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GATE 2022 CE (CE2) – Question 46

Structural Analysis · Deflections of determinate beams, frames and trusses · 2 marks · Multiple select

Which of the following statement(s) is/are correct?

  1. 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.
  2. 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.
  3. If a structure is acted upon by two force system Pa and Pb , in equilibrium separately, the external virtual work done by a system of forces Pb during the deformations caused by another system of forces Pa is equal to the external virtual work done by the Pa system during the deformation caused by the Pb system.
  4. 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.

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Show answer and explanation

Correct answer: (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.; (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.; (C) If a structure is acted upon by two force system Pa and Pb , in equilibrium separately, the external virtual work done by a system of forces Pb during the deformations caused by another system of forces Pa is equal to the external virtual work done by the Pa system during the deformation caused by the Pb system.

Explanation

For linear elasticity, strain energy expressed in displacements satisfies $\partial U/\partial\delta_i=P_i$; expressed in loads it satisfies $\partial U/\partial P_i=\delta_i$. These establish A and B.

C is Maxwell–Betti reciprocal work. In the conjugate-beam method shear represents real-beam **slope**, while moment represents deflection, making D false. **Answer: A,B,C**.