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.

Space truss

E = 30,000 ksi

L = 10 ft

P = 10 kips

All members are L4x4x5/16

Hand Calculation

Calculate average deflection at top:

δ_avg = (0.45255 in + 0.43535 in)/2 = 0.444 in

Stiffness of truss:

k = P/δ_avg = 20.0 kip/0.444 in = 45.5 k/in

Mass:

M = w/g = 20.0/386.4 = 0.05175 k-sec²/in

Frequency:

f = \frac{\sqrt{k / m}}{2 π} = \frac{\sqrt{45.05 / 0.05175}}{2 π} = 4.696 Hz

Comparison

Table 1. Comparison of results
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 CONNECT Edition\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    0.45368   0.06897   0.00000   0.00000   0.00000   0.00000
      6    1    0.43644  -0.06897   0.00000   0.00000   0.00000   0.00000