# Thermal stress calculator

Tanvesh
Masters in Structural Engineering | Research Interest - Artificial Intelligence and Machine learning in Civil Engineering | Youtuber | Teacher | Currently working as Research Scholar at NIT Goa

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## What is thermal stress and strain?

The stress and strain induced in a restrained element due to change in ambient temperature is called as thermal stress and strain.

### Formula for thermal stress

\sigma=\alpha \Delta T E

Where \alpha = coefficient of thermal expansion.

\Delta T = Change in temperature.

E = Youngs modulus of the material.

The free expansion due to thermal stress is given as:-

\delta=\alpha \Delta T L

### Thermal stress definition

The thermal stress due to change in temperature is induced in the element only when the element is restrained from free expansion.

Thermal stress does not depend upon the area of cross section and shape of the cross section.

## Thermal stress in composite bars

### Bars in parallel (case 2)

If \alpha 1 < \alpha 2 then:

\alpha_{1} \Delta T L_{1}+\frac{\sigma_{1} L_{1}}{E_{1}}=\alpha_{2} \Delta T L_{2}-\frac{\sigma_{2} L_{2}}{E_{2}}

When two bars are connected parallel to each other and clamped at the end, then the material with lesser coefficient of thermal expansion increases more than its required free expansion and the material with higher thermal coefficient has expansion lesser than its free expansion.

### Bars in series (case 3)

If \alpha 1 < \alpha 2 then:

\alpha_{1} \Delta T L_{1}+\alpha_{2} \Delta T L_{2}=\frac{\sigma_{1} L_{1}}{E_{1}}+\frac{\sigma_{2} L_{2}}{E_{2}}

When two bars are connected in series, their net deflection is 0. For this the free expansion of two rods is equated to the induced deflection by stresses developed in the rods.

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