Abstract
The present work focuses on the 2-D formulation of a nonlinear beam model for slender structures that can exhibit large rotations of the cross sections while remaining in the small-strain regime. Bernoulli-Euler hypothesis that plane sections remain plane and perpendicular to the deformed beam centerline is combined with a linear elastic stress-strain law.The formulation is based on the integrated form of equilibrium equations and leads to a set of three first-order differential equations for the displacements and rotation, which are numerically integrated using a special version of the shooting method. The element has been implemented into an open-source finite element code to ease computations involving more complex structures. Numerical examples show a favorable comparison with standard beam elements formulated in the finite-strain framework and with analytical solutions.
Highlights
Slender fiber- or rod-like components represent essential constituents of mechanical systems in many fields of application such as civil, mechanical and biomedical engineering
The results indicate that comparable accuracy can be obtained if the present method replaces traditional finite elements by the same number of integration segments located within one single finite element
We present an efficient formulation for a geometrically nonlinear beam model that accounts for arbitrarily large rotations of the beam sections and the effect of curvature on the change of distance between end sections measured along the chord
Summary
Slender fiber- or rod-like components represent essential constituents of mechanical systems in many fields of application such as civil, mechanical and biomedical engineering. Simo developed a dynamic formulation for Reissner’s beam [5] and together with Vu-Quoc [6] initiated the finite element implementation. He introduced the useful concept of a geometrically exact beam, based on recasting Reissner’s theory in a form which is valid for any magnitude of displacements and rotations. A geometrically nonlinear beam model is formulated for the 2-D case This model is applicable when the strains remain in a limited range while the rotations of beam sections can become arbitrarily large, and it properly accounts for the effect of curvature on the change of distance between end sections measured along the chord. They are combined with the kinematic equations and generalized material equations (linking internal forces to the curvature and centerline extension) and numerically integrated
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