Example 5a.1
Given: A steppedshaft (polar moment of inertias J_{1},
J_{2}) made of two materials (shear moduli G_{1}, G_{2}).
The shaft is fixed (no rotation) at the top. Torques are applied about
the shaft's axis: 3T (counterclockwise about the +yaxis at Pt. B) and
T (clockwise about the +yaxis at Pt. E).
Req'd: Determine the total angle of twist of the bar, q_{total}.
Sol'n:
Step 1. Equilibrium.
Solve for the reaction at Pt. A: R = 2T (clockwise about +yaxis, a negative
torque).
Step 2. Twist/Torque of each Segment.
Break up the bar into segments over which all the values (torque, crosssection
and modulus) are constant.
The top segment, AB, has length, a, Polar Moment of
Inertia, J_{1}, modulus, G_{1}, and supports torque 2T
(a negative torque on the positive face of AB).
With Crosssection A fixed, the rotation of Section B with
respect to fixed Section A is:
Taking each of the other segments with constant torque, section
and modulus:
Step 3. Compatibility.
The total angle of twist:

Free
Body Diagrams
of shaft segments.
Angular deflection
of steppedshaft.
