# Restricted Thermal Expansion - Introduced Force and Stress

## Stress and force when thermal expansion a pipe, beam or similar is restricted.

Linear expansion due to change in temperature can be expressed as

*dl = α l _{o} dt (1)*

*where *

*dl = elongation (m, in)*

* α = temperature expansion coefficient (m/mK, i n/in^{ o}F)*

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

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

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 (Pa (N/m^{2}), psi)*

*σ = stress (Pa (N/m ^{2}), psi)*

### 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 (Pa (N/m

^{2}), psi)

### 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}, in^{2})*

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

A *DN150 Std. (6 in)* steel pipe with length *50 m (1969 in)* is heated from *20 ^{o}C (68^{o}F)* to

*90*. The expansion coefficient for steel is

^{o}C*(194*^{o}F)*12 10*. The modulus of elasticity for steel is

^{-6}m/mK (6.7 10^{-6}in/in^{o}F)*200 GPa (10*

^{9}N/m^{2}) (29 10^{6}psi (lb/in^{2})).make 3D models with The Engineering ToolBox Sketchup Extension

Expansion of unrestricted pipe:

*dl = ( 12 10^{-6} m/mK) (50 m) ((90^{o}C) - (20^{o}C))*

* = 0.042 m*

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)*

* = 168 MPa*

**Note**! - if there is pressure in the pipe - the axial and circumferential (hoop) stress may be added to restricted temperature expansion stress by using vector addition.

The outside diameter of the pipe is *168.275 mm (6.63 in)* and the wall thickness is *7.112 mm (0.28 in)*. The cross-sectional area of the pipe wall can then be calculated to

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

* = 3598 mm ^{2}*

* = 3.6 10 ^{-3} m^{2}*

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

*F = (168 10^{6} N/m^{2}) (3.6 10^{-3} m^{2})*

* = 604800 N*

* = 604 kN*

##### The calculation in Imperial units

Expansion of unrestricted pipe:

*dl = (6.7 10^{-6} in/in^{o}F) (1669 in) ((194^{o}F) - (68^{o}F))*

* = 1.4 in*

Stress in restricted pipe:

*σ _{dt} = (29 10^{6} lb/in^{2}) (6.7 10^{-6} in/in^{o}F) ((194^{o}F) - (68^{o}F))*

* = 24481 lb/in ^{2} (psi)*

Cross sectional area:

*A = π ((6.63 in) / 2)^{2} - π ((6.63 in) - 2 (0.28 mm)) / 2)^{2}*

* = 5.3 in ^{2}*

Axial force acting at the ends:

*F = (24481 lb/in^{2}) (5.3 in^{2})*

* = 129749 lb*

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

When two materials with different temperature expansion coefficients are connected - as typical with concrete and steel reinforcement, or in district heating pipes with PEH insulation etc. - temperature changes introduces tensions.

This can be illustrated with a PVC plastic bar of *10 m* reinforced with a steel rod.

The free expansion of the PVC bar without the reinforcement - with a temperature change of *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 with a temperature change of *100 ^{o}C* - 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 *

* = 11.8 MPa*

The Tensile Yield Strength of PVC is approximately *55 MPa*.

### Thermal Expansion Axial Force Calculator

This calculator can be used to calculate the axial force caused by an object with restricted temperature expansion. The calculator is generic and can be used for both metric and imperial units as long as the use of units are consistent.

*Length of restricted object (m, inches)*

* Area of restricted object (m ^{2}, in^{2})*

* Temperature difference ( ^{o}C, ^{o}F)*

* Young's modulus (GPa, 10 ^{9} psi)*

* Expansion coefficient (10 ^{-6} m/mK, 10^{-6} in/in^{o}F) *