Earthquakes and Modeling

- Chris Van Horn and Kyle Eli

Modeling Building Vibrations

- By Chris Van Horn

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

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

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

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

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)

Natural Frequency

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

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

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

The Differential Equations

- Finally we have three differential equations for

a three story building

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)

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

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

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

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

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

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

References

- http//www.shodor.org/reneeg/weave/module1/m1intr

o.html - http//quake.wr.usgs.gov/research/index.html

Earthquake Loss Modeling

- Kyle Eli

HAZUS

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

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

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

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

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

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

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

HAZUS, NYC Earthquake

HAZUS, NYC Earthquake

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HAZUS, NYC Earthquake

HAZUS, NYC Earthquake

Bridge Seismic Fragility

- How do you determine damage to a structure such

as a bridge? - Fragility curves
- Direct losses
- Indirect losses

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

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

Bridge Seismic Fragility

Bridge Seismic Fragility

Bridge Seismic Fragility

Bridge Seismic Fragility

- Probability of failure
- Fragility curve

Bridge Seismic Fragility

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

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

Bridge Seismic Fragility

- Damage States
- Use states defined in HAZUS

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