What is strain energy due to bending?

Short Answer:

The strain energy due to bending is the energy stored in a beam or structural member when it is subjected to a bending moment. This energy develops because of the internal stresses and deformations produced within the material as it bends.

In simple words, when a beam bends under load, its fibers above the neutral axis are compressed while those below are stretched. This deformation stores energy in the beam known as bending strain energy. The energy can be recovered when the load is removed if the material remains within its elastic limit.

Detailed Explanation :

Strain Energy Due to Bending

When a beam or any structural element is subjected to external loads, internal stresses develop within the material to resist these loads. As a result of these stresses, the beam deforms slightly, bending into a curved shape. During this process, the internal work done by the applied loads is stored within the beam as strain energy.

In the case of bending, the upper fibers of the beam are compressed, and the lower fibers are stretched, while the middle layer (neutral axis) remains unstressed. The energy stored in the beam due to this bending action is called the strain energy due to bending.

This concept is essential in strength of materials because it helps in understanding how materials store and release energy under load and how much deformation or deflection a beam will undergo when subjected to bending.

Definition of Strain Energy Due to Bending

Strain energy due to bending can be defined as:

“The energy stored in a beam or flexural member due to the bending stresses produced by external loads acting on it.”

This energy is stored as potential energy within the beam and is completely recoverable when the external load is removed, as long as the stresses remain within the elastic limit of the material.

Basic Concept of Bending

When a beam bends under an external moment (bending moment ), the material fibers experience different types of stresses:

  • Compressive stress acts on the upper fibers (shortened).
  • Tensile stress acts on the lower fibers (elongated).
  • Neutral axis separates the tensile and compressive zones where stress is zero.

Due to these stresses, small amounts of strain develop in the fibers, and this strain results in the storage of strain energy per unit volume.

Derivation of Strain Energy Due to Bending

Let us consider a simply supported beam of length  subjected to a bending moment  at a section.

At any small element of the beam of length ,

  • The bending moment =
  • The curvature of the beam =

From the bending equation,

where,
= Bending moment,
= Moment of inertia of beam section,
= Young’s modulus,
= Radius of curvature,
= Bending stress at a distance  from the neutral axis.

Rearranging for curvature:

Now, strain at any fiber located at a distance  from the neutral axis is given by:

The stress at the same fiber is:

The strain energy per unit volume (u) is:

Substituting :

Now, the strain energy stored in the small volume element  (where  is the width of the beam) is:

Integrating over the entire cross-section:

But,

Hence,

This gives the total strain energy due to bending for the entire beam.

Final Expression

Where,
= Total strain energy due to bending (Joules),
= Bending moment at a section (N·m),
= Young’s modulus of the material (N/m²),
= Moment of inertia of the cross-section (m⁴),
= Length of the beam (m).

Strain Energy Per Unit Volume

For uniform bending, the strain energy per unit volume is:

This equation shows that strain energy increases with the square of the stress, meaning that doubling the bending stress quadruples the energy stored.

Physical Meaning

The strain energy due to bending represents the potential energy stored within the beam because of elastic deformation.

  • When the load is applied, the beam bends and stores energy.
  • When the load is removed, the beam tries to return to its original shape, releasing the stored energy.

If the stresses go beyond the elastic limit, the beam undergoes permanent deformation, and part of the energy is lost as plastic deformation.

Applications of Strain Energy Due to Bending

  1. Design of Beams and Bridges:
    Engineers calculate strain energy to ensure beams can handle applied loads safely without excessive deflection or failure.
  2. Resilience Calculation:
    It helps in finding the resilience (energy stored per unit volume) of materials, which indicates their ability to absorb energy before failure.
  3. Dynamic Loading:
    Used to study how structures respond to shocks, impacts, and vibrations.
  4. Spring Design:
    The concept is applied in designing leaf and helical springs which store energy through bending action.
  5. Failure Analysis:
    Helps in predicting how much energy a structure can store before it fails or yields.

Importance of Strain Energy in Engineering

  • It provides insight into the strength and flexibility of materials.
  • It helps determine the deflection of beams under load.
  • It aids in designing energy-absorbing devices like springs and dampers.
  • It ensures materials operate safely within the elastic limit.
Conclusion

The strain energy due to bending is the elastic potential energy stored in a beam or structural member when subjected to bending moments. It depends on the bending moment, modulus of elasticity, and moment of inertia of the beam section. Mathematically, it is given by

This principle helps engineers design safe and efficient structures by analyzing how much energy a beam can store and how it behaves under different loading conditions. Understanding strain energy is fundamental in structural design, spring mechanics, and energy absorption applications.