Shear Center of a J section
Shear Center of a J section
(OP)
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/getfile.aspx?folder=c...
jpg image:
http://files.engineering.com/getfile.aspx?folder=8...
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/getfile.aspx?folder=c...
jpg image:
http://files.engineering.com/getfile.aspx?folder=8...






RE: Shear Center of a J section
As a guess for an open section I might work out the shear centre by taking the first moment of area of all the bt3/3s, or very least as a validation of some famcy method, dont just use this approach though!
RE: Shear Center of a J section
I added the picture attachment. Can you give an the example calculation using the first moment of area of all the bt3/3s?
I have been able to find a solution with FEM but like to understand the theory better. Thank you for your help.
RE: Shear Center of a J section
Why not try the analysis i suggested for a simpler shape and compare it to those from standard tables? Have a look through your softwares manual on section property calculations to see if theres anything useful in there.
Also, not going to do your work for you sorry! 😉
RE: Shear Center of a J section
If you want to go really deep, this is the most in depth torsion reference that I know of: Link. I had to wait a good while for one to show up on eBay.
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Shear Center of a J section
Then calculate the "centroid" of the section in a similar manner to working out the elastic neutral axis, but using J instead of A i.e. Shear centre = sum of J* distance to centroid/ sum of J.
Disclaimer: For lititgious reasons, I don't officially endorse using this exact method for actual design purposes. No doubt theres a winning ticket to willy wonkas fudge factory lurking within its "derivation" but rather the method i would first compare for known shapes with tabulated values!
I cant see the section, but maybe the solution is in your load path. Are there any clever ways you can avoid torsion in your system? If not, why not use a RHS section or similar which has a much better torsional resistance. Alternatively if i needed a quick solution you could try taking the torsion as a warping couple in the top and bottom flanges.
RE: Shear Center of a J section
1) SC vertical location would be closer to the upper flange about in proportion to the weighted average of the upper and lower flange lengths.
2) SC horizontal location would be left of the vertical web as it's on the web for a zee and left of the web for a cee.
I've not heard of UK's method but it sounds as though it would generate a consistent result.
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Shear Center of a J section
You should have four forces, two for the top flange in opposite directions, one for the web (shear in the web is V) and one for the bottom flange. The shear center is the point where ∑F.d = 0 where F is the shear in each section and d is distance to shear center.
BA
RE: Shear Center of a J section
BAretired,
I tried the method you described and even rotated the cross section to its principal axis first but was not able to match the solution I got from my FEM (x= -0.32743, y= -0.19332 if a=1.000" and t=0.100"). I used Timoshenko's integration method where one expresses the static area moment and shear stress as a function of dimension s (distance along the flange centerlines). With Timoshenko's method I was able to match the known solutions for a an I and C section but mentioned J section remains a project in work....
Note that I applied shear loads in the FEM and observed no rotation, confirming mentioned shear center location. Not sure what I am missing.
Ukbridge,
I will try to add a jpg file to the original post and will try your method shortly. Thx.
RE: Shear Center of a J section
I did not calculate a y value.
BA
RE: Shear Center of a J section
I sppreciate it if you could post a scan of your analysis. Thx.
RE: Shear Center of a J section
RE: Shear Center of a J section
IES Software out of Bozeman, MT has a program called Shapebuilder. You can download a working copy for 30 days and run your shape in it and get all the values as well as the Shear Center in either Metric or Imperial dimensions. This would allow you to check you numbers.
www.iesweb.com
Jim
RE: Shear Center of a J section
I follow this stepwise procedure with appropriate corrections (in particular, pg 11 equations are wrong)
http://www.cranerepairengineer.com/Torsional%20War...
RE: Shear Center of a J section
I now find I of the section to be 0.34769 in4
The c.g. is 0.6724 below the X axis.
Draw a line diagram representing the center of the three elements.
Label top flange A, B and C
Label bot. flange D, E
QB = 3(0.1)0.6224 = 0.18672in3
QD = 0.95(0.1)1.2776 = 0.12137in3
Assume V = 100#
qB = VQ/I = 100*0.18672/0.34769 = 53.7 psi
FBA = 53.7/3 * 1/2 = -8.95#
FBC = 2*53.7/3 *2/2 = 35.8
Ftop flg = 35.8 - 8.95 = 26.85#
qD = 100 * 1.2137/034769 = 34.9 psi
Fbot. flg = 34.9 * 0.95/2 = -16.58#
Fweb = 100#
Solving for x,
x = [26.85(0.6224) + 16.58(1.2776)]/100 = 0.379" left of Y axis, or -0.379"
A value for y could be obtained in a similar way with V acting from left to right.
BA
RE: Shear Center of a J section
BA
RE: Shear Center of a J section
RE: Shear Center of a J section
The long form - [ Vx ( 1/(Ixx Iyy - Ixy^2) x INT (Ixx X - Ixy Y ) dA ] becomes Vx INT (X dA) / Iyy only when Ixy = 0 (bending about the principal axis) INT X dA = Qyy, INT Y dA = Qxx
RE: Shear Center of a J section
I only put the pretty stuff up. The actual calculations are a disaster and a half long Excel Spreadsheet following the procedure in my first link, which I am still fixing (important but not critical). Someday when I'm happy with it(ie never), I will get it into C++
RE: Shear Center of a J section
RE: Shear Center of a J section
My next step is to grab some Z purlin properties and see if I can match them. Next time I'll get it right.
RE: Shear Center of a J section
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.
RE: Shear Center of a J section
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
RE: Shear Center of a J section
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
RE: Shear Center of a J section
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.