What is the First Moment of Area for Calculating Shearing Stress at a Point?

[member 392791]

Hello,

I am having confusion when calculating the first moment of area, Q, to solve for the shearing stress at point a in 13.9, I use the area which is above the point, but for 13.28 its to the right of the point. It seems they are both subjected to a vertical shear, so I don't think that has anything to do with it.

 

[SteamKing]

It comes down to the shape of the section and the point where you want to calculate the shear stress.

In general, thin walled structures and sections can be analyzed for shear stress using a technique called shear flow. The shear flow is calculated using the following formula:

q = VQ / I, where

q = shear flow
V = shear force
Q = first moment of area
I = second moment of area for the whole section

The shear stress at any point is equal to the shear flow divided by the thickness of the material, so

τ = q / t = VQ / It

In calculating the shear flow of an I beam, one starts with a shear flow q = 0 at the ends of the flanges. As you travel from the free edge of the flange towards the junction between the flange and the web, the shear flow q increases. This is because the first moment of area Q is calculated for that portion of the area between the point of interest in the flange and the end of the flange.

Once you reach the junction of the flange with the web, Q equals the first moment of area of the entire flange area. Going down the web, Q increases until it reaches its maximum value at the centroid of the section, which is where the maximum shear stress occurs.

This is why the two methods of establishing Q for the two problems appear to be so different.

http://kisi.deu.edu.tr/mehmet.aktas/Dersnotlari/6.pdf

You could calculate Q for Prob. 13.28 using the area above the middle of the thickness of the flange, but the resulting shear stress represents only an average value for the entire width of the flange. By using the shear flow method, you calculate the average shear stress across the thickness of the flange, which is probably a more pertinent value.

 

[member 392791]

Ok, so suppose now that point a is on the left side of the flange. Does that mean the area I calculate is to the left of point a or to the right of point a?

 

[SteamKing]

Ok, so suppose now that point a is on the left side of the flange. Does that mean the area I calculate is to the left of point a or to the right of point a?

You always start from the nearest point where the shear flow q = 0, in this case the edge of the left flange.

 

[member 392791]

ok great, thanks a lot