Question: Which of the following is the use of Castigliano’s first theorem?
A Calculating internal force in a structure
B Determining the displacement at a specific point due to applied loads
C Analyzing the strain energy in a structure
D Finding the support reactions of a structure
Answer: Determining the displacement at a specific point due to applied loads
| Theorem | Castigliano’s First Theorem | Castigliano’s Second Theorem |
| Basic Idea | Castigliano’s first theorem relates to the partial derivative of the strain energy with respect to an external load to the displacement at the point of application of the load. | Castigliano’s second theorem relates to the partial derivative of the strain energy with respect to a displacement component to the internal force at the point of application of the displacement. |
| Mathematical Formulation | δU = ∂U/∂F * δF | δU = ∂U/∂δ * δP |
| Use Case | Used to find displacements or rotations at a specific point in a structure when external forces are applied. | Used to find internal forces (such as shear forces, bending moments, or axial forces) at a specific point in a structure when a displacement is imposed. |
| Application | Applicable for solving displacement-related problems, e.g., deflections due to applied loads. | Applicable for solving force-related problems, e.g., determining internal forces in structures due to imposed displacements. |