1. **Problem 1: Maximum point load on simply supported beam** Given: - Beam dimensions: 356 mm × 171 mm × 56.7 kg/m - Effective span, $L = 12\,000$ mm = 12 m - Bending stress limit, $\sigma = 164$ MPa **Step 1: Calculate the section modulus $Z$** The bending stress formula is: $$\sigma = \frac{M}{Z}$$ where $M$ is the bending moment and $Z$ is the section modulus. **Step 2: Calculate the maximum bending moment for a central point load $P$ on a simply supported beam:** $$M = \frac{P L}{4}$$ **Step 3: Calculate the self-weight of the beam:** Weight per meter $w_s = 56.7$ kg/m Convert to Newtons: $w_s = 56.7 \times 9.81 = 556.53$ N/m **Step 4: Calculate the section modulus $Z$ from bending stress and moment:** Rearranged formula: $$Z = \frac{M}{\sigma}$$ **Step 5: Calculate the maximum moment $M$ from allowable bending stress and $Z$:** Since $Z$ is not given explicitly, we use the relation: $$M = \sigma Z$$ **Step 6: Calculate the maximum point load $P$ considering self-weight:** Total load includes point load $P$ and distributed load $w_s$. The maximum moment due to self-weight is: $$M_s = \frac{w_s L^2}{8}$$ Total moment at center: $$M = \frac{P L}{4} + M_s$$ Set $M = \sigma Z$ and solve for $P$: $$\sigma Z = \frac{P L}{4} + \frac{w_s L^2}{8}$$ $$P = \frac{4}{L} \left( \sigma Z - \frac{w_s L^2}{8} \right)$$ **Step 7: Calculate $Z$ from beam properties:** Given the beam weight per meter and dimensions, approximate $Z$ or use standard tables (not provided here). For this problem, assume $Z$ is known or calculated from cross-section. **Final answer:** Calculate $P$ using the above formula with known $Z$. --- 2. **Problem 2: Maximum uniformly distributed load on girder beam** Given: - Beam length $L = 12$ m - Bending stress limit $\sigma = 162$ MPa - Beam cross-section: compound I-section with plates - Density of steel $\rho = 7\,870$ kg/m³ **Step 1: Calculate the self-weight of the compound beam** Calculate volume per meter length from cross-section dimensions and thicknesses, then multiply by density and gravity to get self-weight $w_s$ in N/m. **Step 2: Calculate section modulus $Z$ for the compound section** Use geometric properties of plates and I-section to find $Z$ (not detailed here). **Step 3: Maximum bending moment for uniformly distributed load $w$ on simply supported beam:** $$M = \frac{w L^2}{8}$$ **Step 4: Total load $w$ includes self-weight $w_s$ and external load $w_u$:** $$w = w_u + w_s$$ **Step 5: Set bending stress limit:** $$\sigma = \frac{M}{Z} = \frac{w L^2}{8 Z}$$ Rearranged to find $w$: $$w = \frac{8 \sigma Z}{L^2}$$ **Step 6: Calculate maximum uniformly distributed load $w_u$:** $$w_u = w - w_s$$ **Final answer:** Calculate $w_u$ using the above formula with known $Z$ and $w_s$. --- **Note:** Detailed numerical calculations require the exact section modulus $Z$ and volume calculations from the beam dimensions, which are not fully provided here.