Castigliano's method, named after Carlo Alberto Castigliano, is a method for determining the displacements of a linear-elastic system based on the partial derivatives of the energy. He is known for his two theorems. The basic concept may be easy to understand by recalling that a change in energy is equal to the causing force times the resulting displacement. Therefore, the causing force is equal to the change in energy divided by the resulting displacement. Alternatively, the resulting displacement is equal to the change in energy divided by the causing force. Partial derivatives are needed to relate causing forces and resulting displacements to the change in energy.
Шаблон:Bulleted list
Examples
For a thin, straight cantilever beam with a load P at the end, the displacement <math>\delta</math> at the end can be found by Castigliano's second theorem :
Cantilever BeamCantilever Beam with a Point Load at Free End
<math display="block">\delta = \frac{\partial U}{\partial P}</math>
<math display="block">\delta = \frac{\partial}{\partial P} \int_0^L {\frac{M^2(x)}{2EI} dx} = \frac{\partial}{\partial P} \int_0^L{\frac{(Px)^2}{2EI} dx} </math>
where <math>E</math> is Young's modulus, <math>I</math> is the second moment of area of the cross-section, and <math>M(x)=Px</math> is the expression for the internal moment at a point at distance <math>x</math> from the end. The integral evaluates to:
<math display="block">\begin{aligned}
\delta &= \int_0^L {\frac{P x^2}{EI}dx} \\
&= \frac{PL^3}{3EI}.
\end{aligned}</math>
The result is the standard formula given for cantilever beams under end loads.
