Bending and vibration analysis of composite sandwich plates

00417949(95)00357-6

BENDING

Compurers& Brumm~s Vol. 60. No. I. pp 103-I12. 1996 Copyright I” 1996 Elsevier Smncc Ltd Printed in Great Britain. All nghts reserved 0045.7949196 115.00 + 0.00

AND VIBRATION ANALYSIS SANDWICH PLATES

OF COMPOSITE

L. J. Lee and Y. J. Fan Institute

of Aeronautics

and Astronautics,

National Cheng Kung Republic of China

(Received

University,

Tainan,

Taiwan,

70101,

17 January 1995)

Abstract-A finite element analysis of composite

sandwich plates is studied in this paper. The face plates of composite sandwich structures are modeled based on the Mindlin’s plate theory. The displacement fields of the sandwich core material are linearly interpolated in terms of the displacements of two face plates. According to this assumption, the transverse shear strains are linearly varied while the transverse normal strain is constant through the core thickness. Finite element analysis based on the Mindlin formulation is adopted using a nine-node isoparametric element. Static and free vibrational problems have been solved to investigate the effect of considering the transverse normal deformation of the core. It was found that when the sandwich plate is subjected to a concentrated load, the aforementioned normal deformation of the core should not be neglected. For the case of free vibration, natural frequencies decrease when the core is considered to be flexible in the transverse normal direction while the natural mode shapes do not have prominent change.

1. INTRODUCHON

Sandwich applications

construction for more

has been than

used

sandwich constructions by finite element modeling. He combined built-up elements, i.e., a plane elasticity element for the core, and a beam element with eccentric nodes for facings, together for allowing the analysis of diverse structural forms. Kanematsu et al. [7] studied the bending and vibration of rectangular sandwich plates by the Ritz method. They also conducted experiments through holographic techniques to obtain the deflection and vibrational modes. All the papers quoted above, except Ref. (61, did not take account of the effect of transverse normal deformation of the core. Reference [6] considered this effect of the sandwich beams using a plane elasticity element for the core. But as sandwich plate problems are addressed, the method proposed by O’Connor is cumbersome and results in very large degrees of freedom in finite element equations. The purpose of this paper is to develop a finite element formulation to include the transverse normal deformation, in addition to the transverse shear stresses transmission, in the core. The faces of the sandwich construction are modeled by plate element which is based upon the Mindlin’s assumptions and lamination theory. The displacements of the core are linearly interpolated in terms of the nodal variables of two faces. Only those stress components in the transverse direction are considered in the core. It is found that transverse normal deformation needs to be considered if a concentrated load is applied to the sandwich plate. Meanwhile, the natural frequencies of the sandwich decrease when the core is flexible in the transverse normal direction, although its corresponding mode shapes do not change.

in aeronautical

40 years, since it offers the

possibility of achieving a high bending stiffness for small weight penalty. Recently, new attention is focused on using sandwich structures in the primary structures of an aircraft, such as the skin of the wing, the vertical fin torque box, the aileron and the spoiler, all can be making use of this type of construction. The main reason for the increasing application of the sandwich structure in aircraft is the introduction of new materials, such as advanced composite materials for the faces, which offer long awaited properties of both high specific stiffness and strength. Furthermore, new materials for the core are available such as nomex and plastic foams. Many researchers have studied the behavior of sandwich structures. Reissner [I, 21 derived the governing equations for both small and finite transverse deflections of isotropic sandwich plates. He assumed that the face layers behave like membranes and the face parallel stresses in the core are negligible. Since then, many papers have been published on various aspects of sandwich theory. Later, Liaw and Little [3] solved the bending problems of multilayered sandwich plates based on the theory developed by Reissner. Azar [4] extended Liaw and Little’s results to the same problems, but with orthotropic facings. Khatua and Cheung [5] investigated multilayered sandwich constructions by taking into account the bending stiffness of face layers and release the condition of common shear angle for all cores. Recently, O’Connor [6] proposed a method for the analysis of 103

L. J. Lee and Y. J. Fan

104 2. GENERAL THEORY OF COMPOSITE

uyx, y)

LAMINATES

The composite sandwich plate studied in this paper is composed of two composite laminated faces and a soft core. The faces are considered to be thin and anisotropic. Mindlin’s plate hypothesis in conjunction with the laminated plate theory is employed to model the faces. The core material can be either isotropic or orthotropic. In any case, the face-parallel stresses in the core are negligible. A finite element method based on the aforementioned assumptions is derived to analyze the sandwich plate. Detailed steps are described in the following sections.

2.1. Displacement

and strain jields

= ux”(X,y) - z”‘(b’$(X,y)

v(‘~(x, y) = uf(x, y) - z”@\“(x, y)

w(‘)(x, y = wb’yx y) 1

i = 1, 2

(1)

,

in where ug), @, w$) are mid-plane displacements the x-, y- and z-directions of face plate i, and 4’:’ and I@ are rotations of the cross-sections of face i perpendicular to the x- and y-axes, respectively. In addition, the displacements in the core are assumed to vary linearly through the thickness, therefore, it can be expressed in terms of the displacements of two face plates. Thus

Let the parenthesized superscripts i = I, 2 denote quantities associated with faces 1 and 2 of the sandwich plate and subscript c be expressed as for the core. Refering to Fig. 1. the displacement field of any arbitrary point within two face plates can be expressed as follows:

+ (up + [

up, + (P4g)

- t”‘f#Jl”) 4

2

1

-_

--I-4

Deformed cross-section

__

e

Face plate 2 I

_T __-----___---

Undeformed

t(2)

cross-section

-------------

:! h

Core

+I)

-------------

Face plate

Fig. 1. Deformation

of sandwich

plate.

I

Bending and vibration analysis of composite sandwich plates (42’ - #‘)

v, =

WC =

+ (,‘*)4p

h

+ f”‘&l’)

2h

(wf’ - w!y) z + (wlf’ + WC’) 2 ’ h

1z

2.2. Constitutive

relations for faces and the core

By integrating the stresses and the stresses multiplied by z across the thickness of each individual face, we obtain the laminate constitutive relations

(2)

where h --
105

h <--, 2

t and h are the thicknesses of faces and the core, respectively. The strains of faces and the core become

(51

=

[;

;

;;I

{;I,

(6)

where NC’),MC’?,Q(” are the in-plane stress resultants, stress moments, and transverse shears of face i, respectively. The submatrices A”‘, B”), DC”and H”’ are the respective in-plane, bending-in-plane, bending and transverse shear stiffness matrices. Since the face-parallel stresses in the core are negligible, there are only the transverse stress components left. The core material is assumed to be homogeneous and isotropic if plastic foam is used. In this case, the stress-strain relations are

(3)

and (4)

where

In eqn (7) G, and E, are shear and Young’s moduh, of the core material, respectively. If the honeycomb core is addressed, it can be modeled as an orthotropic material. In such a case, the stress-strain relations are expressed as

where G,,,, G,., and E, are effective core properties which can be related to the cell geometry and actual material properties of hexagonal cell construction as follows [8]: & = k, (t, id)& G,,, = &(tcld)G, G,:, = k, (t, ld)G, and

PC= k,(t,ld)p,

(5)

and are the in-plane strains, bending strains transverse shear strains of face i, respectively. The explicit expressions for {cc} are listed in the Appendix.

(9)

where G,, E, and p are the actual shear, Young’s moduli and mass density of the honecomb material, rC is core cell wall thickness, and d is diameter of a circle inscribe the cell. The values of k2, k3 and k, depend on the types of honeycomb construction. Reference [8] gave 8/3, 5/3 and 1 for Kaechele and 8/3, 4/3 and S/l5 for MIL HDBK-23, respectively. The last eqn of (9) gives the effective mass density of the honeycomb core which will be employed to

106

L. J. Lee and Y. J. Fan

calculate the kinetic energy of the honeycomb the next section.

core in

For simplicity,

eqn (13) may be written

as

{II”‘} = [N]{a(‘)}, 3. FINITE

ELEMENT

3.1. Total potential

FORMULATION

energy and kinetic energy

The total potential energy of the composite sandwich system consists of the strain energy stored in the sandwich plate and the work done of the external loading,

(15)

Substitution of eqns (3) (4), (7) and (8) into eqn (IO), using the displacement expression from eqn (13) and integrating over the thickness of each individual component, the total potential energy of the sandwich plate can be expressed for the case of equal thickness for face plates, i.e. (t”’ = P),

(16)

{u”‘}~{E(‘)) df’

where Ug’ and UCare the total potential energy of the faces and the core, respectively, and can be written as in the following expressions:

(10)

r/i) = 1

{a”‘}T[B,]T[A’i’][B,] {a”‘} dA sA

where

{,,“‘}’

= [,(I)

D(~)

+ i

{ac’b}TIB,]TIBc’l][Bf] {a”‘} dA

+ A

{a(‘)}T[B,][B(‘)][Bl]{a(“} dA

u,(~)],

In eqn (lo), the first two terms on the side are the strain energies stored in the and the core, respectively, while the last work done due to the external loadings plates. The kinetic energy of the composite plate is

right hand face plates term is the on the face

+ 5

A {a(‘)}T[B,]T[Dcf~][B,]T{a(“}dd s

+ :

sandwich

A {a’” }TIB,]TIHcr)] [B,] {a} dA s

+ {P)‘(a(“)

(17)

p’“{,$“}T{,$“} df/

T= ; i

and

,-I s 1

(11)

U, = i

{a(“}TIN]TIDC,][N]{ao)}

dA

sA where pCOthe mass density

{il,}T= [ti, The Lagrangian system is

of face (i) and d,

L of the

1P + i JA {a”‘}T[B,]T[DC,][N]{ac’)} dA

Li’,]. composite

L=U-T.

sandwich

(12)

+ i

+ i

A {a”‘}TIB,]TIDC,][B,l~a”‘}

dA

s

3.2. Finite element discretization The plate displacement {u(‘)} within an element is given as a function of n discrete nodal displacements, {U(i)}T= i: N,D]{aji)j ,=I

{a”‘}T[N]T[DC,]T[Bp] {a”)} dA

i = 1,2,

+ i

{a(‘) )T[B,]T[~C,][N] {a(*)} dA

(13)

where IV,are the shape functions, [I] is a 5 x 5 identity matrix and {a)‘)} are the nodal displacements,

A {ac’1}T[N]T[DCI][N]{a(2)) dA s

sA + f

1 {a”‘}T[Bp]T[DC,][N] {a”)} dA s

+ f

A {ac’))T[Bp]T[DC,][B,] {ac2)} dA s

Bending and vibration analysis of composite sandwich plates

+i

(a(2’}T[NJT[DC,]~

of the sandwich variables,

{ac2’} dA

IA + k

107

plate system with respect to the nodal

A {a(2’}T[B,]T[DC8][N]{a(2)} dA s

+ i

dA

A ja’z’)T[N]T[DC,][B,](a(2’1 i

+ f

In the above equation, the subscripts and/or superscripts “1” and “2” are denoted to respresent the degrees of freedom lying on face plates 1 and 2, respectively. For example, the submatrices M,, , M,, are mass matrices for the degrees of freedom on face plates 1 and 2, respectively, while M,, is the coupling mass matrix between face plates 1 and 2. A similar meaning is true for stiffness submatrices. Explicit expressions for M,,, M,,, M,,, K,,, K,2 and K,, are listed in the Appendix. In the numerical integration using Gauss quadrature to obtain the stiffness matrix of Mindlin plate element, a famous phenomenon called “shear locking” has been found by many authors 191. A normal 3 x 3 in flexure and reduced 2 x 2 in shear integration was used to eliminate the shear locking phenomenon in nine-node isoparametric plate element for the present study.

{a(2’}T[B,]T[DC,][Bp]{a(2’} dA. sA (18)

In terms of nodal displacement energy is expressed as

T=

{a(‘)), the kinetic

i Ta+T,, i= I

(19)

where TC and r, are the kinetic energies of the faces and core, respectively, 1 T’” = _ p 2

{P’“}[N]T@‘“][N] {icO} dA

(20)

and

4. RESULTS AND DISCUSSIONS

T, =;

{B”‘)T[N]T[S,][N]{&“‘}

Both static and free vibration analyses are studied in this paper. Part of the results presented herein will be compared with those given by Kanematsu et aI. [7].

dA

sA A (~(“}[NIT[s,I[NI(~(~‘}

dA

s +;

4. I. Static analysis In order to test the accuracy of the developed finite element program, rectangular sandwich plates (450 x 300 mm) composed of CFRP faces and AL 3/8-5056-0007 honeycomb core were used for static analysis. Material properties used in the analysis are same as those in Ref. [7]. The elastic properties of the carbon/ epoxy laminar are given as

A {sc”}[N]T[S2][N]{~(2’} dA s

+k

j8’2’}T[~]T[~,~[N~~~(*)} d,4. sA

(21)

In eqns (17)-(21), the explicit expression of matrices [I$], P,l, PSI, [BJ, [J&l, , PC,1 and [S,] . . [S,] are listed in the Appendix. A nine-node isoparametric plate element based on Mindlin’s plate assumption and lamination theory was employed to model faces of the sandwich structures. Following routine finite element procedures, the finite element equations of motion can be obtained by minimization of the Lagrangian L = T - U

E, = 105 GPa VI2= 0.327 p = 1.60

E, = 8.74 GPa Cl* = G,, = Gz3 = 4.56 GPa

x

IO3kg m-’

ply thickness

(23)

Table 1. Central deflection for sandwich plates under uniform load Edge condition

Mesh division in quarter plate

SPI

Clamped

5x3

0.05172

(C)

Simply supported (SS)

Central deflection (mm) SP2 SP3

Series solution*

0.05040

0.055 14 005524 0.05400

5x3 6x4 Series soIution*

0.1208 0.1213 0.1173

0.1773 0.1774 0.1829

6x4

*Based on six term approximation

t = 0.125 mm.

0.05190

of Ritz solution

[7].

SP4

0.078 12

0.06210

0.07834 0.07720

0.06216 0.06130

0.1729 0.1729 0. I794

0.2137 0.2138 0.2206

L. J. Lee and Y. J. Fan

108 The core is assumed properties are: G,;, = 103 MPa E,, = 103 MPa

to be homogeneous

and its

I

0 (C) Concentrated load O (SS) 0 (C) Uniformly A (SS) I distributed load

G,., = 62.1 MPa p, = I .60 x 10’ kg rnm3. (24)

There are four different types (designated as SPl, SP2, SP3, SP4) of stacking sequences adopted for the faces of sandwich plate. Each sandwich plate has the same face plate which is laid symmetrically about the mid-plane of the core. Stacking sequences are [30], , [0], , [30/-30/30] and [O/90/0], respectively, for SPl, SP2, SP3 and SP4 plates. The thickness of the core is 10 mm for plates SPI and SP2, and 7 mm for plates SP3 and SP4. A uniformly distributed load of p = 1010 Pa is applied vertically on these sandwich plates. The deflections at the center of the sandwich plates were calculated. The sandwich plates were either simply supported (SS) or clamped (C) along all edges. The boundary conditions are

12 --

IO-

d

8

6-

z

4__& 2 _..

0

0.2

0.4

0.6

0.8

I .o

‘”

Fig. 2. Relative

displacement of face plates moduli ratio for SP2.

vs normalized

y = 0, b : uo, = MO* = 00,= t‘o2= w#),= U’(Q = hr = h2 = 0 (25) for all edges clamped

and

x = 0,a: u(), = voz = wO, = wOr= &, = &* = 0

for the simply supported case in the present method. Results obtained from the present study, along with those in Ref. [7], are listed in Table 1. It has to be pointed out that the transverse normal deformation in the core is not considered in Ref. [7], thus, E, in the present method was assumed to be a large value, say 103 x lo5 GPa to simulate the incompressibility of the core in the normal direction in calculating deflection at the center. Comparisons between the two results are very good. The difference between the two solutions are not more than 1.5% for clamped edges and 3.5% for simply supported case. The two methods are based on different theories of plate, clamped boundary conditions, eqn (25), imposed herein are essentially the same as in Ref. [7], while the simply supported boundary conditions, eqn (26), are only an approximation of those prescribed in [7]. Therefore, the present method predicts center deflection more closely to those by the Ritz method [7] for clamped plates than for simply supported plates. As the moduli for the core decreases, the core becomes more flexible and allows for larger transverse normal and shear deformations. In order to investigate how the moduli affect the face plate displacements, the honeycomb core is used herein since, from eqn (9), all its transverse moduli can be expressed as a function of r = t,/d. The two aforementioned plates, SP2 and SP4 are analyzed as r = t,/d changes for demonstration.

The moduli listed in eqn (24) are considered to be reference values. When the sandwich plates are subject to a uniformly distributed vertical load, the defections at the center on both faces are same for both SP2 and SP4. But if the sandwich plates are under the application of a concentrated load, the deflections at the central points of both faces are different. This difference increases as r = t,/d decreases. Figures 2 and 3 show this phenomenon. In these figures, the abscissa denotes the normalized ratio rn = r’/r, where r is the ratio of t,/d for the reference core material, while r’ is

0

0.2

0.4

0.6

0.8

I .o

‘n

Fig. 3. Relative

displacement of face plates vs normalized moduli ratio for SP4.

Bending and vibration analysis of composite sandwich plates

109

O (0 Concentrated O (SS) I load 0 (C) Uniformly A (SS) I distributed W -z-zL

W’ L’

load

2 3

50 --

$

40 --

TI30 -20 IO 0 -~

0

0.2

0.4

0.6

0.8

I.0

1.2

-101 0

L’/L (WYW) Fig. 4. Relative displacement of face plates vs variation of plate size for SP2.

of the core with reduced t,/d. The coordinate resents the relative displacement d, defined as

M’(2)

I

I

I

I

0.4

0.6

0.8

I .o

x 100%,

where w(r) and NJ(~)are the center displacements of the loaded and unloaded faces, respectively. It is seen that under uniform load d, is zero for both SP2 and SP4, no matter its edges are clamped or simply supported, and d, remain zero as rn decreases. It means that under distributed load, there is no relative deformation between the two face plates. However, the relative displacement d, are 7.1% and 3.6% for SP2 with clamped (C) and simply supported boundary (SS) conditions, respectively, at r, = 1, as rn decreases from 1 to 0.1, d, increases to 17.5 and 13.9%. As for SP4, d, are 2.2 and 4.5% for C and SS conditions at rn = 1, increases to 7.9 and 9.8%, respectively, at rn = 0.1. The increasing rate for SP2 is larger than SP4, because SP2 has a thicker core. The relative displacement of the centers on two faces due to concentrated load is also affected by the

Fig. 5. Relative displacement of face plates vs variation of plate size for SP4.

size of the sandwich plate. Consider the same type of sandwich plates SP2 and SP4. If the ratio of length L to width W is kept constant (L/W = 1.5) while L and W change simultaneously, the relative displacement d, increases from 3.6 to 31% when simply supported and from 7.1 to 49.5% when clamped for SP2 as the plate reduces to l/4 (L’/L = W’/ W = 0.5) of its original size. For plate SP4, dr increases from 2.2 to 19% and from 4.5 to 29.5%, respectively, for simply supported and clamped conditions. The size effect is shown in Figs 4 and 5. It is noted that d, has a tendency to increase exponentially as plate size reduced. Figures 4 and 5 also show that d, is almost zero under uniform load as the size of the plate changes. From the analysis described above, the transverse normal deformation of the core should be included during the stress analysis of a sandwich construction under concentrated load, since it results in different deflections on both faces. The difference of displacement between two face plates yields different stress distributions which is important in the analysis of foreign object impact problems [ 1 I].

Table 2. Natural frequencies for the first five modes of sandwich plates

I

2

Mode no. 3

4

5

SPI

Ref. [7] Present method

720 707

1181 1150

1463 1424

1683 1627

2074 1990

SP2

Ref. [7] Present method

701 691

1215 1200

1401 1353

1768 1715

2017 1997

SP3

Ref. [7] Present method

567 558

1024 997

1115 1090

I528 1478

1670 1604

SPI

Ref. [7] Present method

637 628

1032 1007

1313 1272

1568 1517

1658 I593

Specimen

I

L’IL (W’IW)

rep-

w(1) _ w,(2)

d, =

I 0.2

Note: 9 x 6 mesh devision in full plate is used in the present analysis.

L. J. Lee and Y. J. Fan

110

4.2. Free vibration In the absence of forcing terms, eqn (18) is reduced to a natural vibration problem,

M,, M,,

ral frequency w can be determined following eigenvalue problem:

by solving

the

M,,M,,_ (28)

The general written as

solution

($1

After substituting

of the

= { $}

equation

ezwr.

is

(29)

eqn (29) into eqn (28), the natu-

z (a)

above

first

mode

(707

(~7 third

mode

(1424

The corresponding eigenvector, i.e. the natural mode shape, could then be obtained accordingly. The natural frequencies of the lowest five modes for SPI, SP2, SP3 and SP4 clamped on all four edges iteration are calculated using the subspace method [9]. The results are listed in Table 2, together with the series solution obtained in Ref. [7] for comparison. It is noted that all the frequencies calculated by the present formulation are less than those

Hz)

Hz)

’

! (e)

(b) second

mode

( I I SOHz)

(d) fourth

mode

(1627 Hz)

\-----~~-. l’it’th

mode

(IWO Hz)

Fig. 6. First five natural mode shapes for SPI with clamped edges, (a) first mode (707 Hz), (b) second mode (1150 Hz), (c) third mode (1424 Hz), (d) fourth mode (1627 Hz), (e) fifth mode (1990 Hz).

Bending from

Ref.

[7].

The

reason

is

and vibration that

the

analysis

effect

of

normal deformation of the core is included in the analysis. It reduces the stiffness of the sandwich system and hence reduces the natural frequencies. The first five vibration mode shapes of all four sandwich plates were carefully investigated by plotting the displacement contour lines for both face plates. The contour lines of the two faces do not show any difference between each other. Figure 6 exhibits the first five natural modes of plate SPl. It is similar to those shown in Ref. [7].

of composite

sandwich

plates

Ill

plates and shells. Int. J. numer. Meth. Engng 3, 275-209 (1971). JO. K. J. Bathe, Finite Element Procedures in Enaineering Analysis. Prentice-Hall, Englewood Cliffs, NJ (1982). 11 L. J. Lee, K. Y. Huang and Y. J. Fan, Dynamic response of composite sandwich plate subjected to impact. To be presented in ICCM/8, 14-19 July, 1991, Honolulu, Hawaii.

APPENDIX

Strain field in the core The strains based on the displacement eqn (2) in the transverse direction are

field given

by

5. CONCLUSIONS

Composite sandwich plates were studied in this work. The composite face plates were modeled by Mindlin’s plate assumptions in conjunction with the lamination theory. The core material is considered to be either isotropic, e.g. rigid foam, or orthotropic, e.g. honeycomb. In any case, the core is allowed to carry both shear and normal stresses in the transverse direction. Static and free vibration analyses were investigated using the finite element method which was formulated based upon the aforementioned assumptions. It is found that the transverse deflections of the two face plates are different under the application of a concentrated load. In the study of foreign object impact on a composite sandwich plate is a problem of this kind. In addition, the vibrational analysis shows that all the natural frequencies calculated in present method are lower than those methods which consider the core transmission only for the transverse shear forces. Thus the frequencies obtained from the present method are more conservative than other methods.

(Al)

+h(f+~)]+z(~-~)}

Matrices in equations (17)-(2 1) The matrices

used in eqns (17)--(21) are defined as follows:

PaI = LB,,, where a is representative

[B,,] =

Bending and vibration of CFRP-faced rectangular sandwich plates. Compos. Strucf. 10, 145-163 (1988). 8. J. R. Vinson, Optimum design of composite honeycomb sandwich panels subjected to uniaxial compression. AIAA J. 24, 1690-1696 (1986). 9. 0. C. Zienkiewicz, R. L. Taylor and J. M. Too, Reduced integration technique in general analysis of CAS w--E

0

0 ahi, _aU

=I 0

R,l

Pril=

. Bad,

for r, s, f, p. aN! ax

REFERENCES

I. E. Reissner, On bending of elastic plates. Q. appl. Math. 5, 5548 (1947). 2. E. Reissner, Finite deflections of sandwich plates. J. Aeronauf. Sci. July, 43540 (1948). 3. B. D. Liaw and R. W. Little, Theory of bending multilayer sandwich plates. AZAA J. 5,301-304 (1967). 4. J. J. Azar, Bending theory of multilayer orthotropic sandwhich plates. AZAA J. 6, 21662169 (1968). 5. T. P. Khatua and Y. K. Cheung, Bending and vibration of sandwich beams and plates. Inr. J. numer. Merh. Engng 6, 1 l-24 (1973). 6. D. J. O’Connor, A finite element package for the analysis of sandwich construction. &mpos: Struct. 8, 143-161 (1987). 7. H. H. ‘Kandmatsu, Y. Hirano and H. Iyama,

(A2)

0%

000

$

0

0

0

alv, ax

0

0

0

-N,

0

ay

0 Of&N,

0

0

00-Z

0

00

0

0

0

3

0

0

2

[B,,l =

I

0

-T!!!!

0

ay

ay 0

-2 0

1

OOdN’OO ay

L. J. Lee and Y. J. Fan

112 4G,

0

0

- 2tG,,

0

4G,,

0

0

0

0

4E,

0

0

0

t2G,,

0

0

0

1‘G,.,,

wc,1=;

-2rG,,, 0

-21G,:,

O -2G,,

-21G,:,

(A5)

0

0 rG,, 0

0

0

0

lG,,

1

646)

647)

-4G,,

0

0

0

- 4G,,:,

0

0

- 21G,,

0

0

-4E,

0

0

2%,

0

0

2tG,.,,

0

[DC,l=&

0

1

(A9)

0

(Al’3

0

2rG,

0

lZGTZC

0

0

I 2GvTc

where

4G,.,

0

0

2rG,:c

o

o

4E,

0

0

0 21G,.,

0 0

12G, 0

0

f = hr=p,/12,

M,l=PJ’d’l+

2fG,, 0

+?.x_,

N’L%IN dA

(A17)

NTts,l[Nl dA, sA

6418)

[K,,l = [Kb”+ [K;l+ Kl + Kl+ bl

DGI + W;,l + W;J

e

0

0

-g

e

0

0

-g

0

0

e

0

0

-g

0

0

f

0

P,l*DGlN s [&I= $,lT

0

-gO

0

f_

VU=

[S,] =

NPC,lPl 5A

Kl =

0

ogo

0

e

0 0

0

0

e

0

Of0

[K;] = [K$

0 0 f_

[&I= $,I

-_g

0

&? 0

6414)

WI = ;

[&I=

(AIS)

P,lTPWPl

s

s NoI = & [.%I=

dA d.4

[B,lTIDGIP,l dA sI AP Kl = ; j PITW,IN dA A

e

-_g LO

WI =

6413)

Kl

= [&I+ t&l + [VI + [K;l

= [$‘I + Kl+

0

o-

6416)

where [ME’] is the face plate mass matrix.

Ll (Al-3

[WI = fPc,l

[S,] =

sA

Pbl = [M~‘l+

(All)

-

c NT[~,lPld~

JA

[MuI =; W,l=;

g = p,ht/6.

Mass and stiffness submatrices in equation (22)

-

0

(‘48)

0

e = hpJ3,

WC,1 = 2W21 4G,,,

-2rG,,,

PITPC,Wl

dA

dA

K, I= [&IT P&l = f&l, where (Kg’] is the face plate stiffness matrix.

(Al9)

6420) (‘421)