## Analyse the state of stress at the critical section, Civil Engineering

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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, σx = Mymax/I = 5×106×40/2.016×106 = 99.206 N/mm2

Maximum shear stress τxy  = 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

τmax =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)

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