Thermal Expansion - Stress and Force
Stress and force when thermal expansion is restricted
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Linear expansion due to change in temperature can be expressed as
dl = α lo dt (1)
where
dl = elongation (m)
α = temperature expansion coefficient (m/mK)
lo = initial length (m)
dt = temperature difference (oC)
The strain - or deformation - for an unrestricted expansion can be expressed as
ε = dl / lo (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/m2)
σ = stress (N/m2, Pa)
Thermal Stress
When restricted expansion is "converted" to stress - then (1), (2) and (3) can be combined to
σdt = E ε
= E dl / lo
= E α lo dt / lo
= E α dt (4)
where
σdt = stress due to change in temperature (N/m2)
Axial Force
The axial force acted by the restricted bar due to change in temperature can be expressed as
F = σdt A
= E α dt A (5)
where
F = axial force (N)
A = cross-sectional area of bar (m2)
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 20oC to 90oC. The expansion coefficient for steel is 12 10-6 m/mK. The modulus of elasticity for steel is 200 GPa (200 109 N/m2).
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If the expansion of the pipe is restricted - the stress created due to the temperature change can be calculated as
σdt = (200 109 N/m2) (12 10-6 m/mK) ((90oC) - (20oC))
= 168 106 N/m2 (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)
= 3598 mm2
= 3.6 10-3 m2
The force acting at the end of the pipe when it is restricted can be calculated as
F = (168 106 N/m2) (3.6 10-3 m2)
= 604800 N
= 604 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 oC - can be calculated from (1) to
dlPVC = (50.4 10-6 m/mK) (10 m) (100 oC)
= 0.054 m
The free expansion of the steel rod can be calculated from (1) to
dlsteel = (12 10-6 m/mK) (10 m) (100 oC)
= 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
σPVC = (2.8 109 Pa) (0.054 m - 0.012 m) / (10 m)
= 11.8 106 Pa
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