Free vibration analysis of penta, hepta-gonal shaped plates

Pergamon

0045-7949(94)00423-4

Com/~urers & SrrucruresVol. 62, No. 2, pp. 395-407, 1997 Copyri@tt0 1996 ElswicrScienceLtd Printedin Great Britain.All tightsresewed 004%7949/97 s17.00 + 0.w

FREE VIBRATION ANALYSIS OF PENTA, HEPTA-GONAL SHAPED PLATES S. S. A. Gbaxi, F. A. Barki and H. M. Safwat Department of Engineering Mathematics, University of Alexandria, Alexandria (21544), Egypt (Received 28 December 1993) Abstract-Lagrange’s equations of motion coupled with the finite element technique are applied to analyze the free vibration of pentagonal and heptagonal plates. Firstly, it is assumed that the elastic deformation of the plates is due to bending only. A high-precision 18 degrees of freedom triangular plate bending element, T18, is used. Results for isotropic, as well as orthotropic, plates, which have combinations of clamped, simply-supported and free edge conditions, are presented. Comparisons indicate that the results are in good agreement with those available in the literature. The effects of the composite iilament angle for orthotropic and laminated plates are discussed. In the second part, the transverse shear effect is accounted for and a higher-order T36 finite element is employed. Results for isotropic and orthotropic plates which have fully clamped and fully simply supported edges are obtained. The effects of the plate thickness-to-length ratio on the natural frequency coefficients are discussed. Copyright 0 1996 Elsevier Science Ltd.

INTRODUCTION

In the last decades, many analytical and numerical methods for studying vibrations of plates of various shapes and with different boundary conditions have been proposed. Most of the analytical methods concerning this subject are to be found in [l, 21. The finite element and the boundary element methods are of the well developed computer-oriented methods, which have gained great popularity because of their generality and high accuracy. The purpose of the present work is to study the behaviour of free vibration of plates which have pentagonal or heptagonal planforms. The finite element technique will be used. A limited amount of literature which concerns the free vibration of plates which have high order polygonal boundary shapes is available. For plates with materials of isotropic characteristics, Laura et al. [3] applied the complex-variable theory to the determination of the fundamental frequency coefficient of fully simply-supported and fully clamped plates. The transformed equations of motion were approximately solved by using Gale&in’s method. Yu [4] analyzed the same problem by conformally transforming the many-sided plate onto a unit circle and using the Rayleigh-Ritz method. Other analytical methods were proposed by Conway [5], Walkinshaw et al. [6] and Narita et al. [7J. For polygonal plates which are composed of orthotropic materials, the free vibrational analysis was investigated, analytically, by Laura et al. [8], Narita [9, lo] and Bhat [l l] (whose results are given only for isosceles and right-angled triangular plates) under the restricting assumption of rectangular orthotropy.

It is important to point out that, in most of the previously published works, the analysis was limited to the determination of the fundamental natural frequency and it was tedious to extend the proposed analytical methods to predict the behaviour of the higher modes of vibration. Conversely, it was difficult to manipulate the problem in the cases of combined edge conditions and with general orthotropic characteristics. It is also of a great interest to mention that, in most of the previous investigations concerning the free vibration analysis of plates which have exotic boundary shapes, the effects of the transverse shear on natural frequencies were neglected. A popular approach to include the transverse shear deformation effects on the bending of elastic plates was proposed by Reissner [12]. However, a problem known as ‘shear locking’ is encountered when this approach is employed. One approach which totally avoids the shear locking problem is to consider the total transverse displacement of the plate as the sum of the displacement due to bending alone and that due to shear deformation alone. Such a superposition approach was first suggested by Kapur [13] in the formulation of a beam element and it was extended to plate static and dynamic analysis by Gallagher [14], Yang [15] and Chang et al. [16] in the formulation of triangular finite elements. Since the orthotropic plates of non-rectangular and non-circular shapes are widely used nowadays in modem technology, such as nuclear, ocean and space engineering and also in electronic packages, their accurate free vibration analysis becomes a stringent prerequisite for their design. So, it is of great

395

396

S. S.

A. Ghazi et al.

importance to employ mathematical techniques by which some of the restricting assumptions which were proposed to facilitate the analysis could be removed. The finite element technique is considered as one of the powerful tools for achieving such goals. In the first part of the present work, it is considered that the plate elastic deformations are due to bending only. A high-precision 18 degrees of freedom T18 element is used. This element is conforming and suitable for a great variety of structural boundary shapes. The history of the development of this element is found in [17-201. The simplest and fastest formulation, which will be followed here, is that found in [20]. In the second part, the transverse shear effect is accounted for. A higher order T36 finite element [16] is employed. The total transverse displacement of this element is expressed as the sum of the displacement due to bending and that due to shear deformation. The double-sized overall stiffness and mass matrices are reduced in size, as if only the total displacement was considered. Before undergoing a series of computational work. the results for isotropic and specially orthotropic plates are compared with those available in the literature. Comparisons indicate the accuracy and validity of the present technique. The frequency coefficients of plates which have some complex combinations of rigidly clamped, simply-supported and free edge conditions are presented for isotropic, as well as orthotropic and laminated plates. The effects of the fiber angle and of the plate thickness-to-length ratio are discussed. PART 1: ANALYSIS OF THE MOTION THE T18 ELEMENT

within the T18 element could be expressed in the following form: w = {A}‘{a},

where A is a column vector, its elements are those of a complete quintic polynomial expressed in terms of the area coordinates l and <, i.e

a is a column vector which consists of 21 a, interpolation functions to be determined: a2 a3

{a}‘={a,

{(%1’= {W u’,.! U’,,, w,,,,

(3)

Carrying out the multiplication in eqn (2) and substituting for the <, c coordinates of each of the three vertices in the general expressions of the nodal degrees of freedom, 18 equations in the 21 ai unknowns are obtained. The remaining three equations result from suppressing the normal slope of the three midsides points of the element. The solution of the 21 equations, after transforming from 5, [ into x, y quantities, is put in the following matrix form:

[QlW3

(4)

where Q is a (21 x 18) transformation matrix related to the element geometry, it is given explicitly in [20] with the element Q(4, 1) = 0, while the correct value of this element is 10.0, and {q} is a column vector which contains the element’s 18 degrees of freedom. Combining eqns (2) and (4) one obtains the following expression of w: w=

(1) where L is the Langrangian function defined as L = T - U; T is the total kinetic energy of the system; U is the total strain energy; q, and q; are the generalized coordinates and the generalized velocities respectively and the dot means, as usual, differentiation with respect to the time t. The Lagrangian L will be constructed based on the finite element technique. In this part of the analysis, the T18 element will be used. The x and y coordinates and all the deformations are non-dimensionalized by a characteristic length (a) which, for regular polygons, is the plate side length. The transverse displacement field w

WX?., w,,.i1, xi=1,2,3

{a) =

Consider a whole plate of any shape which freely vibrates as a holonomic conservative elastic system, its motion will be governed by Lagrange’s equations of motion:

a2,};

the superscript T denotes the transpose of a matrix or a vector. The six degrees of freedom of each vertex of the triangular element are the deflection, the slopes and the curvatures

BY USING

Formulation

(2)

WTIQl{s).

(5)

According to Kirchhoffs hypothesis, the strain energy U of the element is given by the following integration over the triangular area A:

U=fD,, ss

P-VPlWIdxdy,

(6)

A

where H is the curvature vector,

D is the elasticity matrix normalized by the y-axis bending rigidity, D,, . For especially orthotropic

Vibration analysis of shaped plates materials, the quantities D,, = E,,h3/12(1 - vIzv2,), D,, = D,, EJE,, and Dg3= G,,,h’/12 are flexural rigidities, in which E,, and Ez are Young’s moduli in the x and y directions, respectively, and GX,,is the shear modulus, v,* and v2, are Poisson’s ratios and h is the plate thickness. For filamentary materials composed of generally orthotropic laminates, the elements of the elasticity matrix are those defined in [211. The curvature vector H transforms x, y coordinates into 5, [ derivatives, w is substituted from eqn (5) and after carrying out integrations of the standard type, I-C

f(mn)=

r”‘[“drd[

=

m!n!

(m+n+2)!’

397

whole plate into eqn (1) and ensuring that the equilibrium and the compatability conditions are satisfied at all the external nodes within the subdivided plate. In the case of free vibration and with the coupling of the assumption of simple harmonic motion, they could be expressed in the following matrix form: ([fbl

- ~2Pwlb~ = W?

(12)

where K* is the overall bending stiffness matrix, M is the overall mass matrix, ~b is a column vector contains the generalized degrees of freedom which represent the generalized coordinates and 1 is the non-dimensionalized natural frequency coefficient defined by

the following expression of the element strain energy is obtained as

where Fb is a (21 x 21) symmetric matrix depending on the element elastic and geometrical properties. This matrix is given explicitly in [20] by a long subroutine. A very simple and compact subroutine for the generation of Fbwill be given in the Appendix. From eqn (7), the element bending stiffness matrix is readily found to be D22

Kl. = os

[Q1’[F~1[Q1.

(8)

The kinetic energy of the element is given by

T = fa4ph

w ‘2dxdy,

(9)

where p is the mass per unit area of the element. Substituting for w from eqn (5) and after some manipulations, the element kinetic energy may be derived in the form

f~4~h(2~)~~~T[QlT[~l,Ql~,‘~~w

where the (21 x 21) symmetric matrix B is given by [B] =

From eqn (lo), the element consistent mass matrix is given by

(11)

Equations of motion and boundary conditions The non-constrained equations of motion are derived by the substitution of the Lagrangian L of the

(13)

By multiplying both sides by R;’ , the generalized eigenvalue problem (13) is reduced to the following standard eignevalue problem: (14)

where B is the dynamical matrix, I is the identity matrix and p2 = 1/12. The polygonal plates considered here have pentagonal and heptagonal forms, the edges of which are combinations of rigidly clamped, simply-supported and free edge conditions. The boundary conditions for each type of support are as follows: Clamped edge. Since w and (&/an) are constants along the edge (w = @w/an) = 0), the following conditions also must be satisfied for all the external nodes belonging to that edge:

aw a% ah -=-== 0, as as2 asan

1 I-’ {A} {A}Td{d[. ss0 0

WI, = ~4~4W[QlTPl [Ql.

([f&l- ~2wfl)Mb~ = PI.

cm - B‘rmib 1 = w,

A

T=

where o is the natural frequency of the vibration. Since the plates under consideration have inclined rectilinear edges, with respect to the x and y axes, some of the associated boundary conditions, which represent the constraints, will be expressed as linear relations between some of the generalized coordinates. These relations will be substituted into eqn (12) to obtain the following unsymmetric generalized eigenvalue problem:

Wa)

where s is the edgewise direction coordinate and n is that notial to the edge. The two conditions @w/as) = @w/an) = 0 are equivalent to the following: aw -=-

aw=O

ax ay

S. S. A. Ghazi ef al.

398

and the two conditions (d2w/8s2) = (d2w/ds an) = 0 are transformed into the following relations:

s=c’e

-=

cz”

C?*w

a*W

w

ax ay

d.Y2’

(15c)

The bending moment Mn is evaluated, for each edge, in terms of M,, M,,, MxY and the angle of inclination of that edge, which, in turn, are substituted in terms of the material elastic constants and the curvatures. From the coupling of the two conditions, a*w/as, = 0 and M,, = 0, one obtains:

where C is a constant related to the edge’s angle of inclination, it takes the following values:

a*w ---T- -

ay

c=o

a*w c, __ axay’

azw

ax2-

-C

2

2 axay’

(16~)

for the edge AB

= -cot 0,

for the edge BC

= cot 8,

for the edge AE

where C = +2C’D,,

1

for the edge CD

= cot 0,

= -cot Q2 for the edge ED and EG = cot 8,

for the edge DH

= -cot e3

for the edge GH

Simply -supported edge.

as

M,=O,

as*

ax

?Y

and D, (i, j = 1,2, 3) are the elements of the elasticity matrix D. The relations (16~) are substituted in eqn (12) by analogy to those of (15~). Free edge. M,=O,

V”=Qn-%=O.

(16b)

UW

The definitions of V,, Q, and M,, are found in [22]. Since the second natural boundary condition V, = 0 leads to relations between the third-order partial derivatives of w, which are not contained in the generalized unknowns vector qb, only the first condition of (17a) will be satisfied here. The transformation of this condition leads to the following relation:

(17b)

(16a)

where M, is the bending moment per unit length associated with the cross-section whose outward normal is n. The condition dw/as = 0 is transformed into the following relation:

aw -=+c”

’

Cz = + 2c - CT,

and the angles 0,) O2and & are of stationary values according to the plate planform [Fig. l(a) and (b)]. The two conditons in eqns (15~) are substituted into eqns (12) by modifying the columns in the matrices Kb and M, which correspond to the degrees of freedom, Pw/ay*, for each node belonging to a clamped edge, by adding to each column the following modification factor: C’jv,} + C{v2}, where v, and v2 are the columns which correspond to the two degrees of freedom d*w/ax* and d2w/axay, of that node, respectively. The rows and the columns which correspond to the prescribed degrees of freedom d2w/ax2 and a*w/&ay are then deleted.

aw a*W ~~~-...._-_~,

T 2CD,, - 6C*D,, - 2D,, f 4CD,,

- C4D,, T 2C’D,, + D,, T 2CDz3

where C’ =

c, =

-C*D,,

-D,,

T 2CD,,

C2D12+ D2*+ 2CD2, ’ 2C2D,, - 2D2, f 4CD,, C2D,2 + D,, k 2CD,,

(a)

Fig. 1. Some polygonal plates. (a) Pentagonal plate, N = 3. (b) Heptagonal plate. (c) Triangular plate.

Vibration analysis of shaped plates

399

Table 1. Convergence of fundamental frequency coethcient L = oo2m plates (v = 0.3)

of isotropic

Order of the polygon

(5) N

NW

2 3 4 5

20 45 80 125

Ref. [3] ]41 [81 HOI

(7)

Clamped

Simply supported

18.098 19.588 19.779 19.808

10.417 10.869 10.972 10.990

19.955 20.868 19.850 20.010

11.007 11.531 11.151 -

N

NE

2 3 4

28 63 112

Clamped

Simply SUpported

8.793 9.099 9.047

4.799 5.043 4.986

9.061 9.292 9.620 9.127

5.061 5.133 -

t NE is the total number of elements.

Numerical results and discussion

To check the convergence and the accuracy of the present numerical results before undergoing a series of computational work, the cases of fully clamped/simply-supported, isotropic/specially orthotropic, pentagonal/heptagonal plates are examined. The results introduced for comparisons are the first mode solution values obtained by using the matrix iteration method [23]. One advantage of this method is that computational errors do not bring wrong results, since any error in one of the premultiplications of the trial vector by the dynamical matrix does not have a persistent damaging effect because the resulting vector in error could be looked upon as a new trial vector. Comparisons of fundamental frequency coefficient of isotropic plates between approximate results previously published in the litrerature and those obtained by the present technique are presented in Table 1. For the pentagonal plate, the results corresponding to four different mesh divisions which comprise a minimizing sequence are listed. As could be shown, monotonic convergence is achieved through increasing the number of elements for both cases of clamped and simply-supported plates. In contrast, the method yields good accuracy for the divisions N = 4 and N = 5. For the heptagonal plate, monotonic convergence is somewhat affected through changing the division from N = 3 to N = 4; this is because the size of the problem is nearly doubled through this changing process. Rounding errors may

be the cause of this phenomenonon. However, the differences between results obtained by these two divisions are very small. For all the following cases of study concerning the heptagonal plate, results corresponding to the two mesh divisions are obtained and the smallest are tabulated. It is known that an exact solution of the problem under consideration is, at best, difficult to be achieved. All the previous solutions introduced for comparisons here are approximate solutions and the degree of accuracy of each one depends upon the method used. In [3], Galerkin’s method is used after transforming the actual polygonal domain onto a circular domain in the complex plane. The general functional behaviour of the deflection w is then assumed in some way, so as to approximately satisfy the transformed differential equation and only the transformed essential boundary conditions. The natural boundary conditions are not satisfied. The accuracy and the convergence of the method are dependent upon the skill and the experience of the analyst in choosing the approximating functions. The solution in [3] in the case of a clamped square plate is approximately 1.4% higher than the exact solution, while it is nearly equal to the exact solution in the case of a simply supported square plate. The generalization of the method to the manipulation of complex cases such as plates which have different combinations of edge conditions is almost impossible. The well known Rayleigh-Ritz method was used in [4] by Yu. Also, the essential boundary conditions only

Table 2. Comparison between approximate methods and the exact solution for isotropic square plates (v = 0.3) Method of solution Present: F.E. [3] Galerkin [4] Rayleigh-Ritz Exact

Edge condition

Clamped

Present: F.E. Simply

i&ct

SUPPOrted -_

1,

1,

1,

L

1,

35.996 36.494 38.977 35.99

73.488 73.80

73.590 73.80

109.00 108.27

132.25 132.81

19.733 19.741 21.318 19.739

49.304 49.347

49.337 49.347

78.744 78.957

98.591 98.697

et al.

S. S. A. Ghazi

400 Table

3. Comparison

of

fundamental

1 = wa2&@&

frequency coefficient orthotropic plates

Order

D,lH, !

I

kef. I2

[S] [lOI

PI

2

I

12.09 12.54

i

I

D, IHx, 1 *

Clamped

Simply supported

10.121 10.621 10.196

5.47

6.889 7.412 6.971

3.739

-

I

30.37 30.76 30.74

16.24 17.31

2

II

13.88 14.82 13.96

7.39

2

18.49 18.66 10.70

10.19 10.45

2

2

8.45 9.05 8.54

4.68

1101

1101

Dx IHx,

14.898 15.23 15.21

[81

PI

22.09 22.36

(7) Simply supported

2

VOI 2

2

Clamped

8.114 8.51 _

could be satisfied. The convergence of the method could be ensured by employing minimizing sequences. For both cases of clamped and simplysupported square plate, the solutions in [4] are approximately 8.3% higher than the exact solution. It is expected that, for higher order polygonal plates, the differences between the solutions in both [3] and [4] and the true solution will be larger, since the accuracy deteriorates as the order of the polygon increases. In [8], the Rayleigh-Ritz method was used by Laura without examination of the case of isotropic square plate. The truncation error in [8] is expected to be large, since the authors retained only the first two terms of the finite series, which was assumed to express the deflection. The extension of the method to the complex cases of different combinations of edge conditions is not available for high order polygonals. For the analytical method proposed in [lo], the convergence is dependent upon the number of terms of the assumed double infinite series which are employed in the calculation. The method is inconve-

2

nient for computer programming and it is restricted to dealing only with clamped plates. In the present finite element technique, all the boundary conditions, without any restrictions, could be substituted during or after the assembly process. In general, the solution converges to the exact solution as the number of elements is increased and monotonic convergence is achieved by using certain divisions which comprise minimizing sequences. For the case of a square plate, the solution converges quickly to the exact solution, as could be shown from Table 2. It is expected that the finite element solution, for higher order polygonal plates, will be more accurate than solutions obtained by other approximate methods. The generality of the method for analyzing plates which have any complex combinations of edge conditions is straightforward. In Table 3, the fundamental frequency coefficients, i = wa=J&ii,,, for different sets of orthotropic parameters, are listed. For the case of clamped pentagonal, the present results are in

Table 4. Comparison of results of isotropic, isosceles, triangular plates which some combinations of edge conditions (d/a = b/a = 0.5, v = 0.3)

Wze condition

have

Source of results Present

ccc

1251 1261

186.80 187.58 186.81

311.64 315.57

389.82 389.64

474.54 486.02

552.50 556.01

Present [251 [271

98.66 98.696 98.68

197.12 197.39

256.11 256.79 256.6

333.49 335.67 335.6

392.48 395.42

CSF

Present t:ss;

47.86 47.85 47.84

125.78 125.5 125.5

170.53 171.7 171.5

241.87 243.0 243.4

288.52 289.5 291.6

SCF

Present [251

63.32 63.57 63.56

143.96 145.6 145.6

188.04 189.2 188.7

261.03 266.4 267.4

307.62 313.4 312.0

35.97 35.94 35.94

96.01 95.9 95.9

147.62 147.5 147.2

193.97 194.9 194.2

247.54 247.6 247.0

sss

1281 Present CFS

specially

of the polygon

(5) DxIH,

of

Vibration analysis of shaped plates

401

Table 5. Comparison of results of isotropic, isosccles, triangular plates which have some combinations of edge conditions (d/u = 0.5, b/a = 0.3, v = 0.3)

We condition CSF SCF CFS

source of reSUlt3 Present

1, 81.89

1, 205.51

1, 316.07

14 419.60

1, 479.58

[251

82.48

207.10

323.30

437.20

496.60

Present (251

143.72 144.80

266.63

380.90

510.73

536.68

271.70

387.40

528.60

550.30

Present [251

54.02

133.57

136.10

329.04 329.50

355.38

54.89

218.89 221.00

reasonable agreement with those given in both [8] and [lo], while, for clamped heptagonal, they are in good agreement with those. given in [lo] only. The accuracy of the results of [8] deteriorates as the order of the polygon increases. For simply-supported pentagonal plate it is believed that the present results are more accurate than those available in [8], since they are obtained without satisfaction of the natural boundary condition M. = 0. The case of orthotropic simplysupported heptagonal plate is analyzed here, to the best knowledge of the writers, for the first time. In all tables which follow, the symbols C, S and F are used to denote clamped, simply supported and free edges, respectively, with the first indicating the condition along side (l), the second along side (2), etc. The numbering of the plate sides is shown in Fig. l(a)-(c). Also, in all the following cases of study, the resulting unsymmetric eigenvalue problem is solved by employing the available Eiseback routines [24]. Only the first five frequency coefficients obtained from full mode solutions are tabulated. To check the validity and the accuracy of the present technique for analyzing high order polygonal plates which have some complex combinations of edge conditions, the cases of isotropic, isosceles, triangular plates (whose results ae available in numerous pieces of literature) are examined. The mesh division used in subdividing the plate is N = 10, which gives a total of 100 elements. First, the case of an isosceles, right, triangular plate of aspect ratio b/a = 0.5 is investigated. The results are shown in Table 4 for five different combinations of edge conditions. As can be noticed, the results, especially for the fundamental fequency coefficients, which are of great importance, are in good agreement with those previously published in the literature. For higher

368.10

modes of vibrations, the maximum discrepancy between the present results and those given in [25] is about 2.36% and occurring in the second antisymmetric mode in the case of a fully clamped triangle. In Tables 5 and 6, the results of moderately flat and moderately narrow isosceles triangles of aspect ratios 0.3 and 1, respectively, and which have three different combinations of edge conditions, are listed. For the plate with an aspect ratio of unity, the results are in good agreement with those published in [25] for all modes of vibrations and for the three different combinations of edge conditions. The differences between the present results and those given in [25] are noticeable for the more acutely angled plate with an aspect ratio of 0.3. The present solution gives values of the frequency coefficients which are lower than those indicated in [25] for all modes of vibrations and for the three cases of different combinations. The maximum percentage difference between the two sets of results is about 4% and this occurs in the fourth mode of vibration of the C-S-F case. The frequency coefficients of isotropic/orthotropic pentagonal plates which have some combinations of clamped and simply-supported edge conditions are shown in Tables 7 and 8. As can be noticed, the results are bounded by those corresponding to the fully clamped and the fully simply-supported plates, for all modes of vibration. Conversely, the results converge to those of a certain bound as the number of edges which have similar conditions to that bound increases. The pronounced effect of the boundary conditions on the frequency coefficients could be cleared as follows. The rows and the columns of the stiffness and the mass matrices, which cause the dominant effect of the element energies and thus of the system energies, are those corresponding to the

Table 6. Comparison of results of isotropic, isoceles, triangular plates (d/a = 0.5, b/u = 1,v = 0.3)

Jsw

condition CSF

Present 1251

23.29 23.00

64.13 63.52

75.14 74.19

122.37 121.7

147.21 146.3

SCF

Present P5l

21.29 21.28

59.94 59.92

73.46 73.60

116.14 116.40

145.08 145.50

CFS

Present i25.l

25.32 25.08

72.23 71.54

74.64 74.14

136.05 136.0

148.21 148.0

402

S. S. A. Ghazi

et al.

Table 7. Frequency coefficients of an isotropic pentagonal plate with some combinations of clamped and simply supported edges Edge condition ccccc scccc ccssc ssccs cssss sssss

*I 19.8084 18.2092 15.8880 14.1232 12.7331 10.9896

4 40.4754 35.9307 33.8680 21.0974 28.2437 27.6566

lower order terms of the quintic polynomial, by which the deflection is assumed to be varying over the element domain, i.e. the rows and the columns which correspond to w, &v/ax and aw/ay. The effects of the higher order terms dZw/ax2, a*w/dx dy and a*w/a$ are small. If the edge is free, there are no constraints imposed on the three degrees of freedom, w, &~/ax and awl@, thus the flexibility of the system is large and the frequencies are very small. In the case of a simply-supported edge, only the first of the three degrees of freedom, i.e. w, is deleted. The stiffness of the system is somewhat increased and the frequencies are, in turn, increased. For the clamped edge, the three degrees of freedom vanish along the edge and the stiffness of the system is greatly increased. The substitution in the Rayleigh quotient indicates that the increase of the frequency is due to the increase of the system stiffness. In general, any constraints imposed on the system tend to reduce the number of degrees of freedom and render the system stiffer, thus increasing the values of the natural frequencies. In Tables 9 and 10, the frequency coefficients of an isotropic pentagonal plate, which has some complex combinations of free, simply-supported and clamped edge conditions, are indicated. It is important to point out here that, during the analysis, any corner node between two edges which have different conditions is considered to be belonging to the more flexible edge, in order to obtain solutions which represent lower bounds to the approximate solutions. Results for such cases are not available in the literature. Orthotropic pentagonal plates with various fiber angles (0) and two different boundary conditions are then analyzed. The material properties are based on the same values as those used in the preceding tables. Figures 2 and 3 show the variation of frequency coefficient ratio (A/&) for a range of 0 from 0” up to 180”. For clamped plates, both the first and the second modes monotonically decrease with an

4 40.9051 40.4828 36.3140 33.7357 3 1.8807 27.6902

ccccc scccc ccssc ssccs cssss sssss

I, 41.3643 38.7215 34.603 1 24.4739 19.3892 18.1849

A2 62.9857 60.1083 56.9855 40.3790 40.1959 35.9279

4 65.9915 62.5021 58.6654 54.8030 51.7236 48.8848

increase of H in the range from 8 = 0” up to 0 = 30”. Beyond this range, both the modes are insensitive to 0, where a nearly flat response is indicated. For the third mode, a small fluctuation of the value of (A/&) about unity occurs across the entire range of 8. In the case of a simply-supported plate, the variation of (A/&) for the first mode is somewhat different to that for the second and third modes in the range of 0 from 0” to 15”, since an increase of (A/&) of the first mode happens in this range. For the remaining range of 0, the behaviour is nearly the same for the three modes. The sensitivity of the three modes to 0 is concentrated in the neighbourhood of @= 60” and 0 = 120”, since a sudden drop in (A/A,) is caused at those values of 0. Finally, results for four-ply, symmetrically laminated plates are given in Tables 11 and 12, for four different laminates and two different boundary conditions. The elastic properties of each lamina are those previously used. Comparisons between results in Tables 8 and 11 show that, for the clamped pentagonal, the variation of 0 of the two inner layers caused a decrease in the fundamental frequency coefficients of the laminated plates compared to that of a unidirectional plate and this decrease takes its maximum value at 0 = 90”. For higher modes of vibration, a noticeable increase of the frequency coefficients is caused, due to the increase of 6’of the two inner layers. In the case of simply-supported plates, a pronounced monotonic increase and modification in the values of all frequency coefficients is caused as a result of the increase of 0 of the two inner layers. For the laminated, clamped, heptagonal plate, as is shown from Table 12, a small modification in the values of 1, is caused as a result of changing the fiber angles of the two inner layers from 0 = 0” up to Q = 90”. For higher modes of vibration (with the exception of the fourth mode for 0 = 30,45 and 60”) this modification in the values of 1. is more

Table 8. Frequency coefficients of an orthotropic combinations of clamped and simply-supported G,,/E*, = 0.25, Y = 0.3) Edge condition

14 65.3846 61.3542 58.0105 54.4746 50.9105 48.7896

1, 85.7218 82.7301 78.5356 68.5909 63.3020 60.6376

pentagonal with some edges (E,, /E2z = 10,

*q 86.8795 84.5014 79.9433 71.5825 68.4307 63.0800

A, 115.495 111.959 104.045 97.2552 86.2000 85.3537

Vibration analysis of shaped plates Table 9. Frequency wefficients of isotropic pentagonal with otle edge free (v =0.3) Edge condition FCCCC FSCCC FCSCS

FSCSC FCSSS FSSSS

A,

A,

1,

1,

1,

16.47 13.04 12.65 12.65 11.35 8.66

25.53 23.31 21.76 21.76 19.79 17.88

39.95 34.42 32.65 32.65 31.11 26.76

44.41 43.19 40.44 40.44 36.86 35.32

55.71 49.93 48.50 Bs.50 46.15 41.15

3rd mode

6

In the case of simply-supported plates, a sudden increase in the value of Iz, takes place at 0 = 45”, while, for other values of 8, the fundamental frequencies of laminated plates are lower than that of a unidriectional plate. For higher modes of vibration a noticeable increase of the frequencies is obtained for all values of 8. pronounced.

PART 2: TRANSVERSE

403

1.0

5 5 e Ll.

1st mode ~~-x-Y.-x-Y-Y-.Y-Y-~~~

AX

0.4 ’ ’ ’ 1 ’ ’ ’ ’ ’ ’ ’ 1 0 30 60 90 120 150 180 Fibre angle (9) Fig. 2. E%ectof the fiber angle on frequencv coefficient for clamped, orthotropic pentagonal (I, is the-value of I for 6 = 00).

SHEAR F#FECT

To include the transverse shear effect on vibration behaviour, a higher-order T36 finite element is employed. The total transverse displacement of this element is expressed as the sum of the displacement due to bending alone and that due to shear deformation alone, i.e. w=w,+w,,

(2W

The element strain energy due to both bending and shear is evaluated from the following integration

U= q

dy

d

(If9

where the subscripts b and s denote bending shear respectively. Following [lq, the two displa~men~ expressed as follows:

(Hf’[D](H}dx

and are where y is the transverse shear strains vector,

wb = W=l~

1

(194

ws = Witr

>

WW

where

A, a are previously defined and /I is analogous to a and contains the shear interpolation functions. The 12 degrees of freedom for each vertex are

G is the shear modulus matrix normalized by G,, . The shear wrrection factors d, and gr are included in shear moduli as follows: G,, = 8, G,,h and 3.5

3rd mode

3.0

After determining the elements of both a and /I, the displacement fields are given by

irt mode

Table 10. Frequency coe&ients of an isotropic pentagonal with two edges free (v = 0.3) Edge condition CFCCF CFCSF CFSCF SFSCF SFCSF SFSSF

A,

4

2,

4

&

13.52 11.31 11.31 8.47 8.47 7.16

15.04 14.48 14.98 11.93 11.93 10.64

21.18 20.06 20.06 18.92 18.92 17.55

36.56 32.62 32.62 27.66 27.66 26.18

38.32 37.38 37.38 32.40 32.40 29.74

Pibre

angle (0)

Fig. 3. E&et of the i&r angle on frequency wetlicient for sitnpiy-supported, orthotropic pentagonal (2, is the value of 1 for 0 = 0).

S. S. A. Ghazi

404 Table Type support

of

11. Frequency

Model

of laminated

2.2

*I

pentagonal

plates

A,

A,

A,

[O/901,

40.9659 41.1625 4 1.0725 40.8081

62.3063 63.4076 64.8991 66.8817

83.2750 92.1572 94.1414 94.8904

91.0363 92.6718 99.5247 106.6108

116.571 125.986 137.759 138.199

[O/301, [O/451, [O/601, [O/901,

19.3478 20.1816 20.5495 20.5958

37.4568 39.1227 40.9910 43.1319

62.4025 63.0799 63.4544 63.6724

64.8612 68.1330 72.4069 79.1832

87.9922 90.1050 91.1013 189.7585

Clamped :;z;

W'i Simply supported

coefficients

et al.

where the algorithms for generating the matrices F, and B,,, are given explicitly in [16].

Gz2 = o?,G,,h, where G,, = G,, will be assumed for the shear stiffness. The element kinetic energy is given by

Equations of motion and boundary conditions

T = fa4ph

(I+; + $)dxdy.

As previously mentioned, the non-constrained equations of motion are derived by the substitution of the Lagrangian L of the whole system into eqn (1) and satisfying the equilibrium and compatibility conditions at all the exernal nodes within the subdivided plate. In the case of free vibration and with the inclusion of simple harmonic motion assumption, they would be expressed in the following matrix form:

(22)

Substituting from eqns (20) into the expressions of the strain and kinematic energies and carrying out the required coordinate transformations, the element bending, shear stiffness matrices and consistent mass matrix are found to be

where K,,, KS are the overall bending and shear stiffness matrices and M is the overall consisent mass submatrix. The column vectors q6 and e contain the bending and shear generalized unknowns. The size of

a2D2, Kl, = -t2Aj [QITF’JIQl WI, = a4ph(2A)[QlT[B,,I[Q1. Table Type of support Clamped

Simply supported

Table

Clamped

12. Frequency

Model

(23)

coefficients

/I,

of laminated

A:

heptagonal

A3

plates

*4

I‘i

[O/01* [O/301,

18.2937 18.3868

26.9426 27.7513

39.9347 41.5572

45.3227 45.0094

57.2864 59.2842

;;z:;

P/901:

18.5219 18.5264 18.4782

28.5964 29.4900 30.4989

43.4532 45.0339 44.4405

45.1949 45.8490 49.3749

59.8709 59.9668 58.9617

[O/01, [O/301, [O/451, [O/601, [O/901,

9.9950 9.8316 11.7587 8.8528 9.2655

16.5926 17.4029 18.2249 19.0401 19.8903

28.5798 29.5557 31.1963 28.9470 29.0438

31.5521 31.1319 34.803 1 33.1242 36.0036

39.8014 41.1080 42.3000 42.3089 41.6311

shear effect on frequency

coefficients

of isotropic

13. Transverse

0.01 0.05 0.1 0.15 0.2

pentagonal

19.7968 19.5186 18.7959 17.7428 16.5163

40.4248 39.6175 37.3589 34.2904 30.9216

40.8612 39.9993 37.5975 34.3589 31.0019

65.1907 63.2823 58.1958 51.8102 45.5276

65.8853 63.9288 58.7151 52.1638 45.6987

10.9701 10.9167 10.7841 10.5734 10.2981

27.5653 27.2854 26.462 1 25.2397 23.7797

27.6220 27.3542 26.5648 25.3882 23.9754

48.5271 47.6826 45.2994 41.9458 38.2834

48.5967 47.7332 45.2996 42.0084 38.4029

Vibration analysis of shaped plates

405

Table 14. Transverse shear effect on freauencv coefficients of isotronic heutanonal

Clamped

0.01 0.05 0.1 0.15 0.2

9.0446 8.9949 8.8446 8.6091 8.3078

18.7389 18.5680 18.0614 17.2982 16.3702

18.8273 18.6538 18.1394 17.3653 16.4256

30.8192 30.4038 29.2036 27.4712 25.4801

30.8257 30.4093 29.2036 27.4712 25.4801

Simply supported

0.01 0.05 0.1 0.15 0.2

4.9854 4.9768 4.9501 49065 4.8474

12.7205 12.6634 12.4896 12.2153 11.8599

12.7355 12.6797 12.5095 12.2406 11.8916

22.8419 22.6586 22.1122 21.2830 20.2640

22.8421 22.6608 22.1216 21.3025 20.2947

eqn (24) could be reduced to the same size as one of eqns (12) by eliminating e from the two sets of equations represented by (24). After the elimination process, one obtains

The boundary conditions which will be incorporated into eqns (25) are those previously indicated in Part 1, for each case of edge condition. Numerical results and discussion

In the following analysis the shear correction factors 07,and 072are both assumed to be 516 and the tabulated results for each case are obtained by using the same mesh division which was used in thin plate theory solution. Isotropic plates. Tables 13 and 14, show the first fully five frequency coefficients of isotropic clamped/simply supported, pentagonal/heptagonal plates for a range of different thickness to length ratio from 0.01 up to 0.2. For plates under consideration, a three-dimensional elasticity solution is not available for comparison. Comparison between the present results and those corresponding to thin plate theory solution (Table 7) indicates the fact that increasing the plate thickness leads to the decreasing of the plate frequencies for all vibration modes. This is clearly seen from eqns (25), since the elements in the matrix [K,] are several orders of magnitude higher than those in the matrix [&,I when the plate is thin, thus the

effect of shear deformation is small. As the plate thickness increases, the magnitudes of elements in the matrix [K,] arc close to or even lower than those in the matrix [&] and the effect of shear deformation becomes noticeable. Another fact can be observed, which is that the effect of the transverse shear and the plate thickness to length ratio on frequency coefficients are affected by both the plate planform shape and the plate boundary conditions. For example, the thin plate theory solution for the fundamental frequency coefficient of a clamped isotropic square plate [16] is 1, = 36.005 and for a thick square of 0.2 thickness to length ratio is Ii = 27.584. The percentage difference of the value of 1, is about 23.39%. For clamped pentagonal and heptagonal plates, the corresponding percentage differences are 16.66% and 9.69% respectively. For simply-supported plates, the corresponding results of square, pentagonal and heptagonal are 9.67%, 6.14% and 2.77% respectively. This phenomenon could be explained as follows. For the same side length a, the number of constraints per unit area decreases as the number of plate sides increases. The stiffness of the system, in turn, is decreased. The measure of the transverse shear effect, as could be shown from eqn (25) is the matrix resulting from the multiplication K;‘Kb, which is a function of the system stiffness. The smaller the magnitudes of the elements of this matrix, the smaller the transverse shear effect on the system. In Tables 15 and 16, the frequency coefficients of clamped/simply-supported, pentagonal/heptagonal

Table 15. Transverse shear effect on frequency coefficients of orthotropic pentagonal plate (EJE, = 10, G,JE,, = G,,/E, = 0.25, vXY = 0.3) Type of support Clamped

Simply supported

hia 0.01 0.05 0.1 0.15 0.2

1, 41.2281 38.2907 31.9231 25.9215 21.3047

AZ 62.7990 58.7578 49.7863 40.9552 33.9150

1, 85.2971 76.3774 58.9686 45.0838 35.7427

A, 86.6311 81.2024 68.9435 56.8273 47.1844

A, 114.977 104.146 82.1155 62.6887 49.2145

0.01 0.05 0.1 0.15 0.2

18.1732 17.8995 17.1168 16.0113 14.7683

35.8916 35.0529 32.7586 29.7504 26.6417

60.5405 58.3349 49.9568 41.1709 34.1710

62.8957 58.8915 52.69108 45.9823 39.7465

85.0835 79.2524 66.5192 54.3387 44.8389

S. S. A. Ghazi et al.

406

Table 16. Transverse shear et&t on frequency coefficients of orthotropic heptagonal plate (E,/E, = 10, G,,/E, = G,/Er = 0.25, vXy= 0.3) Type of support Clamped

0.01 0.05 0.1 0.15 0.2

4 18.2698 17.7190 16.2587 14.4424 12.6726

4 26.9130 26.2277 24.3648 21.9527 19.5019

39.9363 38.6107 35.1806 28.8229 23.7491

45.1813 42.1137 35.3883 31.2211 27.4308

57.1831 54.4644 46.3778 38.2560 31.7391

Simply supported

0.01 0.05 0.1 0.15 0.2

9.9909 9.8942 9.6090 9.1839 8.6732

16.5852 16.4097 15.8950 15.1341 14.2302

28.5596 28.0842 26.7361 24.1290 20.9570

31.5029 30.3867 27.5244 24.8586 22.7820

39.7458 38.4759 35.1694 31.1419 27.2723

hia

plates made from orthotropic material are listed. As the transverse shear effect for an orthotropic plate is very large compared with that for

can be noticed,

an isotropic one. For example, for an isotropic clamped pentagonal plate of thickness to length ratio of 0.2 the percentage difference in 2, due to the transverse shear effect is about 16.66%, while, for an orthotropic clamped pentagonal plate of the same thickness to length ratio, it is about 48.49%. This is because of the great increase of the plate stiffness due to the orthotropic characteristics.

CONCLUSlON

The finite element technique is extended to the free vibration analysis of regular polygonal plates which have supported, inclined edges, where some of the boundary conditions are expressed as linear relations between some of the nodal variables. The accuracy of the solution has been demonstrated by comparisons between the present results and most of those published before. The transverse shear effect is included, for the first time, in the free vibration analysis of such plates. The benefit of the present approach is included in its generality and applicability to any complex configuration of edge conditions, any material properties and many cases which are very difficult to manipulate analytically. REFERENCES 1. L. Meirovitch, Analytical Methodr in Vibrations. Macmillan, New York (1967). 2. A. W. Liessa, Vibration ofplates, NASA SP-160 (1969). 3. P. A. Shahady, R. Passarelli and P. A. Laura, Application of complex-variable theory to the determination of the fundamental frequency of vibrating plates. J. Acous. Sot. Amer. 42, 806-809 (1967). 4. J. C. M. Yu, Application of conformal mapping and variational method to study the natural frequencies of nolvnonal nlates. J. Acous. Sot. Amer. 49, 781-785 ii9ii). _ 5. H. D. Conway, The bending, buckling and flexural vibrations of simply supported polygonal plates by point . _ __.matching. ASME. J. Appl. Mech. 28, 288-291 (IY60).

6. D. S. Walkinshow and J. S. Kennedy, On forced response of polygonal plates. Ingenier Archiev. 38, 358-369 (1963). 7. T. Irie, G. Yamada and Y. Narita, Free vibration of clamped polygonal plates. Bull. JSME 21, 16961702 (1978). 8. P. A. Laura, L. E. Luisoni and G. S. Sarmiento, A Method for the determination of the fundamental frequency of orthotropic plates of polygonal boundary shape. J. Sound. Vibr 70, 77-84 (1980). 9. Y. Narita, K. Maruyama and M. Sonoda, Transverse vibration of clamped trapezoidal plates having rectangular orthotropy. J. Sound Vibr. 85, 315-322 (1982). 10. Y. Narita, Flexural vibrations of clamped polygonal and circular plates having rectangular orthotropy. ASME. J. Appl. Mech. SO, 691692 (1983). 11. R. B. Bhat, Flexural vibrations of polygonal plates using characteristic polynomials in two variables. J. Sound Vibr. 114, 65-71 (1987). 12. E. Reissner, The effects of the transverse shear deformation on the bending of elastic plates. ASME. J. Appl. Mech. 12, 69-77 (1945). 13. K. K. Kapur, Vibration of a Timoshenko beam using finite element approach. J. Acoust. Sot. Amer. 40, 1058-1063 (1966). 14. G. R. Bhashyam and R. H. Gallagher, An approach to the inclusion of transverse shear deformation infinite element plate bending analysis. Cornput. Struct. 19, 35540 (1984). 15. A. T. Chen and T. Y. Yang, A 36 D.O.F. symmetrically laminated triangular element with shear -deformation and rotary inertia. J. Comp. Mater. 22, 341-359 (1989). 16. L. E. Chung and T. T. Chang. Transverse shear effect on vibation%f laminated plate using higher-order plate element. Comput. Struct. 39, 335-340 (1991). 17. J. H. Argyris, I. Fried and D. W. Scharpf, The TUBA family of plate elements for the matrix displacement method. Aeronaut. J. RAeS 72, 701-709 (1968). 18. K. Bell, A Retined triangular plate bending element. Int. J. Numer. Meth. Engng 1, 101-122 (1969). 19. G. R. Cowper, E. Kosko, G. M. Lindberg and M. D. Olsen, Static and dynamic applications of a high precision traingular plate bending element. AZAA. J. 7, 1957-1965 (1969). 20. C. Jeyachandrabose and J. Kirkhope, Explicit formulation for the high precision traingular plate-bending element. Comput_&ct. 19, 511-519 (1984). 2 1. J. R. Vinson and T. W. Chou. Comoosite Materials and Their Use in Structures. Applied Science, London (1975). 22. S. Timoshinko and S. W. Krieger, Theory of Plates and Shells. McGraw-Hill, New York (1959). 23. J. H. Wilkinson, The Algebraic Eigenvalue Problem. Oxford University Press, Oxford (1965).

Vibration analysis of shaped plates 24. B. T. Smith, Matrix Eigensystem Routines EZSPACK Guide, 2nd Bdn. Springer, Berlin (1976). 25. C. S. Kim and S. M. Dickinson, The free flexural vibration of isotropic and orthotropic general triangular shaped plates. J. Sound. Vibr. 141, 383403 (1992). 26. J. R. Kuttler and V. G. Sigillito, Upper and lower bounds for frequencies of trapezoidal and triangular plates. J. Sound. Vibr. 78, 585-590 (1981). 27. D. J. Gorman, Accurate analytical solution for the vibration

of

the simply

supported

triangular

plate.

AZAA J. 27, 647-651 (1989).

28. C. S. Kim and S. M. Dickinson, The free flexural vibration of right triangular isotropic and orthotropic plates. J. Sound. V&r. 141, 291-311 (1990).

APPENDIX Subroutine to generate the Fb matrix. The required inputs are the elements of the (3 x 3) symmetric matrix E which is defined in eqn (3) in [20].

SUBROUTINE FMAT (Ell,E12,E13,E22,E23,E33) COMMON/TWO/F (21,21) DO 10 1=1,21 CALL POWERS(I,MI,NI) DO 10 J=1,21 CALL POWERS(J,MJ,NJ) IPl=MI+MJ-4 JOl=NI+NJ Fi=FAC(IPl)*FAC(JQl)/FAC(IPl+JQ1+2) Tl=MI*MJ’(MI-l)*(MJ-1) IP2=MI+MJ-2 JQ2=NI+NJ-2 F2=FAC(IP2)*FAC(JQ2)/FAC(IP2+JQ2+2) T2=MI*NJ*(MI-l)*(NJ-l)+NI*MJ*(NI-l)*(MJ-1) IP3=MI+MJ-3 JO3=NI+NJ-1 F%=FAC(IP3)*FAC(JQ3)/FAC(IP3+JQ3+2) T3=MI’(MI-1)*MJ*NJ+MIfNI*MJ1(MJ*(MJ-1) IPQ=MI+MJ JQ4=NI+NJ-4 F4=FAC (IP4) lFAC (JQ4) /FAC (IP4+JQ4+2) T4=NIf(NI-l)*NJ*(NJ-1) IPS=MI+MJ-1 JOS=NI+NJ-3 F:=FAC(IP5)*FAC(JQS)/FAC(IPS+JQS+2) TS=NI*(NI-l)fMJfNJ+MI*NI*NJ*(NJ-1)

10

C

110

120

30

SUBROUTINE POWERS (Kl ,MJ, NJ) INTEGER Kl,MJ,NJ J=l L=l IF(Kl.LE.L)GO TO 120 J=J+ 1 L=L+J GO TO 110 L=L-J J-J-l MJ=J- (Kl-L-1) NJ=J-MJ RETURN END REAL FUNCTION INTEGER I FAC=l .O DO 30 J=Z,I FAC=FAC* J

RETURN

END

FAC (I)

407