Thermal Switch Design Project fast

[naevitar77]

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?

 

[nvn]

naevitar77: Excellent work, so far, except for one minor mistake. Pcr should be, Pcr = -4*Ea*Ia*(pi/L)^2. You erroneously omitted the negative sign here. Try it again.

By the way, numbers less than 1 should always have a zero before the decimal point. E.g., 0.25, not .25. See the international standard for writing units[/color] (ISO 31-0[/color]).

 

[naevitar77]

naevitar77: Excellent work, so far, except for one minor mistake. Pcr should be, Pcr = -4*Ea*Ia*(pi/L)^2. You erroneously omitted the negative sign here. Try it again.

The handout my professor passed out on this project defines Pcr without a negative sign. Why is it suppose to be negative?

 

[nvn]

Pcr is negative because it is compressive force. Your professor listed the magnitude.

 

[naevitar77]

That makes sense. Thank you so much nvn. You're a life saver!