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Earthquakes and Modeling

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Title: Earthquakes and Modeling


1
Earthquakes and Modeling
  • Chris Van Horn and Kyle Eli

2
Modeling Building Vibrations
  • By Chris Van Horn

3
Building Vibrations
  • How a three story building responds to
    earthquakes
  • Can be described with three second order
    differential equations
  • In this model mass, stiffness, and damping will
    be taken into account

4
Vibrations of Single Story
  • System behaves similar to a Spring-Mass-Dampening
    system
  • The roof of the building oscillates so we have
    usual exchange of Kinetic and Potential energy

5
Energy Exchange
  • Potential energy is stored by the elastic
    deformation of the walls
  • Kinetic energy is the energy of the structures
    mass in motion
  • When unforced free vibration each type of energy
    at a max when other at min

6
Natural Frequency
  • So kinetic energy at max when displacement at 0
    and potential energy at max when velocity at 0
  • Setting max kinetic energy equal to max potential
    energy can find natural frequency
  • If building allowed to oscillate freely will do
    so at natural frequency
  • If ground motion at same frequency as natural
    frequency building will resonate

7
Vibrations of Multi Story Buildings
  • X(t) replaced with x a vector of the displacement
    for each story
  • Introduce a stiffness matrix K, and mass matrix M
  • N x N for a N-story building
  • Symmetric
  • Positive definite (K building not free
    floating, M every floor has a positive mass)

8
Natural Frequency
                 
                       
           
9
Natural Frequency
  • Need to solve our equation
  • One solution is when the amplitude equals zero
  • The other is when the determinate is equal to
    zero
  • For an N story building there will be N different
    frequencies for which the determinate will be
    zero
  • For every natural frequency there is a position
    vector that the bottom equation holds
  • Called eigenvectors or mode shapes of the
    building
  • Resonance will happen if any natural frequency is
    matched

10
Shear Forces
  • When one floor moves laterally with respect to
    the floor below it, the columns bend, creating
    lateral "shear" forces
  • F kx
  • K is shear stiffness constant and x is
    displacement

11
Forces on Mass 1
  • Mass 1 displaced distance x1 with respect to the
    ground
  • Forces from the columns below the mass
  • Forces from the columns above the mass
  • Inertial forces
  • Acceleration of mass with respect to the ground
    plus the acceleration of the ground

12
The Differential Equations
  • Finally we have three differential equations for
    a three story building

13
Damping
  • If there was no damping once a building started
    shaking it would not stop shaking
  • Sources of building damping
  • Air drag of building moving through air
  • Columns the building columns absorb some energy
  • Structural yielding if an element gives way can
    cause significant damping can be controlled
    (good) or uncontrolled (bad)

14
Proportional Damping
  • Damping matrix proportional to the Mass and
    stiffness matrix
  • Units of elements in damping matrix
    Force/length/time
  • Can be described with a diagonal matrix

15
Model Examination
  • Will examine our model in 3 situations
  • Free vibration in response to initial
    displacement
  • Vibration resulting from sinusoidal ground
    accelerations
  • Vibration resulting from random ground
    accelerations

16
One Story Free Vibration
  • We guess a function and insert in to our
    differential equation. We solve the differential
    equation, then using those results we can use our
    original function to find answers

17
Multi-Story Free Vibration
  • Use same strategy that we used for a single story
    building
  • Solving the determinate for lambda in terms of c,
    m, and k not possible
  • Since we have values for c, m, and k we can still
    come up with a solution

18
Response to Sinusoidal Ground
  • If ground motion is sinusoidal building will
    eventually oscillate at same frequency as the
    ground
  • If ground motion close to natural frequency, then
    building may oscillate at both frequencies,
    called beat phenomenon
  • At some points they cancel each other out at
    others they add together

19
Random Ground Motions
  • Random ground motion can be thought of as a
    summation of several sinusoidal ground motions,
    all with slightly different frequencies and with
    different phase angles
  • the response to random ground motion as the
    summation of the responses to each of the
    sinusoidal ground motions, individually
  • If the random ground motion includes frequencies
    at or near a natural frequency of the building,
    then the building will respond strongly at that
    natural frequency

20
References
  • http//www.shodor.org/reneeg/weave/module1/m1intr
    o.html
  • http//quake.wr.usgs.gov/research/index.html

21
Earthquake Loss Modeling
  • Kyle Eli

22
HAZUS
  • Hazards U.S. Multi-Hazard (HAZUS-MH)
  • Nationally applicable
  • Earthquakes
  • Floods
  • Hurricane winds

23
HAZUS (contd)
  • Developed by National Institute of Building
    Sciences (NIBS) for FEMA.
  • Committees of experts for each type of natural
    disaster
  • Works with modern GIS software
  • ArcGIS
  • Takes into account various impacts
  • Physical damage
  • Economic loss
  • Social impacts

24
HAZUS Earthquake Model
  • Forecasts damage and loss to buildings,
    infrastructure, and populations that may result
    from earthquakes
  • Used for emergency preparedness, response, and
    recovery planning
  • Works with GIS software to display graphical maps
    of earthquake hazards and potential damage
  • Can work with data sets from national to local
  • Allows custom models for special conditions

25
HAZUS Earthquake Model (contd)
  • Features
  • Building classification
  • Damage estimates for a variety of building types
  • Structure, contents, and interior
  • Debris quantities, shelter needs, fire,
    casualties
  • Direct and indirect economic losses

26
HAZUS Earthquake Model (contd)
  • Uses
  • Formulate policy to reduce loss
  • Estimate resources for disaster relief
  • Improve emergency response planning
  • Plan for clean-up and technical assistance
  • Estimate displaced households and shelter
    requirements

27
HAZUS Case Study
  • Earthquake loss estimation for the New York City
    area
  • One of the most detailed applications of HAZUS
  • Risk and loss characterization for Manhattan
  • Required a complete building inventory
  • Location, height, square footage, structural
    type, structural material, age, quality of
    construction, and seismic design level
  • Detailed geotechnical soil characterization
  • Simulations provided a large variety of useful
    information

28
HAZUS, NYC Earthquake
  • NYC has moderate potential for earthquakes
  • Assets worth nearly 1 trillion
  • Fragile structures
  • New construction not designed for earthquake
    survivability until 1996

29
HAZUS, NYC Earthquake
30
HAZUS, NYC Earthquake
31
(No Transcript)
32
HAZUS, NYC Earthquake
33
HAZUS, NYC Earthquake
34
Bridge Seismic Fragility
  • How do you determine damage to a structure such
    as a bridge?
  • Fragility curves
  • Direct losses
  • Indirect losses

35
Bridge Seismic Fragility
  • Fragility Curves
  • Developed from
  • Empirical data from past earthquakes
  • Expert opinions
  • Analytical methods
  • Useful for
  • Retrofit prioritization
  • Assessing vulnerability
  • Post-earthquake evaluation
  • Route planning

36
Bridge Seismic Fragility
  • Fragility Function
  • S Response measure of bridge or bridge
    component
  • LS Limit state or damage level of bridge or
    bridge component
  • IM ground motion intensity measure
  • Y realization of the chosen ground motion
    intensity measure

37
Bridge Seismic Fragility
38
Bridge Seismic Fragility
39
Bridge Seismic Fragility
40
Bridge Seismic Fragility
  • Probability of failure
  • Fragility curve

41
Bridge Seismic Fragility
42
Bridge Seismic Fragility
  • Bridge Modeling
  • A high quality model is needed
  • Non-trivial task
  • Many structural properties taken into account
  • All vulnerable components should be considered
  • Much prior work considered only columns/piers
  • Uncertainties
  • Generate varied samples
  • Simpler models are better, but accuracy must be
    maintained
  • A full three dimensional model may be
    advantageous
  • Can be extremely computationally expensive

43
Bridge Seismic Fragility
  • Seismic Demand Analysis
  • Combine a suite of ground motions with a suite of
    bridge samples
  • Pairs are analyzed with finite-element analysis
    software
  • For each pair, response quantities such as column
    curvature and bearing/abutment deformation are
    plotted against ground motion intensity

44
Bridge Seismic Fragility
  • Damage States
  • Use states defined in HAZUS

45
References
  • http//www.fema.gov/plan/prevent/hazus/index.shtm
  • http//128.205.131.101/techdocs/news/7NCEE/paper_T
    antala_et_al_7ncee.pdf
  • http//mae.ce.uiuc.edu/Education/Student/Graduate/
    SCOJ/V3N2/Nielson.pdf
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