What is strain energy due to bending?

Short Answer:

Strain energy due to bending is the energy stored in a beam or structural member when it is subjected to bending moments within the elastic limit. When an external load causes bending, internal stresses and strains are developed, and the work done by the external forces is stored in the beam as strain energy.

In simple words, when a beam bends under a load, it deforms slightly, and this deformation stores elastic potential energy. The energy stored per unit length or volume due to bending is known as strain energy due to bending, and it plays an important role in designing beams, bridges, and structural members for strength and flexibility.

Detailed Explanation :

Strain Energy Due to Bending

When a beam or a structural member is loaded by external forces, it bends, causing some of its fibers to elongate while others shorten. The top fibers (in a simply supported beam) are under compression, while the bottom fibers are under tension. Between these two layers lies a neutral axis where there is no stress or strain.

The work done by the external loads during bending is stored in the material as strain energy due to bending. As long as the stress in the material remains within the elastic limit, this energy is recoverable when the load is removed.

This concept is essential in understanding the elastic behavior of beams, deflection calculations, and design of flexible components like leaf springs and cantilevers.

Definition

The strain energy due to bending can be defined as:

“The energy stored per unit length or volume of a beam due to bending moments acting on it within the elastic limit is known as strain energy due to bending.”

It represents the elastic potential energy stored because of the internal bending stresses developed in the beam.

Basic Concept

When a beam bends, the bending moment causes normal stresses to act across its cross-section. The strain energy is due to these stresses. Each small element of the beam stores a small amount of energy, and the total strain energy is obtained by summing up all such small contributions along the beam’s length.

The total strain energy (U) stored in the beam can be expressed as:

where,

  •  = total strain energy due to bending,
  •  = bending moment at any section of the beam,
  •  = Young’s modulus or modulus of elasticity,
  •  = moment of inertia of the beam cross-section,
  •  = small length of the beam.

This is the general formula for strain energy due to bending in any beam subjected to elastic deformation.

Derivation of Expression

Consider a small element of a beam of length  subjected to a bending moment .
For the beam section:

  • The normal stress at a distance  from the neutral axis is:

where  is the moment of inertia of the cross-section.

  • The corresponding strain is given by Hooke’s Law:

The strain energy stored in a small volume element  (of area  and length ) is:

Substitute the expressions for  and :

 

To find the total strain energy for the small length , integrate over the entire cross-section of the beam:

But,

Hence,

Now, for the whole length  of the beam, integrate over :

This gives the total strain energy due to bending.

Special Cases

  1. For Constant Bending Moment (M constant):
  1. For Varying Bending Moment (M changes along the length):
    Divide the beam into small elements, calculate  for each, and integrate as:

Units

  • In SI system: Joules (J)
  • In CGS system: Ergs
  • Since strain energy is energy stored, its units are the same as work (N·m or J).

Physical Meaning

The formula shows that strain energy due to bending depends on:

  • The square of the bending moment (M²) → more bending moment means more stored energy.
  • The modulus of elasticity (E) → stiffer materials (higher E) store less energy.
  • The moment of inertia (I) → larger cross-section reduces bending stresses and hence the stored energy.

It also helps determine deflection using the Castigliano’s theorem, which states that the partial derivative of strain energy with respect to load gives deflection at the point of load application.

Example

A simply supported beam of length  carries a central load of .
Given, , .

Bending moment at any distance  is:

M = \frac{W x}{2} \] for \( 0 \le x \le \frac{L}{2} \). Strain energy stored in half of the beam: \[ U = 2 \int_0^{L/2} \frac{M^2}{2 E I} dx

Substitute values:

 

Hence, the total strain energy stored in the beam due to bending is 1.04 × 10⁻⁶ Joules.

Importance of Strain Energy Due to Bending

  1. Used in Deflection Calculations:
    The strain energy method helps determine deflection in beams and frames accurately.
  2. Design of Beams:
    Ensures beams have sufficient flexibility without yielding under bending stresses.
  3. Spring and Leaf Spring Design:
    Helps estimate energy absorption capacity under bending.
  4. Failure Prevention:
    Ensures that bending stresses remain within the elastic limit to avoid permanent deformation.
  5. Dynamic Load Resistance:
    Helps predict behavior under impact or fluctuating bending loads.
Conclusion

The strain energy due to bending is the elastic potential energy stored in a beam when it bends under external loads. It is expressed as . This concept is vital in analyzing and designing beams, shafts, and springs that experience bending moments. It ensures that the structure can store energy safely within the elastic limit, resist deformation, and recover its original shape upon unloading. Understanding strain energy due to bending helps engineers design efficient and durable structural components.