Castigliano's first theorem is being used in structural analysis for finding deflection of an elastic structure based on the strain energy of the structure. Castigliano's theorem can be applied when the structure’s supports are unyielding, and the temperature of the structure is constant.

Castiglione’s first theorem is a fundamental principle in structural analysis. It is applicable under certain conditions, which are discussed below.

Conditions for the applicability of Castiglione’s first theorem:

1. Principle of superposition:
Castiglione’s first theorem is applicable only if the principle of superposition is valid. This means that the response of a structure to a set of loads can be obtained by adding the responses of the structure to each of the loads applied separately.

2. Linearly elastic behavior:
Castiglione’s first theorem is applicable only for linearly elastic structures. This means that the material of the structure should behave elastically, which implies that the stress-strain relationship is linear.

3. Statically determinate structures:
Castiglione’s first theorem is applicable only for statically determinate structures. This means that the number of unknown reactions should not exceed the number of equations of equilibrium.

Explanation:

Castiglione’s first theorem states that the displacement at any point in a linearly elastic structure, due to a set of external loads, can be expressed as a sum of the products of the partial derivatives of the deflection function with respect to the loads and the corresponding partial derivative of the deflection function with respect to the coordinates of the point.

In other words, Castiglione’s first theorem provides a method to calculate the deflection at any point in a structure due to a set of external loads. The theorem is based on the principle of superposition, which allows the deflection due to each load to be calculated separately and then added to obtain the total deflection.

However, it is important to note that Castiglione’s first theorem is applicable only under the conditions discussed above. If any of these conditions are not satisfied, the theorem may not be applicable or may provide inaccurate results.