Short Answer:

The strain energy due to bending is the energy stored in a beam or structural member when it bends under the action of external loads. This energy is stored in the material because of the internal stresses and strains developed within the beam during bending.

When the external load is removed, this stored energy is released, allowing the beam to return to its original shape if the material remains within the elastic limit. The amount of strain energy depends on the bending moment, the material’s modulus of elasticity, and the geometry of the beam section.

Detailed Explanation:

Strain Energy due to Bending

When a beam or any structural member is subjected to a bending moment, it experiences both tensile and compressive stresses on opposite sides of its neutral axis. These stresses produce strains within the material, and as a result, the beam stores a certain amount of strain energy. This stored energy is known as the strain energy due to bending.

In simple terms, it is the potential energy accumulated in a material as it resists deformation under bending loads. This concept is very important in mechanics of materials, as it helps engineers analyze the behavior of beams and structures under various loading conditions, predict deflection, and ensure safety against failure.

The strain energy is entirely recoverable as long as the beam remains within its elastic limit. If the applied stress exceeds the elastic limit, permanent deformation occurs, and the energy cannot be fully recovered.

Derivation of Strain Energy due to Bending

Consider a small element of a beam subjected to bending under a bending moment .

The bending stress at a distance  from the neutral axis is given by:

Where:

•  = Bending stress at fiber distance
•  = Bending moment at the section
•  = Moment of inertia of the beam’s cross-section about the neutral axis

The strain corresponding to this stress is:

For a small volume element , the strain energy stored is:

Substitute the expressions for  and :

Now, integrate over the entire cross-sectional area :

Since , we get:

Hence, the strain energy per unit length of the beam is:

The total strain energy stored in the beam of length  is:

This equation shows that the total strain energy due to bending depends on the square of the bending moment, the modulus of elasticity, and the moment of inertia of the beam’s cross-section.

Factors Affecting Strain Energy due to Bending

1. Bending Moment (M):
   The strain energy increases with the square of the bending moment. Larger bending moments lead to greater energy storage in the beam.
2. Material Property (E):
   The modulus of elasticity () represents the stiffness of the material. Materials with higher  values store less strain energy for the same bending moment.
3. Cross-sectional Shape (I):
   The moment of inertia () indicates the beam’s resistance to bending. Beams with larger  values store less strain energy for the same bending load.
4. Length of Beam (L):
   Longer beams under the same load will have different bending moment distributions, affecting total strain energy.
5. Loading Conditions:
   The type of loading (point load, distributed load, etc.) affects the bending moment distribution and, consequently, the strain energy.

Applications of Strain Energy due to Bending

1. Deflection Calculations:
   The concept of strain energy is used in the Castigliano’s theorem, which relates deflection to the rate of change of strain energy with respect to load.
2. Design of Springs and Flexible Components:
   Beams and leaf springs are designed considering the amount of strain energy they can store without permanent deformation.
3. Failure Analysis:
   Strain energy helps determine whether a structure can safely absorb energy without exceeding its elastic limit.
4. Vibration and Impact Studies:
   Structures that undergo impact loading rely on strain energy concepts to assess how energy is absorbed and dissipated.
5. Structural Optimization:
   Engineers use strain energy concepts to design lighter and stronger beams that can store energy efficiently under bending loads.

Example:

For a simply supported beam with a central point load  and span , the maximum bending moment is .

The total strain energy due to bending is:

This equation helps in calculating how much energy is stored within the beam due to bending under load.

Conclusion:

The strain energy due to bending is the elastic energy stored in a beam when subjected to bending loads. It is essential for understanding deflection, stress distribution, and structural safety. The formula  shows that strain energy depends on bending moment, material properties, and cross-sectional geometry. This concept plays a major role in the design and analysis of beams, bridges, frames, and other load-bearing structures to ensure strength and reliability.