Castigliano’s Theorem for Beam Deflection
Castigliano’s theorem is a powerful tool for analyzing beam deflections in structures. Let’s explore its application and solve a problem using this theorem.
The Theorem:
Castigliano’s theorem relates the displacement at a load application point to the partial derivatives of the strain energy within the structure.It’s particularly useful for analyzing simply supported beams with point loads.
Problem Scenario:
Consider a simply supported beam with a central point load.The beam has symmetric reaction forces at the supports, each bearing half of the load.
We want to find the deflection at the point where the load is applied.
Steps to Solve:
Calculate the bending moment at any point along the beam using the symmetric reactions.Express the strain energy due to bending based on the bending moment along the beam, integrated over its length.
For this symmetric beam, compute the strain energy over half its length and then double it to account for the entire beam.
The deflection at the center of the beam depends on:
Load magnitude.
Cube of the beam’s length.
Inverse of the product of the moment of inertia and the elastic modulus of the beam’s cross-section.
