# V. Modal Frequencies of a Simply Supported Beam

To find the natural frequencies of vibration for a simply supported beam.

## Reference

Roark’s Formulas for Stress and Strain, Warren C. Young, McGraw Hill, 6th edition.

## Problem

Find the first five flexural natural frequencies of the simple beam. Neglect shear deformation and rotary inertia.

### Model for dynamic beam no. 7

 E = 10,000 ksi

 density = 0.1 lb/in3

 Ax = 2.0 in2

 Ix = 0.6667 in4

 L = 20 in

## Hand Calculations

Weight

wweight = Ax * density = 2.0 (0.1) = 0.2 lb/in

wmass = 0.2 /(386.4) = 0.000518

From Table 36, Item 1b of the reference:

$f n = k n 2 π E I w l 4 = k n 2 π 10 ( 10 ) 6 ( 0.6667 ) 0.000518 ( 20 ) 4 = 45.16 ⋅ k n$
Table 1. Modal stiffness and natural frequencies
Mode kc Frequency (Hz)
1 9.87 445.7
2 39.5 1,783.7
3 88.8 4,010.0
4 158 7,134.9
5 247 11,154

## Comparison

Table 2. Comparison of results
Frequency, f1 (Hz) 445.7 445.495 none
Frequency, f2 (Hz) 1,783.7 1,781.968 none
Frequency, f3 (Hz) 4,010.0 4,009.310 none
Frequency, f4 (Hz) 7,134.9 7,127.074 none
Frequency, f5 (Hz) 11,154 11,133.978 none

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

STAAD PLANE : NATURAL FREQUENCIES OF A S.SUPPORTED BEAM
START JOB INFORMATION
ENGINEER DATE 14-Sep-18
END JOB INFORMATION
*
*  REFERENCE: W.C.YOUNG., "ROARK'S FORMULAS FOR STRESS &amp; STRAIN", 6TH ED.
*  CASE 1B, TABLE 36, PAGE 714
*
UNIT INCHES POUND
JOINT COORDINATES
1 0 0 0; 2 1 0 0; 3 2 0 0; 4 3 0 0; 5 4 0 0; 6 5 0 0; 7 6 0 0; 8 7 0 0;
9 8 0 0; 10 9 0 0; 11 10 0 0; 12 11 0 0; 13 12 0 0; 14 13 0 0;
15 14 0 0; 16 15 0 0; 17 16 0 0; 18 17 0 0; 19 18 0 0; 20 19 0 0;
21 20 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; 11 11 12; 12 12 13; 13 13 14; 14 14 15; 15 15 16; 16 16 17;
17 17 18; 18 18 19; 19 19 20; 20 20 21;
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 1e+07
POISSON 0.33
DENSITY 0.1
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 ALL
MEMBER PROPERTY AMERICAN
1 TO 20 PRIS AX 2 IZ 0.666667
SUPPORTS
1 21 FIXED BUT MZ
2 TO 20 FIXED BUT FY MZ
CUT OFF MODE SHAPE 0
CUT OFF FREQUENCY 12000
SELFWEIGHT Y -1
MODAL CALCULATION REQUESTED
PERFORM ANALYSIS
FINISH


               CALCULATED FREQUENCIES FOR LOAD CASE       1
MODE            FREQUENCY(CYCLES/SEC)         PERIOD(SEC)
1                     445.495                  0.00224
2                    1781.968                  0.00056
3                    4009.310                  0.00025
4                    7127.074                  0.00014
5                   11133.978                  0.00009
MODAL WEIGHT (MODAL MASS TIMES g) IN POUN         GENERALIZED
MODE           X             Y             Z              WEIGHT
1       0.000000E+00  3.228953E+00  0.000000E+00    2.000000E+00
2       0.000000E+00  5.965927E-28  0.000000E+00    2.000000E+00
3       0.000000E+00  3.469944E-01  0.000000E+00    2.000000E+00
4       0.000000E+00  7.262741E-31  0.000000E+00    2.211146E+00
5       0.000000E+00  1.165685E-01  0.000000E+00    2.000000E+00
MASS PARTICIPATION FACTORS
MASS  PARTICIPATION FACTORS IN PERCENT
--------------------------------------
MODE    X     Y     Z     SUMM-X   SUMM-Y   SUMM-Z
1     0.00  84.97   0.00    0.000   84.972    0.000
2     0.00   0.00   0.00    0.000   84.972    0.000
3     0.00   9.13   0.00    0.000   94.104    0.000
4     0.00   0.00   0.00    0.000   94.104    0.000
5     0.00   3.07   0.00    0.000   97.171    0.000