# Castigliano's Theorem: Problem Solving

JoVE Core
Mechanical Engineering
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JoVE Core Mechanical Engineering
Castigliano's Theorem: Problem Solving
##### Vorheriges Video27.7: Castigliano's Theorem
Consider a simply supported beam PQ of length L, carrying a point load F at the center. What will be the deflection at the center of the beam? Here, the reactions at both ends of the beam are equal, and each is half of the central load. Consider the segment PC and choose any point at a distance x from the end P. At this point, the moment due to the reaction at point P is the load at point P times the distance. A partial differentiation of the moment equation with respect to the load at end P is half of x.  A similar analysis can be conducted for the segment QC of the beam. Using Castigliano's theorem, the deflection at point C is determined by the partial derivatives of the strain energy due to the applied load. This equation can be simplified by considering the two segments of beam PC and QC. Performing the integration over half of the length of the beam gives an expression for the deflection at the center of the beam.

## Castigliano's Theorem: Problem Solving

The deflection of a simply supported beam that carries a central point load can be analyzed using structural mechanics principles, particularly by applying Castigliano's theorem. This theorem relates the displacement at the load application point to the partial derivatives of the strain energy in the structure. The simply supported beam with a point load at its center has symmetric reaction forces at the supports, each bearing half of the load. The bending moment at any point along the beam is derived from these reactions, calculated over the distance from the nearest support.

Castigliano's theorem indicates that the deflection at the point where the load is applied is determined by differentiating the strain energy of the beam with respect to the load. Strain energy is calculated based on the bending moment along the beam, integrated over its length. For this beam, the strain energy due to bending is computed from the square of the bending moment expression, integrated along half the beam's length.

Since the beam is symmetric, this value is doubled to account for the entire beam, and the deflection at the center of the beam is computed. It is dependent on the load magnitude, the cube of the beam's length, and inversely proportional to the product of the moment of inertia and the elastic modulus of the beam's cross-section.