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As the simplified theory of beam deflection, taught in typical mechanical engineering courses (in Germany its called first order theory), is limited to small deflections, I want to look into the generalised theory of beam deflection including large deflections and forces acting upon the deformed beam etc. (third order theory)

I do know that this is not solvable analytically, as these more general formulations are nonlinear ODEs/PDEs. Would love to implement a solver myself for some simple examples.

Sadly I didn't find any textbooks treating this topic and the papers I found don't do any form of derivation.

Could someone either explain the matter on the example of a standard cantilever beam or link a textbook that does that? cantilever beam

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    $\begingroup$ For a start, look into "Mechanics of Materials (CH7, 2nd)", by Gere & Timoshenko. $\endgroup$
    – r13
    Jul 25, 2023 at 15:51

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The first order solution to beam deflections assumes trigonometric simplifications (i.e., sin theta is approximately theta for small theta). So the first step on the way to large theta is to eliminate all these approximations and rewrite the constituitive equations without them.

Now note that the x-axis projection of a cantilever beam that is bent by a point load on the end gets shortened for large deflections. this reduces the effective lever arm length for the point load, an effect which is neglectable for small deflections. now you rewrite the deflection equation, including this factor.

And, yes- the act of including all these effects will make the differential equations not have closed-form solutions.

Note that a beam undergoing large deflections nonetheless "solves" these equations on its own in the sense that a well-defined (large) load produces a well-defined (large) deflection.

In practice, the best path forward for large deflections is actually finite-element analysis.

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Sure, I wrote a solution here. It's basically a very simple FE model.

https://smath.com/wiki/GetFile.aspx?File=Examples/finger4.zip

Runs in Smath Studio

As you say the general problem is not solvable analytically, the ODEs are easy to write but have to be solved numerically, as in many or even most real world cases.

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