A new higher order shear deformation theory for static, vibration and buckling responses of laminated plates with the isogeometric analysis

Accepted Manuscript A new higher order shear deformation theory for static, vibration and buckling responses of laminated plates with the isogeometric analysis Peng Shi, Chunying Dong, Fuzhao Sun, Wenfu Liu, Qiankun Hu PII: DOI: Reference:

S0263-8223(18)31294-7 https://doi.org/10.1016/j.compstruct.2018.07.080 COST 9998

To appear in:

Composite Structures

Received Date: Revised Date: Accepted Date:

10 April 2018 21 June 2018 20 July 2018

Please cite this article as: Shi, P., Dong, C., Sun, F., Liu, W., Hu, Q., A new higher order shear deformation theory for static, vibration and buckling responses of laminated plates with the isogeometric analysis, Composite Structures (2018), doi: https://doi.org/10.1016/j.compstruct.2018.07.080

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A new higher order shear deformation theory for static, vibration and buckling responses of laminated plates with the isogeometric analysis Peng Shi1, Chunying Dong2, Fuzhao Sun1, Wenfu Liu1, Qiankun Hu1 1

School of Mechanical and Energy Engineering, Huanghuai University, Zhumadian, Henan 463000, China;

2

School of Aerospace Engineering, Beijing Institute of Technology, Beijing, 100081, China.

Abstract A new hyperbolic tangent shear deformation theory (HTSDT) for the static, free vibration and buckling analysis of laminated composite plates is presented. In the present theory, shear stresses disappear at the top and bottom surfaces of the plates and shear correction factors are no longer required. Weak forms of the static, free vibration and buckling analysis for laminated composite plates based on the HTSDT are then derived and are numerically solved using the isogeometric analysis (IGA). The proposed formulation requires C1 continuity generalized displacements, whereas the basis functions used in IGA can perfectly fulfill this requirement. Based on the available solutions in the literature, the present method shows high accuracy and efficiency when numerical examples are solved. Keywords: Hyperbolic tangent shear deformation theory, Isogeometric analysis, Laminated composite plates.

1

1 Introduction Composite materials provide high performance and reliability due to high strength-to-weight and high stiffness-to-weight ratios, excellent fatigue strength, corrosion resistance, and most importantly the design flexibility also known as tailoring the materials for desired applications [1]. As a result, composite structures, such as composite plates, will continue to be widely used for many years in the engineering fields such as naval, aerospace, automotive and construction industries [2]. Hence, it is imperative to develop efficient plate theories and analysis methods to predict the short and long-term behavior of the multilayer composite materials under a variety of loading and environmental conditions. In the open literature, two main different approaches are used to study laminated composite structures [3-7], i.e., equivalent single layer (ESL) theories and layer-wise (LW) or Zig-Zag (ZZ) theories. In the ESL theories, laminated structures are assumed to be composed of one layer, thus reducing the three dimensional (3D) problem to a two dimensional (2D) one. Among ESL theories, there are many classes. The classical plate theory (CPT), based on Kirchhoff’s hypothesis, is inadequate to describe the accurate behavior of laminated composite structures because the transverse shear deformation is neglected. Therefore, it is necessary to develop more refined and sophisticated plate theories. The concept of shear deformation and its importance were first discussed by Reissner [8], who extended the kinematics of the classical laminated plate theory by including a gross transverse shear deformation in its kinematics assumption, and considered that each normal to the medium surface maintains straight but is not perpendicular to the deformed configuration. Mindlin [9] introduced a correction factor, which was evaluated by comparison with an exact elasticity solution, into the shear stress resultants to account for a uniform shear stress through the thickness of the plate. Generally, the correction factor is less accurate and would violate surface conditions. This method of considering shear deformation is called the Reissner-Mindlin plate theory, or the first order shear deformation theory (FSDT). Based on the FSDT, Phung-Van et al. [10] presented an edge-based smoothed three-node Mindlin plate 2

element for static and free vibration analyses of plates. Phung-Van et al. [11] also studied the static response, free vibration and dynamic control of composite plates integrated with piezoelectric sensors and actuators by the cell-based smoothed discrete shear gap method [12]. Liu and Paavola [13] described a general analytical sensitivity analysis method for the composite laminated plates and shells. The first derivatives of engineering constants of lamina with respect to fibre volume fractions were computed based on the micromechanics theory. The first derivatives of extensional, bending and bending-extensional coupling stiffnesses of laminate with respect to fibre volume fractions and fibre orientations were derived using CPT and FSDT of laminate. Based on the finite element method (FEM), the first derivatives of total stiffness and mass matrices were computed, then they were employed to conduct sensitivity analysis for static and vibration responses of composite laminated plate and shell. The merit of the proposed method is that it is easy to implement sensitivity analysis using the commercial finite element analysis software or make one’s own sensitivity analysis program [13]. However, the accuracy of FSDT cannot be effectively determined [14] because of the dependency of the shear correction factor on the lamination sequence, loading conditions, and boundary conditions, as discussed by Pai [15]. To account for certain higher order effects such as warping of the cross-section, the FSDT needs to be improved to obtain more accurate shear stress distribution. As a result, various higher order shear deformation theories (HSDT) have been developed as described below. The polynomial terms were used to account for approximately parabolic distribution of shear stresses through the thickness of the plate. Earlier third order shear deformation theory can be found in the works of Ambartsumyan [16], who proposed a transverse shear strain shape function to study plate deformation. The polynomial higher order theory expressed in terms of shear strain function was proposed by Kaczkowski [17], Panc [18] and Reissner [19]. The transverse shear stress boundary condition at the top and bottom of the plate can be satisfied. The higher order theory, based upon the principle of stationary potential energy, resulting in eleven second order 3

partial differential equations to determine the eleven functions in the assumed displacement model, was proposed by Lo et al. [20, 21]. The displacement model consisted of polynomials in thickness direction, and its slope with respect to the thickness direction was continuous. This kind of displacement model has been widely used with modifications to satisfy transverse shear stress boundary conditions. By using the sub-set of the model in Ref. [20], and neglecting the strain energy due to transverse normal stress, Levinson [22], Murthy [23] and Reddy [24] proposed the well-known parabolic third order shear deformation theory (TSDT), which was deduced from the power expansion of the displacement field. Based on the TSDT, Phung-Van et al. [25] used a cell-based smoothed discrete shear gap method [12] for static and free vibration analyses of functionally graded plates. Phung-Van et al. [26] also presented a cell-based smoothed three-node Mindlin plate element based on the C0-type TSDT for geometrically nonlinear analysis of laminated composite plates. Another interesting application of the TSDT can be found in the work of Akbarzadeh et al. [27]. By using the TSDT based hybrid Fourier-Galerkin method, they studied the static, vibration and buckling characteristics of moderately-thick laminate plates with manufactured gaps and overlaps. They claimed that shear deformation had a more severe impact on the structural responses of a variable stiffness than a constant stiffness plate. They also found that gaps can deteriorate the structural performance, while overlaps can improve it. In addition, the FSDT and TSDT can also be combined together to model the composite sandwich plate. In the work of Nasihatgozara and Khalili [28], the effect of different boundary conditions on the free vibration response of a sandwich plate was studied. The face sheets were considered as laminated composites which followed the FSDT and the core was modeled based on the HSDT. The higher order displacement and rotation terms were expended in the Taylor's series. The motion equations were derived considering the continuity boundary conditions between the layers based on the energy method and Hamilton's principle. The frequencies and mode shapes of the structure were obtained using the differential quadrature method. 4

Besides the polynomial functions, various non-polynomial functions were developed as the shear functions for the HSDTs. Trigonometric sine shear deformation theory (TSSDT) with the introduction of the sine function was firstly developed by Levy [29]. After nearly one century, it was corroborated and assessed by Stein [30], to study the buckling behaviors of simply supported thick plates. The sine function based shear deformation theory was later developed and extensively used by Touratier [31], recently employed by Singh and Singh [32]. Compared to TSDT, the sine function is much richer than the third order power expansion, which can obtain much accurate transverse shear stress results. Additionally, Grover et al. [33] used the trigonometric secant function as the shear function, to study the structural behavior of laminated composite and sandwich plates. Trigonometric tangent shear deformation theory (TTSDT) was proposed by Mantari et al. [34], and was used to study isotropic, laminated composite and sandwich plates. The results show that the TTSDT model was in close agreement with the shear deformation theories of Reddy [24] and Touratier [31] for analyzing the static behaviors of isotropic, composite laminated and sandwich plates. The trigonometric tangent shear function was also employed and later developed by Singh and Singh [32], to study free vibration and buckling responses of laminated and braided composite plates. Additionally, inverse trigonometric tangent shear deformation theories (ITTSDT) were developed by Thai et al. [35], Suganyadevi and Singh [36], for static, buckling, and free vibration analyses of laminated composites and sandwich plates. Besides, a new inverse trigonometric Cotangent shear deformation theories (ITCSDT) were developed by Grover et al.[33]. Recently, based on inverse trigonometric tangent shear strain function, Adhikari and Singh [37] developed a new higher order Quasi 3D theory to encapsulate the dynamic response of laminated composite plates. Hyperbolic sine shear deformation plate theory (HSSDT) was proposed by Soldatos [38], and was unified by Soldatos and Timarci [39]. The main advantage of this unified theory was the ability to change the transverse strain distribution. Unified shear deformation theory was used for vibration and buckling problems of composite 5

laminated plates and shells by Timarci and Soldatos [40], Aydogdu and Timarci [41]. Based on the model of Soldatos [38], EI Meiche et al. [42] developed a new hyperbolic sine shear deformation theory for buckling and vibration of functionally graded sandwich plates. The hyperbolic sine shear function was also employed and further developed by Grover et al. [43] with its inverse form, to study the static and buckling characteristics of laminated composite and sandwich plates. Besides, hyperbolic tangent shear deformation plate theorys (HTSDTs) [44, 45] were developed for the laminated composite analysis. Akavci and Tanrikulu [44] presented two new HTSDTs by using the hyperbolic tangent and secant functions as the transverse shear deformation functions respectively. Mahi et al. [45] developed a hyperbolic tangent shear deformation theory, in which a combination of hyperbolic tangent function and cubic function of the thickness coordinate was used as the shear function. Exponential shear deformation plate theory (ESDT) was presented by Karama et al. [46] by using an exponential function as the shear stress function. The exponential function was found to be much richer than trigonometric sine function in their Fourier development series, because an exponential function has all even and odd powers in its expansion unlike sine function, which has only odd power. Aydogdu [47] presented a comparison of various shear deformation theories, including TSDT, TSSDT, HSSDT, and ESDT for bending, buckling and vibration of rectangular symmetric cross-ply plates with simply supported edges. Results showed that the deflections and stresses were best predicted by ESDT when compared with three-dimensional elasticity solutions. The TSDT and HSSDT gave the free vibration frequencies and buckling loads more accurately when compared with other shear deformation theories [47]. Aydogdu [48] developed Karama et al.’s ESDT [46], by firstly introducing a transverse shear stress parameter, which was determined by an inverse method using 3D solution results so that errors between 2D solution and 3D solution results were minimized. In fact, Karama et al.’s model [46] is a special case of Aydogdu’s model [48] when the transverse shear stress parameter is equal to exp. Based on this [48], Mantari et al. [49] presented a more precise ESDT with a different transverse shear 6

stress parameter. In their works [49], the determination of parameters was given in detail. The authors [49] claimed that their model performed better than all the higher order shear deformation theories in which the static and dynamic behaviors of multilayered sandwich and composite plates and shells were analyzed. Nguyen et al. [50] reviewed the shear functions used in the HSDTs in the literature and presented a unified frame work on the HSDTs, for modeling and analysis of laminated composite plates. The authors [50] unified all HSDTs in a unique formulation by a polynomial form, and proposed a new higher shear deformation models systematically based on a unified formulation. In addition, the effect of thickness stretching was taken into account. Recently, Candiotti and Mantari [51] presented best theory diagrams (BTDs) constructed from various non-polynomial theories for the static analysis of thick and thin symmetric and asymmetric cross ply laminated plates. The BTD was a curve that provides the minimum number of unknown variables necessary for a fixed error or vice versa. The plate theories that belong to the BTD were obtained by means of the axiomatic/asymptotic method (AAM). The results suggested that the plate models obtained from the BTD using non-polynomial terms can improve the accuracy obtained from Maclaurin expansions for a given number of unknown variables of the displacement field. Although the existing HSDTs provide a better prediction compared with the CPT and FSDT, they are much more complicated and perhaps computationally expensive than the CPT and FSDT because additional dependent unknowns may be introduced into the theory. Thai et al. [52] proposed a new simple HSDT which involved one unknown and one governing equation as in the CPT, but it was capable of accurately capturing shear deformation effects. The displacement field was based on a two variable refined plate theory [53] in which the transverse displacement was partitioned into the bending and shear parts. Based on the equilibrium equations of 3D elasticity theory, the relationship between the bending and shear parts was established. Therefore, the number of unknowns was reduced from two to one. Nguyen et al. [54] proposed a general three-variable plate theory. The strong forms, weak form of the three-variable 7

plate theory including the non-classical boundary conditions were variationally derived so that the theory can be further utilized for general plate problems such as inhomogeneous, laminated composite or sandwich plates. It should be noted that, in the proposed theories [52, 54], the C3 continuity generalized displacements are needed. The ESL theories give a sufficiently accurate description of the global laminate response and in-plane stresses but it is not accurate to capture the transverse shear stresses on the layer interfaces since they cannot ensure the inter-laminar stress continuity and ZZ requirements of laminated plates. As a result, the LW theory was proposed in the late 1970s, which considered independent degrees of freedom for each layer. The LW theory of Reddy [55] may be the most widely used LW theory for composite and sandwich plate analysis. Some interesting LW or ZZ theories were proposed by Mau [56], Srinivas [57], Chou and Corleone [58], Di Sciuva [59], Murakami [60], Ren [61], Carrera [62-64], Ferreira [65] and Demasi [66]. A historical review of such theories in the analysis of multilayered plates and shells can be found in [67]. Additionally, using higher order LW and ZZ shear deformation theories can yield more accurate results, as can be seen in the works of Arya et al. [68], Ferreira et al. [69], Karama et al. [46, 70], Thai et al. [71], Shimpi and Ghugal [72], Lee et al. [73], Pandey and Pradyumna [74], Phung-Van et al. [75]. LW or ZZ theories may provide a better representation of inter-laminar stresses and moderate severe cross-sectional warping. Thus, they allow to analyze the local behavior of laminated structures when needed, but high computational efforts [76] have to be paid. The analytical methods [20, 24, 31, 77, 78] were widely used to solve the static, vibration and buckling problems of laminated composite plates, but limited to the simple geometries, boundary conditions and load cases. To obtain a general application, the finite element methods (FEM) [62, 74, 79-81] and mesh-free methods [82-86] were often employed. However, the geometry was discretized in an approximate form, that would lead to the inaccurate solutions. The isogeometric analysis (IGA) method, which was proposed by Hughes et al. [87], can be capable to overcome the above mentioned shortcoming. The IGA has several advantages over standard FEM, e.g. the smoothness 8

with arbitrary continuity order, exact representation of shapes even at the coarsest level of discretization, simple and systematic refinement strategy, and more accurate modeling of complex geometries. In recent years, IGA has been used for composite plates analysis with various theories, such as FSDT based composite plates [88, 89], LW theory based composite plates [71, 90, 91], HSDT based composite plates [35, 92-94], and geometric nonlinear composite plates [95-97]. Also, the IGA has been employed successfully for the analysis of functionally graded nanoplates [98-102] and smart piezoelectric composite plates [103-105]. From the literature, in the ESL theories, it is observed that the non-polynomial HSDTs are generally based on trigonometric, exponential, hyperbolic functions and their combination forms, but the HSDT based on hyperbolic tangent shear function is quite few. To the authors’ best knowledge, they were only proposed by Akavci and Tanrikulu [44] with the sum of a hyperbolic tangent function and a linear function of the thickness coordinate, and by Mahi et al. [45] with the sum of a hyperbolic tangent function and a cubic function of the thickness coordinate. The cubic term in the shear function of Mahi et al. [45] may be unnecessary since the Taylor’s expansion of hyperbolic tangent has included a cubic term. Both of the shear functions [44, 45] were expressed in relatively complicated forms. It motives us to develop a new hyperbolic tangent shear function, which is simple but with high accuracy. Meanwhile, it can also enlarge the library of shear functions for the non-polynomial HSDTs thus interested readers could have more choices to select a HSDT. In the present work, a new non-polynomial higher order shear deformation theory based upon hyperbolic tangent function is developed. The theory is assessed with the NURBS based isogeometric analysis for the static, free vibration and buckling responses of laminated composite plates. The proposed shear deformation theory satisfies the transverse shear stress conditions on the top and bottom surfaces of the plate beforehand. Hence, the requirement of shear correction factor vanishes. Generalized displacements are constructed using the NURBS basis functions that can yield higher order continuity and fulfill easily the requirement of C 1-continuity of the 9

higher order shear deformation model. The static, free vibration and buckling analysis are studied and various numerical examples are presented to show high effectiveness of the present method for laminated composite plates. Present results are compared with different plate models, including exact 3D elasticity, ESL and LW theories, and various methods, such as analytical or semi-analytical methods, finite element methods, mesh-free methods and other numerical solutions. 2 NURBS basis functions The fundamentals of IGA are stated in this section based on Hughes et al.’s paper [87]. Given a knot vector which is a sequence in a non-decreasing order of parameter values, written as: Ξ  {1 , 2 ,

, n p 1}, i  i 1 , i  1,

,n p

(1)

Where ξi is the i-th knot, n is the number of basis functions and p is the polynomial order. By repeating the first and last knots p times, the knot vector becomes an open one and the endpoints are interpolatory. The basis function is C∞ continuous inside a knot span and is Cp-m continuous at a knot location, where m is the multiplicity of the knot. The associated B-spline basis functions for a given degree p, are defined recursively over the parametric domain by the knot vector. For p = 0,

1 Ni0 ( )   0

if i    i 1 otherwise

(2)

For p ≥ 1, Nip ( ) 

i  p 1     i Nip 1 ( )  N p 1 ( ) i  p  i i  p 1  i 1 i 1

(3)

The non-uniform rational B-splines (NURBS) surface is defined by: n

m

S ( , )   Rijpq  ,  Pij

(4)

i 1 j 1

where Pij is the 2D control point. Rijpq  ,  is the multiplication of univariate NURBS, i.e.:

10

R ( , )  pq ij

Nip ( ) M qj ( )ij n

m

 Nkp ( )M lq ( )kl

(5)

k 1 l 1

where Nip ( ) and M qj ( ) denote the B-spline basis functions defined by the knot vectors   {1 , 2 ,

, n p 1} and   {1 ,2 ,

,m q 1} , and p and q are basis

function orders. ij is the corresponding weight. 3 Formulations of the problem 3.1 Displacement field The geometry of a laminated composite plate can be seen in Fig. 1, in which a, b and h are plate length, width and thickness, respectively. In the Cartesian coordinates system, the xy-plane is the mid-plane of the plate, and the positive z-axis is upward from the mid-plane. For the development of the present shear deformation theory, the following displacement field is assumed, as:

U ( x, y, z )  u ( x, y )  z w( x, y ) x  f  z   ( x, y ), V ( x, y, z )  v( x, y )  z w( x, y ) y  f  z   ( x, y ),

(6)

W ( x, y, z )  w( x, y ) where U, V and W are displacements at a generic point of the plate along the x, y and z directions, respectively. u, v and w are displacement components of a point on the mid-plane of the plate along the x, y and z directions, respectively, and α and β are rotations with respect to the y and x axes, respectively. f  z  is the transverse shear shape function. Thus the considered basic field variables for each control point in this formulation are u, v, w, α and β, as

u  u v w   

T

(7)

It should be noted that the present HTSDT is simple in the sense that it contains the same dependent unknowns as in the FSDT. With the isogeometric analysis, the displacement field is expressed by the NURBS

11

basis functions and the displacement field of each control point u ij , as: n

m

u   Rijpq  ,  uij

(8)

i 1 j 1

where Rijpq  ,  is the NURBS basis function, m and n are the numbers of control points along directions  and  . And

uij   uij vij wij ij ij 

T

(9)

where uij , vij , wij ,  ij and  ij are the displacements and rotations of each control point. In Eq. (6), the shear shape function is presented as f  z  which determines the distribution of the transverse shear strains and stresses along the plate thickness. This distribution function is chosen as an odd function, which leads to a symmetric through-thickness distribution of transverse shear strains and stresses, with respect to the mid-plane of the plate. In the present theory, a hyperbolic tangent function is proposed, as:

f  z  = g( z )  z

(10)

where g( z ) 

h 2  tanh  z  , 2 h 

=

1 cosh 21

(11)

Several shear stress shape functions proposed by other researchers are historical reviewed and listed in Table 1. Another recent review can be found in [50]. Several shape functions and their derivations through the plate thickness are illustrated in Fig. 2. It can be seen that the zeros derivations of shear functions at the top and bottom surfaces of the plate are obtained. From Eqs. (10) and (11), it can also be seen that, compared with the shear functions proposed by Mahi et al. [45], and by Akavci and Tanrikulu [44], present hyperbolic tangent shear function has a simple form. It is easy to compute its derivation and integration for the next analysis. 12

3.2 Strain displacement relations The strain vector ε under the assumption of small deflection and small rotation is given as:

ε   xx  yy  xy  yz  xz    εp εs  T

T

(12)

where εp and εs are in-plane strain vector and transverse shear strain vector, respectively. The in-plane strains are expressed as:

εp   xx  yy  xy   ε0  zκ 0  f  z  κ1

(13)

  xx0   u x   0    0 ε    yy    v y ,  0   u y  v x       xy    xx0     2 w x 2      κ 0    yy0      2 w y 2  ,  0       2  2 w xy   xy    1xx   x    1    1 κ    yy   f  z    y   1    y   x       xy 

(14)

T

where:

and the transverse shear strains are given in the following form:

εs   yz  xz   T

s df  z    yz  df  z      s    dz   xz  dz   

(15)

One can see that:

df  z   h  x, y , z     0 dz  2

(16)

Thus at the top and bottom surfaces of the plates, zeros shear stresses σyz and σxz can be satisfied. From Eqs. (6) and (14) one can see that, based on the present theory with five degrees of freedom of each control point, the proposed formulation requires C1 continuity generalized displacements, whereas the basis functions in IGA can perfectly fulfill this requirement. It is clearly more efficient than those HSDT FEMs [32, 37, 43] 13

with seven degrees of freedom for each node and C0 continuity basis functions. Besides, in dealing with higher continuity requirements i.e. C3 continuity, the IGA is also superior to other methods for shear deformation plate theory. It has been successfully implemented in a three-variable shear deformation plate formulation [54] in which the C3 continuity generalized displacements are needed, whereas the higher continuity of NURBS basis functions can achieve this requirement excellently. It is one of the notable advantages of the IGA over other methods. Based on Eqs. (14) and (15), the strain vector ε can be expressed by the following form with 11 components as:

ε  Z  xx0  yy0  xy0  xx0  yy0  xy0  1xx  1yy  1xy  yzs  xzs 

T

(17)

with the matrix Z, which is a function of thickness coordinate, as:

0 0  1 0 0 z 0 0 f ( z ) 0 0   0 0   0 1 0 0 z 0 0 f (z) 0 Z   0 0 1 0 0 z 0 0 f ( z) 0 0     0 0 0 0 0 0 0 0 0 df ( z ) dz 0  0 0 0 0 0 0 0 0 0 0 df ( z ) dz  

(18)

Using Eqs. (14), (15) and (17), the strain-displacement relations can be written as:

ε  ZLu

(19)

where L can be expressed as:

        L         

x 0 y

0  y  x

0

0

0

0

0 0 0 0 0 0

0 0 0 0 0 0

     -  2 x 2 0 0  -  2 y 2 0 0  -2  2 xy 0 0   0  x 0  0 0  y   0  y  x  0 0 1   0 1 0   0 0 0

3.3 Hooke’s law 14

0 0 0

0 0 0

(20)

For a plate of constant thickness, and under the assumption that each layer possesses a plane of elastic symmetry parallel to the mid-plane, the constitutive equation for the kth layer can be written as follows: σ k  Qk ε k

(21)

where:

σ k   xx  yy  xy  yz  xz 

(22)

kT

here normal stress σzz is neglected as in all other shear deformable plate theories with five degrees of freedom, and:

ε k   xx  yy  xy  yz  xz 

kT

(23)

and:

 Q11 Q12 Q16 0   Q12 Q22 Q26 0 k Q   Q16 Q26 Q66 0 0 0 0 Q44  0 0 0 Q45 

0   0  0  Q45   Q55 

k

(24)

where Qij is a transformed material constant of the kth layer, which can be found in [2]. 3.4 Weak form For the static analysis, the weak form of a laminated composite plate under distributed transverse load can be briefly expressed as: h2



D h 2

 ε TσdzdD    wpdS  0

(25)

S

where p is the transverse loading per unit area. D and S are the mid-surface of the plate and the surface on which the transverse load acts, respectively. Substituting Eqs. (19) and (21) into Eq. (25), can lead to: h2



D h 2

 uT LT ZTQZLudzdD    wpdS  0 S

Integrating along the plate thickness firstly, one can get: 15

(26)

  u L CLudD    wpdS  0 T

T

D

(27)

S

where C is the material constant matrix, which is given by: h2



C

ZTQZdz

(28)

B E 0  D F 0  F H 0  0 0 A s 

(29)

h 2

i.e.:

A  B C E  0 where:

A , B , D , E , F , H  ij

ij

ij

ij

ij

ij

h2



 1, z, z , f  z  , zf  z  , f  z   Q 2

2

ij

dz, i, j  1, 2, 6

(30)

h 2

and: h2

A  s ij



h 2

 df  z     Qij dz, i, j  4, 5  dz  2

(31)

Substituting Eq. (8) into Eq. (26), the formulation for the static analysis can be obtained as:

Kd  f

(32)

where K is the stiffness matrix. f is the load vector. d is the displacement field vector of the plate. The stiffness matrix can be written as:

Κ   BkTCBk dD D

in which:

16

(33)

 Rijpq x  0   pq  Rij y  0   0  Bk   0   0   0  0   0   0 

0 Rijpq Rijpq 0 0 0 0 0 0 0 0

  y 0 0 0   x 0 0 0   -  2 Rijpq x 2 0 0  -  2 Rijpq y 2 0 0  -2  2 Rijpq xy 0 0   pq 0 Rij x 0   0 0 Rijpq y   0 Rijpq y Rijpq x   0 0 Rijpq   0 Rijpq 0  0

0

0

(34)

The load vector f can be written as: f   p  0 0 Rijpq 0 0  dS T

(35)

S

For the free vibration analysis, the weak form of a composite plate can be written as:

  u L CLudD    u mudD  0 T

T

T

D

(36)

D

where:

u   u v w - w x - w y 0   0  ,

(37)

 m1 m 2 m 4    m   m2 m3 m5  m m m  5 6  4

(38)

T

where: h2

 m1 , m2 , m3 , m4 , m5 , m6    I 

1, z, z , f  z  , zf  z  , f  z  2

2

dz

(39)

h 2

where ρ is the material density. I is the 3×3 identity matrix. Substituting Eq. (8) into Eq. (36), the formulation for the free vibration analysis can be obtained as:

17

K   M d  0

(40)

2

where ω is the angular frequency. M is the mass matrix, which can be expressed as:

M   BTMmBM dD

(41)

D

where:

 Rijpq  0  0  0 B M   0 0  0  0 0  

0

0

0

Rijpq

0

0

0

Rijpq

0

0

- Rijpq x

0

0

- Rijpq y

0

0

0

0

0

0

Rijpq

0

0

0

0

0

0

0   0  0  0 0  0  0  Rijpq  0  

(42)

For the buckling analysis, the weak form of a composite plate can be expressed as:

  u L CLudD     wσ wdD  0

(43)

 w x  w     w y 

(44)

T

T

T

D

0

D

where:

and σ 0 is the in-plane pre-buckling stress, i.e.:   xx0  xy0   σ  0   0  xy yy   0

(45)

Substituting Eq. (8) into Eq. (43), the formulation for the buckling analysis can be obtained as:

 K  cr G  d  0

(46)

where λcr is the critical buckling load parameter. G is the geometric matrix, as:

G   BGT σ 0BG dD D

18

(47)

where:

0 BG   0 

0 Rijpq x 0 0 Rijpq y 0

0  0 

(48)

4 Results and discussion Using the proposed mathematical model and solution methodology, a generalized computer program is coded in MATLAB. Various examples are presented for the static, free vibration and buckling analysis of laminated composite plates. In the present IGA, the cubic NURBS element is considered without loss of the generality. Numerical integration is performed with 4×4 Gauss points. Convergence studies are firstly conducted. In order to check the accuracy and efficiency of the present theory, numerous results are compared with the exact 3D elastic solutions and other available results in the literature. The material properties used in this paper are as follows: Material 1: E1 = 25E2, G12 = G13 = 0.5E2, G23 = 0.2E2, ν12 = 0.25, ρ = 1. Material 2: E1 = 40E2, G12 = G13 = 0.6E2, G23 = 0.5E2, ν12 = 0.25, ρ = 1. Material 3: E1 = 181, E2 = 10.3, G12 = G13 = 7.17, G23 = 2.87, ν12 = 0.28, ρ = 1. The boundaries are defined as follows: for simply supported boundaries: u, w,   0 at y  0, b v, w,   0 at x  0, a

(49)

u, v, w,  ,  , w,n  0

(50)

for clamped boundaries:

It should be noticed that, in the present IGA, the Dirichlet boundary conditions u, v, w, α and β can be easily and directly imposed as in the standard FEM. The normal slop w,n can be directly enforced by assigning zeros of the transverse displacement adjacent to the boundary control points [107] without any additional variables. Thus, it is very simple for implementing the essential boundary condition for derivation of displacements compared with other numerical methods. Finally, the boundary conditions are inserted by modifying the diagonal term of the global stiffness 19

matrix as unity, as in the standard FEM. 4.1 Static analysis Let us firstly consider a four-layer [0°/90°/90°/0°] laminated square plate subjected to a sinusoidal load p under simply supported boundary conditions. The Material 1 is used in this example. The plate length-to-thickness ratios are a/h = 4, 10, 20 and 100, respectively. The sinusoidal distributed load is defined as: x   y  p  p0 sin   sin    a   b 

(51)

The normalized displacement and stresses are defined as:

w

100 E2 h3  a b  h2 a b h w , , 0 ;    xx  , ,  ; xx   4 2 p0 a p0b 2 2  2 2 2

 yy 

h2 h2 h a b h   , , ;    xy  0, 0,  ; yy  xy  2 2 p0b p0b 2 2 2 4 

 yz 

h h a   b   yz  , 0, 0  ;  xz   xz  0, , 0  p0b p0b 2   2 

(52)

The convergence study is firstly conducted to assess the effectiveness of the present formulation by using 6×6, 10×10, 14×14, 18×18 and 22×22 cubic elements, respectively. To obtain denser and denser IGA meshes gradually, the k-refinement algorithm (as can be seen in [108], A5.5) is adopted, hence the control points are computed automatically, and they are distributed non-uniformly along the edges of the plate. With the k-refinement, the parametrization of the plate is linear i.e., the determinant of the Jacobian keeps constant. The 10×10 and 18×18 IGA meshes are illustrated in Fig. 3 as two examples, and their knot vectors, weights and control point coordinates are given in the Appendix. In this convergence study, the plate length-to-thickness ratio is chosen as a/h = 4. Table 2 shows the convergence of the normalized deflection and stresses obtained through the present model. Results gained by different laminated composite plate theories, including the exact 3D theory [77], ESL theories [24, 31-34, 36, 49, 109], and LW theories [65, 71, 110, 111], are included. Besides, the IGA results [35, 71] are given 20

in Table 2. The relative error percentages compared with the exact 3D solutions (simplified as Diff.%) are included. It can be seen that, the present results match well with the exact 3D solutions and other published results, especially the displacement and in-plane stresses. One can also see that, the deflection solution converges very fast. Converged stress results can be obtained by 18×18 IGA meshes since the differences of stresses obtained by 18×18 meshes and 22×22 meshes are not significant. Hence, 18×18 meshes will be used for further static studies. The normalized displacement and stresses results with different plate length-to-thickness ratios (a/h = 4, 10, 20 and 100) are given in Table 3. Present solutions are compared with those of other theories such as the exact 3D elastic theory by Pagano [77], close form TSDT by Reddy [24], TTSDT by Mantari et al. [34], TTSDT by Singh and Singh [32], LW theory and RBF-PS discretizations with optimal shape parameter by Ferreira et al. [110], refined laminated plate theory accounting for the TSDT by Wang and Shi [111], generalized LW TSDT of Reddy [24] with IGA by Thai et al. [71], and ITTSDT with IGA by Thai et al. [35]. The relative error percentages compared with the exact 3D solutions are included. It can be seen that the present theory with the IGA gains adequately accurate results for all plate length-to-thickness ratios. Results are much close to Thai et al.’s work [35] using the IGA. Additionally, the normalized deflection and in-plane stresses match the exact 3D solutions excellently. For plates with length-to-thickness ratio a/h = 4, the present central deflection shows better than other results [24, 32, 34, 71, 110, 111] except for that from Ref. [35]. For other plate length-to-thickness ratios, the proposed theory performs better in terms of σxz than Thai et al.’s theory [35]. In addition, using the IGA, stresses distribution through the plate thickness obtained by the present theory, Reddy’s TSDT [24] and Touratier’s TSSDT [31], are illustrated in Fig. 4. Good agreements can be seen. Secondly, a three-layer [0°/90°/0°] laminated, rectangular plate simply supported on all edges and subjected to a sinusoidal load is considered. The Material 1 is used in this example. The plate length-to-thickness ratio are a/h = 4, 10, 20, 50 and 100, 21

respectively. The normalized displacement and stresses are defined in Eq. (52). Pagano [77] studied this problem with the exact 3D elastic theory. Reddy [24], Mantari el al. [49], Karama et al. [109] and Touratier [31] solved this problem with various HSDT based on the ESL model. Present results and those from literature are given in Table 4. The relative errors (compared with 3D solutions) and average errors (simplified as Diff.*%) are also included. It can be seen that, for different plate thicknesses, present results agree excellently with those from the ESL theories, and presented much near to the exact solutions [77]. Additionally, for all plate length-to-thickness ratios, from the relative errors one can see that, the present theory performs best for the transverse shear stress σxz. Furthermore, for thick plates (a/h = 4 and 10), the present theory can obtain minimum average errors, best central deflections and best in-plane shear stresses. Therefore, it can be concluded that the present HTSDT with the IGA can effectively and precisely capture the deflection and stresses of the laminated composite plates, especially for thick plates. 4.2 Free vibration analysis A four-layer [0°/90°/90°/0°] laminated square plate subjected to simply supported boundary conditions is considered. The Material 2

is used. The plate

length-to-thickness ratio a/h = 5 is chosen. The normalized natural frequencies are defined as: 1

 a2     2        h  E2 

(53)

The convergence study is performed by using 6×6, 10×10, 14×14 and 18×18 IGA meshes, respectively. The first ten natural frequencies are given in Table 5. It can be observed that the frequencies converged fast and monotonously. Using 18×18 IGA meshes can obtain the converged first ten natural frequencies. For the first natural frequency, 14×14 IGA meshes is adequate for the convergence. Additionally, Noor [78] solved this problem with the exact 3D elasticity theory. The first natural frequency of this example proposed by Noor [78] is also given in Table 5. It can be seen that the present result is very close to Noor’s solution [78]. 22

Next, for this example, the effect of modulus ratios (E1/E2 = 3, 10, 20, 30 and 40) is considered. The first natural frequencies obtained by 14×14 IGA meshes are given in Table 6. The natural frequencies are normalized by Eq. (53). Besides Noor’s exact 3D elasticity solutions [78], exact TSDT solutions can be found in Reddy’s book [2]. Based on the ESL model, various results are available in the literature, such as: Reddy’s refined TSDT [24] FE results by Phan and Reddy [112], TSSDT and TTSDT FE results by Singh and Singh [32], ESDT FE results by Mantari et al. [49], TSDT results by Khdeir and Librescu [113], FSDT using the moving least squares differential quadrature method results by Liew et al. [114], FSDT wavelet results by Ferreira et al. [115], TSDT IGA results by Thai et al. [35], HSDT IGA results by Nguyen-Xuan et al. [92] , ZZ and LW theory results by Rodrigues et al. [116], Chalak et al. [117] and Thai et al. [71]. By comparing present results with those available solutions obtained by various plate theories and methods, one can see that remarkable agreements can be achieved. Present results are very close to those of Mantari et al. [49]. From the IGA results comparison, one can find that present frequencies are higher than the LW Reddy’s TSDT solution but little lower than the ESL TSDT results. Besides, the relative error percentages Diff.% compared with the exact 3D solutions of Noor [78] are included in Table 6. One can see that, all of the methods perform excellently for the first natural frequencies. The differences between the present results and the exact 3D solutions are very small. For different modulus ratios (E1/E2 = 10, 20, 30 and 40), the relative error percentages are less than 1%. Next, for this example, the influence of various plate length-to-thickness ratios is studied. The modulus ratio is fixed as E1/E2 = 40. Present first natural frequencies normalized by Eq. (53) are given in Table 7. It can be seen that compared with those proposed by Zhen and Wanji [118] with a global-local TSDT, by Matsunaga [119] with a global-local HSDT, by Cho et al. [120] with a higher order individual-layer theory, and by Thai et al. [35], Nguyen-Xuan et al. [92] and Thai et al. [71] incorporating the IGA, good agreements can be observed. To study different boundary conditions, a four-layer [0°/90°/90°/0°] laminated 23

square plate with the clamped boundary conditions of all edges is considered next. The material 3 is used in this case. In Table 8, the first six natural frequencies normalized by Eq. (53) are reported by considering the plate length-to-thickness ratios as 5, 10 and 20, respectively. Results reported by Chalak et al. [117], Kulkarni and Kapuria [121] based on TSDT ESL model and TSDT ZZ model respectively, are considered for the comparison. It can be observed that the present model performs well although the frequencies are slightly higher than the referenced results [117, 121]. Another example is considered, i.e. a three-layer laminated [0°/90°/0°] plate with the four clamped edges and various length-to-width as well as length-to-thickness ratios, is studied. Material 3 is used here. The natural frequencies are normalized as: 1

 b2    h  2     2      D0 

with D0 

E2 h3 12 1  122 

(54)

The first natural frequencies for various values of a/b and b/h are given in Table 9. Present results are compared with those reported by Liew [122] with the global p-Ritz method, by Zhen and Wanji [118] with the FEM, by Shi et al. [123] with the FSDT Galerkin method and by Thai et al. [71] with the IGA by using generalized LW TSDT of Reddy [24], TSSDT of Arya et al. [68] and ESDT of Karama et al. [46], respectively. It can be seen that present solutions agree excellently with other solutions obtained by different methods with various shear deformation theories for a clamped laminated composite plate. The first six mode shapes and natural frequencies with a/b = 1 and b/h = 10 are shown in Figure 5. It can be concluded that the present theory with the IGA performs excellently to solve the free vibration problem of laminated composite plates. 4.3 Buckling analysis The buckling behavior of simply supported symmetric cross-ply laminates under the action of uniaxial in-plane compressive load is investigated. The plate length-to-thickness ratio a/h is fixed to 5. The effect of degree of orthotropy is analyzed on the non-dimensional buckling parameter for three-layer [0°/90°/0°], four-layer [0°/90°/90°/0°],

five-layer

[0°/90°/0°/90°/0°] 24

and

nine-layer

[0°/90°/0°/90°/0°/90°/0°/90°/0°] laminates comprising equal thickness plies. The Material 2 is used. An eigenvalue problem for buckling analysis using the present theory is solved to obtain the critical buckling load, and the non-dimensional form is used to present the results, as:

a2   cr E2 h3

(55)

Table 10 includes the buckling load parameters of the present results and the 3D elastic solutions by Reissner [8] and Noor [124], refined TSDT solutions by Putcha and Reddy [125], solutions using Reddy’s TSDT and collocation with radial basis functions by Ferreira et al. [126], solutions obtained by an inverse hyperbolic shear deformation theory by Grover et al. [43] and Singh and Singh [127], TSSDT and TTSDT solutions by Singh and Singh [32]. Excellent agreements can be seen in Table 10. The present results are very close to the 3D elastic solutions [124]. Additionally, by comparing those solutions with the elastic 3D solutions, the relative errors Diff.% and average errors Diff.*% are calculated and given in Table 10. It can be observed that, present theory shows the best performance with the least average errors. Except for the case of a five layers plate with E1/E2 = 30 and 40, present solutions have the minimum relative errors. Next, for a simply supported four-layer [0°/90°/90°/0°] laminated square plate subjected to the uniaxial compression load, the effect of the length-to-thickness ratios is considered. Material 2 is used in this example. Table 11 summarizes the critical buckling load normalized by Eq. (55) of the present method , and other methods such as the FEM based on TSDT by Reddy and Phan [128] and by Chakrabarti and Sheikh [79], the IGA based on ITTSDT by Thai et al. [35], the IGA based on Reddy’s TSDT [24] by Thai et al. [93], and the IGA based on the LW theory with Karama’s ESDT [109] by Thai et al. [71]. It can be seen that for various plate length-to-thickness ratios, excellent agreements are maintained. Finally, a three-layer [0°/90°/0°] laminated simply supported plate subjected to the biaxial compression load with Material 2 is considered. Various length-to-thickness 25

ratios and elastic modulus ratios are studied in this example. Table 12 and Table 13 show the critical buckling loads normalized by Eq. (55) with respect to various modulus ratios and length-to-thickness ratios, respectively. The obtained results are compared with those of the FEM based on TSDT [113], the analytical method based on TSDT by Fares and Zenkour [129], the mesh-free method based on TSDT by Liu et al. [130], the IGA based on the ITTSDT by Thai et al. [35], the IGA based on Reddy’s TSDT [24] by Thai et al. [93], and the IGA based on the LW theory with Karama’s ESDT [109] by Thai et al. [71]. The present method shows excellent performance compared to other methods and theories for various modulus ratios and length-to-thickness ratios. Therefore, it can be concluded that the present higher order shear deformation theory with the IGA can excellently lead to highly reliable results for the buckling loads of laminated composite plates. 5 Conclusions In the present work, a new hyperbolic tangent higher order shear deformation theory is developed. The proposed shear deformation theory satisfies the transverse shear stress conditions on the top and bottom surfaces of the plate without the requirement of shear correction factor. The theory is assessed with the NURBS based isogeometric analysis for the static, free vibration and buckling analysis of laminated composite plates. Generalized displacements are constructed using the NURBS basis functions that can yield higher order continuity and fulfill easily the requirement of C1 continuity of the higher order shear deformation model. The static, free vibration and buckling analysis are conducted and various numerical examples are presented. Present results are compared with those obtained from various plate models, including exact 3D elasticity, ESL and LW theories, and various methods, such as analytical methods, finite element methods, mesh-free methods and other numerical solutions. It has shown that the present new hyperbolic tangent shear deformation theory incorporating the IGA is highly effective and accurate for the static, free vibration and buckling analysis of laminated composite plates. Additionally, Due to its simplicity and accuracy, it is considered that the present theory can be extended as a standard tool to cover sandwich 26

plates, multilayered shells, higher order LW shear deformation theories and other numerical calculations, such as the finite element methods and mesh-free methods. Also, it is possible that the present theory can be applied to the geometrically nonlinear analysis of laminated plates. However, from the examples of the static analysis it can be observed that there are still considerable differences between 2D and 3D solutions for transverse shear stresses, possibly due to the discontinuity of the shear stress at each layer interfaces. The results may be further improved by considering the continuity of transverse shear stresses between the layer interfaces by combination of the present theory with a LW theory. 6 Acknowledgements The work presented here was supported by Huanghuai University and the National Natural Science Foundation of China (No. 11672038 and No. 61404057). We are thankful to Dr. R.K. Kapania for his suggestions. The authors would also like to thank Yang Wang, Yang Bai and Yanpeng Gong for their assistance with the code debugging.

27

Appendix Information of the IGA meshes modeled by 10×10 cubic elements: knot vectors and weights:     {0, 0, 0, 0, 1 10, 2 10, 3 10, ... , 9 10, 1, 1, 1, 1}.

(A1)

11  12  13  ...  1313  1.

(A2)

Coordinates of control points Pij can be seen in Table A1. Table A1 Coordinates of control points Pij of 10×10 IGA meshes i

1

j

2

3

4

5

…

11

12

13

1

(0,0)

(a/30,0)

(a/10,0)

(2a/10,0)

(3a/10,0)

…

(9a/10,0)

(29a/30,0)

(a,0)

2

(0,b/30)

(a/30,b/30)

(a/10,b/30)

(2a/10,b/30)

(3a/10,b/30)

…

(9a/10,b/30)

(29a/30,b/30)

(a,b/30)

3

(0,b/10)

(a/30,b/10)

(a/10,b/10)

(2a/10,b/10)

(3a/10,b/10)

…

(9a/10,b/10)

(29a/30,b/10)

(a,b/10)

4

(0,2b/10)

(a/30,2b/10)

(a/10,2b/10)

(2a/10,2b/10)

(3a/10,2b/10)

…

(9a/10,2b/10)

(29a/30,2b/10)

(a,2b/10)

5

(0,3b/10)

(a/30,3b/10)

(a/10,3b/10)

(2a/10,3b/10)

(3a/10,3b/10)

…

(9a/10,3b/10)

(29a/30,3b/10)

(a,3b/10)

…

…

…

…

…

…

…

…

…

…

11

(0,9b/10)

(a/30,9b/10)

(a/10,9b/10)

(2a/10,9b/10)

(3a/10,9b/10)

…

(9a/10,9b/10)

(29a/30,9b/10)

(a,9b/10)

12

(0,29b/30)

(a/30,29b/30)

(a/10,29b/30)

(2a/10,29b/30)

(3a/10,29b/30)

…

(9a/10,29b/30)

(29a/30,29b/30)

(a,29b/30)

13

(0,b)

(a/30,b)

(a/10,b)

(2a/10,b)

(3a/10,b)

…

(9a/10,b)

(29a/30,b)

(a,b)

Information of the IGA meshes modeled by 18×18 cubic elements: knot vectors and weights:     {0, 0, 0, 0, 1 18, 2 18, 3 18, ... , 17 18, 1, 1, 1, 1}.

(A3)

11  12  13  ...  2121  1.

(A4)

Coordinates of control points Pij can be seen in Table A2. Table A2 Coordinates of control points Pij of 18×18 IGA meshes i

1

j

2

3

4

5

…

19

20

21

1

(0,0)

(a/54,0)

(a/18,0)

(2a/18,0)

(3a/18,0)

…

(17a/18,0)

(53a/54,0)

(a,0)

2

(0,b/54)

(a/54,b/54)

(a/18,b/54)

(2a/18,b/54)

(3a/18,b/54)

…

(17a/18,b/54)

(53a/54,b/54)

(a,b/54)

3

(0,b/18)

(a/54,b/18)

(a/18,b/18)

(2a/18,b/18)

(3a/18,b/18)

…

(17a/18,b/18)

(53a/54,b/18)

(a,b/18)

4

(0,2b/18)

(a/54,2b/18)

(a/18,2b/18)

(2a/18,2b/18)

(3a/18,2b/18)

…

(17a/18,2b/18)

(53a/54,2b/18)

(a,2b/18)

5

(0,3b/18)

(a/54,3b/18)

(a/18,3b/18)

(2a/18,3b/18)

(3a/18,3b/18)

…

(17a/18,3b/18)

(53a/54,3b/18)

(a,3b/18)

…

…

…

…

…

19

(0,17b/18)

(a/54,17b/18)

(a/18,17b/18)

(2a/18,17b/18)

20

(0,53b/54)

(a/54,53b/54)

(a/18,53b/54)

21

(0,b)

(a/54,b)

(a/18,b)

…

…

…

…

…

(3a/18,17b/18)

…

(17a/18,17b/18)

(53a/54,17b/18)

(a,17b/18)

(2a/18,53b/54)

(3a/18,53b/54)

…

(17a/18,53b/54)

(53a/54,53b/54)

(a,53b/54)

(2a/18,b)

(3a/18,b)

…

(17a/18,b)

(53a/54,b)

(a,b)

28

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Table 1: Several shear functions proposed in the literature. Source Ambartsumian [16] (1958)

Shear functions

z  h2 z 2  f  z     2 4 3 

Kaczkowski [17] (1968) Panc [18] (1975)

f  z 

5z  4 z 2  1   4  3h 2 

Reissner [19] (1975) Levinson [22] (1980) Murthy [23] (1981)

 4z2  f  z   z 1  2   3h 

Reddy [24] (1984) Levy [29] (1877) Stein [30] (1986)

f  z 

 sin   h

 z 

h

Touratier [31] (1991) Soldatos [38] (1992)

z 1 f  z   h sinh    z cosh   h 2

Arya et al. [68] (2002)

  f  z   sin  z  h 

Karama et al. [46] (2003)

f  z   ze2 z / h

Akavci and Tanrikulu [44] (2008)

f  z 

Akavci and Tanrikulu [44] (2008)

  z2         f  z   z sec h  2   z sec h   1  tanh     4  2  4   h 

Aydogdu [48] (2009)

f  z   z 2 z / h

Mantari et al. [49] (2011)

f  z   2.852 z / h z

El Meiche et al. [42] (2011)

  sinh  z   z  h  f  z    cosh    1 2

2

3 h  z  3 z 1 tanh    sec h 2   2 h 2 2

2

ln 

,   0

2

h

42

 

Mantari et al. [106] (2012)

   cos z   f  z   sin  z  e 2  h   z 2h h 

Mantari et al. [34] (2012)

1  mh  f  z   tan  mz   mz sec2  , m  5h  2 

Grover et al. [43] (2013)

2rz  rz  f  z   sinh 1    , r 3  h  h r2  4

Grover et al. [33] (2013)

 rz   r  r  r  f  z   z sec    z sec   1  tan    , r  0.1 h  2  2  2 

Grover et al. [33] (2013)

4rz  rh  f  z   cot 1    , r  0.46  z  h 4r 2  1

Thai et al. [35] (2013)

2  f  z   h arctan  z   z h 

Nguyen-Xuan et al. [92] (2013)

f  z 

Mahi et al. [45] (2015)

h 4z3  2z  f  z   tanh    2 2 2  h  3h cosh (1)

Suganyadevi and Singh [36] (2016)

f  z 

Singh and Singh [32] (2017)

 mz  1 f  z   tan    2 z cosh   , m  5  h  2

Singh and Singh [32] (2017)

 f  z   sin  h

1

43

7 2 2 z  2 z3  4 z5 8 h h

h 1  rz  z tan    2 2 2 , r  2.5 r  h  r z h  1

  z z  2h

Table 2: Convergence study of the normalized displacement and stresses of a simply supported square [0°/90°/90°/0°] laminated plate (a/h = 4) under a sinusoidal load. Source

w

Diff.%

 xx

Diff.%

 yy

Diff.%

 xy

Diff.%

Diff.%

0.2910

 xz

Diff.%

Pagano [77]

1.9540

0.7200

0.6660

Reddy [24]

1.8939 3.08 0.6806

5.47 0.6463

2.96 0.0450 3.64

0.2390 17.87 0.2109 21.89

Mantari et al. [34]

1.8940 3.07 0.6640

7.78 0.6310

5.26 0.0440 5.78

0.2390 17.87 0.2060 23.70

Mantari et al [49]

1.9210 1.69 0.7140

0.83 0.6360

4.50 0.0470 0.64

0.2550 12.37 0.2300 14.81

Singh and Singh [32]

1.9088 2.31 0.7204

0.06 0.6370

4.35 0.0473 1.28

0.2800

Suganyadevi and Singh [36]

1.9244 1.51 0.7303

1.43 0.6385

4.13 0.0475 1.71

0.2529 13.09 0.2710

0.37

Grover et al. [33]

1.9262 1.42 0.7210

0.14 0.6386

4.11 0.0471 0.86

0.2442 16.08 0.2654

1.70

Karama et al. [109]

1.9193 1.78 0.7004

2.72 0.6367

4.40 0.0459 1.71

0.2532 12.99 0.2264 16.15

Touratier [31]

1.9088 2.31 0.6830

5.14 0.6349

4.67 0.0450 3.64

0.2462 15.40 0.2162 19.93

Ferreira [65]

1.9075 2.38 0.6432 10.67 0.6228

6.49 0.0441 5.57

—

—

0.2166 19.78

Ferreira et al. [110]

1.9091 2.30 0.6429 10.71 0.6265

5.93 0.0443 5.14

—

—

0.2173 19.52

Wang and Shi [111]

1.9073 2.39 0.7361

2.24 0.6994

5.02 0.0435 6.85

—

—

0.2110 21.85

Thai et al. [71]

1.9152 1.99 0.7565

5.07 0.6765

1.58 0.0452 3.21

—

—

0.2385 11.67

Thai et al. [35]

1.9258 1.44 0.7164

0.50 0.6381

4.19 0.0467 0.00

0.2624

9.83 0.2396 11.26

6×6 IGA Meshes

1.9222 1.63 0.6590

8.47 0.5770 13.36 0.0465 0.43

0.2238 23.09 0.2340 13.33

10×10 IGA Meshes

1.9233 1.57 0.6926

3.81 0.6158

7.54 0.0464 0.64

0.2453 15.70 0.2330 13.70

14×14 IGA Meshes

1.9234 1.57 0.7003

2.74 0.6266

5.92 0.0464 0.64

0.2513 13.64 0.2329 13.74

18×18 IGA Meshes

1.9234 1.57 0.7034

2.31 0.6310

5.26 0.0464 0.64

0.2538 12.78 0.2328 13.78

22×22 IGA Meshes

1.9234 1.57 0.7050

2.08 0.6322

5.08 0.0464 0.64

0.2551 12.34 0.2328 13.78

44

0.0467

 yz

0.2700

3.78 0.2318 14.15

Table 3: The normalized displacement and stresses of a simply supported square [0°/90°/90°/0°] laminated plate under a sinusoidal load. Source

a/h

w

Diff.%

 xx

Diff.%

 yy

Diff.%

 xy

Diff.%

 yz

Diff.%

 xz

Diff.%

Pagano [77]

1.9540

0.7200

0.6660

0.0467

0.2910

0.2700

Reddy [24]

1.8939

3.08 0.6806

5.47 0.6463

2.96 0.0450

3.64 0.2390

17.87 0.2109

21.89

Mantari et al. [34]

1.8940

3.07 0.6640

7.78 0.6310

5.26 0.0440

5.78 0.2390

17.87 0.2060

23.70

Singh and Singh [32]

1.9088

2.31 0.7204

0.06 0.6370

4.35 0.0473

1.28 0.2800

3.78 0.2318

14.15

Ferreira et al. [110]

1.9091

2.30 0.6429

10.71 0.6265

5.93 0.0443

5.14

—

—

Wang and Shi [111]

1.9073

2.39 0.7361

2.24 0.6994

5.02 0.0435

6.85

—

Thai et al. [71]

1.9152

1.99 0.7565

5.07 0.6765

1.58 0.0452

3.21

—

Thai et al. [35]

1.9258

1.44 0.7164

0.50 0.6381

4.19 0.0467

Present

1.9234

1.57 0.7034

2.31 0.6310

Pagano [77]

0.7430

0.5590

0.4030

Reddy [24]

0.7149

3.78 0.5589

Mantari et al. [34]

0.7150

3.77 0.5450

Singh and Singh [32] 10 Ferreira et al. [110]

0.7224 0.7303

Wang and Shi [111] Thai et al. [71]

4

0.2173

19.52

—

0.2110

21.85

—

0.2385

11.67

0.00 0.2624

9.83 0.2396

11.26

5.26 0.0464

0.64 0.2538

12.78 0.2328

13.78

0.0276

0.1960

0.3010

0.02 0.3974

1.39 0.0273

1.09 0.1530

21.94 0.2697

10.40

2.50 0.3880

3.72 0.0270

2.17 0.1530

21.94 0.2640

12.29

2.77 0.5608

0.32 0.3880

3.72 0.0278

0.72 0.1860

5.10 0.3118

3.59

1.71 0.5487

1.84 0.3966

1.59 0.0273

1.09

—

—

0.2993

0.56

0.7368

0.83 0.5609

0.34 0.4077

1.17 0.0274

0.72

—

—

0.3002

0.27

0.7358

0.97 0.5642

0.93 0.4034

0.10 0.0275

0.36

—

—

0.3238

7.57

Thai et al. [35]

0.7272

2.13 0.5552

0.68 0.3937

2.31 0.0273

1.09 0.1704

13.06 0.3133

4.09

Present

0.7258

2.31 0.5479

1.99 0.3888

3.52 0.0273

1.09 0.1643

16.17 0.3035

0.83

Pagano [77]

0.5170

0.5430

0.3090

0.0230

0.1560

0.3280

Reddy [24]

0.5061

2.11 0.5523

1.71 0.3110

0.65 0.0233

1.30 0.1230

21.15 0.2883

12.10

Mantari et al. [34]

0.5070

1.93 0.5390

0.74 0.3040

1.62 0.0230

0.00 0.1230

21.15 0.2820

14.02

Singh and Singh [32] 20 Ferreira et al. [110]

0.5091

1.53 0.5463

0.61 0.3062

0.91 0.0233

1.30 0.1650

5.77 0.3355

2.29

0.5113

1.10 0.5407

0.42 0.3073

0.55 0.0230

0.00

—

—

0.3256

0.73

Wang and Shi [111]

0.5138

0.62 0.5433

0.06 0.3098

0.26 0.0231

0.43

—

—

0.3279

0.03

Thai et al. [71]

0.5128

0.81 0.5437

0.13 0.3085

0.16 0.0230

0.00

—

—

0.3517

7.23

Thai et al. [35]

0.5098

1.39 0.5412

0.33 0.3058

1.04 0.0229

0.43 0.1366

12.44 0.3372

2.80

Present

0.5093

1.49 0.5348

1.51 0.3021

2.23 0.0229

0.43 0.1317

15.58 0.3264

0.49

Pagano [77]

0.4347

0.5390

0.2710

0.0214

0.1410

0.3390

Reddy [24]

0.4343

0.09 0.5507

2.17 0.2769

2.18 0.0217

1.40 0.1120

20.57 0.2948

13.04

Mantari et al. [34]

0.4350

0.07

0.539

0.00 0.2710

0.00 0.0210

1.87 0.1120

20.57 0.2890

14.75

0.4341

0.14 0.5422

0.59 0.2690

0.74 0.0215

0.47 0.1249

11.42 0.3162

6.73

0.4348

0.02 0.5391

0.02 0.2711

0.04 0.0214

0.00

—

—

0.3359

0.91

Wang and Shi [111]

0.4355

0.18 0.5387

0.06 0.2710

0.00 0.0214

0.00

—

—

0.3389

0.03

Thai et al. [71]

0.4346

0.02 0.5381

0.17 0.2706

0.15 0.0214

0.00

—

—

0.3627

6.99

Thai et al. [35]

0.4345

0.05 0.5380

0.19 0.2705

0.18 0.0213

0.47 0.1229

12.84 0.3467

2.27

Present

0.4344

0.07 0.5320

1.30 0.2675

1.29 0.0214

0.00 0.1186

15.89 0.3355

1.03

Singh and Singh [32] 100 Ferreira et al. [110]

45

Table 4: The normalized displacement and stresses of a simply supported rectangular [0°/90°/0°] laminated plate (b/a = 3) under a sinusoidal load. Source

a/h

Pagano [77]

4

10

20

w

Diff.%

2.8200

 xx

Diff.%

1.1000

 yy 0.1190

Diff.%

 xy

Diff.%

0.0281

 yz 0.0334

Diff.%

 xz

Diff.%Diff.*%

0.3870

Reddy [24]

2.6411 6.34 1.0356 5.85 0.1028 13.61 0.0263 6.41 0.0348 4.19 0.2724 29.61 11.00

Mantari et al. [49]

2.6841 4.82 1.1180 1.64 0.1030 13.45 0.0274 2.49 0.0360 7.78 0.3020 21.96

8.69

Karama et al. [109]

2.6838 4.83 1.0974 0.24 0.1038 12.77 0.0272 3.20 0.0360 7.78 0.2982 22.95

8.63

Touratier [31]

2.6660 5.46 1.0670 3.00 0.1030 13.45 0.0268 4.63 0.0355 6.29 0.2850 26.36

9.86

Present

2.6936 4.48 1.1034 0.31 0.1028 13.61 0.0274 2.49 0.0358 7.19 0.3068 20.72

8.13

Pagano [77]

0.9190

Reddy [24]

0.8622 6.18 0.6924 4.50 0.0398 8.51 0.0115 6.50 0.0170 11.84 0.2859 31.93 11.58

Mantari et al. [49]

0.8800 4.24 0.7080 2.34 0.0400 8.05 0.0118 4.07 0.0180 18.42 0.3260 22.38

9.92

Karama et al. [109]

0.8768 4.59 0.7043 2.86 0.0403 7.36 0.0117 4.88 0.0175 15.13 0.3194 23.95

9.79

Touratier [31]

0.8700 5.33 0.6980 3.72 0.0401 7.82 0.0116 5.69 0.0172 13.16 0.3020 28.10 10.64

Present

0.8808 4.16 0.6993 3.54 0.0400 8.05 0.0118 4.07 0.0174 14.47 0.3306 21.29

Pagano [77]

0.6100

Reddy [24]

0.5937 2.67 0.6407 1.43 0.0289 3.34 0.0091 2.15 0.0139 16.81 0.2880 33.64 10.01

Mantari et al. [49]

0.5994 1.74 0.6450 0.77 0.0290 3.01 0.0092 1.08

0.014 17.65 0.3290 24.19

8.07

Karama et al. [109]

0.5977 2.02 0.6437 0.97 0.0290 3.01 0.0092 1.08 0.0142 19.33 0.3227 25.65

8.67

Touratier [31]

0.5960 2.30 0.6420 1.23 0.0290 3.01 0.0091 2.15 0.0141 18.49 0.3050 29.72

9.48

Present

0.5988 1.84 0.6366 2.06 0.0287 4.01 0.0092 1.08 0.0142 19.33 0.3344 22.95

8.54

Pagano [77]

0.5200

50 Mantari et al. [49]

0.7250

0.6500

0.6280

0.0435

0.0299

0.0259

0.0123

0.0093

0.0084

0.0152

0.0119

0.0110

0.4200

9.26

0.4340

0.4390

0.5198 0.04 0.6270 0.16 0.0260 0.39 0.0084 0.00 0.0130 18.18 0.3300 24.83

7.27

Present

0.5187 0.25 0.6188 1.46 0.0254 1.93 0.0084 0.00 0.0132 20.00 0.3355 23.58

7.87

Pagano [77]

0.5080

Reddy [24]

0.5070 0.20 0.6240 0.00 0.0253 0.00 0.0083 0.00 0.0129 19.44 0.2886 34.26

8.98

0.5083 0.06 0.6240 0.00

100 Mantari et al. [49]

0.6240

0.0253

0.0083

0.0108

0.4390

0.025 1.19 0.0083 0.00 0.0130 20.37 0.3310 24.60

7.70

Touratier [31]

0.5070 0.20 0.6240 0.00 0.0253 0.00 0.0083 0.00 0.0131 21.30 0.3060 30.30

8.63

Present

0.5072 0.16 0.6162 1.25 0.0250 1.19 0.0083 0.00 0.0131 21.30 0.3356 23.55

7.91

46

Table 5: Convergence study of the normalized natural frequencies of a simply supported square [0°/90°/90°/0°] laminated plate (a/h = 5). Mode 1 2 3 4 5 6 7 8 9 10

6×6 10.8262 12.1673 12.1673 17.9449 22.5258 24.3350 24.3350 26.7298 27.7752 34.1720

IGA meshes 10×10 14×14 10.8248 10.8246 12.1673 12.1673 12.1673 12.1673 17.9038 17.8999 22.3418 22.3231 24.3347 24.3347 24.3347 24.3347 26.5489 26.5308 27.0408 26.9640 33.4555 33.3812

47

18×18 10.8246 12.1673 12.1673 17.8991 22.3192 24.3347 24.3347 26.5270 26.9488 33.3664

Exact 3D [78] 10.7520

Table 6: The normalized first natural frequencies of a simply supported square [0°/90°/90°/0°] laminated plate (a/h = 5) with various modulus ratios. E1/E2

Source 3

Diff.%

10

Diff.%

8.2103

20

Diff.%

30

Diff.%

40

Diff.%

Noor [78]

6.6815

9.5603

10.2720

10.7520

Reddy [2]

—

—

8.2982

1.07

9.5671

0.07 10.3260 0.53

10.8540

0.95

Ferreira et al. [115]

—

—

8.2793

0.84

9.5375

0.24 10.2889 0.16

10.8117

0.56

Liew et al. [114]

—

—

8.2924

1.00

9.5613

0.01 10.3200 0.47

10.8490

0.90

Phan and Reddy [112]

6.5597

1.82

8.2718

0.75

9.5263

0.36 10.2720 0.00

10.7870

0.33

Singh and Singh [32] TSSDT

6.5872

1.41

8.2672

0.69

9.5286

0.33 10.2514 0.20

10.7674

0.14

Singh and Singh [32] TTSDT

6.5896

1.38

8.2658

0.68

9.5348

0.27 10.2546 0.17

10.7352

0.16

Mantari et al. [49]

6.5650

1.74

8.2860

0.92

9.5520

0.09 10.3050 0.32

10.8260

0.69

Khdeir and Librescu [113]

—

—

8.2940

1.02

9.5439

0.17 10.2840 0.12

10.8530

0.94

Rodrigues et al. [116]

—

—

8.4142

2.48

9.6629

1.07 10.4013 1.26

10.9054

1.43

Chalak et al. [117]

—

—

8.3456

1.65

9.5703

0.10 10.2976 0.25

10.7984

0.43

Thai et al. [35]

—

—

8.2944

1.02

9.5650

0.05 10.3206 0.47

10.8428

0.84

Nguyen-Xuan et al. [92]

—

—

8.2979

1.07

9.5717

0.12 10.3305 0.57

10.8557

0.96

—

—

8.2792

0.84

9.5454

0.16 10.2308 0.40

10.7329

0.18

6.5648

1.75

8.2865

0.93

9.5521

0.09 10.3045 0.32

10.8246

0.68

Thai et al. [71] Present

48

Table 7: The normalized first natural frequencies of a simply supported square [0°/90°/90°/0°] laminated plate (E1/E2 = 40) with various length-to-thickness ratios. Source

a/h 2

4

Cho et al. [120]

5.9230

—

10.6730 15.0660 17.5350 18.0540 18.6700 18.8350

Matsunaga [119]

5.3211

9.1988

10.6876 15.0721 17.6369 18.0557 18.6702 18.8352

Zhen and Wanji [118]

5.4300

9.2406

10.7294 15.1658 17.8035 18.2404 18.9022 19.1566

Thai et al. [35]

—

9.3781

10.8428 15.1552 17.6677 18.0766 18.6760 18.8367

Nguyen-Xuan et al. [92]

—

9.3937

10.8557 15.1612 17.6697 18.0780 18.6763 18.8368

Thai et al. [71]

—

9.3757

10.8298 15.1249 17.6521 18.0657 18.6728 18.8359

5.5774

9.3606

10.8246 15.1397 17.6609 18.0720 18.6746 18.8364

Present

5

49

10

20

25

50

100

Table 8: The normalized first six natural frequencies of a clamped square [0°/90°/90°/0°] laminated plate with various length-to-thickness ratios. a/h

5

10

20

Mode

Chalak et al. [117]

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

11.7013 18.8459 19.5542 24.6992 28.3035 29.2470 17.9487 28.9435 32.6460 40.2419 45.2807 50.7245 23.9124 36.9889 49.8195 58.2399 59.7848 75.7765

Kulkarni and Kapuria [121] TSDT ESL 11.9516 19.0208 20.3969 25.4967 28.5190 30.9182 18.2744 28.9047 33.8184 41.0769 44.9767 52.7390 24.1130 36.7473 51.1651 59.1384 59.0253 75.9905

50

Kulkarni and Kapuria [121] TSDT ZZ 12.0965 19.3528 21.1266 26.2179 29.0997 33.1200 18.1118 29.0729 33.5629 41.0151 45.6649 52.7698 23.8689 36.6889 50.2908 58.4074 59.3371 75.6414

Present 12.5049 19.9816 22.0378 27.2460 30.1630 34.2258 18.7170 29.5857 35.5372 42.8360 46.3139 56.0660 24.3218 37.0512 52.3800 59.6290 60.4346 77.4942

Table 9: The normalized first natural frequencies of a clamped [0°/90°/0°] laminated plate with various length-to-width and length-to-thickness ratios. a/b b/h Liew [122] Zhen and Wanji [118] Shi et al. [123] Thai et al. [71] TSDT Thai et al. [71] TSSDT Thai et al. [71] ESDT Present

10

1 20

100

7.4110 7.4840 7.4510 7.7082 7.7308 7.7615 7.7286

10.9530 11.0030 11.0150 11.1029 11.1123 11.1281 10.9405

14.6660 14.6010 14.4830 14.4386 14.4389 14.4399 14.0024

51

10

2 20

100

4.1410 4.1190 4.1640 4.1489 4.1420 4.1365 4.0423

4.7790 4.8130 4.8380 4.7802 4.7778 4.7760 4.6407

5.1050 5.1440 5.2500 5.0902 5.0901 5.0901 4.9327

Table 10: The normalized buckling loads of simply supported square plates with various laminates and modulus ratios. Layers

3

4

5

9

Source

E1/E2 3

Diff.%

10

Diff.%

9.7621

20

Diff.%

15.0191

30

Diff.%

Diff.%

Diff.*%

Noor [124]

5.3044

Putcha and Reddy [125]

5.3933

1.68

9.9406

1.83

15.2980

1.86

19.6740

1.92

23.3400

2.01

1.86

Ferreira et al. [126]

5.3872

1.56

9.8331

0.73

14.8975

0.81

19.8942

3.06

22.1513

3.19

0.27

Grover et al. [43]

5.3949

1.71

9.8503

0.90

14.9415

0.52

18.9750

1.70

22.2700

2.67

0.46

Singh and Singh [127]

5.3920

1.65

9.8368

0.77

14.9001

0.79

17.5089

9.30

19.7186

13.82

4.30

Singh and Singh [32] TSSDT

5.4002

1.81

9.8771

1.18

14.9984

0.14

17.8442

7.56

20.2231

11.62

3.27

Singh and Singh [32] TTSDT

5.4121

2.03

9.9115

1.53

15.0018

0.12

17.6452

8.59

20.8743

8.77

2.78

Present

5.3523

0.90

9.7784

0.17

14.8288

1.27

18.8238

2.49

23.2980

1.82

0.17

Reissner [8]

5.2944

Grover et al. [43]

5.4002

2.00

9.9740

2.17

15.3969

2.52

19.8413

2.78

23.5790

3.05

2.50

Singh and Singh [127]

5.3987

1.97

9.9543

1.97

15.3240

2.03

19.7062

2.08

23.3700

2.14

2.04

Singh and Singh [32] TSSDT

5.4060

2.11

9.9811

2.24

15.3916

2.48

19.8249

2.70

23.5475

2.91

2.49

Singh and Singh [32] TTSDT

5.4125

2.23

10.0126

2.57

15.4852

3.10

19.9974

3.59

23.8725

4.33

3.17

Present

5.3567

1.18

9.8943

1.35

15.2589

1.60

19.6539

1.81

23.3436

2.02

1.59

Noor [124]

5.3255

Putcha and Reddy [125]

5.4096

1.58

10.1500

1.90

16.0080

2.27

20.9990

2.60

25.3080

2.91

2.25

Ferreira et al. [126]

5.4041

1.48

10.0890

1.29

15.7913

0.89

20.5914

0.61

24.6901

0.40

0.93

Grover et al. [43]

5.4163

1.71

10.1390

1.79

15.9287

1.76

20.8263

1.76

25.0344

1.80

1.76

Singh and Singh [127]

5.4127

1.64

10.0988

1.39

15.7918

0.89

20.5644

0.48

24.6324

0.16

0.91

Singh and Singh [32] TSSDT

5.4174

1.73

10.1168

1.57

15.8449

1.23

20.6700

1.00

24.8053

0.86

1.28

Singh and Singh [32] TTSDT

5.4280

1.92

10.2108

2.51

16.0410

2.48

20.9958

2.59

25.4594

3.52

2.61

Present

5.3716

0.87

10.0525

0.93

15.7734

0.77

20.6017

0.66

24.7431

0.61

0.77

Noor [124]

5.3352

Putcha and Reddy [125]

5.4313

1.80

10.1970

1.55

16.1720

1.61

21.3150

1.69

25.7900

1.76

1.68

Ferreira et al. [126]

5.4092

1.39

10.1767

1.34

16.1063

1.20

21.1918

1.10

25.6088

1.05

1.22

Grover et al. [43]

5.4202

1.59

10.2100

1.68

16.1911

1.73

21.3413

1.81

25.8298

1.92

1.75

Singh and Singh [127]

5.4176

1.54

10.1854

1.43

16.1093

1.22

21.1838

1.06

25.5847

0.95

1.24

Singh and Singh [32] TSSDT

5.4226

1.64

10.2045

1.62

16.1602

1.54

21.2753

1.50

25.7218

1.49

1.56

Singh and Singh [32] TTSDT

5.4417

2.00

10.2087

1.66

16.2580

2.15

21.5299

2.71

26.2531

3.59

2.42

Present

5.3762

0.77

10.1300

0.88

16.0556

0.88

21.1499

0.90

25.5828

0.94

0.87

9.7621

19.3040

40

15.0191

9.9603

19.3040

15.6527

10.0417

22.8807

20.4663

15.9153

52

22.8807

24.5929

20.9614

25.3436

Table 11: The normalized buckling loads of a simply supported square [0°/90°/90°/0°] laminated plate with various length-to-thickness ratios. Source Reddy and Phan [128] Chakrabarti and Sheikh [79] Thai et al. [35] Thai et al. [93] Thai et al. [71] Present

5

10

a/h 20

12.4440 11.5120 — — — 12.0515

23.8490 23.4280 23.3152 23.2082 23.4801 23.3436

31.7370 31.6590 31.6975 31.6325 31.7253 31.6099

53

50

100

35.1000 35.2970 35.3595 35.3497 35.3601 35.3260

35.6450 35.8940 35.9565 35.9612 35.9562 35.9466

Table 12: The normalized biaxial buckling loads of a simply supported square [0°/90°/0°] laminated plate with various modulus ratios. Source Khdeir and Librescu [113] Fares and Zenkour [129] Thai et al. [35] Thai et al. [93] Thai et al. [71] Present

2

10

E1/E2 20

2.3640 2.3430 — — — 2.3349

4.9630 4.9160 4.9130 4.9766 4.9145 4.9040

5.5160 7.4490 7.4408 7.5429 7.4418 7.4332

54

30

40

9.0560 8.8200 8.7550 8.9383 8.8473 8.7649

10.2590 9.9750 9.8795 10.1046 9.9626 9.8998

Table 13: The normalized biaxial buckling loads of a simply supported square [0°/90°/0°] laminated plate with various length-to-thickness ratios. Source Khdeir and Librescu [113] Liu et al. [130] Thai et al. [35] Thai et al. [93] Thai et al. [71] Present

2

5

a/h 10

1.4650 1.4570 1.4316 1.4399 1.5539 1.5496

5.5260 5.5190 5.3236 5.4465 5.4708 5.4018

10.2590 10.2510 9.8795 10.1046 9.9626 9.8998

55

15

20

12.2260 12.2390 11.9978 12.2570 12.0412 12.0006

13.1850 13.1640 13.0239 13.3037 13.0419 13.0157

Figure 1: Geometry of a laminated composite plate.

56

0.5 0.4 0.3 0.2

z/h

0.1 Present Reddy [24] Touratier [31] Soldatos [38] Karama et al. [46] Thai et al. [35] Mantari et al. [49] Nguyen-Xuan et al. [92] Grover et al. [43]

0.0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

f (z) (a) Shear functions through plate thickness.

0.5 0.4 0.3 0.2

z/h

0.1 Present Reddy [24] Touratier [31] Soldatos [38] Karama et al. [46] Thai et al. [35] Mantari et al. [49] Nguyen-Xuan et al. [92] Grover et al. [43]

0.0 -0.1 -0.2 -0.3 -0.4

-0.5 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

df (z)/dz (b) Derivatives of shear functions through plate thickness. Figure 2: Several shear functions and their derivatives in the literature.

57

(a) 10×10

(b) 18×18 Figure 3: IGA meshes and control net of a simply supported square plate using cubic 58

elements. ○:control points, □: simply supported control points.

59

0.50

z/h

0.25

0.00

Present a/h = 4 Present a/h = 10 Present a/h = 100 Reddy a/h = 4 Reddy a/h = 10 Reddy a/h = 100 Touratier a/h = 4 Touratier a/h = 10 Touratier a/h = 100

-0.25

-0.50 -0.8

-0.6

-0.4

-0.2

0.0

xx

0.2

0.4

0.6

0.8

(a) Distribution of  xx through plate thickness.

0.50

z/h

0.25

0.00

Present a/h = 4 Present a/h = 10 Present a/h = 100 Reddy a/h = 4 Reddy a/h = 10 Reddy a/h = 100 Touratier a/h = 4 Touratier a/h = 10 Touratier a/h = 100

-0.25

-0.50 -0.8

-0.6

-0.4

-0.2

0.0

yy

0.2

0.4

0.6

(b) Distribution of  yy through plate thickness. 60

0.8

0.50 Present a/h = 4 Present a/h = 10 Present a/h = 100 Reddy a/h = 4 Reddy a/h = 10 Reddy a/h = 100 Touratier a/h = 4 Touratier a/h = 10 Touratier a/h = 100

z/h

0.25

0.00

-0.25

-0.50 -0.06

-0.04

-0.02

0.00

xy

0.02

0.04

0.06

(c) Distribution of  xy through plate thickness.

0.50

z/h

0.25 Present a/h = 4 Present a/h = 10 Present a/h = 100 Reddy a/h = 4 Reddy a/h = 10 Reddy a/h = 100 Touratier a/h = 4 Touratier a/h = 10 Touratier a/h = 100

0.00

-0.25

-0.50 0.00

0.05

0.10

0.15

yz

0.20

0.25

(d) Distribution of  yz through plate thickness. 61

0.30

0.50

z/h

0.25 Present a/h = 4 Present a/h = 10 Present a/h = 100 Reddy a/h = 4 Reddy a/h = 10 Reddy a/h = 100 Touratier a/h = 4 Touratier a/h = 10 Touratier a/h = 100

0.00

-0.25

-0.50 0.0

0.1

0.2

0.3

xz

0.4

0.5

0.6

0.7

(e) Distribution of  xz through plate thickness. Figure 4: Distribution of stresses through the thickness of a simply supported square [0°/90°/90°/0°] laminated plate under a sinusoidal load.

62

(a) 1  7.7286

(b) 2  10.4193

(c) 3  15.1095

(d) 4  15.4191

(e) 5  17.0237

(f) 6  20.3299 Figure 5: First six mode shapes and normalized natural frequencies of a clamped 63

square [0°/90°/0°] laminated plate.

64