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naevitar77
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Homework Statement
There are three vertical metal strips (Steel, Aluminum, Steel) between two horizontal rigid bodies
alphaa = 12.5 E-6 /degF -- coefficient of thermal expansion
alphas = 6.6 E-6 /degF
Ea = 10 E6 psi -- Young's Modulus
Es = 30 E6 psi
The original dimensions of the switch are
ta = .0625 in -- thickness of aluminum
wa = .25 in -- width of aluminum
ts = .0625 in
ws = .125 in
L = 4 in -- length of all metals
With these dimensions, the switch will activate with a 180 degF temperature increase
By only changing the dimensions of the aluminum strip, I have to modify the switch to activate with 100 degF temperature increase
Homework Equations
δ = P*L/(E*A) + alpha*ΔT*L
Pcr = 4*pi^2*Ea*Ia / L^2 -- critical axial compressive load for the aluminum to buckle
Ia = wa*ta^3/12 -- minimum second moment of inertia
The Attempt at a Solution
I tried setting the deformation of the aluminum equal to that of the steel
δa = δs
Pa*L/(Ea*Aa) + alphaa*ΔT*L = Ps*L/(Es*As) + alphas*ΔT*L
from a free body diagram, I got Ps = -.5*Pa
Pa*L/(Ea*Aa) + alphaa*ΔT*L = -Pa*L/(2*Es*As) + alphas*ΔT*L
Pa*(1/(Ea*Aa) + 1/(2*Es*As)) = ΔT*(alphas - alphaa)
solving for Pa and simplifying using (1/a + 1/b)^-1 = ab/(a+b)
Pa = ΔT*(alphas - alphaa)*(2*Ea*Aa*Es*As / (Ea*Aa + 2*Es*As))
substituting Pcr = Pa
pi^2*Ea*wa*ta^3/(3*L^2) = ΔT*(alphas - alphaa)*(2*Ea*Aa*Es*As / (Ea*Aa + 2*Es*As))
now solving this for the temperature..
ΔT = pi^2*ta^2/(3*L^2) * (Ea*Aa + 2*Es*As)/(2*Es*As(alphas - alphaa))
when I plug in the variables for the 180 degF switch from above, I am getting
ΔT = -181.5 degF and I do not see anything wrong in my algebra; my units still come out to be degF.
Can someone tell me what I am doing wrong? Am I approaching this the right way?
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