# Thermal Stress Calculator

Table of contents

What is thermal stress? — Thermal load on a structureHow to calculate thermal stressExample: Using the thermal stress calculatorCoefficient of thermal expansionFAQsThe 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:

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:where:

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

## How to calculate thermal stress

To calculate thermal stress:

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

In the `Results`

part, you can also 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**. Take the

`20 °C`

**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:

- Enter the
**coefficient of linear thermal expansion**, $\alpha = 17 \times 10^{-6} \text{ K}^{-1}$. - Fill in the
**Young's modulus**of the material, $E = 110 \text{ GPa}$. - Insert the
**initial temperature**, $T_i = 20 \text{ °C}$. - Enter the
**final temperature**, $T_f = 50 \text{ °C}$. - Using the
**thermal stress**formula:

## 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 |

Lead | 13 | 29 |

Titanium | 116 | 8.6 |

Tungsten | 405 | 4.5 |

Concrete | 27 | 10 |

### 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:

- Find the
**initial and final temperature**of the material. **Subtract**the**initial temperature**from the**final temperature**to obtain the**temperature difference**.**Multiply**the**temperature difference**with the**coefficient of thermal expansion**.**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
- Temperature gradient, ΔT.

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:

**Deformation**in steel railway tracks.**Cracks**in concrete footpaths due to heat.**Slack**in power transmission lines due to heat.**Deformation**in gas pipelines.**Expansion**of metals during welding to form a joint.