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Eng February, Eng May, To avoid this problem, the current position of particles is limited as:. Because of the drawback, Khatibinia et al.
Using the optimization problem, Eq. It is impossible to indirectly compute w from Eq. Therefore, the dual problem shown in Eq. Based upon the Suykens et al. Step 1. Step 2. Step 3. Step 4. The wavelet function is mathematically written as follows:. According to Zhang et al. Therefore, according to Eqs. Selecting appropriate values of these parameters is important for obtaining the excellent predicting performance.
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 et al. In the RBDO procedure, nonlinear time-history 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 ]. 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. This shortcoming of FPSO can be dealt with selecting an adequate initial swarm [ 65 ].
Other particles of the initial swarm, i. Then, the FPSO algorithm is used using this initial swarm.
In other words, creating an ANFIS model with a minimum number of fuzzy rules can eliminate a well-known drawback. Therefore, for overcoming of this drawback, Khatibinia et al. The algorithm flow of the proposed model is shown in Fig. The proposed method is executed in the following steps [ 65 ]:. This parameters is used for optimizing the center of rules and membership functions for the input and output data.
The FIS uses Gaussian function and linear function for membership function of input and output, respectively. In this work, two RC frame structures shown in Fig. 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 m, respectively. The soil is also assumed to have plane strain condition with a constant thickness of 5. 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. To calculate the total expected cost of repair, first, the cumulative distribution is obtained using MCS and the proposed meta-model for the response DI overall adjusting a Beta distribution. Then, the density function is assigned by the derivative of the cumulative distribution. The target values of reliability indices corresponding to the three performance levels Table 1 considered in RBDO of RC structures are equal to 1.
The concrete material parameters shown in Table 5 are considered for the cover of column cross-sections. The other parameters of soil layers depend on their shear-wave velocities. Thus, in the process, first, the shear-wave velocities of soil layers are randomly selected; then, the other parameters are computed based on the shear-wave velocity. The PGA value is also taken into account as follows [ 13 , 22 ]:. The values of these parameters are shown in Table 6. 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. Due to the effect of decreasing gravity, the actual value of the gravitational constant, G t , depends on the actual age of the universe. In order to consider the stochastic nature of the optimization process, ten independent optimization runs are performed and the best solution is considered as the final results.
Six-storey RC frame is shown in Fig. In the frame, the length of each bay and the height of stories are 5 m and 3 m, respectively. The groups of structural elements are presented in Fig. Therefore, the training optimal parameters of the meta-model associated with the mean and the standard deviation of seismic responses are shown in Table 7.
In order to validate the performance and accuracy of the proposed meta-model, relative root-mean-squared error, i. Optimal parameters of the meta-model for training the mean and the standard deviation of the seismic responses. Performance associated with the mean and the standard deviation of the seismic responses. As given in Table 8 , the proposed meta-model trained for the mean and the standard deviation of seismic responses has proper performance generality. The cross-section of beams and columns are selected from Types 2 and 3, which are shown in Table 4.
The optimum designs of the RC frame are listed in Table 9. The convergence histories of the optimum objective function are shown in Fig. As can be seen in Fig. Nine-storey RC frame is shown in Fig. The members of the structure are divided into six groups for the columns and six groups for the beams. After training database using the presented WWLS-SVM optimal parameters of the meta-model associated with the mean and the standard deviation of seismic responses are shown in Table Therefore, the meta-model is reliably employed to predict the necessary responses during the RBDO process.
In this example, the cross-section 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 In general, the optimum design of structures depends on a number of parameters that are inherently uncertain. Reliability-based 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.
Haftka, R. The following examples are given to illustrate the HL-RF method and the transformation of non-Gaussian variables. Generally, the confidence interval of any parameter includes two parts: the confidence level and margin of error. It can be verified that the generated vectors J 1 , J 2 , The only difference between Example 4. Ha-rok Bae of Caterpillar Inc.
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 soil-structure interaction SSI effects. 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:. Therefore, the proposed meta-model, as a substitute for the nonlinear dynamic analysis of SSI system, can estimate the reliability index through MCS with a small computational cost. Numerical results of training and testing the meta-model indicated that performance generality of the meta-model was higher in comparison to WLS-SVM. Hence, the proposed meta-model can predict the nonlinear dynamic analysis of SSI system in terms of accuracy and flexibility.
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.