earthquake geotechnical engineering lecture 2 – strong motion parameters
TRANSCRIPT
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EARTHQUAKE GEOTECHNICAL ENGINEERING
Lecture 2 – Strong Motion Parameters
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Contents
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Just a reminder…………..
Frequency, f (aka temporal frequency) number of occurrences of a repeating event per unit
time SI Unit: Hertz
Period, T time for 1 cycle Inverse of f SI unit: second
Angular Frequency, ω Measure of rotation rate SI unit: radians/second
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1.0 Strong Motion Parameters Earthquake Description
3 components translation 3 components rotation (typically neglected)
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1.0 Strong Motion Parameters Amplitude
Peak Acceleration Peak Velocity Peak Displacement
Frequency Content Duration
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1.1 Amplitude
Peak Horizontal Acceleration Associated with high frequency Largest value from accelogram Vector sum of 2 orthogonal
components Dynamic force in structures
Peak Vertical Acceleration Associated with intermediate frequency PVA = 2/3 PHA Usually less critical
Peak Displacement Associated with lower frequency components Difficult to determine, long period, noise Less common
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1.1 Amplitude
Effective Acceleration (Newark and Hall) “That acceleration …most closely related to
structural response and damage potential… less than peak free-field ground acceleration. Function of
size of loaded area, frequency content of excitation, which in turn
depends on closeness to earthquake source, weight, embedment, damping characteristics and stiffness of the structure and
foundation”
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1.1 Amplitude
a) Peak at high frequency – little effect on buildings with lower frequency
b) Same peak as (a), but more cycles
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1.1 Amplitude
Sustained Maximum Acceleration and Velocity (Nuttli 1979) Defined for 3 (or 5) cycles as the 3rd (or 5th)
highest value of acceleration in time history Effective Design Acceleration
Many definitions Eg 1.25 x 3rd highest peak acceleration,
after filtering (Kennedy, 1980)
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1.2 Frequency
Damage response – frequency sensitive Earthquakes – range of frequencies Determine frequency content of time
history… how?
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1.2.1 Frequency - Fourier Analysis
Joseph Fourier (1768 – 1830), France•Son of a tailor, orphaned at 9•rejected from Science course due parentage, so studied maths•French Revolution•Made Governor of Egypt by Napolean•Translation of Rosetta Stone•Propagation of Heat•Discovered Green House effect
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1.2.1 Frequency – Fourier Analysis
•Many applications in science and engineering eg Fourier analysis has many scientific applications – in physics, signal processing, imaging (JPEG files), probability, differential equations, acoustics, oceanography, sonar………& earthquake engineering !•http://www.fourier-series.com/
•Every period can be approximated by sum of simple harmonic terms
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1.2.1 Frequency – Fourier Analysis General Form
Fourier Coefficients
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1.2.1 Frequency – Fourier Analysis Other forms utilising
trigonometric identities
Euler’s formula
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1.2.1 Frequency – Fourier Analysis Example
Using this form
Coefficients
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1.2.1 Frequency – Fourier Analysis Fourier Amplitude Spectrum
Fourier Phase Spectrum
Amplitude vs Frequency
Phase angle vs Frequency
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1.2.1 Frequency – Fourier Analysis
Earthquake Engineering Applications
Loading Response
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1.2.1 Frequency – Fourier Analysis
Time Domain to Frequency Domain Domain argument
Earthquake Geotechnical Engineering Application Example 1 Dimensional Response Analysis
Time domain
Frequency domain
Frequency domain
Time domain
Frequency domain
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1.2.1 Frequency – Fourier Analysis Fourier Transform Continuous variant of Fourier Series Function that converts signals from “time
domain” to “frequency domain”
Finite data points– Discrete Fourier Transform Fast Fourier Transform http://www.physik.uni-kl.de/fileadmin/
beigang/Vorlesungen/WS_07_08/Fourier_Transforms_Rick_Trebino.pdf
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1.2.2 Frequency – Amplitude Spectra
Fourier Amplitude Spectrum Clearly distinguish frequency content
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1.2.2 Frequency – Amplitude Spectra
Earthquake Characteristic Shape
Corner Frequency – inv proportional to cube root of seismic moment (larger earthquake, lower freq)
Cutoff frequency – constant for geographic region
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1.2.2 Frequency - more parameters
Phase Spectra Gives time of peak harmonic motion Influences variation of ground motion with time
Predominant Period Period of vibration corresponding to Fourier amplitude
spectrum
Bandwidth Gives information on dispersion Level where power of spectrum = ½ max = Fourier Amplitude
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1.2.2 Frequency – more parameters
Power Spectral density Gives intensity of ground motion Either time domain or frequency domain
Central Frequency Frequency where power spectral density
greatest Shape Factor
Dispersion of power spectral density
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1.3 - Duration
Eq damage related to duration Damage depends on load reversal Strong-motion portion of accelogram Bracket duration
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1.4 – Other parameters
Arias Intensity Strength of ground motion
g = accel due gravityTd = duration of signal above threshold
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1.5 Predictive Equations
Estimation of Amplitude, Freq, Duration etc parameters
Include Attenuation Equations (PHA, PVA) aka Ground Motion Prediction Equations (GMPE) Y = f(M,R,Pi)
Ground motion , Y Magnitude, M Distance, R Other parameters (eg source, wave path, site cond), Pi
Regression analysis Recorded earthquakes
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1.5 Predictive Equations
Functional Form
1. Peak values log normally distributed, ln Y
2. Magnitude defined as log of peak motion, therefore ln Y proportional to M
3. Body wave attenuation 1/R, Surface waves
4. Larger M larger fault rupture areagreater distance R
5. Energy exponentially decreases through material damping
6. Source or site Characteristics
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1.5 Predictive Equations
Peak Acceleration Campbell (1981)
PHA Sites < 50km fault rupture M 5.0 to 7.7 World wide data
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1.5 Predictive Equations
Campbell and Bozorgnia (1994) M 4.7 to 8.1 F - 0 for strike slip& normal, 1 for reverse,
reverse-oblique, thrust SSR = 1 soft-rock sites, SHR =1 hard rock
sites, SSR = SHR = 0 alluvium sites
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1.5 Predictive Equations
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2.1 Modes of Vibration
Mode
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2.1 Modes of Vibration
Building Modes
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2.1 Modes of Vibration
Alcoa Building (26 storey), San FranciscoFund. Period of Vibration:N-S = 1.67secE-W= 2.21 secTorsional (vert axis) = 1.12
San Francisco Bridge (main spain1.3km)Fund Period of Vibration:Transverse=18.2 secVertical=10.9 secLongitudional=3.81secTorsional=4.43 sec
Transamerica Building (60 storey), San FranciscoFund. Period of Vibration:N-S =2.9secE-W=2.9sec
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2.1 Modes of Vibration
Most structures have multiple modes of vibration Each mode has a period Fundamental mode
aka natural period of vibration longest period (lower freq)
Higher the mode, the shorter the period (higher freq) Buildings Rule of Thumb
Fundamental period ≈ No of storeys / 10 (moment frame)
http://www.youtube.com/watch?v=cDjcpPgHrbo
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2.2 Degrees of Freedom
Motion of a ship (6 degrees) Buildings
MDOF – multi degree of freedom SDOF – single degree of freedom
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2.3 SDOF
Single Degree of Freedom Systems Buildings often modelled as SDOF
systems Dynamic Response of SDOF systems Solve differential equations for different
conditions
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2.3.1- Condition 1 - Undamped Free Vibration
Undamped free vibration, c = 0, Q(t) =0 Equation of motion
Solve the differential equation Initial conditions Solution where
Natural circular freq of vibration Natural Period of vibration
Natural cyclic freq of vibration
Note:Natural freq depends on:• mass• stiffness
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2.3.2 Condition 2 - Viscously Damped Free Vibration
Damped (underdamped), free vibration, c>0, Q(t)=0 Equation of motion Solution
where ζ = damping ratio Natural period of damped vibration Natural period
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2.3.3 – Condition 3 Undamped Forced Vibration (sinusoidal)
Undamped, forced, c=0, Q(t)>0 Equation of Motion Response
Harmonic force, with forcing freq, ω Response, with ω < ωn
Response, with ω = ωn
Resonant Frequency1. Forcing Freq when ratio of Response
amplitude/Initial Amplitude = max2. For undamped system
• Resonant Freq = ωn
• Response amp/initial amp ratio is unbounded
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2.3.4 Condition 4 - Viscously Damped Forced Vibration (Sinusoidal)
Undamped, forced, c>0, Q(t)>0 Equation of Motion Response
Harmonic force, with forcing freq, ω Response, with ω < ωn
Response, with ω = ωn, s Response, with ω = ωn, various damping ratios
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2.2.5 – SDOF solution subject to varying force (with time)
Solution for SDOF equation of motion with varying force / ground acceleration eg
Numerical finite difference or “time-stepping” procedures http://en.wikipedia.org/wiki/Finite_difference_method
Solution Determined at discrete time instances Depends on prior history Equation of Motion
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3.0 Response Spectrum
Core Concept in Earthquake Engineering Characterises ground motion effect on structures Plot of peak response of all possible SDOF (linear) systems to a particular ground
motion Each plot is for SDOF system of fixed damping ratio Deformation, Velocity, Acceleration Response Spectra At T = 0 response equal to peak ground response
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3.0 Response Spectra
Examples
Example 1Response spectra for various damping ratios
Example 2 Design response spectra for various damping ratios
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3.0 Response Spectra
Examples
Example 3 –Design response spectra envelope
Example 4 – Design response spectra for various soil categories
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3.0 Response Spectra
Examples (cont’d)
Example 5 Inelastic Design Response Spectra with Ductility
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3.0 Response Spectra
Construct a response spectra Define ground acceleration (∆t = 0.02s) Select Tn, ζ for SDOF system
Solve for u(t) using numerical method Determine peak response uo
Spectral Ordinates D=uo, V=(2π/Tn)D, A=(2π/Tn)2D
Design spectra obtained by statistical analysisTripartite Plot, ζ = 0.02
Design spectra, ζ=0.05
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3.0Response Spectra
Deformation, Pseudo Velocity, Pseudo Response Pseudo Velocity Pseudo Acceleration Pseudo vel, pseudo accel approximate
system vel, accel
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4.0 Summary
Parameters to describe Earthquake Motion Amplitude Freq Duration
Fourier Analysis Fundamental Mode SDOF Response Spectra