This paper aims to present a novel computational technique for using reliability-based design while taking into account the effect of geometrically and materially nonlinear imperfect analysis. Consequently, a new bi-directional evolutionary structural optimization scheme is developed. A comparison is made between perfect geometrically and materially nonlinear analysis and imperfect geometrically and materially nonlinear analysis topology optimization designs for both deterministic and probabilistic analysis. In the case of probabilistic analysis, relevant parameters such as volume fraction (including manufacturing imprecisions), material properties, and geometrical imperfections (for stability calculations) are assumed to be random variables that follow a normal distribution to represent the uncertainties. The considered numerical examples have successfully illustrated that the proposed method can find the optimal topology for a reliability-based design using perfect geometrically and materially nonlinear analysis and imperfect geometrically and materially nonlinear analysis. Additionally, the results of the topology optimization according to the mean stress and the final optimized shapes have been influenced by introducing a reliability-based design, considering the reliability index as a bound governing the process.

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.

In this paper, a novel computational (optimization) model is presented to control the plastic behaviour of reinforced concrete haunched beams using complementary strain energy of residual forces formed inside the steel reinforcing bars. For this purpose, a numerical model validation process was held and then two non-linear optimization problems were outlined. In these optimization problems, different objective functions were considered applying the optimal elasto-plastic analysis and design of haunched reinforced concrete beams aiming to find the maximal loading or the minimum volume of the steel used to reinforce the beams as the plastic deformations are controlled by using constraints on the complementary strain energy of the residual internal forces of the steel bars. Moreover, the effect of these constraints on different haunch angle beams is studied. The applied method is described in terms of nonlinear mathematical programming and providing solutions when the plastic reserves of the body are not fully exhausted. It is worth mentioning that in this study a concrete damage plasticity constitutive model is applied in the numerical problems and calibrated in accordance with the data gained from laboratory tests. The optimal solutions of the nonlinear mathematical problems were calculated by MATLAB programming codes written by the authors taking into consideration different objective functions and equality and inequality constraints for each case. Finally, by performing a parametric study, the different optimization problems showed how beams behaved differently under different complementary strain energy limit values shifting from elastic into elasto-plastic state and then reaches the fully plastic state where results showed different comparisons taking into consideration the effect of the different complementary strain energy limit values on the maximum applied load, geometry of the beam and steel volume used to reinforce the beams. Thus, complementary strain energy limit value is used to control the plastic deformation inside steel bars during loading progress where avoiding the formation of the plastic deformation in the steel bars would reflect on the general behaviour of the haunched reinforced concrete beams.

This paper presents elasto-plastic limit analysis of reliability-based geometrically nonlinear topology optimization. For this purpose, by reason of uncertainties the volume fraction is considered randomly during optimization. Thus reliability-based design has been considered for solving the problems. To perform reliability-based topology optimization design, the Monte-Carlo simulation method has been applied to calculate the probability of failure, thus the reliability index. Besides, bi-directional evolutionary structural optimization (BESO) method is used to consider the effect of geometrically nonlinear design for elasto-plastic analysis. Plastic behavior is controlled by applying a bound on the plastic limit load multipliers using limit analysis. The adequacy of the proposed method is exhibited by three 2D benchmark problems. 2D models of L-shape beam and U-shaped plate are considered for reliability-based design and geometrically nonlinear analysis topology optimization in case of elastic material. Additionally, 2D and 3D elasto-plastic material models have been considered to demonstrate that the proposed method can find the optimal topology of elasto-plastic models for reliability-based design and geometrically nonlinear analysis.