Appl. Math. Mech. -Engl. Ed., 32(11), 1407–1422 (2011) DOI 10.1007/s10483-011-1511-9 c Shanghai University and Springer-Verlag Berlin Heidelberg 2011

Applied Mathematics and Mechanics (English Edition)

Convergence and exact solutions of spline finite strip method using unitary transformation approach∗ J. KONG1 , D. THUNG2 (1. Division of Building Science and Technology, City University of Hong Kong, Hong Kong, P. R. China; 2. School of Creative Media, City University of Hong Kong, Hong Kong, P. R. China)

Abstract The spline ﬁnite strip method (FSM) is one of the most popular numerical methods for analyzing prismatic structures. Eﬃcacy and convergence of the method have been demonstrated in previous studies by comparing only numerical results with analytical results of some benchmark problems. To date, no exact solutions of the method or its explicit forms of error terms have been derived to show its convergence analytically. As such, in this paper, the mathematical exact solutions of spline ﬁnite strips in the plate analysis are derived using a unitary transformation approach (abbreviated as the U-transformation method herein). These exact solutions are presented for the ﬁrst time in open literature. Unlike the conventional spline FSM which involves assembly of the global matrix equation and its numerical solution, the U-transformation method decouples the global matrix equation into the one involving only two unknowns, thus rendering the exact solutions of the spline ﬁnite strip to be derived explicitly. By taking Taylor’s series expansion of the exact solution, error terms and convergence rates are also derived explicitly and compared directly with other numerical methods. In this regard, the spline FSM converges at the same rate as a non-conforming ﬁnite element, yet involving a smaller number of unknowns compared to the latter. The convergence rate is also found superior to the conventional ﬁnite diﬀerence method. Key words

spline ﬁnite strip, U-transformation, plate, symmetry

Chinese Library Classification O242.21, TU33 2010 Mathematical Subject Classification 74K20

1

Introduction

The unitary transformation (U-transformation) method was originally developed by Chan et al.[1] for the exact analysis of periodic structures. It was subsequently extended by Cai et al.[2] to bi-periodic systems. The success of the method relies on a complex unitary matrix U . When applied to the transformation of a circulant matrix, it completely diagonalizes the circulant matrix. Such circulant matrices exist in many branches of science and engineering and in the context of structural engineering. They correspond to stiﬀness matrices of periodic structures. As such, the U-transformation method provides a mathematical tool for decoupling a periodic or cyclic symmetric structural system and making it possible for obtaining the corresponding ∗ Received Dec. 30, 2010 / Revised Jul. 16, 2011 Project supported by the Division Research Grant from City University of Hong Kong (No. DRG 13/08-09) Corresponding author J. KONG, Ph. D., E-mail: bsjkong@cityu.edu.hk

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J. KONG and D. THUNG

exact solution in an explicit form. The method has been successfully applied to a wide range of structural systems with rotational and linear periodicity like continuous beams and plates, folded plates, trusses, grillages, cable networks, and spring-mass systems. Extension to stress concentration problems of plates was recently achieved by Yang et al.[3] . The ﬁnite strip method (FSM) has received signiﬁcant attention in the research community and the civil engineering industry worldwide since its ﬁrst introduction in 1968 by Professor Cheung[4] , and reviews of the subjects have been published elsewhere[5–8] . Among diﬀerent versions of ﬁnite strips, the classical FSM, ﬁrst developed by Cheung[4] , has attracted a lot of attention due to the fact that the governing stiﬀness matrix equation is decoupled, signiﬁcantly simplifying the problem and reducing the computational eﬀorts. The classical FSM, however, is only limited to structures with simply-supported ends. For problems with general boundary conditions, the spline FSM can be much more versatile, providing a complementary method to the classical ﬁnite strips. Although the accuracy and the rate of convergence of the classical FSM were studied by Smith and Allen[9] and Li[10] , the eﬃcacy of spline ﬁnite strips was only demonstrated in the literature by comparing numerical results with analytical solutions or other numerical solutions. Explicit mathematical derivation of the rate of convergence and the exact solution form of spline ﬁnite strips are not available in the literature, to the best of the authors’ knowledge. This paper attempts to provide such a derivation using the U-transformation approach for the particular case of plate bending with simply-supported edges, of which the results provide important insights into the rate of convergence and eﬃcacy of the spline FSM. To ﬁrst illustrate the application of the U-transformation to problems involving spline functions, a one-dimensional problem of bending, vibration, and buckling of simple beams is considered. The beam deﬂection is approximated by the cubic B3 -spline function. A linear periodic system is then formed, and the corresponding circulant stiﬀness matrix is established. The U-transformation is used to diagonalize the matrix and thereby obtain the exact solution in an explicit form. Taylor’s series expansion of the exact solution would reveal the error term and the rate of convergence of the cubic B3 -spline approximation. Following the same procedure, the two-dimensional problem of bending, vibration, and buckling of a simply-supported rectangular plate is considered. The plate is divided into cubic B3 -spline ﬁnite strips[4] . Explicit solutions are then obtained using the U-transformation. The rates of convergences are subsequently compared with those of a non-conforming element[1] and the ﬁnite diﬀerence method[11–12] .

2

U-transformation

Given N numbers K1,j , j = 1, 2, · · · , N , a circulant matrix K is a matrix whose rows are of the following forms: ⎡ ⎤ K1,1 K1,2 K1,3 · · · K1,N ⎢ K1,N K1,1 K1,2 · · · K1,N −1 ⎥ ⎢ ⎥ ⎢ K1,N −1 K K · · · K1,N −2 ⎥ 1,N 1,1 (1) K=⎢ ⎥. ⎢ ⎥ .. .. .. .. ⎣ ⎦ . . . . K1,2 K1,3 K1,4 · · · K1,1 Circulant matrices have a very important property. The unitary matrix U of order N is given by U ≡ [U1 , U2 , · · · , UN ],

(2a)

1 Um = √ [1, eimϕ , ei2mϕ , · · · , ei(N −1)mϕ ]T , N

(2b)

where

Convergence and exact solutions of spline ﬁnite strip method

1409

and ϕ=

2π N

and i =

√ −1

(2c)

diagonalize the entire circulant matrix. Denoting U as the complex conjugate of U , we have T

U KU = diag(k1 , k2 , · · · , kN ),

(3a)

where kr =

N 

K1,j ei(j−1)rϕ ,

(3b)

j=1

and T

U U = I.

3

(3c)

Cubic spline functions and U-transformation for beams

Consider a simply-supported, prismatic beam of span l, ﬂexural stiﬀness EI, and mass per unit length m under a general load q(x) (see Fig. 1). The beam is divided into n cubic B3 spline sections of equal section length h = l/n. To form a linear periodic system, hypothetical spans are added to both sides of the original span, and for easy presentation, only two adjacent hypothetical spans are shown in Fig. 2. Assume that the deﬂection proﬁle of the two adjacent spans is made anti-symmetrical with respect to the original span by imposing anti-symmetrical loads thereon, that is, q(x) = −q(−x) = −q(2l − x)

for x ∈ [0, l].

(4)

Fig. 1

Simply supported beam of span l that is subject to load q(x) and divided into cubic spline sections with equal section length

Fig. 2

Two hypothetical spans are appended to original simply-supported span in the middle. Deﬂection proﬁles of hypothetical spans are anti-symmetrical to original span

Obviously, the resulting anti-symmetrical deﬂection proﬁle satisﬁes automatically the boundary conditions at both ends of the original span. By the same token, this anti-symmetry can be made repeatedly for all extended hypothetical spans on both sides, and the resulting deﬂection proﬁle of the original span repeats in every alternate span (see Fig. 3).

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J. KONG and D. THUNG

Fig. 3

Linear periodic system of beam. Each period is divided into 2n cubic spline sections

Next, consider two contiguous spans of the aforesaid linear periodic system with 2n numbers of cubic B3 -spline sections. Assume that the deﬂection is approximated by the cubic B3 -spline functions φi (x)[4] . By Euler’s beam theory, the elements of the stiﬀness matrix can be written as follows: L 2 d φi d2 φj Ki,j = EI dx, (5) dx2 dx2 0 where L = 2l. In general, each cubic local spline function φi (x) couples with only seven consecutive splines (see Fig. 4). As such, row i of the circulant stiﬀness matrix of the periodic system can be written as follows: Ki,i−3 wi−3 + Ki,i−2 wi−2 + Ki,i−1 wi−1 + Ki,i wi + Ki,i+1 wi+1 + Ki,i+2 wi+2 + Ki,i+3 wi+3 = Fi , where Ki,i−3 =

6EI 36h3

= Ki,i+3 , Ki,i−2 = 0 = Ki,i+2 , Ki,i−1 = − 54EI 36h3 = Ki,i+1 , Ki,i = Ki,j = 0 if

Fig. 4

|i − j|  4.

(6) 96EI 36h3 ,

and (7)

Cubic spline functions with equal section length. Each spline function φi couples with seven consecutive splines φi−3 to φi+3

In Eq. (6), the deﬂection variable at each spline knot is denoted by wj . The closed form integration of spline functions in Eq. (5) can easily be done[4] . For the uniformly distributed load p, the load vector entry Fi in Eq. (7) is given by L Fi = p φi (x)dx, (8) 0

where F1 = 0 = Fn+1 = F2n+1 . Upon integration, it can be shown that F2 =

22hp = Fn = −Fn+2 = −F2n . 24

(9)

Convergence and exact solutions of spline ﬁnite strip method

1411

Otherwise, Fi = hp. Applying the U-transformation in Eq. (2), we can write each deﬂection variable wj in terms of generalized coordinates qr and the associated symmetry mode ei(j−1)rϕ , that is, wj =

N  1 √ ei(j−1)rϕ qr , N r=1

(10)

where N = 2n. Using Eqs. (3), (6), and (7), the stiﬀness matrix can be diagonalized as kr =

N 

K1j ei(j−1)rϕ

j=1

= K1,1 ei(1−1)rϕ + K1,2 ei(2−1)rϕ + K1,3 ei(3−1)rϕ + K1,4 ei(4−1)rϕ + K1,N ei(N −1)rϕ + K1,N −1 ei(N −2)rϕ + K1,N −2 ei(N −3)rϕ = K1,1 + K1,2 eirϕ + K1,3 e2irϕ + K1,4 e3irϕ + K1,N e−irϕ + K1,N −1 e−2irϕ + K1,N −2 e−3irϕ 96 − 54eirϕ + 0e2irϕ + 6e3irϕ − 54e−irϕ + 0e−2irϕ + 6e−3irϕ 36h3 96 − 108 cos(rϕ) + 12 cos(3rϕ) = . 36h3 =

(11)

The load vector is transformed using the following complex conjugate modes: N 1  −i(j−1)rϕ fr = √ e Fj N j=1 n N  hp  −i(j−1)rϕ 2 = √ e − e−i(j−1)rϕ − (e−i(2−1)rϕ + e−i(n−1)rϕ ) 24 N j=2 j=n+2 2 + (e−i(n+1)rϕ + e−i(N −1)rϕ ) 24 n n hp  −i(j−1)rϕ  −i(n+j−1)rϕ 2 = √ e − e + (e−irϕ − eirϕ )(cos(mπ) − 1) 24 N j=2 j=2 ⎧ n n ⎪ hp  −i(j−1)rϕ  −i(j−1)rϕ 2 ⎪ ⎨√ if r = odd, e + e + (−2i sin(rϕ))(−2) 24 N j=2 = j=2 ⎪ ⎪ ⎩ 0 if r = even ⎧ n ⎪ 2hp  −i(j−1)rϕ hp 1 ⎪ ⎨√ (i sin(rϕ)) if r = odd, e −1 + √ N j=1 N3 = (12) ⎪ ⎪ ⎩ 0 if r = even.

For the odd number r, the ﬁrst term inside the bracket forms a geometric series, which can be shown as n  j=1

e−i(j−1)rϕ − 1 =

−i sin(rϕ) . 1 − cos(rϕ)

Using the diagonalized stiﬀness matrix in Eq. (11) and the transformed load vector in

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J. KONG and D. THUNG

Eq. (12), the generalized coordinates qr (r = 1, 3, 5, · · · , N −1) can be determined by solving (96 − 108 cos(rϕ) + 12 cos(3rϕ))qr hp 1 2hp i sin(rϕ) +√ (i sin(rϕ)). = −√ 36h3 (1 − cos(rϕ)) N N3

(13)

Hence, qr = −

i sin(rϕ)(5 + cos(rϕ)) h4 p √ . EI N (1 − cos(rϕ))(8 − 9 cos(rϕ) + cos(3rϕ))

(14)

The deﬂection variable can be easily found as wj =

=

N −1 

i sin(rϕ)(5 + cos(rϕ)) 1 −h4 p √ ei(j−1)rϕ √ (1 − cos(rϕ))(8 − 9 cos(rϕ) + cos(3rϕ)) N EI N r=1,3,5,··· N −1 ei(j−1)rϕ i sin(rϕ)(5 + cos(rϕ)) −h4 p  . EIN r=1,3,5,··· (1 − cos(rϕ))(8 − 9 cos(rϕ) + cos(3rϕ))

(15)

Assume that the beam is divided into even number of spline sections, i.e., n is even. Then, the deﬂection at the center of the original span is given by 1 (w n + 4w n2 +1 + w n2 +2 ) 6 2 N −1 −irϕ  sin rπ + 4 + eirϕ ) h4 p 2 sin(rϕ)(5 + cos(rϕ))(e = EIN r=1,3,5,··· 6(1 − cos(rϕ))(8 − 9 cos(rϕ) + cos(3rϕ))

wx= l = 2

=

h4 p EIN

N −1 

sin rπ 2 sin(rϕ)(5 + cos(rϕ))(2 + cos(rϕ)) . 3(1 − cos(rϕ))(8 − 9 cos(rϕ) + cos(3rϕ)) r=1,3,5,···

(16)

Assume that f (r) =

sin rπ 2 sin(rϕ)(5 + cos(rϕ))(2 + cos(rϕ)) . (1 − cos(rϕ))(8 − 9 cos(rϕ) + cos(3rϕ))

(17)

It can be shown that f (r) = f (N − r).

(18)

Hence, the actual deﬂection at the center of the beam is wx= l = 2

n−1  sin rπ sin(rϕ)(5 + cos(rϕ))(2 + cos(rϕ)) h4 p 2 . 3EIn r=1,3,5,··· (1 − cos(rϕ))(8 − 9 cos(rϕ) + cos(3rϕ))

(19)

Expanding the right-hand side of Eq. (19) into Taylor’s series, we have wx= l = 2

l4 p 3EI

n−1  r=1,3,5,···

sin

rπ 12 4 + O(n ) . 2 r5 π 5

(20)

As n goes to inﬁnity, the ﬁrst term reveals that the deﬂection does converge to the analytical beam solution. ∞  5pl4 l4p rπ 12 wx= l = , (21) = sin 5 5 2 3EI r=1,3,5,··· 2 r π 384EI

Convergence and exact solutions of spline ﬁnite strip method

1413

while the second term indicates that the cubic spline solution converges to the exact analytical beam solution at an asymptotic rate of n−4 . It is noteworthy that, by considering a generally distributed load q(x) acting on any arbitrary spline section, the same linear periodic system can be formed by imposing the corresponding anti-symmetrical loads on the contiguous spans. Then, load vectors can be formed by integrating the spline functions involved and subsequently transformed as in Eq. (12). The exact deﬂection solution can be obtained following the same approach given in Eqs. (13)–(21). By the principle of superposition, the exact solution for any arbitrary distributed load acting on the beam can be found. By the same token, for vibration problems, the corresponding mass matrix of the periodic beam can be written as follows: L Mij = m φi φj dx. (22) 0

By transformation in a manner similar to the stiﬀness matrices, it can be shown that mr =

N 

M1j ei(j−1)rϕ

j=1

(2416 + 1191eirϕ + 120e2irϕ + 1e3irϕ + 1191e−irϕ + 120e−2irϕ + 1e−3irϕ)mh 5040 (2416 + 2382 cos(rϕ) + 240 cos(2rϕ) + 2 cos(3rϕ))mh . = 5040

=

(23)

The natural frequency ω of the beam can be found from Eqs. (11) and (23) as follows: (8 − 9 cos(rϕ) + cos(3rϕ))qr EI (1208 + 1191 cos(rϕ) + 120 cos(2rϕ) + cos(3rϕ))qr mh . = ω2 3h3 2520 Hence, ω2 =

840(8 − 9 cos(rϕ) + cos(3rϕ)) EI . 4 mh 1208 + 1191 cos(rϕ) + 120 cos(2rϕ) + cos(3rϕ)

(24)

Expanding it into Taylor’s series, we obtain the explicit solution of the natural frequency and the corresponding error term ω2 =

r8 ϕ8 140EI r4 ϕ4 + + O(ϕ10 ) , 4 mh 140 100800

(25)

ignoring higher-order terms, which can be rewritten as ω2 =

EI (rπ)4 EI(rπ)4 r4 π 4 . + m l4 ml4 720n4

(26)

As such, it is clear that the eigenvalue converges to the analytical solution of the simple beam at an asymptotic rate of n−4 . For buckling of a beam under the axial compression λP , the corresponding geometric matrix can be written as L dφi dφj Gij = λP dx. (27) dx dx 0

1414

J. KONG and D. THUNG

Transforming in a manner similar to the stiﬀness matrices, we obtain the following diagonalized geometric matrix: gr =

N 

G1j ei(j−1)rϕ

j=1

(240 − 45eirϕ − 72e2irϕ − 3e3irϕ − 45e−irϕ − 72e−2irϕ − 3e−3irϕ )λP 360h (240 − 90 cos(rϕ) − 144 cos(2rϕ) − 6 cos(3rϕ))λP . = 360h

=

(28)

The buckling load factor λ can be found from Eqs. (11) and (28), that is, (8 − 9 cos(rϕ) + cos(3rϕ))qr EI λ(40 − 15 cos(rϕ) − 24 cos(2rϕ) − cos(3rϕ))qr P . = 3 3h 60h Hence, λ=

EI 20(8 − 9 cos rϕ + cos(3rϕ)) . P h2 40 − 15 cos(rϕ) − 24 cos(2rϕ) − cos(3rϕ)

(29)

Expanding it into Taylor’s series, we obtain 10EI r2 ϕ2 r6 ϕ6 8 + + O(ϕ ) , P h2 10 7200

(30)

EI r2 π 2 EI r2 π 2 r4 π 4 + . P l2 P l2 720n4

(31)

λ= which can be simpliﬁed to λ=

By taking r = 1, the buckling load factor converges to the Euler buckling load of the simple beam at an asymptotic rate of n−4 . As compared with the convergence of the usual cubic Hermite beam elements[1] , it can be seen from Eqs. (20), (26), and (31) that the convergence rate of deﬂection, frequency, and buckling load of the cubic spline functions is the same as the beam elements yet involving only about the half number of unknowns (if the number of beam elements is the same as that of spline sections). In addition, in comparison with the ﬁnite diﬀerence method[11] , both methods involve about the same number of unknowns (if the same number of sections is used in both methods), but the ﬁnite diﬀerence method only converges at a much slower rate of n−2 .

4

Spline finite strip and U-transformation

Consider the two-dimensional problem of plate bending with simply-supported edges. Assume that the plate is divided into m numbers of identical spline ﬁnite strips across the width and n numbers of spline sections (with identical section length h = l/n) along the length (see Fig. 5). Following the same argument as in previous section, hypothetical plate panels with anti-symmetrical deﬂection proﬁles are assumed to be extended in both directions, as shown in Fig. 6, such that a periodic plate system of four contiguous squarely aligned panels with 2n spline sections along the x -axis and 2m numbers of strips along the y-axis is formed (see Fig. 7). With the conventional thin-plate theory, the elements of the stiﬀness matrix for a strip with thickness t can be written as[4] ki,j = BiT DB j dydx, (32)

Convergence and exact solutions of spline ﬁnite strip method

1415

Fig. 5

Simply-supported plate is divided into m spline ﬁnite strips and n number of spline sections. Each ﬁnite strip has two nodal lines 1 and 2

Fig. 6

Hypothetical panels are appended to the original plate along two orthogonal directions

Fig. 7

Four contiguous panels are divided into 2m strips across the width and 2n spline sections along span

where ⎤ d2 N (y) φ (x) i ⎥ ⎢ dy 2 ⎥ ⎢ 2 ⎥ ⎢ d φ (x) i ⎥, ⎢ Bi = ⎢ −N (y) ⎥ dx2 ⎥ ⎢ ⎣ dN (y) dφ (x) ⎦ i 2 dy dx ⎡

and the isotropic material matrix D is given by ⎤ ⎡ ⎡ 1 ν 0 D11 Et3 ⎦ = ⎣ D21 ⎣ ν 1 0 D= 12(1 − ν 2 ) 0 0 1−ν 0 2

(33)

D12 D22 0

⎤ 0 0 ⎦. D33

In Eq. (33), the cubic beam shape functions are denoted by the row vector N (y).

(34)

1416

J. KONG and D. THUNG

Explicitly, ki,j can be written as ki,j = D11 I22

L

φi (x)φj (x)dx + D12 I20

0

+ D21 I02

L

0

+ 4D33 I11

L

0

φi (x)

d2 φi (x) φj (x)dx + D22 I00 dx2

L

0

L

0

d2 φj (x) dx dx2

d2 φi (x) d2 φj (x) dx dx2 dx2

dφi (x) dφj (x) dx, dx dx

(35)

where Iαβ =

d

0

dα T dβ N (y) N (y)dy, dy α dy β

(36)

and L = 2l, and d is the width of each ﬁnite strip. Explicit forms for the strip stiﬀness matrix are obtained using the software Mathematica. Similar to the previous one-dimensional beam problem, the strip stiﬀness matrix equation with inherent sparsity is given by ki,i−3 δi−3 + ki,i−2 δi−2 + ki,i−1 δi−1 + ki,i δi + ki,i+1 δi+1 + ki,i+2 δi+2 + ki,i+3 δi+3 = fi ,

(37)

where ⎧ 2416D11I22 h 240(D12I21 + D21 I12 ) 96D22 I00 960D33I11 ⎪ ⎪ − + , + ⎪ ki,i = 3 ⎪ 5040 360h 36h 360h ⎪ ⎪ ⎪ ⎪ ⎪ 1191D11 I22 h 45(D12 I21 + D21 I12 ) 54D22 I00 180D33 I11 ⎪ ⎪ + − = ki,i−1 , − ⎨ ki,i+1 = 3 5040 360h 36h 360h (38) ⎪ 120D11 I22 h 72(D12 I21 + D21 I12 ) 0D22 I00 288D33 I11 ⎪ ⎪ k − + = ki,i−2 , = − ⎪ ⎪ ⎪ i,i+2 5040 360h 36h3 360h ⎪ ⎪ ⎪ ⎪ D11 I22 h 3(D12 I21 + D21 I12 ) 6D22 I00 12D33 I11 ⎪ ⎩k + + = ki,i−3 , − i,i+3 = 3 5040 360h 36h 360h   and ki,j = 0, if i − j   4. In Eq. (37), the strip displacement variable vector at the spline knot i is denoted by δi = [w1 , θ1 , w2 , θ2 ]T i . Each ki,j of Eq. (35) is a 4×4 matrix, and can be partitioned according to nodal lines 1 and 2 as   ka kb ki,j = . (39) kd kc i,j The global stiﬀness matrix of the periodic plate system with 2m strips can be assembled in the usual manner with elements denoted by Ki,j as follows: ⎡

Ki,j

⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣

ka + kc

kb

0

···

kd 0 .. .

ka + kc kd .. .

kb ka + kc

0 kb

kb

0

···

kd

where Ki,j = 0 if |i − j|  4.

··· ···

kd .. . 0 .. . ka + kc

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

,

i,j

(40)

Convergence and exact solutions of spline ﬁnite strip method

1417

Corresponding to Ki,j , ζj denotes the nodal line variable vector at the spline knot j, that is, ζj = [w1 , θ1 , w2 , θ2 , · · · , wk , θk , · · · , w2m , θ2m ]T j .

(41)

As the system is periodic in both directions, the U-transformation is applied successively along the span and across the width, or in essence, a double U-transformation is applied as 

wk θk

 j

  M N   1 i(j−1)rϕ i(k−1)sτ q1 1 √ √ = e , e q2 rs M N r=1 s=1

(42)

π , N = 2n, and M = 2m. where ϕ = πn , τ = m It can be shown that the global stiﬀness matrix can be diagonalized as

krs =

M N  

K1j ei(j−1)rϕ ei(k−1)sτ

j=1 k=1

=

M 

(K1,1 ei(1−1)rϕ + K1,2 ei(2−1)rϕ + K1,3 ei(3−1)rϕ + K1,4 ei(4−1)rϕ + K1,N ei(N −1)rϕ

k=1

+ K1,N −1 ei(N −2)rϕ + K1,N −2 ei(N −3)rϕ )ei(k−1)sτ =

M 

(K1,1 + K1,2 eirϕ + K1,3 e2irϕ + K1,4 e3irϕ + K1,N ei(−1)rϕ + K1,N −1 ei(−2)rϕ

k=1

+ K1,N −2 ei(−3)rϕ )ei(k−1)sτ =

M 

(K1,1 + 2K1,2 cos(rϕ) + 2K1,3 cos(2rϕ) + 2K1,4 cos(3rϕ))ei(k−1)sτ

k=1

= ((ka + kc )1,1 + 2(ka + kc )1,2 cos(rϕ) + 2(ka + kc )1,3 cos(2rϕ) + 2(ka + kc )1,4 cos(3rϕ)) · ei(1−1)sτ + ((kb )1,1 + 2(kb )1,2 cos(rϕ) + 2(kb )1,3 cos(2rϕ) + 2(kb )1,4 cos(3rϕ))ei(2−1)sτ + ((kd )1,1 + 2(kd )1,2 cos(rϕ) + 2(kd )1,3 cos(2rϕ) + 2(kd )1,4 cos(3rϕ))ei(M−1)sτ = (kra + krc ) + krb eisτ + krd e−isτ ,

(43)

where ⎧ kra + krc = (ka + kc )1,1 + 2(ka + kc )1,2 cos(rϕ) + 2(ka + kc )1,3 cos(2rϕ) ⎪ ⎪ ⎪ ⎨ + 2(ka + kc )1,4 cos(3rϕ), ⎪ krb = (kb )1,1 + 2(kb )1,2 cos(rϕ) + 2(kb )1,3 cos(2rϕ) + 2(kb )1,4 cos(3rϕ), ⎪ ⎪ ⎩ krd = (kd )1,1 + 2(kd )1,2 cos(rϕ) + 2(kd )1,3 cos(2rϕ) + 2(kd )1,4 cos(3rϕ).

(44)

Assume that the original plate is subject to a uniformly distributed load p over the entire surface. Then, the strip load vector is given by fi = p

0

d

0

L

φi (x)N T (y)dxdy.

(45)

Assembling for the periodic plate system, we obtain components of the global load vector Fk,1 = 0 = Fk,n+1 = Fk,N +1 ,

(46)

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J. KONG and D. THUNG

Fk,2 = Fk,n = −Fk,n+2 = −Fk,N

⎧ 22  d2 T ⎪ ⎪ ⎪ hp 0, if k = 1, ⎪ ⎪ 24 6 ⎪ ⎪  ⎪ 2 T ⎪ ⎪ ⎨ 22 hp 0, − d if k = m + 1, 6 = 24 ⎪ 22 ⎪ ⎪ hp[d, 0]T if k = 2, 3, · · · , m, ⎪ ⎪ 24 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩− 22 hp[d, 0]T if k = m + 2, m + 3, · · · , 2m. 24

(47)

Otherwise,

Fk,j

⎧  d2 T ⎪ ⎪ hp 0, if k = 1, ⎪ ⎪ 6 ⎪ ⎪ ⎪   ⎨ d2 T if k = m + 1, = hp 0, − 6 ⎪ ⎪ ⎪ ⎪ hp[d, 0]T if k = 2, 3, · · · , m, ⎪ ⎪ ⎪ ⎩ −hp[d, 0]T if k = m + 2, m + 3, · · · , 2m.

(48)

In Eqs. (46)–(48), Fk,j represents the load vector components on the nodal line k and the spline knot j (see Fig. 5). Applying the U-transformation to the load vector yields frs

N M 1   −i(j−1)rϕ −i(k−1)sτ 1 = √ √ e e Fk,j N M j=1 k=1 ⎧ ⎪ ⎪ ⎪

2hp −i sin(rϕ) 1 ⎪ hp 1 ⎪ ⎨√ +√ (i sin(rϕ)) √ = N 1 − cos(rϕ) N3 M ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0, otherwise.

⎤ −2i sin(sτ ) ⎢ 1 − cos(sτ ) d⎥ ⎢ ⎥ ⎣ ⎦ 2d2 6

if

r, s = odd,

(49)

Using the transformed stiﬀness matrix in Eqs. (43) and (44) and the transformed load vector in Eq. (49), we can write   q ((kra + krc ) + krb eisϕ + krd e−isϕ ) 1 q2 rs ⎡ ⎤ −2i sin(sτ )   d hp 1 1 ⎢ 1 − cos(sτ ) ⎥ 2hp −i sin(rϕ) ⎥, +√ i sin(rϕ) √ ⎢ = √ (50) ⎦ N 1 − cos(rϕ) N3 M ⎣ 2d2 6 where solutions q1 and q2 can be obtained in a closed form. Assume that both m and n are even numbers. The deﬂection variables at the center of the plate become         w m2 +1 w m2 +1 w m2 +1 w = + + θ x= l ,y= H θ m2 +1 n θ m2 +1 n +1 θ m2 +1 n +2 2

2

2

=−

M−1 1  3 s=1,3,5,···

2

N −1 

2

1 1 sπ rπ √ √ sin sin (2 + cos(rϕ)) 2 2 N M r=1,3,5,···



 q1 . q2

(51)

Convergence and exact solutions of spline ﬁnite strip method

1419

It can be shown that (the details are given in Appendix A), wx= l ,y= H = − 2

2

n−1 m−1  1 1 sπ 4  rπ √ √ sin sin (2 + cos(rϕ))q1 , 3 s=1,3,5,··· r=1,3,5,··· N M 2 2

(52)

and θx= l ,y= H = 0. 2

(53)

2

By expanding q1 and q2 into Taylor’s series and considering only the ﬁrst three terms, we obtain the deﬂection at the center w=

 n−1 m−1   192(1 − ν 2 )p  sπ 1 rπ sin sin  r2 Eπ 6 t3 2 2 rs l2 + s=1,3,5,··· r=1,3,5,··· +

  s2 2 H2



s 2 r2 s2 4 r 2 r 2 s2 4 d 2 h + 2 + + 2 l2 H2 l l2 H2 + Hs 2 )4 HL H

p(1 − ν 2 )( rl )( Hs ) 2

15Eπ 2 t3 ( rl2

+ higher order terms.

(54)

The ﬁrst term corresponds to the classical Navier’s solution of thin plates[13] , while the second error terms reveal that the deﬂection converges at the asymptotic rates of h4 and d4 as n approaches inﬁnity. As previously described in Section 3, by considering a general load q(x, y) acting on an arbitrary spline section of a ﬁnite strip, the same linear periodic system can be formed by imposing the corresponding, anti-symmetrical loads on the contiguous plate panels. Then, the load vectors can be formed by integrating the strip shape functions and spline functions involved and subsequently transformed as in Eq. (49). The exact deﬂection solution can be obtained following the same approach as given in Eqs. (49)– (54). By the principle of superposition, the exact solution for any arbitrarily distributed load acting on the plate can be found. Similar to vibration problems, the strip mass matrices can be easily obtained as follows: mi,j = ρt N T (y)N (y)φi (x)φj (x)dydx (55) with the inherent sparsity as the stiﬀness matrix, that is, ⎧ 2416I00 ρt ⎪ , ⎪ mi,i = ⎪ ⎪ 5040 ⎪ ⎪ ⎪ ⎪ 1191I00 ρt ⎪ ⎪ = mi,i−1 , ⎨ mi,i+1 = 5040 120I00 ρt ⎪ ⎪ ⎪ = mi,i−2 , = m ⎪ ⎪ i,i+2 5040 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ mi,i+3 = I00 ρt = mi,i−3 , 5040

(56)

which can be partitioned as  mi,j =

ma md

mb mc

 . i,j

(57)

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J. KONG and D. THUNG

Similar to Eq. (43), it can be shown that the global mass matrix can be diagonalized as follows: mrs =

M N  

M1j ei(j−1)rϕ ei(k−1)sτ = (mra + mrc ) + mrb eisτ + mrd e−isτ ,

(58)

j=1 k=1

where mra + mrc = (ma + mc )1,1 + 2(ma + mc )1,2 cos(rϕ) + 2(ma + mc )1,3 cos(2rϕ) + 2(ma + mc )1,4 cos(3rϕ), mrb = (mb )1,1 + 2(mb )1,2 cos(rϕ) + 2(mb )1,3 cos(2rϕ) + 2(mb )1,4 cos(3rϕ), mrd = (md )1,1 + 2(md )1,2 cos(rϕ) + 2(md )1,3 cos(2rϕ) + 2(md )1,4 cos(3rϕ). From Eqs. (43), (44), and (58), the frequencies and mode shapes can be found by solving the 2×2 eigenvalue problem:       q1 (kra + krc ) + krb eisϕ + krd e−isϕ − ω 2 (mra + mrc ) + mrb eisτ + mrd e−isτ = 0. (59) q2 rs The eigenvalue can be found directly by solving the quadratic equation in ω 2 , and expanding the root into Taylor’s series, we obtain π 4 D r2 s2 2 D

πs 8 4 πr 8 4 ω2 = (60) + + d + h + higher order terms. ρt H 2 l2 720ρt l H For a plate subject to the in-plane uniform compression λσx along the x -axis, the geometric matrix is given by gi,j = λσx N T (y)N (y)φi (x)φj (x)dydx (61) with the same inherent sparsity ⎧ 240σx I00 −45σx I00 ⎪ ⎨ gi,i = , gi,i+1 = = gi,i−1 , 360h 360h (62) ⎪ ⎩ gi,i+2 = −72σx I00 = gi,i−2 , gi,i+3 = −3I00 = gi,i−3 . 360h 360h Following the same approach as for vibration problems, the geometric matrix can be diagonalized using the double U-transformation. The resulting 2×2 eigenvalue problem can be directly solved for the buckling load factor λ

sπ 8

rπ 8 π 4 D r2 s2 2 D 4 λσx = rπ 2 + + d + h4 2 ( H ) H2 l2 720( rπ l H H) + higher order terms.

(63)

Apparently, from the ﬁrst term of Eqs. (60) and (63), the spline ﬁnite strip solution does converge to the analytical frequencies and the buckling load, respectively[13] . The second term of these equations reveals the error of spline strip solutions and its rate of convergence. By comparing the spline ﬁnite strip results in Eqs. (54), (60), and (63) with the convergence rates of the deﬂection, frequency, and buckling load of the well-known, non-conforming, rectangular plate element with 12 degrees of freedom[1] , it is now clear that the former converges at a much faster rate of m−4 (assume that m = n) with fewer unknowns, while the latter only converges at the rate of m−2 . The same conclusion on the convergence rate can also be drawn when compared with the ﬁnite diﬀerence method[12] ; the latter only converges at the same asymptotic rate as the non-conforming ﬁnite elements.

Convergence and exact solutions of spline ﬁnite strip method

5

1421

Conclusions

Exact solutions of the spline ﬁnite strips in bending, vibration, and buckling analyses of thin plates are derived using the U-transformation. As the number of spline sections and the number of strips approach inﬁnity, the converged solutions agree with the analytical solutions. Explicit forms of error terms and rates of convergence are also derived and compared with ﬁnite elements and ﬁnite diﬀerence methods. It is found that the spline FSM converges at a much faster rate than that of the conventional non-conforming ﬁnite elements, yet involving a smaller number of unknowns. The same result also applies to the convergence in comparison with the conventional ﬁnite diﬀerence method. The method is currently extended to general folded plate structures with higher-order plate theories and arbitrarily distributed loads. Acknowledgements

The authors would like to thank Mr W. Y. Lee for his assistance.

References [1] Chan, H. C., Cai, C. W., and Cheung, Y. K. Exact Analysis of Structures with Periodicity Using U-Transformation, World Scientiﬁc, Singapore (1998) [2] Cai, C. W., Liu, J. K., and Chan, H. C. Exact Analysis of Bi-Periodic Structures, World Scientiﬁc, Singapore (2002) [3] Yang, Y., Liu, J. K., and Cai, C. W. Analytical solutions to stress concentration problem in plates containing rectangular hole under biaxial tensions. Acta Mechanica Solida Sinica, 21(5), 411–419 (2008) [4] Cheung, Y. K. Finite strip method in the analysis of elastic plates with two opposite ends simply supported. Proceedings of the Institution of Civil Engineers, 40, 1–7 (1968) [5] Cheung, Y. K. and Tham, L. G. Finite Strip Method, CRC Press, Boca Raton, Florida (1998) [6] Loo, Y. C. and Cusens, A. R. The Finite Strip Method in Bridge Engineering, Cement and Concrete Association, Slough (1978) [7] Cheung, M. S., Li, W., and Chidiac, S. E. Finite Strip Analysis of Bridges, E & FN Spon, London/ New York (1996) [8] Friedrich, R. Finite strip method: 30 years a bibliography (1968–1998). Engineering Computations, 17(1), 92–111 (2000) [9] Smith, S. S. and Allen, M. B. Error analysis of the ﬁnite strip method for parabolic equations. Journal of Numerical Methods for Partial Diﬀerential Equations, 9(6), 667–690 (1993) [10] Li, Y. The U-Transformation and the Hamiltonian Techniques for the Finite Strip Method, Ph. D. dissertation, the University of Hong Kong (1996) [11] Yang, Y., Cai, C. W., and Liu, J. K. Convergence studies on static and dynamic analysis of beams using the U-transformation method and ﬁnite diﬀerence method. Journal of Structural Engineering and Mechanics, 31(4), 383–392 (2009) [12] Liu, J. K., Yang, Y., and Cai, C. W. Convergence studies on static and dynamic analysis of plates using the U-transformation method and ﬁnite diﬀerence method. Journal of Sound and Vibration, 289, 66–76 (2006) [13] Szilard, R. Theories and Applications of Plate Analysis: Classical, Numerical, and Engineering Methods, John Wiley & Sons, New York (2004) Appendix A Let f (r, s) and g(r, s) be 8 rπ sin < f (r, s) = sin 2 : g(r, s) = sin rπ sin 2

sπ (2 + cos(rϕ))q1 , 2 sπ (2 + cos(rϕ))q2 . 2

(A1)

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J. KONG and D. THUNG

If n and m are even, then it can be shown that 8 > < f (2n − r, s) = f (r, s), f (r, 2m − s) = f (r, s), > : f (2n − r, 2m − s) = f (r, s), and

8 > < g(2n − r, s) = f (r, s), g(r, 2m − s) = −g(r, s), > : g(2n − r, 2m − s) = −g(r, s).

(A2)

(A3)

As such, wx= l ,y= H 2

=−

1 3

2

M −1 X

N−1 X

sπ 1 1 rπ √ √ sin sin (2 + cos(rϕ))q1 2 2 N M

s=1,3,5,··· r=1,3,5,···

1 =− √ 3 NM 1 =− √ 3 NM 1 =− √ 3 NM 4 =− √ 3 NM

M −1 X

N−1 X

f (r, s)

s=1,3,5,··· r=1,3,5,··· m−1 X

n−1 X

(f (r, s) + f (2n − r, s) + f (r, 2m − s) + f (2n − r, 2m − s))

s=1,3,5,··· r=1,3,5,··· m−1 X

n−1 X

(f (r, s) + f (r, s) + f (r, s) + f (r, s))

s=1,3,5,··· r=1,3,5,··· m−1 X

n−1 X

f (r, s),

(A4)

s=1,3,5,··· r=1,3,5,···

θx= l ,y= H 2

1 =− 3

2

M −1 X

s=1,3,5,··· r=1,3,5,···

1 =− √ 3 NM 1 =− √ 3 NM 1 =− √ 3 NM = 0.

N−1 X

M −1 X

rπ sπ 1 1 √ √ sin sin (2 + cos(rϕ))q2 2 2 N M N−1 X

g(r, s)

s=1,3,5,··· r=1,3,5,··· m−1 X

n−1 X

(g(r, s) + g(2n − r, s) + g(r, 2m − s) + g(2n − r, 2m − s))

s=1,3,5,··· r=1,3,5,··· m−1 X

n−1 X

(g(r, s) + g(r, s) − g(r, s) − g(r, s))

s=1,3,5,··· r=1,3,5,···

(A5)

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Author's personal copy ARTICLE IN PRESS Journal of Sound and Vibration 326 (2009) 671–686

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Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi

Vibration of isotropic and composite plates using computed shape function and its application to elastic support optimization Jackson Kong  Division of Building Science and Technology, City University of Hong Kong, Hong Kong

a r t i c l e in fo

abstract

Article history: Received 8 December 2008 Received in revised form 5 May 2009 Accepted 5 May 2009 Handling Editor: L.G. Tham Available online 13 June 2009

Vibration of plates with various boundary and internal support conditions is analyzed, based on classical thin-plate theory and the Rayleigh–Ritz approach. To satisfy the support conditions, a new set of admissible functions, namely the computed shape functions, is applied to each of the two orthogonal in-plane directions. Similar to conventional ﬁnite element shape functions, parameters associated with each term of the proposed functions represent the actual displacements of the plates, thus making the method easily applicable to a wide range of support conditions, including continuous or partial edge supports and discrete internal supports. The method can also be applied to plates consisting of rectangular segments, like an L-shape plate, which sub-domains can be formulated using the computed shape functions and subsequently assembled in the usual ﬁnite element manner. Unlike many other admissible functions proposed in the literature, however, the computed shape functions presented herein are C1—continuous and involve no complicated mathematical functions; they can be easily computed a priori by means of a continuous beam computer program and only the conventional third-order beam shape functions are involved in subsequent formulation. In all the examples given herein, only a few terms of these functions are sufﬁcient to obtain accurate frequencies, thus demonstrating its computational effectiveness and accuracy. The method is further extended to the study of optimal location and stiffness of discrete elastic supports for maximizing the fundamental frequency of plates. Unlike rigid point supports with inﬁnite stiffness, which optimal locations have been studied by many researchers, only discrete supports with a ﬁnite stiffness is considered in this paper. The optimal location and stiffness of discrete supports are determined for isotropic plates and laminated plates with various stacking sequences, which results are presented for the ﬁrst time in literature. & 2009 Elsevier Ltd. All rights reserved.

1. Introduction Vibration of isotropic and composite plates with various support conditions have been studied intensively in the past few decades, based on various plate theories, analytical and numerical methods. To date, numerical methods applied in this area can be classiﬁed into three major categories, namely (1) the ﬁnite element method including the recent meshless methods [1]; (2) the ﬁnite strip method and (3) the Rayleigh–Ritz approach using various types of admissible functions that satisfy the boundary and internal support conditions a priori. For plates without awkward geometry and relatively

 Tel.: +852 2788 9005; fax: 852 2788 9716.

E-mail address: bsjkong@cityu.edu.hk 0022-460X/\$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2009.05.022

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be X

½NðxÞfam gk

for m ¼ 1; 2; . . . ; p; p ¼ be þ 3

(1a)

k¼1

where {am}k denotes the nodal displacements and rotations of beam element k of the mth computed shape function Xm(x). X1(x) and Xp(x) correspond to the two computed shape functions with unit rotation at one end and zero rotation at the other. The remaining computed shape functions correspond to those with unit deﬂection at one node and zero deﬂection at the others. Referring to Fig. 1b and assuming that xn denotes the location of the nth node, the following conditions are satisﬁed by the COMSFUN: For m ¼ 2,3,yp1 X m ðxn Þ ¼ 1 ¼0

if m  1 ¼ n

where n ¼ 1; 2; . . . ; p  2

if m  1an

(1b)

and dX m ðxÞ ¼0 dx

at x ¼ 0

dX 1 ðxÞ ¼ 1; dx dX 1 ðxÞ ¼ 0; dx

x ¼ Lx

(1c)

dX p ðxÞ ¼0 dx

at x ¼ 0;

(1d)

dX p ðxÞ ¼1 dx

at x ¼ Lx ;

(1e)

and

For m ¼ 1 or p

Author's personal copy ARTICLE IN PRESS J. Kong / Journal of Sound and Vibration 326 (2009) 671–686

673

1 0.8

X Z

Lx

0.6 0.4 0.2 0

1

-0.2 Node 1

Y

Node 2

Node 3

Node 4

1.0

4

Node 6

Node 1 Unit Deflection Node 6 Unit Deflection

Node 1 Unit rotation Node 6 Unit rotation

3

Node 5

0.8

2 0.6 1 0.4

0 -1

0.2

-2 0 -3 -4 Node 1

Node 2

Node 3

Node 4

Node 5

-0.2 Node 1

Node 6

Node 2

Node 3

Node 4

Node 5

Node 6

Fig. 1. (a) A plate strip is divided into ﬁve elements for computing eight shape functions; (b) computed shape functions (COMSFUN) corresponding to unit deﬂection at each of the interior nodes; (c) COMSFUN for unit rotation at either end (d) COMSFUN for unit deﬂection at either end.

and X 1 ðxn Þ ¼ 0 ¼ X p ðxn Þ

for all n ¼ 1; . . . ; p  2

(1f)

It is apparent that the COMSFUN only comprises third-order beam functions with C1-continuity. For the sake of clarity, consider only the vertical (out-of-plane) displacement ﬁeld at mid-plane, i.e. w(x, y) of a rectangular plate segment. Using COMSFUN as the trial function in each of the two orthogonal directions, we have:

wðx; yÞ ¼

p X q X m

X m ðxÞY n ðyÞwmn

(2)

n

where Xm(x) and Yn(y) denote the mth COMSFUN and the nth COMSFUN along X- and Y-direction, respectively. p and q are the number of COMSFUN in the respective direction. The displacement parameter associated with the product Xm(x)Yn(y) is denoted by wmn which represents the actual vertical displacement component or its derivatives at the corresponding nodes. To clarify the representation of the displacement parameters, a unit square plate, as shown in Fig. 2, is divided into ﬁve beam segments with eight number of computed shape functions in each direction (two for the unit rotations at the end and six for the unit nodal deﬂections). The displacement parameters wmn of the plate segment in Fig. 2 represents (1) the vertical displacement for all interior nodes; (2) the displacement and its normal derivatives for all edge nodes and (3) the displacement and its normal and cross-derivatives for the four corner nodes. Such physical representations facilitate the direct implementation of point supports at the nodal locations. For application to isotropic thin plates, the vertical displacement ﬁeld as given by Eq. (2) can be directly applied to formulate the stiffness and mass matrices, based on thin-plate theory. In case of laminated plates with general stacking sequences (see Fig. 3), however, in-plane displacements need to be taken into account, as described in the following section.

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J. Kong / Journal of Sound and Vibration 326 (2009) 671–686

Y w xy=w11, w x=w12 w y=w21, w=w22

w x=w13 w=w23

w x=w14 w=w24

w x=w15 w=w25

w x=w16 w=w26

w xy=w18, w x=w17 w=w27 , w y=w28,

w y=w31, w=w32

w=w33

w=w34

w=w35

w=w36

w=w37 w y=w38

w y=w41, w=w42

w=w43

w=w44

w=w45

w=w46

w=w47 w y=w48

w y=w51, w=w52

w=w53

w=w54

w=w55

w=w56

w=w57 w y=w58

w y=w61, w=w62

w=w63

w=w64

w=w65

w=w66

w=w67 w y=w68

w y=w71, w=w72 w xy=w81, X w x=w82

w=w73 w x=w83

w=w74 w x=w84

w=w75 w x=w85

w=w76 w x=w86

w=w77 , w y=w78, w xy=w88, w x=w87

Fig. 2. Correlation between displacement parameters wmn and the actual displacement and its derivatives at the nodes. w ¼ nodal deﬂection; wx ¼ qw=qx and wy ¼ qw=qy represent the rotation about Y- and X- axis, respectively; wxy ¼ q2 w=qxqy denote the nodal cross-derivative (twist).

3. Stiffness and mass matrices By neglecting transverse normal and shear deformation, the displacement ﬁeld of a thin plate with thickness t can be written in terms of three independent displacement variables, that is: 9 @wðx; yÞ > > Uðx; y; zÞ ¼ uðx; yÞ  z > > @x > = qwðx; yÞ Vðx; y; zÞ ¼ vðx; yÞ  z > qy > > > > Wðx; y; zÞ ¼ wðx; yÞ ;

(3)

where U(x, y, z), V(x, y, z) and W(x, y, z) represent the displacement of any point on the plate along the X-, Y- and Z-direction, respectively. Displacements on the mid-plane, i.e. z ¼ 0, are denoted by u(x, y), v(x, y) and w(x, y). Expanding the mid-plane displacements in terms of COMSFUN gives: 9 > X m ðxÞY n ðyÞumn > > > > > m¼1 n¼1 > > > p P q = P vðx; yÞ ¼ X m ðxÞY n ðyÞvmn > m¼1 n¼1 > > > > p P q > P > wðx; yÞ ¼ X m ðxÞY n ðyÞwmn > > ; uðx; yÞ ¼

p P q P

(4)

m¼1 n¼1

where the wmn represents the nodal out-of-plane displacement component or its derivatives, as explained in previous section and Fig. 2. Following the same representation, parameters umn and vmn denote the nodal in-plane displacement components or its derivatives in the respective direction. p and q are the number of COMSFUN in the X- and Y-direction, respectively.

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675

Y

Z

zi+1 zi

x2

Fiber Direction x1

θ

Mid plane z=0

X Fig. 3. A laminated composite plate with ﬁber direction deﬁned by angle y which varies from lamina to lamina. x1 is taken as the ﬁber direction, x2 being transverse to the ﬁber direction in the plane of the lamina. Total thickness of the plate is t. z0 ¼ t/2 (bottom) and znL ¼ t/2 (top) where nL ¼ number of layers. z ¼ 0 deﬁnes the mid-plane. For layer i, z 2 ½zi ; ziþ1 .

Assuming small deformation, the three in-plane strain components ex, ey and gxy can be expressed in terms of the midplane displacements as: 8 9 8 9 > q2 wðx; yÞ > > > quðx; yÞ > > > > > > > > > > > > 8 9 > > > > qx2 q x > > > >  > > > > x > > > > > 2 < = < < q wðx; yÞ > = = qvðx; yÞ y ¼ f g ¼ z (5) 2 q y > > > > qy :g > ; > > > > > > > > > > > > > xy > > quðx; yÞ qvðx; yÞ > > > > > q2 wðx; yÞ > > > > > þ : > > >2 > qy qx ; : qxqy ; Substituting Eq. (4) into Eq. (5) gives: f g ¼

p X q X

(6)

½Bmn fdgmn

m¼1 n¼1

where the strain–displacement matrix [B]mn can be written as: 2 dX 6 mY n; 0; 6 dx 6 6 dY n 6 ½Bmn ¼ 6 ; 0; Xm 6 dy 6 6 4 X dY n ; dX m Y ; m n dy dx

3 d2 X m z Yn 7 7 dx2 7 7 2 d Yn 7 7 zX m dy2 7 7 dX m dY n 7 5 2z dxdy

(7)

and the nodal displacement vector is given by: fdgmn ¼ fumn ; vmn ; wmn gT For a laminated composite plate with various stacking sequence (Fig. 3), it matrices ½Kip and ½Mip of layer i with density ri can be written as 3 2 2 ½Ki1111 . . . ½Mi1111 ... ½Ki11pq 7 6 6 7 6 ... 6 ... ... ... 7 6 6 ... ½Mip ¼ 6 . . . ½Kip ¼ 6 . . . ... 7 . . . ½Kimnrs 7 6 6 5 4 4 i i ... ½Kpqpq ½Kp111 . . . ½Mip111 with

Z Z Z

½Kimnrs ¼ ½Mimnrs ¼

ziþ1 zi

Z Z Z

ziþ1

zi

can be shown that the stiffness and mass ... ...

... ...

...

½Mimnrs

...

...

½BTmn ½Di ½Brs dz dy dx

½UTmn ri ½Urs dz dy dx;

z 2 ½zi ; ziþ1 

3 ½Mi11pq 7 ... 7 7 ... 7 7 5 i ½Mpqpq

(8)

(9)

(10)

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where integration is carried out for z 2 ½zi ; ziþ1  of layer i with density ri (see Fig. 3) and 3 2 dX m Yn 7 0; z 6 XmY n; dx 7 6 6 dY n 7 ½Umn ¼ 6 7 X m Y n ; zX m 7 6 0; dy 5 4 0; 0; Xm Y n

(11)

The material matrix [D]i for layer i in the global coordinate is given by: ½Di ¼ ½Ti ½d½TTi

(12a)

where [d]i is the orthotropic material matrix of layer i in the local coordinate, and [T]i is the corresponding transformation matrix [11], that is: 2 3 0 d11 d12 6 0 7 (12b) ½di ¼ 4 d21 d22 5 0 0 d66 with d11 ¼

E1 ; 1  m12 m21

d12 ¼

m12 E2 ; 1  m12 m21

d22 ¼

E2 ; 1  m12 m21

d66 ¼ G21 ;

m21 ¼ m12 E2 =E1

(12c)

and 2 6 ½Ti ¼ 6 4

cos2 yi

sin2 yi

sin2 yi

cos2 yi

cos yi sin yi

 cos yi sin yi

2 cos yi sin yi

3

7 2 cos yi sin yi 7 5 cos2 yi  sin2 yi

(12d)

With reference to Fig. 3, the material constants E1, E2 represent the longitudinal modulus along the ﬁber direction x1 and its transverse direction x2, respectively. m12 is the Poisson’s ratio and G12 denotes the in-plane shear modulus. Yi is the inclination of ﬁber direction x1 of layer i to the global X-axis. All integration is done analytically using the software Mathematica. Summing up contributions from each layer gives the ﬁnal stiffness matrix and mass matrices. It is noteworthy that the computation of ½Kimnrs involves integration of the COMSFUN in each direction, which procedures are reported in [2–4]. 4. Elastic support stiffness matrix For a vertical point support with zero mass and stiffness k located at (xs, ys), its stiffness matrix can be written as: 2

s1111 6 6 ... 6 ½K s  ¼ k6 6 ... 4 sp111

...

...

...

...

...

smnrs

...

...

s11pq

3

7 ... 7 7 7 ... 7 5

spqpq

¼ k½S

(13)

where each term of the spring stiffness matrix [S] is given by smnrs ¼ X m ðxs ÞY n ðys ÞX r ðxs ÞY s ðys Þ which can be easily obtained by evaluating the respective COMSFUN Xm(x) and Yn(y) at the spring location (xs, ys). In case the support is located directly at a node, its stiffness is directly added to the corresponding diagonal location in the plate stiffness matrix. In this study, only single or multiple supports with equal stiffness is considered, and for the latter, the total support stiffness matrix can be directly added together as: ½K s  ¼ k

no: ofX springs

½Si ¼ k½ST 

i¼1

where no: ofX springs

½Si ¼ ½ST 

i¼1

It is noteworthy that the rank of [Ks] is equal to the number of point supports.

(14)

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5. Optimum support stiffness and location Once the stiffness and mass matrices for plates and elastic point supports are established, the governing equation for the whole system can be written as: ½K p þ K s fdg ¼ o2 ½M p fdg

(15)

where the subscript p and s denote the plate and spring component of the system stiffness matrix and {d} represents the nodal displacement vector of the system. Previous researches reveal that the frequencies of a plate can be signiﬁcantly increased by adding rigid point supports (i.e. supports with inﬁnitely large stiffness) at the optimal locations. According to the Courant–Fisher theorem [7], if m numbers of rigid point supports are added, the maximum frequency that the fundamental frequency o1 can be raised is bounded by the (m+1)th frequency (i.e. om+1) of the original plate. As such, the optimal location of rigid point supports for maximizing the fundamental frequency of a plate is located on the nodal lines of the (m+1)th mode of the original plate, and the optimal location can be found by solving the following optimization problem: max minðo2 ðxs ; ys ÞÞ subject to ðxs ; ys Þ 2 ONL

(16)

where ONL and (xs, ys) represent the nodal lines of the (m+1)th mode of the original plate and the locations of the additional rigid point supports, respectively. The fundamental frequency reaches its maximum value om+1, should the optimal solutions of (16) exist. In practice, however, it is difﬁcult, if not impossible, to construct a support with inﬁnite stiffness. In addition, it was shown that [9,10] the optimal location of elastic point supports varies with their stiffness. As such, in order to raise the frequencies of plates, practicing engineers would be more interested in determining the minimum support stiffness required and the corresponding optimal locations of the supports. Assuming that it is intended to raise the fundamental frequency of a plate to the (m+1)th frequency om+1, of the original plate by adding m numbers of elastic supports with the same stiffness k, one can ﬁrst rearrange Eqs. (14) and (15) as: ½½K p  o2mþ1 M p  þ k½ST fdg ¼ f0g

(17)

which can be considered as an eigenvalue problem for the support stiffness k. Matrices on the left hand side are known quantities for given support locations. The number of positive, real and ﬁnite eigenvalues, if exist, is equal to the rank of [Ks] (i.e. the number of supports, m) and the maximum of which corresponds to the minimum stiffness of the supports required to raise the fundamental frequency to the speciﬁed frequency. As such, to determine the optimal location of supports, one can minimize the support stiffness directly [9]: min maxðkðxs ; ys ÞÞ subject to ðxs ; ys Þ 2 ONL

(18)

Assuming that the nodal line of vibration modes ðONL Þ can be located, both optimization problems as given in (16) and (18) can be solved using direct search optimization tools available on Matlab. Based on the proposed numerical model and the direct minimization of support stiffness, optimal locations of elastic point supports are determined for isotropic and composite laminated plates with various support conditions, as given in the following examples. 5.1. Numerical examples Examples are given below to demonstrate the application of the proposed COMSFUN to the vibration of plates and elastic support optimization of isotropic and composite laminated plates. The ﬁrst two examples involve a square plate with relatively complex boundary conditions that illustrate the accuracy and convergence characteristics of the proposed method by comparing its results with those obtained using commercial ﬁnite element software (SAP and ABAQUS). Both isotropic and composite laminated plates with unsymmetric stacking sequence are given. To demonstrate its versatility, L-shape plates and a square plate with a line crack are presented in Example 3. Optimal locations of elastic point supports for isotropic cantilevered plates are presented in Example 4 and compared with results given in [10] using conventional ﬁnite elements. Extension to isotropic plates with two clamped edges is also presented, followed by composite laminated plates with various stacking sequences in Example 5. Example 1. A square plate of isotropic material, uniform thickness and the following geometric and material properties is analyzed: Young’s modulus ¼ 73:1  109 N=m2 ; density ¼ 2821 kg=m3 ; Poisson’s ratio ¼ 0:3; Thickness ¼ 0:00328 m; length ¼ 0:305 m.

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Convergence of COMSFUN is ﬁrst investigated using a plate with all edges clamped. Results are compared with FEM (using the software SAP) based on a mesh of 900 thin-plate elements, as shown in Table 1. Very good agreement can be observed between the two sets of results; with eleven numbers of COMSFUN in each direction (a total of 121 degrees of freedom), the lowest 10 frequencies can be predicted with only about 1% difference from SAP. To predict the fundamental frequency, only 7  7 numbers of COMSFUN (with 49 degrees of freedom) is sufﬁcient to generate highly accurate result. Based on the aforesaid convergence results, the same square plate is analyzed with different combinations of the following ﬁve types of supports: (1) (2) (3) (4) (5)

free edge (F); clamped edge (C); point supported edge (i.e. a free edge with a vertical, rigid point support at the middle); partially supported edges (P) (a free edge with half of it being free and the other half simply supported); rigid point supports.

In addition, an interesting case of a square plate with a diagonally line support is also analyzed. Frequencies obtained are compared with the analytical results and the Ritz’s results of Young [14] and Abrate [15], respectively. In all cases, all supports are directly imposed at the corresponding nodes. Results in Tables 2, 3 and 6 demonstrate the accuracy of the proposed method; all results of the lowest 12 frequencies compare very favorably with the FEM results. Example 2. To demonstrate the application to plates with membrane-bending coupling, unsymmetric cross-ply composite laminates with two layers of the following orthotropic material are analyzed: E1 ¼ 40; E2 ¼ 1; G12 ¼ 0.2; m12 ¼ 0.25; density ¼ 1.0. (refer to Eq. (12b) for deﬁnition of symbols) Length of plate ¼ 100, thickness of each layer ¼ 0.5; Fiber direction with respect to the global X-axis y: 0/90 Results of the lowest 12 frequencies using the proposed COMSFUN (p ¼ q ¼ 11 is used which amounts to a total of 363 degrees of freedom) are ﬁrst compared with ABAQUS thin-plate ﬁnite element results for two cases: either all edges simply supported or clamped. A very ﬁne ABAQUS mesh with 2500 elements was used to accurately predict the coupling effect. In all cases, very good agreement between the two sets of results are observed from Table 4, although the COMSFUN model only involves less than 5% of the total degrees of freedom of the ABAQUS model. The same plate is analyzed for a more complicated support condition, namely, two adjacent edges being clamped, the third edge being point-supported at the middle and the remaining edge being partially supported (half of it being free and the other half simply supported). In-plane and out-of-plane displacement components are restrained at the point support and the partially supported edge. For the clamped edge, in-plane, out-of-plane displacements and the out-of-plane rotations are restrained. Our results compare very favorably with ABAQUS’s results, as shown in Table 4. Example 3. Having established the accuracy and validity of the proposed method, L-shape cantilever plates with equal or unequal legs are used to demonstrate that the method can be extended to plates comprising rectangular segments. Each of the plates in Fig. 4 is divided into three rectangular segments, with the number of subdivision as shown thereon. For the equal-leg cantilever plate, ﬁve beam elements are used to generate eight COMSFUN in each direction for each identical Table 1 Convergence of the ﬁrst 12 frequencies (rad/s) for a square plate with all edges clamped. Mode

1 2 3 4 5 6 7 8 9 10 11 12

FEM frequencies

COMSFUN frequencies

SAP

7  7a

Differenceb (%)

8  8a

Differenceb (%)

9  9a

Differenceb (%)

11 11a

Differenceb (%)

1948 3967 3967 5813 7111 7148 8835 8835 11 380 11 380 11 687 12 944

1958 4041 4041 5963 7385 7409 9277 9277 12 578

0.55 1.86 1.86 2.59 3.86 3.65 5.01 5.01 10.53

1956 4006 4006 5909 7340 7373 9147 9147 11840 11840 12 252 13 639

0.44 0.97 0.97 1.65 3.23 3.15 3.53 3.53 4.04 4.04 4.83 5.37

1955 3995 3995 5892 7230 7263 9039 9039 11883 11883 12 070 13 582

0.39 0.69 0.69 1.36 1.67 1.62 2.31 2.31 4.42 4.42 3.28 4.93

1955 3988 3988 5881 7166 7199 8979 8979 11 550 11 550 11 977 13 257

0.36 0.52 0.52 1.17 0.77 0.72 1.63 1.63 1.49 1.49 2.48 2.42

For the case with 7  7 terms of COMSFUN, only nine active degrees of freedom remains after applying the support conditions. a Number of terms used for COMSFUN ¼ p  q. b Percentage difference from the SAP results.

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Table 2 First 12 frequencies (rad/s) for a square plate with various supports. Mode

1

2

3

4

5

6

7

8

9

10

11

12

CFFFa SAP COMSFUN Difference (%)

188.20 188.61 0.22

458.85 462.23 0.74

1147.10 1156.47 0.82

1464.80 1477.27 0.85

1663.50 1682.05 1.11

2894.50 2944.22 1.72

3299.60 3329.29 0.90

3434.00 3485.18 1.49

3794.00 3857.87 1.68

4944.70 5051.23 2.15

5163.40 5276.66 2.19

6366.70 6486.91 1.89

CCFFa SAP COMSFUN Difference (%)

375.08 374.28 0.21

1290.30 1290.21 0.01

1437.80 1443.01 0.36

2564.40 2577.46 0.51

3376.80 3400.14 0.69

3534.40 3549.06 0.41

4586.00 4632.95 1.02

4739.20 4799.20 1.27

6516.50 6581.17 0.99

6666.50 6741.94 1.13

6842.60 6971.26 1.88

7704.80 7858.59 2.00

CCCFa SAP COMSFUN Difference (%)

1297.10 1281.49 1.20

2156.40 2156.60 0.01

3428.50 3391.75 1.07

4119.90 4158.16 0.93

4338.50 4340.44 0.04

6231.80 6308.69 1.23

6628.20 6576.41 0.78

7200.30 7315.89 1.61

7548.00 7574.43 0.35

9178.90 9387.99 2.28

9415.10 9563.75 1.58

10 892.00 10 900.81 0.08

CCCPa SAP COMSFUN Difference (%)

1707.90 1713.97 0.36

3256.10 3265.91 0.30

3428.50 3391.75 1.07

4338.50 4340.44 0.04

5230.10 5201.12 0.55

6231.80 6308.69 1.23

6846.10 6867.94 0.32

7411.00 7464.72 0.72

8742.80 8905.44 1.86

9178.90 9387.99 2.28

10 421.00 10 574.65 1.47

10 892.00 10 900.81 0.08

CCCSa SAP COMSFUN Difference (%)

1713.40 1718.94 0.32

3284.20 3282.82 0.04

3767.90 3770.00 0.06

4728.30 4681.65 0.99

5829.10 5872.64 0.75

6757.80 6782.21 0.36

7125.10 7181.04 0.79

8402.00 8514.94 1.34

8915.90 9086.69 1.92

10 404.00 10 537.75 1.29

11121.00 11 299.78 1.61

11 300.00 11 498.15 1.75

CCCS+PSb SAP COMSFUN Difference (%)

3006.20 3014.19 0.27

3764.40 3765.06 0.02

4135.20 4194.58 1.44

4798.80 4764.40 0.72

6058.40 6100.53 0.70

6784.40 6815.50 0.46

8300.10 8417.28 1.41

8786.30 8949.79 1.86

9886.90 10 192.20 3.09

10 429.00 10 568.96 1.34

11 244.00 11 370.79 1.13

12 068.00 12 221.58 1.27

a C ¼ clamped edge; F ¼ free edge; S ¼ partially supported edge; P ¼ point-supported edge. CCFF ¼ two adjacent edges clamped and the remaining two free. b CCCS+PS ¼ CCCS and a point support at the center of the plate. Number of terms used for COMSFUN ¼ 11 11 in all cases.

Table 3 First 12 frequencies for a square plate with various supports. Mode

1

2

3

4

5

6

7

8

9

10

11

12

CCCP SAP COMSFUN Difference (%)

1707.90 1713.97 0.36

3256.10 3265.91 0.30

3428.50 3391.75 1.07

4338.50 4340.44 0.04

5230.10 5201.12 0.55

6231.80 6308.69 1.23

6846.10 6867.94 0.32

7411.00 7464.72 0.72

8742.80 8905.44 1.86

9178.90 9387.99 2.28

10 421.00 10 574.65 1.47

10 892.00 10 900.81 0.08

CCCS SAP COMSFUN Difference (%)

1713.40 1718.94 0.32

3284.20 3282.82 0.04

3767.90 3770.00 0.06

4728.30 4681.65 0.99

5829.10 5872.64 0.75

6757.80 6782.21 0.36

7125.10 7181.04 0.79

8402.00 8514.94 1.34

8915.90 9086.69 1.92

10 404.00 10 537.75 1.29

11121.00 11 299.78 1.61

11 300.00 11 498.15 1.75

CCCS+Pt Suppa SAP COMSFUN Difference (%)

3006.20 3014.19 0.27

3764.40 3765.06 0.02

4135.20 4194.58 1.44

4798.80 4764.40 0.72

6058.40 6100.53 0.70

6784.40 6815.50 0.46

8300.10 8417.28 1.41

8786.30 8949.79 1.86

9886.90 10 192.20 3.09

10 429.00 10 568.96 1.34

11 244.00 11 370.79 1.13

12 068.00 12 221.58 1.27

 C ¼ clamped edge; S ¼ partially supported edge; P ¼ point-supported edge. CCCS ¼ three adjacent edges clamped and the remaining one partially supported. a CCCS and one point support at the center of the plate. Number of terms used for COMSFUN ¼ 11 11 in all cases.

segment; for the unequal-leg cantilever plate, however, different numbers of COMSFUN are used in each segment provided that compatibility is maintained across the interface. Stiffness and mass matrices for each segment are formed and they are then assembled for the entire plate in the conventional ﬁnite element manner, while maintaining compatibility of deﬂection and slope across the interfaces. As shown in Fig. 4, the plate is clamped on one edge with or without point supports, and the lowest 10 frequencies are compared with ABAQUS using a ﬁne mesh of 1200 elements. In all cases, the COMSFUN results agree closely with the ABAQUS results with less than 1.5% difference, as shown in Table 5. Another interesting problem involves a simply supported plate with a line crack along its center line, see Fig. 5. Due to symmetry, half of the plate is analyzed using COMSFUN. Rotations and vertical deﬂections are restrained along the

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Table 4 First 12 frequencies (  103 rad/s) for an un-symmetric cross-ply square plate with different support conditions. Mode

1

2

3

4

5

6

7

SSSS COMSFUN ABAQUS Difference (%)

1.5749 1.5755 0.04

3.2011 3.2033 0.07

3.2011 3.2033 0.07

4.3665 4.3681 0.04

7.2745 7.2527 0.30

7.2912 7.2692 0.30

7.9529 7.9298 0.29

CCCC COMSFUN ABAQUS Difference (%)

2.3721 2.3737 0.07

4.9204 4.9222 0.04

4.9204 4.9222 0.04

6.6088 6.6081 0.01

9.2768 9.2123 0.70

9.2844 9.2199 0.70

CCSP COMSFUN ABAQUS Difference (%)

1.0451 1.0320 1.21

1.8609 1.8355 1.38

3.7902 3.6969 2.52

4.0741 4.0355 0.96

4.8298 4.7696 1.26

5.5337 5.4720 1.13

8

9

10

11

12

7.9529 7.9298 0.29

10.4500 10.4077 0.41

12.1320 11.9058 1.90

12.1320 11.9058 1.90

12.6562 12.4298 1.82

10.3420 10.2788 0.61

10.3420 10.2788 0.61

13.1287 13.0228 0.81

15.6741 15.0964 3.83

15.6741 15.0964 3.83

16.3738 15.8129 3.55

6.6031 6.4928 1.70

7.0491 6.9176 1.90

8.5697 8.5279 0.49

9.2349 9.1370 1.07

9.9785 9.8978 0.81

10.2045 10.0656 1.38

Number of terms used for COMSFUN ¼ 11 11 in all cases.

a

a

Y

Interface

a

Interface

a

X b

a

Y

c

I

d

III

II

X Fig. 4. (a) An equal-leg L-shape plate divided into three identical square segments. a ¼ 100 units. Continuity of slope and deﬂection are maintained across their interfaces. p ¼ q ¼ 8 for each segment. (b) An unequal leg L-shape plate with three unequal rectangular segments. a ¼ c ¼ 100, b ¼ 160, d ¼ 60. p ¼ q ¼ 8 for segment I, p ¼ 8, q ¼ 13 for segment II, p ¼ 9, q ¼ 8 for segment III. The plates are clamped at X ¼ 0 with or without point supports, which are denoted by X in the ﬁgure. E ¼ 40; m ¼ 0.25; density ¼ 1; thickness ¼ 1.

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Table 5 First 10 frequencies (  103 rad/s) for L-shape cantilevered plates in Fig. 4. Mode

1

2

3

4

5

6

7

8

9

10

0.7331 0.7321 0.13

0.9654 0.9625 0.31

1.3117 1.3057 0.46

2.3450 2.3411 0.17

2.9051 2.9060 0.03

3.7156 3.7062 0.26

4.3150 4.3116 0.08

4.7060 4.6826 0.50

5.0101 5.0032 0.14

Equal-leg cantilever+one point support Fig. 4(a) COMSFUN 0.3074 0.8972 1.1876 ABAQUS 0.3059 0.8949 1.1787 Difference (%) 0.49 0.26 0.75

1.7536 1.7440 0.55

2.9022 2.9020 0.01

3.4933 3.4638 0.85

3.7251 3.7127 0.33

4.5602 4.5368 0.51

4.8398 4.8187 0.44

5.0190 5.0091 0.20

Unequal-leg cantilever Fig. 4(b) COMSFUN 0.2431 ABAQUS 0.2422 Difference (%) 0.41

1.1276 1.1176 0.90

1.6097 1.5922 1.10

2.3402 2.3102 1.30

3.9585 3.9060 1.13

4.4107 4.3627 1.11

4.9566 4.8872 1.42

5.2842 5.2168 1.29

6.0391 5.9524 1.46

Unequal-leg cantilever+six point supports Fig. 4(b) COMSFUN 2.1749 2.7185 3.2968 ABAQUS 2.1643 2.7197 3.2645 Difference (%) 0.49 0.04 0.99

3.3877 3.3624 0.75

4.8249 4.7586 1.39

5.1455 5.0817 1.25

5.8749 5.8059 1.19

6.5667 6.4704 1.49

7.3726 7.2629 1.50

8.6545 8.5354 1.40

Equal-leg cantilever Fig. 4(a) COMSFUN 0.1992 ABAQUS 0.1989 Difference (%) 0.16

0.7583 0.7516 0.88

a

b

Interface

Crack length = a/2

b

Fig. 5. A rectangular plate with a central line crack.

Table 6 First ﬁve frequencies parameters l2 ¼ oa2 (rt/D)1/2 for an isotropic square plate with simply-supported edges and a diagonally line support. Number of terms used for COMSFUN ¼ 11 11. Frequency l2 Mode

COMSFUN

Abrate [15]

Analytical [14]

1 2 3 4 5

50.015 66.432 100.095 121.708 129.886

49.3480 66.020 98.696 122.552 128.305

49.348 65.80 98.696 121.95 128.300

separation line for the symmetric modes and anti-symmetric modes, respectively. In both cases, the crack line remains free. COMSFUN results are compared with analytical results [12,13] and good agreement among the three sets of results can be observed from Table 7. Example 4. The isotropic square plate as given in Example 1 is analyzed again with elastic spring supports. Only one of the edges is fully clamped with the remaining edges free. This plate was previously studied by Wang et al. [10] for support optimization. The ﬁrst three fundamental frequencies obtained using COMSFUN are compared with Wang’s FEM results, and excellent agreement can be observed from Table 8. The second mode of this cantilever plate corresponds to a torsional mode about its axis of symmetry (see Fig. 6). As such, an elastic support is placed at the center of the free edge opposite to the clamped edge, and its minimum stiffness required to raise the fundamental frequency to the second frequency of the original plate (o2) is determined by using the eigenvalue Eq. (17). The minimum stiffness obtained using the COMSFUN

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Table 7 Comparison of normalized frequencies l2 ¼ oa2 (rt/D)1/2 for the ﬁrst three anti-symmetric and symmetric modes of a cracked plate. Modes

COMSFUNa

Yu [13]

Stahl et al. [12]

Difference (%)

A1 A2 A3 S1 S2 S3

72.02 169.61 196.87 41.90 72.72 124.20

73.43 168.6 197.9 40.46 72.78 123.5

73.52 168.6 198 40.32 72.77 123.5

2.04 0.60 0.57 3.92 0.07 0.56

D ¼ Et3/12(1m2). a COMSFUN used: symmetric half of the plate was analyzed with p ¼ 7 along the shorter span and q ¼ 11 along the direction of the cracked center line.

1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0

20

40

60

80

100 100

80

60

40

20

0

Fig. 6. Second mode of a cantilever (CFFF) plate. Table 8 Comparison of frequencies (Hz), minimum support stiffness (ksL2/D) and the corresponding support location (x/L for cantilever 2 or y/L for cantilever 3) for a cantilevered square plate with L ¼ 0.305 m. The plate is clamped at x ¼ 0. D ¼ Et3/12(1m2). Frequency Mode

COMSFUN

Wang [10]

Difference (%)

1 2 3

30.018 73.566 184.058 Support stiffness COMSFUN 24.08951 23.69508 9.32034

30.005 73.555 184.394

0.04 0.02 0.18 Support Location COMSFUN

Problema Cantilever 1 Cantilever 2 Cantilever 3

Wang [10] 23.96060 23.63130 9.32620

0.97071 0.28993

Wang [10] 0.97340 0.28400

a cantilever 1 ¼ point support located at the middle of the free edge (i.e. at x/L ¼ 1, y/L ¼ 0.5); cantilever 2 ¼ support located along the axis of symmetry (i.e. along y/L ¼ 0.5); cantilever 3 ¼ two supports located symmetrically about the center of the free edge (i.e. they are located along x/L ¼ 1). Number of terms used for COMSFUN ¼ 11 11.

model agrees very well with Wang’s FEM model, see Table 8 (cantilever 1). This location, however, does not correspond to the optimal location. On this axis of symmetry, positive eigenvalue of Eq. (17) does not exist for supports located from the clamped edge up to x ¼ 0.1525 m, thus indicating that even a rigid support located within this region is unable to raise the fundamental frequency to o2. A plot of the minimum support stiffness required, from x ¼ 0.25 m up to the free edge, is given in Fig. 7, which clearly shows that the optimum location is located close to the free edge, but not exactly thereon. The ﬁnal optimum location and the corresponding minimum support stiffness are given in Table 8 (cantilever 2), which compare well with Wang’s ﬁndings.

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8 7.8 7.6

Spring Siffness

7.4 7.2 7 6.8 6.6 6.4 6.2 6 0.25

0.26

0.27

0.28

0.29

0.3

Distance along axis of symmetry from x=0.25m to x=0.305m Fig. 7. Variation of minimum spring support stiffness (  104 N/m) along the axis of symmetry of the cantilever plate.

1

0

-1 0 0.1 0.2 0.3

0.3

0.25

0.2

0.15

0.1

0.05

0

Fig. 8. Second mode of a plate with two adjacent edges clamped and the remaining edges free.

2 1.8

Spring Stiffness

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0.22

0.23

0.24

Distance along diagonal Fig. 9. Variation of minimum spring support stiffness (  106 N/m) along the diagonal (xO2) of the CCFF plate.

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Alternatively, to raise the fundamental frequency of the cantilevered plate to o2, two elastic supports are placed symmetrically along the free edge. The optimal locations of the supports are obtained directly by minimizing the support stiffness. Our results compare favorably with those of Wang, as indicated in Table 8 (cantilever 3).

Table 9 Minimum support stiffness (ksL2/D) for simply supported, composite laminates. Antisymmetric angle-ply

Unsymmetric unbalanced laminate

Frequencya

Frequencya

Angle

Mode 1 o1

Mode 2 o2

Support stiffness

Angle

Mode 1 o1

Mode 2 o2

Support stiffness

90/90 75/75 60/60 45/45 30/30 15/15 0/0

1.8543 1.6570 1.7291 1.8389 1.7291 1.6570 1.8543

2.2303 2.3847 3.1285 3.7296 3.1285 2.3847 2.2303

16.1952 31.0737 95.4378 257.1598 95.4378 31.0737 16.1952

0/90 0/75 0/60 0/45 0/30 0/15 0/0

1.5749 1.5745 1.6030 1.6466 1.6500 1.7328 1.8543

3.2011 3.0582 2.8975 2.7319 2.4957 2.2761 2.2303

91.2621 75.6899 63.9214 51.9092 35.9773 22.6669 16.1952

2 ). L ¼ 100; D ¼ E1t3/12(1m12 a First two natural frequencies (  103 rad/s) of the original plate.

10

90

Spring Stiffness

9

80

Frequency difference

8

70

7

60

6

50

5

40

4

30

3

20

2

10

1

0

Non-dimensional frequency

Non-dimensional stiffness

100

0 0

15

30

45

60

75

90

Ply Angle 300

12 Frequency difference

Non-dimensional Stiffness

250

10

200

8

150

6

100

4

50

2

0

Non-dimensional frequency

Spring Stiffness

0 0

15

30

45

60

75

90

Ply angle Fig. 10. Variation of non-dimensional spring support stiffness (ksL2/D) and differences in non-dimensional frequency (o2o1)L2(rt/D)0.5 against the ply 2 ). angle (y in degree) for (a) unbalanced, unsymmetric laminates (0/y); (b) anti-symmetric angle-ply laminates (y/y). L ¼ 100; r ¼ 1, D ¼ E1t3/12(1m12

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The same isotropic square plate is analyzed again with two adjacent edges fully clamped and the remaining edges free. Unlike the previous case, the second mode of this plate corresponds to a torsional mode about its diagonal, as shown in Fig. 8. A plot of the minimum support stiffness along the diagonal indicates that the optimum location can only be found in the region as shown in Fig. 9, beyond which positive eigenvalues of Eq. (17) does not exist. By directly minimizing the support stiffness, the exact location of x ¼ 0.2347 m, y ¼ 0.2347 m on the diagonal and the corresponding minimum spring stiffness of ksL2/D ¼ 241.5082 can be found. In this case, no result is available in the literature for comparison. Example 5. A two-layer, composite laminated square plate with geometry and material properties being the same as those deﬁned in Example 2 is analyzed. In addition to unsymmetric cross-ply laminates, anti-symmetric angle-ply (y/y) and unsymmetric, unbalanced (0/y) laminates are also considered. The plate is simply supported around all edges with both in-plane and out-of-plane displacements being restrained. An elastic spring is imposed at the center of the plate and the objective is to determine its minimum stiffness required to raise the fundamental frequency of the supported plate to the second natural frequency of the original plate. Results for both cases are summarized in Table 9 and Fig. 10, which clearly demonstrates that the variation of minimum spring stiffness required for various ply angles follows closely the corresponding change in frequency (o2o1) required. 6. Conclusions Vibration of isotropic and composite laminated plates with various boundary and internal support conditions is analyzed in a uniﬁed manner using the computed shape function and thin-plate theory. Similar to the conventional ﬁnite element shape functions, parameters associated with each term of the proposed functions represent the actual displacements of the plates, thus making the method easily applicable to a wide range of support conditions, as demonstrated in the examples given. The method is also applied to analyze L-shape plates, which can be sub-divided into rectangular segments and assembled in the usual ﬁnite element manner. It is noteworthy that, unlike other admissible functions proposed in the literature, the computed shape functions presented herein are C1—continuous and involve no complicated mathematical functions or implementation. In all the given examples, only several terms of these functions are sufﬁcient to obtain very accurate results for the lowest 10 frequencies, thus demonstrating its computational effectiveness and rapid convergence. Taking advantage of the computational efﬁciency of the proposed method, it is also applied to determine the optimal location and stiffness of discrete elastic supports in maximizing the fundamental frequency of isotropic plates and composite plates. For isotropic plates with one edge clamped, the minimum stiffness and optimal locations determined are veriﬁed against available results from the literature. In addition, the method was used to analyze plates with two adjacent clamped edges and also simply supported composite plates with various stacking sequences; no results are available in the literature for comparison in these cases. In this study, only examples of plates that allow easy location of the nodal lines are given; the method is currently being extended to general cases where the nodal lines of vibration cannot be easily identiﬁed.

Acknowledgement The work described in this paper was fully supported by a grant from City University of Hong Kong (Project no. 7001774). References [1] Y. Chen, J.D. Lee, A. Eskandarian, Meshless Methods in Solid Mechanics, Springer, New York, NY, 2006. [2] Y.K. Cheung, J. Kong, A new ﬁnite strip for analyzing deep beams and shear walls, Communications in Numerical Methods in Engineering 11 (1995) 643–653. [3] Y.K. Cheung, J. Kong, The application of a new ﬁnite strip to the free vibration of rectangular plates of varying complexity, Journal of Sound and Vibration 181 (2) (1995) 341–353. [4] Y.K. Cheung, J. Kong, Vibration and buckling of thin-walled structures by a new ﬁnite strip, Thin-Walled Structures 21 (4) (1995) 327–343. [5] Y. Xiang, C.M. Wang, S. Kitipornchai, Optimal locations of point supports in plates for maximum fundamental frequency, Structural Optimization 11 (3/ 4) (1996) 170–177. [6] B. Akesson, N. Olhoff, Minimum stiffness of optimally located supports for maximum value of beam eigenfrequencies, Journal of Sound and Vibration 120 (1988) 457–463. [7] R. Courant, D. Hilbert, Methods of Mathematical Physics, Vol. 1, Interscience, New York, 1953. [8] Y. Narita, The effect of point constraints on transverse vibration of cantilever plates, Journal of Sound and Vibration 102 (1985) 305–313. [9] K.M. Won, Y.S. Park, Optimal Support positions for a structure to maximize its fundamental natural frequency, Journal of Sound and Vibration 213 (5) (1998) 801–812. [10] M.I. Friswell, D. Wang, The minimum support stiffness required to raise the fundamental natural frequency of plate structures, Journal of Sound and Vibration 301 (2007) 665–677. [11] J.N. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC Press, Boca Raton, FL, 2004. [12] B. Stahl, L.M. Keer, Vibration and Stability of cracked rectangular plates, International Journal of Solids and Structures 8 (1972) 69–91.

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[13] W.L. Cleghorn, S.D. Yu, J.S. Xu, R.G. Fenton, Free vibration analysis of rectangular plates with a linear crack along the central axis, Proceedings of the ASME Winter Annual Meeting, Chicago, IL, November 5–11, 1994. [14] P.G. Young, S.M. Dickinson, On the free ﬂexural vibration of rectangular plates with straight or curved internal line supports, Journal of Sound and Vibration 162 (1) (1993) 123. [15] S. Abrate, Vibration of composite plates with internal supports, Journal of Mechanical Science 36 (11) (1994) 1027–1043.

International Journal of Solids and Structures 37 (2000) 1483±1499

www.elsevier.com/locate/ijsolstr

Free vibration of thick, layered rectangular plates with point supports by ®nite layer method D. Zhou a, Y.K. Cheung b,*, J. Kong c a

School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210014, P.R. China b Department of Civil and Structural Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong c Maunsell Consultants Asia Ltd, Hong Kong Received 6 June 1998; in revised form 29 October 1998

Abstract In this paper, the free vibration of thick, isotropic and laminated composite rectangular plates with point supports is analyzed by the ®nite layer method. A new set of two-dimensional basic functions, which satis®es the kinematic boundary conditions at the edges of the plate and the zero-displacement conditions at point supports, is developed to describe the variation of three-dimensional displacements in the plane of a thin ®nite layer. Onedimensional linear or quadratic shape functions are adopted to describe the variation of the displacements through the thickness layer. The governing eigenvalue equation of the plate is derived via the conventional displacement method. Numerical results for the three-dimensional vibration of rectangular plates with point supports are presented herein for the ®rst time. The eigenfrequencies of simply-supported rectangular plates with a central pointsupport are studied in detail by considering the variations of aspect ratio, side-to-thickness ratio, properties of materials, number of laminates and stacking sequences. Comparison with known thin-plate and Mindlin-plate solutions is carried out to verify the applicability and accuracy of the proposed method. # 1999 Elsevier Science Ltd. All rights reserved.

1. Introduction The rectangular plate is one of the most commonly used structural elements in civil, aeronautical and marine engineering. The vibration frequencies are important parameters for the dynamic analysis of structures. A close scrutiny among the references on dynamic analysis of structural elements reveals that to date, most investigations are about thin plates (Leissa, 1969), while study on vibration of thick plates has received little attention because of the diculty in expressing the three-dimensional displacement ®eld. The diculty of three-dimensional analysis renders the rapid development of re®ned plate theories * Corresponding author. Fax: +852-2559-5337. 0020-7683/00/\$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 6 8 3 ( 9 8 ) 0 0 3 1 6 - 3

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(Mindlin, 1951; Reddy, 1984; Hanna and Leissa, 1994). Applying these theories, one can reduce the dimension of problems from three to two by taking certain averages of some parametric quantities, such as membrane forces, bending moments and shear forces, over the smaller dimension (thickness). The Mindlin plate theory (Mindlin, 1951) is formulated by introducing the concept of a shear correction factor to account for the in¯uence of shear deformation on the dynamic properties of the plate. Numerical studies using the Mindlin plates theory (Dave, 1978; Liew et al., 1993) can be found in the literature. However, it should be noted that although the analytical accuracy can be improved by using higher-order theories, the local variation of through-thickness displacements of the plate cannot be exactly represented and thus results in errors which increase with the thickness of the plate. For very thick plates, a three-dimensional elasticity theory is necessary to obtain accurate results. Because of the complexity of the problem, closed-form exact solutions exist only for simply-supported thick rectangular plates (Srinivas et al., 1970). In most cases, approximate analytical and/or numerical methods have to be adopted. It is well-known that the ®nite element method is an applicable tool to such a problem. However, it requires discretisation in every dimension of the problem and therefore, will require more unknowns for approximation than some other methods. This certainly results in the increase in cost and the requirement of a super-computer. Cheung and Chakrabarti (1972) used the ®nite layer method and Fan and Sheng (1992) used the analytical method to investigate the free vibration of thick, layered rectangular plates. Leissa and Zhang (1983) and Liew et al. (1993, 1994) used polynomials as trial functions to study the three-dimensional free vibration of isotropic thick rectangular plates by the Rayleigh±Ritz method. In some practical applications, such as ¯oor slabs, bridge decks and solar panels, interior and edge point-supports are often placed at some locations of the plate to limit the displacements and to achieve a better distribution of stresses and/or to satisfy special architectural and functional requirements. The eects of point supports on the dynamic characteristics of plates have been an interesting subject for many researchers. For thin rectangular plates with point supports, some pioneering studies have been carried out by Fan and Cheung (1984) using the ®nite strip method, Mizusawa and Kajita (1987) using the spline element method, Kim and Dickinson (1987) using polynomials as trial functions in the Rayleigh±Ritz method and Gorman and Singal (1991) using the analytical superposition method. Recently, Liew et al. (1994) used a set of pb-2 shape functions to study the free vibration of Mindlin plates with point supports. However, no information is currently available for free vibration of threedimensional thick plates with point supports. The ®nite layer method (Cheung and Tham, 1997) is used in this paper to investigate the free vibration of thick, layered rectangular plates with point supports. A new set of basic functions is constructed in two parts with the ®rst part being a set of static beam functions under sinusoidal loads, while the second is for beams under point loads. These functions are developed to describe variation of the three-dimensional displacements in the plane of a thin ®nite layer, while a one-dimensional linear or quadratic shape function is adopted to describe the variation of the displacements through the thickness of the layer. This set of basic functions satis®es the geometric boundary conditions at the edges of the plate and the zero-displacement conditions at the point supports. A simply-supported rectangular plate with a central point-support is taken as an example of numerical application. The in¯uence of aspect ratio, side-to-thickness ratio and various structural parameters on the eigenfrequencies of plates is examined in detail. Some numerical data are tabulated and compared with other thin-plate and Mindlin-plate results and the accuracy has been con®rmed by convergence studies. 2. Two sets of static beam functions In order to study the free vibration of thick rectangular plates with point supports, two sets of one-

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dimensional static beam functions have to be developed. They will together form a set of twodimensional basic functions representing the variation of displacements in the plane of a thin ®nite layer.

2.1. The static beam functions under a series of sinusoidal loads Consider a beam subject to a series of static sinusoidal loads distributed along the length, the complete solution (Zhou, 1996) may be written as zx 

1 X i1

Qi fi x

1a

fi x  B0i  B1i x  B2i x2  B3i x3  sin ipx

1b

where zx is the de¯ection of the beam, Qi denotes the amplitude of the ith sinusoidal load component, x 0 E x E 1 is the non-dimensional coordinate along the beam and Bji  j  0, 1, 2, 3 are the unknown constants which can be determined by the boundary conditions of the beam (for a beam without rigid body motions) as shown in Table 1. For a beam with rigid body motions, the coecients Bji  j  0, 1, 2, 3 cannot be determined directly by the boundary conditions. In this case, the rigid body modes should be added to the static beam functions. For example, the static beam functions for a F-S beam should be selected as f1 x  1 ÿ x, fi  1 x  B0i  B1i x  B2i x2  B3i x3  sin ipx,

ie1

1c

where Bji ( j = 0, 1, 2, 3) are those for the F-C beam. For an S-F beam, the static beam functions should be selected as f1 x  x, fi1 x  B0i  B1i x  B2i x2  B3i x3  sin ipx, i e 1

1d

Table 1 The coecients of static beam functions under sinusoidal loads Boundary condition

B0i

B1i

B2i

B3i

S-S C-C C-F C-S F-C S-C

0 0 0 0 ÿ1i ipip2 =3  1 0

0 ÿip ÿip ÿip ÿÿ1i ipip2 =2  1 ipÿ1i =2

0 ipÿ1i  2 ÿip3 ÿ1i =2 3ip/2 0 0

0 ipÿ1i  1 ip3 ÿ1i =6 ÿip/2 ip3 ÿ1i =6 ÿipÿ1i =2

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Table 2 The coecients of static beam functions under point-loads Boundary condition

A0k

A1k

A2k

A3k

S-S C-C C-F C-S F-C S-C

0 0 0 0 1 ÿ xk 2 2  xk  0

1 ÿ xk xk 2 ÿ xk  0 0 0 ÿ31 ÿ xk 2 31 ÿ xk 2 xk =2

0 3xk 1 ÿ xk 2 3xk ÿ31 ÿ xk xk xk ÿ 2=2 0 0

ÿ1 ÿ xk  ÿ1 ÿ xk 2 1  2xk  ÿ1 1 ÿ xk x2k ÿ 2xk ÿ 2=2 0 ÿ1 ÿ xk 2 2  xk =2

where Bji  j  0, 1, 2, 3 are those for the C-F beam. Finally for an F-F beam, the static beam functions should be selected as f1 x  1, f2 x  x

or f2 x  1 ÿ x,

fi2 x  B0i  B1i x  B2i x2  B3i x3  sin ipx, i e 1

1e

where Bji  j  0, 1, 2, 3 are those for a C-F beam (or an F-C beam). 2.2. The static beam functions under a series of point-loads Consider a beam acted upon by P static point-loads, the complete solution (Zhou, 1994) may be written as zx 

P X Pk fk x,

2a

k1

3 ÿ fk x  A0k  A1k x  A2k x2  A3k x3  x ÿ xkj Ux ÿ xk 

Fig. 1. A thick rectangular plate with point supports.

2b

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where Ux ÿ xk  is the step function, Pk denotes the magnitude of the kth point-load and Aik i  0, 1, 2, 3 are the unknown constants which can be determined by the boundary conditions of the beam (for a beam without rigid body motions) as shown in Table 2. 3. Finite layer formulation A rectangular thick laminated composite plate with point supports is shown in Fig. 1. The plate is divided into a number of ®nite layers through the thickness. Each individual rectangular layer has two (L02) or three (H03) nodal surfaces. A suitable set of displacement functions is selected as ux, y, z  

ux, y, z  

wx, y, z  

I J X X i1 j1 I J X X i1 j1 I J X X i1 j1

  @ Wij x, y =@ x Nz fagij

3a

  @ Wij x, y =@ y Nz b ij

3b

  Wij x, y  Nz fd gij

3c

where Wij x, y are the basic functions formed by the two sets of static beam functions, Nz denotes the one-dimensional linear (L02) or quadratic (H03) shape functions, I and J are truncated order of the displacement functions. The in-plane displacements along the x- and y-axes are de®ned by ux, y, z and ux, y, z, respectively. The displacement unknowns are denoted by fagij , fbgij and fdgij . Using the above three equations, the strain±displacement relationships are derived as 3 2 @ 2 Wij N  0 0 7 6 7 6 @x 2 7 6 2 7 6 9 8 @ Wij 7 6   N 0 0 7 6 > > e x > > @ y2 7 6 > > > > 9 8 > 6 > 7  > > > ey > 7> fag > 6 > > dN > > > > 7 6 > > > 0 0 Wij > > > > 7> I J 6 < X X ez = dz 7<  = 6 7 b 6    : 4 feg  7> > > i  1 j  16 > gyz > @Wij dN @Wij > > > 7 6 > > > > N  7> 0 > > > 6 > > > > 7: fg g ; 6 @ y dz @y > > g > > xz i, j 7 6 > > > > 7 6 @W  dN  > @Wij ; :g > 7 6 ij xy N  7 0 6 7 6 @ x dz @x 7 6 7 6 @ 2W 2 @ Wij 5 4 ij N  N  0 @ x @y @x @ y The stress±strain relationships are fsg  fsx

sy

sz

tyz

txz

txy gT  D feg

5

where D is the property matrix for materials as given in Appendix A. Applying the well-known

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Fig. 2. The relations between point-loaded beams and the point-supported plate.

displacement method, the global stiness and mass matrices can be easily formed by assembling the layer stiness and mass matrices, as given in Appendix B, for each individual ®nite layer. Finally, eigenfrequencies and corresponding mode shapes can be extracted using standard procedures of eigenvalue analysis.

4. Basic functions The basic functions Wij x, y which describe the variation of three-dimensional displacements of each ®nite layer in the x±y coordinate plate, must satisfy the prescribed geometrical boundary conditions of the plate, including the zero-displacement conditions at point supports. It is obvious that the conventional admissible functions such as the vibrating beam functions cannot be directly applied to this problem because of the existence of point supports. On the other hand, using the continuous beam vibrating functions as proposed by Cheung and Delcourt (1977) required rather lengthy computation and is therefore, inconvenient. Here the basic functions selected comprise of two parts, namely, the ®rst part being the product of the one-dimensional static beam functions fi x, cj Z under sinusoidal loads, while the second part is the product of the one-dimensional static beam function fk x, gk Z under the kth point load. The basic functions can be written as Wij x, Z  fi xcj Z  

P X k1

Rijk fk xgk Z 

6

where x  z=a, Z  y=b and P is the number of the point supports. fi x is the static beam function under the ith sinusoidal load component, as given by eqns (1b±e), which satis®es the corresponding boundary conditions of the plate in the x-direction but disregarding the point supports. fk x is the beam function under the kth point-load, as given by eqn (2b), which satis®es the corresponding boundary conditions of the plate in the x-direction and treating all point supports as point-loads. It should be noted that for beams with rigid body motions as shown in Fig. 2, boundary conditions of the type C-F

D. Zhou et al. / International Journal of Solids and Structures 37 (2000) 1483±1499

1489

or F-C should be used. The same principle applies to functions cj Z and gk Z in the Z-direction. The unknown coecients Rijk k  1, 2, . . . , P  in eqn (6) are determined by the zero-displacement conditions of the plate at the P point-supports, as demonstrated below: P X ÿ  Rijk fk xl gk Zl   0, 7a Fij xl , Zl  fi xl cj Zl   k1

where xl , Zl  is the coordinate of the lth point-support of the plate. Consequently, the following linear simultaneous equations can be obtained 32 3 2 3 2 ÿfi x1 cj Z1  f1 x1 g1 Z1  f2 x1 g2 Z1  . . . fP x1 gP Z1  Rij1 76 7 6 6 7 R 7 7 6 f x g Z 7 6 x c Z ÿf   7 2 6 6 1 2 1  2  f2 x2 g2 Z2  . . . fP x2 gP Z2  76 ij2 i j 2 7 6 76 7 6 6 7 7b  7 7, 6 6 .. .. .. .. .. 7 6 .. 76 7 6 6 7 . . . . . 74 . 5 6 7 6 5 5 4 4 f1 xP g1 ZP  f2 xP g2 ZP  . . . fP xP gP ZP  RijP ÿfi xP cj ZP  for every pair of i, j. The solution of the above equation may be easily obtained as follows 2

Rij1

3

2

f1 x1 g1 Z1 

f2 x1 g2 Z1 

7 6 6 6 Rij2 7 6 f1 x2 g1 Z2  f2 x2 g2 Z2  7 6 6 76 6 6 .. .. 6 .. 7 6 6 . 7 6 . . 5 4 4 f1 xP g1 ZP  f2 xP g2 ZP  RijP

...

fP x1 gP Z1 

3ÿ1 2

ÿfi x1 cj Z1 

3

7 6 7 7 6 fP x2 gP Z2  7 7 6 ÿfi x2 cj Z2  7 7 6 7 7 6 7: .. .. .. 7 6 7 . . . 7 6 7 5 4 5 . . . fP xP gP ZP  ÿfi xP cj ZP 

...

8

On closer examination of eqns (3c) and (6), one can easily observe that the out-of-plane displacement wx, y, z vanishes at the point supports across the thickness. This implies that a point support is equivalent to imposing rigid-line constraint to the vertical displacement wx, y, z across the thickness. Obviously, for a plate without point supports (Zhou, 1996), all Rijk k  1, 2, . . . , P  are equal to zero. Furthermore, because the coecient matrix of Rijk k  1, 2, . . . , P  is independent of the summing variables i and j, only one inverse calculation to the coecient matrix in eqn (10) is required when solving the coecients Rijk k  1, 2, . . . , P  for all i and j. As a result, the computational cost is greatly reduced.

5. Numerical studies The ®nite layer method developed inp previous sections is applied to compute the non-dimensional 2 1=2 rt= D , for thick, laminated rectangular plates and l  frequency parameters, l  o b=2 11 D22  p   o b2 rt=D for isotropic thin plates, where o is the circular frequency D11  E1 t3 =121 ÿ v12 v21  and D22  E2 t3 =121 ÿ v12 v21 . It is obvious that D11  D22  D  Et3 =121 ÿ v2  for isotropic plates. For laminated plates, the properties of material (Noor, 1973) are taken as follows: E1 =E2  40; G12 =E2  0:6; G23 =E2  0:5; G13  G23 ; v12  v13  0:25. The ®bre orientations of dierent laminae alternate between 0 and 908 with respect to the x-axis. The in¯uences of the plate aspect ratios l  a=b, side-tothickness ratios t=b, stacking sequences for laminated plates are examined. These results are, to the best of the author's knowledge, presented for the ®rst time in open literature. It is noteworthy that for simply-supported plates with a central point-support, the vibration modes can be classi®ed into four distinct categories, namely, double symmetric (SS) modes, symmetric±antisymmetric modes (SA),

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antisymmetric±symmetric modes (AS) and double antisymmetric modes (AA). Each of these categories is separately determined and thus, results in a smaller set of eigenfrequency equations. However, since the eigenfrequencies of the SA, AS and AA modes in this case are just the same as those of plates without the point support (Cheung and Chakrabarti, 1972; Liew et al., 1993), only the SS modes are computed. In the following examples, all ®nite layers are taken to be the same thickness and 24 Gaussian integration points are used for the integral computations in the x±y plane. For isotropic plates, v  0:3 is assumed. 5.1. Convergence and comparison The ®nite layer approach gives an upper-bound solution to the exact value. A convergence study is ®rst carried out so as to ensure that the solutions to the problem are convergent and to establish the required number of terms in the three-dimensional displacement functions for obtaining satisfactory accuracy. In Table 3, convergence patterns of the ®rst eight eigenfrequencies of the symmetric± symmetric mode for an isotropic homogeneous, simply-supported square plate with a central pointsupport are presented. It can be seen that the eigenfrequencies converge monotonically from above as the number of terms of basic functions and the number of L02 layers increase. A careful scrutiny of the convergence table reveals that the terms of the basic functions in the x±y plane play a more dominant role in the convergence and accuracy than the number of the layers in the z-direction both for thin and thick plates. The convergence rate for the thin plate t=b  0:01 is slightly faster than that for the thick plate t=b  0:02. In general, the comparison of the present results with those (Kim and Dickinson, 1987) obtained by the thin plate theory for the thin plate t=b  0:01 are better than those (Liew et al., 1994) obtained by the Mindlin plate theory for the thicker plate t=b  0:02. However, the dierence is rather small and the maximum error is less than 2.1% for all cases. Moreover, from the table it is shown that using higher-order interpolation functions in the thickness direction of the layer can further improve the computational accuracy. In Table 4, a comparative study of the ®rst ®ve eigenfrequencies of thin square plates t=b  0:01 with a corner-support (where symmetry does not exist) and with four corner-supports (where symmetry exists but not utilized in the computations) are given. The terms of displacement functions in the x- and y-directions and the number of L02 layers in the z-direction are taken as 5  5  5. Comparison of the present results with those obtained by the thin plate theory (Mizusawa and Kajita, 1987; Kim and Dickinson, 1987) shows that good agreement is observed for all cases. 5.2. Numerical examples From the convergence studies, it is found that the 7  7 terms of the displacement functions in the x± y plane and ®ve L02 ®nite layers in the z-direction are sucient to obtain satisfactory results for both thin plates and thick plates and they are used throughout the following computations. The non-dimensional eigenfrequencies of symmetric±symmetric modes for the simply-supported isotropic rectangular plate with a central point-support are given in Table 5. The in¯uences of aspect ratio and side-to-thickness ratio on the eigenfrequencies are studied. It is observed that for a plate with a prescribed aspect ratio, the non-dimensional eigenfrequencies, l, decrease as the side-to-thickness ratio, t/b, increases, especially for the higher modes. Conversely, for a plate with a prescribed thickness ratio, the non-dimensional eigenfrequencies decrease as the aspect ratio, a/b, increases. The second set of results is for a skew-symmetric rectangular laminate with a central point-support. It consists of two plies with 0/908 stacking sequences. In this case, symmetry of vibrating modes of the

D. Zhou et al. / International Journal of Solids and Structures 37 (2000) 1483±1499

1491

Table 3 p Convergence study of non-dimensional eigenfrequencies, l  o b=22 rt=D for isotropic homogeneous square thick plates with simply-supported edges and a central point-support Thickness ratio t/b

Terms in x, y, z

Mode number SS-1

SS-2

SS-3

SS-4

SS-5

SS-6

SS-7

SS-8

0.01

444 445 554 555 664 665 666 775 Kim and Dickinson (1987)

13.77 13.74 13.65 13.62 13.56 13.53 13.52 13.38 13.29

24.80 24.74 24.80 24.74 24.80 24.74 24.71 24.74 24.67

37.83 37.74 37.60 37.52 37.45 37.36 37.32 37.25 37.05

53.41 53.29 53.80 52.96 52.87 52.75 52.68 52.60

64.32 64.15 64.32 64.15 64.32 64.15 64.06 64.15

75.85 75.66 75.50 75.32 75.29 75.11 75.01 74.96

83.99 83.78 83.99 83.78 83.99 83.78 83.66 83.78

106.2 106.0 104.9 104.6 104.1 103.9 103.7 103.3

0.1

444 445 554 555 664 665 666 775 7  7  3a Liew et al. (1994)

11.88 11.85 11.69 11.65 11.54 11.50 11.48 11.39 11.31 11.40

21.58 21.50 21.58 21.50 21.58 21.50 21.46 21.50 21.37 21.26

29.97 29.85 29.62 29.50 29.36 29.24 29.17 29.03 28.82 29.42

36.46 36.46 36.11 36.11 35.88 35.88 35.88 35.71 35.71

40.23 40.07 39.88 39.72 39.63 39.47 39.38 39.28 38.99

45.35 45.35 45.23 45.23 45.15 45.15 45.15 45.10 45.10

48.05 47.86 48.05 47.86 48.05 47.86 47.74 47.86 47.50

53.81 53.57 53.45 53.21 53.19 52.96 52.83 52.76 52.35

0.2

444 445 554 555 664 665 666 775 7  7  3a Liew et al. (1994)

16.72 16.65 16.72 16.65 16.72 16.64 16.60 16.64 16.51 16.29

18.23 18.23 18.05 18.05 17.94 17.94 17.94 17.85 17.85

21.24 21.13 20.96 20.85 20.74 20.64 20.58 20.47 20.29 20.60

22.62 22.62 22.56 22.56 22.52 22.52 22.52 22.49 22.49

27.60 27.45 27.38 27.24 27.24 27.09 27.00 26.98 26.72

32.54 32.35 32.54 32.35 32.49 32.35 32.24 32.35 32.01

32.81 32.80 32.62 32.62 32.54 32.49 32.49 32.40 32.40

a

9.125 9.115 8.962 8.919 8.817 8.775 8.751 8.663 8.590 8.512

The quadratic interpolation in the z-direction is used for each layer.

plate still exists. The thickness of the two laminates is not identical. The thickness of the 0 and the 908 ply is taken as 3/5 and 2/5 of the total thickness of the plate, respectively. The ®rst eight nondimensional eigenfrequencies for the symmetric±symmetric modes are given in Table 6. The ®nal set of results is for a symmetric±symmetric rectangular laminate with a central pointsupport. It consists of three plies with 0/90/08 stacking sequences. It is obvious that symmetry also exists for such a plate. The thickness of each of the two outer 08 plies is taken as 2/5 of the total thickness, while the thickness of the middle 908 ply is taken as 1/5 of the total thickness. The ®rseight nondimensional eigenfrequencies for the symmetric±symmetric modes are listed in Table 7 with dierent aspect ratio and side-to-thickness ratio. It should be pointed out that the accuracy of the ®nite layer analysis can be improved by using quadratic (H03) instead of the linear (L02) interpolations. In Table 8 a comparative study is given for

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D. Zhou et al. / International Journal of Solids and Structures 37 (2000) 1483±1499

Table 4 p The ®rst ®ve non-dimensional eigenfrequencies, l  o b2 rt=D, of isotropic square thin plates with point supports at corners and dierent boundary conditions at the edges Bound. con.

Methods

l1

l2

l3

l4

l5

Present Mizusawa (1987) Kim (1987)

15.59 15.12 15.17

24.51 23.70 23.92

40.32 39.37 39.39

55.52 53.53 54.16

64.98 62.54 62.85

Present Mizusawa (1987) Kim (1987)

12.17 11.94 11.94

21.73 21.06 21.18

35.64 35.01 35.02

48.42 47.24 47.40

60.28 57.92 58.14

Present Mizusawa (1987) Kim (1987)

9.724 9.608 9.6079

17.54 17.32 17.316

30.83 30.60 30.596

44.41 43.65 43.652

52.71 51.04 51.051

Present Mizusawa (1987)

5.427 5.312

16.31 15.86

22.45 21.71

29.95 29.29

44.50 43.39

Present Mizusawa (1987)

7.246 7.111

15.85 15.77

15.85 15.77

20.07 19.60

39.92 38.43

Table 5 p The ®rst eight non-dimensional eigenfrequencies of symmetric±symmetric mode, l  o b=22 rt=D, for isotropic thick plates with simply-supported edges and a central point-support Aspect ratio a/b

Thickness ratio t/b

Mode number SS-1

SS-2

SS-3

SS-4

SS-5

SS-6

SS-7

SS-8

1.0

0.10 0.15 0.20 0.25 0.30

11.38 9.913 8.663 7.648 6.825

21.50 18.95 16.64 14.29 11.90

29.03 23.81 17.85 14.69 13.08

35.71 24.19 20.47 17.65 14.91

39.28 30.03 22.49 17.95 15.46

45.10 32.23 26.98 23.05 20.05

47.86 38.97 32.35 25.92 21.57

52.76 42.17 32.40 26.02 21.60

1.5

0.10 0.15 0.20 0.25 0.30

7.520 6.857 6.208 5.625 5.117

15.26 13.43 11.83 10.50 9.404

22.37 19.50 14.89 11.91 9.927

26.24 19.86 17.00 14.94 12.12

29.79 22.69 19.13 15.28 13.26

32.76 25.53 19.64 17.15 15.15

38.31 27.13 22.86 18.52 15.41

40.54 30.92 23.18 19.64 17.15

2.0

0.10 0.15 0.20 0.25 0.30

5.453 5.132 4.781 4.437 4.115

11.61 10.36 9.210 8.229 7.409

17.80 15.82 13.36 10.68 8.903

21.83 17.81 14.00 12.44 11.13

25.66 19.09 16.68 14.08 11.73

26.72 22.14 17.60 14.68 12.43

29.39 23.48 18.71 14.94 13.04

34.97 24.97 19.14 16.71 14.76

D. Zhou et al. / International Journal of Solids and Structures 37 (2000) 1483±1499

1493

Table 6 p The ®rst eight non-dimensional eigenfrequencies of symmetric-symmetric mode, l  o b=22 rt= D11 D22 1=2 , for skew-symmetric rectangular laminates with simply-supported edges and a central point-support Aspect ratio a/b

Thickness ratio t/b

Mode number SS-1

SS-2

SS-3

SS-4

SS-5

SS-6

SS-7

SS-8

1.0

0.10 0.15 0.20 0.25 0.30

7.286 5.777 4.789 4.090 3.568

14.60 11.35 9.215 7.746 6.679

18.00 13.55 10.83 9.019 7.723

22.95 17.20 13.72 11.39 9.075

28.27 20.52 16.00 11.70 9.771

30.44 22.11 16.43 13.55 10.80

32.67 23.99 17.39 14.08 11.52

35.86 24.03 18.95 14.33 12.15

1.5

0.10 0.15 0.20 0.25 0.30

5.193 4.345 3.695 3.200 2.816

10.01 7.973 6.604 5.627 4.897

15.67 11.88 9.502 7.916 6.786

16.80 12.93 10.48 8.799 7.580

19.75 14.86 11.87 9.869 8.158

23.66 17.82 13.99 10.46 8.503

26.37 19.59 14.25 11.85 9.979

29.48 20.09 15.54 12.85 10.14

2.0

0.10 0.15 0.20 0.25 0.30

4.059 3.541 3.085 2.709 2.405

7.647 6.218 5.211 4.475 3.916

11.95 9.493 7.823 6.6398 5.758

15.68 11.75 9.374 7.802 6.687

16.72 12.79 10.30 8.607 7.377

18.26 13.98 11.29 9.442 7.628

20.88 15.72 12.47 9.580 8.132

23.83 17.48 12.66 10.47 8.955

Table 7 p The ®rst eight non-dimensional eigenfrequencies of symmetric±symmetric mode, l  o b=22 rt= D11 D22 1=2 , for symmetric rectangular laminates with simply-supported edges and a central point-support Aspect ratio a/b

Thickness ratio t/b

Mode number SS-1

SS-2

SS-3

SS-4

SS-5

SS-6

SS-7

SS-8

1.0

0.10 0.15 0.20 0.25 0.30

7.328 5.920 4.896 4.157 3.609

13.19 10.37 8.580 7.328 6.398

20.88 15.10 11.76 9.637 8.161

22.12 16.94 13.66 11.40 9.768

25.73 19.13 15.39 12.88 10.27

30.94 23.00 16.76 12.92 11.14

35.19 23.13 18.27 15.08 11.33

35.59 25.29 19.40 15.25 12.72

1.5

0.10 0.15 0.20 0.25 0.30

5.657 4.681 3.937 3.379 2.954

9.216 7.425 6.164 5.265 4.599

13.95 11.11 9.244 7.914 6.914

19.76 14.59 11.29 9.219 7.792

20.73 15.42 12.31 10.18 8.528

22.45 16.62 13.26 10.43 8.678

25.22 17.92 13.38 11.38 9.847

27.14 18.69 14.86 12.36 10.57

2.0

0.10 0.15 0.20 0.25 0.30

5.105 4.195 3.513 3.010 2.630

7.102 5.910 4.988 4.293 3.761

10.50 8.493 7.086 6.080 5.328

14.45 11.55 9.622 8.238 7.190

18.93 14.42 11.13 9.055 7.524

20.50 14.85 11.50 9.121 7.639

21.27 15.42 11.75 9.666 8.216

22.88 15.44 12.48 10.48 9.739

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D. Zhou et al. / International Journal of Solids and Structures 37 (2000) 1483±1499

Table 8 The convergence and comparison study of ®nite layer method for three-dimensional eigenvalues l  o a2 =p2 rt=D2 of an isotropic thick square plate with SSSS boundary conditions, t/a = 0.5, v = 0.3 Method

Terms in x, y, z

SS-1

SS-2

SS-3

2  2  2a 223 224 225

1.2630 1.2598 1.2592 1.2591

1.8451 1.8451 1.8451 1.8451

2.9351 2.9330 2.9326 2.9325

Linear interpolation

225 228 2  2  10 2  2  15

1.2709 1.2638 1.2621 1.2604

1.8451 1.8451 1.8451 1.8451

2.9439 2.9371 2.9355 2.9338

Rayleigh±Ritzb

449

1.2590

1.8451

2.9335

a

The sequence of the terms is number of terms in x- and y-direction; number of ®nite layers in z-direction. From Liew, K.M., Hung, K.C., Lim, M.K., 1993. A continuum three-dimensional vibration analysis of thick rectangular plates. Int. J. Solids Struct. 30 (24), 3357±3379. b

an isotropic thick plate with SSSS support conditions for linear interpolation (L02) and quadratic interpolation (H03). It can be seen that the results of the 2  2  3 H03 analysis are already nearly exact and are better than those of the 2  2  15 L02 analysis. It should be noted that be condensing the degrees-of-freedom associated with the H03 middle nodal surface there is very little dierence in the amount of computational eorts between the L02 and H03 analysis. 6. Concluding remarks A new set of two-dimensional basic functions has been developed by superimposing a set of static beam functions under sinusoidal loads to another set of beam functions under point loads. Unlike existing basic functions for vibration analysis of plates, this set of basic functions satis®es not only the geometric boundary conditions at the edges of the plate but also the zero out-of-plane de¯ection at the point supports. This new set of functions is combined with the ®nite layer method for the free vibration analysis of isotropic and laminated composite rectangular plates with point supports. Numerical results are compared with the thin-plate results for plates with dierent arrangement of point supports and good agreement is observed in all cases. To demonstrate the in¯uence of aspect ratio, side-to-thickness ratio, material properties and stacking sequences on the vibrational behaviour of the plates with point supports, a simply-supported plate with a central point support is taken as an example to study idetail. Results for isotropic thick plates and laminated composite thick plates with two and three plies are summarized. To the best of the authors' knowledge, the information provided herein for threedimensional vibration of thick plates with point supports is presented for the ®rst time. Appendix A The property matrix D for the composite materials with the ®ber orientation angle y with respect to the x-axis is

D. Zhou et al. / International Journal of Solids and Structures 37 (2000) 1483±1499

2

D11 6 6 D12 6 6 6 D13 D   6 6 60 6 6 60 4 D16

D12

D13

0

0

D22

D23

0

0

D23

D33

0

0

0

0

D44

D45

0

0

D45

D55

D26

D36

0

0

D16

1495

3

7 D26 7 7 7 D36 7 7 7 0 7 7 7 0 7 5 D66

where D11  Q11 m4  2Q12  2Q66 m2 n2  Q22 n4 , ÿ  D12  Q11  Q22 ÿ 4Q66 m2 n2  Q12 m4  n4 , D13  Q13 m2  Q23 n2 , D16  ÿmn3 Q22  m3 nQ11 ÿ mnm2 ÿ n2 Q12  2Q66 , D22  Q11 n4  2Q12  2Q66 m2 n2  Q22 m4 , D23  Q13 n2  Q23 m2 , D33  Q33 , D26  ÿm3 nQ22  mn3 Q11  mnm2 ÿ n2 Q12  2Q66 , D36  Q13 ÿ Q23 mn, D44  Q44 m2  Q55 n2 , D45  Q55 ÿ Q44 mn, D55  Q55 m2  Q44 n2 , 2

D66  Q11  Q22 ÿ 2Q12 m2 n2  Q66 m2 ÿ n2  in which, m  cos y , n  sin y  and Q11  E11 1 ÿ v23 v32 =D,

Q22  E22 1 ÿ v31 v13 =D,

Q33  E33 1 ÿ v12 v21 =D,

Q44  G23 ,

Q66  G12 ,

Q55  G13 ,

Q12  v12  v32 v13 E22 =D,

Q13  v13  v12 v23 E22 =D,

Q23  v23  v21 v13 E33 =D,

D  1 ÿ v12 v21 ÿ v23 v32 ÿ v31 v13 ÿ 2v21 v32 v13 : In the above equations, E11 and E22 are the Young's moduli parallel and perpendicular to the ®bers, respectively and E33 is the Young's modulus in the thickness direction of the plate, G23 , G13 and G12 are the shear moduli of elasticity, v12 , v21 , v13 , v31 , v32 and v23 are the Poisson's ratios.

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Appendix B The layer stiness matrix 2 K 1,1,1,1 . . . 6 6 .. .. 6 . . 6 6 6 K 1,J,1,1 . . . K   6 6 6 K 2,1,1,1 . . . 6 6 .. .. 6 6 . . 4 K I,J,1,1 . . . 2

M 1,1,1,1

6 6 .. 6 . 6 6 6 M 1,J,1,1 M   6 6 6 M 2,1,1,1 6 6 .. 6 6 . 4 M I,J,1,1

K  and mass matrix M  are written in the form of, respectively, 3 K 1,1,1,J K 1,1,2,1 . . . K 1,1,I,J 7 7 .. .. .. .. 7 . . . . 7 7 K 1,J,1,J K 1,J,2,1 . . . K 1,J,I,J 7 7 7 K 2,1,1,J K 2,1,2,1 . . . K 2,1,I,J 7 7 7 .. .. .. .. 7 7 . . . . 5 K I,J,1,J K I,J,2,1 . . . K I,J,I,J

. . . M 1,1,1,J .. .

M 1,1,2,1 .. .

.. .

. . . M 1,J,1,J

M 1,J,2,1

. . . M 2,1,1,J

M 2,1,2,1

.. .

.. .

.. .

...

M I,J,1,J

M I,J,2,1

. . . M 1,1,I,J

3

7 7 7 7 7 . . . M 1,J,I,J 7 7 7 . . . M 2,1,I,J 7 7 7 .. .. 7 7 . . 5 . . . M I,J,I,J .. .

.. .

where 2

K11  K12 

6 K ijkl  6 4 K21  K22  K31  K32  2 6 M ijkl  6 4

K13 

3

7 K23  7 5 , i, k  1, 2, . . . , I, K33 

j, l  1, 2, . . . , J,

ijkl

M11 

0

0

0

M22 

0

0

0

M33 

3 7 7 , i, k  1, 2, . . . , I, 5

j,l  1, 2, . . . , J,

ijkl

in which,    2 @ 2 Wij @ 2 Wkl @ Wij @ 2 Wkl T T K11 ijkl  D11 dx dy N  N  dz  D16 dx dy N  N  dz @x2 @x2 @x 2 @x @ y T       2 @Wij @ Wkl dN dN @ Wij @ 2 Wkl T dz  D16 dx dy dx dy N  N  dz  D55 dz dz @ x @x @x @ y @ x 2   2 @ Wij @ 2 Wkl T dx dy N  N  dz,  D66 @x @y @ x @y

D. Zhou et al. / International Journal of Solids and Structures 37 (2000) 1483±1499

@ 2 Wij @ 2 Wkl @ 2 Wij @ 2 Wkl T T     dx dy N  N  dz dx dy N N dz  D 16 @ x 2 @y2 @x 2 @x @ y T      2  @Wij @ Wkl dN dN @ Wij @ 2 Wkl T dx dy dz  D26 dx dy N  N  dz  D45 @x @y @x @ y @y2 dz dz   2 @ Wij @ 2 Wkl T dx dy N  N  dz,  D66 @x @y @ x @y

K12 ijkl  D12

 T     @ 2 Wij @Wij @ Wkl dN T dN N  dz K13 ijkl  D13 dz  D45 Wkl dx dy N  dx dy dz dz @x2 @x @y T      2  @Wij @ Wkl dN @ Wij T dN N  dz  D36 dz, dx dy Wkl dx dy N   D55 dz dz @x @y @ x @y

2 @ 2 Wij @ 2 Wkl @ Wij @ 2 Wkl T T     dx dy N N dz  D26 dx dy N  N  dz @y2 @ x 2 @ y2 @x @ y T      2  @Wij @ Wkl dN dN @ Wij @ 2 Wkl T dz  D16 dx dy dx dy N  N  dz  D45 dz dz @y @x @x @ y @ x 2   2 @ Wij @ 2 Wkl T dx dy N  N  dz,  D66 @x @y @ x @y

K21 ijkl  D12

2  @ 2 Wij @ 2 Wkl @ Wij @ 2 Wkl T   N T N  dz   dx dy dx dy N dz  D N 26 @y2 @y2 @ y2 @x @ y T       2 @Wij @ Wkl dN dN @ Wij @ 2 Wkl N T N  dz dx dy dz  D26 dx dy  D44 @y @x @x @ y @y2 dz dz   2 @ Wij @ 2 Wkl T dx dy N  N  dz,  D66 @x @y @ x @y

K22 ijkl  D22

 T     @ 2 Wij @Wij @ Wkl dN T dN N  dz K23 ijkl  D23 dx dy dz  D44 Wkl dx dy N  @y2 @y @ y dz dz T       2 @Wij @ Wkl dN @ Wij T dN N  dz  D36 dx dy Wkl dx dy N  dz,  D45 @y @ y @ x @y dz dz

 @ 2 Wkl K31 ijkl  D13 Wij dx dy @x2    @Wij @ Wkl T dx dy N   D45 @y @x

T  @ 2 Wkl dN N  dz  D36 N  dz dx dy Wij dz @x @y      dN @ Wij @ Wkl T dN dx dy N  dz  D55 dz, @ x @x dz dz

dN dz

T

1497

1498

D. Zhou et al. / International Journal of Solids and Structures 37 (2000) 1483±1499

 @ 2 Wkl K32 ijkl  D23 Wij dx dy @y2    @Wij @ Wkl T dx dy N   D44 @y @ y

T  @ 2 Wij dN N  dz  D36 N  dz dx dy Wij @x @ y dz      dN @ Wij @ Wkl T dN dx dy N  dz  D45 dz, @x @y dz dz

dN dz

T

   dN @ Wij @Wkl T K33 ijkl  D33 dz  D44 dx dy N  N  dz Wij Wkl dx dy dz @ y @y     @Wij @ Wkl @Wij @Wkl T T dx dy N  N  dz  D45 dx dy N  N  dz  D45 @y @x @x @y   @Wij @ Wkl T dx dy N  N  dz,  D55 @ x @x 

T 

@ Wij @ Wkl T dx dy N  N  dz, @ x @x

@ Wij @ Wkl T dx dy N  N  dz, @y @ y

M11 ijkl  r

M22 ijkl  r

M33 ijkl  r

dN dz

T Wij Wkl dx dy N  N  dz:

In the above equations, the double integrations are carried out over the entire surface of the plate and integration through the thickness of each layer is done separately, r is the density of the material.

References Cheung, Y.K., Chakrabarti, S., 1972. Free vibration of thick, layered rectangular plates by ®nite layer method. Journal of Sound and Vibration 21, 277±284. Cheung, Y.K., Delcourt, C., 1977. Buckling and vibration of thin, ¯at-walled structures continuous over several spans. Paper No. 7959, Part 2, Proc. Institution of Civil Engineers, pp. 39±103. Cheung, Y.K., Tham, L.G., 1997. Finite Strip Method. CRC Press, LLC. Dave, D.J., 1978. Finite strip models for vibration of Mindlin plates. Journal of Sound and Vibration 59, 441±452. Fan, S.C., Cheung, Y.K., 1984. Flexural free vibrations of rectangular plates with complex support conditions. Journal of Sound and Vibration 93, 81±94. Fan, J.R., Sheng, H.Y., 1992. The exact solution of laminated thick rectangular plates with clamped edges. Acta Mechanica Sinica 24, 574±582. Fan, J.R., Ye, J.Q., 1990. An exact solution for the statics and dynamics of laminated thick plates with orthotropic layers. International Journal of Solids and Structures 26, 655±662. Gorman, D.J., Singal, R.K., 1991. Analytical and experimental study of vibrating rectangular plates on rigid point supports. AIAA 29, 838±844. Hanna, N.F., Leissa, A.W., 1994. A higher order shear deformation theory for the vibration of thick plates. Journal of Sound and Vibration 170, 545±555. Kim, C.S., Dickinson, S.M., 1987. Flexural vibration of rectangular plates with point supports. Journal of Sound and Vibration 117, 249±261. Leissa, A.W., 1969. Vibration of Plates, Washington, D.C.: Oce of Technology Utilization, NASA. NASA SP-160.

D. Zhou et al. / International Journal of Solids and Structures 37 (2000) 1483±1499

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Leissa, A.W., Zhang, Z.D., 1983. On the three-dimensional vibrations of the cantilevered rectangular parallelepiped. International Journal of Solids and Structures 73, 2013±2021. Liew, K.M., Hung, K.C., Lim, M.K., 1993. A continuum three-dimensional vibration analysis of thick rectangular plates. International Journal of Solids and Structures 30, 3357±3379. Liew, K.M., Hung, K.C., Lim, M.K., 1994. Three-dimensional plates: variance of simple support conditions and in¯uence of inplane inertia. International Journal of Solids and Structures 31, 3233±3247. Liew, K.M., Xiang, Y., Kitipornchai, S., 1993. Transverse vibration of thick rectangular platesÐI comprehensive sets of boundary conditions. Computers and Structures 49, 31±58. Liew, K.M., Xiang, Y., Kitipornchai, S., Lim, M.K., 1994. Vibration of Mindlin plates on point supports using constraint functions. ASCE Journal of Engineering Mechanics 120, 499±513. Mindlin, R.D., 1951. In¯uence of rotatory inertia and shear in ¯exural motion of isotropic, elastic plates. ASME Journal of Applied Mechanics 18, 1031±1036. Mizusawa, T., Kajita, T., 1987. Vibration of skew plates resting on point supports. Journal of Sound and Vibration 115, 243±251. Noor, A.K., 1973. Free vibrations of multilayered composite plates. AIAA 11, 1038±1039. Reddy, J.N., 1984. A simple higher-order theory for laminated composite plates. ASME Journal of Applied Mechanics 51, 745± 752. Srinivas, S., Joga Rao, C.V., Rao, A.K., 1970. An exact analysis for vibration of simple-supported homogeneous and laminated thick rectangular plates. Journal of Sound and Vibration 12, 187±199. Zhou, D., 1987. The application of a type of new admissible functions to the vibration of rectangular plates. Computers and Structures 52, 199±204. Zhou, D., 1996. Natural frequencies of rectangular plates using a set of static beam functions in Rayleigh±Ritz method. Journal of Sound and Vibration 189, 81±87.

Journal of Sound and Vibration (1995) 184(4), 639–649

VIBRATION OF SHEAR-DEFORMABLE PLATES WITH INTERMEDIATE LINE SUPPORTS: A FINITE LAYER APPROACH J. Kong and Y. K. Cheung Department of Civil and Structural Engineering, University of Hong Kong, Hong Kong (Received 19 July 1993, and in ﬁnal form 14 March 1994) Free vibration of isotropic and laminated composite plates is studied by the ﬁnite layer method. In this method, a plate is treated as a three-dimensional elastic body and the displacement ﬁeld is approximated by a set of admissible trial functions in the plane of the plate. One-dimensional linear shape functions are adopted to simulate the variation of displacements through the thickness. To satisfy the boundary and internal conditions, a set of modiﬁed beam vibration functions is employed as the trial functions. Stiﬀness and mass matrices are formulated via the conventional displacement method and the eigenequation is then solved. Numerical examples are given to demonstrate the application of the method.

7 1995 Academic Press Limited

1. INTRODUCTION

The ﬁnite layer method [1] pioneered by Y. K. Cheung, is an eﬃcient analysis tool for three-dimensional rectangular structures of layered construction. The method consists of using a two-dimensional analytical solution in the plane of a three-dimensional structure and a one-dimensional ﬁnite element shape function through the thickness. The method has been used to analyze the structural and/or thermal behaviour of rectangular sandwich and isotropic plates with simply supported edges [2, 3], and vibration of rectangular plates with various homogeneous boundary conditions [1]. In addition, the method was employed to study semi-inﬁnite elastic bodies such as the static analysis and consolidation of layered soil [4, 5], interaction between an elastic slab and its foundation [6, 7]. Recently, an inﬁnite layer method [8] was developed for the analysis of pile foundations. The method was also extended to study the vibration of thick-walled circular cylindrical tubes with various end supports [1]. In most of these ﬁnite layer studies, either harmonic series or beam vibration functions were adopted as the trial functions for analyzing structures with supports around the edges. No internal supports were considered. In this study, rectangular shear-deformable plates with intermediate line supports are analyzed. To cater for the transverse shearing and normal deformations, the plates are treated as three-dimensional elastic bodies. To satisfy the boundary and internal conditions, a beam vibration function is augmented by a polynomial [9]. The modiﬁed beam vibration functions are employed as the trial functions for subsequent ﬁnite layer formulation. (Recent developments of trial functions for thin-plate analysis can be found in references [10–13].) A number of numerical examples are provided for reference. They can be used to compare with other simpliﬁed two-dimensional plate theories or numerical models. 639 0022–460X/95/290639 + 11 \$11.00/0

7 1995 Academic Press Limited

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640

2. THE DISPLACEMENT FIELD

In Figure 1 is shown a rectangular thick plate with qx and qy intermediate line supports parallel to the x- and y-axes respectively. It is divided into a number of ﬁnite layers through the thickness. The displacement ﬁeld of each ﬁnite layer can be written as m

n

u(x, y, z) = s s X(x)i,x Y(y)j [N(z)]{a}ij ,

(1a)

i=k j=l

m

n

v(x, y, z) = s s X(x)i Y(y)j,y [N(z)]{b}ij , i=k j=l

m

n

w(x, y, z) = s s X(x)i Y(y)j [N(z)]{g}ij , i=k j=l

(1b, c) where k = 1 + qy , l = 1 + qx and [N(z)] denotes the one-dimensional linear shape functions. The in-plane displacements along the x- and y-axes are represented by u(x, y, z) and v(x, y, z) respectively. The displacement unknowns are denoted by {a}ij , {b}ij and {g}ij . The trial functions and their ﬁrst derivatives are represented by X(x)i , X(x)i,x , Y(y)j and Y(y)j,y . It is important to recognize that the ﬁrst term of the trial function depends on the number of intermediate line supports in the corresponding direction. Furthermore, it is noteworthy that X(x)i,x and Y(y)j,y are incorporated in the expressions for u and v respectively, so that a consistent interpolation can be achieved and shear locking can be avoided when dealing with thin-plate problems [14]. This can be easily observed by writing down the explicit expressions for the transverse shear strains: exz = s s (X(x)i,x Y(y)j [N(z)],z {a}ij + X(x)i,x Y(y)j [N(z)]{g}ij ), i

j

eyz = s s (X(x)i Y(y)j,y [N(z)],z {b}ij + X(x)i Y(y)j,y [N(z)]{g}ij ). i

j

The trial functions for each of the transverse shear strain components are consistent in the sense that they are of equal order in the x- and y-directions. The inconsistency between the ﬁnite element shape function in the z-direction becomes relatively insigniﬁcant when the thickness-span ratio becomes small. Application of these displacement functions to thin-plate vibration will be demonstrated in the numerical examples. Although the ﬁnite layer approach was developed in the 1970’s, it is interesting to recognize that the basic concept coincides with that of the generalized layerwise laminate plate theory which was recently proposed by Reddy and his co-workers [15].

Figure 1. A rectangular thick plate with intermediate line supports. · · · · ·, Lines of supports.

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3. MODIFIED BEAM VIBRATION FUNCTIONS

First, consider the modiﬁed beam vibration functions for the out-of-plane displacement component w, in the x-direction. The functions are required to satisfy the boundary conditions at x = 0 and x = Lx and the zero deﬂection conditions at the intermediate line supports parallel to the y-axis. The functions can be written as (x)i + X (x)i , X(x)i = X

r

where

X (x)i = s lip x p,

(2, 3)

p=0

(x)i is the ith term of the conventional lip are constants to be determined, r = qy + 3 and X beam vibration function which satisﬁes the corresponding boundary conditions; see Table 1. To determine the constants lip of the augmented polynomial, the four boundary conditions at both ends are imposed on equation (3). The remaining qy constants are found by imposing zero deﬂection, for the qy internal supports, on equation (2). The same procedure is repeated for each term of the function. To illustrate this procedure, the following matrix equation is written down for the case of simply supported ends with an intermediate line support at x = xq parallel to the y-axis: 1 0 0 0 K G 2 3 1 L L L x x x G 0 G0 0 2 G 0 0 2 6Lx G k 1 xq xq2 xq3

0 L G Lx4 G 0 G 12Lx2G xq4 G l

0 l F J G i0 J G F G G l 0 i1 G G G G g li2 h = g 0 h . G li3 G G 0 G G f li4 G j G f−X(xq )i G j

The ﬁrst four equations refer to the boundary conditions at the ends, while the last equation refers to the condition of zero deﬂection at the intermediate line support. The same procedure applies to the y-direction. After the constants of the augmented polynomials in both directions have been found, the modiﬁed beam functions are substituted into the displacement functions given in equation (1). On a closer examination of equation (1), one can then easily observe that the cross-section of the plate at the corresponding line support is actually assumed to be a rigid in-plane diaphragm; see Table 1 Beam vibration functions (SS = simply supported) Boundary conditions

Beam vibration functions

SS − SS

i = sin (mi x/Lx ) X

X(0)i = 0, X(0)i,xx = 0,

X(Lx )i = 0 X(Lx )i,xx = 0

Clamped–clamped X(0)i = 0,

mi = ip

X(Lx )i = 0

X(0)i,x = 0, X(Lx )i,x = 0

i = sin (mi x/Lx ) − sinh (mi x/Lx ) X − fi (cos (mi x/Lx ) − cosh (mi x/Lx )) mi = (i + 0·5)p fi = (sin mi − sinh mi )/(cos mi − cosh mi )

SS–clamped X(0)i = 0,

X(Lx )i = 0

X(0)i,xx = 0, X(Lx )i,x = 0

i = sin (mi x/Lx ) − fi sin (mi x/Lx ) X mi = (i + 0·25)p fi = sin mi /sinh mi

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Table 2 Boundary and internal conditions of plates Displacement constraints at the corresponding cross-section

Location and type of supports For edges parallel to the y-axis Simply supported Clamped

v = 0 = w and sx = 0 u=v=w=0

For edges parallel to the x-axis Simply supported Clamped

u = 0 = w and sy = 0 u=v=w=0

For intermediate line supports parallel to the y-axis

v=0=w

For intermediate line supports parallel to the x-axis

u=0=w

Table 2. It should also be noted that the values for mi in Table 1 are only approximations for the true values for the clamped–clamped case and the SS–clamped case, and will be poorer approximations for the lower modes. This is attributed to the fact that the beam functions given in Table 1 represent only the ﬂexural vibration mode shapes of the appropriate thin beams. After having deﬁned the displacement ﬁeld, the stiﬀness and mass matrices are formulated via the conventional displacement formulation as documented in reference [1]. The eigenequation is then solved by the method of subspace iteration [16]. 4. NUMERICAL EXAMPLES

4.1. preliminary assessment: simply supported laminated plates The fundamental frequencies of symmetric and skew-symmetric square laminates have been computed by using one of term of the trial functions in each of the two in-plane directions. They can be compared with the three-dimensional ﬁnite diﬀerence solution of Noor [17]. The ﬁbre orientations of the diﬀerent laminas alternate between 0° and 90° with respect to the x-axis, and in the symmetric laminates the 0° layers are the outer surfaces. The total thicknesses of the aligned layers are the same, whereas the individual Table 3 Fundamental frequencies of square laminates; v¯ = (w/h)(r/E2 )1/2; skew-symmetric, ﬁbre orientation 0°/90° for two layers and 0°/90°/0°/90° for four layers; symmetric, ﬁbre orientation 0°/90°/0° for three layers and 0°/90°/0°/90°/0° for ﬁve layers Number of material layers

Frequency, v¯ ZXXXXXXCXXXXXXV Noor [17] Present

Skew-symmetric 2

0·34250

4 Symmetric 3 5

Number of ﬁnite layers†

0·42719

0·33900 0·33758 0·42379

2 4 4

0·43006 0·45374

0·42666 0·44681

4 5

† Total number of ﬁnite layers through the plate thickness.

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Figure 2. A 2 × 2 continuous panel.

Figure 3. A one-way, three-span continuous panel.

layers are taken to be of equal thickness. Geometric and material constants are as follows: span–thickness ratio = 5; E1 /E2 = 40; G12 /E2 = 0·6; G13 = G23 ; G23 /E2 = 0·5; n12 = 0·25 = n13 ; density r = 1·0. In Table 3, one can easily observe the good agreement between the two sets of results. In most cases, the present ﬁnite layer results are about 1 percent less than those given by Noor. 4.2. continuous rectangular plates A series of rectangular plates with diﬀerent arrangements of intermediate line supports are shown in Figures 2–5. The plates are made of isotropic material or laminated composite material. In cases of laminates, the material constants used are the same as those given in the preceding section, except that G12 = G13 is adopted in the following examples. In all

Figure 4. A one-way, four-span continuous panel.

Figure 5. A 3 × 3 continuous panel.

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cases, the ratio between the shorter span and the thickness is taken as 5 for thick plates and 100 for thin plates. Support conditions along the sides of the plate are indicated by four letters than deﬁne the type of support used along the x = 0, x = Lx , y = 0 and y = Ly sides respectively. The letters S and C denote, respectively, simply supported and clamped boundary conditions. For the 2 × 2 continuous panel in Figure 2, a = b and a = b = 0·5, and for the 3 × 3 continuous panel in Figure 5, a = b, a = 0·35 and b = 0·7, except for Table 4 Frequencies of continuous thin plates

Mode sequence

Frequency, V ZXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXV 1 2 3 4 5 6

Two-way, two-span (Figure 2) S–S–S–S 2  3  3† 81·04 244 81·04 355 79·83 555 79·22 Leissa [18] 78·96

101·04 99·24 98·03 97·40 94·49

101·04 99·24 98·03 97·40 94·49

118·55 115·04 113·8 113·18 108·22

202·20 202·20 199·18 197·61 197·39

202·20 202·20 199·18 197·61 197·39

19·82 19·82 19·74

22·16 22·16 21·60

28·26 27·20 26·00

49·52 49·52 49·35

49·52 49·52

51·15 51·15

21·92 21·63

27·10 26·00

30·77 28·95

50·82 50·45

54·13

55·42

20·41 20·22

24·37

29·70 28·07

49·92 49·63

51·24 50·69

52·45

21·16 20·81

24·83 23·65

30·21 27·12

49·52 49·35

49·52 49·98

Two-way, three-span (Figure 5) C–C–S–S 555 186·38 214·62 Kim [20] 184·30 212·14

235·67 231·35

255·37 252·17

259·65 255·60

291·16 286·95

One-way, three-span S–S–S–S 544 555 Azimi [19] S–S–C–C 555 Azimi [19] S–S–C–S 555 Azimi [19]

(Figure 3)

One-way, four-span (Figure 4) S–S–S–S 555 19·82 Azimi [19] 19·74

C–S–C–S 555 Kim [20]

191·98 190·69

230·85 226·87

231·18 227·18

260·47 259·99

260·60 265·88

266·09 265·93

C–C–C–C 555 Kim [20]

201·77 198·55

248·35 243·27

248·35 243·32

288·35 282·31

302·05 297·20

302·05 297·20

S–S–S–S 555 Zhou [9]

140·08 139·09

172·03 169·04

179·33 175·60

207·19 202·06

217·65 210·78

247·97 239·84

C–C–C–S 555 Kim [20]

196·92 193·78

235·06 229·59

244·51 237·58

264·14 254·80

277·35 268·75

299·00 288·79

† Number of ﬁnite layers  number of terms in x-direction  number of terms in y-direction.

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Table 5 Frequencies of continuous thick plates

Mode sequence

Frequencies, V ZXXXXXXXXXXXCXXXXXXXXXXXV 1 2 3

Two-way, two-span (Figure 2) S–S–S–S 5  3  3† 555 566

56·25 56·25 56·25

64·80 63·26 62·57

64·80 63·26 62·57

One-way, three-span (Figure 3) S–S–S–S 533 553 564

17·63 17·63 17·63

19·37 19·37 19·21

23·64 22·63 22·63

S–S–C–C 533 553 564

19·26 19·13 19·13

22·78 22·49 22·41

24·92 24·62 24·62

S–S–C–S 533 553 564

18·08 18·08 18·04

21·11 20·81 20·78

25·00 24·03 24·00

One–way, four-span (Figure 4) S–S–S–S 555 553 564

17·63 17·63 17·63

18·65 18·64 18·64

21·28 21·28 20·84

Two-way, three-span (Figure 5) C–C–S–S 533 555 566

102·44 101·48 101·36

111·27 109·66 109·47

112·10 110·57 110·47

C–S–C–S 533 555 566

104·93 103·63 103·59

114·34 112·73 112·45

114·50 112·91 112·65

C–C–C–C 533 555 566

107·25 105·55 105·55

116·00 114·65 114·62

116·05 114·68 114·64

S–S–S–S 533 555 566

97·57 97·51 97·25

107·59 105·73 105·44

107·64 105·76 105·47

C–C–C–S 533 555 566

106·05 104·56 104·54

114·70 113·57 113·50

115·64 113·84 113·60

† Number of ﬁnite layers  number of terms in x-direction  number of terms in y-direction.

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the case with S–S–S–S, for which the locations of the intermediate line supports on the x- and y-axes are replaced by (0·3b, 0·65b) and (0·45b, 0·85b) respectively. For isotropic plates, the natural frequencies are normalized as V 2 = v 2rhb 4/D, where D = Eh 3/12(1 − n 2) and n = 0·3, while, for composite plates, V 2 = v 2rhb 4/(D11 D22 )1/2. For isotropic thin plates, the lowest six frequencies are given in Table 4 with diﬀerent conditions of support. Results obtained by the present methods are compared with those of Leissa [18], Azimi et al. [19] and Kim and Dickinson [20]. (Results given in reference [18] are for plates with no intermediate supports. The appropriate values are extracted from

Table 6 Frequencies of continuous symmetric laminates

Mode sequence

Frequencies, V ZXXXXXXXXXXXCXXXXXXXXXXXV 1 2 3

Two-way, two-span, S–S–S–S (Figure 2) Three layers 4  3  3† 35·08 455 35·08 466 35·08

38·09 37·55 37·22

38·72 37·68 37·33

36·36 36·36 36·36

39·27 38·61 38·33

39·87 38·87 38·45

One-way, three-span, S–S–S–S (Figure 3) Three layers 433 14·54 453 14·54 464 14·54

15·71 15·71 15·53

18·18 17·15 17·15

15·25 15·25 15·25

16·41 16·41 16·25

18·81 17·85 17·85

One-way, four-span, S–S–S–S (Figure 4) Three layers 533 14·54 533 14·54 564 14·54

15·25 15·24 15·24

16·84 16·84 16·84

15·25 15·25 15·25

15·95 15·95 15·95

17·51 17·51 17·51

Two-way, three-span, S–S–S–S (Figure 5) Three layers 433 54·18 455 54·15 466 54·01

57·88 57·27 57·12

58·78 57·37 57·28

59·12 58·31 58·17

59·94 58·61 58·51

Five layers 533 555 566

Five layers 533 553 564

Five layers 533 553 564

Five layers 533 555 566

55·50 55·47 55·33

† Number of ﬁnite layers  number of terms in x-direction  number of terms in y-direction.

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the work by examining square plates with various combinations of clamped and SS boundary conditions. This would not induce signiﬁcant diﬀerences in the natural frequencies for the actual continuous support conditions.) The diﬀerences among these sets of results are less than 4 percent. It should be noted that the plates are divided into ﬁve ﬁnite layers in most cases. The lowest three frequencies of thick isotropic plates and laminated composite plates with identical geometric arrangements are summarized in Tables 5–7. These sets of results are presented for the ﬁrst time in the open literature, and therefore no comparison with

Table 7 Frequencies of continuous skew-symmetric laminates

Mode sequence

Frequencies, V ZXXXXXXXXXXCXXXXXXXXXXV 1 2 3

Two-way, two-span, S–S–S–S (Figure 2) Two layers 4  3  3† 32·03 455 32·03 466 32·03

36·01 35·30 35·00

36·01 35·30 35·00

35·08 35·08 35·08

38·37 37·58 37·25

38·37 37·58 37·25

One-way, three-span, S–S–S–S (Figure 3) Two layers 433 11·66 453 11·66 464 11·66

12·72 12·72 12·63

15·04 14·42 14·42

14·47 14·47 14·47

15·46 15·46 15·34

17·56 16·81 16·81

One-way, four-span S–S–S–S (Figure 4) Two layers 533 11·66 553 11·66 564 11·66

12·29 12·29 12·29

18·40 13·79 13·79

14·47 14·47 14·47

15·07 15·07 15·07

16·42 16·42 16·42

Two-way, three-span, S–S–S–S (Figure 5) Two layers 433 51·79 455 51·77 466 51·68

56·52 55·71 55·59

56·53 55·71 55·59

58·19 57·22 57·10

58·19 57·22 57·10

Four layers 533 555 566

Four layers 533 553 564

Four layers 533 553 564

Four layers 533 555 566

54·05 54·02 53·89

† Number of ﬁnite layers  number of terms in x-direction  number of terms in y-direction.

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those obtained by other methods can be made. An increasing number of terms was used to obtain a converged solution in all cases. It can be seen that the frequency parameters of thick isotropic plates are lower than those of thin plates. This decrease can be partly attributed to the increased ﬂexibility due to transverse shear and normal deformation of thick plates. The rotational inertia also contributes to this decrease. 5. CONCLUSIONS

A ﬁnite layer method is used to analyze vibration of plates with intermediate line supports. The plates are treated as three-dimensional elastic bodies. A modiﬁed vibration function is adopted as the trial function for computing the natural frequencies of isotropic and laminated composite plates, and the results are compared with available solutions for the cases of thin plates. For the cases of thick isotropic and laminated composite plates, results are presented for the ﬁrst time in the open literature. Research is currently being undertaken to examine the diﬀerence between the results obtained by the present ﬁnite layer method and the exact continuum approach or other plate theories [21–24]. REFERENCES 1. Y. K. Cheung 1976 Finite Strip Method in Structural Analysis. Oxford: Pergamon Press. 2. Y. K. Cheung, L. G. Tham and K. P. Chong 1982 Computers and Structures 15, 131–134. Buckling of sandwich plate by ﬁnite layer method. 3. K. P. Chong, B. Lee and P. A. Lavdas 1984 Thin-walled Structures 2, 75–95. Analysis of thin-walled structures by ﬁnite strip and ﬁnite layer methods. 4. J. R. Booker and J. C. Small 1986 International Journal of Numerical and Analytical Methods in Geomechanics 10, 415–430. Finite layer analysis of viscoelastic layered materials. 5. Y. K. Cheung and S. C. Fan 1979 Proceedings of the Third International Conference on Numerical Methods in Geomechanics, Aachen 3, 1129–1135. Analysis of pavements and layered foundation by ﬁnite layer method. 6. Y. K. Cheung, L. G. Tham and K. F. Man 1989 Proceedings East Asia – Paciﬁc Conference on Structural Engineering and Construction, Chiang Mai, 715–720. Analysis of circular footing resting on layered soil. 7. Y. K. Chow, S. Swaddiwudhipong and K. F. Phoon 1989 Computational Geotechnology 8, 65–86. Finite strip analysis of strip footings: horizontal loading. 8. Y. K. Cheung, L. G. Tham and D. J. Guo 1988 Geotechnique 38, 415–431. Analysis of pile group by inﬁnite layer method. 9. Zhou Ding 1993 Research Report, Department of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing. Eigenfrequencies of line supported rectangular plates. 10. C. S. Kim 1986 Ph.D. Thesis, University of Western Ontario, Canada. The vibration of beams and plates studies using orthogonal polynomials. 11. K. M. Liew Ph.D. Thesis, National University of Singapore, Singapore. The development of 2-D orthogonal polynomials for vibration of plates. 12. Y. Xiang Ph.D. Thesis, The University of Queensland, Australia. The numerical developments in solving the buckling and vibration of Mindlin plates. 13. E. H. Foster 1991 M.S. Thesis, University of Missouri–Rolla. Optimization of laminated composite plates. 14. G. Prathap and G. R. Bhashyam 1982 International Journal of Numerical Methods in Engineering 18, 195–210. Reduced integration and the shear-ﬂexible beam element. 15. D. H. Robbins and J. N. Reddy 1993 International Journal of Numerical Methods in Engineering 36, 655–677. Modelling of thick composites using a layerwise laminate theory. 16. K. J. Bathe 1982 Finite Element Procedures in Engineering Analysis. Englewood Cliﬀs, New Jersey: Prentice-Hall. 17. A. K. Noor 1973 American Institute of Aeronautics and Astronautics Journal 11, 1038–1039. Free vibrations of multilayered composite plates. 18. A. W. Leissa 1973 Journal of Sound and Vibration 31, 257–293. The free vibration of rectangular plates.

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19. S. Azimi, J. F. Hamilton and W. Soedel 1984 Journal of Sound and Vibration 93, 9–29. The receptance method applied to the free vibration of continuous rectangular plates. 20. C. S. Kim and S. M. Dickinson 1987 Journal of Sound and Vibration 114, 129–142. The ﬂexural vibration of line supported rectangular plate systems. 21. S. Srinivas, C. V. Rao and A. K. Rao 1970 Journal of Sound and Vibration 12, 187–199. An exact analysis for vibration of simply supported homogeneous and laminated thick rectangular plates. 22. K. M. Liew, K. C. Hung and K. M. Lim 1993 International Journal of Solids and Structures 30, 3357–3379. A continuum three-dimensional vibration analysis of thick rectangular plates. 23. K. M. Liew and K. Y. Lam 1991 Journal of Sound and Vibration 147, 255–264. Vibration analysis of multispan plates having orthogonal straight edges. 24. K. M. Liew and C. M. Wang 1992 Mechanics of Structures and Machines 20, 281–292. Vibration analysis of plates by the pb-2 Rayleigh–Ritz method: mixed boundary conditions, reentrant corners, and internal curved supports.

Journal of Sound and Vibration (1995) 184(4), 611–626

A REDUCTION METHOD FOR THE ANALYSIS OF NON-LINEAR FORCED VIBRATION OF PLATES J. Kong and Y. K. Cheung Department of Civil and Structural Engineering, University of Hong Kong, Hong Kong (Received 30 September 1993, and in ﬁnal form 21 February 1994) The non-linear forced vibration of plates is investigated by using an implicit time integration scheme and a reduction method. The plate is ﬁrst discretized into generalized ﬁnite strips with the non-linear strains of von Ka´rma´n type included. A set of basis vectors is generated and transformation on to the subspace spanned by these vectors is then performed in such a way that the time-stepping dynamic analysis is totally carried out within the subspace. Because of the reduction of number of unknowns, a signiﬁcant reduction in computing time can be achieved as compared with that for the analysis of the complete system. An example of a plate with diﬀerent time-varying loads is given to demonstrate the potential of the method. 7 1995 Academic Press Limited

1. INTRODUCTION

The understanding of geometrically non-linear, forced vibration of plates with time-varying loads is important in many engineering designs of structures. Because of the membrane action of an elastic plate undergoing large deﬂection, the non-linear stiﬀening behaviour has to be taken into account. For large amplitude vibration of a plate, the response can be obtained by using an implicit time integration scheme with equilibrium iterations for non-linearity. Within each time step, the dynamic incremental equilibrium equation is established and solved by using equilibrium iterations. However, this process consumes a lot of computational time in the evaluation of the tangent stiﬀness matrix and the calculation of the out-of-balance internal force vector required for equilibrium iterations. For preliminary analysis or study, a reduction of computational eﬀort is highly desirable and frequently necessary. To this end, a reduction method can be applied. This method is aimed at savings in computational costs by using a smaller number of generalized variables to represent the original problem with a larger number of degrees of freedom. It is assumed that the vector of discrete fundamental unknowns can be, with suﬃcient accuracy, represented as a linear combination of a small number of global approximation functions (or basis vectors). The coeﬃcients of the linear combination are determined by solving a reduced system of equations which is the projection of the full system of equations on to the subspace spanned by the basis vectors. The eﬀectiveness of this approach depends, to a great extent, on the proper selection of the reduced basis vectors. By using this method, a compromise between the prediction of complex behaviour and the computational cost can be achieved. The method is, therefore, suitable for preliminary analysis or study of structures for which only a micro- or minicomputer is required. 611 0022–460X/95/290611 + 16 \$12.00/0

7 1995 Academic Press Limited

612

j. kong and y. k. cheung 2. LITERATURE REVIEW

A number of reduction methods for geometrically non-linear dynamic analysis have been proposed since the mid-1970s [1–7]. Most of these studies are based on either the tangent stiﬀness method (TSM) or the pseudo-force method (PFM). In the TSM, the change of basis is performed at each time step or at a certain predeﬁned interval using the mode shapes corresponding to the instantaneous system matrices. The updating of mode shapes (by solving the corresponding eigenproblem) during time-stepping analysis was a time-consuming process until, very recently, an eﬃcient updating procedure was proposed by Chen et al. [8]. Regardless of this eﬃcient updating procedure, it still suﬀers from the drawback that the incompatibility between the old and the new basis introduces a truncation error in velocities and accelerations every time a basis change is made. This error produces a continuously growing lack of equilibrium in the complete system that aﬀects the ﬁnal results [9]. In the PFM, a single set of mode shapes, based on the linear system matrices, is used throughout the analysis. The linear system matrices are employed throughout the complete response calculation, and the non-linearities are taken as a pseudo-force on the right side of the dynamic equilibrium equations. This approach is eﬃcient because it requires neither the solution of the equilibrium equation nor the updating of the tangent stiﬀness matrix at each time step. However, the method is applicable only to weakly non-linear systems. Recently, Idelsohn and Cardona [9, 10] and Chang and Engblom [11] proposed a new approach which is based on a set of mode shapes or Ritz vectors together with their path derivatives as the basis vectors for non-linear dynamic analysis of beams and arches. The idea of path derivatives was originally developed by Noor using the concept of perturbation for non-linear static analysis [12]. An evaluation of this approach was recently conducted by Kapania and Bynn [13]. By combining the perturbation method with the harmonic balance technique, the steady state response of non-linear free vibration of plates was investigated by Noor et al. [14]. Another interesting approach proposed by Chan and Yao [15] and Das et al. [16] involves a number of displacement vectors which are collected at diﬀerent times during the ﬁrst half cycle of the complete system analysis. These vectors are orthonormalized with respect to the mass matrix by using the Gram–Schmidt process. Subsequent analysis is then carried out in the subspace spanned by these vectors. It should be noted that most of the previous studies dealt with simple line-type structures such as a single-span beam, an arch or a frame. A very limited amount of work has been devoted to the analysis of plates [13], shells [4] or general three-dimensional structures [7]. Moreover, no attempt in previous studies has been made to use reduction method for analyzing plates subjected to loadings with diﬀerent time-varying functions. Other researchers such as Bathe and Gracewshi [17] and Sheu et al. [18] have suggested the use of substructuring for reducing computational eﬀort while Liu and Lin [19] employed known elastic solutions for the dynamic analysis of elastic–plastic beams and plates. 3. THE PRESENT APPROACH

The eﬃciency and accuracy of the reduced basis technique in non-linear dynamic problems depends on: (i) the number of vectors required to simulate accurately the response; (ii) the eﬃciency of generating the basis vectors; (iii) the frequency of updating the basis vactors; and (iv) the eﬃciency of the algorithm for transforming the complete system to the reduced system. In this paper, the Ritz vectors are adopted because they have been shown to be accurate, eﬃcient and relatively easy to generate when compared with vibration mode shapes for linear forced-vibration analysis [20]. Their performance in

non-linear forced vibrations of plates

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non-linear dynamic analysis is examined in this work. To maximize the eﬃciency of the method, a single set of vectors is generated and then used throughout the complete response analysis. To this end, a set of Ritz vectors is ﬁrst generated by using the linear elastic stiﬀness matrix. Non-linearity is only partially taken into account by updating the tangent stiﬀness matrix and the eﬀective load vector during time-stepping analysis. Depending on the degree of non-linearity and the quantities of interest, these linear Ritz vectors are augmented with another set of Ritz vectors computed from the tangent stiﬀness matrix based on the static displacement corresponding to the maximum load intensity of the time-varying load. As a result, the coupling of bending and membrane actions (i.e., the non-linear eﬀect) can be introduced into the basis vectors before the time-stepping analysis. In this case, a non-linear static analysis of the plate with the maximum amplitude of the time-varying load applied on it is needed. (This can be justiﬁed because in practice, a non-linear static analysis may very often, precede a non-linear dynamic analysis in order to gain insight into the non-linear response of the system to a given spatial distribution of the applied loading. The present method attempts to extract useful information from the non-linear static analysis.) The eﬃciency of the method can be further enhanced by transforming the full system equations to the subspace prior to the time-stepping analysis so that the updating of the tangent stiﬀness matrix and the internal force vector can be done completely within the subspace. A brief description of the formulation of the present approach is given in the following sections. To demonstrate the potential of the method, a square plate with diﬀerent time-varying loads is analyzed.

4. GEOMETRICALLY NONLINEAR FORMULATION

A generalized rectangular spline ﬁnite strip, which is capable of modelling isotropic and laminated composite plates has been developed [21–23]. In this paper, the element is used to discretize an isotropic thin plate subjected to a time-varying external load; see Figure 1. A brief review of the element is as follows. The generalized spline ﬁnite strip is formulated by using a modiﬁed form of the third order plate theory originally developed by Reddy [24]. With reference to a Cartesian

Figure 1. A plate with an externally applied load. Loading P(x, y, t) = P(x, y)T(t); T(t) = time varying function, P(x, y) = spatial distribution.

Figure 2. The non-linear stiﬀening behaviour of a plate, K0 = initial stiﬀness matrix; KT = tangent stiﬀness matrix corresponding to the displacement at the maximum load level.

j. kong and y. k. cheung

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co-ordinate system, the displacement ﬁeld of the modiﬁed third order plate theory can be written as u1 (x, y, z) = u − zw,x + z[1 − (4/3)(z/h)2]lx , u2 (x, y, z) = v − zw,y + z[1 − (4/3)(z/h)2]ly ,

u3 (x, y) = w(x, y),

(1)

where u, v and w denote the displacements of a point (x, y) on the mid-plane. The transverse shear strains exz and eyz at the mid-plane are represented by lx and ly respectively. The transverse normal strain is neglected in the analysis. Since a linear elastic material is assumed, the displacement ﬁelds satisfy the zero traction conditions on the top and bottom surfaces of the plate and contain the same number of independent variables as the Mindlin plate theory. The displacement functions of the generalized spline ﬁnite strip with s sections can be written as s+1

u(x, y) = s [M(x)]fi (y){z}i ,

s+1

v(x, y) = s [M(x)]fi (y){j}i ,

i = −1

i = −1 s+1

w(x, y) = s [N(x)]fi (y){a}i , i = −1 s+1

lx (x, y) = s [M(x)]fi (y){b}i ,

s+1

ly (x, y) = s [M(x)]fi (y){g}i ,

i = −1

(2)

i = −1

where [M(x)] and [N(x)] are the one-dimensional linear shape functions and the Hermite cubic polynomial respectively. fi (y) is the ith member of the B3 spline series and {z}, {j}, {a}, {b} and {g} are the unknown parameters. By taking the von Ka´rma´n non-linear strains into consideration, and using the total Lagrangian formulation [25–27], the following dynamic incremental equilibrium equation, with m unknowns, at time t + Dt can be derived: M t + Dt{d }k + t[KL + KN + Ks ] d{d}k = t + Dt{F} − t + Dt{R}k − 1, or M t + Dt{d }k + tKT d{d}k = t + Dt{F} − t + Dt{R}k − 1,

(3)

where {d} = {d} + d{d} and {R} = [KE ] {d} . The left superscript represent the time of the conﬁguration in which the quantity occurs and the right superscripts denote the iteration count. t + Dt{F} is the external load vector, t + Dt{R}k − 1 is the internal resistance equivalent to the element stresses that correspond to the displacements t + Dt{d}k − 1, and t + Dt{d}k is the displacement vector at the end of iteration k and time t + Dt. When the iteration count k = 1, t + Dt{d}0 = t{d}, t + Dt{R}0 = t{R} and d{d}1 = d{d}. The matrices KL and KN represent the small displacement stiﬀness matrix and the large displacement matrix respectively. The geometric matrix Ks depends on the current inplane stresses sx , sy and sxy . The sum of these matrices KT is the so-called tangent stiﬀness matrix. Since a linear elastic material is considered in this study, the internal resistance {R} can be expressed in terms of a secant stiﬀness matrix KE . The derivation of the mass matrix is given in reference [23]. The inertia term on the left side of the equilibrium equation drops out in case of static analysis, and the standard Newton–Raphson iteration with load incrementation is employed to solve the static equilibrium equation. A residual force convergence criterion is used in the static analysis. All integrals are carried out over the original volume of each section of a strip with three integration points in each of the X and Y directions. Details of the formulation are given in reference [21–23]. t + Dt

k

t + Dt

k−1

k

t + Dt

k−1

t + Dt

k − 1 t + Dt

k−1

non-linear forced vibrations of plates

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5. REDUCED SYSTEM OF EQUATIONS

A single set of Ritz vectors is ﬁrst generated. It consists of two subsets of vectors, namely (i) p linear vectors computed from the initial linear elastic stiﬀness matrix, and (ii) q non-linear vectors computed from the tangent stiﬀness matrix based on the non-linear displacement corresponding to the maximum load intensity of the time-varying load. These vectors are used throughout the response analysis. To generate the non-linear basis vectors, a non-linear static analysis of the plate is ﬁrst performed with the maximum amplitude of the time-varying load applied on it; see Figure 2. The algorithm for generating Ritz vectors

Figure 3. The response of the square plate subjected to a sinusoidal load of maximum amplitude 10 q0 . t = 0·002 s per time step. (a) In-plane displacement response; (b) vertical displacement response. p = number of linear vectors or Ritz vectors computed from the initial linear elastic stiﬀness matrix. q = number of non-linear vectors or Ritz vectors computed from the tangent stiﬀness matrix at maximum load level. For (a): q, Full; t, p = 4 and q = 4. For (b) q, full; +, p = 2; e, p = 3; r, p = 4; ×, p = 8; t, p = 4 and q = 4.

j. kong and y. k. cheung

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can be found in reference [20]. The displacement vector with m unknowns can be approximated by the n (n = p + q) basis vectors via the transformation n

{d} 2 [G]{c} = s Gi ci ,

(4)

i=1

where [G] = [G1 , G2 . . . , Gn ] is an m × n matrix of basis vectors with rank n and {c} is the column vector of generalized variables with n entries (m w n). By substituting equation (4) into the dynamic equilibrium equation (3) and setting the residual to be orthogonal to the subspace spanned by the basis vectors (i.e., a Galerkin approach applied to the discrete

f

Figure 4. The response of the square plate subjected to a half-sine load of maximum amplitude 10 q0 . t = 0·002 s per litre time step. (a) In-plane displacement response; (b) vertical displacement response. Key as Figure 3.

non-linear forced vibrations of plates

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Figure 5. The response of the square plate subjected to a sinusoidal load of maximum amplitude 50 q0 . t = 0·001 s per time step. (a) In-plane displacement response; (b) vertical displacement response. Key as Figure 3.

dynamic equilibrium equations [28]), the following reduced system of equations can be obtained: k t + Dt {F*} − t + Dt{R*}k − 1. (5) M*t + Dt{c }k + tK* T d{c} = T T T Here M* = [G]TM[G], K* T = [G] KT [G], {F*} = [G] {F} and {R*} = [G] {R}, with the superscripts denoting time and iteration omitted for clarity. Details of the transformation are described in the next section.

6. TRANSFORMATION

It is well-known that for geometrically nonlinear analysis of plates, the strain– displacement equations can be written in the condensed form {e} = [BL + B1 (d)]{d} = [B(d)]{d}, where BL denotes the linear part. The non-linear matrix B1 (d) is a function of the current displacement {d}. By taking the variation with respect to {d}, it can be shown

j. kong and y. k. cheung

618

Figure 6. The response of the square plate subjected to a half-sine load of maximum amplitude 50 q0 . t = 0·001 s per time step. (a) In-plane displacement response; (b) vertical displacement response. Key as Figure 3.

that the incremental strain–displacement equation is given by d{e} = [BL +  (d)]d{d}, where BN (d) = 2B1 (d). A linear elastic material is assumed in this BN (d)]d{d} = [B study. If D is the elasticity matrix with in-plane components Dp and transverse shear component Ds , that is, Dp 0 D= , 0 Ds

\$

%

the components of the tangent stiﬀness matrix can be written as

g g

KL = BLT DBL dV, KN = [BLT Dp BN (d) + BNT (d)Dp BL + BNT (d)Dp BN (d)] dV,

non-linear forced vibrations of plates

g

0

1

sx sxy (w,x , w,y ) dV, sxy sy while the secant stiﬀness matrix is given by Ks = (w,x , w,y )T

619 (6)

g

KE = [BLT DBL + BNT (d)DP BL + BLT Dp B1 (d) + BNT (d)Dp B1 (d)] dV. To update the tangent and secant stiﬀness matrices during the time-stepping analysis, they can be evaluated ﬁrst in the complete system and then transformed back to the subspace by using the transformation matrix [G]. This procedure is expensive in terms of computing time. In this study, however, the tangent and secant stiﬀness matrices are transformed immediately after the basis vectors are established. For a linear elastic material, this can be easily done prior to the time-stepping analysis. The procedure of this transformation is as follows.

Figure 7. The response of the square plate subjected to a step load of maximum amplitude 50 q0 . t = 0·001 s per time step. (a) In-plane displacement response; (b) vertical displacement response. Key as Figure 3.

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Figure 8. The response of the square plate subjected to a step-change load of maximum amplitude 50 q0 . t = 0·001 s per time step. (a) In-plane displacement response; (b) vertical displacement response. Key as Figure 3.

T The linear stiﬀness matrix is transformed in the usual manner: K* L = [G] KL [G]. By substituting equation (4) into equation (6), the large displacement matrix (the secant stiﬀness matrix can be transformed in a similar manner) can be written as the summation of two parts: that is, n

T T K* N = [G] KN [G] = [G] s i=1 n

n

+[G]T s s i=1 j=1 n

=s i=1

6 g [G]T

g

g

[BLT Dp BN (Gi )ci + BNT (Gi )Dp BL ci ] dV[G]

[BNT (Gi )Dp BN (Gj )ci cj ] dV[G]

7

[BLT Dp BN (Gi ) + BNT (Gi )Dp BL ] dV[G] ci

non-linear forced vibrations of plates n

n

+s s i=1 i=1 n

6 g

7

[G]T [BNT (Gi )Dp BN (Gj )] dV[G] ci cj n

n

= s (Q1i ci + Q2i ci ) + s s (Q3ij ci cj ), i=1

621

i=1 j=1

with the ﬁrst and second parts being linear and quadratic functions of the current displacement, respectively. The matrices Q1i , Q2i and Q3ij represent the components of the reduced large displacement matrix. These matrices are computed and stored after the basis vectors are generated. They will be retrieved during the time-stepping analysis and, by multiplying by the corresponding generalized variables, the updating of the large displacement matrix can be done without going back to the full system of equations. For the geometric matrix, the current in-plane stresses can be written explicitly in terms of in-plane strain: that is, {s}p = Dp {e}p , where the right subscript denotes in-plane components. As the von Ka´rma´n non-linear strains are adopted in this study, the in-plane

Figure 9. The response of the square plate subjected to a triangular load of maximum amplitude 50 q0 . t = 0·001 s per time step. (a) In-plane displacement response; (b) vertical displacement response. Key as Figure 3.

622

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Figure 10. The response of the square plate subjected to a exponential load of maximum amplitude 50 q0 . t = 0·001 s per time step. (a) In-plane displacement response; (b) vertical displacement response. Key as Figure 3.

strain can be further subdivided into two parts: namely, a linear function and a quadratic function of the current displacements. By using a procedure similar to that for the large displacement matrix, the transformation and updating of the geometric matrix can be accomplished. It is noteworthy that for external loads with diﬀerent time-varying functions, but identical maximum amplitude and spatial distribution, the same set of vectors and transformed matrices can be used for the time-stepping analysis. 7. THE NEWMARK TIME INTEGRATION

By using Newmark time integration in the time domain, equation (5) can be re-written as 2 k t + Dt {F*} − t + Dt{R*}k − 1 (tK* T + 4/(Dt) M*)d{c} =

− M*[(t + Dt{c}k − 1 − t{c}) − (4/(Dt))t{c } − t{c }].

non-linear forced vibrations of plates

623

During the time-stepping analysis, the tangent stiﬀness matrix is updated only after a user-speciﬁed number of time steps and remains unchanged during equilibrium iterations within each time step. For equilibrium convergence, a displacement criterion is adopted in this study. 8. NUMERICAL EXAMPLE

The present method was coded with Fortran 77 and implemented on a PC-486 compatible microcomputer. In order to test its accuracy and eﬃciency, a square plate under a uniformly distributed load with diﬀerent time-varying functions was analyzed. (As mentioned earlier in the Literature Review section, no attempt in previous studies has been made to use the reduction method for analyzing plates subjected to loadings with diﬀerent time-varying functions.) Details of the plate are given in Table 1. Half of the plate is divided into four strips and eight sections. To establish conﬁdence in this model, the ﬁrst few natural frequencies were predicted and compared with the analytical solution. It is apparent, from Table 2, that the spline ﬁnite strip mesh gives reliable frequencies. The time increment used in the analysis are, respectively, Dt = 0·002 s and 0·001 s for the maximum load amplitudes of 10 q0 and 50 q0 . These time increments are able to give an accurate, stable and eﬃcient solution for full system analysis. Zero initial conditions (displacement, velocity and acceleration) were assumed. In all cases, the tangent stiﬀness matrices were updated on every ﬁfth time step. To compare the reduced basis solutions with the full system solutions, the vertical displacement (w) at the centre of the plate and the in-plane displacement (v) at a quarter-span along the centreline of the plate were monitored. The comparison of computing times is measured in terms of time reduction: percentage reduction in computing time = (1 − T1 /T2 ) × 100, where T1 is the computing time for a reduced basis analysis, which includes the time required to generate the basis vectors, to carry out the Table 1 Conﬁguration and loading of the thin isotropic plate Span = 243·8 cm,−6thickness = 0·635 cm, Young’s modulus = 7·031 × 105 kg/cm2, n = 0·25, density = 2·547 × 10 kgs2/cm4; q(x, y, t) = q0 f(t), q0 = 4·882 × 10−4 kg/cm2; Fundamental period of vibration = 0·1905 s Boundary condition: simply supported edges with unyielding supports in the horizontal plane Six diﬀerent types of time-varying functions are considered: 1. Sinusoidal load f(t) = sin (pt/t1 ), 0 E t Q a, t1 = 0·1257 s

6

7

0 E t Q t1 t1 E t Q a

sin (pt/t1 ), 0,

t1 = 0·1257 s

f(t) =

f(t) = 1,

f(t) =

6

f(t) =

6

0 E t Q t1 , t1 E t Q a

7

t1 = 0·15 s

f(t) =

6

0 E t E t1 t1 E t Q a

7

t1 = 0·15 s

0EtQa

7

1, 0 E t Q t1 , 0, t1 E t Q a 1 − t/t1 , 0,

(1 − t/t1 )exp(−lt/t1 ), 0,

t1 = 0·15 s

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Table 2 Comparison of natural frequencies l = va 2(rh/D)0·5

Mode 1 2 5 6 7

Frequencies, l ZXXXXXXXXCXXXXXXXXV Spline ﬁnite strip Analytical 19·74 49·35 98·77 98·80 128·24

19·74 49·35 98·69 98·69 128·30

non-linear forced vibrations of plates

625

9. CONCLUSIONS

A non-linear dynamic analysis of a thin isotropic plate has been presented in this paper. For diﬀerent load intensities and time-varying functions with low frequency content, the accuracy and eﬃciency of the reduction method have been demonstrated. For moderate load amplitude, satisfactory results can be obtained with Ritz vectors computed from the linear elastic stiﬀness matrix alone, provided that vertical displacement is the only quantity of interest. The computing time increases rapidly as the number of basis vectors increases, and hence the eﬃciency of the reduction method depends very much on the number basis vectors required to simulate accurately the response. The eﬃciency and accuracy of the present method are due to (i) the good performance of Ritz vectors, (ii) the inclusion of non-linear eﬀects in the Ritz vectors by performing a non-linear static analysis a priori, (iii) the fact that a single set of vectors is used throughout the complete response analysis, and (iv) the special transformation technique. Extension of the method to other types of structures with similar nonlinear behaviour (e.g., non-linear stiﬀening or softening without limit points) and loadings with high frequency content is also feasible. REFERENCES 1. N. F. Morris 1977 Computers and Structures 7, 65–72. The use of modal superposition in nonlinear dynamics. 2. P. Lukkunaprasit 1980 Earthquake Engineering and Structural Dynamics 8, 237–250. Dynamic response of an elastic–viscoplastic system in modal co-ordinates. 3. B. Mohraz, F. E. Elghadamsi and C. J. Chang 1991 Earthquake Engineering Structural Dynamics 20, 471–481. An incremental mode-superposition for nonlinear dynamic analysis. 4. C. Byun and R. K. Kapania 1992 Computers and Structures 44, 255–262. Nonlinear transient response of imperfect hyperbolic shells using a reduction method. 5. P. Leger and S. Dussault 1992 Earthquake Engineering and Structural Dynamics 21, 163–176. Non-linear seismic response analysis using vector superposition methods. 6. B. P. Jacob and N. F. F. Ebecken 1992 Computers and Structures 45(2), 333–347. Adaptive reduced integration method for nonlinear structural dynamic analysis. 7. Y. T. Lin and T. C. Sun 1993 Journal of Vibration and Acoustics 115, 397–402. Mode superposition analysis of viscously damped nonlinear structural systems using an incremental algorithm. 8. S. H. Chen, T. Xu and Z. S. Liu 1992 Computers and Structures 45(3), 553–556. Nonlinear frequency spectrum in nonlinear structural analysis. 9. S. R. Idelsohn and A. Cardona 1985 Computer Methods in Applied Mechanics and Engineering 49, 253–279. A reduction method for nonlinear structural dynamic analysis. 10. S. R. Idelsohn and A. Cardona 1985 Computers and Structures 20, 203–210. A load-dependent basis for reduced nonlinear structural dynamics. 11. C. Chang and J. J. Engblom American Institute of Aeronautics and Astronautics Journal 29, 613–618. Nonlinear dynamical response of impulsively loaded structures: a reduced basis approach. 12. A. K. Noor 1982 Computer Methods in Applied Mechanics and Engineering 34, 955–985. On making large nonlinear problems small. 13. R. K. Kapania and C. Byun 1993 Computational Mechanics 11, 65–82. Reduction methods based on eigenvectors and Ritz vectors for nonlinear transient analysis. 14. A. K. Noor, C. M. Andersen and J. M. Peters 1993 Computer Methods in Applied Mechanics and Engineering 103, 175–186. Reduced basis technique for nonlinear vibration analysis of composite panels. 15. A. S. L. Chan and M. S. Yao 1991 International Journal of Space Structures 6, 159–177. Finite element nonlinear dynamic structural analysis using the reduced basis method. 16. S. K. Das, S. Uktu and B. K. Wada 1990 Computing Systems in Engineering 1, 577–589. Use of the reduced basis technique in the inverse dynamics of large space cranes. 17. K. J. Bathe and S. Gracewski 1981 Computers and Structures 13, 699–707. On nonlinear dynamic analysis using substructuring and mode superposition.

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18. C. H. Sheu, G. De Roeck, M. Van Laethem and P. Geyskens 1990 Computers and Structures 35, 593–601. Application of the substructuring technique to non-linear dynamic structural analysis. 19. S. C. Liu and T. H. Lin 1979 Earthquake Engineering and Structural Dynamics 7, 147–159. Elastic–plastic dynamic analysis of structures using known elastic solutions. 20. P. Leger and E. Wilson 1987 Engineering Computation 4, 309–318. Generation of load dependent Ritz transformation vectors in structural dynamics. 21. Y. K. Cheung and J. Kong 1993 Computers and Structures 47(2), 189–192. Linear elastic stability analysis of shear-deformable plates using a modiﬁed spline ﬁnite strip method. 22. J. Kong and Y. K. Cheung 1993 Computers and Structures 46(6), 985–988. Application of the spline ﬁnite strip to the analysis of shear-deformable plates. 23. J. Kong and Y. K. Cheung 1993 (Submitted to Journal of Thin-walled Structures) A generalized spline ﬁnite strip for the analysis of plates. 24. J. N. Reddy 1984 Journal of Applied Mechanics 51, 745–752. A simple higher-order theory for laminated composite plates. 25. K. J. Bathe 1982 Finite Element Procedures in Engineering Analysis. Englewood Cliﬀs, New Jersey: Prentice-Hall. 26. K. Washizu 1982 Variational Methods in Elasticity and Plasticity. Oxford: Pergamon Press. 27. O. C. Zienkiewicz 1977 The Finite Element Method. London: McGraw-Hill; third edition. 28. J. Kong and Y. K. Cheung 1993 Proceedings of the Second Asian-Paciﬁc Conference on Computational Mechanics, Sydney, Australia, 1267–1273. Reduced-basis technique and its application.

Journal of Sound and Vibration (1995) 181(2), 341–353

THE APPLICATION OF A NEW FINITE STRIP TO THE FREE VIBRATION OF RECTANGULAR PLATES OF VARYING COMPLEXITY Y. K. Cheung and J. Kong Department of Civil and Structural Engineering, University of Hong Kong, Hong Kong (Received 30 September 1993, and in ﬁnal form 14 January 1994) A new ﬁnite strip method is applied to the vibration of rectangular plates with complicated boundary and internal support conditions. Based on the classical thin plate theory and displacement formulation, the new ﬁnite strip element is conforming with C1 -continuity. The displacement function of the strip element is expressed as the product of cubic Hermitian functions in the transverse direction and a set of computed static modes in the longitudinal direction. By means of a continuous beam computer program, these longitudinal modes can be computed a priori. Numerical examples are given to demonstrate the versatility, accuracy and eﬃciency of the element.

1. INTRODUCTION

The classical ﬁnite strip method, pioneered by Cheung [1], is an eﬃcient analysis tool for structures with regular geometry and simple boundary conditions. It is a hybrid Ritz approach which combines the versatility of the ﬁnite element method and rapid convergence of the Ritz method by selecting suitable trial functions a priori. Since its introduction in 1967, the method has undergone extensive developments: namely (1) the development of diﬀerent types of longitudinal interpolation, (2) application to material and/or geometric non-linear structures and (3) combination with geometric mapping for the analysis of structures with irregular geometry. For an ideal rectangular, thin plate, bending, ﬁnite strip element based on the displacement method, the element should be C1 -continuous such that a plate with abruptly changing thickness can be treated. When analyzing a plate with point loads, the element should be able to give rapidly converged solutions. In addition, the strip element should be able to take care of complicated support conditions in an easy and uniﬁed manner. Although it is not necessary, easy implementation of the method is highly desirable. Based on these criteria, the present ﬁnite strip element has been developed. The displacement function of the strip element is expressed as the product of cubic Hermitian functions in the transverse direction and a set of computed static modes in the longitudinal direction. Unlike classical ﬁnite strips, the present ﬁnite strip employs a series of computed static modes instead of a series of analytical vibration modes. The method is simple in terms of its basic concept and computer implementation. It involves no complicated functions or mathematics. The element has been successfully applied to the static analysis of plates with abruptly changing rigidity and subjected to point loads [2–4]. This paper describes the use of this new ﬁnite strip for the study of vibrations of rectangular plates with thickness changes and various support conditions. 341 0022–460X/95/120341 + 13 \$08.00/0

7 1995 Academic Press Limited

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Figure 1. (a) A strip with varying longitudinal rigidity; (b) an equivalent beam of the strip in (a); (c) computed static mode Ym (y).

2. LONGITUDINAL INTERPOLATION

A ﬁnite strip element with free ends and varying rigidity along the y-axis is shown in Figure 1(a). In Figure 1(b), a unit width of the strip is taken out as a beam with the same variation of the longitudinal rigidity as the original strip. The beam is divided into p beam elements, which are not necessarily of equal length. Among the p + 1 joints, a number of them, say r, are assigned as nodes. Notice that the joint at each end of the beam is always assigned as a node. Depending on the actual kinematic boundary conditions at each end of the ﬁnite strip, appropriate boundary conditions are imposed at the ends of this beam: see Table 1. In a manner similar to that for Lagrangian shape functions, a unit deﬂection is imposed to one of these nodes while the remaining nodes are constrained with zero deﬂection (see Figure 1(c)). This procedure is then applied to each of the nodes and a total of r computed static modes can be obtained. These static modes correspond to the exact

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343

Table 1 Treatment of boundary conditions Kinematic boundary conditions (2) (3)

(1) Clamped Sliding–clamped Simply supported Free

Sliding–clamped Sliding–clamped Free Free

Clamped Clamped Simply supported Simply supported

(1) Actual boundary condition at one end (y = 0, say) of a strip: clamped, w = w,y = 0; sliding–clamped, w,y = 0; simply supported, w = 0; free, w \$ 0, w,y \$ 0. (2) Boundary condition of the beam at y = 0 when computing the static mode for the corresponding end node: sliding–clamped, w,y = 0; free, w \$ 0, w,y \$ 0. (3) Boundary condition of the beam at y = 0 when computing the static modes for each of the internal nodes and the end node at y = b: clamped, w = w,y = 0; simply supported, w = 0; Note that w,y = 1w/1y. For example, if a strip is clamped at the end y = 0, then a sliding clamped support is assigned at the corresponding end of the beam when computing the static mode for the end node at y = 0. When computing the static modes for the remaining nodes, the beam is fully clamped at the end y = 0.

deﬂection patterns of a beam subjected to a set of point loads at the nodes. They can be easily obtained by means of a continuous beam program. The amount of computing eﬀort required is insigniﬁcant. In terms of the usual beam shape functions, a static mode can be written as p

Ym (y) = s [N(y)]{am }k , k=1

with Ym (ym ) = 1 for m = 1, 2, . . . , r, where r E p + 1, and Ym (yn ) = 0 for n \$ m. Also, [N(y)] are the usual beam shape functions and {am }k is the displacement vector of the joints at the ends of beam element k which corresponds to the longitudinal mode Ym . The summation is carried out with due regard to the joint arrangement. These static modes are identical for all strips with the same variation of longitudinal rigidity. They take into account the presence of point loads and abruptly changing thickness. They can be easily obtained by means of a continuous beam computer program. The amount of computing eﬀort required is insigniﬁcant. It is noteworthy that C1 -continuity is embedded in these static modes. It is also interesting and important to recognize that each of these computed static modes represents, in structural engineering terms, the inﬂuence line for the reaction force at the corresponding node. As a consequence, one can easily derive the following properties: r

s Ym (y) = 1 m=1

r

and

s Ym (y)ym = y. m=1

3. FORMULATION

The displacement function of a ﬁnite strip element can be written as r

r

m=1

m=1

W(x, y) = s [N(x)]Ym (y){d}m = s [Qm ]{d}m , where {d}m = {W1 , u1 , W2 , u2 }m , ui = (1W/1x)i , [Qm ] = [N(x)]Ym (y), and [N(x)] are the usual beam shape functions in the x-direction. It can be easily shown that for the present

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Figure 2. A typical strip with ﬁve master nodes on each nodal line.

strip element with no supports, the rigid body criterion is satisﬁed. The present strip element, with two degrees of freedom per node, is conforming with C1 -continuity. Unlike all other ﬁnite strips, the unknowns {d}m of the present strip represent the actual nodal displacements (W) and rotations (1W/1x) (see Figure 2). As a consequence, support conditions involving vertical deﬂections and/or rotations about the y-axis can be implemented in the same fashion as is done for the usual ﬁnite element method, or, in other words, they can be treated a posteriori. However, for rotations about the x-axis, 1w/1y, at either end of the ﬁnite strip, they are taken into account when forming the computed shape functions: that is, they are treated a priori. By following the standard displacement formulation, the stiﬀness matrix and mass matrix can be easily derived. For the stiﬀness matrix, the strain-displacement matrix can be written as r

{e} = s [B]m {d}m , m=1

where [B]m = {−[Qm ]xx , −[Qm ]yy , 2[Qm ]xy }T. It can be shown that the stiﬀness matrix is given by [S]12 [S]11 ··· [S]1r ··· [S]mn , ··· [S] = ··· [S]r1 [S]r2 [S]rr

&

'

where

g

[S]mn = [B]Tm [D][B]n dx dy and [D] is the elasticity matrix. In addition, the consistent mass matrix can be written as

&

[M]12

[M]11

[M] = where

··· [M]r1

··· [M]mn

[M]r2

[M]1r ···

···

[M]rr

'

,

g

[M]mn = [Q]Tmr[Q]n dx dy, and r is the density of the material. A thorough derivation of [S]mn and [M]mn can be found in reference [1].

application of new finite strip method

345

The major diﬀerence between the present ﬁnite strip and the classical ﬁnite strips (or the spline ﬁnite strips) is that the computation of the integrals in the Y-direction for the present ﬁnite strip involves no numerical integration. The integrals are computed as follows: I1 =

g g g g g

b

0

I2 =

b

b

b

k=1 p

lk

k=1 p

b

0

k=1 p

Y'm Y'n dy = s {am }Tk k=1

[N0(y)]T[N(y)] dy {an }k ,

0

lk

[N(y)]T[N0(y)] dy {an }k ,

0 lk

[N0(y)]T[N0(y)] dy {an }k ,

0 lk

% % % % %

[N(y)]T[N(y)] dy {an }k ,

0

Y0m Y0n dy = s {am }Tk

0

I5 =

p

\$g \$g \$g \$g \$g

lk

Ym Y0n dy = s {am }Tk

0

I4 =

k=1

Y0m Yn dy = s {am }Tk

0

I3 =

p

Ym Yn dy = s {am }Tk

[N'(y)]T[N'(y)] dy {an }k .

0

The explicit form of the integration of the beam shape functions can be found in reference [5]. Assembly of the ﬁnite strips is carried out in the standard manner. By incorporating a continuous beam computer program with a standard ﬁnite strip program, the method can be easily implemented. In-depth formulations are reported elsewhere [2–4]. 4. NUMERICAL EXAMPLES

In the following examples, the study of the free vibration of rectangular plates with diﬀerent support conditions is reported. In most cases, results obtained by the present method are compared with classical ﬁnite strip solutions and the spline ﬁnite strip solutions (all as quoted in the paper by Leung and Au [6]). In all cases, unless otherwise speciﬁed,

Figure 3. Rectangular plates of various homogeneous boundary conditions. ——, Free (F); ––––, simply supported (S); clamped (C). (a) S–S–S–S; (b) C–S–S–S; (c) C–F–C–F; (d) F–S–F–S; (e) C–C–F–C; (f) S–S–F–S.

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Table 2 Frequencies of square plates with various homogeneous boundary conditions; li = vi a 2(rt/D)1/2, n = 0·3

Mode sequence

Dimensionless frequencies, li ZXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXV 1 2 3 4 5 6 7 8

S–S–S–S (Figure 3(a)) Leissa [7] 19·74 Finite strip 19·74 Spline strip 19·74 Present strip 19·74

49·35 49·32 49·36 49·36

49·35 49·34 49·38 49·38

78·96 78·91 78·98 78·98

98·70 98·64 98·80 98·80

98·70 98·68 99·21 99·28

128·30 128·17 128·40 128·41

128·30 128·22 128·72 128·79

C–S–S–S (Figure 3(b)) Leissa [7] 23·65 Finite strip 23·62 Spline strip 23·65 Present strip 23·65

51·67 51·62 51·71 51·71

58·65 58·65 58·67 58·67

86·13 86·16 86·17 86·18

100·27 100·35 100·78 100·86

113·23 113·24 113·40 113·42

133·79 134·00 134·22 134·27

140·85 140·94 141·01 141·02

C–F–C–F (Figure Leissa [7] Finite strip Spline strip Present strip

3(c)) 22·27 22·29 22·33 22·25

26·53 27·08 27·14 26·53

43·66 44·76 44·80 43·65

61·47 61·53 61·72 61·45

67·55 68·29 68·46 67·59

79·90 81·08 81·07 80·18

89·55 89·67 87·82

120·73 122·40 120·83

F–S–F–S (Figure 3(d)) Leissa [7] 9·63 Finite strip 9·87 Spline strip 9·80 Present strip 9·63

16·13 16·91 17·02 16·13

36·73 38·11 37·90 36·72

38·95 39·49 39·38 38·99

46·74 47·72 48·03 46·78

70·74 74·03 72·82 70·78

75·28 76·49 76·37 75·33

87·99 88·94 89·16 88·64

C–C–F–C (Figure Leissa [7] Finite strip Spline strip Present strip

3(e)) 24·02 24·11 24·35 23·95

40·04 40·66 40·66 40·04

63·49 63·61 63·90 63·44

76·76 77·32 77·41 76·82

80·71 81·77 81·66 80·76

116·80 118·60 118·21 116·93

122·87 124·62 123·95

135·11 135·29 134·80

S–S–F–S (Figure 3(f)) Leissa [7] 11·68 Finite strip 11·92 Spline strip 12·00 Present strip 11·68

27·76 28·38 28·37 27·76

41·20 41·62 41·73 41·23

59·07 59·94 60·01 59·09

61·86 62·94 62·45 61·88

90·29 91·07 91·55 90·93

94·48 96·18 95·77 94·54

108·92 109·99 110·42 109·50

the present ﬁnite strips in the y-direction of the plates and the nodes on each nodal line are of equal distance apart. 4.1. square plates with various homogeneous boundary conditions A number of square plates with various homogeneous boundary conditions as shown in Figure 3 were analyzed. A mesh of eight strips and seven nodes was used. In all cases, the strips run in the y-direction. The results for the lowest eight frequencies are shown in Table 2. The present ﬁnite strip results are compared with results for classical ﬁnite strip meshes of ten strips and four terms and for spline ﬁnite strip meshes of eight strips with subdivisions of six sections. Rayleigh–Ritz solutions of Leissa [7] are also given. Good agreement is observed in all cases. 4.2. 2 × 2 continuous panels with various conditions In Figure 4 is shown a series of 2 × 2 continuous panels for which classical ﬁnite strip solutions are available. By assigning a node at the location of the transverse internal line

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Figure 4. 2 × 2 continuous panels with various boundary conditions. Key as for Figure 3. (a) S–S–S–S; (b) C–S–S–S; (c) C–C–S–C; (d) C–C–S–S; (e) C–S–S–F; (f) C–C–S–F; (g) C–F–C–F.

support (parallel to the x-axis), continuous plates with various boundary conditions can be solved. Zero vertical deﬂections at the intermediate line supports are imposed on the corresponding nodes in the global stiﬀness matrix. Eight strips and nine nodes were used in all cases. The dimensionless frequencies obtained are given in Table 3 and compared with analytical solutions of Leissa [7], Kim and Dickinson [8] and Liew and Lam [9] for the ﬁrst case, and the classical ﬁnite strip solutions for the rest. Good agreement is observed in all cases. 4.3. 3 × 3 continuous panels with various conditions Continuous plates with three panels in each direction were also analyzed (see Figure 5). The boundary conditions considered are the same as for the previous 2 × 2 continuous panels. Twelve strips and thirteen nodes were used in all cases. The present ﬁnite strip results are compared with those given by Wu and Cheung [10], obtained by using the continuous ﬁnite strip method. The dimensionless frequencies obtained are tabulated in Table 4 and good agreement is again observed. 4.4. square plates with mixed boundary conditions As obtained by using a mesh of eight strips and seven nodes, the lowest three frequencies of the plates (see Figure 6) are summarized in Table 5 and compared with classical ﬁnite

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Table 3 Frequencies of 2 × 2 continuous panels; li = vi l 2 (rt/D)1/2, n = 0·3; for ﬁnite strip solution, four beam modes and four strips between two supports were used

Mode sequence

Dimensionless frequencies li ZXXXXXXXXXXXXXCXXXXXXXXXXXXXV 1 2 3 4 5 6

S–S–S–S (Figure 4(a)) Leissa [7] Kim and Dickinson [8] Liew and Lam [9] Finite strip Present strip

19·74 19·74 19·74 19·74 19·74

23·63 23·98 23·71 23·67 23·65

23·63 23·98 23·71 23·68 23·66

27·06 27·70 27·10 27·11 27·07

49·35 49·35 49·37 49·35 49·48

49·35 49·35 49·37 49·74 51·77

C–S–S–S (Figure 4(b)) Finite strip Present strip

20·82 20·82

24·61 24·58

27·16 27·15

30·24 30·19

50·38 50·22

52·22 52·37

C–C–S–C (Figure 4(c)) Finite strip Present strip

24·61 24·58

29·84 29·74

30·25 30·21

34·69 34·56

54·52 54·63

57·60 57·49

C–C–S–S (Figure 4(d)) Finite strip Present strip

21·86 21·85

28·00 27·96

28·01 27·97

33·13 33·04

52·84 52·98

53·34 53·11

C–S–S–F (Figure 4(e)) Finite strip Present strip

13·58 13·65

21·38 21·46

22·36 22·34

28·41 28·38

31·65 31·58

36·18 36·65

C–C–S–F (Figure 4(f)) Finite strip Present strip

13·60 13·66

21·40 21·47

26·04 26·00

31·53 31·47

32·58 32·49

37·56 37·37

C–F–S–F (Figure 4(g)) Finite strip Present strip

13·17 13·23

14·09 14·15

21·11 21·18

21·73 21·80

28·75 28·71

34·03 33·93

strip solutions and spline ﬁnite strip solutions. Good agreement is observed in all cases. 4.5. square plates with point supports The free vibrations of a variety of point supported plates as shown in Figure 7 were studied. The conditions of zero vertical deﬂections at the point supports are easily imposed in the same fashion as is done for the usual ﬁnite element method. The results obtained by using the present method with strips and seven nodes are given in Table 6 and compared with the spline ﬁnite strip solutions. Agreement between the two sets of results is excellent. 4.6. square plates with step change in thickness The fundamental frequencies of two square plates with simply supported edges and clamped edges were analyzed by the present method with a mesh of eight strips and seven nodes (see Figure 8). Results are given in Table 7 and compared with the Kantorovich solutions of Cortinez and Laura [11], ﬁnite element results [12] and the exact solution of Chopra [13]. The present ﬁnite strip results are in good agreement with the ﬁnite element results, while the Kantorovich method, which makes use of a one-term approximation solution, slightly over-estimated the frequencies.

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Figure 5. The 3 × 3 continuous panel.

5. CONCLUSIONS

A new ﬁnite strip element has been described and applied in this paper. By using a series of computed static modes in the longitudinal direction, the ﬂexural vibration with complicated support conditions can be solved in an easy and uniﬁed manner. Because of the reduced number of unknowns at each node, there is a considerable saving in computing eﬀort with respect to the ﬁnite element method. The present ﬁnite strip possesses eﬃciency and accuracy comparable to those of the classical ﬁnite strip and the spline ﬁnite strip method. The basic concept of the method is relatively simple and requires no complicated mathematical functions. In addition, the computer implementation of the method is simple

Table 4 Frequencies of 3 × 3 continuous panels; li = vi l 2(rt/D)1/2, n = 0·3; for ﬁnite strip solution, four beam modes and four strips between two supports were used

Mode sequence

Dimensionless frequencies, li ZXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXV 1 2 3 4 5 6

S–S–S–S Finite strip Present strip

19·74 19·74

21·64 21·61

21·62 21·61

23·39 23·34

26·06 26·01

26·09 26·02

C–S–S–S Finite strip Present strip

20·24 20·23

22·08 22·06

23·72 23·66

25·35 25·27

26·46 26·39

28·12 28·10

C–C–S–C Finite strip Present strip

22·08 22·06

25·38 25·27

26·46 26·40

29·31 29·18

29·42 29·32

29·52 29·49

C–C–S–S Finite strip Present strip

20·72 20·71

24·12 24·07

24·14 24·08

27·17 27·07

28·49 28·44

28·52 28·46

C–S–S–F Finite strip Present strip

12·82 12·87

17·17 17·23

20·87 20·85

22·57 22·64

24·28 24·21

25·00 24·95

C–C–S–F Finite strip Present strip

12·82 12·87

17·17 17·23

22·57 22·55

22·57 22·64

25·81 25·72

27·69 27·62

C–F–S–F Finite strip Present strip

12·74 12·80

12·89 12·95

17·12 17·18

17·23 17·29

22·53 22·60

22·61 22·68

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Figure 6. Rectangular plates with mixed boundary conditions. Legend as for Figure 3.

because it involves only the incorporation of a continuous beam program into a standard ﬁnite strip program. In the present ﬁnite strip, all computed static modes are coupled together. Because only a few computed static modes are usually suﬃcient to give acceptable results, the problem of full coupling would not be a major drawback of the method. The Table 5 Frequencies of square plates with various mixed boundary conditions; li = vi a 2(rt/D)1/2, n = 0·3; for spline ﬁnite strip solution, nine strips and six sections were used in the ﬁrst three cases, while eight strips and six sections were used in the last case

Mode sequence

Dimensionless frequencies, li ZXXXXXXXCXXXXXXXV 1 2 3

Figure 6(a) Finite strip Spline strip Present strip

22·5 22·73 22·82

50·15 50·18

56·23 56·42

Figure 6(b) Finite strip Spline strip Present strip

25·9 26·37 26·62

52·23 52·19

61·78 62·29

Figure 6(c) Finite strip Spline strip Present strip

28·1 28·65 28·90

61·06 61·02

62·48 62·99

Figure 6(d) Finite strip Spline strip Present strip

27·63 27·83 28·13

52·39 52·41 52·77

66·24 66·25 66·95

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351

Figure 7. Rectangular plates with various point supports. (a) Corner supported rectangular plate; (b) rectangular plate supported at corners and mid-points of sides; (c) rectangular plate supported at mid-points of sides; (d) rectangular plate simply supported at periphery and at centre of plate; (e) rectangular plate simply supported at two adjacent sides and point supported at opposite corner; (f) rectangular plate clamped at two adjacent sides and point supported at opposite corner: ×, point support; other symbols as for Figure 3.

Figure 8. A square plate with step change in thickness; c/a = 0·5, h2 /h1 = 0·6.

y. k. cheung and j. kong

352

Table 6 Frequencies of square plates with various point supports; li = vi a 2(rt/D)1/2, n = 0·3; for spline ﬁnite strips solutions, eight strips and six sections were used

Mode sequence

Dimensionless frequencies, li ZXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXV 1 2 3 4 5 6

Figure 7(a) Spline strip Present strip

7·11 7·14

15·77 15·78

15·77 15·83

19·58 19·64

38·43 38·62

44·37 44·41

Figure 7(b) Spline strip Present strip

17·85 17·98

34·89 35·19

34·89 35·22

38·43 38·62

60·12 60·82

68·51 68·97

Figure 7(c) Spline strip Present strip

13·47 13·48

17·85 17·98

18·79 18·88

18·79 19·02

26·92 27·16

51·73 51·42

Figure 7(d) Spline strip Present strip

49·35 49·36

49·35 49·38

52·78 53·27

78·96 78·98

98·71 99·04

128·32 128·41

Figure 7(e) Spline strip Present strip

9·61 9·63

17·32 17·35

30·60 30·63

43·66 43·80

51·09 51·19

64·40 64·45

Figure 7(f) Spline strip Present strip

15·17 15·21

23·93 23·97

39·40 39·43

54·16 54·33

62·83 62·94

77·46 77·52

Table 7 Fundamental frequencies of square plates with step change in thickness; l = va 2(rt/D)1/2, n = 0·3 Boundary conditions

Kantorovich [11]

FEM [12]

Present strip

Exact [13]

Simply supported Clamped

16·11 28·90

15·38 28·30

15·52 28·32

15·46

accuracy and the eﬃciency of the strip element have been demonstrated in the numerical examples given. REFERENCES 1. Y. K. Cheung 1976 Finite Strip Method in Structural Analysis, Oxford: Pergamon Press. 2. Y. K. Cheung, F. T. K. Au and J. Kong 1992 Proceedings of the International Conference on Computational Methods in Engineering, Singapore, 1140–1145. Structural analysis by ﬁnite strip using computed shape functions. 3. Y. K. Cheung and J. Kong Computers and Structures (to appear). Finite strip analysis of ﬂat slab using computed shape functions. 4. Y. K. Cheung and J. Kong 1993 Proceedings of the Fourth East Asia Paciﬁc Conference on Structural Engineering and Construction (to appear). Application of a new ﬁnite strip to the analysis of plates with varying rigidity. 5. A. Y. T. Leung and W. K. Wong 1988 Thin-Walled Structures 6, 81–108. Frame mode method for thin-walled structures. 6. A. Y. T. Leung and F. T. K. Au 1990 Computers and Structures 37, 717–729. Spline ﬁnite elements for beam and plates.

application of new finite strip method

353

7. A. W. Leissa 1973 Journal of Sound and Vibration 31, 257–293. The free vibration of rectangular plates. 8. C. S. Kim and S. M. Dickinson 1987 Journal of Sound and Vibration 114, 129–142. The ﬂexural vibration of line supported rectangular plate systems. 9. K. M. Liew and K. Y. Lam 1991 Journal of Sound and Vibration 147, 255–264. Vibration analysis of multi-span plates having orthogonal straight edges. 10. C. I. Wu and Y. K. Cheung 1974 Earthquake Engineering and Structural Dynamics 3, 3–14. Frequency analysis of rectangular plates continuous in one or two directions. 11. V. H. Cortinez and P. A. A. Laura 1990 Journal of Sound and Vibration 137, 457–461. Analysis of vibrating rectangular plates of discontinuously varying thickness by means of the Kantorovich extended method. 12. H. C. Sanzi, A. Bergmann, R. Carnicer and P. A. A. Laura 1989 Journal of Sound and Vibration 135, 161–165. Numerical experiments on vibrating rectangular plates with discontinuously varying cross-section. 13. I. Chopra 1974 International Journal of Mechanical Science 16, 337–344. Vibration of stepped thickness plates.

Thin-WalledStructures21 (1995) 327 343 ÂŠ 1995 Elsevier Science Limited Printed in Great Britain. All rights reserved 0263-8231/95/\$9.50

ELSEVIER

Vibration and Buckling of Thin-Walled Structures by a New Finite Strip

Y. K. C h e u n g & J a c k s o n Kong Department of Civil and Structural Engineering, University of Hong Kong, Hong Kong (Received 6 April 1994; accepted 9 M a y 1994)

A B S T R A CT This paper presents the application of a new finite strip to the analysis of folded-plate structures. The displacement function of aft at shell finite strip is made up of two parts, namely, the two in-plane displacement interpolations and the out-of-plane displacement interpolation. Each of the three displacement components is interpolated by a set of computed shape functions in the longitudinal direction and, as usual, one-dimensional shape functions in the transverse direction. Only standard beam shape .functions are involved in the longitudinal computed shape functions. When compared with other finite strips, the present finite strip is relatively simple in dealing with boundary and internal support conditions. In addition, the method can be easily implemented by incorporating a standard finite strip program with a continuous-beam program. The computation of the stiffness matrix involves no numerical integration. To verify the accuracy and efficiency of the new finite strip, a few numerical experiments are conducted in which the present.finite strip results are compared with those using other finite strips and/or finite elements for the vibration and buckling of folded-plate structures with varying complexity.

INTRODUCTION For the analysis o f prismatic thin-walled structures, the finite strip method is one of the best choices available to an engineer.l"2 Since the first paper of finite strip appeared in 1968, 3 intense interest has been focused on the development of finite strip analysis. The state of the art has been recently 327

328

Y. K. Cheung, Jackson Kong

summarized in Cheung, 4-6 Chong e t a / . 7's and Graves Smith. 9 It is wellknown that the method has several major advantages relative to the standard finite element method, namely, (1) the ease of data input, (2) smaller number of unknowns, (3) rapid convergence, (4) elimination of the problem of in-plane rotation and (5) the ease of implementation. In addition, the basic concept of the method is simple and involves no complicated mathematics. Early versions of the method encompass the singlespan and the continuous-span finite strip methods. Except for single-span finite strip with two simply-supported opposite ends, the early versions of the finite strips were later superseded by the spline finite strip developed by Cheung and his co-workers, l~12 In the spline finite strip method, the analytical vibration functions of beam are replaced by the C2-continuous B3-spline functions. These functions are actually the analytical static functions of an Euler beam subjected to multi-point loads. Major advantages of this method include (i) different boundary conditions can be treated with the same function and (ii) rapid and non-oscillatory convergence for point-loaded plates. For continuous-span structures with multispans and/or internal discrete supports, the kinematic conditions are taken into account a priori by modifying the corresponding local splines. Alternatively, by transforming the generalized displacement parameters to physical displacement variables at the spline knots, the boundary conditions can be directly imposed on the corresponding spline knots in the usual finite element manner. ~3 These two procedures, although conceptually simple, require tedious implementation. It also destroys the sparsity of the tightly-banded spline functions. Another commonly used approach for dealing with boundary conditions is to treat the supports, in a compound finite strip manner, as springs with very large stiffness. The stiffness contributions of these springs are then directly added to the stiffness matrix of the attached strip before assembly. However, the large stiffness of the spring supports would greatly increase the maximum eigenvalues of the global stiffness matrix and may therefore cause trouble in dynamic analysis. ~4 A new finite strip for the analysis of folded-plates is proposed in this paper. Each of the three displacement components in a finite strip is interpolated by a set of computed shape functions in the longitudinal direction and, as usual, one-dimensional shape functions in the transverse direction. Only standard beam shape functions are involved in the longitudinal computed shape functions. When compared with other finite strips, the present finite strip is much simpler in dealing with boundary and internal support conditions; they can be applied directly by restraining the corresponding nodal displacement variables. In addition, the method can be easily implemented by incorporating a standard finite strip

Vibration and buckling o f thin-walled structures by a new finite strip

329

program with a continuous-beam program. The computation o f the stiffness matrix involves no numerical integration. To verify the accuracy and efficiency o f the proposed finite strip, a few numerical experiments are conducted in which the present strip results are compared with those using other finite strips a n d / o r finite elements.

COMPUTED SHAPE FUNCTIONS In Fig. 1 a beam with constant bending rigidity is divided into p beam elements which are not necessarily o f equal length. A n u m b e r of joints, say r, are assigned as nodes. Similar to the concept of Lagrangian shape functions, a unit deflection is imposed to one of these nodes while the deflections of the remaining nodes are constrained to vanish (Fig. l b). This procedure is then applied to each of the nodes and a total of r computed static modes can be obtained. These static modes correspond to the exact deflection o f a beam subjected to a set of point loads at the nodes. They can be easily obtained by means o f a continuous-beam computer program. The a m o u n t of computing effort required is insignificant. Because o f the similarity between the Lagrangian shape functions and the static modes, they are given the name of C O M p u t e d Shape FUNctions, or abbreviated as C O M S F U N hereon. A displacement v a r i a b l e f ( y ) , can be written in terms o f the C O M S F U N ,

f(y) = ÂŁ wiY,(y)

(1)

i=1

A prismatic beam ---,. <_ . . . . . . . . . . Ca

b

. . . . . . . .

b c 0 ---F:-

0

*

y

.

.

.

d

-

-

o e o joint

.

element k of length Ik A

Y2

~

e

â&#x20AC;˘

node

1 1 n o d e 2 at y = '/2

Fig. 1. A prismatic beam is divided into beam elements. A number of joints are assigned as node for computing COMSFUN.

Y.K. Cheung, Jackson Kong

330

It is noteworthy that each of the unknown parameters, wi, represent the actual displacement at node i. In terms of the usual beam shape functions, the C O M S F U N can be expressed in the form P

Ym(Y) = ~

[N(Y)I{~,,}k

(2)

k--I

with

Ym(Ym) 1 m = 1 , 2 , . . . , r ; r < p + 1 Ym(y,)=O n C m =

where [N(y)] are the usual beam shape functions and {O~m}kis the displacement vector of the joints of beam element k for the mth C O M S F U N ym. The summation is carried out with due regard to the joint arrangement. In the preceding discussions, no rotational restraint is applied to either end of the beam. Put in other words, the boundary condition at each end of the beam is either simply supported or free when computing the corresponding shape functions for the internal nodes or end node, respectively. As a result, no restraint is imposed on the first derivative of the resulting C O M S F U N . For the in-plane displacement components of a flat shell finite strip, the first derivative of the in-plane displacement variables is not involved in the kinematic boundary conditions and therefore the aforementioned C O M S F U N can be adopted directly as the corresponding longitudinal trial functions. For the plate-bending component with a transverse clamped or sliding clamped edge however, the first derivative (with respect to the longitudinal coordinate) of the out-of-plane displacement variable vanishes at the corresponding end of the strip. Accordingly, the rotation at the corresponding end of the beam must be constrained to vanish. For plate-bending with a simply-supported or free edge, no rotational restraint is needed at the corresponding end of the beam. FORMULATION By referring to the local co-ordinate system shown in Fig. 2, the displacement function of a flat-shell finite strip element can be written as

u(x, y) = ~

[M(x)] f~,,( Y){~}m

(3a)

[M(x)]

(3b)

//12 1

v(x,),)

= m=l

for in-plane displacements

Vibration and buckling of thin-walled structures by a new finite strip Y

z Z

331

S i

x

Local

X

Global

Â˘ i

â&#x20AC;˘

i

â&#x20AC;˘

i

1 2 nodal line

Fig. 2. A flat-shell finite strip and its coordinate systems.

and w(x, y) = ~

[N(x)] Ym(Y){?}m

(3C)

m--I

for out-of-plane displacement where Ym(Y) and Ym(Y) denote, respectively, the C O M S F U N for the inplane and out-of-plane displacement components. [M(x)] and [N(x)] are one-dimensional linear shape functions and beam shape functions in the local x-direction. Notice that the same number of nodes and identical nodal spacings are involved in the in-plane and out-of-plane displacement functions. It is clear that the present strip contains four displacement variables at each node. Having defined the displacement functions of a flat-shell finite strip, one can then derive the strip stiffness, mass and geometric matrices in local coordinates using the conventional displacement formulation. All integrations of the matrices are carried out separately for the transverse and longitudinal directions. It is worthy of discussion that the integrals of longitudinal C O M S F U N involve no numerical integration; they are computed by summing up the transformed integrals of the product of beam shape functions, that is (for illustration), I~ =

Ii

Ym I1. dy =

{am}/ k=l

[J2

]

[N( y)lr[N( y)] dy {:~}k

(4)

332

Y. K. Cheung, Jackson Kong

The explicit form of integration of the beam shape functions can be easily found using symbolic computation. The unknowns {CO}m,{fl},n and {7}m in eqns (3a-c) represent the physical nodal displacements in local coordinates, that is = (u,. u2)

{Y}m ----(Wl, 01, W2,02)m T

with Oi = (Ow/Ox)i

These local displacement unknowns can be collectively arranged in the form of {d}.,

:

and then transformed to the global co-ordinate system by the standard transformation, giving {d}~ =

[i,l 0] 0

It] {D}m = [TI{D}m

(5)

where [t] =

cos0~

01 sin0~

--sina 0

0 0

i]

cos 0

(6)

and {D}m -----nodal displacement unknowns in global coordinates. Using the transformation matrix [T], the stiffness, mass and geometric matrices can be transformed in the usual manner and subsequently assembled. All boundary conditions are then imposed directly on the corresponding nodal displacement variables. Subspace iteration was used to extract eigen-pairs from the resulting eigen-value equation.

N U M E R I C A L EXAMPLES The main purpose of the following examples is to verify the validity of the new flat-shell finite strip and to assess its accuracy and efficiency by comparing its solution with those obtained by finite element/finite strip method and/or other numerical methods. Free vibration frequencies and buckling loads of numerous folded-plate structures with varying complexity are computed using the proposed finite strip. In all cases, unless otherwise specified, the nodes on each nodal line are of equal distance apart.

Vibration and buckling o f thin-walled structures by a new finite strip

333

Vibration of cantilever folded-plates Three types of cantilever folded-plates are demonstrated in this example to show the accuracy and efficiency of the present flat-shell finite strip in analyzing folded-plate structures with significant bending action in their component fiat plates. The lowest five frequencies of the cantilever channel shown in Fig. 3 are computed using eight strips and six nodes. The finite s[rip results are compared with finite element solutions 15 using a mesh of 160 four-node shell elements, see Table 1. The finite strip results are within 1% difference from the finite element results (except for mode 4 with a difference of 1.6%). It is noteworthy that the number of unknowns for the finite strip method is only 1% of the finite element analysis. The first four mode

a/I

0

/

/ clamped

x Fig. 3. A cantilever channel. E = 1.0, v = 0.3, a = 50, b = 152, c = 150, density = 1-0, thickness = 3.2.

TABLE 1 Frequencies o f a Cantilever Channel

Mode i

1 2 3 4 5

Natural frequencies

Finite strip

Finite element

0.5260 0.9043 1-1963 1.4993 1.8402

0-5230 0.9041 1.1988 1.4758 1.8480

}

F

,

,

,

.,

m

r

,

,

,,<---~-~L~-~-}Z--~,.'C-<,7, - ~ ~'C7',,

&

1:

~r

~j...__~

iI

r

?f~

,

,

J

~

/' rL

.

__i

,l' ,

'"

~j

D

[

(1)

b

,,J

Vibration a n d b u c k l i n g o f t h i n - w a l l e d s t r u c t u r e s b y a n e w f i n i t e strip

335

~X

Ă&#x2014; D~

0 2

b2

Fig. 5. Cross-sections of one-fold and two-fold folded plates. Span = L, thickness - 0.02L. F o r one-fold folded plates, bt --- b 2 = 0.5L. For two-fold folded plates, bl = b2 = b3 ~ L / 3 , oq = c~2 = ~.

shapes of the channel are also given in Fig. 4 from which one can observe the significant bending action in each of the flat plates. In addition to the cantilever channel, vibration of one-fold folded plates and two-fold folded plates with different crank angles are considered (see Fig. 5). Using the present finite strip, the lowest five frequencies are compared with the finite element-transfer matrix solutions of Liu j6 and summarized in Table 2. For one-fold folded plates, six strips and six nodes are used, while nine strips and six nodes are used for the two-fold folded plates. Good agreement is observed; the maximum difference is within 1%. The number of degrees of freedom in these examples are only 168 and 240 for the one-fold and two-fold folded plates, respectively. Vibration and buckling o f a continuous-span 1-beam ( E = 192, v - 0-3)

The free vibration and buckling under uniform compressive stresses of continuous beams are considered. Although the beams can be analyzed by the usual beam elements, they are treated as folded plates in this example. The cross-section of the beam, which was modelled with five finite strips, is an I-section made up of two equal flanges of width 0.5 and web of 1.0 and a constant thickness of 0.1 throughout. In contrast to the cantilever folded-plates in previous examples, significant membrane action is expected in the component flat plates of the beams. The boundary conditions at the end supports and at the intermediate support are modelled as rigid inplane diaphragm. These conditions can be easily imposed, in the usual

336

Y. K. Cheung, Jackson Kong TABLE 2

Frequencies of Cantilever Folded Plates with Different Crank Angles. 2i = o~iL[p(l - v2)/E] I/2. v = 0.3 Crank angle, a (degrees)

Mode i

One-foldfolded plates 90 1 2 3 4 5

Frequencies ).i Present strip Liu 16

0.0491 0.0976 0-1788 0.2096 0.3508

0.0491 0.0971 0.1786 0.2084 0.3558

120

1 2 3 4 5

0.0491 0.0947 0.1789 0.2076 0-2933

0.0491 0.0943 0.1787 0-2065 0-2971

150

1 2 3 4 5

0-0492 0.0809 0-1790 0-1908 0.2207

0.0491 0.0812 0.1787 0.1912 0.2210

Two-foldfolded plates 90 1 2 3 4 5

0.1245 0.1258 0.2598 0-2710 0.3222

0.1249 0.1260 0.2579 0-2692 0.3286

120

1 2 3 4 5

0.0980 0.1249 0.2589 0.2643 0.2927

0.1000 0.1241 0-2571 0.2630 0.2986

150

1 2 3 4 5

0.0675 0.1145 0.2078 0.2407 0.2585

0.0687 0.1145 0.2109 0.2415 0.2571

finite element manner, by restraining the appropriate degrees of freedom a t t h e c o r r e s p o n d i n g l o c a t i o n s in t h e g l o b a l s t i f f n e s s m a t r i x . R e s u l t s a r e c o m p a r e d w i t h b e a m t h e o r y ~7 in T a b l e 3. A g r e e m e n t b e t w e e n t h e t w o sets o f r e s u l t s is f o u n d t o b e s a t i s f a c t o r y , a s o n l y a m a x i m u m o f 2 % d i f f e r e n c e is f o u n d .

Vibration and buckling o f thin-walled structures by a new finite strip

337

TABLE3 VibrationandBucklingofContinuousBeams Present finite str@

Vibration frequencies two-span continuous beam mode 1 0.5718 mode 2 0-8903 mode 3 2.2488 three-span continuous beam mode 1 0.2265 mode 2 0.5803

Beam theor) J7

0.5692 0.8896 2.2770 0.2224 0-5692

Buckling stresses model mode 2 mode 3

two-span continuous beam 0.8281 1.6760 three-span continuous beam 0.4323

0.8210 1.6773 0-4350

For the two-span beam L~ = L2 = 5 where Li = length of span i. For the three-span beam L~ = L3 = 5 where L2 = 10 Spanwise nodal arrangement: 3 nodes within each span and one node at each support, that is, 9 nodes and 13 nodes per nodal line for the two-span and three-span beams, respectively. All nodes within each span are of equal distance apart. It is w o r t h m e n t i o n i n g t h a t in cases where the internal d i a p h r a g m s at the s u p p o r t s are flexible, the in-plane stiffness can be t a k e n into a c c o u n t by dividing the d i a p h r a g m s into the usual t w o - d i m e n s i o n a l finite elements or b o u n d a r y elements. A f t e r a p p l y i n g c o n d e n s a t i o n , the stiffness m a t r i x o f the d i a p h r a g m s , c o r r e s p o n d i n g to the degrees o f f r e e d o m o f the c o n n e c t ing n o d e s between the d i a p h r a g m s a n d the f o l d e d - p l a t e structure, can be directly s u p e r i m p o s e d o n t o the global stiffness matrix. U n l i k e o t h e r finite strips, n o t r a n s f o r m a t i o n o f the d i a p h r a g m stiffness m a t r i x is r e q u i r e d as the n o d a l degrees o f f r e e d o m between the d i a p h r a g m s a n d the f o l d e d - p l a t e s t r u c t u r e are fully c o m p a t i b l e .

Buckling of a continuous-span stiffened panel T o d e m o n s t r a t e the efficiency a n d a c c u r a c y o f the present finite strip with respect to the semi-analytical finite strip, a t w o - s p a n c o n t i n u o u s stiffened panel is e x a m i n e d . Details o f the s t r u c t u r e are given in Fig. 6. In the s t u d y c o n d u c t e d by D e l c o u r t & C h e u n g 18 using the semi-analytical finite strip m e t h o d , a three-strip idealization was a d o p t e d because o f the repetitive

Y.K. Cheung, Jackson Kong

338

l

I_

L~-

F

y

J

-F A

A

E : 206 k N / m m 2

A

I

1.1=0-3

I = 4000 mm b ~ 7540 mm

.1

z

15

1

I

1 1

l_ ! -

l

1

!

l

1

1

1

A

b

b' ~ 580

mm

t

~

I0

=

12 mm

n,n,te

1 w~b

O)

-~

mm

160 mm

(b) C l

lz

x

0 A

B

0 ~ ~,=

, 0

OatAandD OatB

D

(c) Fig. 6. Buckling of continuous stiffened panel; (a) longitudinal and transverse section of the panel, (b) buckling pattern (transverse direction), (c) three strips idealization.

buckling patterns. Because of symmetry about the intermediate support, only one span, with different span length, is studied. The present finite strip results using three strips and nine nodes are compared with the semianalytical finite strip using three strips and nine terms of the vibrating beam series. Only 1% difference is found between the two sets of results in Table 4. The present finite strip, therefore, possesses comparable efficiency and accuracy with respect to the semi-analytical finite strip in this example. Vibration of a four-stiffener panel

A simply-supported four-stiffener panel is given in Fig. 7. Although it is more efficient to analyze the panel by the semi-analytical finite strip (with complete decoupling of the stiffness matrix corresponding to each term of TABLE4

BuckiingofaStift~nedPanel

Span (mm) 2770 2900

Criticalstresses (N/mm 2) Present[hlite str~ Delcourt Is 258.8 258.6

256-14 255-75

Vibration and buckling o f thin-walled structures by a new finite strip

339

ff

Fig. 7. Cross-section of a simply-supported four-stiffener panel.

the trigonometric series), it is used to examine the accuracy of the present finite strip for computing higher frequencies of prismatic thin-walled structures. The present results of the lowest twenty frequencies are given in Table 5 together with the 'exact' finite strip solutions due to Wittrick & Williams. 19 Despite the fact that a relatively coarse mesh with twelve strips and six nodes is intentionally used in this example, reasonable results can still be obtained. Except for modes 17, 19 and 20, the differences between the two sets of results are less than 6%. A maximum of 10% difference is found at mode 20. TABLE 5 Frequencies of a Stiffened Panel. 2i = ~oib/(E/p) 1/2 Mode 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Frequencies 2i Wittrick 19 Present strip 0-0286 0.0291 0.0359 0.0362 0.0391 0.0395 0.0410 0-0504 0-0519 0.0555 0.0636 0.0641 0-0642 0.0645 0.0647 0-0649 0.0654 0.0731 0.0768 0.0804

0.0302 0.0307 0.0384 0.0385 0.0413 0.0414 0.0431 0.0530 0.0547 0.0584 0-0669 0-0677 0.0679 0.0684 0.0684 0.0692 0.0706 0.0772 0.0826 0.0888

Y.K. Cheung, JacksonKong

340

Buckling of a NASA panel

Stroud et al. 2° of N A S A have presented a detailed study of the buckling of seven stiffened panels which have diaphragm ends and are subjected to combined longitudinal compression (N,.) and shear (N~,.). Of particular interest is the NASA example 2 (Fig. 8) which has been studied in detail by Stroud et al. using hybrid finite element method, and by Lau & Hancock 21 using the spline finite strip method. This example provides a good test case for evaluating the efficiency and accuracy of the present finite strip. In the finite element analysis, a very fine mesh of four-node assumedstress hybrid element was used to provide a benchmark solution. The panel is divided into two elements along the depth of the stiffener, four elements between stiffeners, and 36 elements along the length, making a total of 1296 elements, 1369 nodes and approximately 8000 unknowns. In the spline finite strip analysis, six spline sections are used longitudinally with a total of 504 unknowns. For the present method, the stiffened panel is divided into 18 strips (two strips between stiffeners and one strip for each stiffener) and seven nodes longitudinally. There are a total of 436 unknowns after imposing the boundary conditions. A comparison of the three sets of results is given in Table 6. The results obtained by the present finite strip model, with only 5.5% of the number of unknowns of the NASA finite element model, are only 3 4 % higher. In addition, the agreement between the two finite strip models is very good. It is worth noting that since the present NASA panel is simply-supported around the edges, one may think of using semi-analytical finite strip

N,y~~ Z'X ~ Y v

~

II

f

v

v

y

1,/

= Nxy

---

and I"laterlal_~_.p~rhes ol ~

~-~o~

~-i-336---~-~

07620m [ Stiffener "l'htckness I.~.7)2mm

pacing

~Stfffener Oepth

Shffened Panel

I S--~,,~hi~ko~-

01270m

I Y°ung"s Hodutus

072¢ • 10S HPa

o03z,)t, mJ_Poisson's Ratio

0.32

Fig. 8. Loading and dimensions for stiffened panel. Simply-supported on all four edges. X

0. L: v = 0 =

w: Y

0, h ; u

0=

w.

Vibration and buckling of thin-walled structures by a new finite strip

341

TABLE 6 Buckling of NASA Panel

Loading Ar,. (kN/m) 70.05 175.13 350.26 875.65 175-13

N~9. (kN/m) 175.13 175.13 175.13 175.13 0.00

NASA 2째

Splmestr& 21

Presentstr&

0.7195 0-6061 0-4444 0-1929 0.9759

0.7371 0.6183 0.4542 0-1977 1-0003

0.7490 0-6304 0-4612 0-1990 1-0070

method which is more efficient for analyzing structures with two simplysupported opposite ends. Although the method has proved to be accurate and efficient for analyzing the buckling of compressed structures, it has some disadvantages for analyzing the buckling of structures loaded in compression and/or shear: (i) Difficulties are experienced when dealing with non-periodic buckling mode. (iii) The buckling analysis of plate assemblies loaded in shear is problematic. 22 (iii) For thin-walled structures loaded in compression, many buckling modes are possible. Since it is not known in advance which mode is critical, it will be necessary to solve the same eigenvalue problem for different terms of Fourier series in order to obtain the minimum buckling load. Especially in the case of structures with complex cross-sections, this repetitive process may be time-consuming. The present method, however, does not suffer from the problems mentioned above, and therefore seems more suitable to study buckling of prismatic thin-walled structures under arbitrary loading.

CONCLUSIONS A flat-shell finite strip is presented for analyzing vibration and buckling problem of prismatic folded plates. The accuracy of present finite strip is assessed by comparing its solutions with those of the semi-analytical finite strip, spline finite strip and finite element methods. It possesses comparable accuracy, efficiency and versatility as those of the spline finite strip. Because of the reduced number of unknowns at each node, there is a considerable saving in computing effort with respect to the finite element method. Moreover, in comparison with the spline finite strip, the present

342

Y. K. Cheung, Jackson Kong

finite strip is much simpler in dealing with b o u n d a r y and internal support conditions; the support conditions can be easily imposed in the usual finite element manner. Also, the method can be easily implemented by incorporating a standard finite strip program with a continuous-beam program. The computation of the stiffness matrix involves no numerical integration because only beam shape functions are involved in the calculations. A drawback of the method, which is c o m m o n to the semi-analytical finite strip (except for the case of two simply-supported opposite edges) or spline finite strip, is that the stiffness matrices corresponding to different terms of the longitudinal trial function are coupled with each other. This disadvantage however, would not become a major drawback of the method because only a few computed shape functions, usually less than 10, are sufficient to give acceptable results, as demonstrated in the numerical examples. REFERENCES I. Cheung, Y. K., Finite Strip Method in Structural Analysis. Pergamon Press, Oxford, 1976. 2. Loo, Y. C. & Cusens, A. R., The Finite Strip Method in Bridge Engineering. Viewpoint Publication (C&CA), Wexham Springs, 1978. 3. Cheung, Y. K., Finite strip method in the analysis of elastic plates with two opposite ends. Proc. Instn. Civ. Engrs, 40 (1968) 1 7. 4. Cheung, Y.K., Recent advances in finite strip method. Keynote lectures, Proc. of the Asian Paci[i'c Conference on Computational Mechanics. Hong Kong, Dec. 1991, Vol. 1, pp. 13-31. 5. Cheung, Y. K., The spline finite strip method in structural mechanics. Keynote paper, Proc. of the Int. Conj. on Mechanics of Solids and Structures. Nanyang Technological University, Singapore, Sept. 1991, pp. 59 92. 6. Cheung, Y. K., The finite strip method for bridge analysis. Keynote lecture, Proc. of the Second Canadian Conjl on Computing in Civil Engineering. Ottawa, Canada, Aug. 1992, pp. 50-78. 7. Wiseman, D. L., Puckett, J. A. & Chong, K. P., Recent developments of the finite strip method. In Dynamics ~[Structures, ed. J. M. Roesset. ASCE, New York, 1987, 292-309. 8. Boresi, A, P. & Chong, K. P., Appro-~imate Solution Methods in Engineering Mechanics. Elsevier Applied Science, London, 1991. 9. Graves Smith, T. R., The finite strip analysis of thin-walled structures, In Developments in Thin-walled Structures--3, ed. J. Rhodes & A. C. Walker, Chapter 6. Elsevier Applied Science, London, 1987. 10. Fan, S. C., Spline finite strip in structural analysis. Ph.D. thesis, University of Hong Kong, Hong Kong, 1982. 11. Li, W. Y., Spline finite strip analysis of arbitrarily shaped plates and shells. Ph.D. thesis, University of Hong Kong, Hong Kong, 1988. 12. Zhu, D. S., Nonlinear static and dynamic analysis of plates and shells using spline finite strip. Ph.D. thesis, University of Hong Kong, Hong Kong, 1988.

Vibration and buckling of thin-walled structures by a new finite strip

343

13. Au, F. T. K. & Cheung, Y. K., Isoparametric spline finite strip for plane structures. Computers and Structures, 48(1) (19930 23-32. 14. Cook, R., Concepts and Applications of Finite Element. McGraw-Hill, New York, 1989. 15. Cosmos User Manual, Structural Research and Analysis Cooperation, Santa Monica, CA 90404. ' 16. Liu, W. H. & Huang, C. C., Vibration analysis of folded plates. Jnl. of Sound and Vibration, 157(10) (1992) 123 37. 17. Timoshenko, S. & Gere, J. M., Theory of Elastic Stability. McGraw-Hill, New York, 1959. 18. Delcourt, C. R. & Cheung, Y. K., Finite strip analysis of continuous folded plates. I A B S E Periodica, (1978) 1-16. 19. Wittrick, W. H. & Williams, F. W., Buckling and vibration of anisotropic or isotropic plate assemblies under combined loadings. Int. J. Mech. Sci., 16 (1974) 209-39. 20. Stroud, W. J., Green, W. H. & Anderson, M. S., Buckling loads for stiffened panels subjected to combined longitudinal compression and shearing loadings: results obtained with PASCO, EAL and STAGS computer programs, NASA TM83194, 1981. 21. Lau, S. W. & Hancock, G. J., Buckling of thin flat-walled structures by a spline finite strip method. Thin-Walled Structures, 4 (1986) 269 94. 22. Mahendran, M. & Murray, N. W., Elastic buckling analysis of thin-walled structures under combined loading using a finite strip method. Thin-Walled Structures, 4 (1986) 329~462.

Thin-Walled Structures 22 (1995) 181-202

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3

ÂŠ 1995 Elsevier Science Limited Printed in Great Britain. All rights reserved 0263-823 !/95/\$9.50

!zl ~

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ELSEVIER

0263-8231(94)00035-2

A Generalized Spline Finite Strip for the Analysis of Plates

J. K o n g & Y. K. Cheung Department of Civil and Structural Engineering, University of Hong Kong, Hong Kong (Received 6 August 1993; accepted 22 May 1994)

ABSTRACT In this paper, the original spline finite strip, which is based upon the thinplate theory, is modified and generalized in the sense that the formulation is based upon a third-order plate theory and is applicable to thin plates and shear-deformable plates (e.g. thick isotropic plates and laminated composite plates). To extend its application to nonlinear analysis, displacement formulation is employed with geometric nonlinearity and small initial imperfection taken into account. To demonstrate the performance of the present finite strip in the analysis of plates, a number of numerical exampies are given.

INTRODUCTION

The increasing use of laminated composite materials in different branches of engineering stimulates the advances in the analysis of laminated beams and plates. The transverse shear moduli of modem composite materials are usually very low compared to the inplane tensile moduli, with the result that transverse shear deformation can be of considerable importance compared to homogeneous isotropic materials. A very large number of new plate theories for linear analysis have been developed. TM Most of these theories concentrate on the modelling of cross-sectional warping, the calculation o f the nonlinear stresses distribution across the thickness and the satisfaction o f the interlaminar stress continuity. F o r the analysis o f the free edge singularity, delamination and fracture problems, a three-dimensional analysis is restored to. In the past, the 181

182

J. Kong, Y. K. Cheung

numerical nonlinear analysis of laminated plates relied heavily on the classical plate theory and the Mindlin plate theory. 5-1째 The Mindlin plate theory gives constant transverse shear strain distributions through the thickness and hence requires the calculation of a shear correction factor. This factor is problem dependent and crucial to the accurate analysis of composite laminates. 11 It can be obtained by comparison with an exact elasticity solution, if available. However, accurate and reliable nonlinear analysis of composite laminates can be accomplished without using the shear correction factor, this can be done by using a third-order plate theory which is capable of modelling cross-sectional warping and parabolic transverse shear strain distributions. Recently, Reddy and co-workers, 12-16 Stein and co-workers 17-19 and Owens e t al. 2째 employed the same third-order displacement field for their nonlinear analysis. Bhimaraddi 21'22 carried out nonlinear analysis based on a modified form of Reddy's third-order displacement field. By introducing two other assumptions, Singh e t al. 23 and Lim e t al. 24 performed nonlinear analysis with another modified form of the third-order theory developed by Reddy. Other contributions on nonlinear higher-order plate theories have been proposed by Yoda and Atluri, 25 Sacco and Reddy 26 and Hinrichson and Palazotto. 27 It is noteworthy that Dawe e t al.9, 10 and Yoda and Atluri 25 employed the finite strip method in their nonlinear analysis. It is well-known that because of the rapid convergence and ease of data input, the finite strip method is suitable for the analysis of prismatic structures such as rectangular plated-structures. Recently, the original spline finite strip, which is based upon the thin-plate theory, has been successfully generalized by combining with Reddy's third-order plate theory28, 29 for linear static and buckling analysis of thick isotropic and laminated plates. 3째'31 To extend the application to nonlinear analysis of plates and to improve the performance of this spline finite strip for thin plate analysis, the model is modified and generalised. On the basis of the Total Lagrangian approach, geometric nonlinear static and transient analysis are carried out with the von Karman nonlinear strains and small initial imperfection taken into account. To demonstrate the performance of this generalised spline finite strip model, a number of numerical examples are given.

KINEMATICS OF PLATE With reference to a Cartesian coordinate system, the displacement field of a plate starts off with

Analysis of plates by spline finite strip

183

u 1 (x, y, z) : u -~- z ~1x -at- z20)x --~ Z36x U2

(X, y, Z) = V + Z ~by + z2coy + Z36y u3 (x, y) = Wo (x, y) + w (x, y)

where u, v and w denote the displacements of a point (x,y) on the mid plane, and ~'x and ~y are the rotations of normals to mid plane about the y and x axes, respectively. The cross-sectional warping is represented by 09x, ~v, fix, 6y. The small initial imperfection is denoted by Wo(x, y). The transverse normal strain is neglected in the analysis. Taking the nonlinear strains and small initial imperfection into consideration, the strain<lisplacement relationships are given by ~x :

1 2 Ul, x + ~ (W, x) "{'- W, x Wo, x 1

2

ey = U2,x + ~(w,y) + w y w0,y

(1) exy : Ul,y "~ U2,x -~- W,x W,y -~- W,x WO,y "-~ W,y WO,x eyz = U2, z -~- W y ~xz : Ul,z + W,x

where partial differentiation is denoted by a comma ( ),x -

o() Ox

The: tangential static conditions on the top and bottom surfaces of a plate with thickness h can be expressed as

tryz = 0 : axz at z = -4- h/2 and

(ayz z) = 0 = (trxz z) at z = + h/2 where Oez and axz denote the transverse shear components of the symmetric second Piola-Kirchoff stress tensor a. By adopting these conditions and assuming linear elastic material, the displacement field of the third-order plate theory (TOP) is finally reduced to

Ul(X,y,z) = u + z [~'x - 4/3 (z/h) 2 (~kx + W,x)] ] U2(.7C, y , Z) :

i~ + Z [~/y - -

4/3 (z/h) 2 (~ky + W,y)] i

(2)

u3(x, y) = Wo(X, y) + w(x, y) which contain the same number of independent variables as the Mindlin plate theory. In addition to the satisfaction of traction-free conditions, this

J. Kong, Y. K. Cheung

184

displacement field contains the same number of independent unknowns as the Mindlin-Reissner plate theory. Spline finite strip formulation based on TOP has been presented in two other papers. 3°'3~ In this work, this displacement field is modified by making the following transformation: Ox + W,x =

=

xzlz=o

Oy + W,y : ,~,y : F,yzlz= 0

It can be easily shown that the transverse shear strains at mid-plane exz and ~yz, are actually represented by 2x and 2y, respectively. The modified displacement field (MODTOP) can now be written as Ul(X,y,z) = u - z W,x + z [1 - 4 / 3 ( z / h ) 2] 2x u2(x,y, z) = v - z W,y + z [1 - 4/3 (z/h) 2] 2y

(3)

u3(x, y) = w0(x, y) + w(x, y) From a theoretical point of view, the displacement fields given by eqns (2) and (3) are, in fact, the same. SPLINE FINITE STRIP For a spline finite strip with m sections, the displacement functions for TOP can be written as m+l

u(x,y) =

[M(x)] i=-1 m+l

v(x,y) = ~

[M(x)] ~b~(y){¢};

i=-1 m+l

w ( x , y ) = y ~ [N(x)] ¢i(y) {~}i

(4)

i=-I m+l

~bx(X,y ) = y ~ [M(x)] ¢i(y) {fl}i i= -1 m+l

~by(x,y) = y ~ [M(x)] ¢iQv) {y}i i=-I

where [M(x)] and [N(x)] are the one-dimensional linear shape functions and the Hermite cubic polynomial, respectively. ¢i (Y) is the ith member of

Analysis of plates by splinefinite strip

185

the B3-spline series and {~}, {~}, {~}, {/~} and {?} are the unknown parameters. It has been shown by many researchers that the transverse shearing strains must be interpolated consistently in order to avoid shear locking for finite element analysis of thin plates. In the spline finite strip model based on TOP (i.e. eqn (4)), it is obvious that the contributions to the transverse shear strains at mid-plane are of different order polynomials. This causes a problem for thin plates where both ~x2 and eyz must tend towards zero. A consistent interpolation scheme would be one in which identical polynomial terms must be used for both the rotations Sx (or qly) and the derivative of the vertical displacement field W,x (or W y). While such an interpolation scheme is possible, it would not be convenient to implement. However, with the modified displacement field this problem does not exist because the transverse shear strains exz and ey2 (or 2x and 2y) can be interpolated consistently by replacing Sx and ~by of eqn (4) by 2x and 2y, respectively. Linear elastic finite element models based on this modified displacement field (MODTOP) have been developed by Hinton et al. 32 and Averill and Reddy. 33 EQUILIBRIUM EQUATION Having defined the kinematics, strain~lisplacement relationships and the displacement functions of a plate strip, the equilibrium equation will be given in this section. Combining eqns (1) or (2) with (3), the strain-displacement equations can be written in the condensed form of {~} = [BL + B1 (3)] {3} = [B(3)] {3}

where

B1 (~) =

1

"W,x + 2Wo,~ 0 0 W,y+2Wo, y W,y + 2Wo,y

Y

[N(x)],x Â˘i(Y)

i=1

W x+2Wo, x Li=I

where B L denotes the linear matrix. The nonlinear matrix B1 (6) is a function of the current displacement {3}. Taking variation with respect to {6}, it can be shown that the incremental strain~lisplacement equation is given by d {~} = [BL + Bo + BN (fi)] d {fi}

= [BL + B~ (3)] d {6} = [B(3)] d {3}

186

J. Kong,

where

BNO) =

o x

Y. K. Cheung

0]

W,y W,y W,x

/â&#x20AC;˘

[g(x)] ~i,y (Y)

li=l

[

W~,x

BoO) =

0

WO,y

[ ~ [U(x)],x4,~y)] i=1

J

LW0,y Wo, x

/i=1 and

m

BN(6 ) :

+w0x 0 1 W,y -~- WO,y L W,y + Wo,y W,x +W0,xJ 0

i=1 [S(x)] ~i,y (y) l I_i=1

J

Using the Total Lagrangian formulation and the incremental-iterative approach, 34-36 the following dynamic incremental equilibrium equation at time t + At can be derived,

Mt+At {~}k + t[KL + KN + K.] d {5} k = t+At{F} -- t+At{R}k-I or

gt+At {~}k + t[KT] d {~}k = t+At{F } _ t+At{R}k-I

(5)

where

t+At{t~}k : t+At{~}k-1 + d {3} a and t + at{ R } k - 1 = t + at[KE]k-1 t + At{(~}k-1 The left superscripts represent the time of the configuration in which the quantity occurs and the fight superscripts denote iteration count, t+ At{F } is the external load vector; t+ a t { R } k - 1 is the internal resistance equivalent to the element stresses that correspond to the displacements t+&{3}k-l; and t+ at{6}k is the displacement vector at the end of iteration k and time t + At. When the iteration count k = 1, t+At{6}0 : t{6}, t+at{R}O = t{R} and d {6} 1 = d{3}.

Analysis of plates by spline J'mite strip

187

The matrices KL and KN represent the small displacement stiffness matrix and the large displacement matrix, respectively, the geometric matrix K~ depends on the current inplane stresses ax, ay and axy. The s u m of these matrices KT is the so-called tangent stiffness matrix. Since linear elastic material is considered in this study, the internal resistance {R} can be expressed in terms of a secant stiffness matrix KE. The details of the stiffness matrices and the mass matrix are given in the Appendix. The inertia term on the left-hand-side of the equilibrium equation drops out in case of static analysis and the standard Newton-Raphson iteration with load incrementation is employed to solve the static equilibrium equation. A residual force convergence criterion is used in the static analysis. All integrals are carried out over the original volume of each section of a strip with three integration points in each of the X and Y directions.

THE NEWMARK TIME INTEGRATION Discretisation of the time domain is carried out by the Newmark tim integration. The relationships between displacement, velocity and acceleration are given by t+at{6} = '{6} + At (t{6} + '+a'{6})/2

t+a,{~} =

t{6 } + At (t{~} + ,+ A,{3})/2

t+,2Xt{(~}k = t+At{(~}k- 1 .11_d {~}k Substituting these equations into (5), we finally obtain (tKt + 4/(A/)2M)d {6}k = t + at{F } _ t + At{R}k-1 _ M[(t+ at{6}k-I -- '{6}) - (4/(At))'{6}

- '{6}]

For nonlinear dynamic analysis, the tangent stiffness matrix is updated only after a user-specified number of time steps and remains unchanged during equilibrium iterations within each time step. For equilibrium convergence, a displacement criterion is adopted in this study.

NUMERICAL EXAMPLES Linear static analysis of thin isotropic plates (v = 0-3) As a preliminary test of the method, a square plate with either simplysupported edges or clamped edges is analyzed by the proposed strip

J. Kong, Y. K. Cheung

188

element. The plate is subjected to an uniformly distributed load. A quadrant of the plate is divided into a number of strips with 4 spline sections in each strip. Vertical deflection at the centre of the plate is normalised with respect to the exact solution. The performance of the strip based on TOP and MODTOP is demonstrated in Fig. 1. For a very thin plate with spanto-thickness ratio of 1000, shear locking can be observed in the spline finite strip model based on TOP. On the contrary, the spline finite strip model based on MODTOP suffers no shear locking and performs much better in this example. Bending moments at the centre based on the two different spline finite strip models (with 4 strips and 4 sections) are given in Table 1.

(a)

I.l I .(1 0.9

+~+

i 0.8

+/

a} 0.7 0.6 0.5

>

0.4

[] Modtop A/H = 1000

0.3

+ T o p A / H = I{)(}(I

o Modtop A/H = 100

(1.2 ILl 1}

T o p A / H = 100

+

[

I

I

I

~

I

I

4

5

6

7

8

9

l0

Number of strips (4 spline sections)

1.1

(b)

1.0 1).9 11.8 +" -~ 0.7

.-

o.6

~a

[] Modtop A/H = 1000

~" 0.5

+ Top A/H = 1000

O Modtop A/H = 100 A T o p A / H = 100

1).4 0.3

-

I

I

I

I

i

I

4

5

6

7

8

9

10

Number of strips (4 spline sections) Fig. 1. N o r m a l i z e d vertical d e f l e c t i o n s at the centre o f a s q u a r e plate w i t h i n c r e a s i n g n u m b e r o f strips. (a) A l l e d g e s are c l a m p l e d , (b) all e d g e s are s i m p l y - s u p p o r t e d .

189

Analysis of plates by spline flnite strip

TABLE 1 Comparisons of Bending Moments at the Centre of a Square Plate (v = 0.3)

Span-to-thickness

Bending moments

Mx TOP 1000 100 Exact

1000 100 Exact Multiplier

MODTOP

TOP

MODTOP

Simply supported 0.1652 0.4816 0.1589 0.4815 0.4708 0.4816 0.4716 0-4815 0-4789 0-4789

0.0340 0.2191

Clamped 0.2331 0.0285 0-2325 0-2332 0.2068 0.2324 0-2291 0.2291 0.1 qa 2

Linear static analysis of shear deformable plates Results o f a n a l y s e s o f m o d e r a t e l y thick i s o t r o p i c plates a n d l a m i n a t e d c o m p o s i t e plates w i t h d i f f e r e n t s p a n - t o - t h i c k n e s s r a t i o are p r e s e n t e d in T a b l e s 2 a n d 3. I n all cases, the p l a t e s are subjected to u n i f o r m l y distribu t e d l o a d s a n d a q u a d r a n t o f e a c h p l a t e is d i v i d e d into 8 strips a n d 4 spline sections. Results given in T a b l e 2 indicate g o o d a g r e e m e n t b e t w e e n

TABLE 2 Results of Bending Moments at the Centres and Twisting Moments at the Corners of Simply Supported Thick Plates

Theory

Deflection (*qa4/D)

Moment (*qa2) Mx

TOP MODTOP Kant 37

0.00490 0.00490 0.00480

TOP MODTOP Kant 37

0.00427 0-00427 0-00424

Twisting moment (*qa2)

My

Span-to-thickness ratio = 5 0-0479 0.0480 0.0480 0.0480 0.0484 0.0485 Span-to-thickness ratio = 10 0-0479 0.0480 0.0480 0.0481 0.0480 0.0480

Mxy 0-0318 0.0329 0.0299 0.0318 0.0325 0.0317

Poisson ratio = 0.3; q = magnitude of uniformly distributed load; D = bending rigidity; a = span length.

J. Kong, Y. K. Cheung

190

TABLE 3 Results of Deflections for a Three-Ply (00/900/0 °) Square Laminated Plate with Simply Supported Edges

Span-to-thcikness ratio

2 4 10 100

Deflection ff~ TOP

MODTOP

Analytical solutions 2s

7.7663 2.9090 1.0898 0.6699

7.7613 2.9078 1-0900 0-6705

7.7661 2.9091 1.0900 0.6705

~ = wh3E2 * lO0/qa4; h = total thickness of plate; El = 2 5 , 1 0 6 G12 = G13 = 0-5 * 106 psi, G23 = 0.2 • 106 psi; v12 = Vl3 = 0.25.

psi, E2 = 106 psi;

the spline finite strip results and those reported by Kant 37 for isotropic plates. For composite laminates, analytical solutions of TOP developed by Reddy 2s are adopted in this example for comparison. It is apparent that there is no significant difference among the three sets of results given in Table 3. In general, there is no major difference between the results of the two spline finite strip models in the analyses of these shear deformable plates.

Free vibration and buckling of thick laminated plates with simply supported edges The natural frequencies and uniaxial buckling loads of laminated plates are given in Table 4. A quadrant is discretized into a number of spline finite strips. For natural frequency, the spline finite strip result based on MODTOP is lower than the three-dimensional solution of Noor 3s' 39 by 7.4%. The analytical buckling loads based on TOP developed by Khdeir 4° are in close agreement with the spline finite strip results.

Free vibration and buckling of isotropic plates Two isotropic square plates with different thickness are simply supported on all edges. The plates are subjected to constant initial stresses. The buckling loads are given in Table 5. Spline finite strip results based on MODTOP are compared with the solutions due to Srinivas and Rao 41 and Timoshenko and Gere. 42 Very good agreement among the three sets of results can be observed. The present strip element based on MODTOP and the modified B3spline function 31 is also applied to the analysis of isotropic square plates

Analysis of plates by spline finite strip

191

TABLE 4 Frequency a n d Buckling L o a d o f Two Square L a m i n a t e d Plates (0o/900/0 째) with Different Thickness

Strips x sections

4 x 4

8x 8

TOP MODTOP N o o r 38 K h d e i r 4째

0.4318 0-4096

Natural frequency 2 0.4319 0-3985 0.4300 0.4105

Buckling coefficient N TOP MODTOP N o o r 39 K h d e i r 4째

22.14 22.35

22.12 22.35 22.88 22.12

a = span; h = total thickness; E1 = 40 E2; Gl2 : G13 = 0.6 E2, G23 = 0.5 E2; v12 : Y13 = 0.25. F o r frequency a/h = 5 a n d 2 -- o9 (ph/E2) 째'5. F o r buckling a/h = 10 a n d N = trx a2/E2 h E.

TABLE 5 Critical Buckling L o a d F a c t o r for Simply S u p p o r t e d Square Plates (v = 0.3). A M e s h with 8 Strips a n d 8 Sections is used to Discretize a Q u a d r a n t o f the Plate

Theory

Critical buckling coefficients k Uniaxial stresses

Biaxial stresses

Span-to-thickness ratio a/ t = 10 MODTOP Srinivas a n d R a o 41 MODTOP T i m o s h e n k o a n d G e r e 42

3.786 3.741

1.893 1.880 Span-to-thickness ratio a/ t = 10 3.996 1.998 4.000 2.000

with simply supported edges and square cut-outs at the centres (Fig. 2). The critical buckling loads and the natural frequencies are compared with the higher-order finite element solutions 43 and the analytical solutions of the Mindlin plate theory (Table 6). 6 The present strip element performs well for the buckling analysis of thin and thick plates and gives reliable natural frequency for thin plates. However, the spline finite strip result for the 1Lhick isotropic plate is lower than the analytical solution based on Mincllin plate theory 6 by 8.5%. The reason for this anomaly is not apparent at this stage.

J. Kong, Y. K. Cheung

192

S.'S.

(a)

S.S.--

.......f Hoe.....t.......

.... S.S.

S.S.

(b)

S.C.

F S.S.

....11

F

I

S.C.

S.S.

Fig. 2. (a) A square plate with a square hole of width 0.5a. Span of plate = a. Exterior boundaries of the plate are simply-supported. (b) The spline finite strip mesh and the corresponding boundary conditions for a quadrant. S.S = simply-supported; S.C. = sliding clamped; F = Free.

TABLE 6

Buckling Loads and Natural Frequencies for Simply Supported Square Plates with Hole (v = 0.3) Uniaxial critical buckling load k S p a n - t o - t h i c k n e s s ratio

MODTOP Tham et al. 43

10 1-823 1-860

Natural frequency 2 S p a n - t o - t h i c k n e s s ratio 5

MODTOP Reddy 6 trc~ = k n 2 Et2/[12(1 - v2)a2]. 2 = m a 2 (p h / D ) 째'5.

19.71 21.54

100 1.987 2.090 100 23-48 23- 52

Analysis of plates by spline finite strip

193

Y

IlL

. . . .

I~- X

Fig. 3. Square clamped and simply-supported plate under uniformly distributed load. v = 0.316.

Nonlinear analysis: Large deflection Two isotropic square plates with different thickness ( a / h = 5 and 100) are subjected to uniform distributed loads. All edges of the plates are either simply-supported or clamped. Information for the thin plate with a / h := 100 is provided in Fig. 3. For presentation of results in Figs 4 and 5, the following normalised quantities are used: Q = qaa/Et 4

and

W = w(O,O)/t

From the results it is clear that the present finite strip results are in reasonable agreement with the Levy's solution for thin plates 44 and higher order element solutions for thick plates. 45 In the nonlinear analysis of thin plate, the difference between the two spline strip models is not significant. In these analyses, only a quadrant of the plate is analysed with four finite strips and four spline sections.

Nonlinear analysis: Postbuckling Having established the credibility of the present method in nonlinear analysis, the postbuckling behaviour of two isotropic square plates with small initial curvatures loaded in edge compression, are analysed. For the finite strip comparison Yamaki's cases I(b) and II(b) are examined, the boundary conditions for the 4 strip mesh (with 4 spline sections) shown in Fig. 6 being as follows: Case I(b) Simply supported: x = i a / 2 u, v, ~bx free, w = ~/y = 0 y = + a / 2 u, ~by free, v constant, w = ~kx = 0

J. Kong, Y. K. Cheung

194 2.2 -

(a)

2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 I 100

I

L

200

I

300

I

400

500

Q 2.6- (b) 2.42.2

+

2.0 1.8 1.6 1.41.2

U+.~ ~

J

1.0

0.8 0.6 0.4 0.2

j5

n Top (4*4) + High order element

/

/ I 10

I

20

I

30

I

k

40

50

I

60

I

70

I

80

I

90

Q Fig. 4. Ix~ad--detlection of (a) a damped thick plate, a/h = 5; (b) a simply-supported thick plate, a/h = 5.

Case II(b) Simply supported/clamped:

x = + a / 2 u, v free, w = ~bx = qJy = 0 y = 4- a / 2 u, d/y free, v constant, w = ~bx = 0 The initial imperfection in each case is given by Case I(b) w0 = 0.1 t cos (rex~a) cos (roy~a) Case II(b) w0 = 0.05t (1 + cos (2nx/a)) ( sin (2roy~a)) The f o l l o w i n g normalised quantities are used for presentation o f results in Fig. 7

Analysis of plates by spline fmite strip

195

1.9 1.8 2.0 1.7 1 (a) 1.6 1.5 1.4 1.3 1.2 1.1-

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

7

o/

I

+ Top (4*4) O Modtop (4*4) I

I

100

200

I

Q

I

300

I

400

500

(b)

6 5 4 3 2

+ Top (4*4) Â˘ Modtop (4*4)

1 I

I

2

I

3

I

I

I

4 5 6 Q (* 1000)

I

7

I

8

I

9

I 10

Fig. 51. Load-deflection of (a) a clamped thin plate, a/h = 100; (b) a simply-supported thin plate, a/h = 100. V = v

a/t 2

and

W = w(O, O)/t for case I(b) W = w(O, a / 4 ) / t for case II(b)

It can be seen from Fig. 7 that the finite strip results are reasonably close to those provided by Yamaki. 46 It is worth noting that a finer mesh is required for the finite strip model based on TOP than that based on M O D T O P to achieve the same accuracy in case II(b). Nonfinear analysis: Forced vibration A square plate with simply supported edges and unyielding supports in the horizontal plane is subjected to an uniformly distributed step load. The

J. Kong, Y. K. Cheung

196

I I I

----* X

s

S ~

S

S

C

S

C

s = simply supported C = clamped

S

Fig. 6. Postbuckling of a square plate loaded in edge compression.

same plate has been examined by m a n y other researchers (for example in Refs 47-49). The geometry of the plate and applied loading are given below: span = 243.8 cm, thickness = 0.635 cm, Young's m o d u l u s = 7.031 x 105 kg/cm 2, density = 2-547 x 10 -6 kgsec2/cm 4, q ( x , y , t) = qo

v = 0.25,

0 < t < oÂ˘

where the m a x i m u m amplitude of load q0 = 4.882 x 10 -4 kg/cm 2 A quarter of the plate is divided into four strips and four sections. Zero initial conditions (displacement, velocity and acceleration) are assumed. In all cases, the tangent stiffness matrices are updated on every fifth time step. The m a x i m u m vertical deflections at the centre of the plate are compared with mixed finite element solutions of A k a y 47 (Table 7). The agreement between the two sets of results are found to be reasonable.

CONCLUSIONS This paper has presented the analyses of rectangular plates using a higherorder spline finite strip formulation. The finite strip was successfully used to solve a n u m b e r of linear and nonlinear static and transient problems. It

Analysis of plates by spline finite strip

197

4.5 - (a) 4.0

3.5 3.0 2.5

~ 2.o [] Yamaki

~ 1.5 > 1.0 0.5

.d~

~ T째oPd~8; ~ ) . 4,

J

2.4 2.2 2.0

I

I

I

I

10

20

30

Uniform

edge compression

40 V

I

50

(b)

--e. 1.8

~ 1.6

+

~ 1.4 "~ 1.2 :~ 1.0 "~ 0.8 0.6 > 0.4 0.2

+

l~

n/ ~ L,+

+

+

+ Top (4*4) O Top (8*8) ~ Modtop (4*4)

+ I 10

I 20

I 30

I 40

I 50

I

60

Uniform edge compression V Fig. 7. Out-of-plane deflection with increasing edge compression. (a) Case II(b). (b) Case I(b).

TABLE 7 Maximum Centre Deflections Obtained with Different Methods and Load Magnitudes

q0 5q0 10q0

Time steps (s)

0.0050 0.0025 0.0020

Maximum centre deflections Finite strip (MODTOP)

Akay 47

0.5961 1.2570 1.6470

0-6014 1.2607 1.6337

198

J. Kong, Y. K. Cheung

has also been applied to the analysis of thin isotropic plates. For the analysis of shear deformable plates, there is no major difference between the performance of the finite strip models based on TOP and MODTOP. However, the finite strip model based on MODTOP is recommended for thin plate analysis. The performance of the finite strip model based on MODTOP for free vibration analysis of thick plates is not satisfactory and further investigation is required. Unlike the semi-analytical or the classical spline finite strip elements, the present finite strip based on modified B3spline function is applicable to the analysis of rectangular plates with cutouts. Extension of the method to analyse the nonlinear behaviour of a prismatic folded-plate structure is feasible. The study of the nonlinear transient dynamic behaviour of plates using the present spline finite strip will be reported in another paper. REFERENCES 1. Kapania, R. K. & Raciti, S., Recent advances in analysis of laminated beams and plates Part I: Shear effects and buckling. AIAA J., 27 (1989) 923-34. 2. Noor, A. K. & Burton, W. S., Assessment of shear deformable theories for multilayered composite plates. Appl. Mech. Rev., 42(1) (1989) 1-12. 3. Noor, A. K., Mechanics of anisotropic plates and shells - - A new look at an old subject. Computers and Structures, 44 (1992) 499-514. 4. Reddy, J. N., A review of refined theories of laminated composite plates. Shock Vib. Dig., 22(7) (1990) 3-17. 5. Chao, W. C. & Reddy, J. N., Analysis of laminated composite shells using a degenerated 3-D element. Int. J. Numer. Meth. Engng, 20 (1984) 19912007. 6. Reddy, J. N., Large amplitude flexural vibration of layered composite plates with cutouts. J. Sound Vib., 83(1) (1982) 1-10. 7. Reddy, J. N., Huang, C. L. & Singh, I. R., Large deflections and large amplitude vibrations of axisymmetric circular plates. Int. J. Numer. Meth. Engng, 17 (1981) 527-41. 8. Librescu, L. & Stein, M., Postbuckling of shear deformable composite fiat panels taking into account geometrical imperfections. AIAA J., 30 (1992) 1352-60. 9. Lain, S. S. E., Dawe, D. J. & Azizian, Z. G., Nonlinear analysis of rectangular laminates under end-shortening using shear-deformable plate theory. (To appear). 10. Dawc D. J., Lam, S. S. E. & Azizian, Z. G., Nonlinear finite strip analysis of rectangular laminates under end-shortening using classical plate theory. Int. J. Numer. Meth. Engng, 35 (1992) 1087-1110. 11. Vlachoutsis, S., Shear correction factors for plates and shells. Int. J. Numer. Meth. Engng, 33 (1992) 1537-52. 12. Reddy, J. N., A small strain and moderate rotation theory of elastic anisotropic plates. J. Appl. Mech., 54 (1987) 623-26.

Analysis of plates by splinefinite strip

199

13. Reddy, J. N., A refined nonlinear theory of plates with transverse shear deformation. Int. J. Solids Struct., 20 (1984) 881-96. 14. Reddy, J. N., A general non-linear third-order theory of plates with moderate thickness. Int. J. Non-linear Mechanics, 25 (1990) 677-86. 15. Palmerio, A. F., Reddy, J. N. & Schmidt, R., On a moderate rotation theory of laminated anisotropic shells - - Part I. Theory. Int. J. Non-linear Mechanics, 25 0990) 687-700. 16. Putcha, N. S. & Reddy, J. N., A refined mixed shear flexible finite element for the nonlinear analysis of laminated plates. Comp. Struct., 22 (1986) 529-38. 17. Librescu, L. & Stein, M., A geometrically nonlinear theory of transversely isotropic laminated composite plate and its use in the postbuckling analysis. Thin-Walled Struct., l l (1991) 177-201. 18. Stein, M. & Bains, N. J. C., Postbuckling behaviour of longitudinally compressed orthotropic plate with three-dimensional flexibility. AIAA Paper 86-0976, 1986. 19. Stein, M., Effects of transverse shear flexibility on postbuckling of plates in shear. AIAA J., 27(5) 0989) 652-55. 20. ()wens, M. E., Palazotto, A. N. & Dennis, S. T., Transverse shear flexibility in laminated plates undergoing large deflections. Computers and Structures, 45 0992) 69-78. 21. Bhimaraddi, A., Buckling and postbuckling behaviour of laminated plates using the generalized nonlinear formulation. Int. J. Mech. Sci., 34 0992) 703-15. 22. Bhimaraddi, A., Nonlinear free vibration of laminated composite plates. J. Engng Mech., ASCE, l l 8 (1992) 174-89. 23. Singh, G., Venkateswara, G. & Iyengar, N. G. R., Nonlinear bending of thin and thick unsymmetrically laminated composite beams using refined finite element model. Computers and Structures, 42 (1992) 471-79. 24. Lira, S. P., Lee, K. H., Chow, S. T. & Senthilnathan, N. R., Linear and nonlinear bending of shear-deformable plates. Computers and Structures, 30 (1988) 945-52. 25. Yoda, T. & Alturi, S. N., Postbuckling analysis of stiffened laminated composite panels, using a higher-order shear deformation theory. Computational Mechanis, 9 0992) 390-404. 26. Sacco, E. & Reddy, J. N., On first- and second-order moderate rotation theories of laminated plate. Int. J. Numer. Meth. Engng, 33 (1992) 1-17. 27. Hinrichson, R. L. & Palazotto, A. N., Nonlinear finite element analysis of thick composite plates using cubic spline functions. AIAA J., 24 (1986) 1836-42. 28. Reddy, J. N., A simple higher-order theory for laminated composite plates. J. Appl. Mech., 51 (1984) 745-52. 29. Reddy, J. N., Energy and Variational Methods in Applied Mechanics. Wiley, New York, 1984. 30. Kong, J. & Cheung, Y. K., Application of the spline finite strip to the analysis of shear deformable plates. Comp. Struct., 46(6) 0993) 985-88. 31. Cheung, Y. K. & Kong, J., Linear elastic stability analysis of shear deformable plates using a modified spline finite strip method. Computers and Structures, 47(2) 0993) 189-92. 32. Hinton, E., Iossifidis, L. & Ren, J. G., Higher order plate bending analysis using the modified Razzaque-Irons triangle. Comp. Struct., 26 (1987) 681-91.

200

J. Kong, Y. K. Cheung

33. Averill, R. C. & Reddy, J. N., An assessment of four-noded plate finite elements based on a generalized third-order theory. Int. J. Numer. Meth. Engng, 33 (1992) 1553-72. 34. Bathe, K. J., Finite Element Procedures in Engineering Analysis. PrenticeHall, New Jersey, 1982. 35. Washizu, K., Variational Methods in Elasticity and Plasticity. Pergammon Press, Oxford, 1982. 36. Zienkiewicz, O. C., The Finite Element Method, 3rd edn. McGraw-Hill, London, 1977. 37. Kant, T., Numerical analysis of thick plates. Comp. Meth. Appl. Mech. Engng, 31 (1982) 1-18. 38. Noor, A. K., Free vibrations of multilayered composite plates. AIAA J., 11 (1973) 1038-39. 39. Noor, A. K., Stability of multilayered composite plates. Fibre Sci. Technol., 8 (1975) 81-89. 40. Khdeir, A. A., Free vibration and buckling of symmetric cross-ply laminated plates by an exact method. J. Sound Vib., 126(3) (1988) 447~51. 41. Srinivas, S. & Rao, A. K., Buckling of thick rectangular plates. AIAA J., 7 (1969) 1645--46. 42. Timoshenko, S. P. & Gere, J. M., Theory of Elastic Stability, 2nd edn. McGraw-Hill, London, 1961. 43. Tham, L. G., Chan, A. H. C. & Cheung, Y. K., Free vibration and buckling analysis of plates by the negative stiffness method. Comp. Struct., 22 (1986) 687-92. 44. Pica, A., Wood, R. D. & Hinton, E., Finite element analysis of geometrically nonlinear plate behaviour using a Mindlin formulation. Comp. Struct., 11 (1980) 203-15. 45. Chung, W. C., Geometrically nonlinear analysis of plates using higher order finite elements. M. Phil. Thesis, University of Hong Kong, 1986. 46. Yamaki, N., Postbuckling behaviour rectangular plates with small initial curvature loaded in edge compression. J. Applied Mech., ASME, 26 (1959) 407-14. 47. Akay, H. U., Dynamic large deflection analysis of plates using mixed finite elements. Computers and Structures, 11 (1980) 1-11. 48. Chen, J. K. & Sun, C. T., Nonlinear transient responses of initially stressed composite plates. Computers and Structures, 21 (1985) 513-20. 49. Reddy, J. N., Geometrically nonlinear transient analysis of laminated composite plates. AIAA J., 21 (1983) 621-29.

APPENDIX Consistent mass matrix

The kinetic energy o f the plate is given by 2 U E = 1/2 { f (u2tt --F u2,tt --F-u2tt) dV}

Analysis of plates by splinefinite strip

201

Substituting eqn (2) into the above equation, and after some algebraic manipulation and simplification, we obtain the following expression for TOP. U E = 1/2 {f[u 2 + v2 + w 2 + 17h3/315(~h2x + ~k2y) - 4 h3/315(~bxW, x + W,x~x + d/yW,y + W,yd/y) + ha/252(W2x + w2y)]dA} p

Similarly, we can derive the following kinetic energy expression for MODTOP, UE = 1/2

{ f [ U 2 q'- V2 21- W 2

-Jr"17h3/315 (22~+ 22)

- ha/15(2xW, x + W,x2x + ~.yW,y + W,y2y) + h3/12(wEx +wEy)]dA}p

The derivation assumes that the density (p) is uniform through the thickness and initial imperfection is ignored, the consistent mass matrix can be easily formed by putting the displacement functions into the corresponding kinetic energy expression. Stiffness matrices

If D is the elasticity matrix with inplane components Dp and transverse shear component Ds, that is

o::[o p o] the components of the tangential stiffness matrix can be written as KL : I BT D

dV

KN ----f [BT Dp B~ (t~) + ~.i T (~) Dp n L -[- B mT N (iS)Dp B~m (~)] d V

I~ : f (W,x, W,y) T

(W,x, W,y)dV trxy

with

try j

(A1)

J. Kong, Y. K. Cheung

202 m+l

W,x = E [N(x)],xÂ˘'(Y) i=1 m+l

w~ = ~ IN(x)] ~,,,~ (y) i=1

while the secant stiffness matrix is given by

COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING, VOl.

12,107- 114 (1996)

VIBRATION OF CONTINUOUS BEAMS USING MODIFIED BEAM VIBRATION FUNCTIONS J. KONG AND Y. K. CHEUNG Department of Civil and Structural Engineering, The University of Hong Kong, Hong Kong

SUMMARY Free vibration of beams with intermediate point supports is studied by the classical Ritz method within the context of Euler beam theory. For the Ritz method, the displacement of a beam is approximated by a set of admissible trial functions which must satisfy the kinematic conditions at the ends and intermediate supports of the beam. To this end, a polynomial is superimposed on the conventional single-span beam vibration functions to form continuous-span or modified beam vibration functions. These modified beam functions are taken as the admissible trial functions for subsequent formulation. Stiffness and mass matrices are formulated using the conventional procedure and the resulting linear eigen-equation can be solved easily. A number of numerical examples are given to demonstrate the accuracy and efficiency of the present method. KEY WORDS

beam; vibration; trial functions; Ritz method

INTRODUCTION Free vibration analysis of continuous beams is frequently necessary in the design of structures such as bridges. There are two commonly used approaches for computing frequencies of continuous beams, namely the analytical method and the finite element method. In the analytical method,' each span of a continuous beam is treated as an individual, and the solution of the well-known fourth-order differential equation for single-span beam vibration is taken as its corresponding mode shapes. By imposing compatibility and equilibrium at the junction between adjacent spans, a non-linear equation involving transcendental functions is finally obtained. Solving this equation by the Newton-Raphson method, the entire frequency spectrum can be obtained. A similar concept has been applied to the development of the finite dynamic element m e t h ~ d . In ~ ' this ~ case, only a few finite dynamic elements are sufficient to model a continuous beam and to generate the whole frequency spectrum by solving the final non-linear eigenvalue matrix equation. In the standard displacement-based finite element method, a relatively large number of elements is required to model the continuous beam when compared with the finite dynamic element method or the analytical method. The method, therefore, involves a larger number of unknowns. Furthermore, the entire frequency spectrum cannot be obtained and only the lower part of the frequency spectrum can be computed with high accuracy. The frequencies, however, can be obtained easily by simply solving a linear eigenvalue matrix equation. In addition, the formulation of the standard beam finite element is very much simpler and only involves simple CCC 1069-8299/96/020107-08

0 1996 by John Wiley & Sons, Ltd.

Received 23 March 1995 Revised I6 August I995

108

J. KONG AND Y.K.CHEUNG

polynomials. Also it is very often in practice that only the lower part of the frequency spectrum or the first few frequencies are required. The standard finite element method, therefore, is still a popular tool for finding the frequencies of continuous beams. In this paper, free vibration of beams with intermediate point supports is analysed by a special set of admissible trial functions which satisfy the kinematic conditions at the ends and intermediate supports of the beam. These functions are formed by superimposing a polynomial, in the form of a power series, onto the conventional single-span beam vibration functions. (The functions were originally developed for thin-plate analysis4 and have been successfully extended and applied to thick-plate and laminated plate analysis.â&#x20AC;&#x2122;) The formulation and implementation of the method is simple. By using a small number of terms of the trial functions, the first few frequencies of a continuous beam can be easily and accurately computed by solving a linear eigenvalue matrix equation with a small number of unknowns. The method, therefore, retains the advantages of the analytical method or the finite dynamic element method and the standard finite element method for practical analysis and design of continuous beam. DISPLACEMENT FUNCTIONS Figure 1 shows a continuous beam with 4 intermediate point supports. The displacement of the beam can be approximated by

i=m

The ith terms of the trial functions and its corresponding unknown parameter are denoted by X ( x ) , and a;,respectively. It is important to recognize that the first term of the trial functions does not necessarily start with 1; the selection of the starting term will be demonstrated in the numerical examples. MODIFIED BEAM VIBRATION FUNCTIONS

For the Ritz method, each term of the trial functions is required to satisfy the boundary conditions at the ends, x = 0 and x = L , and the zero-deflection conditions at the intermediate point supports. In this study, the ith term of the trial functions is defined as X(X);

clamped end

= R(x)i4- X ( X ) ;

simply-supported end

Figure 1. A continuous beam

(W

MODIFIED BEAM VIBRATION FUNCTIONS

109

where

c r

&); =

rzjpxp

p=o

A, = constants to be determined, r = 4 + 3, and X ( x ) ; is the ith term of the conventional singlespan beam vibration functions which satisfies the corresponding boundary conditions at the ends (see Table I). To determine the constants 1, of the augmented polynomial X ( x ) ; , the four boundary conditions at both ends are imposed on equation (2b). The remaining 4 constants are found by imposing zero deflection, for the 4 intermediate supports, on (2a). The same procedure is repeated for each term of the trial functions. To clarify the idea consider a continuous beam with simply supported ends at x = 0, x = L and with an intermediate point support at x = xq. The boundary conditions at both ends can be expressed in the form and X" (0) = 0 = X" ( L );

X ( 0 );= 0 = X ( L )

ag(0);= 0 = g ( L ) , and p ( 0 )

= 0 =X ( L ) ;

(3)

where the second derivative with respect to the spanwise co-ordinate is denoted by ( )" =d2/dx2. For the intermediate point support, we have

Table I. Beam vibration functions Boundary conditions

Beam vibration functions

s-s

Xi= sin(pix/L)

X(O), = 0 , X ( L ) ,= 0 X ( O ) , = 0 , X ( L ) ,= 0

pi = in

c-c

Xi = sin(pix/L) - sinh(pix/l) - @,(cos(pix/L)- cosh(pix/L)) pi = ( i + 0 . 5 ) ~ G j = (sin p i sinh pj)/(cos p i - cosh p i )

s-c

Ti= sin(pjx/L) - G i sinh(pix/L) pi = ( i + 0.25)~

X ( O ) , = 0 , X ( L ) ,= 0 X'(0), = 0 , x ( L ), = 0

-

X ( O ) , = 0, X ( L ) ,= 0 X"(O), = 0, X ( L ) ,= 0

rpi = sin pi/sinh

F-F X(O),=O, X(L),=O X"'(O), = 0 , X ( L ) ,=o

.Ti = sin(pjx/L) + sinh(pix/l) - ~\$~(cos(p~x/L) + cosh(pjx/L)) pi= ( i - 1.5)~ G j = (sin p i- sinh pj)/(cos p i - cosh pi)

C-F X(O),=O, X ( L ) , = O X(O),=O, X(L),=O

= sin(pjx/L) - sinh(pix/L) - @,(cos(pix/L)- cosh(pix/L)) p i = (i - 1 . 5 ) ~ G i = (sin p i + sinh pi)/(cos p i + cosh p i )

S-F

X ( O ) , = 0 , X ( L ) ,= 0 X(O),=O, X ( L ) , = O

pi

z.

= sin(pix/L)

+ G isinh(pix/L)

pi= (i+0*25)~

Gi=sin pi/sinh p i

S = simply supported; F = free; C = clamped

110

J. KONG AND Y.K.CHEUNG

Substituting (2b) into (3) and (4), and using the corresponding single-span beam vibration functions given in Table I, the five unknown coefficients of the augmented polynomial can be determined by solving the following matrix equation:

-

1 0 0 0 0 1 L L2 L3 L4 0 0 2 0 0 0 0 2 6L 12L2 1 xq xq2 x; x9â&#x20AC;&#x153; ~

The first four rows refer to the boundary conditions at the ends while the last one refers to the condition of zero deflection at the intermediate point support. After the constants of the augmented polynomials are found, the modified beam functions are substituted into (l), and we can then proceed to the formulation of the stiffness and mass matrices.

STIFFNESS AND MASS MATRICES Using the conventional Ritz method, one can easily derive the following stiffness and mass matrices of the beam: [Kim m+l

[Klmm

..***-

[Klmn

1 1

[K] = ................. [K], ................. IKIm

[Klnm+l

[Mlmm

[MI,

m+ i

*---**

[Urn

...... [MI,,

, ..................

[MI = .................. [MI

I M 1 n m+1

.*.*..

[MI,,

and

I,XYEZ(x)Xâ&#x20AC;? dx [MI, 1, dx

(6)

(7)

L

[ K ] ,=

(8)

L

=

xip<x>xj

(9)

where EZ(x) and p ( x ) denote, respectively, the varying bending rigidity and weight per unit length along the span. For each individual span of a continuous beam, the modified beam vibration functions are numerically integrated using ten Gaussian points. The resulting linear eigen-equation, which is given by

[ K l l a ) = m2[M11a) can be easily solved for the natural frequency o.

(10)

NUMERICAL EXAMPLES

To assess the accuracy and efficiency of the modified beam vibration functions, we consider the

MODIFIED BEAM VIBRATION FUNCTIONS

111

following continuous beams with various geometric configuration: (i) a three-span continuous beam with uniform support spacing and different end supports. Bending rigidity and mass density are constant along the overall span (ii) a four-span continuous beam with non-uniform support spacing and simply supported ends. Bending rigidity and mass density are constant along the overall span (iii) a two-span continuous beam with non-uniform support spacing and simply supported ends. The beam is non-prismatic in the sense that it is made up of two prismatic spans with different bending rigidity and mass density.

In addition to the increasing number of terms of the trial functions used in each case, the starting term of the trial functions also varies; for a beam with 4 intermediate supports, and if the lowest nf frequencies are sought, then starting term rn = 1 or 1 + q n f s number of terms taken, nt s 2 x nf Note that, in general, more terms of the series ( n t ) can be taken to increase the accuracy of the solutions. However, it will be demonstrated in the following examples that sufficient engineering accuracy can be obtained by using 2 x nf terms of the series. For comparison with analytical solutions,’ the following normalized frequency ( B ) is adopted: w = (B/L)2(El/p)1’2 where L = beam overall length

Based on the results summarized in Tables 11-IV, a few remarks can be made: (i) In all cases but one, 2 x nfterms are sufficient to give an accuracy of less than 1 per cent difference between the present results and the analytical solutions. The number of unknowns involved in these examples is very small, with only 12 unknowns at most. (ii) For the second non-prismatic two-span beam in Table IV, a maximum of 2.7 per cent difference is found at mode 4 when 2 x nf terms are used to compute the frequencies. This relatively larger difference can be attributed to the fact that the influence of the sharp changes of bending rigidity and mass density of the beam are not taken into account in the trial functions. (iii) In general one can observe that the present solution converges rapidly to the analytical solution whatever the starting term of the trial functions is. When the frequencies are computed using only nf terms, however, the higher frequencies in most cases are more accurately predicted by starting the trial function from the term 1 + q. This is expected as the higher modes of a single-span beam approximate more appropriately the mode shapes of the corresponding continuous-span beam, especially for continuous beams with uniform support spacing and constant bending rigidity and mass density along the overall span. For example, for a simply supported, uniform beam, continuous over the two supports at the 1/3 points, the third mode of a single span beam of equal length is ‘exact’. Unfortunately, faster convergence is not guaranteed for all boundary conditions; for the cases of three-span beams with free-free ends or simply supportedfree ends, unreasonable results, which are excluded from Table 11, are obtained by starting the trial functions from the term 1 + 4. In these cases, the first two terms of the corresponding single-span beam vibration modes cannot be neglected in determining the trial functions.

112

J. KONG AND Y.K.CHEUNG

Table II. Frequencies of three-span continuous beams Frequencies Boundary conditions (m,nt)

Mode 1

2

3

4

5

6

9.425 9.425 9.425 9.425 9-425 9.425 9.425 9.425 9.425

10.72 10.69 10.68 10.67 10.69 10.68 10.68 10.67 10.67

13.06 12.96 12.96 12.92 12.96 12.96 12-93 12.91 12-89

18-85 18.85 18.85 18-85 1845 18-85 18.85 18-85

21.50 20.25 20.24 20.17 20.26 20.26 20.19 20.15 20.12

25-34 22.65 22.45 22.45 22.81 22.46 22.46 22.38 22.29

9.789 9.784 9.784 9.783 9.785 9.784 9.783 9.783 9.782

11.84 11.80 11.79 11.79 11.91 11.80 11.79 11.79 11.78

13.93 13.88 13.87 13.83 13.88 13.87 13-93 13.82 13.80

19-29 19.26 19.25 19.24 19.26 19.25 19.24 19.24 19.23

22-77 21.35 21.30 21.27 21.38 21.31 21.28 21.24 21.21

26.20 23.55 23.36 23.33 23.69 23.38 23.34 23.27 23.18

4.63 1 4.628 4.626 4.625 4.63 1 4.626 4.625 4.624

10.72 10.71 10-71 10.71 10.71 10.71 10-71 10.71

12.92 12.90 12.87 12.86 12.92 12-87 12.86 12.85

14-37 14-25 14-23 14.18 14.25 14.24 14-19 14-16

20.84 20.20 20,15 20.14 20.22 20.16 20.14 20-12

30.48 22.42 22.39 22.33 22.42 22-41 22.34 22.29

12.94 12.91 12-90 12.91 12.90 12.89

14.33 14.28 14.26 14-28 14.27 14.19

20.2 1 20-15 20.14 20-16 20.14 20-12

23.73 22.42 22-38 22.42 22.41 22-29

26.78 23.87 23-76 23.98 23.76 23.56

10.68 10.68 10.67 10.68 10.68 10.67

18.85

4.236 4.236 4.236 4.235 4-235

4.953 4.947 4.945 4.945 4.943

10.75 10.74 10.74 10.74 10.74

12.88 12.84 12433 12-83 12.82

14.38 14.21 14.21 14.15 14.12

20.22 20-15 20.14 20.13 20.12

4.615 4.61 1 4.610 4.610 4-609

9.820 9.813 9.812 9-812 9.810

11.84 11.80 11.80 11-79 11.78

13.94 13.82 13.81 13-77 13.74

19.31 19.25 19.25 19-24 19.23

23.58 21.35 21.31 2 1.27 21.21

*Boundary conditions at ends of beams

113

MODIFIED BEAM VIBRATION FUNCTIONS

Table III. Frequencies of a four-span continuous beam with both ends simply supported Frequencies b Mode

(m,nt)

1

2

3

4

(174) (1,6) (1,8> (494) (4,6) (4,8) Exact

9.665 9.658 9.654 9.668 9.643 9.638 9.630

13.68 13.48 13.47 13.49 13.47 13.47 13.42

17.07 16.56 16.43 16.54 16.50 16.49 16.46

21.91 19.19 19.01 19.51 19.17 19.16 19.00

EI, p = constant. L , = 0.1, L, = 0.3, L, = 0.4,L, = 0.2, where L, = length of span iloverall beam length

Table IV. Frequencies of a two-span non-prismatic continuous beam with both ends simply supported Frequencies j? Mode (m,ntl

1

2

3

4

(EI)z/(EI)l =0.6561 = p 2 / p 1 11.71 11.60 11.65 11-57

15.27 14.42 14.52 14.36

(EZ)J(EZ), =0-1296= pJpl 5.73 1 9.893 11.26 5.643 9.831 11.10 5.741 9-901 11.28 5.625 9.820 11.08 5.510 9.733 10-93

17.11 14.92 15.67 14.88 14-53

5.258 5.248 5.254 5.241

9.382 9-371 9-376 9.367

L , = 0-3, L, = 0.7 where Li = length of span i/overall beam length. For calculating normalised frequencies, ( E I ) , and p , are used

CONCLUSION Modified beam vibration functions are used to compute frequencies of continuous beams with intermediate point supports. The method is simple and the functions are relatively easy to generate when compared with the classical continuous-beam vibration functions. The lower frequencies can be accurately computed by simply solving a very small system of linear eigenvalue equations. The method is also versatile in the sense that they can be applied to continuous beams with different boundary conditions, varying bending rigidity and mass density. The accuracy and efficiency of the present method are demonstrated in the given numerical examples. The method is currently being investigated for the vibration of curved beams, and buckling of continuous beams.

114

J. KONG AND Y. K. CHEUNG

REFERENCES 1. D. J. Gorman, Free Vibration Analysis of Beam and Shafts, Wiley, 1975. 2. N. J. Ferguson, ‘The free and forced vibration of structures using the finite dynamic element method’, Ph.D. thesis, University of Virginia, Florida, USA, 1991. 3. A. Y. T. h u n g , Dynamic Stiffness and Substructures. Springer Verlag, London, UK, 1993. 4. Zhou Ding, ‘Eigenfrequencies of line supported rectangular plates’, Research Report, Department of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing, PRC, 1993. 5. J. Kong and Y. K. Cheung, ‘Vibration of shear-deformable plates with intermediate line supports: A finite layer approach’, J . Sound Vib., 184,639-649 (1995).

Pergamon

Copyright Lâ&#x20AC;&#x2122; 1995 Elsevicr Science Ltd Printed in Greal Britain. All rights mcrved Oa45-7949/95 59.50 + 0.00

00457949(95jooo95-x

THREE-DIMENSIONAL FINITE ELEMENT ANALYSIS OF THICK LAMINATED PLATES J. Kong? and Y. K. Cheung Department of Civil and Structural Engineering, University of Hong Kong, Pok Julaus Road, Hong Kong (Received 11 May 1994)

Abstract-A displacement-based, three-dimensional finite element scheme is proposed for analyzing thick laminated plates. In the present formulation, a thick laminated plate is treated as a three-dimensional inhomogeneous anisotropic elastic body. Particular attention is focused on the prediction of transverse shear stresses. The plane of a laminated plate is first discretized into conventional eight-node elements. Various through-thickness interpolation is then defined for different regions of the plate; layerwise local shape functions are used in the regions where transverse shear stresses are of interest, while an ad hoc global-local interpolation is used in the region where only the general deformation pattern is concerned. For satisfying the displacement compatibility between these two regions, a transition zone is introduced. The model incorporates the advantages of the layerwise plate theory and the single-layer plate theory. Details of formulation will be presented together with several numerical examples for demonstrating the proposed scheme.

w(x,Y,z)=w0(~,Y)+w,(x,Y)~z

INTRODUCTION

The increasing use of laminated composite materials in different branches of engineering stimulates the advances in the analysis of laminated beams and plates. Extensive research has been conducted in the theoretical development of new plate theories and, consequently, many interesting approaches have been put forward during the last decade. No attempt is made to review modern plate theories in this brief section. (A literature review in modern plate theories would entail in itself a major piece of work! Comprehensive reviews of the developments in the analysis of laminated plates have been recently summarized by Kapania and Raciti [l], Noor and Burton [2-41, Reddy (51 and Jenielita [6].) Only a very brief description of the relevant plate theories is given below. Most of the existing displacement-based plate theories fall into two major categories, namely, (I) single-layer theory (SLT) and (2) layerwise theory (LT). In the SLT, the displacement components are expanded in power series of the thickness coordinate z, giving

+

w2(x, y) . z*f.

.

The highest degree of the coordinate z in the series determines the order of the corresponding theory. For high-order theories, plates are physically treated as three-dimensional bodies. All six components of strains are taken into consideration. For low-order theories, however, transverse normal strain is usually neglected and only the remaining five-strain components are considered. An example of the low-order theory is the well-known first-order shear deformation theory due to Mindlin and Reissner for which the displacement functions involve five independent variables, namely, three mid-plane displacements and two rotations, that is: 0,

YVz) = U&Y)

+ z . @r(x, Y)

aY,z)=%(x,Y)+z. w,

d&Y)

Y, 2) = w&, Y).

Mindlin plate theory can be further reduced to Kirchoffs thin-plate theory with only three independent variables; U(& Y, 2) = %I(& Y) - z . hJ(x, ~(X,Y,z)=uo(x,Y)--.~wo(â&#x20AC;&#x153;,y)ldY

t To whom all correspondence

W(X,Y, z> = W&Y). 1051

YWX

1052

J. Kong and Y. K. Cheung

The Mindlin and Kirchoff plate theories have been commonly used for the finite element analysis of moderately thick and thin isotropic plates. Kirchoffs plate theory is not capable of dealing with laminated composite plates, except for very large span-tothickness ratio, because the transverse shear moduli of modern composite materials are usually very low compared to the inplane tensile moduli, with the result that transverse shear deformation can be of considerable importance compared to homogeneous isotropic materials. In addition, these plate theories neglect cross-sectional warping and cannot accurately predict the through-thickness transverse shear stresses of laminated plates. Better accuracy in predicting global responses (such as deflections, natural frequencies and buckling loads) and in-plane stresses can be achieved by increasing the order of interpolation across the thickness of the plate. The number of unknowns involved in such an analysis depends on the desired level of accuracy and is independent of the number of material layers. This hierarchical concept was first proposed by Szabo and Sahrmann [7] for the analysis of isotropic plates and shells, and later extended by Surana and Sorem [8], Barboni and Gaudenzi and Matsunaga [IO] for the analysis of laminated composite plates and thick isotropic plates. In this approach, however, all six components of strains are continuous across the interfaces between layers of different materials and the continuity of transverse shear and normal stresses is totally out of the question. Consequently, if the transverse shear stresses are needed, they are obtained indirectly by integrating the equilibrium equations of elasticity [I lll5]. Regardless of its complex implementation, this controversial procedure involves higher-order in-plane differentiation of the displacement function, and therefore its accuracy is questionable for analyzing plates with highly irregular geometry. In the LT, the displacement variation through the thickness of each layer of material is interpolated by means of the standard one-dimensional Lagrangian shape functions. For analyzing thick laminated plates, each layer of material can be modelled by several linear one-dimensional elements [16-181 or a few quadratic elements [ 191 or a high-order hierarchical element [20]. In this approach, only displacement continuity is maintained between layers of different material and the continuity of transverse shear stresses can still be satisfied in an integral sense through the potential energy formulation. Accurate transverse shear stresses can be obtained without integrating the equilibrium equations of elasticity except for the case in which a single one-dimensional linear element is used for each material layer [16]. In addition, extension to the study of nonlinearity [21] and delamination [22] is straightforward. However, it suffers from a serious drawback in that the number of unknowns involved in such a finite element analysis depends on the number of layers of material. The method is, therefore, computationally

expensive for analyzing laminated plates with many layers of materials. A three-dimensional finite element scheme is presented in this paper for the analysis of laminated composite plates with small span-to-thickness ratio. (A thick plate is herein defined as a plane, non-curved structural element bounded by straight or curved edges and treated as a complete three-dimensional elastic body. For a rigorous definition of thick plates, see Ref. [23].) Particular attention is focused on the prediction of transverse shear stresses for thick plates. The surface of a thick plate is first divided into eightnode elements. Various through-thickness interpolation is then defined for different regions of the plate. The philosophy is that layerwise local shape functions are used in the regions where transverse shear stresses are of interest and importance. The layerwise model ensures the continuity of transverse shear stresses to be satisfied in an integral sense through the potential energy functional and hence a realistic prediction of transverse stresses is anticipated. Because of the large number of unknowns involved in a layerwise model, a special through-thickness global-local interpolation is used in the area of the plate where only global responses are of interest. For satisfying the displacement compatibility between these two regions, a transition region is introduced. The method incorporates the advantages of the layerwise plate theory and the single-layer plate theory. Although the proposed method is similar to a conventional 3-dimensional finite element method in terms of interpolation capability and problem size, it maintains a two-dimensional data structure similar to a conventional two dimensional finite element model. The advantages over the conventional threedimensional finite element models are two-fold, namely, the volume of input data is reduced, and the two-dimensional cross-sectional interpolation and the one-dimensional spanwise interpolation can be refined independently of each other without having to reconstruct a three-dimensional finite element mesh. These advantages facilitate practical modelling of plates. The two-dimensional type data structure also allows faster formulation of the element stiffness matrices. Details of the method and its formulation will be presented in the following sections together with a few numerical examples for verifying its accuracy in analyzing plates with large or small span-to-thickness ratio. FORMULATION

Figure 1 depicts a laminated plate with layers of composite materials. To describe the behavior of an element on the X-Y plane, the displacement components within each element is defined by the conventional eight-node shape function [N(x, v)] (other two-dimensional shape functions are equally valid), mathematically,

Three-dimensional

fmite element analysis

A rectangularthiik plate

1053

M(i) = {M,, Mz, M,, M41

(2)

where M,(c) = -9(1/3

- i)(l

M,(C) = 27(1/3 - i)(l

V

2

.Y

=t

:,,*Y

- <)(1/3 + O/l6 - i)(l

+ [)/lb

M,(C) = 27(1/3 + 5)(1 - i)(l

+ [)/lb

C-.x

/â&#x20AC;&#x2122; :

Fig.

M4(5) = -9(1/3

â&#x20AC;&#x2122;

,â&#x20AC;&#x2122;

1*.__.,

- i)(l

+ 1)(1/3 + 0116

X

1. A thick laminated composite plate and discretization into thick-plate elements.

its

(la)

(lb) ,=I

W

=

,\$,[N(X, .Y)l@jicz){rI,,

UC)

where aj(z) represents the jth term of the throughthickness interpolation with {a},, {/?}, and {y }, being the corresponding nodal displacement unknowns. It is worth noting that the general format of the assumed displacement functions is common to the layerwise theory and the single-layer theory. The choice of the through-thickness interpolation, however, is different from each other. The layerwise interpolation across the thickness of the plate is first defined in such a way that parabolic or higher-order variation of transverse shear stresses can be accounted for. To this end, one-dimensional Lagrangian cubic shape functions are used for each material layer. Mathematically, the cubic shape functions for the ith layer of material are given by

coordinate ranging from -1 to 1 for z 6 [z, ~, , zil. It is computationally inefficient to analyze the entire plate with this layerwise model because of the large number of unknowns involved. To clarify the idea, Fig. 2 shows two simple cases where layerwise model is adopted in the vicinity of geometric discontinuity for computing transverse shear stresses. The remaining area is subdivided into two regions, namely, the transition zone and the region in which the global-local interpolation is used. In the area where only the global response is concerned, a set of through-thickness global functions is selected to transform the nodal variables of the layerwise cubic shape functions to a small set of generalized parameters associated with the global functions. In essence, the layerwise cubic shape function is enforced to follow a small number of pre-defined patterns. With a limited sacrifice of accuracy, the number of unknowns is thereby reduced with respect to a full layerwise model. The jth term of the global-local interpolation can be written as and c is the layerwise

cp,(z)=]M(z)1{4},

for

iL

Line

1

III

(3a)

where {S,), is the nodal displacements of layer i computed from the jth term of the global functions. In this study, the following global functions are chosen, Z(z) = {Z,, zz, z,.

I

z~]~,_,,z,l

z,.

.)

(3b)

where

II

: : Line 2 I I

z, = 1 -z/h,

z, = z/h

Z, = sin[(j - 2)nz/h]

Line

1:

: Line 2

Fig. 2. Division of a plate into three regions with different through-thickness interpolation. Region I: layerwise cubic shape function (M); Region II: through-thickness globallocal interpolation (cp); Region III: transition zone (M + cp). Line 1: nodal degrees of freedom which correspond to the through-thickness global-local interpolation of the transition zone elements are restrained in the global stiffness matrix. Line 2: nodal degrees of freedom which correspond to the layerwise cubic shape functions of the transition zone elements are restrained in the global stiffness matrix.

for

j >, 3

and the number of unknowns increases with increasing number of terms taken from eqn (3b). One can easily observe the hierarchical nature of the globallocal interpolation. It is also noteworthy that any other appropriate global functions can be inserted into eqn (3b) without changing other parts of the formulation. To satisfy the displacement compatibility between these two areas, a transition zone is introduced. Unlike conventional transition finite elements, the interpolation function across the thickness of the

1054

J. Kong and Y. K. Cheung Table 1. Interpolation across thickness in different regions Regions

Interpolation across thickness Layerwise cubic shape function, M Global-local interpolation, cp M+cp

I

II III

Number of terms P,” 3nP + 1 r= 3nlf 1 +r

‘p = number of terms in eqn (1) 3p = number of degrees of freedom per node; bnl = number of layers of material in the plate; 5 = number of terms taken from the global functions given in expression (3b) and r < 3rd + 1,

elements in the transition zone is defined by augmenting the layerwise shape function to the global-local interpolation. (The basic features of the three regions are summarized in Table 1.) As a consequence, displacement compatibility can be enforced by restraining the appropriate nodal degrees of freedom on the boundary between the corresponding regions. These restraints can be easily applied in the same way as is done for the boundary conditions. Having defined the displacement functions of the plate, the three-dimensional strain-displacement relation can be written as

functions are actually involved and they can be evaluated explicitly using symbolic computation. Explicit form of the integrals in the thickness direction is given in the Appendix. It should be noted that the assembly process takes into account the various number of nodal degrees of freedom in different regions of the plate. After assembly, the boundary conditions and the compatibility between different regions are applied to the global stiffness matrix. Stresses are calculated at the 2 x 2 integration point and bi-linearly extrapolated to nodal points. NUMERICAL

To establish the accuracy of the present method, we shall consider two symmetric laminated square plates with four (0’/90”), and nine (O”/900/Oo/90”/Oo),~ layers of material. The plates are formed of layers of orthotropic material with the material principal axis 1 oriented at an angle of 0 or 90” to the global X-axis. The total thickness of the 0 and 90” layers are the same, whereas layers at the same orientation have equal thicknesses. Under these conditions, the effective laminate stiffness in the x and y directions are the same. All edges of the plates are simply supported. In the following examples, the top surface loading is defined as

=:,

P

or it) = C [46~1, ,=I

where

@; = d@/dz

q = tj sin(nx/a) sin(rty/a)

and the element stiffness matrix for each of the three regions takes the form

Kl,., . WI

=

m,.,

. .

Kl,.,

[Kim,n

[Kl,m . . . WI,,, where [K],,, =

EXAMPLES

1

(5)

[B]z[D] [B], d V. s”

Integration can be separately performed for the in-plane shape functions and the through-thickness interpolation. The in-plane eight-node shape functions, denoted by N for simplicity, are numerically integrated using 2 x 2 Gaussian points. In the through-thickness direction, however, integration is carried out analytically for the layerwise shape functions and the globallocal interpolation and the coupling between them. No numerical integration in the thickness direction is required because only integrations of cubic shape

where 4 is the maximum intensity of the double sinusoidal load at the centre of the plate. The material properties are defined as follows: x 106psi

E,, = 25 x lo6 psi

Ezz=E,,=l

G,* = G,, = 0.5 x lo6 psi

G,, = 0.2 x lo6 psi

VIZ= v2, = V,)= 0.25 where 1 signifies the direction parallel to the fibers and 2, 3 the transverse direction. By symmetry, only a quarter of the plate is analyzed. Emphasis is placed on the analysis of plates with a small span-to-thickness ratio as most of the low-order plate theories are not capable of giving accurate stress predictions for such plates. For nine-ply laminates, the in-plane displacements follow closely a zig-zag variation through the thickness [24] and, therefore, an additional zig-zag function is

Three-dimensional

finite element

included in the global functions of eqn (3b). Because through-thickness distribution of in-plane stresses depends on the corresponding distribution of in-plane displacements, more accurate in-plane stresses can be predicted by including zig-zag function in eqn (3b); this will be demonstrated in the numerical examples. Suppose that the mid-surface of the laminated plate is located in the Lth layer, the zig-zag function can be written as, for layer k,

The following normalized quantities are defined for the presentation of results:

i3 =

i(z)=

1 (-l)ihi+(-l)k(l ,=I

+c)hk/2-zL

1055

analysis

(6)

7c4Qw/12s4hq

Q = 4G,z + ]E,, + &(I + 2v22)1/(1- v,2vz,)

where

s=a/h L-l

c

zL=

I= I

Z= z/h - l/2.

(-l)â&#x20AC;&#x2DC;hi+(-l)Lz,

and zd is the distance between the laminate midsurface and the lower surface of the L th layer, hi the thickness of layer i, [ the layer natural coordinate which ranges from - 1 to + 1. The zig-zag function for the nine-ply laminate is given in Fig. 3. Stresses are calculated and extrapolated from the Gaussian points of the eight-node elements. Results are compared with the exact elasticity solutions of Pagan0 and Hatfield [25].

In a few instances (small s) the maximum values of transverse shear stresses do not occur at mid-surface, hence in these cases there are two entries in the respective columns. The upper value gives the function at mid-surface, while the lower number represents the maximum value. Assessment of layerwise model (LM)

Each plate is first analyzed using the present method with the layerwise cubic shape functions alone for the

10

9 8

7

6

I

-1

I

-0.6 -0.8

0

-0.4 Zig-Z09

Fig.

3. The zig-zag

4

0.2

-0.2

function

I

1

0.6 0.4

0.8

function

for a nine-ply

laminate

with thickness

= IO.

1056

J. Kong

Table 2. Comparison Mesh”

6, -,-,+2

664 444 224 663 443 223 exact

664 444 exact

of results for a four-ply

au

1

2 -2

6,. ” a +- 1 2’2’-4

and Y. K. Cheung

laminate

with s = 2 and 4. Only layerwise

i,, 0,; ,o

1.404 - 0.922 1.424 -0.935 I.518 - I.001 I.404 - 0.922 I.423 -0.935 I.517 - I .ooo 1.388 -0.912

0.844 -0.803 0.856 -0.814 0.917 -0.872 0.841 -0.805 0.853 -0.816 0.913 -0.873 0.835 -0.795

s=2 0.154 (0.265) 0.157 (0.271) 0.168 (0.291) 0.155 (0.268) 0.157 (0.271) 0.168 (0.291) 0.153 (0.264)

0.729 -0.693 0.739 -0.702 0.720 - 0.684

0.670 -0.673 0.679 -0.683 0.663 -0.666

S=4 0.222 (0.224) 0.225 (0.227) 0.219 (0.222)

f,_ p-o,0

cubic shape function o,o,

i,,.

* 2

0.289 (0.300) 0.293 (0.304) 0.313 (0.326) 0.306 (0.320) 0.310 (0.329) 0.332 (0.338) 0.295 (0.298)

-0.0872 0.068 I -0.0885 0.0690 - 0.0949 0.0739 -0.0872 0.0680 -0.0884 0.0690 - 0.0948 0.0738 -0.0863 0.0673

0.292

-0.0472 0.0464 -0.0479 0.0470 -0.0467 0.0458

0.296 0.292

I

is used

M;!FO 2’ 2’ 11.764 I I .764 II.754 II.763 II.763 II.689 II.767

4.491 4.491 4.491

“Mesh: first two digit denotes the number of elements in each of the in-plane direction. The third digit denotes the number of one-dimensional cubic element in the thickness coordinate. Values in bracket are the maximum of the corresponding transverse stress.

entire plate. Convergences of the layerwise model are summarized in Tables 2 and 3. One can easily observe that a mesh of 4 x 4 elements in the X-Y plane together with a cubic element per material layer is sufficient to give accurate in-plane and transverse shear stresses; all stresses are within 3% difference to the exact elasticity solutions for both cases of four and nine-ply laminates. Furthermore, the deflections also converge rapidly to the exact solutions.

eight-nodes elements and a cubic element per material layer. Increasing number of terms are then taken from the through-thickness global functions and used throughout the entire plate. The center deflections of the resulting global-local model are compared with the solutions of the corresponding Iayerwise model and the exact solutions, see Table 4. Considering the results of the four-ply laminates obtained by taking five terms from the global functions, one can see that more than 60% reduction in the number of unknowns is achieved with less than 3% loss in accuracy of the corresponding deflections. For the nine-ply laminates, nine terms of the global functions are sufficient to

Assessment qf global-local model (GLM) To examine the accuracy of the global-local model. the plates are first discretized into a mesh of 4 x 4 Table 3. Comparison

of results I

for a nine-ply

Mesh”

5. a2’2’_2 a f-

669

I.274 -0.876 1.291 -0.888 1.368 -0.951 1.260 -0.866

I.063 -0.834 1.078 -0.846 I.154 -0.905 I.051 -0.824

0.692 -0.656 0.701 -0.666 0.684 -0.649

0.635 -0.619 0.644 -0.627 0.628 -0.612

449 229 exact

669 449 exact

6,. ;,;A5

2

laminate r,;

with s = 2 and 4. Only layerwise 0, ”2

,0

s=2 0.206 (0.225) ‘0.209’ (0.228) 0.224 (0.245) 0.204 (0.224) s=4 0.226 0.229 0.223

cubic shape function

is used

a a

5,: ;, 0, 0

M’ - -,o 2’2

0.199 (0.210) ‘0.196’ (0.21 I) 0.213 (0.255) 0.194 (0.21 I)

-0.0730 0.0541 -0.0740 0.0548 -0.0795 0.0587 -0.0722 0.0534

12.288

0.226 (0.228) 0.229 (0.23 I) 0.223 (0.225)

-0.0341 0.0332 -0.0346 0.0336 -0.0337 0.0328

4.079

12.288 12.280 12.288

4.079 4.079

“Mesh: first two digit denotes the number of elements in each of the in-plane direction. The third digit denotes the number of one-dimensional cubic element in the thickness coordinate. Values in brackets are the maximum of the corresponding transverse stress.

1057

Three-dimensional finite element analysis Table 4. Convergence of center deflections of the global-local model Center deflection E s=4

s=2 Pâ&#x20AC;?

GLM

5 9 13

11.432 Il.475 11.764

LM exact

11.764 11.767

5 9 12

11.547 12.188 12.216

LM exact

12.288 12.288

% error

GLM

% error

Four -ply laminate 2.8 4.395 2.5 4.413 0.0 4.491

% reduction

2.1 1.7 0.0

62 31 0

3.8 0.8 0.6

82 68 51

4.491 4.491 Nine-ply laminate 6.0 3.924 0.8 4.047 0.6 4.054

4.079 4.079

% = number of terms taken from expression [eqn (3)]. GLM = LM = layerwise model. % error = percentage difference between corresponding LM(4 x 4 mesh with one cubic element per material = percentage reduction in the number of unknowns of the GLM corresponding LM.

global-local model, the GLM and the layer). % reduction with respect to the

Table 5. Comparison of results for laminated plates with s = 100. A mesh of 4 x 4 with only through-thickness global-local interpolation

2 4

exact 2 4 5

exact

+ 0.550 10.553 kO.539

Four-ply laminate kO.276 +0.277 kO.271

TO.0216 TO.0219 TO.0213

0.993 1.007 I.008

& 0.550 kO.553 kO.553 kO.539

Nine-pty laminate &O&O kO.443 + 0.443 + 0.43 1

TO.0216 TO.0218 TO.0218 TO.0213

0.991 1.005 1.005 1.008

â&#x20AC;&#x153;p = number of terms taken from expression (3).

retain an accurate prediction of deflections with less than 1% loss in accuracy and the number of unknowns is reduced by more than 60% in comparison with a complete layerwise model. To further verify the implementation of the globallocal model, the stresses of the four-ply laminate are obtained by using 13 terms of the global functions. It is found that they are identical to those using four layerwise cubic shape functions; this reveals the fact that no constraint is effectively applied to the layerwise model as an identical number of unknowns is used in both cases. For laminated plates with s = 100, we anticipate that the first two linear terms of the global throughthickness functions are sufficient to give accurate deflections as well as in-plane stresses. As demonstrated in Table 5, the present results, including deflections and stresses, are within 2% difference from the exact solutions. Assessment

of the combined

adopted. The remaining portion is divided into the transition zone and the zone where the global-local model is empfoyed. Comparison of stresses are summarized in Tables 6-8. For the four-ply laminates with s = 2 and s = 4, very good results of transverse shear stresses are obtained by using five terms in the global-local interpolation zone; a maximum of

model

A plate is divided into three different regions in which different interpolation across the thickness is used, as shown in Fig. 4. In the area where transverse shear stresses are required, a layerwise model is

Fig. 4. Division of a square plate into different regions for calculating the transverse shear stresses T, at I and I,, at 2.

1058

J. Kong and Y. K. Cheung

Table 6. Comparison of results for a four-ply laminate with s = 2 and s = 4. A mesh of 4 x 4 using the combined P”

5 exact

5 exact

z,,

a

1

f )* -

2’2

2

au

d,. -,-,*22

1.265 - 0.849 I.388 -0.912

0.713 -0.710 0.835 -0.795

0.721 -0.686 0.720 -0.684

0.641 -0.647 0.663 -0.666

1

4

i,, 0,; ,o s=2 0.157 (0.267) 0.153 (0.264) s=4 0.222 (0.222) 0.219 (0.222)

E!“O 2’2’

f,., ; ,o,o 0.281 (0.296) 0.295 (0.298)

-0.0861 0.0673 -0.0863 0.0673

0.287

-0.0476 0.0466 - 0.0467 0.0458

0.292

model

11.503 II.767

4.422 4.49 1

‘p = number of terms taken from expression (3b)

Table 7. Comparison of results for a nine-ply laminate with s = 2 and s = 4. A mesh of 4 x 4 using the combined model I aa 2 1 6, ” a f*!“lJ 8,. -,-,+fy... o,;,o <,, ;,o,o fVJ 0, 0, + j P” 2’2’-2 2 2 -5 2’ 2’ s=2 9 1.237 1.126 0.209 0.198 -0.0720 12.215 -0.863 -0.908 (0.234) (0.210) 0.0524 exact 1.260 1.051 0.204 0.194 -0.0722 12.288 -0.866 -0.824 (0.224) (0.21 I) 0.0534 s =4 9 0.229 0.228 4.059 0.690 0.658 -0.0340 -0.641 -0.654 0.0330 exact 0.223 0.223 4.079 0.684 0.628 -0.0337 -0.612 -0.649 0.0328 ‘p = number of terms taken from expression (3b).

Table 8. Comparison of results for four- and nine-ply laminates with s = 100. A mesh of 4 x 4 using the combined model 1 aa 1 aa 2 @!!O cTv p,-,+q -,-,+fy: 0, ;, 0 f,.: ;,o, 0 fTP O,O, + 2 P” 2’2’ 2 2 -2 2 2 -5 Four -ply

4 exact

kO.553 kO.539

kO.277 kO.271

4 5 exact

* 0.553 kO.553 kO.539

& 0.443 + 0.443 io.431

0.339 0.339 Nine -ply 0.281 0.258 0.259

0.145 0.138

TO.0219 f0.0213

1.007 1.008

0.242 0.230 0.219

f0.0218 TO.0218 TO.0213

1.004 1.004 1.005

% = number of terms taken from expression (3b).

5% difference is found between the present method and the exact solutions. Reasonable in-plane stresses are also obtained. A 39% reduction in the effective number of degrees of freedom (i.e. the number of degrees of freedom after applying the compatibility restraints on the interfaces between different zones) is achieved when comparing the combined model with respect to the corresponding complete layerwise

model. For the nine-ply laminates, the agreement of the solutions obtained by the present method and the analytical method is reasonably good, see Table 7. In these cases, a 49% reduction in the effective number of degrees of freedom is achieved. Results of laminates with s = 100 are also presented

in Table 8. The prediction of both in-plane and transverse stresses by the combined model is reasonably accurate. To demonstrate the effect of zig-zag function in predicting in-plane stresses for the nine-ply laminates, comparison between the two sets of results obtained

Table 9. Comparison of results of nine-ply laminates for the effects of zig-zag functions on in-plane stresses. A mesh of 4 x 4 using the combined model 1 1 2 P* #,a a *ey ff “, +f,,. o,o, f 2 2’2’ 2 2’2 5 s=2 -0.0720 1.237 1.126 9 - 0.908 0.0524 -0.863 0.914 - 0.0697 9** 1.123 0.0501 -0.676 -0.713 - 0.0722 1.051 exact 1.260 0.0534 -0.824 -0.866 s=4 9 -0.0340 0.658 0.690 0.0330 -0.641 -0.654 -0.0339 9** 0.633 0.576 0.0329 -0.558 -0.597 -0.0337 0.628 exact 0.684 0.0328 -0.612 - 0.649 s = 100 4 TO.0218 + 0.443 kO.553 TO.0219 20.442 4** kO.553 TO.0213 kO.431 kO.539 exact *p = number of terms taken from expression (3b). **Results obtained without using zig-zag function.

Three-dimensional

(4

finite element

analysis

0.7

0.6

0

I

I

I

I

I

I

I

I

I

I

1

2

3

4

5

6

7

8

9

10

Z-coordlnote 0

(b)

+

GLM

Combired

0.7

0.6

05

a4

03 : E : ;;

0.2

0.1

0

-0

I

0

I

I

I

I

I

I

I

I

I

I

I

2

3

4

5

6

7

8

9

10

Z-coordinate 0

GLM

+

Combined

Fig. 5. Distribution of transverse shear stresses through the thickness of the four-ply h = 10). (a)z,YI, (b) zVz. CM =combined model (subdivision into three regions) interpolation is used for the entire plate.

laminate (a/h = 2, HR = global-local

J. Kong (a)

and Y. K. Cheung

0.6

0.5

0.4

03

t 0

-0

f

1 0

I

-

I

I

I

I

I

I

I

I

I

I

1

2

3

4

5

6

7

8

9

10

7

a

9

10

Z-coordinate 0

tb)

+

GLM

Combined

06

0.5

04

0.3

VI : : > v1

0.2

0.1

0

-0

1

0

1

2

5

4

3

6

Z-coordlnote a Fig. 6. Distribution

Gi-M

+

Combined

of transverse shear stresses through the thickness of the nine-ply h = 10). (a) z.~:.-.(b) ql. Legend same as Fig. 5.

laminate

(a/h = 2,

Three-dimensional

by using the same combined model with or without the zig-zag function is given in Table 9. It is apparent that the inclusion of zig-zag function makes a positive influence on predicting in-plane stresses, especially for the thick laminates with s = 2 and s = 4. For s = 100, however, no significant difference between the two sets of results can be found because, in this case, the classical â&#x20AC;&#x153;plane sections remain planeâ&#x20AC;? assumption is sufficiently accurate to describe the variation of inplane displacements through the thickness of plates. In other words, the in-plane displacements follow a straight-line variation through the thickness. To examine the distribution of transverse shear stresses obtained by the combined model and the global-local model, results for the four and nine-ply laminates with s = 2 are given in Figs 5 and 6. It can be seen that results of the global-local model exhibit severe stress jumps across the interfaces between material layers. On the top and bottom surfaces of the plates, moreover, the global-local model shows significant residual transverse shear stresses which violate the traction-free conditions on these surfaces. The combined model, however, performs much better in terms of the traction-free conditions and interfacial stress jump. Similar observations can be made for other span/thickness ratio. The combined model is, therefore, more reliable in predicting the continuous transverse shear stresses distribution in laminated plates. CONCLUSIONS It has been demonstrated that approximate three-dimensional solutions for a laminate can be obtained by using a combined layerwise-single layer model. Reasonable prediction of in-plane stresses and transverse shear stresses can be made and, at the same time, a compromise between the computational cost and accuracy is achieved. Additionally, integration in the thickness coordinate can be done separately and analytically. This facilitates the computer implementation of the method. The accuracy and application of the present method are demonstrated by solving simple problems of square laminated plates. For general use, one should replace the eight-node in-plane shape functions with Lagrangian shape functions as they are not affected by angular distortions and therefore the more reliable elements for analyzing plates with complicated geometry [26]. For rectangular and parallelogram plates, however, serendipity elements are more cost effective as they have the same accuracy with the Langrangian elements but without the additional degrees of freedom associated with the internal nodes. Although very small problems are dealt with in this paper, the savings in computational cost (reduction in number of unknowns) would be much more significant for complicated plates with only a small area where transverse shear stresses are of interest. In this case, the number of unknowns

finite element

1061

analysis

involved in a full layerwise model is extremely large and the present approach would be more cost-effective.

REFERENCES

and S. Raciti, Recent advances in I. R. K. Kapania analysis of laminated beams and plates. part I: shear effects and buckling. AIAA J. 27, 923-934 (1989). 2. A. K. Noor and W. S. Burton, Assessment of shear deformable theories for multilayered composite plates. Appl. mech. Rev. 42, I-12 (1989). 3. A. K. Noor and W. S. Burton, Assessment of computational models for multilayered composite shells. Appl. mech. Rev. 43, 67791 (1990). of anisotropic plates and 4. A. K. Noor, Mechanics shells-a new look at an old subject. Comput. Strucf. 44, 499-514 (1992). 5. J. N. Reddy, A review of refined theories of laminated composite plates. Shock Vibr. Dig. 22, 3-17 (1990). assumptions of refined 6. G. Jemielita, On kinematical theories of plates: A survey. J. appl. Mech. 57, 1088 1091 (1990). 7. B. A. Szabo and G. L. Sahrmann, Hierarchic plate and shell models based on p-extension. Int. J. numer. Merh. Engng 26, 185551881 (1988). 8. K. S. SUrana an R. M. Sorem, Completely hierarchical p-version curved shell element for laminated plates and shells. Comput. Mech. 7, 237-251 (1991). 9. R. Barboni and P. Gaudenzi, A class of Câ&#x20AC;? finite elements for the static and dynamic analysis of laminated plates. Compuf. Struct. 44, 1169-l I78 (1992). The application of a two-dimensional IO. H. Matsunaga, higher-order theory for the analysis of a thick elastic plate. Comput. Struct. 45, 6333648 (1992). II. K. H. Lo, R. M. Christensen and E. M. Wu, Stress solution determination for high order plate theory. Comput. Struct. 14, 655-662 (1978). 12. R. A. Chaudhuri and P. Seide, An approximate semianalytical method for the prediction of interlaminar shear stresses in an arbitrarily laminated thick plate. Comput. Struct. 25, 627-636 (1986). 13. R. A. Chaudhuri, An equilibrium method for prediction of transverse shear stresses in a thick laminated plate. Comput. S~ruct. 23, 139-146 (1986). 14. M. R. Lajczok, New approach in the determination of interlaminar shear stresses from the results of MSC/ Nastran. Comput. Struct. 24, 651-656 (1986). 15. C. Byun and R. K. Kapania, Prediction of interlaminar stresses in a laminated plate using global orthogonal interpolation polynomials. AIAA J. 30, 2740-2749 (1992). I6 J. N. Reddy and E. J. Barbero, A plate bending element based on a generalized laminate plate theory. Inf. J. numer. Meth. Engng 28, 2215-2292 (1989). I7 S. Srinivas, A refined analysis of composite laminates. J. Sound Vibr. 30, 495-507 (1973). I8 D. R. J. Owen and 2. H. Li, A refined analysis of laminated plates by finite element displacement methods -I. Fundamentals and static analysis. Comput. Srrurt. 26, 907-914 (1987). 19. D. H. Robbins and J. N. Reddy, Modelling of thick composites using a layerwise laminate theory. Int. J. numer. Meth. Engng 36, 655-677 (1993). 20. F. J. Orth and K. S. Surana, p-Version two-dimensional curved beam element using piecewise hierarchical approximation for laminated composites. Comput. Mech. 11, 141-161 (1993). 21. J. N. Reddy and M. Savoia, Layerwise shell theory for postbuckling of laminated circular cylindrical shells, AIAA J. 30, 2148-2154 (1992). 37 J. Lee, Z. Gurdal and 0. H. Griffin Jr, Layerwise approach for the bifurcation problem in laminated

_A.

1062

23.

24.

25.

26.

J. Kong

and Y. K. Cheung

composites with delaminations. AIAA J. 31, 331-338 (1993). 1. Babuska and L. Li, The problem of plate modelling: theoretical and computational results. Comput. Meth. appl. Mech. Engng 100, 249-273 (1992). H. Murakami, A Laminated composite plate theory with improved in-plane responses. ASME J. appl. Mech. 53, 661-666 (1986). N. J. Pagan0 and S. J. Hatfield, Elastic behaviour of multilayered bidirectional composites. AIAA J. 10, 931-933 (1972). N. S. Lee and K. J. Bathe, Effects of element distortions on the performance of isoparametric elements. In!. J. numer. Meb. Engng 36, 3553-3576 (1993).

s

I?,[Ml’TIM]’ dz = 1/4Oh,

I

r I48

-189

54

sym.

432

-297 432

-13 -18954

148 where [Ml’ = d[M]/dz. Integration of the product of the global-local interpolation can be written as the summation of the transformed integral of layerwise cubic shape functions, that is

Jh On substituting

APPENDIX Integration of the product of layerwise cubic functions can be written as, for the ith layer,

eon (3a), we obtain

shape I,,=

5

,=I

f&1,

WlT[Mldz{4L

s h,

where

s s

[M]‘[M]dz

= h,/l

r I28

99

-36

fym,

648

,:;

‘91 -;9”

h,

[M]‘T[M]dz

h,

I28

t r -40 = I 8(1:74 t L 7

-57 8; -24

24 -“A 57

where z, denotes the integral of the product of the k th and Ith terms of the global-local interpolations. To illustrate the coupling integral of the product of the jth term of the global-local interpolation and the layerwise shape function, we have

-71

,=I

_:‘: 401

with the summation nodal arrangement

s

b’lTWl dz,

h,

being carried out with due regard to the in the thickness coordinate.

My Journal Publications

200 pages of my publications in structures and computational mechanics

My Journal Publications

200 pages of my publications in structures and computational mechanics