고체역학 (Gere) — Stresses in Beams (Basic Topics): Problem 5.5-23

5.5-23 A beam ABC with an overhang from B to C supports a uniform load of 3 kN/m throughout its length (see figure). The beam is a channel section with dimensions as shown in the figure. The moment of inertia about the z axis (the neutral axis) equals 210 cm\(^4\). (a) Calculate the maximum tensile stress \(\sigma_t\) and maximum compressive stress \(\sigma_c\) due to the uniform load. (b) Find the required span length \(a\) that results in the ratio of larger to smaller compressive stress being equal to the ratio of larger to smaller tensile stress. Assume that the total length \(L = a + b = 6\) m remains unchanged. Given values: \(q = 3 \text{ kN/m}\) \(L = 6 \text{ m}\) \(a = 4 \text{ m}\), \(b = 2 \text{ m}\) \(I_z = 210 \text{ cm}^4 = 2.1 \times 10^6 \text{ mm}^4\) Distance from neutral axis to top fiber, \(c_{top} = 17 \text{ mm}\) Distance from neutral axis to bottom fiber, \(c_{bot} = 63 \text{ mm}\)