# EX. US-28 Calculation of Modes and Frequencies of a Bridge

This example demonstrates the input required for obtaining the modes and frequencies of the skewed bridge shown in the figure below. The structure consists of piers, pier-cap girders and a deck slab.

This problem is installed with the program by default to C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\Sample Models\US\US-28 Calculation of Modes and Frequencies of a Bridge.std when you install the program.

### Example Problem No. 28

`    STAAD SPACE FREQUENCIES OF VIBRATION OF A SKEWED BRIDGE`

Every STAAD input file has to begin with the word STAAD. The word SPACE signifies that the structure is a space frame and the geometry is defined through X, Y and Z axes. The remainder of the words forms a title to identify this project.

`    IGNORE LIST`

Further below in this file, we will call element lists in which some element numbers may not actually be present in the structure. We do so because it minimizes the effort involved in fetching the desired elements and reduces the size of the respective commands. To prevent the program from treating that condition (referring to elements which do not exist) as an error, the above command is required.

`    UNIT METER KN`

The units for the data that follows are specified above.

```    JOINT COORDINATES
1 0 0 0; 2 4 0 0; 3 6.5 0 0; 4 9 0 0; 5 11.5 0 0; 6 15.5 0 0;
11 -1 10 0 25 16.5 10 0
REPEAT ALL 3 4 0 14```

For joints 1 through 6, the joint number followed by the X, Y and Z coordinates are specified first.

Next, using the coordinates of joints 11 and 25 as the basis, joints 12 through 24 are generated using linear interpolation.

Following this, using the data of these 21 joints (1 through 6 and 11 through 25), 63 new joints are generated. To achieve this, the X coordinate of these 21 joints is incremented by 4 meters and the Z coordinate is incremented by 14 meters, in 3 successive operations.

The REPEAT ALL command is used for the generation. Details of this command is available in TR.11 Joint Coordinates Specification . The results of the generation may be visually verified using STAAD.Pro's graphical viewing facilities.

```    MEMBER INCI
1 1 13 ; 2 2 15 ; 3 3 17 ; 4 4 19 ; 5 5 21 ; 6 6 23
26 26 34 ; 27 27 36 ; 28 28 38 ; 29 29 40 ; 30 30 42 ; 31 31 44
47 47 55 ; 48 48 57 ; 49 49 59 ; 50 50 61 ; 51 51 63 ; 52 52 65
68 68 76 ; 69 69 78 ; 70 70 80 ; 71 71 82 ; 72 72 84 ; 73 73 86```

The member connectivity data (joint numbers between which members are connected) is specified for the 24 columns for the structure. The above method, where the member number is followed by the 2 node numbers, is the explicit definition method. No generation is involved here.

```    101 11 12 114
202 32 33 215
303 53 54 316
404 74 75 417```

The member connectivity data is specified for the pier cap beams for the structure. The above method is a combination of explicit definition and generation. For example, member 101 is defined as connected between 11 & 12. Then, by incrementing those nodes by 1 unit at a time (which is the default increment), the incidences of members 102 to 114 are generated. Similarly, we create members 202 to 215, 303 to 316, and, 404 to 417.

```    DEFINE MESH
A JOINT 11
B JOINT 25
C JOINT 46
D JOINT 32
E JOINT 67
F JOINT 53
G JOINT 88
H JOINT 74```

The next step is to generate the deck slab which will be modeled using plate elements. For this, we use a technique called mesh generation. Mesh generation is a process of generating several "child" elements from a "parent" or "super" element. The above set of commands defines the corner nodes of the super-element. Details of the above can be found in TR.14.2 Element Mesh Generation.

Note that instead of elaborately defining the coordinates of the corner nodes of the super-elements, we have taken advantage of the fact that the coordinates of these joints (A through H) have already been defined or generated earlier. Thus, A is the same as joint 11 while D is the same as joint 32. Alternatively, we could have defined the super-element nodes as A -1 10 0 ; B 16.5 10 0 ; C 20.5 10 14 ; D 3 10 14 ; etc.

```    GENERATE ELEMENT
MESH ABCD 14 12
MESH DCEF 14 12
MESH FEGH 14 12```

The above lines are the instructions for generating the "child" elements from the super-elements. For example, from the super-element bound by the corners A, B, C and D (which in turn are nodes 11, 25, 46 and 32), we generate a total of 14X12=168 elements, with 14 divisions along the edges AB and CD, and 12 along the edges BC and DA. These are the elements which make up the first span.

Similarly, 168 elements are created for the 2nd span, and another 168 for the 3rd span.

It may be noted here that we have taken great care to ensure that the resulting elements and the piercap beams form a perfect fit. In other words, there is no overlap between the two in a manner that nodes of the beams are at a different point in space than nodes of elements. At every node along their common boundary, plates and beams are properly connected. This is absolutely essential to ensure proper transfer of load and stiffness from beams to plates and vice versa. The tools in the user interface may be used to confirm that beam-plate connectivity is proper for this model.

```    START GROUP DEFINITION
MEMBER
_GIRDERS 101 TO 114 202 TO 215 303 TO 316 404 TO 417
_PIERS 1 TO 6 26 TO 31 47 TO 52 68 TO 73
ELEMENT
_P1 447 TO 450 454 TO 457 461 TO 464 468 TO 471
_P2 531 TO 534 538 TO 541 545 TO 548 552 TO 555
_P3 615 TO 618 622 TO 625 629 TO 632 636 TO 639
_P4 713 TO 716 720 TO 723 727 TO 730 734 TO 737
_P5 783 TO 786 790 TO 793 797 TO 800 804 TO 807
_P6 881 TO 884 888 TO 891 895 TO 898 902 TO 905
END GROUP DEFINITION```

The above block of data is referred to as formation of groups. Group names are a mechanism by which a single moniker can be used to refer to a cluster of entities, such as members. For our structure, the piercap beams are being grouped to a name called GIRDERS, the pier columns are assigned the name PIERS, and so on. For the deck, a few selected elements are chosen into a few selective groups. The reason is that these elements happen to be right beneath wheels of vehicles whose weight will be used in the frequency calculation.

```    MEMBER PROPERTY
_GIRDERS PRIS YD 0.6 ZD 0.6
_PIERS PRIS YD 1.0```

Member properties are assigned as prismatic rectangular sections for the girders, and prismatic circular sections for the columns.

```    ELEMENT PROPERTY
YRA 9 11 TH 0.375```

The plate elements of the deck slab, which happen to be at a Y elevation of 10 metres (between a YRANGE of 9 metres and 11 metres) are assigned a thickness of 375 mms.

```    UNIT KNS MMS
DEFINE MATERIAL START
ISOTROPIC CONCRETE
E 21.0
POISSON 0.17
DENSITY 2.36158e-008
ALPHA 5e-006
DAMP 0.05
G 9.25
TYPE CONCRETE
STRENGTH FCU 0.0275
END DEFINE MATERIAL
CONSTANTS
MATERIAL CONCRETE ALL```

The Modulus of elasticity (E) is set to 21000 N/sq.mm for all members. The keyword CONSTANTS has to precede this data. Built-in default value for Poisson's ratio for concrete is also assigned to ALL members and elements.

```    UNIT KNS METER
CONSTANTS
DENSITY 24 ALL```

Following a change of units, density of concrete is specified.

```    SUPPORTS
1 TO 6 26 TO 31 47 TO 52 68 TO 73 FIXED```

The base nodes of the piers are fully restrained (FIXED supports).

```    CUT OFF MODE SHAPE 65
```

Theoretically, a structure has as many modes of vibration as the number of degrees of freedom in the model. However, the limitations of the mathematical process used in extracting modes may limit the number of modes that can actually be extracted. In a large structure, the extraction process can also be very time consuming. Further, not all modes are of equal importance. (One measure of the importance of modes is the participation factor of that mode.) In many cases, the first few modes may be sufficient to obtain a significant portion of the total dynamic response.

Due to these reasons, in the absence of any explicit instruction, STAAD calculates only the first 6 modes. This is like saying that the command CUT OFF MODE SHAPE 6 has been specified.

If the inspection of the first 6 modes reveals that the overall vibration pattern of the structure has not been obtained, one may ask STAAD to compute a larger (or smaller) number of modes with the help of this command. The number that follows this command is the number of modes being requested. In our example, we are asking for 65 modes by specifying CUT OFF MODE SHAPE 65.

```    UNIT KGS METER
SELFWEIGHT X 1.0
SELFWEIGHT Y 1.0
SELFWEIGHT Z 1.0
* PERMANENT WEIGHTS ON DECK
YRA 9 11 PR GX 200
YRA 9 11 PR GY 200
YRA 9 11 PR GZ 200
* VEHICLES ON SPANS - ONLY Y & Z EFFECT CONSIDERED
_P1 PR GY 700
_P2 PR GY 700
_P3 PR GY 700
_P4 PR GY 700
_P5 PR GY 700
_P6 PR GY 700
_P1 PR GZ 700
_P2 PR GZ 700
_P3 PR GZ 700
_P4 PR GZ 700
_P5 PR GZ 700
_P6 PR GZ 700```

The mathematical method that STAAD uses is called the eigen extraction method. Some information on this is available in G.17.3 Dynamic Analysis . The method involves 2 matrices - the stiffness matrix, and the mass matrix.

The stiffness matrix, usually called the [K] matrix, is assembled using data such as member and element lengths, member and element properties, modulus of elasticity, Poisson's ratio, member and element releases, member offsets, support information, etc.

For assembling the mass matrix, called the [M] matrix, STAAD uses the load data specified in the load case in which the MODAL CAL REQ command is specified. So, some of the important aspects to bear in mind are :

1. The input you specify is weights, not masses. Internally, STAAD will convert weights to masses by dividing the input by "g", the acceleration due to gravity.

2. If the structure is declared as a PLANE frame, there are 2 possible directions of vibration - global X, and global Y. If the structure is declared as a SPACE frame, there are 3 possible directions - global X, global Y and global Z. However, this does not guarantee that STAAD will automatically consider the masses for vibration in all the available directions.

You have control over and are responsible for specifying the directions in which the masses ought to vibrate. In other words, if a weight is not specified along a certain direction, the corresponding degrees of freedom (such as for example, global X at node 34 hypothetically) will not receive a contribution in the mass matrix. The mass matrix is assembled using only the masses from the weights and directions you specify.

In our example, notice that we are specifying the selfweight along global X, Y and Z directions. Similarly, a 200 kg/sq.m pressure load is also specified along all 3 directions on the deck.

But for the truck loads, we choose to apply it on just a few elements in the global Y and Z directions only. The reasoning is something like - for the X direction, the mass is not capable of vibrating because the tires allow the truck to roll along X. Remember, this is just a demonstration example, not necessarily what you may want to do.

The point we want to illustrate is that if you want to restrict a certain weight to certain directions only, all you need to do is not provide the directions in which those weights cannot vibrate in.

3. As much as possible, provide absolute values for the weights. STAAD is programmed to algebraically add the weights at nodes. So, if some weights are specified as positive numbers and others as negative, the total weight at a given node is the algebraic summation of all the weights in the global directions at that node and the mass is then derived from this algebraic resultant.

`    MODAL CALCULATION REQUESTED`

This is the command which tells the program that frequencies and modes should be calculated. It is specified inside a load case. In other words, this command accompanies the loads that are to be used in generating the mass matrix.

Frequencies and modes have to be calculated also when dynamic analysis such as response spectrum or time history analysis is carried out. But in such analyses, the MODAL CALCULATION REQUESTED command is not explicitly required. When STAAD encounters the commands for response spectrum (see example 11) and time history (see examples 16 and 22), it automatically will carry out a frequency extraction without the help of the MODAL .. command.

`    PERFORM ANALYSIS`

This initiates the processes which are required to obtain the frequencies. Frequencies, periods and participation factors are automatically reported in the output file when the operation is completed.

`    FINISH`

## Input File

``````STAAD SPACE FREQUENCIES OF VIBRATION OF A SKEWED BRIDGE
IGNORE LIST
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 4 0 0; 3 6.5 0 0; 4 9 0 0; 5 11.5 0 0; 6 15.5 0 0;
11 -1 10 0 25 16.5 10 0
REPEAT ALL 3 4 0 14
MEMBER INCI
1 1 13 ; 2 2 15 ; 3 3 17 ; 4 4 19 ; 5 5 21 ; 6 6 23
26 26 34 ; 27 27 36 ; 28 28 38 ; 29 29 40 ; 30 30 42 ; 31 31 44
47 47 55 ; 48 48 57 ; 49 49 59 ; 50 50 61 ; 51 51 63 ; 52 52 65
68 68 76 ; 69 69 78 ; 70 70 80 ; 71 71 82 ; 72 72 84 ; 73 73 86
101 11 12 114
202 32 33 215
303 53 54 316
404 74 75 417
DEFINE MESH
A JOINT 11
B JOINT 25
C JOINT 46
D JOINT 32
E JOINT 67
F JOINT 53
G JOINT 88
H JOINT 74
GENERATE ELEMENT
MESH ABCD 14 12
MESH DCEF 14 12
MESH FEGH 14 12
START GROUP DEFINITION
MEMBER
_GIRDERS 101 TO 114 202 TO 215 303 TO 316 404 TO 417
_PIERS 1 TO 6 26 TO 31 47 TO 52 68 TO 73
ELEMENT
_P1 447 TO 450 454 TO 457 461 TO 464 468 TO 471
_P2 531 TO 534 538 TO 541 545 TO 548 552 TO 555
_P3 615 TO 618 622 TO 625 629 TO 632 636 TO 639
_P4 713 TO 716 720 TO 723 727 TO 730 734 TO 737
_P5 783 TO 786 790 TO 793 797 TO 800 804 TO 807
_P6 881 TO 884 888 TO 891 895 TO 898 902 TO 905
END GROUP DEFINITION
MEMBER PROPERTY
_GIRDERS PRIS YD 0.6 ZD 0.6
_PIERS PRIS YD 1.0
ELEMENT PROPERTY
YRA 9 11 TH 0.375
UNIT KNS MMS
DEFINE MATERIAL START
ISOTROPIC CONCRETE
E 21.0
POISSON 0.17
DENSITY 2.36158e-008
ALPHA 5e-006
DAMP 0.05
G 9.25
TYPE CONCRETE
STRENGTH FCU 0.0275
END DEFINE MATERIAL
CONSTANTS
MATERIAL CONCRETE ALL
SUPPORTS
1 TO 6 26 TO 31 47 TO 52 68 TO 73 FIXED
CUT OFF MODE SHAPE 65
UNIT KGS METER
SELFWEIGHT X 1.0
SELFWEIGHT Y 1.0
SELFWEIGHT Z 1.0
* PERMANENT WEIGHTS ON DECK
YRA 9 11 PR GX 200
YRA 9 11 PR GY 200
YRA 9 11 PR GZ 200
* VEHICLES ON SPANS - ONLY Y & Z EFFECT CONSIDERED
_P1 PR GY 700
_P2 PR GY 700
_P3 PR GY 700
_P4 PR GY 700
_P5 PR GY 700
_P6 PR GY 700
_P1 PR GZ 700
_P2 PR GZ 700
_P3 PR GZ 700
_P4 PR GZ 700
_P5 PR GZ 700
_P6 PR GZ 700
MODAL CALCULATION REQUESTED
PERFORM ANALYSIS
FINISH
``````

```                                                                  PAGE NO.    1
****************************************************
*                                                  *
*           Version  22.04.00.**                   *
*           Proprietary Program of                 *
*           Bentley Systems, Inc.                  *
*           Date=    APR 21, 2020                  *
*           Time=    15:42:25                      *
*                                                  *
*  Licensed to: Bentley Systems Inc                *
****************************************************
1. STAAD SPACE FREQUENCIES OF VIBRATION OF A SKEWED BRIDGE
INPUT FILE: US-28 Calculation of Modes and Frequencies of a Bridge.STD
2. IGNORE LIST
3. UNIT METER KN
4. JOINT COORDINATES
5. 1 0 0 0; 2 4 0 0; 3 6.5 0 0; 4 9 0 0; 5 11.5 0 0; 6 15.5 0 0
6. 11 -1 10 0 25 16.5 10 0
7. REPEAT ALL 3 4 0 14
8. MEMBER INCI
9. 1 1 13 ; 2 2 15 ; 3 3 17 ; 4 4 19 ; 5 5 21 ; 6 6 23
10. 26 26 34 ; 27 27 36 ; 28 28 38 ; 29 29 40 ; 30 30 42 ; 31 31 44
11. 47 47 55 ; 48 48 57 ; 49 49 59 ; 50 50 61 ; 51 51 63 ; 52 52 65
12. 68 68 76 ; 69 69 78 ; 70 70 80 ; 71 71 82 ; 72 72 84 ; 73 73 86
13. 101 11 12 114
14. 202 32 33 215
15. 303 53 54 316
16. 404 74 75 417
17. DEFINE MESH
18. A JOINT 11
19. B JOINT 25
20. C JOINT 46
21. D JOINT 32
22. E JOINT 67
23. F JOINT 53
24. G JOINT 88
25. H JOINT 74
26. GENERATE ELEMENT
27. MESH ABCD 14 12
28. MESH DCEF 14 12
29. MESH FEGH 14 12
30. START GROUP DEFINITION
31. MEMBER
32. _GIRDERS 101 TO 114 202 TO 215 303 TO 316 404 TO 417
33. _PIERS 1 TO 6 26 TO 31 47 TO 52 68 TO 73
34. ELEMENT
35. _P1 447 TO 450 454 TO 457 461 TO 464 468 TO 471
36. _P2 531 TO 534 538 TO 541 545 TO 548 552 TO 555
37. _P3 615 TO 618 622 TO 625 629 TO 632 636 TO 639
38. _P4 713 TO 716 720 TO 723 727 TO 730 734 TO 737
FREQUENCIES OF VIBRATION OF A SKEWED BRIDGE              -- PAGE NO.    2
39. _P5 783 TO 786 790 TO 793 797 TO 800 804 TO 807
40. _P6 881 TO 884 888 TO 891 895 TO 898 902 TO 905
41. END GROUP DEFINITION
42. MEMBER PROPERTY
43. _GIRDERS PRIS YD 0.6 ZD 0.6
44. _PIERS PRIS YD 1.0
45. ELEMENT PROPERTY
46. YRA 9 11 TH 0.375
47. UNIT KNS MMS
48. DEFINE MATERIAL START
49. ISOTROPIC CONCRETE
50. E 21.0
51. POISSON 0.17
52. DENSITY 2.36158E-008
53. ALPHA 5E-006
54. DAMP 0.05
55. G 9.25
56. TYPE CONCRETE
57. STRENGTH FCU 0.0275
58. END DEFINE MATERIAL
59. CONSTANTS
60. MATERIAL CONCRETE ALL
61. SUPPORTS
62. 1 TO 6 26 TO 31 47 TO 52 68 TO 73 FIXED
63. CUT OFF MODE SHAPE 65
64. UNIT KGS METER
66. SELFWEIGHT X 1.0
67. SELFWEIGHT Y 1.0
68. SELFWEIGHT Z 1.0
69. * PERMANENT WEIGHTS ON DECK
71. YRA 9 11 PR GX 200
72. YRA 9 11 PR GY 200
73. YRA 9 11 PR GZ 200
74. * VEHICLES ON SPANS - ONLY Y & Z EFFECT CONSIDERED
76. _P1 PR GY 700
77. _P2 PR GY 700
78. _P3 PR GY 700
79. _P4 PR GY 700
80. _P5 PR GY 700
81. _P6 PR GY 700
82. _P1 PR GZ 700
83. _P2 PR GZ 700
84. _P3 PR GZ 700
85. _P4 PR GZ 700
86. _P5 PR GZ 700
87. _P6 PR GZ 700
88. MODAL CALCULATION REQUESTED
89. PERFORM ANALYSIS
FREQUENCIES OF VIBRATION OF A SKEWED BRIDGE              -- PAGE NO.    3
P R O B L E M   S T A T I S T I C S
-----------------------------------
NUMBER OF JOINTS        579  NUMBER OF MEMBERS      80
NUMBER OF PLATES        504  NUMBER OF SOLIDS        0
NUMBER OF SURFACES        0  NUMBER OF SUPPORTS     24
Using 64-bit analysis engine.
SOLVER USED IS THE IN-CORE ADVANCED MATH SOLVER
TOTAL      PRIMARY LOAD CASES =     1, TOTAL DEGREES OF FREEDOM =    3330
TOTAL LOAD COMBINATION  CASES =     0  SO FAR.
** WARNING: PRESSURE LOADS ON ELEMENTS OTHER THAN PLATE ELEMENTS
ARE IGNORED. ELEM.NO.   101
** WARNING: PRESSURE LOADS ON ELEMENTS OTHER THAN PLATE ELEMENTS
ARE IGNORED. ELEM.NO.   101
** WARNING: PRESSURE LOADS ON ELEMENTS OTHER THAN PLATE ELEMENTS
ARE IGNORED. ELEM.NO.   101
***NOTE: MASSES DEFINED UNDER LOAD#       1 WILL FORM
THE FINAL MASS MATRIX FOR DYNAMIC ANALYSIS.
EIGEN METHOD   : SUBSPACE
-------------------------
NUMBER OF MODES REQUESTED              =    65
NUMBER OF EXISTING MASSES IN THE MODEL =  1665
NUMBER OF MODES THAT WILL BE USED      =    65
***  EIGENSOLUTION : ADVANCED METHOD ***
FREQUENCIES OF VIBRATION OF A SKEWED BRIDGE              -- PAGE NO.    4
CALCULATED FREQUENCIES FOR LOAD CASE       1
MODE            FREQUENCY(CYCLES/SEC)         PERIOD(SEC)
1                       1.648                  0.60681
2                       2.622                  0.38132
3                       2.906                  0.34409
4                       3.783                  0.26436
5                       4.108                  0.24345
6                       4.423                  0.22608
7                       4.561                  0.21927
8                       4.725                  0.21162
9                       5.080                  0.19684
10                       7.277                  0.13742
11                       7.328                  0.13647
12                       7.454                  0.13416
13                      10.418                  0.09599
14                      10.818                  0.09244
15                      11.260                  0.08881
16                      11.377                  0.08790
17                      11.672                  0.08567
18                      11.945                  0.08372
19                      12.028                  0.08314
20                      12.209                  0.08191
21                      12.619                  0.07925
22                      13.823                  0.07234
23                      14.807                  0.06754
24                      14.920                  0.06702
25                      15.294                  0.06539
26                      17.489                  0.05718
27                      17.664                  0.05661
28                      17.937                  0.05575
29                      19.923                  0.05019
30                      20.116                  0.04971
31                      20.724                  0.04825
32                      20.817                  0.04804
33                      21.024                  0.04756
34                      21.340                  0.04686
35                      21.633                  0.04623
36                      22.002                  0.04545
37                      22.290                  0.04486
38                      23.393                  0.04275
39                      23.738                  0.04213
40                      24.235                  0.04126
41                      24.881                  0.04019
42                      25.690                  0.03893
43                      26.275                  0.03806
44                      26.700                  0.03745
45                      27.124                  0.03687
46                      27.610                  0.03622
47                      28.063                  0.03563
48                      29.272                  0.03416
49                      29.866                  0.03348
50                      30.118                  0.03320
51                      31.309                  0.03194
FREQUENCIES OF VIBRATION OF A SKEWED BRIDGE              -- PAGE NO.    5
CALCULATED FREQUENCIES FOR LOAD CASE       1
MODE            FREQUENCY(CYCLES/SEC)         PERIOD(SEC)
52                      31.832                  0.03142
53                      32.014                  0.03124
54                      32.312                  0.03095
55                      32.906                  0.03039
56                      33.204                  0.03012
57                      34.467                  0.02901
58                      35.270                  0.02835
59                      35.522                  0.02815
60                      35.766                  0.02796
61                      36.318                  0.02753
62                      36.919                  0.02709
63                      38.921                  0.02569
64                      39.189                  0.02552
65                      39.905                  0.02506
MODAL WEIGHT (MODAL MASS TIMES g) IN KGS          GENERALIZED
MODE           X             Y             Z              WEIGHT
1       1.291934E+02  1.123418E+00  1.185768E+06    1.205497E+06
2       1.089784E+06  2.961298E+00  2.290601E+02    1.083499E+06
3       2.111306E-01  2.767724E+03  2.799804E+00    5.416754E+05
4       1.488365E+00  3.916563E+04  6.302102E+00    1.988939E+05
5       2.650657E-01  4.869465E+02  5.916391E+02    1.348431E+05
6       5.553985E+02  7.047738E+02  2.743600E+02    8.887365E+04
7       1.872127E+01  3.523236E+05  3.255370E+00    7.402632E+04
8       5.441681E+00  2.711803E+05  8.530684E+00    7.667142E+04
9       5.838996E+03  1.713501E+03  2.233780E+03    6.923635E+04
10       4.436158E+00  1.508718E+03  4.951104E-01    4.324923E+04
11       4.522846E+00  6.614279E+02  1.224734E+00    4.305570E+04
12       2.322126E-01  4.156161E+02  7.878947E-01    3.944772E+04
13       3.968726E+01  2.882564E+00  6.818221E+03    1.532777E+05
14       2.085091E-01  1.298117E+02  1.097372E+02    7.154577E+04
15       2.752974E-02  4.303212E+03  1.232103E+02    4.541365E+04
16       7.726809E+00  7.931786E+01  1.710543E+02    5.083780E+04
17       9.187165E+00  2.295488E+00  5.600056E-01    5.231609E+04
18       1.289274E+02  1.227798E+01  1.533801E+02    8.841806E+04
19       3.797408E-01  3.151484E+01  4.203883E+00    4.601297E+04
20       9.958856E+00  7.524730E+01  1.835670E+01    7.466321E+04
21       8.006146E-01  2.974809E+00  4.620779E+00    7.378575E+04
22       1.242891E+02  1.356224E-01  1.727536E+00    4.087678E+05
23       6.938982E+00  4.623798E+01  8.257328E+01    6.864460E+04
24       4.653175E-01  1.849773E+02  6.876842E+00    5.597259E+04
25       8.352727E+00  1.093998E+01  2.047659E+02    4.884299E+04
26       3.879229E-01  2.187085E+03  5.145218E-03    2.890765E+04
27       8.086885E+00  3.229583E+02  9.429697E-01    3.202062E+04
FREQUENCIES OF VIBRATION OF A SKEWED BRIDGE              -- PAGE NO.    6
28       2.443751E+00  1.315502E+04  2.609740E-01    2.997395E+04
29       6.642359E+02  6.512258E+03  5.785467E+01    3.489561E+04
30       3.896515E+01  4.236802E+04  5.583570E+00    3.273818E+04
31       2.124308E+02  2.362562E+02  6.791406E-01    2.856038E+04
32       2.501307E+01  4.154716E+04  1.610248E-03    3.529900E+04
33       9.944393E+00  2.188227E+03  7.002218E-01    3.488091E+04
34       1.354775E+01  1.273812E+02  9.929674E+00    3.662762E+04
35       2.684736E+00  5.035591E+03  2.202823E+00    4.255654E+04
36       3.584198E-01  7.349881E+00  1.032835E+00    5.119778E+04
37       1.674724E+00  2.647461E+04  8.112263E-01    4.340583E+04
38       1.234580E+00  8.274586E+01  1.831619E-02    1.919433E+04
39       1.177019E-02  3.952054E+02  4.526474E-01    1.820400E+04
40       2.510682E-03  2.913923E+01  2.643008E-04    4.236454E+05
41       1.752055E+00  1.750469E+03  1.098303E+01    4.948152E+04
42       3.172853E-01  5.576266E+04  1.029036E-01    7.029313E+04
43       4.088524E+00  7.230284E+02  1.647904E+00    7.840900E+04
44       6.792844E-03  6.810530E+02  2.323051E-02    7.748353E+04
45       1.286032E-02  2.969021E+02  6.276731E-04    1.698166E+05
46       3.915110E-01  1.842050E+03  1.016859E-02    2.296592E+04
47       1.222720E+01  2.250049E+00  1.763239E+00    1.408108E+04
48       5.796924E-03  4.479735E+03  7.592468E-02    2.885324E+04
49       3.062960E+00  1.622740E+01  2.516519E-01    4.734347E+04
50       5.381654E-01  2.109875E-02  3.715996E+00    4.275013E+04
51       6.988027E-03  1.414464E+02  1.037723E-01    2.721877E+04
52       3.120314E-01  2.679203E+00  8.601323E-01    4.346363E+04
53       7.712285E-01  1.227296E+03  2.159809E-01    3.644744E+04
54       1.092693E+01  6.309261E+01  4.335824E+00    4.837668E+04
55       4.389927E-01  2.654929E+02  5.449158E+00    2.579368E+04
56       1.697410E-01  8.013763E+02  1.519167E-02    3.565397E+04
57       5.812394E-01  1.003674E+02  1.297763E+00    3.819235E+04
58       6.155775E+00  1.623267E+03  1.175780E+01    4.182908E+04
59       3.892790E+00  1.515229E+01  1.764828E+01    3.953446E+04
60       7.019893E+00  5.144657E+02  4.912786E+01    5.200588E+04
61       5.075761E+01  1.707431E+03  3.625510E+01    1.190568E+04
62       2.640553E+01  3.154571E+03  1.681321E+01    1.455738E+04
63       1.733228E-01  5.382845E+02  4.096147E-01    7.325642E+04
64       1.024425E-01  3.773902E+03  1.746056E-01    6.231664E+04
65       7.508947E+00  2.059276E+03  1.501086E+01    3.298603E+04
FREQUENCIES OF VIBRATION OF A SKEWED BRIDGE              -- PAGE NO.    7
MASS PARTICIPATION FACTORS
MASS  PARTICIPATION FACTORS IN PERCENT
--------------------------------------
MODE    X     Y     Z     SUMM-X   SUMM-Y   SUMM-Z
1     0.01   0.00  99.04    0.012    0.000   99.042
2    99.14   0.00   0.02   99.152    0.000   99.061
3     0.00   0.23   0.00   99.152    0.232   99.061
4     0.00   3.27   0.00   99.152    3.503   99.062
5     0.00   0.04   0.05   99.152    3.544   99.111
6     0.05   0.06   0.02   99.202    3.602   99.134
7     0.00  29.43   0.00   99.204   33.030   99.134
8     0.00  22.65   0.00   99.205   55.681   99.135
9     0.53   0.14   0.19   99.736   55.824   99.322
10     0.00   0.13   0.00   99.736   55.950   99.322
11     0.00   0.06   0.00   99.737   56.005   99.322
12     0.00   0.03   0.00   99.737   56.040   99.322
13     0.00   0.00   0.57   99.740   56.040   99.891
14     0.00   0.01   0.01   99.740   56.051   99.901
15     0.00   0.36   0.01   99.740   56.411   99.911
16     0.00   0.01   0.01   99.741   56.417   99.925
17     0.00   0.00   0.00   99.742   56.417   99.925
18     0.01   0.00   0.01   99.754   56.418   99.938
19     0.00   0.00   0.00   99.754   56.421   99.938
20     0.00   0.01   0.00   99.754   56.427   99.940
21     0.00   0.00   0.00   99.755   56.428   99.940
22     0.01   0.00   0.00   99.766   56.428   99.940
23     0.00   0.00   0.01   99.766   56.431   99.947
24     0.00   0.02   0.00   99.767   56.447   99.948
25     0.00   0.00   0.02   99.767   56.448   99.965
26     0.00   0.18   0.00   99.767   56.630   99.965
27     0.00   0.03   0.00   99.768   56.657   99.965
28     0.00   1.10   0.00   99.768   57.756   99.965
29     0.06   0.54   0.00   99.829   58.300   99.970
30     0.00   3.54   0.00   99.832   61.839   99.970
31     0.02   0.02   0.00   99.852   61.859   99.971
32     0.00   3.47   0.00   99.854   65.329   99.971
33     0.00   0.18   0.00   99.855   65.512   99.971
34     0.00   0.01   0.00   99.856   65.522   99.971
35     0.00   0.42   0.00   99.856   65.943   99.972
36     0.00   0.00   0.00   99.856   65.944   99.972
37     0.00   2.21   0.00   99.856   68.155   99.972
38     0.00   0.01   0.00   99.857   68.162   99.972
39     0.00   0.03   0.00   99.857   68.195   99.972
40     0.00   0.00   0.00   99.857   68.197   99.972
41     0.00   0.15   0.00   99.857   68.343   99.973
42     0.00   4.66   0.00   99.857   73.001   99.973
43     0.00   0.06   0.00   99.857   73.061   99.973
44     0.00   0.06   0.00   99.857   73.118   99.973
45     0.00   0.02   0.00   99.857   73.143   99.973
46     0.00   0.15   0.00   99.857   73.297   99.973
47     0.00   0.00   0.00   99.858   73.297   99.973
48     0.00   0.37   0.00   99.858   73.671   99.973
49     0.00   0.00   0.00   99.859   73.673   99.973
50     0.00   0.00   0.00   99.859   73.673   99.973
51     0.00   0.01   0.00   99.859   73.685   99.973
52     0.00   0.00   0.00   99.859   73.685   99.973
53     0.00   0.10   0.00   99.859   73.787   99.973
54     0.00   0.01   0.00   99.860   73.793   99.974
55     0.00   0.02   0.00   99.860   73.815   99.974
56     0.00   0.07   0.00   99.860   73.882   99.974
FREQUENCIES OF VIBRATION OF A SKEWED BRIDGE              -- PAGE NO.    8
MASS  PARTICIPATION FACTORS IN PERCENT
--------------------------------------
MODE    X     Y     Z     SUMM-X   SUMM-Y   SUMM-Z
57     0.00   0.01   0.00   99.860   73.890   99.974
58     0.00   0.14   0.00   99.860   74.026   99.975
59     0.00   0.00   0.00   99.861   74.027   99.977
60     0.00   0.04   0.00   99.861   74.070   99.981
61     0.00   0.14   0.00   99.866   74.212   99.984
62     0.00   0.26   0.00   99.868   74.476   99.985
63     0.00   0.04   0.00   99.868   74.521   99.985
64     0.00   0.32   0.00   99.868   74.836   99.985
65     0.00   0.17   0.00   99.869   75.008   99.987
90. FINISH
FREQUENCIES OF VIBRATION OF A SKEWED BRIDGE              -- PAGE NO.    9
*********** END OF THE STAAD.Pro RUN ***********
**** DATE= APR 21,2020   TIME= 15:42:27 ****
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```

## Understanding the output

After the analysis is complete, look at the output file. (This file can be viewed by selecting the Analysis Output tool in the View group on the Utilities ribbon tab).

1. Mode number and corresponding frequencies and periods

Since we asked for 65 modes, we obtain a report, a portion of which is as shown:

Table 1. Calculated Frequencies for Load Case 1
Mode Frequency

(Cycles/Sec)

Period

(Sec)

Accuracy
1 1.636 0.61111 1.344E-16
2 2.602 0.38433 0.000E+00
3 2.882 0.34695 8.666E-16
4 3.754 0.26636 0.000E+00
5 4.076 0.24532 3.466E-16
6 4.373 0.22870 6.025E-16
7 4.519 0.22130 5.641E-16
8 4.683 0.21355 5.253E-16
9 5.028 0.19889 0.000E+00
10 7.189 0.13911 8.916E-16
11 7.238 0.13815 0.000E+00
12 7.363 0.13582 0.000E+00
2. Participation factors in Percentage

Table 2. Mass Participation Factors in Percent
Mode X Y Z ΣX ΣY ΣZ
1 0.01 0.00 99.04 0.012 0.000 99.042
2 99.14 0.00 0.02 99.151 0.000 99.061
3 0.00 0.23 0.00 99.151 0.229 99.062
4 0.00 3.27 0.00 99.151 3.496 99.062
5 0.00 0.04 0.05 99.151 3.536 99.112
6 0.05 0.04 0.02 99.202 3.575 99.135
7 0.00 26.42 0.00 99.204 30.000 99.135
8 0.00 25.59 0.00 99.204 55.587 99.136
9 0.53 0.15 0.19 99.735 55.740 99.326
10 0.00 0.13 0.00 99.736 55.871 99.326
11 0.00 0.06 0.00 99.736 55.927 99.326
12 0.00 0.04 0.00 99.736 55.969 99.326

In the explanation earlier for the CUT OFF MODE command, we said that one measure of the importance of a mode is the participation factor of that mode. We can see from the above report that for vibration along Z direction, the first mode has a 99.04 percent participation. It is also apparent that the 7th mode is primarily a Y direction mode with a 26.42% participation along Y and 0 in X and Z.

The ΣX, ΣY and ΣZ columns show the cumulative value of the participation of all the modes up to and including a given mode (Corresponding to the SUMM-X, SUMM-Y, and SUMM-Z reported in the output, respectively). One can infer from those terms that if one is interested in 95% participation along X, the first 2 modes are sufficient.

But for the Y direction, even with 10 modes, we barely obtained 60%. The reason for this can be understood by an examination of the nature of the structure. The deck slab is capable of vibrating in several low energy and primarily vertical direction modes. The out-of-plane flexible nature of the slab enables it to vibrate in a manner resembling a series of wave like curves. Masses on either side of the equilibrium point have opposing eigenvector values leading to a lot of cancellation of the contribution from the respective masses. Localized modes, where small pockets in the structure undergo flutter due to their relative weak stiffness compared to the rest of the model, also result in small participation factors.

3. After the analysis is completed, select Post-processing from the mode menu. This screen contains facilities for graphically examining the shape of the mode in static and animated views. The Dynamics page on the left side of the screen is available for viewing the shape of the mode statically. The Animation option of the Results menu can be used for animating the mode. The mode number can be selected from the Loads and Results tab of the Diagrams dialog box which opens when the Animation option is chosen. The size to which the mode is drawn is controlled using the Scales tab of the Diagrams dialog box.