Pure bending of a prismatic beam under constant bending moment If a constant-section (prismatic) beam is subjected to a uniform bending moment throughout its length (no shear force), the deflected shape of the centroidal axis is:

Introduction / Context:
Pure bending refers to a region in a beam where the bending moment is constant and the shear force is zero. Understanding the resulting curvature and deflected shape under pure bending is fundamental to beam theory and the derivation of flexure formulas.

Given Data / Assumptions:

• Prismatic beam of constant E (modulus of elasticity) and I (second moment of area).
• Uniform bending moment M along the entire length; shear force V = 0.
• Small deflection Euler–Bernoulli assumptions.

Concept / Approach:

The curvature κ of a beam is given by κ = M / (EI) under small deflection theory. If M, E, and I are constants, κ is constant. A curve with constant curvature is a circular arc. Therefore, the neutral axis of the beam bends into a part of a circle under pure bending.

Step-by-Step Solution:

Verification / Alternative check:

Differential equation of the elastic curve: d^2y/dx^2 = M/(EI) = constant integrates to y being a quadratic in x only when slope is small; however, the geometric curve corresponding to constant curvature is circular to first order.

Why Other Options Are Wrong:

• Parabolic or catenary shapes arise under different loading (e.g., UDL for parabolic moment diagram; cable under uniform self-weight for catenary).
• “None” or “straight line” contradict constant nonzero curvature.

Common Pitfalls:

• Confusing the parabolic bending-moment diagram under UDL with the geometric deflected shape.

Final Answer:

A circular arc.