Earthquakes and Modeling

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Earthquakes and Modeling

• Chris Van Horn and Kyle Eli

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Modeling Building Vibrations

• By Chris Van Horn

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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

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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

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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

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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

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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)

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Natural Frequency
                 
                       
           
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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

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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

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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

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The Differential Equations

• Finally we have three differential equations for
  a three story building

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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)

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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

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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

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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

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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

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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

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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

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References

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

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Earthquake Loss Modeling

• Kyle Eli

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HAZUS

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

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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

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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

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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

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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

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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

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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

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HAZUS, NYC Earthquake
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HAZUS, NYC Earthquake
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HAZUS, NYC Earthquake
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HAZUS, NYC Earthquake
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Bridge Seismic Fragility

• How do you determine damage to a structure such
  as a bridge?
• Fragility curves
• Direct losses
• Indirect losses

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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

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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

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Bridge Seismic Fragility
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Bridge Seismic Fragility
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Bridge Seismic Fragility
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Bridge Seismic Fragility

• Probability of failure
• Fragility curve

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Bridge Seismic Fragility
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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

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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

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Bridge Seismic Fragility

• Damage States
• Use states defined in HAZUS

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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