Abstract
This paper presents a p-version geometrically nonlinear formulation (GNL) based on total Lagrangian approach for a three node two dimensional curved beam element. The hierarchical element approximation functions and the corresponding nodal variables are derived directly from the Lagrange family of interpolation functions. The resulting element displacement approximation is hierarchical and can be of arbitrary and different polynomial orders in the longitudinal and the transverse directions of the beam element and ensures C0 continuity. The element geometry is described by the coordinates of the nodes located on the axis of the beam (middle surface) and the nodal vectors describing the top and bottom surfaces of the element.
The element properties are established using the principle of virtual work and the hierarchical element displacement approximation. In formulating the properties of the element complete two dimensional stresses and strains are considered hence the element is equally effective for very slender as well as extremely deep beams. Incremental equations of equilibrium are derived and solved using the standard Newton-Raphson method. The total load is divided into increments, and for each increment of load, equilibrium iterations are performed until each component of the residuals is within a preset tolerance. Numerical examples are presented to show the accuracy, efficiency and advantages of the present formulation. The results obtained from the present formulation are compared with those reported in the literature.
The formulation presented here removes virtually all of the drawbacks present in the existing GNL beam finite element formulations and has many additional benefits. First, the currently available GNL beam formulations are based on fixed order of approximation for the displacements and thus are not hierarchical and have no provision for changing the order of approximation for the displacements u and v. Secondly, the element displacement approximations in the existing formulations are either based on linearized displacement field for which a true Lagrangian formulation is not possible and the incremental load step size is severely limited or are based on nonlinear nodal rotation function approach in which case even though the description of displacement for the element is exact but additional complications arise due to the noncummutative nature of nonlinear nodal rotation functions. The p-version displacement approximation used here does not involve traditional nodal rotations that have been used in the existing beam formulations, thus the difficulties associated with their use are not present in this formulation.
A total lagrangian formulation for the geometrically nonlinear analysis of structures using finite elements. Part I. Two-dimensional problems: Shell and plate structures
