  This paper is focused on continuation techniques and Newton-like methods
suggested for numerical determination of safety factors within stability
assessment. Especially, we are interested in the stability of slopes and
related limit load and shear strength reduction methods. We build on
computational plasticity and the finite element method, but we mainly work on
an algebraic level to be the topic understandable for broader class of
scientists and our algorithms more transparent. The presented algorithms are
based on the associated plasticity to be more robust. For non-associated
models, we use Davis-type approximations enabling us to apply the associated
approach. A particular attention is devoted to the Mohr-Coulomb
elastic-perfectly plastic constitutive problem. On this example, we explain
some important features of the presented methods which are beyond the algebraic
settings of the problems. We also summarize the Mohr-Coulomb constitutive
solution and some implementation details.
