Numerical analysis of the counter-intuitive dynamic behavior of the elastic-plastic pin-ended beams under impulsive loading with regard to linear hardening effects

Abstract

The counter-intuitive behavior where the permanent deflection of elastic-plastic beam come to rest in the opposite direction of the impulsive loading, normally appears and disappears abruptly, in certain small ranges of loading and structural parameters. One of the most important issues in the study of this phenomenon is the determination of the influence of different parameters. This work is aimed to study the effects of hardening in counter-intuitive dynamic behavior of elastic-plastic pin-ended beams under impulsive loading. This has been done by developing the proposed Galerkin numerical model and presenting a novel algorithm. The Galerkin method as well as the commercial finite element code ANSYS/LS-DYNA is applied to study this phenomenon. In order to account for the hardening effects in Galerkin method, a new algorithm is proposed. The time history curves for mid-span of the beam is studied in detail and the region of the occurrence of the counter-intuitive behavior is determined. Furthermore, using the finite element software, energy diagrams of the beam are also derived. It has been found that the counter-intuitive behavior is a phenomenon, which is very sensitive to loading, therefore it may appear with a little change in the amount of loading. The results also show that although both methods predict one continuous region of loading for the occurrence of this phenomenon in elastic-perfectly plastic beams, still there are two continuous distinct regions of loading, when considering the hardening effects, for this phenomenon. In addition, this anomalous behavior would occur in the proper ratios of kinetic to internal energy and when considering the linear hardening effects, the possibility of the occurrence of the counter-intuitive behavior exists in a wider domain of energy ratio.

Difference equation vs differential equation on different scales

“…Equation (1) can be easily solved by a numerical method (Li and He, 2019; Alizadeh et al , 2018; Liu, 2019; Pirmohammadi, 2019; Bilige and Han, 2018), and it can be solved by many analytical methods, for example, the variational method, the variational iteration method (He and Latifizadeh, 2020; Anjum and He, 2019; He et al , 2014), the homotopy perturbation method (Yu et al , 2019; Ren et al , 2019; Ban and Cui, 2018; Wu and He, 2018) and Taylor series method (He and Ji, 2019b; He et al , 2019; He, 2019a; Khan, 2018). In this paper, we will apply the exp-function method (He, 2012; He, 2013; He and Wu, 2006; Jia et al , 2015) to study the solution properties at different scales.…”