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A FREE VIBRATION ANALYSIS OF ANISOTROPIC RECTANGULAR PLATES WITH VARIOUS BOUNDARY CONDITIONS
Journal of Sound and Vibration (1995) 187(5), 757–770
A FREE VIBRATION ANALYSIS OF ANISOTROPIC RECTANGULAR PLATES WITH VARIOUS BOUNDARY CONDITIONS C. M, H. M, T. S T. H Department of Structural Engineering, Faculty of Engineering, University of Nagasaki, Nagasaki 852, Japan (Received 19 July 1994, and in final form 13 September 1994) An approximate method in which discrete Green functions are used is described for analyzing the free vibration of anisotropic rectangular plates with various boundary conditions. The discrete Green functions are obtained by transforming the differential equations involving Dirac’s delta functions into integral equations and numerically integrating them. By employing these discrete Green functions, the differential equations governing the free vibration of anisotropic rectangular plates are transformed into the equivalent boundary integral equations, which are then solved to obtain the eigenvalues and eigenmodes. Comparisons with existing results confirm the accuracy of the numerical solutions obtained by the method. Results are presented for anisotropic rectangular plates with a number of combinations of clamped and simply supported edges and various angles of fiber orientation. 7 1995 Academic Press Limited
1. INTRODUCTION
The problem of free vibration of anisotropic rectangular plates with various boundary conditions has been analyzed by many researchers. For example, Huffington and Hoppmann [1] used Levy’s method to analyze an orthotropic plate with seven types of boundary conditions, these being combinations of clamped, simply supported, free and elastically supported. As concerns combinations of simply supported and clamped edges, Hearmon [2] studied orthotropic plates by using Rayleigh’s method and Farsa et al. [3] studied orthotropic and anisotropic plates by using the differential quadrature method. In what follows here, an approximate method for analyzing the free vibration of anisotropic rectangular plates employing discrete Green functions is described. The discrete Green functions are obtained by transforming the differential equations involving Dirac’s delta functions into integral equations and numerically integrating them. Next, by employing these discrete Green functions, the fundamental differential equations governing the free vibration of anisotropic rectangular plates are transformed into equivalent boundary integral equations. Then the eigenvalues and eigenmodes can be obtained. In order to confirm the accuracy of the numerical solutions obtained by the proposed method, the natural frequencies of orthotropic rectangular plates with various boundary conditions have been calculated, and the results compared to those in the literature. As an application of the proposed method, the natural frequencies of anisotropic rectangular plates under any combination of simply supported and clamped edges have been calculated, and are presented and discussed. 757 0022–460X/95/450757+14 $12.00/0
7 1995 Academic Press Limited
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2. DISCRETE GREEN FUNCTION
Consider an anisostropic rectangular plate which is referred to an x, y, z system of rectangular co-ordinates, with the position of the origin O of the x, y, z system at the corner of the middle plane of the plate, as shown in Figure 1. The surfaces of the plate are at z=2h/2, where h is the thickness of the plate. The transverse deflection and the rotations of the middle plane are denoted by w and ux , uy . If Qy and Qx are the transverse shear forces, Mxy the twisting moment and My and Mx the bending moments, then the fundamental differential equations for an anisotropic rectangular plate subjected to a unit concentrated force P are 1Q x 1Q + y=−Pd(x−x0 )d(y−y0 ), 1x 1y
1M x 1M xy + =Q x , 1x 1y
1M y 1M xy + =Q y , 1y 1y (1a–c)
0
1
(1d)
0
1
(1e)
0
1
(1f)
1w¯ 1w¯ y +kA45 x , +u +u 1y 1x
M x=D11
1u x 1u 1u x 1u +D12 y+D16 + y , 1x 1y 1y 1x
M y=D12
1u x 1u 1u x 1u +D22 y+D26 + y , 1x 1y 1y 1x
M xy=D16
1u x 1u 1u x 1u +D26 y+D66 + y , 1x 1y 1y 1x
Q y=kA44
0
Q x=kA45
0
1
0
1
(1g)
1
0
1
(1h)
1w¯ 1w¯ +u y +kA55 +u x . 1y 1x
Here d (· ·) is Dirac’s delta function, k is the shear factor, Dij=(h 3/12)(Q ij /ET ) is the ij /ET ) is the in-plane rigidity (i, j=4, 5), flexural rigidity (i, j=1, 2, 6), Aij=a 2h(Q Q 11=Q11 cos4 u+2(Q12+2Q66 ) cos2 u sin2 u+Q22 sin4 u,
12=Q12 (cos4 u+sin4 u)+(Q11+Q22−4Q66 ) cos2 u sin2 u, Q
Figure 1. The anisotropic rectangular plate and co-ordinate system, and the L–T orthotropy axis.
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Figure 2. Discrete points on the plate.
Q 22=Q11 sin4 u+2(Q12+2Q66 ) cos2 u sin2 u+Q22 cos4 u,
16=(Q11−Q12−2Q66 ) cos3 u sin u+(Q22−Q12−2Q66 ) cos u sin3 u, Q 26=(Q11−Q12−2Q66 ) cos u sin3 u+(Q22−Q12−2Q66 ) cos3 u sin u, Q 66=(Q11+Q22−2Q12−2Q66 ) cos2 u sin2 u+Q66 (cos4 u+sin4 u), Q 44=Q44 cos2 u+Q55 sin2 u, Q
Q 45=(Q55−Q44 ) cos u sin u,
55=Q44 sin u+Q55 cos2 u, Q 2
Q11=EL /(1−nLT nTL ),
Q22=ET /(1−nLT nTL ),
Q12=nTL EL /(1−nLT nTL )=nLT ET /(1−nLT nTL ) Q66=GLT ,
Q44=Q22 /2,
Q55=GLT ,
u is the angle of fiber orientation with respect to the x axis, EL is the modulus of elasticity in the L-direction, ET is the modulus of elasticity in the T-direction, GLT is the shear modulus of elasticity in the L- and T-directions, nLT is the Poisson ratio corresponding to uniaxial stress in the L-direction, and nTL is the Poisson ratio corresponding to uniaxial stress in the T-direction, with L denoting the longitudinal and T the transverse directions, respectively. By using the non-dimensional expressions
y , X1={a 2/D0 (1−nLT nTL )}Q
X2={a 2/D0 (1−nLT nTL )}Q x ,
xy , X3={a/D0 (1−nLT nTL )}M y , X4={a/D0 (1−nLT nTL )}M X8=w¯ /a,
X5={a/D0 (1−nLT nTL )}M x ,
D ij=(12/h3ET )Dij ,
A ij=(12a2/hh02 ET )Aij ,
X6=u y ,
X7=u x ,
h=x/a,
z=y/b,
where D0=ET h03 /12(1−nLT nTL ) is the standard flexural rigidity, the differential equations of the non-dimensional normal functions X1—X8 can be rewritten as 8
s [F1ts (1Xs /1z)+F2ts (1Xs /1h)+F3ts Xs ]+P d(h−h0 )d(z−z0 )d1t=0,
(2)
s=1
where the quantities Fkts are given in Appendix 1 and, t=1, 2, . . . , 8, d1t is Kronecker’s delta. We now divide the rectangular plate vertically into m equal-length parts and horizontally into n equal-length parts, as shown in Figure 2, and consider the plate as a group of discrete
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points which are the intersections of the vertical and horizontal dividing lines. (The rectangular area, 0EhEhi and 0EzEzj , corresponding to an arbitrary intersection (i, j) as shown in Figure 2, is denoted by [i, j] the intersection (i, j) denoted by w + is called the main point of the area [i, j], and the intersections denoted by w are called the inner dependent points, and the intersections denoted by W are called the boundary dependent points. By integrating equations (2) over the area [i, j], the following integral equations are obtained: 8
6 g
s F1ts s=1
hi
[Xs (h, zj )−Xs (h, 0)] dh+F2ts
0
g
zj
[Xs (hi , z)−Xs (0, z)] dz
0
gg hi
+F3ts
0
zj
7
Xs (h, z) dh dz +P d(h−h0 ) d(z−z0 ) d1t=0.
0
(3)
By numerically integrating equations (3), simultaneous equations for the unknown quantities Xsij , which are the dimensionless shear forces, twisting moment, bending moments, rotations and deflections at the main point (i, j) of the area [i, j], are obtained as 8
6
i
j
i
k=0
l=0
j
s F1ts s bik [Xskj−Xsk0 ]+F2ts s bjl [Xsil−Xs0l ]+F3ts s s bik bjl Xskl s=1
k=0 l=0
7
+P d(h−h0 ) d(z−z0 ) d1t=0.
(4)
The solutions Xpij of the simultaneous equations (4) are expressed as 8
6
i
j
k=0
l=0
Xpij= s Apt s bik [Xtk0−Xtkj (1−dik )]+Bpt s bjl [Xt0l−Xtil (1−djl )] t=1
i
j
7
i
j
+Cpt s s bik bjl Xtkl (1−dik djl ) − s s bik bjl Ap1 P kl , k=0 l=0
(5)
k=0 l=0
where p=1, 2, . . . , 8, t=1, 2, . . . , 8, i=1, 2, . . . , m, j=1, 2, . . . , n, dij is Kronecker’s delta, bik=aik /m, bjl=ajl /n, and Apt , Bpt and Cpt are given in Appendix 2. The coefficients bif and bjg are the weight coefficients of the numerical integration. The trapezoidal rule of approximate numerical integration has been used here, and therefore the values of aif and ajg are aif=1−(d0f+dif )/2 and ajg=1−(d0g+djg )/2. In equations (5), the quantities Xpij at the main point (i, j) of the area [i, j] are related to the quantities Xtf0 and Xt0g at the boundary dependent points of the area and the quantities Xtfj , Xtig and Xtfg at the inner dependent points of the area. With spreading of the area [i, j] according to the regular order [1, 1], [1, 2], . . . , [1, n], [2, 1], [2, 2], . . . , [2, n], . . . , [m, 1], [m, 2], . . . , [m, n], a main point of a smaller area becomes one of the inner dependent points of the subsequent larger areas. Whenever one obtains the quantities Xpij at the main point (i, j) of the area [i, j] by using equations (5) in the above-mentioned order, one can eliminate the quantities Xtfj , Xtig and Xtfg at the inner dependent points of the following larger areas by substituting the obtained results into the corresponding terms of the right sides of equations (5). By repeating this process, the equations Xpij at the main point are related to only the quantities Xrf 0
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(r=1, 3, 4, 6, 7, 8) and Xs0g (s=2, 3, 5, 6, 7, 8) at the boundary dependent points. The result is that 6
Xpij= s d=1
6
i
j
f=0
g=0
7
s apijfd Xrf 0+ s bpijgd Xs0g +qpij ,
(6)
where apijfd , bpijgd and qpij are given in Appendix 3. Equation (6) can be recognized as the discrete solution of the fundamental differential equations (1a)–(1h). The discrete Green functions related to deformation are X8ij (=w¯ /a).
3. FREQUENCY EQUATIONS
The fundamental differential equations governing the free vibration of rectangular anisotropic plates are 1Qx 1Qy 1 2w + =rh 2 , 1x 1y 1t
0
1Mx 1Mxy + =Qx , 1x 1y
1
1ux 1u 1ux 1uy , +D12 y+D16 + 1x 1y 1y 1x
Mx=D11
My=D12
1My 1Mxy + =Qy , 1y 1x
0
(7a–c)
1
1ux 1u 1ux 1uy , +D22 y+D26 + 1x 1y 1y 1x (7d, e)
Figure 3. The variation of the first five natural frequency parameters l 2 with the number of divisions for an orthotropic rectangular plate with all edges clamped.
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T 1 The extrapolated values (number of divisions: m=n=12, 16) compared with the results of reference [2]; C, clamped edge; S, simply supported edge
Boundary condition
Mode
D1 /D2=D3 /D2=5.0 ZXXXXXXXCXXXXXXXV Present Reference [2]
First Second Third Fourth Fifth
66·239 107·131 156·058 166·422 204·830
67·203 109·375 158·685 171·132 209·688
First Second Third Fourth Fifth
39·253 76·552 108·081 129·681 156·691
39·478 77·084 108·566 130·935 157·914
First Second Third Fourth Fifth
46·858 94·282 113·387 157·196 171·678
46·921 93·967 114·670 157·692 172·478
First Second Third Fourth Fifth
61·420 92·941 142·425 152·917 194·617
61·758 94·431 146·009 153·819 196·812
First Second Third Fourth Fifth
63·544 99·480 153·832 154·380 199·446
64·454 101·368 157·914 156·677 203·087
First Second Third Fourth Fifth
42·321 84·155 110·038 141·962 162·420
43·120 84·954 112·108 143·652 164·997
First Second Third Fourth Fifth
54·678 99·155 132·651 160·401 185·465
55·708 101·385 135·127 164·851 189·763
First Second Third Fourth Fifth
48·826 83·797 135·469 129·031 174·307
49·203 85·354 138·885 129·667 175·999
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T 2 Convergence and accuracy of fundamental frequency of anisotropic plate with four clamped edges (l 2=va 2zrh0 /D0 ) Number of divisions u ZXXXXXXXCXXXXXXXV (degrees) 6×6 8×8 10×10 12×12 16×16 Ex. v,† 12–16 Reference [3] 0 5 10 15 20 25 30 35 40 45
82·935 82·604 81·641 80·101 78·232 76·154 74·152 72·490 71·389 71·004
79·336 79·013 78·075 76·614 74·787 72·810 70·931 69·394 68·394 68·047
77·762 77·442 76·513 75·069 73·272 71·336 69·508 68·021 67·058 66·725
76·930 76·611 75·686 74·251 72·468 70·553 68·751 67·291 66·346 66·020
75·985 75·799 74·878 73·451 71·682 69·787 68·010 66·575 65·648 65·329
74·770 74·755 73·839 72·422 70·671 68·802 67·057 65·654 64·751 64·441
Reference [4]
75·273 74·958 74·016 72·602 70·843 69·021 67·293 65·911 65·031 64·748
75·305 — — 72·571 — — 67·073 — — 64·434
† Ex. v, extrapolated value.
0
1
1u 1u 1ux 1uy + Mxy=D16 x+D26 y+D66 , 1x 1y 1y 1x
Qx=kA45
0
Qy=kA44
1
0
0
1
0
1
1w 1w +uy +kA45 +ux , 1y 1x
1
1w 1w +uy +kA55 +ux , 1y 1x
(7f–h)
where r is the mass density of the material. T 3 The extrapolated (m=n=12, 16) versus the angle of fiber orientation with respect to the x axis for anisotropic rectangular plate (C, clamped; S, simply supported) D1 /D2=5.0, D3 /D2=1·0 Frequency ZXXXXXCXXXXXV ZXXXXXXXXXXXXCXXXXXXXXXXXXV Boundary condition u (degrees) First mode Second mode Third mode Fourth mode Fifth mode 0 15 30 45
57·145 55·998 53·618 52·470
86·030 86·958 90·926 94·635
142·568 136·540 123·939 115·859
137·151 139·857 143·122 143·913
160·710 162·383 167·899 181·167
0 15 30 45 60 75 90
53·060 51·571 47·707 42·823 38·427 35·762 35·943
70·481 71·752 74·256 76·387 77·081 73·703 73·316
110·235 114·100 120·187 123·277 121·750 121·183 126·397
139·413 134·179 119·677 103·433 93·138 93·715 96·955
153·302 152·456 150·781 154·959 152·406 135·748 128·665
0 15 30 45
40·815 40·833 40·759 40·824
68·181 71·268 75·145 78·286
116·246 112·141 103·506 98·537
117·904 119·942 122·841 123·969
135·276 137·470 146·909 158·957
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Figure 4. The first five natural frequency parameters versus the angle of fiber orientation with respect to the x-axis for an anisotropic rectangular plate with all edges clamped.
By multiplying equation (7a) by w¯ and integrating over the intervals [0, a] and [0, b], one obtains the integral equations
gg0 b
0
a
0
1
1Qx 1Qy w¯ dx dy= + 1x 1y
gg b
0
a
−rhv 2ww¯ dx dy.
(8)
0
Figure 5. The first three natural frequency parameters versus the angle of fiber orientation with respect to the x-axis for an anisotropic rectangular plate with two opposite edges simply supported and the other two edges clamped.
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Figure 6. The first five natural frequency parameters versus the angle of fiber orientation with respect to the x-axis for an anisotropic rectangular plate with two adjoining edges simply supported and the other two edges clamped.
Integration by parts of the left side of equation (8) yields the integral equation
gg0 b
0
a
0
=
1
1Qx 1Qy + w¯ dx dy 1x 1y
g
b
[Qx (a, y)w¯ (a, y)−Qx (0, y)w¯ (0, y)] dy
0
g
a
+
gg0 b
[Qy (x, b)w¯ (x, b)−Qy (x, 0)w¯ (x, 0)] dx−
0
a
Qx
0
0
1
1w¯ 1w¯ +Qy dx dy. 1x 1y
(9)
Substituting equations (1g) and (1h) into the third term of the right side of equation (9) results in
gg0 b
a
−
0
=
g
b
0
Qx
1
1w¯ 1w¯ +Qy dx dy 1x 1y
[Mx (a, y)u x (a, y)−Mx (0, y)u x (0, y)] dy
0
+
g
a
0
[Mxy (x, b)u x (x, b)−Mxy (x, 0)u x (x, 0)] dx
766 +
. .
g
a
g
b
[My (x, b)u y (x, b)−My (x, 0)u y (x, 0)] dx
0
+
[Mxy (a, y)u y (a, y)−Mxy (0, y)u y (0, y)] dy
0
gg 0 b
−
0
−
a
0
A 1 A45 QQ − 2 44 QQ 2 −A44 A45 x y A45 −A44 A45 x x k A45
1
A55 A QQ + 2 45 QQ dx dy 2 A45 −A44 A45 y y A45 −A44 A45 y x
gg0 b
−
0
a
0
Mx
1
1u x 1u 1u 1u +Mxy x+Mxy y+My y dx dy. 1x 1y 1x 1y
In the same way, by using equations (1a)–(1h), one can obtain the equation Pw(x0 , y0 )=
g
b
{Qx (a, y)w¯ (a, y)−Qx (0, y)w¯ (0, y)} dy
0
+
g
a
g
b
g
a
g
a
g
b
g
a
g
a
g
b
{Qy (x, b)w¯ (x, b)−Qy (x, 0)w¯ (x, 0)} dx
0
+
{Mx (a y)u x (a, y)−Mx (0, y)u x (0, y)} dy
0
+
{Mxy (x, b)u x (x, b)−Mxy (x, 0)u x (x, 0)} dx
0
+
{My (x, b)u y (x, b)−My (x, 0)u y (x, 0)} dx
0
+
{Mxy (a, y)u y (a, y)−Mxy (0, y)u y (0, y)} dy
0
−
{uy (x, b)M y (x, b)−uy (x, 0)M y (x, 0)} dx
0
−
{uy (x, b)M xy (x, b)−uy (x, 0)M xy (x, 0)} dx
0
−
0
{ux (a, y)M x (a, y)−ux (0, y)M x (0, y)} dy
(10)
g
b
g
b
g
a
−
767
{ux (a, y)M xy (a, y)−ux (0, y)M xy (0, y)} dy
0
−
{w(a, y)Q x (a, y)−w(0, y)Q x (0, y)} dy
0
−
{w(x, b)Q y (x, b)−w(x, 0)Q y (x, 0)} dx+
0
gg b
0
a
rhv 2ww¯ dx dy.
(11)
0
Consider the following boundary conditions: simply supported edge—My=ux=w=0 along y=0, b, and Mx=uy=w=0 along x=0, a; clamped edge—uy=ux=w=0 along y=0, b, and uy=ux=w=0 along x=0, a. Thus, for any combination of these conditions, equation (11) becomes the integral equation
gg b
Pw(x0 , y0 )=
0
a
rhv 2w(x, y)w¯ (x0 , y0, x, y) dx dy.
(12)
0
By using the non-dimensional expressions l 4=rh0 v 2a 4/D0 ,
h(x, y)=h(x, y)/h0 , m=b/a,
h=x/a,
w˜=w/a,
G=(D0 /Pa 2 )w¯ ,
z=y/b
here l is the non-dimensional eigenvalue, the integral equation (12) can be rewritten as
gg 1
w˜ (h0, z0 )=
0
1
mh (h, z) l4w˜ (h, z) G(h0 , z0 , h, z) dh dz.
(13)
0
4. NUMERICAL RESULTS
In order to test the convergence of numerical solutions by the present method, the first five frequency prameters l 2 were calculated for orthotropic and anisotropic rectangular plates with any combinations of simply supported and clamped edges. The specific values in this analysis are as follows: D1=EL h 3/12(1−nLT nTL ), D3=D4+2D5 , D4=EL h 3nTL /12(1−nLT nTL ), D5=GLT h 3/12, D2=ET h 3/12(1−nLT nTL ), l 2=va 2zrh0 D0 , k=5/6, b/a=1·0. 4.1. For an orthotropic rectangular plate with the stiffness ratios D1 /D2=D3 /D2=5·0, the variation of the first five natural frequency parameters l 2 with the number of area divisions with all edges clamped is shown in Figure 3. It is found from Figure 3 that the numerical solutions by the present method show good convergence. The extrapolated values (i.e., the calculated values when m=n=12, 16) for orthotropic rectangular plates with some combinations of simply supported and clamped edges are shown in Table 1, together with the analytical solution results obtained by Hearmon [2]. It is found that the extrapolated values agree with the analytical solutions. 4.2. For an anisotropic rectangular plate with the stiffness ratios D1 /D2=5·0 and D3 /D2=1·0, the convergence and accuracy of the fundamental frequency factor l 2 for the all edges
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clamped case as obtained by the present method is shown in Table 2. The extrapolated natural frequency parameters of the anisotropic rectangular plate for some combinations of simply supported and clamped edges as shown in Table 3. Next, as an application of the proposed method, the variations of the natural frequency parameters and deflection modes versus the angle of fiber orientation with respect to the x-axis for anisotropic rectangular plates are shown in Figures 4–6. Figure 4 is for the case of an anisotropic clamped rectangular plate, Figure 5, for the case of an anisotropic plate with two opposite edges simply supported and the other two edges clamped, and Figure 6 for the case of an anisotropic plate with two adjoining edges simply supported and the other two edges clamped. It is found from these figures that the natural frequency parameters and deflection modes change with the variation of the angle of fiber orientation to the x axis for the anisotropic rectangular plate. 5. CONCLUSIONS
An approximate method for analyzing the free vibration of rectangular anisotropic plates with various boundary conditions has been proposed. This method is based on an application of the discrete Green function and numerical solution of integral equations. These discrete Green functions give the solutions at discrete points. The natural frequency parameters obtained approach those of the analytical solutions as one increases the number of discrete points. REFERENCES 1. N. J. H J and W. H. H II 1958 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 25, 389–395. On the transverse vibrations of rectangular orthotropic plates. 2. R. F. S. H 1959 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 26, 537–540. The frequency of flexural vibration of rectangular orthotropic plates with clamped or supported edges. 3. J. F, A. R. K and C. W. B 1993 Computers and Structures 46, 465–477. Fundamental frequency analysis of single specially orthotropic, generally orthotropic and anisotropic rectangular layered plates by the differential quadrature method. 4. J. M. W 1987 Structural Analysis of Laminated Anisotropic Plates. Lancaster, Pennsylvania, Technomic.
APPENDIX 1
F111=F123=F134=1,
11 m, F247=D
F212=F225=F233=−F322=−F331=m,
F146=D 12 ,
F147=D 16 ,
F246=D 16 m,
−F345=−F354=−F363=I,
12 m, F257=D
F156=D 22 ,
44 , F178=kA 16 m, F267=D
F166=D 26 ,
F157=D 26 , F376=kA 44 m,
F167=D 66 ,
45 m, F278=F377=F386=kA −F371=−F382=K, m=b/a,
={a/D0 (1−nTL nLT )}P, P
ij=12(a/h0 )2Q ij /ET A
(i, j=4, 5),
F256=D 26 m,
F266=D 66 m,
F188=kA 45 ,
F288=F387=kA 55 m, other Fkts=0, I=m(h0 /h)3, D ij=Q ij /ET
K=mh0 /h,
(i, j=1, 2, 6).
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APPENDIX 2
F Ap1J F J gp1 G G G G 0 G Ap2G G G gp2 G Ap3G G G G Ap4G G G gp3 h, Apt= g h = g 0 G Ap5G G G G Ap6G G D12 gp4+D22 gp5+D26 gp6 G G A G G D g +D g +D g G G p7G G 16 p4 26 p5 66 p6 G f Ap8j f kA44 gp7+kA45 gp8 j F Bp1J F 0 G G G mgp1 G Bp2G G mgp3 G Bp3G G G Bp4G G 0 Bpt= g h = g mgp2 G Bp5G G G Bp6G G m(D16 gp4+D26 gp5+D66 gp6 ) G B G G m(D g +D g +D g ) 11 p4 12 p5 16 p6 G p7G G m(kA 45 gp7+kA 55 gp8 ) f Bp8j f
J G G G G h, G G G G j
F Cp1J F J mgp3+Kgp7 G G G G mgp2+Kgp8 G Cp2G G G G Cp3G G Igp6 G G Cp4G G G Igp5 g h g h, Cpt= = Igp4 G Cp5G G G G Cp6G G −m(kA44 gp7+kA45 gp8 ) G G C G G −m(kA g +kA g ) G 45 p7 55 p8 G p7G G G 0 f Cp8j f j r11=bii ,
[gpt ]=[rpt ]−1 ,
r12=mbjj ,
r25=mbjj , r33=mbjj ,
r34=bii ,
r22=−mbii bjj ,
r23=bii ,
r31=−mbii bjj ,
r45=−Ibii bjj ,
r46=D 12 bii+D 16 mbjj ,
r47=D 16 bii+D 11 mbjj , r54−=Ibii bjj ,
r56=D 22 bii+D 26 mbjj ,
r57=D 26 bii+D 12 mbjj ,
r63=−Ibii bjj ,
r66=D 26 bii+D 66 mbjj ,
r67=D 66 bii+D 16 mbjj ,
r71=−Kbii bjj ,
r76=kA 44 mbii bjj ,
r77=kA 45 mbii bjj ,
r78=k(A 44 bii+A 45 mbjj ),
r82=−Kbii bjj ,
r86=kA 45 mbii bjj ,
r87=kA 55 mbii bjj ,
r88=k(A 45 bii+A 55 mbjj ).
. .
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APPENDIX 3
6
8
apijfd= s t=1
i
i
j
k=0
l=0
s bik Apt [atk0fd−atkjfd (1−dik )]+ s bjl Bpt [at0lfd−atilfd (1−djl )]
7
j
+ s s bik bjl Cpt atklfd (1−dik djl ) , k=0 l=0
8
bpijgd= s t=1
6
i
j
k=0
l=0
s bik Apt [btk0gd−btkjgd (1−dik )]+ s bjl Bpt [bt0lgd−btilgd (1−djl )]
i
7
j
+ s s bik bjl Cpt btklgd (1−dik djl ) , k=0 l=0
6
8
qpij= s t=1
i
i
j
k=0
l=0
s bik Apt [qtk0−qtkj (1−dik )]+ s bjl Bpt [qt0l−qtil (1−djl )]
7
j
i
j
+ s s bik bjl Cpt qtkl (1−dik djl ) − s s bik bjl Ap1 P kl , k=0 l=0
ari0it=1,
a10j01=a j ,
k=0 l=0
a40j03=a j ,
a2i0k1=(T 51 /T 52 )b ik ,
a2i0k5=−(mk/KT 52 )b ik , a2i0k6=−(mk/KT 52 )gik ,
a5i0k2=−(S16 /S11 )b ik ,
a5i0k3=−(S12 /S11 )b ik ,
a5i0k5=(1/D S11 )gik , bs0jjt=1,
b2i001=a¯ i ,
b5i003=a¯ i ,
b10jl6=(k/KT 61 )gjl ,
b10jl1=(T 62 /T 61 )b jl ,
b40jl2=−(S26 /S22 )b jl ,
b10jl4=(mk/KT 61 )b jl ,
b40jl3=−(S12 /S22 )b jl ,
S22 )gjl , b40jl4=(1/mD [Sij ]=[D ij ]−1
(i, j=1, 2, 6),
T 51=A 45 /(A −A 44 A 55 ), 2 45
T 52=A 44 /(A 245−A 44 A 55 ), T 61=A 55 /(A 44 A 55 −A 245 ), T 62=A 45 /(A 44 A 55−A 245 ), D =(h0 /h)3,
a¯ i=(−1)i,
bik=dik+(−1)i+1d0k ,
g¯ ik=4m(−1)i+k/(1+dik+d0k ), r=t+d2t+d3t+2(d4t+d5t+d6t ),
s=t+d1t+d2t+2(d3t+d4t+d5t+d6t ).
A FREE VIBRATION ANALYSIS OF ANISOTROPIC RECTANGULAR PLATES WITH VARIOUS BOUNDARY CONDITIONS C. M, H. M, T. S T. H Department of Structural Engineering, Faculty of Engineering, University of Nagasaki, Nagasaki 852, Japan (Received 19 July 1994, and in final form 13 September 1994) An approximate method in which discrete Green functions are used is described for analyzing the free vibration of anisotropic rectangular plates with various boundary conditions. The discrete Green functions are obtained by transforming the differential equations involving Dirac’s delta functions into integral equations and numerically integrating them. By employing these discrete Green functions, the differential equations governing the free vibration of anisotropic rectangular plates are transformed into the equivalent boundary integral equations, which are then solved to obtain the eigenvalues and eigenmodes. Comparisons with existing results confirm the accuracy of the numerical solutions obtained by the method. Results are presented for anisotropic rectangular plates with a number of combinations of clamped and simply supported edges and various angles of fiber orientation. 7 1995 Academic Press Limited
1. INTRODUCTION
The problem of free vibration of anisotropic rectangular plates with various boundary conditions has been analyzed by many researchers. For example, Huffington and Hoppmann [1] used Levy’s method to analyze an orthotropic plate with seven types of boundary conditions, these being combinations of clamped, simply supported, free and elastically supported. As concerns combinations of simply supported and clamped edges, Hearmon [2] studied orthotropic plates by using Rayleigh’s method and Farsa et al. [3] studied orthotropic and anisotropic plates by using the differential quadrature method. In what follows here, an approximate method for analyzing the free vibration of anisotropic rectangular plates employing discrete Green functions is described. The discrete Green functions are obtained by transforming the differential equations involving Dirac’s delta functions into integral equations and numerically integrating them. Next, by employing these discrete Green functions, the fundamental differential equations governing the free vibration of anisotropic rectangular plates are transformed into equivalent boundary integral equations. Then the eigenvalues and eigenmodes can be obtained. In order to confirm the accuracy of the numerical solutions obtained by the proposed method, the natural frequencies of orthotropic rectangular plates with various boundary conditions have been calculated, and the results compared to those in the literature. As an application of the proposed method, the natural frequencies of anisotropic rectangular plates under any combination of simply supported and clamped edges have been calculated, and are presented and discussed. 757 0022–460X/95/450757+14 $12.00/0
7 1995 Academic Press Limited
. .
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2. DISCRETE GREEN FUNCTION
Consider an anisostropic rectangular plate which is referred to an x, y, z system of rectangular co-ordinates, with the position of the origin O of the x, y, z system at the corner of the middle plane of the plate, as shown in Figure 1. The surfaces of the plate are at z=2h/2, where h is the thickness of the plate. The transverse deflection and the rotations of the middle plane are denoted by w and ux , uy . If Qy and Qx are the transverse shear forces, Mxy the twisting moment and My and Mx the bending moments, then the fundamental differential equations for an anisotropic rectangular plate subjected to a unit concentrated force P are 1Q x 1Q + y=−Pd(x−x0 )d(y−y0 ), 1x 1y
1M x 1M xy + =Q x , 1x 1y
1M y 1M xy + =Q y , 1y 1y (1a–c)
0
1
(1d)
0
1
(1e)
0
1
(1f)
1w¯ 1w¯ y +kA45 x , +u +u 1y 1x
M x=D11
1u x 1u 1u x 1u +D12 y+D16 + y , 1x 1y 1y 1x
M y=D12
1u x 1u 1u x 1u +D22 y+D26 + y , 1x 1y 1y 1x
M xy=D16
1u x 1u 1u x 1u +D26 y+D66 + y , 1x 1y 1y 1x
Q y=kA44
0
Q x=kA45
0
1
0
1
(1g)
1
0
1
(1h)
1w¯ 1w¯ +u y +kA55 +u x . 1y 1x
Here d (· ·) is Dirac’s delta function, k is the shear factor, Dij=(h 3/12)(Q ij /ET ) is the ij /ET ) is the in-plane rigidity (i, j=4, 5), flexural rigidity (i, j=1, 2, 6), Aij=a 2h(Q Q 11=Q11 cos4 u+2(Q12+2Q66 ) cos2 u sin2 u+Q22 sin4 u,
12=Q12 (cos4 u+sin4 u)+(Q11+Q22−4Q66 ) cos2 u sin2 u, Q
Figure 1. The anisotropic rectangular plate and co-ordinate system, and the L–T orthotropy axis.
759
Figure 2. Discrete points on the plate.
Q 22=Q11 sin4 u+2(Q12+2Q66 ) cos2 u sin2 u+Q22 cos4 u,
16=(Q11−Q12−2Q66 ) cos3 u sin u+(Q22−Q12−2Q66 ) cos u sin3 u, Q 26=(Q11−Q12−2Q66 ) cos u sin3 u+(Q22−Q12−2Q66 ) cos3 u sin u, Q 66=(Q11+Q22−2Q12−2Q66 ) cos2 u sin2 u+Q66 (cos4 u+sin4 u), Q 44=Q44 cos2 u+Q55 sin2 u, Q
Q 45=(Q55−Q44 ) cos u sin u,
55=Q44 sin u+Q55 cos2 u, Q 2
Q11=EL /(1−nLT nTL ),
Q22=ET /(1−nLT nTL ),
Q12=nTL EL /(1−nLT nTL )=nLT ET /(1−nLT nTL ) Q66=GLT ,
Q44=Q22 /2,
Q55=GLT ,
u is the angle of fiber orientation with respect to the x axis, EL is the modulus of elasticity in the L-direction, ET is the modulus of elasticity in the T-direction, GLT is the shear modulus of elasticity in the L- and T-directions, nLT is the Poisson ratio corresponding to uniaxial stress in the L-direction, and nTL is the Poisson ratio corresponding to uniaxial stress in the T-direction, with L denoting the longitudinal and T the transverse directions, respectively. By using the non-dimensional expressions
y , X1={a 2/D0 (1−nLT nTL )}Q
X2={a 2/D0 (1−nLT nTL )}Q x ,
xy , X3={a/D0 (1−nLT nTL )}M y , X4={a/D0 (1−nLT nTL )}M X8=w¯ /a,
X5={a/D0 (1−nLT nTL )}M x ,
D ij=(12/h3ET )Dij ,
A ij=(12a2/hh02 ET )Aij ,
X6=u y ,
X7=u x ,
h=x/a,
z=y/b,
where D0=ET h03 /12(1−nLT nTL ) is the standard flexural rigidity, the differential equations of the non-dimensional normal functions X1—X8 can be rewritten as 8
s [F1ts (1Xs /1z)+F2ts (1Xs /1h)+F3ts Xs ]+P d(h−h0 )d(z−z0 )d1t=0,
(2)
s=1
where the quantities Fkts are given in Appendix 1 and, t=1, 2, . . . , 8, d1t is Kronecker’s delta. We now divide the rectangular plate vertically into m equal-length parts and horizontally into n equal-length parts, as shown in Figure 2, and consider the plate as a group of discrete
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points which are the intersections of the vertical and horizontal dividing lines. (The rectangular area, 0EhEhi and 0EzEzj , corresponding to an arbitrary intersection (i, j) as shown in Figure 2, is denoted by [i, j] the intersection (i, j) denoted by w + is called the main point of the area [i, j], and the intersections denoted by w are called the inner dependent points, and the intersections denoted by W are called the boundary dependent points. By integrating equations (2) over the area [i, j], the following integral equations are obtained: 8
6 g
s F1ts s=1
hi
[Xs (h, zj )−Xs (h, 0)] dh+F2ts
0
g
zj
[Xs (hi , z)−Xs (0, z)] dz
0
gg hi
+F3ts
0
zj
7
Xs (h, z) dh dz +P d(h−h0 ) d(z−z0 ) d1t=0.
0
(3)
By numerically integrating equations (3), simultaneous equations for the unknown quantities Xsij , which are the dimensionless shear forces, twisting moment, bending moments, rotations and deflections at the main point (i, j) of the area [i, j], are obtained as 8
6
i
j
i
k=0
l=0
j
s F1ts s bik [Xskj−Xsk0 ]+F2ts s bjl [Xsil−Xs0l ]+F3ts s s bik bjl Xskl s=1
k=0 l=0
7
+P d(h−h0 ) d(z−z0 ) d1t=0.
(4)
The solutions Xpij of the simultaneous equations (4) are expressed as 8
6
i
j
k=0
l=0
Xpij= s Apt s bik [Xtk0−Xtkj (1−dik )]+Bpt s bjl [Xt0l−Xtil (1−djl )] t=1
i
j
7
i
j
+Cpt s s bik bjl Xtkl (1−dik djl ) − s s bik bjl Ap1 P kl , k=0 l=0
(5)
k=0 l=0
where p=1, 2, . . . , 8, t=1, 2, . . . , 8, i=1, 2, . . . , m, j=1, 2, . . . , n, dij is Kronecker’s delta, bik=aik /m, bjl=ajl /n, and Apt , Bpt and Cpt are given in Appendix 2. The coefficients bif and bjg are the weight coefficients of the numerical integration. The trapezoidal rule of approximate numerical integration has been used here, and therefore the values of aif and ajg are aif=1−(d0f+dif )/2 and ajg=1−(d0g+djg )/2. In equations (5), the quantities Xpij at the main point (i, j) of the area [i, j] are related to the quantities Xtf0 and Xt0g at the boundary dependent points of the area and the quantities Xtfj , Xtig and Xtfg at the inner dependent points of the area. With spreading of the area [i, j] according to the regular order [1, 1], [1, 2], . . . , [1, n], [2, 1], [2, 2], . . . , [2, n], . . . , [m, 1], [m, 2], . . . , [m, n], a main point of a smaller area becomes one of the inner dependent points of the subsequent larger areas. Whenever one obtains the quantities Xpij at the main point (i, j) of the area [i, j] by using equations (5) in the above-mentioned order, one can eliminate the quantities Xtfj , Xtig and Xtfg at the inner dependent points of the following larger areas by substituting the obtained results into the corresponding terms of the right sides of equations (5). By repeating this process, the equations Xpij at the main point are related to only the quantities Xrf 0
761
(r=1, 3, 4, 6, 7, 8) and Xs0g (s=2, 3, 5, 6, 7, 8) at the boundary dependent points. The result is that 6
Xpij= s d=1
6
i
j
f=0
g=0
7
s apijfd Xrf 0+ s bpijgd Xs0g +qpij ,
(6)
where apijfd , bpijgd and qpij are given in Appendix 3. Equation (6) can be recognized as the discrete solution of the fundamental differential equations (1a)–(1h). The discrete Green functions related to deformation are X8ij (=w¯ /a).
3. FREQUENCY EQUATIONS
The fundamental differential equations governing the free vibration of rectangular anisotropic plates are 1Qx 1Qy 1 2w + =rh 2 , 1x 1y 1t
0
1Mx 1Mxy + =Qx , 1x 1y
1
1ux 1u 1ux 1uy , +D12 y+D16 + 1x 1y 1y 1x
Mx=D11
My=D12
1My 1Mxy + =Qy , 1y 1x
0
(7a–c)
1
1ux 1u 1ux 1uy , +D22 y+D26 + 1x 1y 1y 1x (7d, e)
Figure 3. The variation of the first five natural frequency parameters l 2 with the number of divisions for an orthotropic rectangular plate with all edges clamped.
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762
T 1 The extrapolated values (number of divisions: m=n=12, 16) compared with the results of reference [2]; C, clamped edge; S, simply supported edge
Boundary condition
Mode
D1 /D2=D3 /D2=5.0 ZXXXXXXXCXXXXXXXV Present Reference [2]
First Second Third Fourth Fifth
66·239 107·131 156·058 166·422 204·830
67·203 109·375 158·685 171·132 209·688
First Second Third Fourth Fifth
39·253 76·552 108·081 129·681 156·691
39·478 77·084 108·566 130·935 157·914
First Second Third Fourth Fifth
46·858 94·282 113·387 157·196 171·678
46·921 93·967 114·670 157·692 172·478
First Second Third Fourth Fifth
61·420 92·941 142·425 152·917 194·617
61·758 94·431 146·009 153·819 196·812
First Second Third Fourth Fifth
63·544 99·480 153·832 154·380 199·446
64·454 101·368 157·914 156·677 203·087
First Second Third Fourth Fifth
42·321 84·155 110·038 141·962 162·420
43·120 84·954 112·108 143·652 164·997
First Second Third Fourth Fifth
54·678 99·155 132·651 160·401 185·465
55·708 101·385 135·127 164·851 189·763
First Second Third Fourth Fifth
48·826 83·797 135·469 129·031 174·307
49·203 85·354 138·885 129·667 175·999
763
T 2 Convergence and accuracy of fundamental frequency of anisotropic plate with four clamped edges (l 2=va 2zrh0 /D0 ) Number of divisions u ZXXXXXXXCXXXXXXXV (degrees) 6×6 8×8 10×10 12×12 16×16 Ex. v,† 12–16 Reference [3] 0 5 10 15 20 25 30 35 40 45
82·935 82·604 81·641 80·101 78·232 76·154 74·152 72·490 71·389 71·004
79·336 79·013 78·075 76·614 74·787 72·810 70·931 69·394 68·394 68·047
77·762 77·442 76·513 75·069 73·272 71·336 69·508 68·021 67·058 66·725
76·930 76·611 75·686 74·251 72·468 70·553 68·751 67·291 66·346 66·020
75·985 75·799 74·878 73·451 71·682 69·787 68·010 66·575 65·648 65·329
74·770 74·755 73·839 72·422 70·671 68·802 67·057 65·654 64·751 64·441
Reference [4]
75·273 74·958 74·016 72·602 70·843 69·021 67·293 65·911 65·031 64·748
75·305 — — 72·571 — — 67·073 — — 64·434
† Ex. v, extrapolated value.
0
1
1u 1u 1ux 1uy + Mxy=D16 x+D26 y+D66 , 1x 1y 1y 1x
Qx=kA45
0
Qy=kA44
1
0
0
1
0
1
1w 1w +uy +kA45 +ux , 1y 1x
1
1w 1w +uy +kA55 +ux , 1y 1x
(7f–h)
where r is the mass density of the material. T 3 The extrapolated (m=n=12, 16) versus the angle of fiber orientation with respect to the x axis for anisotropic rectangular plate (C, clamped; S, simply supported) D1 /D2=5.0, D3 /D2=1·0 Frequency ZXXXXXCXXXXXV ZXXXXXXXXXXXXCXXXXXXXXXXXXV Boundary condition u (degrees) First mode Second mode Third mode Fourth mode Fifth mode 0 15 30 45
57·145 55·998 53·618 52·470
86·030 86·958 90·926 94·635
142·568 136·540 123·939 115·859
137·151 139·857 143·122 143·913
160·710 162·383 167·899 181·167
0 15 30 45 60 75 90
53·060 51·571 47·707 42·823 38·427 35·762 35·943
70·481 71·752 74·256 76·387 77·081 73·703 73·316
110·235 114·100 120·187 123·277 121·750 121·183 126·397
139·413 134·179 119·677 103·433 93·138 93·715 96·955
153·302 152·456 150·781 154·959 152·406 135·748 128·665
0 15 30 45
40·815 40·833 40·759 40·824
68·181 71·268 75·145 78·286
116·246 112·141 103·506 98·537
117·904 119·942 122·841 123·969
135·276 137·470 146·909 158·957
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764
Figure 4. The first five natural frequency parameters versus the angle of fiber orientation with respect to the x-axis for an anisotropic rectangular plate with all edges clamped.
By multiplying equation (7a) by w¯ and integrating over the intervals [0, a] and [0, b], one obtains the integral equations
gg0 b
0
a
0
1
1Qx 1Qy w¯ dx dy= + 1x 1y
gg b
0
a
−rhv 2ww¯ dx dy.
(8)
0
Figure 5. The first three natural frequency parameters versus the angle of fiber orientation with respect to the x-axis for an anisotropic rectangular plate with two opposite edges simply supported and the other two edges clamped.
765
Figure 6. The first five natural frequency parameters versus the angle of fiber orientation with respect to the x-axis for an anisotropic rectangular plate with two adjoining edges simply supported and the other two edges clamped.
Integration by parts of the left side of equation (8) yields the integral equation
gg0 b
0
a
0
=
1
1Qx 1Qy + w¯ dx dy 1x 1y
g
b
[Qx (a, y)w¯ (a, y)−Qx (0, y)w¯ (0, y)] dy
0
g
a
+
gg0 b
[Qy (x, b)w¯ (x, b)−Qy (x, 0)w¯ (x, 0)] dx−
0
a
Qx
0
0
1
1w¯ 1w¯ +Qy dx dy. 1x 1y
(9)
Substituting equations (1g) and (1h) into the third term of the right side of equation (9) results in
gg0 b
a
−
0
=
g
b
0
Qx
1
1w¯ 1w¯ +Qy dx dy 1x 1y
[Mx (a, y)u x (a, y)−Mx (0, y)u x (0, y)] dy
0
+
g
a
0
[Mxy (x, b)u x (x, b)−Mxy (x, 0)u x (x, 0)] dx
766 +
. .
g
a
g
b
[My (x, b)u y (x, b)−My (x, 0)u y (x, 0)] dx
0
+
[Mxy (a, y)u y (a, y)−Mxy (0, y)u y (0, y)] dy
0
gg 0 b
−
0
−
a
0
A 1 A45 QQ − 2 44 QQ 2 −A44 A45 x y A45 −A44 A45 x x k A45
1
A55 A QQ + 2 45 QQ dx dy 2 A45 −A44 A45 y y A45 −A44 A45 y x
gg0 b
−
0
a
0
Mx
1
1u x 1u 1u 1u +Mxy x+Mxy y+My y dx dy. 1x 1y 1x 1y
In the same way, by using equations (1a)–(1h), one can obtain the equation Pw(x0 , y0 )=
g
b
{Qx (a, y)w¯ (a, y)−Qx (0, y)w¯ (0, y)} dy
0
+
g
a
g
b
g
a
g
a
g
b
g
a
g
a
g
b
{Qy (x, b)w¯ (x, b)−Qy (x, 0)w¯ (x, 0)} dx
0
+
{Mx (a y)u x (a, y)−Mx (0, y)u x (0, y)} dy
0
+
{Mxy (x, b)u x (x, b)−Mxy (x, 0)u x (x, 0)} dx
0
+
{My (x, b)u y (x, b)−My (x, 0)u y (x, 0)} dx
0
+
{Mxy (a, y)u y (a, y)−Mxy (0, y)u y (0, y)} dy
0
−
{uy (x, b)M y (x, b)−uy (x, 0)M y (x, 0)} dx
0
−
{uy (x, b)M xy (x, b)−uy (x, 0)M xy (x, 0)} dx
0
−
0
{ux (a, y)M x (a, y)−ux (0, y)M x (0, y)} dy
(10)
g
b
g
b
g
a
−
767
{ux (a, y)M xy (a, y)−ux (0, y)M xy (0, y)} dy
0
−
{w(a, y)Q x (a, y)−w(0, y)Q x (0, y)} dy
0
−
{w(x, b)Q y (x, b)−w(x, 0)Q y (x, 0)} dx+
0
gg b
0
a
rhv 2ww¯ dx dy.
(11)
0
Consider the following boundary conditions: simply supported edge—My=ux=w=0 along y=0, b, and Mx=uy=w=0 along x=0, a; clamped edge—uy=ux=w=0 along y=0, b, and uy=ux=w=0 along x=0, a. Thus, for any combination of these conditions, equation (11) becomes the integral equation
gg b
Pw(x0 , y0 )=
0
a
rhv 2w(x, y)w¯ (x0 , y0, x, y) dx dy.
(12)
0
By using the non-dimensional expressions l 4=rh0 v 2a 4/D0 ,
h(x, y)=h(x, y)/h0 , m=b/a,
h=x/a,
w˜=w/a,
G=(D0 /Pa 2 )w¯ ,
z=y/b
here l is the non-dimensional eigenvalue, the integral equation (12) can be rewritten as
gg 1
w˜ (h0, z0 )=
0
1
mh (h, z) l4w˜ (h, z) G(h0 , z0 , h, z) dh dz.
(13)
0
4. NUMERICAL RESULTS
In order to test the convergence of numerical solutions by the present method, the first five frequency prameters l 2 were calculated for orthotropic and anisotropic rectangular plates with any combinations of simply supported and clamped edges. The specific values in this analysis are as follows: D1=EL h 3/12(1−nLT nTL ), D3=D4+2D5 , D4=EL h 3nTL /12(1−nLT nTL ), D5=GLT h 3/12, D2=ET h 3/12(1−nLT nTL ), l 2=va 2zrh0 D0 , k=5/6, b/a=1·0. 4.1. For an orthotropic rectangular plate with the stiffness ratios D1 /D2=D3 /D2=5·0, the variation of the first five natural frequency parameters l 2 with the number of area divisions with all edges clamped is shown in Figure 3. It is found from Figure 3 that the numerical solutions by the present method show good convergence. The extrapolated values (i.e., the calculated values when m=n=12, 16) for orthotropic rectangular plates with some combinations of simply supported and clamped edges are shown in Table 1, together with the analytical solution results obtained by Hearmon [2]. It is found that the extrapolated values agree with the analytical solutions. 4.2. For an anisotropic rectangular plate with the stiffness ratios D1 /D2=5·0 and D3 /D2=1·0, the convergence and accuracy of the fundamental frequency factor l 2 for the all edges
. .
768
clamped case as obtained by the present method is shown in Table 2. The extrapolated natural frequency parameters of the anisotropic rectangular plate for some combinations of simply supported and clamped edges as shown in Table 3. Next, as an application of the proposed method, the variations of the natural frequency parameters and deflection modes versus the angle of fiber orientation with respect to the x-axis for anisotropic rectangular plates are shown in Figures 4–6. Figure 4 is for the case of an anisotropic clamped rectangular plate, Figure 5, for the case of an anisotropic plate with two opposite edges simply supported and the other two edges clamped, and Figure 6 for the case of an anisotropic plate with two adjoining edges simply supported and the other two edges clamped. It is found from these figures that the natural frequency parameters and deflection modes change with the variation of the angle of fiber orientation to the x axis for the anisotropic rectangular plate. 5. CONCLUSIONS
An approximate method for analyzing the free vibration of rectangular anisotropic plates with various boundary conditions has been proposed. This method is based on an application of the discrete Green function and numerical solution of integral equations. These discrete Green functions give the solutions at discrete points. The natural frequency parameters obtained approach those of the analytical solutions as one increases the number of discrete points. REFERENCES 1. N. J. H J and W. H. H II 1958 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 25, 389–395. On the transverse vibrations of rectangular orthotropic plates. 2. R. F. S. H 1959 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 26, 537–540. The frequency of flexural vibration of rectangular orthotropic plates with clamped or supported edges. 3. J. F, A. R. K and C. W. B 1993 Computers and Structures 46, 465–477. Fundamental frequency analysis of single specially orthotropic, generally orthotropic and anisotropic rectangular layered plates by the differential quadrature method. 4. J. M. W 1987 Structural Analysis of Laminated Anisotropic Plates. Lancaster, Pennsylvania, Technomic.
APPENDIX 1
F111=F123=F134=1,
11 m, F247=D
F212=F225=F233=−F322=−F331=m,
F146=D 12 ,
F147=D 16 ,
F246=D 16 m,
−F345=−F354=−F363=I,
12 m, F257=D
F156=D 22 ,
44 , F178=kA 16 m, F267=D
F166=D 26 ,
F157=D 26 , F376=kA 44 m,
F167=D 66 ,
45 m, F278=F377=F386=kA −F371=−F382=K, m=b/a,
={a/D0 (1−nTL nLT )}P, P
ij=12(a/h0 )2Q ij /ET A
(i, j=4, 5),
F256=D 26 m,
F266=D 66 m,
F188=kA 45 ,
F288=F387=kA 55 m, other Fkts=0, I=m(h0 /h)3, D ij=Q ij /ET
K=mh0 /h,
(i, j=1, 2, 6).
769
APPENDIX 2
F Ap1J F J gp1 G G G G 0 G Ap2G G G gp2 G Ap3G G G G Ap4G G G gp3 h, Apt= g h = g 0 G Ap5G G G G Ap6G G D12 gp4+D22 gp5+D26 gp6 G G A G G D g +D g +D g G G p7G G 16 p4 26 p5 66 p6 G f Ap8j f kA44 gp7+kA45 gp8 j F Bp1J F 0 G G G mgp1 G Bp2G G mgp3 G Bp3G G G Bp4G G 0 Bpt= g h = g mgp2 G Bp5G G G Bp6G G m(D16 gp4+D26 gp5+D66 gp6 ) G B G G m(D g +D g +D g ) 11 p4 12 p5 16 p6 G p7G G m(kA 45 gp7+kA 55 gp8 ) f Bp8j f
J G G G G h, G G G G j
F Cp1J F J mgp3+Kgp7 G G G G mgp2+Kgp8 G Cp2G G G G Cp3G G Igp6 G G Cp4G G G Igp5 g h g h, Cpt= = Igp4 G Cp5G G G G Cp6G G −m(kA44 gp7+kA45 gp8 ) G G C G G −m(kA g +kA g ) G 45 p7 55 p8 G p7G G G 0 f Cp8j f j r11=bii ,
[gpt ]=[rpt ]−1 ,
r12=mbjj ,
r25=mbjj , r33=mbjj ,
r34=bii ,
r22=−mbii bjj ,
r23=bii ,
r31=−mbii bjj ,
r45=−Ibii bjj ,
r46=D 12 bii+D 16 mbjj ,
r47=D 16 bii+D 11 mbjj , r54−=Ibii bjj ,
r56=D 22 bii+D 26 mbjj ,
r57=D 26 bii+D 12 mbjj ,
r63=−Ibii bjj ,
r66=D 26 bii+D 66 mbjj ,
r67=D 66 bii+D 16 mbjj ,
r71=−Kbii bjj ,
r76=kA 44 mbii bjj ,
r77=kA 45 mbii bjj ,
r78=k(A 44 bii+A 45 mbjj ),
r82=−Kbii bjj ,
r86=kA 45 mbii bjj ,
r87=kA 55 mbii bjj ,
r88=k(A 45 bii+A 55 mbjj ).
. .
770
APPENDIX 3
6
8
apijfd= s t=1
i
i
j
k=0
l=0
s bik Apt [atk0fd−atkjfd (1−dik )]+ s bjl Bpt [at0lfd−atilfd (1−djl )]
7
j
+ s s bik bjl Cpt atklfd (1−dik djl ) , k=0 l=0
8
bpijgd= s t=1
6
i
j
k=0
l=0
s bik Apt [btk0gd−btkjgd (1−dik )]+ s bjl Bpt [bt0lgd−btilgd (1−djl )]
i
7
j
+ s s bik bjl Cpt btklgd (1−dik djl ) , k=0 l=0
6
8
qpij= s t=1
i
i
j
k=0
l=0
s bik Apt [qtk0−qtkj (1−dik )]+ s bjl Bpt [qt0l−qtil (1−djl )]
7
j
i
j
+ s s bik bjl Cpt qtkl (1−dik djl ) − s s bik bjl Ap1 P kl , k=0 l=0
ari0it=1,
a10j01=a j ,
k=0 l=0
a40j03=a j ,
a2i0k1=(T 51 /T 52 )b ik ,
a2i0k5=−(mk/KT 52 )b ik , a2i0k6=−(mk/KT 52 )gik ,
a5i0k2=−(S16 /S11 )b ik ,
a5i0k3=−(S12 /S11 )b ik ,
a5i0k5=(1/D S11 )gik , bs0jjt=1,
b2i001=a¯ i ,
b5i003=a¯ i ,
b10jl6=(k/KT 61 )gjl ,
b10jl1=(T 62 /T 61 )b jl ,
b40jl2=−(S26 /S22 )b jl ,
b10jl4=(mk/KT 61 )b jl ,
b40jl3=−(S12 /S22 )b jl ,
S22 )gjl , b40jl4=(1/mD [Sij ]=[D ij ]−1
(i, j=1, 2, 6),
T 51=A 45 /(A −A 44 A 55 ), 2 45
T 52=A 44 /(A 245−A 44 A 55 ), T 61=A 55 /(A 44 A 55 −A 245 ), T 62=A 45 /(A 44 A 55−A 245 ), D =(h0 /h)3,
a¯ i=(−1)i,
bik=dik+(−1)i+1d0k ,
g¯ ik=4m(−1)i+k/(1+dik+d0k ), r=t+d2t+d3t+2(d4t+d5t+d6t ),
s=t+d1t+d2t+2(d3t+d4t+d5t+d6t ).