Torsion in a circular bar: a bar of circular cross-section is fixed at A and subjected to a torque T at its free end B. What is the angle of twist at the fixed end A relative to the undeformed axis?

Introduction / Context:
For a prismatic circular shaft under Saint-Venant torsion, the angle of twist varies linearly along the length when torque is applied at a free end and the other end is fixed. Understanding boundary conditions is essential—rotations at fixed supports are zero by definition.

Given Data / Assumptions:

• Shaft is prismatic, isotropic, and linearly elastic.
• End A is fully fixed against rotation; end B is free and subjected to torque T.
• Uniform shear modulus G and polar moment of inertia J.

Concept / Approach:
For length L, the angle of twist between sections x apart is θ(x) = (T * x) / (G * J) (for constant T along the shaft). Setting x = 0 at the fixed end A gives θ(0) = 0, while at x = L (free end B) θ(L) = T L / (G J), the maximum twist.

Step-by-Step Solution:
Define x from A (fixed) to B (free): θ(x) = T x / (G J).At A (x = 0) → θ_A = 0.At B (x = L) → θ_B = T L / (G J) (maximum).Therefore, the twist at A is zero.

Verification / Alternative check:
Boundary condition of a fixed end precludes rotation; shear strain distribution also goes to zero at the fixed end for the displacement field.

Why Other Options Are Wrong:

• Non-zero angles at the fixed end contradict boundary conditions.
• “Indeterminate” is incorrect; G and J affect magnitude, not the zero rotation at a fixed boundary.

Common Pitfalls:
Confusing slope/rotation diagrams with bending; forgetting that twist is referenced to the fixed end.

Final Answer:
Zero