Calculating Thermal Beam Deflection for Space Boom Arm Design

[dmalwcc89]

Hey I am working with a group on a project where we basically have to design a boom arm for use in space. We've got most of the project figured out but there is a section where we have to determine the beam's deflection if the top side of the beam reached a 300 degree temperature while the bottom side was fixed at zero.

Obviously I've seen the equation delta=(thermal constant*length^2*temp change)/thickness but that is reresentative of when the two beam ENDS are differing temperatures, not the two beam SIDES as in my case.

My question is what equation would I be looking at to determine this information?

 

[nvn]

I tried a quick derivation (too lengthy to show here), and got a cantilever tip deflection of y = 0.5*alpha*(L^2)(Tb - Tt)/h, where alpha = coefficient of thermal expansion (CTE), L = cantilever length, h = cantilever depth, Tb = temperature of cantilever bottom fiber, and Tt = temperature of cantilever top fiber.

 

[Mapes]

I get the same answer as nvn, by assuming a bending strain of $\epsilon=\alpha\Delta T/2$ on the top and $-\alpha\Delta T/2$ on the bottom, plus an axial strain of $\alpha\Delta T/2$ to keep the bottom strain-free. For narrow beams, the amount of bending strain would corresponds to an effective stress of $\sigma=E\epsilon=\alpha\Delta TE/2$, or an effective bending moment of $M=\alpha\Delta TIE/h$. This applied moment would cause a deflection of $\delta=\alpha\Delta T L^2/2h$, assuming small deflections. (This may be the same derivation nvn used.)