In the proposed probabilistic topology optimization method, the minimum penalized weight design of the structure is subjected to probabilistic compliance and side constraints. The calculation of the compliance value is based on the assumption that the magnitude of the loads has uncertainties so that their joint normal distribution functions, mean values, and covariances are known and a minimum probability value is given. Applying an appropriate approximation, the original problem is replaced by a deterministic one. By the use of the optimality conditions, an updated iterative design formulation is presented. Several new topology design problems are calculated with randomly given loads.

The aim of this paper is to introduce a new type of stochastic optimal topology design method where the loads are given randomly. The standard stochastic mathematical programming problem is based on penalized minimum weight design procedure subjected to a probabilistic compliance constraint due to the loading uncertainties. If the probability of the compliance constraint is given by the use of recommendation of Prekopa this probabilistic expression can be substituted with an equivalent one and used as a deterministic constraint in the original problem. In the numerical method, optimality criteria procedure is applied. The application is illustrated by numerical examples.

A novel computational model is proposed in this paper considering reliability analysis in the modelling of reinforced concrete beams at elevated temperatures, by assuming that concrete and steel materials have random mechanical properties in which those properties are treated as random variables following a normal distribution. Accordingly, the reliability index is successfully used as a constraint to restrain the modelling process. A concrete damage plasticity constitutive model is utilized in this paper for the numerical models, and it was validated according to those data which were gained from laboratory tests. Detailed comparisons between the models according to different temperatures in the case of deterministic designs are proposed to show the effect of increasing the temperature on the models. Other comparisons are proposed in the case of probabilistic designs to distinguish the difference between deterministic and reliability-based designs. The procedure of introducing the reliability analysis of the nonlinear problems is proposed by a nonlinear code considering different reliability index values for each temperature case. The results of the proposed work have efficiently shown how considering uncertainties and their related parameters plays a critical role in the modelling of reinforced concrete beams at elevated temperatures, especially in the case of high temperatures.