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5040Frequency (Hz)30201000 1000 2000 3000 4000ModeFigure 5.1Natural frequency and mode numberThe natural frequencies of the first 31 modes with significant modal mass-participantfactors are listed in Table 5.1 along with a brief description of each dominant motion andits respective modal participating mass ratios (MPMR). Here, the significant modes aredefined as those with an MPMR of 2% or higher in any single direction. In Table 5.1,UX, UY, and UZ represent the motions in traffic or longitudinal direction, lateral ortransverse direction, and vertical direction, respectively. In addition to translationalmotions, the rotations about the global X, Y, and Z axes are also included in the table andnoted as RX, RY, and RZ, respectively. The MPMR value provides a measure of howimportant a particular mode of vibration is for the overall response to the accelerationloads in each of the three global directions. It helps ensure a significant and requirednumber of vibration modes are included in seismic response analysis. It can be seen fromTable 5.1 that the accumulated MPMR exceeds 70% in all directions. In particular, theMPMR value is higher than 90% in transverse direction and in its associated rotation inthe transverse-vertical plane or about the longitudinal axis. Additionally, the MPMRvalue also exceeds 90% for the rotational component in the plane of deck or about thevertical axis. The fact that a less mass participation is observed in longitudinal directionis likely attributable to the presence of expansion joints in the bridge, where some parts ofthe structure vibrate independently of the remaining structure. This speculation issupported by the existence of so many local and insignificant modes excluded from Table5.1.The first 30 mode shapes of vibration are presented in Figures 5.2 to 5.31. Additionalmode shapes with significant participating mass ratios are plotted in Figures 5.32 to 5.35.It can be seen that many modes of low frequencies correspond to the coupled motion invertical and longitudinal directions. This observation indicates that the bridge structure ismost flexible in the vertical direction. Indeed, the fundamental frequency of 0.339 Hz54
mainly corresponds to the vertical movement of the main bridge together with the slightlongitudinal motion at towers as shown in Figure 5.2.Table 5.1First 31 natural frequencies with high mass participationNo. ModeFreq.Mass participating factor (%)Description(Hz)UX UY UZ RX RY RZ1 1 0.339 UZ 0.01 0.00 2.50 0.00 0.50 0.002 6 0.625 UY 0.02 10.6 0.00 12.4 0.00 2.543 8 0.689 UZ 0.03 0.00 7.16 0.00 1.46 0.004 10 0.828 UZ 0.05 0.01 2.06 0.02 0.39 0.005 12 0.853 UY+UX 0.10 12.2 0.00 22.1 0.01 3.216 67 1.136 UX 5.23 0.01 0.00 0.01 0.13 0.017 115 1.237 UX+ UY 24.0 0.11 0.01 0.07 0.48 0.088 117 1.243 UX+ UY 6.66 0.14 0.01 0.25 0.13 0.029 121 1.252 UX 2.88 0.00 0.00 0.00 0.07 0.0010 181 1.651 UY 0.08 4.19 0.00 3.06 0.00 1.8511 303 2.167 UY+UX 1.95 2.26 0.00 1.97 0.02 1.5512 366 2.303 UY+UX 0.36 2.76 0.00 2.50 0.01 1.6113 367 2.308 UY+UX 0.94 3.08 0.00 2.74 0.07 1.4614 390 2.389 UY+UX+UZ 0.27 2.14 0.17 1.81 0.23 1.2815 392 2.398 UZ+UY 0.03 0.61 1.24 0.55 2.89 0.4416 412 2.432 UY+UX+UZ 0.41 3.29 0.38 2.94 0.78 2.0017 424 2.485 UZ 0.01 0.02 2.84 0.02 5.08 0.0418 568 3.187 UY+UX 0.10 4.50 0.01 3.66 0.02 7.9819 637 3.382 UY 0.01 1.13 0.00 0.92 0.01 2.2420 691 3.549 UY+UX 0.22 1.32 0.00 1.05 0.02 3.0121 759 3.862 UY+UX 0.19 3.48 0.00 2.75 0.00 7.1822 760 3.874 UY+UX 0.11 0.69 0.00 0.52 0.00 3.1723 1759 7.874 UY 0.01 1.91 0.00 1.08 0.01 3.3124 1852 8.189 UY 0.02 1.04 0.06 0.58 0.04 2.0725 1907 8.395 UY+UZ 0.02 0.99 0.15 0.48 0.12 2.4926 1961 8.834 UZ 0.01 0.00 3.90 0.00 1.40 0.0027 1969 8.853 UZ 0.02 0.01 13.1 0.00 4.73 0.0328 1989 8.980 UZ 0.09 0.00 2.32 0.00 0.83 0.0129 2277 10.76 UZ 0.01 0.00 2.23 0.00 5.04 0.0030 2528 12.86 UZ 0.03 0.00 5.09 0.00 3.29 0.0031 2694 14.09 UZ 0.01 0.00 1.67 0.00 3.99 0.00SUM 77.1 97.5 83.8 97.8 72.1 96.3UX=longitudinal motion (HN3); UY=transverse motion (HN2);Note UZ=vertical motion (HNZ); RX, RY, RZ and R =rotations about X, Y, Zaxis and rocking, respectively.55
Page 1 and 2:
Organizational Results Research Rep
Page 3 and 4:
TECHNICAL REPORT DOCUMENTATION PAGE
Page 5 and 6:
Executive SummaryOpen to traffic on
Page 7 and 8:
8. All cables behave elastically un
Page 9 and 10:
Table of ContentsExecutive summary.
Page 11 and 12:
List of Figure CaptionsFigure 1.1 A
Page 13 and 14:
Figure 5.30 29 th mode shape (1.032
Page 15 and 16:
1. Introduction1.1. GeneralCable-st
Page 17 and 18: Figure 1.1Artist rendering of the B
Page 19 and 20: Expansionand lateralearthquakerestr
Page 21 and 22: 3. Evaluate the model by conducting
Page 23 and 24: 2. Automatic Retrieval of Peak Acce
Page 25 and 26: 2.2.3. Map/Find stationsClicking th
Page 27 and 28: 2.2.5. Displaying seismogramsTo dis
Page 29 and 30: Figure 2.8Directory chooser2.2.6. S
Page 31 and 32: 3.2. Seismic instrumentation networ
Page 33 and 34: The main bridge from Bent 1 to Pier
Page 35 and 36: (a) South side of deck at support o
Page 37 and 38: (a) North side of deck at middle of
Page 39 and 40: The seismic responses at the top (T
Page 41 and 42: Polyreference Complex Exponential (
Page 43 and 44: are evaluated and presented in Figu
Page 45 and 46: 0.353.50.330.252.5Amplitude0.20.15A
Page 47 and 48: 1 x 10-4 Frequency(Hz)Amplitude0.90
Page 49 and 50: Amplitude0.20.180.160.140.120.10.08
Page 51 and 52: Amplitude0.20.180.160.140.120.10.08
Page 53 and 54: 1.20.80.400 100 200 300 400 500 600
Page 55 and 56: 4. Finite Element Modeling of Bill
Page 57 and 58: Since the cable stays are connected
Page 59 and 60: Table 4.3Initial property of cables
Page 61 and 62: Table 4.4Spring and damping coeffic
Page 63 and 64: (a) View of the entire bridge(b) Vi
Page 65 and 66: 4.6. RemarksA detailed 3-D FE model
Page 67: knω = (5.4)nm nSimilarly, when dam
Page 71 and 72: thFigure 5.10 9 mode shape (0.740Hz
Page 73 and 74: thFigure 5.19 18 mode shape (0.947H
Page 75 and 76: Figure 5.27 26 th mode shape (0.977
Page 77 and 78: When the bridge deck moves up and d
Page 79 and 80: 1.1Frequency (Hz)0.90.70.5BC Case 1
Page 81 and 82: 1.1Frequency(Hz)0.90.70.5Plus 0Plus
Page 83 and 84: no difference among the three cases
Page 85 and 86: The FE model of the bridge was vali
Page 87 and 88: 10.6TestFE model0.2-0.20 100 200 30
Page 89 and 90: 1.20.80.40-0.4TestFE model0 100 200
Page 91 and 92: differences is that the exact locat
Page 93 and 94: (a) Vertical acceleration time hist
Page 95 and 96: The FE model of the cable-stayed br
Page 97 and 98: Displacement (m)0.30.20.10-0.1-0.2X
Page 99 and 100: Station D1 (before amplification) a
Page 101 and 102: ending of the tower as shown in Fig
Page 103 and 104: symmetric about the centerline of t
Page 105 and 106: Tensile stress (MPa)800750700650600
Page 107 and 108: 7. Conclusions and RecommendationsB
Page 109 and 110: The vast arrays of acceleration dat
Page 111 and 112: 14. Cunha A, Caetano and Delgado T.
Page 113 and 114: 42. Song, W., Giraldo, D.F., Clayto
Page 115 and 116: Figure A.2 Vertical stiffness and d
Page 117 and 118: A.2 Group Interaction FactorThe cro
Page 119 and 120:
For a group of piles with identical
Page 121:
Missouri Department of Transportati
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