Transcript Modeling Building Vibrations

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 SpringMass-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 structure’s
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/weav
e/module1/m1intro.html
• http://quake.wr.usgs.gov/research/ind
ex.html
Earthquake Loss Modeling
Kyle Eli
HAZUS
• Hazards U.S. Multi-Hazard (HAZUS-MH)
– Nationally applicable
– Earthquakes
– Floods
– Hurricane winds
HAZUS (cont’d)
• 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 (cont’d)
• 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 (cont’d)
• 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
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  P[S  LS | IM  y]
• 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
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•
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http://www.fema.gov/plan/prevent/hazus/index.shtm
http://128.205.131.101/techdocs/news/7NCEE/paper_Tantala_et_al_7ncee.pdf
http://mae.ce.uiuc.edu/Education/Student/Graduate/SCOJ/V3N2/Nielson.pdf