Castigliano's Theorem: Problem Solving

Castigliano's theorem relates deflection to the partial derivatives of strain energy with respect to the applied load. For a simply supported beam with a central point load, you differentiate the strain energy expression with respect to the load magnitude. The strain energy is calculated from the bending moment squared, integrated over the beam's length. This partial derivative yields the deflection at the load application point.

For a simply supported beam carrying a point load F at its center, the reaction forces at both ends are equal and symmetric. Each support reaction equals half of the central load, or F/2. This symmetry simplifies the analysis because the bending moment distribution is identical on both sides of the center point.

The bending moment at any point along the beam is calculated from the reaction force multiplied by the distance from the nearest support. For a beam segment from support P to center C, the moment equals the reaction at P times the distance x from P. The moment expression changes at the center, creating a symmetric distribution that peaks at the beam's midpoint.

Integration is performed over half the beam length because the simply supported beam with a central load exhibits symmetry about its midpoint. The strain energy contribution from each half is identical. After integrating over half the length, the result is doubled to account for the entire beam, simplifying calculations while maintaining accuracy.

The center deflection depends on the load magnitude, the cube of the beam length, and inversely on the product of the moment of inertia and elastic modulus. Larger loads and longer beams increase deflection, while stiffer materials and larger cross-sectional moments of inertia reduce it. This relationship reflects how beam stiffness and geometry control deformation.

Castigliano's theorem requires partial differentiation of the bending moment expression with respect to the applied load. For a simply supported beam, differentiating the moment equation yields a coefficient that, when integrated with the original moment expression, produces the strain energy derivative. This mathematical operation converts the moment distribution into a deflection value.

Elastic strain energy represents the deformation energy stored in the beam due to bending. Castigliano's theorem establishes that the deflection at any point equals the partial derivative of total strain energy with respect to the load at that point. For a simply supported beam, calculating elastic strain energy from bending moments and differentiating it yields the center deflection directly.