This paper reviews and derives finite element incremental formulations for non-linear static and dynamic analysis, focusing on large displacements, large strains, and material non-linearities. The aim is to provide a consistent summary, comparison, and evaluation of these formulations, which have been implemented in the NONSAP program. The formulations are categorized into total Lagrangian and updated Lagrangian approaches, both based on continuum mechanics principles. The paper discusses the numerical solution of the equations of motion using isoparametric finite element discretization and presents specific matrices required for the formulations. It demonstrates the applicability and differences in the formulations through the solution of static and dynamic problems involving large displacements and strains. The paper also compares the linear and non-linear strain-displacement transformation matrices, Cauchy stress matrices, and constitutive relations for elastic, hyperelastic, and hypoelastic materials. The results show that the same numerical results can be obtained using either formulation, provided the appropriate constitutive relations are used. The paper concludes with a detailed discussion on the equilibrium iteration process and the step-by-step integration algorithm.This paper reviews and derives finite element incremental formulations for non-linear static and dynamic analysis, focusing on large displacements, large strains, and material non-linearities. The aim is to provide a consistent summary, comparison, and evaluation of these formulations, which have been implemented in the NONSAP program. The formulations are categorized into total Lagrangian and updated Lagrangian approaches, both based on continuum mechanics principles. The paper discusses the numerical solution of the equations of motion using isoparametric finite element discretization and presents specific matrices required for the formulations. It demonstrates the applicability and differences in the formulations through the solution of static and dynamic problems involving large displacements and strains. The paper also compares the linear and non-linear strain-displacement transformation matrices, Cauchy stress matrices, and constitutive relations for elastic, hyperelastic, and hypoelastic materials. The results show that the same numerical results can be obtained using either formulation, provided the appropriate constitutive relations are used. The paper concludes with a detailed discussion on the equilibrium iteration process and the step-by-step integration algorithm.