Dynamic multiplier for falling mass

The dynamic multiplier (impact factor) is the ratio of the maximum dynamic deflection (or stress) to the static deflection (or stress) produced by the same load:

$$\text{Dynamic multiplier} = \frac{\delta_{max}}{\delta_{st}}$$

Mass falling on the mid-span of a simply supported beam

Let a weight $W$ fall from a height $h$ on to the mid-span of a simply supported beam of span $L$ and flexural rigidity $EI$. Let $\delta$ be the maximum mid-span deflection and neglect the beam's weight and energy losses.

Static deflection of the beam under $W$ at mid-span:

$$\delta_{st} = \frac{WL^3}{48EI}$$

So the beam stiffness at mid-span is $k = \dfrac{W}{\delta_{st}} = \dfrac{48EI}{L^3}$, and the equivalent dynamic load is $P = k\delta$.

Energy balance: loss of potential energy of the falling weight = strain energy stored in the beam.

$$W(h + \delta) = \tfrac12 P\delta = \tfrac12 k\delta^2 = \tfrac12\frac{W}{\delta_{st}}\delta^2$$ $$\delta^2 - 2\delta_{st}\delta - 2\delta_{st}h = 0$$ $$\delta = \delta_{st}\left[1 + \sqrt{1 + \frac{2h}{\delta_{st}}}\right]$$

Therefore the dynamic multiplier is

$$\boxed{\frac{\delta}{\delta_{st}} = 1 + \sqrt{1+\frac{2h}{\delta_{st}}}}$$

The maximum bending moment and stress are multiplied by the same factor, since the structure is linear. If $h = 0$ (suddenly applied load) the multiplier is 2. If $h \gg \delta_{st}$, then $\delta \approx \sqrt{2h\,\delta_{st}}$.