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Transverse vibration of thick rectangular plates—III. Effects of multiple eccentric internal ring supports
Computers & Sfrucrures Vol. 49, No. 1. pp. 55457, 1993 Printed in Great Britain.
TRANSVERSE PLATES-III.
0
004s7949/93 %.cQ + 0.00 1993 Pcrgamon Press Lid
VIBRATION OF THICK RECTANGULAR 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.
I. INTRODUCTION
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
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 complicated 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 twodimensional 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 performed 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 comprehensive 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 conclusions 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.
(1) and
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 coordinate = x/a; q is the non-dimensional&d coordinate = 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.
2. FORMULATION
OF GOVERNING EQUATION
EIGENVALUE
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 constraint 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]
(3)
The geometric boundary conditions and the conditions 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)
e
Fig. I. Geometry and coordinate systems of rectangular Mindlin plates with multiple internal ring supports.
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~, pr1, 64
where I’, is the boundary equation of the jth supporting 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) a,=
1 if the jth edge is clamped (C) or simply supported (S).
(6b)
W)
The basic functions for the rotations can be expressed as
(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.
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)
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
+xl = ,Q
‘_
irj(t, rt)I”
@a) l
Ci,= 0 if the jth edge is free (F) or simply supported (S) in x-direction;
(8b) l
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).
(9)
11@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-dimensional frequency parameter 1 = (wb*/x*) &@ for all cases.
cases l-3 are square plates with an internal ring support of free, simply supported and fully clamped boundary conditions [Fig. 2(i)], respectively; cases 4-6 are free square plates with two, three and four internal ring supports [Figs 2(ii), (iii) and (iv)], respectively; 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 solutions. 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 number 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’) ,&@, having an internal ring support (a/b = 0.5)
for square Mindlin plates
Mode sequences
___-tlb
P,
1
2
0.001
7 9 11 13 15 17 18
PI
10.1844 10.0775 10.0393 10.0269 10.0240 10.0236 10.0236 9.747
14.0738 13.3171 13.0474 12.9072 12.8688 12.8668 12.8575 11.95
14.0738 13.3172 13.0474 12.9072 12.8688 12.8668 12.8575 11.96
15.1007 13.7890 13.6212 13.5761 13.5139 13.4716 13.4716 13.00
0.20
7 9 11 13 15 17 18
6.6046 6.5838 6.5776 6.5753 6.5736 6.5714 6.5699
7.6705 7.5580 7.5282 7.5046 7.4752 7.4488 7.4477
7.6705 7.5580 7.5282 7.5046 7.4752 7.4488 7.4477
7.9218 7.7766 7.7292 7.6682 7.6290 7.6198 7.6122
12.0947 11.9025
0.001
7 9 11 13 15 17 18
PI
11.8581 11.8463 11.8380 11.8380 11.75
20.5377 19.2183 18.8273 18.6717 18.6218 18.6217 18.6134 17.60
20.5377 19.2183 18.8273 18.6717 18.6218 18.6217 18.6134 17.60
20.9114 19.8817 19.7544 19.6053 19.4912 19.4437 19.4436 18.41
7 9 11 13 15 17 18
6.8826 6.8594 6.8455 6.8281 6.8086 6.8078 6.8022
8.6391 8.5623 8.5299 8.5215 8.4989 8.4748 8.4730
8.6391 8.5623 8.5299 8.5215 8.4989 8.4749 8.4730
8.7471 8.7042 8.6641 8.5972 8.5550 8.5538 8.5462
3
4
5
6
7
8
17.8518 15.9917 15.7687 15.7262 15.6284 15.5596 15.5596 14.56
22.3042 20.1707 19.9505 19.8135 19.7242 19.7088 19.7088 -
38.3570 24.05 11 21.2934 20.7719 20.6977 20.6518 20.5990 -
38.3570 24.05 11 21.2934 20.7719 20.6977 20.6518 20.5990
8.5330 8.2556 8.1662 8.0549 7.9822 7.9552 7.9358
10.3290 10.0995 9.9956 9.9267 9.9022 9.8725 9.8406
10.9316 10.6838 10.5193 10.4399 10.3974 10.3772 10.3731
10.9316 10.6838 10.5193 10.4399 10.3974 10.3772 10.3731
25.1632 2 1.2666 20.8263 20.6840 20.5244 20.4630 20.4630 18.59
31.4595 28.0831 27.4036 26.9096 26.7806 26.7803 26.7431 -
458.7991 30.4255 27.4036 26.9096 26.7806 26.7803 26.7431
458.7991 30.4255 27.9874 27.7522 27.6482 27.6437 27.6433 -
9.1647 9.1180 9.0634 8.9584 8.8887 8.8866 8.8724
10.8371 10.7709 10.6661 10.6037 10.5636 10.5627 10.5322
11.3408 11.1100 11.0561 11.0102 10.9466 10.9485 10.9468
11.3408 11.1100 11.0561 11.0102 10.9644 10.9485 10.9468
Square plates with SSSS boundary conditions
Square plates with CCCC boundary conditions
0.20
11.8727
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 RayleighRitz 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 = 0.001and 0.20 have been considered. In Table 2, the convergence patterns of the first eight frequency parameters are tabulated with an increasing 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.
63
Transverse vibration of thick rectangular plates-III
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
Square plates with a/!~ = 0.125 11.0094 7.3013 5.7783 5.6328 5.4717 5.439 1 5.4287 5.3083
18.2692 9.6886 6.2238 5.965 1 5.6180 5.5846 5.5100 5.3786
18.2692 9.6886 6.2238 5.9651 5.6180 5.5846 5.5100 5.3786
21.6210 10.6062 7.3190 6.4801 6.0938 5.8874 5.8633 5.7769
35.7275 16.7626 9.5759 7.5514 6.8900 6.6507 6.6141 6.4345
43.7113 18.9302 10.3150 9.6394 7.6715 7.5602 7.3018 7.2208
94.6969 36.2482 10.6867 10.2212 7.6715 7.5602 7.3018 7.2208
94.6969 36.2482 10.6867 10.2212 8.0847 7.8148 7.7794 7.7105
17 18 19
3.6018 3.3817 3.1858 3.1476 3.0111 3.0055 2.9205 2.9164
3.6018 3.3817 3.2168 3.1476 3.0488 3.0429 3.0185 3.0177
3.6824 3.4538 3.2168 3.1820 3.1088 3.0642 3.0598 3.0473
4.1243 3.5715 3.2950 3.2712 3.1088 3.0642 3.0598 3.0473
4.6339 3.8436 3.3134 3.3057 3.1636 3.1606 3.1316 3.1302
5.3195 4.3671 4.eOO4 3.8676 3.7106 3.6343 3.6286 3.5843
5.3195 4.367 1 4.0004 3.8676 3.7106 3.6343 3.6286 3.5843
5.4272 4.5334 4.3930 4.3624 4.2798 4.2792 4.2453 4.2435
7 9 I1 13 15 17 18 19
23.6672 14.8666 12.8002 12.6377 12.2447 12.1755 12.1707 12.1496
29.6042 19.1914 13.9655 13.5259 13.1501 12.9820 12.9552 12.9344
29.6042 19.1914 13.9655 13.5259 13.1501 12.9820 12.9552 12.9344
32.2294 19.3449 15.0556 14.3829 13.8902 13.7932 13.7130 13.7015
65.9843 36.7412 22.0606 19.0812 17.6149 17.5204 17.5011 17.4723
72.7658 39.5971 22.4763 20.5854 18.1482 18.0019 17.7986 17.7946
307.4878 62.7433 23.3152 20.8539 18.1482 18.0019 17.7986 17.7946
307.4878 62.7433 23.3152 20.8539 18.9515 18.2650 18.2489 17.9880
7 9 11 13 15 17 18 19
7.3905 7.1609 7.0907 7.087 1 7.0557 7.0549 7.0499 7.0482
7.4800 7.3011 7.1485 7.1376 7.1037 7.0959 7.0950 7.0900
7.4809 7.3011 7.1485 7.1376 7.1037 7.0959 7.0950 7.0900
7.5364 7.4120 7.1975 7.1920 7.1425 7.1408 7.1306 7.1282
9.0663 8.5119 8.1820 8.1679 8.0657 8.0636 8.0171 8.0143
9.4226 8.6927 8.3513 8.2436 8.1660 8.1234 8.1207 8.0793
9.4846 8.6927 8.3513 8.2436 8.1660 8.1234 8.1207 8.0793
9.4846 8.9327 8.4011 8.3726 8.2267 8.2196 8.2115 8.2089
7
0.001
0.20
9 11 13 15 17 18 19 7 9 11 13 15
Square plates with a/b = 0.375
0.001
0.20
Note. p, denotes pIT p2 and ~3. 3.2. Numerical results Numerical
calculations
have been carried
out for
rectangular plates continuous over one, two, three and four internal ring supports. The first eight frequency parameters obtained from the calculation Table 3. Frequency parameters, 1 = (ob2/x2) m,
are presented in Tables 3-8 with an increasing mode sequence. In Tables 3-5, approximate frequency parameters are presented for square plates continuous over a single ring support with free, simply supported and
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
0.25
0.001 0.10 0.20
1.3065 1.1502 0.8690
1.3232 1.1502 0.8690
1.3232 1.2618 1.1529
1.6105 1.4205 1.2426
2.4551 2.1643 1.8663
3.9290 3.5072 2.9456
3.9290 3.5072 2.9456
5.3349 4.6928 3.7990
0.50
0.001 0.10 0.20
1.7596 1.7010 1.5644
2.1489 1.9602 1.6177
2.1489 1.9602 1.6177
2.3393 2.0454 1.6571
4.1059 3.5067 2.6914
5.0509 4.3335 3.4388
5.0509 4.3335 3.4388
6.4786 5.5794 4.4758
0.75
0.001 0.10 0.20
2.4963 2.3871 2.1563
3.7315 3.3635 2.7587
3.7315 3.3635 2.7587
4.0653 3.5611 2.8110
6.1728 5.3269 4.1617
9.1680 7.5957 5.3971
9.2461 7.6458 5.5741
9.2461 7.6458 5.5741
1.00
0.001 0.10 0.20
2.0033 1.9299 1.7682
5.3334 4.8724 4.0558
5.3334 4.8724 4.0058
7.9761 6.8866 5.3223
9.1142 7.6670 5.7313
10.3828 8.4316 6.0999
10.3828 8.4316 6.0999
10.5948 9.0149 6.8423
64
K. M. L~EWef 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
0.25
0.001 0.10 0.20
7.1339 6.1740 4.7204
8.1276 6.763 1 5.0740
8.1276 6.763 1 5.0740
9.2837 7.7075 5.8766
12.0386 9.6043 7.0552
14.6027 1I.6175 8.3826
14.6027 Il.6175 8.3826
17.4622 13.6630 9.5921
0.50
0.001 0.10 0.20
10.0236 8.6068 6.5699
12.8575 10.3808 7.4477
12.8575 10.3808 7.4477
13.4716 10.6919 7.6122
15.5596 11.8219 7.9358
19.7088 14.5969 9.8406
20.5990 15.3036 10.3731
20.5990 15.3036 10.3731
0.001 0.10 0.20
6.0445 5.2OQ4 3.9770
12.8307 10.3036 7.3406
12.8307 10.3036 7.3406
20.1314 15.4326 10.6011
21.8817 16.0341 10.6945
22.5502 16.8415 I 1.4080
24.97 11
0.75
11.7702
24.97 1I 17.9230 11.7702
1.00
0.001 0.10 0.20
3.9363 3.0322 2.3609
8.2042 6.2454 4.7125
8.2042 6.2454 4.7125
13.1352 9.7392 7.0790
13.8809 10.1408 7.2227
15.3637 II.1629 7.9065
19.7433 13.9747 9.6113
19.7433 13.9747 9.6113
clamped boundary conditions. In each table, the listed numerical values provide important information to study the effect of ring dimension on the vibration frequency parameters. The data listed in Table 3 show that for the free square plate with a ring support of diameter a/6 = 0.25, 0.50,0.75, 1.0, the m~imum fundamental frequency response occurs with a ring support of d/b z 0.75 for all relative thickness ratios. However the maximum values for the higher modes occur at d/b z 1.0. 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 FT0.50 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 Table 5. Frequency parameters, 1 = (wb2/n2)m,
17.9230
frequency parameters in Tables 3-5 is necessary. It can be seen that the square plate with fully clamped boundaries (Table 5) always gives the highest frequency response when compared to the values for square plates with free (Table 3) and simply supported (Table 4) boundaries, keeping d/b and t/b constant. It may be concluded that assigning a higher constraint at the supporting edge will increase the flexural rigidity of the structure. This leads to a higher frequency response. 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 commonly used in offshore structures, and these data may therefore be used to those engineers dealing with offshore structures. The detailed geometry and dimensions of these plate systems are shown in Figs 2(ii), (iii) and (iv) (Cases 4-6).
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
10.8006
0.10 0.20
a.4604 5.8267
11.9565 8.9832 6.0370
1I .9565 8.9832 6.0370
12.9061 9.6735 6.6415
16.4243 11.6477 7.6794
f8.8567 13.4564 8.8989
19.8567 13.4564 8.8989
22.0745 15.4875 10.0913
0.50
0.001 0.10 0.20
11.8380 9.5063 6.8022
18.6134 13.1843 8.4730
18.6134 13.1843 8.4730
19.4436 13.4400 8.5462
20.4630 14.0315 8.8724
26.7431 17.4927 10.5322
26.7431 17.7162 10.9468
27.6433 17.7162 10.9468
0.75
0.001 0.10 0.20
6.2836 5.2370 4.0129
13.2995 10.3624 7.3598
13.2995 10.3624 7.3598
21.6738 15.8391 10.6416
22.45 I 1 16.0513 10.7087
24.9142 17.6300 11.6073
31.8184 20.9201 12.6594
31.8184 20.9201 12.6594
1.00
0.001 0.10 0.20
3.9923 3.4648 2.8327
8.3138 6.7965 5.1063
8.3138 6.7965 5.1063
13.3396 10.3144 7.3826
14.0052 10.7035 7.4803
15.5615 11.7734 8.2088
19.9763 14.4961 9.7996
19.9763 14.4961 9.79%
0.25
Transverse vibration of thick rectangular plates-111 Table 6. Frequency parameters, 1 = (&/n~
65
&@, 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
0.125
0.001 0.10 0.20
1.2011 0.9754 0.7028
1.3183 1.1907 1.0379
3.8720 3.0996 2.3428
3.9061 3.4256 2.7465
4.5812 3.7464 2.7815
5.2669 4.2842 3.2436
5.5822 4.3211 3.2528
5.7127 4.3958 3.3043
0.25
0.001 0.10 0.20
1.4208 1.2494 0.9998
1.4406 1.3211 1.1722
4.473 1 3.9183 3.0522
4.6168 3.9583 3.1621
5.4776 4.7565 3.6742
5.9825 5.0550 3.9316
8.1012 6.4624 4.6224
8.2943 6.5396 4.6784
0.375
0.001 0.10 0.20
1.6453 1.5102 1.2712
1.7162 1.5410 1.3281
5.3583 4.6755 3.6998
5.5893 4.7919 3.7062
6.695 1 5.7870 4.5280
7.1285 6.0008 4.5956
11.3014 8.8441 6.2687
11.7710 9.2277 6.6670
0.50
0.001 0.10 0.20
1.9651 1.7889 1.5418
2.0815 1.8784 1.5671
6.4685 5.5818 4.3486
6.8694 5.8292 4.4310
7.7404 6.7122 5.2856
8.7166 7.3003 5.4814
10.7117 8.8145 6.4592
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 Mode sequences rib
1
2
3
4
5
6
7
8
0.125
0.001 0.10 0.20
1.9939 1.5860 1.1997
2.0429 1.7872 1.4959
4.6069 3.6247 2.6578
4.8532 3.9432 3.0057
5.6822 4.2109 3.0520
5.7904 4.3427 3.2311
7.2750 5.0044 3.6022
7.7004 6.0402 4.4283
0.25
0.001 0.10 0.20
2.3741 2.0948 1.6335
2.4575 2.1088 1.7824
5.7051 4.7539 3.6051
5.7916 4.7824 3.6882
8.0569 6.4733 4.6449
8.3371 6.5188 4.6469
9.8706 7.5562 5.1709
10.2753 7.9395 5.7313
0.375
0.001 0.10 0.20
2.9875 2.6341 2.1648
3.1629 2.7453 2.1781
7.2572 5.9706 4.4965
7.3208 5.9923 4.5065
12.3078 9.9586 7.0712
13.0976 10.2467 7.1144
13.3422 10.5156 7.3939
14.5764 11.0000 7.5721
0.50
0.001 0.10 0.20
3.9822 3.4572 2.7149
4.2745 3.6747 2.8708
9.3715 7.5341 5.5389
9.5801 7.5740 5.5553
11.2685 9.1061 6.5837
12.1390 9.3221 6.6550
12.2762 9.6171 6.9016
19.1347 14.1144 9.3963
J/b
Tables 6-8 present the data for square plates (a/b = 1.0) while Tables 9 and 10 show the data for rectangular plates (u/b = 1.5). These sets of frequency parameters provide useful information for studying the effect of ring supports on the vibratory characteristics of these plate systems. A comparison of the data
listed in these tables reveals that an increase in the number of ring supports leads to a higher frequency response when ii’/b and t/b ratios are kept constant. This is due to the increase in the flexural rigidity of the plate systems as the number of internal ring supports 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
0.125
0.001 0.10 0.20
5.4083 4.1837 2.9164
5.3786 4.2285 3.0177
5.3786 4.2285 3.0473
5.7169 4.3339 3.0473
6.4345 4.3507 3.1302
7.2208 4.9863 3.5843
7.2208 4.9863 3.5843
7.7105 5.7921 4.2435
0.25
0.001 0.10 0.20
7.8175 6.4372 4.6331
8.0878 6.4685 4.6331
8.0878 6.4685 4.6444
8.3526 6.5211 4.6509
10.0911 7.4720 4.9120
10.3642 7.5429 5.1678
10.3642 7.5429 5.1678
10.4470 7.6292 5.5044
0.375
0.001 0.10 0.20
12.1496 9.8793 7.0482
12.9344 10.1665 7.0900
12.9344 10.1665 7.0900
13.7015 10.3855 7.1282
17.4723 12.1965 8.0143
17.7946 12.4426 8.0793
17.7946 12.4426 8.0793
17.9880 12.8518 8.2089
0.50
0.001 0.10 0.20
10.8152 9.0262 6.5711
12.2946 9.1881 6.5194
12.2946 9.1881 6.5794
13.4897 9.5555 6.7756
21.7439 18.1723 11.8257
25.6120 18.2440 11.8313
26.4218 18.3638 11.8514
26.4218 18.3638 11.8514
66
K.
M.
LIEW
et
of.
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
0.5814 0.4992 0.4177
0.5851 0.5010 0.4223
1.1587 0.9598 0.7989
1.2018 0.9649 0.8089
1.8976 1.5799 1.2691
19023 I .6657 1.4331
1.9442 1.6838 1.4622
3.1138 2.7127 2.2907
0.25
0.001 0.10 0.20
0.6700 0.605 1 0.5252
0.6705 0.6051 0.5258
1.3920 1.1949 1.0051
1.3982 1.1953 1.0051
2.1074 1.8661 1.5891
2.1115 1.8697 1.6146
2.4510 2.0307 1.6908
3.3827 2.9814 2.5130
0.375
0.001 0.10 0.20
0.8055 0.7461 0.6593
0.8063 0.747 1 0.6603
1.8788 1.6512 1.3785
1.8805 1.6513 1.3193
2.5255 2.2365 1.9047
2.5459 2.2389 1.9095
3.7276 2.8585 2.3048
3.8941 3.3983 2.8111
0.50
0.001 0.10 0.20
0.9774 0.9174 0.8185
0.9783 0.9174 0.8187
2.5734 2.3054 I .9320
2.5171 2.3066 1a9329
3.5768 3.1361 2.5485
3.5787 3.1386 2.5514
4.8699 4.1541 3.3236
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
0.125
0.25
tib
1
2
3
4
5
6
7
8
0.001
1.0181 0.7950 0.6554
1.0476 0.8058 0.6658
1.6009 1.3085 1.0601
1.6231 1.3142 1.0997
1.6662 1.3232 1.1093
2.0138 1.5444
3.2929
1.2543
2.8040 2.3441
3.3236 2.8056 2.3544
1.2888 1.0686 0.8874
1.3189
1.9321 1.6173
0.8908
I.8936 1.6166 1.3520
1.3547
2.0591 1.7176 1.4019
2.4244 1.9661 1.5976
3.645s 3.1231 2.5989
3.7214 3.1268 2.6039
1.9183 1.6443 1.3471
I.9371 1.6463 I .3495
2.5256 2.2113 1.8128
2.6141 2.2114 1.8158
3.8732 2.5516 2.0690
3.3508 2.8053 2.2426
4.4398 3.7099 2.9977
4.5081 3.7112 3.0004
3.4310 2.9371 2.3162
3.4375 2.9399 2.3199
4.1550 3.4775 2.7336
4.1620 3.4824 2.7378
5.1448 4.5534 3.4639
6.1895 4.8074 3.6183
6.1960 4.9436 3.7857
6.6049 4.9494 3.7912
0.10 0.20 0.001 0.10 0.20 0.001
0.375
0.10 0.20 0.001
0.50
0.10 0.20
1.0702
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 response depends on the boundary conditions. higher flexural rigidity will be obtained by assigning 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 increasing relative thickness ratio t/b. This is due to the effects of shear deformation and rotatory inertia.
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 proofreading 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. REFERENT
1. K. M. Liew, Y. Xiang and S. Kitipomchai, Transverse vibration of thick rectangular plates--I. Comprehensive sets of boundary conditions. Comput. Struct. 49, l-29 (1993). 2. K. M. Liew, Y. Xiang and S. Kitipornchai, Transverse vibration of thick rectangular plates-II. Inclusion of oblique internal line supports. Comput. Struet. 49, 31-58 (1993). 3. A. W. Leissa, Vibration ofpfates. NASA SP-160(1969). 4. A. W. Leissa, Recent research in plate vibrations: complicating effects. Shock Vibr. Digest 9(11), 21-35 (1977). 5. A. W. Leissa, Plate vibration and research, 1976-1980: complicating effects. Shock Vibr. Digest 13(11), 19-36 (1981). 6. A. W. Leissa, Recent research in plate vibrations, 1981-1985, part II: complicating effects. Shock Vibr. Digest 19(3), lo-24 (1987). 7. K. Nagaya, Vibration of a solid rectangular plate on an eccentric annular elastic support. J. Sound Vibr. 88, 535-545 (1983).
61
Transverse vibration of thick rectangular plates-III 8. K. Nagaya, Free vibration of a solid plate of arbitrary shape lying on an arbitrarily shaped ring support. J. Sound Vibr. 94, 71-85 (1984). 9. K. M. Liew, K. Y. Lam and S. T. Chow, Free vibration analysis of rectangular plates using orthogonal plate function. Compuf. Struct. 34, 79-85 (1990). 10. K. M. Liew, Vibration of eccentric ring and line sup ported circular plates carrying concentrated masses. J. Sound Vibr. 156. 99-107 (1992). 11. K. M. Liew, Response of blat, of arbitrary shape subject to static loading. AXE J. Engng Mech. 118, 1783-1794 (1992). 12. K. M. Liew, On the use of pb-2 Rayleigh-Ritz method for free-flexural vibration of triangular plates with curved internal supports. J. Sound Vibr. 165, 329-340
(1993).
13. K. M. Liew and C. M. Wang, pb-2 Rayleigl-Ritz method for general plate analysis. Engrrg Srrucr. 15, 55-60 (1992).
14. K. M. Liew and C. M. Wang, Vibration analysis of plates by pb-2 Rayleigh-Ritz method: mixed boundary conditions, reentrant corners and internal supports. J. Mech. Struct. Machines 20, 281-292
(1992).
15. K. M. Liew and M. K. Lim, Transverse vibration of trapezoidal plates of variable thickness: symmetric trapezoids. J. Sound Vibr. 165, 4547 (1993). 16. K. M. Liew, Y. Xiang, C. M. Wang and S. Kitipomchai, Vibration of thick skew plates based on Mindlin shear deformation plate theory. J. Sound Vibr. 167 (1993). 17. C. M. Wang, K. M. Liew, Y. Xiang and S. Kitipomchai, Buckling ofrectangular Mindlin plates with internal line SUDDO~~S.
Int. J. Solids Struct. 30. l-17 (19921.
18. R.-b. Mindlin, Influence of rotatory inertia and shear in flexural motion of isotropic, elastic plates. ASME J. appl. Mech. 18, 1031-1036 (1951). 19. R. D. Mindlin and H. Deresiewicz, Thickness-shear and flexural vibrations of a circular disk. J. appl. Ph@cs 25, 1329-1332 (1954).
TRANSVERSE PLATES-III.
0
004s7949/93 %.cQ + 0.00 1993 Pcrgamon Press Lid
VIBRATION OF THICK RECTANGULAR 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.
I. INTRODUCTION
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
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 complicated 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 twodimensional 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 performed 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 comprehensive 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 conclusions 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.
(1) and
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 coordinate = x/a; q is the non-dimensional&d coordinate = 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.
2. FORMULATION
OF GOVERNING EQUATION
EIGENVALUE
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 constraint 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]
(3)
The geometric boundary conditions and the conditions 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)
e
Fig. I. Geometry and coordinate systems of rectangular Mindlin plates with multiple internal ring supports.
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
where I’, is the boundary equation of the jth supporting 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) a,=
1 if the jth edge is clamped (C) or simply supported (S).
(6b)
W)
The basic functions for the rotations can be expressed as
(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.
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)
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
+xl = ,Q
‘_
irj(t, rt)I”
@a) l
Ci,= 0 if the jth edge is free (F) or simply supported (S) in x-direction;
(8b) l
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).
(9)
11@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-dimensional frequency parameter 1 = (wb*/x*) &@ for all cases.
cases l-3 are square plates with an internal ring support of free, simply supported and fully clamped boundary conditions [Fig. 2(i)], respectively; cases 4-6 are free square plates with two, three and four internal ring supports [Figs 2(ii), (iii) and (iv)], respectively; 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 solutions. 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 number 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’) ,&@, having an internal ring support (a/b = 0.5)
for square Mindlin plates
Mode sequences
___-tlb
P,
1
2
0.001
7 9 11 13 15 17 18
PI
10.1844 10.0775 10.0393 10.0269 10.0240 10.0236 10.0236 9.747
14.0738 13.3171 13.0474 12.9072 12.8688 12.8668 12.8575 11.95
14.0738 13.3172 13.0474 12.9072 12.8688 12.8668 12.8575 11.96
15.1007 13.7890 13.6212 13.5761 13.5139 13.4716 13.4716 13.00
0.20
7 9 11 13 15 17 18
6.6046 6.5838 6.5776 6.5753 6.5736 6.5714 6.5699
7.6705 7.5580 7.5282 7.5046 7.4752 7.4488 7.4477
7.6705 7.5580 7.5282 7.5046 7.4752 7.4488 7.4477
7.9218 7.7766 7.7292 7.6682 7.6290 7.6198 7.6122
12.0947 11.9025
0.001
7 9 11 13 15 17 18
PI
11.8581 11.8463 11.8380 11.8380 11.75
20.5377 19.2183 18.8273 18.6717 18.6218 18.6217 18.6134 17.60
20.5377 19.2183 18.8273 18.6717 18.6218 18.6217 18.6134 17.60
20.9114 19.8817 19.7544 19.6053 19.4912 19.4437 19.4436 18.41
7 9 11 13 15 17 18
6.8826 6.8594 6.8455 6.8281 6.8086 6.8078 6.8022
8.6391 8.5623 8.5299 8.5215 8.4989 8.4748 8.4730
8.6391 8.5623 8.5299 8.5215 8.4989 8.4749 8.4730
8.7471 8.7042 8.6641 8.5972 8.5550 8.5538 8.5462
3
4
5
6
7
8
17.8518 15.9917 15.7687 15.7262 15.6284 15.5596 15.5596 14.56
22.3042 20.1707 19.9505 19.8135 19.7242 19.7088 19.7088 -
38.3570 24.05 11 21.2934 20.7719 20.6977 20.6518 20.5990 -
38.3570 24.05 11 21.2934 20.7719 20.6977 20.6518 20.5990
8.5330 8.2556 8.1662 8.0549 7.9822 7.9552 7.9358
10.3290 10.0995 9.9956 9.9267 9.9022 9.8725 9.8406
10.9316 10.6838 10.5193 10.4399 10.3974 10.3772 10.3731
10.9316 10.6838 10.5193 10.4399 10.3974 10.3772 10.3731
25.1632 2 1.2666 20.8263 20.6840 20.5244 20.4630 20.4630 18.59
31.4595 28.0831 27.4036 26.9096 26.7806 26.7803 26.7431 -
458.7991 30.4255 27.4036 26.9096 26.7806 26.7803 26.7431
458.7991 30.4255 27.9874 27.7522 27.6482 27.6437 27.6433 -
9.1647 9.1180 9.0634 8.9584 8.8887 8.8866 8.8724
10.8371 10.7709 10.6661 10.6037 10.5636 10.5627 10.5322
11.3408 11.1100 11.0561 11.0102 10.9466 10.9485 10.9468
11.3408 11.1100 11.0561 11.0102 10.9644 10.9485 10.9468
Square plates with SSSS boundary conditions
Square plates with CCCC boundary conditions
0.20
11.8727
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 RayleighRitz 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 = 0.001and 0.20 have been considered. In Table 2, the convergence patterns of the first eight frequency parameters are tabulated with an increasing 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.
63
Transverse vibration of thick rectangular plates-III
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
Square plates with a/!~ = 0.125 11.0094 7.3013 5.7783 5.6328 5.4717 5.439 1 5.4287 5.3083
18.2692 9.6886 6.2238 5.965 1 5.6180 5.5846 5.5100 5.3786
18.2692 9.6886 6.2238 5.9651 5.6180 5.5846 5.5100 5.3786
21.6210 10.6062 7.3190 6.4801 6.0938 5.8874 5.8633 5.7769
35.7275 16.7626 9.5759 7.5514 6.8900 6.6507 6.6141 6.4345
43.7113 18.9302 10.3150 9.6394 7.6715 7.5602 7.3018 7.2208
94.6969 36.2482 10.6867 10.2212 7.6715 7.5602 7.3018 7.2208
94.6969 36.2482 10.6867 10.2212 8.0847 7.8148 7.7794 7.7105
17 18 19
3.6018 3.3817 3.1858 3.1476 3.0111 3.0055 2.9205 2.9164
3.6018 3.3817 3.2168 3.1476 3.0488 3.0429 3.0185 3.0177
3.6824 3.4538 3.2168 3.1820 3.1088 3.0642 3.0598 3.0473
4.1243 3.5715 3.2950 3.2712 3.1088 3.0642 3.0598 3.0473
4.6339 3.8436 3.3134 3.3057 3.1636 3.1606 3.1316 3.1302
5.3195 4.3671 4.eOO4 3.8676 3.7106 3.6343 3.6286 3.5843
5.3195 4.367 1 4.0004 3.8676 3.7106 3.6343 3.6286 3.5843
5.4272 4.5334 4.3930 4.3624 4.2798 4.2792 4.2453 4.2435
7 9 I1 13 15 17 18 19
23.6672 14.8666 12.8002 12.6377 12.2447 12.1755 12.1707 12.1496
29.6042 19.1914 13.9655 13.5259 13.1501 12.9820 12.9552 12.9344
29.6042 19.1914 13.9655 13.5259 13.1501 12.9820 12.9552 12.9344
32.2294 19.3449 15.0556 14.3829 13.8902 13.7932 13.7130 13.7015
65.9843 36.7412 22.0606 19.0812 17.6149 17.5204 17.5011 17.4723
72.7658 39.5971 22.4763 20.5854 18.1482 18.0019 17.7986 17.7946
307.4878 62.7433 23.3152 20.8539 18.1482 18.0019 17.7986 17.7946
307.4878 62.7433 23.3152 20.8539 18.9515 18.2650 18.2489 17.9880
7 9 11 13 15 17 18 19
7.3905 7.1609 7.0907 7.087 1 7.0557 7.0549 7.0499 7.0482
7.4800 7.3011 7.1485 7.1376 7.1037 7.0959 7.0950 7.0900
7.4809 7.3011 7.1485 7.1376 7.1037 7.0959 7.0950 7.0900
7.5364 7.4120 7.1975 7.1920 7.1425 7.1408 7.1306 7.1282
9.0663 8.5119 8.1820 8.1679 8.0657 8.0636 8.0171 8.0143
9.4226 8.6927 8.3513 8.2436 8.1660 8.1234 8.1207 8.0793
9.4846 8.6927 8.3513 8.2436 8.1660 8.1234 8.1207 8.0793
9.4846 8.9327 8.4011 8.3726 8.2267 8.2196 8.2115 8.2089
7
0.001
0.20
9 11 13 15 17 18 19 7 9 11 13 15
Square plates with a/b = 0.375
0.001
0.20
Note. p, denotes pIT p2 and ~3. 3.2. Numerical results Numerical
calculations
have been carried
out for
rectangular plates continuous over one, two, three and four internal ring supports. The first eight frequency parameters obtained from the calculation Table 3. Frequency parameters, 1 = (ob2/x2) m,
are presented in Tables 3-8 with an increasing mode sequence. In Tables 3-5, approximate frequency parameters are presented for square plates continuous over a single ring support with free, simply supported and
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
0.25
0.001 0.10 0.20
1.3065 1.1502 0.8690
1.3232 1.1502 0.8690
1.3232 1.2618 1.1529
1.6105 1.4205 1.2426
2.4551 2.1643 1.8663
3.9290 3.5072 2.9456
3.9290 3.5072 2.9456
5.3349 4.6928 3.7990
0.50
0.001 0.10 0.20
1.7596 1.7010 1.5644
2.1489 1.9602 1.6177
2.1489 1.9602 1.6177
2.3393 2.0454 1.6571
4.1059 3.5067 2.6914
5.0509 4.3335 3.4388
5.0509 4.3335 3.4388
6.4786 5.5794 4.4758
0.75
0.001 0.10 0.20
2.4963 2.3871 2.1563
3.7315 3.3635 2.7587
3.7315 3.3635 2.7587
4.0653 3.5611 2.8110
6.1728 5.3269 4.1617
9.1680 7.5957 5.3971
9.2461 7.6458 5.5741
9.2461 7.6458 5.5741
1.00
0.001 0.10 0.20
2.0033 1.9299 1.7682
5.3334 4.8724 4.0558
5.3334 4.8724 4.0058
7.9761 6.8866 5.3223
9.1142 7.6670 5.7313
10.3828 8.4316 6.0999
10.3828 8.4316 6.0999
10.5948 9.0149 6.8423
64
K. M. L~EWef 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
0.25
0.001 0.10 0.20
7.1339 6.1740 4.7204
8.1276 6.763 1 5.0740
8.1276 6.763 1 5.0740
9.2837 7.7075 5.8766
12.0386 9.6043 7.0552
14.6027 1I.6175 8.3826
14.6027 Il.6175 8.3826
17.4622 13.6630 9.5921
0.50
0.001 0.10 0.20
10.0236 8.6068 6.5699
12.8575 10.3808 7.4477
12.8575 10.3808 7.4477
13.4716 10.6919 7.6122
15.5596 11.8219 7.9358
19.7088 14.5969 9.8406
20.5990 15.3036 10.3731
20.5990 15.3036 10.3731
0.001 0.10 0.20
6.0445 5.2OQ4 3.9770
12.8307 10.3036 7.3406
12.8307 10.3036 7.3406
20.1314 15.4326 10.6011
21.8817 16.0341 10.6945
22.5502 16.8415 I 1.4080
24.97 11
0.75
11.7702
24.97 1I 17.9230 11.7702
1.00
0.001 0.10 0.20
3.9363 3.0322 2.3609
8.2042 6.2454 4.7125
8.2042 6.2454 4.7125
13.1352 9.7392 7.0790
13.8809 10.1408 7.2227
15.3637 II.1629 7.9065
19.7433 13.9747 9.6113
19.7433 13.9747 9.6113
clamped boundary conditions. In each table, the listed numerical values provide important information to study the effect of ring dimension on the vibration frequency parameters. The data listed in Table 3 show that for the free square plate with a ring support of diameter a/6 = 0.25, 0.50,0.75, 1.0, the m~imum fundamental frequency response occurs with a ring support of d/b z 0.75 for all relative thickness ratios. However the maximum values for the higher modes occur at d/b z 1.0. 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 FT0.50 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 Table 5. Frequency parameters, 1 = (wb2/n2)m,
17.9230
frequency parameters in Tables 3-5 is necessary. It can be seen that the square plate with fully clamped boundaries (Table 5) always gives the highest frequency response when compared to the values for square plates with free (Table 3) and simply supported (Table 4) boundaries, keeping d/b and t/b constant. It may be concluded that assigning a higher constraint at the supporting edge will increase the flexural rigidity of the structure. This leads to a higher frequency response. 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 commonly used in offshore structures, and these data may therefore be used to those engineers dealing with offshore structures. The detailed geometry and dimensions of these plate systems are shown in Figs 2(ii), (iii) and (iv) (Cases 4-6).
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
10.8006
0.10 0.20
a.4604 5.8267
11.9565 8.9832 6.0370
1I .9565 8.9832 6.0370
12.9061 9.6735 6.6415
16.4243 11.6477 7.6794
f8.8567 13.4564 8.8989
19.8567 13.4564 8.8989
22.0745 15.4875 10.0913
0.50
0.001 0.10 0.20
11.8380 9.5063 6.8022
18.6134 13.1843 8.4730
18.6134 13.1843 8.4730
19.4436 13.4400 8.5462
20.4630 14.0315 8.8724
26.7431 17.4927 10.5322
26.7431 17.7162 10.9468
27.6433 17.7162 10.9468
0.75
0.001 0.10 0.20
6.2836 5.2370 4.0129
13.2995 10.3624 7.3598
13.2995 10.3624 7.3598
21.6738 15.8391 10.6416
22.45 I 1 16.0513 10.7087
24.9142 17.6300 11.6073
31.8184 20.9201 12.6594
31.8184 20.9201 12.6594
1.00
0.001 0.10 0.20
3.9923 3.4648 2.8327
8.3138 6.7965 5.1063
8.3138 6.7965 5.1063
13.3396 10.3144 7.3826
14.0052 10.7035 7.4803
15.5615 11.7734 8.2088
19.9763 14.4961 9.7996
19.9763 14.4961 9.79%
0.25
Transverse vibration of thick rectangular plates-111 Table 6. Frequency parameters, 1 = (&/n~
65
&@, 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
0.125
0.001 0.10 0.20
1.2011 0.9754 0.7028
1.3183 1.1907 1.0379
3.8720 3.0996 2.3428
3.9061 3.4256 2.7465
4.5812 3.7464 2.7815
5.2669 4.2842 3.2436
5.5822 4.3211 3.2528
5.7127 4.3958 3.3043
0.25
0.001 0.10 0.20
1.4208 1.2494 0.9998
1.4406 1.3211 1.1722
4.473 1 3.9183 3.0522
4.6168 3.9583 3.1621
5.4776 4.7565 3.6742
5.9825 5.0550 3.9316
8.1012 6.4624 4.6224
8.2943 6.5396 4.6784
0.375
0.001 0.10 0.20
1.6453 1.5102 1.2712
1.7162 1.5410 1.3281
5.3583 4.6755 3.6998
5.5893 4.7919 3.7062
6.695 1 5.7870 4.5280
7.1285 6.0008 4.5956
11.3014 8.8441 6.2687
11.7710 9.2277 6.6670
0.50
0.001 0.10 0.20
1.9651 1.7889 1.5418
2.0815 1.8784 1.5671
6.4685 5.5818 4.3486
6.8694 5.8292 4.4310
7.7404 6.7122 5.2856
8.7166 7.3003 5.4814
10.7117 8.8145 6.4592
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 Mode sequences rib
1
2
3
4
5
6
7
8
0.125
0.001 0.10 0.20
1.9939 1.5860 1.1997
2.0429 1.7872 1.4959
4.6069 3.6247 2.6578
4.8532 3.9432 3.0057
5.6822 4.2109 3.0520
5.7904 4.3427 3.2311
7.2750 5.0044 3.6022
7.7004 6.0402 4.4283
0.25
0.001 0.10 0.20
2.3741 2.0948 1.6335
2.4575 2.1088 1.7824
5.7051 4.7539 3.6051
5.7916 4.7824 3.6882
8.0569 6.4733 4.6449
8.3371 6.5188 4.6469
9.8706 7.5562 5.1709
10.2753 7.9395 5.7313
0.375
0.001 0.10 0.20
2.9875 2.6341 2.1648
3.1629 2.7453 2.1781
7.2572 5.9706 4.4965
7.3208 5.9923 4.5065
12.3078 9.9586 7.0712
13.0976 10.2467 7.1144
13.3422 10.5156 7.3939
14.5764 11.0000 7.5721
0.50
0.001 0.10 0.20
3.9822 3.4572 2.7149
4.2745 3.6747 2.8708
9.3715 7.5341 5.5389
9.5801 7.5740 5.5553
11.2685 9.1061 6.5837
12.1390 9.3221 6.6550
12.2762 9.6171 6.9016
19.1347 14.1144 9.3963
J/b
Tables 6-8 present the data for square plates (a/b = 1.0) while Tables 9 and 10 show the data for rectangular plates (u/b = 1.5). These sets of frequency parameters provide useful information for studying the effect of ring supports on the vibratory characteristics of these plate systems. A comparison of the data
listed in these tables reveals that an increase in the number of ring supports leads to a higher frequency response when ii’/b and t/b ratios are kept constant. This is due to the increase in the flexural rigidity of the plate systems as the number of internal ring supports 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
0.125
0.001 0.10 0.20
5.4083 4.1837 2.9164
5.3786 4.2285 3.0177
5.3786 4.2285 3.0473
5.7169 4.3339 3.0473
6.4345 4.3507 3.1302
7.2208 4.9863 3.5843
7.2208 4.9863 3.5843
7.7105 5.7921 4.2435
0.25
0.001 0.10 0.20
7.8175 6.4372 4.6331
8.0878 6.4685 4.6331
8.0878 6.4685 4.6444
8.3526 6.5211 4.6509
10.0911 7.4720 4.9120
10.3642 7.5429 5.1678
10.3642 7.5429 5.1678
10.4470 7.6292 5.5044
0.375
0.001 0.10 0.20
12.1496 9.8793 7.0482
12.9344 10.1665 7.0900
12.9344 10.1665 7.0900
13.7015 10.3855 7.1282
17.4723 12.1965 8.0143
17.7946 12.4426 8.0793
17.7946 12.4426 8.0793
17.9880 12.8518 8.2089
0.50
0.001 0.10 0.20
10.8152 9.0262 6.5711
12.2946 9.1881 6.5194
12.2946 9.1881 6.5794
13.4897 9.5555 6.7756
21.7439 18.1723 11.8257
25.6120 18.2440 11.8313
26.4218 18.3638 11.8514
26.4218 18.3638 11.8514
66
K.
M.
LIEW
et
of.
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
0.5814 0.4992 0.4177
0.5851 0.5010 0.4223
1.1587 0.9598 0.7989
1.2018 0.9649 0.8089
1.8976 1.5799 1.2691
19023 I .6657 1.4331
1.9442 1.6838 1.4622
3.1138 2.7127 2.2907
0.25
0.001 0.10 0.20
0.6700 0.605 1 0.5252
0.6705 0.6051 0.5258
1.3920 1.1949 1.0051
1.3982 1.1953 1.0051
2.1074 1.8661 1.5891
2.1115 1.8697 1.6146
2.4510 2.0307 1.6908
3.3827 2.9814 2.5130
0.375
0.001 0.10 0.20
0.8055 0.7461 0.6593
0.8063 0.747 1 0.6603
1.8788 1.6512 1.3785
1.8805 1.6513 1.3193
2.5255 2.2365 1.9047
2.5459 2.2389 1.9095
3.7276 2.8585 2.3048
3.8941 3.3983 2.8111
0.50
0.001 0.10 0.20
0.9774 0.9174 0.8185
0.9783 0.9174 0.8187
2.5734 2.3054 I .9320
2.5171 2.3066 1a9329
3.5768 3.1361 2.5485
3.5787 3.1386 2.5514
4.8699 4.1541 3.3236
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
0.125
0.25
tib
1
2
3
4
5
6
7
8
0.001
1.0181 0.7950 0.6554
1.0476 0.8058 0.6658
1.6009 1.3085 1.0601
1.6231 1.3142 1.0997
1.6662 1.3232 1.1093
2.0138 1.5444
3.2929
1.2543
2.8040 2.3441
3.3236 2.8056 2.3544
1.2888 1.0686 0.8874
1.3189
1.9321 1.6173
0.8908
I.8936 1.6166 1.3520
1.3547
2.0591 1.7176 1.4019
2.4244 1.9661 1.5976
3.645s 3.1231 2.5989
3.7214 3.1268 2.6039
1.9183 1.6443 1.3471
I.9371 1.6463 I .3495
2.5256 2.2113 1.8128
2.6141 2.2114 1.8158
3.8732 2.5516 2.0690
3.3508 2.8053 2.2426
4.4398 3.7099 2.9977
4.5081 3.7112 3.0004
3.4310 2.9371 2.3162
3.4375 2.9399 2.3199
4.1550 3.4775 2.7336
4.1620 3.4824 2.7378
5.1448 4.5534 3.4639
6.1895 4.8074 3.6183
6.1960 4.9436 3.7857
6.6049 4.9494 3.7912
0.10 0.20 0.001 0.10 0.20 0.001
0.375
0.10 0.20 0.001
0.50
0.10 0.20
1.0702
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 response depends on the boundary conditions. higher flexural rigidity will be obtained by assigning 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 increasing relative thickness ratio t/b. This is due to the effects of shear deformation and rotatory inertia.
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 proofreading 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. REFERENT
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