A 3-node flat triangular shell element with corner drilling freedoms and transverse shear correction

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2011; 86:1413–1434Published online 23 January 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.3109

A 3-node flat triangular shell element with corner drilling freedomsand transverse shear correction

Yun Zhang1, Huamin Zhou1,∗,†, Jianhui Li1, Wei Feng2 and Dequn Li1

1State Key Laboratory of Materials Processing and Mold Technology, Huazhong University of Science andTechnology, Wuhan 430074, People’s Republic of China

2Shenzhen Institutes of Advanced Technology, Chinese Academy of Science, 1068 Xueyuan Avenue, ShenzhenUniversity Town, Shenzhen 518055, People’s Republic of China

SUMMARY

The formulation, implementation and testing of simple, efficient and robust shell finite elements havechallenged investigators over the past four decades. A new 3-node flat triangular shell element is developedby combination of a membrane component and a plate bending component. The ANDES-based membranecomponent includes rotational degrees of freedom, and the refined nonconforming element method-based bending component involves a transverse shear correction. Numerical examples are carried out forbenchmark tests. The results show that compared with some popular shell elements, the present one issimple but exhibits excellent all-around properties (for both membrane- and bending-dominated situations),such as free of aspect ratio locking, passing the patch test, free of shear locking, good convergence andhigh suitability for thin to moderately thick plates. The developed element has already been adopted in awarpage simulation package for injection molding. Copyright � 2011 John Wiley & Sons, Ltd.

Received 10 April 2010; Revised 27 October 2010; Accepted 10 November 2010

KEY WORDS: flat shell element; triangle; membrane element; plate bending element

1. INTRODUCTION

Shell elements are very efficient for modeling the behavior of shell structures that are thin inone direction and long in the other two directions [1]. There are four types of general shellelements: flat shell elements, curved shell elements, axisymmetric shell elements and degeneratedsolid elements [2]. Among these elements, the flat shell element is the most popular model forfinite element analyses of shell structures, as it is simple, easy to implement and computationallyefficient. There are triangular and quadrilateral elements in common flat shell elements. One ofthe main advantages of triangular shell elements over quadrilateral shell elements is that triangularelements can be used to model an arbitrary-shaped shell, whereas quadrilateral elements can onlybe used to model cylindrical shells and shells of revolution [3]. This paper is concerned with thedevelopment of a flat triangular shell element.

Generally, flat shell elements must possess capabilities of bending and stretching deformation. Inthe case of small deformation, the stretching deformation and bending deformation of a flat elementcan be considered independent of each other and hence it is the superposition of a membraneproblem and a plate bending problem. Accordingly, one approach for developing flat shell elementsis to combine a membrane element and a plate bending element [4, 5]. The main advantage of this

∗Correspondence to: Huamin Zhou, State Key Laboratory of Materials Processing and Mold Technology, HuazhongUniversity of Science and Technology, Wuhan 430074, People’s Republic of China.

†E-mail: [email protected]

Copyright � 2011 John Wiley & Sons, Ltd.

1414 Y. ZHANG ET AL.

approach is that previously available theories and models of membrane and plate bending elementscan be used in the formulation.

The constant strain triangle (CST) element, developed by Turner, Clough and Martin [6, 7], isthe simplest membrane element. The CST element is too stiff, and the stiffness of the assembledCST elements increases as the aspect ratio of elements becomes larger. The linear strain triangle(LST) element, developed by Fraeijs de Veubeke [8], is another widely used element. Because theCST and LST elements lack drilling degrees of freedom, a rotational singularity will occur if anelement patch surrounding a node is coplanar. To overcome this problem, a small fictitious stiffnessfor rotational degree of freedom [9] or membrane elements with rotational degree of freedom canbe employed.

Although drilling freedoms in triangles attracted attention in shell analysis very early [10, 11],a practical element including vertex rotations was presented by Allman until 1984 [12]. Allman’smodel qualifies as simple for commercial use, but suffers from rank deficiency. Bergan andFelippa [13] followed up with another membrane triangle that is not in-plane bending optimal. Theelements above are both nonconforming and can pass displacement-specified patch tests. Allman[14] later proposed a more complete formulation. Since then, other elements with drilling degreesof freedom have been derived [15, 16]. However, most of these elements suffer from aspect ratiolocking problem because the response of these elements is heavily reliant on the aspect ratio[12, 14–16].

Felippa and co-workers [17–19] constructed 9-DOF (degree of freedom) membrane elementswith drilling freedoms using parameterized variational principles, in which the elements werederived within the context of the Assumed Natural DEviatoric Strain (ANDES) formulation. TheANDES combines the free formulation (FF) of Bergan and co-workers [13, 20–24] and a variantof the ANS method due to Park and Stanley [25]. Various elements can be formulated by assigningdifferent free parameters of the ANDES template. For example, Felippa [26] developed an optimalmembrane triangular element (OPT element) with drilling degrees of freedom. Its strain energy wasaccurate for any arbitrary aspect ratio. It has been shown that this element can reach to acceptableresults even in relatively coarse meshes.

More node membrane elements also attracted interest of research recently. For example, Zhangand Kim [5] developed a 6-node triangular membrane element with only two translational degreesof freedom at each node; Kim and Bathe [27] presented a 6-node triangular shell element, using themixed-interpolated-tensorial-component (MITC) approach; Cen et al. [28–30] used the quadrilat-eral area coordinate (QAC) method to developing quadrilateral finite element models; and Cardosoet al. [31, 32] studied the convergence behavior of different one-point quadrature shell elementsand their ability to pass the membrane and bending patch tests.

As for plate bending elements, there has been considerable interest in the development andmany types of triangular plate bending elements have been developed. Clough and Tocher [33]developed a triangular plate bending element by dividing the main triangle into three subtriangles.Bazeley et al. [34] developed conforming and nonconforming plate bending elements by usingshape functions based on the area coordinates. These elements were called after the authors’ initialsas BCIZ. The nonconforming element cannot pass the patch test for arbitrary mesh patterns withthe exception of some regular triangulations, while the conforming element is overstiff, since it isfully conforming.

Batoz et al. [35] developed several effective triangular plate bending elements for the analysisof plates and shells. These elements had two rotational degrees of freedom and one translationaldegree of freedom at each node for a total of 9 degrees of freedom. They presented three typesof plate bending elements: (1) the DKT element based on Discrete Kirchoff Theory assumptions,(2) the HSM element based on the Hybrid Stress Method, to overcome the problems in developmentof pure displacement-based models and (3) the SRI element based on Selective Reduced Integrationscheme that includes transverse shear deformation. By comparing the results obtained for theseelements, they found that the DKT and HSM elements are more effective than the SRI element andthe DKT element gives better results than the HSM element. Hrabok and Hrudey [36] presenteda review of the early plate bending elements as a part of the study on the effectiveness of platebending elements.

Copyright � 2011 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2011; 86:1413–1434DOI: 10.1002/nme

3-NODE FLAT TRIANGULAR SHELL ELEMENT 1415

The DKT (Discrete Kirchoff Triangular) element may possibly be the most influential bendingelement so far. A number of efficient 9-DOF triangular elements based on the discrete Kirchhoffconstraint were developed later, such as the LOOF-DKL by Meek and Tan [37], the DKTP byDhatt et al. [38] and the DKMT by Katili [39]. These elements can converge toward the discreteKirchhoff plate bending elements when the thickness of the plate is very thin. However, in thestrict sense of the word those models cannot be shown to be displacement models because theadditive shear strain does not relate to the displacements of a single element. Batoz et al. [40]reviewed the discrete Kirchhoff flat shell elements for the linear analysis of plates and shells.

The DKT element does not take account of shear deformation and, therefore, is valid only forthin plates. The displacement field of this theory also requires C1 continuity to secure interelementcompatibility. In order to overcome C1 requirement, the Reissner–Mindlin plate theory that onlyrequires C0 continuity was used in the formulation of plate bending so that both thick and thin platescan be analyzed by one element model [41–44], based on the idea of ‘decoupling’ the transverseshear energy for moderately thick plates [45]. The Reissner–Mindlin theory takes account of sheardeformation by decoupling the rotation of plate cross-section from the slope of the deformedmid-surface. The main drawback of this parametric displacement-based model is that a shearlocking phenomenon will occur when the plate becomes thinner. There are several methods toovercome this problem such as reduced or selected integration technique [46, 47], assumed shearstrain method [48–51] and nonconforming element method [52–54].

Chen and Cheung [55, 56] proposed a refined nonconforming element method (RNEM) recently.The RNEM is based on the Reissner–Mindlin plate theory so that one element model can be usedfor both thick and thin plates analyses. Based on this method, they derived quadrilateral plateelements (RDKQM) [57, 58] and triangular thin/thick plate elements (RDKTM) by introducingthe displacement function of the Timoshenko’s beam into the formulation [59, 60]. Later, morerefined elements have been developed and show high performance [61–64]. It has been seen thatthe RNEM method is a promising approach for developing bending elements.

In addition, numerous quadrilateral elements are also used for shell analyses and some freshones are introduced as follows. Pontaza and Reddy [65] presented a finite element formulation forthe bending of thin and thick plates based on least-squares variational principles. High-order nodalexpansions were used to construct the discrete finite element model based on the least-squaresformulation. Brunet and Sabourin [66] presented a 4-node, rotation-free shell element for smalland large deformations. The quadrilateral area coordinates interpolation was used to establishthe required expressions between the rigid-body modes of normal nodal translations and thenormal through thickness bending strains at mid-side. Nguyen-Thanh et al. [67] proposed a 4-nodequadrilateral shell element with smoothed membrane–bending based on the Mindlin–Reissnertheory. The element was a combination of a plate bending and membrane element. Darilmazand Kumbasar [68] presented an 8-node quadrilateral assumed-stress hybrid shell element. Theformulation was based on Hellinger–Reissner variational principle. Ozkul and Ture [69] presentedtwo plate bending elements, based on the Mindlin theory for analysis of both moderately thickand thin plates. These elements have either 4 nodes or 8 nodes with 12 and 24 DOF, respectively.

In short, various flat shell elements have been developed over the past five decades, showinggood performance for certain shell problems. However, more effective triangular shell elementsare still needed [27], and following the high-order element frenzy of the 1970–1980s, the trendtoward simplicity has been back [26]. More attempts are made to develop simple and efficientshell elements. With a view to this point, the ANDES template is very useful for the developmentof membrane elements, capable of generating infinity of element instances by assigning numericvalues to the free parameters. The obvious question is: is there a best one? A comparison of thenew element with the previous one is necessary. On the other hand, the RNEM method is a recentadvance for bending elements. The key issue is how to cater to various special requirements.Furthermore, what is about the combination effects of the new membrane and bending elements?It is known that the behavior of a shell structure is dramatically different depending on whether itis a membrane-dominated or bending-dominated structure [70]. An effective finite element modelshould perform well in both membrane- and bending-dominated situations. But few open papersgave more attention to this issue.



Combining the newly developed membrane and bending elements [26, 60], a 3-node flat trian-gular shell element is presented in this paper, with 6-DOF per node, i.e. three translational andthree rotational degrees of freedom at each node. The element is derived from a 9-DOF membraneelement and a 9-DOF plate bending element. On the whole, this new element exhibits relativesimplicity compared with more node and higher order elements, but provides excellent all-aroundproperties. The membrane component is based on the ANDES template, which employs cornerdrilling freedoms to improve the accuracy of membrane response and to overcome the aspect ratiolocking problem. By introducing the transverse shear correction, the RNEM-based plate bendingcomponent passes the patch test required for the Kirchhoff thin plates, possesses high accuracyfor thin to thick plates, is free of shear locking problem for very thin plate analysis and exhibitsgood convergence.

2. FORMULATION

Considering a flat triangular element with its corners defined by {xi , yi }, i =1,2,3, the coordinatedifferences are

xij = xi −x j , yij = yi − y j (1)

The thickness, area and volume of the element are represented by h, A and V , respectively; the

length of the side i − j is lij =√

x2ij + y2

ij; and �1, �2 and �3 are the area coordinates.

As mentioned earlier, for fully symmetric wall fabrications (e.g. the wall material is homogeneousthrough the thickness), the shell element can be separated into a membrane element and a platebending element, as shown in Figure 1, and correspondingly the element stiffness matrix can alsobe divided into a stretching-stiffness matrix and a bending-stiffness matrix.

For the membrane component, the element equation is written as

K emqe

m =rem (2)

where K em , qe

m and rem are the stiffness matrix, nodal displacements and loads of the membrane

element, respectively.While the element equation of the plate bending component is

K epqe

p =rep (3)

where K ep, qe

p and rep are the stiffness matrix, nodal displacements and loads of the bending

element, respectively.By assembly of the above two equations, the local force–displacement relationship of the shell

element is given by

K eqe =re (4)

Figure 1. The element composition: (a) the flat shell element; (b) the membraneelement; and (c) the plate bending element.



where K e is the integrated stiffness matrix, whose sub-matrix is

K ers =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

km11 km

12 0 0 0 km13

km21 km

22 0 0 0 km23

0 0 k p11 k p

12 k p13 0

0 0 k p21 k p

22 k p23 0

0 0 k p31 k p

32 k p33 0

km31 km

31 0 0 0 km33

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(5)

where kmij and k p

ij are the corresponding elements of K em and K e

p, respectively.

2.1. The membrane component

The degrees of freedom of the membrane component are

{qm}T ={u1 v1 �z1 u2 v2 �z2 u3 v3 �z3} (6)

According to the two-stage fabrication approach [26], the element stiffness is the sum of thebasic stiffness Kb (taking care of consistency) and the higher order stiffness Kh (taking care ofstability and accuracy), as

Km = Kb +Kh (7)

Patch test and template theory require that Kb must share a common form so that elements canpass the individual element test (IET) [71, 72]. From the LST-3/9R configuration, its explicit formcan be written as [13]

Kb = 1

VLDLT (8)

where D is the elasticity matrix, and L is defined as

L = h

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

y23 0 x23

0 x32 y23

1

6�b y23(y13 − y21)

1

6�bx32(x31 −x12)

1

3�b(x31 y13 −x12 y21)

y31 0 x13

0 x13 y31

1

6�b y31(y21 − y32)

1

6�bx13(x12 −x23)

1

3�b(x12 y21 −x23 y32)

y12 0 x21

0 x21 y12

1

6�b y12(y32 − y13)

1

6�bx21(x23 −x31)

1

3�b(x23 y32 −x31 y13)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(9)

where �b is a free parameter. If �b =0, Kb reduces to the stiffness of the CST element. In thiscase, the rows and columns associated with the drilling rotations vanish.

The higher order stiffness is

Kh =∫

�BT DBd� (10)



It comes from the contribution of the hierarchical corner rotations �zi, obtained by subtractingthe average rotation �0 from the nodal drilling rotations �zi, i.e.

�zi =�zi −�0, i =1,2,3, �= [�z1 �z2 �z3]T = T� plp (11)

The corresponding Cartesian strains can then be defined by

ε=Te Q(�i )T� plp (12)

where Te relates the Cartesian strains to the natural strains.The matrix B in the high-order stiffness reads

B =Te(Q1�1 + Q2�2 + Q3�3)T�u (13)

where Te, T�u and Q1−3 are the constant matrices over the element and are expressed as

Te = 1

4A2

⎡⎢⎢⎣

y23 y13l221 y31 y21l2

32 y21 y32l213

x23x13l221 x31x21l2

32 x12x32l213

(y23x31 +x32 y13)l221 (y31x12 +x13 y21)l2

32 (y12x23 +x21 y32)l213

⎤⎥⎥⎦ (14)

T�u = 1

4A

⎡⎢⎣

x32 y32 4A x13 y13 0 x21 y21 0

x32 y32 0 x13 y13 4A x21 y21 0

x32 y32 0 x13 y13 0 x21 y21 4A

⎤⎥⎦ (15)

Q1 = 2A

3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

�1

l221

�2

l221

�3

l221

�4

l232

�5

l232

�6

l232

�7

l213

�8

l213

�9

l213

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, Q2 = 2A

3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

�9

l221

�7

l221

�8

l221

�3

l232

�1

l232

�2

l232

�6

l213

�4

l213

�5

l213

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

Q3 = 2A

3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

�5

l221

�6

l221

�4

l221

�8

l232

�9

l232

�7

l232

�2

l213

�3

l213

�1

l213

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)

The final form of the higher order stiffness can be obtained by using Gauss integration, as

Km = 1

VLDLT + 3

4�0T T

�u K�T�u (17)

where

K� =h(QT4 Enat Q4 + QT

5 Enat Q5 + QT6 Enat Q6) (18)

and

Enat =T Te DTe, Q4 = 1

2 (Q1 + Q2), Q5 = 12 (Q2 + Q3), Q6 = 1

2 (Q3 + Q1) (19)

where �0 and �1 −�9 are an overall scaling coefficient and nine free dimensionless parameters,respectively.



Based on the above formulation, various membrane elements can be derived by specifyingvalues to the free parameters. In order to find the optimal one, a higher order patch test that tunesup the higher order stiffness of triangular elements is employed to determine these parameters. Foran isotropic material, an optimal membrane element can be generated by assigning the followingvalues to the free parameters, as:

�b = 1.5, �0 =0.5(1−4�2), �1,3,5 =1

�2 = 2, �4 =0, �6,7,8 =−1, �9 =−2(20)

where � is Poisson’s ratio.

2.2. The plate bending component

The degrees of freedom of the plate bending component are collected in the nodal vector as

{qp}T ={w1 �x1 �y1 w2 �x2 �y2 w3 �x3 �y3} (21)

Taking the transverse shear correction into account, its stiffness matrix includes a bending part(Kd ) and a shear strain part (Ks), written as

K p = Kd +Ks (22)

2.2.1. The bending part. The rotations �x and �y are used to describe bending strains of theelement. As the procedure of the DKT element, the explicit expression of rotations can be obtainedby eliminating the surplus parameters at the mid-side nodes, as{

�x

�y

}= Nqp (23)

where N is defined as N = [N1 N2 N3] and

Ni =⎡⎣ Pi Pxi Pyi

Qi Qxi Qyi

⎤⎦ (i =1,2,3) (24)

for shape function N1

P1 = 1.5(�12m12 N4/ l12 −�31m31 N6/ l31)

Px1 = −0.75(�12m212 N4 +�31m2

31 N6)+ N1

Py1 = 0.75(�12c12m12 N4 +�31c31m31 N6)

Q1 = 1.5(−�12c12 N4/ l12 +�31c31 N6/ l31)

Qx1 = 0.75(�12c12m12 N4 +�31c31m31 N6)

Qy1 = −0.75(�12c212 N4 +�31c2

31 N6)+ N1

(25)

where �ij =1/(1+12ij), ij =h2/[5l2ij(1−�)]; mij and cij are the direction cosine and sine of the

side i − j , respectively; Ni is the shape function of the 6-node triangular element, as

Ni = (2�i −1)�i (i =1,2,3)

Nk = 4�i� j (k =4,5,6; i j =12,23,31)(26)

Other two shape functions (N2, N3) are obtained by cyclic expressions.It is obvious that the element displacements in Equation (25) will become the displacements of

the DKT element, when h/ lij →0 resulting in ij →0.



By means of differential, the bending strain of the element (ε) can be written as

ε=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

��x

�x

��y

�y

��x

�y+ ��y

�x

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(27)

Substitution of Equation (23) into Equation (27) results in

ε=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

��x

�x

��y

�y

��x

�y+ ��y

�x

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

�Np

�x

�NQ

�y

�Np

�y+ �NQ

�x

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

qp = Bbqp (28)

where∼N =

[NpNQ

].

With the above expression of the bending strain, the bending part of the element stiffness matrixcan be expressed as

Kd =∫

�BT

b Db Bbd� (29)

where Db is the flexural rigidity of the plate.

2.2.2. The shear strain part. The shear strain of the element can be written as

=[

x

y

]=

⎡⎢⎢⎢⎣

�x − �w

�x

�y − �w

�y

⎤⎥⎥⎥⎦ (30)

⎡⎣x

y

⎤⎦ =

[�1 0 �2 0 �3 0

0 �1 0 �2 0 �3

]⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

x1

y1

x2

y2

x3

y3

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(31)

Timoshenko’s beam function is applied in the formulation so as to remove the shear locking.For a strip plate with length L and thickness t , the displacement w and the rotation � at the endof the strip plate (Timoshenko’s beam) can be expressed as

w = (�i +�e�i� j (�i −� j ))wi +(�i� j +�e�i� j (�i −� j ))L/2�i +(� j +�e�i� j (� j −�i ))w j

+(−�i� j +�e�i� j (�i −� j ))L/2�i (32)

� = −(6�i� j/L)�ewi +�i (1−3�e� j )�i +(6�i� j/L)�ew j +� j (1−3�e�i )� j (33)

where �i =1−x/L , � j = x/L , �e =1/(1+12e), e = t2/(5(1−�)L2).



The above functions are used to define the rotation and deflection on the boundaries of theelement. For instance, on the sides 1–2, they read

�s = −(6�1�2/ l12)�12w1 +�1(1−3�12�2)�S1 +(6�1�2/ l12)�12w2 +�2(1−3�12�1)�S2 (34)

ws = (�1 +�12�1�2(�1 −�2))w1 +(�1�2 +�12�1�2(�1 −�2))l12/2�S1 +(�2 +�12�1�2(�2 −�1))w2

+(−�1�2 +�12�1�2(�1 −�2))l12/2�S2 (35)

where �Si is the tangential slope at the node i .And it follows that

�ws

�l= 1

l12(−1+�12(1−6�1�2))w1 +0.5(�1 −�2 +�12(1−6�1�2))�S1 + 1

l12(1−�12(1−6�1�2))w2

+0.5(−�1 +�2 +�12(1−6�1�2))�S2 (36)

The displacement �s −(�ws/�l) shown in Equations (34) and (36) will stay constant on the side,due to

�s − �ws

�l= (1−�12)w1/ l12 +0.5(1−�12)�S1 −(1−�12)w2/ l12 +0.5(1−�12)�S2 (37)

As a result, the shear strains at node i can be expressed by the constant shear strains S j. For

node 1, there exists [S4

S6

]=

[−m12 c12

−m31 c31

][x1

y1

](38)

and [x1

y1

]= 1

−c31m12 +c12m31

[c31 −c12

m31 m12

][S4

S6

](39)

Similarly, other nodal shear strains xj and yj ( j =2,3) can be obtained by cyclic expressions.Finally the shear strain can be written as

⎡⎣x

y

⎤⎦= Ns

⎡⎢⎢⎣

S4

S5

S6

⎤⎥⎥⎦ (40)

where

Ns =[

l3�1 − l2�2 l1�2 − l3�3 l1�1 − l2�3

m3�1 −m2�2 m1�2 −m3�3 m1�1 −m2�3

](41)

where

ci = ci/A1, ci =ci/A2, ci =ci/A3 (42a)

mi = mi/A1, mi =mi/A2, mi =mi/A3 (42b)

A1 = −c31m12 +c12m31, A2 =−c12m23 +c23m12, A3 =−c23m31 +c31m23 (42c)



And Sk (k =4,5,6) are the natural shear strains at mid-side nodes 4, 5, 6 of the element. Thismeans that the shear strains can be expressed as follows:

⎡⎢⎣

s4

s5

s6

⎤⎥⎦=

⎡⎢⎢⎢⎢⎢⎢⎢⎣

�s4 − �ws4

�l

�s5 − �ws5

�l

�s6 − �ws6

�l

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(43)

Substituting �i =�i =0.5 into Equations (34) and (36), the �s4 and �ws4/�l at the mid-side node4 on the sides 1–2 can be obtained as

�s4 = −1.5

l12�12w1 +0.5(1−1.5�12)�s1 + 1.5

l12�12w2 +0.5(1−1.5�12)�s2 (44)

�ws4

�l= − 1

l12(1+0.5�12)w1 −0.25�12�s1 + 1

l12(1+0.5�12)w2 −0.25�12�s2 (45)

where the �sj at the node j( j =1,2) on the sides 1–2 can be expressed as

�S j = [−m1 c1]

⎡⎣�x j

�y j

⎤⎦ ( j =1,2) (46)

�S4 −�wS4/�l can be expressed by

�S4 − �wS4

�l=T4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

w1

�x1

�y1

w2

�x2

�y2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(47)

where T4 = [T�4 −Tw4] and

T�4 = [−1.5�12/ l12 −0.5m12(1−1.5�12) 0.5c12(1−1.5�12) 1.5�12/ l12

−0.5m12(1−1.5�12) 0.5c12(1−1.5�12)] (48)

Tw4 = [−(1+0.5�12)/ l12 0.25m12�12 −0.25c12�12(1+0.5�12)/ l12 0.25m12�12

−0.25c12�12] (49)

Finally, the mid-side node parameters (�S4 −�wS4/�l) of the element can be expressed as

�S4 −�wS4/�l = BS1qp (50)

where

BS1 = [(1−�12)/ l12 −0.5m12(1−�12) 0.5c12(1−�12)

−(1−�12)/ l12 −0.5m12(1−�12) 0.5c12(1−�12) 0 0 0] (51)



Other mid-node parameters (�S j −�wS j /�l( j =5,6)) and BS j ( j =2,3) can be obtained similarly.Therefore, the shear strains of the element can be written as

=

⎡⎢⎢⎢⎢⎣

�x − �w

�x

�y − �w

�y

⎤⎥⎥⎥⎥⎦= Ns

⎡⎢⎢⎢⎢⎢⎢⎢⎣

�S4 − �wS4

�l

�S5 − �wS5

�l

�S6 − �wS6

�l

⎤⎥⎥⎥⎥⎥⎥⎥⎦

= Ns Bsqp = Bsqp (52)

where

BS =

⎡⎢⎣

BS1

BS2

BS3

⎤⎥⎦ (53)

With the above expression of the shear strains, the shear part of the element stiffness matrix reads

Ks =∫

�BT

s Ds Bsd� (54)

where Ds is the elasticity matrix, and Ds = Ds

[10

01

], Ds =5Eh/12(1+�).

3. NUMERICAL EXAMPLES

3.1. Test design

The proposed element in this paper is an 18-DOF flat triangular shell element that is derived froma membrane element and a plate bending element. In order to analyze and assess the performanceof the proposed element, several existing flat triangular shell elements are selected for comparison,as listed in Table I.

Eight numerical examples are presented for testing. These cases are picked out carefully sothat various properties of elements can be covered in the assessment, as listed in Table II. Thetest starts from two membrane-dominated cases, followed by four bending-dominated cases. Andthen two hybrid cases are employed in which both the membrane and bending responses areimportant. The first seven cases are standard benchmark tests because they have exact solutionsfor the convenience of comparison, while the final one is a practical part in industry for testingwhether the present element works well for complicated structures.

3.2. Results

3.2.1. A cantilever under end moment. This is a standard benchmark that focuses on the effectof aspect ratios. As shown in Figure 2, a slender cantilever beam (length=32, height=2) issubjected to an end moment M =100, identifying that the structure falls into the membrane-dominated category. The modulus of elasticity is set to E =768 so that the exact tip deflection

Table I. The existing flat triangular shell element selected for comparison.

No. Name Degrees of freedom Membrane component Plate bending component

1 CST-BCIZ 15 CST BCIZ2 CST-DKT 15 CST DKT3 LST-BCIZ 18 LST∗ BCIZ4 LST-DKT 18 LST∗ DKT

∗Allman’s model with drilling degrees of freedom.



Table II. The numerical examples used for testing.

No. Case Situation Test purpose

1 A cantilever Membrane-dominated Effect of aspect ratios2 A skew plate Membrane-dominated Convergence3 A square with arbitrary mesh pattern Bending-dominated Bending patch test4 A whole clamped plate Bending-dominated Effect of thickness5 A cylindrical shell Bending-dominated Effect of thickness6 An opened spherical dome Bending-dominated Convergence7 A twisted beam Hybrid Hybrid situations8 An injection molded wind board Hybrid Complicated parts and hybrid

situations

Figure 2. A slender cantilever beam under end moment.

Table III. Tip deflections for the cantilever under end moment.

Mesh: x-subdivisions × y-subdivisions

32×2 16×2 8×2 4×2 2×2Element (�=1) (�=2) (�=4) (�=8) (�=16)

CST-BCIZ 61.03 31.29 12.28 4.07 1.07CST-DKT 61.03 31.29 12.28 4.07 1.07LST-BCIZ 88.35 59.67 21.92 4.58 0.21LST-DKT 88.35 59.67 21.92 4.58 0.21Present 99.99 100.05 99.95 99.92 100.12

Exact 100

�= M L/(2E I )=100. Regular meshes ranging from 32×2 to 2×2 are used. The element aspectratios �(�=a/b) vary from 1:1 to 16:1.

Table III reports computed tip deflections (y deflection at point C) for five aspect ratios. Theresults show that the response is totally dependent on the membrane component used, and thatelements with CST or LST component rapidly become overstiff with the aspect ratio increasing.The present element maintains good and perfect accuracy for all aspect ratios (even for the aspectratio as large as 16), i.e. it is free of aspect ratio locking. The tiny error is created by discretizationand computation.

3.2.2. A skew plate. A trapezoidal skew plate that is clamped on one end and subjected to auniformly distributed in-plane bending load is shown in Figure 3, as well as the geometry, material,boundary condition and load. This problem was proposed by Cook [73] and was used by manyliteratures, such as [32, 74–76]. The mesh subdivisions range from 2×2 to 32×32, and eachquadrilateral is assembled with two triangles in the shortest-diagonal-cut layout, as illustrated inFigure 3. Obviously, in-plane shear deformations are predominant in this case, so it represents atest of convergence for the membrane-dominated situation.

The target of test and comparison is to correctly compute the vertical displacement of the freeend mid-point (y deflection at point C). Figure 4 shows the convergence of elements with refining



Figure 3. A skew plate under a distributed in-plane bending load (4×4 meshes).

Figure 4. Vertical displacements at the free end mid-point of the skew plate.

the mesh, for a reference result of 23.91 [74]. It can be seen that the speed of solution convergenceof the proposed formulation is faster than that of the CST and LST elements.

3.2.3. A square with arbitrary mesh pattern. The patch test is a well-known benchmark to validateplate and shell elements. A square with arbitrary mesh pattern for the patch test is depicted inFigure 5, as well as nodal coordinates.

Analytic w field for thin plate bending is given by

w(x, y)=0.001x2+0.002xy+0.0015y2+0.025x +0.03y+0.09 (55)

where the coefficients of the above polynomials have been chosen arbitrarily. The rotation can becalculated under the Kirchhoff–Love assumptions as follows

�x (x, y) = �w

�y=0.002x +0.003y+0.03 (56a)

�y(x, y) = −�w

�x=−(0.002x +0.002y+0.025) (56b)



Figure 5. Mesh and nodal coordinates of the patch test.

Table IV. Displacement w of inner nodes 5–8 for the bending patch test.

Element Node 5 Node 6 Node 7 Node 8

CST-BCIZ 0.2206 0.5087 0.7522 0.5480CST-DKT 0.2180 0.5055 0.7495 0.5455LST-BCIZ 0.2201 0.5082 0.7517 0.5476LST-DKT 0.2180 0.5055 0.7495 0.5455Present 0.2180 0.5055 0.7495 0.5455

Exact 0.2180 0.5055 0.7495 0.5455

The exact transverse displacements and rotations are prescribed as essential boundary conditionsin nodes 1–4. If the calculated displacements at the inner nodes 5–8 do not exactly match theanalytical displacements, the element does not pass the bending patch test. The calculated resultslisted in Table IV show that the present element, as well as the elements with DKT bendingcomponent, passes the bending patch test, but the elements with BCIZ bending component cannotpass it.

3.2.4. A whole clamped plate. A uniformly loaded whole clamped plate is used to test the shearlocking phenomenon. Obviously, this is a bending-dominated situation. The plate is divided intoirregular meshes (160 elements) with various ranges of thickness/span ratios, as shown in Figure 6.The results in Table V indicate that the present element has very good properties of being freefrom shear locking, even for very thin plates (the thickness/span ratio is 10−30). It is also shownthat when the plate thickness is very small, the results of the present element are the same as thoseof the DKT element. However, the BCIZ bending element is not free of shear locking for thinplates.

3.2.5. A cylindrical shell. The former case tests the effect of the thickness parameter on thebending performance of elements for a plate. Here, it is performed for a curved shell geometry.A cylindrical shell with free edges under a point load is divided into 2400 regular elements (20and 60 subdivisions in the axial and radial directions, respectively). The geometry and materialparameters are shown in Figure 7, and its thickness/radius ratios range from 0.5 to 10−5. Theanalytical and numerical results of deflections at point C are shown in Table VI. It can be seenthat only the present element possesses higher accuracy for both thin and moderately thick plates.

3.2.6. An opened spherical dome. A hemispherical shell (radius=10.0, thickness=0.04) with an18◦-hole at its top is subjected to an outward force (Fx =1) and to an inward force (Fy =−1),as shown in Figure 8. Young’s modulus=6.825×107 and Poisson’s ratio=0.3 are used for theisotropic elastic material. This example is a popular benchmark problem to test the element’sability to represent (nearly) inextensible deformations [77]. The radial deflections at point A for



Figure 6. A uniformly loaded whole clamped plate (160 elements).

Table V. Central deflection of the uniformly loaded whole clamped plate.

Thickness/span ratio 0.6 0.25 0.05 10−2 10−7 10−15 10−30

CST-BCIZ 862.7 235.6 57.4 4.8 0.0 0.0 0.0CST-DKT 128.9 128.9 128.9 128.9 128.9 128.9 128.9LST-BCIZ 862.7 235.6 57.4 4.8 0.0 0.0 0.0LST-DKT 128.9 128.9 128.9 128.9 128.9 128.9 128.9Present 850.3 239.7 148.5 136.4 128.9 128.9 128.9

Exact 126.5 126.5 126.5

Figure 7. A cylindrical shell with free edges under a point load.

different mesh divisions are used to compare with the analytical solution (u A =0.094 [78]). Theconvergence behavior with successive mesh refinements is shown in Figure 9. It can be seen thatthe present model is very effective in this example, in the way that the analytical solution is almostachieved even for the very coarse meshes.

3.2.7. A twisted beam. As shown in Figure 10, this case involves analyses of a clamped 90◦-twistedbeam that is subjected to loading patterns inducing bending and warping over the elements. It hasbecome a classical problem for testing the property of elements in hybrid situations [74, 78, 79],



Table VI. Central deflections of the cylindrical shell.

Case no. 1 2 3 4

Thickness/radius 0.5 0.25 0.05 10−5

Thickness 2.477 1.238 2.477×10−1 4.953×10−5

Load P 1×105 1×105 1×103 5×10−7

CST-BCIZ 0.1520 0.1755 0.06153 0.0CST-DKT 0.01022 0.08248 0.06570 0.07828LST-BCIZ 0.1521 0.1762 0.06199 0.0LST-DKT 0.01026 0.08227 0.06571 0.07828Present 0.1495 0.1794 0.06581 0.07828

Exact 0.1508 0.1800 0.06578 0.07819

Figure 8. A pinched hemispherical shell.

Figure 9. Radial deflections at the point A of the pinched shell.

i.e. both the membrane and bending components of elements affect the response of the structure. Theused parameters are length=12.0, width=1.1, thickness=0.0032, Young’s modulus=2.9×107

and Poisson ratio=0.22. Two conditions of loading are considered: (a) an ‘in-plane’ force P =1(along the width direction); or (b) an ‘out-of-plane’ force Q =1 (along the thickness direction).The mesh subdivisions consist of 2×8 and 2×16 (each quadrilateral is further divided into twotriangles). The deflections along the loading direction at point C are given in Table VII. The resultsshow that for each of the load patterns, the present element is able to converge to the analyticalsolution and produces the best results for the refined mesh (2×16).

3.2.8. An injection molded wind board. Injection molding is one of the widely used polymerprocessing methods, in which the final molded part will become deformed (called warpage ininjection molding) due to the residual stress accumulated during processing. In order to obtain



Figure 10. A clamped 90°-twisted beam.

Table VII. Deflections at the point C of the twisted beam.

Load Q Load P

Element 2×8 2×16 2×8 2×16

CST-BCIZ 4902 5276 1269 1283CST-DKT 4865 5292 1263 1286LST-BCIZ 4864 5272 1264 1285LST-DKT 4837 5271 1247 1288Present 4806 5252 1172 1290

Exact 5256 1294

a high quality and to reduce the cost, it is important to predict warpage of the molded part. Inview of the fact that most injection molded components are thin walled, the warpage simulationis a structural analysis of shells or plates in consideration of the accumulated residual stress [80].The elements mentioned above have also been used to develop a warpage simulation package forinjection molding.

An injection molded wind board is selected to evaluate the performance of elements forcomplicated and practical parts, as shown in Figure 11(a). Its boundary dimension is 395.5mm×515.7mm×134.1mm with thickness ranging from 0.5 to 4.0 mm. The locations marked with char-acters ‘A’, ‘B’ and ‘C’ are measurement positions of the part deflections, using the locations markedwith characters ‘D’, ‘E’ and ‘F’ to determine the measuring datum plane. The major processingconditions are listed in Table VIII. The deflections of numerical simulation and experiments areillustrated in Figure 11(b) and (c), respectively. Table IX shows the comparison of predicted andexperimental deflections at the measured points. The results show that the deflections predicted bythe present element agree with the experimental results better than the others.

3.3. Discussion

Two membrane-dominated, four bending-dominated and two hybrid examples have been carriedout for testing. The examples 1–2 indicate that the present element performs well in membrane-dominated situations, even for very large aspect ratios. The examples 3–6 demonstrate that thepresent element maintains good properties in bending-dominated situations. It can pass the bendingpatch test. And the examples 7–8 show that the present element can also provide high accuracy inhybrid situations.

As for the element convergence, the examples 2 and 6 indicate that the numerical results ofthe proposed element are able to converge to the analytical solution in both membrane-dominatedand bending-dominated situations. Its convergence speed is faster and the results are always betterthan those of others.

With regard to the effect of thickness parameter, the examples 4 and 5 represent the tests for aplate and a curved geometry, respectively. The results show that the present model is indeed fit toboth thin and thick shell structures.

In short, the present element exhibits better all-around properties than others. Its superiority canbe summarized in Table X.



Figure 11. Illustrations of: (a) the part geometry and measurement positions; (b) warpage result of thesimulation; and (c) warpage result of the experiment.

Table VIII. The major processing conditions used in the experiments.

Molding Mold Melt Injection Packing CoolingParameter material temperature temperature time pressure time

Initial value: 136.0 MPa;Value PS 115 50.0◦C 230.0◦C 1.2 s degressive speed: 11.0 MPa/s 25 s

Table IX. Comparison of predicted and experimental deflections.

Deflections (mm)

Element Point A Point B Point C

CST-BCIZ 2.91 3.75 6.08CST-DKT 3.31 3.70 5.69LST-BCIZ 3.74 3.99 6.25LST-DKT 2.65 3.64 5.48Present 7.72 7.25 8.46

Experimental 8.20 7.60 8.82



Table X. Property comparison of the present element and others.

Effect of Bending Effect of Hybrid ComplicatedElement aspect ratios patch test thickness Convergence situation parts

CST-BCIZ Worst Not pass Only for thick Ornery Not good Not goodCST-DKT Worst Pass Only for thin Ornery Not good Not goodLST-BCIZ Poor Not pass Only for thick Good Not good Not goodLST-DKT Poor Pass Only for thin Good Not good Not goodPresent Best Pass Both OK Best Good Good

4. CONCLUSIONS

A new flat triangular shell element has been proposed in this paper and compared with somepopular shell elements. The following conclusions can be drawn:

(1) The derivation of the present element is relatively simple because it is formulated by super-imposition of membrane and bending elements. Another reason is that only corner nodes andphysical degrees of freedom are used without adding mid-side nodes and fictitious degreesof freedom. The development meets the back trend toward simplicity, as stated in Reference[26] ‘simple elements provide the results of engineering accuracy with coarse meshes’.

(2) The combination of the optimal membrane element and RNEM-based bending element iseffective. It performs well in both membrane- and bending-dominated situations so that it isquite fit for complicated parts and loads (such as injection molding), in which a structuremay change from one to the other category with seemingly only small changes in geometryor boundary conditions.

(3) The element with the freedom configuration in this paper is insensitive to the aspect ratio,i.e. shows exact in-plane bending response for any aspect ratio. This is very valuable becausesome elements frequently have very high or low aspect ratios when modeling complicatedparts and mesh distortion often occurs in the deformation.

(4) The RNEM-based element takes the transverse shear correction into account. It possesses highaccuracy for thin and thick plates and solves the shear locking problem, capable of obtainingthe Kirchhoff plate results even for very thin plates. It can probably be considered as the mostefficient and simplest element among the existing Mindlin-based models.

ACKNOWLEDGEMENTS

The authors acknowledge financial supports from the National Natural Science Foundation of China (Grant# 50875095) and the National 863 Program (High-tech R&D) of China (Grant # 2009AA03Z104).

REFERENCES

1. Chapelle D, Bathe KJ. The Finite Element Analysis of Shells: Fundamentals. Springer: Berlin, 2003.2. Yang HTY, Saigal S, Liaw DG. Advances of thin shell finite elements and some applications—version-I. Computers

and Structures 1990; 35:481–504.3. Zhang YX, Cheung YK. Geometric nonlinear analysis of thin plates by a refined nonlinear non-conforming

triangular plate element. Thin-Walled Structures 2003; 41:403–418.4. Gal E, Levy R. Geometrically nonlinear analysis structures using a flat triangular element. Archives of

Computational Methods in Engineering 2006; 13:331–388.5. Zhang YX, Kim KS. Linear and geometrically nonlinear analysis of plates and shells by a new refined non-

conforming triangular plate/shell element. Computational Mechanics 2005; 36:331–342.6. Clough RW. The finite element method—a personal view of its original formulation. In From Finite Elements to

the Troll Platform—the Ivar Holand 70th Anniversary Volume, Bell K (ed.). Tapir Academic Press: Trondheim,1994.

7. Turner MJ, Clough RW, Martin HC, Topp LJ. Stiffness and deflection analysis of complex structures. Journal ofthe Aeronautical Sciences 1956; 23:805–824.



8. Zienkiewicz OC. Displacement and equilibrium models in the finite element method by B. Fraeijs de Veubeke. InStress Analysis, Chapter 9, Zienkiewicz OC, Holister GS (eds). Wiley: New York, 1965; 145–197. InternationalJournal for Numerical Methods in Engineering 2001; 52:287–289.

9. Zienkiewicz OC. The Finite Element Method in Engineering Science (4th edn). McGraw-Hill Inc.: London, 1989.10. Carr AJ. A refined finite element analysis of thin shell structures including dynamic loadings. Ph.D. Dissertation,

Department of Civil Engineering, University of California at Berkeley, Berkeley, 1967.11. Clough RW. Analysis of structural vibrations and dynamic response. In Recent Advances in Matrix Methods

of Structural Analysis and Design, Gallagher RH, Yamada Y, Oden JT (eds). University of Alabama Press:Hunstsville, 1971.

12. Allman DJ. A compatible triangular element including vertex rotations for plane elasticity analysis. Computersand Structures 1984; 19:1–8.

13. Bergan PG, Felippa CA. A triangular membrane element with rotational degrees of freedom. Computer Methodsin Applied Mechanics and Engineering 1985; 50:25–69.

14. Allman DJ. Evaluation of the constant strain triangle with drilling rotations. International Journal for NumericalMethods in Engineering 1988; 26:2645–2655.

15. Rankin CC, Brogan FA, Loden WA, Cabiness H. STAGS user manual. Lockheed Mechanics, Materials andStructures Report P032594, 1998.

16. Yang HTY, Saigal S, Masud A, Kapania RK. A survey of recent shell finite elements. International Journal forNumerical Methods in Engineering 2000; 47:101–127.

17. Alvin K, de la Fuente HM, Haugen B, Felippa CA. Membrane triangles with corner drilling freedoms. I. TheEFF element. Finite Elements in Analysis and Design 1992; 12:163–187.

18. Felippa CA, Militello C. Membrane triangles with corner drilling freedoms. II. The ANDES element. FiniteElements in Analysis and Design 1992; 12:189–201.

19. Felippa CA, Alexander S. Membrane triangles with corner drilling freedoms. III. Implementation and performanceevaluation. Finite Elements in Analysis and Design 1992; 12:203–239.

20. Bergan PG. Finite elements based on energy orthogonal functions. International Journal for Numerical Methodsin Engineering 1980; 15:1141–1555.

21. Bergan PG, Nygard MK. Finite elements with increased freedom in choosing shape functions. InternationalJournal for Numerical Methods in Engineering 1984; 20:643–664.

22. Felippa CA, Bergan PG. A triangular plate bending element based on an energy-orthogonal free formulation.Computer Methods in Applied Mechanics and Engineering 1987; 61:129–160.

23. Felippa CA. Parametrized multifield variational principles in elasticity. II. Hybrid functionals and the freeformulation. Communications in Applied Numerical Methods 1989; 5:79–88.

24. Felippa CA. The extended free formulation of finite elements in linear elasticity. Journal of Applied Mechanics—Transactions of the ASME 1989; 56:609–616.

25. Park KC, Stanley GM. A curved C0 shell element based on assumed natural-coordinate strains. Journal ofApplied Mechanics—Transactions of the ASME 1986; 53:278–290.

26. Felippa CA. A study of optimal membrane triangles with drilling freedoms. Computer Methods in AppliedMechanics and Engineering 2003; 192:2125–2168.

27. Kim DN, Bathe KJ. A triangular six-node shell element. Computers and Structures 2009; 87:1451–1460.28. Cen S, Chen XM, Fu XR. Quadrilateral membrane element family formulated by the quadrilateral area coordinate

method. Computer Methods in Applied Mechanics and Engineering 2007; 196:4337–4353.29. Cen S, Chen XM, Li CF, Fu XR. Quadrilateral membrane elements with analytical element stiffness matrices

formulated by the new quadrilateral area coordinate method (QACM-II). International Journal for NumericalMethods in Engineering 2009; 77:1172–1200.

30. Cen S, Du Y, Chen XM, Fu XR. The analytical element stiffness matrix of a recent 4-node membrane elementformulated by the quadrilateral area co-ordinate method. Communications in Numerical Methods in Engineering2007; 23:1095–1110.

31. Cardoso RPR, Yoon JW. One point quadrature shell elements: a study on convergence and patch tests.Computational Mechanics 2007; 40:871–883.

32. Cardoso RPR, Yoon JW, Valente RAF. A new approach to reduce membrane and transverse shear lockingfor one-point quadrature shell elements: linear formulation. International Journal for Numerical Methods inEngineering 2006; 66:214–249.

33. Clough RW, Tocher JL. Finite element stiffness matrices for analysis of plate bending. Proceedings of the FirstConference on Matrix Methods in Structural Mechanics, Dayton, 1966; 515–545.

34. Bazeley GP, Cheung YK, Irons BM, Zienkiewicz OC. Triangular elements in plate bending, conforming andnon-conforming solutions. Proceedings of the First Conference on Matrix Methods in Structural Mechanics,Dayton, 1966; 547–576.

35. Batoz JL, Bathe KJ, Ho LW. A study of three-noded triangular plate bending elements. International Journalfor Numerical Methods in Engineering 1980; 15:1771–1812.

36. Hrabok MM, Hrudey TH. A review and catalogue of plate bending finite elements. Computers and Structures1984; 19:475–495.

37. Meek JL, Tan HS. A discrete Kirchoff plate bending element with Loof nodes. Computers and Structures 1985;21:1197–1212.



38. Dhatt G, Marcotte L, Matte Y. A new triangular discrete Kirchhoff plate/shell element. International Journalfor Numerical Methods in Engineering 1986; 23:453–470.

39. Katili I. A new discrete Kirchhoff–Mindlin element based on Mindlin–Reissner plate theory and assumed shearstrain fields—Part I: an extended DKT element for thick-plate bending analysis. International Journal forNumerical Methods in Engineering 1993; 36:1859–1883.

40. Batoz JL, Zheng CL, Hammadi F. Formulation and evaluation of new triangular, quadrilateral, pentagonal andhexagonal discrete Kirchhoff plate/shell elements. International Journal for Numerical Methods in Engineering2001; 52:615–630.

41. Batoz JL, Lardeur P. A discrete shear triangular nine d.o.f. element for the analysis of thick to very thin plates.International Journal for Numerical Methods in Engineering 1989; 28:533–560.

42. Batoz JL, Katili I. On a simple triangular Reissner–Mindlin plate element based on incompatible modes anddiscrete constraints. International Journal for Numerical Methods in Engineering 1992; 35:1603–1632.

43. MacNeal RH. Perspective on finite elements for shell analysis. Finite Elements in Analysis and Design 1998;30:175–186.

44. Sofuoglu H, Gedikli H. A refined 5-node plate bending element based on Reissner–Mindlin theory. Communicationsin Numerical Methods in Engineering 2007; 23:385–403.

45. Bergan PG, Wang X. Quadrilateral plate bending elements with shear deformations. Computers and Structures1984; 19:25–34.

46. Hughes TJR, Cohen M, M. H. Reduced and selective integration techniques in finite element analysis of plates.Nuclear Engineering and Design 1978; 46:203–222.

47. Gruttmann F, Wagner W. A stabilized one-point integrated quadrilateral Reissner–Mindlin plate element.International Journal for Numerical Methods in Engineering 2004; 61:2273–2295.

48. Hughes TJR, Tezduyar TE. Finite elements based upon Mindlin plate theory with particular reference to thefour-node bilinear isoparametric element. Journal of Applied Mechanics—Transactions of the ASME 1981;48:587–596.

49. Bathe KJ, Dvorkin EN. A formulation of general shell elements—the use of mixed interpolation of tensorialcomponents. International Journal for Numerical Methods in Engineering 1986; 22:697–722.

50. Castellazzi G, Krysl P. Displacement-based finite elements with nodal integration for Reissner–Mindlin plates.International Journal for Numerical Methods in Engineering 2009; 80:135–162.

51. Hu B, Wang Z, Xu YC. Combined hybrid method applied in the Reissner–Mindlin plate model. Finite Elementsin Analysis and Design 2010; 46:428–437.

52. Kim SH, Choi CK. Improvement of quadratic finite element for Mindlin plate bending. International Journalfor Numerical Methods in Engineering 1992; 34:197–208.

53. Brezzi F, Marini LD. A nonconforming element for the Reissner–Mindlin plate. Computers and Structures 2003;81:515–522.

54. Feng MF, Yang Y, Zhou TX. Nonconforming stabilized combined finite element method for Reissner–Mindlinplate. Applied Mathematics and Mechanics—English Edition 2009; 30:197–207.

55. Chen WJ, Cheung YK. Refined non-conforming quadrilateral thin plate bending element. International Journalfor Numerical Methods in Engineering 1997; 40:3919–3935.

56. Chen WJ, Cheung YK. Refined triangular discrete Kirchhoff plate element for thin plate bending, vibration andbuckling analysis. International Journal for Numerical Methods in Engineering 1998; 41:1507–1525.

57. Chen WJ, Cheung YK. Refined quadrilateral element based on Mindlin/Reissner plate theory. InternationalJournal for Numerical Methods in Engineering 2000; 47:605–627.

58. Zhang YX, Cheung YK, Chen WJ. Two refined non-conforming quadrilateral flat shell elements. InternationalJournal for Numerical Methods in Engineering 2000; 49:355–382.

59. Chen WJ, Cheung YK. Refined 9-Dof triangular Mindlin plate elements. International Journal for NumericalMethods in Engineering 2001; 51:1259–1281.

60. Zengjie G, Wanji C. Refined triangular discrete Mindlin flat shell elements. Computational Mechanics 2003;33:52–60.

61. Chen WJ. Refined 15-DOF triangular discrete degenerated shell element with high performances. InternationalJournal for Numerical Methods in Engineering 2004; 60:1817–1846.

62. Wu Z, Chen RG, Chen WJ. Refined laminated composite plate element based on global–local higher-order sheardeformation theory. Composite Structures 2005; 70:135–152.

63. Wu Z, Chen WJ. Refined global–local higher-order theory and finite element for laminated plates. InternationalJournal for Numerical Methods in Engineering 2007; 69:1627–1670.

64. Wu Z, Chen WJ, Ren XH. Refined global–local higher-order theory for angle-ply laminated plates underthermo-mechanical loads and finite element model. Composite Structures 2009; 88:643–658.

65. Pontaza JP, Reddy JN. Mixed plate bending elements based on least-squares formulation. International Journalfor Numerical Methods in Engineering 2004; 60:891–922.

66. Brunet M, Sabourin F. Analysis of a rotation-free 4-node shell element. International Journal for NumericalMethods in Engineering 2006; 66:1483–1510.

67. Nguyen-Thanh N, Rabczuk T, Nguyen-Xuan H, Bordas SPA. A smoothed finite element method for shell analysis.Computer Methods in Applied Mechanics and Engineering 2008; 198:165–177.

68. Darilmaz K, Kumbasar N. An 8-node assumed stress hybrid element for analysis of shells. Computers andStructures 2006; 84:1990–2000.



69. Ozkul TA, Ture U. The transition from thin plates to moderately thick plates by using finite element analysisand the shear locking problem. Thin-Walled Structures 2004; 42:1405–1430.

70. Chapelle D, Bathe KJ. Fundamental considerations for the finite element analysis of shell structures. Computersand Structures 1998; 66:19–36.

71. Felippa CA, Haugen B, Militello C. From the individual element test to finite element templates: evolution ofthe patch test. International Journal for Numerical Methods in Engineering 1995; 38:199–229.

72. Felippa CA, Militello C. Construction of optimal 3-node plate bending elements by templates. ComputationalMechanics 1999; 24:1–13.

73. Cook RD. Improved two-dimensional finite element. Journal of the Structural Division 1974; 100:1851–1863.74. Simo JC, Fox DD, Rifai MS. On a stress resultant geometrically exact shell model. Part II: the linear theory-

computational aspects. Computer Methods in Applied Mechanics and Engineering 1989; 73:53–92.75. Ibrahimbegovic A, Taylor RL, Wilson EL. A robust quadrilateral membrane finite element with drilling degrees

of freedom. International Journal for Numerical Methods in Engineering 1990; 30:445–457.76. Kasper EP, Taylor RL. A mixed-enhanced strain method. Part I: geometrically linear problems. Computers and

Structures 2000; 75:237–250.77. Fish J, Belytschko T. Stabilized rapidly convergent 18 degrees-of-freedom flat shell triangular element.

International Journal for Numerical Methods in Engineering 1992; 33:149–162.78. Sabourin F, Brunet M. Analysis of plate and shell with a simplified three node triangular element. Thin-Walled

Structures 1995; 21:209–223.79. Parisch H. An investigation of a finite rotation four node assumed strain shell element. International Journal for

Numerical Methods in Engineering 1991; 31:127–150.80. Choi DS, Im YT. Prediction of shrinkage and warpage in consideration of residual stress in integrated simulation

of injection molding. Composite Structures 1999; 47:655–665.


A 3-node flat triangular shell element with corner drilling freedoms and transverse shear correction

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