Question 4.SP.9: A horizontal load P is applied as shown to a short section o...

Properties of Cross Section. The following data are taken from Appendix C.

\begin{aligned} & \text { Area: } A=7.46  \mathrm{in}^{2} \\ & \text { Section moduli: } S_{x}=24.7  \mathrm{in}^{3} \quad S_{y}=2.91  \mathrm{in}^{3} \end{aligned}

Force and Couple at \boldsymbol{C}. We replace \mathbf{P} by an equivalent force-couple system at the centroid C of the cross section.

M_{x}=(4.75 \text { in. }) P \quad M_{y}=(1.5 \text { in. }) P

Note that the couple vectors \mathbf{M}_{x} and \mathbf{M}_{y} are directed along the principal axes of the cross section.

Normal Stresses. The absolute values of the stresses at points A, B, D, and E due, respectively, to the centric load \mathbf{P} and to the couples \mathbf{M}_{x} and \mathbf{M}_{y} are

\begin{gathered} \sigma_{1}=\frac{P}{A}=\frac{P}{7.46  \mathrm{in}^{2}}=0.1340 P \\ \sigma_{2}=\frac{M_{x}}{S_{x}}=\frac{4.75 P}{24.7  \mathrm{in}^{3}}=0.1923 P \\ \sigma_{3}=\frac{M_{y}}{S_{y}}=\frac{1.5 P}{2.91  \mathrm{in}^{3}}=0.5155 P \end{gathered}

Superposition. The total stress at each point is found by superposing the stresses due to \mathbf{P}, \mathbf{M}_{x}, and \mathbf{M}_{y}. We determine the sign of each stress by carefully examining the sketch of the force-couple system.

Largest Permissible Load. The maximum compressive stress occurs at point E. Recalling that \sigma_{\text {all }}=-12  \mathrm{ksi}, we write

\sigma_{\text {all }}=\sigma_{E} \quad-12  \mathrm{ksi}=-0.842 P \quad P=14.3  \mathrm{kips}