The stepped steel shaft carries the torque, T. Determine the maximum magnitude o

Question

The stepped steel shaft carries the torque, T. Determine the maximum magnitude of T if the working shear stress is 14 MPa and the rotation of the free end is limited to 3.5°. Use G = 83 GPa for steel. The larger section is 80 mm in diameter and 3 m long; the smaller section is 60 mm in diameter and 4 m long

The stepped steel shaft carries the torque, T. Determine the maximum magnitude of T if the working shear stress is 14 MPa and the rotation of the free end is limited to 3.5 degree. Use G = 83 GPa for steel. The larger section is 80 mm in diameter and 3 m long; the smaller section is 60 mm in diameter and 4 m long

Explanation / Answer

Here there are two limiting cases. one is for angle of twist and one is for torsional shear stress.

we need to find limiting values in both cases and choose the least value.

(a) T based on angle of twist

Angle of twist, = T*L/JG

J = pi/32 * D^4

= 1 + 2

1 = T * L1/(J1G)

2 = T * L2/(J2G)

L1 = 3 m , J1 = pi/32 * 0.08^4

L2 = 4 m, J2 = pi/32 * 0.06^4

3.5 *pi/180 = T * 3/(pi/32 * 0.08^4 *83 * 10^9) + T * 4/(pi/32 * 0.06^4 *83 * 10^9)

T = 3.5 * pi/180 /(3/(pi/32 * 0.08^4 *83 * 10^9) + 4/(pi/32 * 0.06^4 *83 * 10^9)) = 1303.44 N.m

(b) T based on stress

torsional shearing stress = T * r/J

for first part of cylinder,

14 * 10^6 = T * 0.04/(pi/32 * 0.08^4)

T = 14 * 10^6* (pi/32 * 0.08^4)/0.04 = 1407.43 N.m

for second part of cylinder

14 * 10^6 = T * 0.03/(pi/32 * 0.06^4)

T =  14 * 10^6* (pi/32 * 0.06^4)/0.03 = 593.76 N.m

minimum T must be considered.

so

T = 593.76 N.m

The minimmum case is occurs for shear. i.e, shearing stress increases above allowable limit before allowable angle of twist is acheived.

Maximum Torque, T = 593.76 N.m

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