Nonlinear Beam Kinematics by Decomposition of the Rotation Tensor

Abstract

A simple matrix expression is obtained for the strain components of a beam in which the displacements and rotations are large. The only restrictions are on the magnitudes of the strain and of the local rotation, a newly-identified kinematical quantity. The local rotation is defined as the change of orientation of material elements relative to the change of orientation of the beam reference triad. The vectors and tensors in the theory are resolved along orthogonal triads of base vectors centered along the undeformed and deformed beam reference axes, so Cartesian tensor notation is used. Although a curvilinear coordinate system is natural to the beam problem, the complications usually associated with its use are circumvented. Local rotations appear explicitly in the resulting strain expressions, facilitating the treatment of beams with both open and closed cross sections in applications of the theory. The theory is used to obtain the kinematical relations for coupled bending, torsion, extension, shear deformation, and warping of an initially curved and twisted beam.

Hardening/softening behavior and reduced order modeling of nonlinear vibrations of rotating cantilever beams

“…(20), (21) and (23a) with an integration over the cross section, using the above Eqs. (25a-cac) and integrating the result by parts, one obtains the following expression for the work of internal forces:…”

“…A few other works, devoted to finite-element formulations, use the same strain measure, but with a stress measure and a constitutive law derived from the standard 3D Kirchhoff-Saint-Venant law [26,62]. (b) A second family of works explicitly use the Biot strain tensor (also called Biot-Jaumann), defined by E B = U − 1 with U the standard right stretch tensor, according to a linear constitutive law with the energetically conjugated stress tensor [21,28,30,45,55,78]. (c) Finally, a third family of works, that includes our formulation, is based on a standard derivation of the equations from the 3D nonlinear continuum mechanics laws, the only assumptions being (1) Timoshenko or Euler-Bernoulli kinematics; (2) the consistent linearization of the GreenLagrange strain tensor; and (3) a linear constitutive law between Green-Lagrange strains and second Piola-Kirchhoff stresses [25,26].…”

“…The discretization procedure is based on the weak form (22), the simplified Green-Lagrange strains (15a-c) and the linear constitutive laws (21), with all details gathered in 'Appendix 2'. The following discretized nonlinear dynamic problem is obtained for all time t:…”

Geometrically Nonlinear Theory of Composite Beams with Deformable Cross Sections

“…Both energy densities are given by K c , respectively, where is the material density and c is the fourth-order tensor of elastic material constants (compliances). is the local Jaumann-Biot-Cauchy strain tensor given in a mixed-bases projection, as in Danielson and Hodges [25],…”

Simulation of a Moving Elastic Beam Using Hamilton's Weak Principle

“…The engineering strain is defined in [18], using generalized force and moment strains. Force strains, due to axial and shear forces, are represented by γ.…”