# Optimal placement of two support points along a wooden beam

Assuming a beam that is supported at 2 points as per attached image, what would be Lc (length center) and Lo (length overhang) for maximum strength. (or load bearing capability)

For actual material and values, assume that the beam is a douglas fir 4" x 8", the total length is 20 ft, and the load on the beam is uniform.

Let's call the distance from the end of the beam to the supports $$C$$ and the middle part $$B$$ and the load $$w$$.
We need to place the supports at a distance from the end that the moment at the support is half the moment of a simply supported beam with a span of $$B$$ at mid-span. This way the maximum negative and positive moments are equal and $$w$$ is maximum.
\begin{align} M_c &= \frac{wC^2}{2} \\ M_B &= \frac{wB^2}{8} \\ \frac{wB^2}{8} &= 2 \frac{wC^2}{2} \\ wB^2 &= 8 wC^2 \\ B^2 &= 8C^2 \\ B &= \sqrt8 \cdot C \\ B &= 2.828 \cdot C \end{align}
Therefore the point C is at $$\dfrac{1}{2+2.828}$$ of the length of the beam.