The composite shaft ABCD is fixed between walls at A (x=0) and D (x=3L). The shaft has a solid steel core of length 3L and round cross section of uniform radius R. In the left third of the shaft (segment AB of length L) the steel core is surrounded by a copper sleeve of inner radius R perfectly bonded to the core. The outer radius of the copper sleeve is 2R.
A uniform distributed torque per unit length of magnitude q[N⋅m/m] acts only on the central segment of the shaft, BC of length L, as indicated in the figure.
The shear modulus of the steel core is Gcore=G0, while the shear modulus of the copper sleeve is Gsleeve=(1/2)G0.
The given KNOWN quantities are L[m], R[m], q[N·m/m] and G0[Pa] (enter this as G_0).
1) Obtain a symbolic expression for the x-component of the reaction torque at D in terms of q and L:
TxD=
2) Obtain expressions for the twist rate dφdx(x), (in terms of q, x, L, G0, and R), and obtain the position x0 along the shaft where the twist rate goes to zero (dφdx(x0)=0), (in terms of L).
If you have factors of π in your answer, enter π as pi.
for0≤x<L,dφdx(x)=
unanswered
forL<x≤2L,dφdx(x)=
unanswered
for2L≤x≤3L,dφdx(x)=
unanswered
dφdx(x0)=0atx0=
unanswered
3) Obtain a symbolic expression for the maximum absolute value of the shear stress in the shaft, τmax, (in terms of q, L, and R).
If you have factors of π in your answer, enter π as pi.
τmax=
4) Obtain symbolic expressions for the maximum value of the rotation field φ(x) along the shaft φmax (in terms of q, L, G0, and R), and the position along the shaft where the maximum rotation occurs, xφmax (in terms of L).
If you have factors of π in your answer, enter π as pi.
φmax=
unanswered
xφmax=