Journal of Applied Mathematics

Volume 2013 (2013), Article ID 123465, 14 pages

http://dx.doi.org/10.1155/2013/123465

Research Article

Analysis of Laminated Shells by Murakami’s Zig-Zag Theory and Radial Basis Functions Collocation

D. A. Maturi,¹ A. J. M. Ferreira,1,2 A. M. Zenkour,1,3 and D. S. Mashat¹

¹Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

²Faculdade de Engenharia da Universidade do Porto, Porto, Portugal

³Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafr El-Sheikh 33516, Egypt

Received 10 July 2013; Accepted 15 September 2013

Academic Editor: Song Cen

Copyright © 2013 D. A. Maturi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The static and free vibration analysis of laminated shells is performed by radial basis functions collocation, according to Murakami’s zig-zag (ZZ) function (MZZF) theory . The MZZF theory accounts for through-the-thickness deformation, by considering a ZZ evolution of the transverse displacement with the thickness coordinate. The equations of motion and the boundary conditions are obtained by Carrera’s Unified Formulation and further interpolated by collocation with radial basis functions.

1. Introduction

The efficient load-carrying capabilities of shell structures make them very useful in a variety of engineering applications. The continuous development of new structural materials leads to the ever increasingly complex structural designs that require careful analysis. Although analytical techniques are very important, the use of numerical methods to solve shell mathematical models of complex structures has become an essential ingredient in the design process.

The most common mathematical models used to describe shell structures may be classified into two classes according to different physical assumptions: the Koiter model [1], based on the Kirchhoff hypothesis, and the Naghdi model [2], based on the Reissner-Mindlin assumptions that take into account the transverse shear deformation. But these theories are not adequate to describe the so-called zig-zag (ZZ) effect in sandwich structures or layered composites, due to the discontinuity of mechanical properties between faces and core at the interfaces; see Figure 1 (to trace accurate responses of sandwich structures, see the books by Zenkert [3] and Vinson [4]).

Figure 1: Scheme of the zig-zag assumptions for a three-layered laminate.

The ZZ effect can be captured by the layerwise theories which typically assume independent degrees of freedom per layer. Unfortunately the computation can be prohibitive. The layerwise theories are reviewed in Burton and Noor [5], Noor et al. [6], Altenbach [7], Librescu and Hause [8], Vinson [9], and Demasi [10]. In order to overcome the computational cost of the layerwise theories, Murakami [11] proposed a zig-zag function (ZZF) that is able to reproduce the slope discontinuity. Equivalent Single Layer models with only displacement unknowns can be developed on the basis of ZZF. A review of the application of ZZF in plates and shells was presented by Carrera [12–16] and some relevant papers on the analysis of sandwich structures were presented in [17–19].

The most common numerical procedure for the analysis of the shells is the finite element method [20–24]. It is known that the phenomenon of numerical locking may arise from hidden constrains that are not well represented in the finite element approximation and, in the scientific literature, it is possible to find many methods to overcome this problem [25–30]. The present paper, that performs the bending and free vibration analysis of laminated shells by collocation with radial basis functions, avoids the locking phenomenon. A radial basis function, , is a spline that depends on the Euclidian distance between distinct data centers , also called nodal or collocation points. We use the so-called unsymmetrical Kansa method that was introduced by Kansa [31]. The use of radial basis function for the analysis of structures and materials has been previously studied by numerous authors [32–46]. The authors have recently applied the RBF collocation to the static deformations of composite beams and plates [47–49]. One of the authors has already combined Reddy's theory with radial basis functions in [50]. In fact, Reddy's theory is quite efficient for laminated (monolithic) composite plates or shells but not as efficient (or adequate) for sandwich structures because of the very high difference on material properties from the skins and the core. Reddy's theory does not allow the thickness stretching, but our formulation is more general and allows any expansion in the thickness direction. This is where the present paper shows more interest.

In this paper for the first time how the Unified Formulation can be combined with radial basis functions to the analysis of thin and thick laminated shells, using Murakami's zig-zag function, allowing for through-the-thickness deformations, is investigated. The quality of the present method in predicting static deformations and free vibrations of thin and thick laminated shells is compared and discussed with other methods in some numerical examples.

2. Applying the Unified Formulation to MZZF

The Unified Formulation (UF) proposed by Carrera [13, 51–53], also known as CUF, is a powerful framework for the analysis of beams, plates, and shells. This formulation has been applied in several finite element analyses, either by using the Principle of Virtual Displacement, or by using Reissner's mixed variational theorem. The stiffness matrix components, the external force terms, or the inertia terms can be obtained directly with this UF, irrespective of the shear deformation theory being considered.

In this section Carrera's Unified Formulation is briefly reviewed. How to obtain the fundamental nuclei, which allows the derivation of the equations of motion and boundary conditions, in weak form for the finite element analysis and in strong form for the present RBF collocation, is shown.

2.1. The MZZF Theory

Let us consider a sandwich plate (translation to shells becomes evident later in the paper) composed of three perfectly bonded layers, with being the thickness coordinate of the whole plate while is the layer thickness coordinate; see Figure 1. and are length and thickness of the square laminated plate, respectively. The adimensional layer coordinate is further introduced ( is the thickness of the kth layer). Murakami's zig-zag function was defined according to the following formula [11]:

has the following properties.

(1)It is a piecewise linear function of layer coordinates . (2) has unit amplitude for the whole layers. (3)The slope assumes opposite sign between two adjacent layers. Its amplitude is layer thickness independent.

A possible FSDT theory has been investigated by Carrera [14] and Demasi [15], ignoring the through-the-thickness deformations:

A refinement of FSDT by inclusion of ZZ effects and transverse normal strains was introduced in Murakami's original ZZF, defined by the following displacement field:

where and are the bottom and top -coordinates at each layer. The additional degrees of freedom and have a meaning of displacement, and their amplitude is layer independent.

2.2. Governing Equations and Boundary Conditions in the Framework of Unified Formulation

Shells are bidimensional structures in which one dimension (in general the thickness in direction) is negligible with respect to the other two in-plane dimensions. Geometry and the reference system are indicated in Figure 3. The square of an infinitesimal linear segment in the layer and the associated infinitesimal area and volume are given by

where the metric coefficients are

denotes the -layer of the multilayered shell; and are the principal radii of curvature along the coordinates and , respectively. and are the coefficients of the first fundamental form of ( is the boundary). In this work, the attention has been restricted to shells with constant radii of curvature (cylindrical, spherical, and toroidal geometries) for which .

Although one can use the UF for one-layer, isotropic shell, a multilayered shell with layers is considered. The Principle of Virtual Displacement (PVD) for the pure-mechanical case reads

where and are the integration domains in plane and direction, respectively. Here, indicates the layer and the transpose of a vector, and is the external work for the th layer. means geometrical relations and constitutive equations.

The steps to obtain the governing equations are

(i)substitution of the geometrical relations (subscript ), (ii)substitution of the appropriate constitutive equations (subscript ), (iii)introduction of the Unified Formulation.

Stresses and strains are separated into in-plane and normal components, denoted, respectively, by the subscripts and . The mechanical strains in the th layer can be related to the displacement field via the geometrical relations:

The explicit form of the introduced arrays is as follows

The 3D constitutive equations are given as

with

According to the Unified Formulation by Carrera, the three displacement components , , and and their relative variations can be modelled as

with Taylor expansions from the first up to the 4th order:  , , , and if an Equivalent Single Layer (ESL) approach is used.

Resorting to the displacement field in (3), we choose vectors for displacement , , and . We then obtain all terms of the equations of motion by integrating through-the-thickness direction.

It is interesting to note that, under this combination of the Unified Formulation and RBF collocation, the collocation code depends only on the choice of , in order to solve this type of problems. We designed a MATLAB code that just by changing can analyse static deformations, free vibrations, and buckling loads for any type of shear deformation theory. An obvious advantage of the present methodology is that the tedious derivation of the equations of motion and boundary conditions for a particular shear deformation theory is no longer an issue, as this MATLAB code does all that work for us.

In Figure 2 the assembling procedures are shown on layer for ESL approach.

Figure 2: Assembling procedure for ESL approach.

(a)

(b)

Figure 3: Geometry and notations for a multilayered shell (doubly curved).

Substituting the geometrical relations, the constitutive equations, and the Unified Formulation into the variational statement PVD, for the th layer, one has

At this point, the formula of integration by parts is applied

where matrix is obtained applying the gradient theorem

with being the components of the normal to the boundary along the direction . After integration by parts and the substitution of CUF, the governing equations and boundary conditions for the shell in the mechanical case are obtained:

where and depend on the boundary geometry:

The normal to the boundary of domain is

where and are the angles between the normal and the directions and , respectively.

The governing equations for a multilayered shell subjected to mechanical loadings are

where the fundamental nucleus is obtained as

And the corresponding Neumann-type boundary conditions on are

where

And are variationally consistent loads with applied pressure.

2.3. Fundamental Nuclei

The following integrals are introduced to perform the explicit form of fundamental nuclei:

The fundamental nucleus is reported for doubly curved shells (radii of curvature in both and directions; see Figure 3):

The application of boundary conditions makes use of the fundamental nucleus in the form:

One can note that all the equations written for the shell degenerate into those for the plate when . In practice we set the radii of curvature to .

2.4. Dynamic Governing Equations

The PVD for the dynamic case is expressed as

where is the mass density of the th layer and double dots denote acceleration.

By substituting the geometrical relations, the constitutive equations, and the Unified Formulation, we obtain the following governing equations:

In the case of free vibrations one has

where is the fundamental nucleus for the inertial term. The explicit form of that is

where the meaning of the integral has been illustrated in (25). The geometrical and mechanical boundary conditions are the same of the static case. Because we consider the static case only, the mass terms will be neglected.

3. The Radial Basis Function Method

3.1. The Static Problem

Radial basis functions (RBFs) approximations are mesh-free numerical schemes that can exploit accurate representations of the boundary, are easy to implement, and can be spectrally accurate. In this section the formulation of a global unsymmetrical collocation RBF-based method to compute elliptic operators is presented.

Consider a linear elliptic partial differential operator and a bounded region in with some boundary . In the static problems we seek the computation of displacement from the global system of equations

where and are linear operators in the domain and on the boundary, respectively. The right-hand sides of (32) represent the external forces applied on the plate or shell and the boundary conditions applied along the perimeter of the plate or shell, respectively. The PDE problem defined in (32) will be replaced by a finite problem, defined by an algebraic system of equations, after the radial basis expansions.

3.2. The Eigenproblem

The eigenproblem looks for eigenvalues () and eigenvectors that satisfy

As in the static problem, the eigenproblem defined in (33) is replaced by a finite-dimensional eigenvalue problem, based on RBF approximations.

3.3. Radial Basis Functions Approximations

The radial basis function approximation of a function is given by

where , is a finite set of distinct points (centers) in . The most common RBFs are

where the Euclidian distance is real and nonnegative and is a positive shape parameter. Hardy [54] introduced multiquadrics in the analysis of scattered geographical data. In the 1990s Kansa [31] used multiquadrics for the solution of partial differential equations. Considering distinct interpolations and knowing , , we find by the solution of a linear system

where , , and .

3.4. Solution of the Static Problem

The solution of a static problem by radial basis functions considers nodes in the domain and nodes on the boundary, with a total number of nodes . We denote the sampling points by , , and , . At the points in the domain we solve the following system of equations:

or

where

At the points on the boundary, we impose boundary conditions as

or

where

Therefore, we can write a finite-dimensional static problem as

By inverting the system (43), we obtain the vector . We then obtain the solution using the interpolation (34).

3.5. Solution of the Eigenproblem

We consider nodes in the interior of the domain and nodes on the boundary, with . We denote interpolation points by , , and , . At the points in the domain, we define the eigenproblem as

or

where

At the points on the boundary, we enforce the boundary conditions as

or

Equations (45) and (48) can now be solved as a generalized eigenvalue problem

where

3.6. Discretization of the Equations of Motion and Boundary Conditions

The radial basis collocation method follows a simple implementation procedure. Taking (11), we compute

This vector is then used to obtain solution , by using (5). If derivatives of are needed, such derivatives are computed as

and so forth.

In the present collocation approach, we need to impose essential and natural boundary conditions. Consider, for example, the condition , on a simply supported or clamped edge. We enforce the conditions by interpolating as

Other boundary conditions are interpolated in a similar way.

3.7. Free Vibrations Problems

For free vibration problems we set the external force to zero and assume harmonic solution in terms of displacement, , as

where is the frequency of natural vibration. Substituting the harmonic expansion into (49) in terms of the amplitudes , , , , , , , , and , we may obtain the natural frequencies and vibration modes for the plate or shell problem, by solving the eigenproblem

where collects all stiffness terms and collects all terms related to the inertial terms. In (55) are the modes of vibration associated with the natural frequencies defined as .

4. Numerical Examples

All numerical examples consider a Chebyshev grid and a Wendland function, defined as

where the shape parameter () was obtained by an optimization procedure, as detailed in Ferreira and Fasshauer [55].

4.1. Spherical Shell in Bending

A laminated composite spherical shell is here considered, of side and thickness , to be composed of layers oriented at and . The shell is subjected to a sinusoidal vertical pressure of the form

with the origin of the coordinate system located at the lower left corner on the midplane and the maximum load (at center of shell).

The orthotropic material properties for each layer are given by

The in-plane displacement, the transverse displacement, the normal stresses, and the in-plane and transverse shear stresses are presented in normalized form as

The shell is simply supported on all edges.

In Table 1 we compare the static deflections for the present shell model with results of Reddy shell formulation using first-order and third-order shear deformation theories [56]. We consider nodal grids with , , and points. We consider various values of and two values of (10 and 100). Results are in good agreement for various ratios with the higher-order results of Reddy and Liu [56].

Table 1: Nondimensional central deflection, , and variation with various numbers of grid points per unit length, , for different ratios, for .

Method

5
10
20
50
100

10
Present ()
6.8510
7.0818
7.1429
7.1609
7.1637
7.1649

10
Present ()
6.8516
7.0822
7.1433
7.1612
7.1640
7.1653

10
Present ()
6.8516
7.0822
7.1433
7.1612
7.1640
7.1653

10
HSDT [56]
6.7688
7.0325
7.1016
7.1212
7.1240
7.125

10
FSDT [56]
6.4253
6.6247
6.6756
6.6902
6.6923
6.6939

100
Present ()
1.0245
2.3658
3.5167
4.0714
4.1652
4.1975

100
Present ()
1.0250
2.3667
3.5181
4.0728
4.1667
4.1990

100
Present ()
1.0250
2.3669
3.5183
4.0731
4.1669
4.1992

100
HSDT [56]
1.0321
2.4099
3.617
4.2071
4.3074
4.3420

100
FSDT [56]
1.0337
2.4109
3.6150
4.2027
4.3026
4.3370

10
Present ()
6.7737
7.0012
7.0614
7.0791
7.0819
7.0831

10
Present ()
6.7742
7.0015
7.0618
7.0795
7.0822
7.0834

10
Present ()
6.7742
7.0015
7.0618
7.0795
7.0822
7.0834

10
HSDT [56]
6.7865
7.0536
7.1237
7.1436
7.1464
7.1474

10
FSDT [56]
6.3623
6.5595
6.6099
6.6244
6.6264
6.6280

100
Present ()
1.0190
2.3583
3.5125
4.0702
4.1647
4.1972

100
Present ()
1.0194
2.3593
3.5138
4.0717
4.1662
4.1987

100
Present ()
1.0195
2.3594
3.5140
4.0719
4.1664
4.1989

100
HSDT [56]
1.0264
2.4024
3.6133
4.2071
4.3082
4.3430

100
FSDT [56]
1.0279
2.4030
3.6104
4.2015
4.3021
4.3368

4.2. Free Vibration of Spherical and Cylindrical Laminated Shells

We consider nodal grids with , , and points. In Tables 2 and 3 we compare the nondimensionalized natural frequencies from the present Murakami theory for various cross-ply spherical shells, with analytical solutions done by Reddy and Liu [56] who considered both the first-order (FSDT) and the third-order (HSDT) theories. The first-order theory overpredicts the fundamental natural frequencies of symmetric thick shells and symmetric shallow thin shells. The present radial basis function method is compared with analytical results by Reddy and Liu [56] and shows excellent agreement.

Table 2: Nondimensionalized fundamental frequencies of cross-ply laminated spherical shells, , and laminate ().

Method

5
10
20
50
100

10
Present ()
12.0527
11.8889
11.8474
11.8357
11.8340
11.8335

Present ()
12.0523
11.8886
11.8471
11.8355
11.8338
11.8332

Present ()
12.0522
11.8885
11.8470
11.8355
11.8338
11.8332

HSDT [56]
12.040
11.840
11.790
11.780
11.780
11.780

100
Present ()
31.2170
20.5742
16.8698
15.6744
15.4960
15.4361

Present ()
31.2072
20.5679
16.8648
15.6698
15.4915
15.4316

Present ()
31.2059
20.5672
16.8642
15.6693
15.4910
15.4311

HSDT [56]
31.100
20.380
16.630
15.420
15.230
15.170

Table 3: Nondimensionalized fundamental frequencies of cross-ply laminated spherical shells, , and laminate ().

Method

5
10
20
50
100

10
Present ()
11.9831
11.8192
11.7777
11.7660
11.7643
11.7638

Present ()
11.9827
11.8190
11.7774
11.7658
11.7641
11.7635

Present ()
11.9827
11.8190
11.7774
11.7658
11.7641
11.7635

HSDT [56]
12.060
11.860
11.810
11.790
11.790
11.790

100
Present ()
31.1343
20.5420
16.8595
15.6721
15.4950
15.4355

Present ()
31.1244
20.5357
16.8545
15.6675
15.4905
15.4310

Present ()
31.1231
20.5350
16.8540
15.6671
15.4901
15.4306

HSDT [56]
31.020
20.350
16.620
15.420
15.240
15.170

Table 4 contains nondimensionalized natural frequencies obtained using the the present Murakami theory for cross-ply cylindrical shells with lamination schemes [0/90/0], [0/90/90/0]. Present results are compared with analytical solutions done by Reddy and Liu [56] who considered both the first-order (FSDT) and the third-order (HSDT) theories. The present radial basis function method is compared with analytical results by Reddy and Liu [56] and shows excellent agreement.

Table 4: Nondimensionalized fundamental frequencies of cross-ply cylindrical shells, .

Method
[0/90/0]

[0/90/90/0]

5
Present ()
20.4956
11.7774
20.5298
11.8524

Present ()
20.4839
11.7770
20.5201
11.8521

Present ()
20.4829
11.7770
20.5189
11.8521

FSDT [56]
20.332
12.207
20.361
12.267

HSDT [56]
20.330
11.850
20.360
11.830

10
Present ()
16.8475
11.7672
16.8583
11.8382

Present ()
16.8409
11.7669
16.8522
11.8380

Present ()
16.8401
11.7669
16.8516
11.8380

FSDT [56]
16.625
12.173
16.634
12.236

HSDT [56]
16.620
11.800
16.630
11.790

20
Present ()
15.8006
11.7646
15.8039
11.8346

Present ()
15.7955
11.7644
15.7990
11.8344

Present ()
15.7950
11.7644
15.7985
11.8344

FSDT [56]
15.556
12.166
15.559
12.230

HSDT [56]
15.55
11.79
15.55
11.78

50
Present ()
15.4945
11.7639
15.4955
11.8336

Present ()
15.4899
11.7637
15.4909
11.8334

Present ()
15.4895
11.7637
15.4905
11.8334

FSDT [56]
15.244
12.163
15.245
12.228

HSDT [56]
15.24
11.79
15.23
11.78

100
Present ()
15.4503
11.7638
15.4510
11.8335

Present ()
15.4457
11.7636
15.4464
11.8333

Present ()
15.4453
11.7636
15.4460
11.8333

FSDT [56]
15.198
12.163
15.199
12.227

HSDT [56]
15.19
11.79
15.19
11.78

Plate
Present ()
15.4355
11.7638
15.4361
11.8335

Present ()
15.4310
11.7635
15.4316
11.8332

Present ()
15.4306
11.7635
15.4311
11.8332

FSDT [56]
15.183
12.162
15.184
12.226

HSDT [56]
15.170
11.790
15.170
11.780

In Figure 4 we illustrate the first 4 vibrational modes of cross-ply laminated spherical shells, , for a laminate (), using a grid of points, for , . The modes of vibration are quite stable.

(a)

(b)

(c)

(d)

Figure 4: The first 4 vibrational modes of cross-ply laminated spherical shells,  , for laminate () using a grid of points, for , .

In Figure 5 we illustrate the first 4 vibrational modes of cross-ply laminated spherical shells, , for a laminate (), using a grid of points, for , . Again the modes of vibration are quite stable.

(a)

(b)

(c)

(d)

Figure 5: The first 4 vibrational modes of cross-ply laminated spherical shells, , for laminate () using a grid of points, for , (plate).

5. Concluding Remarks

In this paper Murakami's theory was implemented for the first time for laminated orthotropic elastic shells through a multiquadric discretization of equations of motion and boundary conditions. The multiquadric radial basis function method for the solution of shell bending and free vibration problems was presented. Results for static deformations and natural frequencies were obtained and compared with other sources. This meshless approach demonstrated that is very successful in the static deformations and free vibration analysis of laminated composite shells. Advantages of radial basis functions are absence of mesh, ease of discretization of boundary conditions and equations of equilibrium or motion, and very easy coding. We show that the static displacement and stresses and the natural frequencies obtained from present method are in excellent agreement with analytical solutions.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. HiCi/1433/363-4. The authors, therefore, acknowledge with thanks DSR technical and financial support.

References

W. T. Koiter, “On the foundations of the linear theory of thin elastic shells,” Koninklijke Nederlandse Akademie van Wetenschappen, vol. 73, pp. 169–195, 1970.

P. M. Naghdi, “A survey of recent progress in the theory of elastic shells,” Applied Mechanics Reviews, vol. 9, pp. 365–368, 1956.

D. Zenkert, An Introduction to Sandwich Structures, Chamelon Press, Oxford, UK, 1995.

J. R. Vinson, The Behavior of Sandwich Structures of Isotropic and Composite Materials, Technomic Publishing Co., 1999.

W. S. Burton and A. K. Noor, “Assessment of computational models for sandwich panels and shells,” Computer Methods in Applied Mechanics and Engineering, vol. 124, no. 1-2, pp. 125–151, 1995.

A. K. Noor, W. S. Burton, and C. W. Bert, “Computational models for sandwich panels and shells,” Applied Mechanics Reviews, vol. 49, no. 3, pp. 155–199, 1996.

H. Altenbach, “Theories for laminated and sandwich plates. A review,” Mechanics of Composite Materials, vol. 34, no. 3, pp. 243–252, 1998.

L. Librescu and T. Hause, “Recent developments in the modeling and behavior of advanced sandwich constructions: a survey,” Composite Structures, vol. 48, no. 1, pp. 1–17, 2000.

J. R. Vinson, “Sandwich structures,” Applied Mechanics Reviews, vol. 54, no. 3, pp. 201–214, 2001.

L. Demasi, “2D, quasi 3D and 3D exact solutions for bending of thick and thin sandwich plates,” Journal of Sandwich Structures and Materials, vol. 10, no. 4, pp. 271–310, 2008.

H. Murakami, “Laminated composite plate theory with improved in-plane responses,” Journal of Applied Mechanics, vol. 53, no. 3, pp. 661–666, 1986.

E. Carrera, “Historical review of Zig-Zag theories for multilayered plates and shells,” Applied Mechanics Reviews, vol. 56, no. 3, pp. 287–308, 2003.

E. Carrera, “Developments, ideas, and evaluations based upon Reissner's Mixed Variational Theorem in the modeling of multilayered plates and shells,” Applied Mechanics Reviews, vol. 54, no. 4, pp. 301–328, 2001.

E. Carrera, “On the use of the Murakami's zig-zag function in the modeling of layered plates and shells,” Computers and Structures, vol. 82, no. 7-8, pp. 541–554, 2004.

L. Demasi, “∞³ Hierarchy plate theories for thick and thin composite plates: the generalized unified formulation,” Composite Structures, vol. 84, no. 3, pp. 256–270, 2008.

S. Brischetto, E. Carrera, and L. Demasi, “Improved bending analysis of sandwich plates using a zig-zag function,” Composite Structures, vol. 89, no. 3, pp. 408–415, 2009.

H. Hu, S. Belouettar, E. M. Daya, and M. Potier-Ferry, “Evaluation of kinematic formulations for viscoelastically damped sandwich beam modeling,” Journal of Sandwich Structures and Materials, vol. 8, no. 6, pp. 477–495, 2006.

Y. Frostig, “Bending of curved sandwich panels with a transversely flexible core-closed-form high-order theory,” Journal of Sandwich Structures and Materials, vol. 1, no. 1, pp. 4–41, 1999.

Y. Frostig and O. Rabinovich, “Behavior of unidirectional sandwich panels with a multi-skin construction or a multilayered core layout-higher-order approach,” Journal of Sandwich Structures and Materials, vol. 2, pp. 181–213, 2000.

D. Chapelle and K. J. Bathe, The Finite Element Analysis of Shells—Fundamentals, Springer, Berlin, Germany, 2003.

W. Flügge, Stresses in Shells, Springer, Berlin, Germany, 1960.

A. Scordelis and K. S. Lo, “Computer analysis in cylindrical shells,” Journal of American Concrete Institute, vol. 61, pp. 561–593, 1964.

J. N. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC Press, Boca Raton, Fla, USA, 2nd edition, 2003.

E. Carrera, “Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking,” Archives of Computational Methods in Engineering, vol. 10, no. 3, pp. 215–296, 2003.

J. Pitkäranta, “The problem of membrane locking in finite element analysis of cylindrical shells,” Numerische Mathematik, vol. 61, no. 4, pp. 523–542, 1992.

C. Chinosi, L. Della Croce, and T. Scapolla, “Numerical results on the locking for cylindrical shells,” Computer Assisted Mechanics and Engineering Sciences, vol. 5, no. 1, pp. 31–44, 1998.

O. C. Zienkiewicz, R. L. Taylor, and J. M. Too, “Reduced integration technique in general analysis of plates and shells,” International Journal for Numerical Methods in Engineering, vol. 3, no. 2, pp. 275–290, 1971.

K. J. Bathe and E. N. Dvorkin, “A formulation of general shell elements—the use of mixed interpolation of tensorial components,” International Journal for Numerical Methods in Engineering, vol. 22, no. 3, pp. 697–722, 1986.

D. N. Arnold and F. Brezzi, “Locking-free finite element methods for shells,” Mathematics of Computation, vol. 66, no. 217, pp. 1–14, 1997.

H. Hou-Cheng, “Membrane locking and assumed strain shell elements,” Computers and Structures, vol. 27, no. 5, pp. 671–677, 1987.

E. J. Kansa, “Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics. I. Surface approximations and partial derivative estimates,” Computers & Mathematics with Applications, vol. 19, no. 8-9, pp. 127–145, 1990.

Y. C. Hon, M. W. Lu, W. M. Xue, and Y. M. Zhu, “Multiquadric method for the numerical solution of a biphasic mixture model,” Applied Mathematics and Computation, vol. 88, no. 2-3, pp. 153–175, 1997.

Y. C. Hon, K. F. Cheung, X. Z. Mao, and E. J. Kansa, “Multiquadric solution for shallow water equations,” Journal of Hydraulic Engineering, vol. 125, no. 5, pp. 524–533, 1999.

J. G. Wang, G. R. Liu, and P. Lin, “Numerical analysis of Biot's consolidation process by radial point interpolation method,” International Journal of Solids and Structures, vol. 39, no. 6, pp. 1557–1573, 2002.

G. R. Liu and Y. T. Gu, “A local radial point interpolation method (LRPIM) for free vibration analyses of 2-D solids,” Journal of Sound and Vibration, vol. 246, no. 1, pp. 29–46, 2001.

J. G. Wang and G. R. Liu, “A point interpolation meshless method based on radial basis functions,” International Journal for Numerical Methods in Engineering, vol. 54, no. 11, pp. 1623–1648, 2002.

J. G. Wang and G. R. Liu, “On the optimal shape parameters of radial basis functions used for 2-D meshless methods,” Computer Methods in Applied Mechanics and Engineering, vol. 191, no. 23-24, pp. 2611–2630, 2002.

X. L. Chen, G. R. Liu, and S. P. Lim, “An element free Galerkin method for the free vibration analysis of composite laminates of complicated shape,” Composite Structures, vol. 59, no. 2, pp. 279–289, 2003.

K. Y. Dai, G. R. Liu, K. M. Lim, and X. L. Chen, “A mesh-free method for static and free vibration analysis of shear deformable laminated composite plates,” Journal of Sound and Vibration, vol. 269, no. 3–5, pp. 633–652, 2004.

G. R. Liu and X. L. Chen, “Buckling of symmetrically laminated composite plates using the element-free galerkin method,” International Journal of Structural Stability and Dynamics, vol. 2, pp. 281–294, 2002.

K. M. Liew, X. L. Chen, and J. N. Reddy, “Mesh-free radial basis function method for buckling analysis of non-uniformly loaded arbitrarily shaped shear deformable plates,” Computer Methods in Applied Mechanics and Engineering, vol. 193, no. 3–5, pp. 205–224, 2004.

Y. Q. Huang and Q. S. Li, “Bending and buckling analysis of antisymmetric laminates using the moving least square differential quadrature method,” Computer Methods in Applied Mechanics and Engineering, vol. 193, no. 33–35, pp. 3471–3492, 2004.

L. Liu, G. R. Liu, and V. B. C. Tan, “Element free method for static and free vibration analysis of spatial thin shell structures,” Computer Methods in Applied Mechanics and Engineering, vol. 191, no. 51-52, pp. 5923–5942, 2002.

S. Xiang, K. M. Wang, Y. T. Ai, Y. D. Sha, and H. Shi, “Analysis of isotropic, sandwich and laminated plates by a meshless method and various shear deformation theories,” Composite Structures, vol. 91, no. 1, pp. 31–37, 2009.

S. Xiang, H. Shi, K. M. Wang, Y. T. Ai, and Y. D. Sha, “Thin plate spline radial basis functions for vibration analysis of clamped laminated composite plates,” European Journal of Mechanics A, vol. 29, no. 5, pp. 844–850, 2010.

C. M. C. Roque, A. J. M. Ferreira, and R. M. N. Jorge, “Free vibration analysis of composite and sandwich plates by a trigonometric layerwise deformation theory and radial basis functions,” Journal of Sandwich Structures and Materials, vol. 8, no. 6, pp. 497–515, 2006.

A. J. M. Ferreira, “A formulation of the multiquadric radial basis function method for the analysis of laminated composite plates,” Composite Structures, vol. 59, no. 3, pp. 385–392, 2003.

A. J. M. Ferreira, “Thick composite beam analysis using a global meshless approximation based on radial basis functions,” Mechanics of Advanced Materials and Structures, vol. 10, no. 3, pp. 271–284, 2003.

A. J. M. Ferreira, C. M. C. Roque, and P. A. L. S. Martins, “Analysis of composite plates using higher-order shear deformation theory and a finite point formulation based on the multiquadric radial basis function method,” Composites Part B, vol. 34, no. 7, pp. 627–636, 2003.

A. J. M. Ferreira, C. M. C. Roque, and R. M. N. Jorge, “Modelling cross-ply laminated elastic shells by a higher-order theory and multiquadrics,” Computers and Structures, vol. 84, no. 19-20, pp. 1288–1299, 2006.

E. Carrera, “C⁰ reissner-mindlin multilayered plate elements including zig-zag and interlaminar stress continuity,” International Journal for Numerical Methods in Engineering, vol. 39, no. 11, pp. 1797–1820, 1996.

E. Carrera and B. Kröplin, “Zigzag and interlaminar equilibria effects in large-deflection and postbuckling analysis of multilayered plates,” Mechanics of Composite Materials and Structures, vol. 4, no. 1, pp. 69–94, 1997.

E. Carrera, “Evaluation of layerwise mixed theories for laminated plates analysis,” AIAA Journal, vol. 36, no. 5, pp. 830–839, 1998.

R. L. Hardy, “Multiquadric equations of topography and other irregular surfaces,” Geophysical Research, vol. 176, pp. 1905–1915, 1971.

A. J. M. Ferreira and G. E. Fasshauer, “Computation of natural frequencies of shear deformable beams and plates by an RBF-pseudospectral method,” Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 1–3, pp. 134–146, 2006.

J. N. Reddy and C. F. Liu, “A higher-order shear deformation theory of laminated elastic shells,” International Journal of Engineering Science, vol. 23, no. 3, pp. 319–330, 1985.