Bending Strain Energy: Formula and Definition

1. What is Strain Energy?

Imagine compressing a simple coil spring. The work you do to compress it is stored in the spring as potential energy. If you release the spring, it expands and releases that stored energy.

A beam behaves in a similar way. When you apply loads to a beam, it bends and deforms. As it deforms, it stores energy internally. This stored energy, resulting from the material's elastic deformation (strain), is called strain energy. When the loads are removed, a properly designed beam will "spring back" to its original shape, releasing this energy.

For our purposes, we are primarily concerned with the energy stored due to bending, which we call bending strain energy.

To see a quick and clear introduction to this idea, please watch the following short clip.

2. The Formula for Bending Strain Energy

As you just saw in the video, the bending strain energy (U) stored in a beam is calculated using a specific formula. This is the central formula for this part of our course.

The total bending strain energy U in a beam is given by the integral:

$U={\int }_0^L\frac{M^2}{2EI}dx$

Let's break down this important expression.

To fully understand this formula, you need to be familiar with each term. The following reading provides a clear table defining each component.

Let's summarize the key components:

• M (Bending Moment): This is the internal bending moment expression, M(x), that you learned to derive in the previous lesson. Its value changes along the length of the beam, which is why we need to integrate. The M² term indicates that the energy stored increases with the square of the moment—a heavily loaded section stores significantly more energy.
• E (Modulus of Elasticity): This is a material property. It measures the material's stiffness. Steel, for example, has a much higher E than aluminum.
• I (Moment of Inertia): This is a geometric property of the beam's cross-sectional shape. It measures how the shape resists bending. A tall, deep I-beam has a much larger I than a thin, flat plate.
• EI (Flexural Rigidity): This combined term represents the beam's overall resistance to bending. A high EI value means the beam is very stiff. Notice that EI is in the denominator, which makes sense: a stiffer beam (high EI) will store less strain energy for the same bending moment M, because it deforms less.
• ∫ ... dx (Integral): This tells us to sum the quantity M²/(2EI) for every infinitesimal slice dx along the entire length L of the beam.

3. Deriving the Formula (Conceptual)

While a deep theoretical derivation isn't our primary goal, it's helpful to have an intuition for where the formula comes from.

The strain energy stored in a small segment dx of the beam is equal to the work done to bend that segment. The work done by a moment is analogous to the work done by a force (Work = Force × Distance). For a moment, it is:
Work = (1/2) × Moment × Angle of Rotation.

The small rotation dθ of our segment dx is given by dθ = (M/EI) dx. So, the small amount of energy dU in the segment is:
$dU=\frac{1}{2}Md\theta =\frac{1}{2}M\left(\frac{M}{EI}dx\right)=\frac{M^2}{2EI}dx$

When we integrate dU over the whole beam, we get the total strain energy formula. If you're interested in a slightly more detailed walkthrough of this derivation, the "Prepp" resource you just looked at has a good explanation in the section just before the revision table.

4. Application with Multiple Segments

In the last lesson, you saw that a beam with multiple loads requires multiple M(x) expressions. For example, a beam with a point load at its center has one equation M_1(x) for the first half and a different equation M_2(x) for the second half.

To find the total strain energy for such a beam, you simply calculate the energy for each segment and add them up:

$U_{total}={\int }_{segment1}\frac{M_1(x{)}^2}{2EI}dx+{\int }_{segment2}\frac{M_2(x{)}^2}{2EI}dx+\dots$

This is a critical concept for the problems we will solve later. Your ability to correctly define the M(x) expressions and their corresponding integration limits is the most important skill for this entire topic.

While we are focusing only on bending strain energy, it's useful to see it in a broader context. Structures can also store energy from axial loads (tension/compression) and torsion (twisting). The following resource provides a summary table showing the energy formulas for these different loading types.

Conclusion

This lesson introduced the core concept of bending strain energy. You have learned its definition, the formula used to calculate it, and the meaning of each term within that formula.

Key Takeaways:

• Strain Energy (U) is energy stored within a body due to its elastic deformation.
• The formula for bending strain energy is the cornerstone of our work:
  $U=\int \frac{M^2}{2EI}dx$
• To calculate U, you must first find the bending moment expression M(x), which can vary for different segments of a beam.
• The beam's resistance to bending is its flexural rigidity (EI).

We now have all the foundational pieces in place. You know how to find reactions, how to write M(x), and you now have the formula for strain energy U.

In our next lesson, we will finally introduce Castigliano's Second Theorem. This theorem provides a powerful method to link strain energy directly to the deflection of a beam, bringing us one step closer to our goal of analyzing indeterminate structures.