beam bending with large deflections 1

[mmist]

i need to design a flat spring that can be approximated as a cantelivered beam. my problem is that i'm designing for 45 deggrees deflection whereas the simplified equations are only usable with small deflections. where would i look to find the appropriate formulas?

thanks

 

[Qshake]

Texts on Elasticity would be helpful...

You need to include the higher order terms that are normally dropped from the rigorous analysis based on the small displacement theory.

Most texts on advanced mechanics of materials or elasticity will develop these principles to show the higher order terms and when there are and are not relevant.

Regards,
Qshake

Eng-Tips Forums:Real Solutions for Real Problems Really Quick.

 

[prex]

In beams the assumption of sections remaining flat is still generally good for large deflections, so this is not a source of departure from the classical theory.
Another cause of departure is when the ratio r/h between radius of curvature and section height becomes small, but really small (typically less than 10 for more than a few per cent additional error).
So I assume that up to this point your cantilever is still adhering to the classical cubic deflection law (for a concentrated end load), assuming of course the material is behaving linearly.
But now comes the most difficult part.
An assumption that is no more valid at large deflections is the equality r=1/y'' that should be replaced by r=(1+y'²)^3/2/y''. Another point is that the lever arm of the end load in the deflection equation is no more x (origin at beam end), but something like
[∫](1+y'²)^-1/2dx
I'm not aware of any books tabulating the results for such a formulation, that should lead to a differential equation solvable only by numerical methods (not extremely difficult to do, though).

prex

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