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An improved higher order zigzag theory for the static analysis of laminated sandwich plate with soft core
Finite Elements in Analysis and Design 44 (2008) 602 -- 610
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An improved higher order zigzag theory for the static analysis of laminated sandwich plate with soft core Mihir K. Pandit a , Abdul H. Sheikh b , Bhrigu N. Singh c,∗ a Department
of Ocean Engineering and Naval Architecture, Indian Institute of Technology, Kharagpur 721302, India of Civil, Environmental and Mining Engineering, The University of Adelaide, SA 5005, Australia c Department of Aerospace Engineering, Indian Institute of Technology, Kharagpur 721302, India b School
A R T I C L E
I N F O
Article history: Received 18 August 2007 Received in revised form 11 January 2008 Accepted 24 February 2008 Available online 7 April 2008 Keywords: Finite element analysis Laminates Plate theory Sandwich Static analysis
A B S T R A C T
An improved higher order zigzag theory is proposed for the static analysis of laminated sandwich plate with soft compressible core. The variation of in-plane displacements is assumed to be cubic for both the face sheets and the core and transverse displacement is assumed to vary quadratically within the core while it remains constant through the faces. The core is considered to behave as a three-dimensional elastic medium to incorporate the effect of transverse normal deformation. A computationally efficient C 0 finite element is also proposed for this model. Numerical examples of laminated composite and sandwich plate are provided for different thickness ratios and aspect ratio to illustrate the accuracy of the present formulation by comparing the present results with the three-dimensional elasticity solutions. Some new results are also presented. The performance of the present model is excellent in calculating displacements and stresses for a wide range of sandwich plate problems with transversely flexible core. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Composite laminate and sandwich have been widely used to build large proportions of aerospace, underwater and automotive structures mainly because of their high strength-to-weight and stiffnessto-weight ratios. Laminated composite structures are made up of layers of orthotropic materials that are bonded together. The properties of a laminate can be tailored to meet specific design requirement by altering the material, or orientation, or both, of each layer. On the other hand, the sandwich construction is a special type of laminated structures in which the inner layers are replaced by low strength thicker core and the outer layers are high strength laminated faces. The effect of shear deformation in laminated composite plates is significant due to its low shear modulus compared to extensional rigidity as well as variation of material properties between the layers. The situation is much more severe in sandwich construction where a wide variation of material properties is found between the core and face layers. Actually, simple plate theories, i.e. single layer plate theories [1--4] are not sufficient to accurately predict the behavior of the sandwich plate, as the core and face sheets deform in a different manner due to a wide variation of their material properties. Considering this aspect and to model some other features of thick
∗
Corresponding author. Tel.: +91 3222 283026; fax: +91 3222 255303. E-mail address: [email protected] (B.N. Singh).
0168-874X/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.finel.2008.02.001
laminates in a better manner, a number of layer wise plate theories [5--9] have been proposed. These plate theories are the most accurate as the variation of in-plane displacement along the plate thickness can be represented appropriately allowing the desired shear strain jump at the layer interfaces. But the problem with these theories is that the number of nodal unknowns becomes very large in a multilayered laminate as it depends on the number of layers. To retain the advantage of single layer theories as well as layer wise theories a new type of plate theory has been proposed [10--12]. In these theories, the in-plane displacements have a piecewise linear variation across the plate thickness (zigzag). Later, Di Sciuva [13], Bhaskar and Varadan [14] and Cho and Parmerter [15] considered the variation of in-plane displacements to be the superposition of a linearly varying piecewise field on an overall higher order variation of in-plane displacements. In this context, a further improvement has been done by Cho and Parmerter [16] who have combined the Reddy's simple higher order plate theory [4] and the layer-wise zigzag plate theory of [13]. This plate theory possesses all the merits of the above mentioned theories. Therefore, these plate theories are most appropriate for the analysis of sandwich plates as the discontinuity in transverse shear strain at the interfaces is more prominent in comparison to laminated plates. In addition to transverse shear strain jump at core face sheet interfaces, transverse normal strain within the core becomes significant in some situations where the plate thickness/span ratio becomes quite large and core material is transversely soft or flexible.
M.K. Pandit et al. / Finite Elements in Analysis and Design 44 (2008) 602 -- 610
603
z znu nu
hf wu
αxu
nu-1
zn-1u
nu-1
zn-2u h/2
2
αxu1 u
1
x
z1u
-2
hf
z1l αxl
1 l
αxl
zn-1l nl-1
znl
lower) and H(z − ziu ) and H(−z + zjl ) are the unit step functions.
In fact transverse flexibility of low-strength core seriously affects the overall behavior of sandwich construction and also it is crucial for sandwich structures which are subjected to localized loads or concentrated loads. So the conventional modeling techniques adopted by different researchers [17--21] are not sufficient to accurately predict the sandwich construction with soft core. A considerable amount of literature is available on the static analysis of the sandwich plate without taking into account the effect of transverse normal deformation for the core [22--26]. However, a limited study is available on the analysis of the sandwich plate considering the effect of transverse normal deformation for the core [27--34]. Thus, the effect of transverse normal deformation of the core should be taken into account in order to accurately predict the behavior of the sandwich plate. In the present study, an attempt has been made to propose an improved plate model for the analysis of laminated sandwich plate considering the effect of the transverse normal deformation of the core. The in-plane displacement fields have been assumed as a combination of a linear zigzag model with different slopes in each layer and a cubically varying function over the entire thickness. The out of plane displacement has been assumed to be quadratic within the core and constant throughout the faces. The plate model is implemented with a computationally efficient C 0 finite element developed for this purpose and applied to solve a number of sandwich plate problems. 2. Mathematical formulation The in-plane displacement field (Fig. 1) is similar to that of Cho and Parmarter [16] as follows: n u −1
(z − ziu )H(z − ziu )ixu
i=1
+
V = v + zy +
+
j=1
for core
= wu
for upper faces
= wl
for lower faces,
(3)
where wu , w and wl are the values of the transverse displacement at the top layer, the middle layer and the bottom layer of the core, respectively, and l1 , l2 and l3 are Lagrangian interpolation functions in the thickness co-ordinate. The stress--strain relationship of an orthotropic layer/lamina (say kth layer) having any fiber orientation with respect to structural axes system (x.y.z) may be expressed as ⎫ ⎡ ⎫ ⎧ ⎤ ⎧ x ⎪ x ⎪ Q 11 Q 12 Q 13 Q 16 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢Q ⎪ ⎪ ⎪ ⎪ ⎥ ⎪ ⎪ ⎪ Q Q Q 0 0 y ⎪ ⎢ 12 y ⎪ ⎪ ⎪ 22 23 26 ⎥ ⎪ ⎪ ⎪ ⎬ ⎢Q ⎨ ⎪ ⎬ ⎨ ⎪ ⎥ Q Q Q 0 0 z z 13 23 33 36 ⎥ =⎢ or ⎥ ⎢ xy ⎪ 0 0 ⎥ ⎪ xy ⎪ ⎪ ⎪ ⎢ Q 16 Q 26 Q 36 Q 66 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ 0 ⎪ ⎪ ⎪ ⎪ 0 0 0 Q 55 Q 54 ⎦ ⎪ ⎪ ⎪ xz ⎪ ⎪ ⎪ xz ⎪ ⎭ ⎩ ⎭ ⎩ yz 0 0 0 0 Q 54 Q 44 k yz {} = [Q k ]{},
(4)
where {}, {} and [Q k ] are the stress vector, the strain vector and the transformed rigidity matrix of kth lamina, respectively [35]. Utilizing the condition of zero transverse shear stress or zero transverse shear strain at the top and bottom surfaces of the plate and imposing the condition of the transverse shear stress continuity at the interfaces between the layers, x , y , x , y , ixu , ixl , iyu and iyl may be expressed as
(5)
nu −1 1 2 xl xl · · · xll where {} = {x x y y 1xu 2xu · · · xu
(1)
n u −1 i=1
nl −1
W = l1 wu + l2 w + l3 wl
n −1
j (z − zjl )H(−z + zjl )xl + x z 2 + x z 3 ,
j=1
The transverse displacement (Fig. 2) is assumed to vary quadratically over the core thickness and constant over the face sheets and it may be expressed as
{} = [A]{},
nl −1
where u and v denote the in-plane displacement of any point on the midplane, x and y are the rotations of the normal to the midplane about y- and x-axes, respectively, nu and nl are number of upper and lower layers, respectively, x , y , x and y are the higher order unknowns, ixu , ixl , iyu and iyl are the slopes of ith layer (upper and
Fig. 1. General lamination scheme and displacement configuration.
U = u + zx +
Fig. 2. Variation of transverse displacement (W) through the thickness of laminated sandwich plate.
z2
zn-2l -nl-1 -nl
wl
x
-1 h/2
hc
w
z2u
properties. Using the above equation, the in-plane displacement field as given in Eqs. (1) and (2) may be expressed as follows:
(z − ziu )H(z − ziu )iyu j
(z − zjl )H(−z + zjl )yl + y z 2 + y z 3 ,
1yu nl −1 T nu −1 1 2 2 yu · · · yu yl yl · · · yl } , {} = {u v w x y wu,x wu,y T wl,x wl,y } and the elements of [A] are dependent on the material
(2)
U = b1 u + b2 v + b3 w + b4 x + b5 y + b6 wu,x + b7 wu,y + b8 wl,x + b9 wl,y ,
(6)
604
M.K. Pandit et al. / Finite Elements in Analysis and Design 44 (2008) 602 -- 610
V = c1 u + c2 v + c3 w + c4 x + c5 y + c6 wu,x + c7 wu,y + c8 wl,x + c9 wl,y ,
3. Finite element implementation (7)
where the coefficients bi 's and ci 's are function of thickness coordinates, unit step functions and material properties. The displacement field as expressed in Eqs. (6) and (7) contains first order derivatives of wu and wl , which usually requires C 1 continuous shape functions for finite element approximation. In order to avoid the usual difficulties associated with satisfaction of C 1 continuity, wu,x , wu,y , wl,x and wl,y are also considered as independent field variables like wu and wl . The concept is similar to that adopted by Shankara and Iyengar [36] to develop a C 0 continuous plate element based on simple third order plate theory of Reddy [4]. For the sake of convenience, the derivatives of wu and wl are expressed as follows: wu,x = ux ,
wu,y = uy ,
wl,x = lx ,
wl,y = ly .
(8)
With this, the in-plane displacement field may be rewritten as U = b1 u + b2 v + b3 w + b4 x + b5 y + b6 ux + b7 uy + b8 lx + b9 ly ,
(9)
V = c1 u + c2 v + c3 w + c4 x + c5 y + c6 ux + c7 uy + c8 lx + c9 ly .
(10)
The generalized displacement vector { } for the present plate model can now be written with help of Eqs. (3), (9) and (10) as { } = {u v w x y ux uy lx ly wu wl }T . Using linear strain--displacement relation and Eqs. (1)--(5), the strain vector {} may be expressed in terms of unknowns (for the structural deformation) as
jU jV jW jU jV jU jW jV jW T + + + jx jy jz jy jx jz jx jz jy {} = [H]{},
{} =
In the present problem, a nine-noded quadrilateral C 0 isoparametric element with 11 degrees of freedom (u, v, w, x , y , ux , uy , lx , ly , wu and wl ) per node is employed. Fig. 3 shows the node numbering and natural co-ordinates of the element. Using this finite element, the generalized displacement vector at any point may be expressed as =
n i=1
Ni i ,
(16)
where = {u v w x y ux uy lx ly wu wl }T as defined earlier, i is the displacement vector corresponding to node i, Ni is the shape function [37] associated with the node i and n is the number of nodes per element, which is nine in the present study. With the help of Eq. (16), the strain vector {} that appeared in Eq. (11) may be expressed in terms of containing nodal degrees of freedom as {} = [B]{ },
(17)
where [B] is the strain--displacement matrix in the Cartesian coordinate system. The elemental potential energy as given in Eq. (12) may be rewritten with the help of Eqs. (13)--(17) as 1 { }T [B]T [D][B]{ } dx dy − { }T [N w ]T q dx dy e 2 1 = { }T [Ke ]{ } − { }T {Pe }, 2
=
(18)
where
or (11)
[Ke ] =
where
{Pe } =
[B]T [D][B] dx dy,
(19)
[N w ]T q dx dy.
(20)
{} = [u v w x y ux uy lx ly wu wl u,x The equilibrium equation can be obtained by minimizing in the above equation with respect to { } as
v,x w,x x,x y,x ux,x uy,x lx,x ly,x wu,x wl,x u,y v,y w,y x,y y,y ux,y ux,y lx,y ly,y wu,y wl,y ]T
e as given
[Ke ]{ } = {Pe },
and the elements of [H] are functions of z and unit step functions which are given in Appendices A. With the quantities found in the above equations, the total potential energy of the system under the action of transverse load may be expressed as = Us − Wext. , (12) e
(21)
where [Ke ] is the element stiffness matrix and {Pe } is the nodal load vector.
η
7 (-1, 1)
6 (0, 1)
5 (1, 1)
where U is the strain energy and Wext. is the work done by the external transverse load. Using Eqs. (4) and (11), the strain energy is given by Us =
n 1 1 {}T [Q k ]{} dx dy dz = {}T [D]{} dx dy, 2 2
(13)
8 (-1, 0)
k=1
where [D] =
n
[H]T [Q k ][H] dz.
9 (0, 0)
4 (1, 0)
2 (0, -1)
3 (1, -1)
ξ
(14)
k=1
The work done by a distributed transverse static load of intensity q(x, y) is Wq dx dy. (15) Wext. =
1 (-1, -1)
Dofs: u, v, w, x, y , ux, uy, lx, ly, wu , wl. Fig. 3. Node numbers and natural co-ordinates of a nine-noded isoparametric element.
M.K. Pandit et al. / Finite Elements in Analysis and Design 44 (2008) 602 -- 610
Integrations of the Eqs. (19) and (20) are carried out numerically following Gauss quadrature integration rule. A reduced integration technique is adopted for the evaluation of stiffness matrix in order to avoid any possible numerical disturbances such as shear locking, which may appear with a full integration scheme. After evaluating the stiffness matrices for all elements, they have been assembled together to form the overall stiffness matrix [K] of the plate. The storage of [K] has been done in single array following skyline storage technique. 4. Numerical results and discussion
Table 1 Material properties used for the core and face sheets Location
Elastic properties
Core Face
E1
E2
E3
G12
G13
G23
12
0.04E 25E
0.04E E
0.5E --
0.016E 0.5E
0.06E 0.5E
0.06E 0.2E
0.25 0.25
Table 2 Non-dimensional central deflection (w) of a simply supported rectangular sandwich plate with laminated facings (0/90/C/0/90) under sinusoidal load References
(x , y , z , xy ) = (xz , yz ) =
h2 q0 a2
(x , y , z , xy ),
h (xz , yz ), q0 a
w=
100wE2 h3 q0 a4
.
4.1. Simply supported rectangular laminated sandwich plate under distributed load of sinusoidal variation A rectangular sandwich (0/90/C/0/90) plate (Fig. 4) is subjected to a distributed load of intensity q = q0 sin( x/a) sin( y/b). It has a total thickness of h where the thickness of the core is 0.8h and that of each ply in the top and bottom face sheets is 0.05h. The plate is
Aspect ratio (b/a) 1.0
Present (4 × 4)a Present (6 × 6) Present (8 × 8) Present (10 × 10) Present (12 × 12) 3-D elasticity [28] % Error a Mesh
2.0
h/a = 0.1
h/a = 0.2
h/a = 0.1
h/a = 0.2
1.7270 1.7256 1.7253 1.7252 1.7252 1.7272 0.11
4.2577 4.2528 4.2520 4.2518 4.2517 4.2447 0.16
3.1923 3.1897 3.1893 3.1892 3.1891 3.1944 0.16
7.3795 7.3711 7.3697 7.3693 7.3692 7.3727 0.04
size.
Sandwich plate (0/90/C/0/90) h/a = 0.20 0.6
0.4
Normalised depth
In order to demonstrate the accuracy and applicability of the present formulation, several examples of the laminated composite plates and sandwich plates subjected to static load have been analyzed and the results obtained have been compared with the published results. The restrained degrees of freedom for different types of boundary conditions such as clamped and simply supported conditions are as follows. Boundary line parallel to x-axis: Simply supported condition: u = w = x = ux = lx = wu = wl = 0. Clamped condition: u = v = w = x = y = ux = uy = lx = ly = wu = wl = 0. Boundary line parallel to y-axis: Simply supported condition: v = w = y = uy = ly = wu = wl = 0. Clamped condition: u = v = w = x = y = ux = uy = lx = ly = wu = wl = 0. The following non-dimensional quantities used in the present analysis are defined as
605
0.2
Present 3-D Elasticity [28]
0.0
-0.2
-0.4
-0.6 -1.2
-0.9
-0.6 -0.3 0.0 0.3 0.6 0.9 1.2 Non-dimensional in-plane normal stress
1.5
Fig. 5. Variation of non-dimensional in-plane normal stress across the depth of a sandwich plate.
y
n
b
2 1 2 3
m a
Fig. 4. A rectangular plate having a mesh of m × n.
x
analyzed by taking aspect ratio b/a = 1.0 and 2.0 and thickness ratios h/a = 0.1 and 0.2 using different mesh sizes. The material properties used for the core and each laminated face sheets are given in Table 1. The non-dimensional values of central deflection w( 2a , 2b , 0) obtained in the present analysis are presented with those obtained from the three-dimensional elasticity solution [28] in Table 2. As regard to three-dimensional elasticity solution, it is relevant to mention that the numerical results based on elasticity solution used here for the comparison are not readily available in the paper of Pagano [28]. Therefore, these results are generated by writing a separate code based on the analytical solution [28]. The present results are in very good agreement with the elasticity solution. The table also shows that the convergence of the results with mesh refinement is excellent. The variations of non-dimensional in-plane normal stress x = x h2 /(q0 a2 ) at the plate center through the thickness of the plate, inplane shear stress xy =xy h2 /(q0 a2 ) at a corner of the plate obtained in the present analysis (mesh size: 10 × 10, b/a = 1 and h/a = 0.2)
606
M.K. Pandit et al. / Finite Elements in Analysis and Design 44 (2008) 602 -- 610
are plotted in Figs. 5 and 6 with the three-dimensional elasticity solution [28]. The figures show that the present results are in good agreement with the elasticity solution. 4.2. Simply supported square sandwich plate under sinusoidal load A square sandwich (0/C/0) plate, simply supported at all four edges and subjected to distributed load of intensity q = q0 sin( x/a) sin( y/b) is analyzed by taking thickness ratios, h/a = 0.01, 0.02, 0.05 and 0.25. The thickness of each face is 0.1h. The material properties are as given in Table 1. The deflection, the normal stresses and the shear stresses at some important points
0.50
Sandwich plate (0/90/C/0/90) h/a = 0.2
Normalised depth
0.25 3-D Elasticity [28] Present 0.00
-0.25
-0.50 -0.06
0.02 -0.04 -0.02 0.00 0.04 Non-dimensional in-plane shear stress
0.06
Fig. 6. Variation of non-dimensional in-plane shear stress across the depth of a sandwich plate.
obtained by the present formulation are presented in Table 3 along with the three-dimensional elasticity solution of Pagano [28]. The study has been performed for different mesh divisions to show the convergence characteristics. In general present results are close to the elasticity solution with excellent convergence. 4.3. Sandwich plate having different boundary conditions under distributed load of sinusoidal variation A sandwich plate (0/90/C/0/90) subjected to a distributed load of intensity q = q0 sin( x/a) sin( y/b) is analyzed in this example. It has a total thickness of h where the thickness of the core is 0.8h and that of each ply in the top and bottom face sheets is 0.05h. The study has been made for two types of boundary conditions. These are SCSC, i.e. two opposite edges simply supported and other two edges clamped and CCCC, i.e. all edges clamped. The analysis is performed for different thickness ratios (h/a = 0.01, 0.05, 0.10, 0.20 and 0.50). The material properties used for the core and each laminated face sheets are given in Table 1. The non-dimensional values of central deflection w( 2a , 2b , 0) and the stresses calculated at the important points are presented in Table 4. It may be noted from the results that the percentage increase in the transverse displacement for higher thickness ratios, i.e. from 0.20 to 0.50 is much higher than those for lower thickness ratios, i.e. from 0.01 to 0.05 for both the boundary conditions. The major cause behind it may be that for higher thickness ratios the effect of transverse flexibility of the core and shear deformation effects is more pronounced as compared with that of the lower thickness ratios. Moreover, core occupies major portion (80%) of the plate thickness and so, core compressibility plays an important role for highly thick plates (h/a = 0.5) to get high values of transverse displacement which is reflected in the present results. Same kind of observations may also be noted for the stress values except the transverse shear stress (yz ) which decreases as the thickness
Table 3 Non-imensional deflection (w) and stresses (x , y , xz , yz , xy ) at the important points of a simply supported square sandwich (0/C/0) plate under sinusoidal load h/a
References
w
a b , ,0 2 2
x
a b h , , 2 2 2
y
a b h , , 2 2 2
xz 0,
b ,0 2
yz a , 0, 0 2
xy 0, 0,
h 2
0.01
Present (4 × 4) Present (6 × 6) Present (8 × 8) Present (10 × 10) Present (12 × 12) 3-D Elasticity [28]
0.8921 0.8918 0.8917 0.8917 0.8917 0.8923
1.0643 1.0828 1.1089 1.1092 1.1093 1.098
0.0532 0.0542 0.0545 0.0546 0.0547 0.0550
0.3323 0.3379 0.3398 0.3407 0.3412 0.324
0.0315 0.03212 0.0323 0.0323 0.0324 0.0297
0.0423 0.0430 0.0433 0.0434 0.0434 0.0433
0.02
Present (4 × 4) Present (6 × 6) Present (8 × 8) Present (10 × 10) Present (12 × 12) 3-D Elasticity [28]
0.9346 0.9343 0.9342 0.9341 0.9341 0.9348
1.0658 1.0843 1.0905 1.0933 1.0948 1.099
0.0551 0.0561 0.0564 0.0565 0.0566 0.0569
0.3314 0.3370 0.3389 0.3398 0.3403 0.323
0.0324 0.0330 0.0331 0.0332 0.0333 0.0306
0.0432 0.4403 0.0442 0.0444 0.0445 0.0446
0.05
Present (4 × 4) Present (6 × 6) Present (8 × 8) Present (10 × 10) Present (12 × 12) 3-D Elasticity [28]
1.2263 1.2257 1.2254 1.2254 1.2254 1.2264
1.0763 1.0949 1.1012 1.1040 1.1055 1.110
0.0676 0.0687 0.0691 0.0693 0.0694 0.0700
0.3255 0.3311 0.3329 0.3338 0.3342 0.317
0.0381 0.0388 0.0390 0.0391 0.0392 0.0361
0.0496 0.0504 0.0507 0.0508 0.0509 0.0511
0.10
Present (4 × 4) Present (6 × 6) Present (8 × 8) Present (10 × 10) Present (12 × 12) 3-D Elasticity [28]
2.2025 2.2006 2.2003 2.2002 2.2002 2.2004
1.1179 1.1373 1.1438 1.1467 1.1483 1.153
0.1058 0.1076 0.1082 0.1084 0.1086 0.1104
0.3076 0.3128 0.3146 0.3154 0.3158 0.300
0.0555 0.0564 0.0567 0.0569 0.0570 0.0527
0.06913 0.0703 0.0707 0.0709 0.0709 0.0707
0.25
Present (4 × 4) Present (6 × 6) Present (8 × 8) Present (10 × 10) Present (12 × 12) 3-D Elasticity [28]
7.6658 7.6571 7.6557 7.6553 7.6552 7.5962
1.4820 1.5073 1.5158 1.5197 1.5218 1.556
0.2440 0.2481 0.2495 0.2502 0.2506 0.2595
0.2453 0.2495 0.2509 0.2516 0.2520 0.239
0.1126 0.1145 0.1151 0.1154 0.1156 0.1072
0.1430 0.1454 0.1462 0.1466 0.1468 0.1437
M.K. Pandit et al. / Finite Elements in Analysis and Design 44 (2008) 602 -- 610
607
Table 4 Non-dimensional deflection (w) and stresses (x , y , xz , yz , xy ) at the important points square sandwich (0/90/C/0/90) plate with different boundary conditions h/a
w
Boundary conditions
a b , ,0 2 2
x a b h , , 2 2 2
y
a b h , , 2 2 2
xz 0,
b ,0 2
yz a , 0, 0 2
xy h 0, 0, 2
0.01
SCSC CCCC
0.3453 0.2286
0.4077 0.4270
0.0326 0.0228
0.0778 0.2189
0.3086 0.2189
0.0036 0.0005
0.05
SCSC CCCC
0.6052 0.4296
0.5850 0.4275
0.0334 0.0236
0.1061 0.1828
0.2527 0.1828
0.0107 0.0018
0.10
SCSC CCCC
1.3026 1.0489
0.8310 0.4597
0.0372 0.0279
0.1418 0.1587
0.1967 0.1586
0.0189 0.0036
0.20
SCSC CCCC
3.8087 3.4521
1.1415 0.6170
0.0561 0.0470
0.1683 0.1396
0.1539 0.1394
0.0292 0.0074
0.50
SCSC CCCC
19.5512 18.3454
2.3071 1.8156
0.2047 0.1902
0.1691 0.1227
0.1296 0.1217
0.0667 0.0215
Table 5 Non-dimensional deflection (w) and stresses (x , y , xz , yz , xy ) at the important points of a clamped square sandwich (0/C/0) plate under uniformly distributed load h/a
References
w
h a b , ,± 2 2 2
x a b h , ,± 2 2 2
y
a b h , ,± 2 2 2
xz 0,
b ,0 2
yz
xy
a , 0, 0 2
0, 0, ±
h 2
0.01
Present
0.2897 0.2897
0.5398 −0.5398
0.0099 −0.0099
0.5429
0.1764
−0.0025 0.0025
0.02
Present
0.3549 0.3549
0.5478 −0.5478
0.0131 −0.0131
0.5138
0.1806
−0.0040 0.0040
0.05
Present
0.7793 0.7789
0.5754 −0.5754
0.0371 −0.0371
0.4368
0.2020
−0.0089 0.0089
0.10
Present
2.0090 2.0022
0.6346 −0.6346
0.0951 −0.0952
0.3367
0.2372
−0.0171 0.0163
0.25
Present
8.2090 7.9482
1.1809 −1.1819
0.1919 −0.1923
0.2242
0.2630
−0.0401 0.0314
Table 6 Non-dimensional deflection (w) and normal stresses (x , y , z ) of a square sandwich (0/C/0) plate under concentrated load at the plate center
x
y
z
4.9408 4.8921
0.0276 −0.0269
0.0486 −0.0479
0.0278 0.0016
Upper face Lower face
5.7204 5.3474
0.0284 −0.0259
0.0525 −0.0503
0.1040 0.0056
0.05
Upper face Lower face
11.2558 8.0221
0.0311 −0.0201
0.0623 −0.0523
0.5440 0.0326
0.10
Upper face Lower face
31.2576 15.6250
0.0343 −0.0087
0.0667 −0.0452
1.6745 0.0684
h/a
Location
0.01
Upper face Lower face
0.02
w
a b , , ±0.5h 2 2
ratio increases in the case of SCSC boundary condition. Moreover, for the CCCC boundary condition the transverse shear stress (xz ) also decreases in addition to yz . It is also seen that the transverse shear stresses are of the same order of magnitude for all each thickness ratio considered for CCCC boundary condition.
a b , , ±0.4h 2 2
a b , , ±0.4h 2 2
a b , , ±0.4h 2 2
some important points obtained by the present formulation are presented in Table 5. It is easily seen from the table that transverse displacement becomes unequal at the upper and lower facesheets of the sandwich plate for the higher thickness ratios. Also, the in-plane normal and shear stresses developed at the top and bottom of the plate are of different values for plates with higher thickness ratios which is expected.
4.4. All edges clamped square sandwich plate under uniformly distributed load A square sandwich (0/C/0) plate, all four edges clamped and subjected to uniformly distributed load, is analyzed by taking different thickness ratios, h/a=0.01, 0.02, 0.05, 0.10 and 0.25. The thickness of each face is 0.1h. The material properties are as given in Table 1. The non-dimensional deflection, the in-plane normal and shear stresses at the upper and lower faces and the transverse shear stresses at
4.5. Simply supported square sandwich plate under concentrated load at the plate center In this problem a single core square sandwich (0/C/0) plate is subjected to concentrated load at the plate center. The study has been made for thickness ratios, h/a = 0.01, 0.02, 0.05 and 0.10. The thickness of each face is 0.1h. The material properties are as given
608
M.K. Pandit et al. / Finite Elements in Analysis and Design 44 (2008) 602 -- 610
in Table 1. The deflections and the transverse normal stresses at the upper and lower faces at the plate center have been computed and are presented in Table 6. It is observed from the results that the deflections are unequal in upper and lower faces and the values of the transverse normal stress are significant in magnitude in comparison to the in-plane normal stress values. The difference in the values of transverse displacements increases rapidly at higher thickness ratios which justifies its higher order variation. It may be easily noted from the stress values that the transverse normal stress is comparable in magnitude with the in-plane normal stresses. Even for higher thickness ratios it becomes larger than the in-plane normal stresses indicating its significance on the sandwich plate bending.
b3 = A13 z 2 + A23 z 3 +
n u −1 i=1
nl −1
+
j=1
A(j+nu +3)3 (z − zjl )H(−z + zjl ),
b4 = z + A14 z 2 + A24 z 3 +
n u −1 i=1
nl −1
+
j=1
n u −1 i=1
5. Conclusion
nl −1
+
An improved higher order zigzag theory is presented for the analysis of sandwich plate with soft core. The present zigzag theory satisfies shear free conditions at the top and bottom of the plate, cubic variation of the in-plane displacements, quadratic variation of the transverse displacement and interlaminar shear stress continuity at the layer interfaces. A computationally efficient finite element formulation for the improved sandwich plate model is also presented. A nine noded isoparametric element with 11 dof per node is adopted. The results obtained for the deflection and the stresses of a laminated composite and sandwich plate show excellent performance of the present formulation. Convergence of the results is very good indicating reasonably less number of elements required to get the desired results. The effect of the transverse normal deformation has been shown for soft core sandwich plate under concentrated load. Some new results for the sandwich plates have been presented for future reference in this field of study.
j=1
n u −1 i=1
+
j=1
n u −1 k=5 i=1
nl −1
j=1
n u −1 i=1
nl −1
+
j=1
0 ⎢ 0 ⎢ ⎢ 0 [H] = ⎢ ⎢ 0 ⎢ ⎣b
1,z
c1,z b5 0 0 c5 0 0
0 0 0 0 b2,z c2,z b6 0 0 c6 0 0
0 0 l2,z 0 b3,z c3,z b7 0 0 c7 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
b4,z c4,z
b5,z c5,z
b6,z c6,z
b7,z c7,z
b8,z c8,z
b9,z c9,z
b8 0 0 c8 0 0
b9 0 0 c9 0 0
0 0 0 0 l1 0
0 0 0 0 l3 0
0 c1 0 b1 0 0
0 c2 0 b2 0 0
0 c3 0 b3 0 l2
0 c4 0 b4 0 0
0 c5 0 b5 0 0
0 0
0 0
l1,z 0 0 0
l3,z 0 0 0
0 c6 0 b6 0 0
0 c7 0 b7 0 0
b1 0 0 c1 0 0 0 c8 0 b8 0 0
b2 0 0 c2 0 0 0 c9 0 b9 0 0
A(i+4)8 (z − ziu )H(z − ziu )
A(j+nu +3)8 (z − zjl )H(−z + zjl ),
b3 0 0 c3 l2 0 0 0 0 0 0 l1
A(i+4)7 (z − ziu )H(z − ziu )
A(j+nu +3)7 (z − zjl )H(−z + zjl ),
b8 = A18 z 2 + A28 z 3 +
Appendix A. Elements of matrix H
A(i+4)6 (z − ziu )H(z − ziu )
A(j+nu +3)6 (z − zjl )H(−z + zjl ),
b7 = A17 z 2 + A27 z 3 +
+
A(i+4)5 (z − ziu )H(z − ziu )
A(j+nu +3)5 (z − zjl )H(−z + zjl ),
b6 = A16 z 2 + A26 z 3 + nl −1
A(i+4)4 (z − ziu )H(z − ziu )
A(j+nu +3)4 (z − zjl )H(−z + zjl ),
b5 = A15 z 2 + A25 z 3 +
⎡
A(i+4)3 (z − ziu )H(z − ziu )
b4 0 0 c4 0 0 ⎤
0 0⎥ ⎥ 0⎥ ⎥, 0⎥ ⎥ 0⎦ l3
where
b1 = 1 + A11 z 2 + A21 z 3 +
n u −1 i=1
nl −1
+
j=1
A(j+nu +3)1 (z − zjl )H(−z + zjl ),
b2 = A12 z 2 + A22 z 3 +
n u −1 i=1
nl −1
+
j=1
A(i+4)1 (z − ziu )H(z − ziu )
A(i+4)2 (z − ziu )H(z − ziu )
A(j+nu +3)2 (z − zjl )H(−z + zjl ),
b9 = A19 z 2 + A29 z 3 +
n u −1 i=1
nl −1
+
j=1
A(j+nu +3)9 (z − zjl )H(−z + zjl ),
c1 = A31 z 2 + A41 z 3 +
n u −1 i=1
nl −1
+
j=1
A(i+4)9 (z − ziu )H(z − ziu )
A(i+nu +n +2)1 (z − ziu )H(z − ziu ) l
A(j+2nu +n +1)1 (z − zjl )H(−z + zjl ), l
M.K. Pandit et al. / Finite Elements in Analysis and Design 44 (2008) 602 -- 610
n u −1
c2 = 1 + A32 z 2 + A42 z 3 +
i=1 nl −1
+
j=1
l
n u −1
j=1
A(j+2nu +n +1)3 (z − zjl )H(−z + zjl ), n u −1 i=1
+
j=1
A(i+nu +n +2)4 (z − ziu )H(z − ziu ) l
A(j+2nu +n +1)4 (z − zjl )H(−z + zjl ), l
c5 = z + A35 z 2 + A45 z 3 +
n u −1 i=1
nl −1
+
j=1
j=1
n u −1
j=1
n u −1
j=1
n u −1
j=1
U Us v V w wu wl W Wext. ixu , iyu ixl ,iyl x , y , x , y {} {} x , y
A(i+nu +n +2)8 (z − ziu )H(z − ziu ) l
lx , ly
,
element stiffness matrix overall stiffness matrix Lagrangian interpolation functions in the thickness co-ordinate number of upper and lower layers, respectively shape function associated with the node i element load vector intensity of applied load transformed rigidity matrix of kth lamina displacement of any point at mid-plane along x-direction displacement of any point along x-direction strain energy displacement of any point at mid-plane along y-direction displacement of any point along y-direction transverse displacement at the middle of the core transverse displacement at the top layers of the core transverse displacement at the bottom layers of the core displacement at any point along z-direction work done under external transverse load change of slopes at the upper ith interface between ith and (i + 1)th layer change of slopes at the lower ith interface between ith and (i + 1)th layer higher order unknown generalized displacement vector strain vector stress vector rotation of the normal to the mid-plane about y- and x-axes, respectively rotation of the normal to the top face sheet and core interface about y- and x-axes, respectively rotation of the normal to the bottom face sheet and core interface about y- and x-axes, respectively natural co-ordinate
References
l
n u −1 i=1
nl −1
nu , nl Ni {Pe } q [Q k ] u
ux , uy
A(j+2nu +n +1)8 (z − zjl )H(−z + zjl ),
c9 = A39 z 2 + A49 z 3 +
+
l
l
i=1
+
A(i+nu +n +2)7 (z − ziu )H(z − ziu )
A(j+2nu +n +1)7 (z − zjl )H(−z + zjl ),
c8 = A38 z 2 + A48 z 3 + nl −1
l
l
i=1 nl −1
A(i+nu +n +2)6 (z − ziu )H(z − ziu )
A(j+2nu +n +1)6 (z − zjl )H(−z + zjl ),
c7 = A37 z 2 + A47 z 3 +
+
l
l
i=1 nl −1
A(i+nu +n +2)5 (z − ziu )H(z − ziu )
A(j+2nu +n +1)5 (z − zjl )H(−z + zjl ),
c6 = A36 z 2 + A46 z 3 +
+
[Ke ] [K] l1 , l2 , l3
l
c4 = A34 z 2 + A44 z 3 + nl −1
A(i+nu +n +2)3 (z − ziu )H(z − ziu ) l
i=1 nl −1
l
A(j+2nu +n +1)2 (z − zjl )H(−z + zjl ),
c3 = A33 z 2 + A43 z 3 +
+
A(i+nu +n +2)2 (z − ziu )H(z − ziu )
609
A(i+nu +n +2)9 (z − ziu )H(z − ziu ) l
A(j+2nu +n +1)9 (z − zjl )H(−z + zjl ) l
and Aij 's are the elements of the matrix A containing Q ij 's. Appendix B. List of notations a b [B] E G h H(z − ziu ),
length of the plate breadth of the plate strain--displacement matrix Young's modulus shear modulus thickness of the plate unit step functions
[H]
matrix which is a function of z and unit step functions
H(−z + zjl )
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[16] M. Cho, R.R. Parmerter, Efficient higher order composite plate theory for general lamination configurations, AIAA J. 31 (7) (1993) 1299--1306. [17] E. Reissner, On bending of elastic plates, Q. Appl. Math. 5 (1947) 55--68. [18] F.J. Plantema, Sandwich Construction, Wiley, New York, 1966. [19] H.G. Allen, Analysis and Design of Structural Sandwich Panels, Pergamon Press, Oxford, 1969. [20] B. Liaw, R.W. Little, Theory of bending multilayer sandwich plates, AIAA J. 5 (2) (1967) 301--304. [21] J.J. Azar, Bending theory for multilayer orthotropic sandwich plates, AIAA J. 6 (10) (1968) 2166--2169. [22] G.M. Foile, Bending of clamped orthotropic sandwich plates, ASCE J. Eng. Mech. 96 (1970) 243--261. [23] J.M. Whitney, Stress analysis of thick laminated composite and sandwich plates, J. Compos. Mater. 6 (1972) 525--538. [24] T.P. Khatua, Y.K. Cheung, Bending and vibration of multilayer sandwich beams and plates, Int. J. Numer. Methods Eng. 6 (1973) 11--24. [25] G.R. Monforton, I.M. Ibrahim, Analysis of sandwich plates with unbalanced cross ply faces, Int. J. Mech. Sci. 17 (1975) 227--238. [26] B.N. Pandya, T. Kant, Higher-order shear deformable theories for flexure of sandwich plates---finite element evaluations, Int. J. Solids Struct. 24 (12) (1988) 1267--1286. [27] N.J. Pagano, Exact solution of composite laminates in cylindrical bending, J. Compos. Mater. 3 (1969) 398--411.
[28] N.J. Pagano, Exact solution of rectangular bi-directional composites and sandwich plates, J. Compos. Mater. 4 (1970) 20--34. [29] D.J. O'Connor, A finite element package for the analysis of sandwich construction, Compos. Struct. 8 (1987) 143--161. [30] L.J. Lee, Y.J. Fan, Bending and vibration analysis of composite sandwich plates, Comput. Struct. 60 (1) (1996) 103--112. [31] S. Oskooei, J.S. Hansen, Higher order finite element for sandwich plates, AIAA J. 38 (3) (2000) 525--533. [32] Y. Frostig, M. Baruch, O. Vinley, I. Sheinman, High-order theory for sandwichbeam behaviour with transversely flexible core, J. Eng. Mech. 118 (5) (1992) 1026--1043. [33] O.T. Thomsen, Analysis of local bending effects in sandwich plates with orthotropic face layers subjected to localised loads, Compos. Struct. 25 (1993) 511--520. [34] T. Kant, K. Swaminathan, Analytical solutions for the static analysis of laminated composite and sandwich plates based on a higher order refined theory, Compos. Struct. 56 (2002) 329--344. [35] R.M. Jones, Mechanics of Composite Materials, McGraw-Hill, New York, 1975. [36] C.A. Shankara, N.G.R. Iyengar, A C0 element for the free vibration analysis of laminated composite plates, J. Sound Vib. 191 (5) (1996) 721--738. [37] R.D. Cook, D.S. Malkus, M.E. Plesha, R.J. Witt, Concepts and Applications of Finite Element Analysis, Wiley, New York, 2005.
Contents lists available at ScienceDirect
Finite Elements in Analysis and Design journal homepage: w w w . e l s e v i e r . c o m / l o c a t e / f i n e l
An improved higher order zigzag theory for the static analysis of laminated sandwich plate with soft core Mihir K. Pandit a , Abdul H. Sheikh b , Bhrigu N. Singh c,∗ a Department
of Ocean Engineering and Naval Architecture, Indian Institute of Technology, Kharagpur 721302, India of Civil, Environmental and Mining Engineering, The University of Adelaide, SA 5005, Australia c Department of Aerospace Engineering, Indian Institute of Technology, Kharagpur 721302, India b School
A R T I C L E
I N F O
Article history: Received 18 August 2007 Received in revised form 11 January 2008 Accepted 24 February 2008 Available online 7 April 2008 Keywords: Finite element analysis Laminates Plate theory Sandwich Static analysis
A B S T R A C T
An improved higher order zigzag theory is proposed for the static analysis of laminated sandwich plate with soft compressible core. The variation of in-plane displacements is assumed to be cubic for both the face sheets and the core and transverse displacement is assumed to vary quadratically within the core while it remains constant through the faces. The core is considered to behave as a three-dimensional elastic medium to incorporate the effect of transverse normal deformation. A computationally efficient C 0 finite element is also proposed for this model. Numerical examples of laminated composite and sandwich plate are provided for different thickness ratios and aspect ratio to illustrate the accuracy of the present formulation by comparing the present results with the three-dimensional elasticity solutions. Some new results are also presented. The performance of the present model is excellent in calculating displacements and stresses for a wide range of sandwich plate problems with transversely flexible core. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Composite laminate and sandwich have been widely used to build large proportions of aerospace, underwater and automotive structures mainly because of their high strength-to-weight and stiffnessto-weight ratios. Laminated composite structures are made up of layers of orthotropic materials that are bonded together. The properties of a laminate can be tailored to meet specific design requirement by altering the material, or orientation, or both, of each layer. On the other hand, the sandwich construction is a special type of laminated structures in which the inner layers are replaced by low strength thicker core and the outer layers are high strength laminated faces. The effect of shear deformation in laminated composite plates is significant due to its low shear modulus compared to extensional rigidity as well as variation of material properties between the layers. The situation is much more severe in sandwich construction where a wide variation of material properties is found between the core and face layers. Actually, simple plate theories, i.e. single layer plate theories [1--4] are not sufficient to accurately predict the behavior of the sandwich plate, as the core and face sheets deform in a different manner due to a wide variation of their material properties. Considering this aspect and to model some other features of thick
∗
Corresponding author. Tel.: +91 3222 283026; fax: +91 3222 255303. E-mail address: [email protected] (B.N. Singh).
0168-874X/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.finel.2008.02.001
laminates in a better manner, a number of layer wise plate theories [5--9] have been proposed. These plate theories are the most accurate as the variation of in-plane displacement along the plate thickness can be represented appropriately allowing the desired shear strain jump at the layer interfaces. But the problem with these theories is that the number of nodal unknowns becomes very large in a multilayered laminate as it depends on the number of layers. To retain the advantage of single layer theories as well as layer wise theories a new type of plate theory has been proposed [10--12]. In these theories, the in-plane displacements have a piecewise linear variation across the plate thickness (zigzag). Later, Di Sciuva [13], Bhaskar and Varadan [14] and Cho and Parmerter [15] considered the variation of in-plane displacements to be the superposition of a linearly varying piecewise field on an overall higher order variation of in-plane displacements. In this context, a further improvement has been done by Cho and Parmerter [16] who have combined the Reddy's simple higher order plate theory [4] and the layer-wise zigzag plate theory of [13]. This plate theory possesses all the merits of the above mentioned theories. Therefore, these plate theories are most appropriate for the analysis of sandwich plates as the discontinuity in transverse shear strain at the interfaces is more prominent in comparison to laminated plates. In addition to transverse shear strain jump at core face sheet interfaces, transverse normal strain within the core becomes significant in some situations where the plate thickness/span ratio becomes quite large and core material is transversely soft or flexible.
M.K. Pandit et al. / Finite Elements in Analysis and Design 44 (2008) 602 -- 610
603
z znu nu
hf wu
αxu
nu-1
zn-1u
nu-1
zn-2u h/2
2
αxu1 u
1
x
z1u
-2
hf
z1l αxl
1 l
αxl
zn-1l nl-1
znl
lower) and H(z − ziu ) and H(−z + zjl ) are the unit step functions.
In fact transverse flexibility of low-strength core seriously affects the overall behavior of sandwich construction and also it is crucial for sandwich structures which are subjected to localized loads or concentrated loads. So the conventional modeling techniques adopted by different researchers [17--21] are not sufficient to accurately predict the sandwich construction with soft core. A considerable amount of literature is available on the static analysis of the sandwich plate without taking into account the effect of transverse normal deformation for the core [22--26]. However, a limited study is available on the analysis of the sandwich plate considering the effect of transverse normal deformation for the core [27--34]. Thus, the effect of transverse normal deformation of the core should be taken into account in order to accurately predict the behavior of the sandwich plate. In the present study, an attempt has been made to propose an improved plate model for the analysis of laminated sandwich plate considering the effect of the transverse normal deformation of the core. The in-plane displacement fields have been assumed as a combination of a linear zigzag model with different slopes in each layer and a cubically varying function over the entire thickness. The out of plane displacement has been assumed to be quadratic within the core and constant throughout the faces. The plate model is implemented with a computationally efficient C 0 finite element developed for this purpose and applied to solve a number of sandwich plate problems. 2. Mathematical formulation The in-plane displacement field (Fig. 1) is similar to that of Cho and Parmarter [16] as follows: n u −1
(z − ziu )H(z − ziu )ixu
i=1
+
V = v + zy +
+
j=1
for core
= wu
for upper faces
= wl
for lower faces,
(3)
where wu , w and wl are the values of the transverse displacement at the top layer, the middle layer and the bottom layer of the core, respectively, and l1 , l2 and l3 are Lagrangian interpolation functions in the thickness co-ordinate. The stress--strain relationship of an orthotropic layer/lamina (say kth layer) having any fiber orientation with respect to structural axes system (x.y.z) may be expressed as ⎫ ⎡ ⎫ ⎧ ⎤ ⎧ x ⎪ x ⎪ Q 11 Q 12 Q 13 Q 16 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢Q ⎪ ⎪ ⎪ ⎪ ⎥ ⎪ ⎪ ⎪ Q Q Q 0 0 y ⎪ ⎢ 12 y ⎪ ⎪ ⎪ 22 23 26 ⎥ ⎪ ⎪ ⎪ ⎬ ⎢Q ⎨ ⎪ ⎬ ⎨ ⎪ ⎥ Q Q Q 0 0 z z 13 23 33 36 ⎥ =⎢ or ⎥ ⎢ xy ⎪ 0 0 ⎥ ⎪ xy ⎪ ⎪ ⎪ ⎢ Q 16 Q 26 Q 36 Q 66 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ 0 ⎪ ⎪ ⎪ ⎪ 0 0 0 Q 55 Q 54 ⎦ ⎪ ⎪ ⎪ xz ⎪ ⎪ ⎪ xz ⎪ ⎭ ⎩ ⎭ ⎩ yz 0 0 0 0 Q 54 Q 44 k yz {} = [Q k ]{},
(4)
where {}, {} and [Q k ] are the stress vector, the strain vector and the transformed rigidity matrix of kth lamina, respectively [35]. Utilizing the condition of zero transverse shear stress or zero transverse shear strain at the top and bottom surfaces of the plate and imposing the condition of the transverse shear stress continuity at the interfaces between the layers, x , y , x , y , ixu , ixl , iyu and iyl may be expressed as
(5)
nu −1 1 2 xl xl · · · xll where {} = {x x y y 1xu 2xu · · · xu
(1)
n u −1 i=1
nl −1
W = l1 wu + l2 w + l3 wl
n −1
j (z − zjl )H(−z + zjl )xl + x z 2 + x z 3 ,
j=1
The transverse displacement (Fig. 2) is assumed to vary quadratically over the core thickness and constant over the face sheets and it may be expressed as
{} = [A]{},
nl −1
where u and v denote the in-plane displacement of any point on the midplane, x and y are the rotations of the normal to the midplane about y- and x-axes, respectively, nu and nl are number of upper and lower layers, respectively, x , y , x and y are the higher order unknowns, ixu , ixl , iyu and iyl are the slopes of ith layer (upper and
Fig. 1. General lamination scheme and displacement configuration.
U = u + zx +
Fig. 2. Variation of transverse displacement (W) through the thickness of laminated sandwich plate.
z2
zn-2l -nl-1 -nl
wl
x
-1 h/2
hc
w
z2u
properties. Using the above equation, the in-plane displacement field as given in Eqs. (1) and (2) may be expressed as follows:
(z − ziu )H(z − ziu )iyu j
(z − zjl )H(−z + zjl )yl + y z 2 + y z 3 ,
1yu nl −1 T nu −1 1 2 2 yu · · · yu yl yl · · · yl } , {} = {u v w x y wu,x wu,y T wl,x wl,y } and the elements of [A] are dependent on the material
(2)
U = b1 u + b2 v + b3 w + b4 x + b5 y + b6 wu,x + b7 wu,y + b8 wl,x + b9 wl,y ,
(6)
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M.K. Pandit et al. / Finite Elements in Analysis and Design 44 (2008) 602 -- 610
V = c1 u + c2 v + c3 w + c4 x + c5 y + c6 wu,x + c7 wu,y + c8 wl,x + c9 wl,y ,
3. Finite element implementation (7)
where the coefficients bi 's and ci 's are function of thickness coordinates, unit step functions and material properties. The displacement field as expressed in Eqs. (6) and (7) contains first order derivatives of wu and wl , which usually requires C 1 continuous shape functions for finite element approximation. In order to avoid the usual difficulties associated with satisfaction of C 1 continuity, wu,x , wu,y , wl,x and wl,y are also considered as independent field variables like wu and wl . The concept is similar to that adopted by Shankara and Iyengar [36] to develop a C 0 continuous plate element based on simple third order plate theory of Reddy [4]. For the sake of convenience, the derivatives of wu and wl are expressed as follows: wu,x = ux ,
wu,y = uy ,
wl,x = lx ,
wl,y = ly .
(8)
With this, the in-plane displacement field may be rewritten as U = b1 u + b2 v + b3 w + b4 x + b5 y + b6 ux + b7 uy + b8 lx + b9 ly ,
(9)
V = c1 u + c2 v + c3 w + c4 x + c5 y + c6 ux + c7 uy + c8 lx + c9 ly .
(10)
The generalized displacement vector { } for the present plate model can now be written with help of Eqs. (3), (9) and (10) as { } = {u v w x y ux uy lx ly wu wl }T . Using linear strain--displacement relation and Eqs. (1)--(5), the strain vector {} may be expressed in terms of unknowns (for the structural deformation) as
jU jV jW jU jV jU jW jV jW T + + + jx jy jz jy jx jz jx jz jy {} = [H]{},
{} =
In the present problem, a nine-noded quadrilateral C 0 isoparametric element with 11 degrees of freedom (u, v, w, x , y , ux , uy , lx , ly , wu and wl ) per node is employed. Fig. 3 shows the node numbering and natural co-ordinates of the element. Using this finite element, the generalized displacement vector at any point may be expressed as =
n i=1
Ni i ,
(16)
where = {u v w x y ux uy lx ly wu wl }T as defined earlier, i is the displacement vector corresponding to node i, Ni is the shape function [37] associated with the node i and n is the number of nodes per element, which is nine in the present study. With the help of Eq. (16), the strain vector {} that appeared in Eq. (11) may be expressed in terms of containing nodal degrees of freedom as {} = [B]{ },
(17)
where [B] is the strain--displacement matrix in the Cartesian coordinate system. The elemental potential energy as given in Eq. (12) may be rewritten with the help of Eqs. (13)--(17) as 1 { }T [B]T [D][B]{ } dx dy − { }T [N w ]T q dx dy e 2 1 = { }T [Ke ]{ } − { }T {Pe }, 2
=
(18)
where
or (11)
[Ke ] =
where
{Pe } =
[B]T [D][B] dx dy,
(19)
[N w ]T q dx dy.
(20)
{} = [u v w x y ux uy lx ly wu wl u,x The equilibrium equation can be obtained by minimizing in the above equation with respect to { } as
v,x w,x x,x y,x ux,x uy,x lx,x ly,x wu,x wl,x u,y v,y w,y x,y y,y ux,y ux,y lx,y ly,y wu,y wl,y ]T
e as given
[Ke ]{ } = {Pe },
and the elements of [H] are functions of z and unit step functions which are given in Appendices A. With the quantities found in the above equations, the total potential energy of the system under the action of transverse load may be expressed as = Us − Wext. , (12) e
(21)
where [Ke ] is the element stiffness matrix and {Pe } is the nodal load vector.
η
7 (-1, 1)
6 (0, 1)
5 (1, 1)
where U is the strain energy and Wext. is the work done by the external transverse load. Using Eqs. (4) and (11), the strain energy is given by Us =
n 1 1 {}T [Q k ]{} dx dy dz = {}T [D]{} dx dy, 2 2
(13)
8 (-1, 0)
k=1
where [D] =
n
[H]T [Q k ][H] dz.
9 (0, 0)
4 (1, 0)
2 (0, -1)
3 (1, -1)
ξ
(14)
k=1
The work done by a distributed transverse static load of intensity q(x, y) is Wq dx dy. (15) Wext. =
1 (-1, -1)
Dofs: u, v, w, x, y , ux, uy, lx, ly, wu , wl. Fig. 3. Node numbers and natural co-ordinates of a nine-noded isoparametric element.
M.K. Pandit et al. / Finite Elements in Analysis and Design 44 (2008) 602 -- 610
Integrations of the Eqs. (19) and (20) are carried out numerically following Gauss quadrature integration rule. A reduced integration technique is adopted for the evaluation of stiffness matrix in order to avoid any possible numerical disturbances such as shear locking, which may appear with a full integration scheme. After evaluating the stiffness matrices for all elements, they have been assembled together to form the overall stiffness matrix [K] of the plate. The storage of [K] has been done in single array following skyline storage technique. 4. Numerical results and discussion
Table 1 Material properties used for the core and face sheets Location
Elastic properties
Core Face
E1
E2
E3
G12
G13
G23
12
0.04E 25E
0.04E E
0.5E --
0.016E 0.5E
0.06E 0.5E
0.06E 0.2E
0.25 0.25
Table 2 Non-dimensional central deflection (w) of a simply supported rectangular sandwich plate with laminated facings (0/90/C/0/90) under sinusoidal load References
(x , y , z , xy ) = (xz , yz ) =
h2 q0 a2
(x , y , z , xy ),
h (xz , yz ), q0 a
w=
100wE2 h3 q0 a4
.
4.1. Simply supported rectangular laminated sandwich plate under distributed load of sinusoidal variation A rectangular sandwich (0/90/C/0/90) plate (Fig. 4) is subjected to a distributed load of intensity q = q0 sin( x/a) sin( y/b). It has a total thickness of h where the thickness of the core is 0.8h and that of each ply in the top and bottom face sheets is 0.05h. The plate is
Aspect ratio (b/a) 1.0
Present (4 × 4)a Present (6 × 6) Present (8 × 8) Present (10 × 10) Present (12 × 12) 3-D elasticity [28] % Error a Mesh
2.0
h/a = 0.1
h/a = 0.2
h/a = 0.1
h/a = 0.2
1.7270 1.7256 1.7253 1.7252 1.7252 1.7272 0.11
4.2577 4.2528 4.2520 4.2518 4.2517 4.2447 0.16
3.1923 3.1897 3.1893 3.1892 3.1891 3.1944 0.16
7.3795 7.3711 7.3697 7.3693 7.3692 7.3727 0.04
size.
Sandwich plate (0/90/C/0/90) h/a = 0.20 0.6
0.4
Normalised depth
In order to demonstrate the accuracy and applicability of the present formulation, several examples of the laminated composite plates and sandwich plates subjected to static load have been analyzed and the results obtained have been compared with the published results. The restrained degrees of freedom for different types of boundary conditions such as clamped and simply supported conditions are as follows. Boundary line parallel to x-axis: Simply supported condition: u = w = x = ux = lx = wu = wl = 0. Clamped condition: u = v = w = x = y = ux = uy = lx = ly = wu = wl = 0. Boundary line parallel to y-axis: Simply supported condition: v = w = y = uy = ly = wu = wl = 0. Clamped condition: u = v = w = x = y = ux = uy = lx = ly = wu = wl = 0. The following non-dimensional quantities used in the present analysis are defined as
605
0.2
Present 3-D Elasticity [28]
0.0
-0.2
-0.4
-0.6 -1.2
-0.9
-0.6 -0.3 0.0 0.3 0.6 0.9 1.2 Non-dimensional in-plane normal stress
1.5
Fig. 5. Variation of non-dimensional in-plane normal stress across the depth of a sandwich plate.
y
n
b
2 1 2 3
m a
Fig. 4. A rectangular plate having a mesh of m × n.
x
analyzed by taking aspect ratio b/a = 1.0 and 2.0 and thickness ratios h/a = 0.1 and 0.2 using different mesh sizes. The material properties used for the core and each laminated face sheets are given in Table 1. The non-dimensional values of central deflection w( 2a , 2b , 0) obtained in the present analysis are presented with those obtained from the three-dimensional elasticity solution [28] in Table 2. As regard to three-dimensional elasticity solution, it is relevant to mention that the numerical results based on elasticity solution used here for the comparison are not readily available in the paper of Pagano [28]. Therefore, these results are generated by writing a separate code based on the analytical solution [28]. The present results are in very good agreement with the elasticity solution. The table also shows that the convergence of the results with mesh refinement is excellent. The variations of non-dimensional in-plane normal stress x = x h2 /(q0 a2 ) at the plate center through the thickness of the plate, inplane shear stress xy =xy h2 /(q0 a2 ) at a corner of the plate obtained in the present analysis (mesh size: 10 × 10, b/a = 1 and h/a = 0.2)
606
M.K. Pandit et al. / Finite Elements in Analysis and Design 44 (2008) 602 -- 610
are plotted in Figs. 5 and 6 with the three-dimensional elasticity solution [28]. The figures show that the present results are in good agreement with the elasticity solution. 4.2. Simply supported square sandwich plate under sinusoidal load A square sandwich (0/C/0) plate, simply supported at all four edges and subjected to distributed load of intensity q = q0 sin( x/a) sin( y/b) is analyzed by taking thickness ratios, h/a = 0.01, 0.02, 0.05 and 0.25. The thickness of each face is 0.1h. The material properties are as given in Table 1. The deflection, the normal stresses and the shear stresses at some important points
0.50
Sandwich plate (0/90/C/0/90) h/a = 0.2
Normalised depth
0.25 3-D Elasticity [28] Present 0.00
-0.25
-0.50 -0.06
0.02 -0.04 -0.02 0.00 0.04 Non-dimensional in-plane shear stress
0.06
Fig. 6. Variation of non-dimensional in-plane shear stress across the depth of a sandwich plate.
obtained by the present formulation are presented in Table 3 along with the three-dimensional elasticity solution of Pagano [28]. The study has been performed for different mesh divisions to show the convergence characteristics. In general present results are close to the elasticity solution with excellent convergence. 4.3. Sandwich plate having different boundary conditions under distributed load of sinusoidal variation A sandwich plate (0/90/C/0/90) subjected to a distributed load of intensity q = q0 sin( x/a) sin( y/b) is analyzed in this example. It has a total thickness of h where the thickness of the core is 0.8h and that of each ply in the top and bottom face sheets is 0.05h. The study has been made for two types of boundary conditions. These are SCSC, i.e. two opposite edges simply supported and other two edges clamped and CCCC, i.e. all edges clamped. The analysis is performed for different thickness ratios (h/a = 0.01, 0.05, 0.10, 0.20 and 0.50). The material properties used for the core and each laminated face sheets are given in Table 1. The non-dimensional values of central deflection w( 2a , 2b , 0) and the stresses calculated at the important points are presented in Table 4. It may be noted from the results that the percentage increase in the transverse displacement for higher thickness ratios, i.e. from 0.20 to 0.50 is much higher than those for lower thickness ratios, i.e. from 0.01 to 0.05 for both the boundary conditions. The major cause behind it may be that for higher thickness ratios the effect of transverse flexibility of the core and shear deformation effects is more pronounced as compared with that of the lower thickness ratios. Moreover, core occupies major portion (80%) of the plate thickness and so, core compressibility plays an important role for highly thick plates (h/a = 0.5) to get high values of transverse displacement which is reflected in the present results. Same kind of observations may also be noted for the stress values except the transverse shear stress (yz ) which decreases as the thickness
Table 3 Non-imensional deflection (w) and stresses (x , y , xz , yz , xy ) at the important points of a simply supported square sandwich (0/C/0) plate under sinusoidal load h/a
References
w
a b , ,0 2 2
x
a b h , , 2 2 2
y
a b h , , 2 2 2
xz 0,
b ,0 2
yz a , 0, 0 2
xy 0, 0,
h 2
0.01
Present (4 × 4) Present (6 × 6) Present (8 × 8) Present (10 × 10) Present (12 × 12) 3-D Elasticity [28]
0.8921 0.8918 0.8917 0.8917 0.8917 0.8923
1.0643 1.0828 1.1089 1.1092 1.1093 1.098
0.0532 0.0542 0.0545 0.0546 0.0547 0.0550
0.3323 0.3379 0.3398 0.3407 0.3412 0.324
0.0315 0.03212 0.0323 0.0323 0.0324 0.0297
0.0423 0.0430 0.0433 0.0434 0.0434 0.0433
0.02
Present (4 × 4) Present (6 × 6) Present (8 × 8) Present (10 × 10) Present (12 × 12) 3-D Elasticity [28]
0.9346 0.9343 0.9342 0.9341 0.9341 0.9348
1.0658 1.0843 1.0905 1.0933 1.0948 1.099
0.0551 0.0561 0.0564 0.0565 0.0566 0.0569
0.3314 0.3370 0.3389 0.3398 0.3403 0.323
0.0324 0.0330 0.0331 0.0332 0.0333 0.0306
0.0432 0.4403 0.0442 0.0444 0.0445 0.0446
0.05
Present (4 × 4) Present (6 × 6) Present (8 × 8) Present (10 × 10) Present (12 × 12) 3-D Elasticity [28]
1.2263 1.2257 1.2254 1.2254 1.2254 1.2264
1.0763 1.0949 1.1012 1.1040 1.1055 1.110
0.0676 0.0687 0.0691 0.0693 0.0694 0.0700
0.3255 0.3311 0.3329 0.3338 0.3342 0.317
0.0381 0.0388 0.0390 0.0391 0.0392 0.0361
0.0496 0.0504 0.0507 0.0508 0.0509 0.0511
0.10
Present (4 × 4) Present (6 × 6) Present (8 × 8) Present (10 × 10) Present (12 × 12) 3-D Elasticity [28]
2.2025 2.2006 2.2003 2.2002 2.2002 2.2004
1.1179 1.1373 1.1438 1.1467 1.1483 1.153
0.1058 0.1076 0.1082 0.1084 0.1086 0.1104
0.3076 0.3128 0.3146 0.3154 0.3158 0.300
0.0555 0.0564 0.0567 0.0569 0.0570 0.0527
0.06913 0.0703 0.0707 0.0709 0.0709 0.0707
0.25
Present (4 × 4) Present (6 × 6) Present (8 × 8) Present (10 × 10) Present (12 × 12) 3-D Elasticity [28]
7.6658 7.6571 7.6557 7.6553 7.6552 7.5962
1.4820 1.5073 1.5158 1.5197 1.5218 1.556
0.2440 0.2481 0.2495 0.2502 0.2506 0.2595
0.2453 0.2495 0.2509 0.2516 0.2520 0.239
0.1126 0.1145 0.1151 0.1154 0.1156 0.1072
0.1430 0.1454 0.1462 0.1466 0.1468 0.1437
M.K. Pandit et al. / Finite Elements in Analysis and Design 44 (2008) 602 -- 610
607
Table 4 Non-dimensional deflection (w) and stresses (x , y , xz , yz , xy ) at the important points square sandwich (0/90/C/0/90) plate with different boundary conditions h/a
w
Boundary conditions
a b , ,0 2 2
x a b h , , 2 2 2
y
a b h , , 2 2 2
xz 0,
b ,0 2
yz a , 0, 0 2
xy h 0, 0, 2
0.01
SCSC CCCC
0.3453 0.2286
0.4077 0.4270
0.0326 0.0228
0.0778 0.2189
0.3086 0.2189
0.0036 0.0005
0.05
SCSC CCCC
0.6052 0.4296
0.5850 0.4275
0.0334 0.0236
0.1061 0.1828
0.2527 0.1828
0.0107 0.0018
0.10
SCSC CCCC
1.3026 1.0489
0.8310 0.4597
0.0372 0.0279
0.1418 0.1587
0.1967 0.1586
0.0189 0.0036
0.20
SCSC CCCC
3.8087 3.4521
1.1415 0.6170
0.0561 0.0470
0.1683 0.1396
0.1539 0.1394
0.0292 0.0074
0.50
SCSC CCCC
19.5512 18.3454
2.3071 1.8156
0.2047 0.1902
0.1691 0.1227
0.1296 0.1217
0.0667 0.0215
Table 5 Non-dimensional deflection (w) and stresses (x , y , xz , yz , xy ) at the important points of a clamped square sandwich (0/C/0) plate under uniformly distributed load h/a
References
w
h a b , ,± 2 2 2
x a b h , ,± 2 2 2
y
a b h , ,± 2 2 2
xz 0,
b ,0 2
yz
xy
a , 0, 0 2
0, 0, ±
h 2
0.01
Present
0.2897 0.2897
0.5398 −0.5398
0.0099 −0.0099
0.5429
0.1764
−0.0025 0.0025
0.02
Present
0.3549 0.3549
0.5478 −0.5478
0.0131 −0.0131
0.5138
0.1806
−0.0040 0.0040
0.05
Present
0.7793 0.7789
0.5754 −0.5754
0.0371 −0.0371
0.4368
0.2020
−0.0089 0.0089
0.10
Present
2.0090 2.0022
0.6346 −0.6346
0.0951 −0.0952
0.3367
0.2372
−0.0171 0.0163
0.25
Present
8.2090 7.9482
1.1809 −1.1819
0.1919 −0.1923
0.2242
0.2630
−0.0401 0.0314
Table 6 Non-dimensional deflection (w) and normal stresses (x , y , z ) of a square sandwich (0/C/0) plate under concentrated load at the plate center
x
y
z
4.9408 4.8921
0.0276 −0.0269
0.0486 −0.0479
0.0278 0.0016
Upper face Lower face
5.7204 5.3474
0.0284 −0.0259
0.0525 −0.0503
0.1040 0.0056
0.05
Upper face Lower face
11.2558 8.0221
0.0311 −0.0201
0.0623 −0.0523
0.5440 0.0326
0.10
Upper face Lower face
31.2576 15.6250
0.0343 −0.0087
0.0667 −0.0452
1.6745 0.0684
h/a
Location
0.01
Upper face Lower face
0.02
w
a b , , ±0.5h 2 2
ratio increases in the case of SCSC boundary condition. Moreover, for the CCCC boundary condition the transverse shear stress (xz ) also decreases in addition to yz . It is also seen that the transverse shear stresses are of the same order of magnitude for all each thickness ratio considered for CCCC boundary condition.
a b , , ±0.4h 2 2
a b , , ±0.4h 2 2
a b , , ±0.4h 2 2
some important points obtained by the present formulation are presented in Table 5. It is easily seen from the table that transverse displacement becomes unequal at the upper and lower facesheets of the sandwich plate for the higher thickness ratios. Also, the in-plane normal and shear stresses developed at the top and bottom of the plate are of different values for plates with higher thickness ratios which is expected.
4.4. All edges clamped square sandwich plate under uniformly distributed load A square sandwich (0/C/0) plate, all four edges clamped and subjected to uniformly distributed load, is analyzed by taking different thickness ratios, h/a=0.01, 0.02, 0.05, 0.10 and 0.25. The thickness of each face is 0.1h. The material properties are as given in Table 1. The non-dimensional deflection, the in-plane normal and shear stresses at the upper and lower faces and the transverse shear stresses at
4.5. Simply supported square sandwich plate under concentrated load at the plate center In this problem a single core square sandwich (0/C/0) plate is subjected to concentrated load at the plate center. The study has been made for thickness ratios, h/a = 0.01, 0.02, 0.05 and 0.10. The thickness of each face is 0.1h. The material properties are as given
608
M.K. Pandit et al. / Finite Elements in Analysis and Design 44 (2008) 602 -- 610
in Table 1. The deflections and the transverse normal stresses at the upper and lower faces at the plate center have been computed and are presented in Table 6. It is observed from the results that the deflections are unequal in upper and lower faces and the values of the transverse normal stress are significant in magnitude in comparison to the in-plane normal stress values. The difference in the values of transverse displacements increases rapidly at higher thickness ratios which justifies its higher order variation. It may be easily noted from the stress values that the transverse normal stress is comparable in magnitude with the in-plane normal stresses. Even for higher thickness ratios it becomes larger than the in-plane normal stresses indicating its significance on the sandwich plate bending.
b3 = A13 z 2 + A23 z 3 +
n u −1 i=1
nl −1
+
j=1
A(j+nu +3)3 (z − zjl )H(−z + zjl ),
b4 = z + A14 z 2 + A24 z 3 +
n u −1 i=1
nl −1
+
j=1
n u −1 i=1
5. Conclusion
nl −1
+
An improved higher order zigzag theory is presented for the analysis of sandwich plate with soft core. The present zigzag theory satisfies shear free conditions at the top and bottom of the plate, cubic variation of the in-plane displacements, quadratic variation of the transverse displacement and interlaminar shear stress continuity at the layer interfaces. A computationally efficient finite element formulation for the improved sandwich plate model is also presented. A nine noded isoparametric element with 11 dof per node is adopted. The results obtained for the deflection and the stresses of a laminated composite and sandwich plate show excellent performance of the present formulation. Convergence of the results is very good indicating reasonably less number of elements required to get the desired results. The effect of the transverse normal deformation has been shown for soft core sandwich plate under concentrated load. Some new results for the sandwich plates have been presented for future reference in this field of study.
j=1
n u −1 i=1
+
j=1
n u −1 k=5 i=1
nl −1
j=1
n u −1 i=1
nl −1
+
j=1
0 ⎢ 0 ⎢ ⎢ 0 [H] = ⎢ ⎢ 0 ⎢ ⎣b
1,z
c1,z b5 0 0 c5 0 0
0 0 0 0 b2,z c2,z b6 0 0 c6 0 0
0 0 l2,z 0 b3,z c3,z b7 0 0 c7 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
b4,z c4,z
b5,z c5,z
b6,z c6,z
b7,z c7,z
b8,z c8,z
b9,z c9,z
b8 0 0 c8 0 0
b9 0 0 c9 0 0
0 0 0 0 l1 0
0 0 0 0 l3 0
0 c1 0 b1 0 0
0 c2 0 b2 0 0
0 c3 0 b3 0 l2
0 c4 0 b4 0 0
0 c5 0 b5 0 0
0 0
0 0
l1,z 0 0 0
l3,z 0 0 0
0 c6 0 b6 0 0
0 c7 0 b7 0 0
b1 0 0 c1 0 0 0 c8 0 b8 0 0
b2 0 0 c2 0 0 0 c9 0 b9 0 0
A(i+4)8 (z − ziu )H(z − ziu )
A(j+nu +3)8 (z − zjl )H(−z + zjl ),
b3 0 0 c3 l2 0 0 0 0 0 0 l1
A(i+4)7 (z − ziu )H(z − ziu )
A(j+nu +3)7 (z − zjl )H(−z + zjl ),
b8 = A18 z 2 + A28 z 3 +
Appendix A. Elements of matrix H
A(i+4)6 (z − ziu )H(z − ziu )
A(j+nu +3)6 (z − zjl )H(−z + zjl ),
b7 = A17 z 2 + A27 z 3 +
+
A(i+4)5 (z − ziu )H(z − ziu )
A(j+nu +3)5 (z − zjl )H(−z + zjl ),
b6 = A16 z 2 + A26 z 3 + nl −1
A(i+4)4 (z − ziu )H(z − ziu )
A(j+nu +3)4 (z − zjl )H(−z + zjl ),
b5 = A15 z 2 + A25 z 3 +
⎡
A(i+4)3 (z − ziu )H(z − ziu )
b4 0 0 c4 0 0 ⎤
0 0⎥ ⎥ 0⎥ ⎥, 0⎥ ⎥ 0⎦ l3
where
b1 = 1 + A11 z 2 + A21 z 3 +
n u −1 i=1
nl −1
+
j=1
A(j+nu +3)1 (z − zjl )H(−z + zjl ),
b2 = A12 z 2 + A22 z 3 +
n u −1 i=1
nl −1
+
j=1
A(i+4)1 (z − ziu )H(z − ziu )
A(i+4)2 (z − ziu )H(z − ziu )
A(j+nu +3)2 (z − zjl )H(−z + zjl ),
b9 = A19 z 2 + A29 z 3 +
n u −1 i=1
nl −1
+
j=1
A(j+nu +3)9 (z − zjl )H(−z + zjl ),
c1 = A31 z 2 + A41 z 3 +
n u −1 i=1
nl −1
+
j=1
A(i+4)9 (z − ziu )H(z − ziu )
A(i+nu +n +2)1 (z − ziu )H(z − ziu ) l
A(j+2nu +n +1)1 (z − zjl )H(−z + zjl ), l
M.K. Pandit et al. / Finite Elements in Analysis and Design 44 (2008) 602 -- 610
n u −1
c2 = 1 + A32 z 2 + A42 z 3 +
i=1 nl −1
+
j=1
l
n u −1
j=1
A(j+2nu +n +1)3 (z − zjl )H(−z + zjl ), n u −1 i=1
+
j=1
A(i+nu +n +2)4 (z − ziu )H(z − ziu ) l
A(j+2nu +n +1)4 (z − zjl )H(−z + zjl ), l
c5 = z + A35 z 2 + A45 z 3 +
n u −1 i=1
nl −1
+
j=1
j=1
n u −1
j=1
n u −1
j=1
n u −1
j=1
U Us v V w wu wl W Wext. ixu , iyu ixl ,iyl x , y , x , y {} {} x , y
A(i+nu +n +2)8 (z − ziu )H(z − ziu ) l
lx , ly
,
element stiffness matrix overall stiffness matrix Lagrangian interpolation functions in the thickness co-ordinate number of upper and lower layers, respectively shape function associated with the node i element load vector intensity of applied load transformed rigidity matrix of kth lamina displacement of any point at mid-plane along x-direction displacement of any point along x-direction strain energy displacement of any point at mid-plane along y-direction displacement of any point along y-direction transverse displacement at the middle of the core transverse displacement at the top layers of the core transverse displacement at the bottom layers of the core displacement at any point along z-direction work done under external transverse load change of slopes at the upper ith interface between ith and (i + 1)th layer change of slopes at the lower ith interface between ith and (i + 1)th layer higher order unknown generalized displacement vector strain vector stress vector rotation of the normal to the mid-plane about y- and x-axes, respectively rotation of the normal to the top face sheet and core interface about y- and x-axes, respectively rotation of the normal to the bottom face sheet and core interface about y- and x-axes, respectively natural co-ordinate
References
l
n u −1 i=1
nl −1
nu , nl Ni {Pe } q [Q k ] u
ux , uy
A(j+2nu +n +1)8 (z − zjl )H(−z + zjl ),
c9 = A39 z 2 + A49 z 3 +
+
l
l
i=1
+
A(i+nu +n +2)7 (z − ziu )H(z − ziu )
A(j+2nu +n +1)7 (z − zjl )H(−z + zjl ),
c8 = A38 z 2 + A48 z 3 + nl −1
l
l
i=1 nl −1
A(i+nu +n +2)6 (z − ziu )H(z − ziu )
A(j+2nu +n +1)6 (z − zjl )H(−z + zjl ),
c7 = A37 z 2 + A47 z 3 +
+
l
l
i=1 nl −1
A(i+nu +n +2)5 (z − ziu )H(z − ziu )
A(j+2nu +n +1)5 (z − zjl )H(−z + zjl ),
c6 = A36 z 2 + A46 z 3 +
+
[Ke ] [K] l1 , l2 , l3
l
c4 = A34 z 2 + A44 z 3 + nl −1
A(i+nu +n +2)3 (z − ziu )H(z − ziu ) l
i=1 nl −1
l
A(j+2nu +n +1)2 (z − zjl )H(−z + zjl ),
c3 = A33 z 2 + A43 z 3 +
+
A(i+nu +n +2)2 (z − ziu )H(z − ziu )
609
A(i+nu +n +2)9 (z − ziu )H(z − ziu ) l
A(j+2nu +n +1)9 (z − zjl )H(−z + zjl ) l
and Aij 's are the elements of the matrix A containing Q ij 's. Appendix B. List of notations a b [B] E G h H(z − ziu ),
length of the plate breadth of the plate strain--displacement matrix Young's modulus shear modulus thickness of the plate unit step functions
[H]
matrix which is a function of z and unit step functions
H(−z + zjl )
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