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.     

A higher-order facet quadrilateral composite shell element

A C⁰ finite element formulation of flat faceted element based on a higher-order displacement model is presented for the analysis of general, thin-to-thick, fibre reinforced composite laminated plates and shells. This theory incorporates a realistic non-linear variation of displacements through the shell thickness, and eliminates the use of shear correction coefficients. The discrete element chosen is a nine-noded quadrilateral with five and nine degrees of freedom per node. A comparison of results is also made with the 2-D thin classical and 3-D exact analytical results, and finite element solutions with 9-noded first-order element. © 1997 John Wiley & Sons, Ltd.     

A new one-point quadrature,general non-linear quadrilateral shell element with physical stabilization

A new four-node shell element with a single-point quadrature to be used for explicit time integration is presented in this paper. The physical stabilization is applied, which enables explicitly evaluating the stabilizing forces on basis of the general degenerated shell formulation and which does not require any input parameters. An optimized choice of the moduli is performed in order to compute the stabilized forces for non-linear material so that the element's behaviour is improved with respect to similar physical stabilization elements. The cost efficiency of the element is demonstrated by the numerical examples in comparison with a fully integrated 4-node element. The new element is then implemented in and validated with the RADIOSS crash program. © 1998 John Wiley & Sons, Ltd.     

A geometrically non-linear tensorial formulation of a skewed quadrilateral thin shell finite element

A 48-degree-of-freedom (d.o.f.) skewed quadrilateral thin shell finite element, including the effect of geometrical non-linearity, is formulated and appropriate numerical procedures are adopted for the development of an efficient approach for the static and dynamic analysis of general thin shell structures. The element surface is described by a variable-order polynomial in curvilinear co-ordinates. The displacement functions are described by bicubic Hermitian polynomials in curvilinear co-ordinates. The directions of the curvilinear co-ordinates at each nodal point are uniquely defined to coincide with the directions of the boundaries of the element. In the present case of a skewed quadrilateral with non-orthogonal curvilinear coordinates, the coupling terms of the metric tensor and curvature tensor of the surface no longer vanish, such as in the case of orthogonal co-ordinates. The tensor form is used in the setup of the shape functions, geometric derivatives, stiffness matrix and computer code. This allows for the treatment of shells with irregular shapes and variable curvatures. To evaluate the efficiency and accuracy of this formulation, a systematic list of examples is chosen: (i) linear and non-linear static analysis of square and rhombic plates, cylindrical and spherical shells; (ii) linear vibrations of trapezoidal flat and curved plates; (iii) large amplitude vibrations of a rhombic plate. For the square plate and cylindrical and spherical shell, shewed element meshes with various distortion angles are used to study the effect of the distortion angles on the accuracy of the results and to demonstrate the versatility of the present element. All results are compared with alternative available solutions including those obtained using regular rectangular meshes. Pinched thin cylindrical and spherical shells are studied using different skewed meshes and various Gauss integration meshes, and no membrane locking phenomenon is observed.     

A 9-node co-rotational quadrilateral shell element

A new 9-node co-rotational curved quadrilateral shell element formulation is presented in this paper. Different from other existing co-rotational element formulations: (1) Additive rotational nodal variables are utilized in the present formulation, they are two well-chosen components of the mid-surface normal vector at each node, and are additive in an incremental solution procedure; (2) the internal force vector and the element tangent stiffness matrix are respectively the first derivative and the second derivative of the element strain energy with respect to the nodal variables, furthermore, all nodal variables are commutative in calculating the second derivatives, resulting in symmetric element tangent stiffness matrices in the local and global coordinate systems; (3) the element tangent stiffness matrix is updated using the total values of the nodal variables in an incremental solution procedure, making it advantageous for solving dynamic problems. Finally, several examples are solved to verify the reliability and computational efficiency of the proposed element formulation.     

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.     

A continuum-based shell theory for non-linear applications

The paper introduces a non-linear shell theory, which provides a complete three-dimensional state of stress. Since the theory is derived from simple three-dimensional continuum mechanics, it is very easy to understand. As an example, the theory is applied to quadrilateral shell elements, which provide only displacement degrees of freedom located at nodes on the outer surfaces and one degree of freedom at the middle surface. It is proposed to eliminate this degree of freedom on element level, so that the elements have the same layout as the equivalent brick elements, but have a better behaviour in bending, have stress resultants and are cheaper with respect to computational effort. The advantages with respect to implementation in a finite element program, as well as in special applications, are obvious. However, well-known conditioning problems in thin shell applications must be expected. Therefore emphasis is put on this issue in the example problems. It is shown that the elements can give acceptable answers in engineering applications and offer a potential for material non-linear applications, which will be considered in a forthcoming paper.     

An isogeometric locking‐free NURBS‐based solid‐shell element for geometrically nonlinear analysis

In this work, we develop an isogeometric non‐uniform rational B‐spline (NURBS)‐based solid‐shell element for the geometrically nonlinear static analysis of elastic shell structures. A single layer of continuous 3D elements through the thickness of the shell is considered, and the order of approximation in that direction is chosen to be equal to two. A complete 3D constitutive relation is assumed. The objective is to develop a highly accurate low‐order element for coarse meshes. We propose an extension of the mixed method of Bouclier et al. [11] to deal with locking in the context of large rotations and large displacements. The main idea is to modify the interpolation of the average through the thickness of the stress components. It is also necessary to stabilize the element in order to avoid the occurrence of spurious zero‐energy modes. This was achieved, for the quadratic version, through the adjunction of artificial elementary stabilization stiffnesses. The result is an element of order 2, which is at least as accurate as standard NURBS shell elements of order 4. Linear and nonlinear test calculations have been carried out along with comparisons with other published NURBS and classical techniques in order to assess the performance of the element. Copyright © 2014 John Wiley & Sons, Ltd.     

A triangular finite shell element based on a fully nonlinear shell formulation

This work presents a fully nonlinear six-parameter (3 displacements and 3 rotations) shell model for finite deformations together with a triangular shell finite element for the solution of the resulting static boundary value problem. Our approach defines energetically conjugated generalized cross-sectional stresses and strains, incorporating first-order shear deformations for an inextensible shell director (no thickness change). Finite rotations are treated by the Euler–Rodrigues formula in a very convenient way, and alternative parameterizations are also discussed herein. Condensation of the three-dimensional finite strain constitutive equations is performed by applying a mathematically consistent plane stress condition, which does not destroy the symmetry of the linearized weak form. The results are general and can be easily extended to inelastic shells once a stress integration scheme within a time step is at hand. A special displacement-based triangular shell element with 6 nodes is furthermore introduced. The element has a nonconforming linear rotation field and a compatible quadratic interpolation scheme for the displacements. Locking is not observed as the performance of the element is assessed by several numerical examples, which also illustrate the robustness of our formulation. We believe that the combination of reliable triangular shell elements with powerful mesh generators is an excellent tool for nonlinear finite element analysis.Fellowship funding from FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) and CNPq (Conselho Nacional de Pesquisa), together with the material support and stimulating discussions in IBNM (Institut für Baumechanik und Numerische Mechanik), are gratefully acknowledged in this work.     

非协调实体退化厚/薄板单元

在八节点三维实体等参元基础上,将板理论基本假定引入弹性应力-应变矩阵,并为局部坐标系下的挠度位移分量附加非协调位移项,构造了非协调实体退化板单元。板假定的引入用以克服单元厚度泊松自锁,非协调项的引入可缓解单元剪切自锁。通过修正常应变矩阵,确保单元通过分片检验。分域积分方法的采用,使单元可用于层合板结构分析。算例表明:单元对等厚度的厚/薄板均适用。     

An assumed strain formulation of efficient solid triangular element for general shell analysis

An efficient assumed strain triangular solid element is developed for the analysis of plate and shell structures. The finite element formulation is based on the two‐field assumed strain formulation with two independent fields of assumed displacement and assumed strain. The assumed strain field is carefully selected to alleviate the shear locking effect without triggering undesirable spurious kinematic modes. The curvilinear surface of shell structures is modelled with flat facet elements to obviate the membrane locking effect. The patch tests are successfully passed, and numerical test involving various example problems demonstrates the validity and efficiency of the present element. Copyright © 2000 John Wiley & Sons, Ltd.     

Free and forced vibration analysis of laminated composite plates and shells using a 9-node assumed strain shell element

The natural frequencies of isotropic and composite laminates are presented. The forced vibration analysis of laminated composite plates and shells subjected to arbitrary loading is investigated. In order to overcome membrane and shear locking phenomena, the assumed natural strain method is used. To develop a laminated shell element for free and forced vibration analysis, the equivalent constitutive equation that makes the computation of composite structures efficient was applied. The Mindlin-Reissner theory which allows the shear deformation and rotary inertia effect to be considered is adopted for development of nine-node assumed strain shell element. The present shell element offers significant advantages since it consistently uses the natural co-ordinate system. Results of the present theory show good agreement with the 3-D elasticity and analytical solutions. In addition the effect of damping is investigated on the forced vibration analysis of laminated composite plates and shells.     

A resultant 8-node solid-shell element for geometrically nonlinear analysis

A new resultant force formulation of 8-node solid element is presented for the linear and nonlinear analysis of thin-walled structures. The global, local and natural coordinate systems were used to accurately model the shell geometry. The assumed natural strain methods with plane stress concept were implemented to remove the various locking problems appearing in thin plates and shells. The correct warping behavior in the very thin twisted beam test was obtained by using an improved Jacobian transformation matrix. The 2 × 2 Gauss integration scheme was used for the calculation of the element stiffness matrix. From the computational viewpoint, the present solid element is very efficient for a large scale of nonlinear modeling. A lot of numerical tests were carried out for the validation of the present 8-node solid-shell element and the results are in good agreement with references.An erratum to this article can be found at     

A new nine‐node solid‐shell finite element using complete 3D constitutive laws

The solid‐shell element presented in this paper has nine nodes: eight are classically located at the apexes and are fitted with three translational DOFs whereas the ninth is sited at the center and is endowed with only one DOF; a displacement along the ‘thickness’ direction. Indeed, to be used for modeling thin structures under bending effects, this kind of finite element has a favored direction where several integration points are distributed. Besides, there is solely one ‘in‐plane’ quadrature point to avoid locking phenomena and prohibitive CPU costs for large nonlinear computations. Because a reduced integration is not enough to completely prevent transverse shear locking, a shear–strain field is assumed. Compared with the other eight‐node ‘solid‐shell' bricks, the presence of a supplementary node has a main aim: getting a linear normal strain component which, along with a full three‐dimensional constitutive strain–stress behavior, allows to achieve similar results in bending cases as those obtained with the usual plane stress state hypothesis. For that, the ninth node DOF plays the role of an extra parameter essential for a quadratic interpolation of the displacement in the thickness direction. The advantage is that this DOF has a physical meaning and, for instance, a strength equivalent to a normal pressure can be prescribed. With a suitable nodal numbering, the band width is not significantly increased and meshes can easily be generated because the extra nodes are always located at element centers. To emphasize the peculiar features of such an element, a set of examples (linear and nonlinear) is carried out. Numerous comparisons with other elements show pretty good results in bending dominating problems while adding the event of a normal stress component in sheet metal forming simulations with double side contact. Copyright © 2012 John Wiley & Sons, Ltd.     

A coupled analysis method for structures with independently modelled finite element subdomains

A new method for analysing plate and shell structures with two or more independently modelled finite element subdomains is presented, assessed, and demonstrated. This method provides a means of coupling local and global finite element models whose nodes do not coincide along their common interface. In general, the method provides a means of coupling structural components (e.g., wing and fuselage) which may have been modelled by different analysts. In both cases, the need for transition modelling, which is often tedious and complicated, is eliminated. The coupling is accomplished through an interface for which three formulations are considered and presented. These formulations are: collocation, discrete least-squares, and hybrid variational. Several benchmark problems are analysed and it is shown that the hybrid variational formulation provides the most accurate solutions.     

A fully nonlinear multi-parameter shell model with thickness variation and a triangular shell finite element

This work presents a fully nonlinear multi-parameter shell formulation together with a triangular shell finite element for the solution of static boundary value problems. Our approach accounts for thickness variation as additional nodal DOFs, using a director theory with a standard Reissner-Mindlin kinematical assumption. Finite rotations are exactly treated by the Euler-Rodrigues formula in a pure Lagrangean framework, and elastic constitutive equations are consistently derived from fully three-dimensional finite strain constitutive models. The corresponding 6-node triangular shell element is presented as a generalization of the T6-3i triangle introduced by the authors in [3].Fellowship funding from FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) and CNPq (Conselho Nacional de Pesquisa), together with the material support and stimulating discussions in IBNM (Institut für Baumechanik und Numerische Mechanik), are gratefully acknowledged in this work.Received December 2003     

Free vibration of composite and sandwich laminates with a higher-order facet shell element

A simple C⁰ isoparametric finite element formulation based on a shear deformable model of higher-order theory using a higher-order facet shell element is presented for the free vibration analysis of isotropic, orthotropic and layered anisotropic composite and sandwich laminates. This theory incorporates a realistic non-linear variation of displacements through the shell thickness, and eliminates the use of shear correction coefficients. The validity and efficiency of the present formulation is established by obtaining solutions to a wide range of problems and comparing them with the available three-dimensional closed-form and finite element solutions. In addition, other plate and shell solutions of different kind and available in the literature are also compiled and tabulated for the sake of completeness. The parametric effects of degree of orthotropy, length-to-thickness ratio, plate aspect ratio, number of layers and fibre orientation upon the frequencies and mode shapes are discussed.     

Three-dimensional finite element for analysing thin plate/shell structures

A three-dimensional (3-D) hexahedron finite element is presented for the analysis of thin plate/shell structures. The element employs an explicit algebraic definition of six uniform (continuum) strains, six rigid body modes and classical Lagrange-Germain-Kirchhoff thin plate bending modes. Nine additional stiffness factors are used to control higher-order hourglass modes. The element may be used for plate/shell analyses where the flat plate assumptions are appropriate. Also it can easily be adapted to form transition elements to lower order 2-D elements, or to higher-order 3-D continuum elements. The stiffness matrix satisfies the geometric isotropy requirement, passes the patch test, and gives essentially identical response to either applied transverse corner forces or to twisting moments applied on the corner, a requirement of Kirchhoff's corner conditions for a classical thin plate. Several examples are presented to demonstrate the performance of this finite element.     

Three-node macro triangular shell element based on the assumed natural strains

 Finite element models with simple triangular geometry facilitate pre-processing for the finite element analysis. In the present study, a robust shear deformable triangular shell element formulation is presented for general plate and shell analysis, using three-node mesh discretization. The present formulation is developed on the basis of the assumed natural strain (ANS) formulation to attenuate the shear locking effect. Furthermore, this study proposes macro triangular element scheme by dividing a three-node triangular mesh into three parts to produce three individual ANS triangular elements and then merges these by condensing out the virtual center node. The performance of the macro ANS element is highly enhanced by reducing the number of sampling locations of three sub triangular element, but still utilizes three-node mesh discretization same as does the mesh used for three-node triangular elements. The macro ANS element has invariant stiffness, possesses no commutable zero energy mechanism. Numerical tests are presented to illustrate the high performance nature of the macro ANS element in general shell analysis. In particular, the numerical study demonstrates that the macro ANS element completely removes the shear locking effect that has been detrimental in shear deformable linear triangular elements so far. Received: 30 November 2001 / Accepted: 12 July 2002 This research was supported by the Ministry of Science and Technology through National Research Laboratory Programs under contract number 00-N-NL-01-C-026. The authors also thank for the financial support from the Brain Korea 21 Project in 2001.     

A physically and geometrically nonlinear scaled‐boundary‐based finite element formulation for fracture in elastomers

This paper is devoted to the formulation of a plane scaled boundary finite element with initially constant thickness for physically and geometrically nonlinear material behavior. Special two‐dimensional element shape functions are derived by using the analytical displacement solution of the standard scaled boundary finite element method, which is originally based on linear material behavior and small strains. These 2D shape functions can be constructed for an arbitrary number of element nodes and allow to capture singularities (e.g., at a plane crack tip) analytically, without extensive mesh refinement. Mapping these proposed 2D shape functions to the 3D case, a formulation that is compatible with standard finite elements is obtained. The resulting physically and geometrically nonlinear scaled boundary finite element formulation is implemented into the framework of the finite element method for bounded plane domains with and without geometrical singularities. The numerical realization is shown in detail. To represent the physically and geometrically nonlinear material and structural behavior of elastomer specimens, the extended tube model and the Yeoh model are used. Numerical studies on the convergence behavior and comparisons with standard Q1P0 finite elements demonstrate the correct implementation and the advantages of the developed scaled boundary finite element. Copyright © 2014 John Wiley & Sons, Ltd.