A solid steel shaft with a diameter of 50 mm and a length of 500 mm is loaded in torsion. Determine the maximum torque that can be applied to the shaft before yielding occurs according to the Von Mises criterion, given that the steel has a yield strength of 400 Mpa.

To determine the maximum torque that can be applied to the steel shaft before yielding occurs according to the von Mises criterion, we need to calculate the maximum allowable shear stress using the yield strength of the steel.

Given: Diameter of the steel shaft = 50 mm = 0.05 m Length of the steel shaft = 500 mm = 0.5 m Yield strength of the steel = 400 MPa = 400,000,000 Pa

To calculate the maximum allowable shear stress (τ_max), we first need to determine the polar moment of inertia (J) for a solid shaft.

Polar moment of inertia (J) = (π/32) * (diameter)^4 J = (π/32) * (0.05 m)^4 = 6.25 × 10^(-8) m^4

Next, we can calculate the maximum allowable shear stress using the yield strength of the steel.

τ_max = (Yield strength) / √(2 * J)

τ_max = 400,000,000 Pa / √(2 * 6.25 × 10^(-8) m^4) τ_max = 400,000,000 Pa / √(1.25 × 10^(-7) m^4) τ_max ≈ 40,000,000 Pa

Now, we can calculate the maximum torque (T_max) that can be applied to the shaft.

T_max = τ_max * J / (diameter / 2)

T_max = 40,000,000 Pa * (6.25 × 10^(-8) m^4) / (0.05 m / 2) T_max ≈ 50 Nm

Therefore, based on the given information and using the von Mises criterion, the maximum torque that can be applied to the steel shaft before yielding occurs is approximately 50 Nm.