Twierdzenie Menabrea-Castigliano

By using the Castigliano's theorem to determine generalized displacements at the location and in the direction of occurrence of statically indeterminate reactions, where these displacements do not exist, we obtain the principle of minimum energy.

\begin{aligned} \frac{\partial U}{\partial X_{i}}=0 \end{aligned}

where:

U - elastic energy of the system as a function of external loads and statically indeterminate reactions of the system,

\(X_i\) - statically indeterminate (excessive) reaction of the system.

This theorem is called Menabre's theorem and it is formulated as follows:

In a linear-elastic system rigidly supported, the partial derivative of the total elastic energy of the system with respect to the excessive - statically indeterminate quantity is equal to zero.