Abstract :
This paper investigates the formulation of co-rotational flat facet triangular elements for the numerical analysis of instability phenomena in shell structures. The elements have three nodes with six degrees of freedom at each node. The term ‘co-rotational’ relates here to the provision of a local system that continuously rotates and translates with the element. Following mainly Nour-Omid and Rankin [B. Nour-Omid and C.C. Rankin, Finite rotation analysis and consistent linearization using projectors, Comput. Methods Appl. Mech. Engrg. 93 (1991) 353–384], the definition of an element resorts to a change of variables from the local frame to the global one. This is done through the use of a projector matrix which relates the variations of the local displacements to the variations of the global ones, by extracting the rigid body modes from the latter. The main difference from the original formulation lies in the parameterization of 3D finite rotations. In contrast to the paper by Nour-Omid and Rankin, a parameterization based on the rotational vector is here adopted and thus, an additional change of variables has to be performed. As a result, the rotational variables become additive and the necessity of a special updating procedure is avoided. The main feature of the adopted formulation is its independence of the local assumptions used to derive the internal forces and tangent stiffness in local coordinates. For a certain class of elements (i.e. elements with the same number of nodes and degrees of freedom) the main co-rotational framework is the same. Using this property, three types of local formulations are considered. A set of carefully chosen test problems is used in order to assess the performances of the three element types.
