V. Natural Frequency of a Simply Supported Beam

To find the fundamental frequency of vibration for a simply supported beam with a uniform mass.

Reference

Thomson, W.T., Vibration Theory and Applications, Prentice-Hall, Inc., 1965. Also compared with ANSYS^® Finite Element Software.

Problem

Find the fundamental frequency, f, of a simply supported beam of uniform cross-section.

Model for dynamic beam no. 3

E = 30,000 ksi

I = 1.3333 in⁴

A = 4 in²

w = 1.124 lb/in

L = 80 in

Comparison

Table 1. Comparison of results
Result Type	Theory	ANSYS	STAAD.Pro	Difference
Frequency, f (Hz)	28.766	28.767	28.7438 (Rayleigh)	none
Frequency, f (Hz)	28.766	28.767	28.761 (Eigensolution)	none

STAAD Input

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

STAAD PLANE :FUNDAMENTAL FREQUENCY OF A SIMPLY SUPPORTED BEAM
START JOB INFORMATION
ENGINEER DATE 14-Sep-18
END JOB INFORMATION
*
*  REFERENCE:  THOMSON, W.T., "VIBRATION THEORY AND APPLICATIONS",
*      PRENTICE HALL INC., ENGLEWOODS, NEW JERSEY, 1965
*
INPUT WIDTH 72
UNIT INCHES POUND
JOINT COORDINATES
1 0 0 0; 2 20 0 0; 3 40 0 0; 4 60 0 0; 5 80 0 0;
MEMBER INCIDENCES
1 1 2; 2 2 3; 3 3 4; 4 4 5;
MEMBER PROPERTY AMERICAN
1 TO 4 PRIS AX 4 IZ 1.33333
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 3e+07
POISSON 0.290909
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
1 5 PINNED
CUT OFF FREQUENCY 100
CUT OFF MODE SHAPE 3
LOAD 1 DYNAMIC ANALYSIS (NATURAL FREQUENCY)
MEMBER LOAD
1 TO 4 UNI GY 1.124
CALCULATE RAYLEIGH FREQUENCY
MODAL CALCULATION REQUESTED
PERFORM ANALYSIS
FINISH

STAAD Output

               CALCULATED FREQUENCIES FOR LOAD CASE       1
       MODE            FREQUENCY(CYCLES/SEC)         PERIOD(SEC)
         1                      28.761                  0.03477
         2                     114.242                  0.00875
         3                     242.560                  0.00412
            MODAL WEIGHT (MODAL MASS TIMES g) IN POUN         GENERALIZED
      MODE           X             Y             Z              WEIGHT
         1       0.000000E+00  6.551152E+01  0.000000E+00    4.496000E+01
         2       0.000000E+00  1.870800E-31  0.000000E+00    4.496000E+01
         3       0.000000E+00  1.928479E+00  0.000000E+00    4.496000E+01
 MASS PARTICIPATION FACTORS 
                     MASS  PARTICIPATION FACTORS IN PERCENT
                     --------------------------------------
           MODE    X     Y     Z     SUMM-X   SUMM-Y   SUMM-Z
             1     0.00  97.14   0.00    0.000   97.140    0.000
             2     0.00   0.00   0.00    0.000   97.140    0.000
             3     0.00   2.86   0.00    0.000  100.000    0.000
   **********************************************************
   *                                                        *
   * RAYLEIGH FREQUENCY FOR LOADING     1 =   28.74379 CPS  *
   * MAX DEFLECTION =  0.01499 INCH GLO Y,  AT JOINT     3  *
   *                                                        *
   **********************************************************