# V. Modal Response of a Beam

Find the natural frequencies for a beam and compare theoretical answers to the STAAD.Pro solution.

## Reference

Timoshenko, S., Vibration Problems in Engineering, Third Edition, D. Van Nostrand Company, Inc., 1955, page 322

## Problem

The first five natural frequencies and the associated mode shapes are computed for the flexural motion of a simply supported beam.

L = 20 in

### Finite element model

The simply supported beam is divided into twenty spanwise beam elements. At nodes 1 and 21, all degrees of freedom except the rotation about the Z axis are restrained. For the remaining nodes, only the translation along Y and the rotation about Z are permitted.  Both shear deformation and rotary inertia have been excluded from the model. The mass matrix is a diagonal matrix.

## Cross-section Properties

Rectangular Section: 1 inch Width x 2 inch Depth

Area = 2 in2

$J = b 3 a 16 { 16 3 + 3.36 ( b a ) [ 1 − 1 12 ( b a ) 4 ] }$
where
 a = 2 b = 1 J = 0.4578 inch4 I2 = 1×23/12 inch4 I3 = 2×13/12 inch4

## Theoretical Results

The natural bending frequencies, for a uniform beam with hinged ends, are given by:

$f n = π m 2 2 l 2 E I g A γ$
where
 fn = natural frequency for mode n, in cycles per second l = span of the beam E = elastic modulus I = cross-section moment of inertia g = gravitational constant A = cross-section area γ = weight density
The parameters used in the frequency equation are:
• l = 20 in
• E = 10x106 psi
• I = 0.6667 in4
• g = 386.4 in/s2
• A = 2.0 in2
• γ = 0.1 lbs/in3

from which:

 fn = n2x 445.686

## Comparison

The table below shows the natural frequencies computed from the theoretical equation and the subspace iteration method available within STAAD.Pro.  Frequencies are in cycles per second.

Table 1. Comparison of results
1 445.686 445.506 negligible
2 1,782.74 1,782.012 negligible
3 4,011.17 4,009.410 negligible
4 7,130.97 7,127.250 negligible
5 11,142.1 11,134.253 negligible

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

STAAD PLANE
START JOB INFORMATION
ENGINEER DATE 14-Sep-18
END JOB INFORMATION
* Natural modes of a simple beam
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;
MEMBER PROPERTY AMERICAN
1 TO 20 PRIS AX 2 IZ 0.6667
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 1e+07
POISSON 0.3
DENSITY 0.1
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 ALL
CUT OFF MODE SHAPE 5
SUPPORTS
1 21 FIXED BUT MZ
2 TO 20 FIXED BUT FY MZ
SELFWEIGHT X 1
SELFWEIGHT Y 1
MODAL CALCULATION REQUESTED
PERFORM ANALYSIS
FINISH


               CALCULATED FREQUENCIES FOR LOAD CASE       1
MODE            FREQUENCY(CYCLES/SEC)         PERIOD(SEC)
1                     445.506                  0.00224
2                    1782.012                  0.00056
3                    4009.410                  0.00025
4                    7127.250                  0.00014
5                   11134.253                  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  6.313263E-28  0.000000E+00    2.000000E+00
3       0.000000E+00  3.469944E-01  0.000000E+00    2.000000E+00
4       0.000000E+00  3.242349E-30  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