Problem $1$:

A laminated wood beam on simple supports is built up by gluing together $2in.x4in.$ and $3in.x4in.$ boards $($actual dimension$)$ to form a solid beam $4in.x8in.$ cross$-$section, as shown un figure below. The beam is $12$ ft long and is subjected to a load P at the one$-$third point along the beam. The wood has a weight density of $35l\frac{b}{f}{t}^3.$ The allowable shear stress in the wood is $175$ psi, the allowable bending stress in the wood is $1650$ psi, and allowable shear stress in the glue joints is $62$ psi. Determine:

a$.$ The allowable load $P$ based on the wood shear strength.

b$.$ The allowable load $P$ based on the wood bending strength.

c$.$ The allowable load $P$ based on the glue shear strength.