# Simply supported beam with two symmetric point loads

## Summary

### Displacement where $$x = l$$

\begin{align} v(l) = \frac{PL^{3}}{6EI} \bigg[ 3 \bigg( \frac{l}{L} \bigg)^{2} - 4\bigg(\frac{l}{L}\bigg)^{3}\bigg] \end{align}

### Symbols

Symbol Description
$$v(x)$$ Displacement
$$x$$ Distance from left hand support
$$l$$ Distance from support to point load
$$L$$ Span
$$P$$ Load

## Derivation

Define deflection $$v(x)$$ as the superposition $$v_{1}(x) + v_{2}(x)$$ where $$v_{1}(x)$$ and $$v_{2}(x)$$ are given by [simply-supported-beam-offset-point-load] where $$x_{0}$$ is $$l$$ and $$L - l$$ respectively.

## Results

### Deflection where $$x = l$$

\begin{align} v(l) &= v_{1}(l) + v_{2}(l) \\ &= \frac{l - L}{EI}P\bigg( \frac{l^{3}}{6L} + \frac{l^{3}}{6L} - \frac{l^{2}}{3} \bigg) + \frac{(L - l) - L}{EI}P\bigg( \frac{l^{3}}{6L} + \frac{ (L - l)^{2} l }{6L} - \frac{(L - l)l}{3} \bigg) \\ &= \frac{P}{6EIL} \bigg( 2(l - L) l^{3} - 2(l - L)l^{2}L - l^{4} - (L - l)^{2} l^{2} - 2(l - L)l^{2}L \bigg) \\ &= \frac{P}{6EIL} \bigg( 2(l - L) l^{3} - 4(l - L)l^{2}L - l^{4} - (L^{2} + l^{2} - 2lL)l^{2} \bigg) \\ &= \frac{P}{6EIL} \bigg( 2l^{4} - 2l^{3}L - 4l^{3}L + 4l^{2}L^{2} - l^{4} - l^{2}L^{2} - l^{4} + 2l^{3}L \bigg) \\ &= \frac{PL^{3}}{6EI} \bigg[ 3 \bigg( \frac{l}{L} \bigg)^{2} - 4\bigg(\frac{l}{L}\bigg)^{3}\bigg] \end{align}