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Calculating the deflection on curved beams

  1. Sep 28, 2011 #1
    1. The problem statement, all variables and given/known data

    Hi guys,

    I need some help. I haven’t done mechanics for a few years and am a little rusty but have a task for work to verify FEA (which hasn’t been done yet).

    I have a semi circular beam that is bolted down on its ends. Mid way there is vertical force of 100kN pulling the semi circle. I need to find the maximum deflection in the vertical and horizontal direction that this force causes, and if possible the deflection at any angle.

    The beam rectangular, has an internal radius of 200mm, a centroid radius of 250, is 100mm thick and 100mm wide.


    2. Relevant equations

    I have looked in Roark’s formula for stress and strain, and found a half semi circle example on p276 (7th edition). I hope I can assume that it is a thin beam as I have been unable to find anything for thick beams other than the example a few pages after.

    By using symmetry you can half the semi circle and work out 3 unknowns at the freely supported end. As there is symmetry, am I right to assume that there is no horizontal force meaning the formulas are simpler. I also assumed that due to symmetry that there will be no moment (due to 2 opposing moments) giving me formulas of

    Max Deflection in x direction VRc^3/2EI
    Max Deflection in y direction Pi(VRc^3)/4EI

    However looking at
    http://school.mech.uwa.edu.au/~dwright/DANotes/MST/thin/thin.html#thin
    I believe that this is incorrect and that I need to include a moment in my equation which now becomes

    Max Deflection in x direction (2/pi-1/2)VRc^3/EI.
    Max Deflection in y direction (Pi/4-2/Pi)VRc^3/EI

    My deflections are now much smaller than the previous ones.

    I am now confused as to which formula to use after finding example 3 at
    http://www.codecogs.com/reference/engineering/materials/curved_beams.php
    which gave me the same answers as my first attempt at roarks deflections.


    3. The attempt at a solution

    Roarks 1st attempt

    X = 0.226mm
    Y = 0.356mm


    Codeworks

    Horizontal Deflect = Integral (My/EI) ds
    VRc^3/EI*Integral 0-pi/2 Sin Theta * (1-cos theta)

    VRc^3/EI *.-cos.theta - 0.25(Cos.2.theta) in range 0 -pi/2

    Max horiz =0.226mm

    Vertical deflection
    Intergral 0-pi/2 (V*RcsinTheta * Rc Sin Theta/EI)RC Delta theta

    VRc^3/EI*Integral 0-pi/2 1/2(1-Cos.2.theta). Delta theta
    VRc^3/EI*[1/2(theta -sin.theta.cos.theta] in range 0 -pi/2

    Max Vert 0.356mm
    I used these calcs to get the deflection at any angle by putting in the angle of theta i wanted

    From School Mech
    Max Deflection in x direction (2/pi-1/2)VR^3/EI. 0.0619mm
    Max Deflection in y direction (Pi/4-2/Pi)VR^3/EI=0.0674mm


    I am really confused and stuck as I don’t know which the right one is.
    Any help will be greatly appreciated

    Phil
     
    Last edited: Sep 28, 2011
  2. jcsd
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