Strength of materials
Menabrea-Castigliano theorem
- Introduction to statically indeterminate systems
- Definition of statically indeterminate systems
- Problem of excess unknowns
- Equations of static equilibrium
- Energy methods as a solution tool
- Menabrea-Castigliano theorem
- Theoretical foundations
- Relation to Castigliano's theorem
- Differentiation of elastic energy
- Determining redundant reactions
- Energy minimum theorem
- Derivation of the formula
- Notations and symbols
- Elastic energy of the system
- Statically indeterminate reactions
- Formulation of Menabrea's theorem
- Linear-elastic systems
- Condition for zeroing the partial derivative
- Application in practice
Introduction to statically indeterminate systems
The analysis of statically indeterminate systems is one of the key issues in materials strength and structural mechanics. In such systems, the number of unknown reactions exceeds the number of available static equilibrium equations, which requires the use of more advanced computational methods. Energy methods, based on the principles of energy conservation, are particularly effective in solving this type of problem.
Menabrea-Castigliano theorem
Theoretical foundations
The Menabrea-Castigliano theorem is an extension of the Castigliano theorem, used to calculate displacements in statically indeterminate systems. It is based on differentiating the elastic energy of the system with respect to the statically indeterminate reactions. Thanks to this theorem, it is possible to determine reactions that cannot be established from static equilibrium equations.
The theorem of minimum energy
Using the Castigliano theorem to determine generalized displacements at the location and direction of the occurrence of redundant (statically indeterminate) reactions, where these displacements do not exist, we obtain the theorem of minimum energy:
\[ \frac{\partial U}{\partial X_{i}}=0 \]where:
- U – elastic energy of the system as a function of external loads and statically indeterminate reactions of the system,
- \(X_i\) – statically indeterminate (redundant) reaction of the system.
Formulation of the Menabrea theorem
This theorem is named the Menabrea theorem and we formulate it as follows:
In a linear-elastic system with rigid supports, the partial derivative of the elastic energy of the entire system with respect to the redundant – statically indeterminate quantity is equal to zero.