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# Deriving the elastica

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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/dsThe 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+dx22)sinθ=dy/√dy2+dx23)cosθ=dx/√dy2+dx24)1/ρ=dθ/ds5)dsinθ/ds=cosθ.dθ/ds=cosθ/ρIf 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 http://files.engineering.com/getfile.aspx?folder=071cd9ff-d476-4baf-a218-c4
 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?
 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. http://files.engineering.com/getfile.aspx?folder=645b7a8b-f8b7-4dde-bea2-5c
 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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