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
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
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Theoretical Results
The natural bending frequencies, for a uniform beam with hinged ends, are given by:
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- 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.
STAAD Input
The file C:\Users\Public\Public Documents\STAAD.Pro 2023\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
LOAD 1
SELFWEIGHT X 1
SELFWEIGHT Y 1
MODAL CALCULATION REQUESTED
PERFORM ANALYSIS
FINISH
STAAD Output
CALCULATED FREQUENCIES FOR LOAD CASE 1 MODE FREQUENCY(CYCLES/SEC) PERIOD(SEC) 1 445.506 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