This may be a strange question, but;

From a stress, and code stress, standpoint; what is the physical significance of the node at the junction of the branch and header legs?

I understand that the piping model is made up of basically structural members. In that sense, it is easy to visualize the junction. But with a tee, the intersection of the centerlines at the junction node is in the middle of tee with no real "physical" connection.

The nodes at the connecting welds at each end of the header runs and the branch run, in the case of forged welding tee, are obvious; but if the node at the junction exceeds the allowable what is the physical significance of this? What part of the tee is over stresses? Or, if the junction node exceeds the allowable, but the weld nodes do not, for whatever reason, is that a problem?

B31.3, fig 319.4.4B seems to be concerned with the forces, in-plane and out-of-plane moments at the weld faces of the fitting, and not at the junction.

Also, the junction node seems to have two different sets of forces and moments. One set from the pipe run through the header at the junction and another set from the branch at the junction. It would seem that these two sets of forces and moments would be additive, or the same.???

Any help in clarifying my confusion would be greatly appreciated.

You're right, there is nothing at that junction point.

The flexibility analysis that you are performing is based on 3D Beam Elements - infinitely thin sticks. That node point is the intersection of the two header elements with the branch element. The forces and moments computed at this point are representative of the loads in the pipe elements at that location. For Code Compliance, the bending moments are increased by the SIF to address the hot-spot stress somewhere on the fitting.

From basic statics (equilibrium), the forces and moments of all elements framing into a node should sum to zero.

Thank you Richard.

Regarding the second question on the two sets of forces and moments for the junction node - are the bending moments & forces additive? If not, then why not

Yes they are additive and should sum to zero, but at a tee you have three elements, not two.