Presentation of the approach =========================== Introduction ------------ Taking into account large displacements, whether they result from a rigid body movement or from any transformation of the studied structure, introduces additional nonlinearity (in addition to that introduced by behavioral nonlinearity for example). This non-linearity means that the initial configuration of the structure and the final configuration (or "deformed") can no longer be confused as is usually the case for the treatment of problems in small disturbances. For finite structural elements (*i.e.* beams, plates, shells), an additional problem arises: rotations no longer have a vector character, so we can no longer transform them as vectors when passing from the local coordinate system to the global coordinate system. In fact, when displacement vectors are expressed, in a local coordinate system of a finite element (coordinate system attached to SEG2 of the beam in which all the calculations are carried out to simplify the writing of the various terms), it is quite physical to transform them into a global coordinate system attached to the structure. When we are interested in structural elements, they also carry degrees of freedom of rotation; when these rotations are "small", we can show that they identify with vectors and therefore transform them as displacements. This is no longer the case when the rotations are "large" and this results in the non-commutative nature of the rotations. In the absence of a description taking into account this particularity, the continuity of the degrees of freedom at the nodes of the elements is then no longer ensured. The geometrically accurate element POU_D_T_GD ------------------------------------------ The finite beam element POU_D_T_GD, already integrated into the code for many years, models a Timoshenko beam in large displacements in the sense that the displacement field used in the formulation (during the 3D → beams transition) is written exactly. In particular, it takes into account the exact rotation operator between two configurations of the element (i.e. no simplifying assumptions about the movements are made). The behavior, which is only elastic, is always written in small deformations; this element had in fact been introduced to deal with large "almost rigid" movements of a dynamic structure (movement of the spacers between the conductors [1] _ electrical lines from an overhead line). This element is formulated with a relatively complex Total Lagrangian approach, the exact treatment of large rotations requiring the use of quaternion theory to correctly update the displacements. The behavior is directly formulated on the generalized efforts (without going through the constraints) and does not really lend itself to a multi-fiber extension which by nature works on the constraints [2] _ . The POU_D_TGM multi-fiber element -------------------------------- It is a Timoshenko beam element with warping of the cross section and multi-fiber approach to account for the progression of plasticity in the section. This element already had a generic option 'PETIT_REAC' (now removed) that could update the geometry at each iteration of the stepwise resolution algorithm. This made it possible to deal with problems in large movements under the hypothesis of small rotations. However, the absence of geometric rigidity in the formulation makes convergence very difficult in the case of instability, even for flat problems. Moreover, we are not exploiting all the possibilities of updating geometry (see :ref:`4.4.1 `). A short history ------------------- It was from the 70s that the geometric nonlinear analysis of structures formed by beams developed with the use of the finite element method. The analysis first focused on plane structures, then on three-dimensional structures with Bathe's seminal work in 1979. It was in fact at this time that a formulation called "*Updated Lagrangian"* (UL) appeared for the first time, i.e. a formulation that updated the geometry of the structure at each iteration of the resolution algorithm, making it possible to obtain a simplified formulation while remaining robust. Subsequently, numerous works were inspired by those of Bathe and enriched them, in particular with the treatment of the thorny problem of large rotations highlighted by Argyris. We can also cite the work of Yang and McGuire as well as that of Conci. Another important point in nonlinear structural analysis is the need to properly estimate the residue in order to obtain accurate results. However, in the case of beam elements, the geometric interpolation functions are of the first order (the beam elements are not iso-parametric). This leads to errors in calculating the residue if the structure is deformed a lot and if not enough elements are used. One solution obviously consists in increasing the number of finite elements, especially for curved structures. However, this can become expensive and authors have therefore proposed techniques called "*force recovery* [3] _ *"* to get around this problem and allow only one finite element to be used per beam. For this reason, some like Conci have separated rigid body movement and movement resulting in non-zero deformation work. This approach has resulted in a new formulation that is also referred to as a "co-rotational" formulation. Others have undertaken to correct the value of the residue by adding an additional term to the formulation. Approach -------- The approach described here is based on an exhaustive bibliographic study and more particularly on the work of two theses on the nonlinear analysis of structures formed by beams. The element selected is based on a Lagrangian Formulation Updated at each Iteration (FLAI) [4] _ and we assume "moderate" rotations by iteration so that we can simply reduce ourselves to the case where the rotations are commutative up to the second order (hence the term "moderate" as opposed to "small" *i.e.* of the first order). The approach detailed in the following section can be broken down as follows: 1. As with all structural elements, hypotheses on kinematics are formulated to determine a three-dimensional field of displacement at any point in the section from the degrees of freedom considered. 2. The principle of virtual works is written in the framework of a Lagrangian Formulation Updated at each Iteration. We are in small deformations. 3. We introduce the displacement field and the interpolation functions (which interpolate the generalized displacements according to the unknowns at the nodes) to obtain the expressions of the tangent matrix that we will break down into several terms (each referring to the different phenomena that we want to take into account). 4. Taking into account large rotations of the structure is based on a modification of the initial field of movement, assuming moderate rotations between two iterations. .. [1] Spacers are used to ensure minimal spacing between the conductors in a bundle .. [2] There are other approaches to introduce plasticity but with the power of current computers, the multi-fiber approach seems to be the most appropriate .. [3] literally recovering strength, that is, estimating the internal work of the structure .. [4] It is the French equivalent of the term UL introduced above