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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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