## Introduction

The previous article treated on thermal expansion in solids and liquids, and how engineers have to factor this scientific principle into their designs. The construction of buildings and roads can have disastrous consequences if this effect is ignored.

The picture shown here, courtesy of the Imperial College of London, is akin to ones gleefully displayed in certain introductory physics texts. Here, we see an example of railroad track warping directly attributed to thermal stress. Damages to brick structures can also be quite severe.

## Why Thermal Stress Occurs

Thermal stress occurs under heat or cold. Structures susceptible to it, such as roads, buildings, and railroad tracks, have beams or slabs of materials that are afixed into positions that are rigid. This positioning may make it difficult for the materials to expand or contract. Long structures may have localized effects, in that it is not unusual for one section to be buckled or warped while the others remain relatively intact. We have seen an example of this illustrated in the railroad track picture above.

## Calculation of Thermal Stresses

Fortunately, it is possible to calculate the magnitude of these stresses or forces.

Young’s modulus is used to find help find the force or stress on an object subject to simple tension or compression. To recall, tension occurs when an object is pulled or stretched, and compression occurs when an object is squeezed or pushed in. Engineers give the elastic modulus the symbol E. E varies according to the material, not to its shape or size. In SI units, E is 200×10^{9} N/m^{2} for steel, 20×10^{9 }N/m^{2 }for concrete, and for brick it is 14×10^{9 }N/m^{2}.

The force per unit area in terms of Young’s modulus and the dimensionless strain ΔL/L_{0} is:

1) F/A = E ΔL/L_{0}

In SI units, where force is measured in Newtons and area in meters, E is in N/m^{2}. From the first equation in the aforementioned article on thermal expansion, we know that the linear thermal expansion is:

2) ΔL = L_{0}αΔT or ΔL /L_{0 }= αΔT

Substituting the term for ΔL /L0 from 2 into equation 1 we find:

3) F/A = EαΔT = αEΔT

This equation allows us to calculate what may possibly happen to our material under thermal stress if it is not allowed to contract expand properly. We use our calculations and compare it with ultimate strength of material parameters for tensile, compressive, and/or shear strength to find if a substance will buckle or fracture under thermal stress. For a review of these terms, it is recommended that you read the Bright Hub article on materials engineering.

Thermal stress is the reason why engineers build bridges with expansion joints, and why concrete sidewalks and bridges are built with expansion spaces. Lack of proper spacing between concrete slabs to effects such as cracks, as you can see here.

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## For Discussion

Reinforced concrete is concrete where steel bars or fibers are used to strengthen the material. The coefficient of linear thermal expansion for concrete and steel is approximately the same at room temperature – in SI units α for both is 25 x10^{-6}/C^{0}. Why is this important?

## Resources

- Young’s Modulus for Materials in SI Units
- Young’s Modulus for Materials In BTU Units
- Coefficients of linear expansion for certain materials
- Bright Hub Article On Using Pro-Mechanica to Calculate Thermal Loads

## References

Physics for Scientists and Engineers by Douglas Giancolli

Fundamentals of Physics by Halliday, Resnick, and Walker

Conceptual Physics by Paul Hewitt