On the classical shell model underlying bilinear degenerated shell finite elements: general shell geometry

We study the shell models arising in the numerical modelling of shells by geometrically incompatible finite elements. We build a connection from the so‐called bilinear degenerated 3D FEM to the classical 2D shell theory of Reissner–Naghdi type showing how nearly equivalent finite element formulations can be constructed within the classical framework. The connection found here facilitates the mathematical error analysis of the bilinear elements based on the degenerated 3D approach. In particular, the connection reveals the ‘secrets’ that relate to the treatment of locking effects within this formulation. Copyright © 2002 John Wiley & Sons, Ltd.     

A bilinear shell element based on a refined shallow shell model

A four‐node shell finite element of arbitrary quadrilateral shape is developed and applied to the solution of static and vibration problems. The element incorporates five generalized degrees of freedom per node, namely the three displacements of the curved middle surface and the two rotations of its normal vector. The stiffness properties of the element are defined using isoparametric principles in a local co‐ordinate system with axes approximately parallel to the edges of the element. The formulation is based on a modern, refined variant of the shallow shell models found from the classical books on shell theory. In addition, the bending behavior of the element is improved with numerical modifications, which include mixed interpolation of the membrane and transverse shear strains. The numerical experiments show that the element is able to compete in accuracy with the highly reputable bilinear elements of the commercial codes ABAQUS and ADINA. The new formulation even outperforms its commercial rivals in problems with strong layers such as vibration problems or problems with concentrated loads. Copyright © 2009 John Wiley & Sons, Ltd.     

The mathematical shell model underlying general shell elements

We show that although no actual mathematical shell model is explicitly used in ‘general shell element’ formulations, we can identify an implicit shell model underlying these finite element procedures. This ‘underlying model’ compares well with classical shell models since it displays the same asymptotic behaviours—when the thickness of the shell becomes very small—as, for example, the Naghdi model. Moreover, we substantiate the connection between general shell element procedures and this underlying model by mathematically proving a convergence result from the finite element solution to the solution of the model. Copyright © 2000 John Wiley & Sons, Ltd.     

A benchmark study of reduced‐strain shell finite elements: quadratic schemes

We study experimentally the accuracy and reliability of some low‐order shell finite element schemes based on modifying the standard displacement formulation by reduced‐strain expressions. We focus on quadrilateral elements with a quadratic displacement approximation. Three benchmark problems with different asymptotic behaviour in the limit of zero shell thickness is used in the experiments. Following the error analysis of a reduced‐strain scheme, we study two components of the total error, the approximation error and the consistency error. We demonstrate that the performance of the methods is both case and mesh dependent. When a bending dominated problem is solved, none of the methods studied can avoid the usual worst‐case locking effect of the approximation error on general meshes. For a membrane dominated problem the total error is typically dominated by the consistency error which often convergences slowly. Copyright © 2000 John Wiley & Sons, Ltd.     

On using degenerated solid shell elements in adaptive refinement analysis

Three different degenerated shell elements are studied in an adaptive refinement procedure for the solution of shell problems. The stress recovery procedure expressed in a convective patch co‐ordinate system is used for the construction of continuous smoothed stress fields for the a posteriori error estimation. The performance of the stress recovery procedure, the error estimator and the adaptive refinement strategy are tested by solving three benchmark shell problems. It is found that when adaptive refinement is used, the adverse effects of boundary layers and stress singularities are eliminated and all the elements tested are able to achieve their optimal convergence rates. It is also found that the accuracy of the shell elements increases with the number of polynomial terms included in the stress and strain approximations. In addition, if complete Lagrangian polynomial terms are used, the element will be less sensitive to shape distortion than the one in which only complete polynomial terms are employed. Copyright © 1999 John Wiley & Sons, Ltd.     

Bilinear finite elements for shells: Isoparametric quadrilaterals

We study the accuracy and reliability of the lowest‐order bilinear shell finite element schemes. Our approach is based mainly on a simplified shallow shell model analogous to the Reissner–Mindlin model of plate bending. The numerical models are constructed by modifying the strain expressions within the usual energy principle so that error analysis in the energy norm framework is possible. Our theoretical predictions supported by numerical experiments indicate that the performance of the low‐order methods is both mesh and case dependent. Copyright © 2007 John Wiley & Sons, Ltd.     

Anisotropic mixed finite elements for elasticity

In this paper, we present a family of mixed finite elements, which are suitable for the discretization of slim domains. The displacement space is chosen as Nédélec's space of tangential continuous elements, whereas the stress is approximated by normal–normal continuous symmetric tensor‐valued finite elements. We show stability of the system on a slim domain discretized by a tensor product mesh, where the constant of stability does not depend on the aspect ratio of the discretization. We give interpolation operators for the finite element spaces, and thereby obtain optimal order a priori error estimates for the approximate solution. All estimates are independent of the aspect ratio of the finite elements. Copyright © 2011 John Wiley & Sons, Ltd.     

Adding transverse normal stresses to layered shell finite elements for the analysis of composite structures

This work presents a formulation developed to add capabilities for representing the through thickness distribution of the transverse normal stresses, σ_z, in first and higher order shear deformable shell elements within a finite element (FE) scheme. The formulation is developed within a displacement based shear deformation shell theory. Using the differential equilibrium equations for two-dimensional elasticity and the interlayer stress and strain continuity requirements, special treatment is developed for the transverse normal stresses, which are thus represented by a continuous piecewise cubic function. The implementation of this formulation requires only C⁰ continuity of the displacement functions regardless of whether it is added to a first or a higher order shell element. This makes the transverse normal stress treatment applicable to the most popular bilinear isoparametric 4-noded quadrilateral shell elements.

To assess the performance of the present approach it is included in the formulation of a recently developed third order shear deformable shell finite element. The element is added to the element library of the general nonlinear explicit dynamic FE code DYNA3D. Some illustrative problems are solved and results are presented and compared to other theoretical and numerical results.     

On non-linear dynamics of shells: implementation of energy-momentum conserving algorithm for a finite rotation shell model

Continuum and numerical formulations for non-linear dynamics of thin shells are presented in this work. An elastodynamic shell model is developed from the three-dimensional continuum by employing standard assumptions of the first-order shear-deformation theories. Motion of the shell-director is described by a singularity-free formulation based on the rotation vector. Temporal discretization is performed by an implicit, one-step, second-order accurate, time-integration scheme. In this work, an energy and momentum conserving algorithm, which exactly preserves the fundamental constants of the shell motion and guaranties unconditional algorithmic stability, is used. It may be regarded as a modification of the standard mid-point rule. Spatial discretization is based on the four-noded isoparametric element. Particular attention is devoted to the consistent linearization of the weak form of the initial boundary value problem discretized in time and space, in order to achieve a quadratic rate of asymptotic convergence typical for the Newton–Raphson based solution procedures. An unconditionally stable time finite element formulation suitable for the long-term dynamic computations of flexible shell-like structures, which may be undergoing large displacements, large rotations and large motions is therefore obtained. A set of numerical examples is presented to illustrate the present approach and the performance of the isoparametric four-noded shell finite element in conjunction with the implicit energy and momentum conserving time-integration algorithm. © 1998 John Wiley & Sons, Ltd.     

A note on finite element implementation of sandwich shell homogenization

A finite element (FE) implementation for sandwich shell through‐thickness homogenization is presented. The homogenization is performed within the analysis constitutive procedure and is suitable for the FE analysis of sandwich shells using explicit time‐integration scheme. Copyright © 2000 John Wiley & Sons, Ltd.     

Investigation and elimination of nonlinear Poisson stiffening in 3d and solid shell finite elements

We show that most geometrically nonlinear three-dimensional shell elements and solid shell elements suffer from a previously unknown artificial stiffening effect that only appears in geometrically nonlinear problems, in particular in the presence of large bending deformations. It can be interpreted as a nonlinear variant of the well-known Poisson thickness locking effect. We explain why and under which circumstances this phenomenon appears and propose concepts to avoid it.     

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.     

EAS concept for higher‐order finite shell elements to eliminate volumetric locking

The deficiency of volumetric locking phenomena in finite elements using higher‐order shell element formulations based on Lagrangean polynomials and a linear finite shell kinematics cannot be avoided by the existent enhanced assumed strain (EAS) concept established for low‐order elements. In this paper a consistent modification of the EAS concept is proposed to extend its applicability to higher‐order shell elements. This modification, affecting the transversal normal strain for polynomial orders p>1, eliminates pathological modes caused by volumetric locking. The efficiency of the proposed extended EAS method is demonstrated by means of eigenvalue analyses and two representative numerical examples. Copyright © 2008 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.     

Testing the asymptotic behaviour of shell elements—Part I: the classical benchmark tests

The asymptotic behaviour is considered to be one of the most demanding levels of benchmark testing for shell elements. In the present paper, the asymptotic behaviour of classical benchmark problems is analytically and numerically investigated. The aim is to examine the possibility of using the classical benchmark tests for testing the asymptotic behaviour of shell elements. Appropriate analytical approaches are introduced to investigate the asymptotic behaviour of the classical benchmark problems. The reformulated four‐node shell element (RFNS) is employed in the numerical analyses. It is shown that the classical benchmark tests, in addition to testing the reliability and robustness of shell elements, also represent strong challenging tests for the asymptotic behaviour of shell elements. In the course of the numerical investigation of the asymptotic behaviour of the classical benchmark problems, the reliability and efficiency of the RFNS element already established by means of the classical configuration of the benchmark tests is re‐confirmed in all cases of the corresponding asymptotic test configurations. Copyright © 2002 John Wiley & Sons, Ltd.     

Shell finite elements with different through‐the‐thickness kinematics for the linear analysis of cylindrical multilayered structures

The present paper considers the linear static analysis of composite cylindrical structures by means of a shell finite element with variable through‐the‐thickness kinematic. The refined models used are grouped in the Unified Formulation by Carrera (CUF), and they permit to accurately describe the distribution of displacements and stresses along the thickness of the multilayered shell. The shell element has nine nodes, and the mixed interpolation of tensorial components method is employed to contrast the membrane and shear locking phenomenon. Different composite cylindrical shells are analyzed, with various laminations and thickness ratios. The governing equations are derived from the principle of virtual displacement in order to apply the finite element method. The results, obtained with different theories contained in the CUF, are compared with both the elasticity solutions given in the literature and the analytical solutions obtained using Navier's method. From the analysis, one can conclude that the shell element based on the CUF is very efficient, and its use is mandatory with respect to the classical models in the study of composite structures. Copyright © 2012 John Wiley & Sons, Ltd.     

On hybrid stress,hybrid strain and enhanced strain finite element formulations for a geometrically exact shell theory with drilling degrees of freedom

The paper is concerned with the finite element formulation of a recently proposed geometrically exact shell theory with natural inclusion of drilling degrees of freedom. Stress hybrid finite elements are contrasted by strain hybrid elements as well as enhanced strain elements. Numerical investigations and comparison is carried out for a four-node element as well as a nine-node one. As far as the four-node element is concerned it is shown that the stress hybrid element and the enhanced strain one are equivalent. The hybrid strain formulation corresponds to the hybrid stress formulation only in shear dominated problems, that is the case of the plate. © 1998 John Wiley & Sons, Ltd.