Free vibration and buckling analysis of cross-ply laminated composite plates using Carrera's unified formulation based on Isogeometric approach

Computers and Structures 183 (2017) 38–47

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Free vibration and buckling analysis of cross-ply laminated composite plates using Carrera’s unified formulation based on Isogeometric approach Amirhadi Alesadi, Marzieh Galehdari, Saeed Shojaee ⇑ Department of Civil Engineering, Shahid Bahonar University, Kerman, Iran

a r t i c l e

i n f o

Article history: Received 12 October 2016 Accepted 19 January 2017 Available online 6 February 2017 Keywords: NURBS basis functions Free vibration Buckling Composite laminates Carrera’s unified formulation

a b s t r a c t In this paper, The Isogeometric approach (IGA) and Carrera’s Unified Formulation (CUF) are employed for free vibration and linearized buckling analysis of laminated composite plates. The non-uniform rational B-spline (NURBS) basis functions utilized in IGA, are employed as higher order smooth functions to approximate field solution leading to enhance precision of analysis. CUF presents an effective formulation to employ any order of Taylor expansion to analyze two-dimensional plate models. Higher order NURBS basis functions attenuate the shear locking properly and higher order theories supposed by CUF are free from Poisson locking phenomenon and they do not need the use of any shear correction factor. Therefore, combining IGA and CUF ends in a suitable methodology to analyze laminated plates. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Over the past decades, multilayered plates utilization has increased in various industries such as aerospace, automotive and ship vehicles. As the name suggests, multilayered plates are made of distinct layers of boasting different materials with dissimilar features. The success of this class of structures can be supposed to be due to a variety of features they are characterized by, including high structural resistance, high heat resistance, isolation and so on. To analyze above mentioned structures, different threedimensional methods based on elasticity solutions are presented in [1–4]. Although these analytical methods provide accurate solutions, they are limited to simple geometries, boundary conditions and loadings. To compensate these limitations, two-dimensional theories are proposed. Two-dimensional plate theories were classically first introduced by Kirchhoff [5] and then by Reissner [6] and Mindlin [7]. Although Reissner and Mindlin theory, also called first-order shear deformation theory (FSDT), considers shear deformations, it assumes constant displacement field in the thickness direction and linear distribution of the out-of-plane shear stresses. Therefore, transverse shear stresses have to be modified by shear correction factors. In fact, FSDT does not satisfy shear stress free conditions at top and bottom surfaces of plates. Nevertheless, the ⇑ Corresponding author. E-mail address: [email protected] (S. Shojaee). http://dx.doi.org/10.1016/j.compstruc.2017.01.013 0045-7949/Ó 2017 Elsevier Ltd. All rights reserved.

precise evaluation of shear correction factor for composite plates is difficult. To compensate these drawbacks of FSDT, various higher shear deformation theories (HSDTs) have been proposed and they are well documented in several review articles, such as [8] and [9]. Employing higher orders of expansions especially along the thickness direction, not only provides more realistic mathematical models, but also obviates Poisson locking and the need for shear correction factors. HSDTs are usually formulated by axiomatic assumptions and based on fixed-order expansions of the generalized unknowns. In addition, to reach a desirable agreement in analysis, selecting a more suitable model among existing various methods is not really easy to achieve. Moreover, as a result of any changes in theory, the adaptation of the governing equations and the framework of relevant finite element codes will be inevitable. Thus, the process will be much time-consuming. To compensate for this limitation, unified formulations are proposed in literature to allow employing different orders of HSDTs in the same framework. Beside CUF which is applied in present study, some different unified formulations are mentioned as follow. Carrera introduced a class of 2D theories using a compact notation in [10] which was later named as CUF in literature. Mentioned compact notation makes it easy to expand displacement field to an arbitrary order of expansion. To this end, a single 3
3 matrix named ‘‘fundamental nuclei’’ has been utilized. This fundamental nuclei can be used to represent the variable description approaches such as Equivalent Single Layer (ESL) and Layer-wise (LW) models [11–16]. The theoretical foundations of the unified method is

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A. Alesadi et al. / Computers and Structures 183 (2017) 38–47

presented in the comprehensive article by Carrera [17]. CUF has also been used in the formulation of beam and shell theories recently, see [18–24]. Also, a three-dimensional HellingerReissner mixed variational principle to derive an arbitrary order shear and normal deformable plate theory has been proposed by Batra et al. in [4,25] where the equations for the plate theory are expressed in a compact form by taking Legendre polynomials in z direction as the basis functions. Batra’s formulation has also been used in [26,27]. Furthermore, Williams proposed a general GlobalLocal approach to the development of comprehensive, multiscale plate and shell theories to analyze the laminated structures [28–30]. Mentioned theory is presented based on the use of a two length scale displacement formulation where the layer displacement field is assumed to be composed of both global and local components. The order and functional forms for the global and local components of the displacement field are arbitrary. Also, Carrera’s unified formulation has been generalized by Demasi in [31] as the generalized unified formulation (GUF) in the framework of displacement-based theories to analyze composite laminated plates. In this formulation each displacement variable can be analyzed, independently. In addition, GUF has been developed into Reissner’s mixed variational theorem in [32–36] where each of the displacement variables and out-of-plane stresses is independently treated and different orders of expansions for the different unknowns can be considered. Considering mentioned interesting features of HSDTs and unified formulations, various combinations of these approaches have been presented in numerous articles such as [37–46]. Although HSDTs propose a more precise mathematical model of the structure mechanics, they still suffer from shear locking phenomenon which in fact is a problem due to the adopted approximate numerical method (e.g. FEM) and related convergence. Over the past decades, several methods such as reduced integration, selective integration and the mixed interpolation of tensorial components (MITC) technique have been proposed to reduce the effect of shear locking which are also considered in CUF in [47,48]. Within the framework of a weak form solution scheme, a very well-known method to reduce the shear locking is increasing the dimension of the basis functions space, which means either increasing the order or the number of the shape functions. NURBS functions were first used by Hughes et al. [49] as basis functions to approximate solution field. Afterwards, NURBS-based approaches have been developed in a wide range of research areas such as fluid–structure interaction [50,51], shell analysis [52,53], structural analysis [54,55], fracture mechanics [56,57]. Also, Shojaee et al. have utilized this approach in different fields in [58–60]. Furthermore, a numerical overview of IGA has been presented in [61]. A synthesis of IGA and CUF using a fixed order hybrid displacement assumption has been proposed in [62]. Moreover, a numerical approach based on the Generalized Differential Quadrature method using IGA is presented in [63] and a synthesis of IGA and CUF is presented in [64] to analyze laminated composite plates and shells. In this paper NURBS functions are adopted along with CUF, whose hierarchical capabilities allow one to adopt different approximation function indistinctly. NURBS (or B-Splines) functions are employed in this article due to their interesting features. Besides a precise geometric modeling, the NURBS functions show unique properties in analysis. The order of the NURBS can be applied as a free parameter in analysis and it can be counted as one of their most obvious features. Given this specification, higher order NURBS functions can be utilized to reduce the effect of shear locking phenomenon. One of the significant characteristics of the CUF is that it enables the development of analysis to different fields such as buckling and free vibration analysis in an easier and more applicable way espe-

cially in higher order theories. Therefore, CUF has been combined with NURBS basis functions to free vibration and buckling analysis of cross-ply laminated plates using higher order theories. Buckling phenomena is associated with a process whereby a given state of a deformable structure suddenly changes its shape. Triggered by a varying external load, this change in configuration often happens in a catastrophic way (i.e. the structure is destroyed at the end of the process) [65]. Two types of buckling exist, nonlinear collapse and bifurcation buckling. Nonlinear collapse is predicted by means of a nonlinear stress analysis. The other case, bifurcation buckling, refers to a different kind of failure, the onset of which is predicted by means of an eigenvalue analysis [66]. Indeed, A simplified method to perform buckling analysis can be devised by interpreting the critical load as the load at which more than one infinitesimally adjacent equilibrium configuration exists (bifurcation point) [67]. If a linear initial equilibrium path is also assumed, linearized stability analysis reduces the determination of the critical load to a linear eigenvalue problem (Euler’s method) [68]. This simplified approach can be conveniently applied to flat plates because the critical equilibrium configuration shows a gradual geometry change when the load passes through the critical level. A precise evaluation of the 2D approximations about buckling of composite plate/shell models based on CUF and referring to analytical solutions was presented in [67]. To provide accurate buckling results for laminated plates, the classic Euler method for the prediction of bifurcation loads of composite multilayered anisotropic plates based on finite element method and CUF is applied in [69]. In this paper, the classic Euler method using CUF based on NURBS basis functions is employed to buckling analysis of laminated composite plates and results are compared with existing analytical/numerical solutions. 2. Governing equations via CUF Carrera presents a unified formulation which is independent of shape functions, expansion type and order of expansion. In this paper, NURBS basis functions and Taylor expansion are employed based on CUF. Also, ESL models have been utilized as variable description approach. CUF defines fundamental nucleus (FNs) to produce finite element matrices, i.e. for each finite element matrix such as stiffness, mass and initial stress matrix an independent FN is defined. They are dealt with briefly in the coming sections. The readers can refer to [15–17] for more details of CUF. CUF defines the displacement field in a compact form as in Eq. (1):

uk ðx; y; zÞ ¼ F s ðzÞuks ðx; yÞ; duk ðx; y; zÞ ¼ F s ðzÞduks ðx; yÞ

s; s ¼ 0; 1; . . . ; N

ð1Þ

where k denotes the layer, uk (x, y, z) is the 3D displacement vector for layer k, duk is the related virtual variation, Fs and Fs are the thickness functions depending only on z direction (Fs = zs and Fs = zs), uks and duks are the generalized displacement vectors of the variations and the virtual variables, respectively, s and s are sum indexes and N is the number of terms of the theory expansion. Using Taylor expansion, the displacement field can be expanded to any arbitrary order. The typical distribution of the displacements for different orders of expansions is shown in Fig. 1. 2.1. Fundamental nucleus According to principle of virtual variations (PVD), Eq. (2) is defined:

dLint ¼ dLext  dLine

ð2Þ

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A. Alesadi et al. / Computers and Structures 183 (2017) 38–47

The assembly of the matrices consists of four loops on indexes i, j, s and s, and an FN is calculated for each combination of these indexes. A representation of this procedure related to Isogeometric approach is shown in Fig. 2. The diagram shows the simple procedure of building a matrix of the control point at first, then the global stiffness matrix by applying FNs. Mentioned procedures remain exactly the same for the mass matrix and also for the initial stress matrix in buckling analysis.

z N=1

N=2

N=3

N=4

x Fig. 1. Distribution of displacements according to linear and higher order theories in the unified formulation.

where dLint is virtual variation of the strain energy, dLext denotes virtual variation of the work of external loads which is zero in the case of free vibration and linearized buckling analysis and dLine is the virtual variation of the work of inertial loads. Stress (r) and strain (e) components are arranged as in Eq. (3):

n

rk ¼ rkxx rkyy skxy rkzz skxz skyz n

ek ¼ ekxx ekyy ckxy ekzz ckxz ckyz

o

o

ð3Þ

Linear strain–displacement relations and Hooke’s law can be written as:

ek ¼ buk rk ¼ C k ek

ð4Þ

where b is operator matrix and Ck is material matrix for cross-ply laminates which are available in [17]. Then, the virtual variation of the strain energy is considered as:

Z Z

d Lint ¼

X

A

d ek rk dX dz T

s; s ¼ 0; . . . ; N and i; j ¼ 1; . . . ; Nc ð6Þ

where i and j denote summation over the NURBS shape functions and Nc is the number of control points. Fs and Fs are thickness functions, Ri and Rj are NURBS basis functions and uksi and duksj are nodal unknowns. Representing Eq. (6) into Eq. (5), a 3
3 matrix will be generated which is called fundamental nucleus of the stiffness matrix:

2

K kxx

K kxy

6 k k K ssij ¼ 6 4 K yx K kzx

K kxz

3

K kyy

7 K kyz 7 5

K kzy

K kzz

ð7Þ

The first component of FN is presented as follow. All FN’s components are presented in Appendix.

Z

K kxx

¼

Z

Z

Ri;x Rj;x dX F s F s dz þ A Z Z k þ C 66 Ri;y Rj;y dX F s F s dz

C k11

X

X

C k55

X

Z

Ri Rj dX A

F s;z F s;z dz ð8Þ

A

Z

¼

M kyy

¼

M kzz

Z

¼q

k

F s F s dz A

Z

v

X

Ri Rj dX

In which qk is the mass density of layer k.

d ek rk dv ¼ dLext T

ð10Þ

The governing equations in the nonlinear static undamped case [71–73] can be obtained by using the PVD in Eq. (10), FE discretization Eq. (6), constitutive coefficients equation and nonlinear strain-displacements relations (e.g. the von Kármán ones, in Eq. (12)):

Ksu ¼ P

ð11Þ

2

2

3

@ x @x 0 2 7 6 @2 y 7 6 0 @y 7 6 2 7 6 7 6 b ¼ 6 @y @x @x@y 7 7 6 0 0 @z 7 6 7 6 4 @z 0 @x 5

0

@z

ð12Þ

@y

where u is the vector of nodal primary unknowns, Ks is the structure’s secant stiffness matrix that depends on u and P is the vector of nodal loads. Eq. (11) consists of a nonlinear system of algebraic equations, which solution ends in many equilibrium paths of the given structural problems. Solving the stability equations below, bifurcation points which are representing buckling, will be obtained:



K su ¼ P KTu ¼ 0

ð13Þ

where KT is the structure’s tangent stiffness matrix associated to the equilibrium conditions. The solution of Eq. (13) leads to the bifurcation points under the two following conditions: two different states of equilibrium are present; the tangent matrix has zero determinant. In case of in-plane loading (shear or axial), when the plate remains flat the related transverse displacements become zero. As a result, the equilibrium conditions in the first subsystem in Eq. (13) are met by definition. In addition it would be possible to increase the in-plane loading utilizing parameter k in a proportional and uniform manner from the initial conditions. Thus, the tangent matrix can be approximated as follows:

KT ¼ K þ Kr

ð14Þ

where K is the structure’s linear stiffness matrix and K r is the initial stress matrix, which can also be presented in the following form:

An equivalent expression for the mass matrix, produces a diagonal matrix which is presented in Eq. (9).

M kxx

The Principle of Virtual Displacements (PVD) can be written as follows [70]:

ð5Þ

where X indicates the in-plane domain and A is through-thethickness domain. Displacement field and its virtual variation (denoted by d) are approximated using NURBS basis functions as Eq. (6):

uk ðx; y; zÞ ¼ F s ðzÞRi ðx; yÞuksi ; duk ðx; y; zÞ ¼ F s ðzÞRj ðx; yÞduksj ;

3. Buckling analysis

ð9Þ

r K r ¼ kK

ð15Þ

The eigenvalue problem will be as follows:

ðK þ kn K r Þun ¼ 0

ð16Þ

where the eigenvalue kn is the nth buckling factor and un is the corresponding eigenvector. Since plates are considered, only the third diagonal component of K r is different from zero [69]:

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A. Alesadi et al. / Computers and Structures 183 (2017) 38–47

kxx kxy kxz

Control Point

kyx kyy kyz

Fundamental Nuclei

kzx kzy kzz

N

0

Structure 1

Nc

Fig. 2. Assembly procedure of FNs in present study (Nc is the number of control points).

k

Z Z Z ðF s F S Þdz ðRi;x Rj;x ÞdX þ r0yy ðF s F S Þdz ðRi;y Rj;y ÞdX A X A X Z Z þ r0xy ðF s F S Þdz ðRi;x Rj;y ÞdX ð17Þ

K srsij ð3; 3Þ ¼ r0xx

Z

A

X

Subscripts after a comma indicate derivatives and through Eq. (17) the in-plane excitations r0xx ; r0yy ; r0xy can be directly assigned in the model. They can appear singularly or in various combinations. The initial stress matrix K r can be obtained from K srsij expanding the superscripts. Of all probable buckling modes the lowest eigenvalue shows the critical buckling load for the whole laminate of thickness h according to:

N cr ¼ k r0 h

ð18Þ

 Ni;0 ðnÞ ¼

1 if ni 6 n 6 niþ1 0

ð21Þ

otherwise

And for p = 1, 2, 3,. . . they are defined as:

Ni;p ðnÞ ¼

niþpþ1  n n  ni Ni;p1 ðnÞ þ Niþ1;p1 ðnÞ; niþp  ni niþpþ1  niþ1

p ¼ 1; 2; 3; . . . :

ð22Þ

Using Eq. (19) in conjunction with the coordinate of control points ~ i , leads to an equation for a NURBS curve: X

4. NURBS basis functions The NURBS basis functions are employed to approximate the displacement field in this paper. In this section the NURBS functions are briefly illustrated, for further study one can refer to [49,74]. A non-uniform rational B-spline (NURBS) curve is given as follows [49]:

Ni;p ðnÞwi Ri;p ðnÞ ¼ Pn j¼1 N j;p ðnÞwj

ð19Þ

where p is the order of NURBS functions, n is the number of control points, wi are a set of weights corresponding to the control points that must be non-negative Ni;p is the B-spline basis function which is produced by a given knot vector as follows:

N ¼ fn1 ; n2 ; . . . ; nnþpþ1 g ni 6 niþ1 ;

vectors are interpolatory at the ends of the parameter space interval, [n1, nn+p+1], and at the corners of patches in multiple dimensions, but they are not, in general, interpolatory at interior knots. Having a knot vector, the B-spline basis functions are defined recursively starting with piecewise constants (p = 0):

i ¼ 1; 2; . . . ; n þ p

ð20Þ

Knot vectors may be uniform if the knots are equally spaced in the parameter space. If they are unequally spaced, the knot vector is non-uniform. Knot values may be repeated, that is, more than one knot may take on the same value. The multiplicities of knot values have important implications for the properties of the basis. A knot vector is said to be open if its first and last knot values appear p + 1 times. One dimension, basis functions formed from open knot

XðnÞ ¼

n X ~i Ri;p ðnÞX

ð23Þ

i¼1

Rational surfaces are defined analogously in terms of the rational basis functions:

Rp;q i;j ðn; gÞ ¼

n X m X i¼1 j¼1

Ni;p ðnÞM j;q ðgÞwi;j Pn Pm ^i¼1 ^j¼1 N^i;p ðnÞM^j;q ðgÞw^i;^j

0 6 n; g 6 1

ð24Þ

p;q where Ri;j stand for the NURBS basis functions, wij are the corre-

sponding weights and N i;p and Mj;q are the B-spline basis functions defined on the X and H knot vectors, respectively. The important properties of the NURBS basis functions can also be summarized as follows: P  Partition of unity, ni¼1 Ri;p ðnÞ ¼ 1  Non-negativity, Ri;p ðnÞ P 0 Besides the mentioned features of the NURBS functions, the arbitrary order of these functions is a fascinating specification in the analysis. It should be mentioned that the details of numerical

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A. Alesadi et al. / Computers and Structures 183 (2017) 38–47

Fig. 3. Cubic basis functions for given knot vector: X = {0, 0, 0, 0, 0.5, 1, 1, 1, 1}.

derivatives of NURBS functions is available in [74]. Fig. 3 shows an example of cubic basis functions with an open knot vector. 5. Numerical examples To demonstrate validity of present study, several numerical examples, categorized as free vibration and buckling problems, including both single-layer isotropic and cross-ply laminated plates, are investigated. To this end, square plates (a/b = 1) with different side-to-thickness ratios (a/h) and various boundary conditions (BCs), fully clamped (CCCC) and fully simply-supported (SSSS), are considered. In addition, different orders of NURBS basis functions are considered. In all cases, N = 1 to N = 4 models of Taylor expansion are employed and it is inferred that in thin and thick plates third order and fourth order model suffice respectively, for reaching the convergence solution. Also, the integration on the plate reference surface of the NURBS shape functions is performed by Gauss full integration method.

they are compared with Carrera [75] which presented a closed form solutions through CUF on N = 1 to N = 4 models of Taylor expansion. Present study shows that from grid 11
11 results have excellent agreement. On thin plate (a/h = 100) there are some differences between results of cubic and quartic NURBS basis functions which are caused by the shear locking effects. Indeed, in thin plates, quartic NURBS basis functions attenuate the effectiveness of shear locking satisfactorily rather than cubic functions. After that, two isotropic SSSS square plates with a/h = 10 and a/ h = 100 are considered. The first thirteen modes of vibration are calculated and listed in Tables 3 and 4. Results are compared with meshfree method based on radial basis functions by Ferreira et al. [76], 3D-Elasticity and Mindlin closed form solutions [77] and results with meshfree method by Liew et al. [78]. Presented Results in Tables 3 and 4 show excellent agreement with 3D closed form and Mindlin closed form solutions, respectively. And also a CCCC square plate is considered in Table 5 and results have suitable convergence to presented references. 5.2. Free vibration problems of cross-ply laminated plates

5.1. Free vibration problems of isotropic plates The normalized frequencies of isotropic square simplysupported and clamped plates are presented in the following tables. In all cases, a is the side length, h is the thickness of the plate, q = 2800 kg/m3 is the mass density per unit of volume, G is the shear modulus, G = E/2(1 + t), E = 73 GPa is the Young’s modulus and t is the Poisson’s ratio. First of all, an isotropic square plate with given material and BCs is considered and results are shown in Tables 1 and 2. Moreover,

A cross-ply thick laminated plate with given material properties is considered. Density and thickness of all layers of laminate is the same. The following material is used:

EL ¼ 10; 20; 30; 40; ET

GLT ¼ 0:6; ET

GTT ¼ 0:5; ET

mLT ¼ 0:25

ð25Þ

Subscripts L and T denote the normal and transverse directions to the fiber direction in each lamina, which may be oriented at an angle 0 or 90 to the plate axes.

Table 1 qffiffiffi 2 The normalized fundamental frequency x
ah qE for a SSSS isotropic plate with t = 0.34 and cubic NURBS basis functions. a/h

4

10

100

CLT FSDT N=1 N=2 N=3 N=4

6.7318 5.7765 5.7764 5.1397 5.0704 5.0688

7.0120 6.7923 6.7922 5.8706 5.8523 5.8522

7.0688 7.0664 7.0664 6.0572 6.0570 6.0570

Grid 7
7

N=1 N=2 N=3 N=4

5.1370 5.1398 5.0705 5.0689

5.8706 5.8713 5.8530 5.8529

6.1047 6.1048 6.1046 6.1046

Grid 11
11

N=1 N=2 N=3 N=4

5.1368 5.1397 5.0704 5.0688

5.8699 5.8707 5.8523 5.8522

6.0579 6.0580 6.0577 6.0577

Grid 19
19

N=1 N=2 N=3 N=4

5.1368 5.1397 5.0704 5.0688

5.8699 5.8706 5.8523 5.8522

6.0571 6.0572 6.0570 6.0570

Carrera [75]

Present

43

A. Alesadi et al. / Computers and Structures 183 (2017) 38–47 Table 2 qffiffiffi 2 The normalized fundamental frequency x
ah qE for a SSSS isotropic plate with t = 0.34 and quartic NURBS basis functions. a/h

4

10

100

CLT FSDT N=1 N=2 N=3 N=4

6.7318 5.7765 5.7764 5.1397 5.0704 5.0688

7.0120 6.7923 6.7922 5.8706 5.8523 5.8522

7.0688 7.0664 7.0664 6.0572 6.0570 6.0570

Grid 8
8

N=1 N=2 N=3 N=4

5.1368 5.1397 5.0704 5.0688

5.8699 5.8707 5.8523 5.8522

6.0575 6.0575 6.0573 6.0573

Grid 12
12

N=1 N=2 N=3 N=4

5.1368 5.1397 5.0704 5.0688

5.8699 5.8706 5.8523 5.8522

6.0571 6.0571 6.0569 6.0569

Grid 20
20

N=1 N=2 N=3 N=4

5.1368 5.1397 5.0704 5.0688

5.8699 5.8706 5.8523 5.8522

6.0571 6.0571 6.0569 6.0569

Carrera [75]

Present

Table 3 pffiffiffiffiffiffiffiffiffi The normalized natural frequenciesxa q=G for a SSSS isotropic plate with a/h = 10, t = 0.3 and cubic NURBS basis functions. Mode No.

1 2 3 4 5 6 7 8 9 10 11 12 13

Present Grid 11
11

Grid 19
19

N=1

N=2

N=3

N=4

N=1

N=2

N=3

N=4

0.9343 2.2412 2.2412 3.4546 4.2221 4.2221 5.3141 5.3141 6.2832 6.2832 6.7018 7.6674 7.6674

0.9344 2.2417 2.2417 3.4557 4.2237 4.2237 5.3164 5.3164 6.2832 6.2832 7.0140 7.6722 7.6722

0.9315 2.2262 2.2262 3.4213 4.1743 4.1743 5.2423 5.2423 6.2832 6.2832 6.8948 7.5343 7.5343

0.9315 2.2261 2.2261 3.4209 4.1737 4.1737 5.2409 5.2409 6.2832 6.2832 6.8919 7.5306 7.5306

0.9343 2.2411 2.2411 3.4545 4.2199 4.2199 5.3123 5.3123 6.2832 6.2832 6.6801 7.6489 7.6489

0.9344 2.2416 2.2416 3.4555 4.2214 4.2214 5.3146 5.3146 6.2832 6.2832 7.0115 7.6530 7.6530

0.9315 2.2261 2.2261 3.4211 4.1721 4.1721 5.2405 5.2405 6.2832 6.2832 6.8922 7.5149 7.5149

0.9315 2.2260 2.2260 3.4207 4.1714 4.1714 5.2391 5.2391 6.2832 6.2832 6.8893 7.5112 7.5112

Ferreira et al. [76]

3D closed form [77]

Mindlin closed form [77]

Liew et al. [78]

0.930 2.219 2.219 3.406 4.150 4.150 5.213 5.213 6.513 6.513 6.871 7.363 7.366

0.932 2.226 2.226 3.421 4.171 4.171 5.239 5.239 – – 6.889 7.511 7.511

0.930 2.219 2.219 3.406 4.149 4.149 5.206 5.206 6.520 6.520 6.834 7.446 7.446

0.922 2.205 2.205 3.377 4.139 4.139 5.170 5.170 6.524 6.524 6.779 7.416 7.416

Table 4 pffiffiffiffiffiffiffiffiffi The normalized natural frequencies xa q=G for a SSSS isotropic plate with a/h = 100, t = 0.3 and cubic NURBS basis functions. Mode No.

Present Grid 12
12

1 2 3 4 5 6 7 8 9 10 11 12 13

Ferreira et al. [76]

Mindlin closed form [77]

Liew et al. [78]

0.0963 0.2406 0.2406 0.3847 0.4806 0.4806 0.6246 0.6246 0.8156 0.8156 0.8640 0.9592 0.9592

0.0963 0.2406 0.2406 0.3848 0.4809 0.4809 0.6249 0.6249 0.8167 0.8167 0.8647 0.9605 0.9605

0.0961 0.2419 0.2419 0.3860 0.4898 0.4898 0.6315 0.6315 0.8447 0.8447 0.8726 0.9822 0.9822

Grid 20
20

N=1

N=2

N=3

N=4

N=1

N=2

N=3

N=4

0.0963 0.2406 0.2406 0.3848 0.4816 0.4816 0.6254 0.6254 0.8301 0.8301 0.8653 0.9719 0.9719

0.0963 0.2406 0.2406 0.3848 0.4816 0.4816 0.6254 0.6254 0.8302 0.8302 0.8653 0.9720 0.9720

0.0963 0.2406 0.2406 0.3847 0.4815 0.4815 0.6253 0.6253 0.8299 0.8299 0.8650 0.9717 0.9717

0.0963 0.2406 0.2406 0.3847 0.4815 0.4815 0.6253 0.6253 0.8299 0.8299 0.8650 0.9717 0.9717

0.0963 0.2406 0.2406 0.3848 0.4808 0.4808 0.6248 0.6248 0.8165 0.8165 0.8644 0.9602 0.9602

0.0963 0.2406 0.2406 0.3848 0.4808 0.4808 0.6248 0.6248 0.8165 0.8165 0.8644 0.9602 0.9602

0.0963 0.2406 0.2406 0.3847 0.4807 0.4807 0.6246 0.6246 0.8163 0.8163 0.8641 0.9598 0.9598

0.0963 0.2406 0.2406 0.3847 0.4807 0.4807 0.6246 0.6246 0.8163 0.8163 0.8641 0.9598 0.9598

The results of a four-layered plate with variable orthotropic ratios presented in Table 6 show a suitable agreement with meshfree results by Liew et al. [80] based on the FSDT, the analytical values reported in [81,82] and meshfree results by Ferreira et al. [83] based on radial basis functions.

5.3. Buckling problems of isotropic plates The buckling analysis of isotropic plates has been considered by investigating various side-to-thickness ratios under uni-axial compression (UAC) with following materials:

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A. Alesadi et al. / Computers and Structures 183 (2017) 38–47

Table 5 pffiffiffiffiffiffiffiffiffi The normalized natural frequencies xa q=G for a CCCC isotropic plate with a/h = 10, t = 0.3 and cubic NURBS basis functions. Mode No.

Present Grid 11
11

1 2 3 4 5 6 7 8 9 10 11 12 13

Ferreira et al. [76]

Rayleigh–Ritz [79]

Liew et al. [78]

1.5910 3.0389 3.0389 4.2625 5.0247 5.0723 6.0798 6.0798 7.4123 7.4123 7.6798 8.2601 8.3353

1.5940 3.0390 3.0390 4.2650 5.0350 5.0780 – – – – – – –

1.5582 3.0182 3.0182 4.1711 5.1218 5.1594 6.0178 6.0178 7.5169 7.5169 7.7288 8.3985 8.3985

Grid 19
19

N=1

N=2

N=3

N=4

N=1

N=2

N=3

N=4

1.6091 3.0924 3.0924 4.3536 5.1507 5.1977 5.5803 5.5803 7.6731 7.6731 7.911 8.2843 8.5493

1.623 3.1158 3.1158 4.3822 5.1834 5.2324 6.0366 6.0366 7.7188 7.7188 7.9492 8.591 8.6744

1.6058 3.0675 3.0675 4.303 5.0784 5.1272 6.0364 6.0364 7.532 7.532 7.764 8.3806 8.4597

1.6056 3.0669 3.0669 4.3017 5.0764 5.1251 6.0362 6.0362 7.5272 7.5272 7.7587 8.3743 8.4534

1.609 3.0918 3.0918 4.3525 5.1452 5.1921 5.5762 5.5762 7.6312 7.6312 7.9045 8.279 8.5153

1.6178 3.1066 3.1066 4.3707 5.1655 5.2137 6.0325 6.0325 7.6568 7.6568 7.9286 8.5393 8.6192

1.6009 3.059 3.059 4.2922 5.0619 5.11 6.0314 6.0314 7.475 7.475 7.7442 8.3322 8.4086

1.6005 3.0578 3.0578 4.2901 5.0588 5.1069 6.0308 6.0308 7.4683 7.4683 7.7373 8.324 8.4002

Table 6 pffiffiffiffiffiffiffiffiffiffiffi The normalized fundamental frequency of the simply-supported cross-ply laminated square plate [0/90/90/0], ðxa2 =hÞ q=ET with a/h = 5 and cubic NURBS basis functions. Method

Grid

11
11

Liew et al. [80] Exact [81,82] Ferreira et al. [83]

a ¼ b;

11
11 N=1 N=2 N=3 N=4

Present (tTT = 0.49)

E ¼ 73 GPa;

EL/ET

11
11

m ¼ 0:3

ð26Þ

The results are listed in Tables 7 and 8 and are referenced to available exact solutions [84–87] and higher order finite element method by Nali et al. [69]. Considering the results, it is shown that they have excellent agreement. 5.4. Buckling problems of cross-ply laminated plates Considering buckling analysis of cross-ply laminated plates, two cases with various side-to-thickness ratios and different

10

20

30

40

8.2924 8.2982 8.2866

9.5613 9.5671 9.5391

10.320 10.326 10.2676

10.849 10.854 10.7590

8.5479 8.5464 8.3013 8.2989

9.9698 9.9676 9.5466 9.5427

10.8342 10.8319 10.2786 10.2735

11.4401 11.4379 10.7785 10.7725

orthotropic ratios under uni-axial compression is investigated. Material properties are as following:

EL ¼ v ariable; ET

GLT ¼ 0:6; ET

GTT ¼ 0:5; ET

mLT ¼ mTT ¼ 0:25

ð27Þ

First, the results of three-layered plates with variable orthotropic ratios, with different side-to-thickness ratios, given BCs and material properties are presented in Tables 9 and 10 and results have suitable agreement with the analytical values reported by D’Ottavio et al. [67] and higher order finite element method by Nali et al. [69].

Table 7 Buckling load of isotropic SSSS plate under UAC: Nyb2/p2D with cubic NURBS basis functions. Source

a/h = 10

a/h = 100

Brunelle et al. [84] Reddy [85] Doong [86] Matsunaga [87]

3.729 3.787 3.730 3.771

3.997 3.998 3.997 3.998

FSDT- 10
10 FSDT- 20
20 HOT- 10
10 HOT- 20
20

3.871 3.833 3.850 3.810

4.054 4.012 4.057 4.013

N=1 N=2 N=3 N=4

Grid 7
7

3.8213 3.8221 3.7975 3.7974

4.0627 4.0629 4.0626 4.0626

N=1 N=2 N=3 N=4

Grid 11
11

3.8204 3.8211 3.7966 3.7965

3.9992 3.9992 3.9989 3.9989

N=1 N=2 N=3 N=4

Grid 19
19

3.8204 3.8211 3.7966 3.7965

3.9981 3.9981 3.9978 3.9979

Nali et al. [69]

Present

45

A. Alesadi et al. / Computers and Structures 183 (2017) 38–47 Table 8 Buckling load of isotropic SSSS plate under UAC: Nyb2/p2D with quartic NURBS basis functions. Source

a/h = 10

a/h = 100

Brunelle et al. [84] Reddy [85] Doong [86] Matsunaga [87]

3.729 3.787 3.730 3.771

3.997 3.998 3.997 3.998

FSDT- 10
10 FSDT- 20
20 HOT- 10
10 HOT- 20
20

3.871 3.833 3.850 3.810

4.054 4.012 4.057 4.013

N=1 N=2 N=3 N=4

Grid 8
8

3.8204 3.8211 3.7966 3.7965

3.9985 3.9985 3.9983 3.9983

N=1 N=2 N=3 N=4

Grid 12
12

3.8204 3.8211 3.7966 3.7965

3.9981 3.9981 3.9978 3.9978

N=1 N=2 N=3 N=4

Grid 20
20

3.8204 3.8211 3.7966 3.7965

3.9981 3.9981 3.9978 3.9977

Nali et al. [69]

Present

Table 9 Buckling load of [0/90/0] SSSS plate under UAC: Nyb2/ET h3 with EL/ET = 20, Grid 11
11 and cubic NURBS basis functions. a/h = 10

ESL ESL ESL ESL

-

a/h = 100

D’Ottavio et al. [67]

Nali et al. [69]

Present

D’Ottavio et al. [67]

Nali et al. [69]

Present

16.1310 – – 15.3440

15.7988 15.7939 15.0918 15.0912

15.5975 15.5955 14.9103 14.9097

19.9475 – – 19.6551

19.9801 19.9761 19.9636 19.9636

19.6610 19.6610 19.6489 19.6571

N=1 N=2 N=3 N=4

Table 10 Buckling load of [0/90/0] SSSS plate under UAC: Nyb2/ETh3 with a/h = 10, Grid 11
11 and cubic NURBS basis functions. EL/ET = 3

ESL ESL ESL ESL

-

EL/ET = 10

Nali et al. [69]

Present

D’Ottavio et al. [67]

Nali et al. [69]

Present

D’Ottavio et al. [67]

Nali et al. [69]

Present

– 5.3556 5.3060 –

5.5333 5.5239 5.4745 5.4744

5.4525 5.4526 5.3993 5.3993

– 9.9945 9.7720 –

10.2493 10.2449 9.9803 9.9800

10.1064 10.1055 9.8479 9.8477

– 15.6458 15.0551 –

15.7988 15.7939 15.0918 15.0912

15.5975 15.5955 14.9103 14.9097

N=1 N=2 N=3 N=4

EL/ET = 30

ESL ESL ESL ESL

-

EL/ET = 20

D’Ottavio et al. [67]

EL/ET = 40

D’Ottavio et al. [67]

Nali et al. [69]

Present

D’Ottavio et al. [67]

Nali et al. [69]

Present

– 20.4027 19.3785 –

20.3065 20.3012 19.1100 19.1090

20.0692 20.0665 18.9010 18.9000

– 24.4816 23.0021 –

24.0515 24.0460 22.3695 22.3681

23.7919 23.7888 22.1450 22.1436

N=1 N=2 N=3 N=4

Table 11 Buckling load of [0/90/90/0] SSSS plate under UAC: Nxa2/ETh3 with a/h = 10, tTT = 0.49, EL/ET = 40, Grid 11
11 and cubic NURBS basis functions. Expansion order

Present

Neves et al. [88]

Liew et al. [89]

Khdeir and Librescu. [82]

N=1 N=2 N=3 N=4

24.8547 24.8499 23.3535 23.3501

23.2916

23.463

23.453

Second, a four-layered plate is considered and results are listed in Table 11. The results show excellent agreement in comparison with meshfree results by Neves et al. [88] based on radial basis functions, meshfree results by Liew et al. [89] based on the FSDT and the values reported by Khdeir and Librescu [82].

6. Conclusion In this paper the CUF is considered in Isogeometric approach to buckling and free vibration analysis of cross-ply laminated plates. It is shown that employing NURBS basis functions together with

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higher order theories leads to an interesting combination based on the facts that the higher order of NURBS functions reduce the effect of shear locking phenomenon properly and the higher order theories eliminate Poisson locking without using shear correction factor. Indeed, in this study only in the case of N = 1 the modified stiffness coefficients are used to avoid Poisson locking, thus in higher order cases, N > 1, no material matrix modification is needed. The presented numerical results show that cubic and quartic NURBS functions, satisfactorily attenuate the efficacy of shear locking in thin and thick plates, respectively. Considering the results related to various order of Taylor expansion, it is deduced that in thin and thick plates third order and fourth order model suffice respectively, for reaching the convergence solution. Acknowledgement The authors would like to express their sincere appreciation to Professor Erasmo Carrera and his respectable research team specially Dr.Alfonso Pagani for their invaluable tips and comments in the course of the study. Appendix A The equations of FN in the case of cross-ply laminated plates are presented as follow:

R R R Ri;x Rj;x dX A F s F s dz þ C k55 X Ri Rj dX A F s;z F s;z dz R R þ C k66 X Ri;y Rj;y dX A F s F s dz R R R R K kxy ¼ C k12 X Ri;y Rj;x dX A F s F s dz þ C k66 X Ri;x Rj;y dX A F s F s dz R R R R K kxz ¼ C k13 X Ri Rj;x dX A F s;z F s dz þ C k44 X Ri;x Rj dX A F s F s;z dz R R R R K kyx ¼ C k12 X Ri;x Rj;y dX A F s F s dz þ C k66 X Ri;y Rj;x dX A F s F s dz R R R R K kyy ¼ C k22 X Ri;y Rj;y dX A F s F s dz þ C k55 X Ri Rj dX A F s;z F s;z dz R R þ C k66 X Ri;x Rj;x dX A F s F s dz R R R R K kyz ¼ C k23 X Ri Rj;y dX A F s;z F s dz þ C k55 X Ri;y Rj dX A F s F s;z dz R R R R K kzx ¼ C k13 X Ri;x Rj dX A F s F s;z dz þ C k44 X Ri Rj;x dX A F s;z F s dz R R R R K kzy ¼ C k23 X Ri;y Rj dX A F s F s;z dz þ C k55 X Ri Rj;y dX A F s;z F s dz R R R R K kzz ¼ C k33 X Ri Rj dX A F s;z F s;z dz þ C k44 X Ri;x Rj;x dX A F s F s dz R R þ C k55 X Ri;y Rj;y dX A F s F s dz

K kxx ¼ C k11

R

X

ð28Þ

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