988 
As given in Table 8, the proposed metamodel trained for the mean and the standard deviation of seismic responses has proper performance generality. Thus, the approximating performance of the metamodel based on WWLSSVM and GSA is better than the WLSSVM with RBF kernel in predictive ability and precision.
9.1.2. Results of RBDO
In this example, RBDO of the RC frame is performed using DGSA associated with WWLSSVMbased MCS. In the reliability process, the reliability indices,
As shown in Table 9, the optimal solutions of DGSA are better than those of BGSA in terms of the total cost and the number of iterations. The minimum reliability index,
The convergence histories of the optimum objective function are shown in Fig. 9 for DGSA and BGSA models. As can be seen in Fig. 9, DGSA method is more efficient than BGSA method. Optimum designs are found by DGSA and BGSA in 4450 and 5900 required approximate analyses by the metamodel, respectively.









C_{1}  55 
2.67  55 
3.23 
C_{2}  55 
2.64  55 
3.20 
C_{3}  45 
2.56  45 
2.60 
C_{4}  45 
2.33  45 
2.48 
B_{1}  50 
1.98  50 
2.38 
B_{2}  50 
2.13  50 
2.30 
B_{3}  45 
1.81  45 
1.88 
B_{4}  45 
1.69  45 
1.78 

3448  3542  

1080  1045  

4528  4587  
Iterations  89  118  

1.4435  1.6901  

2.4885  2.7342  

2.7986  3.1384 
9.2. Example 2: Ninestorey RC frame
Ninestorey RC frame is shown in Fig. 8(b). In the frame, the length of each bay and the height of stories are 5 m and 3 m, respectively. The members of the structure are divided into six groups for the columns and six groups for the beams. The groups of structural elements are presented in Fig. 8(b).
9.2.1. Training and testing the metamodel
After training database using the presented WWLSSVM optimal parameters of the metamodel associated with the mean and the standard deviation of seismic responses are shown in Table 10. Furthermore, the performance generality of the proposed metamodel and WLSSVM is given in Table 10 in terms of














WWLSSVM  

2.837  3.028  3.127  2.689  3.024  2.682  3.008  3.105  

0.026  0.038  0.035  0.023  0.032  0.024  0.031  0.036  

0.9999  0.9998  0.9998  0.9999  0.9998  0.9999  0.9998  0.9998  
WLSSVM  

5.538  6.346  6.483  6.006  6.305  5.396  5.843  5.579  

0.094  0.138  0.162  0.105  0.1057  0.0987  0.0998  0.0924  

0.9987  0.9985  0.9981  0.9988  0.9987  0.9988  0.9988  0.9988 
The results of Table 10 demonstrate that the metamodel is better than the WLSSVM method in terms of performance generality. Therefore, the metamodel is reliably employed to predict the necessary responses during the RBDO process.
9.2.2. Results of RBDO
As the first example, in this example RBDO of the RC frame is performed using DGSA and BGSA associated with WWLSSVMbased MCS. In this example, the crosssection of beams and columns are selected from Types 1, 2 and 3, which are shown in Table 4. The best optimum designs of the RC frame are listed in Table 11.
As revealed in Table 11, the optimal solutions of DGSA are better than those of BGSA in terms of the total cost and the number of iterations. The minimum reliability index,
The convergence histories of the optimum objective function are shown in Fig. 10 for DGSA and BGSA models. As can be seen in Fig. 10, DGSA method is more efficient than BGSA method. Optimum designs are found by DGSA and BGSA in 4150 and 6150 required approximate analyses by the metamodel, respectively.









C_{1}  65 
2.49  65 
2.90 
C_{2}  60 
2.50  60 
3.08 
C_{3}  55 
2.53  55 
2.87 
C_{4}  50 
2.49  50 
2.79 
C_{5}  45 
2.27  45 
2.68 
C_{6}  40 
2.29  40 
2.89 
B_{1}  55 
1.88  55 
2.18 
B_{2}  55 
1.80  55 
1.98 
B_{3}  50 
1.82  50 
2.01 
B_{4}  50 
1.87  50 
1.98 
B_{5}  45 
1.68  45 
1.79 
B_{6}  45 
1.66  45 
1.85 

5571  5736  

1004  987  

6575  6723  
Iterations  83  123  

1.4985  1.7101  

2.4953  2.7996  

2.8514  3.1837 
10. Conclusions
In general, the optimum design of structures depends on a number of parameters that are inherently uncertain. Reliabilitybased design optimization (RBDO) has been employed as the only method that assesses the influence of uncertain parameters and balance both cost and safety of structures. To account for all necessary uncertain and random parameters in RBDO of RC structures and to achieve the realistic optimum design of RC structures, the uncertain material properties of soil and structure, as well as the characteristics of ground motions should be considered as random parameters. Furthermore, the realistic seismic responses of RC structures can be account by consideration of soilstructure interaction (SSI) effects. In this work, a new discrete gravitational search algorithm (DGSA) and a new metamodeling framework were incorporated for RBDO of RC structures with PerformanceBased Design (PBD) under seismic loading. The objective of the RBDO problem was to minimize the total cost whereas the deterministic constraints and the system reliability index corresponding to each of the performance levels should not exceed a specified target. Based on this study, the following conclusions can be drawn:
To reduce the computational effort and computational cost of the MonteCarlo Simulation (MCS) method, a new metamodel based on a wavelet weighted least squares support vector machine (WWLSSVM) and gravitational search algorithm (GSA) was utilized in the RBDO procedure. Therefore, the proposed metamodel, as a substitute for the nonlinear dynamic analysis of SSI system, can estimate the reliability index through MCS with a small computational cost.
The WWLSSVM and kernel parameters were simultaneously optimized in the proposed metamodel in order to improve performance generality of WWLSSVM. Numerical results of training and testing the metamodel indicated that performance generality of the metamodel was higher in comparison to WLSSVM. Hence, the proposed metamodel can predict the nonlinear dynamic analysis of SSI system in terms of accuracy and flexibility.
The proposed DGSA was presented based on the standard GSA with passive congregation. The passive congregation strategy can be considered as perturbations operator in the optimization procedure. Therefore, the presented DGSA using the passive congregation can transfer information among agents avoiding local minima. Furthermore, the coding and encoding of the position of agents as a time consuming process is omitted in DGSA. To eliminate this drawback, the position of agents was calculated as the integer value. The optimum designs obtained by DGSA were compared with those produced by BGSA model. Numerical examples showed that the proposed DGSA can converge and reach the optimum design more quickly than the BGSA model.
Future extension of current research could include reducing the computations involved in the PBD by replacing MCS with the response surface method or the importance sampling technique. The constraints imposed on the objective function could be also treated as random quantities (see [68]).
Acknowledgments
Our special thanks go to Dr. Eysa Salajegheh (Distinguished Professor of Structural Engineering) in Department of Civil Engineering at Shahid Bahonar University of Kerman, Iran, for his cooperation in this research work.
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