NDLI: Isogeometric rotation-free analysis of planar extensible-elastica for static and dynamic applications

Content Provider	Springer Nature Link
Author	Maurin, Florian Dedè, Luca Spadoni, Alessandro
Copyright Year	2015
Abstract	Finite deformations of planar slender beams for which shear strain can be neglected are described by the extensible- elastica model, where the strain-displacement relation is geometrically exact and the Biot stress–strain relation is linear. However, if the formulation is expressed in terms of displacements without rotation, the kinematics are described by a partial differential equation involving a fourth-order spatial operator, which cannot be approximated by the classical $${\mathcal {C}}^0$$ -continuous FE method in the standard Galerkin framework. In this work, we propose the spatial approximation of such high-order PDE by means of NURBS-based isogeometric analysis (IGA) which allows the use of globally high-order continuous basis functions. The employed IGA approach possesses three advantages: first, it facilitates the encapsulation of the exact geometric representation of the beams in the spatial approximation with fewer discrete points, especially useful for curved structures; second, it allows the discretization of high-order spatial operators; and third, it provides an efficient numerical solution of the discrete problem by using a limited number of degrees of freedom since the employed standard Galerkin formulation does not require rotational degrees of freedom. Yet this approach has not been directly compared to appropriate analytical solutions. To this end, we compare and validate numerical results from FE with the closed-form solutions for a set of static beam problems, including a newly derived solution for an initially curved beam, based on the extensible-elastica theory, by estimating the convergence orders of the errors. We also highlight the advantages of this formulation with the numerical solution of three dynamic problems: the swinging of a pinned beam, the propagation of solitons (nonlinear waves) in post-buckled beams, and snap-through buckling of a pinned beam that is axially buckled before transverse loading.
Starting Page	77
Ending Page	96
Page Count	20
File Format	PDF
ISSN	0924090X
Journal	Nonlinear Dynamics
Volume Number	81
Issue Number	1-2
e-ISSN	1573269X
Language	English
Publisher	Springer Netherlands
Publisher Date	2015-03-08
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Extensible-elastica Isogeometric rotation-free discretization NURBS Nonlinear beams Dynamic snap-through Vibration, Dynamical Systems, Control Mechanics Mechanical Engineering Automotive Engineering
Content Type	Text
Resource Type	Article
Subject	Ocean Engineering Applied Mathematics Control and Systems Engineering Mechanical Engineering Electrical and Electronic Engineering Aerospace Engineering

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