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Homework Help: Beams and bending

  1. Dec 27, 2009 #1
    find the ratio of d/b such that the largest stress in the beam will be minimum
    Capture.JPG

    i know that the maximum stress is

    σmax=Ymax[tex]\frac{M}{I}[/tex]

    and i know that for a rectangle I=bh3/12

    now in the question i am asked to find the ration of the (diameter of the log)/(the width of the rectangle) such that σmax is minimal

    since the rectangle is contained in the circle

    d2=b2+h2
    where h is the height of the rectangle

    σmax=Ymax[tex]\frac{M}{I}[/tex]

    σmax=(h/2)*(12M/(bh2)
    σmax=6M/(b*h2)
    σmax=6M/(b*(d2-b2)

    basically from here i need to find the ratio d/b so that (b*(d2-b2) is maximum,

    but how can i do this??

    d/b=K

    (b*(d2-b2)
    =(bd2-b3)

    =d/b*(d*b2-b4/d)

    but i cant get to the ratio, i feel i am so close but just not getting it
     
    Last edited: Dec 27, 2009
  2. jcsd
  3. Dec 27, 2009 #2

    nvn

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    That is not shear stress. Your comment does not make sense. [EDIT: OK, you edited your post in response to my comment, without letting anyone know, thereby making my above comment appear as nonsense.]

    You seem to not understand what d is, as if you have not looked at the diagram.
     
    Last edited: Dec 27, 2009
  4. Dec 27, 2009 #3
    at first i thought that d was the height of the rectangle, but thatdidnt make sense to me, if that is so what is the circular log given for?? it is not 100% clear to me in the diagram what they mean by d.

    as far as i know that is the equation for stress,

    how would you have gone about solving it??
     
  5. Dec 27, 2009 #4

    nvn

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    Use a straightedge to see what d is.
     
  6. Dec 27, 2009 #5
    so you say that d is the height of the beam, if so why do they need to tell me that the beam is taken from a log,

    if i take the height as d

    I=bd3/12

    σmax=(h/2)*(12M/(bd3)
    σmax=6M/(b*d2)

    now i need b*d2 to be maximum

    but i still cant get to the ratio??

    i supose that b*d2 will be maximum when d/b-->infinity
    but that doesnt seem right at all
     
  7. Dec 27, 2009 #6
    never mind, i think i got it

    b2+d2=D2

    i differentiate the stress equation adn compare to 0 and i get sqrt(2)
     
  8. Dec 27, 2009 #7
    They don't tell you the size of the log, but the ratio r= d/b is constrained by the circular boundary. In my experience, if a problem says to find a ratio, then the first line should be: Let r be the ratio. Then work everything towards equations in r.
     
  9. Dec 27, 2009 #8
    thanks but i got it already sqrt(2)
     
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