# Thermal Stress Calculator

Created by Rahul Dhari
Reviewed by Steven Wooding
Last updated: Sep 24, 2022

The thermal stress calculator tells you the stress on the object due to any thermal load. The thermal stresses are a result of a change in temperature. Thermal loads on a structure include temperature changes that could be due to operating, like components of engines or heat exchangers (learn more about heat exchangers through our LMTD calculator) pipes and valves, being exposed to heat. It could also be a change in temperature due to weather, say, a considerable temperature drop in cold weather or a hot summer day in deserts.

Thermal stress is prevalent in objects like boilers, pipelines, and valves. Exposure to thermal loads could cause changes in dimensions as the structures expand or contract.

Have you noticed cracked footpaths? That occurs due to thermal stress in the concrete. Thermal expansion (see thermal expansion calculator) from heat leads to deformation in steel railway tracks or cracks in brittle materials that can cause catastrophic structural failure. Most engineers give allowances for such designs; for instance, you can observe small gaps between tracks that allow for thermal expansion. This allowance is set considering the coefficient of thermal expansion of stainless steel or any other material used in construction. The applicability of this concept is not limited to construction but it is also valuable for medical sciences to design dental filings and in machinery to devise gears, shafts, coupling, rivets, and boilers to name a few.

To this end, this article explains the thermal expansion formula and thermal stress.

## What is thermal stress? — Thermal load on a structure

Before getting into thermal stress (for mechanical stresses, refer to our stress calculator), let's look at thermal load? — It is the load acting on a structure caused by movements due to thermal expansion. This load is higher in the case of restrained structures that do not account for any movement of parts.

The load causes stress on the structure; that stress is known as thermal stress. In other words, the change in temperature makes the structure expand or contract. This movement causes mechanical stress on the structure, which we call thermal stress. It depends on the expansion rate of the material and the temperature gradient.

Now that you know what thermal stress is, you can use the thermal stress equation to estimate it:

$\quad \sigma_t = E \alpha \Delta T$

where:

• $\sigma_t$Thermal stress;
• $\alpha$Coefficient of thermal expansion;
• $E$Young's modulus; and
• $\Delta T$Change in temperature.

What is temperature gradient? — It is the change in temperature per unit length of a material. It is expressed in the temperature units such as °F or °C and K. For a unit length sized sample, the temperature gradient is:

$\quad \Delta T = T_f - T_i$

where:

• $T_f$ – Final temperature; and
• $T_i$ – Initial temperature.

## How to calculate thermal stress

To calculate thermal stress:

1. Enter the coefficient of linear thermal expansion, $\alpha$.
2. Fill in the Young's modulus of the material, $E$.
3. Insert the initial temperature, $T_i$.
4. Enter the final temperature, $T_f$.
5. The calculator will return the thermal stress.

You can turn on the advanced mode to check the temperature difference, $\Delta T$.

List of materials
You can select the material from the list to directly input the thermal expansions of metals and alloys.

## Example: Using the thermal stress calculator

Estimate the thermal stress in a copper bar if it is heated to a temperature of 50 °C from a temperature of 20 °C. Take the coefficient of thermal expansion of copper as $17 \times 10^{-6} \text{ K}^{-1}$ and Young's modulus as 110 GPa.

To calculate thermal stress in the bar:

1. Enter the coefficient of linear thermal expansion, $\alpha = 17 \times 10^{-6} \text{ K}^{-1}$.
2. Fill in the Young's modulus of the material, $E = 110 \text{ GPa}$.
3. Insert the initial temperature, $T_i = 20 \text{ °C}$.
4. Enter the final temperature, $T_f = 50 \text{ °C}$.
5. Using the thermal stress formula:
\quad \scriptsize \begin{align*} \sigma_t &= 110 \times 10^{9} \times 17 \times 10^{-6} \times (50 - 20) \\ &= 56 \text{ MPa} \end{align*}

## Coefficient of thermal expansion

The table below contains Young's modulus and coefficient of thermal expansion for metals and alloys. You can use the data from the table to find stresses due to thermal expansion in pipes or thermal stress in concrete.

Material

Young's Modulus (GPa)

Linear expansion coefficient (×10⁻⁶/K)

Aluminum

68

23.1

Brass

106

19

Copper

110

17

Gold

77.2

14

Silver

72

18

Gunmetal

103

19.8

Nickel

170

13

13

29

Titanium

116

8.6

Tungsten

405

4.5

Concrete

27

10

## FAQ

### What do you mean by thermal stress?

The stress due to the movements and deformation caused by thermal loads is known as thermal stress. Thermal stress is caused due to the temperature change, either due to the environment or operations. Friction between the components often causes a rise in temperature resulting in thermal stresses. Mathematically, the thermal stress equation is σ = EαΔT.

### How do I calculate thermal stress?

To calculate thermal stress:

1. Find the initial and final temperature of the material.
2. Subtract the initial temperature from the final temperature to obtain the temperature difference.
3. Multiply the temperature difference with the coefficient of thermal expansion.
4. Multiply the result with the material's Young's modulus to obtain the thermal stress.

### What are the factors affecting thermal stress?

The thermal or temperature stress is a function of:

• Young's Modulus, E, of a material;
• Coefficient of linear thermal expansion, α; and

The sign of temperature gradient is crucial to decide whether the thermal stress is compressive (-ve) or tensile (+ve). If the final temperature is higher than the initial temperature for the structure, the temperature gradient is positive, and vice versa.

### What are some examples of thermal expansion?

Some of the examples of thermal expansion are:

1. Deformation in steel railway tracks.
2. Cracks in concrete footpaths due to heat.
3. Slack in power transmission lines due to heat.
4. Deformation in gas pipelines.
5. Expansion of metals during welding to form a joint.
Rahul Dhari
Material properties
Material
Custom ▾
Thermal expansion coefficient (α)
/ K
Young's modulus (Em)
psi
Temperature
Initial temperature (Ti)
°F
Final temperature (Tf)
°F
Results
Thermal stress (σt)
psi
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