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5. Eigensolution and Model Verification of the CablestayedBridge5.1. GeneralCable-stayed bridges, due to their large scale and high flexibility, usually have a longfundamental period, which distinguishes themselves from most of other structures. As aresult, they are sensitive to ambient motions such as traffic and tremor induced vibration.In this section, the natural frequencies and mode shapes of the cable-stayed bridge areevaluated using the FE model developed in the preceding section. They are thencompared with measured data to validate the FE model.5.2. Modal analysisModal analysis in structural dynamics is aimed to determine the natural frequencies andmode shapes of a structure and evaluate its responses under dynamic loading. In mostcases, only a small number of lowest vibration modes dominate the responses of anengineering structure such as cable-stayed bridge.5.2.1. Classical modal analysis theoryThe equation of motion of a linear multiple degree of freedom (MDF) system withoutdamping can be written as:[M ]{ Ut &&()}+[K]{ Ut)} ( = {P(t)} (5.1)in which [M ] and [K] are the mass and stiffness matrices of the structural system,{ Ut)}is ( the displacement vector as a function of time t, and {P( t)}is the external loadvector.The displacement vector {U( t)} of an MDF system can be expanded into a summation ofmodal contributions, i.e.,N{U( t)} = ∑{φ r}q r(t) = [Φ]{q(t)} (5.2)r=1where N is the number of degrees of freedom, {φ r}is the r th mode vector, [Φ] is acollection of all mode vectors, qr ( t) is the rth modaldisplacement, and {q( t)}is ageneralized displacement vector or a collection of modal displacement. By using Eq.(5.2), the coupled equation (5.1) in {U( t)}can be transformed to a set of uncoupledequations with the unknown modal displacement q n(t) after modal orthogonalityconditions have been introduced (Chopra, 2006). That is,mq&& () t + k n n nq()nt = p n() t (5.3)in which m { } Tn= φn[ M]{ φ n} and k { } Tn= φn[ K]{ φ n} are the n th modal mass and stiffness,p n() t = {φ n} T { P()t }is a n th modal force. Eq. (5.3) represents a generalized single degreeof freedom (SDF) system. The natural frequency ω nof the SDF system can be evaluatedby:52
knω = (5.4)nm nSimilarly, when damping is present, the n th modal equation of motion can be derived asm nq&& n() t + c nq& n+ k nq n() t = pn () t(5.5)where c is the damping coefficient of the n mode of vibration. Eq. (5.5) indicates thatnthe n th modal displacement q tn ( ) depends on its corresponding natural frequencyω n ,damping ratio cnth/2 m k n, and the frequency content of external excitation. After thenmodal displacement q (t) has been determined, the contribution of the n thnmode to thedisplacement {U( t)} can be evaluated by Eq. (5.2).5.2.2. Modal analysis of Bill Emerson Memorial cable-stayedbridgeThe natural frequencies and mode vectors of the cable-stayed bridge were determined byan eigensolution method. For time-history analyses, the so-called Ritz-vector method isused. This is because Ritz-vectors can provide a better basis than eigenvectors do whenused for response-spectrum or time-history analysis (Wilson, 1982). The Ritz vectors aregenerated by taking into account the spatial distribution of the dynamic loading, whereasthe direct use of natural mode shapes neglects this very important information.How many modes of vibration must be included in analysis is a practical question. Inbuilding design, a rule of thumb is to have accumulated modal participating mass factorsin all directions of over 90%. For complex 3-D cable-stayed bridge structures, it isextremely difficult to achieve that level of mass participation in all directions. In thisstudy, an effort is made to include an accumulated modal participating mass factor ofover 70% in every direction. To this endeavor, a total of 4000 modes are specified in theinitial analysis of the cable-stayed bridge. The relationship between natural frequency andmode number is depicted in Figure 5.1. It was observed that the natural frequencies of thebridge up to 45.54 Hz were covered. The model was calibrated by slightly increasing themass density of concrete so that the first several natural frequencies can match theexperimental frequencies obtained from the measured responses. This modificationmainly represents the uncertainty of material density and geometry and the effects ofnon-structural components such as bracings, railing supports, and other permanentattachments.53
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: 4.6. RemarksA detailed 3-D FE model
Page 69 and 70: mainly corresponds to the vertical
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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