For a linearly elastic structure, Which of the following principles states that where external forces only cause deformation, the complementary energy is equal to the deformation energy?

Explanation:

Castigliano’s first theorem-

• For linearly elastic structures, where external forces only cause deformations, the complementary energy is equal to the strain energy.
• For such structures, Castigliano’s first theorem may be stated as the first partial derivative of the strain energy of the structure with respect to any particular displacement gives the force causing the deflection at that point
• This first theorem is applicable to linearly or nonlinearly elastic structures in which the temperature is constant and the supports are unyielding.

\(\begin{array}{l} \frac{{\partial U}}{{\partial \Delta }} = P\\ \frac{{\partial U}}{{\partial \theta }} = M \end{array}\)

Castigliano’s second theorem-

• The first partial derivative of the total internal 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.
• The second theorem of Castigliano is applicable to linearly elastic (Hookean material) structures with constant temperature and unyielding supports.

\(\begin{array}{l} \frac{{\partial U}}{{\partial P}} = \Delta \\ \frac{{\partial U}}{{\partial M}} = \theta \end{array}\)