Shear stress at the boundary of beam

[fonseh]

Homework Statement

We all know that $$\tau = VQ/ It $$
how to determine the shear stress at **G** ?
I'm having problem of finding Ay
centroid if the solid that i found earlier is y = 98.5mm

There's a dashed line from G to the bottom . It's making me confused . for Q , should i consider the area either to the left or right of the G ? like the red and orange part

Homework Equations

The Attempt at a Solution

so , my working is
Q = Ay = (250x10^-3)(50x10^-3)( 50+125-98)(10^-3) = 9.63x10^-4

Is my concept correct ?
I'm considering the area above the G .
[/B]

 

[PhanthomJay]

Homework Statement

We all know that $$\tau = VQ/ It $$
how to determine the shear stress at **G** ?
I'm having problem of finding Ay
centroid if the solid that i found earlier is y = 98.5mm

There's a dashed line from G to the bottom . It's making me confused . for Q , should i consider the area either to the left or right of the G ? like the red and orange part

Homework Equations

The Attempt at a Solution

so , my working is
Q = Ay = (250x10^-3)(50x10^-3)( 50+125-98)(10^-3) = 9.63x10^-4

Is my concept correct ?
I'm considering the area above the G . [/B]

Yes, excellent work, that is the best way to do it. If the flanges were thinner, you could use the orange area and calculate the horizontal shear stress in the lower flange, and get the same result, but that doesn't work well for thick flanges, so your method and answer is correct, except you used 98 instead of 98.5 on your calc, no big deal.

 

[fonseh]

why

Yes, excellent work, that is the best way to do it. If the flanges were thinner, you could use the orange area and calculate the horizontal shear stress in the lower flange, and get the same result, but that doesn't work well for thick flanges, so your method and answer is correct, except you used 98 instead of 98.5 on your calc, no big deal.

there's a dashed line over there ? it makes me confused ...

 

[PhanthomJay]

why

there's a dashed line over there ? it makes me confused ...

Yes it is confusing. If you use the area to the right of the dashed line and calculate Q using that area times the distance from its centroid to the horizontal neutral axis , you calculate the horizontal shear stress , but this calculation does not include the vertical shear stress in that flange,which although small, adds to the shear stress value.

 

[fonseh]

Yes it is confusing. If you use the area to the right of the dashed line and calculate Q using that area times the distance from its centroid to the horizontal neutral axis , you calculate the horizontal shear stress , but this calculation does not include the vertical shear stress in that flange,which although small, adds to the shear stress value.

It's not stated in the question , right ? How to know what to find ? vertical shear stress or horizontal shear stress ?

 

[PhanthomJay]

It's not stated in the question , right ? How to know what to find ? vertical shear stress or horizontal shear stress ?

The shear stress in the green area is vertical only, with its complimentary longitudinal shear stress. The shear stress in the orange area is both vertical/longitudinal and horizontal/longitudinal, which, when summed properly, should in theory equal the longitudinal shear stress calculated for the green area at G.

 

[fonseh]

he shear stress in the orange area is both vertical/longitudinal and horizontal/longitudinal, which, when summed properly, should in theory equal the longitudinal shear stress calculated for the green area at G.

Do you mean the sum of both vertical shear stress and horizontal shear stress = vertical shear stress of green part ?

 

[PhanthomJay]

Do you mean the sum of both vertical shear stress and horizontal shear stress = vertical shear stress of green part ?

yes, which is the same as saying that the sum of both the longitudinal shear stress (due to vert shear stress in flange)and long shear stress (due to horiz shear stress in flange) is equal to the long shear stress in green part (due to vert shear stress in green part).

 

[fonseh]

yes, which is the same as saying that the sum of both the longitudinal shear stress (due to vert shear stress in flange)and long shear stress (due to horiz shear stress in flange) is equal to the long shear stress in green part (due to vert shear stress in green part).

How do you know that ? It's a concept or we need to do the caluclation to know it ? Is it possible to know it without calculation ?

 

[PhanthomJay]

Regardless of calculation of shear stress at G, it should be the same no matter which method is used. But it is a lot easier to use the green area for the calc, as it avoids the complications that you get when using the orange area. If the flange was thin, say 5 mm instead of 50 mm, then either method could be used without complication, because there would essentially be no vert shear stress in flange. But the flange is not thin here, so your original method using the green area is best.

 

[fonseh]

Regardless of calculation of shear stress at G, it should be the same no matter which method is used. But it is a lot easier to use the green area for the calc, as it avoids the complications that you get when using the green area. If the flange was thin, say 5 mm instead of 50 mm, then either method could be used without complication, because there would essentially be no vert shear stress in flange. But the flange is not thin here, so your original method using the green area is best.

Can you explain why when the flange is thin , the vertical shear stress in the flange can be ignored ?

 

[PhanthomJay]

well, shear stress is (VQ/It), and Q is A(y_bar) ,and since with a thin flange Q is rather small since A is so small, and since t is rather large since you use the flange width, b, for the t value, then the shear stress is very small since A is small and t is large. look at a wide flange I beam for example, you can see how small is the vert shear stress in the flange, it then sharply increases in the web at the web interface since now t is web thickness instead of b flange width.