Consider a rod BD of uniform cross section which is hit at its end B by a body of mass m moving with a velocity v0. As the rod deforms under the impact, stresses develop within the rod and reach a maximum value σm. After vibrating for a while, the rod will come to rest, and all stresses will disappear. Such a sequence of events is referred to as an impact loading.
In order to determine the maximum value σm of the stress occurring at a given point of a structure subjected to an impact loading, serveral simplifying assumptions are made.
First, assume that the kinetic energy T = (1/2)mv02 of the striking body is transferred entirely to the structure and, thus that the strain energy Um corresponding to the maximum deformation xm is:
Um = (1/2)mv02
This assumption leads to the following two specific requirements:
1. No energy should be dissipated during the impact.
2. The striking body should not bounce off the structure and retain part of its energy. This, in turn, necessitates that the inertia of the structure be negligible, compared to the inertia of the striking body.
In practice, neither of these requirements is satisfied, and only part of the kinetic energy of the striking body is actually transferred to the structure. Thus, assuming that all of the kinetic energy of the striking body is transferred to the structure leads to a conservative design of that structure.
We further assume that the stress-strain diagram obtained from a static test of the material is also valid under impact loading. Thus, for an elastic deformation of the structure, the maximum value of the strain energy can be expressed as:
In the case of the uniform rod of fig?, the maximum stress σm has the same value throughout the rod, and the following is written:
Solving for σm and substituting for Um from Eq the following is written:
Note from the expression obtained that selecting a rod with a large volume V and a low modulus of elasticity E will result in a smaller value of the maximum stress σm for a given impact loading.
In most problems, the distribution of stresses in the structure is not uniform, and Eq? does not apply. It is then convenient to determine the static load Pm which would produce the same strain energy as the impact loading, and compute from Pm the corresponding value σm of the largest stress occurring in the structure.