Free vibration of partially supported triangular plates

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Comourers & Swucrures Vol. 51. No. 2. DD. 143-150. 1994 Copyright 0 1994 Ei&ier Scien& Ltd Printed in Great Britain. Allrightsmewed

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VIBRATION OF PARTIALLY SUPPORTED TRIANGULAR PLATES S. MIRZA and Y. ALIZADEH

Department of Mechanical Engineering, University of Ottawa, 161 Louis Pasteur, Ottawa, Ontario KIN 6N5, Canada (Received I9 Ocrober 1992)

Abstract-The

cracked plate has been length. The effects of the detached investigated in this paper. First-order predict the free vibration frequencies model based on the small deflection this class of problems. These analyses conditions.

idealized in this paper as partially supported with varying supported base length on vibration of these types of structures have been transverse shear-deformation (Mindlin) theory has been used to and modal shapes for isotropic triangular plates. A finite element linear theory has been developed to obtain numerical solutions for involve a wide range of variables, namely: aspect ratios and support

average normal rotation shear in the xz, yz plane

NOTATION length of plate at base, length of plate in cantilever direction detached length of plate (c = 2~‘) detached length from right, left, both sides time, thickness of isotropic plate non-dimensional parameters in plates ratio of detached length to the base length, aspect ratio number of nodes per element, number of elements in plate plate rigidity global Cartesian coordinates displacement in the X, y, and z directions differential with respect to x Young’s modulus, shear modulus, Poisson’s ratio Jacobian matrix element stiffness matrix, global stiffness matrix element mass matrix, general mass matrix isoparametric interpolation matrix eight nodes quadrilateral element partially clamped, free, free boundary of plate clamped, free, free boundary of plate

Y.ry,YZ, ipIg I.5 I$ =b/a

from

transverse

shear strain in xy, g plane isotropic plate with b/a = 1.5 aspect ratio for plate

1. INTRODUCTION Many airplane failures are attributed to the lack of information on the interaction between fracture, vibration and modal displacements of wings due to aerodynamic forces. This necessitates further research investigating the effect of the cracked length on the dynamic behaviour of these type of structures. By idealizing the cracked edge as a partially supported model, one will be able’to obtain crucial information on dynamic cracked triangular plates. A literature survey shows that very little work has been done so far on the partially clamped plates. Some investigations have been carried out and reported on the study of triangular, rectangular and other type of plates with the fully supported or complex supported conditions. Leissa [I, 21, presented a review of vibration of triangular plates. Gorman [3] used the superposition technique for the free vibration analysis of simply supported right triangular plates. Classical methods have also been used by Kim and Dickinson [4] and Bhat [5]. Anderson ef a/. [6] used Ritz technique to study this problem. Experimental results have been provided by a number of other researches [7.8]. Finite element method was employed by Mirza and Bijlani [9], Cowper et crl. [IO] and Utjes et rd. [I I]. The objective of this paper is to investigate the free vibration of partially clamped cantilever plates. The

element curvature vector strain vector, stress vector strain vector for bending, strain vector for shear generalized displacement vector generalized displacement vector at node i flexural stiffness matrix curvature-displacement matrix shear strain-displacement matrix shape function for node i two-dimensional isoparametric coordinates nodal value at node i stress. strain mass density non-dimensional frequency, frequency (rad./ sec.) normal rotation in .r. 3’ planes 143

S. MIRZA and Y. ALIZADEH

144

Y

5

C

F

b------b Fig. 1. Mindlin plate model. displacement type finite element method has been used in this study for analysing natural frequencies and modal shapes of plates. Numerical data have been generated on the effect of partial support (c/a) on the dynamic behaviour of plates.

2. FORMULATION

OF PLATE THEORY

The classical plate theory (Kirchhoff theory) neglects the transverse shear deformation. It has been established earlier that the effect of this shear deformation on thick plates is very important. Thick plate theory or Mindlin plate theory [12] provides a more realistic alternative to Kirchhoff theory which takes into account shear deformation effect. The plate assumptions used in the formulations of plate theory in this study are essentially those introduced by Mindlin [12]. Some of the important assumptions used in the development of the theory are: (1) the deflection of the midsurface of the plate is small, (2) transverse normal stresses are negligible, and (3) the normal to the midsurface of the plates remains straight after deformation but not necessarily normal. Based on assumption (3) the displacement field for a first-order shear deformation theory can be expressed as

4

Fig. 2. Right triangular C-F-F plate. u = Z~,(X,Y, r) U= z&(x,y, t) w = W(X,Y, t),

(1)

where u, v, and w are displacement components in the x, y, and z directions, respectively and OX,0, are the normal rotations in xz and zy planes, respectively, as shown in Fig. 1. The rotations 0, and OF can be expressed as

where 4, and & are the average normal rotation due to transverse shear in the xz and yz plane, i.e.

id& 4,)‘= br.-7Yclr. 3. FINITE ELEMENT

MODEL FOR PLATES

The displacement and normal rotations at any point in a typical element can be expressed as

Fig. 3. Right triangular plate with detached support: (r) detached length from right side. (I) detached length from left side, and (r - I) detached length from both sides.

Vibration Table

of partially

supported

triangular

plates

145

1. Comparison of frequency parameters R = r~a~~#?(pr/~)“~~ for C-F-F triangular plate (v = 0.3). a = 10 in, a/I = 158.73

Source of results

4

a?

a,

n,

b/a = 1 Present (55 elements) Kim and Dickinson [4] FEM (25 elements) [4] FEM (100 elements) [4] Mirza and Bijlani (25 elements) [9] Christensen [ 131 Bhat [5] Experiment [7]

6.162 6.164 6.455 6.122 6.159 6.160 6.173 5.930

23.445 23.457 22.937 23.020 23.061 23.700 23.477 23.400

32.625 32.664 31.473 31.853 33.289 32.540 32.716 32.700

56.203 56.149 55.074 54.88 I 55.915 55.010 56.405 55.900

b/a =2 Present (55 elements) Kim and Dickinson [4] FEM (25 elements) [4] FEM (100 elements) [4] Mirza and Biilani (25 elements) 191

6.641 6.622 7.054 6.608 6.696

28.552 28.435 29.033 28.185 29.499

49.667 49.398 50.534 49.500 51.011

70.036 69.65 1 70.417 69.070 77.759

where

or

where Ai are the nodal values w, OX,and f7,, respectively, at the ith node of the element, and nn is the number of nodes in each element. The N, are shape functions associated with node i and expressed in terms of the natural local coordinate system (5, q), i.e. N, E N,({, 9). In an isoparametric formulation, the coordinates of any point x and y in the element can be described by the same interpolation function N,([, q), and the coordinates of the ith node of the element. The straindisplacement relation can be expressed as

Ni., 0

Ni,,

(6)

The strain tensor can be split in two parts as follows: 1. The flexure strain

Table 2. Comparison of non-dimensional frequency n = ~u~+~@f/O)“~~ for partially supported P-F-F isotropic triangular plate ([email protected]). a = 10.0 in, 4 = b/a = 0.5, a/t = 158.73

Table 3. Comparison of non-dimensional frequency R = UJU~I$~(P~/D)~~~ for partially supported P-F-F isotropic triangular plate ($4 1). a = 10.0 in, 4 = b/a = 1.O, a/t = 158.73

r

r

I

Detached 0.00 0.20 0.40 0.60 0.80 1.00

length 5.4949 4.3819 2.5003 1.3183 0.6978 0.2497

from right side 15.0260 28.3327 12.7836 18.8064 7.1009 14.0991 4.7584 9.7395 3.4070 6.1591 1.7256 3.5009

29.9839 29.3605 28.4317 22.0616 17.8688 11.7934

Detached 0.00 0.20 0.40 0.60 0.80 1.00

length 5.4949 5.4948 5.4773 3.8891 2.1013 0.7818

from left side 15.0260 15.0210 8.9368 5.2435 4.1479 2.0248

29.9839 29.9099 24.5793 18.4809 14.9320 12.4445

I 28.3327 28.3228 14.4378 11.0172 8.5396 6.4374

r - I Detached length from both sides 0.00 5.4949 15.0260 28.3327 0.20 5.1963 14.6031 25.423 I 0.40 4.3819 12.7808 18.8041 0.60 3.393 1 9.3941 15.7397 0.80 2.4943 6.9583 8.9390 1.00 1.4463 4. I746 5.7830

29.9839 29.5869 29.2962 16.2653 13.7767 10.5808

Detached 0.00 0.20 0.40 0.60 0.80 1.00

length from right side 6.1628 23.4524 32.6357 5.7118 23.2154 27.2022 4.5470 16.3487 22.5726 3.2212 10.0867 20.7558 2.0280 6.8295 17.3768 0.7964 3.3326 I 1.6859

56.2332 55.3499 43.9650 36.9024 31.0417 25.3488

Detached 0.00 0.20 0.40 0.60 0.80 1.00

length from left side 6.1628 23.4524 6.1570 23.2970 6.0590 21.2530 5.6280 12.0340 4.5680 06.6020 2.0380 03.2980

32.6357 32.5590 29.2990 21.1320 19.5690 18.7700

56.2332 55.4390 33.2260 32.3380 32.1770 26.9160

from both sides 23.4524 32.6357 23.3837 3 I .2409 23.0503 27.1612 20.7998 22.8810 15.4641 20.9957 8.6740 17.5473

56.2332 55.9632 54.6209 48.1147 30.1222 21.7838

r - I Detached length 0.00 6.1628 0.20 6.0570 0.40 5.7062 0.60 5.1539 0.80 4.43 I2 1.00 3.0609

S. M~RZAand Y. ALEADEH

146

Table 4. Comparison of non-dimensional frequency R = oa*~z(pr/II)O’ for partially supported P-F-F isotropic triangular plate (i& 1.5). a = 10.0 in, 4 = h/a = 1.5, alt = 158.73 c/a r

i

Q,

Q,

Q2

c/a

%

Detached length from right side 40.1730 0.00 6.4697 27.2658 36.8189 0.20 6.2252 26.8185 29.5514 0.40 5.4601 23.6994 26.5725 0.60 4.4028 16.2749 24.3870 0.80 3.1652 10.7873 19.5185 I .oo 1.4074 4.8806

66.8654 65.1707 51.6153 48.5898 41.4429 32.3882

Detached length from left side 0.00 6.4697 27.2638 0.20 6.4550 27.0760 0.40 6.3120 25.6090 0.60 5.8700 19.6710 0.80 4.9530 I 1.8220 1.oo 2.7790 05.5210

40.1730 39.7680 37.03 10 29.3930 26.1640 24.0580

66.8654 66.4080 57.9180 42.6130 41.0870 38.1790

sides 40.1730 39.2575 36.4254 3 1.5567 25.5675 2 1.4750

66.8654 46.3642 64.7461 61.0506 53.7161 42.4543

r - 1 Detached length from both 0.00 4.4697 27.2658 0.20 6.4137 27.f302 0.40 6.2096 26.6688 0.60 5.8307 25.6014 0.80 5.2494 23.5366 1.oo 3.9819 14.7917

Table 5. Comparison of non-dimensional frequency R = w&$*(pt /O)O5 for partially supported P-F-F isotropic triangular plate (ipld,2). a = 10.0 in, 4 = b/a = 2.0, a/r = 158.73

r

Q,

Q3

122

Detached length from right side 0.00 6.6418 28.5620 49.7287 0.20 6.4879 28.0971 47.3072 0.40 5.9444 26.1350 39.8763 0.60 5.1111 21.3438 32.2500 0.80 3.9643 15.0798 28.4440 1.oo i .9309 6.6809 23.3974

70.0671 68.5476 64.3432 58.6410 52.0667 41.5462

I Detached length from left side

where (Bh,} is the curvature-displacement

0.00 0.20 0.40 0.60 0.80 1.00

6.6418 6.6130 6.4540 6.0420 5.2180 3.2280

28.5620 28.3680 27.3860 24.2530 17.0680 08.3980

70.0671 69.7180 68.2370 59.7100 54.4920 50.2350

49.7287 48.9730 45.1130 35.9530 29.5460 25.9940

r - 1 Detached length from both sides

matrix

(8)

0.00 0.20 0.40 0.60 0.80 1.00

6.6418 6.5957 6.4577 6.1689 5.6852 4.5578

28.5620 28.3763 27.9303 26.9803 25.5975 21.6774

49.7287 48.8642 46.6349 42.0038 34.7646 23.7242

70.0671 69.5502 68.1738 65.6086 62.0520 56.2654

The shape functions N, are given in the natural coordinate system. The element stiffness matrix and the mass matrix of the element are given as

2. The shear strain k: where

{&i

niz =,;r i&W,j.

is the shear strain~is~lacem~nt

(91

matrix

where {J) is the Jacobian rigidity matrix.

matrix and {D) is the

120 110 100

90 80 70 60 so 40

0

0.2

0.4

0.6

c/a-r

0.8

I 0

I

I

I

I

0.2

0.4

0.6

0.8

I

da-r-l

Fig 4. Variation of neon-djnlcnsio~l frequency with c,(~, for PmF F isotropic plate. u = IO in. h :u = 0.5. ct!r = 158.73.

Vibration of partially supported triangular plates

L 0

0

0.2 0.4 0.6 0.8

1 1 I I 0.2 0.4 0.6 0.8

c/a-r

0

c/a-l

147

0.2 0.4 0.6 0.8

I

c/a-r-l

Fig. 5. Variation of non-dimensional frequency with c/a, for P-F-F isotropic plate. a = 10 in, h/a = 1.0, a/l = 158.73. Finally, the total stiffness matrix and the total mass matrix are {K}=

f {K}““’ m=l

{M} = f {M}‘“‘, In=, where ne is the number

of elements

(13) in the plate.

4. NUMERICAL RESULTS

The present

formulation

nodes quadrilateral)

is based on a Q-8 (eight

plate bending element which is

considered to have three degrees of freedom (DOF) at each node. Unless otherwise indicated the symbols F, S, C, and P generally denote free, simply, clamped, and partially supported wedges, respectivley. Figure 2 shows a typical finite element grid for a sample problem. The computations in this investigation have been carried out by varying two essential parameters. These are the aspect ratio b/a and unsupported length c/a as shown in Fig. 3. In order to demonstrate the accuracy of the analysis, the results for several degenerate cases of triangular plates with fully supported side, i.e. c/a = 0, are computed and compared with previously published results.

_ 55 50 45

Legend

55

AMode 1 0 Mode2 o Mode3 0 Mode4

50 45 401

40

0

0.2 0.4 0.6 0.8

c/a-r

1

0

0.2 0.4 0.6 0.8

c/a-l

0

0.2 0.4 0.6 0.8

I

c/a-r-l

Fig. 6. Variation of non-dimensional frequency with c/a, for P-F-F isotropic plate. a = 10 in, b/a = 1.5, U/f = 158.73.

148

S. MIRZA and Y. ALIZADEH

and (iii) case c - r - I which have been explained in Fig. 3. Figures 47 show variation of frequency parameter with c/a for the lowest four modes for a P-F-F triangular plate with aspect ratios 0.5, 1.O. I .5, and 2.0. Some sample modal shapes corresponding to these frequencies have been given in Fig. 8.

For generating a database, the non-dimensional frequency parameter used in the tables is defined as

Trial computations for several alternative grid forms and number of elements were conducted and it was concluded that good convergence is achieved with 55 eight-node quadrilateral elements. The data have been generated for the first four frequencies in each case. In computations the length 0, which is along the support, is taken as 10.0 in (25.4 cm) and the thickness t = 0.063 in (0.16 cm). Computations have been carried out for the frequency in its non-dimensional form which is applicable for any kind of isotropic plate. Furthermore the modulus of elasticity and the mass density have been arbitrarily set to unity. Also, Poisson’s ratio is taken to be 0.3.

5. DISCUSSION AND RESULTS

A finite element process has been presented for the free vibration analysis of isotropic plates. On the basis of this study, the isoparametric quadrilateral eight-node element with reduced or selective integration emerges as the best suited for use with Mindlin plate theory. For this technique, integration in the transverse direction could be done numerically and individual sampling points are allowed to behave elastically. Furthermore, the transverse shear effects in thin plates can be effectively eliminated by employing a reduced (2 x 2) Gaussian integration rule. For both the thick and thin plates, two-point quadrature has been found to be satisfactory. Consistent mass matrix has been evaluated by using a 3 x 3 Gaussian quadrature rule. Computations have been carried out for the following three cases by varying the aspect ratio and the non-dimensional crack length c/a: (i) case rm~ detached length from right side, (ii) case I---detached length from left, and (iii) case r - I---detached length from both sides. Non-dimensional parameters were used to generate the general database. The material properties E and p were set to a value of unity making the data presented in the tables here, equally applicable to other materials.

4.1. Fully supported isotropic triangular plates Table 1 shows the comparison of the lowest four frequency parameters for cantilever plates with aspect ratios, b/a = 1 and 2. For both these cases it may be seen that the present results are in close agreement with the published results. This particular case has been essentially generated to provide a check on the computer program and convergence. 4.2. Partially supported cantilever triangular plates The frequency parameters for P-F-F, right triangular plates with aspect ratios (6/a) 0.5, 1.0, 1.5, and 2.0 and for several values of c/a are presented in Tables 2-5. In each table, the frequencies are presented for the three cases, (i) case c - r, (ii) case c - I,

55 50 -

o Mode 3 0 Mode 4

45 40 35 30 25 -

0

0.2

0.4

0.6

0.8

)

0

0.2

Fig, 7. Variation

of non-dimensional

0.4

0.6

0.8

0-

c/a-l

c/a-r frequency

with c/a, for P-F-F

a/t = 158.73.

c/a-r-l isotropic

plate. a = 10 in, b/u = 2.0.

149

Vibration of partially supported triangular plates aode= 1 D= 6.163

Mode= 3 Q= 32.636

Mode= 1

n= 4.547

Mode= 3 R= 22.573

Mode= 2 D= 23.452

Mode= 4 n= 56.233

Mode= 2 R= 16.349

diode= 4 1= 43.965

I

c/a = 0.0.

cla = 0.4.

Mode= 1

Mode= 2 R= 10.087

Mode= 4 a= 36.902

_I_

Ck=2.028

Mode= 3 R= 17.377

Mode= 2 R= 6.830

Mode= 4 CI= 3 1.042

II

cla = 06. Fig. 8. Some typical modal shapes for triangular isotropic plate b/a = 1.O, c - r.

It is concluded that the frequency is more sensitive to the non-dimensional crack length c/a for the case r than for the other two cases reported. As the aspect ratio increases, the frequency becomes less and less sensitive to c/a. This seems to be true for all three cases. It can be seen from Table S and Fig. 7 that for the aspect ratio 2 and for as much as 80% detachment of the support length there is very little change in frequency. It is obvious that when the aspect ratio increases, the plate behaviour becomes similar to the beam response. It is interesting to note that for b/a = 2, the fundamental frequency for the

cla = 08.

fully supported and the point supported triangular plates are about the same. In other words, we may state that when the plate vibrates in the beammode, the base crack has very little or no effect on natural frequency of triangular plates. Another important observation for all three types of detached supports is that the frequency is much more sensitive to c/a for higher modes. A careful study of the mode shapes indicates that, regardless of the plate aspect ratio, this high frequency variation occurs for the twisting modes and the combined bending-twisting modes.

S. MIRZA and Y. ALIZADEH

150 REFERENCES

7.

1. A. W. Leissa, Plate vibration research, 19761980: classical theorv. Shock Vibr. Dinesf 13. II-22 11981).

P. N. Gustafson, W. F. Stokey and C. F. Zorowski, An experimental study of natural vibrations of cantilevered triangular plates. 3. Aeronaur. Sci. 20, 331-337 (1953). 8. R. R. Craig and H. J. Plass. Vibration of hub-bin I mates. .

2. A. W. Leissa, Recent studies in piate vibrations: 1981-1985 Part i. Classical theory. Shock Vibr. Digest

AIAA Jnlj, 1177-l 178 (1965). 9. Mirza and Bijlani, Vibration of triangular plates. AIAA

19, II-18

(1987).

3. D. .I. Gorman, A highly accurate analytical solution for free vibration analysis of simply supported right triangular plates. J. Sound Vibr. 89, 107-118 (1983).

4. Kim and Dickinson, The free flexural vibration of right triangular isotropic and orthotropic plates. J. Sound Vibr. 141, 291-311 (1990). 5. R. B. Bhat, Flexural vibration of polygonal plates using characteristic orthogonal polynomials in two variables. J. Sound Vibr. 114, 65-71 (1987). 6. B. W. Anderson, Vibration of triangular cantilever plates by the Ritz method. 1. Appl. Mech. 365-370 (1954).

Jnl 21, 1472-1475 (1983). 10. G. R. Cowper, E. Kosko, G. M. Lindberg and M. D. Olson, Static and dynamic applications of a highprecision triangular plate bending element. AIAA Jnl7. 1957-1965 (1969).

II. J. C. Utjes, K. Ercoli, P. A. A. Laura and R. D. Santos. Transverse vibration of right triangular plates having the hypotenuse free. J. Sound Vibr. 102,445.-447 (1985). 12. R. D. Mindlin, Influence of rotary inertia and shear of flexural motion of isotropic elastic plates. J. Appl. Mech. 18, 31-38 (1951). 13. R. M. Christensen, Vibration of a 45 right triangular cantilever plate by gridwork method. AlAA Jnl 1, 1790-1795 (1963).