Analyse the state of stress at the critical section:

A prismatic bar of circular section with 80 mm diameter is subjected to bending moment of 5 kN-m and a torque of 7 kN-m. Analyse the state of stress at the critical section.

Solution

Moment of the inertia, I = πD⁴/64 = π×80⁴/64 = 2.016×10⁶ mm⁴

Polar moment of inertia, J = πD⁴/32 = π/32 ×80⁴ = 4.032×10⁶ mm⁴

Maximum bending stress, σ_x = My^max/I = 5×10⁶×40/2.016×10⁶ = 99.206 N/mm²

Maximum shear stress τ_xy  = 7 × 10⁶ × 40/4.032×10⁶ =69.444 N/mm²

Determination of Principal Stress

Thus, we get

σ₁  = 134.944 N/mm²  and σ₂ = -35.737 N/mm²

τ_max =σ1 -σ 2/2 = 85.34 N/mm²

Note

While the cross-section of the bar is hollow circular along with outer and inner diameters D and d respectively the analysis of stresses is to be carried through the similar procedure except for using the expressions,

 I = π/64 (D⁴ -d⁴) and J =π/32 (D⁴ -d⁴)