# Chapter 8Finite Element Analysis for Dynamic Problems

## 8.1 INTRODUCTION

In the previous chapters, we have discussed the static analysis of solids and structures, in which the loads are assumed to be applied slowly such that the structure is in equilibrium at every instant until the full load is applied. When the effect of velocity and acceleration can be ignored due to slow application of loads, it is called a quasi‐static loading. The structure may be considered to be under *quasi‐static* deformation when the inertial forces are orders of magnitude smaller than the internal forces caused by the deformation. The structure gradually deforms to produce internal forces (i.e., stresses) so that the internal forces are in equilibrium with the externally applied load. In structural mechanics, this is called *static equilibrium*. We did not consider the time involved, that is, how quickly the load is applied or how quickly the structure reaches equilibrium with the applied loads. However, when the loads vary rapidly with time, the inertial effects cannot be neglected. In such cases, the quasi‐static assumption is not valid and we need to compute the *dynamic equilibrium*, which is the main topic of this chapter. Dynamic problems are concerned with the study of motion of structures under external loads. When the external loads are time varying, or steady but applied suddenly, it will trigger dynamic effects that cause structural vibration immediately upon the application of the load. If the ...

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