고체역학 (Gere) — Stresses in Beams (Advanced Topics): Problem 6.5-9

6.5-9 A beam with a semicircular cross section of radius \(r\) is subjected to a bending moment \(M\) having its vector at an angle \(\theta\) to the \(z\) axis (see figure). Derive formulas for the maximum tensile stress \(\sigma_t\) and the maximum compressive stress \(\sigma_c\) in the beam for \(\theta = 0, 45^\circ\), and \(90^\circ\). Express the results in the form \(\alpha M/r^3\), where \(\alpha\) is a numerical value.