An equilibrium-based formulation with nonlinear configuration dependent interpolation for geometrically exact 3D beams

This article describes a novel equilibrium-based geometrically exact beam finite element formulation. First, the spatial position and rotation fields are interpolated by nonlinear configuration-dependent functions that enforce constant strains along the element axis, completely eliminating locking phenomena. Then, the resulting kinematic fields are used to interpolate the spatial sections force and moment fields in order to fulfill equilibrium exactly in the deformed configuration. The internal variables are explicitly solved at the element level and closed-form expressions for the internal force vector and tangent stiffness matrix are obtained, allowing for explicit computation, without numerical integration. The objectivity and absence of locking are verified and some important numerical and theoretical aspects leading to a computationally efficient strategy are highlighted and discussed. The proposed formulation is successfully tested in several numerical application examples.     

Numerical integration of the stiff dynamics of geometrically exact shells: an energy‐dissipative momentum‐conserving scheme

This paper presents a new family of time‐stepping algorithms for the integration of the dynamics of non‐linear shells. We consider the geometrically exact shell theory involving an inextensible director field (the so‐called five‐parameter shell model). The main characteristic of this model is the presence of the group of finite rotations in the configuration manifold describing the deformation of the solid. In this context, we develop time‐stepping algorithms whose discrete solutions exhibit the same conservation laws of linear and angular momenta as the underlying physical system, and allow the introduction of a controllable non‐negative energy dissipation to handle the high numerical stiffness characteristic of these problems. A series of algorithmic parameters for the different components of the deformation of the shell (i.e. membrane, bending and transverse shear) fully control this numerical dissipation, recovering existing energy‐momentum schemes as a particular choice of these algorithmic parameters. We present rigorous proofs of the numerical properties of the resulting algorithms in the full non‐linear range. Furthermore, it is argued that the numerical dissipation is introduced in the high‐frequency range by considering the proposed algorithm in the context of a linear problem. The finite element implementation of the resulting methods is described in detail as well as considered in the final arguments proving the aforementioned conservation/dissipation properties. We present several representative numerical simulations illustrating the performance of the newly proposed methods. The robustness gained over existing methods in these stiff problems is confirmed in particular. Copyright © 2002 John Wiley & Sons, Ltd.     

Collapse of thin shell structures—stress resultant plasticity modelling within a co‐rotated ANDES finite element formulation

Due to the very non‐linear behaviour of thin shells under collapse, numerical simulations are subject to challenges. Shell finite elements are attractive in these simulations. Rotational degrees of freedom do, however, complicate the solution. In the present study a co‐rotated formulation is employed. The deformation of the shell is decomposed in to a contribution from large rigid body rotation and a strain producing term. A triangular assumed strain shell finite element is used. Hence, a high performance elastic element is combined with the co‐rotated formulation. In the co‐rotated co‐ordinate system the plasticity is accounted for by a simplifyed Ilyushin stress resultant yield surface. The stress update is determined from the backward Euler difference, and a consistent geometrical and material tangent stiffness is derived. Comparison with other published analysis results show that the present formulation gives acceptable accuracy. Copyright © 1999 John Wiley & Sons, Ltd.     

Total Lagrangian Reissner's geometrically exact beam element without singularities

In this paper, we introduce a new Reissner's geometrically exact beam element, which is based on a total Lagrangian updating procedure. The element has the rotation vector as the dependent variable and the singularity problems at the rotation angle 2π and its multiples are passed by the change of parametrization on the rotation manifold. The beam formulation has several benefits such as all the unknown vectors belong to the same tangential vector space, no need for secondary storage variables, the path‐independence in the static case, any standard time‐integration algorithm may be used, and the symmetric stiffness. Copyright © 2006 John Wiley & Sons, Ltd.     

Finite element of slender beams in finite transformations: a geometrically exact approach

This article is devoted to the modelling of thin beams undergoing finite deformations essentially due to bending and torsion and to their numerical resolution by the finite element method. The solution proposed here differs from the approaches usually implemented to treat thin beams, as it can be qualified as ‘geometrically exact’. Two numerical models are proposed. The first one is a non‐linear Euler–Bernoulli model while the second one is a non‐linear Rayleigh model. The finite element method is tested on several numerical examples in statics and dynamics, and validated through comparison with analytical solutions, experimental observations and the geometrically exact approach of the Reissner beam theory initiated by Simo. The numerical result shows that this approach is a good alternative to the modelling of non‐linear beams, especially in statics. Copyright © 2003 John Wiley & Sons, Ltd.     

A new boundary element formulation for shear deformable shells analysis

A new domain‐boundary element formulation to solve bending problems of shear deformable shallow shells having quadratic mid‐surface is presented. By regrouping all the terms containing shells curvature and external loads together in equilibrium equation, the formulation can be formed by coupling boundary element formulation of shear deformable plate and two‐dimensional plane stress elasticity. The boundary is discretized into quadratic isoparametric element and the domain is discretized using constant cells. Several examples are presented, and the results shows a good agreement with the finite element method. Copyright © 1999 John Wiley & Sons, Ltd.     

Refined hybrid degenerated shell element for geometrically non-linear analysis

Based on a variational principle with relaxed inter-element continuity requirements, a refined hybrid quadrilateral degenerated shell element GNRH6, which is a non-conforming model with six internal displacements, is proposed for the geometrically non-linear analysis. The orthogonal approach and non-conforming modes are incorporated into the geometrically non-linear formulation. Numerical results show that the orthogonal approach can improve computational efficiency while the non-conforming modes can eliminate the shear/membrane locking phenomenon and improve the accuracy. © 1998 John Wiley & Sons, Ltd.     

Assessment of nonlinear exact geometry sampling surfaces solid-shell elements and ANSYS solid elements for 3D stress analysis of piezoelectric shell structures

In this work, the finite rotation exact geometry four-node solid-shell element using the sampling surfaces (SaS) method is developed for the analysis of the second Piola-Kirchhoff stresses in laminated piezoelectric shells. The SaS method is based on choosing inside the layers the arbitrary number of SaS parallel to the middle surface and located at Chebyshev polynomial nodes in order to introduce the displacements and electric potentials of these surfaces as fundamental shell unknowns. The outer surfaces and interfaces are also included into a set of SaS. To circumvent shear and membrane locking, the hybrid-mixed solid-shell element on the basis of the Hu-Washizu variational principle is proposed. The tangent stiffness matrix is evaluated by 3D analytical integration throughout the finite element. This novelty provides a superior performance in the case of coarse meshes. A comparison with the SOLID226 element showed that the developed exact geometry SaS solid-shell element allows the use of load increments, which are much larger than possible with existing displacement-based finite elements. Thus, it can be recommended for the 3D stress analysis of doubly-curved laminated piezoelectric shells because the SaS formulation gives the opportunity to obtain the 3D solutions of electroelasticity with a prescribed accuracy.     

A geometrically exact formulation for thin multi-layered laminated composite plates: Theory and experiment

A geometrically exact approach is employed to formulate the equations of motion of thin multi-layered isotropic and laminated composite plates subject to excitations that cause large strains, displacements, and rotations. The linearization of the obtained semi-intrinsic theory leads to the Mindlin–Reissner theory while an ad hoc truncated kinematic approximation delivers, as a by-product, the Föppl–von Kármán theory of plates. An experimental validation is sought for fully clamped plates which are either of the isotropic single-layered type or of the multi-layered laminated composite type. To this end, nonlinear equilibrium paths are constructed both theoretically and experimentally when the plates are subject to a quasi-statically increasing central point load. The comparisons between the experimentally obtained results and those furnished by the geometrically exact theory as well as by the Föppl–von Kármán (FVK) theory show the high accuracy of the proposed nonlinear theory while the FVK theory becomes increasingly inaccurate at deflection amplitudes of the order of the plates thickness.     

A locking-free hexahedral element for the geometrically non-linear analysis of arbitrary shells

The development of the formulation for a highly adaptable hexahedral shell finite element is presented in this paper. A basic 18-node isoparametric hexahedral element is adopted as the basis of the formulation. Potential strategies to alleviate transverse shear, trapezoidal, thickness and membrane locking are investigated, in several combinations, using a wide variety of geometrically linear benchmarks. The most promising approach is further assessed using geometrically non-linear shell and plate problems. The recommended ANS-formulation performs well against an extensive range of benchmarks, and continues to be accurate at an aspect ratio of 1:10,000.     

An objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy–momentum conserving scheme in dynamics

We present in this paper a new finite element formulation of geometrically exact rod models in the three‐dimensional dynamic elastic range. The proposed formulation leads to an objective (or frame‐indifferent under superposed rigid body motions) approximation of the strain measures of the rod involving finite rotations of the director frame, in contrast with some existing formulations. This goal is accomplished through a direct finite element interpolation of the director fields defining the motion of the rod's cross‐section. Furthermore, the proposed framework allows the development of time‐stepping algorithms that preserve the conservation laws of the underlying continuum Hamiltonian system. The conservation laws of linear and angular momenta are inherited by construction, leading to an improved approximation of the rod's dynamics. Several numerical simulations are presented illustrating these properties. Copyright © 2002 John Wiley & Sons, Ltd.     

Director‐based beam finite elements relying on the geometrically exact beam theory formulated in skew coordinates

In the present work, a new director‐based finite element formulation for geometrically exact beams is proposed. The new beam finite element exhibits drastically improved numerical performance when compared with the previously developed director‐based formulations. This improvement is accomplished by adjusting the underlying variational beam formulation to the specific features of the director interpolation. In particular, the present approach does not rely on the assumption of an orthonormal director frame. The excellent performance of the new approach is illustrated with representative numerical examples. Copyright © 2013 John Wiley & Sons, Ltd.     

An assumed strain triangular curved solid shell element formulation for analysis of plates and shells undergoing finite rotations

A formulation for 36‐DOF assumed strain triangular solid shell element is developed for efficient analysis of plates and shells undergoing finite rotations. Higher order deformation modes described by the bubble function displacements are added to the assumed displacement field. The assumed strain field is carefully selected to alleviate locking effect. The resulting element shows little effect of membrane locking as well as shear locking, hence, it allows modelling of curved shell structures with curved elements. The kinematics of the present formulation is purely vectorial with only three translational degrees of freedom per node. Accordingly, the present element is free of small angle assumptions, and thus it allows large load increments in the geometrically non‐linear analysis. Various numerical examples demonstrate the validity and effectiveness of the present formulation. Copyright © 2001 John Wiley & Sons, Ltd.     

Stress resultant geometrically exact form of classical shell model and vector‐like parameterization of constrained finite rotations

In this work we consider the geometrically exact shell model subjected to finite rotations, making use of rotation vector parameters for handling the corresponding constrained rotation for smooth shells. A modification of such a parameterization which is based on the incremental rotation vector and thus capable of avoiding the singularity problem is also discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

A hierarchical finite element for geometrically non‐linear vibration of doubly curved,moderately thick isotropic shallow shells

A p‐version, hierarchical finite element for doubly curved, moderately thick, isotropic shallow shells is derived and geometrically non‐linear free vibrations of panels with rectangular planform are investigated. The geometrical non‐linearity is due to large displacements, and the effects of the rotatory inertia and transverse shear are considered. The time domain equations of motion are obtained by applying the principle of virtual work and the d'Alembert's principle. These equations are mapped to the frequency domain by the harmonic balance method, and are finally solved by a predictor–corrector method. The convergence properties of the element proposed and the influence of several parameters on the dynamic response are studied. These parameters are the shell's thickness, the width‐to‐length ratio, the curvature‐to‐width ratio and the ratio between curvature radii. The first and higher order modes are analysed. Some results are compared with results published or calculated using a commercial finite element package. It is demonstrated that with the proposed element low‐dimensional, accurate models are obtained. Copyright © 2002 John Wiley & Sons, Ltd.     

An energy‐momentum method for in‐plane geometrically exact Euler–Bernoulli beam dynamics

Nonlinear geometrically exact rod dynamics is of great interest in many areas of engineering. In recent years, the research was focused towards Timoshenko‐type rod theories where shearing is of importance. However, in many general model of mechanisms and spatial deformations, it is desirable to have a displacement‐only formulation, which brings us back to the classical Bernoulli beam. While it is well known for linear analysis, the Bernoulli beam is not as common in geometrically exact models of dynamics, especially when we want to incorporate the rotational inertia into the model. This paper is about the development of an energy‐momentum integration scheme for the geometrically exact Bernoulli‐type rod. We will show that the task is achievable and devise a general framework to do so. Copyright © 2014 John Wiley & Sons, Ltd.     

General descriptions of follower forces derived via a geometrically exact inverse contact algorithm

The main feature of the geometrically exact theory of contact is that all objects, which are necessary for computation, weak form and residual, linearized weak form, and tangent matrices are given in a covariant closed form in the local coordinate system corresponding to the geometry of contact pairs. This allows easily to construct computational algorithms for the normal and tangential follower forces as an inverse contact algorithm. In this case, following the definition of the follower forces as given and not changing in the local coordinate system, we have to modify all objects for the contact taking into account the definition of follower forces instead of constitutive relationships for the contact interfaces. The main feature is that the tangent matrices for both normal and tangential part being split into the rotational and the curvature parts are symmetric for any order of approximation. The following numerical examples are selected in the current article to illustrate the effectiveness of implementation: (1) the modeling of a pure bending with a moment applied either as a pair of single forces or a distributed follower forces (pressure) in both 2D and 3D cases; (2) modeling of inflation of a plate as application of the distributed follower normal forces (pressure); and (3) modeling of twisting of a beam with the rectangular cross‐section as application of the tangential follower forces algorithm. Copyright © 2016 John Wiley & Sons, Ltd.