Calculus and Seismic Analysis November Ken Mark Bechtel Project Manager
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1 Calculus and Seismic Analysis November 2007 Ken Mark Bechel Projec Manager
2 BART Today 5 Lines 104 Miles Underground 37 Miles Aerial 23 Miles Surface 44 Miles 43 Saions 330,000 Daily Ridership 150,000 Daily Trips hrough Transbay Tube $15 Billion Replacemen Cos
3 BART s Hisory 1964 Consrucion Begins on Sae-of-he-Ar Sysem 1972 BART Carries Firs Passenger 1989 Loma Priea Earhquake 50 Miles (80 Km) Souh Minor Damage o BART Major Damage o Roadways
4 Learning from Earhquakes 1985 Meico Ciy Magniude Norhridge Magniude San Fernando Magniude Loma Priea Magniude Kobe, Japan Magniude 6.9 San Fernando, Meico Ciy, and Norhridge Phoos Couresy of he Karl V. Seinbrugge Collecion Earhquake Engineering Research Cener
5 Increasing Earhquake Probabiliy U.S. Geological Survey
6 Bay Area Fauls and Earhquakes Major Faul Lines in BART Service Area Hayward Faul Parallel o BART/ Crosses BART 1868 Las Major Rupure of Souhern Hayward Faul (130- o170-year Reurn Cycle) Four earhquake scenarios mos likely o damage BART: 7.0 Hayward Faul 8.0 San Andreas Faul 6.7 Concord Faul 6.7 Calaveras Faul
7 Findings Poenial Life Safey Issues Primary Vulnerabiliies Transbay Tube Criical Link beween Oakland and San Francisco Aerial Guideways Locaed Throughou Sysem Saions Primarily he 13 Elevaed Saions Evening Peak Delay (Easbound) w/ BART w/o BART Porions could be closed 2.5 years or longer o repair era driving raffic would cause gridlock on Bay Area roads. minues :00 17:00 18:00 19:00 20:00 21:00 22:00 deparure ime
8 Aerial Guideway Vulnerabiliy Toal of 1,918 Aerial Guideway Suppors Foundaions oo Small Poenial for Crumbling of Columns, Similar o Cypress Freeway
9 Seismic Analysis Aerial Srucures can be idealized as simplified 1- degree of freedom (dof) models Buildings and Saions are more complicaed requiring muli-dof models In many cases, he effec of he soil and foundaions are imporan, and he models could become much more complicaed o include soilsrucure ineracion effecs To more accuraely represen seismic response, someimes nonlinear effecs also have o be considered. Bu le s sar wih he simple 1-dof sysem
10 Response of a Single-DOF Sysem Equaion of Moion m && c& k f () Inerial Damping Resising Applied Force Force Force Eernal Force
11 Aerial Srucure Suppor Moion Suppor Moion F F F F spring damper ineria eernal ku cu& m( u&& 0 u&& g ) F Absolue acceleraion m( u&& u&& g ) cu& ku 0 mu&& cu& ku mu&& g Like applied force f()
12 Damped Free Vibraion angle phase ampliude D where De raio damping km c B A e m k m c k c m φ φ β ω βω β β ω β ω βω ω βω 2 1 sin cos 2 1 sin & && & && & &&
13 Deermine Damped Response - Duhamel Inegral Impulse momenum Effec is giving sysem an iniial velociy So from damped free vibraion, he response From his impulse is Then sum all responses as an inegral v m f m f v o v e m f e v d d d d o ω βω ω ω βω ω sin sin ( ) ( ) [ ] τ τ ω τ βω τ ω d e f m d d sin 1 0
14 Life Safey Aerial Guideway Srenghening Conceps Operabiliy Srenghen Foundaions Jacke Columns Add Shear Keys Add Piles (Where Appropriae) Similar Saion Rerofis
15 Response of Muli-Degree of Freedom Srucures Srucure Model Srucures are muli-degree of freedom sysems Srucures have naural frequencies a which he srucure responds very srongly
16 Mode Shapes A hese naural frequencies he displacemen of he srucure has a special shape, called a mode shape 1 s mode 2 nd mode 3 rd mode (Lowes Frequency)
17 Equaions of Moion [ ] {} [ ] {} [ ] {} [ ] {} g u M K C M & & & & & 1 11 Inerial force damping force resising force suppor moion forcing funcion {} {} i i i q φ Displacemen in i h mode mode shape * response parameer facor paricipaion i γ {} {} {} {} u q q q oal g i i i i i i i & & & & & γ ω β ω
18 Hayward Tower
19 Hayward Tower SAP Analysis Mode 2 Mode 3 Mode 4 Mode 5
20 Transbay Tube SF Transiion Srucure
21 3D SASSI Model of SFTS
22 3D ADINA Model of SFTS and TBT
23 FLAC Finie Difference Analysis of Slope Sabiliy
24 Transbay Tube Srenghening Conceps Vibro-Replacemen/Compacion around Tube Secions
25 Cenrifuge Tes o Verify Liquefacion Analysis
26 Comparison of Cenrifuge Tes Resuls
27 BART is Commied o Safeguarding Bay Area Transporaion and Economic Well-Being Quesions & Answers
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