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### Two Bars Different Temperatures Stress Calculator and Equations

Two Bars At Different Temperatures Stress Calculator and Equations

The bars are attached such that the cold bar restrains the expansion of the hot bar. The bars remain straight with no bending.

Eq. 1

${\sigma }_{1}=-{E}_{1}{\alpha }_{1}\left({T}_{1}-{T}_{0}\right){C}_{1}$

Eq. 2

${\sigma }_{2}=-{A}_{1}/{A}_{2}{\sigma }_{1}$

Eq. 3

${C}_{1}=\left[\frac{{E}_{2}{A}_{2}}{{E}_{1}{A}_{1}+{E}_{2}{A}_{2}}\right]\left[\frac{1-{\alpha }_{2}\left({T}_{2}-{T}_{0}\right)}{{\alpha }_{1}\left({T}_{1}-{T}_{0}\right)}\right]$

Figure 1 Two bars at different Temperatures

Where

A1 = Cross section area bar 1
A2 = Cross section area bar 2
E1 = modulus of elasticity bar 1
E2 = modulus of elasticity bar 2
α1 = coefficient of thermal expansion bar 1
α2 = coefficient of thermal expansion bar 1
T0 = initial (reference) temperature
T1 = temperature bar 1
T2 = temperature bar 2
σ1 = stress bar 1
σ2 = stress bar 2

• Bell Helicopter Structural Design Manual, 1977

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