On the classical shell model underlying bilinear degenerated shell finite elements

We study the shell models arising in the numerical modelling of shells by bilinear degenerated shell finite elements. The numerical model of a cylindrical shell obtained by using flat shell elements is given an equivalent formulation based on a classical two‐dimensional shell model. We use the connection between the models to explain how a parametric error amplification difficulty or locking is avoided by some elements. Copyright © 2001 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.     

A two‐node curved axisymmetric shell element based on coupled displacement field

An efficient two‐node curved axisymmetric shell element is proposed. The element with three degrees of freedom per node accounts for the transverse shear flexibility and rotary inertia. The strain components are defined in a curvilinear co‐ordinate frame. The variation of normal displacement (w) along the meridian is represented by a cubic polynomial. The relevant constitutive relations and the differential equations of equilibrium in the meridional plane of the shell are used to derive the polynomial field for the tangential displacement (u) and section rotation (θ). This results in interdependent polynomials for the field variables w, u and θ, whose coefficients are coupled by generalized degrees of freedom and geometric and material properties of the element. These coupled polynomials lead to consistently vanishing coefficients for the membrane and transverse shear strain fields even in the limit of extreme thinness, without producing any spurious constraints. Thus the element is devoid of membrane and shear locking in thin limit of inextensible and shearless bending, respectively. Full Gaussian integration rules are employed for evaluating stiffness marix, consistent load vector and consistent mass matrix. Numerical results are presented for axisymmetric deep/shallow shells having curved/straight meridional geometries for static and free vibration analyses. The accuracy and convergence characteristics of this C⁰ element are superior to other elements of the same class. The performance of the element demonstrates its applicability over a wide range of axisymmetric shell configurations. Copyright © 1999 John Wiley & Sons, Ltd.     

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.     

Analysis of adaptive shell structures using a refined laminated model

This work presents the development of a shell conical panel finite element model, which has the possibility of having embedded piezoelectric actuators and/or sensors patches. A mixed laminated theory is used, which combines an equivalent single layer higher order shear deformation approach for the mechanical behavior with a layerwise representation in the thickness direction to describe the distribution of the electric potential in each of the piezoelectric layers of the finite element. The electrical potential function is represented through a linear variation across the thickness with two electric potential nodes for each piezoelectric layer. Based in this model an active damping scheme applied to laminated shell structures is presented and discussed.     

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.     

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.     

The embedded discontinuous Galerkin method: application to linear shell problems

A new discontinuous Galerkin method for elliptic problems which is capable of rendering the same set of unknowns in the final system of equations as for the continuous displacement‐based Galerkin method is presented. Those equations are obtained by the assembly of element matrices whose structure in particular cases is also identical to that of the continuous displacement approach. This makes the present formulation easily implementable within the existing commercial computer codes. The proposed approach is named the embedded discontinuous Galerkin method. It is applicable to any system of linear partial differential equations but it is presented here in the context of linear elasticity. An application of the method to linear shell problems is then outlined and numerical results are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

A hybrid equilibrium element for folded plate and shell structures

This paper extends hybrid equilibrium formulation concepts, previously used with success for planar problems, to the analysis of folded plates and curved shells. A 2D hybrid equilibrium flat shell quadrilateral element is formulated for linear analysis, where detailed consideration is given to the implication of slope discontinuity when the element is used for non‐planar domains. Benchmark plate bending, folded plate and curved shell problems are modelled using equilibrium and conforming elements for comparison. In models of the latter two problems, torsional moments may be released along lines of slope discontinuity, and the effects of this assumption for the folded plate are studied by analysing a third type of model composed of 3D solid brick elements. The comparisons demonstrate an excellent performance from the new hybrid equilibrium analysis method for folded plates and curved shells. Copyright © 2013 John Wiley & Sons, Ltd.     

A quasi-conforming triangular laminated composite shell element based on a refined first-order theory

A quasi-conforming triangular laminated shell element based on a refined first-order shear deformation theory is presented. The Hu-Washizu variational principle, involving strain and displacement fields as variables, with stresses being considered as Lagrange multipliers, is used to develop the laminate composite shell element. Both strains and displacements are discretized in the element, while displacements alone are discretized at the boundary. The inter-element C ¹ continuity is satisfied a posteriori in a weak form. Due to the importance of rotations and shear deformation in the geometrically non-linear analyses of shells, 7 degrees of freedom per node are chosen, viz. three displacements, two first-derivatives in the in-plane directions of the out-of-plane displacement, and two transverse shear strains at each node. To consider the effect of transverse shear deformation on the global behavior of the laminated composite shell, the Reissner-Mindlin first-order theory, with shear correction factors of Chow and Whitney, is adopted. The transverse shear stresses are obtained through the integration of the 3-D equilibrium equations; and the warping induced by transverse shear is considered in the calculation of the in-plane stresses to improve their accuracy. Numerical examples show that the element has good convergence properties and leads to highly accurate stresses.     

An efficient assumed strain element model with six DOF per node for geometrically non-linear shells

The present paper describes an assumed strain finite element model with six degrees of freedom per node designed for geometrically non-linear shell analysis. An important feature of the present paper is the discussion on the spurious kinematic modes and the assumed strain field in the geometrically non-linear setting. The kinematics of deformation is described by using vector components in contrast to the conventional formulation which requires the use of trigonometric functions of rotational angles. Accordingly, converged solutions can be obtained for load or displacement increments that are much larger than possible with the conventional formulation with rotational angles. In addition, a detailed study of the spurious kinematic modes and the choice of assumed strain field reveals that the same assumed strain field can be used for both geometrically linear and non-linear cases to alleviate element locking while maintaining kinematic stability. It is strongly recommended that the element models, described in the present paper, be used instead of the conventional shell element models that employ rotational angles.     

A rotation‐free shell triangle for the analysis of kinked and branching shells

This paper extends the capabilities of previous BST and EBST rotation‐free thin shell elements to the analysis of kinked and branching surfaces. The computation of the curvature tensor is first redefined in terms of the angle change between the normals at the adjacent elements. This allows to deal with arbitrary large angles between adjacent elements and to treat kinked surfaces. A relative stiffness between elements is introduced to consider non‐homogeneous surfaces. This idea is latter generalized to deal with branching shells. Several linear and non‐linear examples are presented showing that the formulation leads to the correct results. Copyright © 2006 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.     

A four‐noded quadrilateral element for composite laminated plates/shells using the refined zigzag theory

A new bilinear four‐noded quadrilateral element (called quadrilateral linear refined zigzag) for the analysis of composite laminated and sandwich plates/shells based on the refined zigzag theory is presented. The element has seven kinematic variables per node. Shear locking is avoided by introducing an assumed linear shear strain field. The performance of the element is studied in several examples where the reference solution is the 3D finite element analysis using 20‐noded hexahedral elements. Copyright © 2013 John Wiley & Sons, Ltd.     

Corotational non‐linear analysis of thin plates and shells using a new shell element

A new three‐node triangular shell element is developed by combining the optimal membrane element and discrete Kirchhoff triangle (DKT) plate bending element, and is modified for laminated composite plates and shells so as to include the membrane‐bending coupling effect. Using appropriate shape functions for the bending and membrane modes of the element, the ‘inconsistent’ stress stiffness matrix is formulated and the tangent stiffness matrix is determined. Non‐linear analysis of thin‐walled structures with geometric non‐linearity is conducted using the corotational method. The new element is thoroughly tested by solving few popular benchmark problems. The results of the analysis are compared with those obtained using existing membrane elements. Copyright © 2006 John Wiley & Sons, Ltd.     

A bilinear shell element with the cross-reduced integration technique

A bilinear shell element is presented. The element is based on the concepts of laminas and fibres. A cross-reduced integration technique is used to carry out the element integration. In this reduced integration technique, the Gauss points are chosen both on the reference lamina and along the central fibre. Spurious shear strains are averaged on the reference lamina to solve the shear locking problem and to eliminate any zero-energy mode associated with the rotational degrees of freedom. Due to the integration on the reference lamina, zero-energy modes associated with the translational degrees of freedom can also be removed. The behaviour of the element is examined in several numerical examples with various loading conditions and boundary conditions.