Impact Loading Approximate Formulas

Impact and Sudden Loading Approximate Formulas Equations

If it is assumed that the stresses due to impact are distributed throughout any elastic body exactly as in the case of static loading, then it can be shown that the vertical deformation d_i and stress σ_i produced in any such body (bar, beam, truss, etc.) by the vertical impact of a body falling from a height of h are greater than the deformation d and stress s produced by the weight of the same body applied as a static load in the ratio:

d_i / d = σ_i / σ = 1 + ( 1 + 2h / d )^0.5

If h = 0, we have the case of sudden loading, and d_i / d = σ_i / σ, as is usually assumed.

If the impact is horizontal instead of vertical, the impact deformation and stress are given by:

d_i / d = σ_i / σ = ( v² / ( g d ) )^0.5

where,

d = is the deformation the weight of the moving body would produce if applied as a static load in the direction of the velocity,
d_i = Vertical Deformation,
σ_i = produced in any such body (bar, beam, truss, etc.) by the vertical impact,
v = velocity of impact,
h = Height of the falling body

Note:

It is improbable that in any actual case of impact the stresses can be calculated accurately by any of the methods or formulas given above. Equation given, for instance, is supposedly very nearly precise if the conditions assumed are realized, but those conditions- perfect elasticity of the bar, rigidity of the moving body, and simultaneous contact of the moving body with all points on the end of the rod are obviously unattainable. On the one hand, the damping of the initial stress wave by elastic hysteresis in the bar and the diminution of the intensity of that stress wave by the cushioning effect of the actually nonrigid moving body would serve to make the actual maximum stress less than the theoretical value; on the other hand, uneven contact between the moving body and the bar would tend to make the stress conditions nonuniform across the section and would probably increase the maximum stress.

The formulas given are based upon an admittedly false assumption, viz. that the distribution of stress and strain under impact loading is the same as under static loading.

References:

Roark's Formulas for Stress and Strain, Seventh Edition