Satisfying that equation just means the dimensions satisfy the assumptions of the Uniform Force Method. That means there are no moments on the gusset-to-column and gusset-to-beam connections.

There is no rule that says you must use the UFM. You can choose dimensions that don't satisfy the equation, but the gusset will need a moment at the column or beam to satisfy moment equilibrium.

To: 271828
Thank you for your insight. I would to clarify further. So does it mean that I can set my B = B_bar to remove the moment on the column, regardless of the divergence of the line of action of the forces?

I'm not sure. I just know if you set alpha and beta so that equation is satisfied, there will be no moment at the gusset to member connections. I don't usually look much at the lines of convergence.

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Since a and a_bar or B and B_bar can somehow(usually) not equal, is it necessary to the satisfy the equation a-Btan(theta) = eb(tan(theta)) - ec?

It's necessary if you want to execute the uniform force method as your connection design method and, thereby, generate a design that doesn't require your gusset plates to resist in plane moments. In what follows, I've assumed that we're sticking with the UFM. As 271828 mentioned, there are other methods available and, sometimes, those methods will also result in designs that do no require the gussets to resist in plane moment.

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My question is about the convergence of the line of action through the control points of the beam, column, and gusset plate.

1) When one draws the lines of action for a gusset in isolation, as I believe that you have, they should converge at a common point. That said, there is no expectation that they will converge at the centerline of the column.

2) When you consider the forces that your beam connection will impose upon the column in conjunction with those imposed by the gusset, the combination of those actions should balance out and produce no net moment on your column. And I would recommend that you ultimately draw a free body diagram of your connection as a whole as part of your design exercise to verify that this is the case.

I know, this can feel a bit counterintuitive at times. I find that it helps me to remember that:

1) UFM doesn't produce no moment on columns. Rather, it produces a connection design that does not require moment resistance from the columns.

2) Analytically, UFM only produces no column moments beyond the physical limits of the joint. Within the joint, there will be moments. In the context of your question, your brace is producing moment in the column that is eventually rectified by the forces that the beam imposes on the column.