How is deformation distributed in composite bars?

Short Answer:

In a composite bar, deformation is distributed among the different materials based on their elastic properties (E) and cross-sectional areas (A). Since the materials are rigidly joined, they experience the same total deformation (equal strain) under a given load, even though the stresses in each material may differ.

In simple words, when a composite bar made of two or more materials is loaded axially, all materials extend or compress by the same amount. However, each material carries a different portion of the total load according to its stiffness and cross-sectional area.

Detailed Explanation:

Deformation in Composite Bars

Definition:
A composite bar is a structural member made of two or more different materials that are rigidly connected so that they act as a single unit under loading. The materials are chosen to combine the beneficial properties of each, such as high strength, ductility, corrosion resistance, or light weight.

When such a composite bar is subjected to an axial load, each material deforms by the same amount of extension or compression, since they are bonded together. However, because the materials differ in elastic modulus (E) and cross-sectional area (A), the load carried and stress developed in each component are different.

This relationship between equal deformation and different stresses forms the basis of deformation distribution in composite bars.

Condition of Equal Deformation

For a composite bar to act as a single member, the materials must:

  1. Have the same initial length.
  2. Be rigidly bonded together, so no relative motion occurs between them.
  3. Experience the same total deformation (δ) when subjected to a load.

Hence, the axial strain in each material will be the same:

But the stress in each material will be different because of different moduli of elasticity:

So, the ratio of stresses in the materials is equal to the ratio of their elastic moduli:

This proves that deformation (strain) is the same in all materials, while stress (σ) varies with the material’s stiffness.

Derivation for Deformation Distribution in Composite Bar

Consider a composite bar of length , consisting of two materials — material 1 and material 2 — with:

  • Cross-sectional areas  and ,
  • Elastic moduli  and ,
  • Stresses  and ,
  • Carrying a total axial load .

From equilibrium condition:

where

From compatibility condition (equal deformation):

or,

From this, we get:

Substituting this into the equilibrium equation:

 

From which, the stress in material 2 is:

and stress in material 1 is:

The common deformation (δ) of the composite bar is:

Thus, although both materials deform equally (same δ), the stresses and loads shared by them are different.

Distribution of Deformation

From the above, we conclude:

  1. Same Deformation (Equal Strain):
    • Since the materials are joined together, they deform equally.
    • Mathematically:
  1. Different Stress:
    • Stress in each material is proportional to its elastic modulus :
    • Hence, a stiffer material (larger ) carries more load.
  2. Load Distribution:
    • The total external load  is divided between materials in proportion to :
    • So, deformation is same, but load division depends on material stiffness and area.

Example

A composite bar consists of steel and copper rods joined together, both of length .
Given:

Under a load :

Hence, the steel (stiffer material) carries twice the load of copper, although both experience same deformation.

This example shows that deformation is distributed equally but stress and load are shared based on material stiffness.

Applications of Composite Bars

  1. Reinforced Concrete:
    • Steel rods are embedded in concrete to share the load effectively.
    • Both materials deform equally but carry load based on their moduli.
  2. Bimetallic Strips:
    • Two metals with different coefficients of expansion deform differently under temperature change but are bonded to act as one unit.
  3. Machine Shafts:
    • Shafts with composite sections (steel + aluminum) distribute deformation uniformly.
  4. Thermal Stress Applications:
    • Used in cases where temperature-induced deformation must be managed by combining materials with different properties.

Advantages of Composite Bars

  • Efficient utilization of material properties.
  • Improved strength-to-weight ratio.
  • Reduced cost by combining expensive and inexpensive materials.
  • Controlled deformation and improved safety.
  • Useful in high-temperature or heavy-load applications.
Conclusion:

In a composite bar, deformation is uniformly distributed across all materials because they are rigidly bonded and experience equal strain. However, the stresses and loads are not equal; they are distributed according to the elastic modulus (E) and cross-sectional area (A) of each material. The stiffer material carries more load, while the softer one carries less. This principle ensures effective use of materials in engineering applications like reinforced concrete, bimetallic strips, and composite shafts.