How to calculate J and Cw for non-standard shapes? 2

[orreng1]

I have an inverted "T" shape and I need to calculate the J and Cw values. I donot have quick access to the Roark Book, as I understand it is in there. Does anyone know how to do this?

 

[JAE]

For a "T" shape where

b = width of the flange
tf = thickness of the flange
h = height from the tip of the stem to the mid-height of the flange (d - tf/2)
tw = thickness of the stem

J = 1/3 x ((b x tf^3) + (h x tw^3))


Cw = 1/36 x ((b^3 x tf^3)/4 + (h^3 x tw^3))

So a verification -
Let b = 4
tf = .5
h = 6.25
tw = .3

Then I get J = 0.2229

Cw = 0.2387

Cw = 0 for small thicknesses

 

[orreng1]

thanks for the quick reply.

Is a "T" value valid for an inverted T?

Also, I thought the values depended on a "mesh", or matrix...?

Guess I really don't understand how these values are derived.

 

[JedClampett]

I would ignore the effect of Cw. It (torsional warping) will contribute a trivial amount to the capacity. All the torsion will be taken by the St. Venant's stresses. If you have to enter a value for a computer program put in .0001 in^6.

 

[orreng1]

My beams vary but on average are 30" high, web is 3" and flange is 6" x 2". Thinking I should not iognore it for this. What about the inverting of the "T"?

 

[JedClampett]

I would still ignore this effect. The mechanics of torsional warping is very complicated, but an easy way to simulate for an I-beam or channel it is to place equal and opposite horizontal forces of T/2h (where h is the distance between your top and bottom flange). Then analyze the flanges for weak axis bending for these forces.
Now if you try the same thing with a T-section, there is no opposite flange to create the couple that can carry the torsion. Even with your beefy section, the warping will be a small effect.

 

[JAE]

J and Cw are independent of orientation (T vs. Inv. T)

 

[orreng1]

Thanks people, I really appreciate the input. It helped to clear the muddy water on this subject.

 

[UcfSE]

Check the slenderness ratio of the elements composing the T. To use the 1/3 factor in the calc for J the width/thickness ratio for the element needs to be greater than 10 (theoretically infinity). I don't have my book with me right now to verify the actual numbers.

 

[DRW75]

Does anyone know what the generalized form for Cw is for any arbitrary shape? Or could anyone perhaps refer to a good text which describes this?

I find all the given formulae for various shapes fine and dandy, however I would like somthing more along the lines of a derivation.