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 = πD^{4}/64 = π×80^{4}/64 = 2.016×10^{6} mm^{4}
Polar moment of inertia, J = πD^{4}/32 = π/32 ×80^{4} = 4.032×10^{6} mm^{4}
Maximum bending stress, σ_{x} = My^{max}/I = 5×10^{6}×40/2.016×10^{6} = 99.206 N/mm^{2}
Maximum shear stress τ_{xy } = 7 × 10^{6} × 40/4.032×10^{6} =69.444 N/mm^{2}
Determination of Principal Stress
Thus, we get
σ_{1} = 134.944 N/mm^{2} and σ_{2} = -35.737 N/mm^{2}
τ_{max} =σ1 -σ 2/2 = 85.34 N/mm^{2}
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 (D^{4} -d^{4}) and J =π/32 (D^{4} -d^{4})