1. **Problem Statement:** Determine for rods AB and CD made of steel with failure stress $\sigma_{fail} = 510$ MPa and factor of safety $F.S.=1.75$: i. Internal forces in rods AB and CD ii. Allowable normal stress for rods AB and CD iii. Smallest diameter of rods AB and CD to sustain the applied loads The beam AC is pinned at A and C with vertical rods AB and CD at the ends. Loads of 4 kN, 6 kN, and 5 kN act downward at 2 m, 4 m, and 7 m from A respectively. 2. **Step 1: Calculate reactions at supports A and C** Sum of vertical forces: $R_A + R_C = 4 + 6 + 5 = 15$ kN Taking moments about A (counterclockwise positive): $$R_C \times 10 = 4 \times 2 + 6 \times 4 + 5 \times 7 = 8 + 24 + 35 = 67$$ $$R_C = \frac{67}{10} = 6.7 \text{ kN}$$ Then reaction at A: $$R_A = 15 - 6.7 = 8.3 \text{ kN}$$ 3. **Step 2: Internal forces in rods AB and CD** Since rods AB and CD are vertical and pinned at A and C, the internal forces equal the reactions: $$F_{AB} = R_A = 8.3 \text{ kN}$$ $$F_{CD} = R_C = 6.7 \text{ kN}$$ 4. **Step 3: Allowable normal stress** Allowable stress is failure stress divided by factor of safety: $$\sigma_{allow} = \frac{\sigma_{fail}}{F.S.} = \frac{510}{1.75} = 291.43 \text{ MPa}$$ 5. **Step 4: Smallest diameter of rods AB and CD** Normal stress formula: $$\sigma = \frac{F}{A} = \frac{4F}{\pi d^2}$$ Rearranged for diameter $d$: $$d = \sqrt{\frac{4F}{\pi \sigma_{allow}}}$$ Calculate for each rod (convert kN to N and MPa to N/mm²): - For AB: $$F = 8.3 \times 10^3 \text{ N}, \quad \sigma_{allow} = 291.43 \text{ N/mm}^2$$ $$d_{AB} = \sqrt{\frac{4 \times 8300}{\pi \times 291.43}} = \sqrt{36.18} = 6.02 \text{ mm}$$ - For CD: $$F = 6.7 \times 10^3 \text{ N}$$ $$d_{CD} = \sqrt{\frac{4 \times 6700}{\pi \times 291.43}} = \sqrt{29.25} = 5.41 \text{ mm}$$ 6. **Final answers:** - Internal force in AB: 8.3 kN - Internal force in CD: 6.7 kN - Allowable normal stress: 291.43 MPa - Smallest diameter of AB: 6.02 mm - Smallest diameter of CD: 5.41 mm These diameters ensure the rods can safely sustain the applied loads with the given factor of safety.