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Formula for shear stress

  1. Jul 11, 2016 #1
    1. The problem statement, all variables and given/known data
    Refer to the second photo , Q= Ay
    Does the author mean point P is at the center of the region (62.5mm x 100mm) ???The figure is not labelled well i am confused
    Lk22d7F.png
    q2sgFeq.png
    2. Relevant equations


    3. The attempt at a solution
    In the notes , I was told that y is the location of the shear stress is measured from the neutral axis
     
  2. jcsd
  3. Jul 11, 2016 #2
    why the value of y is 62.5 / 2 ?
    I have no idea...
    MNGDDZw.png
     
  4. Jul 11, 2016 #3

    SteamKing

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    If the area of interest is a rectangle measuring 100 mm wide by 62.5 mm deep, where is the vertical centroid located, relative to the neutral axis?
     
  5. Jul 12, 2016 #4
    do you mean y is the distance from the centorid of specific area to neutral axis ?
     
  6. Jul 12, 2016 #5
    why it's 62.5 / 2 in B ? and it's 12.5 +(50/2) in part A ?
     
  7. Jul 12, 2016 #6
    I'm confused where is the location of point P now ....
     
  8. Jul 12, 2016 #7

    SteamKing

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    Because one calculation is for the shear stress for part a) of the problem and the other is for the shear stress in part b).

    Part a) wants to know the shear stress in the beam for a point which is located 50 mm below the top of the beam. Remember, Q must be calculated using the neutral axis as the reference point, which is why the y-bar is 12.5 mm + 50 mm / 2.

    Part b) wants to know the shear stress in the beam, which the box in the illustration indicates occurs at the neutral axis. The y-bar here is 62.5 mm / 2.
     
  9. Jul 12, 2016 #8
    then, for part a , why the value of y isn't 12.5mm? Since it's 12.5mm away from neutral axis.
    neutral axis is located 62.5mm way from the top of beam, why value of y is not 62.5 mm?
     
  10. Jul 12, 2016 #9

    SteamKing

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    Because you are supposed to calculate the first moment Q = A * y-bar for the area of the beam which lies above the line which is 12.5 mm above the neutral axis.

    And it's not y you are dealing with, but y-bar, which is the location of the centroid of the area A.

    Haven't you understood anything about the calculation of the shear stress using the formula ##τ = \frac{V ⋅ Q}{I ⋅ t} ## ?

    Again, it's not y you are dealing with, but y-bar, which is the location of the centroid of the area A above the neutral axis.
     
  11. Jul 12, 2016 #10
    [/QUOTE]

    the maximum shear stress occur at the neutral axis of the beam , am i right? why we need to consider the centroid of area which is above the neutral axis?
     
  12. Jul 12, 2016 #11

    SteamKing

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    the maximum shear stress occur at the neutral axis of the beam , am i right? why we need to consider the centroid of area which is above the neutral axis?[/QUOTE]
    This is getting really circular. I don't know why you can't understand the shear stress formula, when it is laid out and explained.

    The formula for calculating the shear stress is a beam is ##τ = \frac{V ⋅ Q}{I ⋅ t}##

    τ - shear stress
    V - shear force
    Q - first moment of area above the location where the shear stress is calculated.
    I - second moment of area for the entire beam about the N.A.
    t - width of the beam where the shear stress is calculated.

    Now V, I, and t are all pretty easy to find for a given beam.

    The first moment Q can be calculated using Q = A * y-bar, since that is the definition of the first moment of area.

    If the beam cross section is a nice square or rectangle, then y-bar can be determined very easily.
     
  13. Jul 12, 2016 #12
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    Last edited: Jul 12, 2016
  14. Jul 12, 2016 #13
    This is getting really circular. I don't know why you can't understand the shear stress formula, when it is laid out and explained.

    The formula for calculating the shear stress is a beam is ##τ = \frac{V ⋅ Q}{I ⋅ t}##

    τ - shear stress
    V - shear force
    Q - first moment of area above the location where the shear stress is calculated.
    I - second moment of area for the entire beam about the N.A.
    t - width of the beam where the shear stress is calculated.

    Now V, I, and t are all pretty easy to find for a given beam.

    The first moment Q can be calculated using Q = A * y-bar, since that is the definition of the first moment of area.

    If the beam cross section is a nice square or rectangle, then y-bar can be determined very easily.[/QUOTE]
    do you mean the shear stress is to be measured at neutral axis , so the first moment of area is taken to be the area above the neutral axis (100mm)(62.5mm) ?
     
  15. Jul 12, 2016 #14

    SteamKing

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    do you mean the shear stress is to be measured at neutral axis , so the first moment of area is taken to be the area above the neutral axis (100mm)(62.5mm) ?[/QUOTE]
    Yes, for part b) of the problem. The shear stress for part a) is measured at a different location, and its area is slightly different.
     
  16. Jul 12, 2016 #15
    Yes, for part b) of the problem. The shear stress for part a) is measured at a different location, and its area is slightly different.[/QUOTE]
    for part b , could it be area below the neutral axis?
     
  17. Jul 12, 2016 #16

    SteamKing

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    for part b , could it be area below the neutral axis?[/QUOTE]
    The neutral axis of the section splits the section into two parts, such that the first moment of area of the part above the N.A. is the same as the first moment of area of the part below the N.A. This is why the shear stress is a maximum at the N.A. of the cross section.
     
  18. Jul 12, 2016 #17
    The neutral axis of the section splits the section into two parts, such that the first moment of area of the part above the N.A. is the same as the first moment of area of the part below the N.A. This is why the shear stress is a maximum at the N.A. of the cross section.[/QUOTE]
    so,taking moment about an area below or above the neutral axis is acceptable?

    How did the same area below and above neutral axis related to the maximum shear stress?
     
  19. Jul 12, 2016 #18

    SteamKing

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    It doesn't matter if the area is above or below the neutral axis. Q for either area is the same.
    Because Q is the same for the area above or below the neutral axis, that's as large as Q can get for that cross section, hence, the shear stress is a maximum at the N.A.
     
  20. Jul 12, 2016 #19
    What do you mean Q is the same for the area above or below the neutral axis, that's as large as Q can get for that cross sectio
     
  21. Jul 12, 2016 #20

    SteamKing

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    If you don't believe me, work out Q above and below the N.A. for the wooden beam in the problem described in the OP.

    The beam is 125 mm deep and 100 mm wide.

    The N.A. is located 62.5 mm from the top or bottom of the beam.

    A = 62.5 * 100 = 6250 mm2

    y-bar = 62.5 / 2 = 31.25 mm

    Q = A * y-bar = 6250 * 31.25 = 195,312.50 mm3

    The only way Q gets bigger than 195,312.5 mm3 is if the N.A. moves up or down, but this is impossible, since it won't be the N.A. anymore.
     
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