# 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.

### Simply supported, rectangular plate

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
$f i j = λ i j 2 2 π a 2 E h 3 12 γ ( 1 − ν 2 )$
where
 i = number of half-waves in this mode shape along the horizontal axis j = number of half-waves in this mode shape along the vertical axis ν = Poisson’s ratio E = elastic modulus h = plate thickness γ = mass of material per unit area a = length of plate b = width of plate

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

$1 2 π a 2 E h 3 12 γ ( 1 − ν 2 ) = 1 2 π ( 10.0 ) 2 ( 30.0 ⋅ 10 6 ) ( 0.20 ) 3 12 ( 1.460 ⋅ 10 − 4 ) [ 1 − ( 0.3 ) 2 ] = 0.9644 ⁢$

λ2 ij is tabulated from the reference as follows:

Table 1. Values of λ2 ij
Mode Number λ2 ij Number of Half-Waves in Length (i) Number of Half-Waves in Width (j)
1 32.08 1 1
2 61.69 2 1
3 98.70 1 2
4 111.0 3 1
5 128.3 2 2
6 177.7 3 2

## Comparison

Table 2. Comparison of results
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