Solid mechanics (shear stress) question

AI Thread Summary
In a cantilever beam with a rectangular cross-section, the shear stress formula SAy/bI indicates that at the neutral axis, y is zero, leading to confusion about shear stress presence. It is clarified that while normal stress is zero at the neutral axis, shear stress is actually at its maximum due to the opposing forces on either side. The discussion emphasizes the importance of understanding the variables in the shear stress equation, particularly the distinction between the area A and the distance y. The correct interpretation is that y should be measured from the neutral axis to the centroid of the area being analyzed. Ultimately, shear stress is significant at the neutral axis, requiring careful consideration in structural analysis.
milan666
Messages
13
Reaction score
0
Hi, if the shear stress on an element in a solid is SAy/bI, where S is the shear force and y is the distance from the neutral axis, what value of y do i use if the point I'm calculating is on the neutral axis?
 
Physics news on Phys.org
On first glance, you would use zero. But it would help to know the shape of the solid, the type of loading, how you got SAy/bI, and what the other variables mean for me to be confident with this recommendation.
 
Its a cantilever beam, with a rectangular cross-section, and i have to find the shear and normal stresses on a point which is at the neutral axis. If y is zero, then the shear force would be zero. Does that mean that there are no shear stresses at a point on the neutral axis? I can't post the actual problem itself cause I am at uni and that would be plagarism.
 
Last edited:
OK, so what is the loading, why are you using SAy/bI, and what do the other variables mean? If the results of an equation don't match your intuition, it could be sign that you're applying the wrong equation.
 
Thats what we were taught to use. S is the shear force at that point on the beam, A is the area of the cross section, y is the distance form the neutral axis, b is the width of the cross section, I is the second moment of area.
 
milan666 said:
Thats what we were taught to use. S is the shear force at that point on the beam, A is the area of the cross section, y is the distance form the neutral axis, b is the width of the cross section, I is the second moment of area.

Are you sure A is the area of the entire cross section and y the distance from the neutral axis, or is A the area of some region above or below the neutral axis, and y the distance from the centroid of that region to the neutral axis?
 
At the neutral axis, normal stress will be zero (strain is zero at the neutral axis, this is part of how 'neutral axis' is defined).

Also, at the neutral axis, since you have a compressive force on one side and a tensile force on the other, SHEAR stress is a maximum.

"An Introduction to the Mechanics of Solids (second edition with SI units)" by Crandall, dahl, and Lardner lists as Equation 7.27 the following for shear stress in the case of beam described as you do

<br /> \tau_{xy}=\frac{V}{2 I_{zz}} \left[\left(\frac{h}{2}\right)^2-y_1^2\right]<br />

where h is the height, y1 is the distance from the neutral axis, V is the shear force, and Izz is the moment of inertia--or second moment of area, whichever terminology you're used to (at least I think these are right from when I took mech of matl's last semester).

The important take-away is that shear stress will have a maximum at the neutral axis (y1=0), i.e. you'll need more glue to hold it together there than anywhere else, all other things being simple in an isotropic material.
 
Last edited:
Oh ok i got it now, y is the distance from the neutral axis to the centroid of the area where you make the cut. Thanks!
 

Similar threads

Replies
2
Views
4K
Replies
6
Views
676
Replies
1
Views
1K
Replies
3
Views
3K
Replies
1
Views
2K
Replies
6
Views
5K
Back
Top