Answer :
The residual stress in the plate is approximately 33,174 psi.
To calculate the residual stress in the plate, we need to consider the thermal expansion and the temperature difference between the melt temperature and room temperature, as well as the yield stress of the material.
Given:
[tex]Melt temperature (T_m) = 1125 \degree F\\Room temperature (T_r) = 70 \degree F[/tex]
Elastic modulus (E) = 33,031,679 psi
Coefficient of thermal expansion (α) = 0.0000358 /°F
Yield stress ([tex]\sigma_y[/tex]) = 33,174 psi
Left side width (L₁) = 2 inches
First, we calculate the change in temperature (ΔT) between the melt temperature and room temperature:
ΔT =[tex]T_m - T_r[/tex]= 1125 °F - 70 °F = 1055 °F
Next, we calculate the thermal expansion mismatch between the different sections of the plate:
Expansion mismatch = ΔT [tex]\times[/tex]α [tex]\times[/tex]L₁
Expansion mismatch = 1055 °F * 0.0000358 /°F * 2 in = 0.0754 in
To determine if yielding occurs, we compare the stress due to thermal expansion (σ_expansion) with the yield stress (σ_y):
σ_expansion = E [tex]\times[/tex]α[tex]\times[/tex] ΔT
σ_expansion = 33,031,679 psi * 0.0000358 /°F * 1055 °F ≈ 126,019 psi
Since σ_expansion (126,019 psi) is greater than σ_y (33,174 psi), yielding occurs, and the residual stress will be equal to the yield stress:
Residual stress = σ_y = 33,174 psi.
Therefore, the residual stress in the plate is approximately 33,174 psi.
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