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

A_x = 2.0 in²

I_x = 0.6667 in⁴

L = 20 in

Hand Calculations

Weight

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

w_mass = 0.2 /(386.4) = 0.000518

From Table 36, Item 1b of the reference:

f_{n} = \frac{k_{n}}{2 π} \sqrt{\frac{E I}{w l^{4}}} = \frac{k_{n}}{2 π} \sqrt{\frac{10 {(10)}^{6} (0.6667)}{{0.000518 (20)}^{4}}} = 45.16 {\cdot k}_{n}

Table 1. Modal stiffness and natural frequencies
Mode	k_c	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
Result Type	Theory	STAAD.Pro	Difference
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

STAAD Input

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
LOAD 1
SELFWEIGHT Y -1 
MODAL CALCULATION REQUESTED
PERFORM ANALYSIS
FINISH

STAAD Output

               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