V. Modal Response of a Circular Plate

Find the natural frequencies of a circular plate and compare theoretical answers to the STAAD solution.

Reference

Blevins, Robert D., Formulas for Natural Frequency and Mode Shape, Van Nostrand Reinhold Company, 1979, Page 240.

Problem

A flat circular plate is simply supported around the entire perimeter.  The first six modes and their associated natural frequencies are to be computed using the subspace iteration method offered by STAAD. This problem demonstrates that the natural frequencies of an axi-symmetric structure can be accurately computed utilizing a 180 degree model with the appropriate boundary conditions.

Diagram of circular plate

The 180 degree sector was modeled using radial lines at intervals of 15 degrees.  Tangential lines were then located utilizing a relationship such that the aspect ratio of the quad-plate elements was approximately 1.0.  All rotations normal to the plane of the plate were restrained.  In-plane translations for all nodes were restrained because the theoretical solution does not consider in-plane effects. Rotations about the global Y axis for the nodes at X=0.0 were restrained because this is a symmetry boundary. Z translation of all nodes on the outside radius were restrained to provide for the simply-supported condition.

Finite model for half circular plate

It should be noted that the outside edge of the plate is a series of secant lines instead of a true arc.  This will result in a loss of about 1% of the plate’s true mass or about .5% of the mass that is effective for this problem.

Therefore, it is not unlikely that a few natural frequencies will be lower than the theoretical values instead of higher which is typical for a finite element analysis using plate elements.  In addition, a true simple-support condition for this problem would require restraining the component of rotation that is radial to the outside edge.

Theoretical Calculations

From the reference case 2 in Table 11-1, the first six natural frequencies of the plate are described by the following equation:

$f i j = λ i j 2 2 π a 2 E h 3 12 γ ( 1 − ν 2 )$

= dimensionless parameter associated with the mode indices i,j

where
 i = number of nodal diameters in this mode shape j = number of nodal circles in this mode shape not counting the boundary ν = Poisson’s ratio E = elastic modulus h = plate thickness γ = mass of plate per unit area a = radius of plate

The numerical values used for this example are:

 ν = 0.30

 E = 10.0×10P6 psi

 h = 0.10 inches

a = 10.0 inches

with the numerical values used above

$1 2 π a 2 E h 3 12 γ ( 1 − ν 2 ) = 1 2 π ( 10.0 ) 2 ( 10.0 ⋅ 10 6 ) ( 0.10 ) 3 12 ( 2.588 ⋅ 10 − 5 ) [ 1 − ( 0.3 ) 2 ] = 9.467 ⁢$

λ2 ij is tabulated from the reference as follows:

Table 1. Values of λ2 ij
Mode Number λ2 ij Number of Nodal Diameters (i) Number of Nodal Circles (j)
1 4.977 0 0
2 13.94 1 0
3 25.65 2 0
4 29.76 0 1
5(l)   3 0
6 48.51 1 1

(l) not tabulated in the reference

Comparison

Table 2. Comparison of results
1 47.12 46.211 1.9%
2 132.0 130.541 1.1%
3 242.8 240.146 1.1%
4 281.7 283.028 0.5%
5 *   373.197 n/a*
6 459.3 466.579 1.6%

*The reference did not tabulate a value of for the fifth mode of the structure, hence a comparison with the theoretical value of this mode cannot be made.

All anti-symmetric mode shapes for the 360 degree circular plate were captured by the 180 degree model with a phase angle included in the calculation. Some of the difference between the theoretical and STAAD.Pro frequencies is attributed to the loss of mass due to the piecewise secant representation of the outer radius and, since this mass is about 1 percent lower than for a true circular plate, it is not surprising that the first few modes are lower than the theoretical solution.

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\08 Dynamic Analysis\Modal Response of a Circular Plate.STD is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 14-Sep-18
END JOB INFORMATION
* Natural Frequencies of a Circular Plate
UNIT INCHES POUND
JOINT COORDINATES
1 -6.39758e-07 -1 0; 2 0.258818 -0.965926 0; 3 0.5 -0.866026 0;
4 0.707107 -0.707107 0; 5 0.866025 -0.5 0; 6 0.965926 -0.258819 0;
7 1 0 0; 8 0.965926 0.258819 0; 9 0.866025 0.5 0;
10 0.707107 0.707107 0; 11 0.5 0.866026 0; 12 0.258818 0.965926 0;
13 -6.39758e-07 1 0; 14 -1.43562e-06 -2.244 0; 15 0.580789 -2.16754 0;
16 1.122 -1.94336 0; 17 1.58675 -1.58675 0; 18 1.94336 -1.122 0;
19 2.16754 -0.58079 0; 20 2.244 0 0; 21 2.16754 0.58079 0;
22 1.94336 1.122 0; 23 1.58675 1.58675 0; 24 1.122 1.94336 0;
25 0.580789 2.16754 0; 26 -1.43562e-06 2.244 0;
27 -2.23148e-06 -3.488 0; 28 0.902759 -3.36915 0; 29 1.744 -3.0207 0;
30 2.46639 -2.46639 0; 31 3.0207 -1.744 0; 32 3.36915 -0.902761 0;
33 3.488 0 0; 34 3.36915 0.902761 0; 35 3.0207 1.744 0;
36 2.46639 2.46639 0; 37 1.744 3.0207 0; 38 0.902759 3.36915 0;
39 -2.23148e-06 3.488 0; 40 -2.90386e-06 -4.539 0;
41 1.17478 -4.38434 0; 42 2.2695 -3.93089 0; 43 3.20956 -3.20956 0;
44 3.93089 -2.2695 0; 45 4.38434 -1.17478 0; 46 4.539 0 0;
47 4.38434 1.17478 0; 48 3.93089 2.2695 0; 49 3.20956 3.20956 0;
50 2.2695 3.93089 0; 51 1.17478 4.38434 0; 52 -2.90386e-06 4.539 0;
53 -3.77841e-06 -5.906 0; 54 1.52858 -5.70476 0; 55 2.953 -5.11475 0;
56 4.17617 -4.17617 0; 57 5.11475 -2.953 0; 58 5.70476 -1.52859 0;
59 5.906 0 0; 60 5.70476 1.52859 0; 61 5.11475 2.953 0;
62 4.17617 4.17617 0; 63 2.953 5.11475 0; 64 1.52858 5.70476 0;
65 -3.77841e-06 5.906 0; 66 -4.91654e-06 -7.685 0;
67 1.98902 -7.42314 0; 68 3.8425 -6.65541 0; 69 5.43411 -5.43412 0;
70 6.6554 -3.8425 0; 71 7.42314 -1.98903 0; 72 7.685 0 0;
73 7.42314 1.98903 0; 74 6.6554 3.8425 0; 75 5.43411 5.43412 0;
76 3.8425 6.65541 0; 77 1.98902 7.42314 0; 78 -4.91654e-06 7.685 0;
79 -6.39758e-06 -10 0; 80 2.58818 -9.65926 0; 81 5 -8.66026 0;
82 7.07107 -7.07107 0; 83 8.66025 -5 0; 84 9.65926 -2.58819 0;
85 10 0 0; 86 9.65926 2.58819 0; 87 8.66025 5 0; 88 7.07107 7.07107 0;
89 5 8.66026 0; 90 2.58818 9.65926 0; 91 -6.39758e-06 10 0; 1000 0 0 0;
ELEMENT INCIDENCES SHELL
1 1000 1 2; 2 1000 2 3; 3 1000 3 4; 4 1000 4 5; 5 1000 5 6; 6 1000 6 7;
7 1000 7 8; 8 1000 8 9; 9 1000 9 10; 10 1000 10 11; 11 1000 11 12;
12 1000 12 13; 13 1 14 15 2; 14 2 15 16 3; 15 3 16 17 4; 16 4 17 18 5;
17 5 18 19 6; 18 6 19 20 7; 19 7 20 21 8; 20 8 21 22 9; 21 9 22 23 10;
22 10 23 24 11; 23 11 24 25 12; 24 12 25 26 13; 25 14 27 28 15;
26 15 28 29 16; 27 16 29 30 17; 28 17 30 31 18; 29 18 31 32 19;
30 19 32 33 20; 31 20 33 34 21; 32 21 34 35 22; 33 22 35 36 23;
34 23 36 37 24; 35 24 37 38 25; 36 25 38 39 26; 37 27 40 41 28;
38 28 41 42 29; 39 29 42 43 30; 40 30 43 44 31; 41 31 44 45 32;
42 32 45 46 33; 43 33 46 47 34; 44 34 47 48 35; 45 35 48 49 36;
46 36 49 50 37; 47 37 50 51 38; 48 38 51 52 39; 49 40 53 54 41;
50 41 54 55 42; 51 42 55 56 43; 52 43 56 57 44; 53 44 57 58 45;
54 45 58 59 46; 55 46 59 60 47; 56 47 60 61 48; 57 48 61 62 49;
58 49 62 63 50; 59 50 63 64 51; 60 51 64 65 52; 61 53 66 67 54;
62 54 67 68 55; 63 55 68 69 56; 64 56 69 70 57; 65 57 70 71 58;
66 58 71 72 59; 67 59 72 73 60; 68 60 73 74 61; 69 61 74 75 62;
70 62 75 76 63; 71 63 76 77 64; 72 64 77 78 65; 73 66 79 80 67;
74 67 80 81 68; 75 68 81 82 69; 76 69 82 83 70; 77 70 83 84 71;
78 71 84 85 72; 79 72 85 86 73; 80 73 86 87 74; 81 74 87 88 75;
82 75 88 89 76; 83 76 89 90 77; 84 77 90 91 78;
ELEMENT PROPERTY
1 TO 84 THICKNESS 0.1
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 1e+07
POISSON 0.3
DENSITY 0.1
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
* centre of circle
1000 FIXED BUT FZ MX
* Interior nodes
2 TO 12 15 TO 25 28 TO 38 41 TO 51 54 TO 64 67 TO 77 FIXED BUT FZ MX MY
* nodes along circumference
79 TO 91 FIXED BUT MX MY
* nodes along diameter except the two at the ends of the diameter.
1 13 14 26 27 39 40 52 53 65 66 78 FIXED BUT FZ MX
SELFWEIGHT X 1
SELFWEIGHT Y 1
SELFWEIGHT Z 1
MODAL CALCULATION REQUESTED
PERFORM ANALYSIS
FINISH


               CALCULATED FREQUENCIES FOR LOAD CASE       1
MODE            FREQUENCY(CYCLES/SEC)         PERIOD(SEC)
1                      46.211                  0.02164
2                     130.541                  0.00766
3                     240.146                  0.00416
4                     283.028                  0.00353
5                     373.197                  0.00268
6                     466.579                  0.00214
MODAL WEIGHT (MODAL MASS TIMES g) IN POUN         GENERALIZED
MODE           X             Y             Z              WEIGHT
1       0.000000E+00  0.000000E+00  1.053697E+00    4.537237E-01
2       0.000000E+00  0.000000E+00  2.214294E-10    3.944628E-01
3       0.000000E+00  0.000000E+00  1.884687E-07    3.935446E-01
4       0.000000E+00  0.000000E+00  1.275807E-01    1.752589E-01
5       0.000000E+00  0.000000E+00  1.060135E-08    4.222042E-01
6       0.000000E+00  0.000000E+00  7.897355E-10    2.165119E-01
MASS PARTICIPATION FACTORS
MASS  PARTICIPATION FACTORS IN PERCENT
--------------------------------------
MODE    X     Y     Z     SUMM-X   SUMM-Y   SUMM-Z
1     0.00   0.00  86.78    0.000    0.000   86.780
2     0.00   0.00   0.00    0.000    0.000   86.780
3     0.00   0.00   0.00    0.000    0.000   86.780
4     0.00   0.00  10.51    0.000    0.000   97.287
5     0.00   0.00   0.00    0.000    0.000   97.287
6     0.00   0.00   0.00    0.000    0.000   97.287