JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech III ADVANCED STRUCTURAL ANALYSIS Previous Question Paper (August - 2022 - R16)

Castigliano's second theorem states that for a linearly elastic structure, the partial derivative of the total strain energy (U) with respect to an external load (P) applied at a point is equal to the displacement (Δ) of that point in the direction of the load. Mathematically, it is expressed as:

This theorem is widely used to determine deflections in beams, frames, and trusses. It assumes that the material follows Hooke's Law, the temperature remains constant, and the supports are unyielding.

To analyze a truss using Castigliano’s second theorem, follow these steps:

• Identify the member forces in terms of external loads and a redundant force (if the truss is statically indeterminate) or directly in terms of applied loads (if determinate).
• Express the total strain energy of the truss as U = Σ (F²L) / (2AE), where F is the axial force in a member, L is its length, A is the cross-sectional area, and E is the Young's modulus.
• Apply the theorem: to find the deflection at a joint, calculate the partial derivative of the total strain energy with respect to the load applied at that joint.
• If the truss is statically indeterminate, calculate the redundant force (R) by setting the partial derivative of the total strain energy with respect to R to zero (∂U/∂R = 0), representing the compatibility condition of zero displacement at the support.
• Substitute the calculated redundant force back into the expressions for member forces to determine the final internal forces in all members.