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In the present study, the reliability assessment of performance-based optimally seismic designed reinforced concrete (RC) and steel moment frames is investigated. In order to achieve this task, an efficient methodology is proposed by integrating Monte Carlo simulation (MCS) and neural networks (NN). Two NN models including radial basis function (RBF) and back propagation (BP) models are examined in this study. In the proposed methodology, MCS is used to estimate the total exceedence probability associated with immediate occupancy (IO), life safety (LS) and collapse prevention (CP) performance levels. To reduce the computational burden of MCS process, the required nonlinear responses of the generated structures are predicted by RBF and BP models. The numerical results imply the superiority of BP to RBF in prediction of structural responses associated with performance levels. Finally, the obtained results demonstrate the high efficiency of the proposed methodology for reliability assessment of RC and steel frame structures.

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A Reliability-Based Design Optimization (RBDO) framework is presented that accounts for stochastic variations in structural parameters and operating conditions. The reliability index calculation is itself an iterative process, potentially employing an optimization technique to find the shortest distance from the origin to the limit-state boundary in a standard normal space. Monte Carlo simulation (MCs) is embedded into a design optimization procedure by a modular double loop approach, which the self-adaptive version of particle swarm optimization method is introduced as an optimization technique. Double loop method has the advantage of being simple in concepts and easy to implement. First, we study the efficiency of self-adaptive PSO algorithm inorder to solve the optimization problem in reliability analysis and then compare the results with the Monte Carlo simulation. While computationally significantly more expensive than deterministic design optimization, the examples illustrate the importance of accounting for uncertainties and the need for regarding reliability-based optimization methods and also, should encourage the use of PSO as the best of evolutionary optimization methods to more such reliability-based optimization problems.

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The use of fiber reinforced polymer (FRP) U-wrap to rehabilitate concrete beams has increased in popularity over the past few years. As such, many design codes and guidelines have been developed to enable designers to use of FRP for retrofitting reinforced concrete beams. FIB is the only guideline for design which presents a formula for torsional capacity of concrete beams strengthened with FRP. The Rackwitz-Fiessler method was applied to make a reliability assessment on the torsional capacity design of concrete beams retrofitted with U-wrap FRP laminate by this guideline. In this paper, the average of reliability index obtained is 2.92, reflecting reliability of the design procedures. This value is somehow low in comparison to target reliability level of 3.5 used in the guideline calibration and so, optimum resistance factor may be needed in future guideline revisions. From the study on the relation between average reliability index and optimum resistance factor, a value of 0.723 for the optimum resistance factor is suggested.

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The interbasin water transfer is a remedy to mitigate the negative issues of water shortage in arid and semi-arid regions. In  a water transfer project  the receiving basin always  benefits while, the sending basin may suffer. In this study, the project of interbasin water transfer from Dez water resources system in south-west of Iran to the central part of the contrary is 
investigated during a drought period. To this end, a multi-objective optimization model is developed  based  on  the  Non  Dominated  Sorting  Genetic  Algorithm  (NSGA-II).  The optimum trade-off between the water supply benefits into and out of the Dez River basin as well  as  energy  production  is  derived.  Formulating  the  problem  as  a  multi-objective 
optimization provides a better insight into the gains and losses of a water transfer project. Analyzing the case study, revealed that to reach an acceptable level of reliability for meeting the water demands it is no longer possible to generate hydropower energy with high levels of reliability. 

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The paper deals with the reliability–based design optimization (RBDO) of concrete gravity dams subjected to earthquake load using subset simulation. The optimization problem is formulated such that the optimal shape of concrete gravity dam described by a number of variables is found by minimizing the total cost of concrete gravity dam for the given target reliability. In order to achieve this purpose, a framework is presented whereby subset simulation is integrated with a hybrid optimization method to solve the RBDO approach of concrete gravity dam. Subset simulation with Markov Chain Monte Carlo (MCMC) sampling is utilized to estimate accurately the failure probability of dams with a minimum number of samples. In this study, the concrete gravity dam is treated as a two–dimensional structure involving the material nonlinearity effects and dam–reservoir–foundation interaction. An efficient metamodel in conjunction with subset simulation–MCMC is provided to reduce the computational cost of dynamic analysis of dam–reservoir–foundation system. The results demonstrate that the RBDO approach is more appropriate than the deterministic optimum approach for the optimal shape design of concrete gravity dams.

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Monte Carlo simulation (MCS) is a useful tool for computation of probability of failure in reliability analysis. However, the large number of samples, often required for acceptable accuracy, makes it time-consuming. Importance sampling is a method on the basis of MCS which has been proposed to reduce the computational time of MCS. In this paper, a new adaptive importance sampling-based algorithm applying the concepts of first-order reliability method (FORM) and using (1) a new simple technique to select an appropriate initial point as the location of design point, (2) a new criterion to update this design point in each iteration and (3) a new sampling density function, is proposed to reduce the number of deterministic analyses. Besides, although this algorithm works with the position of design point, it does not need any extra knowledge and updates this position based on previous generated results. Through illustrative examples, commonly used in the literature to test the performance of new algorithms, it will be shown that the proposed method needs fewer number of limit state function (LSF) evaluations.

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Structural reliability theory allows structural engineers to take the random nature of structural parameters into account in the analysis and design of structures. The aim of this research is to develop a logical framework for system reliability analysis of truss structures and simultaneous size and geometry optimization of truss structures subjected to structural system reliability constraint. The framework is in the form of a computer program called RBO-S>S. The objective of the optimization is to minimize the total weight of the truss structures against the aforementioned constraint. System reliability analysis of truss structures is performed through branch-and-bound method. Also, optimization is carried out by genetic algorithm. The research results show that system reliability analysis of truss structures can be performed with sufficient accurately using the RBO-S>S program. In addition, it can be used for optimal design of truss structures. Solutions are suggested to reduce the time required for reliability analysis of truss structures and to increase the precision of their reliability analysis.

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In this paper, a procedure has been presented to develop fragility curves of structures equipped with optimal variable damping or stiffness semi-active tuned mass dampers (SATMDs). To determine proper variable damping or stiffness of semi-active devices in each time step, instantaneous optimal control algorithm with clipped control concept has been used. Optimal SATMDs have been designed based on minimization of maximum inter-story drift of nonlinear structure which genetic algorithm(GA) has been used to solve the optimization problem. For numerical analysis, a nonlinear eight-story shear building with bilinear hysteresis material behavior has been used. Fragility curves for the structure equipped with optimal variable damping and stiffness SATMDs have been developed for different performance levels and compared with that of uncontrolled structure as well as structure controlled using passive TMD. Numerical analysis has shown that for most range of intensity measure optimal SATMDs have been effective in enhancement of the seismic fragility of the nonlinear structures which the improvement has been more than passive TMDs. Also, it has been found that, although variable stiffness SATMD shows to be more reliable in lower mass ratios, however in higher mass ratios variable stiffness and damping SATMDs performs similarly to improve reliability of the structure.

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One of the most important problems discussed recently in structural engineering is the structural reliability analysis considering uncertainties. To have an efficient optimization process for designing a safe structure, firstly it is required to study the effects of uncertainties on the seismic performance of structure and then incorporate these effects on the optimization process. In this study, a new procedure developed for incorporating two important sources of uncertainties in design optimization process of steel moment resisting frames, is proposed. The first source is related to the connection parameter uncertainties and the second one to seismic demand uncertainty. Additionally Mont Carlo (MC) simulation and a variance reduction technique (VRT) are utilized to deal with uncertainties and to reduce the corresponding computational cost. In the proposed procedure two design objectives are considered, which are structural weight and collapse prevention reliability index for a moment resisting frame in such a way that leads to a set of optimum designs with minimum weight and less possible amounts of sensitivity to connection parameters uncertainties and spectral acceleration uncertainty as seismic demand variation. Additionally, in this procedure the reliability index is computed considering all FEMA-356 performance acceptance criteria, the approach that has never been investigated in other studies. The efficiency of this approach is illustrated by exhibiting a set of optimum designs, in the form of both objective values and investigating nonlinear behavior of optimum designs compared with non-optimum designs. This procedure is introduced in this paper with emphasize on the collapse limit state and applying pushover analysis for studying the nonlinear behavior of structural elements.

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This paper presents a design process using a course grained parallel genetic algorithm to optimize three-dimensional steel moment frames by considering the axial force and biaxial bending moments interaction in plastic hinge formation. The objective function is to minimize the total weight of the structure subjected to the reliability constraint of the structural system. System reliability analysis is performed through the proposed Modified Latin Hypercube Simulation (M-LHS) Method. For optimization, a 3DSMF-RBO program is written in CSHARP programming language. The reliability analysis results show a large decrease in the number of simulation samples and subsequently a decrease in the execution time of optimization computation. The optimization results indicate that by considering interaction of the axial force and biaxial bending moments in plastic hinge formation rather than the only bending moment, to some extent increases the total weight of the designed structure.

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Due to the random nature of the variables affecting the analysis and design of structures, the reliability method is considered as one of the most important and widely used topics in structural engineering. Despite the simplicity of moment methods, the answer to problems with multiple design points (the point with the highest probability of failure) such as transmission line towers depends a lot on the starting point of the search; and it may converge to the local optima answer which is not desirable. Simulation methods also require a large number of evaluations of the limit state function and increase the volume and time of calculations. Also, the design point is not calculated in most of these methods. In this study, the reliability index of four transmission line towers was calculated with four metaheuristic algorithms in which the limit state function was defined based on the displacement of nodes and the results were compared with the results of Monte Carlo Simulation (MCS) method. For this purpose, the objective function was defined as the geometric distance between the point on the function of the boundary condition to the origin in the standard normal coordinate system and the constraint of the problem (the limit state function) based on the displacement of the nodes. Random variables in these problems consisting of the cross-sectional area of the members, the modulus of elasticity, and the nodal loads.

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Nowadays, the optimal design of structures based on reliability has been converted to an active topic in structural engineering. The Reliability-Based Design Optimization (RBDO) methods provide the structural design with lower cost and more safety, simultaneously. In this study, the optimal design based on reliability of dome truss structures with probability constraint of the frequency limitation is discussed. To solve the RBDO problem, nested double-loop method is considered; one of the loops performs the optimization process and the other one assesses the reliability of the structure. The optimization process is implemented using ECBO and EVPS algorithms and the reliability index is calculated using the Monte Carlo simulation method. Finally, the size and shape reliability-based optimization of 52-bar and 120-bar dome trusses has been investigated.

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Economy and safety are two important components in structural design process and stablishing a balance between them indeed results in improved structural performance specially in large-scale structures including space lattice domes. Topology optimization of geometrically nonlinear single-layer lamella, network, and geodesic lattice domes is implemented using enhanced colliding-bodies optimization algorithm for three different spans and two different dead loading conditions. Collapse reliability index of these optimal designs is evaluated to assess the safety of the structures against overall collapse using Monte-Carlo simulation method. The numerical results of this study indicate that the reliability index of most of the optimally designed nonlinear lattice domes is low and this means that the safety of these structures against overall collapse is questionable.

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The present work deals with optimization and reliability assessment of double layer barrel vaults. In order to achieve the optimization task an improved colliding bodies optimization algorithm is employed. In the first phase of this study, different forms of double layer barrel vaults namely, square-on-square, square-on-diagonal, diagonal-on-diagonal and diagonal-on-square are considered and designed for optimal weight by the improved colliding bodies optimization algorithm. In the second phase, in order to account for the existing uncertainties in action and resistance of the structures, the reliability of the optimally designed double layer barrel vaults is assessed using importance sampling method by taking into account a limit-state function on the maximum deflection of the structures. The results demonstrate that the minimum reliability index of the optimal designs is 0.92 which means that all the optimally designed double layer barrel vaults are reliable and safe against uncertainties.  

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In recent decades, steel was used more than other materials in structural engineering. However, the safety of high-heat steel structures dramatically decreased, due to steel mechanical properties. Therefore, the design process should be done in a way that the structure has the required resistance at high temperatures and during the fire, according to the effect of heat on the performance of steel structures. In this study, the optimal design process of steel structures is considered under the fire load. In the optimal design process, the failure risk of the structure members is considered as a constraint. Therefore, the optimization process requires thermal and structural reliability analysis. A parametric model has been used to analyse the reliability of the structure in the fire limit state. The optimization process is also performed based on the Colliding Bodies Optimization (CBO) algorithm. In order to evaluate the optimal design process, 3 and 6-floors frames have been investigated. The results showed that the members' condition is effective in the structural resistance for the thermal loading. On the contrary, the structure design based on the reliability under the fire load provides a proper prediction from the behaviour of the structure and satisfies the requirements for the common state of design.

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Some structural control systems have been devised to protect structures against earthquakes, which the tuned mass damper (TMD) being one of the earliest. The effect of a tuned mass damper depends on its properties, such as mass, damping coefficient, and stiffness. The parameters of tuned mass dampers need to be tuned based on the main system and applied load. In most of the papers, the parameters of TMDs have been tuned based on the nominal parameters of structures. Also, most of the studies considered the minimization of maximum displacement of structure as the objective function of optimizing the parameters of tuned mass dampers. In this study, according to the Monte Carlo method and using the Mouth Brooding Fish algorithm, TMDs have been optimized based on the reliability of structures regarding the uncertain parameters of buildings, and their efficiency in the reduction of maximum displacement and failure probability of hundreds generated buildings with uncertain parameters, are compared with the efficiency of the displacement-based optimized TMDs. The results show that the TMDs optimized regarding uncertainty have better efficiency in reducing the maximum displacement, and failure probability of buildings than the TMDs optimized regarding nominal parameters of buildings. Also, according to the results, the displacement-based optimized TMDs regarding uncertainty show better efficiency in reducing the failure probability and displacement of the buildings than reliability-based optimized TMDs.
 

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This paper introduces a reliability-based multi-objective design method for spatial truss structures. A multi-objective optimization problem has been defined considering three conflicting objective functions including truss weight, nodal deflection, and failure probability of the entire truss structure with design variables of cross sectional area of the truss members. The failure probability of the entire truss system has been determined considering the truss structure as a series system. To this end, the uncertainties of the applied load and the resistance of the truss members have been accounted by generating a set of 50 random numbers. The limitations of members' allowable have been defined as constraints. To explain the methodology, a 25-bar benchmark spatial truss has been considered as the case study structure and has been optimally designed using the game theory concept and genetic algorithm (GA). The results show effectiveness and simplicity of the proposed method which can provide Pareto optimal solution. These optimal solutions can provide both safety and reliability for the truss structure.
 

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Metaheuristic algorithms are preferred by the many researchers to reach the reliability based design optimization (RBDO) of truss structures. The cross-sectional area of the elements of a truss is considered as design variables for the size optimization under frequency constraints. The design of dome truss structures are optimized based on reliability by a popular metaheuristic optimization technique named Enhanced Vibrating Particle System (EVPS). Finite element analyses of structures and optimization process are coded in MATLAB. Large-scale dome truss of 600-bar, 1180-bar and 1410-bar are investigated in this paper and are compared with the previous studies. Also, a comparison is made between the reliability indexes of Deterministic Design Optimization (DDO) for large dome trusses and Reliability-Based Design Optimization (RBDO).
 

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The main aim of this study, is to evaluate the seismic reliability of steel concentrically braced frame (SCBF) structures optimally designed in the context of performance-based design. The Monte Carlo simulation (MCS) method and neural network (NN) techniques were utilized to conduct the reliability analysis of the optimally designed SCBFs. Multi-layer perceptron (MLP) trained by back propagation technique was used to evaluate the required structural responses and then the total exceedence probability associated with the seismic performance levels was estimated by the MCS method. Three numerical examples of 5-, 10-, and 15-story SCBFs with fixed and optimal topology of braces are presented and their probability of failure was evaluated considering the resistance characteristics and the seismic loading of the structures. The numerical results indicate that the SCBFs with optimal topology of braces were more reliable than those with fixed topology of braces.  
 

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The real-world applications addressing the nonlinear functions of multiple variables could be implicitly assessed through structural reliability analysis. This study establishes an efficient algorithm for resolving highly nonlinear structural reliability problems. To this end, first a numerical nonlinear optimization algorithm with a new simple filter is defined to locate and estimate the most probable point in the standard normal space and the subsequent reliability index with a fast convergence rate. The problem is solved by using a modified trust-region sequential quadratic programming approach that evaluates step direction and tunes step size through a linearized procedure. Then, the probability expectation method is implemented to eliminate the linearization error. The new applications of the proposed method could overcome high nonlinearity of the limit state function and improve the accuracy of the final result, in good agreement with the Monte Carlo sampling results. The proposed algorithm robustness is comparatively shown in various numerical benchmark examples via well-established classes of the first-order reliability methods. The results demonstrate the successive performance of the proposed method in capturing an accurate reliability index with higher convergence rate and competitive effectiveness compared with the other first-order methods.