Two refined non‐conforming quadrilateral flat shell elements

Two refined quadrilateral flat shell elements named RSQ20 and RSQ24 are constructed in this paper based on the refined non‐conforming element method, and the elements can satisfy the displacement compatibility requirement at the interelement of the non‐planar elements by introducing the common displacements suggested by Chen and Cheung. A refined quadrilateral plate element RPQ4 and a plane quadrilateral isoparametric element are combined to obtain the refined quadrilateral flat shell element RSQ20, and a refined quadrilateral flat shell element RSQ24 is constructed on the basis of a RPQ4 element and a quadrilateral isoparametric element with drilling degrees of freedom. The numerical examples show that the present method can improve the accuracy of shell analysis and that the two new refined quadrilateral flat shell elements are efficient and accurate in the linear analysis of some shell structures. Copyright © 2000 John Wiley & Sons, Ltd.     

Geometrically nonlinear analysis of shells by quadrilateral flat shell element with drill,shear, and warping

In this paper, a four‐node quadrilateral flat shell element is proposed for geometrically nonlinear analysis based on updated Lagrangian formulation with the co‐rotational kinematics concept. The flat shell element combines the membrane element with drilling degrees of freedom and the plate element with shear deformation. By means of these linearized elements, a simplified nonlinear analysis procedure allowing for warping of the flat shell element and large rotation is proposed. The tangent stiffness matrix and the internal force recovery are formulated in this paper. Several classic benchmark examples are presented to validate the accuracy and efficiency of the proposed new and more proficient element for practical engineering analysis of shell structures. Copyright © 2016 John Wiley & Sons, Ltd.     

Minimal assumed moments and optimal local coordinates for plate elements based upon the complementary energy functional

In most plate/shell elements founded on the complementary energy functional, the assumed mement spaces are not minimal. The reason of introducing additional moment modes is to build elements with sufficient stiffness. A new method of using incompatible deflections to select the minimal moment modes is attempted. A square 4-node and a square semiLoof plate elements are derived. They are both stable and not overflexible. To generalize the elements for distorted geometry, three different local coordinates are employed. Unexpectedly, the skew coordinates formed by the covariant base vectors of the natural coordinates are least desirable.     

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.     

Four-node mixed Hu–Washizu shell element with drilling rotation

In this paper, enhanced four-node shell elements with six DOFs/node based on the Hu–Washizu (HW) functional are developed for Green strain. The drilling rotation is included through the drilling rotation constraint equation. The key features of the approach are as follows.

1. The shell HW functional is derived from the shell potential energy functional, which is an alternative to the derivation from the three-dimensional HW functional. This method is more versatile as it enables the derivation of the so-called partial HW functionals, with different treatment of the bending/twisting part and the transverse shear part of strain energy.
2. For the membrane part of HW shell elements, a seven-parameter stress, a nine-parameter strain and a two-parameter enhanced assumed displacement gradient enhancement are selected as optimal. The assumed representations of stress and strain are defined in skew coordinates in the natural basis at the element's center. This improves accuracy and has positive theoretical consequences.
3. The drilling rotation constraint equation is treated by the perturbed Lagrange method. The faulty term resulting from the equal-order approximations of displacements and the drilling rotation is eliminated, and one spurious mode is stabilized using the gamma method. The proposed formulation is insensitive to the element's distortions and yields a large radius of convergence in the examples involving in-plane bending.

The performance of 4 four-node shell HW elements, having different bending/twisting and transverse shear parts, is analyzed on several numerical examples. Such aspects are considered as: accuracy, radius of convergence, required number of iterations of the Newton method or the arc-length method and time of computations. The element with 29 parameters (HW29) is selected as the best performer. Copyright © 2011 John Wiley & Sons, Ltd.     

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 family of simple and robust finite elements for linear and geometrically nonlinear analysis of laminated composite plates

A family of simple, displacement-based and shear-flexible triangular and quadrilateral flat plate/shell elements for linear and geometrically nonlinear analysis of thin to moderately thick laminate composite plates are introduced and summarized in this paper.

The developed elements are based on the first-order shear deformation theory (FSDT) and von-Karman’s large deflection theory, and total Lagrangian approach is employed to formulate the element for geometrically nonlinear analysis. The deflection and rotation functions of the element boundary are obtained from Timoshenko’s laminated composite beam functions, thus convergence can be ensured theoretically for very thin laminates and shear-locking problem is avoided naturally.

The flat triangular plate/shell element is of 3-node, 18-degree-of-freedom, and the plane displacement interpolation functions of the Allman’s triangular membrane element with drilling degrees of freedom are taken as the in-plane displacements of the element. The flat quadrilateral plate/shell element is of 4-node, 24-degree-of-freedom, and the linear displacement interpolation functions of a quadrilateral plane element with drilling degrees of freedom are taken as the in-plane displacements.

The developed elements are simple in formulation, free from shear-locking, and include conventional engineering degrees of freedom. Numerical examples demonstrate that the elements are convergent, not sensitive to mesh distortion, accurate and efficient for linear and geometric nonlinear analysis of thin to moderately thick laminates.     

A 4‐node assumed strain quasi‐conforming shell element with 6 degrees of freedom

Quasi‐conforming formulations of 4‐node stress‐resultant shell elements are presented. The element formulations use interrelated displacement–rotation interpolations. The formulation also includes drilling degrees of freedom, which improves membrane behavior and allows the modeling of stiffened plates and shells. The proposed treatment for bending provides very good results in the 4‐node shell element. The stiffness matrices for the present elements are explicitly expressed and the stresses are taken accurately at the nodal points. Compared to elements using Gauss integration, where the stresses are most accurate at the integration points, the extrapolation procedure needed for post‐processing is eliminated in the present shell element. A lot of numerical tests were carried out for the validation of the present 4‐node shell element and the results are in good agreement with references. Copyright © 2003 John Wiley & Sons, Ltd.     

Approximation of general shell problems by flat plate elements

An alternative to the approximation of general thin shell problems by flat plate elements (Parts 1 and 2) is proposed: the middle surface is now approached by curved triangular facets. It consists of a P ₂-Lagrange interpolation of the geometry of the shell, while the displacement field is approximated by triangles of type (1) for the membrane components and by reduced H.C.T. triangles for the bending component. To make the implementation easier, we also consider the addition of the drilling degree of freedom as a sixth independent degree of freedom by node (Part 2). We prove the convergence of the method for arbitrary thin shells.     

广义协调平板型三角形壳元

本文构造了一种具有三个角点十八个自由度的平板三角形壳元GST18。其拉伸与弯曲部分分别由含旋转自由度的三角形膜元和薄板弯曲三角形元组成。广义协调方法的采用,使得该单元的收敛性得到保证。在结点上引入了平面内旋转自由度,从根本上克服了单元共面刚度矩阵出现奇异这一困难。对平面膜元采用了缩减积分方案,使该单元不会产生薄膜闭锁现象。数值算例表明,本文提出的GST18薄壳元是计算精度优于同类单元的可靠、实用的单元。     

基于平面弹性-板弯曲比拟理论的Wilson型平板壳单元

平面弹性—板弯曲比拟关系为充分利用现有的平面弹性单元构造新型板弯曲单元提供了有效的途径。按膜、板两部分平行列式的观点,新方法还可以利用平面弹性单元构造性能良好的新型平板壳单元。按此观点根据著名的平面弹性Wilson元(QM6)的列式方式构造出一个新的十六自由度平板壳单元。单元构造简单,无虚假自由度,数值结果表明具有很好的收敛性和精度。     

Improvements in 3-node triangular shell elements

Low-order triangular finite shell elements are computationally economical and easy to implement, but often exhibit very slow convergence. Two new membrane formulations for triangular shell elements are examined which rectify these drawbacks. The first element is based on the Marguerre shallow shell theory and a strain projection method that eliminates spurious membrane strain energy. Resulting expressions are provided in an explicit form for easy implementation of the element. The second element is based on a linear membrane field governed by normal rotations and reduced quadrature. The difficulties with shell-normal rotations are analysed and a method for omitting these rotations while preserving rigid body motion is presented and tested. Finally, a set of test problems are examined which show the importance of mesh patterns and degrees-of-freedom per node on triangular element performance.     

Shell topology optimization using the layered artificial material model

A general methodology for topology optimization using the finite element method is described for shell structures. Four‐ and nine‐node Reissner–Mindlin shell elements with drilling degrees of freedom are used for the finite element response analysis. The artificial material model is used in the topology optimization and in particular, an isotropic multi‐layer shell model is introduced to allow the formation of holes or stiffening zones. In addition, a single design variable resizing algorithm is implemented based on the existing criterion which is found to be adequate for the artificial material model. Several benchmark tests are presented to show the overall performance of the proposed methodology. The strain energy variation together with the variation of the layout of the structure is monitored. Some detailed examples are provided with comparisons of the use of the four‐ and nine‐node elements and studies of critical solution parameters. Copyright 2000 John Wiley & Sons, Ltd.     

One point quadrature shell elements: a study on convergence and patch tests

One point quadrature shell elements are being widely used in the numerical simulation of shell structures, including sheet forming, because essentially of their computational efficiency. Nowadays, the purpose of using one point quadrature shell elements is not only related to computational efficiency but also because these elements have shown to be simultaneously robust and accurate in the simulation of complex sheet metal forming processes. The main objective of this work is to study the convergence behavior of different one-point quadrature shell elements and their ability to pass the membrane and bending patch tests. For comparison purposes, two new elements include a new formulation for the membrane strain field in order to further improve the membrane behavior of the element developed in previous work of (in Cardoso et al. Comput Meth Appl Mech Eng 191:5177, 2002). The original convective membrane strains of Cardoso et al. (Comput Meth Appl Mech Eng 191:5177, 2002) (in the stabilization matrices only) are thus replaced by new membrane strains, constructed directly at the co-rotational coordinate system (located at the element’s center). It is thus proved that with this new membrane formulation the elements pass now all the patch tests but, for warped (or curved) element geometries, their accuracy is not as good as the original element of (Cardoso et al. in Comput Meth Appl Mech Eng 191:5177, 2002) based on the convective coordinate system. In the numerical results presented in this paper, comprehensive comparison and discussion of these formulations are made for well known linear benchmark examples.     

Shells of revolution with local deviations

A finite element model that is suitable for the analysis of shells of revolution with arbitrary local deviations is presented. The model employs three types of shell elements: rotational, general and transitional. The rotational shell elements, which are most efficient, are used in the region where the shell is axisymmetric. The general shell elements, which can simulate almost any shell geometry, are used in the local region of the deviation. The transitional shell elements connect these two distinctively different types of elements and make it possible to combine them in a single analysis. The form of the global stiffness matrix is somewhat unique in the new model. Non-zero terms are not confined to a narrow band along the diagonal, but occur throughout the matrix. This is due to the following: (1) two different types of nodes, ring nodes and point nodes, are combined in a single analysis; and (2) a locally non-axisymmetric geometry creates a coupling of the Fourier harmonic coefficients of the rotational elements. Yet, the matrix still contains many scattered zero terms that should be considered for numerical efficiency. In this paper an efficient solution procedure that is effective for this situation is developed. The steps include the use of a substructuring technique and separate partial harmonic analysis. A numerical example is presented and compared with existing solutions to demonstrate the capabilities and the efficiency of the new model.     

A displacement scheme with drilling degrees of freedom for plane elements

A scheme of boundary displacements with drilling degrees of freedom for plane elements is presented. The scheme is free from zero displacement modes and allows the development of hybrid finite elements with vertex and mid-side nodes, each node including a drilling degree of freedom besides the translational ones. Four quadrilateral isoparametric hybrid stress elements are implemented, and numerical results for some current test problems are given.     

The C° structural finite elements reformulated

A new formulation was recently proposed by the present author aimed at removing the shear and membrane locking mechanisms from the C° structural elements. The performance achieved was shown to be excellent, completely eliminating all locking problems. In some cases of C° plate and shell element applications; however, the proposed formulation was shown to yield flexible (softer than expected) models. Analysis of this behaviour revealed the presence of an internal moment redistribution mechanism with the classical formulation. The absence of this mechanism from the new formulation was found to be responsible for the potential introduction of softening effects in the elastic finite element models. In the present paper, the internal moment redistribution effect is examined analytically and the key component responsible for its development is isolated. The new formulation, as originally proposed for the C° structural elements, is modified so that the internal moment redistribution mechanism is retained, yet, with all locking mechanisms being rejected. The proposed formulation has been subjected recently to extensive numerical investigation with excellent results.     

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.     

A new enhanced assumed strain quadrilateral membrane element with drilling degree of freedom and modified shape functions

A new four‐node quadrilateral membrane finite element with drilling rotational degree of freedom based on the enhanced assumed strain formulation is presented. A simple formulation is achieved by five incompatible modes that are added to the Allman‐type interpolation. Furthermore, modified shape functions are used to improve the behaviour of distorted elements. Numerical results show that the proposed new element exhibits good numerical accuracy and improved performance, and in many cases, superior to existing elements. In particular, Poisson's locking in nearly incompressible elasticity fades and the element performs well when it becomes considerably distorted even when it takes almost triangular shape. Copyright © 2016 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.