V. Natural Frequency of a 2D Truss
To calculate the Natural frequency of vibration for a two story truss.
Reference
Hand calculation using known formula.
Problem
Find the natural frequency of vibration, f, of the truss.
E = 30,000 ksi
L = 10 ft
P = 10 kips
All members are L4x4x5/16
Hand Calculation
Calculate average deflection at top:
δavg = (0.45255 in + 0.43535 in)/2 = 0.444 in
Stiffness of truss:
k = P/δavg = 20.0 kip/0.444 in = 45.5 k/in
Mass:
M = w/g = 20.0/386.4 = 0.05175 k-sec2/in
Frequency:
Comparison
Result Type | Theory | STAAD.Pro | Difference |
---|---|---|---|
Frequency, f (Hz) | 4.696 | 4.687 | <1% |
STAAD Input
The file C:\Users\Public\Public Documents\STAAD.Pro 2024\Samples \Verification Models\08 Dynamic Analysis\Natural Frequency of a 2D Truss.STD is typically installed with the program.
STAAD TRUSS : A TWO STORY TRUSS
START JOB INFORMATION
ENGINEER DATE 14-Sep-18
END JOB INFORMATION
UNIT FEET KIP
JOINT COORDINATES
1 0 0 0; 2 0 10 0; 3 0 20 0; 4 10 0 0; 5 10 10 0; 6 10 20 0;
MEMBER INCIDENCES
1 1 2; 2 2 3; 3 4 5; 4 5 6; 5 1 4; 6 2 5; 7 3 6; 8 2 4; 9 2 6;
MEMBER PROPERTY AMERICAN
1 TO 9 TABLE ST L40405
DEFINE MATERIAL START
ISOTROPIC STEEL_50_KSI
E 4.176e+06
POISSON 0.3
DENSITY 0.489024
ALPHA 6.5e-06
DAMP 0.03
TYPE STEEL
STRENGTH FY 7200 FU 8928 RY 1.5 RT 1.2
END DEFINE MATERIAL
CONSTANTS
MATERIAL STEEL_50_KSI ALL
SUPPORTS
1 PINNED
4 FIXED BUT FX FZ MX MY MZ
CUT OFF MODE SHAPE 0
LOAD 1
JOINT LOAD
3 6 FX 10
MODAL CALCULATION REQUESTED
PERFORM ANALYSIS
PRINT JOINT DISPLACEMENTS LIST 3 6
FINISH
STAAD Output
CALCULATED FREQUENCIES FOR LOAD CASE 1
MODE FREQUENCY(CYCLES/SEC) PERIOD(SEC)
1 4.687 0.21337
2 34.019 0.02940
MODAL WEIGHT (MODAL MASS TIMES g) IN KIP GENERALIZED
MODE X Y Z WEIGHT
1 1.999221E+01 0.000000E+00 0.000000E+00 1.924051E+01
2 7.793822E-03 0.000000E+00 0.000000E+00 1.924051E+01
MASS PARTICIPATION FACTORS
MASS PARTICIPATION FACTORS IN PERCENT
--------------------------------------
MODE X Y Z SUMM-X SUMM-Y SUMM-Z
1 99.96 0.00 0.00 99.961 0.000 0.000
2 0.04 0.00 0.00 100.000 0.000 0.000
33. PRINT JOINT DISPLACEMENTS LIST 3 6
JOINT DISPLACE LIST 3
: A TWO STORY TRUSS -- PAGE NO. 4
JOINT DISPLACEMENT (INCH RADIANS) STRUCTURE TYPE = TRUSS
------------------
JOINT LOAD X-TRANS Y-TRANS Z-TRANS X-ROTAN Y-ROTAN Z-ROTAN
3 1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
6 1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000