Simply supported beam with two symmetric point loads

Figure 1: Diagram
Figure 1: Diagram

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 \(x = l\)

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