# Thermal Expansion - Stress and Force

## Stress and force with restricted thermal expansion

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Linear expansion due to change in temperature can be expressed as

*dl = α l _{o} dt (1)*

*where *

*dl = elongation (m)*

* α = temperature expansion coefficient (m/mK)*

*l _{o} = initial length (m)*

*dt = temperature difference ( ^{o}C) *

The strain - or deformation - for an unrestricted expansion can be expressed as

*ε = dl / l _{o }(2)*

*where*

*ε =* strain - deformation

The Elastic modulus (*Young's Modulus*) can be expressed as

*E = σ / ε (3)*

*where *

*E =Young's Modulus (N/m ^{2})*

*σ = stress (N/m ^{2}, Pa)*

### Thermal Stress

When restricted expansion is "converted" to stress - then *(1)*, *(2)* and *(3)* can be combined to

*σ _{dt} = E ε *

* = E dl / l_{o }*

* = E α l_{o} dt / l_{o}*

* = E α dt (4)*

*where*

*σ _{dt} =* stress due to change in temperature (N/m

^{2})

### Axial Force

The axial force acted by the restricted bar due to change in temperature can be expressed as

*F = σ _{dt} A *

* = E α dt A (5)*

*where *

*F = axial force (N)*

*A = cross-sectional area of bar (m ^{2})*

#### Example - Heated Steel Pipe - Thermal Stress and Force with Restricted Expansion

A *DN150 Std. (6 in)* steel pipe with length *50 m* is heated from *20 ^{o}C* to

*90*. The expansion coefficient for steel is

^{o}C*12 10*. The modulus of elasticity for steel is

^{-6}m/mK*200 GPa (200 10*

^{9}N/m^{2}).- make 3D models with The Engineering ToolBox Sketchup Extension

If the expansion of the pipe is restricted - the stress created due to the temperature change can be calculated as

*σ _{dt} = (200 10^{9} N/m^{2}) (12 10^{-6} m/mK) ((90^{o}C) - (20^{o}C))*

* = 168 10 ^{6} N/m^{2} (Pa)*

The outside diameter of the pipe is *168.275 mm* and the wall thickness is *7.112 mm*. The cross-sectional area of the pipe wall can then be calculated to

*A = π (((168.275 mm) / 2) ^{2} - ((168.275 mm) - 2 (7.112 mm)) / 2)^{2})*

* = 14396 mm ^{2}*

* = 14 10 ^{-3} m^{2}*

The force acting at the end of the pipe when it is restricted can be calculated as

*F = (168 10^{6} N/m^{2}) (14 10^{-3} m^{2})*

* = 2352000 N*

* = 2352 kN*

### Example - Thermal Tensions in Reinforced or Connected Materials

When two materials with different temperature expansion coefficients are connected - typical with steel reinforcement in concrete, in district heating pipes with PEH insulation etc. - a temperature change will introduce tensions.

This can be illustrated with a case where a steel rod reinforce a PVC plastic bar of *10 m*, the free expansion of the PVC bar without reinforcement - with temperature change *100 ^{o}C* - can be calculated from

*(1)*to

*dl _{PVC} = (50.4 10^{-6} m/mK) (10 m) (100 ^{o}C) *

* = 0.054 m*

The free expansion of the steel rod can be calculated from *(1) *to

*dl _{steel} = (12 10^{-6} m/mK) (10 m) (100 ^{o}C)*

* = 0.012 m*

If we assume that the steel rod is much stronger than the PVC bar (depends on the Young's modulus and the areas of the materials) - the tension in the PVC bar can be calculated from the difference in temperature expansion with (4) as

*σ _{PVC} = (2.8 10^{9} Pa) (0.054 m - 0.012 m) / (10 m) *

* = 11.8 10 ^{6} Pa *

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