Static, free vibration and buckling analyses of stiffened plates by CS-FEM-DSG3 using triangular elements

Computers and Structures 125 (2013) 100–113

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Static, free vibration and buckling analyses of stiffened plates by CS-FEM-DSG3 using triangular elements T. Nguyen-Thoi a,b,⇑, T. Bui-Xuan a, P. Phung-Van b, H. Nguyen-Xuan a,b, P. Ngo-Thanh a a Department of Mechanics, Faculty of Mathematics & Computer Science, University of Science, Vietnam National University HCMC, 227 Nguyen Van Cu, Dist. 5, Hochiminh City, Viet Nam b Division of Computational Mechanics, Ton Duc Thang University, Nguyen Huu Tho St., Tan Phong Ward, Dist. 7, Hochiminh City, Viet Nam

a r t i c l e

i n f o

Article history: Received 24 July 2012 Accepted 24 April 2013

Keywords: Smoothed finite element methods (S-FEM) Eccentricity Stiffened plate Finite element method (FEM) Cell-based smoothed discrete shear gap method (CS-DSG3) Triangular elements

a b s t r a c t The paper presents the static, free vibration and buckling analyses of eccentrically stiffened plates by the cell-based smoothed discrete shear gap method (CS-FEM-DSG3) using triangular elements. In this method, the original plate element CS-DSG3 is combined with a membrane element and stiffened by a thick beam element. The eccentricity between the plate and the beam is included in the formulation of the beam. The compatibility of deflection and rotations of stiffeners and plate is assumed at the contact positions. The accuracy and reliability of the proposed method is verified by comparing its numerical solutions with those of analytical solutions, experimental results and others available numerical results. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Nowadays, the stiffened plates have been used widely in many branches of structural engineering such as aircraft, ships, bridges, buildings, etc. In early investigations, the analytical or semi-analytical methods have been used to analyse these stiffened plates. Ramakrishnan and Kunukkasseril [1] presented an analytical method for free vibration analysis of deck and their results were compared with the experimental results. Mukhopadhyay proposed a semi-analytical method for vibration and stability analyses [2–4] and for bending analysis [5] of concentrically and eccentrically stiffened plates. Chan et al. [6] proposed an exact solution by using the U-transformation method for the static analysis of stiffened plates whose rid-stiffeners were concentrically and periodically placed. However, those models are usually complex or possess inherent drawbacks in the methodology. Later, many different numerical models relied on simpler and more efficient methodologies have been proposed such as finite difference method, finite element method (FEM), boundary element method, meshfree methods, etc. Among them, the FEM shows many advantages compared to

⇑ Corresponding author. Address: Faculty of Mathematics & Computer Science, University of Science, Vietnam National University HCMC, 227 Nguyen Van Cu, Dist. 5, Hochiminh City, Viet Nam. Tel.: +84 942340411. E-mail addresses: [email protected], [email protected] (T. NguyenThoi). 0045-7949/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruc.2013.04.027

others methods. In the FEM, the stiffened plate is often separated into the plate and the stiffener. Then, the plate is modelled by plate elements and the stiffeners are modelled by beam elements. For modelling the plate elements in the stiffened plates, the investigators have used the Kirchhoff thin plate theory as well as the Mindlin–Reissner thick plate theory. Based on the Kirchhoff theory, some typical works can be found in Refs. [7–10]. Rossow and Ibrahimkhail [7] applied the constraint method to the finite element of which approximating polynomials had arbitrary order for static analysis of concentrically and eccentrically stiffened plates. Olson and Hazel [8] presented theoretical and experimental results for free vibration analysis of eccentrically stiffened plates. The natural frequencies of clamped stiffened plates having one stiffener and two stiffeners were predicted by the triangular high-precision conforming element. Barik [9] and Barik and Mukhopadhyay [10] combined the four-node rectangular plane-stress element with the plate-bending ACM element for static, free vibration, and pre-buckling analyses of arbitrary bare and stiffened plates. Based on the Mindlin–Reissner theory, some typical works can be found in Refs. [11–16]. Deb and Booton [11] used an isoparametric stiffened plate element under transverse load, then Mukheriee and Mukhoadhyay [12,13] also used this element for free vibration and buckling analysis. However, the isoparametric plate element is suffered the shear-locking phenomena [14] that was not mentioned in those works. Palani et al. [15] then applied two isoparametric elements (the eight-node isoparametric element

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QS8S1 and the nine-node isoparametric element QS9S1) for static and free vibration analyses of plates/shells with eccentric stiffeners. The comparison between two elements in [15] showed the high performance of QS9S1 compared with QS8S1 and other models. Holopainen [16] proposed a new finite element model for free vibration analysis of eccentrically stiffened plates. In this model, the mixed interpolation of tension components (MITC) is applied for both the plate-bending and stiffener elements to free shear-locking and has a good convergence. Although MITC element could avoid shear-locking, but this element is more complicated than the isoparametric element and it is not really better than the isoparametric element in free vibration analysis of stiffened plates. The Rayleigh–Ritz method is used by Liew et al. [17] and Xiang et al. [18] in order to study the vibratory characteristics of rectangular and skew Mindlin plates with intermediate stiffeners. Liew et al. [19] also developed the Mindlin–Engesser model for the vibration analysis of moderately thick plates with arbitrarity oriented stiffeners by using the Ritz minimization procedure. Recently, Peng et al. [20,21] applied the element-free Galerkin method for static, free vibration, and pre-buckling analysis of stiffened plates and for elastic bending analysis of un-stiffened and stiffened corrugated plates [22]. In addition, there were also a number of different numerical methods used to study the behaviours of thick plate as shown in the literature review by Liew et al. [23], Satsangi [24,25]. In comparison, it is seen that many studies in the literature have concerned with the analysis of stiffened plates using four-node or eight-node Mindlin plate elements, while the literature related to those using three-node triangular Mindlin plate elements is somewhat still limited. This paper hence aims to further contribute a new numerical procedure for the analyses of stiffened plates. The method used here is still the FEM, however the elements used are three-node triangular elements which are different from the four-node or eight-node elements mentioned in Refs. [11–16]. In the other front of the development of numerical methods, Liu and Nguyen-Thoi have integrated the strain smoothing technique [26] into the FEM to create a series of smoothed FEM (S-FEM) [27] such as a cell/element-based smoothed FEM (CS-FEM) [28], a node-based smoothed FEM (NS-FEM) [29], an edge-based smoothed FEM (ES-FEM) [30] and a face-based smoothed FEM (FS-FEM) [31]. Each of these smoothed FEM has different properties and has been used to produce desired solutions for a wide class of benchmark and practical mechanics problems. Several theoretical aspects of the S-FEM models have been provided in Refs. [32,33]. The S-FEM models have also been further investigated and applied to various problems such as plates and shells [34– 40], composite plates [41], piezoelectricity [42], fracture mechanics [43], elasto-plasticity [44], visco-elastoplasticity [45,46], limit and shakedown analysis for solids [47], etc. Among these S-FEM models, the CS-FEM [27,28] shows some interesting properties in the solid mechanics problems. Extending the idea of the CS-FEM to plate structures, Nguyen-Thoi et al. [48] have recently formulated a cell-based smoothed stabilized discrete shear gap element (CS-DSG3) for static, and free vibration analyses of isotropic Mindlin plates by incorporating the CS-FEM with the original DSG3 element [49]. In the CS-DSG3, each triangular element will be divided into three sub-triangles, and in each sub-triangle, the stabilized DSG3 is used to compute the strains. Then the strain smoothing technique on whole the triangular element is used to smooth the strains on these three sub-triangles. The numerical results showed that the CS–DSG3 is free of shear locking and achieves the high accuracy compared to the exact solutions and others existing elements in the literature. This paper hence extends the triangular plate element CS-DSG3 [48] for the analysis of the eccentrically stiffened plates. In the present method, the original element CS-DSG3 is combined with a membrane element and stiffened by a thick beam element. The

eccentricity between the plate and the beam is included in the formulation of the beam. The compatibility of deflection and rotations of stiffeners and plate is assumed at the contact positions. The accuracy and reliability of the proposed method is verified by comparing its numerical solutions with those of analytical methods, experiments, commercial softwares and others available numerical results. 2. Weak form of the stiffened plate problems Consider a stiffened plate as shown in Fig. 1. The middle (neutral) surface of plate Oxy is chosen as the reference plane that occupies a domain X  R2, and the stiffener’s centroid is offset from the Oxy plane a given distance e. Let u and v be the in-plane displacements, and w be the deflection of the stiffened plate on the middle of plate. Let bx and by be the rotations of the normal to the middle surface of the plate around y-axis and x-axis, respectively, with the positive directions defined as shown in Fig. 2. 2.1. Formulation of the plate [16] The displacements and rotations of the plate are defined by

uðx; yÞ ¼ u0 ðx; yÞ þ zbx ðx; yÞ;

v ðx; yÞ ¼ v 0 ðx; yÞ þ zby ðx; yÞ;

ð1Þ

wðx; yÞ: For the general case, the unknown vector of five independent field variables at any point in the problem domain of the Mindlin plates can be written as

u ¼ ½u0 ; v 0 ; w; bx ; by T :

ð2Þ

The membrane, bending and shear strains e0, j and c of the plate are defined, respectively, by

2 3 u0 36 7 0 0 0 0 6 v0 7 u0;x 6 7 6 v 0;y 75 ¼ 64 0 @y@ 0 0 0 75 66 w 77 ¼ LmP u; ¼4 6 7 @ @ u0;y þ v 0;x 0 0 0 4 bx 5 @y @x |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} by Lm 2

e0

3

2

@ @x

P

Fig. 1. A plate stiffened by an x-direction stiffener.

Fig. 2. The local coordinate system on the stiffener.

ð3Þ

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T. Nguyen-Thoi et al. / Computers and Structures 125 (2013) 100–113

3 u0 3 7 @ 0 0 0 @x 0 6 bx;x 6 v0 7 7 6 b 7 60 0 0 0 @ 76 b 6 7 j¼4 5¼4 y;y @x 5 6 w 7 ¼ LP u; 7 6 @ @ 0 0 0 @y @x 4 bx 5 bx;y þ by;x |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} by Lb 2

2

3

2

ð4Þ

eGP

P

2 "

c¼

w;x þ bx

#

w;y þ by

u0

@ @x @ @y

0 0

ð5Þ

1 2

ZZ X

ZZ

eT0 Dm e0 dX þ

jT Db j dX þ X

cT Ds c dX ;

ð6Þ

X

where Dm, Db, and Ds are the material matrices involving with the strains of membrane, bending, and shearing components of plate, respectively, and given by

2

1

Et 6 4m 1  m2 0  1 Etk Ds ¼ 2ð1 þ mÞ 0

Dm ¼

m

0

3

0 7 5;

1

3

Db ¼

1m 2

0 0



1

2

m

1

Et 6m 1 4 12ð1  m2 Þ 0 0

0

1 2

ZZ

ZZ þ X

0 7 5;

ZZ X

_2

1 2

u_ T mP u_ dX;

ð10Þ

X

  t3 t3 : mP ¼ diag qt; qt; qt; q ; q 12 12

ð11Þ

Note that from Eqs. (3)–(5), an elastic strain field eEP which includes all membrane, bending and shear strains e0, j and c of the plate can be defined as

3

2

3

2

3

2

3

ex 0 u0;x zbx;x 6e 7 6 v 0;y 77 66 zby;y 77 66 0 77 6 y 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7; 0 eEP ¼ 6 7 6 cxy 7 ¼ 6 u0;y þ v 0;x 7 þ 6 zðby;x þ bx;y Þ 7 þ 6 7 6 7 6w þb 7 6c 7 6 0 0 5 4 5 4 ;x 4 xz 5 4 x5 cyz w;y þ by 0 0

0

0

@ @x @ @y

0

0

0

0

@ @x @ @y

0

0

0

0 0

0

0

0

0

3 0 2 3 u 07 76 7 76 v 7 7 07 76 G 76 w 7 7 ¼ LP u; 0 76 7 6 74 bx 5 @ 7 @x 5 b y

ð17Þ

@ @y

 t3 t3 s; s ; 12 12

ð18Þ

#

r0x r0xy : r0xy r0y

ð19Þ

Substituting Eq. (17) into Eq. (16), we obtain

1 2

ZZ X

uT ðLGP ÞT r0 LGP u dX:

ð20Þ

eGP is the geometric strain of the plate given by

2

Uðr; zÞ

3

2

ur ðrÞ þ zbr ðrÞ

3 ð21Þ

The elastic strain field of the stiffener is defined as

2

then the total strain of plate eP in the case of the plate subjected to in-plane pre-buckling stresses can be written as

eP ¼ eEP þ eGP ;

In this paper, we assume that the beams are placed parallelly with the axis of plate and the effect of warping to beams is neglected. A stiffener which is parallel with the x-axis as shown in Fig. 1 is considered. For general cases, we consider a stiffener that is placed askew an angle to x-axis. A local coordinate O0 rsz having O0 rs plane coincides with the Oxy plane as shown in Fig. 2. Here, we assume that the displacements of stiffeners and plate are the same at the contact positions. In practise, the lateral displacements of stiffener and the rotations about the z-axis are neglected, so displacements of beam in local coordinate are expressed by the middle surface displacements of plate in the local coordinate as

6 7 6 7 USt ¼ 4 Vðr; zÞ 5 ¼ 4 us ðrÞ þ zbs ðrÞ 5: Wðr; zÞ uz ðrÞ þ sbs ðrÞ

ð12Þ

where

2

3

eGP , r0 and s, respectively, are given by

2.2. Formulation of the stiffener [16]

where u_ is the derivative of u with respect to time and mP is the diagonal matrix given by

2

w;x

ð16Þ

ð9Þ

X

ZZ

ð15Þ

ðeGP ÞT r0 eGP dX;



U GP ¼

or in the matrix form

TP ¼

X

r0 ¼ diag ts; s¼

ð8Þ

ZZ

by;y

"

uT ðLbP ÞT Db LbP u dX

2

rT0 eGP dV:

0 6 w;y 7 6 0 6 7 6 6 7 6 6 6 bx;x 7 6 0 7 ¼6 6b 7 ¼ 6 0 6 x;y 7 6 7 6 6 4 by;x 5 6 40

ð7Þ

qðu þ v þ w_ Þ dX; _2

1 2 2

eGP

The kinetic energy of the plate is computed by

1 TP ¼ 2

V

where the matrices

1m 2

:

T m m uT ðLm P Þ D LP u dX þ X  uT ðLsP ÞT Ds LsP u dX :

ZZ

U GP ¼

3

Substituting Eqs. (3)–(5) into Eq. (6), the elastic strain energy U EP can be rewritten as

U EP ¼

ZZZ

By substituting Eq. (14) into Eq. (15) and integrating on the thickness of plate, the geometric strain energy U GP becomes



ZZ

The geometric strain energy enforced by in-plane pre-buckling  T stresses r0 ¼ r0x r0y s0xy 0 0 is then computed by

U GP ¼

The elastic strain energy of the plate is now written as

U EP ¼

ð14Þ

0

3

7 #6 6 v0 7 1 0 6 7 6 w 7 ¼ Ls u: ¼ P 7 0 0 0 1 6 6 7 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 4 bx 5 LsP by "

2   2 z2 @by 2 3 2  1 @w 2 x þ z2 @b þ 2 @x @x 7 6 2 @x 6 2

2 7 2 7 6 1 @w 2 2 @by @bx z z 6 2 @y þ 2 @y þ 2 @y 7 7 6 ¼6 7: 6 @w @w þ z2 @bx @bx þ z2 @by @by 7 6 @x @y @x @y @x @y 7 7 6 5 4 0

ð13Þ

E St

e

3 2 @U 3 2 3 ur;r þ zbr;r er @r 6 7 6 @U @V 7 6 u þ zb 7 ¼ 4 crs 5 ¼ 4 @s þ @r 5 ¼ 4 5: s;r s;r @W @U u þ sb þ b crz þ z;r s;r r @r @z

ð22Þ

The elastic strain energy of stiffener is given by

U ESt ¼

1 2

Z ZZ l

A

ðeESt ÞT DSt eESt dA dl;

ð23Þ

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T. Nguyen-Thoi et al. / Computers and Structures 125 (2013) 100–113

where A and l are the cross-section area and length of stiffener, respectively, and DSt is the material matrix of the stiffener given by

 DSt ¼ diag E;



E E ¼ diagðE; G; GÞ: ; 2ð1 þ mÞ 2ð1 þ mÞ

ð24Þ

Substituting Eq. (22) into Eq. (23) and integrating over crosssectional area of the stiffener, we obtain

U ESt

Z

1 ¼ 2

2

½GAðuz;r þ br Þ þ GðIs þ

l

Iz Þb2s;r

þ EAður;r þ ebr;r Þ

2

þ EIs b2r;r þ GAðus;r þ ebs;r Þ2  dl;

ð25Þ

where e is the eccentricity between the mid-plane of plate and the centroid of beam as shown in Fig. 1; Is is the second moment of stiffener cross-sectional area about an axis which goes through the centroid of the stiffener and is parallel with the s-axis; and Iz is the second moment of stiffener cross-sectional area about the z-axis. Because the bending and shearing energy in the plane of plate is regardless, so the last term of Eq. (25) is neglected. For simplicity, we assume that the stiffeners were not affected by warping. Therefore, the St Venant’s torsion constant J is used to replace the torsion coefficient G(Is + Iz). To calculate the torsion constant, an approximative formulation J ’ 0.025A4/Ir is used, where Ir is the second moment of stiffener cross-sectional area about an axis which goes through the centroid of the stiffener and is parallel with the r-axis. Thus, Eq. (25) is rewritten as

U ESt ¼

Z

1 2

l

½EAður;r þ ebr;r Þ2 þ EIs b2r;r þ kGAðuz;r þ br Þ2

þ GðIs þ Iz Þb2s;r  dl;

U ESt ¼

1 2

l

ðeESt ÞT DSt eESt dX;

ð27Þ

D ¼ diagðEA; EIs ; kGA; GJÞ:

0 z0 nx

60 6 LESt ¼ n  r6 40

0

0

nx

0

1

nx nr

0

0

0

ny

@ nx @x



T

ð29Þ

or

2@

0

0

e @r@

0

@ @r

0

3

2

ur

0

07 7 7 05

LESt

U ESt ¼

ur

3

2

ny 7 7 ny 7 5 nr

ð33Þ

nx

@ ny @y .

1 2

Z

uTSt ðLESt ÞT DSt LESt uSt dl:

l

ð34Þ

The kinetic energy of the stiffener is computed by

T St ¼

Z ZZ

1 2

l

A

_ T U_ St dA dl: U St

ð35Þ

Again, by substituting Eqs. (21) and (31) into Eq. (35) and integrating over cross-sectional area of the stiffener, we obtain

T St ¼

Z

1 2

u_ TSt AT mSt Au_ St dl;

l

ð36Þ

where

2

1 0

60 6 6 mSt ¼ qA6 60 6e 4 0

0

e

0

1 0

0

e

0

1

0

0

0

0 e2 þ IAs

e

0

0 z e2 þ Is þI A

0

3 7 7 7 7: 7 7 5

eSt ¼ eESt þ eGSt ;

ð37Þ

eGSt

e is the geometric strain given by

62 ¼4

ny

0

0

0

32

6 7 6 n n 0 0 07 7 6 us 7 6 y x 7 6 7 6 7¼6 0 7 0 1 0 0 uSt ¼ 6 u z 7 6 7 6 7 6 7 6 0 0 nx ny 5 4 br 5 4 0 0 0 0 ny nx bs |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} A

@r

2

þ z2

@b 2 3 r

@r

0

7 5:

ð39Þ

0

ZZZ

V

ð30Þ

ðr0St ÞT eGSt dV:

ð40Þ

By substituting Eq. (39) into Eq. (40) and integrating over the cross-section area of stiffener, Eq. (40) becomes

uSt

nx

ð38Þ

G St

2  2 1 @uz

U GSt ¼

In Timoshenko beam theory, a correction factor k = 5/6 was added into the shearing strain energy in Eq. (26). The relationship of the displacement field between the local coordinate and the global coordinate, as shown in Fig. 2, is expressed as

2

3

The geometric strain energy enforced by in-plane pre-buckling stresses r0St ¼ ½ r0x 0 0 T is then computed by

3

6 7 6 us 7 6 7 6 uz 7 ¼ LE uSt : St 6 7 @ 0 @r 1 6 7 4 br 5 @ 0 0 0 0 @r |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl} bs @r

60 6 ¼6 40

z0 n y

in which n  r ¼ þ Substituting Eq. (32) into Eq. (27), the strain energy of stiffener now becomes

ð28Þ

e ¼ ur;r þ ebr;r br;r uz;r þ br bs;r ;

e

ny

where

St

E St

nx

For buckling analysis, the geometric strain is added to the total strain of stiffener as

where

E St

2

ð26Þ

or in matrix form

Z

where

uSt 0

3

6 v St 7 6 0 7 6 St 7 6 w 7 ¼ AuSt ; 7 6 6 bSt 7 4 x 5 bSt y

U GSt ¼

1 2

Z l

ðeGSt ÞT r0St eGSt dX;

ð41Þ

where

3 ur 6 #6 us 7 7 0 6 7 6 uz 7 ¼ LG uSt ; St 6 0 6 7 7 4 br 5 bs 2

" G St

e ¼

uz;r br;r

#

" ¼

0 0

@ @r

0

0 0

0

@ @r

ð31Þ

ð42Þ

and

r0St ¼ diag½Ar0x ; ðIs þ e2 AÞr0x :

uSt

ð43Þ

The geometric strain field in the global coordinate is written as

where nx and ny are the direction cosines and A is the transform matrix. The elastic strain field of the stiffener in the global coordinate is expressed as

e ¼ LGSt AuSt ¼ LGSt uSt :

eESt ¼ LESt AuSt ¼ LESt uSt :

Substituting Eq. (44) into Eq. (41), the geometric strain energy now becomes

ð32Þ

G St

ð44Þ

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T. Nguyen-Thoi et al. / Computers and Structures 125 (2013) 100–113

1 2

U GSt ¼

Z l

uTSt ðLGSt ÞT r0St LGSt uSt dl:

ð45Þ

To simplify the formulation, we assume that the plate is just enforced by a stiffener. Then the total potential energy and kinetic energy of stiffened plate, respectively, are

U ¼ U EP þ U GP þ U ESt þ U GSt ;

ð46Þ

T ¼ T P þ T St :

ð47Þ

When the plate is stiffened by many grips, the total energy will be added by the total energy of strain and kinetic energy of stiffeners.

the discrete displacement field into Eqs. (49), (51) and (52), we have the formulation for static, dynamic and pre-buckling analyses, respectively, as

Kd ¼ f;

ð53Þ

2

ðK  x MÞd ¼ 0;

ð54Þ

ðK  kcr KG Þd ¼ 0;

ð55Þ

where K, M, and KG are the elastic stiffness, mass, and geometric matrices, respectively, x is the angular frequency, kcr is the critical buckling load and d is the displacement vector. In this paper, the stiffness matrices K is formulated by the plate element CS-DSG3 [48] and the geometric matrix KG is formulated by the CS-FEM [51], respectively.

2.3. Weak form of the stiffened plate problem 3.1. Brief on the DSG3 formulation In this paper, we apply the Hamilton’s principle to find the weak form of the problem. The principle is stated that

Z

t2

ðdW þ dT  dUÞ dt ¼ 0;

ð48Þ

t1

where W is the work done by external forces on the stiffened plate. For static analysis, the kinetic energy in Eq. (48) and geometric strain energy are omitted. By substituting Eqs. (8) and (34) into Eqs. (46) and (47) and combining with Eq. (48), the principle becomes

Z

ZZ

h i T m m b T b b s T s s duT b  ðLm P Þ D LP  ðLP Þ D LP  ðL P Þ D LP u dX t1 X  Z  duTSt ðLESt ÞT DSt LESt uSt dl ; dt ¼ 0; ð49Þ t2

l

where b is a body force; and t1, t2 are initial and final times. For dynamic analysis, the work and geometric strain energy in Eq. (48) are eliminated, then the principle in this case becomes

Z

ZZ

h i T m m b T b b s T s s du_ T mP u_  duT ðLm P Þ D LP þ ðLP Þ D LP þ ðLP Þ D LP u dX t1 X Z h i  ð50Þ du_ TSt AT mSt Au_ St  duTSt ðLESt ÞT DSt LESt uSt dl dt ¼ 0: þ t2

l

Hence by integrating the first and the fifth terms by parts and using du(t1) = du(t2), duSt(t1) = duSt(t2), we obtain

Z

ZZ

h i T m m b T b b s T s s €  duT ðLm du mP u P Þ D LP þ ðLP Þ D LP þ ðLP Þ D LP u dX t1 X Z h i  € St  duTSt ðLESt ÞT DSt LESt uSt dl dt ¼ 0: ð51Þ þ duTSt AT mSt Au t2

T

The formulation of the DSG3 [49] is based on the concept ‘‘shear gap’’ of displacement along the edges of the elements. In the DSG3, the shear strain is linear interpolated from the shear gaps of displacement by using the standard element shape functions. The DSG3 element is shear-locking-free and has several superior properties as presented in Ref. [49]. In this paper, we just brief on the stiffness matrix of the DSG3 which is necessary for the formulation of stiffness matrix of the CS-DSG3. S e Using a mesh of Ne triangular elements such that X ¼ Ne¼1 Xe and Xi \ Xj = £ , i – j, the approximation uh = [u, v, w, bx, by]T for a 3-node triangular element Xe shown in Fig. 3 for the Reissner– Mindlin plate can be written, at the element level, as

uhe ¼

3 X NI ðxÞI5 deI ; |fflfflfflffl{zfflfflfflffl} I¼1

ð56Þ

NI ðxÞ

where deI = [uI, vI, wI, bxI, byI]T are the nodal degrees of freedom of uhe associated to node I of the element e; I5 is identity matrix and NI(x), I = 1, 2, 3, are the linear shape functions in a natural coordinate defined by

N1 ¼ 1  n  g;

N2 ¼ n;

N3 ¼ g:

ð57Þ

The membrane strains of plate are given by

eh0 ¼ Bm de ;

ð58Þ

where de = [de1, de2, de3]T is the nodal displacement vector of element and Bm is the constant matrix of derivatives of the shape functions given by

l

For buckling analysis, the work and kinetic energies in Eq. (48) are eliminated, then the principle in this case becomes Z t2 Z Z h i T m m b T b b s T s s G T G duT ðLm P Þ D L P  ðLP Þ D LP  ðL P Þ D LP  ðLP Þ r0 LP udX t1 X  Z h i T  duSt ðLESt ÞT DSt LESt þ ðLGSt ÞT r0St LGSt uSt dl dt ¼ 0: ð52Þ l

Note that this section prepared a general background for next section which presents the formulation of the CS-DSG3 for the stiffened plate. The Section 3 hence will present the ideas for the combination of the original plate element CS-DSG3 [48] with a membrane element and stiffened by a thick beam element Timoshenko. 3. Formulation of CS-DSG3 for Reissner–Mindlin plate In the finite element method, the exact displacement field is approximated by a discrete displacement field. By substituting

Fig. 3. Three-node triangular element and local coordinates in the DSG3.

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T. Nguyen-Thoi et al. / Computers and Structures 125 (2013) 100–113

3

2

where the element stiffness and geometric matrices, KDSG3 and KGPe pe of the DSG3 element are respectively given by

7 6 6 b  c 0 0 0 0 c 0 0 0 0 b 0 0 0 0 7 7 1 6 6 0 d  a 0 0 0 0 d 0 0 0 0 a 0 0 0 7 Bm ¼ 7 2Ae 6 7 6 4 d  a b  c 0 0 0 d c 0 0 0 a b 0 0 0 5 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} Bm1

Bm2

KDSG3 ¼ Pe

Bm3

1 ¼ ½ Bm1 Bm2 Bm3 : 2Ae

jh ¼ Bde ;

ð60Þ

where B contains the derivatives of the shape functions that are constants 3

2

B2

B3

1 ¼ ½ B1 B2 B3  2Ae

ð61Þ

with a = x2  x1, b = y2  y1, c = y3  y1, d = x3  x1 as shown in Fig. 3, and xi ¼ ½ xi yi T , i ¼ 1; 2; 3, are coordinates of three nodes, respectively; Ae is the area of the triangular element, and Bi, i = 1, 2, 3, contains the derivatives of the shape functions of ith node. In order to overcome shear locking, Bletzinger et al. [49] proposed the discrete shear gap method (DSG3) to alter the shear strain field. The altered shear strains are in the form of

ch ¼ Sde ;

ð62Þ

where 2

3

7 1 6 6 0 0 b  c Ae 0 0 0 c ac=2 bc=2 0 0 b bd=2 bc=2 7 S¼ 6 7 2Ae 4 0 0 d  a 0 Ae 0 0 d ad=2 bd=2 0 0 a ad=2 ac=2 5 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} S1

S3

S3

BT Db B dX þ Xe

ZZ

ST Ds S dX Xe

ZZ

KGPe ¼

Xe

ð69Þ

BTG r0 BG dX ¼ BTG r0 BG Ae ;

ð70Þ

and the element mass matrix is

MPe ¼

ZZ

NT mP N dX;

ð71Þ

Xe

MiP ¼

nei X Ai

e

e¼1

3

mP ;

ð72Þ

where nei is the number of elements containing the ith node; Aie is the area of element e containing the ith node. Therefore, the global lumped mass matrix is

MP ¼

Nn X MiP ;

ð73Þ

i¼1

where Nn is the number of nodes. It was suggested [50] that a stabilization term needs to be added to the original DSG3 element to further improve the accuracy of approximate solutions and to stabilize shear force oscillations. Such a modification is achieved by simply replacing Ds in b s ¼ kt3 G 2 I2 , in which he is the longest length of the Eq. (69) by D t 2 þahe edges of the element and a is a positive constant [50]. 3.2. Formulation of CS-DSG3 for the Reissner–Mindlin plate

1 ¼ ½ S1 S2 S3 : 2Ae

ð63Þ

The geometric strains in the element are then obtained by

eGh P ¼ B G de ;

ð64Þ

where BG contains the derivatives of the shape functions that are constants 3

2

60 0 b c 0 0 0 0 c 0 0 0 0 b 0 0 7 7 6 7 6 60 0 d a 0 0 0 0 d 0 0 0 0 a 0 0 7 7 6 6 1 6 0 0 0 b  c 0 0 0 0 c 0 0 0 0 b 0 7 7 BG ¼ 7 6 2Ae 6 0 0 0 d  a 0 0 0 0 d 0 0 0 0 a 0 7 7 6 60 0 0 0 b  c 0 0 0 0 c 0 0 0 0 b 7 7 6 7 6 40 0 0 0 d  a 0 0 0 0 d 0 0 0 0 a 5 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} BG1

¼

ZZ

where N is the shape function matrix of 3-node triangular element. In this work, we used the lumped mass matrix replacing for the consistent mass matrix formulated as Eq. (71). The lumped mass matrix for the ith node has a formulation as

7 6 6 0 0 0 b  c 0 0 0 0 c 0 0 0 0 b 0 7 7 1 6 6 0 0 0 0 d  a 0 0 0 0 d 0 0 0 0 a 7 B¼ 7 2Ae 6 7 6 4 0 0 0 d  a b  c 0 0 0 d c 0 0 0 a b 5 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} B1

Xe

BTm Dm Bm dX þ

¼ BTm Dm Bm Ae þ BT Db BAe þ ST Ds SAe ;

ð59Þ

The curvatures of the deflection in the element are then obtained by

ZZ

1 ½ BG1 BG2 BG3 : 2Ae

BG2

Consider a typical triangular element Xe as shown in Fig. 4. We first divide the element into three sub-triangles D1, D2 and D3 such S as Xe ¼ 3i¼1 Di and Di \ Dj = £ , i – j, by simply connecting the central point O of the triangle with 3 field nodes as shown in Fig. 4. In the CS-DSG3, we assume that the displacement vector deO at the central point O is the simple average of three displacement vectors de1, de2 and de3 of three field nodes

deO ¼

1 ðde1 þ de2 þ de3 Þ: 3

ð74Þ

On the first sub-triangle D1, a linear uDe 1 ¼ ½ue ; v e ; we ; bex ; bey T is now constructed by

uDe 1 ¼ N1 ðxÞdeO þ N2 ðxÞde1 þ N 3 ðxÞde2 ¼

BG3

approximation

3 X D NI ðxÞdI 1 ;

ð75Þ

I¼1

ð65Þ

Substituting Eqs. (58), (60), (62) and (64) into Eqs. (49), (51) and (52), we obtain the element stiffness, mass, and geometric matrices of plate, respectively, in the forms of

KDSG3 ¼ P

Ne X KDSG3 ; Pe

ð66Þ

e¼1

MP ¼

Ne X MPe ;

ð67Þ

e¼1

KGP ¼

Ne X KGPe ; e¼1

ð68Þ

Fig. 4. Three sub-triangles (D1, D2 and D3) created from the triangle 1–2–3 in the CS-DSG3 by connecting the central point O with three field nodes 1, 2 and 3.

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T. Nguyen-Thoi et al. / Computers and Structures 125 (2013) 100–113

2

eD0 1

ZZ

T

D

where d 1 ¼ ½ deO de1 de2  is the vector of nodal degrees of freedom of the sub-triangle D1; and NI(x), I = 1, 2, 3, are shape functions in a natural coordinate defined by Eq. (57). The membrane strains eD0 1 , the curvatures of deflection jD1 , the altered shear strains cD1 and the geometric strains eGP D1 in the sub-triangle D1 are then obtained by

h

deO

3

~e0e ¼

Xe

ZZ

j~ e ¼

3 2 deO i6 7 D1 D1 7 jD1 ¼ bD1 1 bD2 1 bD3 1 6 4 de1 5 ¼ b d ; |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} de2 bD1 2 3 deO 6 7  D1 D1 cD1 ¼ sD1 1 sD2 1 sD3 1 6 de1 7 4 5¼s d ; |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}

Xe

ð76Þ

Ue ðxÞ ¼

j

D1

h

de1

3

i6 7 D1 D D D D 7 þ bm21 13 bm11 þ bm31 13 bm11 6 4 de2 5 ¼ Bm de ; |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} D de3 Bm1 1 D1 b 3 m1

2 3 i de1 6 7 ¼ 13 bD1 1 þ bD2 1 13 bD1 1 þ bD3 1 13 bD1 1 4 de2 5 ¼ BD1 de ; |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} d h

ð78Þ

3 h i de1 6 7 D D D D D ¼ 13 s1 1 þ s2 1 13 s1 1 þ s3 1 13 s1 1 4 de2 5 ¼ SD1 de ; |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} d e3 D S

ð79Þ

Ue ðxÞ dX;

ð83Þ

Dj

3 X

GDj

eP

ZZ

Ue ðxÞ dX;

ð84Þ

Dj

x 2 Xe

0

x R Xe

;

ð85Þ

j~ e ¼

3 1X AD jDj ; Ae j¼1 j

c~e ¼

3 1X AD cDj ; Ae j¼1 j

~eGPe ¼

ð86Þ

3 1X GD AD e j ; Ae j¼1 j P

or in the matrix form by

e e; c e G de ; ~e ¼ e j~ e ¼ Bd Sde ; ~eGPe ¼ B

ð87Þ

e m is the smoothed membrane strain gradient matrix given where B by 3 X D em ¼ 1 B AD Bmj ; Ae j¼1 j

ð88Þ

3 X e¼ 1 AD BDj ; B Ae j¼1 j

ð89Þ

e S is the smoothed shear strain gradient matrix given by

1

3 h i de1 6 7 ¼ 13 bDG11 þ bDG21 13 bDG11 þ bDG31 13 bDG11 4 de2 5 ¼ BDG1 de : |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} d e3 D

3 1X e S¼ AD SDj Ae j¼1 j

2

eGP D1

ZZ

e is the smoothed bending strain gradient matrix given by B

2

cD1

1=Ae

e m de ; ~e0e ¼ B ð77Þ

e3

BD1

cDj

j¼1

3 1X D AD e j ; Ae j¼1 j 0

D

where b , sD1 , and bG1 are, respectively, computed similarly as the matrices Bm, B, S, and BG of the DSG3 in Eqs. (59), (61), (63) and (65) but with two following changes: (1) coordinates of three node xi = [xi, yi]T, i = 1, 2, 3 are replaced by xO, x1 and x2, respectively; and (2) the area Ae is replaced by the area AD1 of sub-triangle D1. Substituting deO in Eq. (74) into Eq. (76), and then rearranging we obtain

eD0 1 ¼

eGh P Ue ðxÞ dX ¼

ð82Þ

Dj

j¼1

~e0e ¼

G

2

3 X

Ue ðxÞ dX;

ZZ

where Ae is the area of the triangular element, then the smoothed ~ e, c ~e and ~eGPe in Eqs. (81)–(84) become strains ~e0e , j

de2 3 2 deO h i6 7 D1 D1 7 ¼ bDG11 bDG21 bDG31 6 4 de1 5 ¼ bG d ; |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} D de2 b 1 D1

ch Ue ðxÞ dX ¼

ZZ

sD1

D bm1 ,

jDj

ð81Þ

where Ue(x) is a given smoothing function that satisfies the unity R property Xe Ue ðxÞ dX ¼ 1. Using the following constant smoothing function

h

eGP D1

3 X

Ue ðxÞ dX; Dj

j¼1

Xe

~eGPe ¼

ZZ

j¼1

jh Ue ðxÞ dX ¼

ZZ

D

e0 j

Xe

c~e ¼

i6 7 D1 D1 7 ¼ bDm11 bDm21 bDm31 6 4 de1 5 ¼ bm d ; |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} D de2 bm1

3 X

eh0 Ue ðxÞ dX ¼

BG

ð80Þ

ð90Þ

e G is the smoothed shear strain gradient matrix given by and B

1

Similarly, by using cyclic permutation, we easily obtain the membrane displacements eD0 2 , eD0 3 , the curvatures of the deflection jD2 , jD3 , the altered shear strains cD2 , cD3 , and the geometric strains eGP D2 , eGP D3 for the second sub-triangle D2 and third sub-triangle D3, respectively. Now, applying the cell-based strain smoothing operation in the CS-FEM [27,28], the constant membrane strains eD0 1 , eD0 2 , eD0 3 , bending strains jD1 , jD2 , jD3 , constant shear strains cD1 , cD2 , cD3 , and geometric strain eGP D1 , eGP D2 , eGP D3 are used to create a smoothed ~ e , a smoothed membrane strains ~e0e , a smoothed bending strains j ~e , and a smoothed geometric strains ~eGPe on the eleshear strains c ment Xe, respectively, such as:

3 X D eG ¼ 1 B AD B j : Ae j¼1 j G

ð91Þ

Therefore the global stiffness and geometric matrices of the CSDSG3 are, respectively, assembled by

eP ¼ K

Ne X

e Pe ; K

ð92Þ

e¼1

eG ¼ K P

Ne X

eG ; K Pe

e¼1

e Pe is the smoothed element stiffness matrix given by where K

ð93Þ

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T. Nguyen-Thoi et al. / Computers and Structures 125 (2013) 100–113

e Pe ¼ K

ZZ Xe

e T Dm B e m dX þ B m

ZZ

e T Db Bd e Xþ B Xe

ZZ

i

e b se ST D SdX Xe

e T Dm B e m Ae þ B e T Db BA e eþe b se ¼B ST D SAe m

ð94Þ

e G is the smoothed element geometric matrix given by and K Pe

eG ¼ K Pe

ZZ

Xe

e T r0 B e G dX ¼ B e T r0 B e G Ae : B G G

where dSt ¼ ½ur ; us ; uz ; br ; bs T is the displacement vector of node ith of the element eth and /i i = 1, 2, are the linear shape functions in a natural coordinate defined by

/1 ¼ ð95Þ

1 ð1  nÞ; 2

KSt ¼

ueSt ¼

2 X i /i I5 dSt ;

ð96Þ

1 ð1 þ nÞ; 2

n 2 ½1; 1:

ð97Þ

By discretizing the domain of stiffener into elements and substituting Eq. (96) into Eqs. (49), (51) and (52), we have the stiffness, mass, and geometric matrices of stiffener, respectively, as

3.3. Formulation of FEM for the Timoshenko beam Next, we use the two-node isoparametric element to approximate the stiffener. The interpolation of displacement field on an element eth in the natural coordinate is

/2 ¼

MSt ¼ KGSt ¼

ne X KeSt ;

ð98Þ

e¼1 ne X

MeSt ;

ð99Þ

KGSte ;

ð100Þ

e¼1 ne X e¼1

i¼1

Fig. 5. A simply supported square plate stiffened by a single stiffener. Table 1 Central deflections wC/(qL4/100D). Mesh

Central deflection CS-DSG3

Rossow [7]

SAP2000 [7]

NASTRAN [7]

Concentric

Eccentric

Concentric

Eccentric

Concentric

Eccentric

Concentric

Eccentric

44 66 88

0.0627 0.0682 0.07

0.0198 0.0216 0.0223

0.0709

0.0213

0.0721

–

–

0.0232

10  10 12  12

0.0709 0.0713

0.0226 0.0227

Fig. 6. Convergence of central deflections of the eccentrically stiffened plate by DSG3 and CS-DSG3.

Fig. 7. Convergence of central deflections of the concentrically stiffened plate by DSG3 and CS-DSG3.

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T. Nguyen-Thoi et al. / Computers and Structures 125 (2013) 100–113

where ne is the number of elements of stiffener, and the element stiffness, mass, and geometric matrices, respectively, are computed by

KeSt ¼ MeSt

Z

¼

KGSte ¼

ð101Þ

e ¼ F; Kd e  x2 MÞd ¼ 0; ðK

ð108Þ

U A mSt AU dl;

ð102Þ

e  kcr K e G Þd ¼ 0; ðK

ð110Þ

ðLGSt UÞT r0St LGSt U dl:

ð103Þ

e where x is the angular frequency; kcr is the critical buckling load; K, e G are the global stiffness, mass and geometric matrices, M, and K respectively. It can be seen that the novelty of the present work is the combination of the original plate element CS-DSG3 [48] with a membrane element and stiffened by a thick beam element Timoshenko. The present combined element CS-DSG3 is hence different from the original element CS-DSG3 [48] which only considers the isotropic Mindlin plates with bending and shear strains. In addition, the present combined element CS-DSG3 also analyses the buckling problems which were not considered in the original CSDSG3 [48].

ðLESt UÞT DSt LESt U dl;

l

Z

T

T

Zl l

For the static, free vibration, and pre-buckling analysis of stiffened plate, we have three formulations, respectively, as

3.4. Formulation of CS-DSG3 for the stiffened plate We now apply the displacement conforming conditions to formulate the overall stiffness, mass, and geometric matrices of the stiffened plate. The displacement conforming conditions are stated as

dSt ¼ Td;

ð104Þ

where T is the transform matrix as presented in [20], dSt is the nodal displacement vector of the stiffener and d is the nodal displacement vector of the stiffened plate. Using Eqs. (73), (92), (93), (98)–(100) and (104), we have the overall stiffness, mass, and geometric matrices for the stiffened plate, respectively, as

e ¼K e P þ TT KSt T; K e G þ TT KG T; eG ¼ K K P

St

T

M ¼ MP þ T MSt T:

ð105Þ ð106Þ ð107Þ

ð109Þ

4. Numerical results In this section, four various numerical examples of static, free vibration and buckling analyses are performed to show the accuracy and stability of the CS-DSG3 compared to those of analytical solutions, experimental results and others available numerical results.

Fig. 8. Geometry and boundary condition of a fully clamped square plate with a single stiffener in one direction.

Fig. 9. Geometry and boundary condition of a fully clamped square plate with two stiffeners in one direction.

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T. Nguyen-Thoi et al. / Computers and Structures 125 (2013) 100–113 Table 2 Frequencies of the clamped square plate reinforced by a single stiffener in a direction. Frequency

CS-DSG3 (16  16)

Olson and Hazel [8]

Mukherjee and Mukhopadhyay [53]

Peng et al. [21]

Experimental

FEM

1 2 3 4 5 6 7 8 9 10

717.9 756.6 997.2 1010.4 1424.5 1430.1 1680.7 1848.3 2029 2032.2

689 725 961 986 1376 1413 1512 1770 1995 2069

718.1 751.4 997.4 1007.4 1419.8 1424.3 1631.5 1853.9 2022.8 2025

711.8 768.2 1016.5 1031.9 1465.2 1476.5 1743.8 1866.3 2109.1 2117

574.11 754.35 846.55 993.47 1293.8 1402.8 1650.3 1805.1 1897.2 1904.4

11

2226.6

2158

2224.9

2264.1

–

12

2250.6

2200

2234.9

2296.3

–

13

2433.4

2347

2400.9

2505.8

–

14

2661.3

2597

2653.9

2779.9

–

15

2679.8

2614

2670.2

2820.9

–

16

2808.4

2784

2802.4

2933.3

–

17

2810.2

2784

2804.6

2943.8

–

18

3248.6

3174

3259

3536.1

–

19 20 21

3252.7 3269.8 3754.3

3174 3332 3660

3265.9 3414.2 3754

3536.1 3649.8 3925.7

– – –

22 23 24

3755 3984.6 4017.2

3730 3780 3913

3754.8 3985.5 4045.9

3941.6 4084.3 4138.8

– – –

18

Peng et.al.

16

Relative error (%)

CS-DSC3

Mukherjee & Mukhopadhyay

14

Olson & Hazell

12 10 8 6 4 2 0

Fig. 10. Comparison of frequencies of the plate stiffened by a single stiffener between CS-DSG3 and previous results.

4.1. A static analysis of square plate stiffened by a single beam Consider a simply supported square plate stiffened by a single stiffener as shown in Fig. 5. The plate is subjected to a uniformly distributed load of 6.89476  103 N/mm2. The material parameters of the plate and beam are given by Young’s modulus 1.1721  105 N/mm2 and Poisson’s ratio m = 0.3. Five uniform discretizations of plate corresponding to the meshes 4  4, 6  6, 8  8, 10  10 and 12  12 elements are used.

1

2

3

4

5 6 Modes

7

8

9

10

Fig. 11. Relative error of ten first frequencies of CS-DSG3 compared with those of the others.

The convergence of central deflections of the stiffened plate by the CS-DSG3 for both cases of the eccentric and concentric plates, together with the results of Rossow [7], and of the commercial softwares SAP2000 and NASTRAN [7], are presented in Table 1. It is seen that the results of the CS-DSG3 give a good convergence to the reference results. In addition, Figs. 6 and 7 compare the convergence of central deflections of the stiffened plate by the DSG3 and the CS-DSG3 for the cases of the eccentric and concentric plates, respectively. It is observed that the results of the CS-DSG3 are more accurate than those of the DSG3 in both cases.

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T. Nguyen-Thoi et al. / Computers and Structures 125 (2013) 100–113

Table 3 Frequencies of the clamped square plate reinforced by two stiffeners in a direction. Frequency

CS-DSG3 (18  18)

Olson and Hazel [8] Experimental

Holopainen [52]

FEM

1

931.3

909

965.3

943.8

2

1239.4

1204

1272.3

1237.9

3

1319.7

1319

1364.3

1331.0

4

1405.5

1506

1418.1

1361.2

5

1567.5

1560

1602.9

1561.5

6

1741.6

1693

1757.1

1706.1

7

1810.1

1807

1854.1

1808.3

8

1985.5

1962

2051.4

1962.9

9

2102.3

2052

2109.4

2057.9

Fig. 12. Comparison of frequencies of the plate stiffened by two stiffeners between CS-DSG3 and other results.

10 11 12 13 14 15 16 17 18 19

2163 2414.5 2540.7 2621.2 2642.8 2646.2 2711.8 3098 3161.6 3174.6

2097 2410 2467 2505 2618 2631 2964 3169 3135 3120

2253.1 2453.8 2566.3 2624.2 2729.6 2731.9 2915.4 3180.1 3242.0 3279.1

2163.7 2392.4 2499.6 2561.0 2625.6 2622.8 2832.0 3104.6 3161.8 3185.3

and plotted in Fig. 10. Moreover, Fig. 11 shows the relative errors of ten first frequencies of these methods. Again, it is observed that the results of the CS-DSG3 agree well with those of Olson and Hazel [8]. In comparison with the experimental results, the results by the CS-DSG3 using triangular elements are even better than Mukherjee’s results by isoparametric eight-node quadrilateral elements and Peng’s results [21] by mesh-free element as shown in Fig. 11.

20

3245.2

3251

3313.3

3228.8

4.3. A free vibration analysis of square plate stiffened by double beams

21

3314.3

3446

3412.3

3317.6

22

3331.6

3745

3635.6

3460.5

23

3742.1

4019

4059.4

3968.8

24

4044.6

4053

4135.1

4025.6

In this example, we consider a fully clamped square plate stiffened by double stiffeners parallel to the x-axis as shown in Fig. 9. The material parameters of the plate and beam are given by Young’s modulus 68.9 GPa, Poisson’s ratio 0.3, and the density q = 2670 kg/m3. A uniform discretization of 18  18 elements is used. Frequencies of the reinforced plate by the CS-DSG3, together with the reference results of Holopainen [52] and experimental frequencies [8] are listed in Table 3 and plotted in Fig. 12. Again, it is observed that the results of the CS-DSG3 using only three-node triangular elements agree well with experimental frequencies and those of Holopainen’s results using the nine-node mixed interpolation of tensorial components (MITC9) for bending plate and stiffeners element.

4.2. A free vibration analysis of square plate stiffened by a single beam We next consider a fully clamped square plate stiffened by a single stiffener parallel to the x-axis as shown in Fig. 8. The material parameters of the plate and beam are given by Young’s modulus 68.7 GPa; Poisson’s ratio m = 0.3 and the density q = 2670 kg/ m3. A uniform discretization of 16  16 elements is used. The frequencies of the reinforced plate by the CS-DSG3, together with the reference results of Olson and Hazel [8], Mukherjee and Mukhopadhyay [53], and Peng et al. [21] are listed in Table 2

4.4. A pre-buckling vibration analysis of square plate stiffened We now consider a pre-buckling vibration analysis of a simple support square plate stiffened by a single concentric stiffener as

Fig. 13. A simple support square plate with a concentric stiffener and an in-plane load

r0x .

T. Nguyen-Thoi et al. / Computers and Structures 125 (2013) 100–113

Fig. 14. Buckling parameters of the stiffened plate with a single stiffener by the CSDSG3 (d = 0.05, c = 5).

Fig. 15. Buckling parameters of the stiffened plate with a single stiffener by the CSDSG3 (d = 0.05, c = 10).

Fig. 16. Buckling parameters of the stiffened plate with a single stiffener by the CSDSG3 (d = 0.05, c = 15).

111

Fig. 17. Buckling parameters of the stiffened plate with a single stiffener by the CSDSG3 (d = 0.05, c = 20).

Fig. 18. Buckling parameters of the stiffened plate with two stiffeners by the CSDSG3 (d = 0.05, c = 10/3).

Fig. 19. Buckling parameters of the stiffened plate with two stiffeners by the CSDSG3 (d = 0.05, c = 5).

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Fig. 20. Buckling parameters of the stiffened plate with two stiffeners by the CSDSG3 (d = 0.05, c = 20/3).

Fig. 21. Buckling parameters of the stiffened plate with two stiffeners by the CSDSG3 (d = 0.05, c = 10).

Fig. 23. Relative errors of buckling parameters of the stiffened plate with two stiffeners by the CS-DSG3 (d = 0.1) compared to Timoshenko’s results.

shown in Fig. 13. The stiffened plate is subjected to an in-plane load r0x . In this case, we neglect the torsion and eccentric effects of stiffener (e = 0 and GJs = 0). The plate has the ratio of two dimens sions L and B as b = L/B. The stiffener’s parameters include c ¼ EI BD As (ratio of stiffness between stiffener and plate), and d ¼ BL (ratio of 3 s hs Et 3 cross area between stiffener and plate), where Is ¼ b12 , D ¼ 12ð1 vÞ and As = bshs. A parametric examination by the CS-DSG3 is conducted to determine the effects of various ratio of stiffness c on the buckling parameters of the stiffened plate. Figs. 14–17 present, respectively, the variation of the buckling parameters of the stiffened plate by a single stiffener with d = 0.05 and various ratio of stiffness, c = 5, 10, 15 and 20. And Figs. 18–21 present, respectively, the results for the case of the stiffened plate by two stiffeners with d = 0.1 and various ratio of stiffness, c = 10/3, 5, 20/3 and 10. It is observed that the results by the CS-DSG3 have a good agreement with the analytical solutions of Timoshenko and Geer [54] and those of mesh-free method proposed by Peng et al. [21]. In almost the cases, the relative errors between the results of CS-DSG3 and those of Timoshenko and Geer [54] are less than 5% as illustrated in Figs. 22 and 23 corresponding to two specific cases: the stiffened plate with a single stiffener (d = 0.05; and the stiffened plate with two stiffeners (d = 0.1).

5. Conclusions In this paper, we extended the CS-DSG3 using triangular elements to analyse the static, free vibration and buckling analyses of the eccentrically stiffened plates. In the present method, the original plate element CS-DSG3 is combined with a membrane element and stiffened by a thick beam element. The eccentricity between the plate and the beam is included in the formulation of the beam. The compatibility of deflection and rotations of stiffeners and plate is assumed at the contact positions. The accuracy and reliability of the proposed method is verified by comparing its numerical solutions with those of analytical solutions, experimental results and others available numerical results. Acknowledgements Fig. 22. Relative errors of buckling parameters of the stiffened plate with a single stiffener by the CS-DSG3 (d = 0.05) compared to Timoshenko’s results.

This research is funded by Vietnam National University Hochiminh City (VNU-HCM) under Grant No. B-2013-18-03.

T. Nguyen-Thoi et al. / Computers and Structures 125 (2013) 100–113

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