Indian Journal of Engineering & Materials Sciences Vol. 10, December 2003, pp. 441-451

Simple design formula to evaluate the first axisymmetric frequency of moderately thick annular plates subjected to a uniform compressive

load at the outer edge

S Renjith & G Venkateswara Rao Structural Engineering Group, Vikram Sarabhai Space Centre, Thiruvananthapuram 695 022, India

Received J April 2003; accepted 29 September 2003

Annular plates are commonly used as structural elements in various engineering applications. In many engineering fields, especially those related with aerospace applications, structural systems are almost fully stressed to improve the structural factor and pay load fraction. Evaluation of the first axisymmetric frequency, which is an important design input, of structural elements in the stressed condition is a relati vely complex task. In this paper, the applicability of a simple and effective design formula developed earlier by one of the authors with an assumption on the mode shapes of vibration and buckling, is studied with reference to an initially stressed moderately thick annular plate. The initial stress is due to a uniform compressive load at the outer edge of the annular plate. The values of the first axisymmetric frequency of the initially stressed moderately thick annular plates obtained by using the design formula, when compared with those obtai ned by the powerful and versatile finite e lement method, for different combinations of boundary conditions at the outer and inner edges, and different ratios of internal to external rad ii show the efficacy of the design formula.

Most structural systems that are used in various fields of engineering like civil, mechanical and aerospace are assemblages of simple structural elements like beams, plates of various configurations and shells. The dynamic characteristics of these structural systems can be predicted from the dynamic characteristics of the simple structural elements using dynamic substructuring techniques I . Quick estimation of their first axisymmetric frequencies (from now on called as the fundamental frequency) will be very useful to the design engineers in the preliminary design phase of the structural elements .

Unlike the structural systems in engineering fields of civil, mechanical, nuclear and aerospace structures have to be designed with lower margins of safety, choosing proper optimum configurations to achieve maximum payload capability. In effect, these structural elements operate with high compressive or tensi le initi al stresses, and hence, the dynamic characterization in the presence of initial stresses, which is an involved task, has (0 be essentially carried out.

Even though the finite element method (FEM)2 is a versatile numerical tool to solve this problem, the presence of initial stresses makes it complex and the solution involves parametric study, the initial stress being the parameter to be varied. The free vibration

analysis is to be carried out every time, which involves complex formulation coupled with large computer times. Alternatively, a simple design formula, available to evaluate the fundamental frequency of structural elements with initial stresses, will be very handy for design engineers in the preliminary design stage.

One of the authors, has developed such a formula, which is very simple and effective, which was applied to obtain the fundamental frequency parameter of initially stressed structural elements like tapered beams, circular and rectangular plates with effects like elastic restraints, shear deformation and rotary inertia3-8.

The efficacy of the formula is studied in this paper, for the case of uniform moderately thick annular plates with different boundary conditions, different values of ratios of the inner and outer radii and initial stress parameter. The numerical results are presented in tabular form for easy comprehension. The present results are compared with those available based on FEM9 for bringing out the effectiveness of the design formula proposed.

Design Formula Consider a uniform annular plate of inner radius 'b'

and outer radius 'a' , thickness h, Young' s modulus E

442 INDIAN J. ENG. MATER. SCI. , DECEMBER 2003

and mass density p subjected to a uniform compressi ve load N,i per unit length at the outer boundary (Fig. 1). The governing matrix equation of the initially stressed annular plate executing harmonic osci llations (following Nayar et al.\ is given by

[K]{8}-AJG]{8}-A f [M]{8}=O . .. (1)

where [In, [G] and [M] are the assembled elastic stiffness, geometric stiffness and mass matrices respectively, Ai is the initial load parameter (defined as Ai = N'i a21D, D is the plate flexural rigidity given by D = Eh3112(l-V), v being the Poisson ratio), At- is the frequency parameter with initial stresses (defined as At- = phu} a 41D) and {8} is the eigenvector.

The degenerate case of the free vibration problem without initial stress is obtained from Eq.(l) as

[K]{8}-A fo[M]{8}=O ... (2)

Eq.(2) can be written as

[M]{8}=(1/A fo )[K]{8} ... (3)

where AfO is the frequency parameter when the initial stresses are not present. Similarly the degenerate case of the buckling problem is obtained from Eq.(l) as

.. . (4)

where Ab is the buckling load parameter.

~r ~/4-Nn I

Fig. I- Moderately thick ann ular plate subjected to uniform compressive radial load at the outer edge

Assuming that the eigenvectors for the stress free annular plate, initially stressed annular plate and for the buckling of the annular plate are the same and substituting Eq.(3) and Eq.(4) in Eq.(1) , we get

... (5)

which implies that

... (6)

From Eq. (6), knowing Ai (radial load parameter for a particular value of N,;), Ab and AfO, the frequency parameter At- of the initially stressed annular plate can be computed. It may be noted here that the mode shapes of the stress free vibration, buckling and initially stressed vibration are exactly or approximately the same for all the boundary conditions considered in the present study, except when the radl ial compressive load is closer to the buckling load, say A;!Ab=O.8, val idating the assumption made. As such for all practical purposes, Eq.(6) can be used to quickly calculate the fundamental frequency of the initially stressed moderately thick annular plates, with a reasonable degree of accuracy and is very much useful for practical/design engineers.

Numerical Results The fundamental frequency parameter (A-.t)1 /2 of the

initially stressed annular plate (Fig. I) is obtained using the proposed design formula, for various values of the radii ratios a ( = bla), various values of the initial stress parameter f3 (=A;!Ab) and several combinations of the boundary conditions at the outer and inner edges. The boundary conditions are denoted as SC and CP, where S, C and F stand for simply supported, clamped and free respectively, the first letter indicating boundary condition at the outer edge and second letter indicating the same at the inner edge. As such, SF denotes that the outer edge of the plate is simply supported while the inner edge is free.

The results obtained using the present formula are presented in Tables 1-8. The corresponding results obtained from the FEM9, by idealizing the plate with eight annular plate elements, are also included in the tables. The percentage deviations of the present results from those obtained by the FEM are also presented in these tables to assess the correctness of

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RENJITH & RAO: ANNULAR PLATES SUBJECTED TO A UNIFORM COMPRESSIVE LOAD

Table l--Comparison of frequency parameter At/f2 for S-S plate

Ab AJD 0.2

(FEM) (FEM) Formula FEM % Formula deviation

0.4

FEM

a.=0.1

% Form u la

deviation

0.6

FEM %

Formula deviation

0.8

FEM

443

% deviation

0.001 18.56 209.9 12.96 12.99 -0.294 11.22 11.29 -0.611 9.162 9.251 -0.960 6.479 6.566 -1.338

0.05 18.24 205.3 12.81 12.85 -0.303 11.10 11.17 -0.627 9.061 9.151 -0.987 6.407 6.497 -1.378

0.10 17.36 192.7 12.42 12.46 -0.325 10.75 10.83 -0.679 8.779 8.874 -1.066 6.208 6.302 -1.492

0.15 16.09 175.1 11.84 11.88 -0.357 10.25 10.33 -0.743 8.368 8.467 -1.171 5.917 6.016 -1.645

0.20 14.60 155.5 11.15 11.20 -0.386 9.66 9.74 -0.811 7.886 7.989 -1.284 5.576 5.679 -1.813

u=0.2

0.001 21.52 281.6 15.01 15 .05 -0.270 13.00 13.07 -0.559 10.61 10.71 -0.870 7.504 7.595 -1.202

0.0521.19 276.2 14.87 14.91 -0.280 12.87 12.95 -0.583 10.51 10.61 -0.9077.433 7.528 -1.263

0.10 20.24 261.5 14.46 14.51 -0.313 12.53 12.61 -0.650 10.23 10.33 -1.013 7.231 7.335 -1.412

0.15 18.85 240.4 13.87 13.92 -0.360 12.01 12.10 -0.749 9.805 9.92 -1.176 6.933 7.049 -1.645

0.20 17.19 216.3 13.15 13.21 -0.416 11.39 11.49 -0.875 9.302 9.43 -1.378 6.577 6.707 -1.939

u=0.3

0.001 27.96 444.3 18.85 18.91 -0.295 16.33 16.43 -0.612 13.33 13.46 -0.950 9.427 9.553 -1.316

0.05 27.46 434.9 18.65 18.71 -0.312 16.15 16.26 -0.646 13.19 13.32 -1.004 9.326 9.458 -1.392

0.10 26.06 409.1 18.09 18.16 -0.358 15.67 15.79 -0.742 12.79 12.94 -1.160 9.046 9.194 -1.611

0.15 24.03 372.9 17.27 17.35 -0.427 14.96 15.09 -0.890 12.21 12.39 -1.399 8.637 8.809 -1.955

0.20 21.65 332.6 16.31 16.40 -0.514 14.13 14.28 -1.08 1 11.54 11.74 -1.708 8.156 8.358 -2.408

u=O.4

0.001 39.06 790.9 25.15 25.24 -0.336 21.78 21.93 -0.686 17.79 17.98 -1.077 12.58 12.77 -1.487

0.05 38.17 769.8 24.82 24.91 -0.360 21.49 21.65 -0.745 17.55 17.75 -1.158 12.41 12.61 -1.602

0.1035.71 713.5 23.89 23.99 -0.429 20.69 20.88 -0.891 16.89 17.13 -1.392 11.95 12.18 -1.937

0.15 32.24 637.6 22.59 22.71 -0.535 19.56 19.78 -1.118 15.97 16.26 -1.760 11.29 11.58 -2.467

0.20 28.30 556.9 21.1 I 21.25 -0.668 18.28 18.54 -1.408 14.93 15.27 -2.236 10.55 10.90 -3.172

u=0.5

0:001 58.52 1603 35.82 35.95 -0.380 31.02 31.26 -0.785 25.33 25.64 -1.218 17.91 18.21 -1.681

0.05 56.65 1544 35.16 35.30 -0.417 30.45 30.71 -0.867 24.86 25.20 -1.349 17.58 17.40 1.009

0.10 51.68 1395 33.4 1 33.59 -0.530 28.93 29.26 -1.103 23.62 24.04 -1.725 16.70 17.12 -2.405

0.15 45.00 1206 31.07 31.28 -0.696 26.90 27.30 -1.461 21.97 22.49 ~2 . 312 15.53 16.06 -3 .265

0.20 38.03 1019 28.56 28.82 -0.910 24.73 25.22 -1.913 20.20 20.83 -3.065 14.28 14.94 -4.402

444

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INDIAN J. ENG. MATER. SCI. , DECEMBER 2003

Table 2-Comparison of freq uency parameter A/12 for C-C plate

{3 Ab ~ 0.2 0.4

(FEM) (FEM) Formu la FEM % % Formula FEM Formula deviation deviation

(1=0.1

0.6

FEM %

Form ula deviation

0.8

FEM %

deviation

0.001 52.90 745.8 24.43 24.53 -0.437 21. 15 2 1.35 -0.911 17.27 17.52 -1.426 12.2 1 12.40 -1.991

0.05 50.95 709.4 23.82 23.94 -0.494 20.63 20.85 -1.034 16.85 17.12 -1.626 11.90 12. 10 -2.281

0.10 45.90 620.5 22.28 22.42 -0.614 19.29 19.55 -1.299 15.75 16.09 -2.070 11.10 11.40 -2.951

0.15 39.41 5 15.0 20.30 20.44 -0.72 1 17.58 17.86 -1.556 14.35 14.73 -2.535 10.10 10.50 -3.701

0.20 32.93 417.4 18.27 18.42 -0.786 15 .82 16.10 -1.732 12.92 13.3 1 -2.893 9.136 9.553 -4.362

a=0.2

0.001 70.47 1199 30.97 31.11 -0.475 26.82 27.09 -0.989 2 1.90 22.24 -1.548 15.48 I5.82 -2. 160

0.05 67.34 1130 30.07 30.24 -0.554 26.Q4 26.35 -1.1 56 2 1.26 2 1.66 -1.8 17 15.04 15.43 -2.547

0.10 59.48 968.7 27.84 28.04 -0.718 24. 11 24.48 - 1.520 19.68 20. 17 -2.424 13.92 14.42 -3.454

0.15 49.80 784.8 25.06 25.28 -0.878 21.70 22. 12 - 1.896 17.72 18.28 -3.09 1 12.53 13.12 -4.525

0.20 40.56 622.0 22.31 22.53 -0.994 19.32 19.75 -2.190 15.77 16.37 -3.663 11.15 U .8 1 -5 .540

(1=0.3

0.00 I 98.29 2057 40.57 40.78 -0.524 35. 13 35.52 -1.090 28.68 29. 18 -1.705 20.28 20.78 -2.378

0.05 92.73 19 11 39.10 39.35 -0.627 33.86 34.3 1 -1.309 27.65 28.23 -2.063 19.55 20.13 -2.895

0.10 79.30 1583 35.58 35.88 -0.843 30.81 31.38 -1.79 1 25. 16 25.90 -2.865 17.79 18.55 -4.101

0.15 63.84 1235 31.43 31.77 -1.05 1 27.22 27.86 -2.276 22.23 23.09 -3 .733 15 .72 16.63 -5.511

0.20 50.07 947.5 27.53 27.87 -1.207 23.84 24.50 -2.667 19.47 20.38 -4.490 13.77 14.78 -6.863

(1=0.4

0.0 01 143 .2 3829 55.35 55.66 -0.571 47.93 48.51 - 1. 19 1 39. 14 39.88 -1.866 27.67 28.4 1 -2.608

0.05 132.4 3475 52.73 53.11 -0.7 16 45.66 46.36 - 1.497 37.28 38.19 -2 .364 26.36 27 .27 -3 .325

0.10 107.8 2735 46.78 47.25 -0.997 40.5 1 41.39 -2.126 33.08 34.25 -3.423 23.39 24.61 -4.943

0. 15 82.13 2029 40.29 40.80 -1.25334.89 35.87 -2.73728.49 29.84 -4.536 20. 14 21.6 1 -6.79 1

0.20 61.28 1495 34.58 35.09 - 1.46 1 29.95 30.95 -3.248 24.45 25.88 -5.5 I 8 17.29 18.91 -8.575

(1=0.5

0.00 I 220.7 7967 79.83 80.33 -0.622 69.14 70.05 - 1.297 56.45 57.62 -2 .034 39.92 41.09 -2.844

0.05 197.2 6960 74.62 75.24 -0.825 64.62 65.76 - 1.735 52.76 54.25 -2.746 37.31 38.82 -3 .88 I

0.10 149.0 5083 63.77 64.54 -1.194 55 .23 56.68 -2.568 45.09 47.06 -4. 180 31.89 33.96 -6.115

0. 15 105.0 3527 53. 12 53 .94 -1.523 46.00 47 .60 -3.356 37.56 39.80 -5.63 1 26.56 29.06 -8.604

0.20 73.41 2476 44.5 1 45.33 -1.815 38.54 40.18 -4.067 31.47 33.84 -6.994 22.25 25.04 -1 1.12

the proposed formula. The values of Ab and Atv obtained by FEM for a given Ai are given in these tables , to fac ilitate the readers to compute AI for any Ai not presented in the tab les.

Discussion The fundamental frequency parameter AI of an

initially stressed S-S annular plate is presented in Table 1. It can be seen from thi s table that the

h/a

RENJITH & RAO: ANNULAR PLATES SUBJECTED TO A UNIFORM COMPRESSIVE LOAD

Table 3-Comparison of frequency parameter )..//2 for S-C plate

(3 Ab A,v 0.2 0.4

(FEM) (FEM) Formula FEM % % Formula FEM Formula de viation deviation

446

Iva }'h A}O (FEM) (FEM)

Form ula

INDIAN 1. ENG. MATER. SCI., DECEMBER 2003

Table 4-Comparison of frequency parameter A/12 for C-S plate

0.2 0.4

FEM %

deviation Formula FEM

u=O.1

{3

% Formula

deviation

0.6

FEM %

deviation Formula

0.8

FEM %

deviation

0.001 43.57 515.4 20.31 20.38 -0.341 17.59 17.71 -0.7 10 14.36 14.52 -1.110 10.15 10.31 -1.546

0.05 42.29 497.6 19.95 20.03 -0.382 17.28 17.42 -0.797 14. 11 14.29 -1.249 9.976 10.15 -1.741

0.10 38.90 451.3 19.00 19.09 -0.479 16.46 16.62 - 1.007 13.44 13.65 -1.593 9.501 9.7 19 -2.249

0.15 34.33 391.5 17.70 17.80 -0.597 15.33 15 .52 -1.274 12.51 12.78 -2.045 8.849 9.117 -2.938

0.2029.49 331.0 16.27 16.39 -0.708 14.09 14.3 1 - 1.531 11.51 11.80 -2 .507 8.136 8.448 -3.685

u=0.2

0.001 52.81 714.9 23.91 23.99 -0.312 20.71 20.85 -0.644 16.9 1 17.08 -1.002 11.96 12. 13 -1.387

0.05 51.18 689.4 23.49 23.57 -0.358 20.34 20.49 -0.743 16.6 1 16.80 -1.159 11.74 11.93 -1.611

0.1046.83 623.8 22.34 22.44 -0.469 19.35 19.54 -0.983 15.80 16.05 -1.551 11.17 11.42 -2.181

0.15 41.00 539.5 20.78 20.90 -0.605 17.99 18.23 -1.288 14.69 15.00 -2.067 10.39 10.71 -2.968

0.20 34.86 454.8 19.07 19.22 -0.734 16.52 16.79 -1.590 13.49 13.85 -2.603 9.537 9.918 -3.839

u=0.3

0.001 70.99 1140 30.20 30.30 -0.319 26.16 26.33 -0.662 21.36 21.58 -1.029 15.10 15.32 -1.425

0.05 68.28 1091 29.54 29.65 -0.378 25.58 25.79 -0.785 20.89 21.15 -1.225 14.77 15.03 -1.702

0.10 61.23 967.6 27.82 27.97 -0.5 15 24.09 24.36 -1.082 19.67 20.02 -1.711 13.91 14.26 -2.415

0.1 5 52.13 816.6 25.56 25.73 -0.676 22.13 22.46 -1.444 18.07 18.50 -2.331 12.78 13 .23 -3.374

0.2043.01 672.1 23 .1 9 23.38 -0.827 20.08 20.45 -1.796 16.40 16.90 -2.962 11.59 12. 13 -4.414

u=O.4

0.001 101.8 2029 40.29 40.43 -0.344 34.89 35.14 -0.709 28.49 28.81 -1.103 20.14 20.46 -1.525

0.05 96.58 1915 39.14 39.30 -0.420 33.89 34.19 -0.875 27.67 28.06 -1.365 19.57 19.95 -1.900

0.10 83.67 1642 36.25 36.47 -0.595 31.39 31.79 -1.258 25.63 26.15 -1.999 18.12 18.65 -2.842

0.15 68.11 1333 32.65 32.91 -0.791 28.28 28.77 -1.704 23.09 23.75 -2.777 16.33 17.02 -4.070

0.20 53.69 1058 29.10 29.38 -0.977 25.20 25 .75 -2.135 20.57 21.33 -3 .554 14.55 15.38 -5.377

u=0.5

0.001 155.6 4093 57.22 57.43 -0.37 1 49.55 49.94 -0.769 40.46 40.95 -1.197 28.61 29.09 -1.657

0.05 144.3 3773 54.94 55.2 1 -0.482 47.58 48.06 -1.005 38.85 39.47 -1.572 27.47 28.09 -2.199

0.10 118.2 3070 49.56 49.91 -0.715 42.92 43.58 -1.520 35.04 35.92 -2.440 24.78 25.68 -3 .512

0.15 89.88 2355 43.40 43.83 -0.965 37.59 38.39 -2 .096 30.69 31.79 -3.456 21.70 22.88 -5 .163

0.20 66.43 1783 37.77 38.23 -1.222 32.71 33.61 -2.683 26.71 27.97 -4.516 18.88 20.30 -6.960

Similar results are given in Table 2 for the case of initially stressed C-C annular plate. The variation of percentage deviation is smooth and monotonic and the maximum deviation is -11.12% for the extreme cases of a = 0.5 and {3 = 0.8, for a thickness ratio of 0.2.

The three mode shapes are matching with each other, but the match is not as good as that of the S-S annular plate, for higher values of the thickness ratio.

Table 3 gives the results for the initially stressed S-C annular plate, indicating the maximum percentage

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RENJITH & RAO: ANNULAR PLATES SUBJECTED TO A UNIFORM COMPRESSIVE LOAD

Table 5-Comparison of frequency parameter AjI2 for F-S plate

{3 Ah Ap 0.2 0.4

(FEM) (FEM) Formu la FEM % % Formula FEM Formula deviation deviation

0.:=0.1

0.6

FEM

0.8 %

Formula FEM de viation

447

% dev iation

0.00 I 3.983 11.91 3.087 3.149 -1.994 2.673 2.788 -4.123 2.183 2.332 -6.396 1.543 1.693 -8.830

0.05 3.97 1 11.82 3.075 3. 138 -2.022 2.663 2.779 -4.180 2.174 2.325 -6.487 1.537 1.689 -8 .959

0.10 3.938 11.57 3.042 3.107 -2.105 2.634 2.754 -4.354 2.151 2.307 -6.763 1.521 1.678 -9.35 1

0.15 3.884 11.18 2.990 3.058 -2.231 2.589 2.715 -4.622 2. 11 4 2.278 -7 .1 95 1.495 1.661 -9 .970

0.20 3.8 11 10.67 2.922 2.993 -2.391 2.530 2.662 -4.968 2.066 2.240 -7.753 1.46 1 1.637 -10.77

a =0.2

0.001 3.535 11.1 5 2.987 3.028 -1.340 2.587 2.660 -2.75 1 2.11 2 2.206 -4.240 1.494 1.586 -5.812

0.05 3.527 11.10 2.980 3.02 1 -1.355 2.581 2.655 -2.783 2.107 2.202 -4.291 1.490 1.583 -5.886

0.10 3.502 10.95 2.960 3.002 -1.399 2.564 2.640 -2.879 2.093 2.191 -4.444 1.480 1.576 -6. 100

0.15 3.46 1 10.72 2.928 2.972 -1.474 2.536 2.6 15 -3.036 2.070 2.1 72 -4.693 1.464 1.565 -6.453

0.20 3.406 10.40 2.885 2.931 -1.574 2.498 2.582 -3.246 2.040 2.148 -5 .026 1.442 1.550 -6.926

a=0.3

0.00 I 3.106 11.71 3.06 1 3.088 -0.860 2.65 1 2.698 - 1.751 2.164 2.224 -2.674 1.530 1.588 -3.634

0.05 3.099 11.67 3.055 3.082 -0.873 2.646 2.694 -1.777 2. 161 2.221 -2.714 1.528 1.586 -3 .689

0.10 3.08 1 11.55 3.039 3.067 -0.908 2.632 2.682 - 1.850 2.149 2.2 12 -2.828 1.520 1.580 -3.847

0.153.051 11.353.0133.042 -0.964 2.6092.662 -1.967 2.1312.197 -3.0131.5071.571 -4.101

0.20 3.010 11.08 2.978 3.009 -1.041 2.579 2.635 -2.128 2.106 2.177 -3 .264 1.489 1.558 -4.450

a =O.4

0.00 1 2.763 13.49 3.285 3.303 -0.546 2.845 2.877 -1.106 2.323 2.362 -1.678 1.642 1.68 1 -2.263

0.05 2.758 13.44 3.280 3.298 -0.557 2.840 2.873 -1.128 2.3 19 2.359 -1.7 11 1.640 1.679 -2.309

0.10 2.745 13.31 3.263 3.283 -0.584 2.826 2.860 -1. 187 2.308 2.350 -1.803 1.632 1.672 -2.434

0.15 2.722 13.10 3.237 3.258 -0.633 2.804 2.840 -1.287 2.289 2.335 -1.957 1.619 1.663 -2.645

0.20 2.69 1 12.82 3.202 3.225 -0.70 1 2.773 2.8 13 -1.420 2.264 2.314 -2. 163 1.601 1.649 -2.929

a =0.5

0.00 I 2.500 16.98 3.686 3.698 -0.338 3. 192 3.2 14 -0.684 2.606 2.633 -1.029 1.843 1.869 -1.38 1

0.05 2.497 16.92 3.679 3.692 -0.350 3. 186 3.209 -0.699 2.602 2.629 -1.057 1.840 1.866 - 1.4 19

0.10 2.486 16.74 3.660 3.673 -0.376 3.169 3.193 -0.755 2.588 2.618 -1.139 1.830 1.858 -1.529

0.15 2.468 16.45 3.628 3.643 -0.417 3. 142 3.168 -0.839 2.565 2.598 -1.270 1.814 1.845 -1.707

0.20 2.444 16.07 3.585 3.602 -0.473 3. 105 3. 135 -0.956 2.535 2.572 - 1.448 1.793 1.828 -1.950

deviation to be around -6.8% for the extreme cases of 0: , [3 and the thickness ratio. As such, it is concluded that the match in the three mode shapes is better than the C-C annular plate.

The C-S plate behaves like the S-C plate, with the

percentage deviation being around -6.96% for a = 0.5, f3 = 0.8 and the thickness ratio = 0.2 (Table 4). The C-S plate behaves similar to the S-C plate and the same argument given for the S-C plate holds good for this case also.

448

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INDIAN J. ENG. MATER. SCI., DECEMBER 2003

Table 6-Comparison of frequency parameter ).//2 for S-F plate

fJ AI> AjD 0.2 0.4

(FEM) (FEM) Formu la FEM % % Formula FEM Formula deviation ' deviation

a=O.1

0.6

FEM

0.8 %

Formula FEM deviation

% deviation

0.00 I 3.983 23.56 4.342 4.342 0.000 3.760 3.760 0.000 3.070 3.070 0.000 2. 171 2.171 0.000

0.05 3.971 23.46 4.332 4.332 0.000 3.752 3.752 0.000 3.063 3.064 0.000 2.166 2.166 0.000

0.10 3.938 23.18 4.306 4.306 0.000 3.729 3.729 0.000 3.045 3.045 0.000 2.153 2. 153 0.000

0.15 3.884 22.73 4.264 4.264 0.000 3.693 3.693 0.000 3.016 3.016 0.000 2.132 2. 132 0.000

0.20 3.8 11 22. 14 4.208 4.208 0.000 3.644 3.644 0.000 2.976 2.976 0.000 2. 104 2.104 0.000

u=0.2

0.00 I 3.535 22.26 4.220 4.220 0.000 3.655 3.655 0.000 2.984 2.984 0.000 2. 110 2. 110 0.000

0.05 3.527 22.17 4.21 2 4.2 12 0.000 3.647 3.647 0.000 2.978 2.978 0.000 2.106 2. 106 0.000

0. 10 3.502 21.92 4.188 4.188 0.000 3.627 3.627 0.000 2.961 2.961 0.000 2.094 2.094 0.000

0.15 3.46 1 2 1.52 4.150 4.150 0.000 3.594 3.594 0.000 2.934 2.934 0.000 2.075 2.075 0.000

0.20 3.406 20.99 4.098 4.098 0.000 3.549 3.549 0.000 2.898 2.898 0.000 2.049 2.049 0.000

a=0.3

0.001 3.106 21.75 4.172 4.172 0.000 3.6 13 3.613 0.000 2.950 2.950 0.000 2.086 2.086 0.000

0.05 3.099 21.67 4.164 4.164 0.000 3.606 3.606 0.000 2.944 2.944 0.000 2.082 2.082 0.000

0. 10 3.08 1 2 1.44 4.14 1 4.141 0.000 3.586 3.586 0.000 2.928 2.928 0.000 2.07 1 2.071 0.000

0. 15 3.05 1 21.05 4.104 4.104 0.000 3.554 3.554 0.000 2.902 2.902 0.000 2.052 2.052 0.000

0.203.010 20.55 4.054 4.054 0.000 3.511 3.511 0.000 2.867 2.867 0.000 2.027 2.027 0.000

a=O.4

0.001 2.763 22.70 4.26 1 4.26 1 0.000 3.690 3.690 0.000 3.013 3.0 13 0.000 2.131 2. 131 0.000

0.05 2.758 22.6 1 4.253 4.253 0.000 3.683 3.683 0.000 3.007 3.007 0.000 2.126 2.126 0.000

0. 102.745 22.35 4.228 4.228 0.000 3.662 3.662 0.000 2.990 2.990 0.000 2.114 2.114 0.000

0.15 2.722 21.93 4.189 4. I 89 0.000 3.628 3.628 0.000 2.962 2.962 0.000 2.095 2.095 0.000

0.20 2.69 1 21.38 4.1 36 4.136 0.000 3.582 3.582 0.000 2.925 2.925 0.000 2.068 2.068 0.000

a=O.S

0.00 I 2.500 25.78 4.541 4.54 1 0.000 3.933 3.933 0.000 3.211 3.2 1 I 0.000 2.270 2.270 0.000

0.05 2.497 25.66 4.53 I 4.531 0.000 3.924 3.924 0.000 3.204 3.204 0.000 2.265 2.265 0.000

0. 102.486 25.32 4.501 4.501 0.000 3.898 3.898 0.000 3.183 3.1 83 0.000 2.25 1 2. 251 0.000

0.15 2.468 24.79 4.453 4.453 0.000 3.856 3.856 0.000 3. 149 3. 149 0.000 2.227 2.227 0.000

0.20 2.444 24.08 4.389 4.389 0.000 3.801 3.80 1 0.000 3. 104 3.104 0.000 2.195 2. 195 0.000

It is interesting to note that the trend of the results is altogether different for the F-S annular plate (Table 5) compared to the plate with the previous boundary conditions. The percentage deviation is maximum for smaller ex and larger {3, the value being around - 10.77% for ex = 0.1 and {3=0.8 and monotonically

decreasing to - 1.95% for ex = 0.5. {3 = 0 .8 and thickness ratio = 0.2. This indicates that the match in the three mode shapes is better for larger values of ex for a given {3, but for very smal l values of ex with larger values of {3 the match is not good.

Very in teresting results are obtained for the case of

h/a

RENJlTH & RAO: ANNULAR PLATES SUBJECTED TO A UNIFORM COMPRESSIVE LOAD

Table 7-Comparison of frequency parameter V12 for C-F plate

Ab Ajl) 0_.2 __ ,....-

(FEM) (FEM) Formula FEM % Formula FEM % Formula FEM deviation de viatio n

0.4 0.6

u=O.1

0.8 % Formula FEM

deviation

449

% deviation

0.001 13.95 103.3 9.088 9.117 -0.3 12 7.871 7.922 -0.652 6.427 6.493 -1.025 4.544 4.611 -1.438

0.05 13.82 101.9 9.028 9.058 -0.327 7.8 18 7.872 -0.681 6.384 6.453 -1.070 4.514 4.583 -1.499

0.10 13.4498.10 8.859 8.890 -0.355 7.672 7.730 -0.7446.264 6.339 -1.174 4.429 4.504 -1.650

0.15 12.85 92.43 8.599 8.634 -0.398 7.447 7.510 -0.838 6.081 6.162 -1.328 4.300 4.382 -1.875

0.20 12.10 85 .58 8.274 8.311 -0.444 7.166 7.234 -0.941 5.85 1 5.940 -1.503 4.137 4.228 -2.143

u =0.2

0.001 13.6 1 108.3 9.310 9.338 -0.299 8.063

0.05 13.49 107.0 9.251 9.280 -0.3 11 8.011

8.113 -0.623 6.583 6.648 -0.973 4.655 4.719 -1.352

6.608 -1.010 4.625 4.691 -1.405 8.063 -0.647 6.541

0.10 13.16 103. 1 9.0829.113 -0.3417.8657.922 -0.709 6.4226.494 -1.110 4.5414.612 -1.547

0.15 12.65 97.32 8.824 8.857 -0.379 7.641 7.703 -0.794 6.239 6.318 -1.248 4.412 4.490 -1.749

0.20 11.99 90.30 8.499 8.535 -0.421 7.361 7.426 -0.886 6.010 6.095 -1.403 4.250 4·.336 -1.981

u=0.3

0.001 14.96 130.5 10.22 10.24 -0.234 8.849 8.892 -0.482 7.225 7.279 -0.746 5.109 5.162 -1.028

0.05 14.84 128.8 10.15 10.17 -0.244 8.790 8.834 -0.504 7.177 7.233 -0.781 5.075 5.1 30 -1.075

0.10 14.48 123.9 9.955 9.982 -0.274 8.621 8.670 -0.565 7.039 7.102 -0.877 4.978 5.039 ·· 1.213

0.15 13.92 116.6 9.658 9.688 -0.310 8.364 8.418 -0.646 6.829 6.899 -1.008 4.829 4.898 -1.402

0.20 13.20 107.8 9.287 9.320 -0.350 8.043 8.102 -0.732 6.567 6.644 -1.153 4.644 4.720 -1.620

u=O.4

0.001 18.56 185.0 12.17 12.19 -0.174 10.54 10.57 -0.355 8.603 8.650 -0.547 6.083 6.129 -0.750

0.05 18.38 182.2 12.07 12.09 -0. 186 10.45 10.49 -0.381 8.536 8.587 -0.586 6.036 6.085 -0.804

0.10 17.88 174.2 11.80 11.83 -0.217 10.22 10.27 -0.443 8.347 8.405 -0.688 5.902 5.959 -0.947

0.15 17.10 162.5 11.40 11.43 -0.254 9.873 9.925 -0.529 8.061 8.128 -0.824 5.700 5.766 -J.l42

0.20 16.11 148.7 10.91 10.94 -0.289 9.446 9.503 -0.606 7.71 2 7.787 -0.957 5.453 5.528 -1.346

a=0.5

0.001 25.76 313.8 15.84 15.86 -0.129 13.72 13.76 -0.264 11.20 11.25 -0.408 7.922 7.966 -0.556

0.05 25.45 307.6 15.69 15.7 1 -0.144 13.58 13.62 -0.295 11.09 11. 14 -0.454 7.843 7.892 -0.622

0.10 24.58 290.5 15 .24 15.27 -0.179 13.20 13.25 -0.369 10.78 10.84 -0.570 7.622 7.682 -0.785

0.1523.23 266.2 14.59 14.63 -0.219 12.64 12.70 -0.454 10.32 10.39 -0.710 7.297 7.370 -0.993

0.20 2 1.55 238.9 13.82 13.86 -0.248 11.97 12.04 -0.524 9.776 9.858 -0.838 6.9 12 6.996 -1. 19 1

an initially stressed S-F annular plate. It can he seen from Table 6 that the percentage deviation is zero for this plate for all values of a, f3 and thickness ratios, indicating that the three mode shapes are exactly the same.

For C-F annular plates (Table 7), the perceniage

deviations involved are small, the maximum being around -2.14% for a = 0.1 , f3 = 0.8 and thickness ratio = 0.2. It can be concluded from the percentage deviation that the three mode shapes are almost the same.

The F-C annular plate behaviour is shown in

450

h/a

INDIAN J. ENG. MATER. SCI., DECEMBER 2003

Table 8~ Comparison of frequency parameter A/12 for F-C plate

Ab AjV 0.2

(FEM) (FEM) Formula FEM % Formula deviation

0.4

FEM

Ct.=0.1

[3

% Formula

deviation

0.6

FEM %

Formula deviation

0.8

FEM %

deviation

0.00 I 4.415 17.97 3.792 3.858 -1.720 3.284 3.405 -3.545 2.681 2.837 -5.478 1.896 2.050 -7.531

0.05 4.402 17.80 3.774 3.841 -1.764 3.268 3.391 -3.631 2.668 2.827 -5.618 1.887 2.045 -7.726

0.104.361 17.31 3.721 3.793 -1.887 3.223 3.353 -3.888 2.631 2.800 -6.017 1.861 2.029 -8.286

0.154.295 16.55 3.639 3.716 -2.071 3.151 3.292 -4.2792.573 2.756 -6.633 1.819 2.003 -9.154

0.204.205 15.59 3.532 3.615 -2.298 3.059 3.2 12 -4.765 2.498 2.697 -7.410 1.766 1.968 -10.26

Ct.=0.2

0.001 5.063 26.85 4.635 4.704 -1.486 4.014 4.140 -3.045 3.277 3.438 -4.688 2.317 2.476 -6.418

0.05 5.046 26.60 4.613 4.684 -1.526 3.995 4.124 -3 .130 3.262 3.427 -4.818 2.306 2.469 -6.600

0.10 4.994 25.88 4.550 4.626 -1.640 3.940 4.077 -3.368 3.217 3.393 -5.193 2.275 2.449 -7. 122

0.154.909 24.76 4.451 4.533 -1.817 3.854 4.004 -3.739 3.147 3.340 -5.776 2.225 2.417 -7.937

0.204.796 23.36 4.323 4.412 -2.037 3.743 3.908 -4.202 3.057 3.269 -6.511 2.161 2.374 -8.977

Ct.=0.3

0.00 I 6.246 44.36 5.957 6.040 -1.360 5.159 5.307 -2.784 4.213 4.400 -4.271 2.979 3.163 -5.829

0.05 6.220 43.90 5.926 6.010 -1.405 5.132 5.284 -2.875 4.190 4.384 -4.415 2.963 3.153 -6.03 1

0.10 6.144 42.56 5.835 5.926 -1.534 5.053 5.217 -3.143 4.126 4.335 -4.830 2.918 3.124 -6.605

0.15 6.020 40.52 5.693 5.794 -1.729 4.931 5.112 -3.550 4.026 4.259 -5.468 2.847 3.077 -7.495

0.20 5.855 37.99 5.513 5.624 -1.971 4.774 4.976 -4.058 3.898 4.159 -6.271 2.756 3.016 -8.623

Ct.=0.4

0.0018.269 81.37 8.068 8.174 -1.2956.987 7.177 -2.645 5.705 5.946 -4.051 4.034 4.270 -5.5 18

0.058.225 80.31 8.016 8.126 -1.3526.942 7.139 -2.761 5.668 5.918 -4.2304.0084.253 -5 .765

0.10 8.096 77.32 7.865 7.985 -1.509 6.811 7.028 -3.086 5.561 5.838 -4.737 3.932 4.204 -6.463

0.15 7.889 72.84 7.633 7.769 -1.745 6.611 6.856 -3 .578 5.398 5.712 -5.505 3.8 17 4.128 -7.535

0.20 7.6 16 67.42 7.344 7.496 -2.031 6.360 6.638 -4.179 5.193 5.551 -6.453 3.672 4.029 -8.865

Ct.=0.5

0.00 I 11.84 169.6 11.65 11.80 -1.267 10.09 10.36 -2.580 8.237 8.576 -3.948 5.825 6.155 -5.368

0.05 11.75 166.7 11.55 11.70 -1.342 10.00 10.28 -2.736 8.165 8.522 -4.187 5.773 6.122 -5.698

0.10 11.50 158.4 11.26 11.43 -1.553 9.749 10.07 -3.173 7.960 8.367 -4.865 5.628 6.028 -6.633

0.15 I LI O 146.5 10.82 11.03 - 1.856 9.3759.746 -3.810 7.6548.131 -5.8625.4125.884 -8.022

0.20 10.58 132.7 10.30 10.54 -2.216 8.922 9.349 -4.565 7.285 7.838 -7.058 5.151 5.705 -9.708

Table 8. The maximum percentage deviation being -10.26% for a = 0.1 and f3 = 0.8, for the thickness ratio of 0.2 indicating that the match in the three mode shapes is not very good for higher values of f3 for a given a and the thickness ratio.

Conclusions A simple design formula developed by one of the

authors earlier is applied to study its effectiveness when applied to evaluate the fundamental frequency of initially stressed moderately thick annular plates

RENJITH & RAO: ANNULAR PLATES SUBJECTED TO A UNIFORM COMPRESSIVE LOAD 451

with various thickness ratios, subjected to a uniform compressive load at the outer edge. Several combinations of boundary conditions are considered at the outer and inner edges of the annular plates in the present study. The numerical results show that the proposed design formula gives sufficiently accurate estimates of the fundamental frequency parameter of moderately thick, initially stressed annular plates in general. But when a free edge boundary condition is considered, either there is a very good match or there is more percentage deviation for the same values of a and f3 when compared with the FEM results, depending on the edge, where the free boundary condition is specified. Based on the enormous data presented in this paper, it can be summarised that, the proposed design formula is very elegant and can be used for quickly estimating the fundamental

frequency of initially stressed, moderately thick annular plates, during the preliminary design phase.

References I Craig R R Jr, Combined experimental ! analytical modeling

of dynamic structural systems. edited by Martinez D R & Miller A K, (American Society of Mechanical Engineers, New York), 67, 1985, 1-30.

2 Zienkiewicz 0 C, The finite element method, (McGraw-Hill , London), 1977.

3 Naidu N R & Rao G V, J Aeronaut Soc India , 52, (2002), 40. 4 Rao G V & Naidu N R, AIM J, 39, (2001), 186. 5 Rao G V, J Soulld Vibrat , 246, (2001), 185. 6 Rao G V & Neetha R, J Vibrat Acoust (ASME) , 124, (2002),

451. 7 Rao G V & Neetha R, J Struct Eng (SERC) , 29, (2002), 177. 8 Rao G V & Neetha R, J Sound Vibrat, 259, (2003), 1265. 9 Nayar L S, Raju K K & Rao G V, J Sound Vibrat, 178,

(1994), 50 I.