One-Dimensional Site Response Analysis

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One-Dimensional Site Response Analysis
What do we mean? One-dimensional Waves
propagate in one direction only
2
One-Dimensional Site Response Analysis
What do we mean? One-dimensional waves
propagate in one direction only Motion is
identical on planes perpendicular to that motion
to infinity
to infinity
3
One-Dimensional Site Response Analysis
What do we mean? One-dimensional waves
propagate in one direction only Motion is
identical on planes perpendicular to that
motion Cant handle refraction so layer
boundaries must be perpendicular to direction of
wave propagation Usual assumption is
vertically-propagating shear (SH) waves
Horizontal surface motion
4
One-Dimensional Site Response Analysis
When are one-dimensional analyses appropriate?
Stiffer with depth
Focus
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One-Dimensional Site Response Analysis
When are one-dimensional analyses appropriate?
Horizontal boundaries waves tend to be
refracted toward vertical
Stiffer with depth
Focus
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One-Dimensional Site Response Analysis
When are one-dimensional analyses appropriate?
Not appropriate here
Stiffer with depth
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One-Dimensional Site Response Analysis
Not here!
When are one-dimensional analyses appropriate?
Retaining structures
Inclined ground surface and/or non-horizontal
boundaries can require use of two-dimensional
analyses
Tunnels
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One-Dimensional Site Response Analysis
Not here!
When are one-dimensional analyses appropriate?
Complex soil conditions
Multiple structures
Localized structures may require use of 3-D
response analyses
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One-Dimensional Site Response Analysis
How should ground motions be applied?
Soil
Not the same!
Rock
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One-Dimensional Site Response Analysis
How should ground motions be applied?
Input (object) motion If recorded at rock
outcrop, apply as outcrop motion (program will
remove free surface effect). Bedrock should be
modeled as an elastic half-space. If recorded in
boring, apply as within-profile motion (recording
does not include free surface effect). Bedrock
should be modeled as rigid.
Object motion
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Methods of One-Dimensional Site Response Analysis
Complex Response Method Approach used in computer
programs like SHAKE Transfer function is used
with input motion to compute surface motion
(convolution) For layered profiles, transfer
function is built layer-by-layer to go from
input motion to surface motion
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Complex Response Method (Linear analysis)
Consider the soil deposit shown to the right.
Within a given layer, say Layer j, the horizontal
displacements will be given by
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Complex Response Method (Linear analysis)
Defining aj as the complex impedance ratio at
the boundary between layers j and j1, the wave
amplitudes for layer j1 can be obtained from the
amplitudes of layer j by solving the previous two
equations simultaneously
Propagation of wave energy from one layer to
another is controlled by (complex) impedance ratio
Wave amplitudes in Layer j
Wave amplitudes in Layer j1
So, if we can go from Layer j to Layer j1, we
can go from j1 to j2, etc. This means we can
apply this relationship recursively and express
the amplitudes in any layer as functions of the
amplitudes in any other layer. We can therefore
build a transfer function by repeated
application of the above equations.
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Complex Response Method (Linear analysis)
Single layer on rigid base H 100 ft Vs 500
ft/sec x 10
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Complex Response Method (Linear analysis)
Single layer on rigid base H 50 ft Vs 1,500
ft/sec x 10
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Complex Response Method (Linear analysis)
Single layer on rigid base H 100 ft Vs 300
ft/sec x 5
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Complex Response Method (Linear analysis)
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Complex Response Method (Linear analysis)
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Complex Response Method (Linear analysis)
Different sequence of soil layers Different
transfer function Different response
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Complex Response Method (Linear analysis)
Another sequence of soil layers Different
transfer function Different response
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Complex Response Method (Linear analysis)
Complex response method operates in frequency
domain Input motion represented as sum of series
of sine waves Solution for each sine wave
obtained Solutions added together to get total
response
Can we capture important effects of nonlinearity
with linear model?
22
Equivalent Linear Approach
Soils exhibit nonlinear, inelastic behavior under
cyclic loading conditions Stiffness decreases
and damping increases as cyclic strain amplitude
increases The nonlinear, inelastic stress-strain
behavior of cyclically loaded soils can be
approximated by equivalent linear properties.
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Equivalent Linear Approach
Soils exhibit nonlinear, inelastic behavior under
cyclic loading conditions Stiffness decreases
and damping increases as cyclic strain amplitude
increases The nonlinear, inelastic stress-strain
behavior of cyclically loaded soils can be
approximated by equivalent linear properties.
x
g(1)
g(1)
Assume some initial strain and use to estimate G
and x
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Equivalent Linear Approach
Soils exhibit nonlinear, inelastic behavior under
cyclic loading conditions Stiffness decreases
and damping increases as cyclic strain amplitude
increases The nonlinear, inelastic stress-strain
behavior of cyclically loaded soils can be
approximated by equivalent linear properties.
x
g(1)
g(1)
Use these values to compute response
25
Equivalent Linear Approach
Soils exhibit nonlinear, inelastic behavior under
cyclic loading conditions Stiffness decreases
and damping increases as cyclic strain amplitude
increases The nonlinear, inelastic stress-strain
behavior of cyclically loaded soils can be
approximated by equivalent linear properties.
gmax
geff
x
g(1)
g(1)
Determine peak strain and effective strain geff
Rg gmax
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Equivalent Linear Approach
Soils exhibit nonlinear, inelastic behavior under
cyclic loading conditions Stiffness decreases
and damping increases as cyclic strain amplitude
increases The nonlinear, inelastic stress-strain
behavior of cyclically loaded soils can be
approximated by equivalent linear properties.
x
g(1)
g(1)
g(2)
g(2)
Select properties based on updated strain level
27
Equivalent Linear Approach
Soils exhibit nonlinear, inelastic behavior under
cyclic loading conditions Stiffness decreases
and damping increases as cyclic strain amplitude
increases The nonlinear, inelastic stress-strain
behavior of cyclically loaded soils can be
approximated by equivalent linear properties.
x
g(1)
g(3)
g(1)
g(2)
g(2)
g(3)
Compute response with new properties and
determine resulting effective shear strain
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Equivalent Linear Approach
Soils exhibit nonlinear, inelastic behavior under
cyclic loading conditions Stiffness decreases
and damping increases as cyclic strain amplitude
increases The nonlinear, inelastic stress-strain
behavior of cyclically loaded soils can be
approximated by equivalent linear properties.
x
geff
geff
Repeat until computed effective strains are
consistent with assumed effective strains
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Equivalent Linear Approach
Advantages Can work in frequency domain Compute
transfer function at relatively small number of
frequencies (compared to doing calculations at
all time steps) Increased speed not that
significant for 1-D analyses Increased speed can
be significant for 2-D, 3-D analyses Equivalent
linear properties readily available for many
soils familiarity breeds comfort/confidence Can
make first-order approximation to effects of
nonlinearity and inelasticity within framework of
a linear model
The equivalent linear approach is an
approximation. Nonlinear analyses are capable of
representing the actual behavior of soils much
more accurately.
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Nonlinear Analysis
Equation of motion must be integrated in time
domain
Wave equation for visco-elastic medium
Divide time into series of time steps
t
Divide profile into series of layers
z
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Nonlinear Analysis
Equation of motion must be integrated in time
domain
Wave equation for visco-elastic medium
tj
Divide time into series of time steps
t
zi
Divide profile into series of layers
vij v (z zi, t tj)
z
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Nonlinear Analysis
Equation of motion must be integrated in time
domain
Wave equation for visco-elastic medium
tj
t
More steps, but basic process involves using wave
equation to predict conditions at time j1 from
conditions at time j for all layers in profile.
zi
z
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Nonlinear Analysis
Equation of motion must be integrated in time
domain
Wave equation for visco-elastic medium
tj
t
More steps, but basic process involves using wave
equation to predict conditions at time j1 from
conditions at time j for all layers in profile.
zi
Can change material properties for use in next
time step.
Changing stiffness based on strain level, strain
history, etc. can allow prediction of nonlinear,
inelastic response.
z
34
Nonlinear Analysis
Equation of motion must be integrated in time
domain
Wave equation for visco-elastic medium
tj
t
More steps, but basic process involves using wave
equation to predict conditions at time j1 from
conditions at time j for all layers in profile.
zi
Can change material properties for use in next
time step.
Changing stiffness based on strain level, strain
history, etc. can allow prediction of nonlinear,
inelastic response.
z
35
Nonlinear Analysis
Equation of motion must be integrated in time
domain
Wave equation for visco-elastic medium
tj
t
More steps, but basic process involves using wave
equation to predict conditions at time j1 from
conditions at time j for all layers in profile.
zi
Can change material properties for use in next
time step.
Changing stiffness based on strain level, strain
history, etc. can allow prediction of nonlinear,
inelastic response.
z
36
Nonlinear Analysis
Equation of motion must be integrated in time
domain
Wave equation for visco-elastic medium
tj
t
More steps, but basic process involves using wave
equation to predict conditions at time j1 from
conditions at time j for all layers in profile.
zi
Can change material properties for use in next
time step.
Changing stiffness based on strain level, strain
history, etc. can allow prediction of nonlinear,
inelastic response.
z
37
Nonlinear Analysis
Equation of motion must be integrated in time
domain
Wave equation for visco-elastic medium
tj
t
More steps, but basic process involves using wave
equation to predict conditions at time j1 from
conditions at time j for all layers in profile.
zi
Step through time
Can change material properties for use in next
time step.
Changing stiffness based on strain level, strain
history, etc. can allow prediction of nonlinear,
inelastic response.
Procedure steps through time from beginning of
earthquake to end.
z
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Nonlinear Behavior
Actual
Approximation
t
t
g
g
In a nonlinear analysis, we approximate the
continuous actual stress-strain behavior with an
incrementally-linear model. The finer our
computational interval, the better the
approximation.
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Nonlinear Approach
Advantages Work in time domain Can change
properties after each time step to model
nonlinearity Can formulate model in terms of
effective stresses Can compute pore pressure
generation Can compute pore pressure
redistribution, dissipation Avoids spurious
resonances (associated with linearity of EL
approach) Can compute permanent strain
permanent deformations
Nonlinear analyses can produce results that are
consistent with equivalent linear analyses when
strains are small to moderate, and more accurate
results when strains are large. They can also do
important things that equivalent linear analyses
cant, such as compute pore pressures and
permanent deformations.
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Equivalent Linear vs. Nonlinear Approaches
What are people using in practice?
Equivalent linear analyses One-dimensional
2-D / 3-D
SHAKE QUAD4, FLUSH
Nonlinear analyses One-dimensional 2-D / 3-D

DESRA, DMOD TARA, FLAC, PLAXIS
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Equivalent Linear vs. Nonlinear Approaches
What are people using in practice?
Equivalent linear analyses One-dimensional
2-D / 3-D
SHAKE QUAD4, FLUSH
Nonlinear analyses One-dimensional 2-D / 3-D

DESRA TARA
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Available Codes
Since early 1970s, numerous computer programs
developed for site response analysis Can be
categorized according to computational procedure,
number of dimensions, and operating system
Dimensions OS Equivalent Linear Nonlinear
1-D DOS Dyneq, Shake91 AMPLE, DESRA, DMOD, FLIP, SUMDES, TESS
1-D Windows ShakeEdit, ProShake, Shake2000, EERA CyberQuake, DeepSoil, NERA, FLAC, DMOD2000
2-D / 3-D DOS FLUSH, QUAD4/QUAD4M, TLUSH DYNAFLOW, TARA-3, FLIP, VERSAT, DYSAC2, LIQCA, OpenSees
2-D / 3-D Windows QUAKE/W, SASSI2000 FLAC, PLAXIS
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Current Practice
Informal survey developed to obtain input on site
response modeling approaches actually used in
practice
Emailed to 204 people Attendees at ICSDEE/ICEGE
Berkeley conference (non-academic) Geotechnical
EERI members 2003 Roster (non-academic)
Survey Respondents WNA WNA ENA ENA Overseas Overseas
Survey Respondents Private Public Private Public Private Public
Number of responses 35 3 6 1 5 5
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Current Practice
Method of Analysis
Of the total number of site response analyses
you perform, indicate the approximate
percentages that fall within each of the
following categories a. One-dimensional
equivalent linear b. One-dimensional
nonlinear c. Two- or three-dimensional
equivalent linear d. Two- or
three-dimensional nonlinear
Method of Analysis WNA WNA ENA ENA Overseas Overseas
Method of Analysis Private (35) Public (3) Private (6) Public (1) Private (5) Public (5)
1-D Equivalent Linear 68 52 86 50 24 5
1-D Nonlinear 11 17 12 0 48 5
2-D/3-D Equiv. Linear 9 28 1 25 6 0
2-D/3-D Nonlinear 12 3 1 25 23 90
One-dimensional equivalent linear analyses
dominate North American practice nonlinear
analyses are more frequently performed overseas
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Nonlinear Behavior
Equivalent linear vs nonlinear analysis how
much difference does it make?
30 m
Vs 300 m/sec
Vs 762 m/sec
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Nonlinear Behavior
Equivalent linear vs nonlinear analysis how
much difference does it make?
Topanga motion scaled to 0.05 g
Weak motion stiff soil
Low strains
Low degree of nonlinearity
Similar response
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Nonlinear Behavior
Equivalent linear vs nonlinear analysis how
much difference does it make?
Topanga motion scaled to 0.05 g
Weak motion stiff soil
Low strains
Low degree of nonlinearity
Similar response
48
Nonlinear Behavior
Equivalent linear vs nonlinear analysis how
much difference does it make?
Topanga motion scaled to 0.05 g
Weak motion stiff soil
Low strains
Low degree of nonlinearity
Similar response
49
Nonlinear Behavior
Equivalent linear vs nonlinear analysis how
much difference does it make?
Topanga motion scaled to 0.05 g
Weak motion stiff soil
Low strains
Low degree of nonlinearity
Similar response
50
Nonlinear Behavior
Equivalent linear vs nonlinear analysis how
much difference does it make?
Topanga motion scaled to 0.20 g
Moderate motion stiff soil
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Nonlinear Behavior
Equivalent linear vs nonlinear analysis how
much difference does it make?
Acceleration
Topanga motion scaled to 0.20 g
Moderate motion stiff soil
Relatively low strains
Relatively low degree of nonlinearity
Similar response
52
Nonlinear Behavior
Equivalent linear vs nonlinear analysis how
much difference does it make?
Topanga motion scaled to 0.20 g
Moderate motion stiff soil
Relatively low strains
Relatively low degree of nonlinearity
Similar response
53
Nonlinear Behavior
Equivalent linear vs nonlinear analysis how
much difference does it make?
Topanga motion scaled to 0.20 g
Moderate motion stiff soil
Relatively low strains
Stiffness starting to vary more significantly
over course of ground motion
Relatively low degree of nonlinearity
Similar response
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Nonlinear Behavior
Equivalent linear vs nonlinear analysis how
much difference does it make?
Topanga motion scaled to 0.50 g
Strong motion stiff soil
Moderate strains
Low moderate degree of nonlinearity
Noticeably different response
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Nonlinear Behavior
Equivalent linear vs nonlinear analysis how
much difference does it make?
Topanga motion scaled to 0.50 g
Strong motion stiff soil
Moderate strains
Low moderate degree of nonlinearity
Noticeably different response
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Nonlinear Behavior
Equivalent linear vs nonlinear analysis how
much difference does it make?
Topanga motion scaled to 1.0 g
Very strong motion stiff soil
Moderate strains
Moderate degree of nonlinearity
Noticeably different response
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Nonlinear Behavior
Equivalent linear vs nonlinear analysis how
much difference does it make?
Topanga motion scaled to 0.50 g
Very strong motion stiff soil
Moderate strains
Moderate degree of nonlinearity
Noticeably different response
58
Nonlinear Behavior
Equivalent linear vs nonlinear analysis how
much difference does it make?
Vs 100 m/sec
16 m
Vs 300 m/sec
14 m
Vs 762 m/sec
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Nonlinear Behavior
Equivalent linear vs nonlinear analysis how
much difference does it make?
Large strain levels (6) near bottom of upper
layer
EL model converges to low G and high x
High-frequency components cannot be transmitted
through over-softened EL model
NL model Stiffness stays relatively high except
for a few large-amplitude cycles
60
Nonlinear Behavior
Equivalent linear vs nonlinear analysis how
much difference does it make?
Large strain levels (6) near bottom of upper
layer
More consistency, but NL model can transmit
high-frequency oscillations superimposed on
low-frequency cycles too much?
EL model converges to low G and high x
High-frequency components cannot be transmitted
through over-softened EL model
NL model Stiffness stays relatively high except
for a few large-amplitude cycles
61
Nonlinear Behavior
Equivalent linear vs nonlinear analysis how
much difference does it make?
Large strain levels (6) near bottom of upper
layer
NL model exhibits stiff behavior following
strongest part of record EL maintains low
stiffness, high damping behavior throughout.
EL model converges to low G and high x
High-frequency components cannot be transmitted
through over-softened EL model
NL model Stiffness stays relatively high except
for a few large-amplitude cycles
62
Nonlinear Behavior
Equivalent linear vs nonlinear analysis how
much difference does it make?
Large strain levels (6) near bottom of upper
layer
EL model converges to low G and high x
High-frequency components cannot be transmitted
through over-softened EL model
NL model Stiffness stays relatively high except
for a few large-amplitude cycles
63
Nonlinear Soil Behavior
Small cycle superimposed on large cycle (after
Assimaki and Kausel, 2002)
High stiffness
Equivalent linear model maintains constant
stiffness and damping higher stiffness
excursions associated with higher frequency
oscillations arent seen.
Low stiffness
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Nonlinear Soil Behavior
Small cycle superimposed on large cycle (after
Assimaki and Kausel, 2002)
Low damping
Equivalent linear model maintains constant
stiffness and damping higher stiffness
excursions associated with higher frequency
oscillations arent seen.
High damping
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Modified Equivalent Linear Approach
High frequencies are associated with smaller
strains High stiffness and low damping are
associated with smaller strains Make stiffness
and damping frequency-dependent
Normalized strain spectra from five motions
Normalized strain spectrum from one motion
Frequency (Hz)
Frequency (Hz)
66
Modified Equivalent Linear Approach
Assimaki and Kausel
Frequency-dependent model
Conventional model
High frequencies oversoftened and overdamped
Excellent agreement with nonlinear model
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Benchmarking of Nonlinear Analyses
Stewart and Kwok PEER study to determine proper
manner in which to use nonlinear analyses
Worked with five existing nonlinear codes hired
developers to run their codes and comment on
results Established advisory committee to
oversee analyses and assist with
interpretation Met regularly with advisory
committee and developers
68
Benchmarking of Nonlinear Analyses
Stewart and Kwok Considered codes
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Benchmarking of Nonlinear Analyses
D-MOD_2 (Matasovic) Enhanced version of D-MOD,
which is enhanced version of DESRA Lumped mass
model Rayleigh damping
Rayleigh
Damping ratio
Stiffness-proportional
Mass-proportional
Frequency
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Benchmarking of Nonlinear Analyses
D-MOD_2 (Matasovic) Enhanced version of D-MOD,
which is enhanced version of DESRA Lumped mass
model Rayleigh damping Newmark b method for time
integration Variable slice width simulating
response of dams, embankments on rock
Decreasing stiffness due to geometry
71
Benchmarking of Nonlinear Analyses
D-MOD_2 (Matasovic) Enhanced version of D-MOD,
which is enhanced version of DESRA Lumped mass
model Rayleigh damping Newmark b method for time
integration Variable slice width simulating
response of dams, embankments on rock Can
simulate slip on weak interfaces Uses MKZ soil
model (modified hyperbola needs Gmax, tmax, a
and s) Can soften backbone curve to model cyclic
degradation
72
Benchmarking of Nonlinear Analyses
D-MOD_2 (Matasovic) Enhanced version of D-MOD,
which is enhanced version of DESRA Lumped mass
model Rayleigh damping Newmark b method for time
integration Variable slice width simulating
response of dams, embankments on rock Can
simulate slip on weak interfaces Uses MKZ soil
model (modified hyperbola needs Gmax, tmax, a
and s) Can soften backbone curve to model cyclic
degradation Uses Masing rules for
unloading-reloading behavior
Need input parameters for MKZ backbone curve
(4) Cyclic degradation (3 for clay, 4 for
sand) Pore pressure generation (4 for clay, 4 for
sand) Pore pressure redistribution/dissipation
(at least 2) Rayleigh damping coefficients
(2) Basic layer properties (density, shear wave
velocity, half-space properties)
73
Benchmarking of Nonlinear Analyses
DEEPSOIL (Hashash) Similar to DMOD-2 (lumped
mass, derives from DESRA-2) More advanced
Rayleigh damping scheme (lower frequency
dependence) TESS (Pyke) Finite difference wave
propagation analysis (not lumped
mass) Cundall-Pyke hypothesis for
loading-unloading behavior Similar backbone curve
to DMOD-2 and DEEPSOIL Inviscid (sort of)
low-strain damping scheme OpenSees (Yang,
Elgamal) Finite element model (1D, 2D, 3D
capabilities) Multi-surface plasticity model (von
Mises yield surface, kinematic hardening,
non-associative flow rule) Full Rayleigh
damping SUMDES Finite element model Bounding
surface plasticity model (Lade-like yield
surface, kinematic hardening, non-associative
flow rule) Simplified Rayleigh damping
74
Benchmarking of Nonlinear Analyses
Recommendations Specification of control
motion For outcropping motion, use recorded
motion with elastic base For motions recorded at
depth, use recorded motion with rigid
base Specification of viscous damping Use full
or extended Rayleigh damping iterate on
selection of control frequencies to match
equivalent linear response for low loading levels
(linear response domain). If not possible, use
full Rayleigh damping with targets at fo and
5fo. Backbone curve parameters Adjust, if
possible, to produce correct shear strength at
large strains Bound nonlinear, inelastic behavior
by running analyses with Backbone curve fit to
match G/Gmax behavior Backbone curve fit to
minimize error in G/Gmax and damping curves
75
Benchmarking of Nonlinear Analyses

• Performance
• Based on validations against vertical array data
• Models produce reasonable results
• Some indication of overdamping at high
  frequencies, overamplification at site frequency
• Variability of predictions due to backbone curves
  and damping models most pronounced at Tlt0.5 sec
  and is significant only for relatively thick
  profiles. Model-to-model variability most
  pronounced at low periods.
• Nonlinearity modeled well up to levels for which
  adequate data is available (generally up to about
  0.2g). Data for stronger shaking being sought
  (centrifuge tests, recent Nigaata earthquake).
• DMOD-2, DEEPSOIL, and OpenSees generally produced
  similar amplification factors and spectral
  shapes TESS produced different response at high
  frequencies (different damping formulation),
  SUMDES results were significantly different than
  all others for deep sites (probably due to
  simplified Rayleigh damping).

76
Nonlinear Behavior Effective Stress Analyses
Wildlife Superstition Hills recordings
77
Nonlinear Behavior Effective Stress Analyses
Wildlife Superstition Hills recordings
78
Nonlinear Behavior Effective Stress Analyses
Wildlife Elmore Ranch recordings
79
Nonlinear Behavior Effective Stress Analyses
Wildlife Superstition Hills recordings
Ground surface record
Low frequency
???
80
Site Effects
Elmore Ranch record no liquefaction
Ratio of wavelet amplitudes variation with
frequency and time
Frequency (Hz)
Time (sec)
81
Site Effects
Elmore Ranch record no liquefaction
Ratio of wavelet amplitudes variation with
frequency and time
Frequency (Hz)
Time (sec)
82
Nonlinear Behavior Effective Stress Analyses
Wildlife Superstition Hills recordings
83
Nonlinear Behavior Effective Stress Analyses
Wildlife Superstition Hills recordings
84
One-Dimensional Site Response Analysis
Summary Must be aware of assumptions Uni-direction
al wave propagation (normal to layer
boundaries) Uni-directional particle motion (no
surface waves) Particularly useful for profiles
with high impedance contrasts Equivalent linear
approach works very well for most cases Material
properties readily available Computations
performed rapidly Nonlinear analyses match
equivalent linear when strains are
small Nonlinear analyses are preferred when
strains are high soft soils and/or strong
shaking Can account for shear strength of
soil Can handle pore pressure generation some
well, some poorly Can predict permanent
deformations for common for 2-D analyses
85
Thank you