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The plastic behavior of strengthened haunched beams utilizing carbon fiber-reinforced polymers (CFRP) was investigated using a probabilistic design that took into account random concrete properties, CFRP properties, and complementary strain energy values, with the reliability index serving as a limiting index, as the proposed method considers a novel method that deals with probabilistic parameters for models with limited plastic behavior designed based on the reliability index. The data used in this research were gathered and evaluated in a recent study on simply supported haunched beams reinforced with carbon fiber-reinforced polymers. The purpose of this research was to use the reliability limitation index for simulated strengthened haunched beams by taking into account randomness in concrete and CFRP properties and the complementary strain energy value, which is considered a plastic behavior controller that provides an illustration of the damage amount within the reinforcement steel bars. The results indicate how randomness affects the behavior of the presented models, which are chosen to have different numbers of CFRP strips. The variable randomness affects load and deflection values where the reliability index value increases as the corresponding load value decrease, reflecting the increased probability of failure in models subjected to higher loading conditions, while tension concrete damage percentages are reflected in the damage pattern presented in the results, showing that as the produced load increases, so does the damage intensity. It is also obvious that the reliability index served as a limitation index while taking concrete characteristics and complementary strain energy as random variables.

This work introduces elasto-plastic analysis of prestressed reinforced concrete beams under different prestressing conditions by limiting the residual plastic behaviour inside the steel bars using complementary strain energy. A non-linear optimal method was used to limit residual plastic deformations in steel bars, including prestressed tendons, used to reinforce beams from two previous research investigations. This was considered by using an optimization approach with an objective function to find the maximum loading while applying constraints on the complementary strain energy of residual internal forces in steel elements to control residual plastic deformations. Thus, an elasto-plastic optimization programme was linked to models simulated by ABAQUS, as concrete was calibrated by the concrete damage plasticity (CDP) model. Some variables were considered regarding the force applied inside prestressed tendons and the number of tendons used inside the models. Thus, limits on complementary strain energy affected load values and model damage where an increase in the permissible strain energy value leads to an increase in the corresponding loading values produced; thus, this produces a higher stress intensity in steel and tension-damaged areas in concrete. Based on these data, many comparisons have been made to determine when beams behaved elastically and how they turned into plastic.

In this research, in order to evaluate the plastic limit load and also plastic design parameters of the long pile foundations subjected to horizontal loads, shakedown method is applied. In carrying out shakedown analysis and design methods, large plastic deformations and residual displacements could develop in the pile foundation which might lead to the failure of the structure. For this reason, complementary strain energy of residual forces proposed as a limit condition to control the plastic deformation of the pile structure. Furthermore, considering the uncertainties (strength, manufacturing, geometry) the limit conditions on the complementary strain energy of residual forces are assumed randomly and the reliability condition was formed by the use of the strict reliability index. The influence of the limit conditions on the plastic limit load and design parameters of the long pile in cohesionless soil subjected to lateral load were investigated and limit curves for shakedown load factors are presented. The numerical results show that the probabilistic given limit conditions on the complementary strain energy of residual forces have significant influence on the load bearing limit and the design parameters of pile foundations. The formulations of the reliability based problems lead to mathematical programming which were carried out by the use of non-linear algorithm.