Development and implementation of geometrically accurate reduced-order models: Convergence properties of planar beams

A geometrically accurate infinitesimal-rotation planar beam element is developed and implemented in this study. The performance of the element, which is suited for developing reduced-order models for both structural and multibody systems (MBS), is evaluated using an eigenvalue analysis. Unlike conventional infinitesimal-rotation finite elements (FE), the new element is compatible with the computer-aided design (CAD) B-spline and NURBS (Non-Uniform Rational B-Spline) representations and allows for a straightforward linear transformation of CAD solid models to FE analysis meshes. The absolute nodal coordinate formulation (ANCF) elements, which are related to B-splines and NURBS by linear mapping, are used as the basis for developing the planar beam element. The new element has a shape function matrix expressed in terms of geometric coefficients obtained using the ANCF position vector gradients in the reference configuration. The change in the position vector gradients is written in terms of infinitesimal rotation coordinates using a velocity transformation that defines constant element mass and stiffness matrices. Using this approach, initially straight and curved configurations can be modeled using the same displacement field. The eigenvalue analysis is used to evaluate the element performance and examine the effect of shear locking on the predicted frequencies. Several elastic force formulations are used to evaluate the convergence characteristics, including the direct displacement method (DDM), general continuum mechanics (GCM) approach, elastic line (EL) approach, and strain split method (SSM). The element performance is compared with the conventional Euler-Bernoulli and Timoshenko elements as well as the analytical solutions.