Nonlinear Free Vibration of Laminated Composite and Sandwich Plates Using Multiquadric Collocations

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ScienceDirect Materials Today: Proceedings 2 (2015) 3049 – 3055

4th International Conference on Materials Processing and Characterization Nonlinear Free Vibration of Laminated Composite and Sandwich Plates Using Multiquadric Collocations a

Manoj Kumar Solanki , Sabin Kumar Mishrab , K K Shuklac and Jeeoot Singhd* a-Department of Mechanical Engineering, B.I.T., KEC-201007, Gaziabad, U.P. India. b- Faculty of engineering at HCT, Ras Al Khaimah, RaʼS Al Khaymah, United Arab Emirates c-Departement of Applied Mechanics, MNNIT, Allahabad,211004, UP, India d*- Department of Mechanical Engineering, B.I.T., Mesra, Ranchi-835215, Jharkhand, India. Abstract

The paper examines the nonlinear free vibration response of shear deformable laminated composite and sandwich plates. The mathematical formulation of the actual physical problem of the plate is presented utilizing shear deformation theories and von Karman nonlinear kinematics. A meshless technique based on multi quadric radial basis function (MQRBF) is used for analysis of the problems. Using standard computational software of MATLAB 7, the eigenvalue problem is solved to get the eigen frequencies and eigen vectors. Several numerical results depicting the non dimensional frequency parameter of the isotropic, orthotropic, laminated and sandwich plates are presented. In order to check the accuracy of the present method, the results are compared with existing one in the literature. The effects of aspect ratio, span to thickness ratio, orthotropy ratio of the material and fiber orientation on the frequency are also studied. © 2014Elsevier The Authors. Ltd. All rights reserved. © 2015 Ltd. AllElsevier rights reserved. Selection and under responsibility of the committee members of the 4th conferenceconference on Materialson the 4th International Selection andpeer-review peer-review under responsibility ofconference the conference committee members ofInternational Processing Processing and Characterization. Materials and Characterization. Keywords: Nonlinear free vibration, Composite, Plates, MQRBF, Meshfree.

* Corresponding author. Tel.: +919431382595

E-mail address: [email protected], [email protected]

2214-7853 © 2015 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of the conference committee members of the 4th International conference on Materials Processing and Characterization. doi:10.1016/j.matpr.2015.07.210

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1. Introduction The significant increase of the industrial use of laminated composite and sandwich plates calls for the development of new numerical tools for the analysis of such high performance structures. Most popular techniques are based on finite differences and finite elements. In the last decade, meshless methods have received much attention because of their flexibility in the construction of finite dimensional subspaces. Most of the work till date on meshless methods using RBFs relates to scattered data approximation, there has recently been an increased interest in their use for solving partial differential equations (PDEs). This approach, which approximates the whole solution of the PDE directly using RBFs, is very attractive due to the fact that this is truly a mesh-free technique. A considerable work has been done on meshfree analysis of composites in the last few years. Kansa [1] introduced the concept of solving PDEs using RBFs. Kansa’s method is an unsymmetrical RBF collocation method based upon the MQ interpolation functions. Radial basis functions (RBFs) is one of the best rescently developed methods that has attracted several researchers in recent years especially in the area of computational mechanics. A review of meshless methods for laminated plated has been presented by Liew et al [2]. Radial basis function was applied by Xiang et al [3-4] for linear flexural and free vibration analysis of the laminated composite and sandwich plates. Recently Ferreira et al [6-7] used wavelets and Wendland radial basis function for buckling analysis of laminated composite plates. Liew and Huang [8] used moving leastsquares differential quadrature for bending and buckling; Liew et al [8-10] used reproducing kernel approximations and meshfree method for buckling analysis of isotropic circular and skew plates. R.Ramanarayanan et al [11] have studied the Heat resistant composite materials for aerospace applications . Gunti Rajesh et al [12] focuses their study on Effect of fibre loading and successive alkali treatments on Tensile properties of short jute fibre reinforced polypropylene composites. 2. Mathematical Formulation A rectangular plate having edge lengths ‘a’ and ‘b’ along x and y axes, respectively and thickness ‘h’ along z axis, whose mid plane is coinciding with xy plane of the coordinate system is considered. Perfect bonding between the layers of laminated composites and sandwich plates is assumed. The displacement at any point in the plate is expressed as: u ( x, y, z, t ) u0 ( x, y, t )  z

ww0 ( x, y, t )  f ( z ) I x ( x, y , t ) wx

(1.1)

ww ( x, y, t )  f ( z ) I y ( x, y , t ) v( x, y, z, t ) v0 ( x, y, t )  z 0 wy

(1.2)

w( x, y, z, t ) w0 ( x, y, t )

(1.3)

Where, u, v and w are the in-plane and transverse displacements of the plate at any point (x, y, z, t) in x, y and z directions, respectively. u0, v0 and w0 are the displacements at mid plane of the plate at any point (x, y) in x, y and z directions, respectively. The functions

Ix and I y

are the higher order rotations of the normal to the mid plane due §

2·

¨ ©

3h ¸¹

to shear deformation about y and x axes, respectively. f ( z ) z ¨1  4 z 2 ¸ represents the transverse shear stress functions determining the distribution of the transverse shear strains and stresses along the thickness. Strain-displacement relations are expressed as:

­H ½ °° xx °° ®H yy ¾ ° ° ¯° J xy ¿°

2 ­ wu w 2 w0 wI 1 § ww · ° 0 z  f  z x  ¨ 0 ¸ wx 2 © wx ¹ ° wx wx 2 ° wI y 1 § ww0 ·2 ° wv0 w 2 w0 z  f z  ¨ ® ¸ 2 wy 2 © wy ¹ wy ° wy ° 2 wI y w w0 wI x ° wu0 wv0 ° wy  wx  2 z wxwy  f  z wy  f  z wx ¯

½ ° ° ° ° ¾ ° ° § ww0 · § ww0 · ° ¨ ¨ ¸° ¸ © wx ¹ © wy ¹ ¿

(2.1)

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­ wf  z ½ Iy ° °° wz ° ® ¾ ° wf  z I ° x° ¯° wz ¿

­° J yz °½ ® ¾ ¯° J zx ¿°

(2.2)

The governing differential equations of plate are obtained using Hamilton’s principle and expressed as: Gu0 :

Gv0 :

3 2 · § w 2u 0  I w w0  I w I x ¸ ¨ I0 1 3 2 2 ¨ wW wxwW wW2 ¸¹ ©

wN xx wN xy  wx wy

wN xy wx



§ w 2v 3 w 2I y · 0  I w w0  I ¨ I0 ¸ 1 3 2 2 ¨ wW wywW wW2 ¸¹ ©

wN yy wy

wN xy · ­ wwo ½ § wN yy wN xy · ­ wwo ½ § wN Gw0 : ¨ xx   ¸® ¸® ¾¨ ¾ wy ¹ ¯ wx ¿ © wy wx ¹ ¯ wy ¿ © wx I0

GI x :

GI y :

w 2 w0 wt

2

(3.1) (3.2)  N xx

3 4 · § w 3u · § 4 0  w v0 ¸  I ¨ w w0  w w0 ¸  I1 ¨ ¨ wxwW2 wywW2 ¸ 2 ¨ wx 2wW2 wy 2wW2 ¸ ¹ © ¹ ©

w 2 w0 wx

2

 N yy

w 2 w0 wy

2

 2 N xy

4 § w 4I · x  w Iy ¸  I4 ¨ 2 2 2 2 ¨ wy wW ¹¸ © wx wW

w 2 w0 wxwy



w 2 M xx wx

2



w 2 M yy wy

2

2

w 2 M xy wxwy

 qz

(3.3)

f f wM xy wM xx   Qxf wx wy

3 2 § w 2u 0  I w w0  I w I x ¨I 4 5 2 2 ¨ 3 wW2 w x wW wW ©

· ¸ ¸ ¹

(3.4)

f wM xy

§ w 2v 3 w 2I y · 0  I w w0  I ¨ I3 ¸ 4 5 2 2 ¨ wW wywW wW2 ¸¹ ©

(3.5)

wx



f wM yy

wy

 Qyf

The combinations of the simply supported (S) and clamped (C) boundary conditions are considered in the analysis and expressed as: Simply Supported: x

0, a : N xx

0, v0

y

0 , b : u0

0, N yy

0, w0 0, w0

0, M xx

0, I y

0, Ix

0, M yy

0 0

(4.1) (4.2)

Clamped: x

0, a : u0

y

0, b : Nxy

0, Nxy

0, w0

0, Ix

0, I y

0

(4.3)

0, v0

0, w0

0, Ix

0, I y

0

(4.4)

3. Solution methodology Radial basis functions are used to discretized the governing differential equations in space domain. A radial basis functions (g) approximate the solution on the principle of interpolation of scattered data over entire domain. The data point in and over boundary are connected through a special kind of polynomial in terms of radial distance between nodes which is known as radial basis function (RBF). For the present analysis, Multiquadrics radial basis functionis taken as: g

 r 2  c2

m

; c

§ a ·

2

§ b ·

¨ ¸ D . ¨¨ ¸¸  ¨ n ¸ © nx ¹ © y ¹

2

(5.1)

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Where,

x  xj   y  y j 2

X Xj

r

2

and ‘m’ and ‘c’ are shape parameters. ‘a’ and ‘b’ are the length and width of

the plate, respectively. nx and ny are number of divisions along the length and width, respectively. α is a constant that governs the value of ‘c’ for interior and boundary nodes. Radial basis function based formulation works on the principle of scattered data over entire domain consisting of domain and boundary nodes. For a two dimensional rectangular domain with any arbitrary boundary having NB boundary nodes and NI domain interior nodes any variable u can be interpolated in the form of radial distance between nodes using following expression: N

u0

¦D g  X  X , c u j

(5.2)

j

j 1

Where, N is total number of nodes, which is equal to the summation of boundary nodes NB and domain interior nodes NI, D uj is unknown coefficient, g X  X j , c is radial basis function, X  X j is distance between the nodes and c is





the shape parameter. Similarily other variables can be defined Singh and Shukla [13]. The discretized governing equations for linear free vibration analysis can be written as: ª ª[ K ] º º ª[M ]º «« L » » ^G`  Z2 « » N u1 [ K ] 0 «¬ ¬ B ¼ N u N ¬ ¼ N u N »¼

Where, [ K ]L [ K ]B

(5.3)

0



 

ªL g X  X , c º i j »¼ NI u N ¬« D ªL g X  X , c º i j «¬ B »¼ NBu N



(5.4)



[M } ª g X i  X j , c º «¬ »¼ NI u N

Or ([K]1+λ[M]1){δ}=0

(5.5)

Using standard eigen value solver for equation (6.5) the frequency is calculated as: [V,D]=eig([K]1,[M]1);

(5.6)

frequency (ω) = D The following iterative procedure is adopted to solve the nonlinear eigen value problem: 1. In initial step linear eigen values and eigen vectors are calculated from Equation.(5.6). 2. Mode shape normalization: The transverse displacement wmax at center of the plate for the selected mode is calculated using equation (5.6) . 3. Then scaling of the selected mode shape is done using δj = (wa/wmax) δj. Here wa is the specified amplitude of motion. 4. Nonlinear stiffness matrix [K] NL is evaluated using normalized and scaled mode shape δj. 5. Stiffness matrix is modified as; [K] = [K]L + [ K] NL 6. Eigen Value Problem is solved to get ω(j+1) and {δj+1} 7. A relative convergence criteria §¨ ©

convergence check).

Z( j 1)  Z( j )

· d Z( j ) ¸¹

(where,  104 is adopted as the stoping criteria for the

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4. Results and Discussions: In this section, large amplitude free vibration behavior of the isotropic, laminated, sandwich and functionally graded plates is addressed. The efficacy of the proposed methodology is examined by comparing the results with those (Znl / Zl ) corresponding to different amplitude available in the literature. The nonlinear to linear frequency ratio ratio (wmax/h) are presented for different cases. 4.1 Isotropic Plate An isotropic plate ( Q =0.3, a/h=10) is taken for the analysis. Table 1 presents the fundamental frequency ratio (Znl / Zl ) for different amplitude to thickness ratios (wmax/h). The present results are compared with the results due to Ganapathi et al. [14] and Chandrashekhar and Ganguli [15]. It can be seen that the present results are in very good agreement.

(Z / Z )

nl l for different values of amplitude ratios of a simply Table-1 Comparison of fundamental frequency ratio supported square isotropic plate

Method

(wmax/h) 0.6 0.8 1.213 1.377

Present (7x7)

0.2 1.021

0.4 1.102

1 1.553

Present (9x9)

1.024

1.106

1.217

1.382

1.559

Present (11x11)

1.025

1.107

1.218

1.383

1.560

Present (13x13)

1.027

1.109

1.220

1.386

1.564

Ganapathi et al [14]

1.0266

1.1019

1.2194

1.3704

1.5414

Chandrashekhar and Ganguli [15]

1.0256

1.0975

1.2103

1.3582

1.5321

4.2 Laminated Plate

(Z / Z )

nl l at various amplitude ratios for a simply supported The results for fundamental nonlinear frequency ratio [0/90/0/90], rectangular, laminated composite plates are obtained and shown in Table 2. The material properties used Q are: E11=40E22, G12=G13=0.6E22, 12 = 0.25, G23=0.5E22, E22=6.92 GPa. The results are compared with the results due to Chandrashekhar and Ganguli [15] and Onkar and Yadav [16]. It is observed that the results agree well.

Table-2 Comparison of fundamental frequency ratio supported [0/90/ 90/0] cross-ply square plate a/b

0.5

1

(Znl / Zl )

for the various values of amplitude ratios of a simply

Source

(wmax/h) 0.3

a/h=100 0.6

0.9

0.3

Chandrashekhar and Ganguli [15]

1.215

1.703

2.351

1.024

1.094

1.203

Onkar and Yadav [16]

1.204

1.680

2.265

1.024

1.095

1.204

Present

1.2092

1.7063

2.3834

1.024

1.095

1.208

Chandrashekhar and Ganguli [15]

-

-

-

1.011

1.039

1.085

Onkar and Yadav [16]

-

-

-

1.009

1.037

1.082

1.1960

1.6180

2.1347

1.012

1.039

1.087

Present

a/h=33.33 0.6 0.9

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4.3 Sandwich Plate Nonlinear free vibration analysis of a three layered sandwich plate with a soft core is carried out. Following properties are used:

U (a) for isotropic face layers: E1=E2=68.9 GPa, Q 12 = 0:30, s = 2.77 x 103 N s2/m4 (b) for core layer: G13=0.134 GPa, G23=0.052 GPa,

Uc

= 0.122 x 103 Kg=m3

(c) Geometric properties: a=1.83 m, b=1.22 m, hs = 4.06x 10-4 m, hc=0.0064 m.

(Z / Z )

nl l for the various values of amplitude ratios of a Table-3 Comparison of fundamental frequency ratio square sandwich plate

Source Chandrashekhar and Ganguli [15]

0.3 1.00297

(wmax/h) 0.6 0.9 1.02289 1.04748

Present

1.00377

1.04157

1.05828

The results obtained by present solution methodology are shown in Table 3 along with the results due to Chandrashekhar and Ganguli [15] in order to show the applicability of the present solution methodology. The present result seems to be in good agreement with the results due to Chandrashekhar and Ganguli [15]. 5. Conclusions Nonlinear free vibration responses of thick to thin isotropic, laminated, sandwich rectangular plates are presented. Multiquadric RBF based meshless method is utilized first time to the best of author’s knowledge for nonlinear free vibration analysis of laminated, sandwich rectangular plates. The results herein agree well with the other numerical and analytical results indicating the efficacy of present solution methodology. References [1] [2] [3] [4] [5] [6] [7] [8]

E.J. Kansa, Multiquadrics a scattere ddat aapproximation scheme with applications to computational fluid dynamics, i:Surface approximations and partial derivative estimates, Comput. Math. Appl. 19 (8/9) (1990) 127–145. Liew K.M. Xin Zhao, Antonio J.M. Ferreira, A review of meshless methods for laminated and functionally graded plates and shells, Composite Structures, 93, 8, 2031-2041 Xiang S, Ke-ming Wang, Yan-ting Ai, Yun-dong Sha, Shi H. Analysis of isotropic, sandwich and laminated plates by a meshless method and various shear deformation theories. Composite Structures 91(2009) 31-37. Xiang S, Ke-ming Wang. Free vibration analysis of symmetric laminated composite plates by trigonometric shear deformation theory and inverse multiquadrics RBF. Thin-Walled Structures 47(2009) 304-310 Castro LMS, Ferreira AJM, Bertoluzza S, Batra RC, Reddy JN. A wavelet collocation method for the static analysis of sandwich plates using a layerwise theory. Composite Structures 92 (2010) 1786–179 Ferreira AJM, Castro LM, Roque CMC, Reddy JN, Bertoluzza S. Buckling analysis of laminated plates by wavelets. Article in Press. Computers and Structures (2011) Ferreira AJM, Roque CMC, Neves AMA, Jorge RMN, Soares CMM, Liew KM. Buckling and vibration analysis of isotropic and laminated plates by radial basis functions. Composites: Part B: Engineering 42 (2011) 592–606 Liew KM, Huang YQ. Bending and buckling of thick symmetric rectangular laminates using the moving least-squares differential quadrature method. International Journal of Mechanical Sciences 45 (2003) 95–114 [9] Liew KM, Chen XL, Reddy JN. Meshfree radial basis function method for buckling analysis of non-uniformly loaded arbitrarily shaped shear deformable plates. Computer Methods in Applied Mechanics Engineering 193 (2004) 205–224 [10] Liew KM, Wang J, Ng TY, Tan MJ. Free vibration and buckling analyses of shear-deformable plates based on FSDT meshfree method Journal of Sound and Vibration 276 (2004) 997–1017.

Manoj Kumar Solanki et al. / Materials Today: Proceedings 2 (2015) 3049 – 3055 [11] R.Ramanarayanan, C. HariVenkateswara Rao and C. Venkateshwara Reddy , Heat resistant composite materials for aerospace applications , Advanced Materials Manufacturing & Characterization 1(1) (2013), 79-82. [12] Gunti Rajesh, Atluri V. Ratna Prasad , Effect of fibre loading and successive alkali treatments on Tensile properties of short jute fibre reinforced polypropylene composites. Advanced Materials Manufacturing & Characterization 3(2) (2013), 528-532. [13] Jeeoot Singh and K. K. ShuklaNon, Linear flexural analysis of laminated composite plates using RBF based meshless method,Composite Structures, 94(5 )(2012) 1714-1720 [14] Ganapathi, M., Varadan, T.K. and Sarma, B.S..(1991). Nonlinear flexural vibrations of laminated orthotropic plates. Computers & Structures.,39(6),685–8. [15] Chandrashekhar, M. and Ganguli, R. , Nonlinear vibration analysis of composite laminated and sandwich plates with random material properties.” International Journal of Mechanical Sciences, 52(2010), 874–891. [16] Onkar, A.K. and Yadav, D , Non-linear free vibration of laminated composite plate with random material properties.” Journal of Sound and Vibration.,272(2004),627–41.

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