Earthquakes and ModelingChris Van Horn and Kyle Eli

Modeling Building VibrationsBy 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 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/weave/module1/m1intro.html

• http://quake.wr.usgs.gov/research/index.html

Earthquake Loss ModelingKyle 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 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

]|[ yIMLSSPFragility =>=

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_Tantala_et_al_7ncee.pdf• http://mae.ce.uiuc.edu/Education/Student/Graduate/SCOJ/V3N2/Nielson.pdf