C6.1 Elastic Strain Energy for Various Loadings

As mentioned in the chapter overview, we’ve covered various types of loadings ranging from axial, torsion, shear and bending. When these loadings are applied, they cause stress and hence strain in the material.

When a material is strained, there is actually strain energy stored in it. It’s like a spring; when you exert a force on a spring (with a certain stiffness), the spring compresses/stretches (i.e. strains), and there is spring energy stored in it. Materials have an inherent stiffness too, and will store energy when you exert a load on it.

When you release the spring, the spring returns to its original undisplaced form. In the same way, materials will return to their original form when you release the loading, and the strain energy stored returns to zero. That is why it’s called elastic strain energy; we’re looking at the energy stored in the material within its elastic region (i.e. able to return to its original form when the load is removed).

C6.1 Elastic Strain Energy for Various Loadings

As mentioned in the chapter overview, we’ve covered various types of loadings ranging from axial, torsion, shear and bending. When these loadings are applied, they cause stress and hence strain in the material.

When a material is strained, there is actually strain energy stored in it. It’s like a spring; when you exert a force on a spring (with a certain stiffness), the spring compresses/stretches (i.e. strains), and there is spring energy stored in it. Materials have an inherent stiffness too, and will store energy when you exert a load on it.

When you release the spring, the spring returns to its original undisplaced form. In the same way, materials will return to their original form when you release the loading, and the strain energy stored returns to zero. That is why it’s called elastic strain energy; we’re looking at the energy stored in the material within its elastic region (i.e. able to return to its original form when the load is removed).

Without further ado, here are the formulas to calculate the different types of strain energy:

Strain energy formula for various loadings

Note:

N, V, M, T are the internal forces in the structure caused by external loadings
- N – internal normal force (units: N)
- V – internal shear force (units: N)
- M – internal bending moment (units: Nm)
- T – internal torque (units: Nm)
A, I, J are the section properties
- A – area of cross-section (units: m² or mm²)
- I – moment of inertia (units: m⁴ or mm⁴)
- J – polar moment of inertia (units: m⁴ or mm⁴)
E, G are the Young’s and shear moduli respectively (units: GPa)
ƒ_s is the shear strain energy is the form factor. You don’t really have to worry about this; usually it’s given to you in the question. Please consult your textbook for more info.
The integration limits (0 to L) is for the integration to be performed along the whole length of the structure, to get the total strain energy stored.

We’ll look at one example now on how to calculate our strain energy.

Without further ado, here are the formulas to calculate the different types of strain energy:

Strain energy formula for various loadings

Note:

N, V, M, T are the internal forces in the structure caused by external loadings
- N – internal normal force (units: N)
- V – internal shear force (units: N)
- M – internal bending moment (units: Nm)
- T – internal torque (units: Nm)
A, I, J are the section properties
- A – area of cross-section (units: m² or mm²)
- I – moment of inertia (units: m⁴ or mm⁴)
- J – polar moment of inertia (units: m⁴ or mm⁴)
E, G are the Young’s and shear moduli respectively (units: GPa)
ƒ_s is the shear strain energy is the form factor. You don’t really have to worry about this; usually it’s given to you in the question. Please consult your textbook for more info.
The integration limits (0 to L) is for the integration to be performed along the whole length of the structure, to get the total strain energy stored.

We’ll look at one example now on how to calculate our strain energy.