# Allowable shear stress in bending

1. Aug 25, 2016

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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2. Aug 26, 2016

### SteamKing

Staff Emeritus
What does σ allowable > Mmax < Ze even mean?

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.

3. Aug 26, 2016

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 ?

4. Aug 26, 2016

### SteamKing

Staff Emeritus
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.