Question

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=

Answer 1

Q2_1_1
TXC=-t_0*L

Q2=1_2
a) 0<=x<L ===>> (2*t_0*L)/(pi*G_0*R4)

b)L<=x<=3L =>>(2*t_0*(2*L-))/pi*G_0*R^4)

c) x=2*L