Pure torsion — identify the statement that is NOT an assumption Which one of the following statements is NOT an assumption of the elementary (pure) torsion theory for a uniform circular shaft?
-
AMaterial is homogeneous, isotropic, and obeys Hooke’s law in shear
-
BCross-sections that are plane and circular before twist remain plane and circular after twist
-
CRadial lines remain straight and rotate as rigid lines about the axis
-
DShear stress is uniform over the cross-section at a given torque
-
EThe rate of twist per unit length is constant for a prismatic shaft under constant torque
Answer
Correct Answer: Shear stress is uniform over the cross-section at a given torque
Explanation
Introduction / Context:Elementary torsion theory underpins the design of shafts. It relies on a set of assumptions that lead to the familiar torsion formula tau = T r / J and angle-of-twist relation.
Given Data / Assumptions:
- Uniform circular shaft subjected to pure torque.
- Saint-Venant torsion conditions.
- No warping for circular sections.
Concept / Approach:The correct assumptions include material linearity and isotropy, plane circular sections remaining plane, straight radial lines, and (for constant T and constant GJ) a constant rate of twist. However, the shear stress is not uniform over the section; it varies linearly with radius: tau(r) = T r / J, being zero at the axis and maximum at the surface.
Step-by-Step Solution:
Recall the torsion formula: tau(r) = T r / J.Observe that tau depends on radius r ⇒ not uniform.Therefore, any statement claiming uniform shear across the section is not an assumption of pure torsion theory.Verification / Alternative check:The torque equilibrium integral T = ∫(tau * r) dA requires tau ∝ r for circular shafts to satisfy compatibility.
Why Other Options Are Wrong:
- Options a, b, c, and e are standard assumptions or results for a prismatic circular shaft under constant torque.
Common Pitfalls:Confusing uniform shear stress (false) with uniform rate of twist (true) for prismatic shafts under constant torque.
Final Answer:Shear stress is uniform over the cross-section at a given torque