FREE VIBRATION ANALYSIS OF RECTANGULAR MINDLIN PLATES WITH ELASTIC RESTRAINTS UNIFORMLY DISTRIBUTED ALONG THE EDGES

Journal of Sound and Vibration (1996) 192(4), 885–904

FREE VIBRATION ANALYSIS OF RECTANGULAR MINDLIN PLATES WITH ELASTIC RESTRAINTS UNIFORMLY DISTRIBUTED ALONG THE EDGES K. N. S  R. C. K Department of Mechanical Engineering

 P. K. D Department of Aerospace Engineering, Indian Institute of Technology, Kharagpur-721302, India (Received 19 October 1994, and in final form 8 September 1995) Free vibration analysis of isotropic Mindlin plates with edges elastically restrained against rotation and translation is carred out by a variational method. The elastic restraints are assumed to be uniformly distributed along the edges. Elastically restrained Timoshenko beam functions are used as constitutive shape functions and the boundary conditions not satisfied remain as boundary terms in the final variational energy expression. The results for reduced problems are compared with those available in the literature and found to be in good agreement. Results showing the effect of elastic restraints on the natural frequencies of a thick plate indicate that the combined variation of rotational and translational restraint is more significant than their independent variations. Variations in mode shapes caused by the changes in the boundary conditions are also presented. 7 1996 Academic Press Limited

1. INTRODUCTION

Plates used in structures under dynamic loads are often thick and the forcing frequency may be higher than the fundamental frequency. In the vibration analysis of such problems rotary inertia and shear deformation must be considered. The shear deformation and rotary inertia effects are included in the Mindlin plate theory [1] which calls for three basic reference quantities (w, cx , cy ) rather than the flexural deflection alone as in thin plate theory. In Mindlin plate theory, a shear factor (k) is introduced to take into account non-uniform shear strain distribution through the plate thickness. This assumption of uniform shear strain distribution is accounted for in higher order plate theories [2], but for isotropic material Mindlin plate theory yields accurate results. Exact solutions for simply supported Mindlin plates have been presented by Mindlin et al. [3]. Later, Levinson [4] reported an exact elasticity solution for free vibrations of a simply supported rectangular plate. Yu and Cleghorn [5] have analyzed clamped Mindlin plates by the superposition method which uses analytical expressions as its building blocks and Mindlin [6] has suggested exact solutions for flexural vibrations of rectangular plates with free edges. However, unless the plate is simply supported, or at least two opposite 885 0022–460X/96/190885 + 20 $18.00/0

7 1996 Academic Press Limited

886

. .   .

edges of it are simply supported, use of an approximate method is unavoidable. Dawe and Roufaeil [7] have applied the Rayleigh–Ritz method taking Timoshenko beam functions, as suggested by Huang [8], to constitute the trial functions for the spatial variation of the independent quantities (w, cx , cy ). Liew and Wang [9] have also adopted a Rayleigh–Ritz method with shape functions (pb-2) generated by sets of two-dimensional polynomials. Reddy [10] has resorted to a finite element method based on a variational principle in his textbook. Carnicer and Alliney [11] have developed a different formulation for general boundary conditions for analysis by means of finite element techniques. Mikami and Yoshimura [12] have applied the collocation method with orthogonal polynomials for rectangular Mindlin plates with varying thickness. Aksu [13] has used a finite difference method based on the variational procedure for frequency analysis of corner-pointsupported Mindlin plates. Mizusawa [14] has used the spline strip method for tapered plates. Complicating effects in the vibration of Mindlin plates are numerous. Leissa [15] has discussed some of them in his review of the literature. Liew et al. [16] have considered thick skew plates. Kobayashi and Sonoda [17] have dealt with rectangular Mindlin plates on elastic foundations. Al-Kaabi and Aksu [18] have analyzed rectangular Mindlin plates with parabolically varying thickness. Recently in a series of papers, Liew et al. [19] have considered the effects of boundary conditions, oblique internal line supports, internal ring supports and in-plane isotropic pressure on transverse vibration of thick rectangular plates. Roufaeil and Dawe [20] have also studied rectangular Mindlin plates subjected to membrane stresses. Lee et al. [21] have observed thick plate effects for rectangular plates with cutouts. The boundary conditions of a real system are seldom classical and hence research studies for non-classical boundaries need to be emphasized. The only work found in this area is due to Chang et al. [22] who have considered orthotropic Mindlin plates with edges elastically restrained against rotation. In the present study, free vibration analysis of isotropic rectangular Mindlin plates having edges elastically restrained against rotation and translation is studied. Results are compared with those of Bapat et al. [23], Sheikh and Mukhopadhyay [24] and Gorman [25] as special cases.

2. ANALYSIS

The calculation of the natural frequencies is carried out by employing a variational method. Elastically restrained Timoshenko beam functions are used as constitutive shape functions. The method for formation of trial functions as the series of products of beam functions is well known and is not repeated here for brevity. Use of beam functions corresponding to the boundary conditions of the two opposite edges of the plate is the intuitively simplest and most established method for analyzing plate problems. This has been successfully and effectively used by Leissa [26], Dawe and Roufaeil [7] and many others (see, e.g., references [15, 20]). Huang [8] presented an early work on the determination of Timoshenko beam functions for a uniform beam with simple end conditions. Abbas [27] has solved the problem of an elastically supported Timoshenko beam by a finite element method. Rao and Mirza [28] have presented exact frequency and normal mode shape expressions for a generally restrained Euler–Bernoulli beam. For a generally restrained Timoshenko beam, Maurizi et al. [29] have provided an exact solution for flexural vibration only. In the present study a generally restrained Timoshenko beam is analyzed both for transverse deflection (u) and cross-sectional rotation (f).

   

887

Figure 1. A generally restrained Timoshenko beam.

2.1.    A generally restrained Timoshenko beam is shown in Figure 1. The uncoupled governing differential equations for the flexural modes of vibration in non-dimensional form are u IV + v 2(q1 + q2 )u0 − v 2(1 − v 2q1 q2 )u = 0,

(1)

f + v (q1 + q2 )f0 − v (1 − v q1 q2 )f = 0,

(2)

IV

2

2

2

where j = x/L,

q1 = EI/kGAL 2,

t = t/z(rAL 4/EI)

q2 = I/AL 2, and

v 2 = V 2rAL 4/EI,

()' = d()/dj,

etc.

(3)

(A list of nomenclature is given in Appendix III.) The general solution of the governing differential equations (1) and (2) is u(j) = C1 cosh vaj + C2 sinh vaj + C3 cos vbj + C4 sin vbj,

(4a)

f(j) = C 1 cosh vaj + C 2 sinh vaj + C 3 cos vbj + C 4 sin vbj,

(4b)

a = (1/z2){z(q1 − q2 )2 + 4/v 2 − (q1 + q2 )}1/2,

(5)

b = (1/z2){z(q1 − q2 )2 + 4/v 2 + (q1 + q2 )}1/2,

(6)

where

for v 2 Q 1/q1 q2 ; Ci and C i (i = 1 to 4) are constants. Otherwise, solutions are given by u(j) = C1 cosh va'j + C2 sinh va'j + C3 cos vbj + C4 sin vbj,

(7a)

f(j) = jC 1 cosh va'j + C 2 sinh va'j + C 3 cos vbj + C 4 sin vbj,

(7b)

Figure 2. A generally restrained Mindlin plate.

. .   .

888

Figure 3. The effect of plate thickness on natural frequencies of a square plate (a) with clamped edges, and (b) with simply supported edges.

where j = z − 1, and a' = (1/z2){(q1 + q2 ) − z(q1 − q2 )2 + 4/v 2}1/2.

(8)

The coefficients C1–C4 and C 1–C 4 in equations (4) and (7) are related by C 1 /C1 = C 2 /C2 = v(a + q1 /a),

C 3 /C3 = v((q1 /b) − b),

C 4 /C4 = v(b − q1 /b)

(9a)

for v 2 Q 1/q1 q2 . Otherwise, the first two ratios are C 1 /C1 = C 2 /C2 = v(a − q1 /a),

(9b)

with C 3 /C3 and C 4 /C4 remaining unchanged. The boundary conditions are f' − SrL f = 0

and

(u' − f) + StL q1 u = 0

at j = 0,

(10a,b)

f' + SrR f = 0

and

(u' − f) − StR q1 u = 0

at j = 1,

(10c,d)

where St = Kt L 3/EI, Sr = Kr L/EI and the second subscripts L and R of St and Sr refer to the left and right ends of the beam, respectively.

   

889

Substituting equations (4) or (7), as the case may be, in the boundary conditions (10), one obtains, for a non-trivial solution, the natural frequencies of the system from =dij = = 0,

i, j = 1 to 4,

(11)

where the elements dij are given in Appendix I. The coefficients Ci and C i (i = 1 to 4) are normalized by C4 and C 4 , respectively, and the normal mode shapes are used subsequently in the Mindlin plate problem. The functions characterizing the normal mode shapes, being the eigenfunctions of the corresponding elastically restrained Timoshenko beam, are orthogonal and form a complete set. They can accommodate any number of possible combinations of edge conditions and thus have added advantages. 2.2.         A generally restrained rectangular Mindlin plate is shown in Figure 2. The edges of the plate are restrained against rotation and translation, the spring stiffnesses being Kri and

Figure 4. (a) The effect of rotational elastic restraint R on the natural frequencies of a square plate for R = R4, K4 = 107 and edges 1, 2 and 3 simply supported; (b) as (a) but for R = R3 = R4, K3 = K4 = 107 and edges 1 and 2 simply supported; (c) as (a) but for R = R2 = R3 = R4, K2 = K3 = K4 = 107 and edge 1 simply supported; (d) as (a) but for R = R1 = R2 = R3 = R4, K1 = K2 = K3 = K4 = 107. ——, h/a=0·01; – – –, h/a = 0·1.

. .   .

890

Figure 5. (a) The effect of elastic restraint R on the natural frequencies of a square plate for R = R4 = K4 and edges 1, 2 and 3 clamped; (b) as (a) but for edges 1, 2 and 3 simply supported. Key as Figure 4.

Kti (i = 1 to 4). Elastic restraint along any particular edge is assumed to be uniform. The expressions for the strain energy (V) and the kinetic energy (T) are well known and are not repeated here. For the conservation of the total energy of the system, dH/dt = 0.

(12)

Substitution of the expressions for the energies in equation (12) and use of the non-dimensional quantities j = x/a,

h = y/b,

q4 = h 2/12a 2,

w = W/a, AR = b/a,

v = Va 2zrh/D, St = Kt a 3/D,

q3 = kGha 2/D,

Sr = Kr a/D

(13)

Figure 6. The effect of elastic restraint R on the fundamental natural frequency of a plate for various aspect ratios and R = R4 = K4; (a) edges 1, 2 and 3 clamped; (b) edges 1, 2 and 3 simply supported. Key as Figure 4.

   

891

Figure 7. The effect of elastic restraint R on the natural frequencies of a square plate for (a) R = R1 = R2 = R3 = R4 = K1 = K2 = K3 = K4; (b) as (a) but for R = K1 = K2 = K3 = K4 and R1 = R2 = R3 = R4 = 10−7.

yield

g g $6 1

0

1

0

0

1 2w 1 2w 1cx 1 1cy 1 1 2w + + 2 2 2 − q3 2 + 1t 1j 1j AR 1h AR 1h

170 1 1w 1t

6

1 2cx 1 2cx (1 + n) 1 2cy (1 − n) 1 2cx 1w − + cx − 2 − − q3 1t 2 1j 2AR2 1h 2 1j 2AR 1j1h

6

1 1 2cy (1 + n) 1 2cx (1 − n) 1 2cy 1 2cy − − 2 − 1t 2 1j 2 AR2 1h 2 2AR 1j1h

+ q4

+ q4

0

170 1 1cx 1t

Figure 8. The effect of elastic restraint R on the fundamental natural frequency of a plate for various aspect ratios and (a) R = R1 = R2 = R3 = R4 = K1 = K2 = K3 = K4; (b) as (a) but for R = K1 = K2 = K3 = K4 and R1 = R2 = R3 = R4 = 10−7.

. .   .

892

Figure 9—continued (caption on page 893).

− q3

+

0

1 1w + cy AR 1h

0

1cx n 1cy + + Sr cx 1c AR 1h

q3

0

10 1 0 10 1cx 1−n + dt 2

1

0 10

10 1% %

0

+

$g $6 0 1

dj dh +

$g $6 0 1

+

170 1% 1cy 1t

1−n 2AR

q3 1 1w S + cy + t w AR AR 1h AR 1 1cx 1cy + AR 1h 1j

1cx 1t

1

1w + c x + St w 1j

1w 1t

10 1% %

1 1cx 1cy + AR 1h 1j

70 1 0

70 1

1cy 1t

1

dh

1w 1 1cy n 1cx Sr + 2 + + c 1t AR 1h AR 1j AR y

0

10 1 1cy 1t

1

dj

0

= 0.

(14)

   

893

Figure 9. Vibration mode shapes for a square simply supported plate: (a) h/a = 0·1; (b) h/a = 0·01.

Let w(j, h, t) = W(j, h)g(t),

N

W(j, h) = s ai Wxi (j)Wyi (h),

(15a)

i=1

cx (j, h, t) = Cx (j, h)g(t),

2N

Cx (j, h) = s

ak cxk (j)Wyk (h),

(15b)

am cym (h)Wxm (j),

(15c)

k=N+1

cy (j, h, t) = Cy (j, h)g(t),

Cy (j, h) =

3N

s m = 2N + 1

. .   .

894

Figure 10—continued (caption on page 896).

where g(t) = e jvt for the motion to be synchronous, and t is the non-dimensional time, given by t/z(rha 4/D). Substitution of equations (15) in equation (14) yields the equation of motion, which is, in matrix form, −v 2[M]{a} + [K]{a} = {8},

(16)

where

[M] =

&

[m11 ] [8] [8] [m22 ] [8] [8]

'

[8] [8] , [m33 ]

&

[K11 ] [K] = [K21 ] [K31 ]

[K12 ] [K22 ] [K32 ]

'

[K13 ] [K23 ] . [K33 ]

(17)

   

895

Figure 10—continued (caption on page 896).

[M] and [K] are symmetric matrices of order 3N × 3N. Each submatrix is of order N × N, where N = r 2, r being the number of modes of the beam functions taken into consideration. [Kij ] = [Kji ]T for i, j = 1, 2, 3 and [8] denotes a null matrix of order N × N. The elements of the various submatrices are given in Appendix II. The deflection at any point (j, h) on the plate can be obtained from equation (15a) by substituting into it the appropriate eigenvector together with the associated shape functions Wxi (j) and Wyi (h).

3. RESULTS AND DISCUSSION

The edges of the plate are indicated by 1–4 as shown in Figure 2 and accordingly the non-dimensional translational and rotational restraints at the edges can be denoted by K1 to K4 and R1 to R4, respectively, for convenience.

896

. .   .

Figure 10. Vibration mode shapes for a square plate with edges 1, 2 and 3 clamped and the remaining one elastically restrained. (a) R = R4 = K4 = 107; (b) as (a) but for R = 100; (c) as (a) but for R = 10−7.

In the following discussion the classical boundary conditions refer to certain values of non-dimensional translational and rotational restraints, as suggested by Bapat et al. [23] and are indicated in Table 1. The results of the convergence study of the current problem are shown in Table 2. Computations were carried out on a CYBER 840 computer system. The CPU time in seconds is also indicated in Table 2. The number of terms chosen in the analysis is 3 × 32 which is based on the convergence shown in Table 2 and the corresponding CPU time. In the present work all numerical problems were solved for k = 0·85 and n = 0·3. The natural frequencies are indicated by v1 , v2 , v3 , etc. Comparisons of results are presented in Tables 3–5. First a test problem was solved to indicate the effect of thickness ratio on the natural frequencies. The effect is shown in Figures 3(a) and 3(b) for a square plate with clamped and simply supported edges respectively. As can be seen in the figures, the variation of

   

897

T 1 Kn and Rn values for simply supported, clamped and free boundary conditions (n = 1 to 4) Boundary condition

Kn

Rn

7

Simply supported Clamped Free

10−7 107 10−7

10 107 10−7

T 2 Convergence study results; AR = 1·0, h/a = 0·01, n = 0·3, k = 0·85 C–C–C–C plate ZXXXXXXXXCXXXXXXXXV 3 × 22 3 × 32 3 × 42 3 × 52

Terms v(1) v(2) v(3) v(4)

36·071 73·592 73·592 108·554

Time

1·870

S–S–S–S plate ZXXXXXXXCXXXXXXXV 3 × 22 3 × 32 3 × 42 3 × 52

35·967 73·345 73·345 108·554

35·967 73·319 73·319 108·148

35·966 73·311 73·311 108·075

19·732 49·304 49·304 78·844

19·732 49·304 49·304 78·844

19·732 49·304 49·304 78·844

19·723 49·304 49·304 78·844

7·569

23·516

59·000

1·765

7·353

23·238

58·457

T 3 Fundamental frequencies for a S–R–S–R plate (AR = 1·0) R = 10−7

R = 1·0

R = 10·0

R = 102

R = 103

R = 104

R = 107

1

A B

9·7452 9·56827

9·9048 9·75601

10·8479 10·7927

16·1945 16·1945

25·9075 25·9118

28·6078 28·6148

28·9314 28·9505

2

A B

9·7452 9·56827

9·8700 9·69744

10·8700 10·3259

15·4845 15·3733

19·2259 19·0785

19·6885 19·6702

19·7320 19·7391

1, R = R2 = K2 = R4 = K4; 2, R = K2 = K4, R2 = R4 = 10−7. A, Present method (h/a = 0·01); B, Bapat et al. [23].

T 4 Natural frequencies for a C–C–C–R plate (AR = 1·0; K4 = R4 = R) Mode

R = 10−7

R = 10·0

R = 102

R = 103

R = 104

R = 107

1

A C

24·016 23·941

24·941 24·902

28·219 28·227

34·429 34·441

35·817 35·825

35·967 35·989

2

A C

40·039 40·016

43·482 43·476

47·143 47·146

63·127 63·091

72·373 72·398

73·345 73·419

3

A C

63·435 63·302

63·956 63·921

65·443 65·548

70·717 70·757

73·106 73·124

73·345 73·439

A, Present results; C, Sheikh and Mukhopadhyay [24].

T 5 Comparison of present results (A) with those of Gorman [25] (D) (taken from his graph; R1 = R2 = R3 = R4 = R, K1 = K2 = K3 = K4 = 3000 A D

R = 10−7

R = 10

R = 25

R = 50

R = 100

R = 500

R = 2700

R = 107

19·43 19·73

27·51 27·2

30·45 30·2

32·00 31·8

32·96 33·0

33·84 33·8

34·04 34·3

34·08 35·98

898

. .   .

frequencies with thickness is more pronounced at higher modes, showing thick plate effects, as discussed earlier. It is also observed that the thick plate effect is more prominent when the plate has clamped boundaries than it is when they are simply supported. In the results shown in Figures 4(a)–8(b), a solid line indicates a thickness ratio of h/a = 0·01, whereas a dashed line indicates a thickness ratio of h/a = 0·1. In Figures 4(a)–(d) is shown the variation of the natural frequencies with the restraint parameter R for (a) one edge restrained (R = R4), (b) two adjacent edges restrained (R = R3 = R4), (c) three edges restrained (R = R2 = R3 = R4) and (d) all edges restrained (R = R1 = R2 = R3 = R4) against rotation. In each case, the translational restraints are fixed (Kn = 107 ) and any edge which is not restrained is simply supported. The variation of the fundamental natural frequencies for a thin plate (h/a = 0·01) shows good agreement with that given by Warburton [30]. For all the four cases, the natural frequencies vary over a certain range of restraint parameter (R varying from 1 to 1000) which remains unchanged for thin and thick plates.

Figure 11—continued (caption on page 900).

   

899

In Figures 5(a) and 5(b), the effect of simultaneous variation of translational and rotational restraints (R = R4 = K4) are shown for a square plate with three unrestrained edges clamped and simply supported, respectively. These figures depict that, when both the rotational and translational restraints are varying, the natural frequencies vary over a wide range of restraint parameter (R varying from 0·1 to 104 ) both for thin and thick plates. In Figures 6(a) and 6(b), the fundamental frequency is plotted against the restraint parameter R (R = R4 = K4) for different aspect ratios for similar boundary conditions as in the previous case. It is observed that for a higher aspect ratio (AR = 2·0), the difference in the natural frequency between thin and thick plates remains almost unchanged as the elastic restraint parameter R is varied. However, for a lower aspect ratio (AR = 0·8) the difference in frequency between thin and thick plate increases as R is increased from 100 to 1000 (approximately) and remains unchanged thereafter.

Figure 11—continued (caption on page 900).

900

. .   .

In Figure 7(a), the effect of simultaneous variation of translational and rotational restraints along all the edges of a square plate is shown (R = R1 = R2 = R3 = R4= K1 = K2 = K3 = K4). In Figure 7(b) only the translational restraints are varied along all the edges (R = K1 = K2 = K3 = K4), while the rotational restraints are kept fixed at the lowest value (10−7 ). For lower values of the restraint parameter R, the plate tends to become free at the boundaries and hence the first three natural frequencies tend to zero. With increase in R, the natural frequencies increase and the difference in frequency between thin and thick plates also increases. In Figures 8(a) and 8(b), the fundamental frequency is plotted against the restraint parameter R for different aspect ratios and for different boundary conditions. In Figures 9(a) and 9(b), the mode shpaes are shown for simply supported plates with thickness ratios 0·1 and 0·01, respectively. It is observed that the vibration modes are not affected by the plate thickness, which may be attributed to the fact that the deflection

Figure 11. Vibration mode shapes for a square plate with all elastically restrained edges. (a) R = R1 = R2 = R3 = R4 = K1 = K2 = K3 = K4 = 103; (b) as (a) but for R = 10; (c) as (a) but for R = 0·1.

   

901

W(x, y) is uncoupled from the two cross-sectional rotations cx (x, y) and cy (x, y). In the remaining figures, the mode shapes are furnished for moderately thick plates (h/a = 0·1) corresponding to the first four eigenfrequencies. In Figures 10(a)–(c), the effect of simultaneous variation of translational and rotational restraints (R = R4 = K4) on the vibration mode shapes are shown for a square plate with three edges clamped. For very high values of R, the restrained edge becomes clamped, whereas for very low values it is free. For an intermediate value (e.g., R = 100), the fourth mode resembles more that of a clamped plate, whereas the first three modes correspond to those with a free edge. In Figures 10(a) and 11(a)–(c), the effect of simultaneous reduction of translational and rotational restraints along all the edges of a square plate is shown. For a relatively lower value of R (e.g., R = 0·1), the first three vibration modes look like rigid body movement as expected for a plate with free edges whereas for a relatively higher value of elastic restraint (e.g., R = 103 ), the vibration modes tend to those of a clamped plate. 4. CONCLUSIONS

For higher modes, the difference between the natural frequency for thick plate (h/a = 0·1) and that of a thin plate (h/a = 0·01) is more pronounced. The effects of elastic restraints on the natural frequencies of the plate show that the combined effects of rotational and translational restraints along a particular edge of the plate is more significant than that of the rotational restraints alone. The simultaneous variation of rotational and translational restraints along all the edges of the plate causes the natural frequencies to vary over the widest range of restraint parameters. When only rotational restraints are varied, the difference in natural frequencies between thin and thick plates remains very small over the range of values of restraint parameters considered. However, the effect of plate thickness on the frequencies is significant for combined variations of rotational and translational restraints. For lower values of elastic restraints at the boundaries of the plate, the natural frequencies of thin and thick plates differ very little up to a certain value of the restraint parameter, beyond which they differ significantly, the difference being more pronounced at higher modes. The vibration modes are not affected by the plate thickness. When translational and rotational restraints are varied simultaneously along all the edges, the first three vibration modes resemble rigid body movement for relatively lower values of R, whereas for a relatively higher value, e.g., R = 103, the vibration modes tend to those of a clamped plate. REFERENCES 1. R. D. M 1951 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 18, 31–38. Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates. 2. J. N. R 1984 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 51, 745–752. A simple higher order theory for laminated composite plates. 3. R. D. M, A. S and H. D 1956 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 23, 430–436. Flexural vibrations of rectangular plates. 4. M. L 1985 Journal of Sound and Vibration 98, 289–298. Free vibrations of a simply supported rectangular plate, an exact elasticity solution. 5. S. D. Y and W. L. C 1993 Transactions of the Canadian Society of Mechanical Engineers 17(2), 243–255. Accurate free vibration analysis of clamped Mindlin plates using the method of superposition.

902

. .   .

6. R. D. M 1986 Mechanics Research Communications 13(6), 349–357. Flexural vibrations of rectangular plates with free edges. 7. D. L. D and O. L. R 1980 Journal of Sound and Vibration 69, 345–359. Rayleigh–Ritz vibration analysis of Mindlin plates. 8. T. C. H 1961 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 28, 579–584. The effect of rotary inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions. 9. K. M. L and C. M. W 1993 Engineering Structures 15, 55–60. pb-2 Rayleigh–Ritz method for general plate analysis. 10. J. N. R 1984 An Introduction to the Finite Element Method. New York: McGraw Hill. 11. R. S. C and S. A 1989 Applied Mechanics Review 42(11), Part 2, s32–s38. A Mindlin–Reissner variational principle to analyze the behavior of moderately thick plates. 12. T. M and J. Y 1984 Computers and Structures 18(3), 425–431. Application of the collocation method to vibration analysis of rectangular Mindlin plates. 13. G. A 1993 Computers and Structures 48(6), 1163–1166. Vibration of Mindlin plates symmetrically column supported at four points. 14. T. M 1993 Computers and Structures 46(3), 451–463. Vibration of rectangular Mindlin plates with tapered thickness by the spline strip method. 15. A. W. L 1987 Shock and Vibration Digest 19(3), 10–24. Recent studies in plate vibrations: 1981–1985, part 2, complicating effects. 16. K. M. L, Y. X, S. K and C. M. W 1993 Journal of Sound and Vibration 168, 39–69. Vibration of thick skew plates based on Mindlin shear deformation plate theory. 17. H. K and K. S 1989 International Journal of Mechanical Science 31(9), 679–692. Rectangular Mindlin plates on elastic foundations. 18. S. A. A-K and G. A 1990 Computers and Structures 34(3), 395–399. Free vibration analysis of Mindlin plates with parabolically varying thickness. 19. K. M. L, Y. X and S. K 1993 Computers and Structures 49(1), 1–78. Transverse vibration of thick rectangular plates—I: comprehensive sets of boundary conditions, II: Inclusion of oblique internal line supports, III: Effects of multiple internal eccentric ring supports, IV: Influence of isotropic in-plane pressure. 20. O. L. R and D. J. D 1982 Journal of Sound and Vibration 85, 263–275. Rayleigh–Ritz vibration analysis of rectangular Mindlin plates subjected to membrane stresses. 21. H. P. L, S. P. L and S. T. C 1992 International Journal of Solids and Structures 29(11), 1351–1359. Effect of transverse shear deformation and rotary inertia on the natural frequencies of rectangular plates with cutouts. 22. J. H. C, T. Y. C and K. C. K 1993 Journal of Sound and Vibration 163, 151–163. Vibration analysis of orthotropic Mindlin plates with edges elastically restrained against rotation. 23. A. V. B, N. V and S. S 1988 Journal of Sound and Vibration 120, 127–140. Simulation of classical edge conditions by finite elastic restraints in the vibration analysis of plates. 24. A. H. S and M. M 1993 Transactions of the American Society of Mechanical Engineers, Journal of Vibration and Acoustics 115(7), 295–302. Transverse vibration of plate structures with elastically restrained edges by the spline/finite strip method. 25. D. J. G 1990 Journal of Sound and Vibration 139, 325–335. A general solution for the free vibration of rectangular plates resting on uniform elastic edge supports. 26. A. W. L 1973 Journal of Sound and Vibration 31, 257–293. The free vibration of rectangular plates. 27. B. A. H. A 1984 Journal of Sound and Vibration 97, 541–548. Vibrations of Timoshenko beams with elastically restrained ends. 28. C. K. R and S. M 1989 Journal of Sound and Vibration 130, 453–465. A note on vibrations of generally restrained beams. 29. M. J. M, R. E. R and P. M. B 1990 Journal of Sound and Vibration 141, 359–362. Free vibration of uniform Timoshenko beams with ends elastically restrained against rotation and translation. 30. G. B. W and S. L. E 1984 Journal of Sound and Vibration 95, 537–552. Vibrations of rectangular plates with elastically restrained edges.

   

903

APPENDIX I

For v Q 1/q1 q2 the elements dij in equation (11) are given by 2

d11 = am C01 ,

d13 = bm C03 ,

d12 = −KrL C02 ,

d21 = C01 (am CHLM + KrR SHLM),

d22 = C02 (am SHLM + KrR CHLM),

d23 = C03 (bm CBM + KrR SBM), d31 = −KtL ,

d14 = −KrL C04 ,

d24 = C04 (−bm SBM + KrR CBM),

d32 = (am − C02 )/q1 ,

d33 = −KtL ,

d41 = SHLM(am − C01 )/q1 + KtR CHLM,

d34 = (bm − C04 )/q1 ,

d42 = CHLM(am − C02 )/q1 + KtR SHLM,

d43 = −SBM(bm + C03 )/q1 + KtR CBM,

d44 = CBM(bm − C04 )/q1 + KtR SBM.

otherwise, d11 = −am C01 ,

d21 = −C01 (am CLM + KrR SLM),

d22 = C02 (−am SLM + KrR CLM), d41 = −SLM(am − C01 )/q1 + KtR CLM,

d42 = CLM(am − C02 )/q1 + KtR SLM,

all other elements remaining unchanged. Here am = av, SHLM = sinh (am ), CHLM = cosh (am ),

bm = bv,

C0i = C i /Ci ,

i = 1 to 4,

SHBM = sinh (bm ),

SLM = sin (am ),

CHBM = cosh (bm ),

CLM = cos (am ),

SBM = sin (bm ), CBM = cos (bm ).

APPENDIX II

The elements of the various submatrices in equation (17) are given by m11ij = s s i

j

gg

m22kl = q4 s s k

l

m33mn = q4 s s m

$ gg

k11ij = s s q3 i

j

$g

+ St

n

(Wxi Wxj Wyi Wyj ) dj dh,

gg

(cxk cxl Wyk Wyl ) dj dh,

gg

(Wxm Wxn cym cyn ) dj dh,

(W'xi W'xj Wyi Wyj ) dj dh +

% $ g 1

(Wxi Wxj Wyi Wyj ) dh

+

0

k21kj = s s q3 k

j

gg

St AR

q3 AR2

gg

(Wxi Wxj W'yi W'yj ) dj dh

%% 1

(Wxi Wxj Wyi Wyj ) dj

(cxk W'xj Wyk Wyj ) dj dh,

0

,

. .   .

904

$ gg

k22kl = s s q3 k

+

l

(1 − n) 2AR2

(cxk cxl Wyk Wyl ) dj dh +

gg

$ g

m

$ gg

m

l

m

+

n

1 AR2

j

q3 AR

gg

(1 − n) 2AR

(Wxm c'xl c'ym Wyl ) dj dh +

$ gg

(Wxm Wxn cym cyn ) dj dh +

gg

%% 1

(cxk cxl Wyk Wyl ) dh

,

0

(Wxm Wxj cym W'yj ) dj dh,

n AR

k33mn = s s q3

(c'xk c'xl Wyk Wyl ) dj dh

(cxk cxl W'yk W'yl ) dj dh + Sr

k31mj = s s

k32ml = s s

gg

(Wxm Wxn c'ym c'yn ) dj dh +

(1 − n) 2

$ g Sr AR

gg

%

(W'xm cxl cym W'yl ) dj dh ,

gg

(W'xm W'xn cym cyn ) dj dh

%% 1

(Wxm Wxn cym cyn ) dj

.

0

In the above expressions, i, j = 1 to N, k, l = N + 1 to 2N and m, n = 2N + 1 to 3N. (') represents differentiation with respect to j for Wx and cx and differentiation with respect to h for Wy and cy . APPENDIX III: NOMENCLATURE a, b, h A, L, I AR D E, G k Kt , Kr R St , Sr u(j, t) u(j) w(j, h, t) Wxi , Wyi f(j) cx (j, h, t) cy (j, h, t) c x i , c yi r, n t, t v, V

plate dimensions area, length and M.O.I. of the beam aspect ratio of the plate, b/a = Eh 3/12(1 − n 2 ), plate rigidity modulus of elasticity and modulus of rigidity, respectively shear strain distribution correction factor translational and rotational restraints at the edges a parameter representing the general restraints of plate non-dimensional translational and rotational restraints non-dimensional tranvsverse deflection of the beam non-dimensional transverse deflection of the beam non-dimensional transevrse deflection of the plate beam functions for describing the spatial variations of W(j, h) rotation due to bending of the beam rotation due to bending along x-axis of the plate rotation due to bending along y-axis of the plate beam functions for describing the spatial variations of the bending slope of the plate mass density and Poisson ratio, respectively non-dimensional and dimensional time non-dimensional and dimensional natural frequencies