State of stress at the critical section:

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

Solution

Moment of the inertia, I =  P D⁴ /64  =  (P * 80⁴ )/ 64 =  2.016 * 10⁶  mm⁴

Polar moment of inertia, J =  P D⁴ /64= (P/32) *80⁴ = 4.032 * 10⁶  mm⁴

Maximum bending stress,  s_x = My_max / I

                                                = 5 *10⁶ * 40 / (2.016 * 10⁶)  =  99.206 N/mm²

Maximum shear stress t_xy =( 7 * 10 * 40 )/ (4.032 * 10⁶  )=  69.444 N/mm²

Determination of Principal Stress

Therefore, we get

s₁  = 134.944 N/mm   and s₂  = - 35.737 N/mm²

t _max =  ( s₁ - s₂)/2   =85.34 N/mm²

Note

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

I =  P/64 (D⁴  - d ⁴ ) and  J  = P/32  (D⁴  - d ⁴ )