고체역학 (Gere) — Torsion: Problem 3.4-21

3.4-21 A nonprismatic bar $ABC$ with a solid circular cross section is loaded by distributed torques (see figure). The intensity of the torques, that is, the torque per unit distance, is denoted $t(x)$ and varies linearly from zero at $A$ to a maximum value $T_0/L$ at $B$. Segment $BC$ has linearly distributed torque of intensity $t(x) = T_0/3L$ of opposite sign to that applied along $AB$. Also, the polar moment of inertia of $AB$ is twice that of $BC$, and the shear modulus of elasticity of the material is $G$. (a) Find the reaction torque $R_A$. (b) Find internal torsional moments $T(x)$ in segments $AB$ and $BC$. (c) Find the rotation $\phi_C$. (d) Find the maximum shear stress $\tau_{max}$ and its location along the bar. (e) Draw the torsional moment diagram (TMD: $T(x), 0 \le x \le L$).