# Shaft Stress Analysis: Python Code Below is a **Python code** for the **stress analysis of a stepped shaft** subjected to gear forces as shown above. The code includes calculations for: - Bending Normal Stress - Torsional Shear Stress - Transverse Shear Stress - Principal Plane Stress - Maximum In-plane Shear Stress - Analytical and Graphical Solutions (Mohr’s Circle) You will need `numpy` and `matplotlib` for this code. --- ```python import numpy as np import matplotlib.pyplot as plt # --- INPUT PARAMETERS (update as needed) --- # Forces (N) F_A = 11000 # Force at Gear A (N) F_B = 800 # Example: Force at Gear B (N) -- change as per your problem theta_A = np.radians(20) # Force angle for A (degrees to radians) theta_B = np.radians(25) # Force angle for B (degrees to radians) # Dimensions (mm to m) d_shaft = .05 # Shaft diameter in meters (50 mm) r_shaft = d_shaft / 2 section_modulus = np.pi * (d_shaft**3) / 32 # Distance from support to gear (m) L_A = .400 # 400 mm L_B = .350 # 350 mm # Torque (Nm) T_A = F_A * L_A * np.cos(theta_A) # Applied torque at A (N·m) T_B = F_B * L_B * np.cos(theta_B) # Applied torque at B (N·m) # Bending Moment (Nm) M_A = F_A * L_A * np.sin(theta_A) M_B = F_B * L_B * np.sin(theta_B) # Transverse Shear (N) V_A = F_A * np.sin(theta_A) V_B = F_B * np.sin(theta_B) # --- STRESS CALCULATIONS --- # 1. Bending Normal Stress (sigma = My/I) I = np.pi * r_shaft**4 / 4 y = r_shaft sigma_bending = (M_A + M_B) * y / I # 2. Torsional Shear Stress (tau = Tc/J) J = np.pi * r_shaft**4 / 2 tau_torsion = (T_A + T_B) * r_shaft / J # 3. Transverse Shear Stress (tau = V*A'/Ib) A_prime = (3/2) * (np.pi * r_shaft**2) # Shear area approx for solid shaft tau_shear = (V_A + V_B) * A_prime / (I * d_shaft) # --- PRINCIPAL STRESSES --- sigma_x = sigma_bending tau_xy = tau_torsion + tau_shear sigma_y = sigma_1 = .5*(sigma_x + sigma_y) + np.sqrt( (.5*(sigma_x - sigma_y))**2 + tau_xy**2 ) sigma_2 = .5*(sigma_x + sigma_y) - np.sqrt( (.5*(sigma_x - sigma_y))**2 + tau_xy**2 ) tau_max_inplane = .5 * (sigma_1 - sigma_2) # --- MOHR'S CIRCLE PLOT --- fig, ax = plt.subplots() center = .5*(sigma_x + sigma_y) radius = np.sqrt( (.5*(sigma_x - sigma_y))**2 + tau_xy**2 ) circle = plt.Circle((center, ), radius, fill=False, color='blue', linestyle='--') ax.add_artist(circle) ax.plot([sigma_x, sigma_y], [tau_xy, -tau_xy], 'ro') ax.set_aspect('equal', 'box') plt.title("Mohr's Circle for Shaft Stress") plt.xlabel("Normal Stress (Pa)") plt.ylabel("Shear Stress (Pa)") plt.grid(True) plt.show() # --- OUTPUT --- print(f"Bending Normal Stress: {sigma_bending:.2f} Pa") print(f"Torsional Shear Stress: {tau_torsion:.2f} Pa") print(f"Transverse Shear Stress: {tau_shear:.2f} Pa") print(f"Principal Stresses: σ1 = {sigma_1:.2f} Pa, σ2 = {sigma_2:.2f} Pa") print(f"Maximum In-plane Shear Stress: {tau_max_inplane:.2f} Pa") ``` --- ## **How to Use** 1. **Change Input Values**: Adjust forces, angles, and dimensions as per your problem. 2. **Run the Code**: The results will output the *stresses* and plot *Mohr’s Circle*. 3. **Interpret Results**: Use the output to assess shaft safety and failure criteria. --- > **Note**: For a precise solution, use the distributed loadings, reactions, and actual shaft geometry as per your detailed engineering problem. This code gives a **framework**—expand as needed for more sections or loading cases. --- **If you need a more specific solution for this exact shaft (with all forces and dimensions fully detailed), let me know!**