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.

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

P_o = 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} = \int_{x_{i}}^{x_{j}} \frac{P (x)}{2} ⅆ x = \frac{1}{2} \int_{x_{i}}^{x_{j}} \frac{4}{l^{2}} (x l - x^{2}) ⅆ x

F_{i} = F_{j} = {| \frac{x^{2}}{l} - \frac{2 x^{3}}{3 l^{2}} |}_{x_{i}}^{x_{j}} = \frac{x_{j}^{2} - x_{i}^{2}}{200} - \frac{x_{j}^{3} - x_{i}^{3}}{60, 000}

Theoretical Solution

The theoretical solution for this example is taken from:

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.

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

f_{i} = \frac{λ_{1}^{2}}{2 π l^{2}} \sqrt{\frac{E I}{m}}

where

m	=	ρA/g
f_i	=	$\frac{λ_{i}^{2}}{2 π {(200)}^{2}} \sqrt{\frac{1.0 {(10)}^{6} (2 / 3)}{0.10 (2.0 / 386.4)}}$ = 4.5156271*10^-1λ_i ²

λ_isatisfies the characteristic equation:

cosλ coshλ - 1 = 0

Table 1. Calculated natural frequency for each mode
i	λ_i	ω_i	f_i
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 \frac{λ_{i} x}{l} - \cos \frac{λ_{i} x}{l} - σ_{i} (\sinh \frac{λ_{i} x}{l} - \sin \frac{λ_{i} x}{l})

Where:

σ_{i} = \frac{\cosh λ_{i} - \cos λ_{i}}{\sinh λ_{i} - \sin λ_{i}}

The response of mode i to a harmonic force:

η_{i} = \frac{\int_{l}^{} ϕ_{i} (x) P (x) ⅆ x}{ω_{i}^{2} \int {ϕ_{i} (x)}^{2} m ⅆ x} - \frac{\sin (ω t - ψ_{i})}{\sqrt{{(1 - \frac{ω^{2}}{ω_{i}^{2}})}^{2} + {(\frac{c ω}{k})}^{2}}}

Where ψ_i is the response phase lag relative to the applied force and c is the damping.

Since c = 0.0, ψ_i = 0.0

Upon substitution and rearranging terms:

η_{i} = \frac{\int_{l}^{} \frac{4 P_{o}}{l^{2}} (x l - x^{2}) ϕ_{i} (x) ⅆ x}{\frac{ρ A}{g} (ω_{i}^{2} - ω^{2}) \int_{l}^{} {ϕ_{i} (x)}^{2} ⅆ x} \sin ω t

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

\frac{4 P_{o}}{l^{2}} \int_{l}^{} x ϕ_{i} (x) ⅆ x - \frac{4 P_{o}}{l^{2}} \int_{l}^{} x^{2} ϕ_{i} (x) ⅆ x = \frac{8 P_{0}}{β_{i}^{2} l} [1 + {(- i)}^{i} - {(- i)}^{i} σ_{i} β_{i} l] - \frac{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, β_i = λ_i/l

So:

\int_{l}^{} ϕ_{i} (x) P (x) ⅆ x = \frac{16 P_{o} l}{λ_{i}^{2}}

From Reference 1, page 457 case 5:

\int_{l}^{} {ϕ_{i} (x)}^{2} ⅆ x = 1

Therefore:

ω = 7.5(2π) = 47.1239 radians/sec

η_{i} = \frac{16 P_{o}}{m λ_{i}^{2} (ω_{i}^{2} - ω^{2})} \sin ω t = \frac{16 (- 1.0) \sin ω t}{\frac{0.10 (2)}{386.4} λ_{i}^{2} (ω_{i}^{2} - 2, 220.661)} = \frac{30, 912 \cdot \sin ω t}{λ_{i}^{2} (ω_{i}^{2} - 2, 220.661)}

Table 2. Mode shapes
i	λ_i	ω_i	η_i(t)	σ_i	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) = \sum_{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)η_i(t)	φ(1/2)η_i(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
Location	Theory	STAAD Advanced Analysis
Node 6 (X = 50 inches)	0.66220	0.65963
Node 11 (X = 100 inches)	1.21011	1.20545

Steady-state analysis requires the STAAD.Pro Advanced Analysis Plus license.

STAAD Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\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 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
LOAD 1
SELFWEIGHT X 1 
SELFWEIGHT Y 1 
SELFWEIGHT Z 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 =      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
    44. BEGIN STEADY FORCE
    45. STEADY FORCE FREQ 7.5 DAMP 1E-10
    46. JOINT LOAD
      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 **************