torsional constant "J" 4

Actually, it would be pretty close to the sum of the parts. Unless, the section is changed from an "open" section to a "closed" section. That's when it really starts to change.

well, you do have four small closed sections ...

it would not be 1/3*sum(b*t^3) ... classic open section

I think it'd be slightly higher than the sum of the four enclosed areas.

another day in paradise, or is paradise one day closer ?

Is your "enhancement" strictly to increase J ??

It looks like each angle forms its own little rectangular tube. Every leg angle would have to be stitch/skip/intermittent welded "enough" .
The effect of the 4 mini-tubes on J would be "just a bit more." (Vince Noir)

I'm guessing There are probably better ways to arrange the same weigh of steel to achieve greater "enhancment."

This method of reinforcement is the one that all steel suppliers and welders ask to do when possible. Is it the most efficient in terms of getting the bulk of the area away from the centroid? No. But it does make for the simplest welds on site. Trying to provide a groove weld for a bar in the radius of a W-beam just sounds like a lot of screwing around on site in typically tight spaces working off of a scissor lift. The angles oriented this way give it a nice 90 degree intersection to just fillet weld.

Sorry, I mis-interpreted your image. You would have 4 smaller closed sections in that overall cross section. So, it would change.

I'd probably calculate it the following way:

a) Include a portion of the flange and web equal to the thickness of the angle in with the angle then calculate this as a "closed tube" J.
b) Use summation of bt^3/3 for the remainder of the wide flange. Or, just neglect this completely if it's low in comparison to A.

If you're really excited about getting a more accurate answer, then you might want to buy a program that does this. There should be many on the market that do this.

Caveat: I do work for a software company. So, realize that my last comment has some implicit bias.

Thank y'all! No the purpose was not to enhance J by any means. The purpose is to enhance the "I" and I would have done it differently by adding a plate on the bottom, but that was not possible due to the existing pipes that are right on the bottom flange and cannot move. In terms of deflection it works fine. I just wanted to have the J value as correct as possible in my series of calculations in MathCAD. I will just use the sum of the J value of all five elements.
Thank you again!

...even solid round bar in the flange-web fillet... easier to place and more effective...

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

There is this excellent and free python tool (I am not the author of the tool I believe his handle on here is handofthelion):
Section Properties

My Personal Open Source Structural Applications:

Open Source Structural GitHub Group:

As per your previous thread with adding the plate, to get an accurate answer you'll need to resort to an FEA based approach to solve for torsion and warping constants. So same program recommendation as Celt83. As per the other thread, if you want a script to return the properties of this type of strengthening happy to provide one.

Because you are creating closed sections adding the sum of the J values will be hugely conservative (massively underestimating J) and J isn't likely to be much higher than the bare beam as the angles will likely have lower relative J to the proportionally larger beam.