Strain Energy Calculations in Engineering

Strain energy is very important in engineering. We use these calculations in the design of different kinds of engineering systems. So, it is very important to know to calculate the strain energies due to different kinds of reasons. Here you can find different kinds of calculations of this phenomenon for different reasons.

Strain Energy Calculations of Pure Bending on Beams

Type 1: If a simple beam is simply supported from each side, and the bending force is not at the center;

Simple beam.

Also, the required equation to calculate strain energy storage on this beam is calculated with this formula;

Strain energy storage on simple beam.

So, the lengths are important in terms of the placement in the equation. 

Type 2: If the considered simply supported beam’s load is at the center of the beam;

Strain energy storage on beam.

So the equation to calculate the strain energy storage on this beam reduces to;

Strain energy formula.

Type 3: If a simply supported beam undergoes uniformly distributed load;

Uniformly distributed load.

So, the equation will be like this;

Shear stress calculation with uniformly ditributed load.

Also here, ‘P’ represents the total load acting on the beam, which is ‘w*l’. 

Type 4: If a cantilevered beam from one side, carries a unitary load at the other edge; 

Simple cantilevered beam.

And also, the equation will be like this;

Strain energy formula for cantilevered beam.

Type 5: Consider the same cantilevered beam which carries a uniformly distributed load on it; 

Cantilevered beam with uniformly distributed load.

Then the equation will be like this again;

Strain energy calculation for cantilevered beam.

Strain Energy Calculation With Volumetric Strain

Equal stress values acting on a stress element.
Equal stress values acting on a stress element(Image Source:D. K. Singh – Strength of Materials-Springer, 2020, pg.419).

You can make this calculation with the calculator below;

Strain Energy Calculator off Volumetric Strains
















The use of the calculator above is very simple. You just need to enter the required values inside the brackets. Then click on the ‘Calculate!’ button to see the result. If you want to do another calculation, just click on the ‘Reset’ button. 

Here, you need to enter the stress value that is applied from three sides of the body. Then enter the elasticity modulus and Poisson’s ratio of the material. And enter the total volume which is the volume of the stress element that undergoes stress. If you are making your calculation for a unit volume, enter the total volume as 1. 

You can enter the values in suggested units in parentheses.

Strain Energy due to the Principal Stresses

Principal stresses on a cubic unit volume.
Principal stresses on a cubic unit volume(Image Source:D. K. Singh – Strength of Materials-Springer, 2020, pg.418).

For the triaxial system, you will have three different principal stresses. And for biaxial systems, you will have two different principal stresses. According to the system that you are dealing with, you need to use proper calculations. 

Stored Strain Energy for Biaxial Systems

In biaxial systems, the third principal stress value which is also denoted as σ3 is neglected. In this case, you need to be calculated the σ1 and σ2 values. You can use the calculator below to calculate the total stored strain energy for your system; 

Strain Energy Calculator Of Biaxial Principal Stresses


















In this calculator, you just need to enter the required values. Then click on the ‘Calculate!’ button to see the result. You can also click on the ‘Reset’ button to perform another calculation. 

In this calculator; ‘σ1’ and ‘σ2’ values are the principal stress values of the biaxial system. Enter the elasticity modulus of the material. The volume that you want to calculate is the total strain energy, which undergoes these principal stresses. Enter the Poisson’s ratio of the material. 

You can use the suggested units in parentheses to obtain the result.

Stored Strain Energy Calculation for Triaxial Systems

The difference of this calculator is that there is an additional ‘σ3’ value because of the triaxial system.

Strain Energy Calculator Of Triaxial Principal Stresses




















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 a system that 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 time. This increment is considered linear. 

Consider 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 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;

Stored strain energy with gradual load.

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 on a gradual basis. So, the total strain energy stored on the bar will be twofold 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. 

Suddenly applied load.
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.

Stress occurence on bar.

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. 

Stored Strain Energy Calculation on Helical Springs

Helical springs are one of the most used spring types in engineering and they are used for axial loads generally. These loads can be both dynamic and static. And in some engineering applications, stored strain energy on the helical spring body can be important. This total strain energy can be dependent on various factors. 

Schematic illustration of a helical spring.
Schematic illustration of a helical spring(Image Source: D. K. Singh – Strength of Materials-Springer, 2020, pg.380).

Here we explain how to calculate the stored strain energy on helical springs and the calculator for it. 

The use of the calculator below is very simple to use. 

Helical Spring Strain Energy Calculator


















In this calculator, you just need to enter the required values inside the corresponding brackets. And click on the ‘Calculate!’ button to calculate the total strain energy result. If you want to make another calculation, click on the ‘Reset’ button to do it. 

The general unit sets are recommended inside the parentheses. You can enter your values according to them.

In this calculator; 

  • Total load is the load value that acts axially to the helical spring. 
  • Diameter is the diameter of the helical spring wire. 
  • The coil radius is the radius of the helical spring coil. 
  • The number of turns is the total number of turns of the helical spring wire that constitutes the helical spring coil.
  • The modulus of rigidity value is related to the wire material. Special for the material of the wire. 

The Strain Energy of Torsion

Strain energy is the total energy on a body that undergoes torsional twisting. It is calculated with the equation below; 

Strain energy formula of twisting.

So, as you see above, the equation of the strain energy of torsion is very simple. In this equation; 

?max is the maximum shear stress occurred on the body that undergoes torsion. 

G is the shear modulus of the material of the body. 

V is the total volume of the body that undergoes torsion. 

Strain Energy of Torsion Calculator














Also, the use of the calculator above to calculate the strain energy of torsion on a body is very basic. You just need to enter the required values inside the brackets then click on the ‘Calculate!’ button to see the result. If you want to make another calculation, click on the ‘Reset’ button then re-enter the values. 

You can enter the values in the unit recommendations inside the parentheses both in SI and US Customary units.

Conclusion

So, these are the general formulations and calculators related to the strain energy calculations

Finally, do not forget to leave your comments and questions below about the strain energy of pure bending on beams.

Your precious feedbacks are very important to us.

The source of the images above: D. K. Singh – Strength of Materials-Springer, 2020, pg.417. No copyright infringement intended. This article is just for educational purposes.

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