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Free vibrations of laminated rectangular plates analyzed by higher order individual-layer theory
Journal of Sound and Vibration (1991) 145(3), 429-442
FREE VIBRATIONS OF LAMINATED RECTANGULAR PLATES ANALYZED BY HIGHER ORDER INDIVIDUAL-LAYER THEORY K. N. CHO, C. W. BERT AND A. G. STRIZ
School of Aerospace and Mechanical Engineering, The University of Oklahoma, Norman, Oklahoma 73019, U.S.A. (Received 3 April 1989, and in final form 5 June 1990)
A higher order plate theory is used in each individual layer to determine the natural frequencies and the relative stress and deflection distributions through the thickness of simply supported rectangular plates. This theory approximates the in-plane and normal displacements by third and second order functions of the thickness co-ordinate, respectively. The theory satisfies the displacement compatibility and stress equilibrium conditions along the interfaces between adjacent layers. Numerical results are calculated for the cases of isotropic and orthotropic homogeneous plates and for symmetric and non-symmetric cross-ply laminates. The present theory predicts more modes of free vibration than other approximate theories. Also, the relative stress and deflection distributions obtained by the present higher order theory are in very good agreement with results by three-dimensional elasticity theory. 1. INTRODUCTION It has been well known that the classical laminated plate theory (CPT), based on the Kirchhoff hypotheses, is inaccurate for even a moderately thick plate with relatively soft transverse shear modulus. The inaccuracy is due to neglecting the transverse shear strains (under the assumption of infinitely large transverse shear modulus), assuming a linear distribution of the in-plane normal strain through the thickness, and neglecting the transverse normal strain. However, most of the advanced composites in use today have a relatively soft transverse shear modulus, and transverse shear deformations play an important role in predicting failure in multi-layered composite structures due to delamination. Early attempts to consider transverse shear deformation in plate theory were made by Reissner [1] in 1944 and Mindlin [2] in 1951 for the cases of bending and free vibration of plates, respectively. However, the transverse shear deformable theory (SDT) releases merely one assumption, the neglect of transverse shear strains, to allow for constant transverse shear strains through the thickness. Therefore, the SDT requires the use of a transverse shear correction factor either implicitly (Reissner) or explicitly (Mindlin). Also, the stress equilibrium equations are necessary to predict more realistic quadratic distributions of the transverse shear strains, but these distributions are contradictory to those in the constitutive relations between transverse shear stresses and strains. Srinivas and his colleagues [3-5] analyzed the bending, vibration and buckling behavior of simply supported thick homogeneous and laminated plates by formulating and solving exactly the problem as one in three-dimensional elasticity theory. Also, Pagano [6, 7] used the same approach for cylindrical bending and for the bending of thick laminated plates. All of these solutions showed considerable non-linearity in the distributions of 429 0022-460x/91/060429 + 14 $03.00/0 © 1991 Academic Press Limited
430
K.N.
CHO E T AL.
in-plane strains through the thickness rather than the linear distributions of the SDT. In 1957 Ambartsumyan [8] is believed to have been the first to account for more realistic distributions of the transverse shear strains with all assumptions of the CPT released. Reissner [9] introduced his higher order plate theory with in-plane and normal higher order displacements, for the special problem of plate bending. Subsequent to these investigations, higher order theories have been developed by many researchers [10-17] to predict more realistic displacement, stress and frequency behavior of a thick laminated plate. According to all of the above-mentioned plate theories, a laminate is considered as one homogeneous plate. Therefore, the transverse shear angle is either not permitted to vary at all from layer to layer [18, 19], or is assumed to vary smoothly through the thickness of the entire laminate. The first attempts to consider each layer in a laminate as a separate beam or plate are due to Kao and Ross [20] for multi-core sandwich beams and due to Swift and Heller [21] for the bending of a laminated beam. Seide [22] expanded the idea to a laminated plate by considering each layer as a Reissner shear-deformable plate. Hinrichsen and Palazotto [23] developed a finite element model in which each layer was represented by in-plane displacements cubic in the thickness co-ordinate (z). Reddy [24, 25] and Reddy et al. [26] treated each layer separately with in-plane displacements linear in z. Also, Valisetty and Rehfield [27] treated each layer separately, using a higher order displacement field which was derived on the basis of the CPT, for the problem of flexural wave propagation in laminated plates. Recently, Cho et al. used the higher order displacement field suggested by Lo and his colleagues in each layer separately for cylindrical bending and the bending of a laminated plate [28], for thermal stresses [29], and for bimodular laminates [30]. This higher order, individual-layer theory satisfies the displacement compatibility and shear stress equilibrium conditions along the interfaces. It differs from the other individual-layer theories mentioned above in that the thickness-direction displacement is quadratic in z. The deflection and stress distributions by this theory were shown to be in excellent agreement with the results by Pagano's elasticity-theory solution [6, 7]. An excellent survey of many theories was presented by Noor and Burton [31]. The objective of the present paper is to improve the natural frequency analysis of laminated rectangular plates by using the Cho et al. higher order, individual-layer theory. Numerical results will be shown and compared with solutions by three-dimensional elasticity theory and by other approximate plate theories for plates with simply supported edge conditions. 2. FORMULATION Consider an N-layer laminated composite, shown in Figure 1, with material symmetry directions coinciding with Cartesian co-ordinate systems (x, y, Zk), such that the zk-axes are perpendicular to the plane defined by x and y. The higher order theory in each discrete layer is based on the following displacement field: Uk(X, y, Zk, t) = Uok(X, y, t) + ZkUlk(X, y, t) + Z2 U2k(X, y, t) + Z3kU3k(X, y, t), Vk(X, y, Zk, t) = Vok(X, y, t) + ZkV,k(X, y, t) + Z2kV2k(X, y, t) + Z3kV3k(X, y, t),
Wk(X,y, zk, t)=Wok(X,y,t)+ZkWlk(X,y,t)+Z2kW2k(x,y,t),
k = 1 , 2 , . . . , N.
(1)
Here Uk, Vk, and Wk are the displacements in the respective x, y and zk directions, subscript k indicates layer number, and U~k, Vik (i = 0, 1, 2, 3), and Wik(i = 0, 1, 2) are defined by unknown displacement functions corresponding to the ith order of the z k
431
FREE VIBRATION OF LAMINATED PLATES fY
;
~ht
-----~zN
Layer N - -
~z2
Layer 2
~'zl
Layer 1
-
./ Figure 1. Geometry of laminated rectangular plate.
co-ordinates and time t. Therefore, there are 11 N unknown functions to be determined. The displacement field assures the non-linearity of the stress distributions through the thickness: third order functions of Zk for the in-plane normal, shear and transverse normal stresses and second order functions for the transverse shear stresses in the kth layer. Neglecting body forces and body moments, one may derive the following force and moment equilibrium equations for each layer from the stress equilibrium equations by integrating through each layer's thickness:
ONxk + o N~yk + ~zklzk=,~/2 -Ox
~,zk
t=~....
/2 = Iok/io~ + I2'~/i2~,
Oy
aNyk aNx,~ 4"' + Zvzklzk=,k/2 -- 7"vzklzk= --tk/2 = IOki)Ok q- I2ki)2k, Oy Ox OMxk ~_0Mvk+~ (rx~k1~=,~/2+ Z~kl~ ~-,~/2)- Q~k = I2kii,k + l, kii3k, Ox oy z Oy
Ox
2
OQxk q OQyk+ O.zklz~=,k/2 _ O.zklz~=_tj2 = Iok#ok + lekff:k, Ox Oy
k = 1, 2 . . . . , N.
(2)
Here a dot denotes a derivative with respect to time, and
ff t~/'2 (Nik, Mik) = ( N v k , M~yk) =
tk/2
(1, Zk)Zxvk dzk, f,~,,2 d h,/2
(1, Zk)Crik dzk, Qik =
fd-tt,/2 ,~/2 ri..k dZk,
i = X, y.
(3)
Also, tk is the thickness of the kth layer, N~k, M~k, and Q;k are the resultant in-plane forces, moments and transverse shear forces in the kth layer, and Nxyk and Mxyk a r e the resultant in-plane shear force and twisting moment in the kth layer. The inertias of the kth layer, Lk (i = 0, 2, 4), are defined by
f tk/2 ( Iok, 12k, I4k ) =
d -tk/2
pk(1, ZZk, Z4k) dZk,
where Ok is the material density of the kth layer.
432
K . N . CHO ET AL.
The constitutive relations between the stresses and strains for the kth orthotropic layer, where the principal axes of the material properties coincide with the geometric Cartesian co-ordinates, may be ex ~ressed in the form 'O'xk ~
"Cll k
C12 k
C13 k
0
0
0
f exk '
O'yk l1
C12 k
C22 k
C23 k
0
0
0
I Eyk
Orzk1
C13 k
C23 k
C33 k
0
0
0
Ezk
Tyzk It
0
0
0
C44 k
0
0
' "Yyzk
"txzk|
0
0
0
0
C55 k
0
~xzk
Txyk ]I
0
0
0
0
0
C66k
~~lxyk
i
(4)
where Cij k ( i , j = 1, 2 . . . . ,6) are the elements of the three-dimensional stiffness matrix [32] corresponding to the kth orthotropic layer. In the present individual-layer theory, each layer is considered as a separate plate. Therefore, the adjacent layers must be matched by the following interface conditions of the displacements and stresses:
Uk(X, y, tk/2, t ) = Uk+1(X, y,--tk+l/2, t), Wk(X, y, tk/2, t) = Wk+1(X, y, - t k + J 2 , t),
Vk(X, y, tk/2, t ) = Vk+I(X , y,--tk+l/2, t), rxzk(X, y, tk/2, t) = Zxz(k+l)(X,y, --tk+l/2, t),
~'yzk(X, y, tk/2, t) = 7yz(k+l)(X, y, --tk+l/2, t), ~'zk(X, y, tk/2, t) = O'~(k+~)(X,y, --tk+l/2, t),
k = 1, 2 . . . . , N - 1.
(5)
It is noted that the above interface conditions are not used for the case of a single-layer plate. On the outer surfaces of the laminate, the following free surface stress conditions must be satisfied for the present free vibration problem:
rxz~(x,y,-t~/2, t ) = 0 ,
"rxzN(X,y, tN/2, t ) = 0 ,
q'yzl(X,y,
ZyzN(X,y, tN/2, t ) = 0 ,
Crz~(X,y,-t,/2, t)=0,
cr~N(x,y, tN/2, t)=0.
- t l / 2 , t)=0, (6)
Here, by satisfying the free transverse normal stress conditions directly on the outer surfaces, the present higher order theory represents a more realistic mathematical model than the SDT and other higher order theories which apply the transverse stresses only in the transverse force equilibrium equation. Then, the governing equations, which are composed of the 5N equilibrium equations and the 6 ( N - l ) interface conditions and the six surface conditions, provide l l N equations to determine the 11N generalized displacement components. Equations (2), (5) and (6) may be expressed in the form of the unknown displacement components, as follows: 5N force equilibrium equations, A llkUOk.x x + A66kUOk.yy q- 2 tkC55kb12k + DllkUEk, xx "~-D66kU2k.yy + (AlE k + A66k)l)Ok, xy + ( D i E k + O66k)l)2k.xy "-[-(A13 k + tkC55k)Wlk, x -.~ IokiiOk + I2ki~2k, (AiEk + A66k )UOk,xy + ( O l 2 k + O66k )UEk,xy ~- AE2k120k.yy -[- A66kVOk.yy
d- 21kC44kl)2k q- D22k122k,yy -b D66kl)2k,xx -b (A23 k q- tkC44k)Wlk,y = Ioki)Ok "~-I2k~2k , I 3 D~ ~kU~k~x + D66kUl k.yy dl-2 tk C55kU3k "~ F11 kU3k.xx "~ F66kU3k,yy _~_(O12 k + D66k)l)lk.xy _1_(FIE k + F66k )l)3k.xy q- (2D13k "4"gtk 1 3 C55k )WEk,x ~" I2ki~lk "q"I4k~3k, (DiE k -1- O66k)Ulk,xy q- (F12 k q- F66k)U3k, xy q- D22kl~lk,yy q- O66kl)lk, xx
.~ f 22k~3k,yy _~ f 66kt~3k.xx "~--~tkC44klA3k 13 13 __ -~ (2D23 k "~glkC44k)W2k.y -- 12ki)lk -[- I4kiJ3k ,
FREE VIBRATION OF LAMINATED
(AfaR + tkC55k)Ulk~x + (3D,3k + ¼takCssk)Ua~x + ( A 2 3 k
433
PLATES
tkC44k)Vlk,y
+
+ (3D23k +¼t3C44k)V3k.y + tk(CsskWog.xx + C.4kWOk,yy)+2tkC33kWzk
+ ~tk (CsskW2~xx + C44kW2~yy) = Iok#Ok+
k = 1, 2 , . . . , N;
Izk(¢Zk,
6 ( N - 1) interface conditions, Uok + ltkUlk +~tki~Ek I 2 13 1 12 I 3 _-- 0 , +glkU3k -- UO(k+l) +~tk+l~l(k+l) --~tk+lU2(k+l) +~tk+ltl3(k+l) 12
l
I 3
1
12
I 2
l)Ok +~lkl)lk +~tkV2k +gtkDak -- DO(k+l) +~tk+l Vl(k+I) --gtk+lU2(k+l)+~tk+lU3ik+l~ = 0, W o k + l t k W l k + l 4lkW2k--Wo(k+l) 2 + 12tk~lWl(k+l)--~tk+lW2(k+l) 1 2 -~0, C55k(Ulk + lkU2k+~lkUak32
+Wok,~+SlkWlk, x+~lkW2k, 12 x)+C55(k+l)(__til(k+l)+tk+lU2(k+l
32
1
12
-- ~tk+l g/3(k+l) -- Wo(k-r 1),x + ~ t k + l W l ( k + l ),x --4lk+l W2(k+l Lx) = 0, 32
1
12
C4ak(Dlk + tkl)2k + 4tkO3k + WOk,y + 2tkWlk,y + 4tkW2k v) + 644(k+l)(--1)l(k-Vl) + l k + l D 2 ( k + l ~
--3t2k+,V3(k+,)- Wo(k+,),y + llk+lWl(k+l),y--112+,W2(k+l),y) 12
1
= O,
13
1 12 13 Cl3k ( UOk,x +~tkgllk, x + 2,tktl2k, x +~lkU3k, x)+C23k(DOk,y +~lkl31k, v +~tkZ)2k,y +gtkD3k, v) +C33k(Wlk.~tkW2k).~_C13(k+l)(__UO(k+l), x + 2tk+llJl(k+t),x 1 12 --4tk+llJ2(k+l),x 1 3
1
l 2
+gtk+lU3(k+l),x ) "~- C23(k+l)(--l)O(k+l),y + 2lk+lVl(k+ll,y --gtk+l V2(k+l),y
+~ta+lV3(k+l),y) + C33(k+l)(--Wltk+l)+ tk+l W2(k+l~), = 0;
k = 1, 2 , . . . , ( N - 1);
six surface conditions, C551(/,/11 --
flU2, ~-3 t2U31 ~- WOl,x - - i t , w,,.~ + l t21w21.x) = O, 32
1
12
C55N(UIN -]- INU2N +~INU3N + WON,.x +~tNWlN, x +~tNW2N, x) -----O, 32 1 C441(D11 - II/)21 "+-~,111)31 Jr" Wol v - 2 l l 32
12 _ W11,y -l'-411W21,y) - 0 , I
12
C44N( DI N "~- IN1J2N + 4t NI)3N + WON v + 2tNWl N,y + 4t N W2N,y ) = O, Ci31(uOl,x
12
1
13
-~tlUll,x+~tlU21,x - g / 1 g/31,x)
+ C231(uOl.y
12
1
13
--~tlVl 1,v. + 4 / 1 D 2 1 , y - s t l / ) 3 1 , v )
+ C 3 3 1 ( w i i - / 1 w21) = 0,
C13N (U0N,x +itNUlN, 1 1 2NU~N,x +gt i 3NU3N,x) x + gt 1
1 2
-J- C23N( ~ON,y "[-~fN1)IN, y 2t-~t N~)2N,y + ~ t 3 U3N, v "~- C33N( W1N + tNW2N ) : O.
(7)
Here Allk
Al2k
A13k
A22k
A23k
A66k ]
Dllk
D12k
D13k
D22k
D23k
D66k
Fl,k
F12k Fl3k
F22k F23k F66kJ
C12k Cl3k C22k C23k C66kJdZk. (8) It is important to note that in the present higher order theory used in each layer, the bending-stretching coupling stiffness does not occur even in general angle-ply laminates due to the homogeneity o f each layer.
K . N . CHO ET AL.
434
3. EXAMPLE PROBLEMS OF HOMOGENEOUS AND LAMINATED RECTANGULAR PLATES WITH SIMPLY SUPPORTING EDGE CONDITIONS The laminated rectangular plate is simply supported without axial force at the edges x -- 0, a and y = 0, b. The following general displacement field, which satisfies the boundary conditions, can be assumed: Uk
( Clok q- alk2k "k a2kZ2k "i" a3k Z3 ) COS ( m~'x / a ) sin ( nzry / b ) e i,~mn ,
Vk = ( bok + blk•k + bEkZk+ 2 b3kZk) 3 sin (m~rx/a) cos ( n ~ y / b ) e i~ t Wg = (Cok + ClkZk + C2kZ~k) sin ( m ~ x / a ) sin (n~ry/b) e Icomn! ,
k = 1, 2 , . . . , N,
(9)
where m and n are integers. Substituting the assumed displacement field (9) into the governing differential equations (7), one may express the equations in the generalized eigenvalue and eigenvector matrix system
[A]
{11 d2
dl N
= to~.[B]
d2
(10)
dl N
where [A] and [B] are 11N x 11N square matrices containing all known elements of the simultaneous equations. Here, the notations d~ (i = 1, 2 , . . . , 11 N ) are used to denote the unknown constants of the assumed displacement field, aik, big (i =0, 1, 2, 3), and Cik (i = 0, 1, 2). The matrix [B] is not invertible since it contains 6 N rows with all zero elements. Thus, this is not a true eigenvalue system. Therefore, a subroutine for the solution of the generalized eigenvalue problem including homogeneous simultaneous algebraic equations [33] is necessary. In this case the IMSL subroutine E I G Z F is used. The accuracy of the present higher order, individual-layer theory was verified by computing the numerical results for the first five natural frequencies of isotropic (u = 0.3) and orthotropic plates, treated as single-layer and four-layer plates. These results are presented in Tables 1 and 3, respectively. The following orthotropic stiffness properties of Aragonite crystals [5] were used: C22/C~1 = 0.543103; C~3/CH = 0.530172; C~2/C~ = 0.23319; C~3/C~ =0.010776; C23/C~ =0.098276; C44/C1~ =0.26681; C55/C~ = 0.159914; C66/C~ = 0.262931. The value of the material density, 0, is arbitrary since the investigations are non-dimensional. The results are compared with those by the 3-D elasticity theory of Srinivas and his co-workers [4, 5] and by the higher order theory of Reddy and Phan [14]. In Tables 1 and 3, it is shown that the present higher order theory predicts more accurate natural frequencies when treating homogeneous plates as multilayer laminated plates with an increasing number of layers, even though the properties are the same in all of the layers. Also, it predicts the natural frequencies for most of the modes which are obtained by 3-D elasticity theory. Tables 2 and 4 document a comparison between the relative stresses and displacements through the half-thickness by the present theory and 3-D elasticity theory. Here, the thickness-direction co-ordinate was normalized by the thickness of a plate. The distributions of the transverse shear stresses obtained by the present higher order theory using the stress equilibrium equations are given inside of the parentheses and are in good agreement with the distributions by 3-D elasticity theory. In Tables 5 and 6 are shown the fundamental natural frequencies of isotropic and orthotropic three-layer laminated plates with various ratios of moduli and densities between the layers. Compared to the 3-D elasticity-theory solution, the results by the CPT show a trend of increasing inaccuracy with increasing ratio of moduli. It is expected
435
FREE VIBRATION OF LAMINATED PLATES TABLE 1
Comparison o f natural frequencies, to = tor~,hx/ p / G , o f isotropic (v = 0"3) homogeneous plates mh/a
nh/a
0.1
0.1
0. l
0.2
0.1
0.2
0.3
0.2
0.2
0.3
0.3
0.3
I-At
I-S:~
Exact [4] Present ( N = 1) Present ( N = 4 ) Reddy and Phan [14]§ Mindlin et al. [34]¶
0.0931 0.0930 0.0931 0.0931 0.0930
0.4443 0.4443 0.4443 . --
Exact Present ( N Present ( N Reddy and Mindlin et
= 1) =4) Phan al.
0.2226 0.2219 0.2224 0.2222 0.2218
Exact Present ( N Present ( N Reddy and Mindlin et
= 1) = 4) Phan aL
Exact Present ( N Present ( N Reddy and Mindlin et
II-A:~
III-A
0.7498 0.7516 0.7499 . . --
3.1729 3.1933 3-1736 . 3.1729
3-2465 3-2756 3.2496
0-7025 0.7025 0.7025 . --
1.1827 1.1897 1.1832 . . --
3"2192 3.2394 3.2200 . 3.2192
3.3933 3.5831 3.3999
0.3421 0.3404 0.3416 0.3411 0.3402
0.8886 0.8886 0.8886 . --
1.4943 1.5066 1.4932 . . --
3.2648 3.2847 3.2656 . 3.2648
3.5298 3.5831 3.5398
= 1) = 4) Phan al.
0-4171 0.4147 0.4164 0.4158 0.4144
0.9935 0.9935 0.9934 . --
1.6654 1.6856 1.6668 . . --
3.2949 3.3147 3.2957 . 3-2949
3.6160 3.6772 3-6282
Exact Present ( N = 1) Present ( N =4) Reddy and Phan Mindlin et al.
0.5239 0.5203 0.5228 0.5221 0.5197
1.1327 1.1328 1.1327 . --
1"8936 1.9241 1.8957 . . --
3"3396 3-3590 3.3403 . 3.3396
Exact Present ( N Present ( N Reddy and Mindlin et
0.6889 0.6830 0-6872 0.6862 0.6821
1.3329 1.3329 1-3329 . --
2.2171 2.2680 2.2207 . . --
3.4126 3.4317 3.4134 . 3.4126
= 1) = 4) Phan al.
II-S
3.2538
3.4112
3.5580
3-6510 3.7393 3.8122 3.7538 3.7842 3.9310 4.0235 3.9522 3.9926
t A and S denote modes which are antisymmetric and symmetric about the mid-plane, respectively. $ Thickness-twist modes. § Only the fundamental natural frequencies were listed in reference [14]. ¶ The numerical results were obtained by using a shear correction factor of k2= 7r2/12. that this t r e n d can be f o u n d for all plate theories which c o n s i d e r a l a m i n a t e d plate as a single h o m o g e n e o u s plate. However, the present higher order, i n d i v i d u a l - l a y e r theory predicts results o f i n c r e a s i n g accuracy for increasing m o d u l i ratios. The f u n d a m e n t a l n a t u r a l frequencies of two-layer a n t i s y m m e t r i c a n d four-layer symmetric cross-ply lamin a t e d plates with various length-to-thickness ratios are s h o w n in T a b l e 7, a n d the results are c o m p a r e d with those by R e d d y a n d P h a n ' s higher order theory [14], the S D T a n d the CPT. The material properties of each layer were E22/E~ 1 = E33/E11 = 40, G23/E22 = O"5, G12/E22 = GI3/E22=0"6 a n d v~2 = v13=0.25. I n Table 7 are s h o w n some discrepancies b e t w e e n the results given by the present theory a n d those of the R e d d y a n d P h a n higher order theory even t h o u g h the two theories predicted very close f u n d a m e n t a l n a t u r a l frequencies for the h o m o g e n e o u s isotropic a n d o r t h o t r o p i c plates in Tables 1 a n d 3. It is t h o u g h t that these d i s c r e p a n c i e s arise from the differences in treating the l a m i n a t e d
436
K.N.
6
~
II
~
CHO
Illl
E T AL.
II
o~
~ 6 6 6~6 6
~ 6~6 6 6 ~ 0
II
~
~ .~ . . .
I I I I I I
~66666
~6666o
6 II
0
m ~
~ 6 6 6 ~
0
Illlll
o 6 ~ 6 6 ~ Y
~~ 6 6-6 6 ~
I I I I I I
~ 6 6 ~ 6 ~
~
~ I I I I I I
I
& 6 6 6 6
. . . . . .
~ 6 6 6 6
. . . . . .
~
0
~
~
I
~ o ~
~ 6 6 6 6 ~
.
IIIIII
~ 6 6 6 6 ~
.
.
.
.
~ ° .
~
N
. ~ a ~ 6 6 6 .
. . . . . . .
'~6666~
. . . . . . .
m
..66666~
~66666~
~
m
v
FREE VIBRATION OF LAMINATED PLATES
437
TABLE 3
Comparison o f natural frequencies, to = to,..hv/ 97 C~1, o f orthotropic homogeneous plates mh/a
nh/a
0.1
0.1
Exact [5] Present ( N Present ( N Reddy and M i n d l i n et
= l) = 4) P h a n [14] aL [34]§
I-A?
I-S~:
II-S
II-A~:
Ill-A§
0.0474 0.0474 0.0474 0.0474 0.0474
0.2170 0.2170 0.2170 ---
0.3941 0.3941 0.3941 ---
1.3077 1.3159 1.3081 1.3086 1.3159
1.6530 1.6646 1.6536 1.6550 1.6646
0.1
0.2
Exact Present ( N = 1) Present ( N = 4 ) Reddy a n d P h a n M i n d l i n et al.
0.1033 0.1033 0.1033 0.1033 0.1032
0.3450 0.3451 0.3450 ---
0.5624 0.5626 0.5624 ---
1.3331 1.3411 1.3335 1.3339 1.3411
1.7160 1.7301 1.7173 1.7209 1.7305
0.2
0.1
Exact Present ( N = 1) Present ( N = 4 ) Reddy and Phan M i n d l i n et aL
0.1188 0.1188 0.1188 0.1189 0.1187
0.3515 0.3515 0.3515 ---
0.6728 0.6728 0.6728 ---
1.4205 1.4285 1.4208 1.4216 1.4285
1.6805 1.6920 1.6812 1.6827 1.6921
0.2
0.2
Exact Present ( N = 1) Present ( N = 4 ) Reddy and Phan M i n d l i n et al.
0.1694 0.1693 0.1694 0.1695 0.1692
0.4338 0.4340 0.4338 ---
0.7880 0.7882 0.7880 ---
1.4316 1.4393 1.4319 1.4323 1.4393
1.7509 1.7649 1.7523 1.7562 1.7655
0.1
0.3
Exact Present ( N Present ( N Reddy and M i n d l i n et
0.1888 0.1885 0.1887 0.1888 0.1884
0.4953 0.4955 0.4953 ---
0.7600 0.7610 0.7601 ---
1.3765 1.3841 1.3768 1.3772 1.3841
1.8115 1-8296 1.8138 1.8210 1.8304
= 1) =4) Phan
aL
0.3
0.1
Exact Present ( N = 1) Present ( N = 4 ) Reddy and Phan M i n d l i n et al.
0.2180 0.2178 0.2180 0-2184 0.2178
0.5029 0.5030 0-5029 ---
0.9728 0.9728 0.9728 ---
1"5778 1.5857 1.5781 1.5789 1.5857
1.7334 1-7448 1.7342 1.7361 1.7450
0.3
0.3
Exact Present ( N = 1) Present ( N = 4 ) Reddy and Phan M i n d l i n et aL
0.3320 0.3313 0.3319 0.3326 0.3310
0.6504 0.6511 0.6505 ---
1.1814 1.1823 1.1815 ---
1.5737 1.5814 1.5741 1.5744 1.5813
1.9289 1.9469 1.9221 1.9395 1.9480
? A a n d S d e n o t e m o d e s w h i c h are antisymmetric a n d symmetric a b o u t the mid-plane, respectively. ? Thickness-twist modes. § The n u m e r i c a l results were o b t a i n e d by using a shear correction factor of k 2 = 5/6. p l a t e as a n h o m o g e n e o u s p l a t e o r c o n s i d e r i n g e a c h l a y e r s e p a r a t e l y . F r o m t h e f a c t s d i s c u s s e d i n r e l a t i o n t o T a b l e s 5 a n d 6, t h e r e s u l t s g i v e n b y t h e p r e s e n t h i g h e r o r d e r theory are considered more accurate than the results by Reddy and Phan.
4. C O N C L U S I O N S The present higher order individual-layer theory in which higher order in-plane and transverse normal displacements are used in each layer may not be more accurate than the Reddy and Phan higher order theory or Mindlin's shear deformable theory with shear
438
K . N . CHO E T A L .
TABLE 4
Comparison of relative stresses and displacements across the thickness of orthotropic homogeneous plates ( mh / a = nh / b = 0.3)
o,x/o,x(0.5) •
~'~
Exact
o-/%(0.5) •
Present
[5]
•
A
Exact
z.,/~xy(0.5) •
Present
[5]
,
~
Exact
u/u(0.5) •
Present
[5]
,
v/v(0.5)
A
Exact
^
Present
[5]
Exact
•
Present
[5]
First antisymmetric thickness mode (I-A) 0.5 1.0 1.0 1.0 1.0 0.4 0.713 0.712 0.754 0.747 0-3 0.487 0.482 0-540 0.530 0-2 0.303 0.297 0.348 0.338 0.1 0.145 0.141 0.170 0.165 0 0 0 0 0
1.0 0.736 0.516 0.328 0.159 0
1"0 0.736 0-514 0.325 0.157 0
1.0 0.696 0.465 0.284 0.134 0
1.0
1.0
1.0
0.695 0.459 0.277 0.130 0
0.764 0.553 0.360 0.177 0
0.765 0.553 0.356 0.177 0
Second antisymmetric thickness mode 0.5 1.0 1.0 1.0 0.4 0.951 0.944 0.951 0.3 0.809 0.792 0.809 0.2 0.588 0.568 0.588 0.1 0.309 0.296 0.309 0 0 0 0
1.0 0.951 0.809 0.588 0.309 0
1.0 0.945 0.794 0.570 0.267 0
1.0 0.951 0.809 0.588 0.309 0
1.0 0.944 0.792 0.568 0.296 0
1.0 0.951 0.809 0.588 0.309 0
1.0 0.943 0.791 0.567 0.296 0
Third antisymmetric thickness mode ( I l l - A ) 0.5 1.0 1.0 1.0 0 0-4 0.984 0.970 0.952 0.967 0.3 0.857 0.826 0.811 0.822 0.2 0.633 0.598 0.591 0.595 0-1 0.336 0.313 0.311 0.311 0 0 0 0 0
1.0 0.979 0.850 0.623 0.332 0
1.0 0.964 0.819 0.591 0.309 0
1.0 0.951 0.870 0.645 0.343 0
1.0 0.974 0.832 0.603 0.316
1.0 0.972 0.840 0.617 0.326 0
1.0 0.959 0.812 0.586 0.306
First symmetric thickness mode (I-S) 0.5 1.0 1.0 1.0 1.0 0-4 0'979 0'980 1"013 1.014 0'3 0.964 0.964 1"023 1"025 0"2 0'953 0"953 1.030 1'033 0" 1 0'947 0.946 1"034 1'038 0 0.953 0.944 1"035 1.040
1-0 1"042 1.072 1.093 1" 106 1"110
1.0 1.039 1.070 1'092 1' 105 1'109
1.0 0"989 0.981 0.976 0"973 0.972
1.0 0"989 0-981 0-975 0.972 0.971
1.0 1'007 1.013 1"017 1"019 1.020
1.0 1.007 1"012 1"016 1"018 1"019
Second symmetric thickness 0"5 1.0 1"0 0.4 1.008 1.006 0.3 1.014 1.010 0-2 1.018 1.014 0.1 1.021 1.016 0 1.022 1.016
1.0 1.009 1.017 1.022 1.025 1-027
1.0 1.007 1.012 1.016 1.018 1.019
1.0 1.007 1.013 1.017 1.020 1.021
1"0 1-005 1-010 1.013 1.015 1.015
1.0 1.012 1.022 1.030 1.034 1.020
1.0 1.009 1.016 1.020 1.023 1.024
(II-A) 1.0 0.963 0.817 0.590 0.308 0
mode (II-S) 1.0 1.0 1.006 1.008 1.011 1.013 1.015 1.018 1.018 1.020 1.018 1.021
t The values in parentheses give the transverse shear stresses by the present higher order theory in using the stress equilibrium equations. ;i: Z~J~',,~(0"3) instead of ~'~J'r~(O). § %,z/%,2(0"3) instead of %,z/r~z(O).
FREE VIBRATION OF LAMINATED PLATES
TABLE ,1
~/o~z(o.1)
-,xJ ~~(o)
439
continued
w/w(O.5)
~-,:/¢v~(o)
A
Exact
Present
[5]
Exact
Present
Exact
[5]
Present
Exact
Present
[5]
[5]
1.0
1.0
0
0
0 0.394 0.674 0.860 0.966 1.0
o 1.335 1.880 1.710 1.0 0
o 1.50 2.00 1.75 1.0 0
o 0.309 0.588 0.809 0.951 1.0
o 0.36 0.64 0.84 0.96 1.0
(o) (0.314) (0.594) (0.818) (0.952) (1.0)
o 0.309 0.588 0.809 0.951 1.0
o 0.36 0.64 0.84 0.96 1.0
(o) (0.314) (0.594) (0-813) (0.952) (1.0)
0 1"310 1'861 1'704 1'0 0
0 1"50 2"00 1"75 1'0 0
0 0'294 0'572 0'799 0"948 1.0
0 0'36 0"64 0.84 0"96 1.0
(0) (0"305) (0"585) (0"808) (0"951) (1.0)
0 0'302 0"580 0"805 0"950 1.0
0 0"36 0"64 0"84 0"96 1.0
(0) (0"311) (0"591) (0"811) (0'952) (1.0)
1"0 0"902 0"771 0"644 0'553 0-521
1"0 0"926 0"868 0-827 0"802 0.793
0 0.387 0.678 0.880 1.0 1.040
0 0.375 0.667 0.875 ~:t~i066 1.0 L~ "492 1.042
-- -
(0.750)
------
(1.0) § ~i0866 (0'875) [~.492 (0)(0"500)
(0) (0.750) (1.0) (0.875)
1-0 0.811 0.615 0.413
1.0 0.80 0.60 0.40
(0.500) (0)
0.207 0
0.20 0
0 0"360 0"653 0"868 1"0 1"044
0 0"375 0"667 :I:~ 1~097 0"875 1"0 1'042
(o) (0.750) (1 .o) (0.875) (0.500) (o)
1.0 0"836 0.649 0.443 0.225 0
1'0 0'80 0'60 0.40 0.20 0
0 1.569 2.048 1.766
0 1.50 2.00 1.75
[ 761
(0.724
LOttO
0 0.36 0.64 0.84 0.96 1-0
(0)t (0-397) (0-677) (0.861) (0.965) (1.0)
0 0.384 0.663 0.853 0.964 1.0
0 0.36 0.64 0.84 0.96 1.0
(0)t (0.387) (0.667) (0.856) (0.965) (1.0)
lo,
foO761
m
w
_
---
__
[0.734
(0.750) (1.0) 1-0 (0.875) § ~0.889
,o oo
D
m
1.0 1.005 1.008 1.010 1.010 1.010 1.0
16.66 30.76 41.93 49.10 51.56
1,0 1.006 1-011 1-015 1.017 1.018 1-0
2.448 3-573 4-377 4.860 5.021
K.N. CHO ET AL.
440
TABLE 5
Comparison of fundamental natural frequencies, to = toll/'2x/P2/02, of isotropic three-layer laminated plates with various ratios ofmoduli and materialdensities (tl =/`3 = 0.1 h,/'2 = 0.8 h, m h / a = n h / b = O . 1 , p l = p s , G I = G3, v l = v3, v 2 = 0 . 3 ) Vl
Pl/P2
Gt/G 2
Exact [4]
Present
% Error
CPT
% Error
0"3 0"3 0-3 0-3 0"3 0"3 0.2
1 1 1 1 2 3 1
1 2 5 15 15 15 1
0"074520 0"089986 0.123048 0.183664 0.167574 0.155082 0.072298
0-074881 0.089954 0.123072 0.183647 0.167558 0"155068 0"072262
(0.48) (-0"036) (-0"020) (-0.009) (-0"010) (-0.009) (-0.05)
0.077054 0.093994 0.132390 0.215642 0-196852 0.182250 0.074666
(2-9) (4.5) (7.6) (17.4) (17.5) (17"5) (3"28)
TABLE 6
Comparison of fundamental natural frequencies, to = tol~ t2v/-'~2/Cl12, of orthotropic threelayer laminated plates with various ratios of moduli and material densities ( /,~ =/'s = O. 1 h, t 2 = 0 . 8 h, mh/ a = n h / b = 0 . 1 , p l = p 3 ) Exact
Pl/P2
C111/C112
[5]
Present
% Error
CPT
% E~or
1 1 1 1 1 3
1 2 5 10 15 15
0.047419 0.057041 0.077148 0.098104 0.112034 0.094548
0.047416 0.057035 0.077147 0.098102 0.112032 0.094546
(-0.006) (-0.011) (-0.001) (-0.002) (-0.002) (-0.002)
0.049666 0.060584 0.085333 0.115328 0.138994 0.117471
(4.7) (6.2) (10.6) (17.6) (24.1) (24.3)
TABLE 7
Comparison of fundamen/,al natural frequencies, to = tolla2x/p2/ E22, of cross-ply three-layer laminated square plates [00/900/900/0 °]
a/h
Present
HSDPT [14]
2 5 10 20 25 50 100
5"923 10-673 15.066 17"535 18"054 18"670 18.835
5.576 10.989 15'270 17"668 18"050 18"606 18~755
SDT 5"492 10"820 15.083 17"583 17"991 18.590 18"751
[0°/90 °]
CPT 15.830 18.215 18.652 18.767 18.780 18.799 18.804
Present
HSDPT [14]
4"810 8"388 10.270 11-016 11"118 11"260 11"296
5"699 9.010 10.449 10-968 11.037 11.132 11"156
SDT 5"191 8.757 10"355 10.941 11"020 11.127 11"155
CPT 8.499 10.584 11.011 11.125 11-139 11.158 11"163
FREE VIBRATION OF LAMINATED
PLATES
441
correction factor in the analysis of fundamental natural frequencies for homogeneous plates. This is due to the fact that the distributions of the transverse shear strains through the thickness are forced to be quadratic. However, the present theory has the advantage of predicting more free vibration modes (as in 3-D elasticity theory) than the other approximate theories. More importantly, though, the natural frequencies given by the present theory are closer to the three-dimensional elasticity-theory results than those of other plate theories for the case of laminated plates. By accounting for the second order distribution of transverse normal displacement through each layer's thickness, rather accurate distributions of the relative transverse normal stresses and the deflection can be predicted. Moreover, for laminated plates, the present higher order individual-layer theory predicts accurate distributions of the relative transverse shear stresses which play an important role in the prediction of failure behavior.
REFERENCES 1. E. REISSNER 1944 Journal of Mathematics and Physics 23, 183-191. On the theory of bending elastic plates. 2. R. D. MINDLIN 1951 American Society of Mechanical Engineers, Journal of Applied Mechanics 18, 31-38. Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates. 3. S. SRINIVAS, A. K. RAO and C. V. JOGA RAO 1969 ZeitschriftJ~r Angewandte Mathematik und Mechanik 49(8), 449-458. Flexure of simply supported thick homogeneous and laminated rectangular plates. 4. S. SRINIVAS, C. V. JOGA RAO and A. K. RAO 1970 Journal of Sound and Vibration 12, 187-195. An exact analysis for vibration of simply-supported homogeneous and laminated thick rectangular plates. 5. S. SRINIVAS and A. K. RAO 1970 International Journal of Solids and Structures 6, 1463-1481. Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates. 6. N. J. PAGANO 1969 Journal of Composite Materials 3, 398-411. Exact solutions for composite laminates in cylindrical bending. 7. N. J. PAGANO 1970 Journal of Composite Materials 4, 20-34. Exact solutions for rectangular bidirectional composites and sandwich plates. 8. S. A. AMBARTSUMYAN 1970 Theory of Anisotropic Plates. Stamford: Technomic. 9. E. REISSNER 1975 International Journal of Solids and Structures 11, 569-573. On transverse bending of plates, including the effect of transverse shear deformation. 10. K. H. Lo, R. M. CHRISTENSEN and E. M. WU 1977 American Society of Mechanical Engineers Journal of Applied Mechanics 44, 663-668. A higher order theory of plate deformation--part 1: homogeneous plates. 11. K. H. Lo, R. M. CHRISTENSEN and E. M. Wu 1977 American Society of Mechanical Engineers, Journal of Applied Mechanics 44, 669-676. A higher order theory of plate deformation--part 2: laminated plates. 12. M. LEVINSON 1980 Mechanics Research Communications 7, 343-350. An accurate, simple theory of the statics and dynamics of elastic plates. 13. J. N. REDDY 1984 American Society of Mechanical Engineers, Journal of Applied Mechanics 51, 745-752. A simple higher-order theory for laminated composite plates. 14. J. N. REDDY and N. D. PHAN 1985 Journal of Sound and Vibration 98, 157-170. Stability and vibration of isotropic, orthotropic and laminated plates according to a higher-order deformation theory. 15. G. Z. VOYIADJIS and M. M. BALUCH 1988 American Society of Civil Engineers, Journal of Engineering Mechanics 114(4), 671-687. Refined theory for thick composite plates. 16. J. L. DOONG and C. P. FUNG Journal of Sound and Vibration 125, 325-339. Vibration and buckling of bimodulus laminated plates according to a higher-order plate theory. 17. N. R. SENTHILNATHAN, S. P. LIM, K. H. LEE and S. T. CHow 1988 Composite Structures 10, 211-229. Vibration of laminated orthotropic plates using a simplified higher-order deformation theory.
442
K.N.
CHO ET AL.
18. P. C. YANG, C. H. NORRIS and Y. STAVSKY 1966 International Journal of Solids and Structures 2, 665-685. Elastic wave propagation in heterogeneous plates. 19. J. M. WHITNEY and N. J. PAGANO 1973 American Society of Mechanical Engineers, Journal of Applied Mechanics 40, 302-304. Shear deformation in heterogeneous anisotropic plates. 20. J. KAO and R. J. ROSS 1968 American Institute of Aeronautics and Astronautics Journal 6, 1583-1585. Bending of multilayer sandwich beams. 21. G. W. SWIFT and R. A. HEELER 1974 Journal of Engineering Mechanics Division, Proceedings American Society of Civil Engineers 100, 267-282. Layered beam analysis. 22. P. SLIDE 1980 Mechanics Today 5, 451-466. An improved approximate theory for the bending of laminated plates. 23. R. L. HINRICHSEN and A. N. PAEAZOTTO 1986 American Institute of Aeronautics and Astronautics Journal 24, 1836-1842. Nonlinear finite element analysis of thick composite plates using cubic spline functions. 24. J. N. REDDY 1987 Communications in Applied Numerical Methods 3, 173-180. A generalization of two-dimensional theories of laminated composite plates. 25. J. N. REDDY 1989 International Journal for Numerical Methods in Engineering 27, 361-382. On refined computational models of composite laminates. 26. J. N. REDDY, E. J. BARBERO and J. L. TEPLY 1988 29th Structures, Structural Dynamics and Materials Conference, Williamsburg, Virginia, AIAA CP882, Paper 88-2322, 937-943. A plate bending element based on a generalized laminated plate theory. 27. R. R. VALISETTY and L. W. REHFIEED 1988 Journal of Sound and Vibration 126, 183-194. Application of ply level analysis to flexural wave propagation. 28. K. N. CHO, C. W. BERT and A. G. STRIZ 1987 Engineering Science Preprint ESP24.87034. New theory for bending of bimodular laminates. 29. K. N. CHO, A. G. STRIZ and C. W. BERT 1989 Journal of Thermal Stresses 12, 321-332. Thermal stress analysis of laminate using higher-order theory in each layer. 30. K. N. CHO, A. G. STRIZ and C. W. BERT 1990 Composite Structures 15, 1-24. Bending analysis of thick bimodular laminates by higher-order individual-layer theory. 31. A. K. NOOR and W. S. BURTON 1989 Applied Mechanics Reviews 42, 1-13. Assessment of shear deformation theories for multilayered composite plates. 32. R. M. JONES 1975 Mechanics of Composite Materials. Washington, D.C.: Hemisphere. 33. C. B. MOLER and G. W. STEWART 1973 Society for Industrial and Applied Mathematics Journal for Numerical Analysis 10, 241-256. An algorithm for generalized matrix eigenvalue problems. 34. R. D. MINDLIN, A. SCHACKNOW and H. DERESIEWICZ 1956 American Society of Mechanical Engineers, Journal of Applied Mechanics 23, 430-436. Flexural vibrations of rectangular plates.
FREE VIBRATIONS OF LAMINATED RECTANGULAR PLATES ANALYZED BY HIGHER ORDER INDIVIDUAL-LAYER THEORY K. N. CHO, C. W. BERT AND A. G. STRIZ
School of Aerospace and Mechanical Engineering, The University of Oklahoma, Norman, Oklahoma 73019, U.S.A. (Received 3 April 1989, and in final form 5 June 1990)
A higher order plate theory is used in each individual layer to determine the natural frequencies and the relative stress and deflection distributions through the thickness of simply supported rectangular plates. This theory approximates the in-plane and normal displacements by third and second order functions of the thickness co-ordinate, respectively. The theory satisfies the displacement compatibility and stress equilibrium conditions along the interfaces between adjacent layers. Numerical results are calculated for the cases of isotropic and orthotropic homogeneous plates and for symmetric and non-symmetric cross-ply laminates. The present theory predicts more modes of free vibration than other approximate theories. Also, the relative stress and deflection distributions obtained by the present higher order theory are in very good agreement with results by three-dimensional elasticity theory. 1. INTRODUCTION It has been well known that the classical laminated plate theory (CPT), based on the Kirchhoff hypotheses, is inaccurate for even a moderately thick plate with relatively soft transverse shear modulus. The inaccuracy is due to neglecting the transverse shear strains (under the assumption of infinitely large transverse shear modulus), assuming a linear distribution of the in-plane normal strain through the thickness, and neglecting the transverse normal strain. However, most of the advanced composites in use today have a relatively soft transverse shear modulus, and transverse shear deformations play an important role in predicting failure in multi-layered composite structures due to delamination. Early attempts to consider transverse shear deformation in plate theory were made by Reissner [1] in 1944 and Mindlin [2] in 1951 for the cases of bending and free vibration of plates, respectively. However, the transverse shear deformable theory (SDT) releases merely one assumption, the neglect of transverse shear strains, to allow for constant transverse shear strains through the thickness. Therefore, the SDT requires the use of a transverse shear correction factor either implicitly (Reissner) or explicitly (Mindlin). Also, the stress equilibrium equations are necessary to predict more realistic quadratic distributions of the transverse shear strains, but these distributions are contradictory to those in the constitutive relations between transverse shear stresses and strains. Srinivas and his colleagues [3-5] analyzed the bending, vibration and buckling behavior of simply supported thick homogeneous and laminated plates by formulating and solving exactly the problem as one in three-dimensional elasticity theory. Also, Pagano [6, 7] used the same approach for cylindrical bending and for the bending of thick laminated plates. All of these solutions showed considerable non-linearity in the distributions of 429 0022-460x/91/060429 + 14 $03.00/0 © 1991 Academic Press Limited
430
K.N.
CHO E T AL.
in-plane strains through the thickness rather than the linear distributions of the SDT. In 1957 Ambartsumyan [8] is believed to have been the first to account for more realistic distributions of the transverse shear strains with all assumptions of the CPT released. Reissner [9] introduced his higher order plate theory with in-plane and normal higher order displacements, for the special problem of plate bending. Subsequent to these investigations, higher order theories have been developed by many researchers [10-17] to predict more realistic displacement, stress and frequency behavior of a thick laminated plate. According to all of the above-mentioned plate theories, a laminate is considered as one homogeneous plate. Therefore, the transverse shear angle is either not permitted to vary at all from layer to layer [18, 19], or is assumed to vary smoothly through the thickness of the entire laminate. The first attempts to consider each layer in a laminate as a separate beam or plate are due to Kao and Ross [20] for multi-core sandwich beams and due to Swift and Heller [21] for the bending of a laminated beam. Seide [22] expanded the idea to a laminated plate by considering each layer as a Reissner shear-deformable plate. Hinrichsen and Palazotto [23] developed a finite element model in which each layer was represented by in-plane displacements cubic in the thickness co-ordinate (z). Reddy [24, 25] and Reddy et al. [26] treated each layer separately with in-plane displacements linear in z. Also, Valisetty and Rehfield [27] treated each layer separately, using a higher order displacement field which was derived on the basis of the CPT, for the problem of flexural wave propagation in laminated plates. Recently, Cho et al. used the higher order displacement field suggested by Lo and his colleagues in each layer separately for cylindrical bending and the bending of a laminated plate [28], for thermal stresses [29], and for bimodular laminates [30]. This higher order, individual-layer theory satisfies the displacement compatibility and shear stress equilibrium conditions along the interfaces. It differs from the other individual-layer theories mentioned above in that the thickness-direction displacement is quadratic in z. The deflection and stress distributions by this theory were shown to be in excellent agreement with the results by Pagano's elasticity-theory solution [6, 7]. An excellent survey of many theories was presented by Noor and Burton [31]. The objective of the present paper is to improve the natural frequency analysis of laminated rectangular plates by using the Cho et al. higher order, individual-layer theory. Numerical results will be shown and compared with solutions by three-dimensional elasticity theory and by other approximate plate theories for plates with simply supported edge conditions. 2. FORMULATION Consider an N-layer laminated composite, shown in Figure 1, with material symmetry directions coinciding with Cartesian co-ordinate systems (x, y, Zk), such that the zk-axes are perpendicular to the plane defined by x and y. The higher order theory in each discrete layer is based on the following displacement field: Uk(X, y, Zk, t) = Uok(X, y, t) + ZkUlk(X, y, t) + Z2 U2k(X, y, t) + Z3kU3k(X, y, t), Vk(X, y, Zk, t) = Vok(X, y, t) + ZkV,k(X, y, t) + Z2kV2k(X, y, t) + Z3kV3k(X, y, t),
Wk(X,y, zk, t)=Wok(X,y,t)+ZkWlk(X,y,t)+Z2kW2k(x,y,t),
k = 1 , 2 , . . . , N.
(1)
Here Uk, Vk, and Wk are the displacements in the respective x, y and zk directions, subscript k indicates layer number, and U~k, Vik (i = 0, 1, 2, 3), and Wik(i = 0, 1, 2) are defined by unknown displacement functions corresponding to the ith order of the z k
431
FREE VIBRATION OF LAMINATED PLATES fY
;
~ht
-----~zN
Layer N - -
~z2
Layer 2
~'zl
Layer 1
-
./ Figure 1. Geometry of laminated rectangular plate.
co-ordinates and time t. Therefore, there are 11 N unknown functions to be determined. The displacement field assures the non-linearity of the stress distributions through the thickness: third order functions of Zk for the in-plane normal, shear and transverse normal stresses and second order functions for the transverse shear stresses in the kth layer. Neglecting body forces and body moments, one may derive the following force and moment equilibrium equations for each layer from the stress equilibrium equations by integrating through each layer's thickness:
ONxk + o N~yk + ~zklzk=,~/2 -Ox
~,zk
t=~....
/2 = Iok/io~ + I2'~/i2~,
Oy
aNyk aNx,~ 4"' + Zvzklzk=,k/2 -- 7"vzklzk= --tk/2 = IOki)Ok q- I2ki)2k, Oy Ox OMxk ~_0Mvk+~ (rx~k1~=,~/2+ Z~kl~ ~-,~/2)- Q~k = I2kii,k + l, kii3k, Ox oy z Oy
Ox
2
OQxk q OQyk+ O.zklz~=,k/2 _ O.zklz~=_tj2 = Iok#ok + lekff:k, Ox Oy
k = 1, 2 . . . . , N.
(2)
Here a dot denotes a derivative with respect to time, and
ff t~/'2 (Nik, Mik) = ( N v k , M~yk) =
tk/2
(1, Zk)Zxvk dzk, f,~,,2 d h,/2
(1, Zk)Crik dzk, Qik =
fd-tt,/2 ,~/2 ri..k dZk,
i = X, y.
(3)
Also, tk is the thickness of the kth layer, N~k, M~k, and Q;k are the resultant in-plane forces, moments and transverse shear forces in the kth layer, and Nxyk and Mxyk a r e the resultant in-plane shear force and twisting moment in the kth layer. The inertias of the kth layer, Lk (i = 0, 2, 4), are defined by
f tk/2 ( Iok, 12k, I4k ) =
d -tk/2
pk(1, ZZk, Z4k) dZk,
where Ok is the material density of the kth layer.
432
K . N . CHO ET AL.
The constitutive relations between the stresses and strains for the kth orthotropic layer, where the principal axes of the material properties coincide with the geometric Cartesian co-ordinates, may be ex ~ressed in the form 'O'xk ~
"Cll k
C12 k
C13 k
0
0
0
f exk '
O'yk l1
C12 k
C22 k
C23 k
0
0
0
I Eyk
Orzk1
C13 k
C23 k
C33 k
0
0
0
Ezk
Tyzk It
0
0
0
C44 k
0
0
' "Yyzk
"txzk|
0
0
0
0
C55 k
0
~xzk
Txyk ]I
0
0
0
0
0
C66k
~~lxyk
i
(4)
where Cij k ( i , j = 1, 2 . . . . ,6) are the elements of the three-dimensional stiffness matrix [32] corresponding to the kth orthotropic layer. In the present individual-layer theory, each layer is considered as a separate plate. Therefore, the adjacent layers must be matched by the following interface conditions of the displacements and stresses:
Uk(X, y, tk/2, t ) = Uk+1(X, y,--tk+l/2, t), Wk(X, y, tk/2, t) = Wk+1(X, y, - t k + J 2 , t),
Vk(X, y, tk/2, t ) = Vk+I(X , y,--tk+l/2, t), rxzk(X, y, tk/2, t) = Zxz(k+l)(X,y, --tk+l/2, t),
~'yzk(X, y, tk/2, t) = 7yz(k+l)(X, y, --tk+l/2, t), ~'zk(X, y, tk/2, t) = O'~(k+~)(X,y, --tk+l/2, t),
k = 1, 2 . . . . , N - 1.
(5)
It is noted that the above interface conditions are not used for the case of a single-layer plate. On the outer surfaces of the laminate, the following free surface stress conditions must be satisfied for the present free vibration problem:
rxz~(x,y,-t~/2, t ) = 0 ,
"rxzN(X,y, tN/2, t ) = 0 ,
q'yzl(X,y,
ZyzN(X,y, tN/2, t ) = 0 ,
Crz~(X,y,-t,/2, t)=0,
cr~N(x,y, tN/2, t)=0.
- t l / 2 , t)=0, (6)
Here, by satisfying the free transverse normal stress conditions directly on the outer surfaces, the present higher order theory represents a more realistic mathematical model than the SDT and other higher order theories which apply the transverse stresses only in the transverse force equilibrium equation. Then, the governing equations, which are composed of the 5N equilibrium equations and the 6 ( N - l ) interface conditions and the six surface conditions, provide l l N equations to determine the 11N generalized displacement components. Equations (2), (5) and (6) may be expressed in the form of the unknown displacement components, as follows: 5N force equilibrium equations, A llkUOk.x x + A66kUOk.yy q- 2 tkC55kb12k + DllkUEk, xx "~-D66kU2k.yy + (AlE k + A66k)l)Ok, xy + ( D i E k + O66k)l)2k.xy "-[-(A13 k + tkC55k)Wlk, x -.~ IokiiOk + I2ki~2k, (AiEk + A66k )UOk,xy + ( O l 2 k + O66k )UEk,xy ~- AE2k120k.yy -[- A66kVOk.yy
d- 21kC44kl)2k q- D22k122k,yy -b D66kl)2k,xx -b (A23 k q- tkC44k)Wlk,y = Ioki)Ok "~-I2k~2k , I 3 D~ ~kU~k~x + D66kUl k.yy dl-2 tk C55kU3k "~ F11 kU3k.xx "~ F66kU3k,yy _~_(O12 k + D66k)l)lk.xy _1_(FIE k + F66k )l)3k.xy q- (2D13k "4"gtk 1 3 C55k )WEk,x ~" I2ki~lk "q"I4k~3k, (DiE k -1- O66k)Ulk,xy q- (F12 k q- F66k)U3k, xy q- D22kl~lk,yy q- O66kl)lk, xx
.~ f 22k~3k,yy _~ f 66kt~3k.xx "~--~tkC44klA3k 13 13 __ -~ (2D23 k "~glkC44k)W2k.y -- 12ki)lk -[- I4kiJ3k ,
FREE VIBRATION OF LAMINATED
(AfaR + tkC55k)Ulk~x + (3D,3k + ¼takCssk)Ua~x + ( A 2 3 k
433
PLATES
tkC44k)Vlk,y
+
+ (3D23k +¼t3C44k)V3k.y + tk(CsskWog.xx + C.4kWOk,yy)+2tkC33kWzk
+ ~tk (CsskW2~xx + C44kW2~yy) = Iok#Ok+
k = 1, 2 , . . . , N;
Izk(¢Zk,
6 ( N - 1) interface conditions, Uok + ltkUlk +~tki~Ek I 2 13 1 12 I 3 _-- 0 , +glkU3k -- UO(k+l) +~tk+l~l(k+l) --~tk+lU2(k+l) +~tk+ltl3(k+l) 12
l
I 3
1
12
I 2
l)Ok +~lkl)lk +~tkV2k +gtkDak -- DO(k+l) +~tk+l Vl(k+I) --gtk+lU2(k+l)+~tk+lU3ik+l~ = 0, W o k + l t k W l k + l 4lkW2k--Wo(k+l) 2 + 12tk~lWl(k+l)--~tk+lW2(k+l) 1 2 -~0, C55k(Ulk + lkU2k+~lkUak32
+Wok,~+SlkWlk, x+~lkW2k, 12 x)+C55(k+l)(__til(k+l)+tk+lU2(k+l
32
1
12
-- ~tk+l g/3(k+l) -- Wo(k-r 1),x + ~ t k + l W l ( k + l ),x --4lk+l W2(k+l Lx) = 0, 32
1
12
C4ak(Dlk + tkl)2k + 4tkO3k + WOk,y + 2tkWlk,y + 4tkW2k v) + 644(k+l)(--1)l(k-Vl) + l k + l D 2 ( k + l ~
--3t2k+,V3(k+,)- Wo(k+,),y + llk+lWl(k+l),y--112+,W2(k+l),y) 12
1
= O,
13
1 12 13 Cl3k ( UOk,x +~tkgllk, x + 2,tktl2k, x +~lkU3k, x)+C23k(DOk,y +~lkl31k, v +~tkZ)2k,y +gtkD3k, v) +C33k(Wlk.~tkW2k).~_C13(k+l)(__UO(k+l), x + 2tk+llJl(k+t),x 1 12 --4tk+llJ2(k+l),x 1 3
1
l 2
+gtk+lU3(k+l),x ) "~- C23(k+l)(--l)O(k+l),y + 2lk+lVl(k+ll,y --gtk+l V2(k+l),y
+~ta+lV3(k+l),y) + C33(k+l)(--Wltk+l)+ tk+l W2(k+l~), = 0;
k = 1, 2 , . . . , ( N - 1);
six surface conditions, C551(/,/11 --
flU2, ~-3 t2U31 ~- WOl,x - - i t , w,,.~ + l t21w21.x) = O, 32
1
12
C55N(UIN -]- INU2N +~INU3N + WON,.x +~tNWlN, x +~tNW2N, x) -----O, 32 1 C441(D11 - II/)21 "+-~,111)31 Jr" Wol v - 2 l l 32
12 _ W11,y -l'-411W21,y) - 0 , I
12
C44N( DI N "~- IN1J2N + 4t NI)3N + WON v + 2tNWl N,y + 4t N W2N,y ) = O, Ci31(uOl,x
12
1
13
-~tlUll,x+~tlU21,x - g / 1 g/31,x)
+ C231(uOl.y
12
1
13
--~tlVl 1,v. + 4 / 1 D 2 1 , y - s t l / ) 3 1 , v )
+ C 3 3 1 ( w i i - / 1 w21) = 0,
C13N (U0N,x +itNUlN, 1 1 2NU~N,x +gt i 3NU3N,x) x + gt 1
1 2
-J- C23N( ~ON,y "[-~fN1)IN, y 2t-~t N~)2N,y + ~ t 3 U3N, v "~- C33N( W1N + tNW2N ) : O.
(7)
Here Allk
Al2k
A13k
A22k
A23k
A66k ]
Dllk
D12k
D13k
D22k
D23k
D66k
Fl,k
F12k Fl3k
F22k F23k F66kJ
C12k Cl3k C22k C23k C66kJdZk. (8) It is important to note that in the present higher order theory used in each layer, the bending-stretching coupling stiffness does not occur even in general angle-ply laminates due to the homogeneity o f each layer.
K . N . CHO ET AL.
434
3. EXAMPLE PROBLEMS OF HOMOGENEOUS AND LAMINATED RECTANGULAR PLATES WITH SIMPLY SUPPORTING EDGE CONDITIONS The laminated rectangular plate is simply supported without axial force at the edges x -- 0, a and y = 0, b. The following general displacement field, which satisfies the boundary conditions, can be assumed: Uk
( Clok q- alk2k "k a2kZ2k "i" a3k Z3 ) COS ( m~'x / a ) sin ( nzry / b ) e i,~mn ,
Vk = ( bok + blk•k + bEkZk+ 2 b3kZk) 3 sin (m~rx/a) cos ( n ~ y / b ) e i~ t Wg = (Cok + ClkZk + C2kZ~k) sin ( m ~ x / a ) sin (n~ry/b) e Icomn! ,
k = 1, 2 , . . . , N,
(9)
where m and n are integers. Substituting the assumed displacement field (9) into the governing differential equations (7), one may express the equations in the generalized eigenvalue and eigenvector matrix system
[A]
{11 d2
dl N
= to~.[B]
d2
(10)
dl N
where [A] and [B] are 11N x 11N square matrices containing all known elements of the simultaneous equations. Here, the notations d~ (i = 1, 2 , . . . , 11 N ) are used to denote the unknown constants of the assumed displacement field, aik, big (i =0, 1, 2, 3), and Cik (i = 0, 1, 2). The matrix [B] is not invertible since it contains 6 N rows with all zero elements. Thus, this is not a true eigenvalue system. Therefore, a subroutine for the solution of the generalized eigenvalue problem including homogeneous simultaneous algebraic equations [33] is necessary. In this case the IMSL subroutine E I G Z F is used. The accuracy of the present higher order, individual-layer theory was verified by computing the numerical results for the first five natural frequencies of isotropic (u = 0.3) and orthotropic plates, treated as single-layer and four-layer plates. These results are presented in Tables 1 and 3, respectively. The following orthotropic stiffness properties of Aragonite crystals [5] were used: C22/C~1 = 0.543103; C~3/CH = 0.530172; C~2/C~ = 0.23319; C~3/C~ =0.010776; C23/C~ =0.098276; C44/C1~ =0.26681; C55/C~ = 0.159914; C66/C~ = 0.262931. The value of the material density, 0, is arbitrary since the investigations are non-dimensional. The results are compared with those by the 3-D elasticity theory of Srinivas and his co-workers [4, 5] and by the higher order theory of Reddy and Phan [14]. In Tables 1 and 3, it is shown that the present higher order theory predicts more accurate natural frequencies when treating homogeneous plates as multilayer laminated plates with an increasing number of layers, even though the properties are the same in all of the layers. Also, it predicts the natural frequencies for most of the modes which are obtained by 3-D elasticity theory. Tables 2 and 4 document a comparison between the relative stresses and displacements through the half-thickness by the present theory and 3-D elasticity theory. Here, the thickness-direction co-ordinate was normalized by the thickness of a plate. The distributions of the transverse shear stresses obtained by the present higher order theory using the stress equilibrium equations are given inside of the parentheses and are in good agreement with the distributions by 3-D elasticity theory. In Tables 5 and 6 are shown the fundamental natural frequencies of isotropic and orthotropic three-layer laminated plates with various ratios of moduli and densities between the layers. Compared to the 3-D elasticity-theory solution, the results by the CPT show a trend of increasing inaccuracy with increasing ratio of moduli. It is expected
435
FREE VIBRATION OF LAMINATED PLATES TABLE 1
Comparison o f natural frequencies, to = tor~,hx/ p / G , o f isotropic (v = 0"3) homogeneous plates mh/a
nh/a
0.1
0.1
0. l
0.2
0.1
0.2
0.3
0.2
0.2
0.3
0.3
0.3
I-At
I-S:~
Exact [4] Present ( N = 1) Present ( N = 4 ) Reddy and Phan [14]§ Mindlin et al. [34]¶
0.0931 0.0930 0.0931 0.0931 0.0930
0.4443 0.4443 0.4443 . --
Exact Present ( N Present ( N Reddy and Mindlin et
= 1) =4) Phan al.
0.2226 0.2219 0.2224 0.2222 0.2218
Exact Present ( N Present ( N Reddy and Mindlin et
= 1) = 4) Phan aL
Exact Present ( N Present ( N Reddy and Mindlin et
II-A:~
III-A
0.7498 0.7516 0.7499 . . --
3.1729 3.1933 3-1736 . 3.1729
3-2465 3-2756 3.2496
0-7025 0.7025 0.7025 . --
1.1827 1.1897 1.1832 . . --
3"2192 3.2394 3.2200 . 3.2192
3.3933 3.5831 3.3999
0.3421 0.3404 0.3416 0.3411 0.3402
0.8886 0.8886 0.8886 . --
1.4943 1.5066 1.4932 . . --
3.2648 3.2847 3.2656 . 3.2648
3.5298 3.5831 3.5398
= 1) = 4) Phan al.
0-4171 0.4147 0.4164 0.4158 0.4144
0.9935 0.9935 0.9934 . --
1.6654 1.6856 1.6668 . . --
3.2949 3.3147 3.2957 . 3-2949
3.6160 3.6772 3-6282
Exact Present ( N = 1) Present ( N =4) Reddy and Phan Mindlin et al.
0.5239 0.5203 0.5228 0.5221 0.5197
1.1327 1.1328 1.1327 . --
1"8936 1.9241 1.8957 . . --
3"3396 3-3590 3.3403 . 3.3396
Exact Present ( N Present ( N Reddy and Mindlin et
0.6889 0.6830 0-6872 0.6862 0.6821
1.3329 1.3329 1-3329 . --
2.2171 2.2680 2.2207 . . --
3.4126 3.4317 3.4134 . 3.4126
= 1) = 4) Phan al.
II-S
3.2538
3.4112
3.5580
3-6510 3.7393 3.8122 3.7538 3.7842 3.9310 4.0235 3.9522 3.9926
t A and S denote modes which are antisymmetric and symmetric about the mid-plane, respectively. $ Thickness-twist modes. § Only the fundamental natural frequencies were listed in reference [14]. ¶ The numerical results were obtained by using a shear correction factor of k2= 7r2/12. that this t r e n d can be f o u n d for all plate theories which c o n s i d e r a l a m i n a t e d plate as a single h o m o g e n e o u s plate. However, the present higher order, i n d i v i d u a l - l a y e r theory predicts results o f i n c r e a s i n g accuracy for increasing m o d u l i ratios. The f u n d a m e n t a l n a t u r a l frequencies of two-layer a n t i s y m m e t r i c a n d four-layer symmetric cross-ply lamin a t e d plates with various length-to-thickness ratios are s h o w n in T a b l e 7, a n d the results are c o m p a r e d with those by R e d d y a n d P h a n ' s higher order theory [14], the S D T a n d the CPT. The material properties of each layer were E22/E~ 1 = E33/E11 = 40, G23/E22 = O"5, G12/E22 = GI3/E22=0"6 a n d v~2 = v13=0.25. I n Table 7 are s h o w n some discrepancies b e t w e e n the results given by the present theory a n d those of the R e d d y a n d P h a n higher order theory even t h o u g h the two theories predicted very close f u n d a m e n t a l n a t u r a l frequencies for the h o m o g e n e o u s isotropic a n d o r t h o t r o p i c plates in Tables 1 a n d 3. It is t h o u g h t that these d i s c r e p a n c i e s arise from the differences in treating the l a m i n a t e d
436
K.N.
6
~
II
~
CHO
Illl
E T AL.
II
o~
~ 6 6 6~6 6
~ 6~6 6 6 ~ 0
II
~
~ .~ . . .
I I I I I I
~66666
~6666o
6 II
0
m ~
~ 6 6 6 ~
0
Illlll
o 6 ~ 6 6 ~ Y
~~ 6 6-6 6 ~
I I I I I I
~ 6 6 ~ 6 ~
~
~ I I I I I I
I
& 6 6 6 6
. . . . . .
~ 6 6 6 6
. . . . . .
~
0
~
~
I
~ o ~
~ 6 6 6 6 ~
.
IIIIII
~ 6 6 6 6 ~
.
.
.
.
~ ° .
~
N
. ~ a ~ 6 6 6 .
. . . . . . .
'~6666~
. . . . . . .
m
..66666~
~66666~
~
m
v
FREE VIBRATION OF LAMINATED PLATES
437
TABLE 3
Comparison o f natural frequencies, to = to,..hv/ 97 C~1, o f orthotropic homogeneous plates mh/a
nh/a
0.1
0.1
Exact [5] Present ( N Present ( N Reddy and M i n d l i n et
= l) = 4) P h a n [14] aL [34]§
I-A?
I-S~:
II-S
II-A~:
Ill-A§
0.0474 0.0474 0.0474 0.0474 0.0474
0.2170 0.2170 0.2170 ---
0.3941 0.3941 0.3941 ---
1.3077 1.3159 1.3081 1.3086 1.3159
1.6530 1.6646 1.6536 1.6550 1.6646
0.1
0.2
Exact Present ( N = 1) Present ( N = 4 ) Reddy a n d P h a n M i n d l i n et al.
0.1033 0.1033 0.1033 0.1033 0.1032
0.3450 0.3451 0.3450 ---
0.5624 0.5626 0.5624 ---
1.3331 1.3411 1.3335 1.3339 1.3411
1.7160 1.7301 1.7173 1.7209 1.7305
0.2
0.1
Exact Present ( N = 1) Present ( N = 4 ) Reddy and Phan M i n d l i n et aL
0.1188 0.1188 0.1188 0.1189 0.1187
0.3515 0.3515 0.3515 ---
0.6728 0.6728 0.6728 ---
1.4205 1.4285 1.4208 1.4216 1.4285
1.6805 1.6920 1.6812 1.6827 1.6921
0.2
0.2
Exact Present ( N = 1) Present ( N = 4 ) Reddy and Phan M i n d l i n et al.
0.1694 0.1693 0.1694 0.1695 0.1692
0.4338 0.4340 0.4338 ---
0.7880 0.7882 0.7880 ---
1.4316 1.4393 1.4319 1.4323 1.4393
1.7509 1.7649 1.7523 1.7562 1.7655
0.1
0.3
Exact Present ( N Present ( N Reddy and M i n d l i n et
0.1888 0.1885 0.1887 0.1888 0.1884
0.4953 0.4955 0.4953 ---
0.7600 0.7610 0.7601 ---
1.3765 1.3841 1.3768 1.3772 1.3841
1.8115 1-8296 1.8138 1.8210 1.8304
= 1) =4) Phan
aL
0.3
0.1
Exact Present ( N = 1) Present ( N = 4 ) Reddy and Phan M i n d l i n et al.
0.2180 0.2178 0.2180 0-2184 0.2178
0.5029 0.5030 0-5029 ---
0.9728 0.9728 0.9728 ---
1"5778 1.5857 1.5781 1.5789 1.5857
1.7334 1-7448 1.7342 1.7361 1.7450
0.3
0.3
Exact Present ( N = 1) Present ( N = 4 ) Reddy and Phan M i n d l i n et aL
0.3320 0.3313 0.3319 0.3326 0.3310
0.6504 0.6511 0.6505 ---
1.1814 1.1823 1.1815 ---
1.5737 1.5814 1.5741 1.5744 1.5813
1.9289 1.9469 1.9221 1.9395 1.9480
? A a n d S d e n o t e m o d e s w h i c h are antisymmetric a n d symmetric a b o u t the mid-plane, respectively. ? Thickness-twist modes. § The n u m e r i c a l results were o b t a i n e d by using a shear correction factor of k 2 = 5/6. p l a t e as a n h o m o g e n e o u s p l a t e o r c o n s i d e r i n g e a c h l a y e r s e p a r a t e l y . F r o m t h e f a c t s d i s c u s s e d i n r e l a t i o n t o T a b l e s 5 a n d 6, t h e r e s u l t s g i v e n b y t h e p r e s e n t h i g h e r o r d e r theory are considered more accurate than the results by Reddy and Phan.
4. C O N C L U S I O N S The present higher order individual-layer theory in which higher order in-plane and transverse normal displacements are used in each layer may not be more accurate than the Reddy and Phan higher order theory or Mindlin's shear deformable theory with shear
438
K . N . CHO E T A L .
TABLE 4
Comparison of relative stresses and displacements across the thickness of orthotropic homogeneous plates ( mh / a = nh / b = 0.3)
o,x/o,x(0.5) •
~'~
Exact
o-/%(0.5) •
Present
[5]
•
A
Exact
z.,/~xy(0.5) •
Present
[5]
,
~
Exact
u/u(0.5) •
Present
[5]
,
v/v(0.5)
A
Exact
^
Present
[5]
Exact
•
Present
[5]
First antisymmetric thickness mode (I-A) 0.5 1.0 1.0 1.0 1.0 0.4 0.713 0.712 0.754 0.747 0-3 0.487 0.482 0-540 0.530 0-2 0.303 0.297 0.348 0.338 0.1 0.145 0.141 0.170 0.165 0 0 0 0 0
1.0 0.736 0.516 0.328 0.159 0
1"0 0.736 0-514 0.325 0.157 0
1.0 0.696 0.465 0.284 0.134 0
1.0
1.0
1.0
0.695 0.459 0.277 0.130 0
0.764 0.553 0.360 0.177 0
0.765 0.553 0.356 0.177 0
Second antisymmetric thickness mode 0.5 1.0 1.0 1.0 0.4 0.951 0.944 0.951 0.3 0.809 0.792 0.809 0.2 0.588 0.568 0.588 0.1 0.309 0.296 0.309 0 0 0 0
1.0 0.951 0.809 0.588 0.309 0
1.0 0.945 0.794 0.570 0.267 0
1.0 0.951 0.809 0.588 0.309 0
1.0 0.944 0.792 0.568 0.296 0
1.0 0.951 0.809 0.588 0.309 0
1.0 0.943 0.791 0.567 0.296 0
Third antisymmetric thickness mode ( I l l - A ) 0.5 1.0 1.0 1.0 0 0-4 0.984 0.970 0.952 0.967 0.3 0.857 0.826 0.811 0.822 0.2 0.633 0.598 0.591 0.595 0-1 0.336 0.313 0.311 0.311 0 0 0 0 0
1.0 0.979 0.850 0.623 0.332 0
1.0 0.964 0.819 0.591 0.309 0
1.0 0.951 0.870 0.645 0.343 0
1.0 0.974 0.832 0.603 0.316
1.0 0.972 0.840 0.617 0.326 0
1.0 0.959 0.812 0.586 0.306
First symmetric thickness mode (I-S) 0.5 1.0 1.0 1.0 1.0 0-4 0'979 0'980 1"013 1.014 0'3 0.964 0.964 1"023 1"025 0"2 0'953 0"953 1.030 1'033 0" 1 0'947 0.946 1"034 1'038 0 0.953 0.944 1"035 1.040
1-0 1"042 1.072 1.093 1" 106 1"110
1.0 1.039 1.070 1'092 1' 105 1'109
1.0 0"989 0.981 0.976 0"973 0.972
1.0 0"989 0-981 0-975 0.972 0.971
1.0 1'007 1.013 1"017 1"019 1.020
1.0 1.007 1"012 1"016 1"018 1"019
Second symmetric thickness 0"5 1.0 1"0 0.4 1.008 1.006 0.3 1.014 1.010 0-2 1.018 1.014 0.1 1.021 1.016 0 1.022 1.016
1.0 1.009 1.017 1.022 1.025 1-027
1.0 1.007 1.012 1.016 1.018 1.019
1.0 1.007 1.013 1.017 1.020 1.021
1"0 1-005 1-010 1.013 1.015 1.015
1.0 1.012 1.022 1.030 1.034 1.020
1.0 1.009 1.016 1.020 1.023 1.024
(II-A) 1.0 0.963 0.817 0.590 0.308 0
mode (II-S) 1.0 1.0 1.006 1.008 1.011 1.013 1.015 1.018 1.018 1.020 1.018 1.021
t The values in parentheses give the transverse shear stresses by the present higher order theory in using the stress equilibrium equations. ;i: Z~J~',,~(0"3) instead of ~'~J'r~(O). § %,z/%,2(0"3) instead of %,z/r~z(O).
FREE VIBRATION OF LAMINATED PLATES
TABLE ,1
~/o~z(o.1)
-,xJ ~~(o)
439
continued
w/w(O.5)
~-,:/¢v~(o)
A
Exact
Present
[5]
Exact
Present
Exact
[5]
Present
Exact
Present
[5]
[5]
1.0
1.0
0
0
0 0.394 0.674 0.860 0.966 1.0
o 1.335 1.880 1.710 1.0 0
o 1.50 2.00 1.75 1.0 0
o 0.309 0.588 0.809 0.951 1.0
o 0.36 0.64 0.84 0.96 1.0
(o) (0.314) (0.594) (0.818) (0.952) (1.0)
o 0.309 0.588 0.809 0.951 1.0
o 0.36 0.64 0.84 0.96 1.0
(o) (0.314) (0.594) (0-813) (0.952) (1.0)
0 1"310 1'861 1'704 1'0 0
0 1"50 2"00 1"75 1'0 0
0 0'294 0'572 0'799 0"948 1.0
0 0'36 0"64 0.84 0"96 1.0
(0) (0"305) (0"585) (0"808) (0"951) (1.0)
0 0'302 0"580 0"805 0"950 1.0
0 0"36 0"64 0"84 0"96 1.0
(0) (0"311) (0"591) (0"811) (0'952) (1.0)
1"0 0"902 0"771 0"644 0'553 0-521
1"0 0"926 0"868 0-827 0"802 0.793
0 0.387 0.678 0.880 1.0 1.040
0 0.375 0.667 0.875 ~:t~i066 1.0 L~ "492 1.042
-- -
(0.750)
------
(1.0) § ~i0866 (0'875) [~.492 (0)(0"500)
(0) (0.750) (1.0) (0.875)
1-0 0.811 0.615 0.413
1.0 0.80 0.60 0.40
(0.500) (0)
0.207 0
0.20 0
0 0"360 0"653 0"868 1"0 1"044
0 0"375 0"667 :I:~ 1~097 0"875 1"0 1'042
(o) (0.750) (1 .o) (0.875) (0.500) (o)
1.0 0"836 0.649 0.443 0.225 0
1'0 0'80 0'60 0.40 0.20 0
0 1.569 2.048 1.766
0 1.50 2.00 1.75
[ 761
(0.724
LOttO
0 0.36 0.64 0.84 0.96 1-0
(0)t (0-397) (0-677) (0.861) (0.965) (1.0)
0 0.384 0.663 0.853 0.964 1.0
0 0.36 0.64 0.84 0.96 1.0
(0)t (0.387) (0.667) (0.856) (0.965) (1.0)
lo,
foO761
m
w
_
---
__
[0.734
(0.750) (1.0) 1-0 (0.875) § ~0.889
,o oo
D
m
1.0 1.005 1.008 1.010 1.010 1.010 1.0
16.66 30.76 41.93 49.10 51.56
1,0 1.006 1-011 1-015 1.017 1.018 1-0
2.448 3-573 4-377 4.860 5.021
K.N. CHO ET AL.
440
TABLE 5
Comparison of fundamental natural frequencies, to = toll/'2x/P2/02, of isotropic three-layer laminated plates with various ratios ofmoduli and materialdensities (tl =/`3 = 0.1 h,/'2 = 0.8 h, m h / a = n h / b = O . 1 , p l = p s , G I = G3, v l = v3, v 2 = 0 . 3 ) Vl
Pl/P2
Gt/G 2
Exact [4]
Present
% Error
CPT
% Error
0"3 0"3 0-3 0-3 0"3 0"3 0.2
1 1 1 1 2 3 1
1 2 5 15 15 15 1
0"074520 0"089986 0.123048 0.183664 0.167574 0.155082 0.072298
0-074881 0.089954 0.123072 0.183647 0.167558 0"155068 0"072262
(0.48) (-0"036) (-0"020) (-0.009) (-0"010) (-0.009) (-0.05)
0.077054 0.093994 0.132390 0.215642 0-196852 0.182250 0.074666
(2-9) (4.5) (7.6) (17.4) (17.5) (17"5) (3"28)
TABLE 6
Comparison of fundamental natural frequencies, to = tol~ t2v/-'~2/Cl12, of orthotropic threelayer laminated plates with various ratios of moduli and material densities ( /,~ =/'s = O. 1 h, t 2 = 0 . 8 h, mh/ a = n h / b = 0 . 1 , p l = p 3 ) Exact
Pl/P2
C111/C112
[5]
Present
% Error
CPT
% E~or
1 1 1 1 1 3
1 2 5 10 15 15
0.047419 0.057041 0.077148 0.098104 0.112034 0.094548
0.047416 0.057035 0.077147 0.098102 0.112032 0.094546
(-0.006) (-0.011) (-0.001) (-0.002) (-0.002) (-0.002)
0.049666 0.060584 0.085333 0.115328 0.138994 0.117471
(4.7) (6.2) (10.6) (17.6) (24.1) (24.3)
TABLE 7
Comparison of fundamen/,al natural frequencies, to = tolla2x/p2/ E22, of cross-ply three-layer laminated square plates [00/900/900/0 °]
a/h
Present
HSDPT [14]
2 5 10 20 25 50 100
5"923 10-673 15.066 17"535 18"054 18"670 18.835
5.576 10.989 15'270 17"668 18"050 18"606 18~755
SDT 5"492 10"820 15.083 17"583 17"991 18.590 18"751
[0°/90 °]
CPT 15.830 18.215 18.652 18.767 18.780 18.799 18.804
Present
HSDPT [14]
4"810 8"388 10.270 11-016 11"118 11"260 11"296
5"699 9.010 10.449 10-968 11.037 11.132 11"156
SDT 5"191 8.757 10"355 10.941 11"020 11.127 11"155
CPT 8.499 10.584 11.011 11.125 11-139 11.158 11"163
FREE VIBRATION OF LAMINATED
PLATES
441
correction factor in the analysis of fundamental natural frequencies for homogeneous plates. This is due to the fact that the distributions of the transverse shear strains through the thickness are forced to be quadratic. However, the present theory has the advantage of predicting more free vibration modes (as in 3-D elasticity theory) than the other approximate theories. More importantly, though, the natural frequencies given by the present theory are closer to the three-dimensional elasticity-theory results than those of other plate theories for the case of laminated plates. By accounting for the second order distribution of transverse normal displacement through each layer's thickness, rather accurate distributions of the relative transverse normal stresses and the deflection can be predicted. Moreover, for laminated plates, the present higher order individual-layer theory predicts accurate distributions of the relative transverse shear stresses which play an important role in the prediction of failure behavior.
REFERENCES 1. E. REISSNER 1944 Journal of Mathematics and Physics 23, 183-191. On the theory of bending elastic plates. 2. R. D. MINDLIN 1951 American Society of Mechanical Engineers, Journal of Applied Mechanics 18, 31-38. Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates. 3. S. SRINIVAS, A. K. RAO and C. V. JOGA RAO 1969 ZeitschriftJ~r Angewandte Mathematik und Mechanik 49(8), 449-458. Flexure of simply supported thick homogeneous and laminated rectangular plates. 4. S. SRINIVAS, C. V. JOGA RAO and A. K. RAO 1970 Journal of Sound and Vibration 12, 187-195. An exact analysis for vibration of simply-supported homogeneous and laminated thick rectangular plates. 5. S. SRINIVAS and A. K. RAO 1970 International Journal of Solids and Structures 6, 1463-1481. Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates. 6. N. J. PAGANO 1969 Journal of Composite Materials 3, 398-411. Exact solutions for composite laminates in cylindrical bending. 7. N. J. PAGANO 1970 Journal of Composite Materials 4, 20-34. Exact solutions for rectangular bidirectional composites and sandwich plates. 8. S. A. AMBARTSUMYAN 1970 Theory of Anisotropic Plates. Stamford: Technomic. 9. E. REISSNER 1975 International Journal of Solids and Structures 11, 569-573. On transverse bending of plates, including the effect of transverse shear deformation. 10. K. H. Lo, R. M. CHRISTENSEN and E. M. WU 1977 American Society of Mechanical Engineers Journal of Applied Mechanics 44, 663-668. A higher order theory of plate deformation--part 1: homogeneous plates. 11. K. H. Lo, R. M. CHRISTENSEN and E. M. Wu 1977 American Society of Mechanical Engineers, Journal of Applied Mechanics 44, 669-676. A higher order theory of plate deformation--part 2: laminated plates. 12. M. LEVINSON 1980 Mechanics Research Communications 7, 343-350. An accurate, simple theory of the statics and dynamics of elastic plates. 13. J. N. REDDY 1984 American Society of Mechanical Engineers, Journal of Applied Mechanics 51, 745-752. A simple higher-order theory for laminated composite plates. 14. J. N. REDDY and N. D. PHAN 1985 Journal of Sound and Vibration 98, 157-170. Stability and vibration of isotropic, orthotropic and laminated plates according to a higher-order deformation theory. 15. G. Z. VOYIADJIS and M. M. BALUCH 1988 American Society of Civil Engineers, Journal of Engineering Mechanics 114(4), 671-687. Refined theory for thick composite plates. 16. J. L. DOONG and C. P. FUNG Journal of Sound and Vibration 125, 325-339. Vibration and buckling of bimodulus laminated plates according to a higher-order plate theory. 17. N. R. SENTHILNATHAN, S. P. LIM, K. H. LEE and S. T. CHow 1988 Composite Structures 10, 211-229. Vibration of laminated orthotropic plates using a simplified higher-order deformation theory.
442
K.N.
CHO ET AL.
18. P. C. YANG, C. H. NORRIS and Y. STAVSKY 1966 International Journal of Solids and Structures 2, 665-685. Elastic wave propagation in heterogeneous plates. 19. J. M. WHITNEY and N. J. PAGANO 1973 American Society of Mechanical Engineers, Journal of Applied Mechanics 40, 302-304. Shear deformation in heterogeneous anisotropic plates. 20. J. KAO and R. J. ROSS 1968 American Institute of Aeronautics and Astronautics Journal 6, 1583-1585. Bending of multilayer sandwich beams. 21. G. W. SWIFT and R. A. HEELER 1974 Journal of Engineering Mechanics Division, Proceedings American Society of Civil Engineers 100, 267-282. Layered beam analysis. 22. P. SLIDE 1980 Mechanics Today 5, 451-466. An improved approximate theory for the bending of laminated plates. 23. R. L. HINRICHSEN and A. N. PAEAZOTTO 1986 American Institute of Aeronautics and Astronautics Journal 24, 1836-1842. Nonlinear finite element analysis of thick composite plates using cubic spline functions. 24. J. N. REDDY 1987 Communications in Applied Numerical Methods 3, 173-180. A generalization of two-dimensional theories of laminated composite plates. 25. J. N. REDDY 1989 International Journal for Numerical Methods in Engineering 27, 361-382. On refined computational models of composite laminates. 26. J. N. REDDY, E. J. BARBERO and J. L. TEPLY 1988 29th Structures, Structural Dynamics and Materials Conference, Williamsburg, Virginia, AIAA CP882, Paper 88-2322, 937-943. A plate bending element based on a generalized laminated plate theory. 27. R. R. VALISETTY and L. W. REHFIEED 1988 Journal of Sound and Vibration 126, 183-194. Application of ply level analysis to flexural wave propagation. 28. K. N. CHO, C. W. BERT and A. G. STRIZ 1987 Engineering Science Preprint ESP24.87034. New theory for bending of bimodular laminates. 29. K. N. CHO, A. G. STRIZ and C. W. BERT 1989 Journal of Thermal Stresses 12, 321-332. Thermal stress analysis of laminate using higher-order theory in each layer. 30. K. N. CHO, A. G. STRIZ and C. W. BERT 1990 Composite Structures 15, 1-24. Bending analysis of thick bimodular laminates by higher-order individual-layer theory. 31. A. K. NOOR and W. S. BURTON 1989 Applied Mechanics Reviews 42, 1-13. Assessment of shear deformation theories for multilayered composite plates. 32. R. M. JONES 1975 Mechanics of Composite Materials. Washington, D.C.: Hemisphere. 33. C. B. MOLER and G. W. STEWART 1973 Society for Industrial and Applied Mathematics Journal for Numerical Analysis 10, 241-256. An algorithm for generalized matrix eigenvalue problems. 34. R. D. MINDLIN, A. SCHACKNOW and H. DERESIEWICZ 1956 American Society of Mechanical Engineers, Journal of Applied Mechanics 23, 430-436. Flexural vibrations of rectangular plates.