Robust C0 high-order plate finite element for thin to very thick structures: mechanical and thermo-mechanical analysis

This paper presents a new C⁰ eight-node quadrilateral finite element (FE) for geometrically linear elastic plates. This finite element aims at modeling both thin and thick plates without any pathologies of the classical plate finite elements (shear and Poisson or thickness locking, spurious modes, etc). A C¹ FE was previously developed by the first author based on the kinematics proposed by Touratier. This new FE can be viewed as an evolution towards three directions: (1) use of only C⁰ FE approximations; (2) modeling of thick to thin structures; and (3) capability in multifield problems. The transverse normal stress is included allowing use of the three-dimensional constitutive law. The element performances are evaluated on some standard plate tests, and comparisons are given with exact three-dimensional solutions for plates under mechanical and thermal loads. Comparisons are made with other plate models using C¹ and semi-C¹ FE approximations as well as with an eight node C⁰ FE based on the Reissner–Mindlin model. All results indicate that the present element is highly insensitive to mesh distortion, has very fast convergence properties and gives accurate results for displacements and stresses. Copyright © 2012 John Wiley & Sons, Ltd.     

A new discrete Kirchhoff-Mindlin element based on Mindlin–Reissner plate theory and assumed shear strain fields—part II: An extended DKQ element for thick-plate bending analysis

This is the second part of a two-part paper on plate bending elements with shear effects included. This paper presents a new four-node, 12-d.o.f. quadrilateral plate bending element valid for the analysis of thick to thin plates. The element called DKMQ, has a proper rank (contains no spurious zero-energy modes), passes the patch test for thin and thick plates in an arbitrary mesh and is free of shear locking. Very good results have been obtained for thin and thick plates by the element. An extended DKT element for thick-plate bending analysis is evaluated in Part I.¹⁹     

应用基于解析试函数的广义协调四边形厚板元分析中厚板的自由振动

自从广义协调元提出以来,多种性能优异的板单元被构造且得到广泛应用,但是将广义协调元用于板的动力分析目前仅局限于薄板,该文将基于解析试函数的广义协调四边形厚板元ATF—PQ4b用于板的动力分析。该单元是将静力条件下中厚板的齐次微分方程的多项式解析解作为试函数,从而推导得到单元刚度矩阵和质量矩阵。笔者编写了相关的MATLA...     

Refined discrete quadrilateral degenerated shell element by using Timoshenko's beam function

A refined discrete degenerated 20‐DOF quadrilateral shell element RQS20 is proposed. The exact displacement function of the Timoshenko's beam is used as the displacement on the element boundary. The re‐constitute method for shear strain matrix is adopted. The proposed element can be used for the analysis of both moderately thick and thin plates/shells, and the convergence for the very thin case can be ensured theoretically. Numerical examples presented show that the new model indeed possesses higher accuracy in the analysis of thin and thick plates/shells, and that it can pass the patch test required for the Kirchhoff thin plate elements. Most important of all, it is free from the membrane and shear locking phenomena for extremely thin plates/shells, on the one hand, and it can also avoid the phenomenon of oscillatory solutions for thick plates/shells case on the other. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

实体退化板单元及其在板的振动分析中的应用

经典板壳单元是由板壳理论构造出来的,而经典的板壳理论是在空间弹性理论的基础上考虑板壳的基本假定得来的。在空间等参数单元的基础上,直接引入板壳的基本假定,修改空间等参数单元的弹性矩阵,从而构造出适合于厚薄板壳分析的20结点实体退化板单元,并将其应用于开口圆柱薄壳的静力分析和厚薄板的固有振动分析。数值算例表明,该单元收敛快,稳定性好,具有较高的精度。此外,该单元还可以用于曲边变厚度板、壳体及层合板的振动分析。     

A simple method for including shear deformations in thin plate elements

A method is described for including the effect of shear deformations in existing thin plate finite elements, and thereby extending their range of application to include moderately thick plates. The method does not add extra degrees of freedom to the final element, so the thick and thin plate elements can be used interchangeably, and the thick plate solution is not appreciably more expensive than the thin plate solution. It is assumed that the shear deformations are constant over the element and, to account for this, two extra internal shear strain variables are added to the element. Various methods for eliminating these internal variables are examined but it is shown to be impossible to simultaneously satisfy both the constant bending moment and constant shear patch tests, except for parallelograms. However, one method gives elements which pass the constant shear patch test and, although failing the constant bending moment patch test for arbitrary geometries, gives errors which are small enough to be neglected in most engineering applications. This method has been applied to a triangular plate element and it is shown that the results obtained with this element converge (for all practical purposes) to the correct thick plate results.     

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.     

A quadrilateral hybrid-Trefftz plate bending element for the inclusion of warping based on a three-dimensional plate formulation

A plate formulation, for the inclusion of warping and transverse shear deformations, is considered. From a complete thick and thin plate formulation, which was derived without ad hoc assumptions from the three-dimensional equations of elasticity for isotropic materials, the bending solution, involving powers of the thickness co-ordinate z, is used for constructing a quadrilateral finite plate bending element. The constructed element trial functions, for the displacements and stresses, satisfy, a priori, the three-dimensional Navier equations and equilibrium equations, respectively. For the coupling of the elements, independently assumed functions on the boundary are used. High accuracy for both displacements and stresses (including transverse shear stresses) can be achieved with rather coarse meshes for thick and thin plates.     

用杂交法改善应力解的新型复合材料层合板单元

本文给出一种基于一阶剪切变形理论(FSDT)新型的、无闭锁的位移型四边形复合材料层合板单元TMQ20的列式;并根据Hellinger-Reissner变分原理,针对位移型复合材料板单元提出了一种新型应力杂交化后处理方法来改善单元计算应力的能力,使位移型单元可以简单和正确地预测层合板的应力,特别是层间横向剪应力的解。数值算例表明,经过改善的TMQ20单元具有位移型和杂交型有限元的双重优点,它不仅自由度少,列式简单,而且对位移和应力都可以得到高精度的结果,适用于从薄到中等厚度的复合材料层合板的计算。本文所提出的杂交化后处理方法的概念适用于改善任何种类的位移型单元的应力解。     

Efficient and accurate four-node quadrilateral C0 plate bending element based on assumed strain fields

Based on the assumed element strain fields and the interrelated displacement-rotation interpolations, a four-node (12 dof) quadrilateral C⁰ finite element, designated as QCCP-2, for the analysis of thick/thin plates is developed in this paper. The four-node C⁰ plate element presented here possesses a linear bending strain field, and the element stiffness matrices are given explicitly. Therefore, the present four-node C⁰ plate element is more efficient and accurate than the existing four-node C⁰ plate elements where the constant strain stiffness matrices are obtained by numerical integration. By the use of the interrelated displacement-rotation interpolations, QCCP-2 is capable of automatically satisfying the Kirchhoff assumption for the case of thin plates. Consequently, QCCP-2 is not only free of shear locking, but also free from the numerical ill-conditioning. Furthermore, QCCP-2 passes the patch test of thin plates. The four-node quadrilateral C⁰ elements presented here can automatically reduce to the corresponding three-node triangular elements. Several numerical examples are given to demonstrate the efficiency and accuracy of the C⁰ plate bending element QCCP-2.     

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.     

厚薄通用板元在厚筏基础中的应用

Timoshenko厚梁理论提供了随梁厚变化的剪应变函数,将其应用于厚板中,得到板剪应变场,此外,假设整个板元的挠度场为不完全四次式,引入广义协调理论,建立了两个变量场的协调方程,从而构建了一个无剪切闭锁的厚薄通用板元。在此基础上,利用最小位能原理,得出考虑转角支撑作用的厚筏基础和Winkler弹性地基的共同作用方程。最后,将其首次应用于实际工程的分析中。     

A new hybrid-mixed variational approach for Reissner–Mindlin plates. The MiSP model

A valuable variational approach for plate problems based on the Reissner–Mindlin theory is presented. The new MiSP (Mixed Shear Projected) approach is based on the Hellinger–Reissner variational principle, with a particular representation of transversal shear forces and transversal shear strains. The approximations of the shear forces are derived from those of the bending moments using the corresponding equilibrium relations. The shear strains are defined in terms of the edge tangential strains that are projected on the element degrees of freedom. Two finite elements are developed on the MiSP approach basis: 3-node triangular element MiSP3 and 4-node quadrilateral element MiSP4. Both elements can be considered as the most simple among the existent mixed elements. A modified MiSP model with a derived 4-node element is also presented. Numerical experiments are presented which show that the MiSP elements do not exhibit shear locking and give excellent results for thick and thin plates. They also pass the patch test for a general triangle and quadrilateral. © 1998 John Wiley & Sons, Ltd.     

梯度复合材料热传导分析的梯度单元法

给出了一种适用于梯度复合材料热传导分析的梯度单元, 采用细观力学方法描述材料变化的热物理属性, 通过线性插值和高阶插值温度场分别给出了4节点和8节点梯度单元随空间位置变化的热传导刚度矩阵。推导了在温度梯度载荷和热流密度载荷作用下, 矩形梯度板的稳态温度场和热通量场精确解。基于该精确解对比了连续梯度模型和传统的离散梯度模型的热传导有限元计算结果, 验证了梯度单元的有效性, 并讨论了相关参数对梯度单元的影响。结果表明, 梯度单元和均匀单元得到的温度场基本一致; 当热载荷垂直于材料梯度方向时, 梯度单元能够给出更加精确的局部热通量场; 当热载荷平行于材料梯度方向时, 4节点梯度单元性能恶化, 8节点梯度单元和均匀单元的计算结果与精确解吻合很好。     

Development of eight-node quadrilateral membrane elements using the area coordinates method

 Two eight-node quadrilateral elements, namely, AQ8-I and AQ8-II, have been developed using the quadrilateral area coordinate and generalized conforming methods. Some appropriate examples were employed to evaluate the performance of the proposed elements. The numerical results show that the proposed elements are superior to the standard eight-node isoparametric element, thereafter called Q8. This is because the former does not only possess the same accuracy as the latter when regular meshes are employed for analysis, but is also very insensitive to mesh distortion, for which the Q8 element can not handle. It has also been demonstrated that the area coordinate method is an efficient tool for developing simple, effective and reliable serendipity plane membrane elements. Received 11 August 1999     

A quadrilateral hybrid stress element for Mindlin plates based on incompatible displacements

An hybrid stress element formulation based on internal, incompatible displacements is used to develop efficient Mindlin plate elements. The 4-node quadrilateral Mindlin plate element is derived from a modified energy functional. Both displacements and stresses are defined in the natural co-ordinate interpolation system. The assumed stress field is obtained by tensor transformation and so chosen as to ensure that the element is co-ordinate invariant and stable. Shear locking is avoided through an appropriate identification of the internal, incompatible displacement field. The role played by incompatible displacements in the formulation of hybrid stress elements for thin and moderately thick plates is discussed. Numerical applications are presented to illustrate the accuracy and reliability of the suggested Mindlin plate element.     

A shear-flexible triangular finite element model for laminated composite plates

Formulation and numerical evaluation of a shear-flexible triangular laminated composite plate finite element is presented in this paper. The element has three nodes at its vertices, and displacements and rotations along with their first derivatives have been chosen as nodal degrees-of-freedom. Computation of element matrices is highly simplified by employing a shape function subroutine, and an optimal numerical integration scheme has been used to improve the performance. The element has satisfactory rate of convergence and acceptable accuracy with mesh refinement for thick as well as thin plates of both homogeneous isotropic and laminated anisotropic materials. The numerical studies also suggest that reliable prediction of the behaviour of laminated composite plates necessitates the use of higher order shear-flexible finite element models, and the proposed finite element appears to have some advantages over available elements.     

The hybrid-stress model for thin plates

The assumed-stress hybrid finite element model is examined for application to the bending analysis of thin plates. A hybrid-stress functional is defined by using a Mindlin-type displacement assumption and including all components of stress. The Euler equations and matrix formulation corresponding to this functional are examined to assess the effects of plate thickness, and a rationale is presented for the selection of stress assumptions so that locking is avoided in the thin plate limit. To illustrate these concepts, a series of linear displacement quadrilateral elements are derived and tested, and the best of these elements is identified for suggested implementation in general-purpose computer programs.     

A new discrete Kirchhoff-Mindlin element based on Mindlin-Reissner plate theory and assumed shear strain fields—part I: An extended DKT element for thick-plate bending analysis

This is the first of a two-part paper on plate bending elements with shear effects included. This paper presents a new three-node, nine-d.o.f. triangular plate bending element valid for the analysis of thick to thin plates. The element, called DKMT, has a proper rank (contains no spurious zero-energy modes), passes the patch test for thin and thick plates in an arbitrary mesh and is free of shear locking. Very good results have been obtained for thin and thick plates' by the element. An extended DKQ element for thick-plate bending analysis is evaluated in Part II.²⁴