Statically Indeterminate Torsion Members:
Parallel linked composite shaft comprises two or more shafts in a manner that the torques on individual shafts is unequal fractions of a single or various applied torque. Such a system is statically indeterminate because additionally to equilibrium, relationship among the deformations or angles of twist shall have to be used to discover torques on separate portions of the composite shaft.
In Figure a composite shaft is illustrated which is built by connecting the portion AB and BC. The two parts are fixed respectively at A and C, & join at B. The join at B is perfect and rigid such that AB and BC may not rotate at B with respect to each other. This means that the rotation of section B with respect to A in part AB and rotation of section B with respect to C in portion BC have to be equal call this rotation, θ. Referring to Figure , you might note that if a torque T is applied at B, resulting in two reactions TA and TC, respectively in parts AB and BC.
T = TA + TC
Also note that
θAB = θCB = θ
with characteristics of shafts as explained in given Figure.
TA l1 / J1 G1 = TC l2 / J 2 G2 =θ
T = J1 G1 / l1 θ, k1 = J1 G1/ l1
TC = J2 G2 / l2 θ, k2= J2G2/ l2
From Eq. (i)
T= q(J1G1/l1 + J1G2/l2)
∴ T/θ = k = k1 + k2
