# 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

1. Problem 1, from the ASME 1972 Program Verification and Qualification Library
2. DeSalvo, G.J., and Swanson, J.A., ANSYS Engineering Analysis System Examples Manual, Swanson Analysis Systems, Inc., 1979, Example Problem No. 2.
3. 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.

### Frame model

• 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)
• 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:

$α = A Q I b = 4 3 R o 3 − R i 3 ( R o 2 − R i 2 ) ( R o − R i )$

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

Table 1. Comparison of results
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.

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
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


               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