将薄板矩形单元扩展为厚薄板通用单元的一般方法

本文提出将已有的薄板矩形单元扩展为无剪切闭锁现象的厚薄板通用单元的一般方法。对于薄板极限情况,这种厚薄板通用单元自动退化为原有的薄板单元,完全无剪切闭锁现象发生。应用此法构造的两个厚薄板通用矩形单元ＡＤＳ１和ＡＤＳ２自由度少,只含有角点挠度和转角,推导简明并具有一般性,能够通过分片检验。数值算例表明这两个单元对薄板和厚板都具有较高的精度,适用于实际工程计算。     

一个有效的厚薄板通用COONS曲面矩形元

本文采用板壳有限元Coons曲面法和挠度位移与转角位移导数相关构造法构造出一个有效的厚薄板通用矩形单元,并直接给出位移场形函数;其推导简单易行,灵活多样,几何意义直观,力学概念清晰,并具有一般性。在薄板极限情况下,本文厚薄板通用单元自动退化为一个性能良好的薄板Coons曲面矩形元,不出现剪切闭锁现象。数值计算表明:本文构造的厚薄板通用单元具有较高的计算精度和良好的收敛性。     

增补挠度场和剪应变场的厚薄板三角形广义协调元

本文提出将薄板三角形单元扩展为厚薄板通用单元的一般方法。对于薄板极限情况，这种厚薄板通用单元自动退化为原有的薄板单元，完全无剪切闭锁现象发生。应用此法构造的厚薄板通用三角形单元GCLLM-T9自由度少，推导简明并具有通用性，能够通过分片检验。数值算例表明单元GCLLM-T9对薄板和厚板都具有较高的精度，适用于实际工程计算。     

引入广义泡状型位移场影响的厚板矩形单元

本文提出了在已有的无闭锁现象的广义协调厚板矩形单元的基础之上，引入广义泡状型位移场的影响，构造厚薄板通用的优质单元的一般方法．应用此法构造出新的单元．数值算例表明，在引入适当的广义泡状位移影响后，单元的计算结果显著得到改善，精度进一步提高．     

对转角场和剪应变场进行合理插值的厚薄板通用四边形单元

本文提出构造厚薄板通用四边形单元的一种合理方法：先按Timoshenko厚梁理论导出单元各边的转角和剪应变公式,然后进行合理插值,导出单元的转角场、曲率场和剪应变场。当板的厚度变小时,厚梁理论自动退化为薄梁理论,各边剪应变以及单元剪应变插值函数自动退化为零,厚板单元自动退化为薄板单元,彻底消除了剪切闭锁现象。此单元对厚板和薄板都给出了高精度的结果。     

一种新型厚薄板通用三角形广义协调元

本文提出了一种假设单元剪切应变场的新方法，基于广义协调理论构造了一个只有９个自由度的三角形厚薄板通用单元ＴＣＧＣ Ｔ９。数值算例表明：该单元具有自由度少，精度高，无剪切闭锁现象等特点，适用于从极薄板到厚板较大的范围。     

采用面积坐标的四边形厚薄板通用单元

本文采用四边形面积坐标,利用假设剪切应变场方法和广义协调理论构造出一个具有12个自由度的四边形厚薄板通用弯曲单元TACQ。基本思路如下：首先从Mindlin厚板理论出发,独立假设剪应变场和挠度场,而转角场则由挠度场和剪应变场导出;其次,单元剪应变场是先按Timoshenko厚梁理论确定单元各边剪应变,然后在单元内进行合理插值导出;第三,单元挠度场是根据单元角点处挠度的点协调条件以及单元各边挠度和法向转角的平均协调条件导出。这个方法有两个特点,（1）由于满足点协调和边协调的广义协调条件,故能保证收敛;（2）由于在薄板情况剪应变退化为零,故不出现剪切闭锁现象。数值算例表明：该单元具有精度高,收敛性和可靠性好,对网格畸变不敏感,无剪切闭锁现象等优点;适用于从极薄板到厚板较大的范围。     

扁壳广义协调曲面矩形元

本文从修正的扁壳胡海昌-鹫律原理泛函出发,引入两方面的广义协调条件(单元边界位移的积分型协调条件,膜应变与位移之间的积分型协调条件),使泛函退化为扁壳势能原理泛函,在此基础上导出一个具有二十个自由度的扁壳曲面矩形元。此单元对厚扁壳和薄扁壳都通用,不出现剪切闭锁和薄膜闭锁现象,具有良好的性能。     

厚薄板振动分析的通用矩形单元

本文将两种已有的厚薄板通用广义协调矩形单元，由静力分析推广到振动分析，求解了前几阶固有频率，数值算例表明：在一定的厚跨比范围内，无论对薄板还是中厚板，单元均有较高的精度。     

对转角场和剪应变场进行合理插值的厚板元

本文提出构造厚板三角形元的一种合理方法，先按Ｔｉｍｓｈｅｎｋｏ厚梁理论导出单元各边的转角和剪应变公式，然后进行合理插值，导出单元的转角场、曲率场和剪应查场。当板的厚度变小时，厚梁理论自动退化为薄梁理论，各边剪应变以及单元剪应变插值函数自动退化为零，厚板单元自动退化为薄板单元，彻底消除了剪切闭锁现象。此单元对厚板和薄板都给出高精度的结果。     

Triangular and rectangular plate elements based on generalized compatibility conditions

Three generalized conforming elements for thin plate bending based on generalized compatibility conditions are derived: a triangular element with 9 d.o.f. (LT element) and two rectangular elements with 12 d.o.f. (LR-1 and LR-2). Their formulation is as simple as the conventional non-conforming element and they can be implemented in routine manner. By satisfying the generalized compatibility condition under constant moments, these new elements can always pass the patch test. Numerical examples show that high accurate result is obtained with fewer degrees of freedom by using these new elements of thin plate.     

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     

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     

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.     

Non-linear analysis of thin-walled structures using plate elements

This paper presents a simple and efficient rectangular thin plate element for elastic geometric non-linear analysis of thin-walled structures. Based on the updated Lagrangian formulation and small strain hypothesis, this element can accurately predict the non-linear pre- and post-buckling load–deflection paths of a plate structure. Using symbolic manipulation, the linear and geometric stiffness matrices for the rectangular thin plate element are derived explicitly, thereby eliminating the need for numerical integration. The element contains 30 degrees of freedom (d.o.f.): 14 d.o.f. for the in-plane (membrane) action and 16 d.o.f, for the out-of-plane (bending) action. Several examples involving the large deflection behaviour of thin-walled structures are presented to demonstrate the accuracy, efficiency and versatility of the method.     

Evaluation of a new quadrilateral thin plate bending element

A review of 4-node, 12 degrees-of-freedom quadrilateral elements for thin plates is presented. A new element called DKQ is discussed. The formulation is based on a generalization of the efficient and reliable triangular element DKT presented in References 1 and 2 and on the rectangular element QC presented in Reference 3. These elements are derived using the so-called discrete Krichhoff technique. A detailed numerical evaluation of the behaviour of the DKQ element for the computation of displacements and stresses for thin plate bending problems is presented and discussed. The DKQ element appears to be a simple and reliable 12 degrees-of-freedom thin plate bending element.     

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.     

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。