V. Steady State Loading on a Beam
Calculate the deflections at two points along the beam at steady-state condition.
Reference
- Blevins, R. D., Formulas for natural Frequency and Mode Shape, Van Nostrand Reinhold, 1979, pp. 108, 455-486.
- Warburton, G. B., The Dynamical Behavior of Structures, Pergamon Press, 1964, pp. 10-15, 85, 86.
- Mechanics Research, Inc. "STARDYNE Verification Manual." 1984.
Problem
Determine the steady-state displacements of the quarter and mid-span points of a fixed-fixed beam subjected to a parabolically varying distributed load operating at a 7.5 Hz frequency.
The solution utilizes the first 8 of 19 modes calculated by STAAD (steady state analysis) for which a value of 1.0(10)-10 times critical damping is assigned. A single forcing frequency equal to 7.5 Hz is specified for the distributed load. This load is distributed to the nodes by calculating the total integrated load for each beam segment and lumping one-half of this force to the respective start and end nodes (i and j).
Theoretical Solution
The natural frequencies of the system are calculated using the equations from Blevins (1) page 108 and Warburon (2) page 85.
λi satisfies the characteristic equation:
i | λi | ωi | fi |
---|---|---|---|
1 | 4.730041 | 63.47865 | 10.10294 |
2 | 7.853205 | 174.9814 | 27.84915 |
3 | 10.99561 | 343.0334 | 54.59546 |
4 | 14.13717 | 567.0517 | 90.24907 |
5 | 17.27876 | 847.0773 | 134.8165 |
6 | 20.42035 | 1183.108 | 188.2975 |
7 | 23.56194 | 1575.144 | 250.6919 |
8 | 26.70354 | 2023.185 | 321.9998 |
The mode shapes are:
Where:
The response of mode i to a harmonic force:
Where ψi is the response phase lag relative to the applied force and c is the damping.
Since c = 0.0, ψi = 0.0
Upon substitution and rearranging terms:
From reference 1, page 466, case c and page 467, case 29:
Since the load is symmetric, this expression is zero for the even modes, 2, 4, 6, etc.;
Therefore, only odd modes 1,3,5, etc. contribute to the result.
And from the reference, βi = λi/l
So:
From Reference 1, page 457 case 5:
Therefore:
i | λi | ωi | ηi(t) | σi | φ(1/4) | φ(1/2) |
---|---|---|---|---|---|---|
1 | 4.730041 | 63.47865 |
|
0.9825022 | 0.8631319 | 1.5881463 |
3 | 10.99561 | 343.0334 |
|
0.9999664 | 1.3708047 | -1.4059984 |
5 | 17.27876 | 847.0773 |
|
0.9999999 | -0.5278897 | 1.4145675 |
7 | 23.56194 | 1575.144 |
|
1.0000000 | -1.3037973 | -1.4141982 |
i | φ(1/4)ηi(t) | φ(1/2)ηi(t) |
---|---|---|
1 | -0.6592727 | -1.2130493 |
3 | -0.0030357 | 0.0031137 |
5 | 0.0000764 | -0.0002048 |
7 | 0.0000293 | -0.0000318 |
Summation | -0.6622028 | -1.2101086 |
Comparison
Steady-state displacement (in.) at location: | Theory | STAAD Advanced Analysis | ||
---|---|---|---|---|
Location | Distance (x, in.) | Node | ||
L/4 | 50 | 6 | 0.66220 | 0.65963 |
L/2 | 100 | 11 | 1.21011 | 1.20545 |
Steady-state analysis requires the STAAD.Pro Advanced Analysis license.
STAAD Input
The file C:\Users\Public\Public Documents\STAAD.Pro 2024\Samples \Verification Models\08 Dynamic Analysis\Steady State Loading on a Beam.STD is included in the STAAD.Pro installation folder.
STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 29-Mar-06
END JOB INFORMATION
* FIXED BEAM SUBJECTED TO A HARMONIC LOAD WITH A PARABOLIC DISTRIBUTION X
* NUMBER OF NODES 21 X
* HIGH NODE NUMBER 21 X
* NODES FULLY RESTRAINED 2 X
* NUMBER OF BEAM ELEMENTS 20 X
* NUMBER OF EIGENVECTORS 19
SET SHEAR
UNIT INCHES POUND
JOINT COORDINATES
1 0 0 0; 2 10 0 0; 3 20 0 0; 4 30 0 0; 5 40 0 0; 6 50 0 0; 7 60 0 0;
8 70 0 0; 9 80 0 0; 10 90 0 0; 11 100 0 0; 12 110 0 0; 13 120 0 0;
14 130 0 0; 15 140 0 0; 16 150 0 0; 17 160 0 0; 18 170 0 0; 19 180 0 0;
20 190 0 0; 21 200 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 AY 0 AZ 0 IX 0.001 IY 0.666667 IZ 0.166667
*********************************************************************
SUPPORTS
2 TO 20 FIXED BUT FX FY MX MY MZ
1 21 FIXED
*********************************************************************
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 1e+07
POISSON 0.3
DENSITY 0.0999194
END DEFINE MATERIAL
CONSTANTS
BETA 90 ALL
MATERIAL MATERIAL1 ALL
*********************************************************************
CUT OFF MODE SHAPE 8
CUT OFF FREQUENCY 330
*********************************************************************
LOAD 1
SELFWEIGHT Y 1
MODAL CALCULATION REQUESTED
PERFORM STEADY STATE ANALYSIS
BEGIN STEADY FORCE
STEADY FORCE FREQ 7.5 DAMP 1e-10
JOINT LOAD
2 FY 1.8666
3 FY 3.5666
4 FY 5.0666
5 FY 6.3666
6 FY 7.4666
7 FY 8.3666
8 FY 9.0666
9 FY 9.5666
10 FY 9.8666
11 FY 9.9666
12 FY 9.8666
13 FY 9.5666
14 FY 9.0666
15 FY 8.3666
16 FY 7.4666
17 FY 6.3666
18 FY 5.0666
19 FY 3.5666
20 FY 1.8666
END
PRINT JOINT DISPLACEMENTS LIST 6 11
FINISH
STAAD Output
P R O B L E M S T A T I S T I C S
-----------------------------------
NUMBER OF JOINTS 21 NUMBER OF MEMBERS 20
NUMBER OF PLATES 0 NUMBER OF SOLIDS 0
NUMBER OF SURFACES 0 NUMBER OF SUPPORTS 21
Using 64-bit analysis engine.
SOLVER USED IS THE IN-CORE ADVANCED MATH SOLVER
TOTAL PRIMARY LOAD CASES = 1, TOTAL DEGREES OF FREEDOM = 95
TOTAL LOAD COMBINATION CASES = 0 SO FAR.
***NOTE: MASSES DEFINED UNDER LOAD# 1 WILL FORM
THE FINAL MASS MATRIX FOR DYNAMIC ANALYSIS.
EIGEN METHOD : SUBSPACE
-------------------------
NUMBER OF MODES REQUESTED = 8
NUMBER OF EXISTING MASSES IN THE MODEL = 19
NUMBER OF MODES THAT WILL BE USED = 8
*** EIGENSOLUTION : ADVANCED METHOD ***
STAAD SPACE -- PAGE NO. 3
CALCULATED FREQUENCIES FOR LOAD CASE 1
MODE FREQUENCY(CYCLES/SEC) PERIOD(SEC)
1 10.103 0.09898
2 27.849 0.03591
3 54.591 0.01832
4 90.230 0.01108
5 134.747 0.00742
6 188.093 0.00532
7 250.166 0.00400
8 320.780 0.00312
MODAL WEIGHT (MODAL MASS TIMES g) IN POUN GENERALIZED
MODE X Y Z WEIGHT
1 0.000000E+00 2.759074E+01 0.000000E+00 1.584634E+01
2 0.000000E+00 1.030463E-21 0.000000E+00 1.763389E+01
3 0.000000E+00 5.287477E+00 0.000000E+00 1.757978E+01
4 0.000000E+00 9.714628E-21 0.000000E+00 1.788088E+01
5 0.000000E+00 2.138447E+00 0.000000E+00 1.901368E+01
6 0.000000E+00 8.978414E-19 0.000000E+00 1.837854E+01
7 0.000000E+00 1.145139E+00 0.000000E+00 1.756698E+01
8 0.000000E+00 2.899899E-17 0.000000E+00 2.004783E+01
MASS PARTICIPATION FACTORS
MASS PARTICIPATION FACTORS IN PERCENT
--------------------------------------
MODE X Y Z SUMM-X SUMM-Y SUMM-Z
1 0.00 72.67 0.00 0.000 72.666 0.000
2 0.00 0.00 0.00 0.000 72.666 0.000
3 0.00 13.93 0.00 0.000 86.591 0.000
4 0.00 0.00 0.00 0.000 86.591 0.000
5 0.00 5.63 0.00 0.000 92.223 0.000
6 0.00 0.00 0.00 0.000 92.223 0.000
7 0.00 3.02 0.00 0.000 95.239 0.000
8 0.00 0.00 0.00 0.000 95.239 0.000
STAAD SPACE -- PAGE NO. 4
47. BEGIN STEADY FORCE
48. STEADY FORCE FREQ 7.5 DAMP 1E-10
49. JOINT LOAD
50. 2 FY 1.8666
51. 3 FY 3.5666
52. 4 FY 5.0666
53. 5 FY 6.3666
54. 6 FY 7.4666
55. 7 FY 8.3666
56. 8 FY 9.0666
57. 9 FY 9.5666
58. 10 FY 9.8666
59. 11 FY 9.9666
60. 12 FY 9.8666
61. 13 FY 9.5666
62. 14 FY 9.0666
63. 15 FY 8.3666
64. 16 FY 7.4666
65. 17 FY 6.3666
66. 18 FY 5.0666
67. 19 FY 3.5666
68. 20 FY 1.8666
69. END
*DIRECTIONS FOR WHICH AMPLITUDE VS. FREQUENCY DATA WAS ENTERED = 0 2 0 0 0 0
*DIRECTIONS FOR WHICH AMPLITUDE VS. PHASE LAG DATA WAS ENTERED = 0 0 0 0 0 0
FORCE DIRECTION NUMBER 2
FREQUENCY AMPLITUDE PHASE ANGLE
1 0.749800E+01 0.100000E+01 0.000000E+00
2 0.750200E+01 0.100000E+01 0.000000E+00
STAAD SPACE -- PAGE NO. 5
8 MODES (EIGENVECTORS) HAVE BEEN SELECTED.
MODE NATURAL FREQUENCY GENERALIZED WEIGHT DAMPING DAMPED FREQUENCY
NO. (HZ) (RAD/SEC) (WEIGHT) (MASS) COEFFICIENT (HZ)
1 1.010292E+01 6.347852E+01 1.584634E+01 4.104327E-02 1.000000E-10 1.010292E+01
2 2.784865E+01 1.749782E+02 1.763389E+01 4.567317E-02 1.000000E-10 2.784865E+01
3 5.459144E+01 3.430081E+02 1.757978E+01 4.553302E-02 1.000000E-10 5.459144E+01
4 9.022966E+01 5.669297E+02 1.788088E+01 4.631289E-02 1.000000E-10 9.022966E+01
5 1.347470E+02 8.466405E+02 1.901368E+01 4.924693E-02 1.000000E-10 1.347470E+02
6 1.880927E+02 1.181821E+03 1.837854E+01 4.760188E-02 1.000000E-10 1.880927E+02
7 2.501660E+02 1.571839E+03 1.756698E+01 4.549988E-02 1.000000E-10 2.501660E+02
8 3.207803E+02 2.015522E+03 2.004783E+01 5.192548E-02 1.000000E-10 3.207803E+02
PARTICIPATION FACTORS FOR EACH MODE
MODE NO. X Y Z MX MY MZ
1 0.000000E+00 0.218577E+04 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00
2 0.000000E+00 -0.126801E-07 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00
3 0.000000E+00 0.382113E+03 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00
4 0.000000E+00 0.398960E-07 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00
5 0.000000E+00 -0.148269E+03 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00
6 0.000000E+00 -0.367234E-06 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00
7 0.000000E+00 -0.829074E+02 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00
8 0.000000E+00 -0.213528E-05 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00
70. PRINT JOINT DISPLACEMENTS LIST 6 11
JOINT DISPLACE LIST 6
STAAD SPACE -- PAGE NO. 6
JOINT DISPLACEMENT (INCH RADIANS) STRUCTURE TYPE = SPACE
------------------
JOINT LOAD X-TRANS Y-TRANS Z-TRANS X-ROTAN Y-ROTAN Z-ROTAN
6 1 0.00000 0.65963 0.00000 0.00000 0.00000 0.01831
11 1 0.00000 1.20545 0.00000 0.00000 0.00000 0.00000