Let's solve **Problem 03.053.a: Failure in section AB**, step by step. --- ## **Given Data** - \( P = 295.\ \text{N} \) (force at C, positive x-direction) - \( Q = 200.\ \text{N} \) (force at E, positive y-direction) - Diameter of bar ABD, \( d = 10.00\ \text{mm} \) - Find: **Precise location of the critical stress element at cross section A as an angle from the vertical y-axis.** --- ## **Step 1: Draw the Free Body Diagram** At section **A**, the bar is subjected to: - An axial force \( P \) (horizontal, x-direction) - A transverse force \( Q \) (vertical, y-direction) --- ## **Step 2: Find Stresses at Section A** ### **A. Axial Stress (\(\sigma_x\))** \[ \sigma_x = \frac{P}{A} \] Where \( A = \frac{\pi}{4}d^2 \). ### **B. Shear Stress (\(\tau_{xy}\))** \[ \tau_{xy} = \frac{Q}{A} \] --- ## **Step 3: Calculate Area \(A\)** \[ A = \frac{\pi}{4} d^2 = \frac{\pi}{4}(10.00\ \text{mm})^2 = \frac{\pi}{4} \times 100 = 78.54\ \text{mm}^2 \] --- ## **Step 4: Calculate Stresses** ### **A. Axial Stress** \[ \sigma_x = \frac{P}{A} = \frac{295.}{78.54} = 3.76\ \text{MPa} \] ### **B. Shear Stress** \[ \tau_{xy} = \frac{Q}{A} = \frac{200.}{78.54} = 2.55\ \text{MPa} \] --- ## **Step 5: Find Critical Plane (Maximum Normal Stress)** The critical location is where the maximum normal stress occurs. For combined axial and shear loading, the critical angle \(\theta\) (from the vertical y-axis) for maximum (or minimum) normal stress is given by: \[ \tan 2\theta = \frac{2\tau_{xy}}{\sigma_x} \] Plug in the values: \[ \tan 2\theta = \frac{2 \times 2.55}{3.76} = \frac{5.10}{3.76} = 1.356 \] So, \[ 2\theta = \arctan(1.356) = 53.9^\circ \] \[ \theta = \frac{53.9^\circ}{2} = 26.95^\circ \] --- ## **Step 6: Final Answer** **The precise location of the critical stress element will be at an angle of \(26.95^\circ\) from the vertical y-axis.** --- ### **Summary Table** | Stress | Value (MPa) | |-----------------|-------------| | \(\sigma_x\) | 3.76 | | \(\tau_{xy}\) | 2.55 | | Critical Angle | \(26.95^\circ\) | --- ### **Final Boxed Answer** \[ \boxed{26.95^\circ} \] **The critical stress element at section A will be at a \(26.95^\circ\) angle from the vertical y-axis.**