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rverspuij1-Visitor
1-Visitor
June 27, 2016
Question

How to model bending beam segments?

  • June 27, 2016
  • 25 replies
  • 13343 views

I have tried to solve differential equations describing the bending behavior of a beam.

The equation is:

EI * w'''''= -q

This is only valid when all the derivatives of w are continuous.

Thus in the case of a point load, a new section is to be defined and conditions are formulated.

(same deflection, same angle, moment equilibrium, shear force equilibrium)

What is going wrong in my file?

    25 replies

    F
    Fred_Kohlhepp23-Emerald I
    23-Emerald I
    June 27, 2016

    I can't read your file, but I suspect this may help.

    V
    VladimirN24-Ruby IV
    24-Ruby IV
    June 28, 2016

    Please find a PDF copy of this worksheet in attachment.

    R
    rverspuij1-VisitorAuthor
    1-Visitor
    June 28, 2016

    You just beat me to it

    F
    Fred_Kohlhepp23-Emerald I
    23-Emerald I
    June 28, 2016

    Your file has several issues:

    • you're setting up two beams, one over L1, the second over L2.  It looks like you want to solve for deflection (w1, w2) over the full length of L1,L2; but w1 shouldn't be defined over L2, and w2 shouldn't be defined over L1.
    • You define q1(x) and q2(x) but don't solve for them.  (Looking at my first response above, q(x) = k  w(x), so your basic equations are not right unless you solve for the q1(x) and q2(x) as functions.  Then you need two more boundary conditions . . .


    R
    rverspuij1-VisitorAuthor
    1-Visitor
    June 28, 2016

    The q1 and q2 are just the loads on the two beams so why do I have to solve them?

    How do I tell mathcad that I want w1(x) solved for x=0...L1 and w2(x) for x=L1..L2?

    F
    Fred_Kohlhepp23-Emerald I
    23-Emerald I
    June 28, 2016

    Where did you get these equations (if not "beam on elastic foundation?)  The basic beam bending equation is second order:

    When I saw your equation, I immediately recalled an elastic foundation (railroad train tracks on the ground, or a pipe line.)  For these cases the ground pushes back--the larger the deflection the stronger the reaction.  This is q(x), but it's not constant.  For my cases q(x) = k * w(x), where k is N/mm^2.  So that is a different problem than what you're working because of the dependence on w(x).

    F
    Fred_Kohlhepp23-Emerald I
    23-Emerald I
    June 29, 2016

    Your technique would work if units were not involved.  (We both showed that.)

    If you fight your way thru with units, putting in the integration constants and using the built-in end boundary conditions to solve for them, you wind up with the same answers.  Mathcad apparently doesn't have the chops to do that directly, however.

    Added a painful unit-tracking solution to the back end.  Sorry for the mess.

    F
    Fred_Kohlhepp23-Emerald I
    23-Emerald I
    June 30, 2016

    You can solve this system this way, but you must non-dimensionalize the problem so Mathcad can deal with it.

    New file attached

    R
    rverspuij1-VisitorAuthor
    1-Visitor
    July 4, 2016

    My approach indeed lacked units but the problem was coupling two beams.

    That problem is not yet solved when using an approach like I suggested in the beginning.

    Your way of adding units is however usefull and interesting, thx!

    F
    Fred_Kohlhepp23-Emerald I
    23-Emerald I
    July 5, 2016

    I do not see how to add a point load to the beam using your method.  A point load produces a step change in beam shear, but beam shear is the integral of q.  You define q as a distributed load and it defines your fourth order derivative equation.  Unless you make a "point load" into a high magnitude short span "distributed load", I don't know how to fit it into your solution.  What about a moment applied?  How do you write those into your method?

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