Calculate the deflections at two points along the beam at steady-state condition.

## Reference

1. Blevins, R. D., Formulas for natural Frequency and Mode Shape, Van Nostrand Reinhold, 1979, pp. 108, 455-486.
2. Warburton, G. B., The Dynamical Behavior of Structures, Pergamon Press, 1964, pp. 10-15, 85, 86.

## 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.

### Model: divide span 20@10.0

E = 10.0x106 psi

L = 200 inches

I = 2/3 in4

A = 2 in2

Po = 0.1 lbf/in3

g = 386.4 in/sec2

The DYNRE2 run utilizes all 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 and lumping one-half of this force to the respective i and j nodes.

$F i = F j = ∫ x i x j P ( x ) 2 ⅆ x = 1 2 ∫ x i x j 4 l 2 ( x l − x 2 ) ⅆ x$
$F i = F j = | x 2 l − 2 x 3 3 l 2 | x i x j = x j 2 − x i 2 200 − x j 3 − x i 3 60 , 000$

## Theoretical Solution

The theoretical solution for this example is taken from:
1. Blevins, R. D., "Formulas for natural Frequency and Mode Shape," Van Nostrand Reinhold, 1979, pp 108, 455-486.
2. Warburton, G. B., "The Dynamical Behavior of Structures," Pergamon Press, 1964, pp. 10-15, 85, 86.

The natural frequencies of the system are calculated using the equations from reference 1 page 108 and reference 2 page 85.

$f i = λ 1 2 2 π l 2 E I m$
where
 m = ρA/g fi = $λ i 2 2 π ( 200 ) 2 1.0 ( 10 ) 6 ( 2 / 3 ) 0.10 ( 2.0 / 386.4 )$ = 4.5156271*10-1λi 2

λisatisfies the characteristic equation:

cosλ coshλ - 1 = 0

Table 1. Calculated natural frequency for each mode
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:

$ϕ i = cosh ⁡ λ i x l − cos ⁡ λ i x l − σ i ( sinh ⁡ λ i x l − sin ⁡ λ i x l )$

Where:

$σ i = cosh ⁡ λ i − cos λ i sinh ⁡ λ i − sin ⁡ λ i$

The response of mode i to a harmonic force:

$η i = ∫ l ϕ i ( x ) P ( x ) ⅆ x ω i 2 ∫ { ϕ i ( x ) } 2 m ⅆ x − sin ⁡ ( ω t − ψ i ) ( 1 − ω 2 ω i 2 ) 2 + ( c ω k ) 2$

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:

$η i = ∫ l 4 P o l 2 ( x l − x 2 ) ϕ i ( x ) ⅆ x ρ A g ( ω i 2 − ω 2 ) ∫ l { ϕ i ( x ) } 2 ⅆ x sin ⁡ ω t$

From reference 1, page 466, case c and page 467, case 29:

$4 P o l 2 ∫ l x ϕ i ( x ) ⅆ x − 4 P o l 2 ∫ l x 2 ϕ i ( x ) ⅆ x = 8 P 0 β i 2 l [ 1 + ( − i ) i − ( − i ) i σ i β i l ] − 8 ( − l ) i P 0 β i 2 l [ 2 − σ i β i l ]$

Since the load is symmetric, this expression is zero for I = 2, 4, 6 …;

Therefore, i = 1,3,5……, and: (-1)i = -1

And from the reference, βi = λi/l

So:

$∫ l ϕ i ( x ) P ( x ) ⅆ x = 16 P o l λ i 2$

From Reference 1, page 457 case 5:

$∫ l { ϕ i ( x ) } 2 ⅆ x = 1$

Therefore:

ω = 7.5(2π) = 47.1239 radians/sec

$η i = 16 P o m λ i 2 ( ω i 2 − ω 2 ) sin ⁡ ω t = 16 ( − 1.0 ) sin ⁡ ω t 0.10 ( 2 ) 386.4 λ i 2 ( ω i 2 − 2 , 220.661 ) = 30 , 912 ⋅ sin ⁡ ω t λ i 2 ( ω i 2 − 2 , 220.661 )$
Table 2. Mode shapes
i λi ωi ηi(t) σi φ(1/4) φ(1/2)
1 4.730041 63.47865 $− 7.63815 ( 10 ) − 1 sin ⁡ ω t$ 0.9825022 0.8631319 1.5881463
3 10.99561 343.0334 $− 2.21457 ( 10 ) − 3 sin ⁡ ω t$ 0.9999664 1.3708047 -1.4059984
5 17.27876 847.0773 $− 1.44744 ( 10 ) − 4 sin ⁡ ω t$ 0.9999999 -0.5278897 1.4145675
7 23.56194 1575.144 $− 2.24623 ( 10 ) − 5 sin ⁡ ω t$ 1.0000000 -1.3037973 -1.4141982
$y ( x , t ) = ∑ i = 1 , 3 , 5 , 7 η i ( t ) ϕ i ( x )$
Table 3. Steady state displacements for 1/4 point and 1/2 point (nodes 6 and 11, respectively)
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

Table 4. Comparison of results
Node 6  (X = 50 inches) 0.66220 0.65963
Node 11 (X = 100 inches) 1.21011 1.20545

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 FY 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 7
CUT OFF FREQUENCY 500
SELFWEIGHT X 1
SELFWEIGHT Y 1
SELFWEIGHT Z 1
MODAL CALCULATION REQUESTED
STEADY FORCE FREQ 7.5 DAMP 1e-10
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


            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 =      38
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              =     7
NUMBER OF EXISTING MASSES IN THE MODEL =    19
NUMBER OF MODES THAT WILL BE USED      =     7
***  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
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  1.241542E-21  0.000000E+00    1.763389E+01
3       0.000000E+00  5.287477E+00  0.000000E+00    1.757978E+01
4       0.000000E+00  5.783031E-23  0.000000E+00    1.788088E+01
5       0.000000E+00  2.138447E+00  0.000000E+00    1.901368E+01
6       0.000000E+00  1.058584E-19  0.000000E+00    1.837854E+01
7       0.000000E+00  1.145139E+00  0.000000E+00    1.756698E+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
45. STEADY FORCE FREQ 7.5 DAMP 1E-10
STAAD SPACE                                              -- PAGE NO.    4
47. 2 FY 1.8666
48. 3 FY 3.5666
49. 4 FY 5.0666
50. 5 FY 6.3666
51. 6 FY 7.4666
52. 7 FY 8.3666
53. 8 FY 9.0666
54. 9 FY 9.5666
55. 10 FY 9.8666
56. 11 FY 9.9666
57. 12 FY 9.8666
58. 13 FY 9.5666
59. 14 FY 9.0666
60. 15 FY 8.3666
61. 16 FY 7.4666
62. 17 FY 6.3666
63. 18 FY 5.0666
64. 19 FY 3.5666
65. 20 FY 1.8666
66. 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
7 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
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 -0.139214E-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 -0.416926E-08  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  0.128546E-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
67. 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
************** END OF LATEST ANALYSIS RESULT **************