Thermal Expansion - Stress and Force

Stress and force when thermal expansion is restricted

Temperature expansion - stress and strain

Linear expansion due to change in temperature can be expressed as

dl = α lo dt                              (1)


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)


ε = strain - deformation

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

E = σ ε                                (3)


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)


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

   E α dt A                                  (5)


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). 

Pipe - temperature expansion

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

Related Topics

  • Temperature Expansion - Thermal expansion of pipes and tubes - stainless steel, carbon steel, copper, plastics and more
  • Mechanics - Forces, acceleration, displacement, vectors, motion, momentum, energy of objects and more
  • Statics - Loads - force and torque, beams and columns

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