Geotechnical Earthquake
Engineering
by
Dr. Deepankar Choudhury Humboldt Fellow, JSPS Fellow, BOYSCAST Fellow
Professor
Department of Civil Engineering
IIT Bombay, Powai, Mumbai 400 076, India.
Email: [email protected]
URL: http://www.civil.iitb.ac.in/~dc/
Lecture – 32
3
Site Response - The Problem
Predicts the response of a soil deposit due to earthquake
excitation
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Site Response
Ideally, a complete ground response analysis should
include:
Rupture mechanism at source of an earthquake
(source)
Propagation of stress waves through the crust to the
top of bedrock beneath the site of interest (path)
How ground surface motion is influenced by the soils
that lie above the bedrock (site)
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In Reality,
Mechanism of fault rupture is very complicated and
difficult to predict in advance
Crustal velocity and damping characteristics are generally
poorly known
Nature of energy transmission between the source and site
is uncertain
Site Response
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Ground Response Analysis
In practice,
Seismic hazard analyses (probabilistic or
deterministic) are used to predict bedrock motions at the
location of the site.
Seismic hazard analyses rely on empirical attenuation
relationships to predict bedrock motion parameters.
Ground response problem becomes one of determining
response of soil deposit to the motion of the underlying
bedrock. IIT Bombay, DC
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Ground Response Analysis
Used to:
Predict ground surface motions, Time histories, Response
spectra, Scalar parameters
Evaluate dynamic stresses and strains, Liquefaction
hazards, Foundation loading
Evaluate ground failure potential, Instability of earth
structures, Response of various geotechnical structures like
retaining wall, earth dam, pile, various foundations etc.
8
Ground Response Analysis
Definitions:
Rock outcropping motion the motion that would occur
where rock outcrops at a surface.
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Ground Response Analysis
Definitions:
Bedrock motion – the motion that occurs at bedrock overlain by a
soil deposit. Differs from rock outcrop motion due to lack of
free surface effect.
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Ground Response Analysis
Definitions:
Free surface motion – the motion that occurs at the surface of a
soil deposit
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Ground Response Analysis
Common situations # 1
Rock outcrop motion is known – usually obtained from attenuation
relationship (based on database of rock outcrop motions)
Free surface motion is to be determined
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Ground Response Analysis
Common situations # 2
Free surface motion is known – usually obtained from attenuation
relationship (based on database of soil outcrop motions)
Free surface motion is to be determined for site with different soil
conditions
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Linear Analysis
A known time history of bedrock (input) motion is
represented as a Fourier series, usually using the FFT
Each term in the Fourier series of the bedrock (input)
motion is then multiplied by the transfer function to
produce the Fourier series of the ground surface (output)
motion
The ground surface (output) motion can then be expressed
in the time domain using the inverse FFT
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Transfer Function
The transfer function determines how each frequency in the
bedrock (input) motion is amplified, or de-amplified by the soil
deposit
A transfer function may be viewed as a filter that acts upon
some input signal to produce an output signal
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Input
Transfer
function
(filter) Output
Transfer Function Evaluation Uniform Undamped Soil on Rigid Rock
16
Assume harmonic base motion,
Then, response should also be harmonic
Wave traveling in
– z direction
(upward)
Wave traveling in
+ z direction
(downward)
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Transfer Function Evaluation Uniform Undamped Soil on Rigid Rock
Displacement :
Stress:
At z = 0 (ground surface)
A = B
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Transfer Function Evaluation
Uniform Undamped Soil on Rigid Rock
, 22
ikz ikz
i te e
u z t A e
, 2 cos i tu z t A kz e
0, 2 1
2 cos cos,
i t
i t
u t AeF
A kH e kHu H t
Defining a transfer function as the ratio of the displacement at
the ground surface to the bedrock displacement
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Transfer Function Evaluation
Uniform Undamped Soil on Rigid Rock
As kH = wH/Vs goes to zero, denominator goes to zero Transfer
function goes to infinity
Natural frequencies
/
2n SV n H
Fundamental period
02 4
SS
HTV
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Transfer Function Evaluation
Uniform Damped Soil on Rigid Rock
How do we handle damping?
Complex shear modulus 2 2
* * */SG V k
1/2* 2 */k G Complex Wave
Number
* * / 1S SV G V i
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*
*1
S
wk k i
V
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Transfer Function Evaluation
Uniform Damped Soil on Rigid Rock
Repeat analysis as before
Transfer function becomes
**
0, 2 1
cos, 2 cos
i t
i t
u t AeF
k Hu H t A k H e
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Transfer Function Evaluation
Uniform Damped Soil on Rigid Rock
*
*
1 1,
coscos
S
F wk H wH
V
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2 22 2
1 1,
cos cos / /S S
F wkH kH wH V wH V
23
Transfer Function Evaluation
Uniform Damped Soil on Rigid Rock
Note:
Natural frequencies still exist
Low natural frequencies strongly amplified
High natural frequencies weakly amplified
Very high frequencies de – amplified
Amplification strongly frequency - dependent
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Transfer Function Evaluation
Uniform Undamped Soil on Elastic Rock
su* *
,s si t k t i t k t
s s s su z t C e D e* *
,r ri t k t i t k t
r r r ru z t C e D e
, 0,s s r ru z H t u z t
, 0,s s r rz H t z t
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Transfer Function Evaluation
Uniform Undamped Soil on Elastic Rock
Maintaining equilibrium and compatibility of displacements
at the boundary, the amplitude of the transfer function can be
written as
2 2 2
1, 0
cos sins z s
F wk H k H
*
*s ss
zr sr
v
v
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Transfer Function Evaluation
Uniform Undamped Soil on Elastic Rock
Note:
Even with no soil damping, resonance cannot occur
Why???
Energy removed from soil layer by transmission into rock
Form of radiation damping
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Transfer Function Evaluation
Uniform Undamped Soil on Elastic Rock
Maintaining equilibrium and compatibility of displacements
at the boundary, the amplitude of the transfer function can be
written as
2 2 2
1, 0
cos sins z s
F wk H k H
*
*s ss
zr sr
v
v
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Transfer Function Evaluation
Uniform Undamped Soil on Elastic Rock
Note:
Even with no soil damping, resonance cannot occur
Why???
Energy removed from soil layer by transmission into rock
Form of radiation damping
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Transfer Function Evaluation
Layered, Damped Soil on Elastic Rock
For layer j
From equilibrium
From compatibility
* *
, j j j jik z ik z i t
j j j ju z t A e B e e
* *
1 1j j j jik h ik h
j j j jA B A e B e
* ** *
1 1 * *
1 1
s j s jik h ik hj j
j j j j
j j
G kA B A e B e
G k
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Transfer Function Evaluation
Layered, Damped Soil on Elastic Rock
If we know response at layer j (Aj and Bj are known), then
we have two equations with two unknowns (Aj+1 and Bj+1)
We can relate Aj+1 and Bj+1 to Aj and Bj by means of
recursive relationships
For layer j
From equilibrium
From compatibility
* *
, j j j jik z ik z i t
j j j ju z t A e B e e
* *
1 1j j j jik h ik h
j j j jA B A e B e
* ** *
1 1 * *
1 1
s j s jik h ik hj j
j j j j
j j
G kA B A e B e
G k
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Transfer Function Evaluation
Layered, Damped Soil on Elastic Rock
Solving for the unknowns
* ** *
1
1 11 1
2 2
j j j jik h ik h
j j j j jA A e B e
* ** *
1
1 11 1
2 2
j j j jik h ik h
j j j j jB A e B e
1 1 1j jA a A1 1 1j jB b B
Or, relating the coefficients to those at the ground surface
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Transfer Function Evaluation
Layered, Damped Soil on Elastic Rock
Then, a transfer function relating the motion in layer i to
the motion in layer j can be written as
If we know the motion at any layer, we can use this
transfer function to compute the corresponding motion at
any other layer
i i
ij
j j
a bF
a b
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Equivalent Linear Approach
The actual nonlinear hysteretic stress – strain behavior of
cyclically loaded soils can be approximated by equivalent linear
properties
Assume some initial strain and use to
estimate G and ξ
Determine peak strain and effective strain maxeff R
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Select properties based on updated strain level
Compute response with new properties and determine
resulting effective shear strain
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Repeat until computed effective strains are consistent with
assumed effective strains
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Non – linear Approach
Solve Wave equation incrementally
, ,i t t i tu uu
t t
2
2
u u
z t t
1, ,i t i t
z z
1, , 1, ,i t i t i t i tu u
z t
Approximate partial derivatives
Finite difference form
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Non – linear Approach
Solve Wave equation incrementally
then
, , 1, ,i t t i t i t i t
tu u
z
Velocity at time t+Δt can be calculated from velocity
and shear stress at time t
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Non-Linear Approach
Solve wave equation incrementally
Start with initial stiffness, Gmax
Compute response for small time step, Δt
Compute shear strain amplitude at end of time step
Use stress-strain model to find Gtan for next time step
Compute shear strain amplitude at end of next time step
Continue stepping through time for entire input motion
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Non – linear Approach
Solve Wave equation incrementally
Nonlinear response is simulated in
incrementally linear fashion
Material damping is taken care by
hysteretic response
Approach requires good model for
description of soil stress – strain
behaviour
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Non-Linear Stress-Strain Models
Two main types
Cyclic nonlinear models
Advanced constitutive models
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Non – Linear Stress – Strain Models
Cyclic nonlinear models
Requires :
• Backbone curve
• Unloading – reloading rules
• Pore pressure model
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Non – Linear Stress – Strain Models
Advanced constitutive models
Requires:
•Yield surfaces
•Hardening rule
•Failure surface
•Flow rule
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Non-Linear Stress-Strain Models
Cyclic nonlinear models Advantages:
Relatively simple
Small number of parameters
Disadvantages:
Simplistic representation of soil behavior
Cannot capture dilatancy effects
Advanced constitutive models Advantages:
Can better represent mechanics of yield, failure
Disadvantages:
Many parameters
Difficult to calibrate
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