고체역학 (Gere) — Stresses in Beams (Advanced Topics): Problem 6.5-12

6.5-12 A UPN 220 channel section has an angle with equal legs attached as shown; the angle serves as a lintel beam. The combined steel section is subjected to a bending moment $M$ having its vector directed along the $z$ axis, as shown in the figure. The centroid $C$ of the combined section is located at distances $x_c$ and $y_c$ from the centroid ($C_1$) of the channel alone. Principal axes $x_1$ and $y_1$ are also shown in the figure and properties $I_{x1}$, $I_{y1}$, and $\theta_p$ are given. Find the orientation of the neutral axis and calculate the maximum tensile stress $\sigma_t$ and maximum compressive stress $\sigma_c$ if the angle is an L $90 \times 90 \times 7$ section and $M = 3.5 \text{ kN}\cdot\text{m}$. Use the following properties for principal axes for the combined section: $I_{x1} = 35.14 \times 10^6 \text{ mm}^4$, $I_{y1} = 4.265 \times 10^6 \text{ mm}^4$, $\theta_p = 7.826^\circ$ (CW), $x_c = 11.32 \text{ mm}$, and $y_c = 21.08 \text{ mm}$.