# A Co-Rotational Meshfree Method for the Geometrically Nonlinear Analysis of Structures
## 1 Introduction  
The geometrically nonlinear analysis of a beam structure undergoing large displacement remains a strong topic of interest, with applications in various fields such as civil engineering over the past few decades. Various formulation strategies and numerical procedures using FEM have been developed to deal with the large deformation problems [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. For the large displacements and finite rotation problems of a geometrically nonlinear analysis, the description of the kinematics can be regarded as the dominant factor. The formulations used to describe the kinematics of beam structures in the literature might be divided into three categories: total Lagrangian (TL) formulation [2, 5, 9, 11, 12, 15], updated Lagrangian (UL) formulation [3, 7, 18], and co-rotational (CR) formulation [4, 8, 10]. The main difference between these formulations lies in their definitions of the reference of the coordinates adopted in FEM. In the total Lagrangian formulation, the initial undeformed configuration is adopted as the reference configuration. Consequently, the system equations are defined with respect to the fixed global coordinate system through the analysis. In contrast, the updated Lagrangian formulation adopted the local coordinate system, updated with the current configuration, as the reference configuration--that is, the system equations were constructed at the current deformed beam configuration. Generally, the system equations expressed in the updated Lagrangian formulation will be simpler than the corresponding equations in the total Lagrangian formulation. However, if the displacement from the current configuration to the last equilibrium configuration is large, it will be in violation of the basic assumption and will lead to unreliable results. To circumvent this problem, the co-rotational formulation was introduced. In the co-rotational framework, there are global coordinates and co-rotational coordinates, kept fixed at the deformed beam segment during the analysis procedure. The co-rotational coordinate rotates and translateswith each element but does not deform with it. Therefore, the total motions of the element can be divided into the rigid body part and the purely deformational part. For the finite element formulation, another important issue is the choice of the primary interpolated variables. No matter which of the above formulations is adopted to describe the kinematics, most of the large deformation analyses performed with the FEM use the conventional displacement approach with the principle of virtual work or minimum potential energy [2; 3; 4; 5]. Both displacements and rotations or rotations alone are taken as the interpolated variables in the finite element implementation. Hence, the choice of the primary interpolated variables is also an important issue in the finite element formulation. In standard finite element formulations, the stress resultants solved by the equilibrium equations, and those computed by the constitutive equations, are not equal since the differentiated quantities are, as a rule, one order less accurate than the quantity itself. When formulations based on displacement and rotation were employed, the strains had to be computed by differentiating toward the assumed kinematic field. The internal forces and moments will be evaluated by the constitutive law with the recovered strain field. However, the accuracy of the differentiated quantities may decline. In particular, employing low-order shape functions to solve the shear flexible models may yield locking problems. Therefore, the performance of the element may deteriorate rapidly while dealing with shear flexible models such as thinner beam problems. However, the numerical effect can be corrected with the aid of reduced integration. Still, the displacement-rotation-based formulations may acquire inaccurate stress fields due to recovery by the differentiation procedure, even if the displacement fields can be solved accurately.