Structural members of a vehicle are designed to increase the energy absorption efficiency and thus to enhance the safety and reliability of a vehicle. The crashworthiness of each member needs to be evaluated at the initial stage of vehicle design for good performance of an assembled vehicle. As the dynamic behavior of structural members is different from the static one, the crashworthiness of vehicle structures has to be assessed by numerical impact analysis considering the dynamic response related to the inertia and strain-rate hardening effect. Impact simulation is usually carried out with the elastic-plastic finite element analysis code such as PAMCRASH or LS-DYNA3D. It inevitably requires tremendous time and efforts to estimate the crashworthiness of structural members even with the explicit methods. An efficient, alternative analysis tool could be an extended limit method for fast evaluation of the crashworthiness of structural members.
Limit analysis has become a useful and efficient numerical tool in the collapse behavior assessment for structural members since the method can easily calculate the plastic collapse load, energy absorption and deformation mode. The conventional limit analysis method seems seldom applied to real complicated structural problems probably because of the limitation of the method. However, a burst of development in limit theories and computer technologies enable limit analysis applied to complicated structural problems. Especially, the limit analysis concept has extended to a class of work-hardening materials from its long conjecture of perfectly plastic materials. Although the algorithm with a simple formulation has the advantage of stable convergence, computational efficiency and easy access to work-hardening materials, the method has been developed only for static collapse behavior of structures. Development of dynamic limit analysis could make it possible to evaluate the crashworthiness of structural members efficiently, accurately and systematically.
In this paper, the limit analysis concept is extended to incorporate with the dynamic equilibrium condition considering the inertia and strain-rate effects instead of the static equilibrium. A dynamic formulation of limit analysis has been derived for sequential incremental analysis dealing with time integration, strain and stress evaluation, strain hardening, strain-rate hardening and thermal softening. The time dependent term in the governing equation is integrated with the WBZ-α method proposed by Wood, Bossak and Zienkiewicz. The dynamic material behavior is described by the Johnson-Cook model in order to consider strain-rate hardening and thermal softening as well as strain hardening. The contact condition is added in order to check the penetration in the complicated structures.
The analysis method developed has been applied to a class of impact analysis of structural members. The high velocity deformation analysis of the plate with a hole has been carried out in order to verify the accuracy and validity of the dynamic limit analysis and WBZ-α method. The impact analysis of a Taylor bar has been performed in order to comparing the rate-dependent constitutive model with the quasi-static constitutive relation. Impact analysis of an S-rail has been performed with the dynamic limit analysis code and the numerical results have been compared with elasto-plastic explicit analysis results by LS-DYNA3D for collapse loads and its deformed shapes as well as the strain distribution. Comparison demonstrates that the dynamic finite element limit analysis can predict the crashworthiness of structural members effectively with less effort and computing time than the commercial codes compared. The impact analysis of S-rails has been conducted with the variation of design parameters such as the thickness and the aspect ratio of cross-section in order to estimate the energy absorption ratio with respect to the design parameters.
Numerical simulations have been carried out with a finite element limit analysis in order to identify forming effects on crash behavior of an S-rail. The formed S-rail contains fabrication histories such as residual stress, work hardening, non-uniform thickness distribution and geometric changes resulted from the forming process. Crashworthiness such as the collapse load, the deformed mode and the energy absorption has been investigated in order to identify forming effects. It is fully demonstrated that the design of auto-body structures needs to consider the forming effects for proper assessment of load-carrying capacity and deformation of the formed structures.
The analysis results demonstrate that the dynamic limit analysis method is an effective and useful tool in the dynamic analysis and prediction of the crashworthiness of structural members.
