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### Bar Restrained With Spring Uniform Heat Applied Calculator

Bar Restrained With Spring and Uniform Heat Applied 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 and spring by the thermal expansion is:

Eq. 1

$P=\frac{-{\alpha}_{1}{L}_{1}(T-{T}_{o}{)}_{1}+{\alpha}_{2}{L}_{2}(T-{T}_{o}{)}_{2}}{1/C+{L}_{1}/{A}_{1}{E}_{1}}$C = spring rate for L_{2} at its final temperature T. This spring may be real or represent another structure.

Figure 1 Restraint with Spring, Uniform Heat

Where

A = area of contact

L_{1} = length rigid bar

L_{2} = length spring

P = load

E_{1} = modulus of elasticity

E_{2} = modulus of elasticity spring

α_{1} = coefficient of thermal expansion bar

α_{2} = coefficient of thermal expansion spring

T = temperature initial

T_{o} = temperature final

- Bell Helicopter Structural Design Manual, 1977

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