Numerical simulation of large displacements of structures subjected to extreme loading events

When the structural resisting forces and stiffness matrices are computed with a finite element model, it is sometimes important to consider their dependence from displacements such as in case of earthquakes, other extreme loading events, or for very flexible structures. Geometric nonlinearity must be taken into account in numerical simulations to obtain correct numerical results. To consider these effects in frame structures, M. Crisfield introduced his corotational formulation in 1990, herein designated as "average nodal rotation" (for reasons that will be made apparent in the thesis). In this formulation, a new independent reference system is created: the basic reference system. The basic reference system characterises the current state of the element and follows the frame element in its deformed configuration as it moves through space. Additionally, by using the corotational formulation, it is possible to separate clearly both sources of nonlinearity: material and geometric. The basic reference system fully characterises the deformed state of the element through the basic displacements, the basic forces and the associated basic stiffness matrix. For the planar case, the implementation of the corotational formulation is simple but complexity increases substantially in the spatial case. The main difficulty comes from the non-commutativity of spatial rotations, i.e. the order of several spatial rotations must be taken into account. Different representations of spatial rotations exist: some have 3 parameters, the vectorial type, while others have 4 parameters, the normalized quaternions. The rotation matrix is a 3 × 3 tensor, related to a rotation angle. The quaternions are hypercomplex numbers introduced by the mathematician W. Hamilton in 1843 and they are used to represent a spatial rotation similarly to complex numbers. These latter representations of spatial rotations will be used in the spatial corotational formulations addressed in this thesis. Three spatial corotational formulations are studied: the "average nodal rotation" of M. Crisfield programmed by R. Souza, a formulation based on the rotation of the initial node and the more recent "double reference system" developed by A. Correia and implemented in structural analysis software "SeismoStruct". For the first approach, M. Crisfield made an approximation by taking an intermediate rotation between both ends of the element. The second formulation is only based on the rotation of the first node of the element. And in the third approach, A. Correia creates a double reference system by taking into account independent reference systems related to both extremities of the element. The present document starts by studying and revising their theoretical background in detail, establishing connections and identifying differences. The formulations are then implemented in a programming language and several numerical examples are carried out to assess their performance.