V. First Modal Frequency of a Cantilever Beam

To calculate the Natural frequency of vibration for a rectangular cantilever beam with a mass at the free end.

Reference

Hand calculation using known formulas.

Problem

Find the natural frequency of vibration, ω, of the cantilever beam.

Cantilever beam with mass at free end

E = 30,000 ksi

h = 12.0 in

b = 6.0 in

P = 10.0 k

L = 120 in

Hand Calculations

Stiffness at free end:

k = 3EI/L³ = 45 k/in

Mass

m = w/g = 10.0 k / (386.4 k-sec²/in) = 0.02588 k-sec²/in

Circular frequency:

ω = \sqrt{\frac{k}{m}} = \sqrt{\frac{45}{0.02588}} = 41.7 rad/sec

= 6.637 cycles/sec

Comparison

Table 1. Comparison of results
Result Type	Theory	STAAD.Pro	Difference
Frequency, f (Hz)	6.637	6.633	none

STAAD Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\08 Dynamic Analysis\First Modal Frequency of a Cantilever Beam.STD is typically installed with the program.

STAAD PLANE A RECTANGULAR CANTILEVER BEAM WITH A MASS AT THE FREE END
START JOB INFORMATION
ENGINEER DATE 14-Sep-18
END JOB INFORMATION
INPUT WIDTH 72
SET SHEAR
UNIT FEET KIP
JOINT COORDINATES
1 0 0 0; 2 5 0 0; 3 10 0 0;
MEMBER INCIDENCES
1 1 2; 2 2 3;
UNIT INCHES KIP
MEMBER PROPERTY AMERICAN
1 2 PRIS YD 12 ZD 6
UNIT FEET KIP
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 4.32e+06
POISSON 0.290909
END DEFINE MATERIAL
UNIT INCHES KIP
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
1 FIXED
CUT OFF MODE SHAPE 1
UNIT FEET KIP
LOAD 1 VERTICAL LOAD
MEMBER LOAD
2 CON Y -10 5
1 2 UNI GY -0.001
MODAL CALCULATION REQUESTED
PERFORM ANALYSIS
*PRINT MODE SHAPES
FINISH

STAAD Output

               CALCULATED FREQUENCIES FOR LOAD CASE       1
       MODE            FREQUENCY(CYCLES/SEC)         PERIOD(SEC)
         1                       6.633                  0.15076
            MODAL WEIGHT (MODAL MASS TIMES g) IN KIP          GENERALIZED
      MODE           X             Y             Z              WEIGHT
         1       0.000000E+00  1.000514E+01  0.000000E+00    1.000299E+01
 MASS PARTICIPATION FACTORS 
                     MASS  PARTICIPATION FACTORS IN PERCENT
                     --------------------------------------
           MODE    X     Y     Z     SUMM-X   SUMM-Y   SUMM-Z
             1     0.00  99.98   0.00    0.000   99.976    0.000