Shear Center of a J section

[FJCV]

Dear fellow engineers,
I am requesting your help figuring out the best way to calculate the shear center of an unsymmetrical cross section as shown in the attachment. I am familiar with the theoretical solutions of Timoshenko for cross sections with at least one axis of symmetry.
I appreciate your help finding solutions for cross sections with multiple flanges without a symmetry axis.

tiff image:

http://files.engineering.com/getfil...94-491e-af7e-adb400b69ec8&file=J-Section.tiff

jpg image:

http://files.engineering.com/getfil...51f-479a-b289-00d38c7d6e60&file=J-Section.jpg

 

[Teguci]

Impressive but wrong in this case. I rotated my principal axes in the wrong direction. With the corrected rotation, I'm getting x0 = -0.329 and y0 = -0.191 which closely matches FJCV's FEM results. I still get Cw = 0.0508 and J = 0.00195

My next step is to grab some Z purlin properties and see if I can match them. Next time I'll get it right.

 

[FJCV]

Hello All,
Thank you all again for your great posts.

Hello BAretired,
Thank you for posting the scan of your hand analysis.
Your notes are easy to follow and the outcome matches the numbers I posted earlier.
I think the small difference is caused by the detail of the corners in the idealization which is hard to prevent. Are you concerned that the sum of horizontal x forces is not zero?

Hello Teguci,
Thank you for posting the example hand calculation for shear center and the bending-torsion constant Cw. Like you, I was originally a bit concerned about the Ixy not being equal to zero.
Can you scan the details of your revised calculation?

I think the best way to verify the end answer is to build a FEM model [in lieu of an small element test]. Apply Fx and Fy forces to the 'candidate' shear center and look for any twisting in the deformed shape. Let us know your results if you decide to investigate your answers further. I welcome any independent confirmation/correction.

 

[BAretired]

Hello FJCV
I would have to think about the problem a bit more. Teguci found that Ixy had an effect on the outcome and I did not consider Ixy, so I suspect my answer may be in error and needs to be reviewed.

Insofar as the sum of the horizontal forces not being zero, I thought about the example of a Z section where, unlike a channel section, the sum of the horizontal forces is not zero.

BA

 

[ukbridge]

This is what I get if I plug it in using your values of 1" and 0.1" for a and the plate thickness respectively.

NB(Your dimensions a and 2a arent consistent i.e. some dims given wrt centreline geometry and some not) so theres probably an small error in there somewhere. I think all our answers correlate well enough for design purposes though i.e. what torque is acting on your section due to the offset shear centre

 

[Teguci]

Here are the bones for my corrected calculation.
1 - Node input
2 - Element connectivity
3 - translate nodes to centroid and calculate Ixx, yy, xy and alpha (rotation to principal)
4 - Calculate ro - dL for each element (note - direction of element is considered)
5 - Calculate preliminary warping starting with 0 at node 1. Element C inherits node 2 result.
6 - Correct and normalize the warping values (the value at node 1 is not 0)
7 - Rotate nodes to principal axes
8 - Calculate Iwx and Iwy for each element
9 - Solve for shear center with regard to rotated centroid
10 - Rotate and translate shear center back to user coordinate system

To get the warping constant continue with (not shown)-

11 - translate nodes W.R.T. shear center
12 - Calculate ro0 dL for each element
13 - Calculate preliminary warping starting with 0 at node 1. Element C inherits node 2 result.
14 - Correct and normalize the warping values (the value at node 1 is not 0)
15 - Cw = 1/3 SUM[(Wni^2 + Wni Wnj + Wnj^2) Ai]

Hope it helps.