Question

The displacement δ_i, in line with force F_i, is given by:

1. First derivative of total energy with respect to Fi
2. First derivative of potential energy with respect to Fi
3. First derivative of internal energy with respect to Fi
4. First derivative of complementary energy with respect to Fi

Answer 1

Correct Answer - Option 4 : First derivative of complementary energy with respect to Fi

Explanation:

Castigliano’s second theorem: 

• The first partial derivative of the complementary strain energy in a structure with respect to the force applied at any point is equal to the deflection at the point of application of that force in the direction of its line of action.

\(\delta_i = \frac{dU}{dF_i}\)

Where U = Complementary strain energy and δ_i = Displacement produced in the line by force F_i

• The second theorem of Castigliano is applicable to linearly elastic structures with constant temperature and unyielding supports.
• Note that in the above statements, force may mean point force or a couple (moment) and displacement may mean translation or angular rotation.