Short Answer:

Thermal stress is the stress developed in a material when it is heated or cooled and is not allowed to expand or contract freely. This happens due to temperature changes that cause internal forces resisting the expansion or contraction of the material.

In simple words, when a solid body is heated, it tends to expand, and when it is cooled, it tends to contract. If this natural expansion or contraction is restricted, internal stresses called thermal stresses are produced inside the material. These stresses can be compressive or tensile depending on the direction of temperature change.

Detailed Explanation:

Thermal Stress

Definition and Meaning:
When a material is subjected to a change in temperature, its length changes. This change in length is called thermal deformation or thermal strain. If the material is free to expand or contract, no stress is developed. However, if the material’s movement is restricted due to external constraints or structural conditions, thermal stresses are produced inside the material to oppose the deformation.

The magnitude of thermal stress depends on the material’s coefficient of thermal expansion, temperature change, and modulus of elasticity. These stresses are very important in engineering design because temperature variations can cause materials to crack, bend, or fail if not properly accounted for.

Derivation of Thermal Stress

Let us derive the expression for thermal stress developed in a bar when it is not allowed to expand or contract freely.

Consider a bar:

Length of bar =
Cross-sectional area =
Modulus of elasticity =
Coefficient of linear expansion =
Temperature change =

Free expansion:

If the bar is free to expand, the increase in length due to temperature rise is:

No stress develops in this case because the expansion is unrestricted.

Restrained condition:

If both ends of the bar are fixed and the bar is not allowed to expand, it will try to expand by the above amount, but due to the restriction, compressive stresses develop inside the material to oppose this expansion.

The thermal stress () is related to the strain prevented. Since the actual expansion is zero, the strain prevented () is equal to the free thermal strain.

Now, according to Hooke’s Law,

Substitute the value of :

This is the formula for thermal stress.

Explanation of Terms:

: Thermal stress (N/m² or Pa)
: Modulus of Elasticity of material (N/m²)
: Coefficient of linear expansion (/°C)
: Change in temperature (°C or K)

This equation shows that the thermal stress is directly proportional to modulus of elasticity, coefficient of thermal expansion, and temperature change.

Types of Thermal Stresses

Tensile Thermal Stress:
When a bar is cooled and prevented from contracting, it develops tensile stress because the bar tries to shorten but is held fixed.
Compressive Thermal Stress:
When a bar is heated and prevented from expanding, it develops compressive stress because it tries to elongate but is restricted.

Thermal Strain

If the expansion or contraction of a body is partially restricted, both strain and stress will develop.
The thermal strain is given by:

It represents the change in length per unit length due to temperature change.

For partially restricted conditions, the effective strain and stress can be found based on the degree of restriction and external loads acting on the body.

Numerical Example

Let us calculate thermal stress developed in a steel bar.

Given:
,
,

Using the formula:

Hence, a compressive thermal stress of 120 MPa will develop if the bar is prevented from expanding when heated by 50°C.

Factors Affecting Thermal Stress

Temperature Change ():
The greater the change in temperature, the higher the thermal stress.
Material Property (E and α):
Materials with higher modulus of elasticity (E) and higher coefficient of expansion (α) develop higher thermal stress.
Boundary Conditions:
- Free ends → no thermal stress.
- Fully fixed ends → maximum thermal stress.
- Partially fixed ends → intermediate stress.
Shape and Size of the Material:
Larger sections may develop uneven stress due to temperature gradients.
Rate of Heating or Cooling:
Rapid temperature change increases the risk of thermal cracking due to non-uniform expansion.

Practical Examples of Thermal Stress

Railway Tracks:
Rails expand in summer and contract in winter. Expansion joints are provided to reduce thermal stress.
Bridges:
Bridge decks expand and contract due to temperature variation. Expansion joints are provided between bridge segments to prevent cracking.
Pipes and Boilers:
Thermal stress occurs in pipes carrying hot fluids. Expansion bends are used to absorb the expansion safely.
Concrete Structures:
Thermal stress develops due to temperature gradients between inner and outer layers, which can cause cracks.
Machine Components:
Shafts, engine blocks, and turbine blades experience thermal stress during operation because of uneven heating.

Importance of Thermal Stress in Engineering Design

Prevention of Failure: Engineers must consider thermal stresses to prevent cracking, buckling, or fracture.
Material Selection: Materials with suitable thermal expansion coefficients are chosen for high-temperature applications.
Expansion Control: Use of expansion joints, flexible supports, and insulating materials minimizes damage.
Safety in Thermal Systems: Ensures safe operation of boilers, reactors, and engines.

Reduction of Thermal Stresses

Allowing Free Expansion:
Provide gaps or joints to allow movement.
Use of Low α Materials:
Materials with small coefficients of expansion (e.g., Invar steel) are used.
Gradual Temperature Change:
Avoid sudden heating or cooling.
Pre-stressing:
Apply pre-compressive stress to balance thermal expansion.
Composite Material Design:
Combine materials to balance thermal effects.

Conclusion:

Thermal stress is the internal stress induced in a body due to temperature changes when expansion or contraction is restricted. It can be compressive or tensile depending on whether the material is heated or cooled. The formula for thermal stress is:

Thermal stresses play a vital role in the design of machines, bridges, pipelines, and concrete structures. To prevent failure, engineers must carefully consider temperature effects, use proper materials, and allow for safe thermal expansion through joints or flexible design.