Limit design of reinforced concrete haunched beams by the control of the residual plastic deformation

Abstract

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.