Strain Energy Calculations Of Different Types Of Direct Mechanical Loads

Strain energy is a very important subject of mechanical engineering and mechanical calculations of different physical systems. In different methods, strain energy that is stored on the model or body must be calculated. And strain energy is the result of different load applications. And stored strain energy is generally used in Castigliano’s theorem. 

Here we explain the general loading conditions that appear in mechanics, and how to calculate the total strain energy stored and Castigliano’s theorem. 

Strain Energy Calculations Of Direct Mechanical Loads

The general formulations of strain energy calculations are generally expressed with theoretical bodies and loading conditions, in which the general mechanical systems can be reduced to the combinations of these loading conditions. If you build the system which constitutes these different loading conditions, you can use Castigliano’s theorem for safety calculations. 

The approach of classical mechanics includes three types of loads; gradually applied load, suddenly applied load, and impact load. 

Gradually Applied Load

Gradually applied load means the load that gradually increases with the passing time. This increment is considered linear. 

Consider that a circular bar that is cantilevered to the ceiling and loaded with load ‘P’. This circular bar has a constant cross-section ‘A’. Say that the length of the bar is ‘l’. 

From the classical mechanics, the tensile stress σ on the bar is ‘P/A’. And the modulus of elasticity of the bar is ‘E’. The stored strain energy with this gradual load is;

In here, ‘V’ is the volume of the bar. The gradual application reflected the equation as ‘1/2’. 

Suddenly Applied Load

In this case, the load ‘P’ is suddenly applied instead of a gradual basis. So, the total strain energy stored on the bar will be twofold of the gradually applied load. 

Impact Energy

This situation is slightly different from the two situations above. Here, a basic theoretical system is devised to explain the impact energy situation. 

Illustration of the system that explained below(Image Source: D. K. Singh – Strength of Materials-Springer, 2020, pg.413).

As you see above, there is a load attached to the bar. And this bar is cantilevered to the ceiling. And there is a collar that the load can suddenly fall on it. Here, we must devise some parameters.

‘h’ is the total height that the load starts to free fall on the collar. ‘P’ is the total load of the body. ‘l’ is the length of the bar. The other parameters are the same as above.

With the equation above, the stress occurrence on the bar can be calculated easily. This is a very useful formulation that the sudden stresses can be calculated with the impact action. 


As you see above the strain energy calculations and the general principles of strain energies of different mechanical loadings are very understandable.

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