For friction to develop, the pipe must slide against a restraint. Call the direction of sliding "X" (perhaps North). For convergence, there must be two consecutive iterations where the non-linear results are within the specified tolerances (such as the friction angle).

In the second (next) iteration, assume the direction of sliding is "Y". If "Y" is within plus or minus the "friction angle variation" of "X", then the sliding direction is assumed converged.

Using a numerical example:

Friction angle variation = 15 degrees.
Iteration 1 Direction = 90 degrees (assume this is East)
Iteration 2 direction = 45 degrees (North-East)

90 - 45 = 45
45 is greater than 15, the tolerance -> no convergence

Iteration 3 direction = 55 degrees (East-North-East)
55 - 45 = 10
10 is less than 15, the tolerance -> (direction) convergence