Determination of ultimate load and possible failure path for solid

In this study, an effective and practical, h-version, enrichment mesh generation, and finiteelement adaptive procedure for the non-linear solution of problems in continuous media is presented.moreover, based on the gradient recovery rule, a general recovery technique is developed to measure errorand refine mesh in general finite element solutions. the recovery technique is simple and cost effectiveto implement. the technique has been formulated for two dimensional problems by employing triangularelements. the formulation is consistent with non-linear formulations which iteratively equilibrate thecontinuous media problems.in the present study, in addition to correlating various norms (such as energy norm, l2 norm for stressand l2 norm for strain), a new norm, namely, deviating stresses norm (called j norm in this study), is alsocorrelated by the authors to estimate the error rate in the finite element method. based on the results ofthis study, the j norm can be used as a tool to estimate the error rate in the finite element method, and todetermine the ultimate load and the possible failure path in continuous domains. for several numericalexamples, the developed algorithms are demonstrated and the resulting meshes are presented