Allowable shear stress in bending

[chetzread]

Homework Statement

I think the notes is wrong . in the notes , it's given that σ allowable > Mmax / Ze

Homework Equations

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 ?

 

[SteamKing]

Homework Statement

I think the notes is wrong . in the notes , it's given that σ allowable > Mmax / Ze

Homework Equations

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 ?

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.

 

[chetzread]

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.

since , $\frac{Mmax}{Ze}$ = caluculated stress , then the $σ\, allowable$ shouldn't exceed the calculated stress , right ? if $σ\, allowable > \frac{Mmax}{Ze}$ , the beam will rupture , right ?

 

[SteamKing]

since , $\frac{Mmax}{Ze}$ = caluculated stress , then the $σ\, allowable$ shouldn't exceed the calculated stress , right ? if $σ\, allowable > \frac{Mmax}{Ze}$ , the beam will rupture , right ?

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.