V. Modal Response of a Rectangular Plate
Find the natural frequencies of a rectangular plate and compare theoretical answers to the STAAD.Pro solution.
Reference
Blevins, Robert D., Formulas for Natural Frequency and Mode Shape, Van Nostrand Reinhold Company, 1979, page 258.
Problem
A flat rectangular plate is simply supported on all four sides. The first six modes and their associated natural frequencies are to be computed for this structure using the subspace iteration method offered by STAAD.Pro. This problem also demonstrates that the mesh refinement can be chosen to accurately calculate modes of interest based on the expected mode shapes.
L = 45 in., W = 30 in., t = 0.2 in.
A plate with an aspect ratio of 1.5 was used so that comparison could be made with theoretical results tabulated for plates in the reference. An equally spaced mesh was utilized in both the x and the y dimensions of the plate. The number of elements in each dimension was determined on the basis of the highest mode of interest. Since the number of half-waves in the sixth mode is 3 in the length dimension and 2 in the width dimension, a node spacing of 3.75 inches results in each half- wave being represented by four elements which means that no element will be expected to deform in double curvature. The simply supported edge condition requires that translation normal to the plane of the plate be restrained for these edge nodes. Rotations normal to the plate were restrained for all nodes.
Theoretical Calculations
From the reference case 16 in Table 11-4, the first six natural frequencies of the plate are described by the following equations:
dimensionless parameter associated with the mode indices i, j where| = | ||
| = | ||
| = | ||
| = | ||
| = | ||
| = | ||
| = | ||
| = |
The numerical values used for this example are:
- ν = 0.30
- E =30.0×106 psi
- h=0.2 inches
- a = 45.0 in
- b = 30.0 in
with the numerical values used above
λ2 ij is tabulated from the reference as follows:
Comparison
| Mode Number | Theoretical | STAAD.Pro | Difference |
|---|---|---|---|
| 1 | 30.94 | 30.599 | 1.1% |
| 2 | 59.49 | 58.724 | 1.3% |
| 3 | 95.18 | 95.063 | negligible |
| 4 | 107.1 | 106.277 | 0.8% |
| 5 | 123.7 | 122.092 | 1.3% |
| 6 | 171.3 | 168.009 | 1.9% |
As noted earlier, the node spacing was based on the highest mode of interest. It follows that the difference between the theoretical and STAAD.Pro frequencies generally increases with increasing mode sequence.
STAAD Input
The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\08 Dynamic Analysis\Modal Response of a Rectangular 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 rectangular plate
UNIT INCHES POUND
JOINT COORDINATES
1 0 0 0; 2 3.75 0 0; 3 7.5 0 0; 4 11.25 0 0; 5 15 0 0; 6 18.75 0 0;
7 22.5 0 0; 8 26.25 0 0; 9 30 0 0; 10 33.75 0 0; 11 37.5 0 0;
12 41.25 0 0; 13 45 0 0; 14 0 3.75 0; 15 3.75 3.75 0; 16 7.5 3.75 0;
17 11.25 3.75 0; 18 15 3.75 0; 19 18.75 3.75 0; 20 22.5 3.75 0;
21 26.25 3.75 0; 22 30 3.75 0; 23 33.75 3.75 0; 24 37.5 3.75 0;
25 41.25 3.75 0; 26 45 3.75 0; 27 0 7.5 0; 28 3.75 7.5 0; 29 7.5 7.5 0;
30 11.25 7.5 0; 31 15 7.5 0; 32 18.75 7.5 0; 33 22.5 7.5 0;
34 26.25 7.5 0; 35 30 7.5 0; 36 33.75 7.5 0; 37 37.5 7.5 0;
38 41.25 7.5 0; 39 45 7.5 0; 40 0 11.25 0; 41 3.75 11.25 0;
42 7.5 11.25 0; 43 11.25 11.25 0; 44 15 11.25 0; 45 18.75 11.25 0;
46 22.5 11.25 0; 47 26.25 11.25 0; 48 30 11.25 0; 49 33.75 11.25 0;
50 37.5 11.25 0; 51 41.25 11.25 0; 52 45 11.25 0; 53 0 15 0;
54 3.75 15 0; 55 7.5 15 0; 56 11.25 15 0; 57 15 15 0; 58 18.75 15 0;
59 22.5 15 0; 60 26.25 15 0; 61 30 15 0; 62 33.75 15 0; 63 37.5 15 0;
64 41.25 15 0; 65 45 15 0; 66 0 18.75 0; 67 3.75 18.75 0;
68 7.5 18.75 0; 69 11.25 18.75 0; 70 15 18.75 0; 71 18.75 18.75 0;
72 22.5 18.75 0; 73 26.25 18.75 0; 74 30 18.75 0; 75 33.75 18.75 0;
76 37.5 18.75 0; 77 41.25 18.75 0; 78 45 18.75 0; 79 0 22.5 0;
80 3.75 22.5 0; 81 7.5 22.5 0; 82 11.25 22.5 0; 83 15 22.5 0;
84 18.75 22.5 0; 85 22.5 22.5 0; 86 26.25 22.5 0; 87 30 22.5 0;
88 33.75 22.5 0; 89 37.5 22.5 0; 90 41.25 22.5 0; 91 45 22.5 0;
92 0 26.25 0; 93 3.75 26.25 0; 94 7.5 26.25 0; 95 11.25 26.25 0;
96 15 26.25 0; 97 18.75 26.25 0; 98 22.5 26.25 0; 99 26.25 26.25 0;
100 30 26.25 0; 101 33.75 26.25 0; 102 37.5 26.25 0; 103 41.25 26.25 0;
104 45 26.25 0; 105 0 30 0; 106 3.75 30 0; 107 7.5 30 0; 108 11.25 30 0;
109 15 30 0; 110 18.75 30 0; 111 22.5 30 0; 112 26.25 30 0; 113 30 30 0;
114 33.75 30 0; 115 37.5 30 0; 116 41.25 30 0; 117 45 30 0;
ELEMENT INCIDENCES SHELL
1 1 2 15 14; 2 2 3 16 15; 3 3 4 17 16; 4 4 5 18 17; 5 5 6 19 18;
6 6 7 20 19; 7 7 8 21 20; 8 8 9 22 21; 9 9 10 23 22; 10 10 11 24 23;
11 11 12 25 24; 12 12 13 26 25; 13 14 15 28 27; 14 15 16 29 28;
15 16 17 30 29; 16 17 18 31 30; 17 18 19 32 31; 18 19 20 33 32;
19 20 21 34 33; 20 21 22 35 34; 21 22 23 36 35; 22 23 24 37 36;
23 24 25 38 37; 24 25 26 39 38; 25 27 28 41 40; 26 28 29 42 41;
27 29 30 43 42; 28 30 31 44 43; 29 31 32 45 44; 30 32 33 46 45;
31 33 34 47 46; 32 34 35 48 47; 33 35 36 49 48; 34 36 37 50 49;
35 37 38 51 50; 36 38 39 52 51; 37 40 41 54 53; 38 41 42 55 54;
39 42 43 56 55; 40 43 44 57 56; 41 44 45 58 57; 42 45 46 59 58;
43 46 47 60 59; 44 47 48 61 60; 45 48 49 62 61; 46 49 50 63 62;
47 50 51 64 63; 48 51 52 65 64; 49 53 54 67 66; 50 54 55 68 67;
51 55 56 69 68; 52 56 57 70 69; 53 57 58 71 70; 54 58 59 72 71;
55 59 60 73 72; 56 60 61 74 73; 57 61 62 75 74; 58 62 63 76 75;
59 63 64 77 76; 60 64 65 78 77; 61 66 67 80 79; 62 67 68 81 80;
63 68 69 82 81; 64 69 70 83 82; 65 70 71 84 83; 66 71 72 85 84;
67 72 73 86 85; 68 73 74 87 86; 69 74 75 88 87; 70 75 76 89 88;
71 76 77 90 89; 72 77 78 91 90; 73 79 80 93 92; 74 80 81 94 93;
75 81 82 95 94; 76 82 83 96 95; 77 83 84 97 96; 78 84 85 98 97;
79 85 86 99 98; 80 86 87 100 99; 81 87 88 101 100; 82 88 89 102 101;
83 89 90 103 102; 84 90 91 104 103; 85 92 93 106 105; 86 93 94 107 106;
87 94 95 108 107; 88 95 96 109 108; 89 96 97 110 109; 90 97 98 111 110;
91 98 99 112 111; 92 99 100 113 112; 93 100 101 114 113;
94 101 102 115 114; 95 102 103 116 115; 96 103 104 117 116;
ELEMENT PROPERTY
1 TO 96 THICKNESS 0.2
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 3e+07
POISSON 0.3
DENSITY 0.282
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 ALL
CUT OFF MODE SHAPE 6
CUT OFF FREQUENCY 1000
SUPPORTS
* Corner nodes
1 13 105 117 FIXED BUT MX MY
* Nodes along y=0 and y=30
2 TO 12 106 TO 116 FIXED BUT MX MY
* Nodes along x=0
14 27 40 53 66 79 92 FIXED BUT MX MY
* Nodes along x=45
26 39 52 65 78 91 104 FIXED BUT MX MY
* Interior nodes
15 TO 25 28 TO 38 41 TO 51 54 TO 64 67 TO 77 80 TO 90 93 TO 102 -
103 FIXED BUT FZ MX MY
LOAD 1
SELFWEIGHT X 1
SELFWEIGHT Y 1
SELFWEIGHT Z 1
MODAL CALCULATION REQUESTED
PERFORM ANALYSIS
FINISH
STAAD Output
CALCULATED FREQUENCIES FOR LOAD CASE 1 MODE FREQUENCY(CYCLES/SEC) PERIOD(SEC) 1 30.599 0.03268 2 58.724 0.01703 3 95.063 0.01052 4 106.277 0.00941 5 122.092 0.00819 6 168.009 0.00595 MODAL WEIGHT (MODAL MASS TIMES g) IN POUN GENERALIZED MODE X Y Z WEIGHT 1 0.000000E+00 0.000000E+00 4.833474E+01 1.915440E+01 2 0.000000E+00 0.000000E+00 9.476868E-17 1.920213E+01 3 0.000000E+00 0.000000E+00 4.308438E-14 1.925532E+01 4 0.000000E+00 0.000000E+00 4.837131E+00 1.927455E+01 5 0.000000E+00 0.000000E+00 3.971700E-13 1.927602E+01 6 0.000000E+00 0.000000E+00 2.329330E-10 1.936312E+01 MASS PARTICIPATION FACTORS MASS PARTICIPATION FACTORS IN PERCENT -------------------------------------- MODE X Y Z SUMM-X SUMM-Y SUMM-Z 1 0.00 0.00 79.15 0.000 0.000 79.146 2 0.00 0.00 0.00 0.000 0.000 79.146 3 0.00 0.00 0.00 0.000 0.000 79.146 4 0.00 0.00 7.92 0.000 0.000 87.066 5 0.00 0.00 0.00 0.000 0.000 87.066 6 0.00 0.00 0.00 0.000 0.000 87.066