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9. Appendix A: Stiffness and Damping CoefficientsThe stiffness and damping of a single pile and a pile group withappropriate interaction factors are evaluated for all the piers in the Illinoisapproach of the Bill Emerson Memorial Cable-stayed Bridge. The Novak’sformulations (1974) were followed with the sign convention as depicted in FigureA.1 and the following assumptions:1. Each pile behaves linear elastically and has a round cross section. For othershapes, an equivalent radius r is determined in each mode of vibration.o2. There is no separation between soil and pile during vibration.(a) Translational and coupled constraintsFigure A.1 Sign conventionA.1 Stiffness and damping factors of a single pile(b) Rotational constantsFor vertical vibration, the stiffness (k z ) and damping factors (c z ) arekz⎡EpA⎤= ⎢ ⎥ f⎣ ro⎦w1⎡E pA ⎤, c z = ⎢ ⎥ f⎣ vs ⎦w 2(A.1)in which E p is the modulus of elasticity of pile, A is the cross section of a singlepile, r o is the radius of a solid pile or an equivalent pile radius, V s is the shearwave velocity of soils along the floating pile, and f w1 and f w2 are the twoparameters that can be obtained from Figure A.2. The torsional stiffness (k ψ ) anddamping factors (c ψ ) can be expressed into⎡GJ⎤k = p pψ ⎢ ⎥ f⎣ ro⎦T ,1⎡GJp p⎤, c = ψ ⎢ ⎥ f (A.2)T ,2⎣ V s ⎦where G p is the shear modulus of the pile, J p is the polar moment of inertia of asingle pile about z axis, and f T,1 and f T,2 are the parameters from Figure A.3.100
Figure A.2 Vertical stiffness and damping parameters of floating piles(Novak and El-Shornouby, 1983)Figure. A.3 Torsional stiffness and damping parameters of RC piles (Novakand Howell, 1977)101
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Organizational Results Research Rep
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TECHNICAL REPORT DOCUMENTATION PAGE
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Executive SummaryOpen to traffic on
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8. All cables behave elastically un
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Table of ContentsExecutive summary.
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List of Figure CaptionsFigure 1.1 A
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Figure 5.30 29 th mode shape (1.032
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1. Introduction1.1. GeneralCable-st
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Figure 1.1Artist rendering of the B
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Expansionand lateralearthquakerestr
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3. Evaluate the model by conducting
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2. Automatic Retrieval of Peak Acce
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2.2.3. Map/Find stationsClicking th
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2.2.5. Displaying seismogramsTo dis
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Figure 2.8Directory chooser2.2.6. S
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3.2. Seismic instrumentation networ
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The main bridge from Bent 1 to Pier
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(a) South side of deck at support o
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(a) North side of deck at middle of
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The seismic responses at the top (T
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Polyreference Complex Exponential (
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are evaluated and presented in Figu
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0.353.50.330.252.5Amplitude0.20.15A
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1 x 10-4 Frequency(Hz)Amplitude0.90
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Amplitude0.20.180.160.140.120.10.08
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Amplitude0.20.180.160.140.120.10.08
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1.20.80.400 100 200 300 400 500 600
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4. Finite Element Modeling of Bill
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Since the cable stays are connected
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Table 4.3Initial property of cables
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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 and 68: knω = (5.4)nm nSimilarly, when dam
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: 42. Song, W., Giraldo, D.F., Clayto
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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