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Two Bars Different Temperatures Stress Calculator and Equations

Heat Transfer Engineering

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

${\displaystyle {\mathrm{\sigma <!--\; \sigma \; -->}}_1=-<!--\; -\; -->E_1{\mathrm{\alpha <!--\; \alpha \; -->}}_1(T_1-<!--\; -\; -->T_0)C_1}$

Eq. 2

${\displaystyle {\mathrm{\sigma <!--\; \sigma \; -->}}_2=-<!--\; -\; -->A_1/A_2{\mathrm{\sigma <!--\; \sigma \; -->}}_1}$

Eq. 3

${\displaystyle C_1=\left[\frac{E_2{A}_2}{E_1{A}_1+E_2{A}_2}\right]\left[\frac{1-<!--\; -\; -->{\mathrm{\alpha <!--\; \alpha \; -->}}_2(T_2-<!--\; -\; -->T_0)}{{\mathrm{\alpha <!--\; \alpha \; -->}}_1(T_1-<!--\; -\; -->T_0)}\right]}$

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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