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In this paper, a new enhanced version of the Particle Swarm Optimization (PSO) is presented. An important modification is made by adding probabilistic functions into PSO, and it is named Probabilistic Particle Swarm Optimization (PPSO). Since the variation of the velocity of particles in PSO constitutes its search engine, it should provide two phases of optimization process which are: exploration and exploitation. However, this aim is unachievable due to the lack of balanced particles’ velocity formula in the PSO. The main feature presented in the study is the introduction of a probabilistic scheme for updating the velocity of each particle. The Probabilistic Particle Swarm Optimization (PPSO) formulation thus developed allows us to find the best sequence of the exploration and exploitation phases entailed by the optimization search process. The validity of the present approach is demonstrated by solving three classical sizing optimization problems of spatial truss structures.

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The purpose of this study is to compare the results of different probabilistic methods such as the perturbation method, Stochastic Finite Element Method (SFEM) and Monte Carlo Method. These methods were used to study the convergence of direct approach for slope stability analysis and are developed for a linear soil behavior. In this study, two dimensional random fields are used and both the First Order Reliability Method (FORM) and Limited Step Length Iteration Method (LSLIM) have been adopted to evaluate the reliability index. The study found that the perturbation method of the second order is easy to apply using the field’s theory because accuracy is reached even with different coefficients of variation of input variables, while the spectral finite element method yields accurate results only for high levels of solution development.

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In this paper, the probabilistic seismic performance of vertically irregular steel buildings, considering soil-structure interaction effects, is evaluated. Various irregular distributions of structural properties, including mass, stiffness and strength along the height of three-dimensional moment resisting steel frames were intended. The finite element model of soil medium was created with solid elements below the structure. The nonlinear material behavior of soil was considered as well. Nonlinear incremental dynamic analysis was performed to evaluate the flexible-base structural performance in the framework of probabilistic performance-based earthquake engineering. According to the median curves of intensity-demand of structures, it is concluded that non-uniform height-wise distribution of lateral resistance properties of steel structures varies the displacement demand and the seismic capacity of the irregular frames, compare to the regular structure. The capacity variation of most irregular frames is more obvious at the nonlinear phase of structural behavior. Due to the foundation flexibility, the damage concentration raises in the bottom floor and the irregularity increase the seismic demands of the lower floors of the system. Among all the irregular steel frames, the average increase of the displacement demand and reduction of the seismic capacity are maximal for the strength and concurrent variation of stiffness and strength irregularity models, respectively. Additionally, mass irregularity shows minor influence in the seismic demand and capacity variations of the steel frames. The predominant influence of stiffness and strength irregularities (soft and weak story) is observed in reduction of the structural ductility factor and the mean annual frequency of exceeding limit states.