Transverse vibration of thick rectangular plates—IV. Influence of isotropic in-plane pressure

Compufers & Structures Vol.49.No. I. pp.69-78,1993 Primedin GreatBritain.

004s7949/93 s6.00+ 0.00 0 1993 Pcrgamon PressLtd

TRANSVERSE VIBRATION OF THICK RECTANGULAR PLATES-IV. INFLUENCE OF ISOTROPIC IN-PLANE PRESSURE K. M. LIEW,~ Y. XIANG$ and S. KITIPORNCHAI$ TDynamics and Vibration Centre, School of Mechanical and Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 2263 $Department of Civil Engineering, The University of Queensland, Brisbane, Australia 4072 (Received 11 Augusr 1992)

Abstract-Part IV of this series of four papers reports a study of the effects of isotropic in-plane pressure on the vibration response of thick rectangular plates. In the theoretical formulation of the governing functional, the effects of transverse shear deformation and rotatory inertia have been incorporated using Mindlin’s first order shear deformation theory. Sets of mathematically complete two-dimensional polynomials have been used as the admissible displacement and rotation functions in the Rayleigh-Ritz procedure to derive the governing eigenvalue equation. In this paper, several rectangular plates with different combinations of boundary conditions and various aspect ratios a/b and relative thickness ratios r/b are considered. Sets of reasonably accurate vibration frequencies are presented for these in-plane loaded rectangular plates in the form of design charts. The results, where possible, are compared with established values from the open literature. Some basic discussion of the effects of shear deformation and rotatory inertia has been included based on trends in the numerical results.

NOTATION

% P 4 UJ r

length of plate width of plate unknown coefficients for transverse deflection w flexural rigidity of plate unknown coefficients for rotation fIX Young’s modulus shear modulus unknown coefficients for rotation 0, stiffness matrix isotropic in-plane pressure factor = Nb2/n2D buckling factor = NC,b2/rr2D isotropic in-plane pressure critical isotropic in-plane pressure degree of complete polynomial for w degree of complete polynomial for OX degree of complete polynomial for 0, variable in double summation maximum kinetic energy of plate thickness of plate strain energy of plate work done by external pressure transverse deflection along z direction longitudinal coordinate vertical coordinate transverse coordinate basic function for w m th polynomial term for w non-dimensional&d coordinate = y/b basic function for 0, n th polynomial term for 0, basic function for 0, Ith polynomial term for 0, shear correction factor frequency parameter = (ob2/n2)m frequency parameter in [lo] = ob 2(1 + v)p/E Poisson’s ratio energy functional rotation along y direction

rotation along x direction plate density per unit volume power for jth edge function angular frequency non-dimensionalized coordinate = x/a

1. INTRODUCTION

Analytical and numerical approaches have been used to solve the buckling and vibration of rectangular plates [l-6]. The effects of isotropic in-plane pressure on the vibration response have also been investigated [4-6]. However, most of these reported works have been based on classical thin plate theory which involves the Kirchhoff assumption that straight lines originally normal to the plate median surface remain straight and normal during the deformation process. This assumption simplified the problem considerably, but errors were introduced for thick plate analysis because the effects of transverse shear deformation and rotatory inertia were ignored. Mindlin [7,8] improved the classical thin plate theory by including the effects of shear deformation and rotatory inertia. Since then many papers have been published using Mindlin’s plate theory [9, lo]. This comprehensive series of four papers [l l-131 examines the free vibration analysis of thick rectangular plates based on Mindlin’s plate theory. The first three parts of the series have considered the vibration of thick rectangular plates with: (i) 21 possible combinations of boundary conditions [l 11, (ii) oblique internal line supports [12] and (iii) eccentric internal ring supports [13]. Part IV (this paper) 69

K. M. LIEW el al.

70

aims to provide sets of useful design charts for the vibration of thick rectangular plates subjected to isotropic in-plane pressure. The pb-2 Rayleigh-Ritz method [ 1l-221 has been further applied to solve these plate problems for a range of boundary conditions. In the approximation process, the trial functions are required for the spatial variation of the displacement, W,and the rotations, 0, and 0,. In this study, sets of mathematically complete two-dimensional polynomials have been used as the admissible transverse deflection and rotations. The data presented here are for thick rectangular plates of various aspect ratios a/b subjected to isotropic in-plane pressure. Three ratios of plate thickness to side length t/b =O.OOl, 0.10 and 0.20 have been considered. Various numbers of degree sets of polynomial terms have been used in obtaining the frequency parameters so that the convergence of the method can be examined in detail for only a selected number of types of plate problems. 2. FORMULATION OF GOVERNING EIGENVALUE EQUATION

Consider a flat, isotropic, thick, rectangular plate (Fig. 1) of uniform thickness t, length a, width b, Young’s modulus E, shear modulus G and Poisson’s ratio v. The plate may have any combination of prescribed supporting edges and is subjected to an isotropic in-plane pressure, N (load per unit length along the plate edge). The problem is to determine the frequencies of the plate. The strain energy, iJ, work done by external pressure, W, and the maximum kinetic energy, T, for the plate in non-dimensionalized orthogonal coordinates (<, 9) are given by [I 1,221

and

T,fa2

[ptw2 + $ pt’(0: + @lab d5 drl

(3)

in which D is the flexural rigidity of plate = Et3/[12( 1 - v2)]; N is the isotropic in-plane pressure; w is the transverse deflection; x is the longitudinal coordinate; y is the vertical coordinate; 0, is the rotation along y direction; 0, is the rotation along x direction; 5 is the non-dimensionalized coordinate = x/a; q is the non-dimensional&d coordinate = y/b; K is the shear correction factor = 5/6; o is the angular frequency; and p is the plate density per unit volume. The total energy functional of the plate can be expressed as l-l=U-W-T.

(4)

The geometric boundary conditions of Mindlin plates can be found in [21]. For Mindlin plates, the transverse deflection and bending slopes may be parameterized by [ 1l]

q=Oi=O

q=Oi=O

where ps, s = 1, 2 and 3, is the degree of the polynomial space; c,,,, d, and e, are the unknown coefficients and

(64

in which r#~,,)Lx,and $,, are the basic functions which must satisfy the geometric boundary conditions given in [21]. The basic function for the deflection can be expressed as

Fig. 1. Rectangular

Mindlin plate subject to isotropic in-plane pressure.

(W

Transverse vibration of thick rectangular plates-IV

where r, is the boundary equation of the jth supporting edge and Q,, depending on the support edge condition, is given by f’?,= 0 if the jth edge is free (F);

(7b)

a, = 1 if the jth edge is clamped (C) or simply supported (S). The basic functions expressed as

for

the

)LX,= jfi, [r,(t;,

rotations

can

(7~) be

n, = 0 if the jth edge is free (F) or simply supported (S) in y-direction;

(8b)

fJ, = 1 if the jth edge is clamped (C) or simply supported (S) in x-direction;

(8~)

and tiy,I=jfi

frj(ttrl)P

(94

Qj = 0 if the jth edge is free (F) or simply supported (S) in x-direction; a,=

1 if the jth edge is clamped (C) or simply supported (S) in y-direction.

(9b)

(9c)

Minimizing the total energy functional shown in eqn (4) with respect to the unknown coefficients leads to[ll]

WI -

WI - ~*Pfl)

F CUO 7

The expressions for the various elements of the stiffness matrix [kT]and mass matrix [M] can be found in [l 11. The elements of the geometric stiffness matrix [L] are given in the Appendix. The buckling factor, k, = N,b*/lr*D, is first determined by setting o = 0 in eqn (10). The frequency 1 =(ob2/n2)@, for the given parameter, isotropic in-plane pressure ratio k/k,, is further obtained by solving the generalized eigenvalue problem defined by eqn (10). 3. NUMERICAL

@aI

SF

= (0).

(10)

71

EXAMPLES AND DISCUSSION

The computer program written for Part I of this paper [l 1] has been modified to include the effect of isotropic in-plane pressure. In this paper, several example plate problems have been chosen to illustrate the capability of the computer software. The eigenvalues obtained from the software have been expressed in terms of non-dimensional frequency parameters 1 = (wb*/n*)&@. Although some of the plates considered have one or two axes of symmetry, no account has been taken of this in calculations. The value of Poisson’s ratio v has been taken to be 0.3 and the shear correction factor K has been set to 5/6. The support conditions for the nine example problems considered here are SSSS, SCSC, SFSF, CCCC, CCSS, CCFF, CFCF, CFFF and SSFF rectangular plates as shown in Fig. 2. 3.1. Convergence and comparison studies Two plate examples have been chosen to demonstrate the convergence of the frequency parameters by varying the number of degree sets of polynomial terms. In Tables 1 and 2, the first three frequency parameters are presented for the simply supported and

F ham 8

Care 9

Fig. 2. Boundary conditions of Mindlin plates analysed. CA.9 WI--F

12

K. M. LIEW ef al.

Table 1. Convergence study of frequency parameters, I = (wb*/n*)m, isotropic in-plane messure--Case

alb

for rectangular Mindlin plates subject to 1: SSSS

k/k,, = -0.5

k/k,,= 0.5

Mode sequences

Mode sequences

tlb

P.

1

2

3

1

2

3

0.001

2 4 6 8 10 12 13

2.6030 2.4502 2.4495 2.4495 2.4495 2.4494 24494

6.4496 5.4931 5.4773 5.4772 5.4771 5.4771 5.4771

6.4496 5.4932 5.4773 5.4773 5.4773 5.4772 5.4772

2.2749 2.1655 2.1650 2.1650 2.1650 2.1650 2.1650

4.8463 4.3382 4.3296 4.3296 4.3296 4.3296 4.3296

4.8463 4.3382 4.3296 4.3296 4.3296 4.3296 4.3296

0.2

2 4 6 8 10 12 13

1.5028 1.4146 1.4145 1.4145 1.4144 1.4144 1.4143

5.4638 4.4899 4.4722 4.4721 4.472 1 4.4720 4.4720

5.4638 4.4899 4.4722 4.4721 4.4721 4.4720 4.4720

1.3148 1.2505 1.2502 1.2502 1.2502 1.2502 1.2502

3.8264 3.3466 3.3371 3.3371 3.3371 3.3371 3.3371

3.8264 3.3466 3.3371 3.3371 3.3371 3.3371 3.3371

0.001

2 4 6 8 10 12 13

1.6533 1.5314 1.5310 1.5310 1.5310 1.5309 1.5309

2.5493 2.2953 2.2914 2.2914 2.2913 2.2913 2.2912

5.5988 4.4579 3.5821 3.5493 3.5488 3.5488 3.5487

1.5085 1.4114 1.4110 1.4110 1.4110 1.4110 1.4110

2.2369 2.0461 2.0431 2.0431 2.0431 2.0431 2.0431

4.2766 3.4967 3.0174 2.9914 2.9970 2.9970 2.9970

0.2

2 4 6 8 10 12 13

0.9546 0.8843 0.8839 0.8839 0.8837 0.8837 0.8836

1.8736 1.6628 1.6584 1.6583 1.6583 1.6583 1.6583

4.9859 3.9204 2.9580 2.9214 2.9209 2.9208 2.9208

0.8715 0.8149 0.8147 0.8147 0.8147 0.8147 0.8147

1.5936 1.4444 1.4411 1.4411 1.4411 1.4411 1.4411

3.6056 2.9400 2.3944 2.3715 2.3711 2.3711 2.3711

Note. pI denotes p, , p2 and p, .

fully clamped rectangular Mindlin plates with an increasing number of degree sets of polynomial terms. The rectangular plates considered are subjected to isotropic in-plane pressure. The results shown in these tables are for rectangular plates with different aspect ratios a/b and relative thickness ratios t/b under two types of isotropic in-plane pressure ratios k/k, = -0.50 and 0.50. It can be seen that the frequency parameters have converged reasonably as the number of degree sets pi =p2 =p, = 13 has been reached. On this basis, the degree sets p, = p2 = p, = 13 have been used for all calculations in this study. A comparison study of the frequency parameters, 1’ (= wbJ_, has been carried out for a fully clamped square plate (a/b = 1.0) with relative thickness ratio t/b = 0.01 and 0.10 subjected to isotropic in-plane pressure k = NbZ/n2D = - 5.0 and 5.0. A similar analysis has been carried out by Roufaeil and Dawe [lo] using the Rayleigh-Ritz method with a set of Timoshenko beam functions. It is noted that the higher order terms in the non-linear part of the Green strain have been taken into account in their paper [lo] but are neglected in the present study. In Table 3, the present results and those of Roufaeil and Dawe are presented. Good agreement

has been obtained thereby confirming the accuracy of the present method. 3.2. Numerical results The critical buckling load factor, k, = N,b2/n2D, has been determined by setting o = 0 in the eigenfunction eqn (10). For the nine plate examples considered here, the critical buckling load factors k, have been obtained, and these are presented in Table 4. In Table 4 the critical buckling load factors have been determined for plates with aspect ratios a/b = 1.0, 1.5 and 2.0 and relative thickness ratios t/b = 0.001, 0.10 and 0.20, respectively. A decrease in critical buckling load factors with increasing relative thickness ratio t/b can be observed in this table. This

is due to the effect of shear deformation. For vibration analysis, N is set to zero and the frequency parameters 1 are obtained by solving the governing eigenvalue equation eqn (10). In this study, only the first three frequency parameters are presented. The nine plate examples considered earlier have been solved with different aspect ratios and relative thickness ratios. The first three

73

Transverse vibration of thick rectangular plates-IV. Table 2. Convergence study of frequency parameters, 1 = (ob2/x*)~, isotropic in-plane pressurc_casC

for rectangular Mindlin plates subject to 4: CCCC

k/k, = -0.5

k/k, = 0.5

Mode sequences

Mode sequences 0

t/b

P,

1

2

3

1

2

3

0.001

2 4 6 8 10 12 13

183.06 4.4612 4.4382 4.4363 4.4360 4.4360 4.4357

797.60 144.88 8.4692 8.4050 8.4015 8.4012 8.4011

797.60 144.88 8.4692 8.4050 8.4015 8.4012 8.4011

3.2805 3.2558 3.2533 3.2532 3.2531 3.2531 3.2531

6.0604 5.4807 5.4678 5.4675 5.4675 5.4675 5.4675

6.0604 5.4807 5.4678 5.4675 5.4675 5.4675 5.4675

0.2

2 4 6 8 10 12 13

105.88 2.5005 2.6003 2.5998 2.5996 2.5996 2.5996

765.21 144.77 6.4478 6.3182 6.3142 6.3138 6.3137

765.2 1 144.77 6.4478 6.3182 6.3142 6.3138 6.3137

1.9277 1.9364 1.9354 1.9353 1.9353 1.9353 1.9353

4.4368 3.7619 3.7436 3.7433 3.7433 3.7433 3.7433

4.4368 3.7619 3.7436 3.7433 3.7433 3.7433 3.7433

0.001

2 4 6 8 10 12 13

131.48 3.0584 3.0291 3.0281 3.0281 3.0280 3.0279

491.62 95.930 3.8730 3.8706 3.8694 3.8693 3.8692

740.04 107.97 5.3360 5.2730 5.2684 5.2682 5.2682

2.4000 2.3589 2.3570 2.3569 2.3569 2.3569 2.3569

3.4954 2.9688 2.9548 2.9544 2.9544 2.9544 2.9544

5.4423 3.9624 3.8870 3.8818 3.8816 3.8816 3.8816

2 4 6 8

76.039 1.7659 1.7781 1.7779 1.7779 1.7779 1.7778

474.78 95.830 2.4186 2.3978 2.3969 2.3966 2.3964

715.78 107.92 3.7875 3.6571 3.6514 3.6511 3.6511

1.4088 1.4066 1.4065 1.4064 1.4064 1.4064 1.4064

2.5406 1.8286 1.7981 1.7974 1.7973 1.7973 1.7973

4.1615 2.6977 2.5655 2.5554 2.5551 2.5551 2.5551

0.2

10

12 13

Note. p, denotes pI, p2 and ~3.

frequency parameters arranged in an increasing mode sequence for these sets of problems are given in Table 5. These data provide important information on the effects of shear deformation and rotatory inertia on the vibration frequency response. In this table, a decrease in frequency parameters is observed with increasing relative thickness ratio t/b for a given a/b ratio and given boundary conditions. Finally, vibration of rectangular plates subjected to isotropic in-plane pressure has been studied. The

eigenvalues obtained from the governing eigenvalue equation for various isotropic in-plane pressure ratios k/k, are the corresponding frequency parameters d of the plates. Design charts, as shown in Figs 3-11, are used to present these results. It is shown that the frequency parameters are expressed in terms of a function of isotropic in-plane pressure ratio k/k,,. The design charts provide the relation between the ratios t/b, k/k, and the frequency parameter 1.

Table 3. Comparison study of frequency parameters, A’ = cob,/-, for square Mindlin plates subject to isotropic in-mane oressure with CCCC boundary conditions Mode sequences

t/b

k

Method

-5

Present

1

2

3

4

5

6

0.2417 0.2419

0.4407 0.4412

0.6177 0.6193

0.7337 0.7344

0.7357 0.7363

0.8992 0.9017

1101

0.0421 0.04210

0.2448 0.2449

0.4141 0.4144

0.5286 0.5287

0.5340 0.5341

0.6899 0.6907

FYesent [lOI

2.2563 2.280

3.9012 3.949

5.2345 5.315

6.0423 6.112

6.0752 6.141

7.1631 7.266

1.7008 1.624

2.8970 2.825

3.6435 3.558

3.7246 3.648

4.6492 4.569

WI 0.01 5 -5

Resent

0.1 5

Present

WI

K. M. LIEW et al.

14 Table 4. Buckling

factors,

k,, = N,,b’/x*D,

for rectangular frequency

Mindlin plates subject to isotropic 0 = 0 in eon (10)

a/b = 1.0 lib

in-plane

a/b = 1.5 tlb

pressure

with angular

a/b = 2.0

lib

We condition

0.001

0.10

0.20

0.001

0.10

0.20

0.001

0.10

0.20

ssss scsc SFSF cccc ccss CCFF CFCF CFFF SSFF

2.oOOO 3.8298 1.0551 5.3036 3.2415 0.5181 2.1461 0.2397 0.2037

1.8932 3.3761 0.9886 4.5419 2.873 1 0.5401 2.2991 0.2363 0.1936

1.6319 2.5186 0.8820 3.2399 2.2218 0.4890

1.4444 3.1615 0.5012 4.1212 2.4367 0.4003 I .0680 0.1054 0.1226

1.3879 3.3531 0.4737 3.6466 2.2301 0.3819 0.9517 0.1045 0.1177

1.2421 2.5329 0.4370 2.7367 1.8219 0.3564 0.7757 0.1029 0.1124

1.2500 3.8246 0.2975 3.9234 2.2212 0.3284 0.5412 0.0588 0.0772

I .2074 3.3761 0.281 I 3.4953 2.0593 0.3178 0.4923 0.0584 0.0744

1.0955 2.5186 0.2616 2.6458 1.7115 0.3021 0.4205 0.0579 0.0715

1.6756 0.2299 0.1829

In Figs 3-l 1, the first three frequency parameters of SSSS, SCSC, SFSF, CCCC, CCSS, CCFF, CFCF, CFFF and SSFF rectangular plates for various a/b, t/b and k/k,, ratios are presented. The results depicted in these figures show the variation of frequency parameters with respect to the isotropic in-plane pressure ratio k/k,,. Given the fixed relative thickness ratio t/b, the frequency parameters increase with increasing tension stress (-k/k,,) and decrease with increasing compression

Table 5. Frequency

parameters,

1 = (ob2/n2)&@,

stress (+ k/k,,). For a constant k/k, value, a decrease in frequency parameters is evident as the relative thickness ratio t/b increases. This is because of the shear deformation and rotatory inertia effects. In general, these effects are more significant for plates under tension stress than compression stress. It may be observed that mode shape shifting occurs in Case 7 (CFCF) when the isotropic in-plane pressure ratio k/k,, changes from 0.5 to 0.8 for all a/b ratios.

for rectangular in eqn (10)

a/b = 1.0 Mode sequences Edge condition

rib

Mindlin

plates with isotropic

a/b = 1.5 Mode sequences

1

1

2

3

1.9999 1.9317 1.7679

5.0000 4.6084 3.8656

5.0000 4.6084 3.8656

1.4445

2.9334 2.7021 2.2655

5.5465 4.9762 4.0379

SFSF

0.9768 0.9565 0.9102

1.6355 1.5593

cccc

3.6463 3.2954 2.6875

ccss

in-plane

pressure

N = 0

a/b = 2.0 Mode sequences

1

2

3

1.3164

2.7778 2.6491 2.3612

4.4445 4.1303 3.5117

I .2500 I .2227 1.1521

7.0242 5.9992 4.5205

2.5374 2.3500 1.9789

3.5567 3.2562 2.7151

5.5465 4.9762 4.0379

2.4130 2.2404 1.8917

1.4280

3.7215 3.4307 2.9521

0.4330 0.4262 0.4162

0.9747 0.9344 0.8735

1.7443 1.6865 1.5553

0.2437 0.2395 0.2362

0.6984 0.6703 0.6314

0.9768 0.9565 0.9102

7.4364 6.2858 4.6907

7.4364 6.2858 4.6907

2.7364 2.5247 2.1197

4.2257 3.7975 3.0725

6.6998 5.7369 4.3217

2.4906 2.3092 1.9495

3.2249 2.9515 2.4524

4.5364 4.0708 3.2901

2.7415 2.5617 2.2024

6.1339 5.4098 4.2591

6.1590 5.4441 4.2979

2.0220 1.9197

1.6972

3.4472 3.1928 2.7046

5.5083 4.9199 3.9279

1.8009 1.7179 1.5314

2.5535 2.4038 2.0940

3.8477 3.5490 2.9864

CCFF

0.7028 0.6762 0.6328

2.4235 2.2438 1.9221

2.6942 2.5049 2.1499

0.5054 0.4906 0.4675

I .3422 1.2757 1.1543

2.3594 2.2119 1.9170

0.4358 0.4252 0.4089

0.9224 0.8852 0.8189

1.8606 1.7625

CFCF

2.2474 2.0904

1.7772

2.6776 2.4342 2.0151

4.4190 3.9055 3.1652

0.9964 0.9611 0.8823

1.3876 1.3023 1.1498

2.7466 2.5509 2.1629

0.5596 0.5466 0.5189

0.9122 0.8648 0.7845

I .5406 1.4736 1.3255

CFFF

0.3549 0.3476 0.3384

0.8634 0.8168 0.7445

2.1579 2.0356 1.7806

0.1621 0.1545 0.1523

0.5270 0.5015 0.4654

0.9680 0.9407 0.8776

0.0964 0.0867 0.0859

0.3777 0.3593 0.3361

0.5449 0.5342 0.5124

SSFF

0.3443 0.3329 0.3222

1.7550 1.6779 I .5358

1.9553 1.8750 1.7141

0.23 I5 0.2217 0.2160

0.9677 0.9377 0.8863

1.6906

0.1690 0.1652 0.1613

0.6429 0.6270 0.6008

1.4881

1.4082

1.6311 1.5057

2

3

1.9999 1.9317

1.7679

3.2590 3.0162 2.7023

2.9333 2.7021 2.2655

3.9606 3.6120 2.998 1

1.5758

1.4358 1.3314

Transverse vibration of thick rectangular plates-IV -

Firrt

mode

;

-----Second 6

mode ;

----

Third mode

.‘..,‘*~‘I”=’

(a/b=

lrotropic

1.6)

In-plane

pressure

ratlo, k/k cr

Fig. 3. Frequency parameter I versus isotropic in-plane pressure ratio k/k, for rectangular Mindlin plates-Case

-

First mode

-----8ocond

:

mode

;

---*-

1: SSSS.

Third mode

~~.,t/b_=O.OOl

j

6

4

3

4

2 2 ,

Isotropic

in-plane

pressure

ratio.

k/k,,

Fig. 4. Frequency parameter I, versus isotropic in-plane pressure ratio k/k,, for rectangular Mindlin plates-Case

-

tlb=O.OOl

:

(a/b = 1 .O)

First mode

I. * ..I*,., 0.6

;

-----Second

mode

;

---*-

Third mode

I....,....

1.0 Isotropic

2: SCSC.

0.6 In-plane

pressure

1.0

ratio, k/kcr

Fig. 5. Frequency parameter A versus isotropic in-plane pressure ratio k/k, for rectangular Mindlin plates-Case

3: SFSF.

K. M. L~EWet al.

76 -

Firrt

-----

;

mod0

-*----

;

Soaond mode

e lo L I cu^ 8 ,r ‘: : i 0 r

6

5 ; 0

4

E f s f! u.

Third modo

0

4

2 2

0 -0.5

-%a

0 I8ottoplc

in-plane

ptessuro

ratio,

0

0.3

Fig. 6. Frequency parameter 1 versus isotropic in-plane pressure ratio k/k, for rectangular Mind~n plates-Case

-

;

-----8raond

laotropic

in-plane

Firot mode

;

-*--*-

ratio,

k/k,,

mode

pressure

First mods

;

-----Second

mode

;

-+-*-*-

4: CCCC.

Third mode

Fig, 7. Frequency parameter 1 versus isotropic in-plane pressure ratio k/k,, for rectangular Mindlin plates-Case

-

I.0

klker

5: CCSS.

Third mode

_ (afb=2.0) Isotropic

In-plane

proowro

ratio,

k/kcr

Fig. 8. Frequency parameter 1 versus isotropic in-plane pressure ratio k/k,, for rectangular Mindlin plates-Case

6: CCFF.

71

Transverse vibration of thick rectangular plates-IV -

Firrt

modr

-----

;

--*-*-

Oroond mod0 ;

Third mode

9

Case

_ Case

7: CFC

OOo.6

OO

-0.5

-0.6 lrotroplc

In-plans

7: CFCF

prowwe

ratio,

1.0

k/k or

Fig. 9. Frequency parameter 1 versus isotropic in-plane pressure ratio k/k, for rectangular Mindlin plates-Case

2.6

,

1

1

,

,

Flrrt mode .

.

I

I

,

.

1

.

-----Second

;

mode

;

---.-

Third mode

0.611

.

0.5 lsotroplc

in-plane

pressure

ratio.

First mode

;

lrotroplc

-----Second

In-plane

;

--*--

ratio,

k/kor

mode

preeoure

1.0

k/kcr

Fig. 10. Frequency parameter rl versus isotropic in-plane pressure ratio k/k, for rectangular Mindlin plates-Case

-

7: CFCF.

8: CFFF.

Third mode

Fig. 11. Frequency parameter A versus isotropic in-plane pressure ratio k/k, for rectangular Mindlin plates-Case

9: SSFF.

K. M. Lt~w et al.

78 4. CONCLUSIONS

This paper deals with the free vibration analysis of thick rectangular plates under isotropic in-plane pressure. A computer program based on the pb-2 Rayleigh-Ritz method has been developed for solving these eigenvalue problems. Mindlin’s plate theory has been employed for consideration of the effects of shear deformation and rotatory inertia. It was employed because of its simplicity in implementation with the pb-2 Rayleigh-Ritz procedure for modelling plates of any combination of boundary conditions. In the whole numerical process, only one single continuum element is used. This is very different from the discretization methods because no boundary is lost during the calculation. Thus it is believed that the present method will provide very accurate solutions. Comprehensive sets of solutions for buckling, vibration and vibration subjected to isotropic in-plane pressure have been presented for several selected thick rectangular plates. Design charts have also been included to present the variation of frequency parameters with respect to the isotropic in-plane pressure ratio k/k,,. These sets of design charts provide an easy reference for engineers and designers. Conclusions have been drawn from observations of the numerical data presented for the thick rectangular plates with different relative thickness ratios. The effects of shear deformation and rotatory inertia on the vibration response have been noted, and a decrease in frequency parameters is evident as the relative thickness ratio t/b increases. Secondly the effects of isotropic in-plane pressure have been examined. The frequency parameters increase with increasing tension stress (-k/k,,) and decrease with increasing compression stress (+ k/k,,) for a fixed relative thickness ratio t Ib. Acknowledgements-This research was supported by the Australian Research Council (ARC) under Project Grant No. ARC 834. The authors wish to thank Mr Warren H. Traves of Gutteridge Haskins and Davey Pty Ltd for proof-reading the manuscript. The first author appreciates the assistance provided by the Department of Civil Engineering, The University of Queensland during his visit to that institution.

7. R. D. Mindlin, Influence of rotatory inertia and shear in flexural motion of isotropic, elastic plates. ASME J. appl. Mech. 18, 1031-1036 (1951). 8. R. D. Mindlin and H. Deresiewin, Thickness-shear and flexural vibrations of a circular disk. J. appl. Physics 25, 1329-1332 (1954). 9. D. J. Dawe and 0. L: Roufaeil, Rayleigh-Ritz vibration analvsis of Mindlin mates. J. Sound Vibr. 69. 345-359 (198b). 10. 0. L. Roufaeil and D. J. Dawe, Rayleigh-Rib vibration analysis of rectangular Mindlin plates subjected to membrane stresses. J. Sound Vibr. Ss, 263-275 (1982). 11. K. M. Liew, Y. Xiang and S. Kitipomchai, Transverse vibration of thick rectangular plates-I. Comprehensive sets of boundary conditions. Comput. Struct. 49, l-29 (1993). 12. K. M. Liew, Y. Xiang and S. Kitipomchai, Transverse vibration of thick rectangular plates-II. Inclusion of oblique internal line supports. Comput. Struct. 49, 31-58 (1993). 13. K. M. Liew, Y. Xiang and S. Kitipomchai, Transverse vibration of thick rectangular plates-III. Effects of multiple internal eccentric ring supports. Comput. Struct. 49, 59-67 (1993). 14. K. M. Liew, K. Y. Lam and S. T. Chow, Free vibration analysis of rectangular plates using orthogonal plate function. Comput.St&. 34 79-85 (1990): _ 15. K. M. Liew. Vibration of eccentric ring and line sunported circular plates carrying concentrated masses. J. Sound Vibr. 156, 99-107 (1992). 16. K. M. Liew, Response of plates of arbitrary shape subject to static loading. ASCE J. Engng Mech. 118, 1783-1794(1992).

17. K. M. Liew, On the use of pb-2 Rayleigh-Ritz method for free-flexural vibration of triangular plates with curved internal supports. J. Sound Vib; la: 329-340 (1993). 18. K. M. Liew and C. M. Wane ob-2 Ravleinh-Ritz method for general plate analysis.%&tg Shuci. 13,55-60 (1993). 19. K. M. Liew and C. M. Wang, Vibration analysis of plates by pb-2 Rayleigh-Ritz method: mixed boundary conditions, reentrant comers and internal supports. J. Mech. Struct. Machines Zo, 281-292 (1992). 20. K. M. Liew and M. K. Lim, Transverse vibration of trapezoidal plates of variable thickness: symmetric trapezoids. J. Sound Vibr. 165, 4567 (1993). 21. K. M. Liew, Y. Xiang, C. M. Wang and S. Kitipomchai, Vibration of thick skew plates based on Mindlin shear deformation plate theory. J. Sound Vibr. 167 (1993). 22. C. M. Wang, K. M. Liew, Y. Xiang and S. Kitipomchai, Buckling of rectangular Mindlin plates with internal line supports. ht. J. Solids Struct. 30, l-17 (1993). APPENDIX

The elements of [L] are given by REFERENCES

1. Handbook of Structural Stability. Column Research Committee of Japan (Ed.), Corona, Tokyo (1971). 2. P. S. Bulson, The Stability of Flat Plates. American Elsevier, New York (1969). 3. J. P. Sit@ and S. S. Dey, Parametric instability of rectangular &es by the energy based tinite di&m& method. Cornout. Meth. at&. Mech. Enrmp 97. l-21 (1992). 4. A. W. Leissa, Re&t research in plate’vibratibns: complicating effects. Shock Vibr. Digest 9(11), 21-35 (1977). 5. A. W. Leissa, Plate vibration research, 1976-1980: complicating effects. Shock Vibr. Digest 13(11), 19-36 (1981). 6. A. W. Leissa, Recent research in plate vibrations, 1981-1985, part II: complicating effects. Shock Vibr. Digest B(3), IO-24 (1987).

i= 1.2, ..,ii;j=l,2

,...,

A

(11)

LdU=O,

i= 1,2, ..,fi;

,...,

A

(12)

L,“=O,

i = 1,2, ..,m;j=l,2

,...,

i

(13)

Lu~=O,

i= 1,2, ..,A;

j=l,2

,...,

R

(14)

L,=O,

i= 1,2, ..,A;

j=1,2

,...,

T

(15)

&=O,

i= 1,2, ..,I:,

j=l,2

,...,

i

(16)

j=1,2

in which and Tare number ^ .fi,^ A____ . . of the polynomial terms for w, vx and t$ 1111,respecttvely.