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Bar Restrained One End and Gap on Other Formula and Calculator

Heat Transfer Engineering

Bar Restrained Against One End and Gap on Other Stress and Force 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 if and how much axial force is applied onto the bar by the thermal expansion is:

Eq. 1


g / L α ( T T o )

P = 0


Eq. 2

g / L < α ( T T o )

P = E A α ( T T o ) + g A E / L

Eq. 2a


σ = E α ( T T o ) + g E / L

Bar full restraint with gap, uniform heat
Bar full restraint with gap, uniform heat


A = area of cross section
L = length
g = gap
P = load
E = modulus of elasticity
α = coefficient of thermal expansion
T = temperature final
To = temperature initial

  • Bell Helicopter Structural Design Manual, 1977