V. Beam Subject to response spectrum
Find the maximum moment due to the time history loading and compare theoretical answers to the STAAD solution.
Reference
- Biggs, John M., Introduction to Structural Dynamics, McGraw Hill, 1964, pp. 256-263
- Blevins, Robert D., Formulas for Natural Frequency and Mode Shape, Van Nostrand-Reinhold, 1979.
Problem
The supports of a simply supported beam are subjected to an acceleration time history. The maximum bending moment in the beam is computed for the first mode of the structure. This problem demonstrates the capabilities of STAAD to calculate the correct modal response of a structure utilizing response spectrum data.
The STAAD model consists of 11 nodes and 10 elastic beam elements. Node 1 is completely restrained with the exception of having rotational freedom in the Z direction, the remaining nodes are restrained except for X and Y displacements and Z rotations. Node 11 is additionally restrained against displacements in the Y direction to provide for the simple support condition . Only the contribution of the first mode of the structure is considered.
Theoretical Solution
Material Properties
From Reference 2, Table 8-1, page 108, the fundamental frequency of the beam is:
The modal participation factor for the fundamental mode is:
Where the first mode shape, φ(x) = sin(πx/1)
The maximum relative modal displacement is given by:
therefore:
The bending moment
Where u for the first mode = A sin(πx/l)
at x=l/2
Comparison
Solution | Theory | STAAD.Pro | Difference |
---|---|---|---|
Bending Moment (kip-inch) | 946.351 | 947.088 | negligible |
STAAD Input
The file C:\Users\Public\Public Documents\STAAD.Pro 2024\Samples \Verification Models\08 Dynamic Analysis\Beam Subject to response spectrum.STD is typically installed with the program.
STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 14-Sep-18
END JOB INFORMATION
* RESPONSE OF A SIMPLY SUPPORTED BEAM TO A SHOCK SPECTRUM
UNIT INCHES POUND
JOINT COORDINATES
1 0 0 0; 2 24 0 0; 3 48 0 0; 4 72 0 0; 5 96 0 0; 6 120 0 0; 7 144 0 0;
8 168 0 0; 9 192 0 0; 10 216 0 0; 11 240 0 0;
MEMBER INCIDENCES
1 1 2; 2 2 3; 3 3 4; 4 4 5; 5 5 6; 6 6 7; 7 7 8; 8 8 9; 9 9 10;
10 10 11;
MEMBER PROPERTY AMERICAN
1 TO 10 PRIS AX 20.4082 IX 40 IY 3.6139 IZ 333.333 YD 14 ZD 1.45777
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 3e+07
POISSON 0.3
DENSITY 3.78672
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 ALL
CUT OFF MODE SHAPE 1
SUPPORTS
1 FIXED BUT MZ
2 TO 10 FIXED BUT FX FY MZ
11 FIXED BUT FX MZ
LOAD 1
SELFWEIGHT X 1
SELFWEIGHT Y 1
SPECTRUM SRSS Y 1 ACC SCALE 386.4 DAMP 0.001
0.15 1.648; 0.17 1.648;
PERFORM ANALYSIS
PRINT MEMBER FORCES LIST 5
FINISH
STAAD Output
CALCULATED FREQUENCIES FOR LOAD CASE 1
MODE FREQUENCY(CYCLES/SEC) PERIOD(SEC)
1 6.069 0.16476
RESPONSE SPECTRUM LOAD 1
RESPONSE LOAD CASE 1
MODAL WEIGHT (MODAL MASS TIMES g) IN POUN GENERALIZED
MODE X Y Z WEIGHT
1 0.000000E+00 1.478714E+04 0.000000E+00 9.273617E+03
SRSS MODAL COMBINATION METHOD USED.
DYNAMIC WEIGHT X Y Z 1.761987E+04 1.669251E+04 0.000000E+00 POUN
MISSING WEIGHT X Y Z -1.761987E+04 -1.905373E+03 0.000000E+00 POUN
MODAL WEIGHT X Y Z 0.000000E+00 1.478714E+04 0.000000E+00 POUN
MODE ACCELERATION-G DAMPING
---- -------------- -------
1 1.64933 0.00100
MODAL BASE ACTIONS
MODAL BASE ACTIONS FORCES IN POUN LENGTH IN INCH
-----------------------------------------------------------
MOMENTS ARE ABOUT THE ORIGIN
MODE PERIOD FX FY FZ MX MY MZ
1 0.165 0.00 24388.86 0.00 0.00 0.00 2926662.97
STAAD SPACE -- PAGE NO. 4
PARTICIPATION FACTORS
MASS PARTICIPATION FACTORS IN PERCENT BASE SHEAR IN POUN
-------------------------------------- ------------------
MODE X Y Z SUMM-X SUMM-Y SUMM-Z X Y Z
1 0.00 88.59 0.00 0.000 88.585 0.000 0.00 24388.86 0.00
---------------------------
TOTAL SRSS SHEAR 0.00 24388.86 0.00
TOTAL 10PCT SHEAR 0.00 24388.86 0.00
TOTAL ABS SHEAR 0.00 24388.86 0.00
34. PRINT MEMBER FORCES LIST 5
MEMBER FORCES LIST 5
STAAD SPACE -- PAGE NO. 5
MEMBER END FORCES STRUCTURE TYPE = SPACE
-----------------
ALL UNITS ARE -- POUN INCH (LOCAL )
MEMBER LOAD JT AXIAL SHEAR-Y SHEAR-Z TORSION MOM-Y MOM-Z
5 1 5 0.00 1931.41 0.00 0.00 0.00 900734.25
6 0.00 1931.41 0.00 0.00 0.00 947088.00