1. The problem statement, all variables and given/known data 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 2. Relevant 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 3. 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?
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 (ISO 31-0).
The handout my professor passed out on this project defines Pcr without a negative sign. Why is it suppose to be negative?