Let's break this problem down step by step: --- # Step 1: **Identify the Problem** Given: - Couple \( M = 18 \) kN·m = **18,000 N·m** - Rectangle section with: - \( h = 70 \) mm = **.070 m** - \( b = 25 \) mm = **.025 m** We are to find **the stress at points A and B**. --- # Step 2: **Find the Moment of Inertia (I)** For a rectangle about its centroid (neutral axis): \[ I = \frac{1}{12} b h^3 \] Plugging in the values (convert mm to m): \[ I = \frac{1}{12} \times .025 \times (.070)^3 \] \[ I = \frac{1}{12} \times .025 \times .000343 \] \[ I = \frac{1}{12} \times .000008575 \] \[ I = .0000007146 \ \text{m}^4 \] --- # Step 3: **Bending Stress Formula** Bending stress at a point is: \[ \sigma = \frac{M y}{I} \] Where: - \( M \): Moment (in N·m) - \( y \): Distance from the neutral axis to the point (in meters) - \( I \): Moment of inertia (in m\(^4\)) --- # Step 4: **Find Distance \( y \) for Points A and B** - **Neutral axis** is at the centroid: at \( \frac{h}{2} \) from either edge. - \( y \) is measured from the centroid: - For **point A**: - From centroid to A: \( y_A = \frac{h}{2} - .020 = .035 - .020 = .015 \) m (since centroid is 35 mm from bottom, A is 20 mm above centroid). - For **point B**: - From centroid to B: \( y_B = -\frac{h}{2} = -.035 \) m (since B is at the very bottom, so it's -35 mm from centroid). --- # Step 5: **Calculate Stress at Each Point** ## At Point A \[ \sigma_A = \frac{M y_A}{I} = \frac{18000 \times .015}{.0000007146} \] \[ \sigma_A = \frac{270}{.0000007146} = 3.78 \times 10^8 \ \text{N/m}^2 = 378 \ \text{MPa} \] ## At Point B \[ \sigma_B = \frac{M y_B}{I} = \frac{18000 \times (-.035)}{.0000007146} \] \[ \sigma_B = \frac{-630}{.0000007146} = -8.82 \times 10^8 \ \text{N/m}^2 = -882 \ \text{MPa} \] --- # Step 6: **Interpretation** - **Positive stress**: tension - **Negative stress**: compression --- ## **Final Answers** **(a) At A:** \[ \boxed{\sigma_A = 378 \ \text{MPa} \ \text{(tension)}} \] **(b) At B:** \[ \boxed{\sigma_B = -882 \ \text{MPa} \ \text{(compression)}} \] --- Let me know if you need the calculations double-checked or more explanation on any step!