(a) Determine an expression for the maximum (cross-sectional) shear stress in th

Question

(a) Determine an expression for the maximum (cross-sectional) shear stress in the tapered shaft as a function of the distance x from the left end.

(b) Determine an expression for the total angle of twist of the shaft, (psi)

3. The solid shaft in the figure below has a shear modulus G and has an outer diameter that varies linearly from d0 at x = 0 to 2d0 at x = L. A circular hole of diameter d0/2 is drilled into the rod along its whole length L, as shown in the insert, giving the resulting tubular shaft a linearly varying wall thickness. It is subjected to a torque T0 at x = 0, and is attached to a rigid wall at x = L. (a) Determine an expression for the maximum (cross-sectional) shear stress in the tapered shaft as a function of the distance x from the left end. (b) Determine an expression for the total angle of twist of the shaft, (psi)

Explanation / Answer

 J = d4/32 - (d0/2)4/32 =[ d0(1+x/L)]4/32- (d0/2)4/32

J = d4/32 - (d0/2)4/32 =(d04 /32) [ (1+x/L)4- 1/16]

0=T/(GJ) dx = (T/G)0L 1/{(d04 /32) [ (1+x/L)4- 1/16]}dx

    =64TL[2tan-1(2)-2tan-1(4)+ln(9/5)]/(Gd04) = 3.06492TL/(Gd04)

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