Specifications for the 6 in. × 6 in. square post shown in Figure P1.40 require that the normal and shear stresses on plane AB not exceed 800 psi and 400 psi, respectively. Determine the maximum load P that can be applied without exceeding the specifications.
The general equations for normal and shear stresses on an inclined plane in terms of the angle θ are
\sigma_{n}=\frac{P}{2 A}(1+\cos 2 \theta) (a)
and
\tau_{n t}=\frac{P}{2 A} \sin 2 \theta (b)
The cross-sectional area of the square post is A=(6 in .)^{2}=36 in .^{2} , and the angle θ for plane AB is 40°.
The normal stress on plane AB is limited to 800 psi; therefore, the maximum load P that can be supported by the square post is found from Eq. (a):
P \leq \frac{2 A \sigma_{n}}{1+\cos 2 \theta}=\frac{2\left(36 in. ^{2}\right)(800 psi )}{1+\cos 2\left(40^{\circ}\right)}=49,078 lbThe shear stress on plane AB is limited to 400 psi. From Eq. (b), the maximum load P based the shear stress limit is
P \leq \frac{2 A \tau_{n t}}{\sin 2 \theta}=\frac{2\left(36 in .^{2}\right)(400 psi )}{\sin 2\left(40^{\circ}\right)}=29,244 lbThus, the maximum load that can be supported by the post is
P_{\max }=29,200 lb =29.2 kips