Journal o f Sound and Vibration (1973) 30(2), 257-260

VIBRATION OF RECTANGULAR PLATES WITH MIXED BOUNDARY CONDITIONS

It is well known that the finite element method of analysis is a versatile numerical technique for solving unorthodox structural and non-structural problems [1 ]. However, there does not seem to be any work in the literature which shows the application of finite elements in solving plate problems with mixed boundary conditions. The purpose of this present letter is to demonstrate the effectiveness of finite elements in solving vibration problems of plates with mixed boundary conditions through three typical examples,

The high precision triangular plate bending element developed by Cowper et al. [2] is used here to idealize the plate. This element has six degrees of freedom per node, namely w, Ow/Ox, Ow/Oy, O2W/OX 2, O2W[OxOy and O2w/Oy 2. For the sake of completeness and clarity, the degrees of freedom to be suppressed in the analysis on an arbitrary edge, y - b, are given below for various classes of boundary conditions:

(1) simply supported: w aw 0 2 w

O; ax ax 2

(2) clamped: w = aw aw a z w a 2 w

- - - 0 ; ax ay ax z - a x a y

(3) free: no condition;

(4) line of symmetry: aw 8 2 w

ay - ax ay - - = 0 ;

aw a" w (5) line of antisymmetry: +v = - - =

ax ax 2 =0 .

Similar conditions can be written on an edge x = a.

L

I<- a ->

SS

ss

Jss

$5

Figure 1. Simply supported square plate partially clamped along one edge from one corner, ct = alL; SS, simply supported; C, clamped.

257

258 LETTERS TO TIlE EDITOR

TABLE I

Fundamental frequency parameter, 2, of a square, partially clamped, shnply supported

plate (see Figure 1)

Keer and Stahl = Present work [3]

0 2-000 2"000 0-1667 2"083 2"042 0"3333 2"206 2"151 0'5 2"326 2"279 0'6667 2"389 2-366 0"8333 2"403 2"394 I 2"403 2"396

The first problem considered is the vibration of a square, simply supported plate, one edge being partially clamped from one corner (see Figure 1). A 6 x 4 mesh is used in the complete plate to solve this problem. The fundamental frequency is calculated for various values of ct, which is the ratio of clamped length to the length of the plate. Table 1 gives the fundamental frequency parameter, defined as 2=toV'-~pt/D]n 2 (where o~ is the circular frequency, p is the mass density, t is the thickness, L is the length and D is the flexural rigidity of the plate), for various values of 0r. For the sake of comparison the results given in reference [3], which are obtained by a continuum procedure, are also included. It can be seen from the table that the agreement between the two methods is good and the maximum difference of the results obtained by these two methods is about 2.5 %.

In the second problem, vibration of a simply supported square plate, partially clamped, symmetrically, from the midpoints of two opposite sides, is considered (see Figure 2) for various values of ct. The fundamental frequency parameter; Y. = 2/2, is presented in Table 2 for various values ofct, as obtained by using a 5 • 5 mesh in each quarter of the plate. The results obtained by the finite element method agree very well with those of continuum method of reference [3], the maximum difference being 2 %.

The third problem is vibration of a square, free-free plate, symmetrically simply supported, partially, at all the corners (see Figure 3), for various values of ~, where ~ is the ratio of the total length ofsimple support of an edge to the length of the plate. The fundamental frequency parameter ~ obtained with a 5 x 5 mesh in each quarter plate is presented in Table 3. It is

1 L

..V_

I<--~

SS C SS

SS

SS

SS C SS

Figure 2. Simply supported square plate partially clamped along two opposite edges, symmetrically from midpoint, ct = alL; SS, simply supported; C, clamped.

LETTERS TO THE EDITOR

TABLE 2

Fundamental frequency parameter, ~, of a square, partially clamped, shnply supported

plate (see Figure 2)

Keer and Stahl = Present work [3]

0 1 "000 1 "000 0-1667 1"411 1"383 0-25 1"450 1"437 0-3333 1"464 1-460 0"4167 1"466 1"466 0-5 1"467 1-467

259

--~ o k--

F L

F SS

F SS

k- L- -".-~

Figure 3. Free-free square plate partially simply supported along all edges, symmetrically from the corners. ~. = 2alL; SS, simply supported; F, free.

TABLE 3

Fundamental frequency parameter, ~, of a square, partially shnply supported, free-free

plate (see Figure 3)

Keer and Stahl Present work [3]

0"1 0-704 0"681 0-2 0"878 0"866 0"3 0-975 0'976 0'4 0999 - - 0"5 1 "000 1.000

clear from the table that the present results are in good agreement with the continuum solution of reference [3], with a maximum difference of about 3-5 ~o.

It can be concluded from the above examples that finite element analysis gives accurate results for problems of rectangular plates with mixed boundary conditions. This method is simple and easy to apply compared to the cumbersome continuum methods for solving problems of the type discussed in this letter.

260 LETTERS TO THE EDITOR

ACKNOWLEDGMENT

The authors take this opportunity to thank Dr C. L. Amba-Rao for his constant encourage- ment and suggestions during the course of this work.

Structural Engineering Division, Space Science attd Technology Centre, Trivandrum, Kerala, India

(Received 30 May 1973)

G. VENKATESXVARA RAO I. S. RAJU T. V. G. K. MURTHY

REFERENCES

1. O. C. ZIENKIEWICZ 1971 The Finite Element Method in Engineering Science. London: McGraw- Hill.

2. G. R. COWPER, E. KOSKO, G. M. LINDnERG and M. D. OLSON 1968 National Research Council o f Canada Aeronautical Report LR-514. A high precision triangular plate-bending element.

3. L. M. KEER and B. STAHL 1972 Transactions of the American Society of Mechanical Engineers, Journal of Applied 3Iechanics 39, 513-520.