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.

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 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;

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

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