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Home > Forums > Structural Engineers > Activities > Structural engineering other technical topics Forum

Deriving the elasticaHelpful Member! 

thread507-322680
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StevenKatzeff (Mechanical)
25 May 12 4:41
Hi.

I would like to derive the classic Bernoulli/Newton elastica curve from first principles.

Bernoulli's/Newton's classic elastica equation is of the form:

1) curvature = dθ/ds

The RHS reforms to the well known elastica equation:

2) dy2/d2x/(1+(dy/dx)2)3/2

But how do I get from 1) to 2)?

Here are the initial steps I took with no success:


1)ds= √dy2+dx2

2)sinθ=dy/√dy2+dx2

3)cosθ=dx/√dy2+dx2

4)1/ρ=dθ/ds

5)dsinθ/ds=cosθ.dθ/ds=cosθ/ρ


If anybody could help me that would be marvelous!
dhengr (Structural)
25 May 12 17:28
A good starting point for you might be to read and understand such books as: “The Mathematical Theory of Elasticity,” by A.E.H. Love; “The Mathematical Theory of Elasticity,” by I.S. Sokolnikoff; “Theory of Elastic Stability,” by S.P. Timoshenko and J.M. Gere; and other Theory of Elasticity text books.
BAretired (Structural)
26 May 12 15:39
Here is a blurb from an old math book:

BA

StevenKatzeff (Mechanical)
30 May 12 2:29
Thanks Everyone.

dhengr - a brief look through Timoshenko's "Theroy of Elasticity" yielded no derivation of the elastica I am considering.
BAretired - thanks for the picture, but they also seem to jump a few crucial steps in deriving the formula that I cannot replicate.

Any more suggestions?
Helpful Member!  StevenKatzeff (Mechanical)
30 May 12 6:01
With a little help from Physics Forum, Wikipedia and BAretired's picture, I finally have the solution. Excuse the bad hand-writing, I was excited...

Thanks everyone.
RogerWW (Structural)
20 Jul 12 18:11
The mathematics of the elastica is thoroughly derived in a book called "The Elastica" by an author named Firsch-Fay and English publisher I believe. RogerWW
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