A space frame structure is subjected to a sinusoidal (dynamic) loading. The commands necessary to describe the sine function are demonstrated in this example. Time History analysis is performed on this model.

This problem is installed with the program by default to C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\Sample Models\US\US-22 Time History Analysis for Sinusoidal Loading.STD when you install the program.

### Example Problem No. 22

```    STAAD SPACE

The word SPACE signifies that the structure is a space frame and the geometry is defined through X, Y, and Z axes. The comment line which begins with an asterisk is an optional title to identify this project.

`    UNIT KIP FEET`

The units for the data that follows are specified above.

```    JOINT COORDINATES
1 0 0 0 ; 2 15 0 0 ; 3 15 0 15 ; 4 0 0 15
5 0 20 0 ; 6 7.5 20 0 ; 7 15 20 0 ; 8 15 20 7.5
9 15 20 15 ; 10 7.5 20 15 ; 11 0 20 15
12 0 20 7.5```

The joint number followed by the X, Y and Z coordinates are specified above.

Note: Semicolons (;) are used as line separators to allow for input of multiple sets of data on one line.
```    MEMBER INCIDENCES
1 1 5 ; 2 2 7 ; 3 3 9 ; 4 4 11 ; 5 5 6 ; 6 6 7
7 7 8 ; 8 8 9 ; 9 9 10 ; 10 10 11 ; 11 11 12 ; 12 12 5
13 6 13 ; 14 13 10 ; 15 8 13 ; 16 13 12```

The members are defined by the joints to which they are connected.

```    UNIT INCH
MEMBER PROPERTIES
1 TO 12 PRIS YD 12 ZD 12```

Members 1 to 12 are defined as PRISmatic sections with width and depth values of 12 inches. The UNIT command is specified to change the units for input from FEET to INCHes.

```    SUPPORTS
1 TO 4 PINNED```

Joints 1 to 4 are declared to be pinned-supported.

```    DEFINE MATERIAL START
ISOTROPIC CONCRETE
E 3150
POISSON 0.17
DENSITY 8.68e-005
ALPHA 5e-006
DAMP 0.05
G 1346.15
TYPE CONCRETE
STRENGTH FCU 4
END DEFINE MATERIAL
CONSTANTS
MATERIAL CONCRETE ALL```

The DEFINE MATERIAL command is used to specify material properties and the CONSTANT is used to assign the material to all members.

```    DEFINE TIME HISTORY
TYPE 1 FORCE
FUNCTION SINE
AMPLITUDE 6.2831 FREQUENCY 60 CYCLES 100
*
ARRIVAL TIMES
0.0
DAMPING 0.075```

There are two stages in the command specification required for a time-history analysis. The first stage is defined above. Here, the parameters of the sinusoidal loading are provided.

Each data set is individually identified by the number that follows the TYPE command. In this file, only one data set is defined, which is apparent from the fact that only one TYPE is defined.

The word FORCE that follows the TYPE 1 command signifies that this data set is for a forcing function. (If you want to specify an earthquake motion, an ACCELERATION may be specified.)

The command FUNCTION SINE indicates that instead of providing the data set as discrete TIME-FORCE pairs, a sinusoidal function, which describes the variation of force with time, is provided.

The parameters of the sine function, such as FREQUENCY, AMPLITUDE, and number of CYCLES of application are then defined. STAAD internally generates discrete TIME-FORCE pairs of data from the sine function in steps of time defined by the default value (refer to TR.31.4 Definition of Time History Load for more information). The arrival time value indicates the relative value of time at which the force begins to act upon the structure. The modal damping ratio for all the modes is set to 0.075.

```    LOAD 1 STATIC LOAD CASE
5 6 7 8 9 10 11 12 UNI GY -1.0```

The above data describe a static load case.

```    LOAD 2 DYNAMIC LOAD CASE
SELFWEIGHT X 1.0
SELFWEIGHT Y 1.0
SELFWEIGHT Z 1.0
8 12 FX 4.0
8 12 FY 4.0
8 12 FZ 4.0
8 12 FX 1 1```

This is the second stage of command specification for time history analysis. The two sets of data specified here are:

1. the weights for generation of the mass matrix
2. the application of the time varying loads on the structure.

The weights (from which the masses for the mass matrix are obtained) are specified in the form of selfweight and joint loads.

Following that, the sinusoidal force is applied using the TIME LOAD command. The forcing function described by the TYPE 1 load is applied on joints 8 and 12 and it starts to act starting at a time defined by the 1st arrival time number.

```    PERFORM ANALYSIS
PRINT ANALYSIS RESULTS
FINI```

The above commands are self explanatory. The FINISH command terminates the STAAD run.

## Input File

``````STAAD SPACE EXAMPLE FOR HARMONIC LOADING GENERATOR
UNIT KIP FEET
JOINT COORDINATES
1 0 0 0 ; 2 15 0 0 ; 3 15 0 15 ; 4 0 0 15
5 0 20 0 ; 6 7.5 20 0 ; 7 15 20 0 ; 8 15 20 7.5
9 15 20 15 ; 10 7.5 20 15 ; 11 0 20 15
12 0 20 7.5
MEMBER INCIDENCES
1 1 5 ; 2 2 7 ; 3 3 9 ; 4 4 11 ; 5 5 6 ; 6 6 7
7 7 8 ; 8 8 9 ; 9 9 10 ; 10 10 11 ; 11 11 12 ; 12 12 5
UNIT INCH
MEMBER PROPERTIES
1 TO 12 PRIS YD 12 ZD 12
SUPPORTS
1 TO 4 PINNED
DEFINE MATERIAL START
ISOTROPIC CONCRETE
E 3150
POISSON 0.17
DENSITY 8.68e-005
ALPHA 5e-006
DAMP 0.05
G 1346.15
TYPE CONCRETE
STRENGTH FCU 4
END DEFINE MATERIAL
CONSTANTS
MATERIAL CONCRETE ALL
DEFINE TIME HISTORY
TYPE 1 FORCE
FUNCTION SINE
AMPLITUDE 6.2831 FREQUENCY 60 CYCLES 100
*
ARRIVAL TIMES
0.0
DAMPING 0.075
5 6 7 8 9 10 11 12 UNI GY -1.0
SELFWEIGHT X 1.0
SELFWEIGHT Y 1.0
SELFWEIGHT Z 1.0
8 12 FX 4.0
8 12 FY 4.0
8 12 FZ 4.0
8 12 FX 1 1
PERFORM ANALYSIS
PRINT ANALYSIS RESULTS
FINI
``````

```                                                                  PAGE NO.    1
****************************************************
*                                                  *
*           Version  22.04.00.**                   *
*           Proprietary Program of                 *
*           Bentley Systems, Inc.                  *
*           Date=    APR 21, 2020                  *
*           Time=    15:42:10                      *
*                                                  *
*  Licensed to: Bentley Systems Inc                *
****************************************************
2. UNIT KIP FEET
3. JOINT COORDINATES
4. 1 0 0 0 ; 2 15 0 0 ; 3 15 0 15 ; 4 0 0 15
5. 5 0 20 0 ; 6 7.5 20 0 ; 7 15 20 0 ; 8 15 20 7.5
6. 9 15 20 15 ; 10 7.5 20 15 ; 11 0 20 15
7. 12 0 20 7.5
8. MEMBER INCIDENCES
9. 1 1 5 ; 2 2 7 ; 3 3 9 ; 4 4 11 ; 5 5 6 ; 6 6 7
10. 7 7 8 ; 8 8 9 ; 9 9 10 ; 10 10 11 ; 11 11 12 ; 12 12 5
11. UNIT INCH
12. MEMBER PROPERTIES
13. 1 TO 12 PRIS YD 12 ZD 12
14. SUPPORTS
15. 1 TO 4 PINNED
16. DEFINE MATERIAL START
17. ISOTROPIC CONCRETE
18. E 3150
19. POISSON 0.17
20. DENSITY 8.68E-005
21. ALPHA 5E-006
22. DAMP 0.05
23. G 1346.15
24. TYPE CONCRETE
25. STRENGTH FCU 4
26. END DEFINE MATERIAL
27. CONSTANTS
28. MATERIAL CONCRETE ALL
29. DEFINE TIME HISTORY
30. TYPE 1 FORCE
32. FUNCTION SINE
33. AMPLITUDE 6.2831 FREQUENCY 60 CYCLES 100
FOR SEQUENTIAL HARMONIC FORCING CURVE NUMBER=             1
NUMBER OF POINTS IN DIGITIZED HARMONIC FUNCTION=       1201
NUMBER OF POINTS PER QUARTER CYCLE OF HARMONIC FUNCTION=  3
FORCE STEP DELTA TIME PER POINT                  1.38889E-03
ENDING TIME FOR THIS DIGITIZED HARMONIC FUNCTION  1.66667E+00
34. *
35. ARRIVAL TIMES
36. 0.0
37. DAMPING 0.075
40. 5 6 7 8 9 10 11 12 UNI GY -1.0
42. SELFWEIGHT X 1.0
43. SELFWEIGHT Y 1.0
44. SELFWEIGHT Z 1.0
46. 8 12 FX 4.0
47. 8 12 FY 4.0
48. 8 12 FZ 4.0
50. 8 12 FX 1 1
51. PERFORM ANALYSIS
P R O B L E M   S T A T I S T I C S
-----------------------------------
NUMBER OF JOINTS         12  NUMBER OF MEMBERS      12
NUMBER OF PLATES          0  NUMBER OF SOLIDS        0
NUMBER OF SURFACES        0  NUMBER OF SUPPORTS      4
Using 64-bit analysis engine.
SOLVER USED IS THE IN-CORE ADVANCED MATH SOLVER
TOTAL      PRIMARY LOAD CASES =     2, TOTAL DEGREES OF FREEDOM =      60
TOTAL LOAD COMBINATION  CASES =     0  SO FAR.
***NOTE: MASSES DEFINED UNDER LOAD#       2 WILL FORM
THE FINAL MASS MATRIX FOR DYNAMIC ANALYSIS.
EIGEN METHOD   : SUBSPACE
-------------------------
NUMBER OF MODES REQUESTED              =     6
NUMBER OF EXISTING MASSES IN THE MODEL =    24
NUMBER OF MODES THAT WILL BE USED      =     6
***  EIGENSOLUTION : ADVANCED METHOD ***
CALCULATED FREQUENCIES FOR LOAD CASE       2
MODE            FREQUENCY(CYCLES/SEC)         PERIOD(SEC)
1                       1.202                  0.83191
2                       1.204                  0.83057
3                       1.451                  0.68908
4                       7.559                  0.13229
5                      11.073                  0.09031
6                      11.670                  0.08569
MODAL WEIGHT (MODAL MASS TIMES g) IN KIP          GENERALIZED
MODE           X             Y             Z              WEIGHT
1       2.299868E+01  5.213994E-29  2.154110E-25    2.280506E+01
2       2.148403E-25  9.984362E-31  2.299902E+01    2.293083E+01
3       2.180358E-21  1.626557E-31  2.281548E-21    3.300665E+01
4       1.122613E-24  1.446952E-29  2.732241E-24    2.026121E+01
5       4.006222E-23  5.345261E-06  4.306780E-20    1.127253E+01
6       1.341889E-20  1.076830E+01  1.183963E-23    1.025517E+01
MASS PARTICIPATION FACTORS
MASS  PARTICIPATION FACTORS IN PERCENT
--------------------------------------
MODE    X     Y     Z     SUMM-X   SUMM-Y   SUMM-Z
1   100.00   0.00   0.00   99.998    0.000    0.000
2     0.00   0.00 100.00   99.998    0.000  100.000
3     0.00   0.00   0.00   99.998    0.000  100.000
4     0.00   0.00   0.00   99.998    0.000  100.000
5     0.00   0.00   0.00   99.998    0.000  100.000
6     0.00  46.82   0.00   99.998   46.821  100.000
A C T U A L  MODAL  D A M P I N G  USED IN ANALYSIS
MODE       DAMPING
1      0.07500000
2      0.07500000
3      0.07500000
4      0.07500000
5      0.07500000
6      0.07500000
TIME STEP USED IN TIME HISTORY ANALYSIS = 0.00139 SECONDS
NUMBER OF MODES WHOSE CONTRIBUTION IS CONSIDERED =    6
TIME DURATION OF TIME HISTORY ANALYSIS =      1.667 SECONDS
NUMBER OF TIME STEPS IN THE SOLUTION PROCESS =     1200
52. PRINT ANALYSIS RESULTS
BASE SHEAR UNITS ARE -- KIP  INCH
MAXIMUM BASE SHEAR  X=  -2.228746E-01  Y=   1.490116E-08  Z=   2.273737E-13
AT TIMES               0.194444           1.519444           0.072222
ANALYSIS RESULTS
JOINT DISPLACEMENT (INCH RADIANS)    STRUCTURE TYPE = SPACE
------------------
JOINT  LOAD   X-TRANS   Y-TRANS   Z-TRANS   X-ROTAN   Y-ROTAN   Z-ROTAN
1    1    0.00000   0.00000   0.00000  -0.01045   0.00000   0.01045
2    0.00000   0.00000   0.00000   0.00000   0.00000  -0.00037
2    1    0.00000   0.00000   0.00000  -0.01045   0.00000  -0.01045
2    0.00000   0.00000   0.00000  -0.00000   0.00000  -0.00037
3    1    0.00000   0.00000   0.00000   0.01045   0.00000  -0.01045
2    0.00000   0.00000   0.00000   0.00000  -0.00000  -0.00037
4    1    0.00000   0.00000   0.00000   0.01045   0.00000   0.01045
2    0.00000   0.00000   0.00000  -0.00000  -0.00000  -0.00037
5    1    0.00118  -0.09524   0.00118   0.02103   0.00000  -0.02103
2    0.06536   0.00008   0.00000  -0.00000   0.00000  -0.00008
6    1    0.00000  -1.56841   0.00118   0.02103   0.00000   0.00000
2    0.06537   0.00000   0.00000   0.00000  -0.00000   0.00004
7    1   -0.00118  -0.09524   0.00118   0.02103   0.00000   0.02103
2    0.06536  -0.00008  -0.00000   0.00000   0.00000  -0.00008
8    1   -0.00118  -1.56841   0.00000   0.00000   0.00000   0.02103
2    0.06587  -0.00008   0.00000   0.00000   0.00000  -0.00008
9    1   -0.00118  -0.09524  -0.00118  -0.02103   0.00000   0.02103
2    0.06536  -0.00008   0.00000  -0.00000  -0.00000  -0.00008
10    1    0.00000  -1.56841  -0.00118  -0.02103   0.00000   0.00000
2    0.06537   0.00000   0.00000   0.00000   0.00000   0.00004
11    1    0.00118  -0.09524  -0.00118  -0.02103   0.00000  -0.02103
2    0.06536   0.00008  -0.00000   0.00000  -0.00000  -0.00008
12    1    0.00118  -1.56841   0.00000   0.00000   0.00000  -0.02103
2    0.06587   0.00008   0.00000   0.00000   0.00000  -0.00008
SUPPORT REACTIONS -UNIT KIP  INCH    STRUCTURE TYPE = SPACE
-----------------
JOINT  LOAD   FORCE-X   FORCE-Y   FORCE-Z     MOM-X     MOM-Y     MOM Z
1    1      5.95    180.00      5.95      0.00      0.00      0.00
2     -0.06     -0.15     -0.00      0.00      0.00      0.00
2    1     -5.95    180.00      5.95      0.00      0.00      0.00
2     -0.06      0.15      0.00      0.00      0.00      0.00
3    1     -5.95    180.00     -5.95      0.00      0.00      0.00
2     -0.06      0.15     -0.00      0.00      0.00      0.00
4    1      5.95    180.00     -5.95      0.00      0.00      0.00
2     -0.06     -0.15      0.00      0.00      0.00      0.00
MEMBER END FORCES    STRUCTURE TYPE = SPACE
-----------------
ALL UNITS ARE -- KIP  INCH     (LOCAL )
MEMBER  LOAD  JT     AXIAL   SHEAR-Y  SHEAR-Z   TORSION     MOM-Y      MOM-Z
1    1     1    180.00     -5.95     5.95      0.00      0.00      -0.00
5   -180.00      5.95    -5.95      0.00  -1428.10   -1428.10
2     1     -0.15      0.06    -0.00      0.00      0.00       0.00
5      0.15     -0.06     0.00      0.00      0.00      13.37
2    1     2    180.00      5.95     5.95      0.00      0.00       0.00
7   -180.00     -5.95    -5.95      0.00  -1428.10    1428.10
2     2      0.15      0.06     0.00      0.00      0.00       0.00
7     -0.15     -0.06    -0.00      0.00     -0.00      13.37
3    1     3    180.00      5.95    -5.95      0.00     -0.00       0.00
9   -180.00     -5.95     5.95      0.00   1428.10    1428.10
2     3      0.15      0.06    -0.00      0.00      0.00       0.00
9     -0.15     -0.06     0.00      0.00      0.00      13.37
4    1     4    180.00     -5.95    -5.95      0.00      0.00       0.00
11   -180.00      5.95     5.95      0.00   1428.10   -1428.10
2     4     -0.15      0.06     0.00      0.00      0.00       0.00
11      0.15     -0.06    -0.00      0.00     -0.00      13.37
5    1     5      5.95     90.00    -0.00     -0.00      0.00    1428.10
6     -5.95     -0.00     0.00      0.00     -0.00    2621.90
2     5     -0.01     -0.15    -0.01     -0.00      0.84     -13.37
6      0.01      0.15     0.01      0.00      0.00      -0.00
6    1     6      5.95     -0.00    -0.00      0.00      0.00   -2621.90
7     -5.95     90.00     0.00     -0.00      0.00   -1428.10
2     6      0.01     -0.15    -0.01     -0.00      0.00       0.00
7     -0.01      0.15     0.01      0.00      0.84     -13.37
7    1     7      5.95     90.00     0.00     -0.00     -0.00    1428.10
8     -5.95     -0.00    -0.00      0.00      0.00    2621.90
2     7     -0.01      0.00     0.02      0.00     -0.84       0.00
8      0.01     -0.00    -0.02     -0.00     -1.41       0.00
8    1     8      5.95      0.00     0.00      0.00     -0.00   -2621.90
9     -5.95     90.00    -0.00     -0.00     -0.00   -1428.10
2     8     -0.01     -0.00    -0.02      0.00      1.41      -0.00
9      0.01      0.00     0.02     -0.00      0.84      -0.00
9    1     9      5.95     90.00    -0.00     -0.00      0.00    1428.10
10     -5.95     -0.00     0.00      0.00      0.00    2621.90
MEMBER END FORCES    STRUCTURE TYPE = SPACE
-----------------
ALL UNITS ARE -- KIP  INCH     (LOCAL )
MEMBER  LOAD  JT     AXIAL   SHEAR-Y  SHEAR-Z   TORSION     MOM-Y      MOM-Z
2     9      0.01      0.15     0.01      0.00     -0.84      13.37
10     -0.01     -0.15    -0.01     -0.00      0.00      -0.00
10    1    10      5.95      0.00    -0.00      0.00      0.00   -2621.90
11     -5.95     90.00     0.00     -0.00      0.00   -1428.10
2    10     -0.01      0.15     0.01      0.00      0.00       0.00
11      0.01     -0.15    -0.01     -0.00     -0.84      13.37
11    1    11      5.95     90.00     0.00     -0.00     -0.00    1428.10
12     -5.95      0.00    -0.00      0.00      0.00    2621.90
2    11      0.01     -0.00    -0.02      0.00      0.84      -0.00
12     -0.01      0.00     0.02     -0.00      1.41      -0.00
12    1    12      5.95      0.00     0.00      0.00     -0.00   -2621.90
5     -5.95     90.00    -0.00     -0.00     -0.00   -1428.10
2    12      0.01      0.00     0.02      0.00     -1.41       0.00
5     -0.01     -0.00    -0.02     -0.00     -0.84       0.00
************** END OF LATEST ANALYSIS RESULT **************
53. FINI
*********** END OF THE STAAD.Pro RUN ***********
**** DATE= APR 21,2020   TIME= 15:42:10 ****