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Allowable shear stress in bending

  1. Aug 25, 2016 #1
    1. The problem statement, all variables and given/known data
    I think the notes is wrong . in the notes , it's given that σ allowable > Mmax / Ze

    2. Relevant equations


    3. The attempt at a solution
    i think it should be σ allowable > Mmax < Ze , am i right ? How could the σ allowable > Mmax / Ze ? If σ allowable > Mmax / Ze , the beam will rupture , right ?
     

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  3. Aug 26, 2016 #2

    SteamKing

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    What does σ allowable > Mmax < Ze even mean?

    Think about this for a minute.

    Ze has units of L3, Mmax has units of force × L, and σ allowable of course has units of ##\frac{force}{L^2}##, so these three quantities cannot be related by a simple inequality as you are thinking.

    The bending stress σ in a beam is given by the formula

    ##σ = \frac{M ⋅ y}{I}##

    The elastic section modulus of the beam Ze is the ratio ##\frac{I}{y}##, which means that ##σ = \frac{M}{Ze}##.
    Therefore, ##σ\, allowable > \frac{Mmax}{Ze}##, since you want the allowable bending stress to be greater than the actual calculated bending stress in the beam.
     
  4. Aug 26, 2016 #3
    since , ##\frac{Mmax}{Ze}## = caluculated stress , then the ##σ\, allowable## shouldnt exceed the calculated stress , right ? if ##σ\, allowable > \frac{Mmax}{Ze}## , the beam will rupture , right ?
     
  5. Aug 26, 2016 #4

    SteamKing

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    No, you have things confused.

    σ allowable is the maximum stress the material of the beam can experience before something bad happens. Typically for steel, σ allowable represents the yield point of the material. The actual stress in the beam is not σ allowable, but is calculated from M / Ze, where M is the bending moment and Ze is the elastic section modulus of the beam.

    In order for a beam to remain intact, σ bending < σ allowable.
     
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