Hybrid displacement function element method: a simple ...cfli/papers_pdf_files/... · Hybrid...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2014; 98:203–234Published online 28 January 2014 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.4632

Hybrid displacement function element method: a simplehybrid-Trefftz stress element method for analysis of

Mindlin–Reissner plate

Song Cen1,2,3,*,† , Yan Shang1,2, Chen-Feng Li4 and Hong-Guang Li5

1Department of Engineering Mechanics, School of Aerospace, Tsinghua University, Beijing 100084, China2High Performance Computing Center, School of Aerospace, Tsinghua University, Beijing 100084, China3Key Laboratory of Applied Mechanics, School of Aerospace, Tsinghua University, Beijing 100084, China

4College of Engineering, Swansea University, Swansea SA2 8PP, UK5Aviation University of Air Force, Changchun, Jilin 130022, China

SUMMARY

In order to develop robust finite element models for analysis of thin and moderately thick plates, a simplehybrid displacement function element method is presented. First, the variational functional of complemen-tary energy for Mindlin–Reissner plates is modified to be expressed by a displacement function F , whichcan be used to derive displacement components satisfying all governing equations. Second, the assumedelement resultant force fields, which can satisfy all related governing equations, are derived from the funda-mental analytical solutions of F . Third, the displacements and shear strains along each element boundaryare determined by the locking-free formulae based on the Timoshenko’s beam theory. Finally, by applyingthe principle of minimum complementary energy, the element stiffness matrix related to the conventionalnodal displacement DOFs is obtained. Because the trial functions of the domain stress approximationsa priori satisfy governing equations, this method is consistent with the hybrid-Trefftz stress element method.As an example, a 4-node, 12-DOF quadrilateral plate bending element, HDF-P4-11ˇ, is formulated. Numeri-cal benchmark examples have proved that the new model possesses excellent precision. It is also a shape-freeelement that performs very well even when a severely distorted mesh containing concave quadrilateral anddegenerated triangular elements is employed. Copyright © 2014 John Wiley & Sons, Ltd.

Received 9 June 2013; Revised 25 November 2013; Accepted 16 December 2013

KEY WORDS: finite element methods; hybrid displacement function element method; hybrid-Trefftz stresselement; fundamental analytical solution; plate bending

1. INTRODUCTION

In the past few decades, significant efforts have been made to construct plate bending elements [1–3]derived from the Mindlin–Reissner theory [4, 5], in which the rotations x , y and the deflectionw are independently defined. The major difficulty encountered in earlier time was shear lockingthat leads to the over-stiffness problem for thin plates. To overcome this challenge, various effec-tive approaches and schemes have been proposed, and they include the classical reduced [6] andselective reduced integral schemes [7], the substitute shear strain technique [8], the stabilization pro-cedure [9, 10], the hybrid element method derived from the modified Hellinger–Reissner principle[11], the mixed interpolated tensorial components technique [12, 13], the discrete Reissner–Mindlin method [14], the linked interpolation technique [15, 16], the discrete shear method [17],the extended discrete Kirchhoff–Mindlin method [18], the mixed shear projected approach [19, 20],

*Correspondence to: Song Cen, Department of Engineering Mechanics, School of Aerospace, Tsinghua University,Beijing 100084, China.

†E-mail: [email protected]

Copyright © 2014 John Wiley & Sons, Ltd.

204 S. CEN ET AL.

the improved shear strain interpolation schemes derived from the formulae of the Timoshenko’sbeam [21–26], the degenerated shell element method [27], the geometrically exact shell elementmethod [28, 29], and so on. Recently, new Mindlin–Reissner elements are still emerging in manyliteratures: Nguyen-Xuan et al. [30] proposed a smoothed curvature method to develop quadrilat-eral elements; Castellazzi et al. [31] used nodal integration method to construct displacement-basedmodels; Hu et al. [32] presented a combined hybrid method; Hansbo et al. [33] formulated newelement with discontinuous rotations; Nguyen-Thanh et al. [34] proposed an alternative alpha finiteelement; Falsone et al. [35] developed a new model using the Kirchhoff-like solution; Nguyen-Thoiet al. [36] proposed a cell-based smoothed discrete shear gap method; Ribaric et al. [37] studiedthe quadrilateral elements using higher-order linked interpolation; Vu-Quac et al. [38] used efficienthybrid-EAS solid element for accurate stress prediction in thick plates; and so on. It is very interest-ing that this topic attracts attentions from so many researchers and has become a platform for testingnew numerical methods [39].

Besides the aforementioned shear locking problem, a good plate bending element should alsosatisfy the general requirement for finite element technologies: (1) tolerance to various mesh dis-tortions and (2) high-precision results for stress/resultant-force solutions, as well as displacement.In 2006, Cen et al. [40] developed a quadrilateral Mindlin–Reissner plate bending element withthe generalized conforming concept. Furthermore, they employed the quadrilateral area coordinates[41, 42] to improve the robustness in distorted meshes for the model, and the hybrid-enhanced post-processing procedure [43] to reduce the precision loss of the resultant force solutions. Comparedwith some other elements, better results were obtained. However, some problems remain unsolvedfrom the outset. For example, the precision will be dramatically damaged with the element shapedegenerates into a triangle, and the hybrid-enhanced post-processing procedure will bring additionalcomputational cost.

Recently, in order to develop plane quadrilateral elements that are immune to mesh distortions,Fu et al. [44] and Cen et al. [45–47] proposed a simple technique named the hybrid stress-function(HSF) element method. Derived from the conventional principle of minimum complementaryenergy, this method makes use of the Airy stress function � that is interpolated by its fundamentalanalytical solutions. The resulting quadrilateral HSF elements can produce high-quality results forall field variables, even for severely distorted meshes containing concave quadrangles or degeneratedtriangles. These new HSF elements have been termed as shape-free elements because they are notsensitive to severely ill-conditioned meshes. From the viewpoint of variational formulations, theHSF element method can also be treated as a modified hybrid stress method because its deriva-tion follows closely the procedure of the first hybrid stress element proposed by Pian [48]. Pian’sfirst element [48] is a special case of the HSF element method because its assumed that stressesare all fundamental analytical solutions of the Airy stress function �. On the other hand, becausethe trial function satisfying all governing equations is employed by the HSF element method, thisapproach may also be viewed as an extension to the well-known hybrid-Trefftz stress elementmethods [49–52]. Since 1977, the hybrid-Trefftz element method has been successfully applied fordeveloping plate bending elements [53–63], among which the simplest lower-order elements are ofparticular interest, such as the 4-node, 12-DOF quadrilateral element QQ21�11 [57], and the 8-node,16-DOF quadrilateral element Q21-13 [59]. But these two elements were proved to possess spuriousmodes [58].

The Mindlin–Reissner plate bending theory features a displacement function F that can be usedto derive the displacement components satisfying all governing equations [64]. In this paper, ahybrid displacement function element method, which makes use of the displacement-functionF , isproposed to develop shape-free plate bending elements for analysis of both thin and moderatelythick plates. First, the variational functional of complementary energy for Mindlin–Reissner platesis modified to be expressed by the displacement function F . Second, the assumed element resultantforce fields, which can satisfy all related governing equations, are derived from the fundamentalanalytical solutions of such F function. Third, the displacements and shear strains along each ele-ment boundary are determined by the locking-free formulae based on Timoshenko’s beam theory[22–24]. Finally, by applying the principle of minimum complementary energy, the element stiff-ness matrix related to the conventional nodal displacement DOFs can be obtained. Because the trial

Copyright © 2014 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2014; 98:203–234DOI: 10.1002/nme

HYBRID DISPLACEMENT FUNCTION ELEMENT METHOD FOR MINDLIN-REISSNER PLATE 205

functions of the domain stress approximations a priori satisfy governing equations, this hybrid dis-placement function element method is consistent with the hybrid-Trefftz stress element method, butsome concepts and techniques employed by the original hybrid stress element method are still keptfor avoiding more mathematical derivations. As an example, a 4-node, 12-DOF quadrilateral platebending element, HDF-P4-11ˇ, has been formulated.

Some classic benchmark examples using traditional and new severely distorted mesh divisions areused to estimate the new model’s performance. Numerical results prove that the proposed model pos-sesses proper rank, successfully passes all patch tests, avoids shear locking, and provides superiorprecision for displacements and resultant forces in analysis of thin and moderately thick plates. Inparticular, it is a shape-free element and performs very well even when a severely distorted meshcontaining concave quadrilateral and degenerated triangular elements is employed.

2. GOVERNING EQUATIONS AND THE DISPLACEMENT FUNCTION

The typical model of a Mindlin–Reissner plate is shown in Figure 1. The mid-surface of the plate isdefined as the x � y plane, and ´ is the thickness direction (�h=2 6 ´ 6 h=2), where h is the platethickness. The transverse deflection along ´ direction is denoted by w, and the rotations in the x�´and y � ´ planes are denoted by x and y , respectively. In the Mindlin–Reissner plate bendingtheory, the transverse deflection and the rotations are independently defined. Then, the displacementcomponents at an arbitrary point of the plate are:

u D �´ x v D �´ y w D w.x; y/

��h

26 ´ 6 h

2

�: (1)

2.1. Governing equations

The equilibrium equations for the plate subjected to a transverse distributed load q are given by

@Mx

@xC@Mxy

@y� Tx D 0

@Mxy

@xC@My

@y� Ty D 0

@Tx

@xC@Ty

@yC q D 0

9>>>>>>=>>>>>>;; (2)

where Mx;My ;Mxy , Tx and Ty are the bending moments, twisting moments, and shear forces perunit length, respectively (refer to Figure 1).

Figure 1. Mindlin–Reissner plate: definitions of the displacements and the resultant forces.


206 S. CEN ET AL.

For linear cases, the strain–displacement relations are as follows

�x D �@ x

@x; �y D �

@ y

@y; �xy D �

�@ x

@yC@ y

@x

�;

�x D@w

@x� x; �y D

@w

@y� y ;

(3)

where �x , �y , and �xy are the curvatures, and �x and �y are the transverse shear strains. FromEquation (3), the equations of strain compatibility can be obtained:

@�x

@y�1

2

@�xy

@xD1

2

@

@x

�@�x

@y�@�y

@x

�

@�y

@x�1

2

@�xy

@yD1

2

@

@y

�@�x

@y�@�y

@x

� : (4)

The constitutive relations are given by:

R D AE ; (5)

where R and E are

R D ŒMx My Mxy Tx Ty �T; (6)

E D Œ�x �y �xy �x �y �T: (7)

For isotropic and linearly elastic plates, the elasticity matrix A is given by

A D

2666664

D �D 0 0 0

�D D 0 0 0

0 0 1��2D 0 0

0 0 0 C 0

0 0 0 0 C

3777775; (8)

D DEh3

12.1 � �2/; C D

5

6Gh; (9)

in which � is Poisson’s ratio, E Young’s modulus, and G D E=[2(1+�)] the shear modulus.The boundary conditions can be classified into four categories:

(i) Fixed boundary: w D Nw; n D N n; and s D N s(ii) ‘Soft’ simply supported (SS1) boundary: w D Nw; Mn D NMn; and Ms D NMs

(iii) ‘Hard’ simply supported (SS2) boundary: w D Nw; Mn D NMn; and s D N s(iv) Free boundary: Tn D NTn; Mn D NMn; and Ms D NMs;

in which s and n are the tangential and normal directions along the boundaries, respectively.

2.2. Displacement function

Hu [64] proved that the solutions of deflection w and rotations x and y can be expressed by afunction F as follows:

w D F �D

Cr2F; x D

@F

@x; y D

@F

@y; (10)

in which

Dr2r2F D q: (11)



Substitution of Equation (10) into Equation (3) yields

�x D �@2F

@x2; �y D �

@2F

@y2�xy D �2

@2F

@x@y;

�x´ D �D

C

@

@x

�r2F

�; �y´ D �

D

C

@

@y

�r2F

�;

(12)

which satisfy the strain compatibility Equation (4). Then, substitution of Equation (12) into Equation(5) yields

Mx D �D

�@2F

@x2C �

@2F

@y2

�

My D �D

�@2F

@y2C �

@2F

@x2

�

Mxy D �D.1 � �/@2F

@x@y

Tx D �D@

@x

�r2F

�

Ty D �D@

@y

�r2F

�

9>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>;

: (13)

It can be easily proved that Equation (13) satisfies the equilibrium Equation (2).So, the displacements derived by the function F (see Equations (10) and (11)), and the resulting

curvatures, shear strains, and resultant forces satisfy all governing equations. This function F isdefined as the displacement-function.

2.3. The fundamental analytical solutions for the displacement function and the resultingresultant forces

The displacement-function F given by Equation (11) can be rewritten as

F D F 0 C F �; (14)

in which F � is the particular solution of F , and

Dr2r2F � D q: (15)

Then, from Equation (13), the corresponding particular solutions of the resultant forces can beobtained as

R� D

8ˆ<ˆˆ:

M �x

M �y

M �xy

T �x

T �y

9>>>>>>>=>>>>>>>;D

8ˆˆ<ˆˆ:

�D�@2F �

@x2C � @

2F �

@y2

�

�D�@2F �

@y2C � @

2F �

@x2

�

�D.1 � �/ @2F �

@x@y

�D @@x

�r2F �

��D @

@y

�r2F �

�

9>>>>>>>>>>=>>>>>>>>>>;

: (16)

In Equation (14), F 0 is the general (homogeneous) solution of F that satisfies the followingbiharmonic equation

r2r2F 0 D 0: (17)


208 S. CEN ET AL.

Then, from Equation (13), the corresponding general solutions of the resultant forces can beexpressed by

R0 D

8ˆ<ˆˆ:

M 0x

M 0y

M 0xy

T 0x

T 0y

9>>>>>>>=>>>>>>>;D

8ˆˆ<ˆˆ:

�D�@2F 0

@x2C � @

2F 0

@y2

�

�D�@2F 0

@y2C � @

2F 0

@x2

�

�D.1 � �/ @2F 0

@x@y

�D @@x

�r2F 0

��D @

@y

�r2F 0

�

9>>>>>>>>>>>=>>>>>>>>>>>;

: (18)

Let

F 0 D

kXiD1

F 0i ˇi D S0F “; (19)

with

S0F D�F 01 F

02 F

03 � � � � � � F

0k

“ D Œˇ1 ˇ2 ˇ3 � � � � � � ˇk�

T; (20)

where F 0i .i D 1 � k/ are k entries of the fundamental analytical solutions for F 0, and ˇi .i D1 � k/ are k unknown coefficients. The F 0i functions should be chosen from second-order terms.x2, xy, and y2) to higher-order terms (see Table I, the first 11 solutions in Cartesian coordinatesare given), and the completeness must be guaranteed. Each entry of these fundamental analytical

Table I. Eleven fundamental analytical solutions for the general part of the displacementfunction and resulting resultant forces.

i 1 2 3 4 5 6 7

�DF 0i x2 xy y2 x3 x2y xy2 y3

M 0xi 2 0 2� 6x 2y 2�x 6�y

M 0yi 2� 0 2 6�x 2�y 2x 6y

M 0xyi 0 1-� 0 0 2(1-�)x 2(1-�)y 0

T 0xi 0 0 0 6 0 2 0

T 0yi 0 0 0 0 2 0 6

i 8 9 10 11

�DF 0i x3y xy3 x4 � y4 6x2y2 � x4 � y4

M 0xi 6xy 6�xy 12

�x2 � �y2

�12.1 � �/

�y2 � x2

�M 0yi 6�xy 6xy �12

�y2 � �x2

�12.1 � �/

�x2 � y2

�M 0xyi 3.1 � �/x2 3.1 � �/y2 0 24.1 � �/xy

T 0xi 6y 6y 24x 0

T 0yi 6x 6x �24y 0



solutions can satisfy the biharmonic Equation (17). Then, from Equation (18), the correspondinggeneral solutions of the resultant forces can be obtained as

R0 D

8ˆ<ˆˆ:

M 0x

M 0y

M 0xy

T 0x

T 0y

9>>>>>>>=>>>>>>>;D OS“ D

2666666664

M 0x1 M 0

x2 M 0x3 : : : M 0

xk

M 0y1 M 0

y2 M 0y3 : : : M 0

yk

M 0xy1 M

0xy2 M

0xy3 : : : M

0xyk

T 0x1 T 0x2 T 0x3 : : : T 0xk

T 0y1 T 0y2 T 0y3 : : : T 0yk

3777777775

8ˆ<ˆˆ:

ˇ1

ˇ2

ˇ1

:::

ˇk

9>>>>>>>=>>>>>>>;; (21)

where M 0xi , M

0yi , M

0xyi , T

0xi and T 0yi (i D 1 � k) are also given by Table I.

The particular solution F � can be easily determined from Equation (15). For example, if q is auniformly distributed transverse load, the following particular solution

F � Dq

48D

�x4 C y4

�(22)

can be employed. Then, from Equations (16), the corresponding particular solutions of the resultantforces can be expressed by

R0 D

8ˆ<ˆˆ:

M �x

M �y

M �xy

T �x

T �y

9>>>>>>>=>>>>>>>;D

8ˆ<ˆˆ:

�q4

�x2 C �y2

��q4

��x2 C y2

�0

�q2x

�q2y

9>>>>>>>=>>>>>>>;: (23)

Therefore, the total resulting resultant force vector (direct domain resultant approximations) is thesum of the particular and the general solution parts:

R D R0 C R� D OS“C R�; (24)

where OS and “ are given by Equation (21).

Figure 2. Shape-free 4-node quadrilateral plate bending element.


210 S. CEN ET AL.

Figure 3. Timoshenko’s beam element.

3. A HYBRID DISPLACEMENT FUNCTION ELEMENT METHOD AND A 4-NODEQUADRILATERAL MODEL WITH ARBITRARY SHAPES

3.1. A hybrid displacement function element method for Mindlin–Reissner plate

The standard complementary energy functional of a Mindlin–Reissner plate element can beexpressed in the following matrix form [64]:

…eC D

ZZAe

1

2RTCRdxdy C

ZSe nM nds C

ZSe sM nsds �

ZsewT nds

D

ZZAe

1

2RTCRdxdy C

ZSe

RTdds D …e�

C C Ve�

C

(25)

with

…e�

C D

ZZAe

1

2RTCRdxdy; Ve

�

C D

ZSe

RTdds (26)

C D A�1 D

2666664

D �D 0 0 0

�D D 0 0 0

0 0 1��2D 0 0

0 0 0 C 0

0 0 0 0 C

3777775

�1

D

2666666664

1

D.1��2/��

D.1��2/0 0 0

��

D.1��2/1

D.1��2/0 0 0

0 0 2D.1��/

0 0

0 0 0 1C0

0 0 0 0 1C

3777777775

(27)

R D

8<ˆ:

M n

M ns

�T n

9>>=>>;; d D

8<ˆ:

n

s

w

9>>=>>;; (28)

where …e�

C and Ve�

C are the complementary energy functional within the element and along thekinematic boundaries (element edges), respectively; C, the elasticity matrix of compliances; R, theresultant force vector given by Equation (6); R is the value of R along the element edges. The vectord denotes the rotations and deflections along element boundaries, and can be expressed by:

d D Nj�

qe; (29)

where qe is the element nodal displacement vector, and Nj�

is the corresponding interpolationfunction matrix. The details of the matrix Nj

�for a 4-node quadrilateral element will be discussed

in Section 3.2.



y

x

(x1, y

1)

(x4, y

4)

(x3, y

3)

(x2, y

2)

(x2, y

2)

(x3, y

3)

(x1, y

1)

(x4, y

4)

4

43

3

2

2

11

s12, l

12s12

, l12

s23, l

23

s34, l

34

s41, l

41

s23, l

23

s34, l

34

s41, l

41

Figure 4. The shear strain �sij and the length lij along each element edge.

Figure 5. Patch tests, geometry, loads, and meshes.

Figure 6. Typical meshes used by a quarter of square plate (c is the central point of plate).


212 S. CEN ET AL.

Table II. Clamped square plate: Dimensionless results of central deflection wc=�qL4=100D

�and moment

Mc=�qL2=10

�obtained by element HDF-P4-11“ (Example 4.3).

Mesh density Analyticalh=L Mesh type 2 � 2 4 � 4 8 � 8 16 � 16 32 � 32 solutions

wc=�qL4=100D

�10�30 � 0:001 Mesh A-regular 0.12390 0.12601 0.12646 0.12652 0.12653

Mesh B-distorted 0.12378 0.12585 0.12643 0.12652 0.12653 0.1265Mesh C-triangular 0.11698 0.12480 0.12616 0.12646 0.12652

0.01 Mesh A-regular 0.12418 0.12630 0.12673 0.12678 0.12678Mesh B-distorted 0.12407 0.12618 0.12670 0.12678 0.12678 0.1267Mesh C-triangular 0.11727 0.12509 0.12644 0.12672 0.12677


Mc=�qL2=10

�10�30 � 0:001 Mesh A-regular 0.22108 0.22844 0.22897 0.22904 0.22905



From Equation (13), substitution of the displacement function F into the resultant force vectorR yields

R D

8ˆ<ˆ:

Mx

My

Mxy

Tx

Ty

9>>>>>=>>>>>;D

8ˆˆ<ˆˆ:

�D�@2F@x2C � @

2F@y2

�

�D�@2F@y2C � @

2F@x2

�

�D.1 � �/ @2F@x@y

�D @@x

�r2F

��D @

@y

�r2F

�

9>>>>>>>>>=>>>>>>>>>;

D QD.F /I (30)

and the element boundary resultant force vector R can be written as

R D

8<ˆ:M n

M ns

�T n

9>>=>>;D

264

l2 m2 2lm 0 0

�lm lm l2 �m2 0 0

0 0 0 �l �m

375

8ˆ<ˆ:

Mx

My

Mxy

Tx

Ty

9>>>>>=>>>>>;D L QD.F /; (31)

where l and m denote the direction cosines of the element boundaries’ outer normal n.After substituting Equations (30) and (31) into Equation (25), the element complementary energy

functional expressed by the displacement function F can be obtained as follows:

…eC D …

e�C C Ve�C D

1

2

“Ae

QD.F /TC QD.F /dxdy CZSe

hL QD.F /

iTNdds: (32)



Table III. SS1 square plate: Dimensionless results of central deflection wc=�qL4=100D

�and moment

Mc=�qL2=10



wc=�qL4=100D

�10�30 � 0:001 Mesh A-regular 0.40646 0.40622 0.40624 0.40624 0.40624



Mc=�qL2=10

�10�30 � 0:001 Mesh A-regular 0.47471 0.47883 0.47886 0.47886 0.47886



Table IV. SS2 square plate: Dimensionless results of central deflection wc=�qL4=100D

�and moment

Mc=�qL2=10



wc=�qL4=100D

�10�30 � 0:001 Mesh A-regular 0.40516 0.40617 0.40623 0.40624 0.40624




Mc=�qL2=10

�10�30 � 0:001 Mesh A-regular 0.47857 0.47884 0.47886 0.47886 0.47886

Mesh B-distorted 0.48653 0.48043 0.47909 0.47889 0.47887Mesh C-triangular 0.46716 0.47921 0.47896 0.47889 0.47887


0.1 Mesh A-regular 0.47289 0.47663 0.47831 0.47873 0.47883Mesh B-distorted 0.47489 0.47763 0.47847 0.47875 0.47884Mesh C-triangular 0.46629 0.47899 0.47912 0.47894 0.47899


214 S. CEN ET AL.

(a) h/L=0.001 (thin plate case)

(b) h/L=0.1 (thick plate case)

Figure 7. Convergence of the central deflections and moments for square plates subjected to uniform load(Clamped BC, Mesh A).

Assume that the trial functions of the resultant force vector R are given by the Equation (24), whichwill be determined by the fundamental analytical solutions for the displacement function F . Then,substitution of Equations (24), (30), and (31) into Equation(32) yields

…eC D

ZZAe

1

2

�OS“C R�

�TC�OS“C R�

�dxdy C

ZSe

�OS“C R�

�TLTNj

�qeds (33)

D1

2

�“TM“C “TM� CM�T“CQ

�C “THqe C Vqe;

where

M DZZ

Ae

OSTC OSdxdy; M� DZZ

Ae

OSTCR�dxdy; Q DZZ

AeR�TCR�dxdy (34)

H DZSe

OSTLTNj�

ds; V DZSe

R�TLTNj�

ds: (35)

From the stationary condition

@…eC

@“D 0 (36)

of the complementary energy, we obtain

“ D �M�1�M� CHqe

�: (37)

Then, substitution of previous relations into Equation (33) yields





Figure 8. Convergence of the central deflections and moments for square plates subjected to uniform load(SS1 BC, Mesh A).

…eC D �

1

2qeTHTM�1Hqe �

1

2M�TM�1M� C

1

2Q �M�TM�1Hqe C Vqe: (38)

By applying the principle of minimum complementary energy once again,

@…eC

@qeD 0; (39)

we obtain

Keqe D Peq; (40)

in which Ke is the element stiffness matrix:

Ke D HTM�1HI (41)

and Peq is the element nodal equivalent load vector caused by the distributed transverse load q,

Peq D VT �HTM�1M�: (42)

Thus, the hybrid displacement function element method for analysis of Mindlin–Reissner plates isestablished. It can be used to develop arbitrary polygonal plate bending elements, and is very easyto be integrated into the standard framework of finite element programs. Because the trial functions


216 S. CEN ET AL.



Figure 9. Convergence of the central deflections and moments for square plates subjected to uniform load(SS2 BC, Mesh A).

Δ

Δ Δ

L/2=5 L/2=5

L/2=

5

L/2=

5

(a) Symmetric (b) Asymmetric

C C

Figure 10. A quarter of clamped square plate for sensitivity test to mesh distortion.

of the domain stress approximations a priori satisfy governing equations, this hybrid displacementfunction element method is consistent with the hybrid-Trefftz stress element method.

3.2. The formulations of a new 4-node quadrilateral element

The proposed method can be used to develop arbitrary polygonal element models. As an example,a 4-node quadrilateral model (one of the most popular element forms) is developed in this section.As shown in Figure 2, a 4-node, 12 DOFs (three DOFs per node) quadrilateral thick plate elementis considered. Different with the usual finite elements, the element shape is quite free: it can be aconvex quadrangle, a degenerated triangle, or even a concave quadrangle. The nodal displacementvector of the element is:

qe D�w1 x1 y1 w2 x2 y2 w3 x3 y3 w4 x4 y4

T: (43)



The locking-free formulae of Timoshenko’s beam theory, which are expressed by the nodal displace-ments [64], are used to determine the shear strains and the displacements along each element edge.Similar techniques have been used in various displacement-based models [22–25, 65]. As shown inFigure 3, the deflection w, rotation s , and shear strain � for the Timoshenko beam element ij aregiven by [64]:

wij D�1 � s C .1 � 2ıij /Z3

wi C

�s � .1 � 2ıij /Z3

wj

Clij

2

�Z2 C .1 � 2ıij /Z3

si �

lij

2

�Z2 � .1 � 2ıij /Z3

sj

(44)

sij D �6

lij.1 � 2ıij /Z2wi C

6

lij.1 � 2ıij /Z2wj

C�1 � s � 3.1 � 2ıij /Z2

si C

�s � 3.1 � 2ıij /Z2

sj

(45)

�ij D ıij� (46)

Table V. Normalized results of the central deflections for symmetric mesh distortion.

MITC4 MISC2 AC-MQ4 S4R CRB1 CRB2 S1 DKQ HDF-P4-11“ Analytical� [12, 30] [30] [40] [67] [68] [68] [68] [69] present solutions

-1.75 — — — — — — — — 0.9421-1.50 — — — failed — — — — 0.9595-1.25 — — — — — — — — 0.9715-1.249 0.7692 0.9099 0.9802 1.0569 1.0917 1.9154 0.8735 1.3391 0.9716-1.00 0.8158 0.9202 1.0032 1.0822 1.0988 1.5296 0.9170 1.3107 0.9790-0.50 0.8957 0.9542 0.9992 1.0704 0.9858 1.0150 0.9557 1.2198 0.98380.00 0.9573 1.0008 0.9842 1.0237 0.9581 0.9581 0.9573 1.1542 0.9794 1.0000�

0.50 0.9842 1.0458 0.9676 0.9755 1.0648 1.0522 0.9209 1.1209 0.97031.00 0.9399 1.0522 0.9352 0.9423 1.0617 1.3020 0.8372 1.1281 0.96301.249 0.8593 1.0174 0.9431 0.9328 0.9976 1.5391 0.7708 1.1051 0.96251.25 — — — — — — — — 0.96251.50 — — — failed — — — — 0.96561.75 — — — — — — — — 0.9738�the standard value of wc=.qL4=100D) is 0.1265

Figure 11. Results for the sensitivity test to symmetric mesh distortion.


218 S. CEN ET AL.

Table VI. Normalized results of the central deflections for asymmetric mesh distortion.

AC-MQ4 S4R CRB1 CRB2 S1 DKQ HDF-P4-11“ Analytical� [40] [67] [68] [68] [68] [69] present solutions

0.00 0.9842 1.0237 0.9581 0.9581 0.9573 1.1542 0.97940.02 0.9842 1.0111 0.9217 0.9209 0.9455 1.1526 0.97930.04 0.9834 0.9755 0.8498 0.8490 0.9123 1.1518 0.97910.06 0.9834 0.9209 0.7889 0.7881 0.8625 1.1510 0.97900.08 0.9834 0.8545 0.7486 0.7478 0.8008 1.1502 0.97870.10 0.9834 0.7818 0.7233 0.7225 0.7328 1.1502 0.9786 1.0000�

0.15 0.9834 0.6016 0.6941 0.6933 0.5660 1.1470 0.98520.20 0.9826 0.4530 0.6862 0.6854 0.4285 1.1447 0.97780.30 0.9818 0.2632 0.6893 0.6885 0.2514 1.1407 0.97710.50 0.9802 0.1091 0.7170 0.7209 0.1067 1.1344 0.97570.80 0.9723 0.0427 0.7692 0.7976 0.0435 1.1289 0.97461.00 0.9613 0.0269 0.7984 0.8648 0.0277 1.1273 0.97461.50 0.9154 0.0111 0.7968 1.0988 0.0119 1.1312 0.97642.00 0.8577 0.0055 0.6055 1.3399 0.0063 1.1455 0.97682.49 0.7992 0.0040 0.3360 1.5549 0.0040 1.1613 0.9680�the standard value of wc=.qL4=100D) is 0.1265

Figure 12. Results for the sensitivity test to asymmetric mesh distortion.

with8ˆˆ<ˆˆˆ:

� D2

lij.�wi C wj / � si � sj

Z2 D s.1 � s/

Z3 D s.1 � s/.1 � 2s/

ıij D6ij

1C 12ij

ij DDd

Cd l2ij

;

(47)

where Dd and Cd denote the beam’s bending and shear stiffness, respectively, and will be replacedby D and C ; lij is the beam’s length.

Along each plate element edge ij, linear variation is assumed for the normal rotation n :

nij D .1 � s/ ni C s nj : (48)



Figure 13. Distorted meshes for a quarter of clamped square plate.

Figure 14. Mesh 4 � 4 for Razzaque’s 60˚ skew plate.

Thus, the deflection, tangential, and normal rotations along each plate element edge can bedetermined by using the Equations (44) to (48), as shown in Figure 4.

The relationship between . n; s/ and . x; y/ along the edge ij are as follows:

² n s

³ij

D1

lij

�yij xij�xij � yij

�² x y

³ij

;yij D yi � yjxij D xi � xj

(49)

Substitution of Equation (49) into the Equations from (44) to (48) yields

wij D�1 � s C .1 � 2ıij /Z3

wi C

�s � .1 � 2ıij /Z3

wj

�1

2

�Z2 C .1 � 2ıij /Z3

�xij xi C yij yi

�

C1

2

�Z2 � .1 � 2ıij /Z3

�xij xj C yij yj

�(50)


220 S. CEN ET AL.

Table VII. Razzaque’s 60˚ skew plate: Normalized results of the central deflection and moment.

Mesh N�N 2 � 2 4 � 4 6 � 6 8 � 8 12 � 12 16 � 16 Reference

(a) Normalized central deflection wc=wref

MITC4 [12] 0.5004 0.8480 0.9278 0.9578 0.9799 —DKMQ [18] 0.8390 0.9685 0.9854 0.9913 0.9953 —MiSP4 [19] 0.6444 0.9137 — 0.9794 — 0.9936MMiSP4 [19] 0.3750 0.8393 — 0.9571 — 0.9858MiSP4+ [20] 0.8829 0.9679 — 0.9912 — 0.9975ARS-Q12 [24] 0.8390 0.9680 0.9854 0.9913 0.9955 0.9969 1.0000�

RDKQM [25] 0.8391 0.9687 0.9855 0.9914 0.9955 —MISC2 [30] 0.4709 0.8464 0.9285 0.9578 0.9787 0.9860DSQ [72] 0.8862 0.9699 — 0.9868 — 0.9907AC-MQ4 [40] 0.9539 0.9950 0.9974 0.9981 0.9985 0.9986HDF-P4-11“ 0.9624 0.9931 0.9947 0.9953 0.9956 0.9957

(b) Normalized central moment My=Mref

MITC4 [12] 0.3952 0.8093 0.9083 0.9480 0.9772 —DKMQ [18] 0.9615 1.0011 1.0022 1.0022 1.0011 —MiSP4 [19] 0.6326 0.9150 — 0.9827 — 0.9977MMiSP4 [19] 0.4888 0.8034 — 0.9440 — 0.9872MiSP4+ [20] 0.7509 0.9474 — 0.9918 — 0.9985ARS-Q12 [24] 0.9642 1.0006 1.0021 1.0017 1.0014 1.0013 1.0000��

RDKQM [25] 0.9336 1.0090 1.0004 0.9993 1.0002 1.0006MISC2 [30] 0.4889 0.8657 0.9391 0.9657 0.9853 0.9923DSQ [70] 0.9854 0.9995 — 1.0021 — 1.0014AC-MQ4 [40] 1.0562 1.1546 1.0736 1.0422 1.0198 1.0118HDF-P4-11“ 1.0003 1.0232 1.0067 1.0028 1.0018 1.0015� the standard reference value is 0:7945 � 10�9 [71]; �� the standard reference value is 0:9589 � 10�3 [71]

sij D �6

lij.1 � 2ıij /Z2wi C

6

lij.1 � 2ıij /Z2wj

�1

lij

�1 � s � 3.1 � 2ıij /Z2

�xij xi C yij yi

�(51)

�1

lij

�s � 3.1 � 2ıij /Z2

�xij xj C yij yj

�

nij D1

lij.1 � s/.�yij xi C xij yi /C

1

lijs.�yij xj C xij yj /: (52)

Therefore, along the element edge ij, the boundary displacement vector can be written as

dij D

8<: n

s

w

9>=>;ij

D Nj�ij

qe (53)

with

Nj�ijD

2664

0 N 12 N 13 0 N 15 N 16 0 N 18 N 19 0 N 1;11 N 1;12

N 21 N 22 N 23 N 24 N 25 N 26 N 27 N 28 N 29 N 2;10 N 2;11 N 2;12

N 31 N 32 N 33 N 34 N 35 N 36 N 37 N 38 N 39 N 3;10 N 3;11 N 3;12

3775 ; (54)

in which all components are null except those associated with the side nodal displacements. Thecorresponding expressions are listed in the Appendix of this paper.



Figure 15. Convergence test for central deflection wc and moment My of Razzaque’s 60˚ skew plate.

Figure 16. Mesh 4 � 4 for Morley’s 30˚ skew plate.

The assumed resultant force vector is given by Equation (24). In order to avoid the spuriouszero modes in the single element stiffness matrix, the number k of the fundamental analytical solu-tions used for the general solution F 0i (given by Equation (19)) must be no less than 9. Here, letk D 11, which means that the first 11 fundamental analytical solutions F 0i .i D 1 � 11/ givenby Table I are used as the trial functions. Therefore, 11 unknown coefficients ˇi are introduced(Equations (19) and (21)). In Equation (21), OS will be a 5�11matrix. From Table I, it can be provedthat the assumed moments and shear forces possess second-order and first-order completeness inCartesian coordinates, respectively.

The shear locking problem will not occur in the present model. The first and main reason isbecause it is a stress-based element. Equation (27) shows the shear deformations tend to zero (andthe thin plate assumption is recovered) for an increasing C (defined by Equation (9)). The second


222 S. CEN ET AL.

Table VIII. Results of deflections and principal moments at the center of Morley’s 30˚ skewplate (L=h D 1000).

Morley’s solutionsMesh N�N 4 � 4 8 � 8 16 � 16 32 � 32 for thin plate[73]

(a) Central deflection wo=�qL4=1000D

�

MITC4 [12] 0.358 0.343 0.343 0.362Q4BL [15] 0.512 0.439 0.429 0.424DKMQ [18] 0.760 0.507 0.443 0.425ARS-Q12 [24] 0.756 0.506 0.442 0.424 0.408Q4-U2 [37] 0.009 0.081 0.203 0.314AC-MQ4 [40] 0.431 0.409 0.405 0.405HDF-P4-11“ 0.462 0.426 0.419 0.416

(b) Central max principal moment Mmax=�qL2=100

�


(c) Central min principal moment Mmin=�qL2=100

�


reason is because the proposed element is a hybrid-Trefftz stress element, meaning that a basiscontaining exact thin plate solutions leads to improved rates of convergence to the thin plate solution.

Here, the new element model is named by HDF-P4-11“ to record that it is a ‘HybridDisplacement Function element for Plate and with 4 nodes and 11“ ’. Once qe is solved, fromEquations (24) to (37), the element resultant force vector can be obtained:

R D �OSM�1�M� CHqe

�C R�: (55)

Within an element, the resultant forces at arbitrary position can be evaluated directly by substitutingits Cartesian coordinates into Equation (55).

For evaluating the matrices in Equations (34) and (35) through the numerical integration method,the global coordinates x and y should be transformed into the forms of local isoparametriccoordinates and �. The transformation relations are given by

x D

4XiD1

N 0i .; �/xi ; y D

4XiD1

N 0i .; �/yi ; (56)

where N 0i .; �/ (i D 1 � 4) are the shape functions for the conventional 4-node isoparametric

elements; and .xi ; yi / (i D 1 � 4) are the global Cartesian coordinates of node i .For evaluating the matrices M and M� given by Equation (34), a standard 3 � 3 Gauss integration

scheme is employed. Here, it should be noted that these area integrals are not transformed intoboundary integrals. Three integration points are needed at each element edge for evaluating H andV given by Equation (35).



4. NUMERICAL EXAMPLES

Seven classic benchmark examples using traditional and new severely distorted mesh divisions areemployed to assess the performance of the proposed element HDF-P4-11“. To illustrate the superi-ority of the new element sufficiently, the results calculated by some other well-known quadrilateralelements are also listed and plotted for comparison.

4.1. Eigenvalues and rank

For various element shapes (such as convex and concave quadrangles and degenerated triangle) fromvery thin to moderately thick plates, the element stiffness matrix of HDF-P4-11“ always producesonly three zero eigenvalues corresponding to three rigid body modes. So, element HDF-P4-11“ hasa proper rank and will not provide spurious zero-energy modes.

4.2. Irons patch tests

Figure 5 shows the Irons patch test problems. Three mesh types A, B, and C are used, in whichMesh B and Mesh C contain seriously distorted elements degenerated into triangle and concavequadrangles, respectively. The patch is divided by five elements, subjected to various line distributedloads along the patch boundaries, and constrained by the boundary conditions for eliminating rigidbody motions. Three different span-thickness ratios 2a/h = 1000, 100,and 10 are considered.

(a) Constant bending moment case (Mn D 1). As shown in Figure 5a, for the rectangular platepatch subjected to moment Mn D 1 along its all edges, the equivalent nodal forces can be

Table IX. Results of deflections and principal moments at the center of Morley’s 30˚ skew plate(L=h D 100).

Morley’s solutionsMesh N�N 4 � 4 8 � 8 16 � 16 32 � 32 for thin plate[73] 3D Solution[74]

(a) Central deflection wo=�qL4=1000D

�

MITC4 [12] 0.359 0.357 0.383 0.404Q4BL [15] 0.513 0.440 0.431 0.427DKMQ [18] 0.757 0.504 0.441 0.423ARS-Q12 [24] 0.754 0.503 0.440 0.423 0.408 0.423Q4-U2 [37] 0.163 0.292 0.374 0.405AC-MQ4 [40] 0.431 0.410 0.407 0.409HDF-P4-11“ 0.463 0.427 0.421 0.420

(b) Central max principal moment Mmax=�qL2=100

�


(c) Central min principal moment Mmin=�qL2=100

�



224 S. CEN ET AL.

Figure 17. Convergence test for central deflections and principle moments of Morley’s 30˚ skew plate.

x

y

45°

C L

L

Figure 18. A 45˚ clamped rhombic skew plate, L D 8, h=L D 0:01, E D 8:736 � 107, and � D 0:3.

calculated directly from the linear normal rotation n given by Equation (48). The computedresults of bending moments Mx. D 1/ and My. D 1/, twisting moment Mxy. D 0/, andshear forces Tx.D 0/ and Ty.D 0/, at any point, are exact for all span-thickness ratio cases.Actually, for the new element, such solutions have been included in the domain approximationbasis defined in Table I, and the degree of the boundary displacement approximation defined



Table X. Normalized center deflection of a clamped 45˚ rhombic skewplate (Figure 18) subjected to uniformly distributed load, Reference

solution [75]: wc D 0:006032.

Mesh ADQ4 [76] QQ21-11[57] HDF-P4-11“

2 � 2 0.948 0.968 1.0184 � 4 0.970 0.985 1.0046 � 6 0.978 0.986 0.9968 � 8 0.989 0.994 0.99910 � 10 0.995 0.997 1.00112 � 12 0.997 0.999 1.001

Figure 19. Circular plate problem.

in Equations (44)–(47) contains the degree of the exact solution and, consequently, thoseof the displacement boundary conditions. Furthermore, the mapping of the geometry of theelement is independent from the domain and boundary bases. Therefore, exact solutions canbe easily obtained by HDF-P4-11“ by using arbitrary distorted meshes. The situations arequite similar for Test (b).

(b) Constant twisting moment case (Mns D 1). As shown in Figure 5b, for the rectangular platepatch subjected to moment Mns D 1 along its four edges, the equivalent nodal forces for anarbitrary thickness h can be obtained from the tangential rotation s defined by Equation (51).In all cases, the numerical results of Mxy. D 1/, Mx. D 0/, My. D 0/ Tx. D 0/, andTy. D 0/ obtained by the element HDF-P4-11“ are exact. The reasons have been explainedin Test (a).

In summary, element HDF-P4-11“ can pass all previously mentioned patch tests. However, itshould be noted that, for the new element HDF-P4-11“, the previous two tests are rather debuggingtests than performance tests.

4.3. Square plate

This is a classical benchmark for testing plate bending elements. Figure 6 gives three mesh typesused for this example, in which only a quarter of the plate is considered. Mesh A and Mesh B areconventional and distorted meshes, respectively, that are often used in other literatures. Mesh C is anew and specially distorted mesh for testing quadrilateral elements in which all elements degenerateinto triangles, and cannot be used by most existing quadrilateral elements.

Three boundary condition (BC) cases are considered, including the clamped BC (w D 0, n D 0,and s D 0), the soft simply supported (SS1) BC (w D 0) and the hard simply supported (SS2)BC (w D 0 and s D 0). The edge length and the thickness of the square plate are denoted byL and h, respectively, and Poisson’s ratio is � D 0:3. The plate, from thick case (h=L D 0:1) tovery thin case

�h=L D 10�30

�, is subjected to a uniform transverse load q D 1. The dimensionless

results of deflections and moments (here, let L D 1 and D D 1) at the plate center are presentedin Tables II to IV. The new element HDF-P4-11“ exhibits excellent performance for both precisionand convergence, and even performs very well in the severely distorted mesh (Mesh C, degeneratedtriangular mesh) where other quadrilateral models cannot work.


226 S. CEN ET AL.

The comparisons of the results (using regular Mesh A) obtained by present element HDF-P4-11“ and some other quadrilateral elements, including MITC4 [12, 30], Q4BL [15], DKMQ[18], ARS-Q12 [24], MISC2 [30], AC-MQ4 [40], PQI [66] S4R [67], and hybrid-Trefftz elementsQQ21 � 11 [57], Q21–13 [59] (8-node, 16-DOF), are plotted in Figures 7 to 9.

4.4. Test for checking the sensitivity problem to mesh distortions

A quarter of thin (h=L D 0:001) clamped square plate is divided by a very coarse mesh (2 � 2).As shown in Figure 10, two distortion modes are considered. In the first distortion mode (symmetricdistortion, see Figure 10a), the central mesh node will be moved along the main diagonal of theplate to the corner node. Results of the deflections at the node C (the central point of the plate),obtained by present element HDF-P4-11“ and some other models, are listed in Table V and plottedin Figure 11. In the second distortion mode (asymmetric distortion, see Figure 10b), the central meshnode will be moved parallel to the plate edge. Results are given in Table VI and plotted in Figure 12.

In the symmetric distortion mesh (Figure 10a), when the absolute value of the distortion param-eter � reaches 1.25, one element will degenerate into a triangle; and when it exceeds 1.25, thatelement will degenerate into a concave quadrangle. Under these situations, most elements will fail,whereas the new element HDF-P4-11“ still performs well. From the numerical results, it can beconcluded that the present element HDF-P4-11“ is the most robust model and quite insensitive tothe severe mesh distortions. Although other hybrid-Trefftz plate bending elements may also possesssuch character, few reports can be found in related literatures.

Table XI. Normalized center deflection wc=wref and moments Mc=Mref of simply-supported (SS1) circular plates subjected to a uniform load.

Mesh N 3 12 48 192 Analytical

(a) h=R D 0:02.h D 0:1/ wc=wref

MITC4 [12] 0.9146 0.9801 0.9951 —DKMQ [18] 0.9573 0.9900 0.9976 —ARS-Q12 [24] 0.9573 0.9897 0.9974 0.9993 1.0000CHRM [32] — 0.9671 0.9923 0.9981 (the referenceAC-MQ4 [40] 1.0145 1.0058 1.0014 1.0002 value is 39831.5)HDF-P4-11“ 1.0242 1.0065 1.0017 1.0004

Mc=Mref


(b) h=R D 0:2.h D 1/ wc=wref


Mc=Mref




Table XII. Normalized center deflection wc=wref and momentsMc=Mref of clamped circularplates subjected to a uniform load.

Mesh N 3 12 48 192 Analytical

(a) h=R D 0:02.h D 0:1/ wc=wref


Mc=Mref


(b) h=R D 0:2.h D 1/ wc=wref


Mc=Mref


4.5. Another test for checking the sensitivity problem to mesh distortions

Reference [70] gave another test to check the quadrilateral element’s ability to deal with the prob-lematic geometry. A quarter of thin square plate with symmetry and clamped BCs and subjectedto a uniformly distributed load is used. The model is the same as example 4.3. Figure 13 gives thenormalized results of the central deflection and moment of the plate.

It can be seen that, even though some meshes contain concave quadrilateral elements, goodresults can still be obtained. This test again confirms that the new quadrilateral element HDF-P4-11“ is quite insensitive to the severely distorted meshes containing degenerated triangle and concavequadrangle.

4.6. Skew plates subjected to uniformly distributed load

(i) A 60˚ skew thin plate (span-thickness ratio of h=L D 0:001), which has been investigatedby Razzaque [71] in 1973, is shown in Figure 14. Two opposite edges of this plate are free,whereas the other two edges are soft simply supported (SS1). The central (at node C, seeFigure 14) transversal deflection w and the bending moment My are calculated. Results arelisted in Table VII and plotted in Figure 15. The new element produces excellent results forboth w and My , even when the very coarse mesh (2 � 2) is used.

(ii) A 30˚ skew plate with SS1 BC, which has been studied by Morley [73] under the thin plateassumptions, is shown in Figure 16. This test is a more critical one because singularity appears


228 S. CEN ET AL.

Table XIII. Distribution of resultant forces along radius of clamped circular plates, results by 48 HDF-P4-11ˇ elements.

r 0 0.625 1.25 1.875 2.5 3.125 3.75 4.375 5

Mr

Thin plate .h=R D 0:02)Solution I1 — 1.93708 1.69573 1.29314 0.72964 -0.00821 -0.87186 -1.91159 -3.13270Solution II2 2.01760 1.93672 1.69387 1.28746 0.71684 0.00409 -0.89902 -1.93441 —Solution A3 2.01760 1.93690 1.69480 1.29030 0.72324 -0.00206 -0.88544 -1.92300 -3.13270

Thick plate .h=R D 0:2)

Solution I1 — 1.92698 1.67899 1.27053 0.70330 -0.02000 -0.90286 -1.95031 -3.17640Solution II2 2.01460 1.94122 1.70741 1.31347 0.75756 0.04202 -0.83050 -1.86269 —Solution A3 2.01460 1.93410 1.69320 1.29200 0.73043 0.01101 -0.86668 -1.90650 -3.17640Analytical 2.03125 1.95068 1.70898 1.30615 0.74219 0.01709 -0.86914 -1.91650 -3.125

M�

Thin plate .h=R D 0:02)Solution I1 — 1.97260 1.83464 1.60432 1.28348 0.87150 0.36906 -0.22450 -0.91517Solution II2 2.01760 1.97000 1.83036 1.59928 1.27732 0.86560 0.36372 -0.23066 —Solution A3 2.01760 1.97130 1.83250 1.60180 1.28040 0.86855 0.36639 -0.22758 -0.91517

Thick plate .h=R D 0:2)Solution I1 — 1.97639 1.84477 1.61991 1.30057 0.88516 0.37944 -0.21152 -0.88304Solution II2 2.01460 1.96001 1.81303 1.57249 1.24083 0.81722 0.29854 -0.31534 —Solution A3 2.01460 1.96820 1.82890 1.59620 1.27070 0.85119 0.33899 -0.26343 -0.88304Analytical 2.03125 1.98486 1.84570 1.61377 1.28906 0.87158 0.36133 -0.24170 -0.9375

Tr

Thin plate .h=R D 0:02)Solution I1 — -0.31066 -0.61989 -0.93597 -1.26855 -1.61698 -1.96685 -2.31307 -2.68790Solution II2 0.00184 -0.30658 -0.60705 -0.88201 -1.12645 -1.37122 -1.62735 -1.94673 —Solution A3 0.00184 -0.30862 -0.61347 -0.90899 -1.19750 -1.49410 -1.79710 -2.12990 -2.68790

Thick plate .h=R D 0:2)

Solution I1 — -0.31109 -0.62091 -0.93258 -1.24825 -1.57111 -1.89035 -2.20774 -2.53030Solution II2 0.00103 -0.30981 -0.61743 -0.92252 -1.21735 -1.50929 -1.80425 -2.10486 —Solution A3 0.00103 -0.31045 -0.61917 -0.92755 -1.23280 -1.54020 -1.84730 -2.15630 -2.53030Analytical 0 -0.3125 -0.625 -0.9375 -1.25 -1.5625 -1.875 -2.1875 -2.5

1. Solution I: direct solutions by equation (55) at inner elements;2. Solution II: direct solutions by equation (55) at outer elements;3. Solution A: smoothed solutions by averaging Solution I and Solution II.

in the bending moment at the obtuse corner. Two span-thickness ratios (L=h D 1000 and100) are considered. The principal bending moments and deflections at the central node O arecalculated. Table VIII, Table IX, and Figure 17 present the dimensionless results obtained bythe new element HDF-P4-11“ and other models. This problem has also been solved as a 3Delastic case by Babuška and Scapolla [74]. Their solution for h=L D 0:01 is more close tonumerical results of the present element HDF-P4-11“.

(iii) A clamped 45˚ rhombic skew thin plate (span-thickness ratio h=L D 0:01) subjected to uni-formly distributed load, which has also been studied by Morley [75], is shown in Figure 18.Table X shows the normalized results for the deflection at the center C of the plate. The resultsobtained by other 4-node models are also given for comparison, including a hybrid stressshell element ADQ4 with 13 stress coefficients [76], and the hybrid-Trefftz element QQ21�11with 11 Trefftz terms and 12 standard DOFs [57]. It can be seen that the present elementHDF-P4-11“ provides the best answers.



Figure 20. Resultant forces (smoothed) obtained by HDF-P4-11“ along the radius of clamped circular platesubjected to uniform load (48 elements).

4.7. Circular plate subjected to uniformly distributed load

Figure 19 shows a circular plate subjected to a uniform load q D 1. The SS1 BC (w D 0) and theclamped BC (w D 0, n D 0, and s D 0) are both considered. The geometries and the meshes forthe plate are given in Figure 19. According to the symmetry, only a quarter of the plate is modeled.The radius is R D 5, Poisson’s ratio is � D 0:3, and Young’s modulus is E D 10:92. Two differentthickness-radius ratio cases (h=R D 0:02 and 0.2) are considered. The analytical solutions are foundin References [19, 20]. Results obtained by the new element HDF-P4-11“ and some other modelsare given by Tables XI and XII.

Table XIII and Figure 20 give the results (obtained by 48 elements) of bending momentsMr ,M�

and shear force Tr along the radius of thin (h=R D 0:02) and thick (h=R D 0:2) clamped circularplates. One can see that the new element can provide good results, including smoothed and directsolutions, coinciding with the exact solutions.

5. CONCLUDING REMARKS

A simple hybrid displacement function element method is established for developing Mindlin–Reissner plate bending elements (valid for both thin and moderately thick plates). Because the trialfunctions of the domain stress approximations a priori satisfy governing equations, it is consistent


230 S. CEN ET AL.

with the hybrid-Trefftz stress element method. Indeed, all advantages of the hybrid-Trefftz stresselement method can be observed in the new element model. The proposed method still follows thestandard procedure for the hybrid stress element method. But its variational formulations are modi-fied to be expressed by a displacement function F (Note: F is not displacement), which can be usedto derive displacement components satisfying all governing equations. Then, the assumed elementresultant force fields, which can satisfy all related governing equations, are derived from the funda-mental analytical solutions of such F . Similar to the hybrid stress element method, the displacementfields within element domain are not directly considered and used, although they can also be derivedfrom F .

Compared with the other hybrid-Trefftz stress element methods, the distinction is that the locking-free formulae from the Timoshenko’s beam theory [64], which have been successfully applied bydisplacement-based plate bending elements, are employed as the deflections, rotations, and shearstrains along element edges. And then, a 4-node, 12 DOFs quadrilateral element HDF-P4-11“ isformulated. This new model has a proper rank and will not provide spurious zero-energy modes (seeSection 4.1), which means that the correct balancing of the domain and boundary approximationsmay be established.

Numerical examples show that element HDF-P4-11“ is free of shear locking, and can producehigh precision results for all field variables. Particularly, it is quite insensitive to various severe meshdistortions, even performs well when a severely distorted mesh containing concave quadrilateral anddegenerated triangular elements is employed. So, it is a robust shape-free FEM.

The proposed method possesses advantages from both analytical and discrete methods, and canbe easily integrated into the standard framework of finite element programs. An interesting futurework is to combine the proposed element HDF-P4-11“ and the shape-free membrane element HSF-Q4�-7“ [45] with drilling DOF, to develop new shape-free quadrilateral shell element model.

APPENDIX

The matrix Nj�ij

in Equation (54) has different expressions along four element edges.

� Along edge 12, the non-zero components of the matrix are

N 12 D �y12

l12.1 � s/; N 13 D

x12

l12.1 � s/; N 15 D �

y12

l12s; N 16 D

x12

l12s;

N 21 D �6

l12.1 � 2ı12/Z2; N 22 D �

x12

l12Œ1 � s � 3.1 � 2ı12/Z2�;

N 23 D �y12

l12Œ1 � s � 3.1 � 2ı12/Z2�; N 24 D

6

l12.1 � 2ı12/Z2;

N 25 D �x12

l12Œs � 3.1 � 2ı12/Z2�; N 26 D �

y12

l12Œs � 3.1 � 2ı12/Z2�;

N 31 D 1 � s C .1 � 2ı12/Z3; N 32 D �x12

2ŒZ2 C .1 � 2ı12/Z3�;

N 33 D �y12

2ŒZ2 C .1 � 2ı12/Z3�; N 34 D s � .1 � 2ı12/Z3;

N 35 Dx12

2ŒZ2 � .1 � 2ı12/Z3�; N 36 D

y12

2ŒZ2 � .1 � 2ı12/Z3�:

� Along edge 23, the non-zero components of the matrix are

N 15 D �y23

l23.1 � s/; N 16 D �

x23

l23.1 � s/; N 18 D �

y23

l23s; N 19 D

x23

l23s;

N 24 D �6

l23.1 � 2ı23/Z2; N 25 D �

x23

l23Œ1 � s � 3.1 � 2ı23/Z2�;

N 26 D �y23

l23Œ1 � s � 3.1 � 2ı23/Z2�; N 27 D

6

l23.1 � 2ı23/Z2;



N 28 D �x23

l23Œs � 3.1 � 2ı23/Z2�; N 29 D �

y23

l23Œs � 3.1 � 2ı23/Z2�;

N 34 D 1 � s C .1 � 2ı23/Z3; N 35 D �x23

2ŒZ2 C .1 � 2ı23/Z3�;

N 36 D �y23

2ŒZ2 C .1 � 2ı23/Z3�; N 37 D s � .1 � 2ı23/Z3;

N 38 Dx23

2ŒZ2 � .1 � 2ı23/Z3�; N 39 D

y23

2ŒZ2 � .1 � 2ı23/Z3�:

� Along edge 34, the components of the matrix are

N 18 D �y34

l34.1 � s/; N 19 D

x34

l34.1 � s/; N 1;11 D �

y34

l34s; N 1;12 D

x34

l34s;

N 27 D �6

l34.1 � 2ı34/Z2; N 28 D �

x34

l34Œ1 � s � 3.1 � 2ı34/Z2�;

N 29 D �y34

l34Œ1 � s � 3.1 � 2ı34/Z2�; N 2;10 D

6

l34.1 � 2ı34/Z2;

N 2;11 D �x34

l34Œs � 3.1 � 2ı34/Z2�; N 2;12 D �

y34

l34Œs � 3.1 � 2ı34/Z2�;

N 37 D 1 � s C .1 � 2ı34/Z3; N 38 D �x34

2ŒZ2 C .1 � 2ı34/Z3�;

N 39 D �y34

2ŒZ2 C .1 � 2ı34/Z3�; N 3;10 D s � .1 � 2ı34/Z3;

N 3;11 Dx34

2ŒZ2 � .1 � 2ı34/Z3�; N 3;12 D

y34

2ŒZ2 � .1 � 2ı34/Z3�

� Along edge 41, the components of the matrix are

N 1;11 D �y41

l41.1 � s/; N 1;12 D

x41

l41.1 � s/; N 12 D �

y41

l41s; N 13 D

x41

l41s;

N 2;10 D �6

l41.1 � 2ı41/Z2; N 2;11 D �

x41

l41Œ1 � s � 3.1 � 2ı41/Z2�;

N 2;12 D �y41

l41Œ1 � s � 3.1 � 2ı41/Z2�; N 21 D

6

l41.1 � 2ı41/Z2;

N 22 D �x41

l41Œs � 3.1 � 2ı41/Z2�; N 23 D �

y41

l41Œs � 3.1 � 2ı41/Z2�;

N 3;10 D 1 � s C .1 � 2ı41/Z3; N 3;11 D �x41

2ŒZ2 C .1 � 2ı41/Z3�;

N 3;12 D �y41

2ŒZ2 C .1 � 2ı41/Z3�; N 31 D s � .1 � 2ı41/Z3;

N 32 Dx41

2ŒZ2 � .1 � 2ı41/Z3�; N 33 D

y41

2ŒZ2 � .1 � 2ı41/Z3�:

ACKNOWLEDGEMENTS

The authors would like to acknowledge the financial supports of the National Natural Science Foundation ofChina (11272181), the Specialized Research Fund for the Doctoral Program of Higher Education of China(20120002110080), and the National Basic Research Program of China (Project No. 2010CB832701).

REFERENCES

1. Zienkiewicz OC, Taylor RL. The Finite Element Method, Vol. 2. Solid Mechanics (5th edn). Butterworth-Heinemann:Oxford, 2000.

2. Bathe KJ. Finite Element Procedures. Prentice Hall: New Jersey, 1996.


232 S. CEN ET AL.

3. Long YQ, Cen S, Long ZF. Advanced Finite Element Method in Structural Engineering. Springer-Verlag GmbHBerlin Heidelberg & Tsinghua University Press: Beijing, 2009.

4. Mindlin RD. Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates. Journal of AppliedMechanics-Transactions of the ASME 1951; 18(1):31–38.

5. Reissner E. The effect of transverse shear deformation on the bending of elastic plates. Journal of Applied Mechanics-Transactions of the ASME 1945; 12(2):69–77.

6. Zienkiewicz OC, Taylor RL, Too JM. Reduced integration technique in general analysis of plates and shells.International Journal for Numerical Methods in Engineering 1971; 3(2):275–290.

7. Hughes TJR, Taylor RL, Kanoknukulchai W. A simple and efficient finite element for plate bending. InternationalJournal for Numerical Methods in Engineering 1977; 11(10):1529–1543.

8. Hinton E, Huang HC. A family of quadrilateral Mindlin plate element with substitute shear strain fields. Computers& Structures 1986; 23(3):409–431.

9. Belytschko T, Tsay CS, Liu WK. A stabilization matrix for the bilinear Mindlin plate element. Computer Methodsin Applied Mechanics and Engineering 1981; 29(3):313–327.

10. Belytschko T, Tsay CS. A stabilization procedure for the quadrilateral plate element with one point quadrature.International Journal for Numerical Methods in Engineering 1983; 19(3):405–419.

11. Lee SW, Wong SC. Mixed formulation finite elements for Mindlin theory plate bending. International Journal forNumerical Methods in Engineering 1982; 18(9):1297–1311.

12. Bathe KJ, Dvorkin EN. A four-node plate bending element based on Mindlin—Reissner plate theory and a mixedinterpolation. International Journal for Numerical Methods in Engineering 1985; 21(2):367–383.

13. 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(3):697–722.

14. Onate E, Zienkiewicz OC, Suarez B, Taylor RL. A general methodology for deriving shear constrained Reissner–Mindlin plate elements. International Journal for Numerical Methods in Engineering 1992; 33(2):345–367.

15. Zienkiewicz OC, Xu ZN, Zeng LF, Samuelsson A. Linked interpolation for Ressiner-Mindlin plate element: Part I-a simple quadrilateral. International Journal for Numerical Methods in Engineering 1993; 36(18):3043–3056.

16. Taylor RL, Auricchio F. Linked interpolation for Reissner-Mindlin plate elements: Part II—a simple triangle.International Journal for Numerical Methods in Engineering 1993; 36(18):3057–3066.

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

18. Katili I. A new discrete Kirchhoff-Mindlin element based on Mindlin-Reissner plate theory and assumed shear strainfields—Part II: an extended DKQ element for thick-plate bending analysis. International Journal for NumericalMethods in Engineering 1993; 36(11):1885–1908.

19. Ayad R, Dhatt G, Batoz JL. A new hybrid-mixed variational approach for Reissner–Mindlin plates. The MiSP Model.International Journal for Numerical Methods in Engineering 1998; 42:1149–1179.

20. Ayad R, Rigolot A. An improved four-node hybrid-mixed element based upon Mindlin’s plate theory. InternationalJournal for Numerical Methods in Engineering 2002; 55:705–731.

21. Ibrahimbegovic A. Quadrilateral finite elements for analysis of thick and thin plates. Computer Methods in AppliedMechanics and Engineering 1993; 110:195–209.

22. Soh AK, Long ZF, Cen S. A Mindlin plate triangular element with improved interpolation based on Timoshenko’sbeam theory. Communications in Numerical Methods in Engineering 1999; 15(7):527–532.

23. Soh AK, Long ZF, Cen S. A new nine DOF triangular element for analysis of thick and thin plates. ComputationalMechanics 1999; 24(5):408–417.

24. Soh AK, Cen S, Long ZF, Long YQ. A new twelve DOF quadrilateral element for analysis of thick and thin plates.European Journal of Mechanics A-Solids 2001; 20(2):299–326.

25. Chen WJ, Cheung YK. Refined quadrilateral element based on Mindlin-Reissner plate theory. International Journalfor Numerical Methods in Engineering 2000; 47:605–627.

26. Zhang HX, Kuang JS. Eight-node Reissner-Mindlin plate element based on boundary interpolation usingTimoshenko beam function. International Journal for Numerical Methods in Engineering 2007; 69(7):1345–1373.

27. Vu-Quoc L, Mora JA. A class of simple and efficient degenerated shell elements–Analysis of global spurious-modefiltering. Computer Methods in Applied Mechanics and Engineering 1989; 74:117–175.

28. Vu-Quoc L, Deng H, Tan XG. Geometrically-exact sandwich shells: the static case. Computer Methods in AppliedMechanics and Engineering 2000; 189(1):167–203.

29. Vu-Quoc L, Deng H, Tan XG. Geometrically-exact sandwich shells: the dynamic case. Computer Methods in AppliedMechanics and Engineering 2001; 190(22–23):2825–2873.

30. Nguyen-Xuan H, Rabczuk T, Bordas S, Debongnie JF. A smoothed finite element method for plate analysis.Computer Methods in Applied Mechanics and Engineering 2008; 197:1184–1203.

31. Castellazzi G, Krysl P. Displacement-based finite elements with nodal integration for Reissner-Mindlin plates.International Journal for Numerical Methods in Engineering 2009; 80(2):135–162.

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

33. Hansbo P, Heintz D, Larson MG. A finite element method with discontinuous rotations for the Mindlin-Reissnerplate model. Computer Methods in Applied Mechanics and Engineering 2011; 200(5–8):638–648.



34. Nguyen-Thanh N, Rabczuk T, Nguyen-Xuan H, Bordas S. An alternative alpha finite element method with discreteshear gap technique for analysis of isotropic Mindlin-Reissner plates. Finite Elements in Analysis and Design 2011;47(5):519–535.

35. Falsone G, Settineri D. A Kirchhoff-like solution for the Mindlin plate model: a new finite element approach.Mechanics Research Communications 2012; 40:1–10.

36. Nguyen-Thoi T, Phung-Van P, Nguyen-Xuan H, Thai-Hoang C. A cell-based smoothed discrete shear gap methodusing triangular elements for static and free vibration analyses of Reissner-Mindlin plates. International Journal forNumerical Methods in Engineering 2012; 91(7):705–741.

37. Ribaric D, Jelenic G. Higher-order linked interpolation in quadrilateral thick plate finite elements. Finite Elements inAnalysis and Design 2012; 51:67–80.

38. Vu-Quoc L, Tan XG. Efficient Hybrid-EAS solid element for accurate stress prediction in thick laminated beams,plates, and shells. Computer Methods in Applied Mechanics and Engineering 2013; 253:337–355.

39. da Veiga LB, Buffa A, Lovadina C, Martinelli M, Sangalli G. An isogeometric method for the Reissner-Mindlin platebending problem. Computer Methods in Applied Mechanics and Engineering 2012; 209:45–53.

40. Cen S, Long YQ, Yao ZH, Chiew SP. Application of the quadrilateral area co-ordinate method: a new element forMindlin-Reissner plate. International Journal for Numerical Methods in Engineering 2006; 66(1):1–45.

41. Long YQ, Li JX, Long ZF, Cen S. Area coordinates used in quadrilateral elements. Communications in NumericalMethods in Engineering 1999; 15(8):533–545.

42. Long ZF, Li JX, Cen S, Long YQ. Some basic formulae for area coordinates used in quadrilateral elements.Communications in Numerical Methods in Engineering 1999; 15(12):841–852.

43. Cen S, Long YQ, Yao ZH. A new hybrid-enhanced displacement-based element for the analysis of laminatedcomposite plates. Computers & Structures 2002; 80(9–10):819–833.

44. Fu XR, Cen S, Li CF, Chen XM. Analytical trial function method for development of new 8-node plane elementbased on the variational principle containing Airy stress function. Engineering Computations 2010; 27(4):442–463.

45. Cen S, Zhou MJ, Fu XR. A 4-node hybrid stress-function (HS-F) plane element with drilling degrees of freedom lesssensitive to severe mesh distortions. Computers & Structures 2011; 89(5–6):517–528.

46. Cen S, Fu XR, Zhou MJ. 8-and 12-node plane hybrid stress-function elements immune to severely distortedmesh containing elements with concave shapes. Computer Methods in Applied Mechanics and Engineering 2011;200(29–32):2321–2336.

47. Cen S, Fu XR, Zhou GH, Zhou MJ, Li CF. Shape-free finite element method: the plane Hybrid Stress-Function(HS-F) element method for anisotropic materials. SCIENCE CHINA Physics, Mechanics & Astronomy 2011;54(4):653–665.

48. Pian THH. Derivation of element stiffness matrices by assumed stress distributions. AIAA Journal 1964; 2(7):1333–1336.

49. Jirousek J, Venkatesh A, Hybrid Trefftz plane elasticity elements with p-method capabilities. International Journalfor Numerical Methods in Engineering 1992; 35:1443–1472.

50. Jirousek J. Variational formulation of two complementary hybrid-Treffrz FE models. Communcations in NumericalMethods in Engineering 1993; 9:837–845.

51. Teixeira de Freitas JA. Formulation of elastostatic hybrid-Trefftz stress elements. Computer Methods in AppliedMechanics and Engineering 1998; 153:127–151.

52. Teixeira de Freitas JA, Wang ZM. Hybrid-Trefftz stress elements for elastoplasticity. International Journal forNumerical Methods in Engineering 1998; 43:655–683.

53. Jirousek J, Leon N. A powful finite element for plate beding. Computer Methods in Applied Mechanics andEngineering 1977; 12(1):77–96.

54. Jirousek J. Basis for development of large finite elements locally sarisfying all field equations. Computer Methods inApplied Mechanics and Engineering 1978; 14(1):65–92.

55. Jirousek J, Guex L. The hybrid-Trefftz finite element model and its application to plate bending. International Journalfor Numerical Methods in Engineering 1986; 23:651–693.

56. Jirousek J. Hybrid-Trefftz plate bending elements with p-method capabilityes. International Journal for NumericalMethods in Engineering 1987; 24:1367–1393.

57. Jirousek J, Wròblewski A, Szybinski B. A new 12 DOF quadrilateral element for analysis of thick and thin plates.International Journal for Numerical Methods in Engineering 1995; 38:2619–2638.

58. Jirousek J, Wròblewski A, Qin QH, He XQ. A family of quadrilateral hybrid-Trefftz p-elements for thick plateanalysis. Computer Methods in Applied Mechanics and Engineering 1995; 127:315–344.

59. Petrolito J. Hybrid-Trefftz quadrilateral elements for thick plate analysis. Computer Methods in Applied Mechanicsand Engineering 1990; 78(3):331–351.

60. Petrolito J. Triangular thick plate elements based on a hybrid-Trefftz approach. Computers & Structures 1996;60(6):883–894.

61. Jin FS, Qin QH. A variational principle and hybrid Trefftz finite-element for the analysis of Reissner plates.Computers & Structures 1995; 56(4):697–701.

62. Jin WG, Cheung YK, Zienkiewicz OC. Trefftz method for Kirchhoff plate bending problems. International Journalfor Numerical Methods in Engineering 1993; 36:765–781.

63. Rezaiee-Pajand M, Karkon M. Two efficient hybrid-Trefftz elements for plate bending analysis. Latin AmericanJournal of Solids and Structures 2012; 9(1):43–67.


234 S. CEN ET AL.

64. Hu HC. Variational Principle of Theory of Elasticity with Applications. Science press, Gordon and Breach, Sciencepublisher: Beijing, 1984.

65. Chen YL, Cen S, Yao ZH, Long YQ, Long ZF. Development of triangular flat-shell element using a new thin-thickplate bending element based on semiLoof constrains. Structural Engineering and Mechanics 2003; 15(1):83–114.

66. Ibrahimbegovic A. Plate quadrilateral finite elements with incompatible modes. Communications in AppliedNumerical Methods 1992; 8(8):497–504.

67. Abaqus 6.9. HTML Documentation. Dassault Systèmes Simulia Corp.: Providence, RI, USA, 2009.68. Weissman SL, Taylor RL. Resultant fields for mixed plate bending elements. Computer Methods in Applied

Mechanics and Engineering 1990; 79(3):321–355.69. Batoz JL, Bentahar M. Evaluation of a new quadrilateral thin plate bending element. International Journal for

Numerical Methods in Engineering 1982; 18(11):1655–1677.70. Castellazzi G, Krysl P. A nine-node displacement-based finite element for Reissner-Mindlin plates based on an

improved formulation of the NIPE approach. Finite Elements in Analysis and Design 2012; 58:31–43.71. Razzaque A. Program for triangular bending elements with derivative smoothing. International Journal for

Numerical Methods in Engineering 1973; 6(3):333–343.72. Lardeur P. Développement et Evaluation de Deux Nouveaux Eléments Finis de Plaques et Coques Composites avec

Influence du Cisaillement Transverse. Thèse de Doctorat, UTC, France, 1990.73. Morley LSD. Skew Plates and Structures, International Series of Monographs in Aeronautics and Astronautics.

Macmillan: New York, 1963.74. Babuška I, Scapolla T. Benchmark computation and performance evaluation for a rhombic plate bending problem.

International Journal for Numerical Methods in Engineering 1989; 28(1):155–179.75. Morley LSD. Bending of clamped rectilinear plates. The Quarterly Journal of Mechanics & Applied Mathematics

1964; 17(3):293–317.76. Aminpour MA. Direct formulation of a hybrid 4-node shell element with drilling degrees of freedom. International

Journal for Numerical Methods in Engineering 1992; 35:997–1013.


Hybrid displacement function element method: a simple ...cfli/papers_pdf_files/... · Hybrid...

Documents

Transcript of Hybrid displacement function element method: a simple ...cfli/papers_pdf_files/... · Hybrid...