The finite element method

Large displacements of the solid, whether or not accompanied by large deformations, make it impossible to calculate mechanical quantities simply on the basis of the initial configuration. It is therefore necessary to take into account the change in geometry throughout the loading history. The main difficulty lies in the impossibility of expressing the strain and stress tensors on the deformed configuration, given that the latter is unknown. Indeed, the deformed configuration is itself the sought-after solution to the non-linear problem. For this reason, we need to consider solid motion during loading, and not just in the final state.

The solid's motion can be tracked using the Lagrangian description, according to two main approaches:

• total Lagrangian description, where the initial configuration is taken as the reference for mechanical quantities (i.e. displacements, strains and stresses);

• the updated Lagrangian description, where the deformed configuration is taken as reference.

In finite element analysis, the Lagrangian description, whether total or updated, is used to express the incremental equilibrium of the structure, and consequently to evaluate the tangent stiffness matrix and nodal forces for each element. Resolution of the structure-wide equilibrium system is performed using incremental and iterative methods, such as the Newton-Raphson and arc-length methods. This incremental description is an indispensable numerical tool for the analysis of a wide range of non-linear behaviors, such as large displacements, large deformations and structural instability. The implementation of these non-linear formulations in finite element software enables the analysis of numerous applications in the fields of structural mechanics and civil engineering.