Gordan Jelenić

Geometrically exact 3D beam theory: implementation of a strain-invariant finite element for statics and dynamics

Geometrically exact 3D beam theory has been used as a basis for development of a variety of finite element formulations. It has recently become apparent that the important requirement of objectivity of adopted strain measures, although provided by the theory itself, does not automatically extend to a finite element formulation. In this paper we present a new finite element formulation of the geometrically exact 3D beam theory, specifically designed to preserve the objectivity of the adopted strain measures. In order to do so the current local rotations are interpolated in a manner similar to that adopted in co-rotational approaches. However, no approximations typical for co-rotational approaches are introduced into the procedure, so in contrast to co-rotational formulations, the present formulation fully preserves the geometric exactness of the theory. A range of numerical examples serves to illustrate the problem and to assess the formulation.

OBJECTIVITY OF STRAIN MEASURES IN THE GEOMETRICALLY EXACT THREE-DIMENSIONAL BEAM THEORY AND ITS FINITE-ELEMENT IMPLEMENTATION

The paper discusses the issue of discretization of the strain–configuration relationships in the geometrically exact theory of three–dimensional (3D) beams, which has been at the heart of most recent nonlinear finite–element formulations. It is demonstrated that the usual discretization procedures for implementing these strain measures within a finite–element framework violate the important property of objectivity: invariance to rigid body rotations. A method is proposed for overcoming this limitation, which paves the way for an objective finite–element formulation of the geometrically exact 3D beam theory. For a two–noded element, this method involves obtaining the relative rotation matrix that rotates one nodal triad onto the other and then interpolating the resulting rotation vector.

A kinematically exact space finite strain beam model - finite element formulation by generalized virtual work principle

The present paper presents a novel finite element formulation for static analysis of linear elastic spatial frame structures extending the formulation given by Simo and Vu-Quoc [A geometrically-exact rod model incorporating shear and torsion-warping deformation, Int. J. Solids Structures 27 (3) (1991) 371-3931, along the lines of the work on the planar beam theory presented by Saje [A variational principle for finite planar deformation of straight slender elastic beams, Internat. J. Solids and Structures 26 (1990) 887-9001. We apply exact non-linear kinematic relationships of the space finite-strain beam theory, assuming the Bernoulli hypothesis and neglecting the warping deformations of the cross-section. Finite displacements and rotations as well as finite extensional, shear, torsional and bending strains are accounted for in the formulation. A deformed configuration of the beam is described by the displacement vector of the deformed centroid axis and an orthonormal moving frame, rigidly attached to the cross-section of the beam. The position of the moving frame relative to a fixed reference frame is specified by an orthogonal matrix, parametrized by the rotational vector which rotates the moving frame from an arbitrary position into the deformed configuration in one step. Also, the incremental rotational vector is introduced, which rotates the moving frame from the configuration obtained at the previous iteration step into the current configuration of the beam. Its components relative to the fixed global coordinate system are taken to be the rotational degrees of freedom at nodal points. Because in 3-D space both the axial and the follower moments are non-conservative, not the variational principle but the principle of virtual work has been introduced as a basis for the finite element discretization. Here we have proposed the generalized form of the principle of virtual work by including exact kinematic equations by means of a procedure, similar to that of Lagrangian multipliers. This makes possible the elimination of the displacement vector field from the principle, so that the three components of the incremental rotational vector field remain the only functions to be approximated in the finite element implementation of the principle. Other researchers, on the other hand, employ the three components of the incremental rotational vector field and the three components of the incremental displacement vector field. As a result, more accurate and efficient family of beam finite elements for the non-linear analysis of space frames has been obtained. A one-field formulation results in the fact that in the present finite elements the locking never occurs. Any combination of deformation states is described equally precisely. This is in contrast with the elements developed in literature, where, in order to avoid the locking, a reduced numerical integration has to be applied, which unfortunately, diminishes the accuracy of the solution. Polynomials have been chosen for the approximation of the components of the rotational vector. In this case the order of the numerical integration can rationally be estimated and the computer program can be coded in such a way that the degree of polynomials need not be limited to a particular value. The Newton method is used for the iterative solution of the non-linear equilibrium equations. In an non-equilibrium configuration, the tangent stiffness matrix, obtained by the linearization of governing equations using the directional derivative, is non-symmetric even for conservative loadings. Only upon achieving an equilibrium state, the tangent stiffness matrix becomes symmetric. Thus, obtained tangent stiffness matrix can be symmetrized without affecting the rate of convergence of the Newton method. For non-conservative loadings, however, the tangent stiffness matrix is always non-symmetric. The numerical examples demonstrate capability of the present formulation to determine accurately the non-linear behaviour of space frames. In numerical examples the out-of-plane buckling loads are determined and the whole pre-and post-critical load-displacement paths of a cantilever and a right-angle frame are traced. These, in the analysis of space beams standard verification example problems, show excellent accuracy of the solution even when employing only one element to describe the displacements of the size of the structure itself, the rotations of 2n, and extensional strains much beyond the realistic values of linear elastic material.

INTERPOLATION OF ROTATIONAL VARIABLES IN NONLINEAR DYNAMICS OF 3D BEAMS

The formulation of dynamic procedures for three-dimensional (3-D) beams requires extensive use of the algebra pertaining to the non-linear character of the rotation group in space. The corresponding extraction procedure to obtain the rotations that span a time step has certain limitations, which can have a detrimental eect on the overall stability of a time-integration scheme. The paper describes two algorithms for the dynamics of 3-D beams, which dier in their manifestation of the above limitation. The rst has already been described in the literature and involves the interpolation of iterative rotations, while an alternative formulation, which eliminates the above eect by design, requires interpolation of incremental rotations. Theoretical arguments are backed by numerical results. Similarities between the conventional and new formulation are pointed out and are shown to be big enough to enable easy transformation of the conventional formulation into the new one. ? 1998 John Wiley & Sons, Ltd.

Dynamic analysis of 3D beams with joints in presence of large rotations

Abstract In this paper we present a way to extend the earlier static master–slave formulation for beams and joints [1] to dynamic problems. The dynamic master–slave approach is capable of (i) handling the problems of linear elasticity in a geometrically non-linear environment, (ii) accounting for the non-linear kinematics of arbitrary types of joints and (iii) performing the numerical time integration while preserving some of the important constants of motion like total energy and the total momenta for Hamiltonian systems in the absence of external loading. The performance of the formulation is demonstrated with the aid of two representative numerical examples.

Enhanced lower-order element formulations for large strains

The paper describes a range of lower-order element formulations that can be applied to both elastic and elasto-plastic large-strain elements. For plane-strain analysis, this process involves four-noded quadrilaterals while the enhancements involve incompatible modes or enhanced strains. One particular new formulation can be considered as either a “co-rotational approach” or a modified form of “Biot stress procedure”.

Some aspects of the non-linear finite element method

Abstract The paper discusses four separate aspects of the non-linear finite element method: (i) An alternative formulation for the static co-rotational technique in conjunction with a simple faceted shell idealisation. (ii) Solution procedures for non-linear dynamics with emphasis on energy conserving techniques (iii) The use of interface elements and fracture energy related softening “stress-strain” curves for modelling mixed mode delamination in “composites” (iv) Hybrid static/dynamic solution procedures. While the topics are separate, there are links and these are explored.

Non-linear ‘master-slave’ relationships for joints in 3-D beams with large rotations

A novel approach is presented for the analysis of spatial beam elements with end releases, in which, for each joint in the structure, an additional (slave) set of kinematic variables is introduced, which is directly related to the existing (master) set of variables at that node. This relationship is established according to the nature of the joint to be modelled and takes into account that the sliding/rotation takes place along/around the axis that is rigidly attached to the structural node and is thus not fixed in space. In this way, the most interesting releases such as revolute, spherical, prismatic and cylindrical joints can be analysed accurately and efficiently. The main concepts are applicable to any spatial beam finite element, provided that a displacement vector and a rotation matrix are defined at both end nodes. The numerical examples, in which a highly deformable space frame with different joints is analysed, demonstrates the accuracy of the proposed procedure and its advantages compared to the ‘penalty’ technique.

Finite element formulation of hyperelastic plane frames subjected to nonconservative loads

Abstract A finite element formulation for finite deformation static analysis of plane hyperelastic frames subjected to nonconservative loads is presented. A rubber-like material is considered for which the behaviour in tension and in compression differs substantially. A new proposal for the strain energy function of rubber at uniaxial stress state is given, convenient for the present deformation analysis. The finite element formulation is based on a new variational principle of the Hu-Washizu type where exact nonlinear kinematic equations of one-dimensional finite-strain beam theory are taken into account. The contribution of the shear deformations to the total potential energy and the initial curvature of the beam are neglected. The functional of the variational principle is expressed in terms of only one function, the rotation of the cross-section of the beam. Thus only one function needs to be approximated in the functional in the finite element implementation of the variational principle. The outstanding accuracy and high efficiency of the method are illustrated by numerical examples. The application of the present method for the analysis of hyperelastic frames subjected to static nonconservative forces is shown, and some results for critical loads for the dynamic instabilities in the form of flutter are given.

A general framework for conservative single-step time-integration schemes with higher-order accuracy for a central-force system

A general framework for algorithms that conserve angular momentum for single-body central-force problems is presented. It is shown that any family of momentum-conserving algorithms can have at most three free parameters, one of which may be used to ensure energy conservation (and hence will be configuration-dependent). Further restrictions can be made that enable the algorithms to recover the orbits of relative equilibria of the underlying physical problem. In addition, the algorithms can be made time-reversible, whilst still leaving two parameters unspecified. The order of accuracy of a general momentum-conserving family is analysed, and it is shown that energy-momentum algorithms that preserve the underlying physical relative equilibria can have unlimited accuracy if the two remaining parameters are appropriately chosen functions of the configuration and the time-step: this does not require any additional degrees of freedom, extra stages of calculation or information from past solutions. Numerical examples are given that show the performance of some representative higher-order schemes when applied to stiff and non-stiff problems, and the issue of Newton-Raphson convergence is discussed.