Instability analysis of thin plates and arbitrary shells using a faceted shell element with loof nodes

The faceted representation is employed in the paper to derive a 24-dof triangular shell element for the instability analysis of shell structures. This element, without the deficiencies of displacement incompatibility, singularity with coplanar elements, inability to model intersections, and low-order membrane strain representation, which are normally associated with existing flat elements, has previously been found by the authors to perform well in linear static shell analyses. The total Lagrangian approach is used in the nonlinear formulation, and the results of the various numerical examples indicate that its performance is comparable to existing nonlinear shell elements. An extrapolation stiffness procedure, which will improve the convergence characteristics of the constant arc length solution algorithm used here, is also presented.     

Insight into 3-node triangular shell finite elements: the effects of element isotropy and mesh patterns

In this paper, we study the convergence characteristics of some 3-node triangular shell finite elements. We review the formulations of three different isotropic 3-node elements and one non-isotropic 3-node element. We analyze a clamped plate problem and a hyperboloid shell problem using various mesh topologies and present the convergence curves using the s-norm. Considering simple bending tests, we also study the transverse shear strain fields of the shell finite elements. The results and insight given are valuable for the proper use and the further development of triangular shell finite elements.     

A conical shell finite element

In the analysis of rocket and missiles structures one frequently encounters cylindrical and cornica' shells. A simple finite element which fits the above configuration is obviously a conical shell finite element. In this paper stiffness matrix for a conical shell finite element is derived using Novozhilov's strain-displacement relations for a conical shell. Numerical integration is carried out to ge. the stiffness matrix. The element has 28 degrees-of-freedom and is nonconforming. An eigenvalue analysis of the stiffness matrix showed that it contains all the rigid body modes (six in this case) adequately, which is one of the convergence criteria. An advantage of this element is that a cylindrical shell, an annular segment flat plate, a rectangular flat plate elements can easily be obtained as degenerate cases. The effectiveness of this element is shown through a variety of numerical examples pertaining to annular plate, cylindrical shell and conical shell problems. Comparison of the present solution is made with the existing ones wherever possible. The comparison shows that the present element is superior in some respects to the existing elements     

A simple and effective element for analysis of general shell structures

A simple flat three-node triangular shell element for linear and nonlinear analysis is presented. The element stiffness matrix with 6 degrees-of-freedom per node is obtained by superimposing its bending and membrane stiffness matrices. An updated Lagrangian formulation is used for large displacement analysis. The application of the element to the analysis of various linear and nonlinear problems is demonstrated.     

A high precision triangular laminated anisotropic cylindrical shell finite element

The stiffness matrix for a high precision triangular laminated anisotropic cylindrical shell finite element has been formulated and coded into a composite structural analysis program. The versatility of the element's formulation enables its use in the analysis of multilayered composite plate and cylindrical shell type structures taking into account actual lamination parameters. The example applications presented demonstrated that accurate predictions of stresses as well as displacements are obtained with modest number of elements.     

A discrete Kirchoff plate bending element with loof nodes

The majority of existing flat shell finite elements suffer from the deficiencies of displacement incompatibility, singularity when the elements are coplanar at a node, inability to model intersections and low-order membrane strain representation. In this paper, a plate bending element, labeled DKL (for Discrete Kirchoff element with Loof nodes), with the same nodal configuration as a triangular Semiloff plate element, but not formulated through the isoparametric concept is presented. This element when superposed with the linear strain triangle results in a faceted shell element free from the abovementioned deficiencies. Various numerical examples are tested using this plate element so as to demonstrate its reliability, accuracy and convergence characteristics.     

Hybrid strain based three node flat triangular shell elements—II. Numerical investigation of nonlinear problems

In a companion paper [M. L. Liu and C. W. S. To, Comput. Struct. 54, 1031–1056 (1995)] theories and incremental formulation of nonlinear shell structures discretized by the finite element method are discussed. The updated Lagrangian formulation and the incremental Hellinger-Reissner variational principle are adopted. The independently assumed fields employed are the incremental displacements and incremental strains. Based on the theory and incremental formulation explicit element stiffness and mass matrices of three node flat triangular shell finite elements are derived. In the present paper the derived element matrices are applied to nine examples. The latter include static and dynamic response analysis of shell structures with geometrical, material, and geometrical and material nonlinearities. The formulation adopted and element matrices derived are found to be accurate, flexible and applicable to various types of shell structures with geometrical and material nonlinearities.     

Explicit formulation of two anisotropic,triangular, thin,shallow shell elements

The derivation and assessment of two anisotropic, triangular, thin, shallow shell elements is presented. Flexure in these elements is based on two simple, effective triangular plate elements. DKT and E-1, so that the resulting stiffness matrices can be obtained explicitly. Numerical examples compare these new shell elements with the high-precision element based on the T-18 bending triangle. FORTRAN listings are provided to assist implementation of the elements.     

Buckling and vibration analysis of laminated composite plate/shell structures via a smoothed quadrilateral flat shell element with in-plane rotations

This paper presents buckling and free vibration analysis of composite plate/shell structures of various shapes, modulus ratios, span-to-thickness ratios, boundary conditions and lay-up sequences via a novel smoothed quadrilateral flat element. The element is developed by incorporating a strain smoothing technique into a flat shell approach. As a result, the evaluation of membrane, bending and geometric stiffness matrices are based on integration along the boundary of smoothing elements, which leads to accurate numerical solutions even with badly-shaped elements. Numerical examples and comparison with other existing solutions show that the present element is efficient, accurate and free of locking.     

A combined scheme of edge-based and node-based smoothed finite element methods for Reissner–Mindlin flat shells

In this paper, a combined scheme of edge-based smoothed finite element method (ES-FEM) and node-based smoothed finite element method (NS-FEM) for triangular Reissner–Mindlin flat shells is developed to improve the accuracy of numerical results. The present method, named edge/node-based S-FEM (ENS-FEM), uses a gradient smoothing technique over smoothing domains based on a combination of ES-FEM and NS-FEM. A discrete shear gap technique is incorporated into ENS-FEM to avoid shear-locking phenomenon in Reissner–Mindlin flat shell elements. For all practical purpose, we propose an average combination (aENS-FEM) of ES-FEM and NS-FEM for shell structural problems. We compare numerical results obtained using aENS-FEM with other existing methods in the literature to show the effectiveness of the present method.     

On the use of deep,shallow or flat shell finite elements for the analysis of thin shell structures

This paper is confined to the study of thin shells. The aim is to summarize the different theories used and to examine the assumptions upon which each of them is based. The intention is to show when it is more suitable to use a particular approximation and to indicate the errors it introduces. Beginning with the general deep shell theory, some simplifications are introduced to obtain the shallow shell theories. The special implications of this theory for the finite element method are also examined. Finally the particular case of flat elements is discussed.     

Medial surface generation using chordal axis transformation in shell structures

This paper describes the generation of chordal surfaces for various shell structures, such as automobile bodies, plastic injection mold components, and sheet metal parts. After a single-layered tetrahedral mesh is generated by the advancing front method, the chordal surface is generated by cutting the tetrahedral mesh. One or more shell elements are generated at each tetrahedral element, and the chordal surface is constructed with triangular or quadrilateral shell elements. This algorithm has been tested on several models with rib structures.     

Advances in the formulation of the rotation-free basic shell triangle

A family of rotation-free three node triangular shell elements is presented. The simplest element of the family is based on an assumed constant curvature field expressed in terms of the nodal deflections of a patch of four elements and a constant membrane field computed from the standard linear interpolation of the displacements within each triangle. An enhanced version of the element is obtained by using a quadratic interpolation of the geometry in terms of the six patch nodes. This allows to compute an assumed linear membrane strain field which improves the in-plane behaviour of the original element. A simple and economic version of the element using a single integration point is presented. The efficiency of the different rotation-free shell triangles is demonstrated in many examples of application including linear and non-linear analysis of shells under static and dynamic loads, the inflation and de-inflation of membranes and a sheet stamping problem.     

Study of trial functions in shell triangular finite elements of quadratic parametric representation

A study is made of trial functions in a curved triangular finite element for the solution of problems in first approximation shell theory. The element geometry is in quadratic parametric representation of the real shell surface while the displacement and stress function trial functions are in cubic parametric representation. Special attention is given to inextensional bending and homogeneous membrane states. Numerical examples are provided for specimen elements with positive, zero and negative Gaussian curvature.Vector equations are listed for the underlying shell theory together with Fortran subroutines for the element geometry, kinematics and statics.     

Towards improving the MITC6 triangular shell element

Effective triangular shell elements are of utmost interest in engineering practice, and the MITC6a element – a 6 node quadratic general shell element of the MITC family – has been shown to significantly reduce the locking phenomena arising in bending dominated behaviours. However, for some specific combinations of midsurface geometry and boundary conditions, the MITC6a element features some non-physical displacement modes with vanishing membrane strain energy. This phenomenon is thoroughly analyzed, and a remedy based on a stabilized bilinear form is proposed. Detailed numerical tests are included and the results demonstrate the good performance of the proposed method both for membrane and bending dominated problems.     

Efficient sensitivity analysis and optimization of shell structures by the ABAQUS code

In this paper, a new efficient sensitivity analysis procedure is presented for the optimization of shell structures without access to the finite element source code. It is devised as a general interface tool to extend existing finite element systems from pure structural analysis to design capability. The implementation is performed based on the ABAQUS code. Kirchhoff flat shell elements are taken into account in the study with the element thickness as design variables. To ensure the performance and the validity of the proposed procedure, satisfactory sensitivity and optimization results are illustrated for numerical examples.     

Improvements in the membrane behaviour of the three node rotation-free BST shell triangle using an assumed strain approach

In this paper an assumed strain approach is presented in order to improve the membrane behaviour of a thin shell triangular element. The so called Basic Shell Triangle (BST) has three nodes with only translational degrees of freedom and is based on a Total Lagrangian Formulation. As in the original BST element the curvatures are computed resorting to the surrounding elements (patch of four elements). Membrane strains are now also computed from the same patch of elements which leads to a non-conforming membrane behaviour. Despite this non-conformity the element passes the patch test. Large strain plasticity is considered using a logarithmic strain–stress pair. A plane stress behaviour with an additive decomposition of elastic and plastic strains is assumed. A hyperplastic law is considered for the elastic part while for the plastic part an anisotropic quadratic (Hill) yield function with non-linear isotropic hardening is adopted. The element, termed EBST, has been implemented in an explicit (hydro-)code adequate to simulate sheet-stamping processes and in an implicit static/dynamic code. Several examples are given showing the good performance of the enhanced rotation-free shell triangle.     

Finite element formulation by nested interpolations: Application to the drilling freedom problem

Inclusion of the drilling freedom, that is the in plane rigid body rotation, in membrane elements has proved a formidable problem. There have been few attempts and these have not been particularly successful. As a consequence, though rigorous treatment of this freedom in shell analysis is sometimes desirable, the governing practice at present is to associate arbitrary stiffness with φ in flat shell elements when this is necessary to avoid singularity at points where flat shell elements are coplanar. In the present paper the drilling freedom is defined in the triangular natural coordinate system and the “method of nested interpolations” is then used to derive a nine freedom membrane element with u, v, φ at each vertex. In this method boundary interpolations are used to redefine the element freedoms so that a complete interpolation within the element subdomain is able to follow and the method has already been successfully applied to a nine freedom plate bending element with good results[1].