Restricted Thermal Expansion  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, in/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 = crosssectional 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^{o}C (194^{o}F). The expansion coefficient for steel is 12 10^{6} m/mK (6.7 10^{6} in/in^{o}F). The modulus of elasticity for steel is 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 crosssectional 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.
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Forces acting on bodies at rest under equilibrium conditions  loads, forces and torque, beams and columns. 
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Thermal expansion and expansion coefficients. Expansion pipes and tubes made of stainless steel, carbon steel, copper, plastics and more.
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