Short Answer:

Complementary energy is the energy stored in a material or structure when it is subjected to external loads, expressed in terms of stresses instead of strains. It is a concept used in elasticity to describe the energy associated with internal forces that develop due to external loads.

In simple words, complementary energy represents the area under the load-deformation curve when the load increases gradually from zero to its final value, assuming the relationship between stress and strain remains linear and elastic. It is helpful in analyzing structures using energy methods such as Castigliano’s theorem.

Detailed Explanation:

Complementary Energy

Complementary energy is an important concept in the study of elasticity and strength of materials. It is related to the internal energy stored in a body due to deformation, similar to strain energy, but expressed differently. When a load is applied to a body, it undergoes deformation, and internal stresses develop within it. These stresses and strains are responsible for storing potential energy inside the body, which can be represented as strain energy or complementary energy depending on whether the energy is expressed in terms of strain or stress.

In the strain energy approach, the energy is written as a function of strain, while in the complementary energy approach, it is expressed as a function of stress. Both represent the same physical quantity — the internal energy stored due to elastic deformation — but the mathematical forms differ depending on which variable is considered independent.

The concept of complementary energy is especially useful when stresses are known but strains are not easily measurable. This is common in many engineering applications such as beam bending, torsion, and structural analysis, where stresses can be found from equilibrium equations.

Definition and Expression

The complementary energy is defined as:

The energy stored in a body when the external load is applied gradually from zero to a final value, expressed in terms of stresses rather than strains.

Let us consider a linear elastic material that obeys Hooke’s law.
For a uniaxial stress–strain relationship, we have:

Where:

•  = stress
•  = strain
•  = modulus of elasticity

The strain energy per unit volume  is given by:

Substituting , we get:

Now, the complementary energy per unit volume  is also equal to:

Substituting Hooke’s law again ():

Hence, for linear elastic materials, the complementary energy and strain energy are equal in magnitude. However, their mathematical expressions differ based on whether they are written in terms of stress or strain.

Complementary Energy in Terms of Strain and Stress

In general, for any elastic body:

• Strain Energy (U):

• Complementary Energy (Uₐ):

Both expressions represent the area under the load–deformation curve, but one is taken with strain as the variable and the other with stress.

Importance of Complementary Energy

1. Useful in Energy Theorems:
   Complementary energy plays a major role in energy-based methods such as Castigliano’s theorem and the theorem of least work, which are used to determine deflections and internal forces in structures.
2. Simplifies Stress-Based Analysis:
   When stresses are known but strains are complex to determine, complementary energy provides a more convenient approach.
3. Applicable in Complex Systems:
   For materials and structures with multiple stress components (like in beams or pressure vessels), complementary energy helps to calculate total stored energy efficiently.
4. Foundation for Reciprocal Theorems:
   The concept forms the basis for Maxwell’s Reciprocal Theorem, which relates displacements at different points in an elastic body.

Graphical Representation

In the stress–strain diagram of a linear elastic material, the area under the curve up to a certain stress represents both the strain energy and complementary energy.

• For strain energy, the area is calculated with strain as the base.
• For complementary energy, the area is calculated with stress as the base.

In both cases, for elastic materials, the two areas are equal and represented by a triangle under the linear portion of the curve.

Example

Consider a steel rod subjected to a gradually increasing tensile load.
Let the stress developed be  and .

Then, the complementary energy per unit volume is:

This means that each cubic meter of material stores 25,000 joules of complementary energy due to the applied stress.

Conclusion

Complementary energy is the energy stored in an elastic material expressed in terms of stress. It is numerically equal to strain energy for linear elastic materials but differs in representation. This concept is essential in mechanical and structural analysis, especially when stresses are more accessible than strains. By using complementary energy, engineers can calculate deflections, internal forces, and reactions efficiently using energy-based methods, making it a vital tool in elasticity and strength of materials.