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

MODELING and ANALYSIS:
Properties of Cross Section. The cross section is shown in Fig. 1, and the following data are taken from Appendix E.

Area: A = 7.46 in²

Section moduli: S_x=24.7  in ^3 \quad S_y=2.91  in ^3

Force and Couple at C. Using Fig. 2, we replace 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 M _x \text { and } 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 P and to the couples M _x \text { and } M _y  are

\begin{aligned} & \sigma_1=\frac{P}{A}=\frac{P}{7.46  in ^2}=0.1340  P \\ & \sigma_2=\frac{M_x}{S_x}=\frac{4.75  P}{24.7  in ^3}=0.1923  P \\ & \sigma_3=\frac{M_y}{S_y}=\frac{1.5  P}{2.91  in ^3}=0.5155  P \end{aligned}

Superposition. The total stress at each point is found by superposing the stresses due to P, M _x \text { and } 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_{ all } = −12 ksi, we write

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