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Uniformly Heated Rectangular Plate Stress Calculator and Equations

Heat Transfer Engineering

Partially Restraint of a Uniformly Heated Rectangular Plate Stress Calculator and Equations

Uniformly Heated Rectangular Plate Restrained in the X and Y Directions by a Frame Plate Calculator and Equations for Stress

Preview Partially Restrained Rectangle Plate in Frame, Uniformly Heated Stress Calculator

Eq. 1

${\displaystyle {\mathrm{\sigma <!--\; \sigma \; -->}}_{x_1}=E_1/(1-<!--\; -\; -->{\mathrm{\mu <!--\; \mu \; -->}}^2)[{\mathrm{ϵ<!--\; ϵ\; -->}}_x+\mathrm{\mu <!--\; \mu \; -->}{\mathrm{ϵ<!--\; ϵ\; -->}}_y-<!--\; -\; -->(1+\mathrm{\mu <!--\; \mu \; -->}){\mathrm{\alpha <!--\; \alpha \; -->}}_1(T_1-<!--\; -\; -->T_0)]}$

Eq. 2

${\displaystyle {\mathrm{\sigma <!--\; \sigma \; -->}}_{y_1}=E_1/(1-<!--\; -\; -->{\mathrm{\mu <!--\; \mu \; -->}}^2)[{\mathrm{ϵ<!--\; ϵ\; -->}}_y+\mathrm{\mu <!--\; \mu \; -->}{\mathrm{ϵ<!--\; ϵ\; -->}}_x-<!--\; -\; -->(1+\mathrm{\mu <!--\; \mu \; -->}){\mathrm{\alpha <!--\; \alpha \; -->}}_1(T_1-<!--\; -\; -->T_0)]}$

Eq. 3

${\displaystyle {\mathrm{\sigma <!--\; \sigma \; -->}}_2=E_2[{\mathrm{ϵ<!--\; ϵ\; -->}}_x-<!--\; -\; -->{\mathrm{\alpha <!--\; \alpha \; -->}}_2(T_2-<!--\; -\; -->T_0)]}$

Eq. 4

${\displaystyle {\mathrm{\sigma <!--\; \sigma \; -->}}_3=E_3[{\mathrm{ϵ<!--\; ϵ\; -->}}_y-<!--\; -\; -->{\mathrm{\alpha <!--\; \alpha \; -->}}_3(T_3-<!--\; -\; -->T_0)]}$

Eq. 5

${\displaystyle \mathrm{\Sigma <!--\; \Sigma \; -->}{F}_x={\mathrm{\sigma <!--\; \sigma \; -->}}_{x_1}{A}_{x_1}+{\mathrm{\sigma <!--\; \sigma \; -->}}_2{A}_2=0}$

Eq. 6

${\displaystyle \mathrm{\Sigma <!--\; \Sigma \; -->}{F}_y={\mathrm{\sigma <!--\; \sigma \; -->}}_{y_1}{A}_{y_1}+{\mathrm{\sigma <!--\; \sigma \; -->}}_3{A}_3=0}$

Eq. 7

${\displaystyle A_{x_1}=bt}$

Eq. 8

${\displaystyle A_2=2w{t}_2}$

Eq. 9

${\displaystyle A_{y_1}=at}$

Eq. 10

${\displaystyle A_3=2v{t}_3}$

Equations 5 and 6 can be solved simultaneously for E_x and E_y. Substituting back into equations 1,2,3, and 4, the stress at the edge of the members can be obtained.

Figure 1 - Partially Restrained Rectangle Plate

Where

E₁ = modulus of elasticity plate 1
E₂ = modulus of elasticity frame plate 2
E₃ = modulus of elasticity frame plate 3
α₁ = coefficient of thermal expansion plate 1
α₂ = coefficient of expansion frame plate 2
α₃ = coefficient of expansion frame plate 3
ε_x = strain along x coordinate
ε_y = strain along y coordinate
T₀ = reference temperature
T₁ = temperature center plate
T₂ = temperature frame plate 2
T₃ = temperature frame plate 3
µ = Poisson's ratio
σ_x1 = stress
σ_x2 = stress
a = width
b = height
w = width of plate frame
v = width of plate frame
t₁ = thickness plate 1
t₂ = thickness plate 2
t₃ = thickness plate 3

• Bell Helicopter Structural Design Manual, 1977

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