2012 csmin3 mindlin plate

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Computational Mechanics

Solids, Fluids, Structures, Fluid-

Structure Interactions, Biomechanics,

Micromechanics, Multiscale Mechanics,

Materials, Constitutive Modeling,

Nonlinear Mechanics, Aerodynamics

ISSN 0178-7675

Comput Mech

DOI 10.1007/s00466-012-0705-y

cell-based smoothed three-node Mindlinlate element (CS-MIN3) for static and free

vibration analyses of plates

T. Nguyen-Thoi, P. Phung-Van,

H. Luong-Van, H. Nguyen-Van &

H. Nguyen-Xuan


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Comput Mech

DOI 10.1007/s00466-012-0705-y

ORIGINAL PAPER

A cell-based smoothed three-node Mindlin plate element(CS-MIN3) for static and free vibration analyses of plates

T. Nguyen-Thoi P. Phung-Van

H. Luong-Van H. Nguyen-Van H. Nguyen-Xuan

Received: 13 August 2011 / Accepted: 23 March 2012

Springer-Verlag 2012

Abstract The cell-based strain smoothing technique is

combined with thewell-known three-node Mindlin plate ele-ment (MIN3) to give a so-called the cell-based smoothed

MIN3 (CS-MIN3) for static and free vibration analyses of

plates. In the process of formulating the system stiffness

matrix of the CS-MIN3, each triangular element will be

divided into three sub-triangles, and in each sub-triangle, the

stabilized MIN3 is used to compute the strains and to avoid

the transverse shear locking. Then the strain smoothing tech-

nique on whole the triangular element is used to smooth the

strains on these three sub-triangles. The numerical exam-

ples demonstrated that the CS-MIN3 is free of shear locking,

passes the patch test and shows four superior properties such

as: (1) be a strong competitor to many existing three-node

triangular plate elements in the static analysis, (2) can give

high accurate solutions for problems with skew geometries

in the static analysis, (3) can give high accurate solutions in

free vibration analysis, (4) can provide accurately the values

of high frequencies of plates by using only coarse meshes.

T. Nguyen-Thoi (B) H. Nguyen-XuanDepartment of Mechanics, Faculty of Mathematics & Computer

Science, University of Science, Vietnam National University

HCM, 227 Nguyen Van Cu, Dist. 5, Hochiminh City, Vietnam

e-mail: [email protected]; [email protected]

T. Nguyen-Thoi P. Phung-Van H. Nguyen-XuanDivision of Computational Mechanics, Ton Duc Thang University,

98 Ngo Tat To St., Ward 19, Binh Thanh Dist., Hochiminh City,

Vietnam

H. Luong-Van

Faculty of Civil Engineering, Hochiminh City University

of Technology (HCMUT), Hochiminh City, Vietnam

H. Nguyen-Van

Faculty of Civil Engineering, Hochiminh City University

of Architecture, Hochiminh City, Vietnam

Keywords ReissnerMindlin plate Shear locking

Finite element method (FEM) Cell-based smoothedthree-node Mindlin plate element (CS-MIN3)

Three-node Mindlin plate element (MIN3)

Strain smoothing technique

1 Introduction

In the past 50years, many of plate bending elements based

on the MindlinReissner theory using the first-order shear

deformation (FSDT)hasbeen proposed by researchers.Such

a large amount of elementscan be found in literatures [110],

and in recent papers [1114]. In formulations of a Mindlin

Reissner plate element using the FSDT, the deflection w and

rotations x , y are independent functions and required at

least to be C0-continuous. Due to the complexity of existing

plate elements, research on simpler, more efficient and inex-

pensive plate elements receives continuously strong interest.

Compared with quadrilateral elements, three-node triangular

plate elements are particularly attractive because of its sim-

plicity, easyin automaticmeshingandre-meshingin adaptive

analysis. These elements usually possess high accuracy and

fast convergence speed for displacement solutions. However

for stresses or internal forces, they usually give the low accu-

racy[1517] and need a post-process to improve the solution

[18,19]. In addition, the main difficulty encountered of these

elements is the phenomenon of shear locking which induces

over-stiffness as the plate becomes progressively thinner.

In order to avoid shear locking, many new numerical

techniques and effective modifications have been proposed

and tested, such as the reduced integration and selective

reduced integration schemes proposed by Zienkiewicz et al.

[20] and Hughes et al. [21,22]; the stabilization procedure

proposed by Belytschko et al. [23,24]; free formulation

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method proposed by Bergan and Wang [25], the substitute

shear strain method proposed by Hinton and Huang [26],

mixed formulation/hybrid elements [2730]; etc. However,

these elements are still subjected to some drawbacks such as

instability due to rank deficiency or low accuracy or com-

plex performance. In order to overcome these drawbacks,

Hughes and Tezduzar [31] introduced an assumed natural

strain (ANS) method which allows defining the shear strainsindependently from the approximation of kinematic vari-

ables. In this method, the shear strain field of a 4-node quad-

rilateral element is interpolated independently by rational

constant shear strains along each element side, and the shear

locking problem will be overcome. It has been proved to

be mathematically valid by Bathe and Brezzi [32,33] and

Brezzi et al. [34]. Based on this method, many success-

ful models were then presented, including the mixed inter-

polated tensorial components (MITC) family proposed by

Bathes group [3238]; the discrete ReissnerMindlin fam-

ily [39,40] and the linked interpolation elements (Q4BL [41]

and T3BL [42]) by Zienkiewiczs group; discrete Kirchhoffelements DKT [43] and DKQ [44] proposed by Batoz and

Lardeur; the discrete KirchhoffMindlin elements DKMQ

[45] and DKMT proposed by Katili [46]; the discrete-shear-

gap (DSG) method proposed by Bletzinger et al. [47]; etc.

Also based on the ANS method, Tessler and Hughes [48]

proposed a three-node Mindlin plate element (MIN3) in

which a complete quadratic deflection field is constrained

by continuous shear edge constraints. The MIN3 is free of

shear locking and produces convergent solutions. However,

the accuracy of the MIN3 element is still worse than those

of some others three-node plate elements [14], especially for

the thick plate.

In theother frontier of developingadvanced finite element

technologies, Liu and Nguyen-Thoi et al. [49] have applied a

strain smoothingtechniqueof meshfreemethods [50] into the

conventional finite element method(FEM) using linear inter-

polations to formulate a series of smoothed FEM (S-FEM)

models named as the cell-based S-FEM (CS-FEM) [5155],

the node-based S-FEM (NS-FEM) [5660], the edge-based

S-FEM (ES-FEM) [6166] and the face-based S-FEM (FS-

FEM) [67,68]. In these S-FEM models, the finite element

mesh is used similarly as in the FEMmodels. However, these

S-FEM models evaluate the weakform based on smoothing

domains created from the entities of the element mesh such

as cells/elements,or nodes,or edges,or faces.These smooth-

ing domains can be located inside the elements (CS-FEM)

or cover parts of adjacent elements (NS-FEM, ES-FEM and

FS-FEM). These smoothing domains are linear independent

and hence ensure stability and convergence of the S-FEM

models.

In the CS-FEM [51,52], the number of supporting nodes

in smoothing domains are as same as those in elements, and

hence the bandwidth of stiffness matrix in the CS-FEM is

similar to that in the FEM. However, due to using the strain

smoothing technique to increase the accuracy in each ele-

ments,the computational cost in theCS-FEM is a littlehigher

than those of the FEM. However, in general, when the effi-

ciency of computation (computation time for the same accu-

racy) in termsof theerrorestimator versus computational cost

is considered, the CS-FEM is still efficient than the counter-

part FEM models. The CS-FEM, however so far, has beendeveloped mainly only for the 4-node quadrilateral elements

[5154] and the improvement of accuracy of solutions com-

pared to those of FEM is still marginal.

This paper hence extends the CS-FEM for triangular ele-

ments and for significant improvement of solutions of plate

analysis. The cell-based strain smoothing technique in the

CS-FEM is combined with the MIN3 [48] to give a so-called

thecell-based smoothed MIN3 (CS-MIN3)forstaticandfree

vibrationanalyses of plates. In theprocess of formulating the

system stiffness matrix of the CS-MIN3, each triangular ele-

ment will be divided into threesub-triangles, andin each sub-

triangle, the stabilized MIN3 is used to compute the strainsand to avoid the transverse shear locking. Then the strain

smoothing technique on whole the triangular element is used

to smooth the strains on these three sub-triangles.

2 Weakform for the ReissnerMindlin plate

Consider a plate under bending deformation. The middle

(neutral) surface of plate is chosenas the reference plane that

occupies a domain R2 as shown in Fig. 1. Let wbe the

deflection, and T = x y be the vector of rotations, inwhich x , y are the rotations of the middle plane aroundy- and x-axis, respectively, with the positive directions

definedas shownin Fig. 1. Theunknown vector of three inde-

pendent field variables at any point in the problem domain

of the ReissnerMindlin plates can be written as

uT =

w x y

(1)

The curvature of the deflected plate and the shear strains

are defined, respectively, as

= Ld (2)

= w + (3)

where = [/x /y]T, and Ld is a differential operator

matrix defined by

Ld =

/x 00 /y

/y /x

(4)

The standard Galerkin weakform of the static equilibrium

equations for the ReissnerMindlin plate can now be written

as [69]:

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Fig. 1 ReissnerMindlin thick plate and positive directions of the displacementw and two rotations x , y

TDbd +

TDsd =

uTbd (5)

where b = [p (x, y) 0 0]T is the distributed load applied on

the plate. The matrix Db is the material matrix related to the

bending deformation, and is given by

Db =Et3

121 2

1 0 1 00 0 (1 ) /2

(6)in which E is the Youngs modulus, t is the thickness of

plate. The matrix Ds is the material matrix related to shear

deformation, and has the form of

Ds = kt

0

0

(7)

with the shear modulus , and the shear correction factor

k = 5/6.

For the free vibrationanalysis of ReissnerMindlin plates,the standard Galerkin weakform can be derived from the

dynamic form of energy principle [69]:

TDbd +

TDs d +

uTmud = 0 (8)

where m is the matrix containing the mass density of the

material and thickness t as

m =

t 0 0

0 t3

120

0 0t3

12

(9)

3 FEM formulation for the ReissnerMindlin plate

Now, discretize the bounded domain into Ne finite ele-

ments such that =Ne

e=1 e and i j = , i = j ,

then the finite element solution uh =

w x yT

of a dis-

placementmodelfor the MindlinReissner plateis expressed

as:

uh =

NnI=1

NI(x) 0 00 NI(x) 0

0 0 NI(x)

dI (10)

where Nn is the total number of nodes of problem domain

discretized, NI(x) is the shape function at node I, dI =

[wI x I y I]T is thedisplacement vector of thenodaldegrees

of freedom (DOF) ofuh associated to node I, respectively.

The bending and shear strains can be then expressed in

the matrix forms as:

=

I

BIdI, s =

I

SIdI (11)

where

BI =

0 NI,x 00 0 NI,y

0 NI,y NI,x

, Si =

NI,x NI 0

NI,y 0 NI

(12)

The discretized system of equations of the MindlinReissner

plate using the FEM for static analysis then can be expressed

as,

Kd = F (13)

where

K =

BTDbB d +

STDs S d (14)

is the global stiffness matrix, and the load vector

F =

pN d + fb (15)

in which fb is the remaining part ofF subjected to prescribed

boundary loads.

For free vibration analysis, we have

(K 2M)d = 0 (16)

where is the natural frequency and M is the global mass

matrix defined by

M =

NTmNTd (17)

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Fig. 2 Three-node triangular element

4 CS-MIN3 formulation for the ReissnerMindlin plate

In this section, the cell-based strain smoothing technique

[51] is combined with the MIN3 [48] to give a so-called the

CS-MIN3 for static and free vibration analyses of plates.

4.1 Brief on the MIN3 formulation [48]

In the MIN3 [48], the rotations are assumed to be linear

through the rotational DOFs at three nodes of the elements,

and the deflection is initial assumed to be quadratic through

the deflection DOFs at six nodes (three node of the elements

and three mid-edge points). Then, by enforcing continuous

shear constraints at every element edge, the deflection DOFs

at three mid-edge points can be removed and the deflection

is now approximated only by vertex DOFs at three nodes of

the elements. The MIN3 element hence can overcome shear-

locking-free and produces convergent solutions [48]. In this

paper, we just brief on the MIN3 formulation which is nec-

essary for the formulation of the CS-MIN3.

Using a mesh of three-node triangular elements as shown

in Fig. 2, two rotations x and y at the element level are

assumed to be linear as

x =

3I=1

NI(x)x I = Nx ; y =

3I=1

NI(x)y I = Ny

(18)

and the initial deflection w is assumed to be quadratic as

w =

6I=1

RIwI = Rwini (19)

where N = [N1(x) N2(x) N3(x)] with NI(x), I = 1, 2, 3,

are the linear shape functions at node I; Tx = [x1 x2 x3]

and Ty = y1 y2 y3 are the rotational DOFs at threenodes of the element; wTini = [w1 w2 w3 w4 w5 w6] is the

deflection DOFs at six nodes (three nodes of the elements

and three mid-edge points as shown in Table 1), and R is the

row vector of quadratic shape functions given by

R = [N1(2N1 1) N2(2N2 1) N3(2N3 1)

4N1N2 4N2N3 4N3N1]

(20)

Now, in order to condense out the mid-edge deflection

DOFs, w4, w5, w6 in w, the continuous shear constraints atevery element edge were enforced by the following differen-

tial relationw,s + n

,s

edges

= 0 (21)

where s denotes the edge coordinate and n is the tangen-

tial edge rotation as shown in Fig. 2. The enforcement of

constraints (21) at three element edges yields

w4 =1

2(w1 + w2)

+

1

8 b3 (x1 x2) + a3 y2 y1w5 =

1

2(w2 + w3)

+1

8

b1 (x2 x3) + a1

y3 y2

w6 =

1

2(w3 + w1)

+1

8

b2 (x3 x1) + a2

y1 y3

(22)

Table 1 Nodal configurationsfor initial (unconstrained) and

constrained displacement field

in the MIN3

Shape functions Initial nodalconfiguration

Continuous shear edge constraints:w,s + n

,s

edges

= 0

Constrained nodalconfiguration

w x , y

Quadratic Linear Three edge constraints

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Fig. 3 Three-node triangular element coordinate description in the

MIN3 element

where a1 = x3 x2, a2 = x1 x3, a3 = x2 x1, b1 =

y2 y3, b2 = y3 y1, b3 = y1 y2 as shown in Fig. 3.

Substituting Eq. (22) into Eq. (19), the deflection, w, is

now approximated only by vertex DOFs at three nodes of the

elements as

w =

3I=1

NIwI +

3I=1

HIx I

+

3

I=1LIy I = Nw + Lx + Hy (23)

where wT = [w1 w2 w3] , H = [H1 H2 H3] , L = [L1 L2 L3]

are the vectors of shape functions, with HI and LI, I =

1, 2, 3, given by

H1 =1

2(a2N3N1 a3N1N2) ;

H2 =1

2(a3N1N2 a1N2N3) ;

H3 =1

2(a1N2N3 a2N3N1) (24)

L1 =

1

2 (b3N1N2 b2N3N1) ;

L2 =1

2(b1N2N3 b3N1N2) ;

L3 =1

2(b2N3N1 b1N2N3) (25)

The element stiffness matrix, KMIN3e , is now given by

KMIN3e =

e

BTDbBd +

e

STDs Sd (26)

where

B =

0 0 N,x0 N,y 0

0 N,x N,y

(27)

S =

N,x L,x + N H,xN,y L,y H,y + N

(28)

It was suggested [70] that a stabilization term needs to be

added to further improve the accuracy of approximate solu-

tions and to stabilize shear force oscillations. Such a modifi-

cation is achieved by simply replacing Ds in Eq. (26) by Ds ,

as follows.

KMIN3e =

e

BTDbBd +

e

STDs Sd (29)

where

Ds =kt3

t2 + h2e 1 0

0 1 (30)in which he is the longest length of the edges of the element

and is a positive constant [71]. In this paper, the stabilized

parameter is fixed at 0.1 for both static and free vibration

analyses.

4.2 Formulation of CS-MIN3

In theCS-MIN3, each triangular element is divided into three

sub-triangles by connecting the central point of the element

to three field nodes, and the displacement vector at the cen-

tral point is assumed to be the average of three displace-ment vectors of field nodes. In each sub-triangles, MIN3 is

used to compute the strains, and then the strain smoothing

technique on whole triangular element is used to smooth

the strains on these three sub-triangles. Hence, the contin-

uous shear constraints in CS-MIN3 will be satisfied at three

element edges, and also at three interior lines connecting

the central point with three field nodes. The CS-MIN3 is

hence softer than MIN3, and shows many superior proper-

ties compared to than the MIN3 and other triangular plate

elements.

Consider a typical triangular element e as shown in

Fig. 4. We now divide the element into three sub-trian-gles 1, 2 and 3 by connecting the central point O

with three field nodes as shown in Fig. 4. The coordi-

nates xO = [x O yO ]T of the central point O are calculated

by

xO =1

3(x1 + x2 + x3) ; yO =

1

3(y1 + y2 + y3) (31)

where xi = [xi yi ]T , i = 1, 2, 3, are coordinates of three

field nodes, respectively.

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sub-triangle

1

23

central point

O1

3

2

Fig. 4 Three sub-triangles (1, 2 and 3) created from the triangle

1-2-3 in theCS-MIN3 by connecting the central point O with three field

nodes 1, 2 and 3

In the CS-MIN3, we assume that the displacement vec-

tor deO at the central point O is the simple average of three

displacement vectors de1, de2 and de3 of three field nodes

deO =

1

3 (de1 + de2 + de3) (32)

On the first sub-triangle 1 (triangle O-1-2), we now con-

struct linear approximation u1e by

u1e = N11 (x)deO + N

12 (x)de1 + N

13 (x)de2 = N

1 d1

(33)

where d1 = [deO de1 de2]T is the vector of nodal DOF of

the sub-triangle 1 and N1 =

N

11 N

12 N

13

is the vec-

tor containing the linear shape functions at nodes O, 1, 2 of

the sub-triangle 1.

The curvatures of the deflection 1 and the altered shear

strains 1 in the sub-triangle 1 are then obtained by

1 =

b

11 b

12 b

13

b1

deOde1

de2

= b1 d1 (34)

1 =

s11 s

12 s

13

s1

deOde1

de2

= s1 d1 (35)

where b1 and s1 are, respectively, computed similarly as

the matrices B and S of the MIN3 in Eqs. (27) and (28)

but with one adjustment: the coordinates of three node xi =[xi yi ]T , i = 1, 2, 3 are replaced by xO , x1 and x2, respec-

tively.

Substituting deO in Eq. (32) into Eqs. (34) and (35), and

then rearranging we obtain

1 =

1

3b

11 + b

12

1

3b

11 + b

13

1

3b

11

B1

de1de2

de3

= B1 de

(36)

1 =

1

3s

11 + s

12

1

3s

11 + s

13

1

3s

11

S1

de1de2

de3

= S1 de

(37)

Similarly, for the second sub-triangle 2 (triangle O-2-3)

and third sub-triangle 3 (triangle O-3-1), the curvatures ofthe deflection

j , the altered shear strains j and matrices

Bj , Sj , j = 2, 3, respectively, can be obtained by cyclic

permutation.

Now, applying the cell-based strain smoothing operation

in the CS-FEM [51,52], the bending strains 1 , 2 , 3

and shear strains 1 , 2 , 3 are, respectively, used to

create a smoothed bending strain e and a smoothed shear

strain e on the element e, such as:

e =

e

h e (x) d =

1

1

e (x) d

+ 2

2

e (x) d + 3

3

e (x) d (38)

e =

e

h e (x) d = 1

1

e (x) d

+2

2

e (x) d + 3

3

e (x) d (39)

where e (x) is a given smoothing function that satisfies at

least unity property e e (x) d = 1. Using the follow-ing constant smoothing function e (x) =

1/Ae x e0 x / e

,

where Ae is the area of the triangular element, the smoothed

bending strain e and the smoothed shear strain e in Eqs.

(38) and (39) become

e =A1

1 + A22 + A3

3

Ae;

e =A1

1 + A22 + A3

3

Ae(40)

where A1 , A2 and A3 are the areas of the sub-triangles

2, 2, and 3, respectively.

Substituting j and j , j = 1, 2, 3, into Eq. (40), the

smoothed bending straine and the smoothed shear strain eare expressed by

e = Bde; e = Sde (41)

where B is thesmoothed bending straingradient matrixgiven

by

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(a) (b) (c)

Fig. 5 Square plate models and their discretizations using triangular elements,a clamped plate, b simply supported plate,c four discretizations of

a quarter of plate using triangular elements

B =A1 B

1 + A2 B2 + A3 B

3

Ae

(42)

and S is the smoothed shear strain gradient matrix given by

S =A1 S

1 + A2 S2 + A3 S

3

Ae(43)

Therefore the global stiffness matrix of the CS-MIN3 are

assembled by

K =

Nee=1

Ke (44)

where Ke is the smoothed element stiffness given by

Ke =

e

BTDbB d +

e

STDs S d = BTDbBAe

+ STDs SAe

(45)

Note that the rank of the CS-MIN3 element is similar to

that of the MIN3 element and hence the temporal stability

of the CS-MIN3 element is ensured. Only three eigenvalues

are always zero (corresponding to the rigid body modes of

the element) for various element shapes of very thin and

thick plates, and the CS-MIN3 element hasno spurious zero-

energy modes as shown in various numerical examples in

Sect. 5.

Also note that the introduction of the central points in the

triangular elements in the CS-MIN3 is only for the inter-

mediate formulation of the element stiffness matrix. In the

final form of element stiffness matrix as shown in Eq. (45),

the nodal displacement vectors of the central points will be

replaced by those of three vertex nodes. Hence no extra DOF

are associated with these central points. This means that the

nodal unknowns in the CS-MIN3 are the same as those in the

MIN3 of the same mesh.

5 Numerical results

In this section, various numerical examples are performed to

show the accuracy and stability of the proposed CS-MIN3

compared to the analysis solutions. For comparison, several

other elements such as DSG3 [47], ES-DSG3 [14], MIN3

[48] and MITC4 [35] have also been implemented in our

package.

5.1 Static analysis

The patch test is performed and CS-MIN3 element passes

the patch test within machine precision.

5.1.1 Square plate

Figure 5a and b describe the models of a square plate (length

L, thickness t) with clamped (w = n = s = 0) and simply

supported (w = s = 0) boundary conditions subjected to a

uniform load q = 1, respectively. The material parameters

are given by Youngs modulus E = 1.092.000 and Poissons

ratio = 0.3. Five uniform discretizations N N of plate

with N=4, 8,10,12 and 16areused and a quarter ofthesedis-

cretizations are plotted in Fig. 5c. The detailed expressions

of analytical solutions can be found by Taylor and Auricchio

[42].

First, a shear locking test using a coarse mesh (mesh

12 12) for the clamped plate is performed and the cen-

tral deflection and central moment are shown in Fig. 6.

The results show that similarly to DSG3, MIN3, ES-DSG3

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(a) (b)

Fig. 6 Results of various methods in the shear locking test of the clamped square plate subjected to uniform load with mesh 12 12, a centraldeflection, b central moment

(a) (b)

Fig. 7 Convergence of results of clamped square plate witht/L = 0.001, a central deflection, b central moment

(b)(a)

Fig. 8 Convergence of results of clamped square plate witht/L = 0.1, a central deflection, b central moment

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Fig. 9 Comparison of the computational time of clamped square plate

with t/L = 0.1

and MITC4, the CS-MIN3 also provides a locking-free

solution when the plate thickness becomes progressively

small.

For the clamped plate, the convergence of the central

deflection and the central moment against the different mesh

densities N N for the thin plate (ratio t/L = 0.001)

and thick plate (ratio t/L = 0.1) are plotted in Figs. 7

and 8, respectively. It is seen that with the same DOF,

the CS-MIN3 is worse than the MITC4 but better than

the DSG3 and MIN3 for both thin and thick plates. Com-

pared to the ES-DSG3, the CS-MIN3 usually gives betterresults for thin plates, but a little worse results for thick

plates.

Figure 9 compares the computational time of methods

for the clamped plate with t/L = 0.1. It is seen that with

the same set of nodes, the CS-MIN3 takes a longer time to

solve than the MIN3 and MITC4, but a little better than the

ES-DSG3.

For the simply supported plate (w = s = 0), the con-

vergence of the central deflection and the central moment

against the different mesh densities N N for the thin plate

(ratio t/L = 0.001) and thick plate (ratio t/L = 0.1) are

plotted in Figs. 10 and 11. It is again confirmed the obtainedcomments as in the above clamped plate.

(a) (b)

Fig. 10 Convergence of results of simply supported square plate(w = s = 0) with t/L = 0.001, a central deflection, b central moment

(b)(a)

Fig. 11 Convergence of results of simply supported square plate(w = s = 0) with t/L = 0.1, a central deflection, b central moment

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(b)(a)

Fig. 12 a A simply supported (w = 0) skew Morleys model, b four discretizations of a quarter of plate using triangular elements

Fig. 13 Convergence of deflection at the central point of Morley plate

(with L/t = 1,000) by various methods

5.1.2 Skew plate

We now consider a rhombic plate with simply supported

(w = 0) boundary conditions subjected to a uniform loadq = 1 as shown in Fig. 12a. This plate was originally stud-

ied by Morley [72]. Geometry and material parameters are

given by length L = 100, thickness t = 0.1, Youngs mod-

ulus E = 10.92 and Poissons ratio = 0.3. Five uniform

discretizations N N of plate with N = 4, 8, 10, 12 and

16 are used and a quarter of these discretizations are plotted

in Fig. 12b.

The convergence of the deflection and principal moments

at the central point by various methods are plotted in Figs. 13

and 14. It is seen that the CS-MIN3 shows remarkably excel-

lent performance compared to the DSG3, MIN3 and even

MITC4, and is a good competitor to the ES-DSG3. These

results hence imply that for problems with skew geometries,

theCS-MIN3 cangive high accurate solutionsand is a strong

competitor to many existing plate elements.

5.2 Free vibration analysis of plates

In this section, we investigate the accuracy and efficiency of

the CS-MIN3 element for analyzing natural frequencies of

plates. The plate may have free (F), simply (S) supported or

clamped (C) edges. The symbol, CFSF, for instance, repre-

sents clamped, free, supported and free boundary conditions

along the edges of rectangular plate. A non-dimensional fre-quencyparameter is oftenused to stand for thefrequencies

andthe obtained results usethe regular meshes. Theresults of

the CS-MIN3 are then compared to analytical solutions and

other numerical results which are available in the literature.

(a) (b)

Fig. 14 Convergence of results of at the central point of Morley plate (withL/t = 1, 000), a max principal moment, b min principal moment

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Fig. 15 Affection of the lumped mass matrix and consistent mass

matrix to the first frequency of the CS-MIN3

Also note that, in dynamic analysis using the CS-MIN3,

we can use the usual consistent mass matrix defined in Eq.

(17) to compute. However, in this paper for computational

efficiency [69], the well-known lumped mass matrix is used.

The reason is that the lumped mass matrix is usually softer

than the consistent mass matrix. For the displacement-based

plate elements, the stiffness matrix K is usually stiffer than

the real stiffness matrix. Using the lumped mass matrix

instead of consistent mass matrix in the displacement-based

plate elements will hence soften the over-stiffness of the sys-

temand will help togivemuchmore accurateeigenfrequency

prediction.

5.2.1 Square plate

We now consider square plates of length L and thick-

ness t. The material parameters are Youngs modulus E =

2.0 1011, Poissons ratio = 0.3 and the density mass

= 8, 000. A non-dimensional frequency parameter =

(2a4t/D)1/4 is used, where D = Et3/(12(1 2)) is the

flexural rigidity of the plate.

First, we analyze two thin and thick SSSS plates corre-

sponding to thickness-to-length t/L = 0.005 and t/L =

0.1. The geometry of the plate is shown in Fig. 5b, and four

uniform discretizations N N of plate with N = 4, 8, 16,and 24 are used and a quarter of these discretizations are

plotted in Fig. 5c.

Figure 15 shows the affection of using of the lumped mass

matrix and the consistent mass matrix by Eq. (17) to the

first frequency of the CS-MIN3. It is seen that the results of

Table 2 Convergence of six

lowest non-dimensional

frequency parameters of a

SSSS thin square plate

(t/L = 0.005)

Meshing Methods Mode sequence number

1 2 3 4 5 6

4 4 DSG3 5.5626 8.8148 11.8281 13.4126 18.1948 19.2897

MIN3 5.0409 8.6812 10.0678 12.9804 17.2128 18.9643

ES-DSG3 4.9168 8.1996 9.4593 11.5035 14.2016 15.0164

MITC4 4.6009 8.0734 8.0734 10.305 15.0109 15.0109

CS-MIN3 4.4965 7.1241 7.2503 9.0931 10.0933 10.1619

8 8 DSG3 4.7327 7.4926 8.2237 10.2755 11.6968 12.4915

MIN3 4.5804 7.4049 7.6488 9.9064 11.2774 11.4598

ES-DSG3 4.5376 7.2981 7.4659 9.6486 10.8937 11.028

MITC4 4.4812 7.2519 7.2519 9.2004 10.7796 10.7796

CS-MIN3 4.4543 7.0536 7.0791 8.975 10.0418 10.0477

16 16 DSG3 4.5131 7.1502 7.3169 9.3628 10.3772 10.4461

MIN3 4.4759 7.1174 7.1704 9.1459 10.2473 10.257

ES-DSG3 4.4641 7.087 7.1193 9.0582 10.1444 10.1489MITC4 4.4522 7.0792 7.0792 8.9611 10.1285 10.1285

CS-MIN3 4.4453 7.031 7.0367 8.9051 9.959 9.9592

24 24 DSG3 4.4718 7.0796 7.1459 9.1026 10.1256 10.1371

MIN3 4.4572 7.0653 7.0879 8.9998 10.0706 10.0724

ES-DSG3 4.4519 7.0511 7.0648 8.9583 10.0231 10.0238

MITC4 4.4458 7.0440 7.0440 8.9094 10.0059 10.0059

CS-MIN3 4.4438 7.0270 7.0295 8.8931 9.94400 9.94400

Exact [73] 4.443 7.025 7.025 8.886 9.935 9.935

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Table 3 Convergence of six

lowest non-dimensional

frequency parameters of a

SSSS thick square plate

(t/L = 0.1)

Meshing Methods Mode sequence number

1 2 3 4 5 6

4 4 DSG3 4.997 8.149 9.4311 11.354 14.129 14.9353

MIN3 4.9129 8.4029 9.3892 12.1445 15.608 16.9309

ES-DSG3 4.7376 7.658 8.4524 10.1882 12.1227 12.7533

MITC4 4.5146 7.6192 7.6192 9.4471 12.2574 12.2574

CS-MIN3 4.4032 6.7790 6.8435 8.3901 9.0714 9.0889

8 8 DSG3 4.4891 7.0697 7.253 9.1263 10.2195 10.3361

MIN3 4.5114 7.2165 7.3557 9.3924 10.6582 10.7313

ES-DSG3 4.4433 6.9495 7.0727 8.8487 9.8575 9.9221

MITC4 4.4025 6.9402 6.9402 8.6082 9.8582 9.8582

CS-MIN3 4.3743 6.7560 6.7712 8.3830 9.2329 9.2341

16 16 DSG3 4.3943 6.8227 6.8587 8.5447 9.4557 9.4616

MIN3 4.4297 6.9612 6.9892 8.7917 9.8071 9.8107

ES-DSG3 4.3846 6.7922 6.8196 8.4744 9.3666 9.3698

MITC4 4.3753 6.7918 6.7918 8.4166 9.3728 9.3728

CS-MIN3 4.3683 6.7470 6.7511 8.3623 9.2256 9.2257

24

24 DSG3 4.3785 6

.7788 6

.7940 8

.4383 9

.3243 9

.3254

MIN3 4.4157 6.9163 6.9282 8.6873 9.6642 9.6648

ES-DSG3 4.3744 6.7651 6.7770 8.4072 9.2845 9.2851

MITC4 4.3694 6.7618 6.7618 8.3757 9.2800 9.2800

CS-MIN3 4.3672 6.7454 6.7472 8.3580 9.2235 9.2235

Exact [73] 4.37 6.74 6.74 8.35 9.22 9.22

(a) (b)

Fig. 16 Six lowest frequencies of SSSS square plates discretized by a uniform mesh 4 4, a thin plate (t/L = 0.005), b thick plate (t/L = 0.1)

the CS-MIN3 using the lumped mass matrix are much more

accurate than those of the CS-MIN3 using the consistent

mass matrix, and even better than those of the MITC4 (using

the consistent mass matrix). In this paper, we hence use the

lumped mass matrix for the CS-MIN3 in all free vibration

analyses with the aim of increasing the accuracy of solutions

and computational efficiency.

Tables 2 and 3 give theconvergenceof six lowest frequen-

cies of thin plate (t/L = 0.005) and thick plate (t/L = 0.1),

respectively. In addition, Figs. 16 and 17 plot the values of

six lowest frequencies of thin plate (t/L = 0.005) and thick

plate (t/L = 0.1) for two uniform meshes 4 4 and 24 24,

respectively. It is observed that the results of CS-MIN3 agree

excellently withtheanalyticalresults [73] and are muchmore

accurate than those of the others elements for both thin and

thick plates, and for both coarse and fine meshes. In particu-

lar, the CS-MIN3 can provide accurately the values of high

frequencies of plates by using only coarse meshes.

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(a) (b)

Fig. 17 Six normalized lowest frequencies (h/exact) of SSSS square plates discretized by a uniform mesh 24 24, a thin plate (t/L = 0.005),b thick plate (t/L = 0.1)

Table 4 Convergence of six lowest non-dimensional frequencyparam-

eters of a CCCC thin square plate (t/L = 0.005)Meshing Methods Mode sequence number

1 2 3 4 5 6

4 4 DSG3 8.4197 12.772 14.9652 17.2579 21.389 21.76

MIN3 7.4097 11.7631 13.4014 16.3547 20.979 21.9197

ES-DSG3 6.9741 10.1934 11.4756 13.0548 15.4035 15.936

MITC4 6.5638 11.5231 11.5231 13.9505 62.6046 62.6054

CS-MIN3 6.1712 8.6783 8.9731 10.3804 11.0673 11.2107

8 8 DSG3 6.7161 9.7867 10.5673 12.9981 14.5306 15.3143

MIN3 6.346 9.3326 9.6896 12.1067 13.632 13.8879

ES-DSG3 6.1982 9.0117 9.2894 11.5616 12.795 13.0357

MITC4 6.1235 9.0602 9.0602 11.0186 12.9981 13.0263

CS-MIN3 6.0475 8.6471 8.7198 10.5863 11.701 11.7459

16 16 DSG3 6.1786 8.8759 9.068 11.2452 12.2182 12.2992

MIN3 6.0818 8.7504 8.8301 10.8435 11.9680 12.0010

ES-DSG3 6.0355 8.6535 8.7081 10.6584 11.743 11.772

MITC4 6.0285 8.6801 8.6801 10.5443 11.7989 11.8266

CS-MIN3 6.0101 8.5862 8.603 10.4502 11.5285 11.5556

24 24 DSG3 6.0728 8.6965 8.7757 10.7808 11.7863 11.8148

MIN3 6.0346 8.6464 8.6806 10.5952 11.6856 11.7114

ES-DSG3 6.0126 8.5999 8.623 10.5029 11.5795 11.6056

MITC4 6.0057 8.6013 8.6013 10.4389 11.5797 11.6077

CS-MIN3 6.0033 8.5742 8.5816 10.4215 11.4936 11.5205

Exact [73] 5.999 8.568 8.568 10.407 11.472 11.498

Next, we analyze two thin and thick CCCC square plates

shown in Fig. 5a. Four uniform meshes and two ratios of

thickness-to-length t/L are similar to those of the SSSS plate

case. Tables 4 and 5 give the convergence of six lowest fre-

quencies of thin plate (t/L = 0.005) and thick plate (t/L =

0.1), respectively. In addition, Figs. 18 and 19 plot the val-

ues of six lowest frequencies of thin plate (t/L = 0.005)

and thick plate (t/L = 0.1) for two uniform meshes 44

and 2424, respectively. Again it is seen that the obtained

Table 5 Convergence of six lowest non-dimensional frequencyparam-

eters of a CCCC thick square plate (t/L = 0.1)Meshing Methods Mode sequence number

1 2 3 4 5 6

4 4 DSG3 6.8748 9.8938 11.0847 12.6362 15.1032 15.6402

MIN3 6.9924 10.7891 12.0792 14.5388 18.102 19.132

ES-DSG3 6.2662 8.7952 9.6625 10.9112 12.6101 13.136

MITC4 6.1612 9.5753 9.5753 11.2543 14.0894 14.1377

CS-MIN3 5.8163 7.8647 8.0481 9.2126 9.6233 9.7156

8 8 DSG3 5.9547 8.3618 8.6293 10.2985 11.3415 11.5397

MIN3 6.114 8.8097 9.0456 11.0107 12.3554 12.5117

ES-DSG3 5.8068 8.0861 8.2701 9.8397 10.76 10.896

MITC4 5.8079 8.2257 8.2257 9.731 10.9921 11.0457

CS-MIN3 5.7180 7.8901 7.9111 9.3563 10.1380 10.1875

16 16 DSG3 5.7616 7.9935 8.0525 9.5772 10.4153 10.4697

MIN3 5.9281 8.3778 8.4288 10.1422 11.1503 11.193

ES-DSG3 5.725 7.9211 7.9627 9.4499 10.2631 10.3126

MITC4 5.7288 7.9601 7.9601 9.423 10.3257 10.3752

CS-MIN3 5.7117 7.8846 7.8965 9.3442 10.1319 10.1803

24 24 DSG3 5.7288 7.9282 7.9537 9.4376 10.251 10.2992

MIN3 5.8963 8.3026 8.3246 9.9873 10.947 10.9855

ES-DSG3 5.7123 7.8952 7.9133 9.379 10.1815 10.2289

MITC4 5.7104 7.9048 7.9048 9.3559 10.1957 10.2446

CS-MIN3 5.707 7.8803 7.8856 9.334 10.1263 10.1744

Exact [73] 5.71 7.88 7.88 9.33 10.13 10.18

comments in the SSSS plates are confirmed for the CCCC

plates.

5.2.2 Skew plate

We now consider the thin (t/L = 0.001) and thick (t/L =

0.2) cantilever rhombic (CFFF) plates. The geometry of the

plate is illustrated in Fig. 20a with skew angle = 60. The

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(a) (b)

Fig. 18 Six lowest frequencies of CCCC square plates discretized by mesh 4 4, a thin plate (t/L = 0.005), b thick plate (t/L = 0.1)

(a) (b)

Fig. 19 Six normalized lowest frequencies (h

/exact) of CCCC square plates discretized by uniform mesh 24 24, a thin plate (t/L = 0.005),b thick plate (t/L = 0.1)

(a) (b)

Fig. 20 a A parallelogram rhombic plate with boundary condition CFFF,b four discretizations of a quarter of plate using triangular elements

material parameters are Youngs modulus E = 2.0 1011,

Poissons ratio = 0.3 and the density mass = 8, 000. Anon-dimensional frequency parameter is used. Four uni-

form discretizations N N of plate with N = 4, 8, 16 and

24 are used and a quarter of these discretizations are plotted

in Fig. 20b. Figures 21 and 22 plot the values of six low-

est frequencies of thin plate (t/L = 0.001) and thick plate

(t/L = 0.2) for two uniform meshes 4 4 and 24 24,

respectively. Again, it is confirmed that the results of CS-

MIN3 agree excellently with those of the semi-analytical pb-

2 Ritz method [74], and are much more accurate than those

DSG3, MIN3 and MITC4 for both coarse and fine meshes,

and is a good competitor of the ES-DSG3. In particular, theCS-MIN3 can provide accurately the values of high frequen-

cies of plates by using only coarse meshes.

6 Conclusions

The cell-based strain smoothing technique is combined with

the well-known MIN3 to give a so-called the CS-MIN3

for static and free vibration analyses of plates. Through

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Comput Mech

(a) (b)

Fig. 21 Six lowest frequencies of CFFF rhombic square plates discretized by mesh 4 4, a thin plate (t/L = 0.001), b thick plate (t/L = 0.2)

(a) (b)

Fig. 22 Six normalized lowest frequencies (h/exact ) of CFFF rhombic square plates discretized by mesh 24 24, a thin plate (t/L = 0.001),b thick plate (t/L = 0.2)

the formulations and numerical examples, some concluding

remarks can be drawn as follows:

The CS-MIN3 uses only three DOFs at each vertex node

without additional DOF. The CS-MIN3 is free of shear

locking and passes the patch test.

The CS-MIN3 only use the triangular elements which is

a clear advantage compared to using four-node quadri-

lateral elements of MITC4. This advantage will be seen

clearer when the geometry domain of plate is skew or

complicated.

For static analysis, the results of the CS-MIN3 agree well

with analytical solutions and results of several other pub-

lished elements in the literature. The CS-MIN3 is much

more accurate than DSG3, MIN3 and is a good competi-

tor to ES-DSG3, MITC4 for some cases.

For free vibration analysis, the CS-MIN3 is stable tem-

porally, agrees well with analytical solutions and shows

somesuperiorproperties. TheCS-MIN3givesmoreaccu-

rate results than the DSG3, MIN3, MITC4 and ES-DSG3

and shows to be a strong competitor to existing compli-

cated quadrilateral plate elements such as the Rayleigh

Ritz method, the pb-2 Ritz method. In particular, the

CS-MIN3 can provide accurately the values of high fre-

quencies of plates by using only coarse meshes.

For the problems with skew geometries, the CS-MIN3

shows to be a strong competitor to others methods.

Note that, the performance of CS-MIN3 is also simple

and only based on elements, the CS-MIN3 hence can be

extended to geometrical nonlinearproblems easily by adding

to a non-linear membrane strain and using the total Lagrang-

ian approach and arc-length technique as performed in the

[53]. Furthermore, CS-MIN3 can be extended easily to the

flat shell element in which each node will have 6 DOF, and

a transformation matrix of coordinates needs to be used to

transformtheglobal coordinatesystemto the localcoordinate

system as performed in [75]. This extension hence highlights

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the advantage of CS-MIN3 which uses only triangular ele-

ments,because thegeometry of shell structures is often much

more complicated than that of plate structures. In addition,

the extension of CS-MIN3 for the implementation of mate-

rial nonlinear behaviour in plate problems will be verified in

coming time.

Acknowledgements This work was supported by Vietnam NationalFoundation for Science and Technology Development (NAFOSTED),

Ministry of Science and Technology, under the basic research program

(Project no.: 107.02.2010.01).

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