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FEA: Stiffness Matrix for Beam Element

  1. May 14, 2012 #1
    I have gone through Beam Element analysis in several books on FEA. 80% of books have a particular Stiffness matrix, while a very small number of books have a matrix which is subtly different in sign.


    Most books have the 2nd matrix in the photo above.
    A couple of books have 1st one.

    Likewise the loading matrix is also slightly different.

    Corresponding to the first one, the loading matrix is F*L/12 [ 6 -L 6 L]T
    While the majority have F*L/12 [ 6 L 6 -L]T

    Which method is correct?
    Last edited: May 14, 2012
  2. jcsd
  3. May 14, 2012 #2


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    The two matrices are using different sign conventions for positive displacements and rotations. if they both take the same direction (up or down) for positive displacements, one will be taking clockwise rotations as positive and the other antoclockwise as positive.

    I can't remember which is which, but the books should say what convention they are using.

    It doesn't matter which system you use so long as you are consistent.
  4. May 14, 2012 #3
    It looks like it's the sign of the BM (not the others). The first matrix uses sagging BM as negative, the 2nd one uses sagging BM as positive.
    That's what I think.

    Unfortunately when I looked through all these books in the library, I didn't think about checking the BM sign convention and I am not going to go to the library for week now. If anyone has a copy of any book which uses the 1st matrix, can they please confirm if the book uses sagging BM negative conventions? That would be helpful.
    Last edited: May 14, 2012
  5. May 14, 2012 #4


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    There are plenty of references on the web, e.g.
    http://comp.uark.edu/~jjrencis/femur/Learning-Modules/Stress-Analysis/One-Dimensional-Elements/Beam-Element/Element_Formulation.html [Broken] has lots of diagrams.
    Last edited by a moderator: May 6, 2017
  6. May 15, 2012 #5
    Both, although the convention when teaching beam deflection problems is that 'downward' deflection is taken as positive. (I think this convention comes from Naval engineering cant remember now though)

    Its just a matter of how the reference coordinate system is defined.
  7. May 15, 2012 #6


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    It doesn't matter whether you choose "up" or "down" as a positive deflection. What matters is now the positive slopes (or moments) are related to positive deflections.

    IMO there doesn't seem much sense in teaching this using a "left handed" coordinate system - but if that's the traditional way it is done, I guess students can expect to be confused later, since every general purpose computer analysis program I know of uses right handed coordiate systems.
  8. May 16, 2012 #7

    Well yes and no really. It does help in a broader sense by introducing students to the idea that you can choose a coordinate system to suit your problem. Its suprising how many students dont realise this.
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