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Deflection equation using Macaulay's

  1. May 15, 2015 #1
    1. The problem statement, all variables and given/known data
    BEAM.PNG

    I need to calculate the deflection equation.
    R1=33kN
    R2=32kN
    E=210GPa
    calculated I=5.4*10^-7 m^4

    2. Relevant equations


    3. The attempt at a solution
    equation.PNG

    Is the equation correct? I'm not sure if I got the u.d.l. right.
     
  2. jcsd
  3. May 15, 2015 #2

    SteamKing

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    Indeed, there is a problem with the UDL. The UDL starts at x = 0, whereas your BM equation has it starting at x = 2 m.

    Also, for completeness, I would add a term for the reaction R2 located at x = 5 m, even though this would not affect any values calculated by the BM expression.
     
  4. May 19, 2015 #3
    Sorry for the late reply. My current solution is attached.
     

    Attached Files:

  5. May 19, 2015 #4

    SteamKing

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    I noticed several things which need fixing.

    1. In your expression for the BM, you have EI y" = 103[Mess].
    When you integrate to find the slope, you wind up with EI y' = 103[Mess + A] and EI y = 103[Mess + Ax + B].
    There's really no need to bring the constants of integration inside the brackets where the other terms are multiplied by 1000 due to the loading of the beam.

    2. When you integrated the terms for the UDL, the expression -5x2 became -1.67x3 and then -0.56x4. I think -1.67 / 4 ≠ -0.56.
    This same error pops up in the term which cuts off the UDL.

    You should look at your deflections again, particularly checking the calculation of A and B. The magnitudes of your calculated deflections (one the order of 0.5 m) seem rather large given the length of this beam. Make sure you haven't misplaced a decimal somewhere.
     
  6. May 20, 2015 #5
    Thanks, in that case one more time! I've corrected the 0.56 to 0.4175 and the value of A is now A=-75.878. Those changes however made the deflection even greater. I've checked everything a couple of times now and I can't see where I when wrong. I've attached the full calcs again. I'm a bit stuck now.
     

    Attached Files:

  7. May 20, 2015 #6

    SteamKing

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    I double checked your deflection calculations, and they are correct, but for a few round off errors. What you have as 0.4175 should be 0.4167, and I get A = -76.00.

    Now that you have included the dimensions of the cross section of the beam, I can see why the deflections are so large. The max. bending stresses for the beam are way beyond the elastic limit for steel (whose E = 210 GPa) and are the same order of magnitude as E. This suggests that a real beam made of steel loaded and supported as shown in the OP would have failed completely.

    In order for the equation M/EI = y" to be valid, the slope of the beam must be small, such that θ << 1, which is not the case here. While this beam is a good exercise for showing how to calculate deflection using McCauley's Method, it is a terrible choice because its deflections are much too large to satisfy the small slope approximation on which elastic beam theory is based.
     
  8. May 21, 2015 #7
    Thanks for your help on that, it took a wee while. I shall round up all calcs to 4 d.p. for a better accuracy.
     
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