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Elastic energy of a bent steel rod

  1. Sep 5, 2013 #1
    1. The problem statement, all variables and given/known data
    [Math. for Physicists, M. Stone Problem 1.4]

    Assume that a rod of length L is only slightly bent into the yz plane and lies close to the z axis, show that the elastic energy can be approximated as
    [tex]U[y]= \int_{0}^{L} \frac{1}{2}YI(y'')^2 dz[/tex]

    2. Relevant equations

    It is given that the elastic energy per unit length of a bent rod , [tex]u=\frac{1}{2}YI/R^{2}[/tex]
    R is the radius of curvature due to the bending, Y is the Young's modulus of the steel and I is the moment of inertia of the rod's cross section about an axis through its centroid and
    perpendicular to the plane in which the rod is bent.

    3. The attempt at a solution

    I don't quite get the picture.
    Does it mean that each infinitesimal piece is a segment of a circle R with a different center? Or should I consider the whole bent rod as a segment of a circle of radius R?

    But still the infinitesimal rod length should be [tex]\sqrt{1+(y')^2} dz[/tex], so how can I get [tex]y''^2 [/tex]?

    Thank you very much!



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 5, 2013 #2

    TSny

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