# Rods in Tension: Stress and Safety Factor Calculations Given: - Each rod diameter \( d = 0.5 \) inch - Load at C: \( F = 4500 \) pounds - Angles: Left rod (A) at \( 35^\circ \), Right rod (B) at \( 60^\circ \) - Two materials: SAE 1040 A steel, 6061-T6 aluminum alloy Let's proceed step by step. --- ## (a) **Calculate the Stresses in Each Rod** ### **Step 1: Resolve Forces in Each Rod** Let tension in rod A be \( T_A \), and in rod B be \( T_B \). **Vertical equilibrium:** \[ T_A \sin(35^\circ) + T_B \sin(60^\circ) = 4500 \] **Horizontal equilibrium:** \[ T_A \cos(35^\circ) = T_B \cos(60^\circ) \] \[ T_A \cos(35^\circ) = T_B \times 0.5 \] \[ T_A = \frac{T_B \times 0.5}{\cos(35^\circ)} \] \[ T_A \approx \frac{T_B \times 0.5}{0.8192} \approx 0.610T_B \] **Plug into vertical equilibrium:** \[ (0.610T_B) \sin(35^\circ) + T_B \sin(60^\circ) = 4500 \] \[ 0.610T_B \times 0.5736 + T_B \times 0.8660 = 4500 \] \[ 0.3499T_B + 0.8660T_B = 4500 \] \[ 1.2159T_B = 4500 \] \[ T_B = \frac{4500}{1.2159} \approx 3702~\text{lbs} \] \[ T_A = 0.610 \times 3702 \approx 2258~\text{lbs} \] ### **Step 2: Calculate Stress in Each Rod** **Area of each rod:** \[ A = \frac{\pi}{4} d^2 = \frac{\pi}{4} (0.5)^2 \approx 0.1963~\text{in}^2 \] **Stress:** \[ \text{Stress}_A = \frac{T_A}{A} = \frac{2258}{0.1963} \approx 11,510~\text{psi} \] \[ \text{Stress}_B = \frac{T_B}{A} = \frac{3702}{0.1963} \approx 18,868~\text{psi} \] --- ## (b) **SAE 1040 A Steel (Repeated Loading): Safety Factor** - **Endurance Limit (\( S_e \))**: For repeated loading, use endurance limit. - Typical \( S_e \) for SAE 1040 A steel ≈ **30 ksi** = 30,000 psi (conservative for rotating bending; actual value may depend on surface/size factors, but this is standard for calculation) ### **Safety Factors:** \[ n_A = \frac{S_e}{\text{Stress}_A} = \frac{30,000}{11,510} \approx 2.6 \] \[ n_B = \frac{30,000}{18,868} \approx 1.6 \] **Are the rods safe?** - **Typical minimum safety factor for fluctuating loads is 2**. - **Rod A:** Safe (\( n_A > 2 \)) - **Rod B:** Marginal/Not safe (\( n_B < 2 \)), risk of fatigue failure. --- ## (c) **6061-T6 Aluminum Alloy (Repeated Loading): Safety Factor** - **Endurance Limit (\( S_e \))**: For aluminum, use fatigue strength at \( 5 \times 10^8 \) cycles. - Typical \( S_e \) for 6061-T6 ≈ **9 ksi** = 9,000 psi ### **Safety Factors:** \[ n_A = \frac{9,000}{11,510} \approx 0.78 \] \[ n_B = \frac{9,000}{18,868} \approx 0.48 \] **Are the rods safe?** - **No, both safety factors are well below 2.** - **Both rods are NOT safe** if made from 6061-T6 aluminum under repeated loading. --- ## **Summary Table** | Rod | Tension (lbs) | Stress (psi) | Steel SF | Steel Safe? | Alu SF | Alu Safe? | |-------|---------------|--------------|----------|-------------|--------|-----------| | **A** | 2258 | 11,510 | 2.6 | Yes | 0.78 | No | | **B** | 3702 | 18,868 | 1.6 | No | 0.48 | No | --- ## **Conclusion** - **(a)** Stresses: Rod A = **11,510 psi**, Rod B = **18,868 psi** - **(b)** SAE 1040 A steel: Rod A is safe, Rod B is not (under repeated load) - **(c)** 6061-T6 aluminum: Neither rod is safe (under repeated load) --- **If you need calculations for static (non-repeated) loading, let me know!**