Numerical procedure for the deflections of buckled elasto-plastic flexible bars

Abstract

In this paper, a numerical procedure to determine the deflections at the free end and along the central line of buckled elasto-plastic flexible bars is introduced. Because of the large deflection of the bar, the nonlinear theory of bending is used to solve the problem at hand. The exact expression of the curvature is used in the moment-curvature relationship for both the elastic (elastica) and the elasto-plastic states of stress. For the elastica problem, a closed-form solution may be found, if any. However, for elasto-plastic flexible bars, where a region of the bar yields, and yielding progresses along the bar, a closed-form solution is not possible because of the complexity of the problem. Therefore, the resulting first-order, nonlinear differential equations, for the problem at hand, are numerically integrated, using the improved Euler method, from the base of the bar where boundary values are known, to the free end where boundary value must initially be guessed. The integration process is repeated until the computed value of deflection at the free end of the bar is close enough to the trial value within acceptable tolerance. Deflections at the free end of the bar in the x and y directions are presented in graphical form. A computer program is developed, and the results are compared to those obtained from the experimental data.