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.     

One point quadrature shell element with through-thickness stretch

A general purpose one point quadrature shell element accounting for through-thickness deformation is developed. In the shell, a complete 3-D constitutive law is introduced, leading to a 7-parameter theory which explicitly accounts for thickness change and also for a linear variation of thickness stretch. An interpolation scheme for the shell director is developed to avoid thickness locking. The developed shell element covers flexible warping behavior by using a local nodal coordinate system at each node, which is updated with second order accuracy. A physical stabilization scheme for zero energy modes is employed based on the decomposition of the strain field into constant and linear terms with respect to the natural coordinates. The rigid body projection is applied to treat rigid body rotations effectively. Linear and nonlinear patch tests including elasto-plasticity and contact are performed and the results are compared with analytical or previously reported results. The results are also compared with those of 5-parameter shell elements, in order to show that there is no significant deterioration in accuracy, especially for thin shell applications.     

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.     

The bending strip method for isogeometric analysis of Kirchhoff–Love shell structures comprised of multiple patches

In this paper we present an isogeometric formulation for rotation-free thin shell analysis of structures comprised of multiple patches. The structural patches are C¹- or higher-order continuous in the interior, and are joined with C⁰-continuity. The Kirchhoff–Love shell theory that relies on higher-order continuity of the basis functions is employed in the patch interior as presented in Kiendl et al. [36]. For the treatment of patch boundaries, a method is developed in which strips of fictitious material with unidirectional bending stiffness and zero membrane stiffness are added at patch interfaces. The direction of bending stiffness is chosen to be transverse to the patch interface. This choice leads to an approximate satisfaction of the appropriate kinematic constraints at patch interfaces without introducing additional stiffness to the shell structure. The attractive features of the method include simplicity of implementation and direct applicability to complex, multi-patch shell structures. The good performance of the bending strip method is demonstrated on a set of benchmark examples. Application to a wind turbine rotor subjected to realistic wind loads is also shown. Extension of the bending strip approach to the coupling of solids and shells is proposed and demonstrated numerically.     

Three dimensional curved shell finite elements for heat conduction

This paper presents a finite element formulation for three dimensional curved shell heat conduction where nodal temperatures and nodal temperature gradients through the shell thickness are retained as primary variables. The three dimensional curved shell geometry is constructed using the coordinates of the nodes lying on the middle surface of the shell and the nodal point normals. The element temperature field is defined in terms of the element approximation functions, nodal temperatures and nodal temperature gradients. The weak formulation of the three dimensional Fourier heat conduction equation is constructed in the Cartesian coordinate system. The properties of the curved shell elements are then derived using the weak formulation and the element temperature approximation. The element formulation permits linear temperature distribution through the element thickness.Distributed heat flux as well as convective boundaries are permitted on all six faces of the element. The element also has internal heat generation as well as orthotropic material capability. The superiority of the formulation in terms of applications, efficiency and accuracy is demonstrated. Numerical examples are presented and comparisons are made with theoretical solutions.     

Transition finite elements with temperature and temperature gradients as primary variables for axisymmetric heat conduction

This paper presents finite element formulation for a special class of elements referred to as “transition finite elements” for axisymmetric heat conduction. The transition elements are necessary in applications requiring the use of both axisymmetric solid elements and axisymmetric shell elements. The elements permit transition from the solid portion of the structure to the shell portion of the structure. A novel feature of the formulation presented here is that nodal temperatures as well as nodal temperature gradients are retained as primary variables. The weak formulation of the Fourier heat conduction equation is constructed in the cylindrical co-ordinate system (r, z). The element geometry is defined in terms of the co-ordinates of the nodes as well as the nodal point normals for the nodes lying on the middle surface of the element. The element temperature field is approximated in terms of element approximation functions, nodal temperatures and the nodal temperature gradients. The properties of the transition elements are then derived using the weak formulation and the element temperature approximation. The formulation presented here permits linear temperature distribution through the element thickness. Convective boundaries as well as distributed heat flux is permitted on all four faces of the element. Furthermore, the element formulation also permits distributed heat flux and orthotropic material behaviour. Numerical examples are presented, first to illustrate the accuracy of the formulation and second to demonstrate its usefulness in practical applications. Numerical results are also compared with the theoretical solutions.     

A smoothed finite element method for shell analysis

A four-node quadrilateral shell element with smoothed membrane-bending based on Mindlin-Reissner theory is proposed. The element is a combination of a plate bending and membrane element. It is based on mixed interpolation where the bending and membrane stiffness matrices are calculated on the boundaries of the smoothing cells while the shear terms are approximated by independent interpolation functions in natural coordinates. The proposed element is robust, computationally inexpensive and free of locking. Since the integration is done on the element boundaries for the bending and membrane terms, the element is more accurate than the MITC4 element for distorted meshes. This will be demonstrated for several numerical examples.     

Three dimensional solid-shell transition finite elements for heat conduction

This paper presents a finite element formulation for a special class of finite elements referred to as ‘Solid-Shell Transition Finite Elements’ for three dimensional heat conduction. The solid-shell transition elements are necessary in applications requiring the use of both three dimensional solid elements and the curved shell elements. These elements permit transition from the solid portion of the structure to the shell portion of the structure. A novel feature of the formulation presented here is that nodel temperatures as well as nodal temperature gradients are retained as primary variables. The element geometry is defined in terms of coordinates of the nodes as well as the nodal point normals for the nodes lying on the middle surface of the element. The temperature field with the element is approximated in terms of element approximation functions, nodal temperatures and nodal temperature gradients. The properties of the transition element are then derived using the weak formulation (or the quadratic functional) of the Fourier heat conduction equation in the Cartesian coordinate system and the element temperature approximation. The formulation presented here permits linear temperature distribution in the element thickness direction.

Convective boundaries as well as distributed heat flux is permitted on all six faces of the elements. Furthermore, the element formulation also permits internal heat generation and orthotropic material behavior. Numerical examples are presented firstly to illustrate the accuracy of the formulation and secondly to demonstrate its usefulness in practical application. Numerical results are also compared with the theoretical solutions.     

A linear quadrilateral shell element with fast stiffness computation

A new quadrilateral shell element with 5/6 nodal degrees of freedom is presented. Assuming linear isotropic elasticity a Hellinger–Reissner functional with independent displacements, rotations and stress resultants is used. Within the mixed formulation the stress resultants are interpolated using five parameters for the membrane forces as well as for the bending moments and four parameters for the shear forces. The hybrid element stiffness matrix resulting from the stationary condition is integrated analytically. This leads to a part obtained by one point integration and a stabilization matrix. The element possesses the correct rank, is free of locking and is applicable within the whole range of thin and thick shells. The in-plane and bending patch tests are fulfilled and the computed numerical examples show that the convergence behaviour of the stress resultants is very good in comparison to comparable existing elements. The essential advantage is the fast stiffness computation due to the analytically integrated matrices.     

MITC elements for a classical shell model

The formulation of nine-node mixed-interpolated shell elements based on a classical mathematical shell theory is presented, taking into account some fundamental considerations for the finite element analysis of shells. The elements are based on the mixed interpolation of tensorial components approach (MITC), but the assumed covariant strain fields are applied only for the membrane and shear components. Two different types of elements are considered, depending on whether or not geometric approximations are included in the formulation. The performance of the proposed elements is illustrated with a well-established test problem––the Scordelis-Lo roof.     

Curved shell elements for heat conduction with p-approximation in the shell thickness direction

A finite element formulation is presented for the curved shell elements for heat conduction where the element temperature approximation in the shell thickness direction can be of an arbitrary polynomial order p. This is accomplished by introducing additional nodal variables in the element approximation corresponding to the complete Lagrange interpolating polynomials in the shell thickness direction. This family of elements has the important hierarchical property, i.e. the element properties corresponding to an approximation order p are a subset of the element properties corresponding to an approximation order p + 1. The formulation also enforces continuity or smoothness of temperature across the inter-element boundaries, i.e. C⁰ continuity is guaranteed.

The curved shell geometry is constructed using the co-ordinates of the nodes lying on the middle surface of the shell and the nodal point normals to the middle surface. The element temperature field is defined in terms of hierarchical element approximation functions, nodal temperatures and the derivatives of the nodal temperatures in the element thickness direction corresponding to the complete Lagrange interpolating polynomials. The weak formulation (or the quadratic functional) of the three-dimensional Fourier heat conduction equation is constructed in the Cartesian co-ordinate space. The element properties of the curved shell elements are then derived using the weak formulation (or the quadratic functional) and the hierarchical element approximation. The element matrices and the equivalent heat vectors (resulting from distributed heat flux, convective boundaries and internal heat generation) are all of hierarchical nature. The element formulation permits any desired order of temperature distribution through the shell thickness.

A number of numerical examples are presented to demonstrate the superiority, efficiency and accuracy of the present formulation and the results are also compared with the analytical solutions. For the first three examples, the h-approximation results are also presented for comparison purposes.     

Geometrically nonlinear formulation for the axi-symmetric transition finite elements

This paper presents a geometrically nonlinear formulation for the axi-symmetric transition finite elements using total lagrangian approach. The basic element is formulated using properties of the axi-symmetric solids and the axi-symmetric shells. A novel feature of the formulation presented here is that the restriction on the magnitude of the rotations for the shell nodes of the transition element is eliminated. This is accomplished by retaining true nonlinear functions of nodal rotations in the definition of the element displacement field. Such transition elements are essential for geometrically nonlinear applications requiring both axi-symmetric solids and the axi-symmetric shells. They ensure proper connection of the axi-symmetric solid portion of the structure to the shell like portion of the structure. It is shown that the selection of different stress and strain components at the integration points does not effect the overall linear response of the element. However, in the geometrically nonlinear formulation, it is necessary to select appropriate components of the stresses and the strains at the integration point for accurate and converging element behavior. Numerical examples are presented to demonstrate such characteristics of the transition elements.     

Finite element analysis of shell structures

Summary A survey of effective finite element formulations for the analysis of shell structures is presented. First, the basic requirements for shell elements are discussed, in which it is emphasized that generality and reliability are most important items. A general displacement-based formulation is then briefly reviewed. This formulation is not effective, but it is used as a starting point for developing a general and effective approach using the mixed interpolation of the tensorial components. The formulation of various MITC elements (that is, elements based on Mixed Interpolation of Tensorial Components) are presented. Theoretical results (applicable to plate analysis) and various numerical results of analyses of plates and shells are summarized. These illustrate some current capabilities and the potential for further finite element developments.     

Superelastic shell structures modelling: Part I: Element formulation

A C⁰ three-node shell finite element belonging to the assumed shear strain elements family is extended to account for large strains when a rotating frame formulation is adopted to describe the material behaviour. Within an incremental method associated with the Newton iterative scheme, a strain measure is defined and interpolated in an intermediate configuration assuming a linear interpolation of the incremental geometric transformation. This strain measure allows the definition, in a rotated configuration, of a constitutive incremental strain obtained from a material cumulated tensorial strain. The proposed approach is validated herein considering several elastic finite strains examples.     

Elastic torsion of thin walled bars of variable cross sections

An application of the finite element method to the theory of thin walled bars of variable cross sections has been presented in this paper. A solution of this problem is based on the linear membrane shell theory with the application of Vlasov's assumptions. A bar is divided into elements along its longitudinal axis and then, a shell mid-surface of the element is approximated by arbitrary triangular Subelements. Nodal displacements of the element are assumed to be polynomials of the third order and the equivalent stiffness matrix is obtained. Calculated nodal displacements enable an analysis of normal and shearing stresses.     

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.     

Geometrically non-linear formulation for the three dimensional solid-shell transition finite elements

This paper presents a geometrically non-linear formulation using total lagrangian approach for the solid-shell transition finite elements. Such transition finite elements are necessary in geometrically non-linear analysis of structures modelled with three dimensional solid elements and the curved shell elements. These elements are an essential connecting link between the solid elements and the shell elements. The element formulation presented here is derived using the properties of the three dimensional solid elements and the curved shell elements. No restrictions are imposed on the magnitude of the nodal rotations. Thus the element formulation is capable of handling large rotations between two successive load increments. The element properties are derived and presented in detail. Numerical examples are also presented to demonstrate their behavior, accuracy and applications in three dimensional stress analysis.

It is shown that the selection of different stress and strain components at the integration points do not effect the overall linear response of the element. However, in geometrically non-linear applications it may be necessary to select appropriate stress and the strain components at the integration points for stable and converging element behavior. Numerical examples illustrate various characteristics of the element.     

Isoparametric axisymmetric shell elements with temperature gradients for heat conduction

This paper presents a finite element formulation for axisymmetric shell heat conduction where temperature gradients through the shell thickness are retained as primary nodal variables. The element geometry is constructed using the coordinates of the nodes lying on the middle surface of the shell and the middle surface nodal point normals. The element temperature field is approximated in terms of element approximation functions, the nodal temperature, and the nodal temperature gradients. The weak formulation of the two-dimensional Fourier heat conduction equation in cylindrical coordinate system is constructed. The finite element properties of the shell element are then derived using the weak formulation and the element temperature field approximation. The formulation permits linear temperature gradients through the shell thickness. Distributed heat flux as well as convective boundaries are permitted on all four faces of the element. Furthermore, the element can also have internal heat generation as well as orthotropic material properties. The superiority of the formulation in terms of efficiency and accuracy is demonstrated. Numerical examples are presented and a comparison is made with the theoretical results.     

A large deformation,rotation-free,isogeometric shell

Conventional finite shell element formulations use rotational degrees of freedom to describe the motion of the fiber in the Reissner–Mindlin shear deformable shell theory, resulting in an element with five or six degrees of freedom per node. These additional degrees of freedom are frequently the source of convergence difficulties in implicit structural analyses, and, unless the rotational inertias are scaled, control the time step size in explicit analyses. Structural formulations that are based on only the translational degrees of freedom are therefore attractive. Although rotation-free formulations using C⁰ basis functions are possible, they are complicated in comparison to their C¹ counterparts. A C^k-continuous, k ? 1, NURBS-based isogeometric shell for large deformations formulated without rotational degrees of freedom is presented here. The effect of different choices for defining the shell normal vector is demonstrated using a simple eigenvalue problem, and a simple lifting operator is shown to provide the most accurate solution. Higher order elements are commonly regarded as inefficient for large deformation analyses, but a traditional shell benchmark problem demonstrates the contrary for isogeometric analysis. The rapid convergence of the quadratic element is demonstrated for the NUMISHEET S-rail benchmark metal stamping problem.     

A torsion bending element for thin-walled beams with open and closed cross sections

A finite element is formulated for the torsion problems of thin-walled beams. The element is based on Benscoter's beam theory, which is valid for open and also closed cross-sections. The non-polynomial interpolation presented in this paper allows the exact static solution to be obtained with only one element. Numerical results are presented for three thin-walled cantilever beams, one with a channel cross-section and the two others with rectangular cross-sections. The influence of the transverse shear strain is investigated and the different models of torsion are compared. For one example, the results obtained with one-dimensional torsion elements are compared with those obtained using shell elements.