V. Modal Response of a 3D Frame
Find the natural frequencies of a space frame. Compare the results from subspace iteration method used by STAAD.Pro with the results from EASE2® and ANSYS®.
Reference
- Problem 1, from the ASME 1972 Program Verification and Qualification Library
- DeSalvo, G.J., and Swanson, J.A., ANSYS Engineering Analysis System Examples Manual, Swanson Analysis Systems, Inc., 1979, Example Problem No. 2.
- Peterson, F.E., EASE2 Elastic Analysis for Structural Engineering Example Problem Manual, Engineering Analysis Corporation, 1981, Example 2.03.
Problem
A three dimensional frame is analyzed for its natural frequencies and the associated mode shapes using the subspace iteration method offered by STAAD.Pro.
-
L1 = 8.625 ft
-
L2 = 10 ft
All dimensions are in inches.
The only entity present in the model is the general purpose three dimensional beam.
Cross-section Properties
All the beam elements have the same cross section:
- Outside radius, Ro = 1.1875 in.
- Inside radius, Ri = 1.0335 in.
- Area, A = π(Ro 2 - Ri 2) = 1.074532 in2
- Moment of Inertia, I = π/4(Ro 4 - Ri 4)
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I = 0.665747 in4
- Torsion constant = twice the inertia, for closed circular sections = 1.331494 in4
The expression for the shear flexibility factor is derived from the ratio of maximum shear stress to the average shear stress:
α = AQ/Ib
Where the four items – A, Q, I, and b are properties of the half cross-section about the centerline. For the circular section they are:
A = π/2(Ro2 - Ri2)
Q = 2/3(Ro3 - Ri3)
I = π/8(Ro4 - Ri4)
I = 2(Ro - Ri)
The final expression for the shear flexibility factor, for a circular tube section is:
α = 1.993620
Hence the shear area is entered as
AY = Cross Section Area / α = 1.074532 / 1.99362 = 0.538985 in2
Comparison
The following table compares Twenty-Four Natural Frequencies
| Mode Number | DeSalvo and Swanson2 | Peterson3 | STAAD.Pro |
|---|---|---|---|
| 1 | 111.52 | 111.53 | 111.236 |
| 2 | 115.95 | 115.96 | 115.801 |
| 3 | 137.60 | 137.61 | 137.169 |
| 4 | 218.02 | 218.03 | 215.799 |
| 5 | 404.23 | 404.23 | 404.270 |
| 6 | 422.70 | 422.72 | 422.642 |
| 7 | 451.72 | 451.75 | 451.599 |
| 8 | 553.99 | 554.07 | 549.012 |
| 9 | 735.70 | 735.81 | 733.589 |
| 10 | 762.32 | 762.44 | 758.563 |
| 11 | 852.57 | 852.72 | 851.354 |
| 12 | 894.08 | 894.26 | 892.281 |
| 13 | 910.21 | 910.40 | 893.060 |
| 14 | 916.98 | 917.18 | 910.829 |
| 15 | 940.02 | 940.02 | 932.286 |
| 16 | 959.98 | 960.27 | 956.391 |
| 17 | 971.15 | 971.44 | 964.012 |
| 18 | 976.92 | 977.22 | 967.267 |
| 19 | 1012.2 | 1012.5 | 981.536 |
| 20 | 1028.4 | 1028.8 | 1009.256 |
| 21 | 1123.6 | 1123.9 | 1070.562 |
| 22 | 1134.5 | 1134.9 | 1123.161 |
| 23 | 1164.1 | 1164.4 | 1149.368 |
| 24 | 1216.7 | 1217.2 | 1199.882 |
In both references by DeSalvo and Swanson2 and Peterson3, the number of dynamic degrees of freedom has been reduced from 42 to 24, by means of the Guyan method. No such reduction is performed in the STAAD.Pro model.
STAAD Input
The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\08 Dynamic Analysis\Modal Response of a 3D Frame.STD is typically installed with the program.
STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 14-Sep-18
END JOB INFORMATION
* Natural Modes of a space frame
UNIT INCHES POUND
JOINT COORDINATES
1 0 0 0; 2 27.25 0 0; 3 0 10 0; 4 27.25 10 0; 5 0 18.625 0;
6 8.625 18.625 0; 7 18.625 18.625 0; 8 27.25 18.625 0; 9 0 18.625 8.625;
10 27.25 18.625 8.625; 11 0 0 17.25; 12 27.25 0 17.25; 13 0 10 17.25;
14 27.25 10 17.25; 15 0 18.625 17.25; 16 8.625 18.625 17.25;
17 18.625 18.625 17.25; 18 27.25 18.625 17.25;
MEMBER INCIDENCES
1 1 3; 2 2 4; 3 3 5; 4 4 8; 5 5 6; 6 6 7; 7 7 8; 8 5 9; 9 8 10; 10 9 15;
11 10 18; 12 11 13; 13 12 14; 14 13 15; 15 14 18; 16 15 16; 17 16 17;
18 17 18;
MEMBER PROPERTY AMERICAN
1 TO 17 -
18 PRIS AX 1.07453 AY 0.538985 AZ 0.538985 IX 1.33149 IY 0.665747 -
IZ 0.665747
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 2.79e+07
POISSON 0.3
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 ALL
CUT OFF MODE SHAPE 24
SUPPORTS
1 2 11 12 FIXED
LOAD 1
JOINT LOAD
3 4 13 14 FX 3.4517 FY 3.4517 FZ 3.4517
6 7 16 17 FX 3.4517 FY 3.4517 FZ 3.4517
9 10 FX 3.4517 FY 3.4517 FZ 3.4517
5 8 15 18 FX 9.7973 FY 9.7973 FZ 9.7973
MODAL CALCULATION REQUESTED
PERFORM ANALYSIS
FINISH
STAAD Output
CALCULATED FREQUENCIES FOR LOAD CASE 1 MODE FREQUENCY(CYCLES/SEC) PERIOD(SEC) 1 111.236 0.00899 2 115.801 0.00864 3 137.169 0.00729 4 215.799 0.00463 5 404.270 0.00247 6 422.642 0.00237 7 451.599 0.00221 8 549.012 0.00182 9 733.589 0.00136 10 758.563 0.00132 11 851.354 0.00117 12 892.281 0.00112 13 893.060 0.00112 14 910.829 0.00110 15 932.286 0.00107 16 956.391 0.00105 17 964.012 0.00104 18 967.267 0.00103 19 981.536 0.00102 20 1009.256 0.00099 21 1070.562 0.00093 22 1123.161 0.00089 23 1149.368 0.00087 24 1199.882 0.00083 MODAL WEIGHT (MODAL MASS TIMES g) IN POUN GENERALIZED MODE X Y Z WEIGHT 1 6.984127E+01 6.404211E-24 1.284664E-24 6.127270E+01 2 1.547059E-22 1.789605E-20 7.005318E+01 5.765801E+01 3 1.064176E-21 1.226078E-19 9.256770E-23 7.213569E+01 4 9.147252E-20 1.045721E-17 1.002150E-20 7.915618E+01 5 4.553346E-19 4.244579E-03 4.985566E-20 1.532638E+01 6 5.447627E-22 1.825470E+01 1.949591E-25 1.469096E+01 7 1.160472E-20 1.375280E-18 5.112414E-03 1.450510E+01 8 1.699531E-22 8.948109E-20 2.966451E-01 1.825927E+01 9 1.055938E-23 9.521535E+00 1.626382E-22 1.177226E+01 10 9.808838E-04 8.794438E-20 3.656936E-22 1.174657E+01 11 1.321191E+00 9.714550E-18 1.273002E-20 1.582128E+01 12 6.082169E-21 1.911260E+00 1.769785E-20 1.624941E+01 13 1.722239E+00 1.063875E-16 1.119734E-19 3.802991E+01 14 4.611355E-20 4.549234E-17 5.528088E-21 2.195002E+01 15 4.392235E-20 6.381431E-17 2.363042E-01 1.628835E+01 16 1.400997E-20 4.122974E+00 1.604295E-20 1.648079E+01 17 8.893063E-19 1.239356E-15 1.393843E-18 1.622824E+01 18 1.975827E-18 2.780487E-15 3.083966E+00 1.640512E+01 STAAD SPACE -- PAGE NO. 4 19 3.491399E-01 1.038217E-15 5.994489E-19 2.835638E+01 20 5.049440E-19 1.122183E+01 4.036566E-19 2.560377E+01 21 3.350978E-18 3.211568E-15 7.101758E-19 3.464248E+01 22 3.635973E-01 3.238417E-15 1.516123E-18 3.137071E+01 23 8.666280E-17 9.060816E-14 3.752768E-17 1.555328E+01 24 6.382000E-02 1.876943E-13 5.827892E-17 2.520961E+01 STAAD SPACE -- PAGE NO. 5 MASS PARTICIPATION FACTORS MASS PARTICIPATION FACTORS IN PERCENT -------------------------------------- MODE X Y Z SUMM-X SUMM-Y SUMM-Z 1 94.76 0.00 0.00 94.756 0.000 0.000 2 0.00 0.00 95.04 94.756 0.000 95.044 3 0.00 0.00 0.00 94.756 0.000 95.044 4 0.00 0.00 0.00 94.756 0.000 95.044 5 0.00 0.01 0.00 94.756 0.006 95.044 6 0.00 24.77 0.00 94.756 24.773 95.044 7 0.00 0.00 0.01 94.756 24.773 95.051 8 0.00 0.00 0.40 94.756 24.773 95.453 9 0.00 12.92 0.00 94.756 37.691 95.453 10 0.00 0.00 0.00 94.758 37.691 95.453 11 1.79 0.00 0.00 96.550 37.691 95.453 12 0.00 2.59 0.00 96.550 40.284 95.453 13 2.34 0.00 0.00 98.887 40.284 95.453 14 0.00 0.00 0.00 98.887 40.284 95.453 15 0.00 0.00 0.32 98.887 40.284 95.774 16 0.00 5.59 0.00 98.887 45.878 95.774 17 0.00 0.00 0.00 98.887 45.878 95.774 18 0.00 0.00 4.18 98.887 45.878 99.958 19 0.47 0.00 0.00 99.360 45.878 99.958 20 0.00 15.23 0.00 99.360 61.103 99.958 21 0.00 0.00 0.00 99.360 61.103 99.958 22 0.49 0.00 0.00 99.854 61.103 99.958 23 0.00 0.00 0.00 99.854 61.103 99.958 24 0.09 0.00 0.00 99.940 61.103 99.958