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Restrained Beam Thermal Expansion Force Axial Stress Calculator

Heat Transfer Engineering

Bar Restrained Against Lengthwise Thermal Expansion Calculator and Equations

A bar rigidly mounted with uniform heat applied will expand and exert force and axial stress onto itself and adjoining surfaces.

When a structural element is subjected to a change in temperature, it will either expand or contract depending on whether the temperature change is an increases or decrease. I the element is restrained, the attempt ar expansion or contraction will induce stresses into the structure.

The formula for determining the axial force applied onto the bar by the thermal expansion is:

Eq. 1

P = A E α ( T T o )


Eq. 2

σ = E α ( T T o )

Bar full restraint, uniform heat
Bar full restraint, uniform heat




A = area of contact
P = load
E = modulus of elasticity
α = coefficient of thermal expansion
T = temperature final
To = temperature initial

  • Bell Helicopter Structural Design Manual, 1977