GATE CE 2017 Set 1 — Question 5

For a beam subjected to transverse loading, the relationship between the shear force ($V$) and the bending moment ($M$) is given by:$$V = \frac{dM}{dx}$$This mathematical relationship implies that the bending moment reaches a local maximum or minimum at points where its first derivative, the shear force, is zero. Therefore, statement (B) is true.According to the Euler-Bernoulli beam theory, the curvature ($\kappa$) of the beam's neutral axis is proportional to the bending moment:$$M = EI\kappa$$where $E$ is the Young's modulus and $I$ is the area moment of inertia. This relationship indicates that the curvature is zero at sections where the bending moment is zero. For a simply supported beam under a uniformly distributed load (UDL), the bending moment is zero at the supports ($x = 0$ and $x = L$). At these same support locations, the shear force is at its maximum absolute value ($|V_{max}| = \frac{wL}{2}$). Consequently, the maximum shear force occurs at the sections where the curvature is zero. Thus, statement (A) is also true.Since both statements (A) and (B) are technically correct for the given conditions, the official GATE answer key accepts both options as correct.