Galerkin Projection for Geometrically Exact Sandwich Beams Allowing for Ply Drop-off

Abstract

A Galerkin projection of the equations of equilibrium for a recent theory of geometrically exact sandwich beams that allow finite rotations and shear deformation in each layer is presented. The continuity of the displacement across the layers is exactly satisfied. The resulting finite element formulation can accommodate large deformation. The number of layers is variable, with layer lengths and thicknesses not required to be the same, thus allowing the modeling of sandwich structures with ply drop-off. Numerical examples are presented which underline the salient features of the formulation. Saint-Venant principle is demonstrated for very short sandwich beams.

Nonlinear dynamics of elastic rods using the Cosserat theory: Modelling and simulation

“…Alternatively, the moving frame can be described by the rotation group SO(3) as the formulations in Vu-Quoc (1988, 1991), Vu-Quoc and Deng (1995) and Vu-Quoc and Ebcioglu (1995). The three successive rotations starts by rotating the basis {e 1 , e 2 , e 3 } an angle w(s, t) about the axis aligned with the director e 2 as shown in Fig.…”

“…Specifying a second vector d 2 orthogonal to the first vector (thereby placing it in the plane of the cross-section) can be used to encode the state of bending and twist along the element. In such a way, the current orientation of each cross-section at s 2 [0, '] is defined by specifying the orientation of a moving basis d 1 , d 2 , d 3 = d 1 · d 2 , see Simo and Vu-Quoc (1988) and Vu-Quoc and Deng (1995) for details. Elastic deformations about the line of centroids are then coded into the rates of change of r and the triad d 1 , d 2 , d 3 of the cross-section at s. Thus a time dependent field of two mutually orthogonal unit vectors along the segment provides three continuous dynamical degrees of freedom that, together with the three continuous degrees of freedom describing the centroid space-curve relative to some arbitrary origin in space (with fixed inertial frame e 1 , e 2 , e 3 ), define a simple Cosserat rod model.…”

“…For the three-dimensional problem, we refer to Vu-Quoc (1988, 1991) for a geometrically exact rod model incorporating shear and torsion-warping deformation. Further studies on the dynamic formulation of sandwich beams have been presented in Vu-Quoc and Deng (1995) and Vu-Quoc and Ebcioglu (1995) based on the geometrically-exact description of the kinematics of deformation. Moreover, Esmailzadeh and Jalili (1998) investigated the parametric response of cantilever Timoshenko beams with lumped mass, but restricted themselves to consideration of nonlinear inertia terms.…”

A co-rotational formulation for 3D beam element using vectorial rotational variables

“…Updated Lagrangian formulation (Bathe and Bolourchi 1979, Cardona and Geradin 1988, Chen and Blandford 1991, Misra et al 2000, Teh and Clarke 1999 and co-rotational formulation (Battini and Pacoste 2002, Crisfield 1990, Crisfield and Moita 1996, Hsiao et al 1987, Hsiao and Lin 2000a, Teh and Clarke 1998, certainly, there also exist some mixed type formulations of them (Jiang and Chernuka 1994, Hsiao and Lin 2000b, Lin and Hsiao 2001. In addition, Simo and Vu-Quoc developed a class of geometrically-exact beam formulation, this formulation demonstrates its computational efficiency in large displacement analyses of frame structures and benefit in solving dynamic problems of flexible beam or beams system subject to large overall motions (Simo and VuQuoc 1986a,b, 1988, Vu-Quoc and Deng 1995, Vu-Quoc and Ebcioglu 1995, 1996, Vu-Quoc and Simo 1987. For convenience, these formulations can also be classified into two groups: formulations with asymmetric element tangent stiffness matrices and formulations with symmetric element tangent stiffness matrices.…”

“…For convenience, these formulations can also be classified into two groups: formulations with asymmetric element tangent stiffness matrices and formulations with symmetric element tangent stiffness matrices. Due to the non-commutativity of spatial rotations, most co-rotational formulations belong to the first group, and the geometrically-exact beam formulation proposed by Simo and Vu-Quoc also falls into this category (Simo and Vu-Quoc 1986a,b, 1988, Vu-Quoc and Deng 1995, Vu-Quoc and Ebcioglu 1995, 1996, Vu-Quoc and Simo 1987. For an asymmetric tangent stiffness matrix, more storage is occupied so as to store all its components.…”