The strategies of resolution of a nonlinear equations system are related with the decomposition of it in several linear systems each of them are solved with the known methods.
A very known strategy of resolution of nonlinear equations systems is the Newton-Raphson methods. In this method an incremental-iterative scheme is utilised. The total load is decomposed in several increments. For each increment an iterative procedure is performed until the convergence criteria is obtained.
There are several methods to control the following of the load-displacement curve of the structure. Each one of these methods control different parameters like load, displacement or arc-length increments.
Some advances procedures like line search, automatic incrementation or automatic arc-length switch in order to improve the speed of calculations can be applied.
Deeper explanations can be found in Bathe (1996), Crisfield (1991).