# Explanation Of The Castigliano’s Theorem

Strain energy is a very important term in classical engineering. And strain energy of bodies is used in many engineering calculations. One of these calculations is Castigliano’s theorem. Here we simply explain Castigliano’s theorem.

## Types Of Strain Energy Storages In Engineering

First of all, the general sources of strain energy inducers must be known, to understand Castigliano’s theorem. These are;

- Strain Energy Induced By Torsional Movements: If there are torsional movements on a body, such as mechanical shafts, strain energy is stored.
- Strain Energy Stored By Volumetric Strains:
- Strain Energy Due To Shear Stresses: Shear stress is another type of mechanical loading. Strain energy is also stored on a body if the shear stresses are prevalent.
- Strain Energy Of Bending: Bending is also a very important mechanical loading type. And strain energies are stored on beams because of these bending loading conditions.
- Strain Energy With Principal Stresses:
- Shear Strain Energy With Principal Stresses: With principal stresses, strain energy induced because of the shear actions can be calculated.
- Strain Energy Stored Because Of Direct Mechanical Loads: The type of application of a mechanical load is also very important in the calculation of stored strain energy. These types of loads can be; gradually applied load suddenly applied load and constant load.

## Castigliano’s Theorem

Castigliano’s theorem states that, if the total stored strain energy is known because of a load in a certain direction, the partial derivative of this stored strain energy with the deflection in the direction of the load, gives the load itself. And it has the equation like this;

Also, if the partial derivative of the total stored strain energy with the applied load in a certain direction is performed, it will give the total deflection in that direction because of the load. Also, this theorem has the equation like this;

## Conclusion

A simple explanation of Castigliano’s theorem can be made like above.

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