Computers & Sfrucrures Vol. 49, No. 1. pp. 55457, 1993 004s7949/93 %.cQ + 0.00 Printed in Great Britain. 0 1993 Pcrgamon Press Lid

TRANSVERSE VIBRATION OF THICK RECTANGULAR PLATES-III. EFFECTS OF MULTIPLE ECCENTRIC

INTERNAL RING SUPPORTS

K. M. Ltnw,t Y. XIANG$ and S. KITIFQRNCHAI$ TDynamics and Vibration Centre, School of Mechanical and Production Engineering,

Nanyang Technological University, Nanyang Avenue, Singapore 2263 IDepartment of Civil Engineering, The University of Queensland, Brisbane, Australia 4072

(Received 29 July 1992)

Ah&me-Part III of this series of four papers presents the free vibration analysis of thick rectangular plates with multiple internal ring supports of arbitrary orientation. The aim of this paper is to investigate the effects of different numbers of rings and boundary conditions on the vibratory response of the plates. The first order shear deformation theory proposed by Mindlin is employed in the theoretical formulation. Through the process the resulting energy functional is minimized using the Bayleigh-Ritz method with sets of admissible mathematically complete two-dimensional polynomials in the displacement (transverse deflection) and rotation functions. This leads to a governing eigenvalue equation which can be solved to determine the vibration frequencies. Rectangular plates resting on multiple eccentric ring supports with different combinations of boundary conditions have been solved to demonstrate the effectiveness and accuracy of the method. In this paper, sets of reasonably accurate vibration frequencies are presented for various plate aspect ratios a/b and relative thickness ratios t/b. For some cases where established literature exists, comparisons have been made to verify the present solution process.

NOTATION

length of plate width of plate unknown coefficients for transverse deflection w flexural rigidity of plate unknown coefficients for rotation 0, Young’s modulus shear modulus unknown coefficients for rotation 0, stiffness matrix mass matraix degree of complete polynomial for w degree of complete polynomial for 6, degree of complete polynomial for 0, variable in double summation maximum kinetic energy of plate thickness of plate strain energy of plate transverse deflection along z direction longitudinal coordinate vertical coordinate transverse coordinate basic function for w mtb polynomial term for w non-dimensionalized coordinate = y/b basic function for 0, nth polynomial term for e, basic function for 6, Ith polynomial term for e, shear correction factor frequency parameter = (ab2/z 2, m Poisson’s ratio energy functional rotation along y direction rotation along x direction plate density per unit volume power for jth edge function angular frequency non-dimensionalized coordinate = x/a

I. INTRODUCTION

Analysis of the free vibration of rectangular plates is important in many branches of engineering because rectangular plate elements are commonly used as basic design elements in structural applications. The purpose of the analysis is to avoid resonance due to external excitations. Various analytical and numerical methods have been in the computational process. A survey of these methods has been reported in Parts I and II of this series of four papers [ 1,2].

In Part III of the series (this paper) [l, 21, a comprehensive vibration study is made on thick rectangular plates with consideration of the effects of multiple internal ring supports of arbitrary orientation. The lack of research results [3-81 in this area was the prime motivation for this investigation. Plate systems are being more widely applied in various branches of modem technology, and examples include structures in aeroframes, ships and submarines, and offshore. This study aims to provide valuable design information to fill the gaps in existing information.

To overcome the limitations and difficulties of analytical methods caused by the presence of com- plicated boundary conditions and the internal ring supports in the plate domain, the pb-2 Rayleigh-Ritz method [ 1,2,9-171 has been used in this study. For simplicity in numerical modelling, Mindlin’s [18, 191 first order shear deformation theory has been employed to account for the effects of rotatory inertia and transverse shear deformation. The admissible shape functions employed in the study are sets of two- dimensional mathematically complete polynomials.

59

60 K. M. LIEW et al.

As previously indicated [ 1,2], these functions are used in the transverse deflection and bending slopes to simulate the mode shapes of vibration. Using the Rayleigh-Ritz procedure by minimizing the energy function with respect to each coefficient in the assumed functions, a governing eigenvalue equation is derived. It is relatively simple to solve this eigenvalue problem numerically.

In this paper, the theoretical treatment of the various boundary conditions and internal supports is discussed. Numerical calculations have been per- formed for thick rectangular plates with multiple ring supports in various configurations. These results are believed to be the first in the open literature since no existing literature has been found. Because of their practical importance, sets of reasonably com- prehensive vibration frequency parameters have been presented for different combinations of boundary conditions, plate aspect ratios, relative thickness ratios and numbers of ring supports. An investigation into the effects of these factors on the vibratory characteristics has been carried out, and several con- clusions have been drawn. The accuracy of the method has been shown through the convergence tests. Numerical comparisons of different methods for very thin plates (relative thickness ratio t/b = 0.001) have been made. Close agreement has been achieved for all cases thus further confirming the accuracy of the present method.

2. FORMULATION OF GOVERNING EIGENVALUE

EQUATION

Consider a flat, isotropic, thick, rectangular plate (Fig. 1) of uniform thickness t, length u, width b, Young’s modulus E, shear modulus G and Poisson’s ratio v. The plate is internally supported by multiple ring supports which impose a zero deflection con- straint in a transverse direction. The problem is to determine the natural frequencies of the plate.

The strain energy, U and the maximum kinetic energy, T for plate in non-dimensionalized orthogonal coordinates (5, q) are given by [l]

e

Fig. I. Geometry and coordinate systems of rectangular Mindlin plates with multiple internal ring supports.

and

(1)

x & drt (2)

in which D is the flexural rigidity of plate = Et’/ [12(1 - v2)]; w is the transverse deflection; x is the longitudinal coordinate; y is the vertical coordinate; 6, is the rotation along y direction; 0, is the rotation along x direction; < is the non-dimensional&d co- ordinate = x/a; q is the non-dimensional&d co- ordinate = y/b; K is the shear correction factor = S/6; w is the angular frequency; and p is the plate density per unit volume. The total energy functional of the plate can be expressed as

I-I=U-T. (3)

The geometric boundary conditions and the con- ditions for internal ring supports of Mindlin plates can be found in the [16, 171. For Mindlin plates, the transverse deflection and rotations may be parameterized by [l]

where ps, s = 1, 2 and 3, is the degree of the polynomial space; c,,,, d, and e, are the unknown coefficients and

I(ly,= (?tl’-WyI (5c)

in which 4,) *XI and $,, are the basic functions which must satisfy the geometric boundary conditions and the conditions for internal ring supports given in

Transverse vibration of thick mngular plates-III 61

[16,17]. The basic function for the deflection can be expressed as

4, =,!J [r,ceJP~, pr<e4>1, 64

where I’, is the boundary equation of the jth support- ing edge, A, the equation of rth internal ring support and f the number of the internal ring supports, while n,, depending on the support edge condition, takes on

LI, = 0 if the jth edge is free (F) (6b)

a,= 1 if the jth edge is clamped (C) or simply supported (S). W)

The basic functions for the rotations can be expressed as

f?, = 0 if the jth edge is free (F) or simply supported (S) in y-direction; (7b)

Qj = 1 if the jth edge is clamped (C) or simply supported (S) in x-direction. (7c)

+xl = ,Q irj(t, rt )I” @a) ‘_

Ci, = 0 if the jth edge is free (F) or simply supported (S) in x-direction; (8b)

n,= 1 if the jth edge is clamped (C) or simply supported (S) in y-direction. (8~)

Minimizing the total energy functional [eqn (3)] with respect to the unknown coefficients leads to [I]

{c) WI - ~2w1) {d} = (0).

11

(9)

@I The expressions for the various elements of the stiffness matrix [a and mass matrix [M] can bc found in [I].

The frequency parameter, Iz = (ob*/n*) m, is obtained by solving the generalized eigenvalue problem defined by eqn (9).

3. NUMERICAL EXAMPLES AND DIRCUSRION

A computer program based on the formulations described in Sec. 2 has been developed. This program has been used to obtain the eigenvalues for several rectangular plates with internal ring supports. The eigenvalues are expressed in terms of the non-dimen- sional frequency parameter 1 = (wb*/x*) &@ for all cases.

(iii)

Fig. 2. Rectangular plates with internal ring supports: (i) single ring support; (ii) hvo ring supports; (iii) three ring

supports and (iv) four ring supports.

The present method can be used to solve rectangular plates with any combination of internal ring supports. Due to space limitations, however, only the following cases are presented here:

l cases l-3 are square plates with an internal ring support of free, simply supported and fully clamped boundary conditions [Fig. 2(i)], respectively;

l cases 4-6 are free square plates with two, three and four internal ring supports [Figs 2(ii), (iii) and (iv)], respectively;

l cases 7 and 8 are free rectangular plates of aspect ratio a/b = 1.5 with three and four internal ring supports [Figs Z(iii) and (iv)], respectively.

The symbols F, S and C, as shown in the figures, denote free, simply supported and clamped supporting edges.

In numerical calculations, Poisson’s ratio v = 0.30 and shear correction factor K = 516 have been used.

3.1. Convergence and comparison studies

Three example problems have been chosen to demonstrate the convergence of the proposed sol- utions. Two examples have been chosen with single ring support with the other having four ring supports.

The first two examples considered were the simply supported and fully clamped square plates with an internal ring support of diameter 2 = 0.506 [Fig. 2(i), Cases 1 and 21. Different relative thickness ratios t/b = 0.001 and 0.20 have been considered. The convergence patterns for the first eight frequency parameters are given in Table 1. For both cases, a monotonic convergence has been observed as the number of degree sets of polynomials increases. The rate of convergence becomes rather slow as the num- ber of degree sets of polynomials increases from 13 to

62 K. M. Ltnw et al.

Table I. Convergence and comparison studies of frequency parameters, A = (wb*/n’) ,&@, for square Mindlin plates having an internal ring support (a/b = 0.5)

Mode sequences ___--

tlb P, 1 2 3 4 5 6 7 8

Square plates with SSSS boundary conditions

7 10.1844 14.0738 14.0738 15.1007 17.8518 22.3042 38.3570 38.3570 9 10.0775 13.3171 13.3172 13.7890 15.9917 20.1707 24.05 11 24.05 11

11 10.0393 13.0474 13.0474 13.6212 15.7687 19.9505 21.2934 21.2934 0.001 13 10.0269 12.9072 12.9072 13.5761 15.7262 19.8135 20.7719 20.7719

15 10.0240 12.8688 12.8688 13.5139 15.6284 19.7242 20.6977 20.6977 17 10.0236 12.8668 12.8668 13.4716 15.5596 19.7088 20.6518 20.6518 18 10.0236 12.8575 12.8575 13.4716 15.5596 19.7088 20.5990 20.5990 PI 9.747 11.95 11.96 13.00 14.56 - -

7 6.6046 7.6705 7.6705 7.9218 8.5330 10.3290 10.9316 10.9316 9 6.5838 7.5580 7.5580 7.7766 8.2556 10.0995 10.6838 10.6838

11 6.5776 7.5282 7.5282 7.7292 8.1662 9.9956 10.5193 10.5193 0.20 13 6.5753 7.5046 7.5046 7.6682 8.0549 9.9267 10.4399 10.4399

15 6.5736 7.4752 7.4752 7.6290 7.9822 9.9022 10.3974 10.3974 17 6.5714 7.4488 7.4488 7.6198 7.9552 9.8725 10.3772 10.3772 18 6.5699 7.4477 7.4477 7.6122 7.9358 9.8406 10.3731 10.3731

Square plates with CCCC boundary conditions

7 12.0947 20.5377 20.5377 20.9114 25.1632 31.4595 458.7991 458.7991 9 11.9025 19.2183 19.2183 19.8817 2 1.2666 28.0831 30.4255 30.4255

11 11.8727 18.8273 18.8273 19.7544 20.8263 27.4036 27.4036 27.9874 0.001 13 11.8581 18.6717 18.6717 19.6053 20.6840 26.9096 26.9096 27.7522

15 11.8463 18.6218 18.6218 19.4912 20.5244 26.7806 26.7806 27.6482 17 11.8380 18.6217 18.6217 19.4437 20.4630 26.7803 26.7803 27.6437 18 11.8380 18.6134 18.6134 19.4436 20.4630 26.7431 26.7431 27.6433

PI 11.75 17.60 17.60 18.41 18.59 - -

7 6.8826 8.6391 8.6391 8.7471 9.1647 10.8371 11.3408 11.3408 9 6.8594 8.5623 8.5623 8.7042 9.1180 10.7709 11.1100 11.1100

11 6.8455 8.5299 8.5299 8.6641 9.0634 10.6661 11.0561 11.0561 0.20 13 6.8281 8.5215 8.5215 8.5972 8.9584 10.6037 11.0102 11.0102

15 6.8086 8.4989 8.4989 8.5550 8.8887 10.5636 10.9466 10.9644 17 6.8078 8.4748 8.4749 8.5538 8.8866 10.5627 10.9485 10.9485 18 6.8022 8.4730 8.4730 8.5462 8.8724 10.5322 10.9468 10.9468

Note. p, denotes p, , p2 and p,.

18. In general, however, p, = pz = p3 = 18 is sufficient

to furnish an acceptable upperbound convergence. A comparison study has been made with the

numerical results published by Nagaya [7,8], and it can be seen that the results are always higher than those solutions given by Nagaya. This is because the modified Fourier expansion collocation method developed by Nagaya [7,8] provides a lower bound value, as can be seen from the convergence study presented in this paper [7]. The present pb -2 Rayleigh- Ritz method provides an upper bound value. The slight discrepancies are probably due to a lack of convergence in the results presented by Nagaya. This can be seen from the convergence information given in his paper [7j.

The third example considered was a free square plate with four internal ring supports of equal diameter [Fig. 2(iv), Case 61. In this convergence study, rings of diameter a = 0.125b and 0.3756 with relative thickness ratios t/b = 0.001 and 0.20 have been considered. In Table 2, the convergence patterns of the first eight frequency parameters are tabulated with an increas- ing number of degree sets of polynomial terms. It can

be seen that the frequency parameters are significantly enhanced with the increasing number of polynomial sets. The degree sets of polynomials p, = p2 = p3 = 19 are generally required to achieve a reasonably good convergence. The rate of convergence is slightly faster for the plate with ring supports of larger diameter. However, all the numerical values presented here are within an acceptable range for practical purposes. Comparisons cannot be made with other studies because there are no published results in the open literature.

The last example problem considered above is one of the most critical cases for numerical convergence. This is because the required degree sets of polynomials increase as number of the internal ring supports increase, as shown in the convergence studies carried out for the three examples considered above. In this study, results are presented for square plates with single and two ring supports using pI = p2 = p3 = 18 and for rectangular plates with three and four ring supports using p, =p2 =p, = 19. This ensures the accuracy of the present numerical values so that they may be employed for practical purposes.

Transverse vibration of thick rectangular plates-III 63

Table 2. Convergence study of frequency parameters, 1 = (wb2/nr) &@, for square Mindlin plates having four internal ring supports with FFFF boundary conditions

Mode sequences

t/h ” 1 2 3 4 5 6 7 8

7 11.0094 18.2692 9 7.3013 9.6886

11 5.7783 6.2238 0.001 13 5.6328 5.965 1

15 5.4717 5.6180 17 5.439 1 5.5846 18 5.4287 5.5100 19 5.3083 5.3786

7 9

3.6018 3.6018 3.3817 3.3817 3.1858 3.2168 3.1476 3.1476 3.0111 3.0488 3.0055 3.0429 2.9205 3.0185 2.9164 3.0177

11 0.20 13

15 17 18 19

7 23.6672 29.6042 29.6042 32.2294 65.9843 72.7658 307.4878 307.4878 9 14.8666 19.1914 19.1914 19.3449 36.7412 39.5971 62.7433 62.7433

I1 12.8002 13.9655 13.9655 15.0556 22.0606 22.4763 23.3152 23.3152 0.001 13 12.6377 13.5259 13.5259 14.3829 19.0812 20.5854 20.8539 20.8539

15 12.2447 13.1501 13.1501 13.8902 17.6149 18.1482 18.1482 18.9515 17 12.1755 12.9820 12.9820 13.7932 17.5204 18.0019 18.0019 18.2650 18 12.1707 12.9552 12.9552 13.7130 17.5011 17.7986 17.7986 18.2489 19 12.1496 12.9344 12.9344 13.7015 17.4723 17.7946 17.7946 17.9880

7 7.3905 7.4800 7.4809 7.5364 9.0663 9.4226 9.4846 9.4846 9 7.1609 7.3011 7.3011 7.4120 8.5119 8.6927 8.6927 8.9327

11 7.0907 7.1485 7.1485 7.1975 8.1820 8.3513 8.3513 8.4011 0.20 13 7.087 1 7.1376 7.1376 7.1920 8.1679 8.2436 8.2436 8.3726

15 7.0557 7.1037 7.1037 7.1425 8.0657 8.1660 8.1660 8.2267 17 7.0549 7.0959 7.0959 7.1408 8.0636 8.1234 8.1234 8.2196 18 7.0499 7.0950 7.0950 7.1306 8.0171 8.1207 8.1207 8.2115 19 7.0482 7.0900 7.0900 7.1282 8.0143 8.0793 8.0793 8.2089

Square plates with a/!~ = 0.125

18.2692 21.6210 35.7275 9.6886 10.6062 16.7626 6.2238 7.3190 9.5759 5.9651 6.4801 7.5514 5.6180 6.0938 6.8900 5.5846 5.8874 6.6507 5.5100 5.8633 6.6141 5.3786 5.7769 6.4345

3.6824 4.1243 4.6339 3.4538 3.5715 3.8436 3.2168 3.2950 3.3134 3.1820 3.2712 3.3057 3.1088 3.1088 3.1636 3.0642 3.0642 3.1606 3.0598 3.0598 3.1316 3.0473 3.0473 3.1302

Square plates with a/b = 0.375

43.7113 94.6969 94.6969 18.9302 36.2482 36.2482 10.3150 10.6867 10.6867 9.6394 10.2212 10.2212 7.6715 7.6715 8.0847 7.5602 7.5602 7.8148 7.3018 7.3018 7.7794 7.2208 7.2208 7.7105

5.3195 5.3195 5.4272 4.3671 4.367 1 4.5334 4.eOO4 4.0004 4.3930 3.8676 3.8676 4.3624 3.7106 3.7106 4.2798 3.6343 3.6343 4.2792 3.6286 3.6286 4.2453 3.5843 3.5843 4.2435

Note. p, denotes pIT p2 and ~3.

3.2. Numerical results are presented in Tables 3-8 with an increasing mode Numerical calculations have been carried out for sequence.

rectangular plates continuous over one, two, three In Tables 3-5, approximate frequency parameters and four internal ring supports. The first eight are presented for square plates continuous over a frequency parameters obtained from the calculation single ring support with free, simply supported and

Table 3. Frequency parameters, 1 = (ob2/x2) m, for square Mindlin plates having an internal ring support with FFFF boundary conditions

Mode sequences

;ilb tlb 1 2 3 4 5 6 7 8

0.001 1.3065 1.3232 1.3232 1.6105 2.4551 3.9290 3.9290 5.3349 0.25 0.10

0.20

0.001 0.50 0.10

0.20

0.001 0.75 0.10

0.20

0.001 1.00 0.10

0.20

1.1502 0.8690

1.7596 1.7010 1.5644

2.4963 2.3871 2.1563

1.1502 0.8690

2.1489 1.9602 1.6177

3.7315 3.3635 2.7587

2.0033 5.3334 1.9299 4.8724 1.7682 4.0558

1.2618 1.1529

2.1489 1.9602 1.6177

3.7315 3.3635 2.7587

5.3334 4.8724 4.0058

1.4205 1.2426

2.3393 2.0454 1.6571

4.0653 3.5611 2.8110

7.9761 6.8866 5.3223

2.1643 1.8663

4.1059 3.5067 2.6914

6.1728 5.3269 4.1617

9.1142 7.6670 5.7313

3.5072 3.5072 4.6928 2.9456 2.9456 3.7990

5.0509 4.3335 3.4388

9.1680 7.5957 5.3971

5.0509 4.3335 3.4388

9.2461 7.6458 5.5741

6.4786 5.5794 4.4758

9.2461 7.6458 5.5741

10.3828 10.3828 10.5948 8.4316 8.4316 9.0149 6.0999 6.0999 6.8423

64 K. M. L~EW ef nl.

Table 4. Frequency parameters, I = (ob2/nZ) m, for square Mindlin plates having an internal ring support with SSSS boundarv conditions

Mode sequences

a/b r/b 1 2 3 4 5 6 7 a

0.001 7.1339 8.1276 8.1276 9.2837 12.0386 14.6027 14.6027 17.4622 0.25 0.10 6.1740

0.20 4.7204

0.001 10.0236 0.50 0.10 8.6068

0.20 6.5699

0.001 6.0445 0.75 0.10 5.2OQ4

0.20 3.9770

0.001 3.9363 1.00 0.10 3.0322

0.20 2.3609

6.763 1 6.763 1 7.7075 9.6043 1 I.6175 5.0740 5.0740 5.8766 7.0552 8.3826

12.8575 12.8575 13.4716 15.5596 19.7088 10.3808 10.3808 10.6919 11.8219 14.5969 7.4477 7.4477 7.6122 7.9358 9.8406

12.8307 12.8307 20.1314 21.8817 22.5502 10.3036 10.3036 15.4326 16.0341 16.8415 7.3406 7.3406 10.6011 10.6945 I 1.4080

8.2042 8.2042 13.1352 13.8809 15.3637 6.2454 6.2454 9.7392 10.1408 II.1629 4.7125 4.7125 7.0790 7.2227 7.9065

clamped boundary conditions. In each table, the listed frequency parameters in Tables 3-5 is necessary. It numerical values provide important information to can be seen that the square plate with fully clamped study the effect of ring dimension on the vibration boundaries (Table 5) always gives the highest frequency parameters. The data listed in Table 3 frequency response when compared to the values for show that for the free square plate with a ring support square plates with free (Table 3) and simply supported of diameter a/6 = 0.25, 0.50,0.75, 1.0, the m~imum (Table 4) boundaries, keeping d/b and t/b constant. fundamental frequency response occurs with a ring It may be concluded that assigning a higher constraint support of d/b z 0.75 for all relative thickness ratios. at the supporting edge will increase the flexural rigidity However the maximum values for the higher modes of the structure. This leads to a higher frequency occur at d/b z 1.0. response.

In Tables 4 and 5, the frequency parameters do not show the same trend for the simply supported and fully clamped square plates. For both cases, the numerical data show that the maximum values for the first three modes occur with a ring support with d/b FT 0.50 regardless of relative thickness ratio. From the fourth and higher modes, however, all of the maxima occur with a ring support with a/b x 0.75. The only exception is that for the sixth mode of the fully clamped square plate, the maximum occurred at d/b a 0.50 for relative thickness ratio t/b = 0.001.

To investigate the effect of boundary constraint on the vibration response, a comparison of the numerical

A decrease in frequency parameters with an increase in relative thickness t/b can be observed from Tables 3-5 if the d/b ratio and boundary conditions are kept constant. This is due to the effects of transverse shear deformation and rotatory inertia.

The first eight frequency parameters for free rectangular plates with two, three and four internal ring supports have been determined and they are listed in Tables 6-10. These plate systems are com- monly used in offshore structures, and these data may therefore be used to those engineers dealing with off- shore structures. The detailed geometry and dimen- sions of these plate systems are shown in Figs 2(ii), (iii) and (iv) (Cases 4-6).

Il.6175 8.3826

20.5990 15.3036 10.3731

24.97 11 17.9230 11.7702

19.7433 13.9747 9.6113

13.6630 9.5921

20.5990 15.3036 10.3731

24.97 1 I 17.9230 11.7702

19.7433 13.9747 9.6113

Table 5. Frequency parameters, 1 = (wb2/n2) m, for square Mindlin plates having an internal ring support with CCCC boundary conditions

Mode sequences

dlb rib 1 2 3 4 5 6 7 a

0.001 0.25 0.10

0.20

0.001 0.50 0.10

0.20

0.001 0.75 0.10

0.20

0.001 1.00 0.10

0.20

10.8006 11.9565 1 I .9565 a.4604 8.9832 8.9832 5.8267 6.0370 6.0370

11.8380 9.5063 6.8022

18.6134 13.1843 8.4730

6.2836 5.2370 4.0129

18.6134 13.1843 8.4730

13.2995 10.3624 7.3598

8.3138 6.7965 5.1063

13.2995 10.3624 7.3598

3.9923 3.4648 2.8327

8.3138 6.7965 5.1063

12.9061 16.4243 f8.8567 9.6735 11.6477 13.4564 6.6415 7.6794 8.8989

19.4436 20.4630 26.7431 13.4400 14.0315 17.4927 8.5462 8.8724 10.5322

21.6738 22.45 I 1 24.9142 15.8391 16.0513 17.6300 10.6416 10.7087 11.6073

13.3396 14.0052 15.5615 10.3144 10.7035 11.7734 7.3826 7.4803 8.2088

19.8567 22.0745 13.4564 15.4875 8.8989 10.0913

26.7431 27.6433 17.7162 17.7162 10.9468 10.9468

31.8184 31.8184 20.9201 20.9201 12.6594 12.6594

19.9763 19.9763 14.4961 14.4961 9.7996 9.79%

Transverse vibration of thick rectangular plates-111 65

Table 6. Frequency parameters, 1 = (&/n~ &@, for square Mindlin plates having two internal rings support with FFFF boundary conditions

Mode sequences

d/b t/b 1 2 3 4 5 6 7 8

0.001 0.125 0.10

0.20

0.001 0.25 0.10

0.20

0.001 0.375 0.10

0.20

1.2011 1.3183 0.9754 1.1907 0.7028 1.0379

1.4208 1.4406 1.2494 1.3211 0.9998 1.1722

3.8720 3.9061 4.5812 3.0996 3.4256 3.7464 2.3428 2.7465 2.7815

4.473 1 4.6168 5.4776 3.9183 3.9583 4.7565 3.0522 3.1621 3.6742

5.2669 4.2842 3.2436

5.9825 5.0550 3.9316

1.6453 1.7162 5.3583 5.5893 6.695 1 7.1285 1.5102 1.5410 4.6755 4.7919 5.7870 6.0008 1.2712 1.3281 3.6998 3.7062 4.5280 4.5956

0.001 1.9651 2.0815 6.4685 6.8694 7.7404 8.7166 0.50 0.10 1.7889 1.8784 5.5818 5.8292 6.7122 7.3003

0.20 1.5418 1.5671 4.3486 4.4310 5.2856 5.4814

5.5822 4.3211 3.2528

8.1012 6.4624 4.6224

11.3014 8.8441 6.2687

10.7117 8.8145 6.4592

5.7127 4.3958 3.3043

8.2943 6.5396 4.6784

11.7710 9.2277 6.6670

11.0934 9.0320 6.5645

Table 7. Frequency parameters, 1 = (wb2/n2) m, for square Mindlin plates having three internal ring supports with FFFF boundary conditions

J/b

0.125

0.25

0.375

0.50

rib 1 2 3

0.001 1.9939 2.0429 4.6069 0.10 1.5860 1.7872 3.6247 0.20 1.1997 1.4959 2.6578

0.001 2.3741 2.4575 5.7051 0.10 2.0948 2.1088 4.7539 0.20 1.6335 1.7824 3.6051

0.001 2.9875 3.1629 7.2572 0.10 2.6341 2.7453 5.9706 0.20 2.1648 2.1781 4.4965

0.001 3.9822 4.2745 9.3715 0.10 3.4572 3.6747 7.5341 0.20 2.7149 2.8708 5.5389

Mode sequences

4 5

4.8532 5.6822 3.9432 4.2109 3.0057 3.0520

5.7916 8.0569 4.7824 6.4733 3.6882 4.6449

7.3208 12.3078 5.9923 9.9586 4.5065 7.0712

9.5801 11.2685 7.5740 9.1061 5.5553 6.5837

6 7 8

5.7904 7.2750 7.7004 4.3427 5.0044 6.0402 3.2311 3.6022 4.4283

8.3371 9.8706 10.2753 6.5188 7.5562 7.9395 4.6469 5.1709 5.7313

13.0976 13.3422 14.5764 10.2467 10.5156 11.0000 7.1144 7.3939 7.5721

12.1390 12.2762 19.1347 9.3221 9.6171 14.1144 6.6550 6.9016 9.3963

Tables 6-8 present the data for square plates listed in these tables reveals that an increase in the (a/b = 1.0) while Tables 9 and 10 show the data for number of ring supports leads to a higher frequency rectangular plates (u/b = 1.5). These sets of frequency response when ii’/b and t/b ratios are kept constant. parameters provide useful information for studying This is due to the increase in the flexural rigidity of the the effect of ring supports on the vibratory character- plate systems as the number of internal ring supports istics of these plate systems. A comparison of the data increases.

Table 8. Frequency parameters, 1 = (ob2/n2) m, for square Mindlin plates having four internal ring supports with FFFF boundary conditions

Mode sequences

alb tlb 1 2 3 4 5 6 7 8

0.001 5.4083 0.125 0.10 4.1837

0.20 2.9164

0.001 7.8175 0.25 0.10 6.4372

0.20 4.6331

0.001 12.1496 0.375 0.10 9.8793

0.20 7.0482

0.001 10.8152 0.50 0.10 9.0262

0.20 6.5711

5.3786 5.3786 4.2285 4.2285 3.0177 3.0473

8.0878 8.0878 6.4685 6.4685 4.6331 4.6444

12.9344 12.9344 10.1665 10.1665 7.0900 7.0900

12.2946 12.2946 9.1881 9.1881 6.5194 6.5794

5.7169 4.3339 3.0473

8.3526 6.5211 4.6509

13.7015 10.3855 7.1282

13.4897 9.5555 6.7756

6.4345 7.2208 7.2208 7.7105 4.3507 4.9863 4.9863 5.7921 3.1302 3.5843 3.5843 4.2435

10.0911 10.3642 10.3642 7.4720 7.5429 7.5429 4.9120 5.1678 5.1678

17.4723 17.7946 17.7946 12.1965 12.4426 12.4426 8.0143 8.0793 8.0793

21.7439 25.6120 26.4218 18.1723 18.2440 18.3638 11.8257 11.8313 11.8514

10.4470 7.6292 5.5044

17.9880 12.8518 8.2089

26.4218 18.3638 11.8514

K. M. LIEW et of. 66

Table 9. Frequency parameters, 1 = (wb’/n*) &@, for rectangular Mindlin plates having three internal ring supports with FFFF boundary conditions (a/b = 1.5)

Mode sequences

d/b tlb I 2 3 4 5 6 7 8

0.001 0.125 0.10

0.20

0.001 0.25 0.10

0.20

0.001 0.375 0.10

0.20

0.001 0.50 0.10

0.20

0.5814 0.5851 1.1587 1.2018 0.4992 0.5010 0.9598 0.9649 0.4177 0.4223 0.7989 0.8089

0.6700 0.6705 0.605 1 0.6051 0.5252 0.5258

1.3920 1.1949 1.0051

1.8788 1.6512 1.3785

1.3982 1.1953 1.0051

0.8055 0.8063 0.7461 0.747 1 0.6593 0.6603

1.8805 2.5255 2.5459 3.7276 1.6513 2.2365 2.2389 2.8585 1.3193 1.9047 1.9095 2.3048

0.9774 0.9783 2.5734 0.9174 0.9174 2.3054 0.8185 0.8187 I .9320

2.5171 2.3066 1 a9329

1.8976 19023 1.5799 I .6657 1.2691 1.4331

2.1074 2.1115 1.8661 1.8697 1.5891 1.6146

1.9442 1.6838 1.4622

2.4510 2.0307 1.6908

3.5768 3.5787 4.8699 3.1361 3.1386 4.1541 2.5485 2.5514 3.3236

3.1138 2.7127 2.2907

3.3827 2.9814 2.5130

3.8941 3.3983 2.8111

4.8772 4.1584 3.3275

Table 10. Frequency parameters, i: = (wb2/n2) m, for rectangular Mindlin plates having four internal ring supports with FFFF boundary conditions (u/6 = 1.5)

Mode sequences

d/b tib 1 2 3 4 5 6 7 8

0.001 0.125 0.10

0.20

0.001 0.25 0.10

0.20

0.001 0.375 0.10

0.20

0.001 0.50 0.10

0.20

1.0181 0.7950 0.6554

1.2888 1.0686 0.8874

1.9183 1.6443 1.3471

3.4310 2.9371 2.3162

1.0476 1.6009 0.8058 1.3085 0.6658 1.0601

1.3189 I.8936 1.0702 1.6166 0.8908 1.3520

I.9371 2.5256 1.6463 2.2113 I .3495 1.8128

3.4375 4.1550 2.9399 3.4775 2.3199 2.7336

4. CONCLUSIONS

This paper considers the free vibration analysis of thick rectangular plates with muitiple eccentric internal ring supports. Several example problems have been analysed. The first eight frequency parameters for these plate problems have been obtained using the pb-2 Rayleigh-Ritz method. Mindlin plate theory has been used to consider the effects of shear defo~a~on and rotatory inertia.

Based on the discussion on the formulation and from the numerical data presented in this paper, the following conclusions can be drawn:

a faster rate of convergence can be achieved with fewer internal ring supports. The rate slows down as the number of internal ring supports increases. the effect of ring dimension on the vibration re- sponse depends on the boundary conditions. higher flexural rigidity will be obtained by assign- ing greater constraint at the supporting edges. higher frequency response will result from an increase in the number of internal ring supports. a decrease in frequency response occurs with in- creasing relative thickness ratio t/b. This is due to the effects of shear deformation and rotatory inertia.

1.6231 1.3142 1.0997

1.9321 1.6173 1.3547

2.6141 2.2114 1.8158

4.1620 3.4824 2.7378

1.6662 1.3232 1.1093

2.0591 1.7176 1.4019

3.8732 2.5516 2.0690

5.1448 4.5534 3.4639

2.0138 1.5444 1.2543

2.4244 1.9661 1.5976

3.2929 3.3236 2.8040 2.8056 2.3441 2.3544

3.645s 3.7214 3.1231 3.1268 2.5989 2.6039

3.3508 4.4398 4.5081 2.8053 3.7099 3.7112 2.2426 2.9977 3.0004

6.1895 6.1960 6.6049 4.8074 4.9436 4.9494 3.6183 3.7857 3.7912

Acknowledgements-This research was supported by the Australian Research Council (ARC) under Project Grant No. ARC 834. The authors wish to thank Mr Warren H. Traves of Gutteridge Haskins and Davey Pty Ltd for proof- reading the manuscript. The first author appreciates the assistance provided by the Department of Civil Engineering, The University of Queensland during his research stays at that institution.

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