Composite Structures xxx (2015) xxx–xxx

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Composite Structures

journal homepage: www.elsevier .com/locate /compstruct

Analysis of composite plates using a layerwise theory and a differentialquadrature finite element method

http://dx.doi.org/10.1016/j.compstruct.2015.07.1010263-8223/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: The Solid Mechanics Research Centre, BeihangUniversity (BUAA), Beijing 100191, China.

E-mail address: liubo68@buaa.edu.cn (B. Liu).

Please cite this article in press as: Liu B et al. Analysis of composite plates using a layerwise theory and a differential quadrature finite element mCompos Struct (2015), http://dx.doi.org/10.1016/j.compstruct.2015.07.101

Bo Liu a,b,⇑, A.J.M. Ferreira b, Y.F. Xing a, A.M.A. Neves b

a The Solid Mechanics Research Centre, Beihang University (BUAA), Beijing 100191, Chinab Departamento de Engenharia Mecânica, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal

a r t i c l e i n f o

Article history:Available online xxxx

Keywords:Composite plateLayerwise theoryDifferential quadrature finite elementmethodBendingHigh accuracy

a b s t r a c t

A layerwise shear deformation theory for composite laminated plates is discretized using a differentialquadrature finite element method (DQFEM). The DQFEM is a weak-form differential quadrature methodthat can provide highly accurate results using only a few sampling points. The layerwise theory proposedby Ferreira is based on an expansion of Mindlin’s first-order shear deformation theory in each layer andresults for a laminated plate with three layers were presented as example in the original paper. This workgeneralized the layerwise theory to plates with any number of layers. The combination of the DQFEMwith Ferreira’s layerwise theory allows a very accurate prediction of the field variables. Laminated com-posite and sandwich plates were analyzed. The DQFEM solutions were compared with various models inliterature and especially showed very good agreements with the exact solutions in literature that wasbased on a similar layerwise theory. The analysis of composite plates based on Ferreira’s layerwise theoryindicates that the DQFEM is an effective method for high accuracy analysis of large-scale problems.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Composite and sandwich plates are one of the most significantapplications of composite materials in industry. Layers are stackedtogether to form thin or thick laminates. When the main emphasisof the analysis is to determine the global response of the laminatedcomponent, equivalent single-layer laminate theories (ESL theo-ries) [1,2] is accurate enough. The classical laminate plate theory,the first-order shear deformation theory [3–7], and higher-ordertheories [8] are commonly used examples of simple ESL theories.In some cases, particularly in sandwich applications, the differencebetween material properties makes it difficult for such theories tofully accommodate the bending behavior. Another set of theoriesthat was introduced back in the 1980s are the layerwise theories,which consider independent degrees of freedom for each layer[9–16]. Layerwise displacement fields provide a much more correctrepresentation of the moderate to severe cross-sectional warpingassociated with the deformation of thick laminates [2]. A veryrecent and comprehensive review of such theories in the analysisof multilayered plates and shells has been presented by Carrera[17]. This work adopts Ferreira’s layerwise theory [1] that is based

on an expansion of Mindlin’s first-order shear deformation theoryin each layer. The displacement continuity at layer’s interface isguaranteed. Also the theory directly produces very accurate trans-verse shear stress, although constant, in each layer.

Most of the spacial discretization techniques thus far have beenbased on finite differences (FDM) and finite elements (FEM). Suchlow order schemes typically use low order basis functions andthe accuracy is improved through mesh refinement. High orderschemes like the hierarchical finite element method (HFEM)[18,19], the radial basis functions (RBFs) [20,21], the mesh freemethods [22,23], the differential quadrature method (DQM) [24],and more recently, the iso-geometric analysis (IGA) [25] and thedifferential quadrature method finite element method (DQFEM)[26–28], successively emerged as highly accurate numerical meth-ods. All works via high order methods yield excellent results due tothe use of the high-order or global basis functions. High ordermethods tend to give accurate results with far fewer degrees offreedom than low order schemes and have made noticeable suc-cess. This work adopts the DQFEM [26,27] that is a weak-form dif-ferential quadrature method in essence. The DQFEM used thedifferential quadrature (DQ) rule and the Gauss–Lobatto quadra-ture rule to directly discretize the potential functional of structuresto obtain the stiffness and mass matrices that are the same as thosein the finite element method. The DQFEM has overcome the limita-tions of the DQM pointed out by Bert and Malik [24], and was

ethod.

( )ixψ

( )1ixψ +

( )1ixψ −

( )iz

( )1iz +

( )1iz −

hi+1

i-1

h

h

i

z

x

Fig. 1. One-dimensional representation of the layerwise kinematics.

2 B. Liu et al. / Composite Structures xxx (2015) xxx–xxx

hoped to be a competitive method with FEM for analysis oflarge-scale problems.

This paper focuses for the first time on the analysis of compositelaminated plates by the differential quadrature finite elementmethod (DQFEM) [26,27] and using a layerwise theory. Thiscombination allows the accurate analysis of isotropic, composite,and sandwich plates of arbitrary shape and boundary conditions.

2. A layerwise theory

The layerwise theory proposed by Ferreira [1] is used in thiswork. The theory is based on the assumption of a first-order sheardeformation theory in each layer and the imposition of displace-ment continuity at the layer’s interfaces. Reference [1] considereda laminated plate with three layer for simplicity. This work gener-alized the theory to plates with any number of layers. The displace-ment of the ith layer, according to first order shear deformationtheory, can be written as

uðiÞ x; y; zð Þ ¼ uðiÞ0 x; yð Þ þ zðiÞwðiÞx

v ðiÞ x; y; zð Þ ¼ v ðiÞ0 x; yð Þ þ zðiÞwðiÞy

wðiÞ x; y; zð Þ ¼ w x; yð Þ

ð1Þ

The continuous of displacement u and v at the layer’s interfaces(see Fig. 1) requires that

uðiþ1Þ0 x; yð Þ ¼ uðiÞ0 x; yð Þ þ hi

2wðiÞx þ

hiþ1

2wðiþ1Þ

x

v ðiþ1Þ0 x; yð Þ ¼ v ðiÞ0 x; yð Þ þ hi

2wðiÞy þ

hiþ1

2wðiþ1Þ

y

ð2Þ

where hi are the ith layer thickness and zðiÞ 2 ½�hi=2;hi=2� are the ithlayer z coordinates. Using the recursion formula Eq. (2), one onlyneeds to consider one layer. For example, in finite element method,after the stiffness and mass matrices of each layer are derived, theglobal stiffness and mass matrices of all layers can be obtained byusing Eq. (2). Since laminated plates always have identical layers,

( )1xψ (1

yψ

Fig. 2. The deformation shape wð1Þx , wð1Þy a

Please cite this article in press as: Liu B et al. Analysis of composite plates usinCompos Struct (2015), http://dx.doi.org/10.1016/j.compstruct.2015.07.101

the stiffness and mass matrices of identical layers only need to becomputed once.

For simplicity, in the following the superscript ðiÞ is omitted,since the formulations for all layers are the same. The strain–dis-placement relations for ith layer are given by

exx

eyy

cxy

cyz

czx

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;¼

Dx 0 zDx 0 00 Dy 0 zDy 0

Dy Dx zDy zDx 00 0 0 1 Dy

0 0 1 0 Dx

26666664

37777775

u0

v0

wx

wy

w

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

ð3aÞ

or

e ¼ Du ð3bÞ

where Dx ¼ @=@x and Dy ¼ @=@y are differential operators. Note thatD ¼ DðzÞ, namely, the differential operator matrix is a function of z.Neglecting rz for each orthotropic layer, the stress–strain relationsin the fiber local coordinate system can be expressed as

r1

r2

s12

s23

s31

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;¼

Q 11 Q12 0 0 0Q 12 Q22 0 0 0

0 0 Q 33 0 00 0 0 Q 44 00 0 0 0 Q 55

26666664

37777775

e1

e2

c12

c23

c31

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

ð4Þ

where subscripts 1 and 2 are the directions of the fiber and in-planenormal to fiber, respectively; subscript 3 indicates the directionnormal to the plate; and the reduced stiffness components, Qij,are given by

Q 11 ¼E1

1� t12t21; Q 22 ¼

E2

1� t12t21; Q 12 ¼ t12Q11

Q 33 ¼ G12; Q 44 ¼ G23; Q55 ¼ G31; t21E1 ¼ t12E2

ð5Þ

in which E1, E2, t12, t21, G12, G23, and G31 are material properties oflamina i.

By performing adequate coordinate transformation, the stress–strain relations in the global x–y–z coordinate system can beobtained as

rxx

ryy

sxy

syz

szx

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;¼

�Q 11�Q12

�Q16 0 0�Q 12

�Q22�Q26 0 0

�Q 16�Q26

�Q66 0 00 0 0 �Q 44

�Q 45

0 0 0 �Q 45�Q 55

26666664

37777775

exx

eyy

cxy

cyz

czx

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

ð6aÞ

or

r ¼ �Qe ð6bÞ

By considering h as the angle between the x and 1 axes, where the 1axis is the first principal material axis, usually connected with fiber

) w

nd w of the square plate with R = 5.

g a layerwise theory and a differential quadrature finite element method.

( )1xσ ( )1

yσ ( )1xyτ

( )1yzτ ( )1

xzτFig. 3. The stress resultants at z ¼ �2h=5 calculated from layer 1 at zð1Þ ¼ h1=2, R = 5.

B. Liu et al. / Composite Structures xxx (2015) xxx–xxx 3

direction, the components Qij can be calculated by adequate coordi-nate transformation (as in [2]).

As in higher-order theories, this layerwise theory does notrequire the use of shear-correction factors. The principle of virtualdisplacements are required to derive the finite element matrices inthis work. The virtual strain energy ðdUÞ and the virtual work doneby applied forces ðdVÞ are given by

dU ¼Z

XrTdedX ¼

ZX

Duð ÞT �Q Dduð ÞdX ð7Þ

and

dV ¼ �Z

XqTdudX ð8Þ

where X denotes the domain of the laminate, and q is the externaldistributed load. Usually Eqs. (7) and (8) are expanded by substitut-ing Eqs. (3) and (6). This process will becomes complex with theincrease of the number of displacements. Therefore, an alternativebut equivalent way will be presented in next section.

Discretize the virtual strain energy and virtual work using finiteelement method, one obtains the finite element formulation for ithlayer as

Table 1Square laminated plate under uniform load (R = 5).

Method N �w �r1x �r2

x �r3x

HSDT [30] 256.13 62.38 46.91 9.38FSDT [30] 236.10 61.87 49.50 9.89CLT 216.94 61.141 48.623 9.78Ferreira [31] 258.74 59.21 45.61 9.12Ferreira [32] 15 257.38 58.725 46.980 9.39HSDT [33] 11 253.671 59.6447 46.4292 9.28HSDT [33] 15 256.239 60.1834 46.8581 9.37HSDT [33] 21 257.110 60.3660 47.0028 9.40Ferreira [1] 11 252.084 58.8628 45.4232 9.88Ferreira [1] 15 255.920 59.6503 46.0366 9.20Ferreira [1] 21 257.523 59.9675 46.2906 9.25Present 5 260.791 62.3697 48.1774 9.63Present 7 258.799 60.2134 46.4949 9.29Present 11 258.828 60.2396 46.5201 9.30Present 15 258.833 60.2466 46.5151 9.30Present 25 258.835 60.2540 46.5098 9.30Exact [29] 258.97 60.353 46.623 9.34

Please cite this article in press as: Liu B et al. Analysis of composite plates usinCompos Struct (2015), http://dx.doi.org/10.1016/j.compstruct.2015.07.101

K ðiÞuðiÞ ¼ qðiÞ ð9Þ

The displacement vector uðiÞ can be written as

uðiÞ ¼

uðiÞ0

v ðiÞ0

wðiÞx

wðiÞy

w

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;

ð10Þ

where uðiÞ0 , v ðiÞ0 , wðiÞx , wðiÞy and w are the discrete values of uðiÞ0 ;vðiÞ0 , wðiÞx ,

wðiÞy and w, respectively. In the present work, only symmetric lami-nates are considered; therefore, u0 and v0 of layer 2 can be dis-carded. For a three-layered laminated plate as considered in [1],

the displacement field uðiÞ0 and v ðiÞ0 of the first and third layers canbe expressed as

uð1Þ0 x; yð Þ ¼ �h1

2wð1Þx �

h2

2wð2Þx

v ð1Þ0 x; yð Þ ¼ �h1

2wð1Þy �

h2

2wð2Þy

ð11Þ

�r1y �r2

y �r3y �s1

xz �s1yz

2 38.93 30.33 6.065 3.089 2.5669 36.65 29.32 5.864 3.313 2.4443 36.622 29.297 5.860 4.5899 3.3862 37.88 29.59 5.918 3.593 3.5936 37.643 27.714 4.906 3.848 2.83958 38.0694 29.9313 5.9863 3.8449 1.965016 38.3592 30.1642 6.0328 4.2768 2.222706 38.4563 30.2420 6.0484 4.5481 2.391046 37.6901 29.4765 5.8953 3.8311 2.531973 38.1408 29.8296 5.9659 3.9773 2.537581 38.3209 29.9740 5.9948 4.0463 2.390155 40.4410 31.6633 6.3327 4.2994 3.537290 38.4624 30.0898 6.0180 4.1297 3.419640 38.4895 30.1130 6.0226 4.1153 3.411830 38.4914 30.1118 6.0224 4.1080 3.402820 38.4951 30.1091 6.0218 4.1076 3.39940 38.491 30.097 6.161 4.3641 3.2675

g a layerwise theory and a differential quadrature finite element method.

Table 2Square laminated plate under uniform load (R = 10).

Method N �w �r1x �r2

x �r3x �r1

y �r2y �r3

y �s1xz �s1

yz

HSDT [30] 152.33 64.65 51.31 5.131 42.83 33.97 3.397 3.147 2.587FSDT [30] 131.095 67.80 54.24 4.424 40.10 32.08 3.208 3.152 2.676CLT 118.87 65.332 48.857 5.356 40.099 32.079 3.208 4.3666 3.7075Ferreira [31] 159.402 64.16 47.72 4.772 42.970 42.900 3.290 3.518 3.518Ferreira [32] 15 158.55 62.723 50.16 5.01 42.565 34.052 3.400 3.596 3.053HSDT [33] 11 153.008 64.7415 49.4716 4.9472 42.8860 33.3524 3.3352 2.7780 1.8207HSDT [33] 15 154.249 65.2223 49.8488 4.9849 43.1521 33.5663 3.3566 3.1925 2.1360HSDT [33] 21 154.658 65.3809 49.9729 4.9973 43.2401 33.6366 3.3637 3.5280 2.3984Ferreira [1] 11 155.037 63.5984 47.4765 4.7476 42.6696 32.7369 3.2737 3.7016 3.3051Ferreira [1] 15 157.374 64.4828 48.1544 4.8154 43.1887 33.1392 3.3139 3.8447 3.2183Ferreira [1] 21 158.380 64.8462 48.4434 4.8443 43.3989 33.3062 3.3306 3.9237 2.8809Present 5 160.497 67.4660 50.4716 5.0472 45.7919 35.2056 3.5206 4.1910 3.5244Present 7 159.371 65.1601 48.7223 4.8722 43.6173 33.4913 3.3491 4.0360 3.4160Present 11 159.396 65.2015 48.7577 4.8758 43.6378 33.5098 3.3510 4.0040 3.3982Present 15 159.403 65.2178 48.744 4.8744 43.6425 33.5061 3.3506 3.9957 3.3862Present 25 159.406 65.2306 48.733 4.8733 43.6494 33.5002 3.3500 3.9957 3.3829Exact [29] 159.38 65.332 48.857 4.903 43.566 33.413 3.500 4.0959 3.5154

Table 3Square laminated plate under uniform load (R = 15).

Method N �w �r1x �r2

x �r3x �r1

y �r2y �r3

y �s1xz �s1

yz

HSDT [30] 110.43 66.62 51.97 3.465 44.92 35.41 2.361 3.035 2.691FSDT [30] 90.85 70.04 56.03 3.753 41.39 33.11 2.208 3.091 2.764CLT 81.768 69.135 55.308 3.687 41.410 33.128 2.209 4.2825 3.8287Ferreira [31] 121.821 65.650 47.09 3.140 45.850 34.420 2.294 3.466 3.466Ferreira [32] 15 121.184 63.214 50.571 3.371 45.055 36.044 2.400 3.466 3.099HSDT [33] 11 113.594 66.3646 49.8957 3.3264 45.2979 34.9096 2.3273 2.1686 1.5578HSDT [33] 15 114.387 66.7830 50.2175 3.3478 45.5427 35.1057 2.3404 2.6115 1.9271HSDT [33] 21 114.644 66.9196 50.3230 3.3549 45.6229 35.1696 2.3446 3.0213 2.2750Ferreira [1] 11 118.298 64.9159 46.8241 3.1216 45.4432 34.2237 2.2816 3.6123 3.8412Ferreira [1] 15 120.077 65.8418 47.5260 3.1684 46.0049 34.6566 2.3104 3.7556 3.6695Ferreira [1] 21 120.988 66.2911 47.8992 3.1933 46.2924 34.8898 2.3260 3.8311 3.2562Present 5 122.549 68.9406 49.8964 3.3264 48.7625 36.8343 2.4556 4.1139 3.5367Present 7 121.737 66.5786 48.1702 3.2113 46.5131 35.0824 2.3388 3.9683 3.4308Present 11 121.764 66.6355 48.2074 3.2138 46.5281 35.0953 2.3397 3.9264 3.4062Present 15 121.774 66.6633 48.1818 3.2121 46.5367 35.0878 2.3392 3.9190 3.3932Present 25 121.777 66.6800 48.1664 3.2111 46.5463 35.0790 2.3386 3.9192 3.3580Exact [29] 121.72 66.787 48.299 3.238 46.424 34.955 2.494 3.9638 3.5768

Table 4Laminated square plate (0�/90�/0�) under sinusoidal load.

Method �w �rxx �ryy �szx �syz

Three-dimensional (Pagano[34])

0.7530 0.590 0.285 0.357 0.1228

Liou and Sun [35] 0.7546 0.580 0.285 0.367 0.127Layerwise linear LD1 Carrera

[36]0.7371 0.5608 0.2740 0.3726 0.1338

Mixed layerwise LM4 Carrera[36]

0.7528 0.5801 0.2796 0.3626 0.1249

Reddy [8] 0.7125 0.5684 – 0.1033 –Ferreira [1], layerwise

(N = 11)0.7399 0.5711 0.2801 0.3560 0.0872

Ferreira [1], layerwise(N = 15)

0.7420 0.5731 0.2808 0.3582 0.0931

Ferreira [1], layerwise(N = 21)

0.7427 0.5738 0.2810 0.3590 0.0953

Present (N = 5) 0.7391 0.5615 0.2758 0.3526 0.0923Present (N = 7) 0.7402 0.5715 0.2806 0.3583 0.0958Present (N = 11) 0.7402 0.5717 0.2807 0.3582 0.0958Present (N = 15) 0.7402 0.5717 0.2807 0.3582 0.0958

4 B. Liu et al. / Composite Structures xxx (2015) xxx–xxx

uð3Þ0 x; yð Þ ¼ h2

2wð2Þx þ

h3

2wð3Þx

v ð3Þ0 x; yð Þ ¼ h2

2wð2Þy þ

h3

2wð3Þy

ð12Þ

Please cite this article in press as: Liu B et al. Analysis of composite plates usinCompos Struct (2015), http://dx.doi.org/10.1016/j.compstruct.2015.07.101

Therefore, the displacement field of the three layers can beexpressed by the global independent displacement fields as

uð1Þ0

v ð1Þ0

..

.

uð3Þ0

v ð3Þ0

..

.

w

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;

¼

�12 h1 0 �1

2h2 0 0 0 00 �1

2 h1 0 �12h2 0 0 0

..

. ... ..

. ... ..

. ... ..

.

0 0 12h2 0 1

2h3 0 00 0 0 1

2h2 0 12h3 0

..

. ... ..

. ... ..

. ... ..

.

0 0 0 0 0 0 1

266666666666664

377777777777775

wð1Þx

wð1Þy

wð2Þx

wð2Þy

wð3Þx

wð3Þy

w

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

9>>>>>>>>>>>>>=>>>>>>>>>>>>>;

ð13Þ

or

u ¼ H�u ð14Þ

Therefore, the final stiffness and mass matrices of the three-layeredlaminated plates can be written as

K ¼ HTK ð1Þ 0 0

0 K ð2Þ 00 0 K ð3Þ

264

375H; q ¼ HT

qð1Þ

qð2Þ

qð3Þ

8><>:

9>=>; ð15Þ

where HT and H are required by the principle of virtual displace-ments shown in Eqs. (7) and (8). It is clear that for layers of identical

g a layerwise theory and a differential quadrature finite element method.

B. Liu et al. / Composite Structures xxx (2015) xxx–xxx 5

property, one only needs to compute the stiffness matrix of onelayer.

3. The differential quadrature finite element method

The differential quadrature finite element method (DQFEM)[26,27] is a weak-form differential quadrature method (DQM)[24]. Namely, the strain energy and work potential of structuresare directly discretized by the DQM together with the Gauss–Lobatto integration to obtain the global stiffness and mass matricesof finite element method. The DQM approximates the nth deriva-tives of a field variable f ðxÞ at point xi by a weighted linear sum as

f ðnÞi ¼XN

j¼1

AðnÞij f j i ¼ 1;2; . . . ;Nð Þ ð16Þ

or

f ðnÞ ¼ AðnÞf ð17Þ

where AðnÞij are the weighting coefficients of the nth order deriva-tives, and N the number of grid points in the x-direction. For thecomputation of weighting coefficients and more details about theDQM, one may refer to the survey paper [24]. The DQM for twodimensional problem can be expressed in similar way as Eq. (16)or (17) [24,26,27].

The Gauss–Lobatto quadrature is the Gauss integration withtwo endpoints fixed, which can be found in mathematics hand-books or in [19]. Here a simple introduction of it is presented tomake the paper self-contained. The Gauss–Lobatto quadrature rulewith precision degree (2n-3) for function f ðxÞ defined at [�1,1] is

Z 1

�1f ðnÞdx ¼

Xn

j¼1

Cjf nj

� �ð18Þ

where the weights Cj of Gauss–Lobatto integration are given by

C1 ¼ Cn ¼2

nðn� 1Þ ; Cj ¼2

nðn� 1Þ½Pn�1ðnjÞ�2j – 1; nð Þ ð19Þ

where nj is the (j� 1)th zero of P0n�1ðnÞ. Liu et al. [19] presented thedetail of computing roots for Legendre polynomials using the recursionformula of Legendre polynomials if more than 40 roots are required.

Denote Að1Þ and Bð1Þ as the two-dimensional DQM weightingcoefficient matrix for first order derivative with respect to x andy, respectively. Using the differential quadrature method, the dif-ferential operator matrix D in Eq. (3) can be discretized as

D ¼

Að1Þ 0 zAð1Þ 0 00 Bð1Þ 0 zBð1Þ 0

Bð1Þ Að1Þ zBð1Þ zAð1Þ 00 0 0 I Bð1Þ

0 0 I 0 Að1Þ

26666664

37777775

ð20Þ

where I is a unit matrix. Formulating the Gauss–Lobatto quadrature

weight in the material constant matrix �Q and denoting it as ~Q , thenthe stiffness matrix of ith layer can be expressed as

K ðiÞ ¼Z hi=2

�hi=2DT ~QDdz ¼

Z hi=2

�hi=2

�K ðiÞðzÞdz ð21Þ

Using the Gauss quadrature rule one only needs two Gauss points todo the integral in Eq. (21). Clearly, simpler than expanding Eq. (7).The obtaining of load vector qðiÞ is simple, since only Gauss–Lobatto quadrature is needed [27].

Please cite this article in press as: Liu B et al. Analysis of composite plates usinCompos Struct (2015), http://dx.doi.org/10.1016/j.compstruct.2015.07.101

4. Numerical examples

4.1. Three-layer square sandwich plate under uniform load

A simply supported square sandwich plate under a uniformtransverse load is considered. This is a classical sandwich exampleof Srinivas [29]. The material properties of the sandwich coreexpressed in the stiffness matrix, �Q core, are expressed as

�Q core¼

0:999781 0:231192 0 0 00:231192 0:524886 0 0 0

0 0 0:262931 0 00 0 0 0:266810 00 0 0 0 0:159914

26666664

37777775

ð22Þ

Skins material properties are related to core properties by a factor Ras follows:

�Q skin ¼ R �Q core ð23Þ

Transverse displacement and stresses are normalized through thefollowing factors:

�w ¼ w a=2; a=2;0ð Þ0:999781hq

�r1x ¼

rð1Þx a=2; a=2;�h=2ð Þ�q

; �r2x ¼

rð1Þx a=2; a=2;�2h=5ð Þ�q

�r3x ¼

rð2Þx a=2; a=2;�2h=5ð Þ�q

; �r1y ¼

rð1Þy a=2; a=2;�h=2ð Þ�q

�r2y ¼

rð1Þy a=2; a=2;�2h=5ð Þ�q

; �r3y ¼

rð2Þy a=2; a=2;�2h=5ð Þ�q

�s1xz ¼

sð2Þxz 0; a=2;0ð Þq

; �s1yz ¼

sð2Þyz a=2;0;0ð Þq

ð24Þ

In order to clearly show how well the boundary conditions are sat-isfied, first the deformation modes of three of the seven indepen-dent variables in the right side of Eq. (13) are shown in Fig. 2. Thestress resultants at z ¼ �2h=5 calculated from layer 1 atzð1Þ ¼ h1=2 are shown in Fig. 3. It is clear that the boundary condi-tions are satisfied very well.

Transverse displacement and stresses for a sandwich plate areindicated in Tables 1–3 and compared with various formulations,where N is the number of sampling points used in each methodfor convergence studies. One can see the good agreement betweenthe present results and the exact results. The present results agreewith the deflection of exact results for 3 significant digits, agreewith the normal stress of exact results for about 2 significant digitsin general, and agree with the shear stress of exact results for onesignificant digit. One can also see the fast convergence rate of theDQFEM from Tables 1–3. Even with a minimum number of sam-pling points, N = 5, the present method still has good accuracy.The present method has about 3–5 significant digits convergedwhen N = 15, while those in literature have only 1–2 significantdigits converged.

4.2. Three-layer (0/90/0) square cross-ply laminated plate undersinusoidal load

A square laminate of length a and thickness h is composed ofthree equally thick layers oriented at (0/90/0). It is simply sup-ported on all edges and subjected to a sinusoidal vertical pressureof the form

pz ¼ P sinpxa

� �sin

pya

� �ð25Þ

g a layerwise theory and a differential quadrature finite element method.

6 B. Liu et al. / Composite Structures xxx (2015) xxx–xxx

where the origin of the coordinate system is located at thelower-left corner on the mid-plane. For this example, there is athree dimensional exact solution by Pagano [34]. Here we comparethe present solution by DQFEM for a/h = 10 with various models,particularly full mixed and hybrid finite element method (FEM)analysis, classical FEM analysis, etc. The material properties are

E1 ¼ 25:0E2; G12 ¼ G13 ¼ 0:5E2

G23 ¼ 0:2E2; t12 ¼ 0:25ð26Þ

The numerical results are presented in Table 4, in a normalizedform, as indicated by the following expressions:

�w ¼ 102w a=2; a=2;0ð Þh3E2

Pa4 ; �rxx ¼rxx a=2; a=2; h=2ð Þh2

Pa2

�ryy ¼ryy a=2; a=2;h=6ð Þh2

Pa2 ; �szx ¼szx 0; a=2;0ð Þh

Pa

�syz ¼syz a=2;0;0ð Þh

Pa

ð27Þ

As can be seen that, the present methodology converges to verygood results, especially agree with the layerwise results ofFerreira [1] very well, since this work are based on the layerwiseformulation of [1]. The convergence of the DQFEM in this case iseven better than the case of Section 4.1, as can be seen that theDQFEM already converged very well when the sampling pointsnumber N = 7.

5. Conclusions

The authors have used the differential quadrature finite ele-ment method (DQFEM) for vibration of isotropic beams, platesand three-dimensional elasticity [26,27]. In all cases, excellentresults that closely agree with exact results were obtained. In thispaper, the analysis of composite laminated plates by the use of theDQFEM [26,27] and using a layerwise theory [1] with independentrotations in each layer is performed for the first time. The layer-wise theory [1] was also generalized to plates with any numberof layers. The stiffness and mass matrices were schematically for-mulated. Composite laminated plates and sandwich plates wereconsidered for testing of the present methodology and the resultsshowed excellent accuracy for all cases. This layerwise theory com-bined with DQFEM discretization is a simple yet very effective andaccurate numerical technique for the analysis of thick or thin com-posite or sandwich laminates and their structures.

Acknowledgments

The financial support of National Natural Science Foundation ofChina (Grant Nos. 11402015, 11372021, 11172028), the special-ized research fund for the doctoral program of higher education(20131102110039), FCT-Fundação para a Ciência e a Tecnologiato Project PTDC/EMS-PRO/2044/2012 and to GrantSFRH/BPD/99591/2014 is gratefully acknowledged.

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g a layerwise theory and a differential quadrature finite element method.