Short Answer:

Complementary energy is the energy associated with the external work done by applied loads during the deformation of a body within the elastic limit. It represents the part of the total elastic potential energy that is recoverable when the external loads are gradually removed.

In simple terms, when a material is loaded, strain energy is stored inside it, and at the same time, an equal amount of energy called complementary energy is supplied by the external forces. Both these energies are equal under elastic conditions and together form the total potential energy of the system.

Detailed Explanation :

Complementary Energy

In the study of elasticity and strength of materials, complementary energy plays an important role in understanding how work is done by external forces and how energy is stored in the material.

When an elastic body is subjected to external loads, two forms of energy are considered:

1. Strain energy – the energy stored internally within the material due to deformation.
2. Complementary energy – the energy supplied externally by the applied loads during the process of loading.

Both these energies are equal in magnitude when the material behaves elastically (that is, follows Hooke’s law). The complementary energy can therefore be considered as the external counterpart of the internal strain energy.

Definition

The complementary energy is defined as:

“The energy represented by the external work done by gradually applied forces during the deformation of an elastic body within the elastic limit is called complementary energy.”

In other words, complementary energy is the work done by external loads to deform a body elastically.

It is also known as external strain energy or work energy, as it is the energy given to the system from outside.

Conceptual Explanation

When an external force  is applied gradually on a material, the material deforms, and the corresponding internal stresses develop to resist this load. During this process, the external load does work, and this work is stored as internal strain energy.

However, the energy point of view can be expressed in two ways:

• The internal strain energy (U) stored within the material.
• The external complementary energy (U’) supplied by the external forces.

Under perfectly elastic conditions,

and both together represent the total elastic potential energy of the body.

Mathematical Expression for Complementary Energy

Consider a member of uniform cross-section subjected to an external load .
Let  be the total deformation of the member under this load.

If the load is applied gradually from 0 to , the load increases linearly with deformation. Therefore, the average load during loading is .

The work done by the load, which is the complementary energy, is given by:

 

From Hooke’s law:

where,

•  = length of the member,
•  = cross-sectional area,
•  = modulus of elasticity.

Substituting this value in the above equation:

 

Now, dividing by the volume  of the member, we get the complementary energy per unit volume:

Since ,

This shows that the complementary energy per unit volume is equal to half the product of stress and strain within the elastic range.

Relation Between Complementary Energy and Strain Energy

For an elastic body following Hooke’s law,

Then,

and

Hence,

That means the strain energy stored in the material is numerically equal to the complementary energy supplied by the external load during elastic deformation.

Therefore,

Graphical Representation

In the stress–strain diagram, the strain energy corresponds to the area under the curve when stress increases with strain, while complementary energy corresponds to the area under the curve when stress decreases during unloading.

Under linear elastic behavior (Hooke’s law), the loading and unloading paths are identical, and thus the two areas are equal, meaning strain energy and complementary energy are equal.

Applications of Complementary Energy

1. Deflection Calculation:
   Complementary energy is used in energy methods like Castigliano’s theorem to find deflection and slope in beams and frames.
2. Elastic System Analysis:
   Helps determine total potential energy in elastic systems.
3. Failure Prediction:
   Used in evaluating energy absorption capacity before failure.
4. Dynamic Loading Problems:
   Helps analyze the energy balance in vibrating systems or components under impact loading.
5. Finite Element Analysis (FEA):
   Complementary energy forms the basis for numerical energy formulations used in computational mechanics.

Units

Since complementary energy is a form of work, its unit is the same as that of energy:

• In SI system: Joules (J)
• In CGS system: Ergs

For per unit volume: Joules per cubic meter (J/m³).

Physical Meaning

The complementary energy gives an idea of how much work external forces perform to deform a body elastically. It indicates the energy exchange between the external load and internal structure during deformation.
If the load is removed, this energy is completely recoverable, meaning the structure returns to its original shape without loss of energy (in the absence of hysteresis or plastic deformation).

Example

A steel bar of cross-sectional area , length , and modulus of elasticity  is subjected to a load of .

Thus, the complementary energy stored in the bar is 6.25 Joules, equal to its strain energy.

Conclusion

The complementary energy is the external work done by gradually applied loads during elastic deformation. It equals the internal strain energy stored in the material, and both together form the total elastic potential energy of a structure. Mathematically,

This concept is widely used in analyzing deformation, deflection, and energy storage in mechanical and structural systems. Understanding complementary energy helps engineers design components that can safely absorb and release energy within their elastic limits.