A new trigonometric layerwise shear deformation theory for the finite element analysis of laminated composite and sandwich plates

Computers and Structures 94–95 (2012) 45–53

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A new trigonometric layerwise shear deformation theory for the finite element analysis of laminated composite and sandwich plates J.L. Mantari, A.S. Oktem, C. Guedes Soares ⇑ Centre for Marine Technology and Engineering (CENTEC), Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal

a r t i c l e

i n f o

Article history: Received 30 June 2011 Accepted 14 December 2011 Available online 16 January 2012 Keywords: Higher order shear deformation theory Trigonometric displacement field Layerwise finite element Laminated composites Sandwich

a b s t r a c t A layerwise finite element formulation of a newly developed higher-order shear deformation theory for the flexure of thick multilayered plates is presented. The proposed trigonometric layerwise shear deformation theory accounts for: (a) non-linear and constant variation of in-plane and transverse displacement respectively through the panel thickness; (b) adequate transverse shear deformation and satisfy transverse shear traction free conditions on the top and the bottom surfaces of the plate. The accuracy of the present code is ascertained by comparing it with the exact solution and various available results in the literature. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Composite structures have an increasing use in many engineering fields such as marine, aerospace, automotive, civil, sport and other applications. This is because of their high performance and reliability due to high strength-to-weight and high stiffness-toweight ratios, excellent fatigue strength, resistance to corrosion (e.g. glass fiber composites), and most importantly the design flexibility also known as tailoring the materials for desired applications. With the increased use of laminated composite and sandwich structures, there is a need to develop efficient and reliable mathematical models, deformation theories, and analysis methods to predict the short and long-term behavior of the multilayer composite structures under a variety of loading and environmental conditions. Many theories have been developed to analyze composite structures. Among equivalent single layer theories, there are mainly three major theories; namely the classical lamination theory (CLT) which is based on the assumptions of Kirchoff’s plate theory [1–6] which neglects the interlaminar shear deformation, the first order shear deformation theory (FSDT) [7–12] assumes constant transverse shear deformation through the entire thickness of the laminate and violates stress free boundary conditions at the top and bottom surfaces of the panel. More accurate theories such as higher order theories (HSDT) assume quadratic, cubic or higher variations of surface-parallel displacements through the entire thickness of the laminates to model the behavior of the structure for thick to thin regions [13–30]. ⇑ Corresponding author. Tel.: +351 218 417607; fax: +351 218 474015. E-mail address: [email protected] (C. Guedes Soares). 0045-7949/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2011.12.003

However, the abovementioned theories may be not sufficient if local effects are important or accuracy in the calculation of the transverse stresses is required. For that reason, more advanced plate theories were developed to include zig–zag effects [22,31,32]. Layerwise models have also an important role when the abovementioned approaches fail to predict local effects [33–35]. This advanced method is an accurate refined method but with the disadvantage of expensive computing time. More detailed literature review may be found for various shear deformation theories and finite element approaches for composite and sandwich plates and shells using equivalent single layer and layerwise theories in excellent research papers presented by Reddy [36,37], Carrera [38,39], Kreja [40] and Altenbach [41]. The welldescribed unified formulation, initially presented by Carrera [31] and recently extended by Demasi [32,35,42–45], describes precisely and clearly the models, types and class of theories. Carrera [46] presented free vibration analyses of layered plates, cylindrical and spherical shells made of isotropic and orthotropic layers for simply supported boundary condition. The transverse normal stress effects were included in the displacement model by allowing different polynomial orders. It was concluded that for the accurate analyses of the vibrational response of highly anisotropic, thick and very thick shells requires layerwise description and interlaminar continuous transverse shear and normal stresses were crucial for the modified classical models. Carrera and Brischetto [47] presented an extensive survey for a variety of plate theories and evaluated the bending and vibration of sandwich structures. The main drawbacks of the equivalent single layer analysis for the soft-core sandwich structures were specified. Particularly, for the high values of face-to-core ratios higher order theories were found

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J.L. Mantari et al. / Computers and Structures 94–95 (2012) 45–53

to be unsuitable and it was stated that even the error can be double by the neglecting the through-thickness deformation effects and CLT and FSDT were failed to analyze the sandwich structures. Recently, Carrera and Petrolo [48] by using Carrera unified formulation (CUF) and an asymptotic expansion method discussed the effectiveness of higher-order terms in refined beam theories. CUF allows to understand and to quantify the importance of the terms used in the displacement field. The effect of each displacement variable in the solution was investigated by comparing the error obtained accounting and removing the variable in the plate equations in another work of Carrera and Petrolo [49]. One important conclusion that can be inferred for the orthotropic plates was the additional term(s) used in the transverse displacement to consider the stretching of the panel highly affects the normal stress (rxx) results in a positive way, but they have no influence on the other considered results. In similar fashion, Carrera et al. [79] by using CUF have found the so-called best plate theory, but this time considering an axiomatic hypothesis method. More detailed information and applications of CUF can be found in the very recent books authored by Carrera et al. [80,81]. Abovementioned CUF approach was used by Carrera et al. [50] for the analysis of various displacement models using the finite elements for static analysis of laminated plates. Errors related to the models were specified in terms of normal and shear stresses. For example the CLT theory was found to be good enough to obtain the normal stress rxx while it was not suitable to detect correctly the transverse displacement (uz) and out-of-plane shear stress (sxz, syz). It was also stated that the layerwise theories and zig– zag theories were required for more precise results for laminated structures. Additional conclusions were also inferred as mentioned in reference [49]. Regarding to trigonometric higher order shear deformation theories, it is well-known that the first trigonometric shape strain function was introduced by Levy [74]. Then, the TPT (trigonometric plate theory) were corroborated and assessed by Stein [75] (after almost one century) and later extensively used by Touratier [51] and co-workers. Later, this HSDT, which introduces the sinus function, was used in several higher order layerwise and zig–zag shear deformation theories [52–55]. Moreover, advanced numerical calculations such as finite element [55,56] and meshless methods [28,57,58] were also implemented by using Touratier theory [51]. Recently, for the first time, Neves et al. [76,77] and Ferreira et al. [78] presented meshless solutions by using CUF with non-polynomial shape strain functions (sinus) to analyze static and dynamic behaviour of classical and advanced (e.g. functionally graded) composites. Therefore, it can be said that there are evidences of the demand of trigonometric shear deformation theories, mainly because they are richer than polynomial functions, simple, more accurate, and the free surface boundary conditions can be guaranteed a priori. With respect to finite elements applied to bending analysis by using the abovementioned theories, FSDT and HSDTs have been explored since 1960s. The FSDT requires only displacement (C0) continuity across the interelement boundaries, which is suitable for the formulation of general finite elements. However, it requests arbitrary shear correction factors and presents shear locking problems for thin plates. Many HSDTs , among the ones developed by Reddy and Liu [17], Touratier [51], Soldatos [59] and Karama et al. [60], account for approximately parabolic transverse shear deformation and satisfy transverse shear traction free conditions on the top and the bottom surfaces of the plate. Moreover, these theories do not require any shear correction factor. However these plate theories have all positive features except one drawback, which is found in a situation when finite element analysis is applied to this plate model, because these theories require continuity of the transverse displacement as well as its derivatives (C1 continuity) along the interelement boundary, which poses difficulties in

developing simple conforming quadrilateral element. A good survey of non-conforming finite element models of HSDTs, such as the one developed by Reddy and Liu [17], can be found in Sheinkh et al. [61,62], Ngo et al. [63] and Kulkarni and Kapuria [64]. Analytical, closed form solutions may predict exactly the displacement and stresses and can be used to check the accuracy of other numerical methods such as the finite element models [65,66]. Pagano [67] provided an exact solution for the problem of rectangular orthotropic sandwich plate subjected to a laterally distributed load. This solution was used as benchmark for the comparison to various numerical methods such as, finite differences, finite element, and other approximation solutions [68]. Srinivas [69] also provided exact solutions, which were extensively used as reference for various numerical calculations, such as meshless methods [28,57,58]. According to the authors knowledge, this paper presents for the first time a tangential layerwise trigonometric shear deformation theory for modelling the laminated composites and sandwiches plates. The new displacement field accounts for approximately parabolic distribution of the transverse shear deformation and satisfies the transverse shear traction free conditions on the top and bottom surfaces of the plate, thus a shear correction factor is not required. By enforcing the free conditions on upper and lower surfaces and interlaminar continuity conditions of the transverse shear stresses, the number of total unknowns does not depend on the number of layers of the laminate. Therefore, the discrete element chosen is a four-nodded quadrilateral with seven-degrees-of-freedom per node. For the finite element formulation, the seven-degrees-offreedom ðuo ; v o ; wo ; @w=@x; @w=@y; hxc ; hyc Þ are assumed and they are independent of each other. Therefore, a C0 Lagrangian isoparametric faceted quadrilateral element to analyze the static behaviour of the general laminated composite and sandwich plates is introduced. The accuracy of the present code is ascertained by comparing it with Srinivas [69] exact solution and with various numerical calculations available in the literature such as the finite element [70,71] and the meshless solutions [28,57,58,72]. 2. Higher order layerwise displacement field In this paper, the equivalent single layer shear deformation theory developed by Mantari et al. [73], which produces similar results as the shear deformation theory presented by Reddy and Liu [17] and Touratier [51], is extended to a layerwise shear deformation theory for the finite element analysis of sandwich and composite laminated plates. The sandwich laminated plate, composed of finite number of orthotropic layers of different thickness is shown in Fig. 1. The inplane displacement continuity of the present layerwise model is guaranteed by obtaining the summations of the continuous mathematical terms present in the model at each local layer coordinate as in Fig. 1. In order to reduce the degrees of freedom, the transverse shear stress continuity is also imposed at the layer interfaces. The displacement field for the inner layers (e.g. h1, h2, etc.) is given as follows:

uin ¼ uo þ

v in

c X hc x hin x @w hk hxk  hc  hin þ zhxin  ðz þ ain Þ @x 2 2 k¼inþ1

þ tan mðz þ ain Þhxc ; c X hc hin y @w ¼ v o þ hyc  hk hyk  hin þ zhyin  ðz þ ain Þ @y 2 2 k¼inþ1

þ tan mðz þ ain Þhyc ; win ¼ wo ðx; yÞ:

ð1a — cÞ

where

ain ¼

c X hc hin hk   : 2 k¼inþ1 2

ð2Þ

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J.L. Mantari et al. / Computers and Structures 94–95 (2012) 45–53

z

z

w h5 h4 h3 h2 h1

tanmz θ 3x

-z w x θ5x

x

θ4x

x o

u x x

u

w

θ3x

x

Layer 5

x

Layer 4

o

x θ2x θ1x

Layer 3

x

u x

Layer 2

x

Layer 1

Fig. 1. Mathematical terms in the modeling of the layerwise displacement field.

hin and hc are the thicknesses of an inner layer and the core in Eq. (2), respectively. The displacement field for the middle layer, considered as the core of the sandwich laminate, is given as:

8 9k > < rxx > =

@w þ zhxc þ tan mðz þ ac Þhxc ; @x @w v c ¼ v o  ðz þ ac Þ þ zhyc þ tan mðz þ ac Þhyc ; @y wc ¼ wo ðx; yÞ: uc ¼ uo  ðz þ ac Þ

> : ð3a — cÞ

where in Eqs. (3a–b), ac = 0. Finally, the displacement field for the outer layers (e.g. h4, h5, etc.), is given below:

uex ¼ uo 

v ex

hc x h þ 2 c

ex1 X

hk hxk þ

k¼c

hex x @w h þ zhxex  ðz þ aex Þ @x 2 ex

þ tan mðz þ aex Þhxc ; ex1 X hc hex y @w ¼ v o  hyc þ hk hyk þ hex þ zhyex  ðz þ aex Þ @y 2 2 k¼c ð4a — cÞ

where x1 hc eX hex hk þ þ : 2 2 k¼c

ð5Þ

hex and hc are the thicknesses of an outer layer and the core in Eqs. (4) and (5), respectively. In Eqs. (1a–c), (3a–c) and (4a–c), uk and vk are the inplane displacements at any point (xk, yk, zk) in layer k 2 {1, 2, . . . , in, . . . , c, . . . ex, . . . , l1, l}, uo and vo denote the inplane displacements of the point (x, y, 0) on the midplane of each layer, wo is the transverse deflection, hyin ; hyex and hxin , hxex are the rotations of the normals to the midplane about the y and x axes, respectively, while hxc and hyc are the core rotations about the x and y axes. As mentioned above, in order to reduce the degrees of freedom, the transverse shear stress continuity is imposed at the layer interfaces and hxin ; hyin ; hxex ; hyex are obtained and given as below:

hxin hyin

¼ ¼

hxex ¼  hyex

¼

!

  hin þ m sec m  þ a in 2 Q in 55 !   Q in1 hin y 2 44 h þ m sec m  þ a in in1 2 Q in 44 !   Q ex1 hex x 2 55 h þ m sec m  þ a ex ex1 2 Q ex 55 !   Q ex1 hex y 2 44 h þ m sec m  þ a ex ex1 2 Q ex 44 Q in1 55

hxin1

2

Q in1 55 Q in 55 Q in1 44 Q in 44

! !  1 hyc ;

Q ex 55 Q ex1 44 Q ex 44

hxc ;

1

Q ex1 55



ryy > sxy ;b

sxz syz

2

Q 11 6 ¼ 4 Q 12

Q 12 Q 22

Q 16

Q 26

"

k ¼ s

Q 55

Q 54

Q 45

Q 44

9k 3k 8 Q 16 > < exx > = 7 ; Q 26 5 eyy > :c > ; Q 66 xy b

#k (

cxz cyx

)k :

ð7a — bÞ

s

in which, r ¼ f rxx ryy sxy sxz syz g and e = {exx, eyy, cxy, cxz, cyz} are the stresses and the linear strain vectors with respect to the laminate coordinate system and Q ij ’s are the plane stresses reduced elastic constants in the global x–y–z coordinate system of the kth lamina, (see Fig. 2). The subscripts b and s stand for bending and shear, respectively. 3. Finite element formulation

þ tan mðz þ aex Þhyc ; wex ¼ wo ðx; yÞ:

aex ¼ 

By performing the transformation rule of stresses/strain between the lamina and the laminate coordinate system as in Fig. 2, the stress-stress–strain relations in the global x–y–z coordinate system can be obtained as,

!  1 hxc ; !  1 hyc : ð6a — dÞ

In the present work, a four-nodded quadrilateral C0 continuous isoparametric element with seven-degrees-of-freedom per node is employed. The generalized displacements included in the present theory can be expressed as follows:

d¼

n X

Ni di :

ð8Þ

i¼1

where d ¼ fuo ; v o ; wo ; @w=@x; @w=@y; hxc ; hyc gT ; di is the displacement vector corresponding to node i, Ni is the shape function associated with the node i and n is the number of nodes per element, which is four in the present study. Considering the Eq. (6), the strain vector {e} can be expressed in terms of d containing nodal degrees of freedom as,

 k

h i

 k

h i

eb ¼ Bkb fdg; cs ¼ Bks fdg:

ð9a — bÞ

where [B] is the strain–displacement matrix in the Cartesian coordinate system. The [B] matrix can be divided in two parts, one which contains the bending terms and other containing the shear terms, as follows:

h i h i h i Bkb ¼ Bkm þ Bkm þ z½Bkb  þ tanðmz þ aÞ Bkb ; h i h i Bks ¼ Bks þ m sec2 ðmz þ aÞ Bks ; a ¼ fain ; ac ; aex g;

ð10a — cÞ Bkb

Bks

where matrices of and (see Eq. 10(a–b)), for the laminate example given in Section 4, are given in Appendix A.

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J.L. Mantari et al. / Computers and Structures 94–95 (2012) 45–53

y y

θc v z, 3

y

3, z

2 α

1

α

Laminate mid-plate

w

b

x h

(1, 2, 3) - Lamina reference axes

u

x

θc

x

a (x, y, z) - Laminate reference axes Fig. 2. Laminate geometry with positive set of lamina/laminate reference axes, displacement components and fiber orientation.

The potential energy Pe of the plate with a mid-surface area X and volume V, with a load vector P corresponding to the seven-degrees-of-freedom of a point on the mid-plane, can be represented as,

The equilibrium equations can be obtained by minimizing Pe with respect to {d} as,

Pe ¼ U s  W ext ;

where [Ke] is the element stiffness matrix and {Pe} is the nodal load vector. In what follows, the problem under consideration is solved for the simply supported boundary conditions and they are given at all four edges as follows:

Us ¼

1 2

Z

1 2

eT rdV þ V

Z

cT sdV

V

l ZZZ  k T h k i k  1X eb Q b eb dxdydz 2 k¼1 l ZZZ  k T h k i k  1X cs Q s cs dxdydz; þ 2 k¼1 Z W ext ¼ wo qdxdy:

¼

½K e fdg ¼ fPe g:

ð19Þ

uðx; 0Þ ¼ uðx; bÞ ¼ v ð0; yÞ ¼ v ða; yÞ ¼ 0; wðx; 0Þ ¼ wðx; bÞ ¼ wð0; yÞ ¼ wða; yÞ ¼ 0; @w=@xðx; 0Þ ¼ @w=@xðx; bÞ ¼ @w=@yð0; yÞ ¼ @w=@yða; yÞ ¼ 0; ð11a — cÞ

hcx ðx; 0Þ ¼ hcx ðx; bÞ ¼ hcy ð0; yÞ ¼ hcy ða; yÞ ¼ 0:

X

ð20Þ

Considering the Eqs. (8) and (9a–b), the potential energy of an element can be rewritten as follows:

Pe ¼ ¼

l l n o n o 1X 1X fdgT K kbe fdg þ fdgT K kse fdg  fdgT fPe g; 2 k¼1 2 k¼1

1 fdgT fK e gfdg  fdgT fPe g; 2

ð12a—bÞ

where

K kbe ¼ K kmm þ K kmm þ K km m þ K kmb þ K km b þ K kbb þ K kbb þ K kb b ð13Þ and

K kse

þ

K kss

Matrices of equations

K kbe

¼

K kss

þ

K ks s ;

ð14Þ

K kse

(see Eqs. (13), (14) are given in the following

and

K kmm ¼ Bkm Q k Bkm f ð1Þ; K kmm ¼ 2 Bkm Q k Bkm f ð1Þ; K km m ¼ Bkm Q k Bkm f ð1Þ; K kmb ¼ 2 Bkm Q k Bkb f ð2Þ; K km b ¼ 2 Bkm Q k Bkb f ð2Þ; K kbb ¼ Bkb Q k Bkb f ð3Þ; K kbb ¼ 2 Bkb Q k Bkb f ð4Þ; K kb b ¼ Bkb Q k Bkb f ð5Þ: K kss ¼ Bks Q k Bks f ð1Þ;

K kss ¼ 2 Bks Q k Bks f ð6Þ; Z

hk

hk1

fPge ¼

ZZ X

A simply supported square sandwich plate is considered under uniform load. The a/h ratio is taken as 10 [69]. It considers a simply supported square sandwich plate under uniform load. The ratio of side to thickness, a/h is taken as 10. The sandwich laminate considers two outside layers (skins) of thickness h1 = h3 = 0.1 h and one inner layer (core) of thickness h2 = 0.8 h. The skin orthotropic properties are obtained by multiplying an integer, R, by the core orthotropic properties, given by

2 Q

K ks s ¼ Bks Q k Bks f ð7Þ;

ð1;tanðmz þ aÞ;z2 ; z tanðmz þ aÞ; tan2 ðmz þ aÞ; m sec2 ðmz þ aÞ; m2 sec4 ðmz þ aÞÞ

½Nwo T qdX:

4. Numerical results and discussion

ð15Þ

ð16Þ f ð1; 2;3; 4; 5; 6; 7Þ ¼

Integrations in Eq. (11a–c) are carried out numerically by Gauss quadrature integration rule. A reduced integration technique is adopted for the calculations of the shear stiffness matrix in order to avoid numerical disturbances such as shear locking, which may appear in a full integration scheme. The stiffness matrix of all the elements is calculated and assembled together to form the overall stiffness matrix [K] of the panel and then the static problem is solved.

core

0:999781 0:231192

0

0

6 0 0 6 0:231192 0:524886 6 ¼6 0 0 0:262931 0 6 6 0 0 0 0:266810 4 0

0

0

0

0 0 0 0

3 7 7 7 7; 7 7 5

0:159914

dz;

The skin properties are obtained by,

ð17Þ

Q skin ¼ RQ core :

ð18Þ

Results are compared with exact results [69] and finite element results [70,71], trigonometric layerwise deformation theory with multiquadrics [57], trigonometric shear deformation theory and

ð21Þ

49

J.L. Mantari et al. / Computers and Structures 94–95 (2012) 45–53 Table 1 Maximum deflection and stresses of a square sandwich plate under uniform load (R = 5). Method

 w

r 1xx

r 2xx

r 3xx

r 1yy

r 2yy

r 3yy

s1xz

s2xz

(a/2, b/2, 0) (a/2, b/2, h/2) (a/2, b/2, 2h/5) (a/2, b/2, 2h/5) (a/2, b/2, h/2) (a/2, b/2, 2h/5) (a/2, b/2, 2h/5) (0, b/2, 0) (0, a/2, 2h/5) Srinivas [69], exact solution Present, layerwise, DOF = 7, N = 21 Present, layerwise, DOF = 7, N = 15 Present, layerwise,DOF = 7, N = 11 Pandya and Kant HSDT [70] Pandya and Kant FSDT [70] Ferreira and Barbosa [71] Roque et al. [57], layerwise, N = 21 Roque et al. [57], layerwise, N = 15 Roque et al. [57], layerwise, N = 11 Ferreira et al. [28], layerwise, N = 21 Ferreira et al. [28], layerwise, N = 15 Ferreira et al. [28], layerwise, N = 11 Xiang et al. (Levinson) [58] Xiang et al. (Touratier) [58] Xiang et al. (Karama) [58]

258.97 257.93 256.71 254.56 258.74 236.10 258.74 259.12 258.72 257.12 257.00 256.50 254.91 253.72 253.99 253.64

60.35 60.22 60.13 60.01 62.38 61.87 59.21 60.34 60.26 59.99 60.40 60.30 60.03 59.95 60.12 60.12

46.62 46.48 46.40 46.23 46.91 49.50 45.61 46.57 46.51 46.31 46.97 46.90 46.69 46.66 47.10 46.70

9.34 9.30 9.28 9.25 9.38 9.90 9.12 9.31 9.30 9.26 9.39 9.38 9.34 9.33 9.42 9.34

38.49 38.49 38.46 38.42 38.93 36.65 37.88 38.55 38.50 38.33 38.46 38.40 38.23 38.19 38.25 38.24

30.10 30.11 30.07 29.99 30.33 29.32 29.59 30.15 30.11 29.98 30.22 30.18 30.04 30.02 30.19 30.02

6.16 6.02 6.01 6.00 6.07 5.86 5.92 6.03 6.02 6.00 6.04 6.04 6.01 6.00 6.04 6.00

4.36 4.12 4.03 3.91 3.09 3.31 3.59 4.54 4.46 4.29 4.55 4.45 4.28 3.64 3.71 3.76

3.27 3.37 3.29 3.19 2.60 2.44 3.59 3.38 3.33 3.22 3.41 3.34 3.23 – – –

r 1yy

r 2yy

r 3yy

s1xz

s2xz

Table 2 Maximum deflection and stresses of a square sandwich plate under uniform load (R = 10). Method

 w

r 1xx

r 2xx

r 3xx

(a/2, b/2, 0) (a/2, b/2, h/2) (a/2, b/2, 2h/5) (a/2, b/2, 2h/5) (a/2, b/2, h/2) (a/2, b/2, 2h/5) (a/2, b/2, 2h/5) (0, b/2, 0) (0, a/2, 2h/5) Srinivas [69], exact solution Present, layerwise, DOF = 7, N = 21 Present, layerwise, DOF = 7, N = 15 Present, layerwise, DOF = 7, N = 11 Pandya and Kant HSDT [70] Pandya and Kant FSDT [70] Ferreira and Barbosa [71] Roque et al. [57], layerwise, N = 21 Roque et al. [57], layerwise, N = 15 Roque et al. [57], layerwise, N = 11 Ferreira et al. [28], layerwise, N = 21 Ferreira et al. [28], layerwise, N = 15 Ferreira et al. [28], layerwise, N = 11 Xiang et al. (Levinson) [58] Xiang et al. (Touratier) [58] Xiang et al. (Karama) [58]

159.38 158.84 158.16 156.98 152.33 131.10 159.40 159.50 159.29 158.32 155.03 154.77 153.83 152.66 153.14 153.36

65.33 65.17 65.06 64.94 64.65 67.80 64.16 65.28 65.22 64.93 65.37 65.29 65.00 65.01 65.05 65.10

48.86 48.68 48.58 48.36 51.31 54.24 47.72 48.28 48.72 48.51 49.82 49.76 49.55 49.68 50.21 49.50

4.90 4.87 4.86 4.84 5.13 4.42 4.77 4.88 4.87 4.85 4.98 4.98 4.96 4.97 5.02 4.95

43.57 43.64 43.61 43.59 42.83 40.10 42.97 43.68 43.64 43.45 43.27 43.22 43.03 42.95 43.02 43.06

33.41 33.49 33.45 33.35 33.97 32.08 42.90 33.52 33.49 33.35 33.60 33.56 33.42 33.39 33.65 33.38

3.50 3.35 3.34 3.33 3.40 3.21 3.29 3.35 3.35 3.34 3.36 3.36 3.34 3.34 3.37 3.34

4.10 3.93 3.84 3.72 3.15 3.15 3.52 4.29 4.22 4.06 4.28 4.21 4.05 3.45 3.64 3.84

3.52 3.53 3.45 3.35 2.60 2.68 3.52 3.67 3.62 3.49 3.67 3.63 3.51 – – –

r 1yy

r 2yy

r 3yy

s1xz

s2xz

Table 3 Maximum deflection and stresses of a square sandwich plate under uniform load (R = 15). Method

 w

r 1xx

r 2xx

r 3xx

(a/2, b/2, 0) (a/2, b/2, h/2) (a/2, b/2, 2h/5) (a/2, b/2, 2h/5) (a/2, b/2, h/2) (a/2, b/2, 2h/5) (a/2, b/2, 2h/5) (0, b/2, 0) (0, a/2, 2h/5) Srinivas [69], exact solution Present, layerwise, DOF = 7, N = 21 Present, layerwise, DOF = 7, N = 15 Present, layerwise, DOF = 7, N = 11 Pandya and Kant HSDT [70] Pandya and Kant FSDT [70] Ferreira and Barbosa [71] Roque et al. [57], layerwise, N = 21 Roque et al. [57], layerwise, N = 15 Roque et al. [57], layerwise, N = 11 Ferreira et al. [28], layerwise, N = 21 Ferreira et al. [28], layerwise, N = 15 Ferreira et al. [28], layerwise, N = 11 Xiang et al. (Levinson) [58] Xiang et al. (Touratier) [58] Xiang et al. (Karama) [58]

121.72 121.37 120.89 120.07 110.43 90.85 121.82 121.88 121.68 120.95 115.46 115.27 114.57 113.09 113.96 114.59

66.79 66.60 66.49 66.36 66.62 70.04 65.65 66.73 66.66 66.38 66.87 66.80 66.51 66.54 66.54 66.62

48.30 48.10 48.00 47.75 51.97 56.03 47.09 48.20 48.15 47.96 50.04 49.99 49.78 50.04 50.68 49.66

3.24 3.21 3.20 3.18 3.47 3.75 3.14 3.21 3.21 3.20 3.34 3.33 3.32 3.34 3.38 3.31

46.42 46.53 46.51 46.51 44.92 41.39 45.85 46.59 46.54 46.33 45.72 45.67 45.47 45.29 45.43 45.55

34.96 35.07 35.03 34.91 35.41 33.11 34.42 35.11 35.07 34.93 35.15 35.11 34.96 34.90 35.28 34.92

2.49 2.34 2.34 2.33 2.36 2.21 2.29 2.34 2.34 2.33 2.34 2.34 2.33 2.33 2.35 2.33

3.96 3.82 3.74 3.63 3.04 3.09 3.47 4.17 4.10 3.95 4.18 4.11 3.95 3.25 3.47 3.71

3.58 3.56 3.48 3.38 2.70 2.76 3.47 3.74 3.71 3.58 3.77 3.73 3.60 – – –

50

J.L. Mantari et al. / Computers and Structures 94–95 (2012) 45–53

0.05

0.05

0.025

0.025

z

z

0

-0.025

0

-0.025

-0.05 -0.25

-0.125

0

0.125

0.25

-0.05

-60 -50 -40 -30 -20 -10

0.05

0.05

0.025

0.025

z

z

0

-0.025

-0.05

0

10

20

30

40

50

60

σ xx

u

0

-0.025

-40

-30

-20

-10

0

10

20

30

-0.05

40

-4

-3.5

-3

-2.5

-2

σ yy

τ

-1.5

-1

-0.5

0

xz

Fig. 3. Displacement and normalized stresses for a sandwich square plate (N = 21, a/h = 10, R = 5).

0.05

0.05

0.025

0.025

z

0

z

-0.025

0

-0.025

-0.05 -0.1

-0.05

0

0.05

0.1

-0.05 -70 -60 -50 -40 -30 -20 -10

u 0.05

0.05

0.025

0.025

z

z

0

10 20 30 40 50 60 70

xx

0

-0.025

-0.025

-0.05 -50

0

σ

-40

-30

-20

-10

0

σ yy

10

20

30

40

50

-0.05 -4

-3.5

-3

-2.5

-2

τ

-1.5

-1

-0.5

0

xz

Fig. 4. Displacement and normalized stresses for a sandwich square plate (N = 21, a/h = 10, R = 15).

multiquadrics [28], higher-order shear formulation with multiquadrics [72] and meshless method and various shear deformation theories [58].

The maximum deflection and stress are non-dimensionalized as follows:

51

J.L. Mantari et al. / Computers and Structures 94–95 (2012) 45–53







 a a 0:999781 a b h 1 a b 2h 1  ¼ w ; ;0  1xx ¼ r1xx ; ;   2xx ¼ r1xx ; ; w ; r ; r ; 2 2 hq 2 2 2 q 2 2 5 q       a b 2h 1 a b h 1 a b 2h 1  1yy ¼ r1yy ; ;  2yy ¼ r1yy ; ; ; r ; r ; r 3xx ¼ r2xx ; ; 2 2 5 q 2 2 2 q 2 2 5 q      a 1 a b 2h 1 a 2h 1 1xz ¼ s2xz 0; ;0 ; s 2xz ¼ s2xz 0; ; r 3yy ¼ r2yy ; ; ; s ; 2 2 5 q 2 q 2 5 q ð22a — iÞ

The results are given in Tables 1–3, where various values of R are considered (5, 10 and 15). The present formulation produces results that are in good agreement with all higher-order formulations and in good agreement with the exact results, for the values of R considered as in given in Tables 1–3 and in Figs. 3 and 4. Table 1 shows that the present results are better in normal and shear stresses (r1xz and r2xz ) than Pandya and Kant [70] and Ferreira and Barbosa [71]. As expected, the presented layerwise finite element formulation is also good agreement with meshless layerwise formulation provided by Roque et al. [57], and gives better results than the equivalent single layer (ESL) finite element formulation provided by Ferreira et al. [28] and Xiang et al. [58]. The ESL meshless formulation used by Ferreira et al. [28] is based on Touratier shear deformation theory [46], and the one developed by Xiang et al. [58] is based on several ESL theories (e.g. Reddy and Liu [17], Touratier [51], Karama et al. [60]). Roque et al. [57] also implemented a layerwise approach based on Touratier shear deformation theory [51]. Table 2 shows that the present model performs better in vertical deflection, normal and shear stresses, as also mentioned in Table 1. This is also visible in Table 3. However, the differences are more pronounced, therefore, it can be inferred that when R increases from 5 to 15, the present model performs much better than Pandya and Kant [70], Ferreira and Barbosa [71], Ferreira et al. [28] and Xiang et al. [58], and in good agreement with Roque et al. [57]. It is important to note that, there are underestimated results reproduced by all the ESL meshless solutions [28,58] for R = 10 and more pronounced for R = 15, as given in Table 3. This can be attributed to the typical difficulty of higher order shear deformation theories in absorbing the distortion of the normal [28]. For such cases, layerwise approaches as the one presented here or the one developed by Roque et al. [57] are recommended. The displacement and normalized stresses for a sandwich square plate for R = 5 and R = 15 are presented in Figs. 3 and 4. It can be noticed in both figures that the differences in mechanical properties between the core and the faces at the boundary interfaces. Figs. 3 and 4 also show the distribution of the transverse shear stress, and it shows good agreement with the exact solution provided by Srinivas [69], which indicates that the present trigonometric layerwise formulation behaves very well. Similar comments were inferred by Roque et al. [57] for the trigonometric layerwise shear deformation theory implemented by Arya et al. [54]. Finally, an assessment of the present theory can be done by considering the normal stress component rz. The present model can further be extended to cover different multilayered structures such as shells. Dynamic and buckling calculations can be also performed.

5. Conclusions A layerwise finite element formulation of a newly developed higher-order shear deformation theory for the flexure of thick multilayered panels is presented. This paper presents for the first time a tangential trigonometric shear deformation theory for modelling the laminates composites and sandwiches. The discrete element

chosen is a four-nodded quadrilateral with seven-degrees-of-freedom per node. The accuracy of the present theory is ascertained by comparing it with various available results in the literature. The numerical results show that the present model performs very well as other existing higher order layerwise deformation theories for analyzing the global and interlaminar mechanical behaviour of multilayered sandwich and composite plates. Therefore, the proposed theory may be extended to other types of multilayered structures such as shells. Additionally, another advanced numerical calculation, such as meshless methods can be implemented based on the present higher-order layerwise shear deformation theory. Acknowledgments The first and second authors have been financed by the Portuguese Foundation of Science and Technology under the contract numbers SFRH/BD/66847/2009 and SFRH/BPD/47687/2008, respectively. Appendix A. Definition of matrices given in Eq. (10a–c)

2 @N

0

@x

6 6 Bkm ¼ 6 0 4

@N @y

@N @y

@N @x

0 0 0 0 0

7 0 0 0 0 07 7; 5 0 0 0 0 0

2

Bkm

0 0 0 Ak @N @x 6 6 0 ¼ 40 0 0 0 0 0 Ak @N @y 2

0 0 0  @N @x

6 0 0 0 0 Bkb ¼ 6 4 0 0 0  @N @y 2

3

0

Bk @N @x

Ak @N @y

0

Ak @N @x

Bk @N @y

0

C k @N @x

 @N @y

0

 @N @x

C k @N @y

@N @x

0 0 0 0 0

6 0 0 0 0 0 Bkb ¼ 6 4 0 0 0 0 0

0 @N @y

0

0

3

7 7 Bk @N @y 5; @N Bk @x 3

0

7 7 C k @N @y 5; @N C k @x

3

@N 7 7 @y 5

ðA:1 a — dÞ

@N @x

and

" Bks ¼

0 0

Bks ¼

m¼

0 0



@N @x @N @y

N

0

CkN

0

0

N

0

CkN

0 0 0 0 0 N

0

0 0 0 0 0

N

p 2h

;

0

# ;

 :

  h2 p ¼ 2mR þ mðR  1Þ sec 2 m ; 2

ðA:2 a — bÞ

ðA:3Þ

Layer 1

a1 ¼ p



h1 þh2 4h

;

2 ; A1 ¼ h1 þh 2

B1 ¼

p h22

ðA:4 a — dÞ þ mh1 ;

C 1 ¼ 2m;

52

J.L. Mantari et al. / Computers and Structures 94–95 (2012) 45–53

Layer 2

a2 ¼ 0; A2 ¼ 0; B2 ¼ 0;

ðA:5 a — dÞ

C 2 ¼ p; Layer 3

a3 ¼ p



h2 þh2 4h

;

3 A3 ¼  h2 þh ; 2

ðA:6 a — dÞ

B3 ¼ p h22  mh3 ; C 3 ¼ 2m:

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