Geometrically Nonlinear Theory of Composite Beams with Deformable Cross Sections

Abstract

A one-dimensional theory of slender structures with heterogeneous anisotropic material distribution is presented. It expands Cosserat's description of beam kinematics by allowing deformation of the beam cross sections. For that purpose, a Ritz approximation is introduced on the cross-sectional displacement field, which defines additional elastic degrees of freedom (finite-section modes) in the one-dimensional model. This results in an extended set of beam dynamic equations that includes direct measures of both the large global displacement and rotations of a reference line, and the small local deformations of the cross sections. Two situations are studied in which this approach provides a simpler alternative to shell models with comparable fidelity. First, we look at the detailed structural response of composite beams with distributed loads. In particular, the case of a composite box beam with embedded piezoelectric actuators is considered. Second, this methodology is applied to study the low-frequency response characterization of composite beams. Numerical results in both cases show that a reduced set of finite-section modes allows a full description of the actual three-dimensional displacement field using a strictly one-dimensional formulation.

Computational Aeroelasticity Framework for Analyzing Flapping Wing Micro Air Vehicles

“…A flow diagram of the process is shown in Figure 1. The structural formulation follows the variational-asymptotic method for the analysis of composite beams [26]: the equations of motion for a slender anisotropic elastic 3-D solid are approximated by the recursive solution of a linear 2-D problem at each cross section [25], and a 1-D geometrically-nonlinear problem along the reference line [24]. This procedure allows the asymptotic approximation of the 3-D warping field in the beam cross sections, which are used with the 1-D beam solution to recover a 3-D displacement field.…”

“…From the resulting 1-D problem, the geometrically-nonlinear dynamic equations of equilibrium along the reference line (as presented in Ref. 24) are written as…”

Computational Aeroelasticity Framework for Analyzing Flapping Wing Micro Air Vehicles

“…The first geometrically-nonlinear structural dynamic solution is based on an asymptotic approach to the equations governing the dynamics of a general 3-D anisotropic slender solid [24,25]. It is implemented in the University of Michigan's Nonlinear Active Beam Solver (UM/NLABS) computer code.…”

Low-Speed Aeroelastic Modeling of Very Flexible Slender Wings with Deformable Airfoils

“…It is therefore based on a 1-D finite-element modeling of the structure, which now includes additional degrees of freedom to allow for sectional deformations. We have called finite-section deformation modes 15 and they arise from a Ritz approximation to the local sectional warping field, such as those shown in Figure 1. The modal amplitudes are the additional degrees of freedom, with associated stiffness and inertia characteristics 17 .…”

“…A mixed-form solution of the structural dynamics equations is used 15 , which solves simultaneously the equation in displacements, forces and momenta (this is particularly relevant for the recovery of the threedimensional displacement field from Eqs. (1) and (6) …”