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EX. UK-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\UK\UK-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
    LOAD 1 FREQUENCY CALCULATION
    SELFWEIGHT X 1.0
    SELFWEIGHT Y 1.0
    SELFWEIGHT Z 1.0
    * PERMANENT WEIGHTS ON DECK
    ELEMENT LOAD
    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
    ELEMENT LOAD
    _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

This terminates the STAAD run.

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 MMS
DEFINE MATERIAL START
ISOTROPIC CONCRETE
E 21.0
POISSON 0.17
DENSITY 2.4e-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
LOAD 1 FREQUENCY CALCULATION
SELFWEIGHT X 1.0
SELFWEIGHT Y 1.0
SELFWEIGHT Z 1.0
* PERMANENT WEIGHTS ON DECK
ELEMENT LOAD
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
ELEMENT LOAD
_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

STAAD Output File

                                                                  PAGE NO.    1
             ****************************************************        
             *                                                  *        
             *           STAAD.Pro CONNECT Edition              *        
             *           Version  22.04.00.**                   *        
             *           Proprietary Program of                 *        
             *           Bentley Systems, Inc.                  *        
             *           Date=    APR 21, 2020                  *        
             *           Time=    15:41: 6                      *        
             *                                                  *        
             *  Licensed to: Bentley Systems Inc                *        
             ****************************************************        
     1. STAAD SPACE FREQUENCIES OF VIBRATION OF A SKEWED BRIDGE
INPUT FILE: UK-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 MMS
    48. DEFINE MATERIAL START
    49. ISOTROPIC CONCRETE
    50. E 21.0
    51. POISSON 0.17
    52. DENSITY 2.4E-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
    65. LOAD 1 FREQUENCY CALCULATION
    66. SELFWEIGHT X 1.0
    67. SELFWEIGHT Y 1.0
    68. SELFWEIGHT Z 1.0
    69. * PERMANENT WEIGHTS ON DECK
    70. ELEMENT LOAD
    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
    75. ELEMENT LOAD
    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                       1.637                  0.61071
         2                       2.604                  0.38399
         3                       2.886                  0.34652
         4                       3.760                  0.26597
         5                       4.083                  0.24494
         6                       4.396                  0.22747
         7                       4.533                  0.22062
         8                       4.696                  0.21293
         9                       5.049                  0.19806
        10                       7.232                  0.13828
        11                       7.281                  0.13734
        12                       7.407                  0.13501
        13                      10.360                  0.09653
        14                      10.758                  0.09296
        15                      11.195                  0.08932
        16                      11.310                  0.08841
        17                      11.605                  0.08617
        18                      11.872                  0.08423
        19                      11.956                  0.08364
        20                      12.134                  0.08241
        21                      12.542                  0.07974
        22                      13.727                  0.07285
        23                      14.719                  0.06794
        24                      14.831                  0.06743
        25                      15.202                  0.06578
        26                      17.387                  0.05751
        27                      17.560                  0.05695
        28                      17.828                  0.05609
        29                      19.795                  0.05052
        30                      19.988                  0.05003
        31                      20.596                  0.04855
        32                      20.688                  0.04834
        33                      20.901                  0.04785
        34                      21.212                  0.04714
        35                      21.502                  0.04651
        36                      21.873                  0.04572
        37                      22.161                  0.04512
        38                      23.248                  0.04301
        39                      23.593                  0.04239
        40                      24.075                  0.04154
        41                      24.730                  0.04044
        42                      25.535                  0.03916
        43                      26.120                  0.03828
        44                      26.537                  0.03768
        45                      26.938                  0.03712
        46                      27.433                  0.03645
        47                      27.884                  0.03586
        48                      29.090                  0.03438
        49                      29.686                  0.03369
        50                      29.927                  0.03341
        51                      31.110                  0.03214
      FREQUENCIES OF VIBRATION OF A SKEWED BRIDGE              -- PAGE NO.    5
               CALCULATED FREQUENCIES FOR LOAD CASE       1
       MODE            FREQUENCY(CYCLES/SEC)         PERIOD(SEC)
        52                      31.643                  0.03160
        53                      31.824                  0.03142
        54                      32.121                  0.03113
        55                      32.701                  0.03058
        56                      33.002                  0.03030
        57                      34.257                  0.02919
        58                      35.053                  0.02853
        59                      35.310                  0.02832
        60                      35.548                  0.02813
        61                      36.095                  0.02770
        62                      36.688                  0.02726
        63                      38.682                  0.02585
        64                      38.951                  0.02567
        65                      39.661                  0.02521
            MODAL WEIGHT (MODAL MASS TIMES g) IN KGS          GENERALIZED
      MODE           X             Y             Z              WEIGHT
         1       1.313672E+02  1.108398E+00  1.201129E+06    1.221103E+06
         2       1.105162E+06  2.912622E+00  2.323269E+02    1.098762E+06
         3       2.073649E-01  2.786898E+03  2.769101E+00    5.493268E+05
         4       1.462758E+00  3.955818E+04  6.224106E+00    2.014392E+05
         5       2.818490E-01  4.828385E+02  5.992502E+02    1.365709E+05
         6       5.598604E+02  6.953900E+02  2.771716E+02    9.013609E+04
         7       1.854187E+01  3.566982E+05  3.225212E+00    7.500848E+04
         8       5.356242E+00  2.742697E+05  8.417095E+00    7.765617E+04
         9       5.904817E+03  1.694393E+03  2.261244E+03    7.013562E+04
        10       4.430377E+00  1.534484E+03  4.903639E-01    4.388408E+04
        11       4.601393E+00  6.655514E+02  1.207885E+00    4.369367E+04
        12       2.289314E-01  4.228225E+02  7.787098E-01    3.999065E+04
        13       3.993157E+01  2.747181E+00  6.896335E+03    1.557242E+05
        14       2.161567E-01  1.315486E+02  1.086174E+02    7.218063E+04
        15       2.743077E-02  4.349509E+03  1.227626E+02    4.601683E+04
        16       7.932399E+00  7.722762E+01  1.715499E+02    5.132691E+04
        17       9.357269E+00  2.437766E+00  6.022200E-01    5.278384E+04
        18       1.310402E+02  1.245065E+01  1.558885E+02    8.975814E+04
        19       3.237358E-01  3.119274E+01  4.475092E+00    4.629568E+04
        20       9.860828E+00  7.537996E+01  1.815535E+01    7.571097E+04
        21       8.020853E-01  2.958744E+00  4.741305E+00    7.486690E+04
        22       1.257488E+02  1.348115E-01  1.747276E+00    4.145462E+05
        23       6.964422E+00  4.654827E+01  8.396615E+01    6.993260E+04
        24       4.645357E-01  1.888568E+02  6.969984E+00    5.688589E+04
        25       8.396620E+00  1.087438E+01  2.082024E+02    4.957080E+04
        26       3.970981E-01  2.273349E+03  6.502642E-03    2.916350E+04
        27       8.158171E+00  3.453441E+02  9.698586E-01    3.234756E+04
      FREQUENCIES OF VIBRATION OF A SKEWED BRIDGE              -- PAGE NO.    6
        28       2.411181E+00  1.344046E+04  2.572347E-01    3.030211E+04
        29       6.757432E+02  6.222985E+03  5.878092E+01    3.533053E+04
        30       3.712278E+01  4.299742E+04  5.310814E+00    3.319031E+04
        31       2.171634E+02  2.603085E+02  6.963874E-01    2.870320E+04
        32       2.478604E+01  4.222519E+04  2.375592E-03    3.571038E+04
        33       9.426945E+00  2.074711E+03  7.006805E-01    3.546272E+04
        34       1.318033E+01  1.391960E+02  9.997939E+00    3.683261E+04
        35       2.655567E+00  5.108468E+03  2.202262E+00    4.277587E+04
        36       3.664638E-01  4.719298E+00  1.036419E+00    5.132263E+04
        37       1.618305E+00  2.671420E+04  8.032944E-01    4.385049E+04
        38       1.289408E+00  8.348047E+01  2.136199E-02    1.948813E+04
        39       1.090681E-02  3.990261E+02  4.500325E-01    1.839953E+04
        40       2.516875E-03  2.838393E+01  1.672893E-04    4.285791E+05
        41       1.800354E+00  1.729541E+03  1.115900E+01    4.982347E+04
        42       3.094704E-01  5.689236E+04  1.057619E-01    7.183174E+04
        43       4.237560E+00  7.117003E+02  1.709904E+00    7.941099E+04
        44       6.985981E-03  7.120621E+02  2.228543E-02    7.795531E+04
        45       1.090046E-02  2.815537E+02  8.068273E-04    1.854435E+05
        46       3.867976E-01  1.895886E+03  9.105723E-03    2.331780E+04
        47       1.245474E+01  2.154164E+00  1.786947E+00    1.432475E+04
        48       6.042358E-03  4.552571E+03  7.449196E-02    2.931614E+04
        49       3.089849E+00  1.585498E+01  2.356090E-01    4.790401E+04
        50       5.917700E-01  3.501022E-02  3.813271E+00    4.319210E+04
        51       7.531865E-03  1.363633E+02  1.039786E-01    2.739737E+04
        52       3.396939E-01  2.796655E+00  9.065380E-01    4.399495E+04
        53       7.633919E-01  1.251482E+03  2.100451E-01    3.694608E+04
        54       1.113975E+01  6.123655E+01  4.429348E+00    4.883033E+04
        55       4.296741E-01  2.619401E+02  5.431140E+00    2.644050E+04
        56       1.724619E-01  8.197998E+02  1.827260E-02    3.670345E+04
        57       5.833701E-01  9.933417E+01  1.283571E+00    3.868065E+04
        58       6.173715E+00  1.663166E+03  1.165825E+01    4.293976E+04
        59       3.968332E+00  1.254915E+01  1.776702E+01    3.983809E+04
        60       7.435025E+00  5.085940E+02  5.033208E+01    5.274970E+04
        61       5.168555E+01  1.721488E+03  3.628858E+01    1.212435E+04
        62       2.675386E+01  3.225244E+03  1.680691E+01    1.478013E+04
        63       1.898740E-01  5.111631E+02  4.371451E-01    7.704926E+04
        64       1.017851E-01  3.878617E+03  1.786230E-01    6.212678E+04
        65       7.558339E+00  2.180679E+03  1.533813E+01    3.321933E+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   99.043
             2    99.14   0.00   0.02   99.153    0.000   99.063
             3     0.00   0.23   0.00   99.154    0.230   99.063
             4     0.00   3.26   0.00   99.154    3.492   99.063
             5     0.00   0.04   0.05   99.154    3.532   99.113
             6     0.05   0.06   0.02   99.204    3.589   99.136
             7     0.00  29.41   0.00   99.206   33.002   99.136
             8     0.00  22.62   0.00   99.206   55.618   99.137
             9     0.53   0.14   0.19   99.736   55.758   99.323
            10     0.00   0.13   0.00   99.736   55.884   99.323
            11     0.00   0.05   0.00   99.737   55.939   99.323
            12     0.00   0.03   0.00   99.737   55.974   99.323
            13     0.00   0.00   0.57   99.740   55.974   99.892
            14     0.00   0.01   0.01   99.740   55.985   99.901
            15     0.00   0.36   0.01   99.740   56.344   99.911
            16     0.00   0.01   0.01   99.741   56.350   99.925
            17     0.00   0.00   0.00   99.742   56.350   99.925
            18     0.01   0.00   0.01   99.753   56.351   99.938
            19     0.00   0.00   0.00   99.754   56.354   99.938
            20     0.00   0.01   0.00   99.754   56.360   99.940
            21     0.00   0.00   0.00   99.754   56.360   99.940
            22     0.01   0.00   0.00   99.766   56.360   99.940
            23     0.00   0.00   0.01   99.766   56.364   99.947
            24     0.00   0.02   0.00   99.766   56.380   99.948
            25     0.00   0.00   0.02   99.767   56.381   99.965
            26     0.00   0.19   0.00   99.767   56.568   99.965
            27     0.00   0.03   0.00   99.768   56.597   99.965
            28     0.00   1.11   0.00   99.768   57.705   99.965
            29     0.06   0.51   0.00   99.829   58.218   99.970
            30     0.00   3.55   0.00   99.832   61.763   99.970
            31     0.02   0.02   0.00   99.852   61.785   99.971
            32     0.00   3.48   0.00   99.854   65.267   99.971
            33     0.00   0.17   0.00   99.855   65.438   99.971
            34     0.00   0.01   0.00   99.856   65.449   99.971
            35     0.00   0.42   0.00   99.856   65.871   99.972
            36     0.00   0.00   0.00   99.856   65.871   99.972
            37     0.00   2.20   0.00   99.856   68.074   99.972
            38     0.00   0.01   0.00   99.856   68.081   99.972
            39     0.00   0.03   0.00   99.856   68.114   99.972
            40     0.00   0.00   0.00   99.856   68.116   99.972
            41     0.00   0.14   0.00   99.857   68.259   99.973
            42     0.00   4.69   0.00   99.857   72.950   99.973
            43     0.00   0.06   0.00   99.857   73.008   99.973
            44     0.00   0.06   0.00   99.857   73.067   99.973
            45     0.00   0.02   0.00   99.857   73.090   99.973
            46     0.00   0.16   0.00   99.857   73.247   99.973
            47     0.00   0.00   0.00   99.858   73.247   99.973
            48     0.00   0.38   0.00   99.858   73.622   99.973
            49     0.00   0.00   0.00   99.858   73.624   99.973
            50     0.00   0.00   0.00   99.858   73.624   99.973
            51     0.00   0.01   0.00   99.858   73.635   99.973
            52     0.00   0.00   0.00   99.858   73.635   99.973
            53     0.00   0.10   0.00   99.859   73.738   99.973
            54     0.00   0.01   0.00   99.860   73.743   99.974
            55     0.00   0.02   0.00   99.860   73.765   99.974
            56     0.00   0.07   0.00   99.860   73.833   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   73.841   99.974
            58     0.00   0.14   0.00   99.860   73.978   99.975
            59     0.00   0.00   0.00   99.861   73.979   99.977
            60     0.00   0.04   0.00   99.861   74.021   99.981
            61     0.00   0.14   0.00   99.866   74.163   99.984
            62     0.00   0.27   0.00   99.868   74.429   99.985
            63     0.00   0.04   0.00   99.868   74.471   99.985
            64     0.00   0.32   0.00   99.868   74.791   99.985
            65     0.00   0.18   0.00   99.869   74.971   99.987
    90. FINISH
      FREQUENCIES OF VIBRATION OF A SKEWED BRIDGE              -- PAGE NO.    9
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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.