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ANALYSIS OF VIBRATING THICK RECTANGULAR PLATES WITH MIXED BOUNDARY CONSTRAINTS USING DIFFERENTIAL QUADRATURE ELEMENT METHOD
Journal of Sound and
ANALYSIS OF VIBRATING THICK RECTANGULAR PLATES WITH MIXED BOUNDARY CONSTRAINTS USING DIFFERENTIAL QUADRATURE ELEMENT METHOD F.-L. LIU CAD/CAM ¸aboratory, Division of Engineering Mechanics, School of Mechanical and Production Engineering, Nanyang ¹echnological ;niversity, Singapore 639798, Singapore AND
K. M. LIEW Center for Advanced Numerical Engineering Simulations, School of Mechanical and Production Engineering, Nanyang ¹echnological ;niversity, Singapore 639798, Singapore (Received 30 November 1998, and in ,nal form 17 March 1999) A free vibration analysis of moderately thick rectangular plates with mixed boundary conditions is presented on the basis of the "rst-order shear deformation plate theory. The di!erential quadrature element method, a highly e$cient and accurate hybrid approach, has been employed. To establish the numerical model, the complex plate domain is "rst decomposed into small simple continuous sub-domains (elements) and the di!erential quadrature method is then applied to each continuous sub-domain to solve the problems. Compatibility conditions are developed for the conjunction nodes on the interface boundaries of elements in order to connect the elements. Convergence and comparison studies are carried out to establish the reliability of the solutions. The "rst eight frequency parameters are predicted for various types of thick rectangular plates with mixed edge constraints. 1999 Academic Press
1. INTRODUCTION
Problems of rectangular plates with mixed edge constraints have been investigated by many researchers adapting di!erent solution techniques. Ota and Hamada [1] predicted the fundamental frequencies of a simply supported rectangular plate partially clamped on the edge using a distributed moment function along the mixed edge. Keer and Stahl [2] studied the same problem by means of Fredholm integral equations of a second kind, and demonstrated the dependence of the fundamental frequencies on the clamped portion. Narita [3] employed a series-type solution approach to solve the problem and obtained the frequency parameters for wide ranges of mixed-edge rectangular plates. Other numerical methods such as the "nite element method [4], superposition method [5], spline "nite strip method [6], spline element method [7], and the Rayleigh}Ritz method [8}10] have also been used to study the free vibration of rectangular plates with discontinuous edge 0022-460X/99/350915#20 $30.00/0
1999 Academic Press
916
F.-L. LIU AND K. M. LIEW
constraints. Laura and Gutierrez tried to predict the fundamental frequency of the simply supported rectangular plates partially clamped along one and two edges by using the global di!erential quadrature method [11]. However, all the research work mentioned above is all con"ned to thin plates, solutions to thick rectangular plates with discontinuous boundary constraints are scarce in the literature. Only one paper was found to have treated the problem of free vibration of the Mindlin plate with mixed boundary constraints [12], but only the fundamental frequencies of the thin plates for two simple cases were calculated. The primary purpose of this paper is to solve the free vibration problems of thick rectangular plates with mixed boundary constraints. A hybrid approach namely the di!erential quadrature element method, which combines the di!erential quadrature method with the domain decomposition method, has been employed. The di!erential quadrature (DQ) method originated with Bellman and his associates [13] and was "rst applied to the structural analysis "eld by Bert and his associates [14}16]. Since then, many studies researches have been done in both the theoretical development and the engineering applications of the method. In all of the applications, this method yielded good to excellent results for only a few discrete points due to the use of the high-order global-based functions in the computational domain. An excellent review paper contributed by Bert and Malik [17] has summarized a detailed literature list on both aspects of the DQ method. Nevertheless, further application of the method has been greatly restricted by its drawback of not being able to be directly employed to solve the problem with discontinuities. To overcome such a drawback, the DQ method has been combined with the domain decomposition method "rst by Striz et al. [18}20] and then by Wang and Gu [21, 22] to solve the static problems of truss and beams and the static and free vibration problems of thin plates. This hybrid approach has been further developed by Han and Liew to solve the one-dimensional bending problem of the axisymmetric shear deformable circular plate [23], and by the present author to solve two-dimensional bending and vibration problems of thick rectangular plates and polar plates having discontinuities [24}28]. In this study, the di!erential quadrature method is combined with the domain decomposition method to deal with the free vibration problems of the moderately thick rectangular plates with mixed boundary constraints. 2. MATHEMATICAL FORMULATIONS
Consider a moderately thick rectangular plate with side lengths a;b as shown in Figure 1. The plate is divided into N elements based on the discontinuities in the # geometry, boundary constraints and materials used. Each element consists of an isotropic material and has uniform thickness and continuous boundary constraints on each edge. 2.1.
EIGENVALUE EQUATION AND BOUNDARY CONSTRAINT CONDITIONS
For a given element l of the plate, the eigenvalue equations for the free vibration problems can be written based on the "rst order shear deformation theory as
VIBRATING THICK RECTANGULAR PLATES WITH MIXED BOUNDARY CONSTRAINTS
917
Figure 1. Con"guration of a thick rectangular plate.
follows [29]:
(1)
(2)
*t o *w *w *w *t V# W " J # # , *y iG *t *x *y *x J
(3)
*t (1!l ) *t (1#l ) *t iG h *w o h *t V# V# W! J J V, J J #t " J J V *x *y *x*y *x 2 2 D 12D *t J J (1#l ) *t *t (1!l ) *t iG h *w o h *t J V# W! J J V, W# J #t " J J W 2 *x*y *x *y 2 *y D 12D *t J J
where w, t and t are the transverse de#ection, the rotation of the normal V W about the y-axis and the rotation of the normal about the x-axis respectively: h , J E , G and l are the thickness of the plate, Young's modulus, shear modulus J J J and Poisson's ratio, respectively; D is the plate #exural rigidity; and i is the J shear correction factor. For free vibration analysis, i is taken to be n/12. Letting w"=(x, y)e SR, t "W (x, y)e SR, t "W (x, y)e SR and V V W W u"X (E/[oa(1!l)] and substituting them into equations (1)}(3), one can obtain the following equations:
(4)
(5)
*W (1!l ) *W (1#l ) *W *= V# J J V# W!F #W "!X/aW , J *x V V *x 2 2 *y *x*y *= (1#l ) *W *W (1!l ) *W V# W!F J W# J #W "!X/aW , J *y W W *x*y *x 2 *y 2
*W !2 *= *= *W V# W " # # X= *y *y i(1!l )a *x *x J
(6)
918
F.-L. LIU AND K. M. LIEW
in which F "6i(1!l )/h . J J J
(7)
The moments and shear forces are expressed as
*t *t V#l W , J *y *x
(8a)
*t *t V# W , M "D l W J J *x *y
(8b)
M "D V J
*t *t 1!l V# W , JD M " J *y VW *x 2
(8c)
*w *w #t , Q "iG h #t . Q "iG h V W J J *y W V J J *x
(9a,b)
The boundary conditions for the edge x"0 can be expressed as follows. Clamped edge (C): w"0, t "0, t "0. V W
(10)
Hard simply supported edge (S): w"0, t "0, W
M "0. V
(11)
Soft simply supported edge (S ): w"0, M "0, M "0. V VW
(12)
Q "0, V
(13)
Free edge (F):
2.2.
M "0, M "0. V VW
RECTANGULAR FREE VIBRATION DQ PLATE ELEMENT
The lth element is further divided into N ;N grid points in the x and V W y directions as shown in Figure 2. Using the DQ procedures [17], the discrete governing equations for free vibration of the lth plate element can be expressed at each discrete point of the inner mesh into (1!l ) ,JW ,JV J CM (W ) !F (W ) C (W ) # GI V IH HK V GK J V GH 2 I K (1#l ) J # 2
,JV ,JV ,JW C CM (W ) !F C (=) "!X/a(W ) , J GI HK W IK GI IH V GH I K I (14a)
VIBRATING THICK RECTANGULAR PLATES WITH MIXED BOUNDARY CONSTRAINTS
919
Figure 2. Arrangement of grid points for element l.
(1#l ) J 2
,JW ,JV ,JW C CM (W ) # CM (W ) GI HK V IK HK W GK I K K
(1#l ) ,JV ,JW J C (W ) !F (W ) !F CM (=) "!X/a(W ) , # GI W IH J W GH J HK GK W GH 2 I K (14b) ,JV ,JW ,JV ,JW C (=) # CM (=) # C (W ) # CM (W ) GI IH HK GK GI V IH HK W GK I K I K !2 " X= , i"1, 2, 3, 2 , N , j"1, 2, 3, 2 , N GH V W i(1!l )a J and l"1, 2, 3, 2 , N #
(14c)
where CL and CM L (r"1, 2, 3, 2 , N ; s"1, 2, 3, 2 , N ) are the weighting PQ PQ V W coe$cients for the nth-order partial derivatives of w, t and t with respect to the V W global co-ordinates x and y. Equations (14a}c) can be further expressed in matrix form as KCdC"jBCdC,
(15)
920
F.-L. LIU AND K. M. LIEW
where j"(ohua/D) is de"ned as the frequency parameter: KC, dC and BC are the element weighting coe$cient matrix, element displacement vector and the element consistent mass matrix, respectively. Also ,t ,t ]2, dC"[w , t ,t ,w ,t , t , ,w V W V W 2 ,V ,W V,V ,W W,V ,W BC"[bC , bC , 2 , bC W , bC , bC ,2, bC W,2, bC V ,2, bC V W ]2, , , , , ,
(16) (17)
where bC , bC ,2, bC W , bC , bC ,2, bC W ,2, bC V , bC V ,2, bC V W are null , , , , , , matrices with 3N;3 elements, and bC ,2, bC W ,2, bC V , , and , \ , \ 2 are matrices with 3N;3 elements, in which only the three elements at bC V , \ ,W\ the corresponding point in each matrix are not equal to zero; the other elements are all zeros. Taking matrix bC , for example, it can be expressed as bC " H ,2, ,W CFFDFFE 0, 0, 0,2, 0, 0, 0
H,W> ,W> ,W>,2, ,W CFFFFFDFFFFFE 0, 0, 0, 0, !1/a, 0,2, 0, 0, 0
H,W> ,W>,2, , CFFFDFFE 0, 0, 0,2, 0, 0, 0
0, 0, 0,2, 0, 0, 0,
0, 0, 0, 0, 0, !1/a,2, 0, 0, 0, 2 0, 0, 0,2, 0, 0, 0, 0, 0, 0, ! , 0, 0,2, 0, 0, 0, i(1!l)a
0, 0, 0,2, 0, 0, 0
.
0, 0, 0,2, 0, 0, 0 (18)
The coe$cients in KC are determined by equations (14a}c). 2.3. COMPATIBILITY CONDITIONS In order to obtain a complete solution for the entire plate, the element weighting coe$cient matrices, the element displacement vectors and the element consistent mass matrices should be assembled into a global system equations for all the nodal points of the plate labelled from 1 to N. Therefore, the compatibility conditions for the conjunction nodes at the interface boundaries of adjacent elements need to be established. This includes the compatibility conditions for both the displacements and the force and moments. Obviously, the displacement compatibility conditions are automatically satis"ed at all the interface conjunction nodes since the same global nodal number is used for each conjunction node. Only the equilibrium conditions for the force and moments are needed to form the compatibility conditions. According to the locations of the conjunction nodes, the compatibility conditions for the free vibration DQ plate elements are given below: E
For conjunction nodes at interface boundaries of two elements: As shown in Figure 3(a), the compatibility conditions for the conjunction nodes at the interface boundaries of two elements l and l , which are connected in the x direction, can be expressed according to the equilibrium conditions as (QJ ) V !(QJ ) "0, (MJ ) V !(MJ ) "0, V , H V H V , H V H
(MJ ) V !(MJ ) "0. VW , H VW H (19a}c)
VIBRATING THICK RECTANGULAR PLATES WITH MIXED BOUNDARY CONSTRAINTS
921
Figure 3. Locations of conjunction nodes on the interface boundaries of adjacent elements: (a) two elements are connected in x direction: (b) two elements are connected in y direction; (c) four elements are connected at node m.
Similarly, for the conjunction nodes at the interface boundaries of two elements l and l connected in the y direction as shown in Figure 3(b), the compatibility conditions are given by (QJ ) W!(QJ ) "0, (MJ ) W!(MJ ) "0, W G , W G W G , W G E
(MJ ) W!(MJ ) "0. VW G , VW G (20a}c)
For conjunction nodes at which four elements meet: If four adjacent elements l , l , l and l are connected at a node m as shown in Figure 3(c), the compatibility conditions for the common node m are (QJ ) V W#(QJ ) V !(QJ ) W!(QJ ) "0, V , , V , V , V (MJ ) V W#(MJ ) V !(MJ ) W!(MJ ) "0, V , , V , V , V or
(21a}c)
(MJ ) V W#(MJ ) V !(MJ ) W!(MJ ) "0 VW , , VW , VW , VW (QJ ) V W!(QJ ) V #(QJ ) W!(QJ ) "0, W , , W , W , W (MJ ) V W!(MJ ) V #(MJ ) W!(MJ ) "0, W , , W , W , W
(22a}c)
(MJ ) V W!(MJ ) V #(MJ ) W!(MJ ) "0 VW , , VW , VW , VW The "nal global matrix forms of equation system for free vibration of the entire plate become Kd"jBb, (23) where K, d and B are the overall weighting coe$cient matrix, global displacement vector, and overall mass matrix of the plate.
922
F.-L. LIU AND K. M. LIEW
Solving the "nal algebraic eigenvalue equation system by the ordinary eigenvalue equation system solver, the solutions to the entire plate can be obtained.
3. NUMERICAL RESULTS AND DISCUSSIONS
Using the method described above, the vibration frequencies of rectangular plates with any arbitrary combination of mixed boundary constraints can be determined easily. In the present study, however, we only focus on six cases as shown in Figure 4(a}f ). The grid points employed in computation are designed by a x " +1!cos[(i!1)n/(N !1)],, i"1, 2, 3,2, N , V V G 2
(24)
b y " +1!cos[( j!1)n/(N !1)],, j"1, 2, 3,2, N H 2 W W
(25)
The Poisson's ratio is taken to be l"0)3. The eigenvalues are expressed in terms of the frequency parameter as j"ua(oh/D.
Figure 4. Con"gurations of rectangular plates with mixed boundary constraints analyzed by present numerical method: (a) a simply supported rectangular plate partially clamped along one edge, (b) a simply supported rectangular plate partially clamped along two opposite edges; (c) a simply supported rectangular plate partially clamped along four edges symmetrically from the corners; (d) a square plate with central portions of edges clampped and the portions around four corners simply supported; (e) a simply supported rectangular plate with central portions of two opposite edges free; (f ) a rectangular plate having simply supported, free and clamped constraint conditions at the same edges.
VIBRATING THICK RECTANGULAR PLATES WITH MIXED BOUNDARY CONSTRAINTS
923
Convergence studies are carried for the example problems (cases a}f ) to establish the grid points required in each element for obtaining the accurate solutions. The convergence patterns of the frequency parameters with the number of grid points in each element are presented in Table 1 for cases a}d and in Table 2 for cases e}f. It can be observed from Tables 1 and 2 that the convergence rate varies for di!erent plate con"gurations. Generally, the convergence rate of the plate which includes more singular points on boundaries is lower than that of a plate having a lesser number of singular points on the boundaries. And the convergence of frequency parameters for plates having partially free boundary conditions on the mixed boundaries (cases e and f ) is slower than that for plates without free boundary conditions on the periphery boundaries. However, for all the cases considered in this paper, a convergence to at least three signi"cant "gures can be achieved when 13;13 grid points are used in each element. Therefore, 13;13 grid points in each element will be used to generate all the results in the following studies to ensure the high accuracy of the numerical solutions. To examine the validity of the numerical method employed, the computed frequency parameters have been compared with the existing solutions available in the open literature. In Table 3, a comparison of the present solutions with those from various sources for cases a}d is presented. Since no solutions have been found for thick plates of cases a}d in the open publications, only the results for the thin plates (h/a"0)01) are compared here with the existing thin plate theory solutions. It is evident from Table 3 that the present solutions for the thin plates (h/a"0)01) agree very well with the analytical solutions established by Ota and Hamada [1], Keer and Stahl [2], Narita [3] and Gorman [5], and the Rayleigh}Ritz solutions given by Liew et al. [9]. However, the results obtained by the "nite element method [4], spline "nite strip method [6] and the spline element method [7] seem to be slightly higher than those given by the present method and the analytical method. Even so, they are well within the range of the acceptable accuracy for engineering applications. Based on the convergence and comparison studies, the present numerical solution approach has been employed to determine the vibration frequencies of the thick rectangular plates having various mixed boundary constraints. The frequency parameters corresponding to the "rst eight modes of free vibration for cases a}f have been computed and presented in Tables 4}9 for rectangular plates of various relative thickness ratios (h/a) and mixed edge constraint ratio (c/a). Some of the results, where possible, are compared with the existing solutions in the literature. For all the cases considered, the dependence of the frequency parameters on both relative thickness ratio and the mixed edge constraint ratio is observed. In Tables 4 and 5, it is observed that as the partially clamped ratio, c/a, approaches the value 1)0, the frequency parameters approach the values of a square plate with one or two edges fully clamped and the other edges simply supported for all the relative thickness varying from h/a"0)01 to 0)2. Also, it is observed that regardless of the relative thickness, the frequency parameters vary only slightly when the partial clamped ratio is within c/a"0)0}0)2 or c/a"0)8}1)0. The signi"cant variations occur, however, in the mid-range of c/a. This phenomenon has been discussed earlier by other researchers for thin plate problems [5, 9]. The
5;5 6;6 7;7 8;8 9;9 10;10 11;11 12;12 13;13 5;5 6;6 7;7 8;8 9;9 10;10 11;11 12;12 13;13
a c/a"0)5
b c/a"0)5
N ;N V W
Case
20)5800 21)0060 21)1124 21)1509 21)1656 21)1747 21)1798 21)1833 21)1857 22)6575 23)1927 23)4377 23)5041 23)5483 23)5652 23)5809 23)5864 23)5941
1 45)5777 45)3880 45)6841 45)7395 45)7404 45)7425 45)7451 45)7475 45)7485 46)7026 47)1896 47)2852 47)2862 47)2919 47)2925 47)2933 47)2938 47)2941
2 84)1521 48)9165 49)5431 49)6484 49)6606 49)6500 49)6602 49)6662 49)6706 84)6325 51)3033 52)3448 52)3253 52)3648 52)3859 52)4047 52)4249 52)4324
3 88)3709 71)1195 71)6598 71)7344 71)7528 71)7347 71)7345 71)7351 71)7359 90)8073 74)3694 75)4754 74)8898 74)9160 74)9229 74)9217 74)9221 74)9222
4 104)048 84)2639 85)6318 85)4580 85)3949 85)4044 85)4061 85)4075 85)4076 111)828 84)7122 86)0588 85)8996 85)8458 85)8532 85)8566 85)8576 85)8588
5
Mode sequence number
127)932 106)086 88)1667 90)1179 89)4480 89)3573 89)2280 89)2487 89)2612 132)023 107)488 91)0545 93)5035 92)3568 92)3010 92)2010 92)2261 92)2482
6
162)738 130)772 107)242 107)596 107)509 107)501 107)489 107)491 107)494 160)864 130)991 109)631 109)394 109)377 109)403 109)409 109)418 109)420
7
167)750 149)536 109)545 110)893 110)194 110)185 110)095 110)099 110)103 167)212 150)397 114)071 116)698 113)956 114)062 114)003 114)002 114)003
8
Convergence of frequency parameters j"ua (oh/D with increasing number of the grid points in each element for a square plate (h/a"0)1) with mixed boundary constraints (cases a}d)
TABLE 1
924 F.-L. LIU AND K. M. LIEW
d c/d"2/3
c c/a"2/3
5;5 6;6 7;7 8;8 9;9 10;10 11;11 12;12 13;13 5;5 6;6 7;7 8;8 9;9 10;10 11;11 12;12 13;13
25)8372 26)1419 26)1550 26)2383 26)2603 26)2898 26)3034 26)3181 26)3262 30)4226 31)4046 31)4080 31)4232 31)4316 31)4344 31)4415 31)4422 31)4467
53)2219 54)2464 54)2643 54)3448 54)3727 54)4062 54)4235 54)4401 54)4505 56)3835 58)8278 58)8566 58)8686 58)8982 58)9043 58)9231 58)9253 58)9366
53)2242 54)2544 54)2647 54)3483 54)3731 54)4080 54)4238 54)4412 54)4506 56)7672 60)2112 60)1536 60)1833 60)1776 60)1934 60)1929 60)2002 60)2002
82)6095 84)1686 84)1607 84)2144 84)2276 84)2552 84)2669 84)2801 84)2878 76)9366 81)4011 81)3033 81)4113 81)3984 81)4393 81)4386 81)4580 81)4582
87)4281 88)5577 88)8201 88)8930 88)9328 88)9705 88)9916 89)0112 89)0229 92)8638 96)9614 97)2763 97)2752 97)3242 97)3314 97)3591 97)3626 97)3785
95)4539 96)3892 96)6394 96)7107 96)7293 96)7612 96)7789 96)7939 96)8046 99)8602 101)994 102)192 102)188 102)179 102)181 102)182 102)181 102)182
117)889 119)263 119)483 119)517 119)529 119)555 119)566 119)579 119)585 111)512 115)053 115)137 115)259 115)247 115)297 115)298 115)323 115)324
117)901 119)265 119)493 119)518 119)533 119)556 119)568 119)579 119)586 114)401 119)639 119)743 119)761 119)754 119)762 119)763 119)766 119)767
VIBRATING THICK RECTANGULAR PLATES WITH MIXED BOUNDARY CONSTRAINTS
925
5;5 6;6 7;7 8;8 9;9 10;10 11;11 12;12 13;13 14;14 5;5 6;6 7;7 8;8 9;9 10;10 11;11 12;12 13;13 14;14
e c/a"1/3
f c /a"1/3 c/b"0)5
N ;N V W
Case
18)4158 18)5178 18)5968 18)6158 18)6434 18)6516 18)6698 18)6717 18)6842 18)6838 24)2577 24)4809 24)5330 24)5784 24)6297 24)6422 24)6742 24)6780 24)6981 24)6994
1 44)1013 41)2398 41)9654 41)9991 42)1940 42)2627 42)3701 42)4129 42)4675 42)4753 43)7685 44)4021 44)6073 44)6854 44)8046 44)8403 44)9041 44)9211 44)9596 44)9682
2 47)9030 44)9147 41)0195 45)0243 45)0650 45)0664 45)0834 45)0827 45)0898 45)0893 55)1378 55)8870 56)0533 56)1111 56)1733 56)2144 56)2483 56)2698 56)2903 56)3029
3 67)8778 67)3257 68)2262 68)5624 68)5857 68)6293 68)6471 68)6686 68)6784 68)6865 68)4028 71)4835 72)2066 72)3105 72)5153 72)6076 72)6947 72)7407 72)7866 72)8124
4 85)0881 68)9756 69)4348 70)9247 70)3221 70)8356 71)1729 71)4087 71)5995 71)7070 73)6311 75)6167 75)9186 76)2283 76)4404 76)6240 76)7552 76)8524 76)9360 76)9437
5
Mode sequence number
87)8688 84)0989 84)6037 84)8291 84)8092 84)8010 84)7951 84)8000 84)8023 84)8049 86)3611 91)3024 93)6428 94)1678 94)6111 94)9493 95)2008 95)3759 95)5246 95)6282
6
115)078 85)9829 87)1892 90)7161 91)7146 91)1472 91)6705 92)0078 92)2459 92)4255 95)2704 96)1577 96)4888 96)4848 95)5001 96)5249 96)5392 96)5513 96)5603 96)5674
7
128)096 108)974 102)019 104)539 104)493 104)440 104)456 104)487 104)531 104)544 106)535 108)120 109)084 109)065 109)078 109)142 109)173 109)203 109)220 109)239
8
Convergence of frequency parameters j"ua (oh/D with increasing number of the grid points in each element for a square plate (h/a"0)1) with mixed boundary constraints (case e}f )
TABLE 2
926 F.-L. LIU AND K. M. LIEW
VIBRATING THICK RECTANGULAR PLATES WITH MIXED BOUNDARY CONSTRAINTS
927
TABLE 3 Comparison studies for frequency parameters j"ua (oh/D of a sqaure plate with various mixed boundary constraints (h/a"0)01, c/a"0)5) Mode sequence number Case
Sources
a
Present Ota and Hamada [1] Keer and Stah [2] Narita [3] Gorman [5] Fan and Cheung [6] Mizusawa and Leonard [7] Liew et al. [9] Laura and Gutierrez [11] Present Ota and Hamada [1] Fan and Cheung [6] Liew et al. [9] Laura and Gutierrez [11] Striz et al. [20] Present Narita [3] Liew et al. [9] Present Ota and Hamada [1] Liew et al. [9]
b
c d
1
2
3
4
5
6
22)42 22)4 22)49 22)63 22)48 22)73 22)71 22)51 21)99 25)59 25)5 26)37 25)71 25)41 26)02 25)31 26)18 25)40 35)66 35)5 35)60
49)84 * * 50)04 * 50)15 50)10 49)95 * 52)04 * 52)23 52)11 * 52)13 57)46 58)70 57)63 71)99 * 71)71
55)48 * * 55)95 * 56)23 56)13 55)72 * 59)75 * 61)78 60)09 * 60)80 57)46 58)70 57)63 72)04 * 71)71
82)13 * * 82)34 * 82)98 82)37 82)29 * 87)95 * * 88)13 * 88)13 96)67 98)58 97)05 102)28 * 101)8
99)47 * * 99)71 * 99)74 99)73 99)69 * 104)34 * * 110)6 * 100)6 100)88 102)0 101)1 125)30 * 124)8
106)53 * * * * * 107)1 * 111)71 * * 112)3 * * 112)88 * 113)3 131)01 * 131)4
frequency parameters decrease as the relative thickness increases, especially pronounced for the higher modes. The numerical results shown in Tables 6 and 7 are the frequency parameters for square plates with partial portions simply supported and other portions clamped along four edges (cases c and d). In Table 6, it is evident that the frequency parameters for case c increase gradually as the clamping ratio c/a increases from c/a"0)0 to 1)0. However, it is evident from Table 7 that there are abrupt changes in the frequency parameters at the onset of the partially clamped conditions [c/a"0)8}1)0 as shown in Figure 4(d)]. When the ratio c/a approaches the value 0)0, the frequency parameters approach those of a fully clamped plate in a more gradual manner. This indicated that the e!ects of the presence of the partially clamped constraint conditions at the middle portions of the plate edges on the frequency parameters are more signi"cant than those at the four corners. Tables 8 and 9 show the frequency parameters for the last two cases (cases e and f ) in which the partial portions of plate edges are free. It is concluded from the results in both tables that all the "rst eight frequency parameters decrease as the lengths of the free edge portions increase. This indicates that the presence of the free boundary conditions reduces the #exural sti!ness of the plate.
0)0 Dawe [30] Leissa [31] 0)2 Liew et al. [9] 0)4 Liew et al. [9] 0)6 Liew et al. [9] 0)8 Liew et al. [9] 1)0 Liew et al. [9] 0)0 0)2 0)4 0)6 0)8 1)0 0)0 Liew et al. [32] 0)2 0)4 0)6 0)8 1)0
0)01
0)2
0)1
Exact
c/a
h/a
19)7319 19)7313 19)7392 20)2695 20)33 21)6637 21)77 23)0250 23)10 23)5810 23)60 23)6324 23)65 19)0584 19)4079 20)5324 21)7427 22)3092 22)3758 17)4291 17)4485 17)6346 18)3584 19)1945 19)6179 19)6708
1 49)3027 49)3026 49)3480 49)3054 49)35 49)4732 49)55 50)4404 50)57 51)4673 51)55 51)6188 51)67 45)4478 45)4490 45)5359 46)1352 46)9120 47)0627 38)0732 38)1519 38)0737 38)1135 38)3941 38)7820 38)8602
2 49)3027 49)3026 49)3480 50)8555 51)06 53)9809 54)23 56)8170 57)04 58)3949 58)51 58)5652 58)65 45)4478 46)3395 48)5688 50)6572 51)9118 52)0899 38)0732 38)1519 38)4883 39)5858 40)6165 41)2401 41)3344
3 78)8410 78)8404 78)9568 80)5396 80)80 82)0791 82)22 82)7062 82)98 85)4168 85)67 85)9723 86)13 69)7167 70)5967 71)7044 71)9763 73)5516 74)0042 55)0024 55)1504 55)3488 55)8043 55)9031 56)4923 56)6667
4 98)5150 98)5144 98)6960 98)5366 98)72 99)1288 99)37 99)5498 99)74 99)8599 100)1 100)0753 100)3 84)9264 84)9340 85)2060 85)4666 85)6124 85)7589 64)9512 65)1453 64)9542 65)0503 65)1385 65)1954 65)2477
5
Mode sequence number
98)5150 98)5144 98)6960 100)0949 100)4 103)8443 104)4 109)1678 109)7 112)5682 112)9 112)9437 113)2 84)9264 85)6756 87)7422 90)7912 92)7783 93)0642 64)9512 65)1453 65)2163 65)9280 66)9742 67)6087 67)6994
6 127)9993 * 128)3049 128)0255 128)3 129)1305 129)6 130)4806 130)9 132)4767 133)0 133)4269 133)8 106)5154 106)5239 106)9654 107)6530 108)5066 109)0722 78)4340 * 78)4371 78)5757 78)7579 79)0455 79)2098
7
8 127)9993 * 128)3049 132)3153 132)9 135)3576 135)8 136)3339 136)9 139)3844 140)0 140)4192 140)9 106)5154 108)3563 110)0657 110)4567 111)9052 112)5269 78)4340 * 78)9999 79)4593 79)5936 79)9985 80)1603
Frequency parameters j"ua (oh/D for a simply supported square plate partially clamped along one edge (case a)
TABLE 4
928 F.-L. LIU AND K. M. LIEW
0)0 Dawe [30] Leissa [31] 0)2 0)4 0)6 0)8 1)0 0)0 0)2 0)4 0)6 0)8 1)0 0)0 Liew et al. [32] 0)2 0)4 0)6 0)8 1)0
0)01
0)2
0)1
Exact
c/a
h/a
19)7319 19)7313 19)7392 20)8153 23)7890 27)1923 28)7760 28)9216 19)0584 19)7602 22)1091 24)9652 26)4648 32)4894 17)4291 17)4485 17)8409 19)3258 21)1744 22)1808 22)3085
1 49)3027 49)3026 49)3480 50)7042 52)0058 52)4360 54)2772 54)6658 45)4478 46)2447 47)2625 47)4769 48)7020 61)9371 38)0732 38)1520 38)4388 38)9257 39)0203 39)5854 39)7559
2 49)3027 49)3026 49)3480 50)9993 56)0268 63)8521 68)7513 69)1927 45)4478 46)4315 49)8941 55)1860 58)7002 61)9371 38)0732 38)1519 38)5372 40)2660 42)7422 44)2736 44)4669
3 78)8410 78)8404 78)9568 82)5448 87)6068 88)3232 93)0406 94)3594 69)7167 71)5730 74)6567 75)0799 77)7063 86)7780 55)0024 55)1504 55)7153 56)8080 56)9672 58)0066 58)3574
4 98)5150 98)5144 98)6960 98)5570 99)7512 100)4066 101)4485 101)9944 84)92637 84)9412 85)4885 85)9314 86)3694 102)2068 64)9512 65)1453 64)9570 65)1498 65)3105 65)4525 65)5675
5
Mode sequence number
98)5150 98)5144 98)6960 101)3438 106)6384 118)2948 127)7640 128)6742 84)9264 86)3209 89)4921 95)7037 100)5394 103)1853 64)9512 65)1453 65)4597 66)6853 68)7717 70)1092 70)2846
6
127)9993 * 128)3049 131)2991 132)4741 135)6685 137)8256 139)7646 106)5154 107)9395 108)4811 110)0295 110)8247 123)5955 78)4340 * 78)8709 79)0287 79)4802 79)7244 80)0060
7
127)9993 * 128)3049 133)4482 143)2852 144)8855 151)4705 154)1989 106)5154 108)8049 113)3701 114)0861 117)1715 123)5955 78)4340 * 79)1373 80)2810 80)4405 81)4712 81)8396
8
Frequency parameters j"ua (oh/D for a simply supported square plate partially clamped along two opposite edges (case b)
TABLE 5
VIBRATING THICK RECTANGULAR PLATES WITH MIXED BOUNDARY CONSTRAINTS
929
0)0 Dawe [30] Leissa [31] 0)2 Liew et al. [9] 0)4 Liew et al. [9] 0)6 Liew et al. [9] 0)8 Liew et al. [9] 1)0 Liew et al. [9] 0)0 0)2 0)4 0)6 0)8 1)0 0)0 Liew et al. [32] 0)2 0)4 0)6 0)8 1)0
0)01
0)2
0)1
Exact
c/a
h/a
19)7319 19)7313 19)7392 20)5706 20)60 23)2656 23)33 27)7453 27)87 33)0430 33)23 35)9375 35)99 19)0584 19)5858 21)5179 24)9088 29)2339 32)4894 17)4291 17)4485 17)7866 19)0892 21)3628 24)2775 26)4534
1 49)3027 49)3026 49)3480 50)6148 50)68 54)5653 54)70 60)9077 61)15 68)7602 69)14 73)2324 73)40 45)4478 46)1783 48)6709 52)7616 57)9557 61)9371 38)0732 38)1519 38)4696 39)7921 41)8623 44)3325 46)1349
2 49)3027 49)3026 49)3480 50)6149 50)68 54)5657 54)70 60)9078 61)15 68)7604 69)14 73)2324 73)40 45)4478 46)1783 48)6712 52)7617 57)9561 61)9371 38)0732 38)1519 38)4696 39)7921 41)8624 44)3327 46)1349
3 78)8410 78)8404 78)9568 82)0756 82)24 91)1024 91)42 101)7207 102)2 107)2746 107)7 107)8890 108)2 69)7167 71)3255 76)3661 82)6074 86)2698 86)7780 55)0024 55)1504 55)7461 57)9598 60)4728 61)7785 61)9297
4 98)5150 98)5144 98)6960 98)5270 98)71 99)2415 99)46 104)2679 104)7 118)2646 119)2 131)1188 131)6 84)9264 84)9294 85)1869 87)3476 94)0125 102)2068 64)9512 65)1453 64)9525 65)0430 65)7928 68)0637 70)5488
5
Mode sequence number
98)5150 98)5144 98)6960 101)2944 101)5 108)4295 108)8 117)6520 118)2 127)2459 128)0 131)7522 132)2 84)9264 86)2389 90)0820 95)0594 100)0377 103)1853 64)9512 65)1453 65)5151 67)0918 68)9603 70)5812 71)5214
6
127)9993 * 128)3049 132)1607 132)6 141)8685 142)4 152)3085 153)1 160)4759 161)3 164)2996 165)0 106)5154 108)3120 113)0899 118)1746 121)7305 123)5955 78)4340 * 79)1265 80)8421 82)4034 83)2685 83)6969
7
127)9993 * 128)3049 132)1632 132)6 141)8833 142)4 152)3176 153)1 160)4766 161)3 164)2996 165)0 106)5154 108)3125 113)0935 118)1782 121)7308 123)5955 78)4340 * 79)1267 80)8431 82)4043 83)2686 83)6969
8
Frequency parameters j"ua (oh/D for a simply supported square plate partially clamped along four edges symmetrically from the corners (case c)
TABLE 6
930 F.-L. LIU AND K. M. LIEW
VIBRATING THICK RECTANGULAR PLATES WITH MIXED BOUNDARY CONSTRAINTS
931
TABLE 7 Frequency parameters j"ua (oh/D for a square plate with central portions of edges clamped and the portions around four corners simply supported (case d) Mode sequence number h/a
c/a
1
2
3
4
5
6
7
8
0)01
0)0 0)2 0)4 0)6 0)8 1)0 0)0 0)2 0)4 0)6 0)8 1)0 0)0 0)2 0)4 0)6 0)8 1)0
35)9375 35)9414 35)8996 35)5013 32)7372 19)7319 32)4894 32)4752 32)3632 31)8048 30)3556 19)0584 26)4534 26)4282 26)2504 25)7163 24)6335 17)4291
73)2324 73)2321 73)0169 71)4384 64)3999 49)3027 61)9371 61)8734 61)4967 59)8738 56)3506 45)4478 46)1349 46)0828 45)7579 44)8150 43)0716 38)0732
73)2324 73)2323 73)0375 71)9516 64)9865 49)3027 61)9371 61)8852 61)5624 60)5993 59)4845 45)4478 46)1349 46)0877 45)7782 45)1897 44)6425 38)0732
107)8890 107)8574 106)9872 102)8593 90)5992 78)8410 86)7780 86)5593 85)4789 82)5614 79)5927 69)7167 61)9297 61)8111 61)138 59)8430 58)7117 55)0024
131)1188 131)1247 130)7174 126)0229 111)4985 98)5150 102)2068 102)2056 101)8436 98)7472 94)2211 84)9264 70)5488 70)5494 70)5434 69)6038 68)0744 64)9512
131)7522 131)7170 131)1311 131)0986 125)5724 98)5150 103)1853 102)9603 102)2495 102)2254 101)9477 84)9264 71)5214 71)4174 70)8201 70)5839 70)5226 64)9512
164)2996 164)1918 162)0256 153)3462 138)7304 127)9993 123)5955 123)1601 121)3052 116)828 112)9509 106)5154 83)6969 83)5593 82)8252 81)4740 80)3515 78)4340
164)2996 164)1929 162)4860 157)8889 146)5304 127)9993 123)5955 123)2376 121)8769 120)0274 119)6739 106)5154 83)6969 83)5705 82)9082 82)2933 82)2045 78)4340
0)1
0)2
TABLE 8 Frequency parameters j"ua (oh/D for a simply supported square plate partially free along central portion of two opposite edges (case e) Mode sequence number h/a
c/a
1
2
3
4
5
0)01
0)2 0)4 0)6 0)8 0)2 0)4 0)6 0)8 0)2 0)4 0)6 0)8
19)7032 19)3223 17)9489 15)3086 18)9688 18)3806 16)7269 13)9729 17)2669 16)5009 14)7502 12)2333
49)0887 46)2526 37)4079 27)4287 44)7273 40)1943 31)0930 22)9320 36)9662 31)8639 24)4214 18)5469
49)2266 49)1094 48)4752 44)5607 45)2506 44)9621 44)0991 38)3113 37)8752 37)5109 35)9174 31)1711
78)6660 78)1793 57)5104 46)2229 69)2345 63)7399 46)5858 41)6353 54)5730 46)1791 36)5996 34)4099
97)3726 82)2857 74)9452 65)0414 81)7609 68)0927 63)7171 54)1729 60)9513 53)2956 49)1241 41)8988
0)1
0)2
6
7
8
98)5017 127)0872 127)7119 98)4244 109)3464 126)1870 84)5854 98)0630 114)0658 78)1043 90)3212 96)2321 84)8962 104)5299 105)8487 83)0393 84)7051 102)7395 70)4994 84)1098 89)1438 67)5186 72)3184 82)0657 64)9314 75)7469 77)8970 59)8742 64)7677 74)7803 54)3304 64)1567 64)2071 52)9609 53)9970 62)4719
932
F.-L. LIU AND K. M. LIEW
TABLE 9 Frequency parameters j"ua (oh/D for a square plate having simply supported, free and clamped constraint conditions at the same edges (case f, c /b"0)5) Mode sequence number h/a
c /a
1
2
3
0)01
0)2 0)4 0)6 0)8 0)2 0)4 0)6 0)8 0)2 0)4 0)6 0)8
28)5202 26)7086 24)1897 20)8682 25)7218 24)0569 21)5997 18)6420 21)7196 20)2634 18)0982 15)7477
53)1143 49)5960 39)6816 28)9284 47)2261 42)7166 32)8643 24)5777 38)3171 33)2580 25)5000 19)9065
68)9142 65)3659 60)8285 47)2733 58)2256 55)0997 48)1098 39)8452 44)0336 42)1687 36)5300 31)9287
0)1
0)2
4
5
6
7
8
86)4119 111)2899 122)0728 133)7538 148)1331 84)7196 90)8904 116)0664 120)7714 134)5360 61)1666 81)9392 88)3379 117)2949 121)9796 57)0899 70)2250 80)9607 96)3591 110)9087 74)5374 88)6812 97)0398 108)7663 114)9896 66)3203 73)8776 85)6998 96)1425 107)2972 52)2427 68)3299 72)5401 92)7412 93)8581 49)0395 58)0647 68)9074 75)4648 89)9096 57)2673 63)0577 68)9444 77)4884 80)6879 46)9952 55)9363 61)0501 68)5758 76)5097 40)6084 51)2475 55)1585 65)3044 67)5281 38)2642 43)9951 53)3836 55)3314 65)4473
The variation trends of the frequency parameters with the relative thickness ratio h/a shown in Tables 6}9 are all very similar to those for cases a and b observed in Tables 4 and 5, i.e. the frequency parameters decrease as the relative thickness ratio increases. 4. CONCLUSION
This paper considers the free vibration of moderately thick rectangular plates with mixed boundary constraints. A new numerical method, the di!erential quadrature element method, was used to perform the analysis. By using this method, the discontinuities of the boundary constraints are isolated by decomposing the whole solution domain into several continuous sub-domains and the accurate solutions are obtained. The convergence and comparison studies were carried out to establish the reliability and accuracy of the numerical results. Several example rectangular plates with various mixed constraint conditions have been analyzed and the numerical results for the "rst eight frequency parameters were presented consequently. The e!ects of the ratio of mixed constraints portion on plate boundaries and those of the plate relative thickness on the frequency parameters of plates have been discussed. Since the solutions for the frequencies of thick plates with various mixed boundary constraints are scarcely reported in the literature, the present solutions should be valuable to engineers and designers for their reference. REFERENCES 1. T. OTA and M. HAMADA 1963 Bulletin of Japan Society of Mechanical Engineering 6, 397}403. Fundamental frequencies of simply supported but partially clamped square plates.
VIBRATING THICK RECTANGULAR PLATES WITH MIXED BOUNDARY CONSTRAINTS
933
2. L. M. KEER and B. STAHL 1972 Journal of Applied Mechanics 39, 513}520. Eigenvalue problems of rectangular plates with mixed edge condtions. 3. Y. NARITA 1981 Journal of Sound and
934
F.-L. LIU AND K. M. LIEW
24. F.-L. LIU and K. M. LIEW 1998 ASME Journal of Applied Mechanics 65, 705}710. Static analysis of Reissner}Mindlin plates by di!erential quadrature element method. 25. F.-L. LIU and K. M. LIEW 1999 International Journal of Solids and Structures, 36, 5101}5123. Di!erential quadrature element method for static analysis of Reissner}Mindlin polar plates. 26. F.-L. LIU 1999 International Journal of Solids and Structures, Rectangular thick plates on Winkler foundation: di!erential quadrature element solutions (in press). 27. F.-L. LIU and K. M. LIEW 1999 ¹ransactions of ASME, Journal of
ANALYSIS OF VIBRATING THICK RECTANGULAR PLATES WITH MIXED BOUNDARY CONSTRAINTS USING DIFFERENTIAL QUADRATURE ELEMENT METHOD F.-L. LIU CAD/CAM ¸aboratory, Division of Engineering Mechanics, School of Mechanical and Production Engineering, Nanyang ¹echnological ;niversity, Singapore 639798, Singapore AND
K. M. LIEW Center for Advanced Numerical Engineering Simulations, School of Mechanical and Production Engineering, Nanyang ¹echnological ;niversity, Singapore 639798, Singapore (Received 30 November 1998, and in ,nal form 17 March 1999) A free vibration analysis of moderately thick rectangular plates with mixed boundary conditions is presented on the basis of the "rst-order shear deformation plate theory. The di!erential quadrature element method, a highly e$cient and accurate hybrid approach, has been employed. To establish the numerical model, the complex plate domain is "rst decomposed into small simple continuous sub-domains (elements) and the di!erential quadrature method is then applied to each continuous sub-domain to solve the problems. Compatibility conditions are developed for the conjunction nodes on the interface boundaries of elements in order to connect the elements. Convergence and comparison studies are carried out to establish the reliability of the solutions. The "rst eight frequency parameters are predicted for various types of thick rectangular plates with mixed edge constraints. 1999 Academic Press
1. INTRODUCTION
Problems of rectangular plates with mixed edge constraints have been investigated by many researchers adapting di!erent solution techniques. Ota and Hamada [1] predicted the fundamental frequencies of a simply supported rectangular plate partially clamped on the edge using a distributed moment function along the mixed edge. Keer and Stahl [2] studied the same problem by means of Fredholm integral equations of a second kind, and demonstrated the dependence of the fundamental frequencies on the clamped portion. Narita [3] employed a series-type solution approach to solve the problem and obtained the frequency parameters for wide ranges of mixed-edge rectangular plates. Other numerical methods such as the "nite element method [4], superposition method [5], spline "nite strip method [6], spline element method [7], and the Rayleigh}Ritz method [8}10] have also been used to study the free vibration of rectangular plates with discontinuous edge 0022-460X/99/350915#20 $30.00/0
1999 Academic Press
916
F.-L. LIU AND K. M. LIEW
constraints. Laura and Gutierrez tried to predict the fundamental frequency of the simply supported rectangular plates partially clamped along one and two edges by using the global di!erential quadrature method [11]. However, all the research work mentioned above is all con"ned to thin plates, solutions to thick rectangular plates with discontinuous boundary constraints are scarce in the literature. Only one paper was found to have treated the problem of free vibration of the Mindlin plate with mixed boundary constraints [12], but only the fundamental frequencies of the thin plates for two simple cases were calculated. The primary purpose of this paper is to solve the free vibration problems of thick rectangular plates with mixed boundary constraints. A hybrid approach namely the di!erential quadrature element method, which combines the di!erential quadrature method with the domain decomposition method, has been employed. The di!erential quadrature (DQ) method originated with Bellman and his associates [13] and was "rst applied to the structural analysis "eld by Bert and his associates [14}16]. Since then, many studies researches have been done in both the theoretical development and the engineering applications of the method. In all of the applications, this method yielded good to excellent results for only a few discrete points due to the use of the high-order global-based functions in the computational domain. An excellent review paper contributed by Bert and Malik [17] has summarized a detailed literature list on both aspects of the DQ method. Nevertheless, further application of the method has been greatly restricted by its drawback of not being able to be directly employed to solve the problem with discontinuities. To overcome such a drawback, the DQ method has been combined with the domain decomposition method "rst by Striz et al. [18}20] and then by Wang and Gu [21, 22] to solve the static problems of truss and beams and the static and free vibration problems of thin plates. This hybrid approach has been further developed by Han and Liew to solve the one-dimensional bending problem of the axisymmetric shear deformable circular plate [23], and by the present author to solve two-dimensional bending and vibration problems of thick rectangular plates and polar plates having discontinuities [24}28]. In this study, the di!erential quadrature method is combined with the domain decomposition method to deal with the free vibration problems of the moderately thick rectangular plates with mixed boundary constraints. 2. MATHEMATICAL FORMULATIONS
Consider a moderately thick rectangular plate with side lengths a;b as shown in Figure 1. The plate is divided into N elements based on the discontinuities in the # geometry, boundary constraints and materials used. Each element consists of an isotropic material and has uniform thickness and continuous boundary constraints on each edge. 2.1.
EIGENVALUE EQUATION AND BOUNDARY CONSTRAINT CONDITIONS
For a given element l of the plate, the eigenvalue equations for the free vibration problems can be written based on the "rst order shear deformation theory as
VIBRATING THICK RECTANGULAR PLATES WITH MIXED BOUNDARY CONSTRAINTS
917
Figure 1. Con"guration of a thick rectangular plate.
follows [29]:
(1)
(2)
*t o *w *w *w *t V# W " J # # , *y iG *t *x *y *x J
(3)
*t (1!l ) *t (1#l ) *t iG h *w o h *t V# V# W! J J V, J J #t " J J V *x *y *x*y *x 2 2 D 12D *t J J (1#l ) *t *t (1!l ) *t iG h *w o h *t J V# W! J J V, W# J #t " J J W 2 *x*y *x *y 2 *y D 12D *t J J
where w, t and t are the transverse de#ection, the rotation of the normal V W about the y-axis and the rotation of the normal about the x-axis respectively: h , J E , G and l are the thickness of the plate, Young's modulus, shear modulus J J J and Poisson's ratio, respectively; D is the plate #exural rigidity; and i is the J shear correction factor. For free vibration analysis, i is taken to be n/12. Letting w"=(x, y)e SR, t "W (x, y)e SR, t "W (x, y)e SR and V V W W u"X (E/[oa(1!l)] and substituting them into equations (1)}(3), one can obtain the following equations:
(4)
(5)
*W (1!l ) *W (1#l ) *W *= V# J J V# W!F #W "!X/aW , J *x V V *x 2 2 *y *x*y *= (1#l ) *W *W (1!l ) *W V# W!F J W# J #W "!X/aW , J *y W W *x*y *x 2 *y 2
*W !2 *= *= *W V# W " # # X= *y *y i(1!l )a *x *x J
(6)
918
F.-L. LIU AND K. M. LIEW
in which F "6i(1!l )/h . J J J
(7)
The moments and shear forces are expressed as
*t *t V#l W , J *y *x
(8a)
*t *t V# W , M "D l W J J *x *y
(8b)
M "D V J
*t *t 1!l V# W , JD M " J *y VW *x 2
(8c)
*w *w #t , Q "iG h #t . Q "iG h V W J J *y W V J J *x
(9a,b)
The boundary conditions for the edge x"0 can be expressed as follows. Clamped edge (C): w"0, t "0, t "0. V W
(10)
Hard simply supported edge (S): w"0, t "0, W
M "0. V
(11)
Soft simply supported edge (S ): w"0, M "0, M "0. V VW
(12)
Q "0, V
(13)
Free edge (F):
2.2.
M "0, M "0. V VW
RECTANGULAR FREE VIBRATION DQ PLATE ELEMENT
The lth element is further divided into N ;N grid points in the x and V W y directions as shown in Figure 2. Using the DQ procedures [17], the discrete governing equations for free vibration of the lth plate element can be expressed at each discrete point of the inner mesh into (1!l ) ,JW ,JV J CM (W ) !F (W ) C (W ) # GI V IH HK V GK J V GH 2 I K (1#l ) J # 2
,JV ,JV ,JW C CM (W ) !F C (=) "!X/a(W ) , J GI HK W IK GI IH V GH I K I (14a)
VIBRATING THICK RECTANGULAR PLATES WITH MIXED BOUNDARY CONSTRAINTS
919
Figure 2. Arrangement of grid points for element l.
(1#l ) J 2
,JW ,JV ,JW C CM (W ) # CM (W ) GI HK V IK HK W GK I K K
(1#l ) ,JV ,JW J C (W ) !F (W ) !F CM (=) "!X/a(W ) , # GI W IH J W GH J HK GK W GH 2 I K (14b) ,JV ,JW ,JV ,JW C (=) # CM (=) # C (W ) # CM (W ) GI IH HK GK GI V IH HK W GK I K I K !2 " X= , i"1, 2, 3, 2 , N , j"1, 2, 3, 2 , N GH V W i(1!l )a J and l"1, 2, 3, 2 , N #
(14c)
where CL and CM L (r"1, 2, 3, 2 , N ; s"1, 2, 3, 2 , N ) are the weighting PQ PQ V W coe$cients for the nth-order partial derivatives of w, t and t with respect to the V W global co-ordinates x and y. Equations (14a}c) can be further expressed in matrix form as KCdC"jBCdC,
(15)
920
F.-L. LIU AND K. M. LIEW
where j"(ohua/D) is de"ned as the frequency parameter: KC, dC and BC are the element weighting coe$cient matrix, element displacement vector and the element consistent mass matrix, respectively. Also ,t ,t ]2, dC"[w , t ,t ,w ,t , t , ,w V W V W 2 ,V ,W V,V ,W W,V ,W BC"[bC , bC , 2 , bC W , bC , bC ,2, bC W,2, bC V ,2, bC V W ]2, , , , , ,
(16) (17)
where bC , bC ,2, bC W , bC , bC ,2, bC W ,2, bC V , bC V ,2, bC V W are null , , , , , , matrices with 3N;3 elements, and bC ,2, bC W ,2, bC V , , and , \ , \ 2 are matrices with 3N;3 elements, in which only the three elements at bC V , \ ,W\ the corresponding point in each matrix are not equal to zero; the other elements are all zeros. Taking matrix bC , for example, it can be expressed as bC " H ,2, ,W CFFDFFE 0, 0, 0,2, 0, 0, 0
H,W> ,W> ,W>,2, ,W CFFFFFDFFFFFE 0, 0, 0, 0, !1/a, 0,2, 0, 0, 0
H,W> ,W>,2, , CFFFDFFE 0, 0, 0,2, 0, 0, 0
0, 0, 0,2, 0, 0, 0,
0, 0, 0, 0, 0, !1/a,2, 0, 0, 0, 2 0, 0, 0,2, 0, 0, 0, 0, 0, 0, ! , 0, 0,2, 0, 0, 0, i(1!l)a
0, 0, 0,2, 0, 0, 0
.
0, 0, 0,2, 0, 0, 0 (18)
The coe$cients in KC are determined by equations (14a}c). 2.3. COMPATIBILITY CONDITIONS In order to obtain a complete solution for the entire plate, the element weighting coe$cient matrices, the element displacement vectors and the element consistent mass matrices should be assembled into a global system equations for all the nodal points of the plate labelled from 1 to N. Therefore, the compatibility conditions for the conjunction nodes at the interface boundaries of adjacent elements need to be established. This includes the compatibility conditions for both the displacements and the force and moments. Obviously, the displacement compatibility conditions are automatically satis"ed at all the interface conjunction nodes since the same global nodal number is used for each conjunction node. Only the equilibrium conditions for the force and moments are needed to form the compatibility conditions. According to the locations of the conjunction nodes, the compatibility conditions for the free vibration DQ plate elements are given below: E
For conjunction nodes at interface boundaries of two elements: As shown in Figure 3(a), the compatibility conditions for the conjunction nodes at the interface boundaries of two elements l and l , which are connected in the x direction, can be expressed according to the equilibrium conditions as (QJ ) V !(QJ ) "0, (MJ ) V !(MJ ) "0, V , H V H V , H V H
(MJ ) V !(MJ ) "0. VW , H VW H (19a}c)
VIBRATING THICK RECTANGULAR PLATES WITH MIXED BOUNDARY CONSTRAINTS
921
Figure 3. Locations of conjunction nodes on the interface boundaries of adjacent elements: (a) two elements are connected in x direction: (b) two elements are connected in y direction; (c) four elements are connected at node m.
Similarly, for the conjunction nodes at the interface boundaries of two elements l and l connected in the y direction as shown in Figure 3(b), the compatibility conditions are given by (QJ ) W!(QJ ) "0, (MJ ) W!(MJ ) "0, W G , W G W G , W G E
(MJ ) W!(MJ ) "0. VW G , VW G (20a}c)
For conjunction nodes at which four elements meet: If four adjacent elements l , l , l and l are connected at a node m as shown in Figure 3(c), the compatibility conditions for the common node m are (QJ ) V W#(QJ ) V !(QJ ) W!(QJ ) "0, V , , V , V , V (MJ ) V W#(MJ ) V !(MJ ) W!(MJ ) "0, V , , V , V , V or
(21a}c)
(MJ ) V W#(MJ ) V !(MJ ) W!(MJ ) "0 VW , , VW , VW , VW (QJ ) V W!(QJ ) V #(QJ ) W!(QJ ) "0, W , , W , W , W (MJ ) V W!(MJ ) V #(MJ ) W!(MJ ) "0, W , , W , W , W
(22a}c)
(MJ ) V W!(MJ ) V #(MJ ) W!(MJ ) "0 VW , , VW , VW , VW The "nal global matrix forms of equation system for free vibration of the entire plate become Kd"jBb, (23) where K, d and B are the overall weighting coe$cient matrix, global displacement vector, and overall mass matrix of the plate.
922
F.-L. LIU AND K. M. LIEW
Solving the "nal algebraic eigenvalue equation system by the ordinary eigenvalue equation system solver, the solutions to the entire plate can be obtained.
3. NUMERICAL RESULTS AND DISCUSSIONS
Using the method described above, the vibration frequencies of rectangular plates with any arbitrary combination of mixed boundary constraints can be determined easily. In the present study, however, we only focus on six cases as shown in Figure 4(a}f ). The grid points employed in computation are designed by a x " +1!cos[(i!1)n/(N !1)],, i"1, 2, 3,2, N , V V G 2
(24)
b y " +1!cos[( j!1)n/(N !1)],, j"1, 2, 3,2, N H 2 W W
(25)
The Poisson's ratio is taken to be l"0)3. The eigenvalues are expressed in terms of the frequency parameter as j"ua(oh/D.
Figure 4. Con"gurations of rectangular plates with mixed boundary constraints analyzed by present numerical method: (a) a simply supported rectangular plate partially clamped along one edge, (b) a simply supported rectangular plate partially clamped along two opposite edges; (c) a simply supported rectangular plate partially clamped along four edges symmetrically from the corners; (d) a square plate with central portions of edges clampped and the portions around four corners simply supported; (e) a simply supported rectangular plate with central portions of two opposite edges free; (f ) a rectangular plate having simply supported, free and clamped constraint conditions at the same edges.
VIBRATING THICK RECTANGULAR PLATES WITH MIXED BOUNDARY CONSTRAINTS
923
Convergence studies are carried for the example problems (cases a}f ) to establish the grid points required in each element for obtaining the accurate solutions. The convergence patterns of the frequency parameters with the number of grid points in each element are presented in Table 1 for cases a}d and in Table 2 for cases e}f. It can be observed from Tables 1 and 2 that the convergence rate varies for di!erent plate con"gurations. Generally, the convergence rate of the plate which includes more singular points on boundaries is lower than that of a plate having a lesser number of singular points on the boundaries. And the convergence of frequency parameters for plates having partially free boundary conditions on the mixed boundaries (cases e and f ) is slower than that for plates without free boundary conditions on the periphery boundaries. However, for all the cases considered in this paper, a convergence to at least three signi"cant "gures can be achieved when 13;13 grid points are used in each element. Therefore, 13;13 grid points in each element will be used to generate all the results in the following studies to ensure the high accuracy of the numerical solutions. To examine the validity of the numerical method employed, the computed frequency parameters have been compared with the existing solutions available in the open literature. In Table 3, a comparison of the present solutions with those from various sources for cases a}d is presented. Since no solutions have been found for thick plates of cases a}d in the open publications, only the results for the thin plates (h/a"0)01) are compared here with the existing thin plate theory solutions. It is evident from Table 3 that the present solutions for the thin plates (h/a"0)01) agree very well with the analytical solutions established by Ota and Hamada [1], Keer and Stahl [2], Narita [3] and Gorman [5], and the Rayleigh}Ritz solutions given by Liew et al. [9]. However, the results obtained by the "nite element method [4], spline "nite strip method [6] and the spline element method [7] seem to be slightly higher than those given by the present method and the analytical method. Even so, they are well within the range of the acceptable accuracy for engineering applications. Based on the convergence and comparison studies, the present numerical solution approach has been employed to determine the vibration frequencies of the thick rectangular plates having various mixed boundary constraints. The frequency parameters corresponding to the "rst eight modes of free vibration for cases a}f have been computed and presented in Tables 4}9 for rectangular plates of various relative thickness ratios (h/a) and mixed edge constraint ratio (c/a). Some of the results, where possible, are compared with the existing solutions in the literature. For all the cases considered, the dependence of the frequency parameters on both relative thickness ratio and the mixed edge constraint ratio is observed. In Tables 4 and 5, it is observed that as the partially clamped ratio, c/a, approaches the value 1)0, the frequency parameters approach the values of a square plate with one or two edges fully clamped and the other edges simply supported for all the relative thickness varying from h/a"0)01 to 0)2. Also, it is observed that regardless of the relative thickness, the frequency parameters vary only slightly when the partial clamped ratio is within c/a"0)0}0)2 or c/a"0)8}1)0. The signi"cant variations occur, however, in the mid-range of c/a. This phenomenon has been discussed earlier by other researchers for thin plate problems [5, 9]. The
5;5 6;6 7;7 8;8 9;9 10;10 11;11 12;12 13;13 5;5 6;6 7;7 8;8 9;9 10;10 11;11 12;12 13;13
a c/a"0)5
b c/a"0)5
N ;N V W
Case
20)5800 21)0060 21)1124 21)1509 21)1656 21)1747 21)1798 21)1833 21)1857 22)6575 23)1927 23)4377 23)5041 23)5483 23)5652 23)5809 23)5864 23)5941
1 45)5777 45)3880 45)6841 45)7395 45)7404 45)7425 45)7451 45)7475 45)7485 46)7026 47)1896 47)2852 47)2862 47)2919 47)2925 47)2933 47)2938 47)2941
2 84)1521 48)9165 49)5431 49)6484 49)6606 49)6500 49)6602 49)6662 49)6706 84)6325 51)3033 52)3448 52)3253 52)3648 52)3859 52)4047 52)4249 52)4324
3 88)3709 71)1195 71)6598 71)7344 71)7528 71)7347 71)7345 71)7351 71)7359 90)8073 74)3694 75)4754 74)8898 74)9160 74)9229 74)9217 74)9221 74)9222
4 104)048 84)2639 85)6318 85)4580 85)3949 85)4044 85)4061 85)4075 85)4076 111)828 84)7122 86)0588 85)8996 85)8458 85)8532 85)8566 85)8576 85)8588
5
Mode sequence number
127)932 106)086 88)1667 90)1179 89)4480 89)3573 89)2280 89)2487 89)2612 132)023 107)488 91)0545 93)5035 92)3568 92)3010 92)2010 92)2261 92)2482
6
162)738 130)772 107)242 107)596 107)509 107)501 107)489 107)491 107)494 160)864 130)991 109)631 109)394 109)377 109)403 109)409 109)418 109)420
7
167)750 149)536 109)545 110)893 110)194 110)185 110)095 110)099 110)103 167)212 150)397 114)071 116)698 113)956 114)062 114)003 114)002 114)003
8
Convergence of frequency parameters j"ua (oh/D with increasing number of the grid points in each element for a square plate (h/a"0)1) with mixed boundary constraints (cases a}d)
TABLE 1
924 F.-L. LIU AND K. M. LIEW
d c/d"2/3
c c/a"2/3
5;5 6;6 7;7 8;8 9;9 10;10 11;11 12;12 13;13 5;5 6;6 7;7 8;8 9;9 10;10 11;11 12;12 13;13
25)8372 26)1419 26)1550 26)2383 26)2603 26)2898 26)3034 26)3181 26)3262 30)4226 31)4046 31)4080 31)4232 31)4316 31)4344 31)4415 31)4422 31)4467
53)2219 54)2464 54)2643 54)3448 54)3727 54)4062 54)4235 54)4401 54)4505 56)3835 58)8278 58)8566 58)8686 58)8982 58)9043 58)9231 58)9253 58)9366
53)2242 54)2544 54)2647 54)3483 54)3731 54)4080 54)4238 54)4412 54)4506 56)7672 60)2112 60)1536 60)1833 60)1776 60)1934 60)1929 60)2002 60)2002
82)6095 84)1686 84)1607 84)2144 84)2276 84)2552 84)2669 84)2801 84)2878 76)9366 81)4011 81)3033 81)4113 81)3984 81)4393 81)4386 81)4580 81)4582
87)4281 88)5577 88)8201 88)8930 88)9328 88)9705 88)9916 89)0112 89)0229 92)8638 96)9614 97)2763 97)2752 97)3242 97)3314 97)3591 97)3626 97)3785
95)4539 96)3892 96)6394 96)7107 96)7293 96)7612 96)7789 96)7939 96)8046 99)8602 101)994 102)192 102)188 102)179 102)181 102)182 102)181 102)182
117)889 119)263 119)483 119)517 119)529 119)555 119)566 119)579 119)585 111)512 115)053 115)137 115)259 115)247 115)297 115)298 115)323 115)324
117)901 119)265 119)493 119)518 119)533 119)556 119)568 119)579 119)586 114)401 119)639 119)743 119)761 119)754 119)762 119)763 119)766 119)767
VIBRATING THICK RECTANGULAR PLATES WITH MIXED BOUNDARY CONSTRAINTS
925
5;5 6;6 7;7 8;8 9;9 10;10 11;11 12;12 13;13 14;14 5;5 6;6 7;7 8;8 9;9 10;10 11;11 12;12 13;13 14;14
e c/a"1/3
f c /a"1/3 c/b"0)5
N ;N V W
Case
18)4158 18)5178 18)5968 18)6158 18)6434 18)6516 18)6698 18)6717 18)6842 18)6838 24)2577 24)4809 24)5330 24)5784 24)6297 24)6422 24)6742 24)6780 24)6981 24)6994
1 44)1013 41)2398 41)9654 41)9991 42)1940 42)2627 42)3701 42)4129 42)4675 42)4753 43)7685 44)4021 44)6073 44)6854 44)8046 44)8403 44)9041 44)9211 44)9596 44)9682
2 47)9030 44)9147 41)0195 45)0243 45)0650 45)0664 45)0834 45)0827 45)0898 45)0893 55)1378 55)8870 56)0533 56)1111 56)1733 56)2144 56)2483 56)2698 56)2903 56)3029
3 67)8778 67)3257 68)2262 68)5624 68)5857 68)6293 68)6471 68)6686 68)6784 68)6865 68)4028 71)4835 72)2066 72)3105 72)5153 72)6076 72)6947 72)7407 72)7866 72)8124
4 85)0881 68)9756 69)4348 70)9247 70)3221 70)8356 71)1729 71)4087 71)5995 71)7070 73)6311 75)6167 75)9186 76)2283 76)4404 76)6240 76)7552 76)8524 76)9360 76)9437
5
Mode sequence number
87)8688 84)0989 84)6037 84)8291 84)8092 84)8010 84)7951 84)8000 84)8023 84)8049 86)3611 91)3024 93)6428 94)1678 94)6111 94)9493 95)2008 95)3759 95)5246 95)6282
6
115)078 85)9829 87)1892 90)7161 91)7146 91)1472 91)6705 92)0078 92)2459 92)4255 95)2704 96)1577 96)4888 96)4848 95)5001 96)5249 96)5392 96)5513 96)5603 96)5674
7
128)096 108)974 102)019 104)539 104)493 104)440 104)456 104)487 104)531 104)544 106)535 108)120 109)084 109)065 109)078 109)142 109)173 109)203 109)220 109)239
8
Convergence of frequency parameters j"ua (oh/D with increasing number of the grid points in each element for a square plate (h/a"0)1) with mixed boundary constraints (case e}f )
TABLE 2
926 F.-L. LIU AND K. M. LIEW
VIBRATING THICK RECTANGULAR PLATES WITH MIXED BOUNDARY CONSTRAINTS
927
TABLE 3 Comparison studies for frequency parameters j"ua (oh/D of a sqaure plate with various mixed boundary constraints (h/a"0)01, c/a"0)5) Mode sequence number Case
Sources
a
Present Ota and Hamada [1] Keer and Stah [2] Narita [3] Gorman [5] Fan and Cheung [6] Mizusawa and Leonard [7] Liew et al. [9] Laura and Gutierrez [11] Present Ota and Hamada [1] Fan and Cheung [6] Liew et al. [9] Laura and Gutierrez [11] Striz et al. [20] Present Narita [3] Liew et al. [9] Present Ota and Hamada [1] Liew et al. [9]
b
c d
1
2
3
4
5
6
22)42 22)4 22)49 22)63 22)48 22)73 22)71 22)51 21)99 25)59 25)5 26)37 25)71 25)41 26)02 25)31 26)18 25)40 35)66 35)5 35)60
49)84 * * 50)04 * 50)15 50)10 49)95 * 52)04 * 52)23 52)11 * 52)13 57)46 58)70 57)63 71)99 * 71)71
55)48 * * 55)95 * 56)23 56)13 55)72 * 59)75 * 61)78 60)09 * 60)80 57)46 58)70 57)63 72)04 * 71)71
82)13 * * 82)34 * 82)98 82)37 82)29 * 87)95 * * 88)13 * 88)13 96)67 98)58 97)05 102)28 * 101)8
99)47 * * 99)71 * 99)74 99)73 99)69 * 104)34 * * 110)6 * 100)6 100)88 102)0 101)1 125)30 * 124)8
106)53 * * * * * 107)1 * 111)71 * * 112)3 * * 112)88 * 113)3 131)01 * 131)4
frequency parameters decrease as the relative thickness increases, especially pronounced for the higher modes. The numerical results shown in Tables 6 and 7 are the frequency parameters for square plates with partial portions simply supported and other portions clamped along four edges (cases c and d). In Table 6, it is evident that the frequency parameters for case c increase gradually as the clamping ratio c/a increases from c/a"0)0 to 1)0. However, it is evident from Table 7 that there are abrupt changes in the frequency parameters at the onset of the partially clamped conditions [c/a"0)8}1)0 as shown in Figure 4(d)]. When the ratio c/a approaches the value 0)0, the frequency parameters approach those of a fully clamped plate in a more gradual manner. This indicated that the e!ects of the presence of the partially clamped constraint conditions at the middle portions of the plate edges on the frequency parameters are more signi"cant than those at the four corners. Tables 8 and 9 show the frequency parameters for the last two cases (cases e and f ) in which the partial portions of plate edges are free. It is concluded from the results in both tables that all the "rst eight frequency parameters decrease as the lengths of the free edge portions increase. This indicates that the presence of the free boundary conditions reduces the #exural sti!ness of the plate.
0)0 Dawe [30] Leissa [31] 0)2 Liew et al. [9] 0)4 Liew et al. [9] 0)6 Liew et al. [9] 0)8 Liew et al. [9] 1)0 Liew et al. [9] 0)0 0)2 0)4 0)6 0)8 1)0 0)0 Liew et al. [32] 0)2 0)4 0)6 0)8 1)0
0)01
0)2
0)1
Exact
c/a
h/a
19)7319 19)7313 19)7392 20)2695 20)33 21)6637 21)77 23)0250 23)10 23)5810 23)60 23)6324 23)65 19)0584 19)4079 20)5324 21)7427 22)3092 22)3758 17)4291 17)4485 17)6346 18)3584 19)1945 19)6179 19)6708
1 49)3027 49)3026 49)3480 49)3054 49)35 49)4732 49)55 50)4404 50)57 51)4673 51)55 51)6188 51)67 45)4478 45)4490 45)5359 46)1352 46)9120 47)0627 38)0732 38)1519 38)0737 38)1135 38)3941 38)7820 38)8602
2 49)3027 49)3026 49)3480 50)8555 51)06 53)9809 54)23 56)8170 57)04 58)3949 58)51 58)5652 58)65 45)4478 46)3395 48)5688 50)6572 51)9118 52)0899 38)0732 38)1519 38)4883 39)5858 40)6165 41)2401 41)3344
3 78)8410 78)8404 78)9568 80)5396 80)80 82)0791 82)22 82)7062 82)98 85)4168 85)67 85)9723 86)13 69)7167 70)5967 71)7044 71)9763 73)5516 74)0042 55)0024 55)1504 55)3488 55)8043 55)9031 56)4923 56)6667
4 98)5150 98)5144 98)6960 98)5366 98)72 99)1288 99)37 99)5498 99)74 99)8599 100)1 100)0753 100)3 84)9264 84)9340 85)2060 85)4666 85)6124 85)7589 64)9512 65)1453 64)9542 65)0503 65)1385 65)1954 65)2477
5
Mode sequence number
98)5150 98)5144 98)6960 100)0949 100)4 103)8443 104)4 109)1678 109)7 112)5682 112)9 112)9437 113)2 84)9264 85)6756 87)7422 90)7912 92)7783 93)0642 64)9512 65)1453 65)2163 65)9280 66)9742 67)6087 67)6994
6 127)9993 * 128)3049 128)0255 128)3 129)1305 129)6 130)4806 130)9 132)4767 133)0 133)4269 133)8 106)5154 106)5239 106)9654 107)6530 108)5066 109)0722 78)4340 * 78)4371 78)5757 78)7579 79)0455 79)2098
7
8 127)9993 * 128)3049 132)3153 132)9 135)3576 135)8 136)3339 136)9 139)3844 140)0 140)4192 140)9 106)5154 108)3563 110)0657 110)4567 111)9052 112)5269 78)4340 * 78)9999 79)4593 79)5936 79)9985 80)1603
Frequency parameters j"ua (oh/D for a simply supported square plate partially clamped along one edge (case a)
TABLE 4
928 F.-L. LIU AND K. M. LIEW
0)0 Dawe [30] Leissa [31] 0)2 0)4 0)6 0)8 1)0 0)0 0)2 0)4 0)6 0)8 1)0 0)0 Liew et al. [32] 0)2 0)4 0)6 0)8 1)0
0)01
0)2
0)1
Exact
c/a
h/a
19)7319 19)7313 19)7392 20)8153 23)7890 27)1923 28)7760 28)9216 19)0584 19)7602 22)1091 24)9652 26)4648 32)4894 17)4291 17)4485 17)8409 19)3258 21)1744 22)1808 22)3085
1 49)3027 49)3026 49)3480 50)7042 52)0058 52)4360 54)2772 54)6658 45)4478 46)2447 47)2625 47)4769 48)7020 61)9371 38)0732 38)1520 38)4388 38)9257 39)0203 39)5854 39)7559
2 49)3027 49)3026 49)3480 50)9993 56)0268 63)8521 68)7513 69)1927 45)4478 46)4315 49)8941 55)1860 58)7002 61)9371 38)0732 38)1519 38)5372 40)2660 42)7422 44)2736 44)4669
3 78)8410 78)8404 78)9568 82)5448 87)6068 88)3232 93)0406 94)3594 69)7167 71)5730 74)6567 75)0799 77)7063 86)7780 55)0024 55)1504 55)7153 56)8080 56)9672 58)0066 58)3574
4 98)5150 98)5144 98)6960 98)5570 99)7512 100)4066 101)4485 101)9944 84)92637 84)9412 85)4885 85)9314 86)3694 102)2068 64)9512 65)1453 64)9570 65)1498 65)3105 65)4525 65)5675
5
Mode sequence number
98)5150 98)5144 98)6960 101)3438 106)6384 118)2948 127)7640 128)6742 84)9264 86)3209 89)4921 95)7037 100)5394 103)1853 64)9512 65)1453 65)4597 66)6853 68)7717 70)1092 70)2846
6
127)9993 * 128)3049 131)2991 132)4741 135)6685 137)8256 139)7646 106)5154 107)9395 108)4811 110)0295 110)8247 123)5955 78)4340 * 78)8709 79)0287 79)4802 79)7244 80)0060
7
127)9993 * 128)3049 133)4482 143)2852 144)8855 151)4705 154)1989 106)5154 108)8049 113)3701 114)0861 117)1715 123)5955 78)4340 * 79)1373 80)2810 80)4405 81)4712 81)8396
8
Frequency parameters j"ua (oh/D for a simply supported square plate partially clamped along two opposite edges (case b)
TABLE 5
VIBRATING THICK RECTANGULAR PLATES WITH MIXED BOUNDARY CONSTRAINTS
929
0)0 Dawe [30] Leissa [31] 0)2 Liew et al. [9] 0)4 Liew et al. [9] 0)6 Liew et al. [9] 0)8 Liew et al. [9] 1)0 Liew et al. [9] 0)0 0)2 0)4 0)6 0)8 1)0 0)0 Liew et al. [32] 0)2 0)4 0)6 0)8 1)0
0)01
0)2
0)1
Exact
c/a
h/a
19)7319 19)7313 19)7392 20)5706 20)60 23)2656 23)33 27)7453 27)87 33)0430 33)23 35)9375 35)99 19)0584 19)5858 21)5179 24)9088 29)2339 32)4894 17)4291 17)4485 17)7866 19)0892 21)3628 24)2775 26)4534
1 49)3027 49)3026 49)3480 50)6148 50)68 54)5653 54)70 60)9077 61)15 68)7602 69)14 73)2324 73)40 45)4478 46)1783 48)6709 52)7616 57)9557 61)9371 38)0732 38)1519 38)4696 39)7921 41)8623 44)3325 46)1349
2 49)3027 49)3026 49)3480 50)6149 50)68 54)5657 54)70 60)9078 61)15 68)7604 69)14 73)2324 73)40 45)4478 46)1783 48)6712 52)7617 57)9561 61)9371 38)0732 38)1519 38)4696 39)7921 41)8624 44)3327 46)1349
3 78)8410 78)8404 78)9568 82)0756 82)24 91)1024 91)42 101)7207 102)2 107)2746 107)7 107)8890 108)2 69)7167 71)3255 76)3661 82)6074 86)2698 86)7780 55)0024 55)1504 55)7461 57)9598 60)4728 61)7785 61)9297
4 98)5150 98)5144 98)6960 98)5270 98)71 99)2415 99)46 104)2679 104)7 118)2646 119)2 131)1188 131)6 84)9264 84)9294 85)1869 87)3476 94)0125 102)2068 64)9512 65)1453 64)9525 65)0430 65)7928 68)0637 70)5488
5
Mode sequence number
98)5150 98)5144 98)6960 101)2944 101)5 108)4295 108)8 117)6520 118)2 127)2459 128)0 131)7522 132)2 84)9264 86)2389 90)0820 95)0594 100)0377 103)1853 64)9512 65)1453 65)5151 67)0918 68)9603 70)5812 71)5214
6
127)9993 * 128)3049 132)1607 132)6 141)8685 142)4 152)3085 153)1 160)4759 161)3 164)2996 165)0 106)5154 108)3120 113)0899 118)1746 121)7305 123)5955 78)4340 * 79)1265 80)8421 82)4034 83)2685 83)6969
7
127)9993 * 128)3049 132)1632 132)6 141)8833 142)4 152)3176 153)1 160)4766 161)3 164)2996 165)0 106)5154 108)3125 113)0935 118)1782 121)7308 123)5955 78)4340 * 79)1267 80)8431 82)4043 83)2686 83)6969
8
Frequency parameters j"ua (oh/D for a simply supported square plate partially clamped along four edges symmetrically from the corners (case c)
TABLE 6
930 F.-L. LIU AND K. M. LIEW
VIBRATING THICK RECTANGULAR PLATES WITH MIXED BOUNDARY CONSTRAINTS
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TABLE 7 Frequency parameters j"ua (oh/D for a square plate with central portions of edges clamped and the portions around four corners simply supported (case d) Mode sequence number h/a
c/a
1
2
3
4
5
6
7
8
0)01
0)0 0)2 0)4 0)6 0)8 1)0 0)0 0)2 0)4 0)6 0)8 1)0 0)0 0)2 0)4 0)6 0)8 1)0
35)9375 35)9414 35)8996 35)5013 32)7372 19)7319 32)4894 32)4752 32)3632 31)8048 30)3556 19)0584 26)4534 26)4282 26)2504 25)7163 24)6335 17)4291
73)2324 73)2321 73)0169 71)4384 64)3999 49)3027 61)9371 61)8734 61)4967 59)8738 56)3506 45)4478 46)1349 46)0828 45)7579 44)8150 43)0716 38)0732
73)2324 73)2323 73)0375 71)9516 64)9865 49)3027 61)9371 61)8852 61)5624 60)5993 59)4845 45)4478 46)1349 46)0877 45)7782 45)1897 44)6425 38)0732
107)8890 107)8574 106)9872 102)8593 90)5992 78)8410 86)7780 86)5593 85)4789 82)5614 79)5927 69)7167 61)9297 61)8111 61)138 59)8430 58)7117 55)0024
131)1188 131)1247 130)7174 126)0229 111)4985 98)5150 102)2068 102)2056 101)8436 98)7472 94)2211 84)9264 70)5488 70)5494 70)5434 69)6038 68)0744 64)9512
131)7522 131)7170 131)1311 131)0986 125)5724 98)5150 103)1853 102)9603 102)2495 102)2254 101)9477 84)9264 71)5214 71)4174 70)8201 70)5839 70)5226 64)9512
164)2996 164)1918 162)0256 153)3462 138)7304 127)9993 123)5955 123)1601 121)3052 116)828 112)9509 106)5154 83)6969 83)5593 82)8252 81)4740 80)3515 78)4340
164)2996 164)1929 162)4860 157)8889 146)5304 127)9993 123)5955 123)2376 121)8769 120)0274 119)6739 106)5154 83)6969 83)5705 82)9082 82)2933 82)2045 78)4340
0)1
0)2
TABLE 8 Frequency parameters j"ua (oh/D for a simply supported square plate partially free along central portion of two opposite edges (case e) Mode sequence number h/a
c/a
1
2
3
4
5
0)01
0)2 0)4 0)6 0)8 0)2 0)4 0)6 0)8 0)2 0)4 0)6 0)8
19)7032 19)3223 17)9489 15)3086 18)9688 18)3806 16)7269 13)9729 17)2669 16)5009 14)7502 12)2333
49)0887 46)2526 37)4079 27)4287 44)7273 40)1943 31)0930 22)9320 36)9662 31)8639 24)4214 18)5469
49)2266 49)1094 48)4752 44)5607 45)2506 44)9621 44)0991 38)3113 37)8752 37)5109 35)9174 31)1711
78)6660 78)1793 57)5104 46)2229 69)2345 63)7399 46)5858 41)6353 54)5730 46)1791 36)5996 34)4099
97)3726 82)2857 74)9452 65)0414 81)7609 68)0927 63)7171 54)1729 60)9513 53)2956 49)1241 41)8988
0)1
0)2
6
7
8
98)5017 127)0872 127)7119 98)4244 109)3464 126)1870 84)5854 98)0630 114)0658 78)1043 90)3212 96)2321 84)8962 104)5299 105)8487 83)0393 84)7051 102)7395 70)4994 84)1098 89)1438 67)5186 72)3184 82)0657 64)9314 75)7469 77)8970 59)8742 64)7677 74)7803 54)3304 64)1567 64)2071 52)9609 53)9970 62)4719
932
F.-L. LIU AND K. M. LIEW
TABLE 9 Frequency parameters j"ua (oh/D for a square plate having simply supported, free and clamped constraint conditions at the same edges (case f, c /b"0)5) Mode sequence number h/a
c /a
1
2
3
0)01
0)2 0)4 0)6 0)8 0)2 0)4 0)6 0)8 0)2 0)4 0)6 0)8
28)5202 26)7086 24)1897 20)8682 25)7218 24)0569 21)5997 18)6420 21)7196 20)2634 18)0982 15)7477
53)1143 49)5960 39)6816 28)9284 47)2261 42)7166 32)8643 24)5777 38)3171 33)2580 25)5000 19)9065
68)9142 65)3659 60)8285 47)2733 58)2256 55)0997 48)1098 39)8452 44)0336 42)1687 36)5300 31)9287
0)1
0)2
4
5
6
7
8
86)4119 111)2899 122)0728 133)7538 148)1331 84)7196 90)8904 116)0664 120)7714 134)5360 61)1666 81)9392 88)3379 117)2949 121)9796 57)0899 70)2250 80)9607 96)3591 110)9087 74)5374 88)6812 97)0398 108)7663 114)9896 66)3203 73)8776 85)6998 96)1425 107)2972 52)2427 68)3299 72)5401 92)7412 93)8581 49)0395 58)0647 68)9074 75)4648 89)9096 57)2673 63)0577 68)9444 77)4884 80)6879 46)9952 55)9363 61)0501 68)5758 76)5097 40)6084 51)2475 55)1585 65)3044 67)5281 38)2642 43)9951 53)3836 55)3314 65)4473
The variation trends of the frequency parameters with the relative thickness ratio h/a shown in Tables 6}9 are all very similar to those for cases a and b observed in Tables 4 and 5, i.e. the frequency parameters decrease as the relative thickness ratio increases. 4. CONCLUSION
This paper considers the free vibration of moderately thick rectangular plates with mixed boundary constraints. A new numerical method, the di!erential quadrature element method, was used to perform the analysis. By using this method, the discontinuities of the boundary constraints are isolated by decomposing the whole solution domain into several continuous sub-domains and the accurate solutions are obtained. The convergence and comparison studies were carried out to establish the reliability and accuracy of the numerical results. Several example rectangular plates with various mixed constraint conditions have been analyzed and the numerical results for the "rst eight frequency parameters were presented consequently. The e!ects of the ratio of mixed constraints portion on plate boundaries and those of the plate relative thickness on the frequency parameters of plates have been discussed. Since the solutions for the frequencies of thick plates with various mixed boundary constraints are scarcely reported in the literature, the present solutions should be valuable to engineers and designers for their reference. REFERENCES 1. T. OTA and M. HAMADA 1963 Bulletin of Japan Society of Mechanical Engineering 6, 397}403. Fundamental frequencies of simply supported but partially clamped square plates.
VIBRATING THICK RECTANGULAR PLATES WITH MIXED BOUNDARY CONSTRAINTS
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2. L. M. KEER and B. STAHL 1972 Journal of Applied Mechanics 39, 513}520. Eigenvalue problems of rectangular plates with mixed edge condtions. 3. Y. NARITA 1981 Journal of Sound and
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F.-L. LIU AND K. M. LIEW
24. F.-L. LIU and K. M. LIEW 1998 ASME Journal of Applied Mechanics 65, 705}710. Static analysis of Reissner}Mindlin plates by di!erential quadrature element method. 25. F.-L. LIU and K. M. LIEW 1999 International Journal of Solids and Structures, 36, 5101}5123. Di!erential quadrature element method for static analysis of Reissner}Mindlin polar plates. 26. F.-L. LIU 1999 International Journal of Solids and Structures, Rectangular thick plates on Winkler foundation: di!erential quadrature element solutions (in press). 27. F.-L. LIU and K. M. LIEW 1999 ¹ransactions of ASME, Journal of