EARTHQUAKE GEOTECHNICAL ENGINEERING Lecture 2 – Strong Motion Parameters

EARTHQUAKE GEOTECHNICAL ENGINEERING

Lecture 2 – Strong Motion Parameters

Contents

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

1.0 Strong Motion Parameters Earthquake Description

3 components translation 3 components rotation (typically neglected)

1.0 Strong Motion Parameters Amplitude

Peak Acceleration Peak Velocity Peak Displacement

Frequency Content Duration

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

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”

1.1 Amplitude

a) Peak at high frequency – little effect on buildings with lower frequency

b) Same peak as (a), but more cycles

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)

1.2 Frequency

Damage response – frequency sensitive Earthquakes – range of frequencies Determine frequency content of time

history… how?

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

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

1.2.1 Frequency – Fourier Analysis General Form

Fourier Coefficients

1.2.1 Frequency – Fourier Analysis Other forms utilising

trigonometric identities

Euler’s formula

1.2.1 Frequency – Fourier Analysis Example

Using this form

Coefficients

1.2.1 Frequency – Fourier Analysis Fourier Amplitude Spectrum

Fourier Phase Spectrum

Amplitude vs Frequency

Phase angle vs Frequency

1.2.1 Frequency – Fourier Analysis

Earthquake Engineering Applications

Loading Response

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

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

1.2.2 Frequency – Amplitude Spectra

Fourier Amplitude Spectrum Clearly distinguish frequency content

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

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

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

1.3 - Duration

Eq damage related to duration Damage depends on load reversal Strong-motion portion of accelogram Bracket duration

1.4 – Other parameters

Arias Intensity Strength of ground motion

g = accel due gravityTd = duration of signal above threshold

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

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

1.5 Predictive Equations

Peak Acceleration Campbell (1981)

PHA Sites < 50km fault rupture M 5.0 to 7.7 World wide data

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

1.5 Predictive Equations

2.1 Modes of Vibration

Mode

2.1 Modes of Vibration

Building Modes

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

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

2.2 Degrees of Freedom

Motion of a ship (6 degrees) Buildings

MDOF – multi degree of freedom SDOF – single degree of freedom

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

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

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

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

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

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

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

3.0 Response Spectra

Examples

Example 1Response spectra for various damping ratios

Example 2 Design response spectra for various damping ratios

3.0 Response Spectra

Examples

Example 3 –Design response spectra envelope

Example 4 – Design response spectra for various soil categories

3.0 Response Spectra

Examples (cont’d)

Example 5 Inelastic Design Response Spectra with Ductility

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

3.0Response Spectra

Deformation, Pseudo Velocity, Pseudo Response Pseudo Velocity Pseudo Acceleration Pseudo vel, pseudo accel approximate

system vel, accel

4.0 Summary

Parameters to describe Earthquake Motion Amplitude Freq Duration

Fourier Analysis Fundamental Mode SDOF Response Spectra