Let's solve the problem step-by-step as requested. **Given Data:** - For the rectangular cross-section beam: - Case 1: \( h = 70.\,\text{mm},\, b = 25.\,\text{mm} \) - Case 2: \( h = 25.\,\text{mm},\, b = 70.\,\text{mm} \) - Crack size: \( a = 7.\,\text{mm} \) - Material properties: - \( S_y = 550.\,\text{MPa} \) - \( K_{Ic} = 69.4025\,\text{MPa}\sqrt{\text{m}} \) - Formulas: - \( K_I = \beta \sigma \sqrt{\pi a} \) - \( \sigma_{bending} = \frac{M c}{I} \) - \( I = \frac{b h^3}{12} \) - \( c = \frac{h}{2} \) ### 1. **Case 1: \( h = 70.\,\text{mm},\, b = 25.\,\text{mm} \)** #### **a) Value of moment \( M \) causing fracture:** - \( a/h = 7/70 = .1 \) - From the graph, for \( a/h = .1 \) and \( L/h = 2.8 \), \( \beta \approx 1.2 \). - \( I = \frac{25 \times 70^3}{12} = \frac{25 \times 343000}{12} = 714583\,\text{mm}^4 \) - \( c = 35\,\text{mm} \) - For fracture: \( K_I = K_{Ic} \) - \( K_{Ic} = \beta \sigma \sqrt{\pi a} \implies \sigma = \frac{K_{Ic}}{\beta \sqrt{\pi a}} \) - \( \sigma = \frac{69.4025}{1.2 \times \sqrt{\pi \times .007}} = \frac{69.4025}{1.2 \times .148} = \frac{69.4025}{.1776} = 390.9\,\text{MPa} \) - \( M = \sigma \frac{I}{c} = 390.9 \times \frac{714583}{35} = 7,983,333\,\text{Nmm} = 7.98\,\text{kN-m} \) #### **b) Value of moment \( M \) causing yield:** - \( \sigma = S_y = 550 \,\text{MPa} \) - \( M = 550 \times \frac{714583}{35} = 11,241,000\,\text{Nmm} = 11.24\,\text{kN-m} \) #### **c) What will occur first: fracture or yield?** - Fracture occurs at \( 7.98\,\text{kN-m} \), yield at \( 11.24\,\text{kN-m} \) - **Fracture will occur first.** --- ### 2. **Case 2: \( h = 25.\,\text{mm},\, b = 70.\,\text{mm} \)** #### **d) Value of moment \( M \) causing fracture:** - \( a/h = 7/25 = .28 \) - From the graph for \( a/h = .28 \) and \( L/h = 2 \), \( \beta \approx 1.35 \) - \( I = \frac{70 \times 25^3}{12} = \frac{70 \times 15625}{12} = 91,146\,\text{mm}^4 \) - \( c = 12.5\,\text{mm} \) - \( \sigma = \frac{K_{Ic}}{\beta \sqrt{\pi a}} = \frac{69.4025}{1.35 \times \sqrt{\pi \times .007}} = \frac{69.4025}{1.35 \times .148} = \frac{69.4025}{.1998} = 347.5\,\text{MPa} \) - \( M = \sigma \frac{I}{c} = 347.5 \times \frac{91,146}{12.5} = 2,533,687\,\text{Nmm} = 2.53\,\text{kN-m} \) #### **e) Value of moment \( M \) causing yield:** - \( M = 550 \times \frac{91,146}{12.5} = 4,011,624\,\text{Nmm} = 4.01\,\text{kN-m} \) #### **f) What will occur first: fracture or yield?** - Fracture at \( 2.53\,\text{kN-m} \), yield at \( 4.01\,\text{kN-m} \) - **Fracture will occur first.** --- ### 3. **g) If moment reverses in (a) and (d), will the crack propagate?** - If the moment reverses, the crack moves from tension to compression side. Cracks do **not** propagate in compression. - **No, the crack will not propagate.** --- ### 4. **h) Compare ratios of \( M_B/M_F \) and conclusions** - \( M_B \): bending moment causing yield - \( M_F \): bending moment causing fracture For case 1: \( \frac{M_B}{M_F} = \frac{11.24}{7.98} = 1.41 \) For case 2: \( \frac{M_B}{M_F} = \frac{4.01}{2.53} = 1.59 \) - **Conclusion:** The wider, shorter section (case 2) has a higher ratio, so it is more likely to fracture before yielding compared to the taller, narrower section. --- ### 5. **i) Compare ratios \( M_{B1}/M_{B2} \) and \( M_{F1}/M_{F2} \)** - \( M_{B1}/M_{B2} = 11.24/4.01 = 2.80 \) - \( M_{F1}/M_{F2} = 7.98/2.53 = 3.15 \) - **Conclusion:** The taller/narrower beam (case 1) is much more resistant to both yielding and fracture in bending than the shorter/wider beam (case 2), but the increase in resistance is more pronounced for fracture. --- ## **Summary Table** | Case | Fracture \(M_F\) (kN-m) | Yield \(M_B\) (kN-m) | Failure Mode First | |------|-------------------------|----------------------|-------------------| | 1 | ~7.98 | ~11.24 | Fracture | | 2 | ~2.53 | ~4.01 | Fracture | --- ## **Key Points** - **Fracture occurs before yielding in both cases.** - **Wider (b), shorter (h) beams are less resistant to both yielding and fracture.** - **Cracks do not propagate in compression.** - **Geometry dramatically affects failure mode and resistance.**