THE FOUR-NODE C0 SHELL ELEMENT REFORMULATED

A reformulated four-node shell element, based on the analysis of the moment redistribution mechanism development by C⁰ plate bending and shell elements, is presented. The moment redistribution mechanism of a finite shell element model is shown to be predominantly activated by the membrane flexural action of the shell. This action is triggered through the membrane strain components which participate in the moment equilibrium equations of the finite element assembly system. An equivalent elastic foundation action, along with the activation of the in-plane twisting stiffness of the shell, may also contribute to the moment redistribution mechanism of the finite shell element model. The proposed shell element formulation aims at retaining the non-spurious contribution of the transverse shear/membrane strain energy to the flexural behaviour of the shell, through the activation of the moment redistribution mechanism. Yet, any potentially spurious, whether locking or kinematic, mechanism is rejected. In warped configurations, the element activates appropriate coupling mechanisms of the bending terms to nodal translations. The so-obtained reformulated four-node shell element exhibits an excellent behaviour without experiencing any locking phenomena or zero-energy modes, while its formulation is kept simple, based on physical considerations. The proposed formulation performs equally well in flat as well as in warped shell element applications.     

A reduced integration solid‐shell finite element based on the EAS and the ANS concept—Geometrically linear problems

In this paper a new reduced integration eight‐node solid‐shell finite element is presented. The enhanced assumed strain (EAS) concept based on the Hu–Washizu variational principle requires only one EAS degree‐of‐freedom to cure volumetric and Poisson thickness locking. One key point of the derivation is the Taylor expansion of the inverse Jacobian with respect to the element center, which closely approximates the element shape and allows us to implement the assumed natural strain (ANS) concept to eliminate the curvature thickness and the transverse shear locking. The second crucial point is a combined Taylor expansion of the compatible strain with respect to the center of the element and the normal through the element center leading to an efficient and locking‐free hourglass stabilization without rank deficiency. Hence, the element requires only a single integration point in the shell plane and at least two integration points in thickness direction. The formulation fulfills both the membrane and the bending patch test exactly, which has, to the authors' knowledge, not yet been achieved for reduced integration eight‐node solid‐shell elements in the literature. Owing to the three‐dimensional modeling of the structure, fully three‐dimensional material models can be implemented without additional assumptions. Copyright © 2009 John Wiley & Sons, Ltd.     

Refined triangular discrete Mindlin flat shell elements

In this paper flat shell elements are formed by the assemblage of discrete Mindlin plate elements RDKTM and either the constant strain membrane element CST or the Allmans membrane element with drilling degrees of freedom LST. The element RDKTM is a robust Mindlin plate element, which can perform uniformly thick and thin plate bending analysis. It also passes the patch test for thin plate bending, and its convergence for very thin plates can be ensured theoretically. The singularity of the stiffness matrix and membrane locking are studied for the present elements. Numerical examples are presented to show that the present models indeed possess properties of simple formulations, high accuracy for thin and thick shells, and it is free from shear locking for thin plate/shell analysis.     

Development of geometrically exact new shell elements based on general curvilinear co‐ordinates

In the present study first‐order shear deformable shell finite elements based on general curvilinear co‐ordinates are proposed. For the development of the present shell elements, a partial mixed variational functional with independently assumed strains is provided in order to avoid the severe locking troubles known as transverse shear and membrane lockings. Bubble functions are included in the shape function of displacement to improve the performance of the developed element. The proposed assumed strain four‐ and nine‐node elements based on the general tensor shell theory provide an efficient linkage framework for shell surface modelling and finite element analysis. In the several benchmark problems, the present shell elements with exact geometric representations demonstrate their performance compared to previously reported results. Copyright © 2002 John Wiley & Sons, Ltd.     

A four‐node,shear‐deformable shell element developed via explicit Kirchhoff constraints

An efficient, four‐node quadrilateral shell element is formulated using a linear, first‐order shear deformation theory. The bending part of the formulation is constructed from a cross‐diagonal assembly of four three‐node anisoparametric triangular plate elements, referred to as MIN3. Closed‐form constraint equations, which arise from the Kirchhoff constraints in the thin‐plate limit, are derived and used to eliminate the degrees‐of‐freedom associated with the ‘internal’ node of the cross‐diagonal assembly. The membrane displacement field employs an Allman‐type, drilling degrees‐of‐freedom formulation. The result is a displacement‐based, fully integrated, four‐node quadrilateral element, MIN4T, possessing six degrees‐of‐freedom at each node. Results for a set of validation plate problems demonstrate that the four‐node MIN4T has similar robustness and accuracy characteristics as the original cross‐diagonal assembly of MIN3 elements involving five nodes. The element performs well in both moderately thick and thin regimes, and it is free of shear locking. Shell validation results demonstrate superior performance of MIN4T over MIN3, possibly as a result of its higher‐order interpolation of the membrane displacements. It is also noted that the bending formulation of MIN4T is kinematically compatible with the existing anisoparametric elements of the same order of approximation, which include a two‐node Timoshenko beam element and a three‐node plate element, MIN3. Copyright © 2000 John Wiley & Sons, Ltd.     

拟协调九结点四边形退化壳单元

本文根据胡－鹫津变分原理和退化壳有限元概念，构造了一个拟协调（或称杂交／混合）九结点四边形退化壳单元ＣＳＨ９．该单元克服了剪切闭锁和膜闭锁现象，整体上无零能模式，通过了分片检验。数值算例表明该单元是相当准确的和有效的。     

A family of ANS four‐node exact geometry shell elements in general convected curvilinear coordinates

The non‐conventional exact geometry shell elements based on the Timoshenko–Mindlin kinematics with five displacement degrees of freedom are proposed. The term ‘exact geometry (EXG)’ reflects the fact that coefficients of the first and second fundamental forms of the reference surface and Christoffel symbols are taken exactly at every Gauss integration point. The choice of only displacements as fundamental shell unknowns gives an opportunity to derive strain–displacement relationships, which are invariant under rigid‐body shell motions in a convected curvilinear coordinate system. This paper presents a newly developed family consisting of three hybrid and one displacement‐based four‐node EXG shell elements. To avoid shear and membrane locking and have no spurious zero energy modes, the ANS concept is employed. The ANS interpolations satisfy exactly the plate compatibility equation for in‐plane strains. As a result, all EXG shell elements developed pass membrane and bending plate patch tests and exhibit a superior performance in the case of distorted coarse mesh configurations. Copyright © 2010 John Wiley & Sons, Ltd.     

An eight‐node hybrid‐stress solid‐shell element for geometric non‐linear analysis of elastic shells

This paper presents eight‐node solid‐shell elements for geometric non‐linear analysis of elastic shells. To subdue shear, trapezoidal and thickness locking, the assumed natural strain method and an ad hoc modified generalized laminate stiffness matrix are employed. A selectively reduced integrated element is formulated with its membrane and bending shear strain components taken to be constant and equal to the ones evaluated at the element centroid. With the generalized stresses arising from the modified generalized laminate stiffness matrix assumed to be independent from the ones obtained from the displacement, an extended Hellinger–Reissner functional can be derived. By choosing the assumed generalized stresses similar to the assumed stresses of a previous solid element, a hybrid‐stress solid‐shell element is formulated. Commonly employed geometric non‐linear homogeneous and laminated shell problems are attempted and our results are close to those of other state‐of‐the‐art elements. Moreover, the hybrid‐stress element converges more readily than the selectively reduced integrated element in all benchmark problems. Copyright © 2002 John Wiley & Sons, Ltd.     

Refined non‐conforming triangular elements for analysis of shell structures

Based on the refined non‐conforming element method, simple flat triangular elements with standard nodal displacement parameters are proposed for the analysis of shell structures. For ensuring the convergence of the elements a new coupled continuity condition at the inter‐element has been established in a weaker form. A common displacement for the inter‐element, an explicit expression of refined constant strain matrix, and an adjustable constant are introduced into the formulation, in which the coupled continuity requirement at the inter‐element is satisfied in the average sense. The non‐conforming displacement function of the well‐known triangular plate element BCIZ [1] and the membrane displacement of the constant strain triangular element CST [2] are employed to derive the refined flat shell elements RTS15, and the refined flat shell elements RTS18 is derived by using the element BCIZ and the Allman's triangular plane element [3] with the drilling degrees of freedom. A simple reduced higher‐order membrane strain matrix is proposed to avoid membrane locking of the element RTS18. An alternative new reduced higher‐order strain matrix method is developed to improve the accuracy of the elements RTS15 and RTS18. Numerical examples are given to show that the present methods have improved the accuracy of the shell analysis. Copyright © 1999 John Wiley & Sons, Ltd.     

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.     

A locking‐free nine‐DOF triangular plate element with incompatible approximation

In this paper, a simple locking‐free triangular plate element, labeled here as Mindlin‐type triangular plate element with nine degrees of freedom (MTP9), is presented. The element employs an incompatible approximation with nine degrees of freedom (DOFs) independent of the nodes and the shape of the triangle to define the displacements u/v/w(which is similar to a general solid element) along the x/y/z axes. It is free of shear locking, has a proper rank, and provides stable solutions for thick and thin plates. Moreover, the paper provides a new way to develop simple and efficient locking‐free thick–thin‐plate/shell elements. A variety of numerical examples demonstrate the convergence, accuracy, and robustness of the present element MTP9. Copyright © 2016 John Wiley & Sons, Ltd.     

An assumed strain triangular curved solid shell element formulation for analysis of plates and shells undergoing finite rotations

A formulation for 36‐DOF assumed strain triangular solid shell element is developed for efficient analysis of plates and shells undergoing finite rotations. Higher order deformation modes described by the bubble function displacements are added to the assumed displacement field. The assumed strain field is carefully selected to alleviate locking effect. The resulting element shows little effect of membrane locking as well as shear locking, hence, it allows modelling of curved shell structures with curved elements. The kinematics of the present formulation is purely vectorial with only three translational degrees of freedom per node. Accordingly, the present element is free of small angle assumptions, and thus it allows large load increments in the geometrically non‐linear analysis. Various numerical examples demonstrate the validity and effectiveness of the present formulation. Copyright © 2001 John Wiley & Sons, Ltd.     

Development of shear locking‐free shell elements using an enhanced assumed strain formulation

The degenerated approach for shell elements of Ahmad and co‐workers is revisited in this paper. To avoid transverse shear locking effects in four‐node bilinear elements, an alternative formulation based on the enhanced assumed strain (EAS) method of Simo and Rifai is proposed directed towards the transverse shear terms of the strain field. In the first part of the work the analysis of the null transverse shear strain subspace for the degenerated element and also for the selective reduced integration (SRI) and assumed natural strain (ANS) formulations is carried out. Locking effects are then justified by the inability of the null transverse shear strain subspace, implicitly defined by a given finite element, to properly reproduce the required displacement patterns. Illustrating the proposed approach, a remarkably simple single‐element test is described where ANS formulation fails to converge to the correct results, being characterized by the same performance as the degenerated shell element. The adequate enhancement of the null transverse shear strain subspace is provided by the EAS method, enforcing Kirchhoff hypothesis for low thickness values and leading to a framework for the development of shear‐locking‐free shell elements. Numerical linear elastic tests show improved results obtained with the proposed formulation. Copyright © 2001 John Wiley & Sons, Ltd.     

Machine locking of degenerated thin shell elements

The degenerated shell element is one of the most efficient elements for analysing shell structures. However, it is known to result in rather stiff models when used in thin element applications. The phenomena associated with this behaviour are known as locking phenomena. This paper analyses the machine locking mechanism developed in thin to very thin Lagrangian and serendipity elements. The machine related locking phenomenon is distinguished from the shear and membrane locking phenomena. A remedy for the pure machine locking problem is developed for the two elements. The proposed remedy is based on the technique of the modified transverse shear modulus. It is also extended to control shear locking. The proposed technique is shown to completely eliminate machine locking. Also, it is shown to effectively alleviate stiffening effects due to the presence of spurious shear strain.     

Analysis of a rotation‐free 4‐node shell element

In this paper, a shell element for small and large deformations is presented based on the extension of the methodology to derive triangular shell element without rotational degrees of freedom (so‐called rotation‐free). As in our original triangular S3 element, the curvatures are computed resorting to the surrounding elements. However, the extension to a quadrilateral element requires internal curvatures in order to avoid singular bending stiffness. The quadrilateral area co‐ordinates interpolation is used to establish the required expressions between the rigid‐body modes of normal nodal translations and the normal through thickness bending strains at mid‐side. In order to propose an attractive low‐cost shell element, the one‐point quadrature is achieved at the centre for the membrane strains, which are superposed to the bending strains in the centred co‐rotational local frame. The membrane hourglass control is obtained by the perturbation stabilization procedure. Free, simply supported and clamped edges are considered without introducing virtual nodes or elements. Several numerical examples with regular and irregular meshes are performed to show the convergence, accuracy and the reasonable little sensitivity to geometric distortion. Based on an updated Lagrangian formulation and Newton iterations, the large displacements of the pinched hemispherical shell show the effectiveness of the proposed simplified element (S4). Finally, the deep drawing of a square box including large plastic strains with contact and friction completes the ability of the rotation‐free quadrilateral element for sheet‐metal‐forming simulations. Copyright © 2005 John Wiley & Sons, Ltd.     

Extended rotation‐free plate and beam elements with shear deformation effects

This paper describes a methodology for extending rotation‐free plate and beam elements to accounting for transverse shear deformation effects. The ingredients for the element formulation are a Hu–Washizu‐type mixed functional, a linear interpolation for the deflection and the shear angles over standard finite elements and a finite volume approach for computing the bending moments and the curvatures over a patch of elements. As a first application of the general procedure, we present an extension of the three‐noded rotation‐free basic plate triangle (BPT) originally developed for thin plate analysis to account for shear deformation effects of relevance for thick plates and composite‐laminated plates. The nodal deflection degrees of freedom (DOFs) of the original BPT element are enhanced with the two shear deformation angles. This allows to compute the bending and shear deformation energies leading to a simple triangular plate element with three DOFs per node (termed BPT+ element). For the thin plate case, the shear angles vanish and the element reproduces the good behaviour of the original thin BPT element. As a consequence the element is applicable to thick and thin plate situations without exhibiting shear locking effects. The numerical solution for the thick case can be found iteratively starting from the deflection values for the Kirchhoff theory using the original thin BPT element. A two‐noded rotation‐free beam element termed CCB+ applicable to slender and thick beams is derived as a particular case of the plate formulation. The examples presented show the robustness and accuracy of the BPT+ and the CCB+ elements for thick and thin plate and beam problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Simple and efficient shear flexible two-node arch/beam and four-node cylindrical shell/plate finite elements

Several simple and accurate C° two-node arch/beam and four-node cylindrical shell/plate finite elements are presented in this paper. The formulation used here is based on the refined theory of thick cylindrical shells and the quasi-conforming element technique. Unlike most C° elements, the element stiffness matrix presented here is given explicitly. In spite of their simplicity, these C° finite elements posseses linear bending strains and are free from the deficiencies existing in curved C° elements such as shear and membrane locking, spurious kinematic modes and numerical ill-conditioning. These finite elements are valid not only for thick/thin beams and plates, but also for arches/straight beams and cylindrical shells/plates. Furthermore, these C° elements can automatically reduce to the corresponding C¹ beam and plate elements and give the C° beam element obtained by the reduced integration as a special case. Several numerical examples indicate that the simple two-node arch/beam and four-node cylindrical shell/plate elements given in this paper are superior to the existing C° elements with the same element degrees of freedom. Only the formulation of the rectangular cylindrical shell and plate element is presented in this paper. The formulation of an arbitrarily quadrilateral plate element will be presented in a follow-up paper³².