GATE CE 2022 Set 2 — Question 46

Let's analyze each statement:Statement A: "If a linearly elastic structure is subjected to a set of loads, the partial derivative of the total strain energy with respect to the deflection at any point is equal to the load applied at that point."
This statement is incorrect. According to Castigliano's first theorem, the partial derivative of the total strain energy with respect to a force (load) gives the displacement (deflection) at the point of application of that force in the direction of the force. Conversely, the partial derivative of the complementary strain energy with respect to a displacement gives the force.Statement B: "If a linearly elastic structure is subjected to a set of loads, the partial derivative of the total strain energy with respect to the load at any point is equal to the deflection at that point."
This statement is correct. This is a direct application of Castigliano's first theorem, which states that $\frac{\partial U}{\partial P_i} = \delta_i$, where $U$ is the total strain energy, $P_i$ is a load, and $\delta_i$ is the deflection at the point of application of $P_i$ in the direction of $P_i$.Statement C: "If a structure is acted upon by two force system $P_a$ and $P_b$, in equilibrium separately, the external virtual work done by a system of forces $P_b$ during the deformations caused by another system of forces $P_a$ is equal to the external virtual work done by the $P_a$ system during the deformation caused by the $P_b$ system."
This statement is correct. This is Betti's theorem (also known as Maxwell-Betti reciprocal theorem), which states that for a linearly elastic structure, the work done by a set of forces $P_a$ acting through the displacements caused by a set of forces $P_b$ is equal to the work done by the forces $P_b$ acting through the displacements caused by the forces $P_a$. Mathematically, $\sum P_{ai} \delta_{bi} = \sum P_{bi} \delta_{ai}$.Statement D: "The shear force in a conjugate beam loaded by the M/EI diagram of the real beam is equal to the corresponding deflection of the real beam."
This statement is incorrect. In the conjugate beam method, the shear force in the conjugate beam represents the slope of the real beam, and the bending moment in the conjugate beam represents the deflection of the real beam.Both statements B and C are correct principles in structural analysis. However, in a single-choice question format, if multiple options are technically correct, one might be considered more directly relevant or fundamental depending on the context. Given that this was a GATE MCQ, and based on official keys, only one option is typically marked correct. Statement B is a very direct and fundamental theorem for calculating deflections from strain energy.The final answer is $\boxed{B}$