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Free vibration analysis of multi-layered composite laminates based on an accurate higher-order theory
Composites: Part B 32 (2001) 535±543
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Free vibration analysis of multi-layered composite laminates based on an accurate higher-order theory M. Ganapathi*, D.P. Makhecha Institute of Armament Technology (Deemed University), Girinagar, Pune 411 025, India Received 11 July 2000; accepted 22 February 2001
Abstract This paper deals with an accurate higher-order theory employing ®nite element procedure for the free vibration analysis of multi-layered thick composite plates. The theory accounts for the realistic variation of in-plane and transverse displacements through the thickness. The accuracy of the present model is veri®ed by comparison with three-dimensional elasticity solutions for the vibration study of the composite laminates. The performance and the applicability of the proposed discrete model are also discussed among developed elements and alternate models, considering different parameters such as ply-angle, degree of orthotropicity, aspect ratio and boundary conditions. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: B. Vibration; Finite element
1. Introduction The application of advanced high stiffness- and strength-to-weight ratios of the composite material systems has played a key role in the success of the aerospace and aircraft industries. However, the analysis of the multi-layered structures is a complex task, compared with conventional single layer metallic structures, due to the exhibition of coupling among membrane, torsion and bending strains; weak transverse shear rigidities; and discontinuity of the mechanical characteristics along the thickness of the laminates. More accurate analytical/numerical analysis based on three-dimensional models may result computationally involved and expensive. Hence, among researchers, there is a growing appreciation of the importance of developing new kinematics for the evolution of accurate two-dimensional theories for the analysis of thick laminates with high orthotropic ratio, leading to less expensive models. In this context, the applications of analytical/numerical methods based on various higherorder theories, not only for the vibrations of thick laminates, but also for the high frequency vibrations of thin composite plates, has recently attracted the attention of several investigators/researchers. Various structural theories proposed for evaluating * Corresponding author. Tel.: 191-20-599-550; fax: 191-20-599-509. E-mail address: [email protected] (M. Ganapathi).
the characteristics of composite laminates under different loading situations have been reviewed and assessed by Noor and Burt [1], Kapania and Raciti [2], Reddy [3], and more recently by Mallikarjuna and Kant [4], Varadan and Bhaskar [5]. It has been concluded that the assumption of displacements as linear functions of the coordinate in the thickness direction has proved to be inadequate for predicting the response of thick laminates. Furthermore, due to the advantage that no shear correction factors are needed and the warping of the cross-section can be accounted for to a certain extent, higher-order theories are widely preferred in the analysis of composite plates. Higher-order displacement ®elds involving higher order expansions of the displacement ®eld in powers of the thickness coordinate yielding quadratic variation of transverse shear strains have been attempted by many researchers [6±13] for better accuracy. A careful look into the results of three-dimensional analysis carried out by Bhaskar et al. [14] for thick laminates has revealed that the in-plane displacements are non-linear through the thickness and exhibit an abrupt discontinuity in slope at any interface, and thickness-stretch/contraction effects in the transverse displacement. Although higher-order theories based on discrete layer approach [15±19] account for slope discontinuity at the interfaces, the number of unknowns to be solved increases with the increase in the number of layers. Recently, Ali et al. [20] have proposed a new higher-order plate theory based on global approximation
1359-8368/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 1359-836 8(01)00028-2
536
M. Ganapathi, D.P. Makhecha / Composites: Part B 32 (2001) 535±543
approach, for multi-layered composite laminates under thermal/mechanical loading, incorporating realistic through the thickness approximations of the in-plane and transverse displacements based on the work given in Ref. [14]. This formulation has proved to give very accurate results for static analysis of symmetric crossply laminates, and this excellent performance of the theory for thick/moderately thick laminates motivated the present extension of the formulation for the vibration analysis of general composite laminates through ®nite element procedure. Here, C 0 eight-noded quadrilateral serendipity plate element with 13 degrees of freedom per node [21] based on the theory given in Ref. [20] is employed for free vibration study of thick composite laminates. The accuracy of the present element with those of other available numerical/analytical models and threedimensional elasticity analysis are compared and discussed. Since the vibration analysis of angle-ply laminates based on three-dimensional elasticity theory is not available in the literature, the present results are compared among the developed elements, and alternate models considering with/without the effects of slope discontinuity and thickness-stretch in the present assumed displacement ®elds. The numerical results also illustrate that the proposed model has a signi®cant in¯uence on the frequencies of higher ¯exural vibration modes. 2. Formulation A composite plate with arbitrary lamination is considered with the co-ordinates x, y along the in-plane directions and z along the thickness direction. The in-plane displacements u k and v k, and the transverse displacement w k for the kth layer, are assumed as uk x; y; z; t u0 x; y; t 1 zux x; y; t 1 z2 bx x; y; t 1 z3 fx x; y; t 1 Sk c x x; y; t
1
a generic point on the reference surface; ux; uy are the rotations of normal to the reference surface about the y and x axes, respectively; w1 ; bx ; by ; G x ; fy are the higher order terms in the Taylor's series expansions, de®ned at the reference surface. c x and c y are generalized variables associated with the zig-zag function, Sk : The zig-zag function, Sk ; as given in Ref. [22], is de®ned by Sk 2 21k zk =hk
where zk is the local transverse coordinate with its origin at the center of the kth layer and hk is the corresponding layer thickness. Thus, the zig-zag function is piecewise linear with values of 21 and 1 alternately at the different interfaces. The `zig-zag' function, as de®ned above, takes care of inclusion of the slope discontinuity of u and v at the interfaces of the laminate as observed in exact three-dimensional elasticity solutions of thick laminated composite structures. The use of such function is more economical than a discrete layer approach of approximating the displacement variations over the thickness of each layer separately. Although both these approaches account for slope discontinuity at the interfaces, in the discrete layer approach the number of unknowns increases with the increase in the number of layers, whereas it remains constant in the present approach. The strains in terms of mid-plane deformation, rotations of normal, and higher order terms associated with displacements are as ( {e}
k
1 z fy x; y; t 1 S c y x; y; t
wk x; y; z; t w0 x; y; t 1 zw1 x; y; t 1 z2 G x; y; t The terms with even power in z in the in-plane displacements and those odd in z occurring in the expansion for w k correspond to stretching problems. But, the terms with odd in z in the in-plane displacements and even in z in the expression for w k represent the ¯exure problems. u 0 ; v0 ; w0 are the displacements of
ebm
) 3
es
The vector {ebm } includes the bending and membrane terms of the strain components and vector {es } contains the transverse shear strain terms. These strain vectors can be de®ned as
vk x; y; z; t v0 x; y; t 1 zuy x; y; t 1 z2 by x; y; t 3
2
{ebm }
8 9 exx > > > > > > > > > < eyy > = > ezz > > > > > > > > :g > ; xy
8 > > > > > <
u;x
> > > > > :
w;z
9 > > > > > =
v;y u;y 1v;x
> > > > > ;
{e0 } 1 z{e1 } 1 z2 {e2 } 1 z3 {e3 } 1 Sk {e4 } ( {es }
gxz gyz
)
(
u; z 1 w; x
4a
)
v; z 1 w;y
{g0 } 1 z{g1 } 1 z2 {g2 } 1 Sk; z {g3 }
4b
M. Ganapathi, D.P. Makhecha / Composites: Part B 32 (2001) 535±543
where
{e0 }
8 > > > > > < > > > > > :
u0;x v0;y w1
8 9 > > > > > > > > > > < = { e1 } > > > > > > > > > > : ;
uX;X uY;Y 2G
( {g0 }
ux 1 w0;x uy 1 w0;y
) {g1 }
(
2bx 1 w1;x
2by 1 w1;y 8 9 ( ) < c x Sk;z = 3fx 1 G ;x {g2 } {g0 } : c Sk ; 3fy 1 G ;y y ;z
The kinetic energy of the plate is given by " # n Zhk 1 1 1 ZZ X k k k k k k T rk {u_ v_ w_ }{u_ v_ w_ } dz dx dy T d 2 k1 hk
9 > > > > > =
7
> > > > > ;
u X;Y 1 uY;X 8 8 9 9 bx;x fx;x > > > > > > > > > > > > > > > > > > > > < < = = by;y fy;y {e2 } {e3 } > > > > 0 0 > > > > > > > > > > > > > > > :b 1 b ; :f 1f > ; x;y y;x x;y y;x 8 9 c x;x > > > > > > > > > > < = c y;y {e4 } > > 0 > > > > > > > :c 1c > ; x;y y;x u0;y 1 v0;x
537
where r k is the mass density of the kth layer. hk, hk11 are the z coordinates of laminate corresponding to the bottom and top surfaces of the kth layer. The strain energy functional U is given by " # n Z hk 1 1 1 ZZ X T U d {s} {e} dz dx dy 8 2 k1 hk Substituting Eqs. (7) and (8) in Eq. (6), one obtains the governing equation for the free vibration of plate as M{d } 1 K{d} {0} (4c)
)
(4d)
The subscript comma denotes the partial derivative with respect to the spatial coordinate succeeding it. The constitutive relations for an arbitary layer k, written in the laminate x; y; z coordinate system can be expressed as {s} {s xx s yy s zz txy txz tyz }T Q k {exx eyy ezz gxy gxz gyz }T 5 where the terms of Q k matrix of ply k are referred to the laminate axes and can be obtained from the Qk corresponding to the ®ber directions with the appropriate transformation, as outlined in the literature [23]. The superscript T refers the transpose of a matrix/vector. The governing equations are obtained by applying Lagrangian equations of motion given by d=dt 2 T 2 U=2d_ i 2 2 T 2 U=2d_ i 0; i 1 to n 6 where T is the kinetic energy; U is the strain energy contributions due to the in-plane and transverse stresses, respectively. {d} {d1 ; d2 ; ¼dl ; ¼dn }T is the vector of the degree of freedoms/generalized coordinates. A dot over the variables represents the partial derivative with respect to time.
9
where M and K are the mass and stiffness matrices. The coef®cients of mass and stiffness matrices involved in Eq. (9) can be rewritten as the product of term having thickness co-ordinate z alone and the term containing x and y. In the present study, while performing the integration, terms having thickness co-ordinate z are explicitly integrated whereas the terms containing x and y are evaluated using full integration with 3 £ 3 points Gauss integration rule. The frequencies and mode shapes are obtained from Eq. (9) using the standard eigen value algorithm. 3. Element description In the present work, a simple, C 0 continuous, eight-noded serendipity quadrilateral shear ¯exible plate element with 13 degrees of freedom (u0, v0, w0, ux ; uy ; w1, bx ; by ; G , f x ; fy ; c x and c y : 13-DOF) is developed. The ®nite element represented as per the kinematics, for Table 1 Alternate eight-noded ®nite element discrete models developed, and available model for comparison purpose Finite element discrete model
Degrees of freedom per node
Q8-HSDT 13 (Present)
u0, v0, w0, u x, u y, w1, b x, by , G , f x , f y, c x, c y u0, v0, w0, u x, u y, b x, b y, f x , f y, c x, c y u0, v0, w0, u x, u y, w1, b x, b y, G , f x, f y u0, v0, w0, u x, u y, b x, b y, f x, f y u0, v0, w0, u x, u y, b x, b y u0, v0, w0, u x, u y u0, v0, w0, u x, u y, w1, b x, b y , G , f x, f y u0, v0, w0, u x, u y, b x, b y, f x, f y u0, v0, w0, u x, u y, f x, f y
Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q8-TSDT 7 Q8-FSDT 5 Q9-HSDT 11 [12] Q9-HSDT 9 [12] Q9-HSDT 9 [9]
538
M. Ganapathi, D.P. Makhecha / Composites: Part B 32 (2001) 535±543
instance see Eq. (1), can be referred as Q8-HSDT 13 with cubic variation. Four more alternate discrete models are proposed, for the comparison purpose, whose displacement ®elds are deduced from the original element by deleting the appropriate degrees of freedom (w1 and G 0; or c 0; or c; w 1 and G 0; or z 2 terms c; w 1 and G 0). In addition to third-order elements as described above, second- and ®rst-order element are deduced from the original proposed elements by neglecting the appropriate terms in the displacement ®elds (z 3 terms, c; w 1 and G 0; dropping all the higher order terms). The above described ®nite elements, and other available models, and corresponding degrees of freedom are shown in Table 1.
4. Results and discussion Numerical experimentation conducted for checking the rank of the element, locking syndrome and patch test has shown that the element is free from spurious rigid modes, locking phenomenon, etc. [21]. A convergence study carried out, based on progressive mesh re®nement, for the free vibration study of composite laminate is given in Table 2 and is found that an 4 £ 4 grid mesh is adequate to model the quarter plates. Although the convergence of the solutions obtained for very thick as well as fairly thick laminates by all the displacement models is monotonic in character, the convergence of lower order models is slower than the higher order models. The convergence study conducted here for fairly thick case matches well with those of Ref. [24]. Since the higher-order theory, in general, is required for accurate analysis for thick laminates, the numerical study is concerned with thick multi-layered anisotropic plates having aspect ratio, S # 50: Furthermore, the ef®cacy of the present ®nite element formulation and theory for the free vibration analysis of thick cases, is tested over the performances of various ®nite element models deduced from the present theory, available ®nite elements, and analytical models based on elasticity and higher-order
theories. For this purpose, a few examples are considered emphasizing the in¯uence of side-to-thickness ratio (S a=h; h is the thickness of the plate), degree of orthotropicity of individual layers E L =ET ; boundary conditions and layer-angle. The material properties, unless speci®ed otherwise, assumed in the present analysis are EL =ET 40;
GLT =ET 0:6;
GTT =ET 0:5;
nLT 0:25
10
where E, G and n are Young's modulus, shear modulus and Poisson's ratio. Subscripts L and T are the longitudinal and transverse directions, respectively, with respect to the ®bers. The ®rst layer corresponds to bottommost layer and the layer-angle is measured from x axis in an anti-clockwise direction and all the layers are of equal thickness. The boundary conditions considered here are Simply-supported condition: Cross-ply : v 0 w 0 uy w 1 G b y f y c y 0 at x 0; a u0 w 0 ux w 1 G b x f x c x 0 at y 0; b Angle-ply : u0 w 0 uy w 1 G b y f y c y 0 at x 0; a v 0 w 0 ux w 1 G b x f x c x 0 at y 0; b
Table 2 p Convergence study for fundamental frequency v v ra2 =ET of nine layered cross ply laminates [24] S a=h
5
2£2 3£3 4£4 6£6 8£8 2£2 3£3 4£4 6£6 8£8
10
a
Grid size
Ref. [24].
Model Q8-HSDT 13
Q8-HSDT 11A
Q8-HSDT 11B
Q8-HSDT 9
Q8-SSDT 7
Q8-FSDT 5
2.313587 2.312446 2.312255 2.312184 2.312172 1.586347 1.584781 1.584517 1.584417 1.584400
2.311024 2.309882 2.309692 2.309621 2.309609 1.585364 1.583798 1.583534 1.583434 1.583417
2.331468 2.330299 2.330104 2.330032 2.330019 1.588952 1.587377 1.587112 1.587011 1.586994
2.328818 2.327648 2.327453 2.327381 2.327369 1.587965 1.586390 1.586124 1.586024 1.586007
2.328818 2.327648 2.327453 2.327381 2.327369 1.587965 1.586390 1.586124 1.586024 1.586007
2.455905 2.454715 2.454519 2.454447 2.454435 1.627079 [1.627] a 1.625402 [1.625] a 1.625124 1.625021 1.625003
M. Ganapathi, D.P. Makhecha / Composites: Part B 32 (2001) 535±543 Table 3 p Non dimensional fundamental frequency v va2 =h r=ET of simply supported two-layered cross-ply square plates EL =ET 40 Model
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q8-SSDT 7 Q8-FSDT 5 Q9-HSDT11 [12] Q9-HSDT9 [12] Elasticity [17] a
Side-to-thickness ratio a=h
Clamped support: u0 v0 w0 ux uy w1 bx by G f x fy c x c y 0
5
10
50
8.6595 8.6496 8.7277 8.7176 8.8008 8.8246 8.7274 8.7172 8.5180 8.5625 a
10.3820 10.3880 10.4340 10.4290 10.4660 10.4710 10.4323 10.4279 10.333
11.2870 11.2870 11.2900 11.2890 11.2920 11.2920 11.2693 11.2689 11.263
Ref. [25].
539
at
x 0; a; y 0; b 11
Here a; b refer the length and width of the plate, respectively. Example 1. Free vibration of simply supported cross-ply laminated square plates with different aspect ratio, S [17] To examine the in¯uence of the variation of the aspect ratio, different values for S are assumed. The solutions evaluated for the fundamental frequencies of the ¯exural vibration are given in Table 3 along with those of various higher-order ®nite elements, layer-wise and
Table 4 p Non dimensional fundamental frequency v v rh2 =ET £ 10 of simply supported cross-ply square plates S 5 No. layer (N)
Model
EL =ET 3
10
20
30
40
2
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q8-SSDT 7 Q8-FSDT 5 Elasticity [25] a
2.4935 2.5478 2.4937 2.5480 2.5177 2.4824 2.5031
2.7886 2.7830 2.7899 2.7843 2.8156 2.7742 2.7938
3.0778 3.1066 3.0858 3.1142 3.1125 3.0802 3.0698
3.2940 3.2897 3.3113 3.3069 3.3391 3.3256 3.2705
3.4638 3.4598 3.4911 3.4870 3.5203 3.5299 3.4250
4
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q9-HSDT11 [12] Q8-HSDT9 [12] Q8-SSDT 7 Q8-FSDT 5 Elasticity [25]
2.6029 2.6547 2.6061 2.6580 2.6059 2.5983 2.6405 2.6004 2.6182
3.2488 3.2408 3.2595 3.2514 3.2594 3.2513 3.3506 3.2871 3.2578
3.7677 3.7796 3.7872 3.7990 3.7871 3.7793 3.9521 3.8706 3.7622
4.0841 4.0771 4.1095 4.1023 4.1094 4.1022 4.3349 4.2415 4.0660
4.3001 4.2936 4.3290 4.3225 4.3289 4.3224 4.6042 4.5007 4.2719
6
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q9-HSDT11 [12] Q8-HSDT9 [12] Q8-SSDT 7 Q8-FSDT 5 Elasticity [25]
2.6264 2.6780 2.6289 2.6805 2.6287 2.6211 2.663 2.6215 2.6440
3.3478 3.3399 3.3547 3.3468 3.3546 3.3467 3.4443 3.3643 3.3657
3.9219 3.9321 3.9342 3.9445 3.9342 3.9267 4.0957 3.9719 3.9359
4.2686 4.2621 4.2849 4.2783 4.2848 4.2782 4.5053 4.3462 4.2783
4.5035 4.4976 4.5225 4.5166 4.5223 4.5164 4.7987 4.6029 4.5091
10
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q9-HSDT11 [12] Q8-HSDT9 [12] Q8-SSDT 7 Q8-FSDT 5 Elasticity [25]
2.6390 2.6905 2.6410 2.6926 2.6409 2.6332 2.6747 2.6321 2.6583
3.4018 3.3941 3.4068 3.3990 3.4066 3.3988 3.4926 3.4022 3.4250
4.0093 4.0189 4.0177 4.0274 4.0176 4.0105 4.1694 4.0201 4.0337
4.3770 4.3709 4.3881 4.3819 4.3880 4.3818 4.5924 4.3952 4.4011
4.6279 4.6224 4.6415 4.6359 4.6398 4.6344 4.8671 4.6508 4.6498
a
Based on higher-order difference scheme.
540
M. Ganapathi, D.P. Makhecha / Composites: Part B 32 (2001) 535±543
Table 5 p Non-dimensional fundamental frequency v va2 =h r=ET for angle-ply two and eight layered 458= 2 458; 458= 2 4584 square plates N
S a=h
Model
EL =ET 3
2
8
a b
10
20
30
40
5
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q8-TSDT 7 Q8-FSDT 5
6.4480 6.4174 6.4480 6.4369 6.4458 6.4291
7.7066 7.6577 7.7066 7.7494 7.8915 7.8006
8.6130 8.5642 8.6130 8.7417 9.1200 8.9285
9.2138 9.1694 9.2138 9.4097 10.0010 9.7200
9.6620 9.6219 9.6620 9.9079 10.681 (10.840 a; 10.692 b) 10.321
10
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q8-TSDT 7 Q8-FSDT 5
7.0751 7.0602 7.0751 7.0705 7.0760 7.0707
8.8408 8.8060 8.8408 8.8686 8.9639 8.9316
10.3080 10.2650 10.3080 10.4150 10.7000 10.6240
11.3880 11.3430 11.3880 11.5800 12.0640 11.9410
12.257 12.213 12.257 12.532 13.208 (13.263 a; 13.207 b) 13.037
20
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q8-TSDT 7 Q8-FSDT 5
7.2628 7.2555 7.2628 7.2663 7.2719 7.2706
9.2186 9.1995 9.2186 9.2528 9.3274 9.3184
10.9800 10.9540 10.9800 11.0690 11.2750 11.2530
12.3680 12.3380 12.3680 12.5140 12.8590 12.8210
13.544 13.513 13.544 13.751 14.233 (14.246 a; 14.228 b) 14.179
5
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q8-TSDT 7 Q8-FSDT 5
6.9894 6.9677 6.9894 6.9739 6.9886 6.9845
9.7175 9.6973 9.7175 9.7154 9.8062 9.7979
11.3490 11.3350 11.3490 11.3630 11.5190 11.4900
12.1890 12.1790 12.1890 12.2120 12.4060 12.3450
12.712 12.705 12.712 12.741 12.961 (12.972 a; 12.967 b) 12.863
10
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q8-TSDT 7 Q8-FSDT 5
7.7612 7.7524 7.7612 7.7546 7.7629 7.7619
11.9370 11.9260 11.9370 11.9350 12.0040 12.0060
15.2850 15.2730 15.2850 15.2930 15.4430 15.4480
17.4360 17.4350 17.4360 17.4540 17.6670 17.6700
18.978 18.969 18.978 19.005 19.267 (19.266 a; 19.274 b) 19.265
20
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q8-TSDT 7 Q8-FSDT 5
8.0040 8.0005 8.0040 8.0014 8.0089 8.0087
12.8080 12.8020 12.8080 12.8070 12.8620 12.8630
17.2030 17.1970 17.2030 17.2070 17.3270 17.3310
20.4220 20.4160 20.4220 20.4330 20.6100 20.6170
22.997 22.990 22.997 23.014 23.243 (23.239 a; 23.236 b) 23.252
Closed form solution based on TSDT 7 [26]. FEM solution of TSDT 7 [9].
three-dimensional elasticity theories. It is worth mentioning here that, for the aspect ratio, S 5; the elasticity solutions are available in both Refs. [17,25]. However, a slight difference in the results presented in Refs. [17,25] is noticed, because of the adoption of different methods of solutions. Furthermore, it can be noted here that the theories employed for developing the elements Q9-HSDT 11, and Q9-HSDT 9 in Ref. [12] are same as those of present elements Q8-HSDT 11B, and Q8-HSDT 9, respectively. Due to these reasons, the results of different elements based on same theory yield almost same results. However, for fairly low aspect ratio cases, the error involved in predicting the solutions using the elements based on the second- and ®rst-order theory is high compared to that of third-order based elements.
Further, in general, the accuracy of the present higherorder theory (Q8-HSDT 13) over other available theories for thick laminates when compared to elasticity solutions can also be noted. It may be concluded from this study that neglecting the higher order terms in transverse displacements (w1 and G , Q8-HSDT 11A) has negligible effect on the results in comparison with those of dropping the zig-zag function (c , Q8-HSDT 11B) in the in-plane displacement ®elds. It can be viewed from this example that, in general, the present model (Q8-HSDT 13) accurately predicts the results for both thick and fairly thin laminates and the zig-zag variation in the in-plane displacement is necessary especially for the thick laminates.
M. Ganapathi, D.P. Makhecha / Composites: Part B 32 (2001) 535±543
541
Table 6 p Non-dimensional fundamental frequency v va2 =h r=ET of four layered angle-ply square plates EL =ET 40; 458= 2 4582 Boundary condition
Model
Side-to-thickness ratio a=h 5
10
20
50
SSSS
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q8-SSDT 7 Q8-FSDT 5 Analytical [13]
11.8192 11.8099 11.8794 11.8685 12.5335 12.6026 12.9280
17.4072 17.3940 17.4811 17.4673 18.3184 18.4424 18.6650 18.320 a
20.9868 20.9746 21.0432 21.0309 21.8067 21.8669 21.9540
22.7349 22.7291 22.7544 22.7487 23.2276 23.2396 23.6520
SSCC
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q8-SSDT 7 Q8-FSDT 5 Analytical [13]
12.3511 12.3401 12.4037 12.3923 13.0575 12.9733 13.4580
19.5351 19.5153 19.6388 19.6188 20.3964 20.4529 20.8870
25.7282 25.7080 25.8201 25.7999 26.4464 26.5366 26.7700
29.5588 29.5443 29.5897 29.5753 29.9060 29.9360 29.9790
CCCC
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q8-SSDT 7 Q8-FSDT 5
13.1969 13.1829 13.2454 13.2321 13.9040 13.7167
21.7727 21.7461 21.9008 21.8743 22.6176 22.6376
29.8091 29.7835 29.9260 29.9003 30.4067 30.5141
35.1984 35.1779 35.2391 35.2186 35.3724 35.4165
a
Ref. [9].
Example 2. Free vibration of simply supported cross-ply laminated square plates S 5 with different E L =ET [25] To analyze the effect of degree of orthotropicity of the layers, EL =ET ( 3, 10, 20, 30 and 40) is varied. There is no change in the other material properties. The results obtained for fundamental frequency are shown in Table 4 along with those of various ®nite elements [12] and three-dimensional elasticity theory [25]. The in¯uence of number of layers is also highlighted in this table. However, for high degree of orthotropicity case, the error involved in predicting the solutions using the elements based on the second- and ®rst-order theory is high compared to the third-order based element as highlighted in Example 1. Furthermore, for low values of EL =ET ; dropping the higher order terms in transverse displacements (w1 and G , Q8-HSDT 11A) can signi®cantly affect the results in comparison with the suppression of zig-zag function (c , Q8-HSDT 11B) in the in-plane displacement. But the zig-zag function has considerable in¯uence on the results for high values of orthotropicity compared to higher order terms in transverse displacements. In general, the effectiveness of the present higher-order theory (Q8HSDT 13) in terms of accuracy can be clearly seen from this study, irrespective of the degree of orthotropicity and the number of layers, while comparing with three-dimensional elasticity solutions. Also, it can be opined here that the lower order displacement models, in general, yields upper bound for the prediction of frequencies whereas model having higher-order terms and zig-zag variation in the in-plane displacements alone slightly under-predicts the results.
Example 3. Free vibration of simply supported angle-ply laminated square plates [26] The performance of present higher-order theory for the free vibration of two- and eight-layered angle-ply cases is demonstrated in Table 5 by varying the aspect ratio and degree of orthotropicity. The in¯uence of zig-zag function, stretching terms in the in-plane and transverse displacements, respectively, and various other higher order terms in the model on the fundamental frequencies can be seen from this table. Further, the results predicted, through the present Q8-TSDT 7 model, are almost same as those of Refs. [9,26] as they are all based on same theory. Also, it is apparent from the results that the dropping of stretching terms (w1 and G , Q8-HSDT 11A) in the transverse displacement signi®cantly affects the accuracy compared to neglecting the zig-zag term (c , Q8-HSDT 11B) in the in-plane displacement ®elds, irrespective of degree of orthotropicity. It is revealed from the detailed analysis that retaining the higher order terms in the transverse displacement is essential for accurately evaluating the solutions for angle-ply cases. However, unlike cross-ply case, it can be seen here that the lower order displacement models, in general, yields upper bound for the prediction of frequencies whereas model having higher-order terms in the in-plane displacements and thickness stretch term in the transverse de¯ection alone slightly under-predict the solutions. Example 4. Free vibration of angle-ply laminated square plates with different boundary conditions [13] The effect of boundary conditions on the fundamental
542
M. Ganapathi, D.P. Makhecha / Composites: Part B 32 (2001) 535±543
Table 7 p Non-dimensional natural frequencies v v rh2 =ET of two-layered square plates Ply-angle
Model
v 1
v 2
v 3
v 4
v 5
08=908
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9
0.34638 0.34598 0.34911 0.34870
0.48670 0.48670 0.48670 0.48670
0.48670 0.48670 0.48670 0.48670
0.70567 0.70409 0.71403 0.71273
0.94847 0.94549 0.95795 0.95555
458= 2 458
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9
0.3865 0.3849 0.3865 0.3963
0.5686 0.5694 0.5686 0.6606
0.7280 0.7260 0.7280 0.7342
0.8896 0.8942 0.8896 0.7350
1.0124 1.0116 1.0124 1.0262
frequencies are brought out here considering four-layered angle-ply laminates for which exact solutions, based on second-order theory, are available. The results based on present Q8-HSDT 13 are tabulated in Table 6 along with those of present element Q8-SSDT 7, and exact solutions using analytical approach [13]. It can be noted here that the theory used for Q8-SSDT 7 is same as that of given in Ref. [13]. As expected, the present results associated with second-order theory match very closely with the analytical/ numerical solutions. Also, for simply supported boundary conditions and S 10 case, the present result is compared with those of Q9-TSDT given in Ref. [9]. However, it is revealed from this table that, for low aspect ratio, the results predicted by the present higher-order theory (Q8-HSDT 13) are different from those of second- and ®rst-order theory (Q8-SSDT 7 and Q8-FSDT 5). Furthermore, it can be opined from this table that the difference in the results among various theories is more while dealing with the clamped boundary conditions compared to other boundary conditions considered here. Thus, the present model (Q8HSDT 13) is ef®cient for any type of boundary conditions. Example 5. Higher mode frequencies of simply supported laminated square plates This example concerns with the in¯uence of various theories on the frequencies of higher ¯exural vibration modes and is shown in Table 7. An 8 £ 8 mesh is used for the modeling of full plate. The numerical results clearly show that the different alternate higher order models deduced here have less in¯uence on the fundamental frequencies. However, for higher modes, the effectiveness of the present model (Q8-HSDT 13) especially for the angle-ply case can be clearly seen in comparison with those of other models considered here. 5. Conclusions A displacement-based C 0 continuous isoparametric eightnoded quadrilateral plate element has been presented here based on a new higher-order laminated theory. The accuracy and effectiveness of the present model over the lower- and other higher-order theories for free vibration analysis of thick composite laminates have been demonstrated
considering problems for which exact/numerical results are available based on elasticity and higher-order theories. The in¯uence of slope discontinuity in thickness direction for in-plane, and the thickness stretching terms in the transverse displacement ®elds on the free vibration characteristics has been brought out. For accurate modeling of thick laminate with high degree of orthotropicity, it can be concluded that, in addition to higher-order terms in the in-plane displacement, zig-zag variation in the in plane displacement through thickness is essential for cross-ply plates whereas thickness stretch term in transverse displacement is required for angle-ply cases. References [1] Noor AK, Burton WS. Assessment of shear deformation theories for multilayered composite plates. ASME J Appl Mech Rev 1989;42:1± 13. [2] Kapania RK, Raciti S. Recent advances in analysis of laminated beams and plates. Part I: Shear effects and buckling. AIAA J 1989;27:923±34. [3] Reddy JN. A review of re®ned theories of composite laminates. Shock Vibr Dig 1990;22:3±17. [4] Mallikarjuna, Kant T. A critical review and some results of recently developed re®ned theories of ®bre reinforced laminated composites and sandwiches. Compos Struct 1993;23:293±312. [5] Varadan TK, Bhaskar K. Review of different laminate theories for the analysis of composite. J Aeronaut Soc India 1997;49:202±8. [6] Lo KH, Christensen RM, Wu EM. A higher-order theory of plate deformation. Part 2: Laminated plates. ASME J Appl Mech 1977;44:669±76. [7] Reddy JN. A simple higher-order theory for laminated composite plates. J Appl Mech 1984;45:745±52. [8] Phan ND, Reddy JN. Analysis of laminated composite plates using a higher-order shear deformation theory. J Numer Meth Engng 1985;21:2201±19. [9] Kant T, Mallikarjuna . A higher-order theory for free vibration of unsymmetrically laminated composite and sandwich plates Ð ®nite element evaluations. Comput Struct 1989;32:1125±32. [10] Shu X, Sun L. An improved simple higher order theory for a laminated composite plates. Comput Struct 1995:50. [11] Shi G, Lam KY, Tay TE. On ef®cient ®nite element modelling of composite beams and plates using higher order theories and an accurate composite beam element. Comput Struct 1998:41. [12] Franco Correia VM, Mota Soares CM, Mota Soares CA. Higher order models on the eigenfrequency analysis and optimal design of laminated composite structures. Compos Struct 1998;39:237±53. [13] Khdeir AA, Reddy JN. Free vibrations of laminated composite plates
M. Ganapathi, D.P. Makhecha / Composites: Part B 32 (2001) 535±543
[14] [15] [16] [17] [18] [19]
using second-order shear deformation theory. Comput Struct 1999;71:617±26. Bhaskar K, Varadan TK, Ali JSM. Thermoelastic solutions for orthotropic and anisotropic composite laminates. Composites 1996;27:415±20. Reddy JN. A generalization of two-dimensional theories of laminated composite plates. Commun Appl Numer Meth 1987;3:173±80. Cho KN, Bert CW, Striz AG. Free vibrations of laminated rectangular plates analyzed by higher order indiviual-layer theory. J Sound Vibr 1991;145:429±42. Nosier A, Kapania RK, Reddy JN. Free vibration analysis of laminated plates using a layer wise theory. AIAA J 1993;31:2335±46. Lee KH, Senthilnathan NR, Lim SP, Chow ST. An improved zigzag model for the bending of laminated composites plates. Compos Struct 1990;15:137±48. Li X, Liu D. Zigzag theory for composite laminates. AIAA J 1995;33:1163±5.
543
[20] Ali JSM, Bhaskar K, Varadan TK. A new theory for accurate thermal/ mechanical ¯exural analysis of symmetrically laminated plates. Compos Struct 1999;45:227±32. [21] Makhecha DP, Ganapathi M, Patel BP. Dynamic analysis of laminated composite plates subjected to thermal/mechanical loads using an accurate theory. Compos Struct 2001;51:221±36. [22] Murukami H. Laminated composite plate theory with improved in-plane responses. ASME J Appl Mech 1986;53:661±6. [23] Jones RM. Mechanics of composite materials. New York: McGrawHill, 1975. [24] Noor AK, Mathers MD. Shear ¯exible ®nite element models of laminated composite plates and shells. NASA TN-D 8044, 1975. [25] Noor AK. Free vibrations of multilayered composite plates. AIAA J 1973;11:1038±9. [26] Reddy JN, Phan ND. Stability and vibration of isotropic, orthotropic and laminated plates according to a higher-order shear deformation theory. J Sound Vibr 1985;98:157±70.
www.elsevier.com/locate/compositesb
Free vibration analysis of multi-layered composite laminates based on an accurate higher-order theory M. Ganapathi*, D.P. Makhecha Institute of Armament Technology (Deemed University), Girinagar, Pune 411 025, India Received 11 July 2000; accepted 22 February 2001
Abstract This paper deals with an accurate higher-order theory employing ®nite element procedure for the free vibration analysis of multi-layered thick composite plates. The theory accounts for the realistic variation of in-plane and transverse displacements through the thickness. The accuracy of the present model is veri®ed by comparison with three-dimensional elasticity solutions for the vibration study of the composite laminates. The performance and the applicability of the proposed discrete model are also discussed among developed elements and alternate models, considering different parameters such as ply-angle, degree of orthotropicity, aspect ratio and boundary conditions. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: B. Vibration; Finite element
1. Introduction The application of advanced high stiffness- and strength-to-weight ratios of the composite material systems has played a key role in the success of the aerospace and aircraft industries. However, the analysis of the multi-layered structures is a complex task, compared with conventional single layer metallic structures, due to the exhibition of coupling among membrane, torsion and bending strains; weak transverse shear rigidities; and discontinuity of the mechanical characteristics along the thickness of the laminates. More accurate analytical/numerical analysis based on three-dimensional models may result computationally involved and expensive. Hence, among researchers, there is a growing appreciation of the importance of developing new kinematics for the evolution of accurate two-dimensional theories for the analysis of thick laminates with high orthotropic ratio, leading to less expensive models. In this context, the applications of analytical/numerical methods based on various higherorder theories, not only for the vibrations of thick laminates, but also for the high frequency vibrations of thin composite plates, has recently attracted the attention of several investigators/researchers. Various structural theories proposed for evaluating * Corresponding author. Tel.: 191-20-599-550; fax: 191-20-599-509. E-mail address: [email protected] (M. Ganapathi).
the characteristics of composite laminates under different loading situations have been reviewed and assessed by Noor and Burt [1], Kapania and Raciti [2], Reddy [3], and more recently by Mallikarjuna and Kant [4], Varadan and Bhaskar [5]. It has been concluded that the assumption of displacements as linear functions of the coordinate in the thickness direction has proved to be inadequate for predicting the response of thick laminates. Furthermore, due to the advantage that no shear correction factors are needed and the warping of the cross-section can be accounted for to a certain extent, higher-order theories are widely preferred in the analysis of composite plates. Higher-order displacement ®elds involving higher order expansions of the displacement ®eld in powers of the thickness coordinate yielding quadratic variation of transverse shear strains have been attempted by many researchers [6±13] for better accuracy. A careful look into the results of three-dimensional analysis carried out by Bhaskar et al. [14] for thick laminates has revealed that the in-plane displacements are non-linear through the thickness and exhibit an abrupt discontinuity in slope at any interface, and thickness-stretch/contraction effects in the transverse displacement. Although higher-order theories based on discrete layer approach [15±19] account for slope discontinuity at the interfaces, the number of unknowns to be solved increases with the increase in the number of layers. Recently, Ali et al. [20] have proposed a new higher-order plate theory based on global approximation
1359-8368/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 1359-836 8(01)00028-2
536
M. Ganapathi, D.P. Makhecha / Composites: Part B 32 (2001) 535±543
approach, for multi-layered composite laminates under thermal/mechanical loading, incorporating realistic through the thickness approximations of the in-plane and transverse displacements based on the work given in Ref. [14]. This formulation has proved to give very accurate results for static analysis of symmetric crossply laminates, and this excellent performance of the theory for thick/moderately thick laminates motivated the present extension of the formulation for the vibration analysis of general composite laminates through ®nite element procedure. Here, C 0 eight-noded quadrilateral serendipity plate element with 13 degrees of freedom per node [21] based on the theory given in Ref. [20] is employed for free vibration study of thick composite laminates. The accuracy of the present element with those of other available numerical/analytical models and threedimensional elasticity analysis are compared and discussed. Since the vibration analysis of angle-ply laminates based on three-dimensional elasticity theory is not available in the literature, the present results are compared among the developed elements, and alternate models considering with/without the effects of slope discontinuity and thickness-stretch in the present assumed displacement ®elds. The numerical results also illustrate that the proposed model has a signi®cant in¯uence on the frequencies of higher ¯exural vibration modes. 2. Formulation A composite plate with arbitrary lamination is considered with the co-ordinates x, y along the in-plane directions and z along the thickness direction. The in-plane displacements u k and v k, and the transverse displacement w k for the kth layer, are assumed as uk x; y; z; t u0 x; y; t 1 zux x; y; t 1 z2 bx x; y; t 1 z3 fx x; y; t 1 Sk c x x; y; t
1
a generic point on the reference surface; ux; uy are the rotations of normal to the reference surface about the y and x axes, respectively; w1 ; bx ; by ; G x ; fy are the higher order terms in the Taylor's series expansions, de®ned at the reference surface. c x and c y are generalized variables associated with the zig-zag function, Sk : The zig-zag function, Sk ; as given in Ref. [22], is de®ned by Sk 2 21k zk =hk
where zk is the local transverse coordinate with its origin at the center of the kth layer and hk is the corresponding layer thickness. Thus, the zig-zag function is piecewise linear with values of 21 and 1 alternately at the different interfaces. The `zig-zag' function, as de®ned above, takes care of inclusion of the slope discontinuity of u and v at the interfaces of the laminate as observed in exact three-dimensional elasticity solutions of thick laminated composite structures. The use of such function is more economical than a discrete layer approach of approximating the displacement variations over the thickness of each layer separately. Although both these approaches account for slope discontinuity at the interfaces, in the discrete layer approach the number of unknowns increases with the increase in the number of layers, whereas it remains constant in the present approach. The strains in terms of mid-plane deformation, rotations of normal, and higher order terms associated with displacements are as ( {e}
k
1 z fy x; y; t 1 S c y x; y; t
wk x; y; z; t w0 x; y; t 1 zw1 x; y; t 1 z2 G x; y; t The terms with even power in z in the in-plane displacements and those odd in z occurring in the expansion for w k correspond to stretching problems. But, the terms with odd in z in the in-plane displacements and even in z in the expression for w k represent the ¯exure problems. u 0 ; v0 ; w0 are the displacements of
ebm
) 3
es
The vector {ebm } includes the bending and membrane terms of the strain components and vector {es } contains the transverse shear strain terms. These strain vectors can be de®ned as
vk x; y; z; t v0 x; y; t 1 zuy x; y; t 1 z2 by x; y; t 3
2
{ebm }
8 9 exx > > > > > > > > > < eyy > = > ezz > > > > > > > > :g > ; xy
8 > > > > > <
u;x
> > > > > :
w;z
9 > > > > > =
v;y u;y 1v;x
> > > > > ;
{e0 } 1 z{e1 } 1 z2 {e2 } 1 z3 {e3 } 1 Sk {e4 } ( {es }
gxz gyz
)
(
u; z 1 w; x
4a
)
v; z 1 w;y
{g0 } 1 z{g1 } 1 z2 {g2 } 1 Sk; z {g3 }
4b
M. Ganapathi, D.P. Makhecha / Composites: Part B 32 (2001) 535±543
where
{e0 }
8 > > > > > < > > > > > :
u0;x v0;y w1
8 9 > > > > > > > > > > < = { e1 } > > > > > > > > > > : ;
uX;X uY;Y 2G
( {g0 }
ux 1 w0;x uy 1 w0;y
) {g1 }
(
2bx 1 w1;x
2by 1 w1;y 8 9 ( ) < c x Sk;z = 3fx 1 G ;x {g2 } {g0 } : c Sk ; 3fy 1 G ;y y ;z
The kinetic energy of the plate is given by " # n Zhk 1 1 1 ZZ X k k k k k k T rk {u_ v_ w_ }{u_ v_ w_ } dz dx dy T d 2 k1 hk
9 > > > > > =
7
> > > > > ;
u X;Y 1 uY;X 8 8 9 9 bx;x fx;x > > > > > > > > > > > > > > > > > > > > < < = = by;y fy;y {e2 } {e3 } > > > > 0 0 > > > > > > > > > > > > > > > :b 1 b ; :f 1f > ; x;y y;x x;y y;x 8 9 c x;x > > > > > > > > > > < = c y;y {e4 } > > 0 > > > > > > > :c 1c > ; x;y y;x u0;y 1 v0;x
537
where r k is the mass density of the kth layer. hk, hk11 are the z coordinates of laminate corresponding to the bottom and top surfaces of the kth layer. The strain energy functional U is given by " # n Z hk 1 1 1 ZZ X T U d {s} {e} dz dx dy 8 2 k1 hk Substituting Eqs. (7) and (8) in Eq. (6), one obtains the governing equation for the free vibration of plate as M{d } 1 K{d} {0} (4c)
)
(4d)
The subscript comma denotes the partial derivative with respect to the spatial coordinate succeeding it. The constitutive relations for an arbitary layer k, written in the laminate x; y; z coordinate system can be expressed as {s} {s xx s yy s zz txy txz tyz }T Q k {exx eyy ezz gxy gxz gyz }T 5 where the terms of Q k matrix of ply k are referred to the laminate axes and can be obtained from the Qk corresponding to the ®ber directions with the appropriate transformation, as outlined in the literature [23]. The superscript T refers the transpose of a matrix/vector. The governing equations are obtained by applying Lagrangian equations of motion given by d=dt 2 T 2 U=2d_ i 2 2 T 2 U=2d_ i 0; i 1 to n 6 where T is the kinetic energy; U is the strain energy contributions due to the in-plane and transverse stresses, respectively. {d} {d1 ; d2 ; ¼dl ; ¼dn }T is the vector of the degree of freedoms/generalized coordinates. A dot over the variables represents the partial derivative with respect to time.
9
where M and K are the mass and stiffness matrices. The coef®cients of mass and stiffness matrices involved in Eq. (9) can be rewritten as the product of term having thickness co-ordinate z alone and the term containing x and y. In the present study, while performing the integration, terms having thickness co-ordinate z are explicitly integrated whereas the terms containing x and y are evaluated using full integration with 3 £ 3 points Gauss integration rule. The frequencies and mode shapes are obtained from Eq. (9) using the standard eigen value algorithm. 3. Element description In the present work, a simple, C 0 continuous, eight-noded serendipity quadrilateral shear ¯exible plate element with 13 degrees of freedom (u0, v0, w0, ux ; uy ; w1, bx ; by ; G , f x ; fy ; c x and c y : 13-DOF) is developed. The ®nite element represented as per the kinematics, for Table 1 Alternate eight-noded ®nite element discrete models developed, and available model for comparison purpose Finite element discrete model
Degrees of freedom per node
Q8-HSDT 13 (Present)
u0, v0, w0, u x, u y, w1, b x, by , G , f x , f y, c x, c y u0, v0, w0, u x, u y, b x, b y, f x , f y, c x, c y u0, v0, w0, u x, u y, w1, b x, b y, G , f x, f y u0, v0, w0, u x, u y, b x, b y, f x, f y u0, v0, w0, u x, u y, b x, b y u0, v0, w0, u x, u y u0, v0, w0, u x, u y, w1, b x, b y , G , f x, f y u0, v0, w0, u x, u y, b x, b y, f x, f y u0, v0, w0, u x, u y, f x, f y
Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q8-TSDT 7 Q8-FSDT 5 Q9-HSDT 11 [12] Q9-HSDT 9 [12] Q9-HSDT 9 [9]
538
M. Ganapathi, D.P. Makhecha / Composites: Part B 32 (2001) 535±543
instance see Eq. (1), can be referred as Q8-HSDT 13 with cubic variation. Four more alternate discrete models are proposed, for the comparison purpose, whose displacement ®elds are deduced from the original element by deleting the appropriate degrees of freedom (w1 and G 0; or c 0; or c; w 1 and G 0; or z 2 terms c; w 1 and G 0). In addition to third-order elements as described above, second- and ®rst-order element are deduced from the original proposed elements by neglecting the appropriate terms in the displacement ®elds (z 3 terms, c; w 1 and G 0; dropping all the higher order terms). The above described ®nite elements, and other available models, and corresponding degrees of freedom are shown in Table 1.
4. Results and discussion Numerical experimentation conducted for checking the rank of the element, locking syndrome and patch test has shown that the element is free from spurious rigid modes, locking phenomenon, etc. [21]. A convergence study carried out, based on progressive mesh re®nement, for the free vibration study of composite laminate is given in Table 2 and is found that an 4 £ 4 grid mesh is adequate to model the quarter plates. Although the convergence of the solutions obtained for very thick as well as fairly thick laminates by all the displacement models is monotonic in character, the convergence of lower order models is slower than the higher order models. The convergence study conducted here for fairly thick case matches well with those of Ref. [24]. Since the higher-order theory, in general, is required for accurate analysis for thick laminates, the numerical study is concerned with thick multi-layered anisotropic plates having aspect ratio, S # 50: Furthermore, the ef®cacy of the present ®nite element formulation and theory for the free vibration analysis of thick cases, is tested over the performances of various ®nite element models deduced from the present theory, available ®nite elements, and analytical models based on elasticity and higher-order
theories. For this purpose, a few examples are considered emphasizing the in¯uence of side-to-thickness ratio (S a=h; h is the thickness of the plate), degree of orthotropicity of individual layers E L =ET ; boundary conditions and layer-angle. The material properties, unless speci®ed otherwise, assumed in the present analysis are EL =ET 40;
GLT =ET 0:6;
GTT =ET 0:5;
nLT 0:25
10
where E, G and n are Young's modulus, shear modulus and Poisson's ratio. Subscripts L and T are the longitudinal and transverse directions, respectively, with respect to the ®bers. The ®rst layer corresponds to bottommost layer and the layer-angle is measured from x axis in an anti-clockwise direction and all the layers are of equal thickness. The boundary conditions considered here are Simply-supported condition: Cross-ply : v 0 w 0 uy w 1 G b y f y c y 0 at x 0; a u0 w 0 ux w 1 G b x f x c x 0 at y 0; b Angle-ply : u0 w 0 uy w 1 G b y f y c y 0 at x 0; a v 0 w 0 ux w 1 G b x f x c x 0 at y 0; b
Table 2 p Convergence study for fundamental frequency v v ra2 =ET of nine layered cross ply laminates [24] S a=h
5
2£2 3£3 4£4 6£6 8£8 2£2 3£3 4£4 6£6 8£8
10
a
Grid size
Ref. [24].
Model Q8-HSDT 13
Q8-HSDT 11A
Q8-HSDT 11B
Q8-HSDT 9
Q8-SSDT 7
Q8-FSDT 5
2.313587 2.312446 2.312255 2.312184 2.312172 1.586347 1.584781 1.584517 1.584417 1.584400
2.311024 2.309882 2.309692 2.309621 2.309609 1.585364 1.583798 1.583534 1.583434 1.583417
2.331468 2.330299 2.330104 2.330032 2.330019 1.588952 1.587377 1.587112 1.587011 1.586994
2.328818 2.327648 2.327453 2.327381 2.327369 1.587965 1.586390 1.586124 1.586024 1.586007
2.328818 2.327648 2.327453 2.327381 2.327369 1.587965 1.586390 1.586124 1.586024 1.586007
2.455905 2.454715 2.454519 2.454447 2.454435 1.627079 [1.627] a 1.625402 [1.625] a 1.625124 1.625021 1.625003
M. Ganapathi, D.P. Makhecha / Composites: Part B 32 (2001) 535±543 Table 3 p Non dimensional fundamental frequency v va2 =h r=ET of simply supported two-layered cross-ply square plates EL =ET 40 Model
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q8-SSDT 7 Q8-FSDT 5 Q9-HSDT11 [12] Q9-HSDT9 [12] Elasticity [17] a
Side-to-thickness ratio a=h
Clamped support: u0 v0 w0 ux uy w1 bx by G f x fy c x c y 0
5
10
50
8.6595 8.6496 8.7277 8.7176 8.8008 8.8246 8.7274 8.7172 8.5180 8.5625 a
10.3820 10.3880 10.4340 10.4290 10.4660 10.4710 10.4323 10.4279 10.333
11.2870 11.2870 11.2900 11.2890 11.2920 11.2920 11.2693 11.2689 11.263
Ref. [25].
539
at
x 0; a; y 0; b 11
Here a; b refer the length and width of the plate, respectively. Example 1. Free vibration of simply supported cross-ply laminated square plates with different aspect ratio, S [17] To examine the in¯uence of the variation of the aspect ratio, different values for S are assumed. The solutions evaluated for the fundamental frequencies of the ¯exural vibration are given in Table 3 along with those of various higher-order ®nite elements, layer-wise and
Table 4 p Non dimensional fundamental frequency v v rh2 =ET £ 10 of simply supported cross-ply square plates S 5 No. layer (N)
Model
EL =ET 3
10
20
30
40
2
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q8-SSDT 7 Q8-FSDT 5 Elasticity [25] a
2.4935 2.5478 2.4937 2.5480 2.5177 2.4824 2.5031
2.7886 2.7830 2.7899 2.7843 2.8156 2.7742 2.7938
3.0778 3.1066 3.0858 3.1142 3.1125 3.0802 3.0698
3.2940 3.2897 3.3113 3.3069 3.3391 3.3256 3.2705
3.4638 3.4598 3.4911 3.4870 3.5203 3.5299 3.4250
4
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q9-HSDT11 [12] Q8-HSDT9 [12] Q8-SSDT 7 Q8-FSDT 5 Elasticity [25]
2.6029 2.6547 2.6061 2.6580 2.6059 2.5983 2.6405 2.6004 2.6182
3.2488 3.2408 3.2595 3.2514 3.2594 3.2513 3.3506 3.2871 3.2578
3.7677 3.7796 3.7872 3.7990 3.7871 3.7793 3.9521 3.8706 3.7622
4.0841 4.0771 4.1095 4.1023 4.1094 4.1022 4.3349 4.2415 4.0660
4.3001 4.2936 4.3290 4.3225 4.3289 4.3224 4.6042 4.5007 4.2719
6
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q9-HSDT11 [12] Q8-HSDT9 [12] Q8-SSDT 7 Q8-FSDT 5 Elasticity [25]
2.6264 2.6780 2.6289 2.6805 2.6287 2.6211 2.663 2.6215 2.6440
3.3478 3.3399 3.3547 3.3468 3.3546 3.3467 3.4443 3.3643 3.3657
3.9219 3.9321 3.9342 3.9445 3.9342 3.9267 4.0957 3.9719 3.9359
4.2686 4.2621 4.2849 4.2783 4.2848 4.2782 4.5053 4.3462 4.2783
4.5035 4.4976 4.5225 4.5166 4.5223 4.5164 4.7987 4.6029 4.5091
10
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q9-HSDT11 [12] Q8-HSDT9 [12] Q8-SSDT 7 Q8-FSDT 5 Elasticity [25]
2.6390 2.6905 2.6410 2.6926 2.6409 2.6332 2.6747 2.6321 2.6583
3.4018 3.3941 3.4068 3.3990 3.4066 3.3988 3.4926 3.4022 3.4250
4.0093 4.0189 4.0177 4.0274 4.0176 4.0105 4.1694 4.0201 4.0337
4.3770 4.3709 4.3881 4.3819 4.3880 4.3818 4.5924 4.3952 4.4011
4.6279 4.6224 4.6415 4.6359 4.6398 4.6344 4.8671 4.6508 4.6498
a
Based on higher-order difference scheme.
540
M. Ganapathi, D.P. Makhecha / Composites: Part B 32 (2001) 535±543
Table 5 p Non-dimensional fundamental frequency v va2 =h r=ET for angle-ply two and eight layered 458= 2 458; 458= 2 4584 square plates N
S a=h
Model
EL =ET 3
2
8
a b
10
20
30
40
5
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q8-TSDT 7 Q8-FSDT 5
6.4480 6.4174 6.4480 6.4369 6.4458 6.4291
7.7066 7.6577 7.7066 7.7494 7.8915 7.8006
8.6130 8.5642 8.6130 8.7417 9.1200 8.9285
9.2138 9.1694 9.2138 9.4097 10.0010 9.7200
9.6620 9.6219 9.6620 9.9079 10.681 (10.840 a; 10.692 b) 10.321
10
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q8-TSDT 7 Q8-FSDT 5
7.0751 7.0602 7.0751 7.0705 7.0760 7.0707
8.8408 8.8060 8.8408 8.8686 8.9639 8.9316
10.3080 10.2650 10.3080 10.4150 10.7000 10.6240
11.3880 11.3430 11.3880 11.5800 12.0640 11.9410
12.257 12.213 12.257 12.532 13.208 (13.263 a; 13.207 b) 13.037
20
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q8-TSDT 7 Q8-FSDT 5
7.2628 7.2555 7.2628 7.2663 7.2719 7.2706
9.2186 9.1995 9.2186 9.2528 9.3274 9.3184
10.9800 10.9540 10.9800 11.0690 11.2750 11.2530
12.3680 12.3380 12.3680 12.5140 12.8590 12.8210
13.544 13.513 13.544 13.751 14.233 (14.246 a; 14.228 b) 14.179
5
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q8-TSDT 7 Q8-FSDT 5
6.9894 6.9677 6.9894 6.9739 6.9886 6.9845
9.7175 9.6973 9.7175 9.7154 9.8062 9.7979
11.3490 11.3350 11.3490 11.3630 11.5190 11.4900
12.1890 12.1790 12.1890 12.2120 12.4060 12.3450
12.712 12.705 12.712 12.741 12.961 (12.972 a; 12.967 b) 12.863
10
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q8-TSDT 7 Q8-FSDT 5
7.7612 7.7524 7.7612 7.7546 7.7629 7.7619
11.9370 11.9260 11.9370 11.9350 12.0040 12.0060
15.2850 15.2730 15.2850 15.2930 15.4430 15.4480
17.4360 17.4350 17.4360 17.4540 17.6670 17.6700
18.978 18.969 18.978 19.005 19.267 (19.266 a; 19.274 b) 19.265
20
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q8-TSDT 7 Q8-FSDT 5
8.0040 8.0005 8.0040 8.0014 8.0089 8.0087
12.8080 12.8020 12.8080 12.8070 12.8620 12.8630
17.2030 17.1970 17.2030 17.2070 17.3270 17.3310
20.4220 20.4160 20.4220 20.4330 20.6100 20.6170
22.997 22.990 22.997 23.014 23.243 (23.239 a; 23.236 b) 23.252
Closed form solution based on TSDT 7 [26]. FEM solution of TSDT 7 [9].
three-dimensional elasticity theories. It is worth mentioning here that, for the aspect ratio, S 5; the elasticity solutions are available in both Refs. [17,25]. However, a slight difference in the results presented in Refs. [17,25] is noticed, because of the adoption of different methods of solutions. Furthermore, it can be noted here that the theories employed for developing the elements Q9-HSDT 11, and Q9-HSDT 9 in Ref. [12] are same as those of present elements Q8-HSDT 11B, and Q8-HSDT 9, respectively. Due to these reasons, the results of different elements based on same theory yield almost same results. However, for fairly low aspect ratio cases, the error involved in predicting the solutions using the elements based on the second- and ®rst-order theory is high compared to that of third-order based elements.
Further, in general, the accuracy of the present higherorder theory (Q8-HSDT 13) over other available theories for thick laminates when compared to elasticity solutions can also be noted. It may be concluded from this study that neglecting the higher order terms in transverse displacements (w1 and G , Q8-HSDT 11A) has negligible effect on the results in comparison with those of dropping the zig-zag function (c , Q8-HSDT 11B) in the in-plane displacement ®elds. It can be viewed from this example that, in general, the present model (Q8-HSDT 13) accurately predicts the results for both thick and fairly thin laminates and the zig-zag variation in the in-plane displacement is necessary especially for the thick laminates.
M. Ganapathi, D.P. Makhecha / Composites: Part B 32 (2001) 535±543
541
Table 6 p Non-dimensional fundamental frequency v va2 =h r=ET of four layered angle-ply square plates EL =ET 40; 458= 2 4582 Boundary condition
Model
Side-to-thickness ratio a=h 5
10
20
50
SSSS
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q8-SSDT 7 Q8-FSDT 5 Analytical [13]
11.8192 11.8099 11.8794 11.8685 12.5335 12.6026 12.9280
17.4072 17.3940 17.4811 17.4673 18.3184 18.4424 18.6650 18.320 a
20.9868 20.9746 21.0432 21.0309 21.8067 21.8669 21.9540
22.7349 22.7291 22.7544 22.7487 23.2276 23.2396 23.6520
SSCC
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q8-SSDT 7 Q8-FSDT 5 Analytical [13]
12.3511 12.3401 12.4037 12.3923 13.0575 12.9733 13.4580
19.5351 19.5153 19.6388 19.6188 20.3964 20.4529 20.8870
25.7282 25.7080 25.8201 25.7999 26.4464 26.5366 26.7700
29.5588 29.5443 29.5897 29.5753 29.9060 29.9360 29.9790
CCCC
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9 Q8-SSDT 7 Q8-FSDT 5
13.1969 13.1829 13.2454 13.2321 13.9040 13.7167
21.7727 21.7461 21.9008 21.8743 22.6176 22.6376
29.8091 29.7835 29.9260 29.9003 30.4067 30.5141
35.1984 35.1779 35.2391 35.2186 35.3724 35.4165
a
Ref. [9].
Example 2. Free vibration of simply supported cross-ply laminated square plates S 5 with different E L =ET [25] To analyze the effect of degree of orthotropicity of the layers, EL =ET ( 3, 10, 20, 30 and 40) is varied. There is no change in the other material properties. The results obtained for fundamental frequency are shown in Table 4 along with those of various ®nite elements [12] and three-dimensional elasticity theory [25]. The in¯uence of number of layers is also highlighted in this table. However, for high degree of orthotropicity case, the error involved in predicting the solutions using the elements based on the second- and ®rst-order theory is high compared to the third-order based element as highlighted in Example 1. Furthermore, for low values of EL =ET ; dropping the higher order terms in transverse displacements (w1 and G , Q8-HSDT 11A) can signi®cantly affect the results in comparison with the suppression of zig-zag function (c , Q8-HSDT 11B) in the in-plane displacement. But the zig-zag function has considerable in¯uence on the results for high values of orthotropicity compared to higher order terms in transverse displacements. In general, the effectiveness of the present higher-order theory (Q8HSDT 13) in terms of accuracy can be clearly seen from this study, irrespective of the degree of orthotropicity and the number of layers, while comparing with three-dimensional elasticity solutions. Also, it can be opined here that the lower order displacement models, in general, yields upper bound for the prediction of frequencies whereas model having higher-order terms and zig-zag variation in the in-plane displacements alone slightly under-predicts the results.
Example 3. Free vibration of simply supported angle-ply laminated square plates [26] The performance of present higher-order theory for the free vibration of two- and eight-layered angle-ply cases is demonstrated in Table 5 by varying the aspect ratio and degree of orthotropicity. The in¯uence of zig-zag function, stretching terms in the in-plane and transverse displacements, respectively, and various other higher order terms in the model on the fundamental frequencies can be seen from this table. Further, the results predicted, through the present Q8-TSDT 7 model, are almost same as those of Refs. [9,26] as they are all based on same theory. Also, it is apparent from the results that the dropping of stretching terms (w1 and G , Q8-HSDT 11A) in the transverse displacement signi®cantly affects the accuracy compared to neglecting the zig-zag term (c , Q8-HSDT 11B) in the in-plane displacement ®elds, irrespective of degree of orthotropicity. It is revealed from the detailed analysis that retaining the higher order terms in the transverse displacement is essential for accurately evaluating the solutions for angle-ply cases. However, unlike cross-ply case, it can be seen here that the lower order displacement models, in general, yields upper bound for the prediction of frequencies whereas model having higher-order terms in the in-plane displacements and thickness stretch term in the transverse de¯ection alone slightly under-predict the solutions. Example 4. Free vibration of angle-ply laminated square plates with different boundary conditions [13] The effect of boundary conditions on the fundamental
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M. Ganapathi, D.P. Makhecha / Composites: Part B 32 (2001) 535±543
Table 7 p Non-dimensional natural frequencies v v rh2 =ET of two-layered square plates Ply-angle
Model
v 1
v 2
v 3
v 4
v 5
08=908
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9
0.34638 0.34598 0.34911 0.34870
0.48670 0.48670 0.48670 0.48670
0.48670 0.48670 0.48670 0.48670
0.70567 0.70409 0.71403 0.71273
0.94847 0.94549 0.95795 0.95555
458= 2 458
Q8-HSDT 13 Q8-HSDT 11A Q8-HSDT 11B Q8-HSDT 9
0.3865 0.3849 0.3865 0.3963
0.5686 0.5694 0.5686 0.6606
0.7280 0.7260 0.7280 0.7342
0.8896 0.8942 0.8896 0.7350
1.0124 1.0116 1.0124 1.0262
frequencies are brought out here considering four-layered angle-ply laminates for which exact solutions, based on second-order theory, are available. The results based on present Q8-HSDT 13 are tabulated in Table 6 along with those of present element Q8-SSDT 7, and exact solutions using analytical approach [13]. It can be noted here that the theory used for Q8-SSDT 7 is same as that of given in Ref. [13]. As expected, the present results associated with second-order theory match very closely with the analytical/ numerical solutions. Also, for simply supported boundary conditions and S 10 case, the present result is compared with those of Q9-TSDT given in Ref. [9]. However, it is revealed from this table that, for low aspect ratio, the results predicted by the present higher-order theory (Q8-HSDT 13) are different from those of second- and ®rst-order theory (Q8-SSDT 7 and Q8-FSDT 5). Furthermore, it can be opined from this table that the difference in the results among various theories is more while dealing with the clamped boundary conditions compared to other boundary conditions considered here. Thus, the present model (Q8HSDT 13) is ef®cient for any type of boundary conditions. Example 5. Higher mode frequencies of simply supported laminated square plates This example concerns with the in¯uence of various theories on the frequencies of higher ¯exural vibration modes and is shown in Table 7. An 8 £ 8 mesh is used for the modeling of full plate. The numerical results clearly show that the different alternate higher order models deduced here have less in¯uence on the fundamental frequencies. However, for higher modes, the effectiveness of the present model (Q8-HSDT 13) especially for the angle-ply case can be clearly seen in comparison with those of other models considered here. 5. Conclusions A displacement-based C 0 continuous isoparametric eightnoded quadrilateral plate element has been presented here based on a new higher-order laminated theory. The accuracy and effectiveness of the present model over the lower- and other higher-order theories for free vibration analysis of thick composite laminates have been demonstrated
considering problems for which exact/numerical results are available based on elasticity and higher-order theories. The in¯uence of slope discontinuity in thickness direction for in-plane, and the thickness stretching terms in the transverse displacement ®elds on the free vibration characteristics has been brought out. For accurate modeling of thick laminate with high degree of orthotropicity, it can be concluded that, in addition to higher-order terms in the in-plane displacement, zig-zag variation in the in plane displacement through thickness is essential for cross-ply plates whereas thickness stretch term in transverse displacement is required for angle-ply cases. References [1] Noor AK, Burton WS. Assessment of shear deformation theories for multilayered composite plates. ASME J Appl Mech Rev 1989;42:1± 13. [2] Kapania RK, Raciti S. Recent advances in analysis of laminated beams and plates. Part I: Shear effects and buckling. AIAA J 1989;27:923±34. [3] Reddy JN. A review of re®ned theories of composite laminates. Shock Vibr Dig 1990;22:3±17. [4] Mallikarjuna, Kant T. A critical review and some results of recently developed re®ned theories of ®bre reinforced laminated composites and sandwiches. Compos Struct 1993;23:293±312. [5] Varadan TK, Bhaskar K. Review of different laminate theories for the analysis of composite. J Aeronaut Soc India 1997;49:202±8. [6] Lo KH, Christensen RM, Wu EM. A higher-order theory of plate deformation. Part 2: Laminated plates. ASME J Appl Mech 1977;44:669±76. [7] Reddy JN. A simple higher-order theory for laminated composite plates. J Appl Mech 1984;45:745±52. [8] Phan ND, Reddy JN. Analysis of laminated composite plates using a higher-order shear deformation theory. J Numer Meth Engng 1985;21:2201±19. [9] Kant T, Mallikarjuna . A higher-order theory for free vibration of unsymmetrically laminated composite and sandwich plates Ð ®nite element evaluations. Comput Struct 1989;32:1125±32. [10] Shu X, Sun L. An improved simple higher order theory for a laminated composite plates. Comput Struct 1995:50. [11] Shi G, Lam KY, Tay TE. On ef®cient ®nite element modelling of composite beams and plates using higher order theories and an accurate composite beam element. Comput Struct 1998:41. [12] Franco Correia VM, Mota Soares CM, Mota Soares CA. Higher order models on the eigenfrequency analysis and optimal design of laminated composite structures. Compos Struct 1998;39:237±53. [13] Khdeir AA, Reddy JN. Free vibrations of laminated composite plates
M. Ganapathi, D.P. Makhecha / Composites: Part B 32 (2001) 535±543
[14] [15] [16] [17] [18] [19]
using second-order shear deformation theory. Comput Struct 1999;71:617±26. Bhaskar K, Varadan TK, Ali JSM. Thermoelastic solutions for orthotropic and anisotropic composite laminates. Composites 1996;27:415±20. Reddy JN. A generalization of two-dimensional theories of laminated composite plates. Commun Appl Numer Meth 1987;3:173±80. Cho KN, Bert CW, Striz AG. Free vibrations of laminated rectangular plates analyzed by higher order indiviual-layer theory. J Sound Vibr 1991;145:429±42. Nosier A, Kapania RK, Reddy JN. Free vibration analysis of laminated plates using a layer wise theory. AIAA J 1993;31:2335±46. Lee KH, Senthilnathan NR, Lim SP, Chow ST. An improved zigzag model for the bending of laminated composites plates. Compos Struct 1990;15:137±48. Li X, Liu D. Zigzag theory for composite laminates. AIAA J 1995;33:1163±5.
543
[20] Ali JSM, Bhaskar K, Varadan TK. A new theory for accurate thermal/ mechanical ¯exural analysis of symmetrically laminated plates. Compos Struct 1999;45:227±32. [21] Makhecha DP, Ganapathi M, Patel BP. Dynamic analysis of laminated composite plates subjected to thermal/mechanical loads using an accurate theory. Compos Struct 2001;51:221±36. [22] Murukami H. Laminated composite plate theory with improved in-plane responses. ASME J Appl Mech 1986;53:661±6. [23] Jones RM. Mechanics of composite materials. New York: McGrawHill, 1975. [24] Noor AK, Mathers MD. Shear ¯exible ®nite element models of laminated composite plates and shells. NASA TN-D 8044, 1975. [25] Noor AK. Free vibrations of multilayered composite plates. AIAA J 1973;11:1038±9. [26] Reddy JN, Phan ND. Stability and vibration of isotropic, orthotropic and laminated plates according to a higher-order shear deformation theory. J Sound Vibr 1985;98:157±70.