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.
Beam with harmonic distributed load
Model: divide span 20@10.0
E = 10.0x106 psi
L = 200 inches
I = 2/3 in4
A = 2 in2
ρ = 0.1 lbf/in3
g = 386.4 in/sec2
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.
where= | ||
= | = 4.5156271×10-1λi 2 |
λi satisfies the characteristic equation:
cosλ coshλ - 1 = 0
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:
ω = 7.5(2π) = 47.1239 radians/sec
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 2023\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