What is Thermal Stress? - Definition & Equation

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  • 0:04 Thermal Stress
  • 1:18 Formulas
  • 3:36 Example
  • 4:33 Lesson Summary
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Lesson Transcript
Instructor: Hassan Alsaud

Earned my B.S. in Civil Engineering back in 2011. Have two years of experience in oil and gas fields and two year as a graduate research assistant. Earned my Master degree in Engineering from Tennessee State University in 2016.

In this lesson, we'll discuss thermal stress, its effects on structural members, and how it is caused. We'll go over the formula for stress caused by thermal expansion and look at how thermal stress is overcome by structural design.

Thermal Stress

One of the properties of metals is that they transfer heat. Physical changes that occur with this transfer include that expansion when the temperature increases and shrinkage when the temperature decreases. This happens in all three dimensions.

Thermal stress occurs as a result of thermal expansion of metallic structural members when the temperature changes. Changes in temperature cause thermal deformation to the structural members. The values of these deformations can be described using the following formula, or relationship:

thermal deformation


  • deltat is the deformation of the structural member due to a change in temperature
  • alpha is the temperature coefficient of expansion, a material property measured in units per Kelvin (K)
  • L is original length of the structural member, measured in feet or meters
  • T is the final temperature measured in units of Kelvin or Celsiuso for the international system and Fahrenheito for the English system

T0 is the initial temperature, again measured in units of K or Co for the international system and Fo for the English system


When a structural member is free to move and expand, there is no stress exerted on it. However, when movement and expansion are restricted, then thermal stress occurs. When motion is restricted in the direction of expansion, the value of the reaction force is equal to the value of the force necessary to compress a beam in the opposite direction, and by the same amount of deformation. We can use the following formula to describe this relationship:

reaction force due to expansion


  • delta is the deflection of the beam due to the reaction force, which is equal to the deflection of the beam due to thermal expansion but in the opposite direction, shown in meters for the S.I. (or international system) and feet for the English system.
  • A is the area of the cross sectional section in f2 or m2.
  • E is the modulus of elasticity of the material from which the beam is manufactured in Pascal (Pa) for the international units system or a or lb/ft2 for the English system.
  • L is the length of the beam in feet or meters.

Putting together our understanding of thermal expansion and the forces involved, we can now solve for thermal stress, represented by the Greek letter sigma, measured in Newtons per square meter or Pascals (Pa).


delta t and delta f

delta t and delta f

And since,

sigma 2

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