Posted by
**Uncle Sam** on
.

The shaft ACB, of constant outer diameter 2R and length 2L, is fixed to walls at both ends A and B. The left half of the shaft (AC) is solid, while the right half (CB) is hollow, with an inner diameter 2√R. The right half, CB, is subjected to a uniform distributed torque per unit length q0[N⋅m/m]. The material of the shaft is linear isotropic elastic with shear modulus G.

Obtain symbolic expressions in terms of R, G, L, q0, and x for the quantities below. (In your answers, leave rationals as fractions and enter q0 and π as q_0 and pi, respectively.)

EQ2_1_1

The x-component of the reaction torques at A and B:

TxA=

TxB=

The axial torque resultant in the bar T(x), and the position x0 along the shaft where this torque resultant goes to zero (T(x0)=0):

for0≤x≤L,T(x)=

forL≤x≤2L,T(x)=

T(x0)=0 at x0=

EQ2_1_3

The maximum absolute value of the shear stress in the shaft (τmax) and its location (rτmax, xτmax):

τmax=

rτmax=

xτmax=

EQ2_1_4

the maximum value of the rotation field φ(x) along the shaft (φmax), and the position along the shaft where the maximum rotation occurs (xφmax):

φmax=

xφmax=