# How to Calculate Thermal Stress ••• MarcBruxelle/iStock/GettyImages

In engineering mechanics classes, the study of thermal stress and its effect on various materials is important. Cold and heat can affect such materials as concrete and steel. If a material is unable to contract or expand when there are temperature differentials, thermal stresses may occur and cause structural problems. To check for problems, such as warping and cracks in concrete, engineers can calculate thermal stress values of different materials and compare them with established parameters.

Find the formula for thermal stress by using the equations for strain and Young’s modulus. These equations are:

Equation 1.) Strain (ϵ)

\epsilon=\alpha \Delta T

Equation 2.) Young’s modulus (E)

E=\frac{\sigma}{\epsilon}

In the strain equation, the term α refers to the linear coefficient of thermal expansion for a given material and ΔT is the temperature difference. Young’s modulus is the ratio that relates stress to strain. (Reference 3)

Substitute the value for Strain from the first equation into the second equation given in step 1 to get Young’s modulus in the following format:

E=\frac{\sigma}{\alpha \Delta T}

The equation in step 2 can also be rearranged to obtain a formula for thermal stress (σ)

\sigma=E\alpha \Delta T

Use the equation in step 3 to calculate the thermal stress in an aluminum rod that undergoes a temperature change or ΔT of 80 degrees Fahrenheit. (Reference 4)

Find Young’s modulus and the thermal expansion coefficient for aluminum from tables found readily in engineering mechanic books, some physics books, or online. These values are E = 10.0×106 psi and α = (12.3×10-6 inch)/ (inch degrees Fahrenheit),(See Resource 1 and Resource 2). Psi stands for pounds per square inch, a unit of measurement.

Substitute the values into the equation given in Step 3. You find that the thermal stress is

\sigma=(10.0\times 10^6\text{ psi})(12.3\times 10^{-6}\frac{\text{in}}{\text{inF}})(80\text{ F})=9840\text{ psi}

#### Tips

• To formulate the equation for thermal stress, it's important to know about the relationships that exist between stress, strain, Young’s modulus, and Hooke’s Law. (See Resource 3)

The linear coefficient of thermal expansion is a measure of how much a material expands for every degree of temperature rise. This coefficient is different for different materials. (See Resource 1)

Young’s modulus is related to the stiffness of a material or its elastic abilities. (Reference 3)

Note that the example in Step 5 is a simple application of this principle. When engineers work on the structural design of buildings, bridges, and roads, many other factors must also be measured and compared to different safety parameters.

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