One-Dimensional Site Response Analysis What do we mean? One-dimensional = Waves propagate in one...
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Transcript of One-Dimensional Site Response Analysis What do we mean? One-dimensional = Waves propagate in one...
One-Dimensional Site Response AnalysisOne-Dimensional Site Response Analysis
What do we mean?
One-dimensional = Waves propagate in one direction only
One-Dimensional Site Response AnalysisOne-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 infinityto infinity
One-Dimensional Site Response AnalysisOne-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
Can’t handle refraction so layer boundaries must be perpendicular to direction of wave propagation
Usual assumption is vertically-propagating shear (SH) waves
Horizontal input motion
Horizontal surface motion
One-Dimensional Site Response AnalysisOne-Dimensional Site Response Analysis
When are one-dimensional analyses appropriate?
Stifferwith
depth
Focus
One-Dimensional Site Response AnalysisOne-Dimensional Site Response Analysis
When are one-dimensional analyses appropriate?
Stifferwith
depth
Horizontal boundaries – waves tend to be refracted
toward vertical
Decreasing stiffness causes refraction of waves
to increasingly vertical path
Focus
One-Dimensional Site Response AnalysisOne-Dimensional Site Response Analysis
When are one-dimensional analyses appropriate?
Stifferwith
depth
Not appropriate
here
Retaining structuresRetaining structures
Dams andembankments
Dams andembankments
TunnelsTunnels
One-Dimensional Site Response AnalysisOne-Dimensional Site Response Analysis
When are one-dimensional analyses appropriate?
Inclined ground surface and/or non-horizontal boundaries can require use
of two-dimensional analyses
Inclined ground surface and/or non-horizontal boundaries can require use
of two-dimensional analyses
Not here!Not here!
Complex soilconditions
Complex soilconditions
Dams innarrow
canyons
Dams innarrow
canyons
Multiple structures
Multiple structures
One-Dimensional Site Response AnalysisOne-Dimensional Site Response Analysis
When are one-dimensional analyses appropriate?
Localized structures may require use of 3-D response analyses
Localized structures may require use of 3-D response analyses
Not here!Not here!
One-Dimensional Site Response AnalysisOne-Dimensional Site Response Analysis
How should ground motions be applied?
Incoming motion
ui
Rock outcropping
motion
2ui
Bedrock motion
ui + ur
Free surface motion
us
Not the same!Not the same!
Soil
Rock
One-Dimensional Site Response AnalysisOne-Dimensional Site Response Analysis
How should ground motions be applied?
Object motion
Free surface motion
us
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.
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
Amplification
De-amplification
Methods of One-Dimensional Site Response AnalysisMethods of One-Dimensional Site Response Analysis
Single elastic layer
Layer j+1
Layer j
z G
G
G
G
G
G
1
N +1
N
j+1
j
2
j+1
1
2
j
N
1
2
j
j+1
N
N +1
1
2
j
j+1
N
N +1
1
2
j
j+1
N
N +1z
h
h
h
h
h
z
z
z
zConsider the soil deposit shown to the right. Within a given layer, say Layer j, the horizontal displacements will be given by
j j jik z
jik z i tu z t A e B e ej j j j,
* *
At the boundary between layer j and layer j+1, compatibility of displacements requires that
j j jik h
jik hA B A e B ej j j j
1 1* *
Continuity of shear stresses requires that
j jj j
j j
ik hj j
ik hA BG k
G kA e B es j s j
1 1
1 1
* *
* *
* *
Complex Response Method (Linear analysis)Complex Response Method (Linear analysis)
Amplitudes of upward- and downward-traveling waves in Layer j
Equilibrium satisfied
No slip
Defining *j as the complex impedance ratio at the boundary between layers
j and j+1, the wave amplitudes for layer j+1 can be obtained from the amplitudes of layer j by solving the previous two equations simultaneously
j j jik h
j jik hA A e B ej j j j
11
21
1
21* ** *
j j jik h
j jik hB A e B ej j j j
11
21
1
21* ** *
Wave amplitudes in Layer j
Wave amplitudes in Layer j+1
So, if we can go from Layer j to Layer j+1, we can go from j+1 to j+2, 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.
Complex Response Method (Linear analysis)Complex Response Method (Linear analysis)
Propagation of wave energy from one layer to another is controlled by
(complex) impedance ratio
Complex Response Method (Linear analysis)Complex Response Method (Linear analysis)
Single layer on rigid base
H = 100 ft
Vs = 500 ft/sec
= 10%
Single layer on rigid base
H = 100 ft
Vs = 500 ft/sec
= 10%
Complex Response Method (Linear analysis)Complex Response Method (Linear analysis)
Single layer on rigid base
H = 50 ft
Vs = 1,500 ft/sec
= 10%
Single layer on rigid base
H = 50 ft
Vs = 1,500 ft/sec
= 10%
Complex Response Method (Linear analysis)Complex Response Method (Linear analysis)
Single layer on rigid base
H = 100 ft
Vs = 300 ft/sec
= 5%
Single layer on rigid base
H = 100 ft
Vs = 300 ft/sec
= 5%
Complex Response Method (Linear analysis)Complex Response Method (Linear analysis)
Complex Response Method (Linear analysis)Complex Response Method (Linear analysis)
Complex Response Method (Linear analysis)Complex Response Method (Linear analysis)
Different sequence of soil layers
Different transfer function
Different response
Complex Response Method (Linear analysis)Complex Response Method (Linear analysis)
Another sequence of soil layers
Different transfer function
Different response
Complex Response Method (Linear analysis)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
Principle of superposition
Linear system
Can we capture important effects of nonlinearity with linear model?
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.
)log( eff)log( eff
Equivalent shear modulusEquivalent shear modulus Equivalent damping ratioEquivalent damping ratio
max/GG
Equivalent Linear ApproachEquivalent Linear Approach
)log( eff)log( eff
max/GG
Assume some initial strain and use to estimate G and Assume some initial strain and use to estimate G and
(1)(1)
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.
Equivalent Linear ApproachEquivalent Linear Approach
)log( eff)log( eff
max/GG
(1)(1)
Use these values to compute responseUse these values to compute response
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.
Equivalent Linear ApproachEquivalent Linear Approach
(t)
t
)log( eff)log( eff
max/GG
(1)(1)
Determine peak strain and effective straineff = R max
Determine peak strain and effective straineff = R max
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.
Equivalent Linear ApproachEquivalent Linear Approach
(t)
t
max
eff
)log( eff)log( eff
max/GG
(1)(1)(2) (2)
Select properties based on updated strain levelSelect properties based on updated strain level
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.
Equivalent Linear ApproachEquivalent Linear Approach
)log( eff)log( eff
max/GG
(1)(1)(2) (2)(3)
(3)
Compute response with new properties and determine resulting effective shear strain
Compute response with new properties and determine resulting effective shear strain
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.
Equivalent Linear ApproachEquivalent Linear Approach
)log( eff)log( eff
max/GG
Repeat until computed effective strains are consistent with assumed effective strains
Repeat until computed effective strains are consistent with assumed effective strains
effeff
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.
Equivalent Linear ApproachEquivalent 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
Equivalent Linear ApproachEquivalent Linear Approach
The equivalent linear approach is an approximation. Nonlinear analyses are capable of representing the actual behavior of soils much more accurately.
The equivalent linear approach is an approximation. Nonlinear analyses are capable of representing the actual behavior of soils much more accurately.
… often, a very good one!
Nonlinear AnalysisNonlinear Analysis
2 3
2 2u u
z t z t
Equation of motion must be integrated in time domain
Wave equation for visco-elastic medium
z
Divide profile into series of
layers
Divide time into series of time steps t
Nonlinear AnalysisNonlinear Analysis
2 3
2 2u u
z t z t
Equation of motion must be integrated in time domain
Wave equation for visco-elastic medium
z
Divide profile into series of
layers
Divide time into series of time steps t
vij = v (z = zi, t = tj)
tj
zi
Nonlinear AnalysisNonlinear Analysis
2 3
2 2u u
z t z t
Equation of motion must be integrated in time domain
Wave equation for visco-elastic medium
z
ttj
zi
, 1/ 2 , ,1
2i j i j i jv v a t
, 1 , , 1/ 21
2i j i j i ju u v t
, 1 , 1/ 2 , 11
2i j i j i jv v a t
More steps, but basic process involves using wave equation to predict conditions at time j+1 from conditions at time j for all layers in profile.
Nonlinear AnalysisNonlinear Analysis
2 3
2 2u u
z t z t
Equation of motion must be integrated in time domain
Wave equation for visco-elastic medium
z
ttj
zi
More steps, but basic process involves using wave equation to predict conditions at time j+1 from conditions at time j for all layers in profile.
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.
Nonlinear AnalysisNonlinear Analysis
2 3
2 2u u
z t z t
Equation of motion must be integrated in time domain
Wave equation for visco-elastic medium
z
ttj
zi
More steps, but basic process involves using wave equation to predict conditions at time j+1 from conditions at time j for all layers in profile.
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.
Nonlinear AnalysisNonlinear Analysis
2 3
2 2u u
z t z t
Equation of motion must be integrated in time domain
Wave equation for visco-elastic medium
z
ttj
zi
More steps, but basic process involves using wave equation to predict conditions at time j+1 from conditions at time j for all layers in profile.
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.
Nonlinear AnalysisNonlinear Analysis
2 3
2 2u u
z t z t
Equation of motion must be integrated in time domain
Wave equation for visco-elastic medium
z
ttj
zi
More steps, but basic process involves using wave equation to predict conditions at time j+1 from conditions at time j for all layers in profile.
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.
Nonlinear AnalysisNonlinear Analysis
2 3
2 2u u
z t z t
Equation of motion must be integrated in time domain
Wave equation for visco-elastic medium
z
ttj
zi
More steps, but basic process involves using wave equation to predict conditions at time j+1 from conditions at time j for all layers in profile.
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.
Step through time
Nonlinear BehaviorNonlinear Behavior
Continuous Linear segments
Actual Approximation
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.
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.
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 ApproachNonlinear Approach
Liquefaction
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 can’t, such as compute pore pressures and 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 can’t, such as compute pore pressures and permanent deformations.
What are people using in practice?
Equivalent Linear vs. Nonlinear ApproachesEquivalent Linear vs. Nonlinear Approaches
Equivalent linear analyses
One-dimensional –
2-D / 3-D –
Nonlinear analyses
One-dimensional –
2-D / 3-D –
SHAKE
QUAD4, FLUSH
DESRA, DMOD
TARA, FLAC, PLAXIS
What are people using in practice?
Equivalent Linear vs. Nonlinear ApproachesEquivalent Linear vs. Nonlinear Approaches
Equivalent linear analyses
One-dimensional –
2-D / 3-D –
Nonlinear analyses
One-dimensional –
2-D / 3-D –
SHAKE
QUAD4, FLUSH
DESRA
TARA
Dimensions OS Equivalent Linear Nonlinear
1-DDOS Dyneq, Shake91 AMPLE, DESRA, DMOD,
FLIP, SUMDES, TESS
Windows ShakeEdit, ProShake, Shake2000, EERA
CyberQuake, DeepSoil, NERA, FLAC, DMOD2000
2-D / 3-DDOS
FLUSH, QUAD4/QUAD4M, TLUSH
DYNAFLOW, TARA-3, FLIP, VERSAT, DYSAC2, LIQCA, OpenSees
Windows QUAKE/W, SASSI2000 FLAC, PLAXIS
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
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)
SurveyRespondents
WNA ENA Overseas
Private Public Private Public Private Public
Number of responses 35 3 6 1 5 5
55 responses
Western North America (WNA)
Eastern North America (ENA)
Overseas
Private firms
Public agencies
Current Practice
Method of Analysis
Method of Analysis
WNA ENA Overseas
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
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
One-dimensional equivalent linear analyses dominate North American practice; nonlinear analyses are more frequently performed overseas
One-dimensional equivalent linear analyses dominate North American practice; nonlinear analyses are more frequently performed overseas
Nonlinear BehaviorNonlinear Behavior
Equivalent linear vs nonlinear analysis – how much difference does it make?
30 m
u(H,t)
u(0,t)
Vs = 300 m/sec
Vs = 762 m/sec
1 m
15 m
29 m
Topanga record (Northridge)
Topanga record
(Northridge)
Ts = 0.4 sec
Nonlinear BehaviorNonlinear 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
Topanga motion scaled to 0.05 g
Weak motion+
stiff soil
Low strains
Low degree of nonlinearity
Similar response
Nonlinear BehaviorNonlinear 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
Nonlinear BehaviorNonlinear 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
Nonlinear BehaviorNonlinear 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
Nonlinear BehaviorNonlinear 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
Nonlinear BehaviorNonlinear Behavior
Equivalent linear vs nonlinear analysis – how much difference does it make?Acceleration
Velocity
Topanga motion scaled to 0.20 g
Moderate motion+
stiff soil
Relatively low strains
Relatively low degree of nonlinearity
Similar response
Nonlinear BehaviorNonlinear Behavior
Equivalent linear vs nonlinear analysis – how much difference does it make?
Equivalent linear overpredicts nonlinear response at certain frequencies – “spurious resonances”
Stress-strain response becoming more complicated
– more variable stiffness and less “elliptical” shape
Topanga motion scaled to 0.20 g
Moderate motion+
stiff soil
Relatively low strains
Relatively low degree of nonlinearity
Similar response
Nonlinear BehaviorNonlinear Behavior
Equivalent linear vs nonlinear analysis – how much difference does it make?
Stiffness starting to vary more significantly over
course of ground motion
Topanga motion scaled to 0.50 g
Strong motion+
stiff soil
Moderate strains
Low – moderate degree of nonlinearity
Noticeably different response
Nonlinear BehaviorNonlinear Behavior
Equivalent linear vs nonlinear analysis – how much difference does it make?
Acceleration
Topanga motion scaled to 0.50 g
Strong motion+
stiff soil
Moderate strains
Low – moderate degree of nonlinearity
Noticeably different response
Nonlinear BehaviorNonlinear 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
Nonlinear BehaviorNonlinear Behavior
Equivalent linear vs nonlinear analysis – how much difference does it make?Acceleration
Substantial softening by EL method causes
underprediction of initial portion of record
Linearity inherent in EL method causes overprediction response in
strongest portion of record
Softening by EL method causes underprediction
Topanga motion scaled to 0.50 g
Very strong motion+
stiff soil
Moderate strains
Moderate degree of nonlinearity
Noticeably different response
Nonlinear BehaviorNonlinear Behavior
Equivalent linear vs nonlinear analysis – how much difference does it make?
Nonlinear BehaviorNonlinear Behavior
Equivalent linear vs nonlinear analysis – how much difference does it make?
14 m Vs = 300 m/sec
Vs = 762 m/sec
16 m Vs = 100 m/sec
u(H,t)
u(0,t)
1 m
15 m
29 m
Nonlinear BehaviorNonlinear 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
High-frequency components cannot be transmitted through
over-softened EL model
NL model: Stiffness stays relatively high except for a few large-amplitude cycles
Acceleration
EL model predicts very soft behavior at beginning of earthquake, before any large strains have developed.
Nonlinear BehaviorNonlinear 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
High-frequency components cannot be transmitted through
over-softened EL model
NL model: Stiffness stays relatively high except for a few large-amplitude cycles
Acceleration
More consistency, but NL model can transmit high-frequency oscillations superimposed on low-frequency cycles – too much?
Nonlinear BehaviorNonlinear 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
High-frequency components cannot be transmitted through
over-softened EL model
NL model: Stiffness stays relatively high except for a few large-amplitude cycles
Acceleration
NL model exhibits stiff behavior following strongest part of record; EL maintains low stiffness, high damping behavior throughout.
Nonlinear BehaviorNonlinear 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
High-frequency components cannot be transmitted through
over-softened EL model
NL model: Stiffness stays relatively high except for a few large-amplitude cycles
Time
Nonlinear Soil BehaviorNonlinear Soil Behavior
Small cycle superimposed on large cycle (after Assimaki and Kausel, 2002)
Low stiffness
High stiffness
Equivalent linear model maintains constant stiffness and damping – higher stiffness excursions associated with higher frequency oscillations aren’t seen.
Time
Nonlinear Soil BehaviorNonlinear Soil Behavior
Small cycle superimposed on large cycle (after Assimaki and Kausel, 2002)
High damping
Low damping
Equivalent linear model maintains constant stiffness and damping – higher stiffness excursions associated with higher frequency oscillations aren’t seen.
High frequencies are associated with smaller strains
High stiffness and low damping are associated with smaller strains
Make stiffness and damping frequency-dependent
Modified Equivalent Linear ApproachModified Equivalent Linear Approach
Normalized strain spectra from five motions
Normalized strain spectra from five motions
Normalized strain spectrum from one motion
Normalized strain spectrum from one motion
Three orders of magnitude
Frequency (Hz) Frequency (Hz)
Assimaki and Kausel
Modified Equivalent Linear ApproachModified Equivalent Linear Approach
Frequency-dependent modelFrequency-dependent model Conventional modelConventional model
High frequencies oversoftened and
overdamped
Excellent agreement with nonlinear model
Benchmarking of Nonlinear AnalysesBenchmarking 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
Benchmarking of Nonlinear AnalysesBenchmarking of Nonlinear Analyses
Stewart and Kwok
Considered codes
D-MOD_2 (Matasovic)Enhanced version of D-MOD, which is enhanced version of DESRALumped mass model
Rayleigh dampingD
ampi
ng r
atio
Frequency
Mass-proportional
Stiffness-proportional
Rayleigh
Benchmarking of Nonlinear AnalysesBenchmarking of Nonlinear Analyses
D-MOD_2 (Matasovic)Enhanced version of D-MOD, which is enhanced version of DESRALumped mass model
Rayleigh dampingNewmark method for time integration
Variable slice width – simulating response of dams, embankments on rock
Benchmarking of Nonlinear AnalysesBenchmarking of Nonlinear Analyses
Decreasing stiffness due to geometry
D-MOD_2 (Matasovic)Enhanced version of D-MOD, which is enhanced version of DESRALumped mass model
Rayleigh dampingNewmark method for time integration
Variable slice width – simulating response of dams, embankments on rockCan simulate slip on weak interfacesUses MKZ soil model (modified hyperbola – needs Gmax, max, and s)
Can soften backbone curve to model cyclic degradation
Benchmarking of Nonlinear AnalysesBenchmarking of Nonlinear Analyses
D-MOD_2 (Matasovic)Enhanced version of D-MOD, which is enhanced version of DESRALumped mass model
Rayleigh dampingNewmark method for time integration
Variable slice width – simulating response of dams, embankments on rockCan simulate slip on weak interfacesUses MKZ soil model (modified hyperbola – needs Gmax, max, and s)
Can soften backbone curve to model cyclic degradationUses 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)
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)
Benchmarking of Nonlinear AnalysesBenchmarking 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 behaviorSimilar backbone curve to DMOD-2 and DEEPSOILInviscid (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
SUMDESFinite element modelBounding surface plasticity model (Lade-like yield surface, kinematic hardening, non-associative flow rule)Simplified Rayleigh damping
Benchmarking of Nonlinear AnalysesBenchmarking of Nonlinear Analyses
Recommendations
Specification of control motionFor outcropping motion, use recorded motion with elastic baseFor motions recorded at depth, use recorded motion with rigid base
Specification of viscous dampingUse 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 parametersAdjust, if possible, to produce correct shear strength at large strainsBound nonlinear, inelastic behavior by running analyses with:
Backbone curve fit to match G/Gmax behaviorBackbone curve fit to minimize error in G/Gmax and damping curves
Benchmarking of Nonlinear AnalysesBenchmarking 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 T<0.5 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).
Benchmarking of Nonlinear AnalysesBenchmarking of Nonlinear Analyses
Nonlinear Behavior – Effective Stress AnalysesNonlinear Behavior – Effective Stress Analyses
Wildlife – Superstition Hills recordings
Nonlinear BehaviorNonlinear Behavior – Effective Stress Analyses– Effective Stress Analyses
Wildlife – Superstition Hills recordings
Nonlinear BehaviorNonlinear Behavior – Effective Stress Analyses– Effective Stress Analyses
Wildlife – Elmore Ranch recordings
Nonlinear BehaviorNonlinear Behavior – Effective Stress Analyses– Effective Stress Analyses
Wildlife – Superstition Hills recordings
Low frequency
High frequency
Ground surface record
???
Site EffectsSite Effects
Elmore Ranch record – no liquefaction
Ratio of wavelet amplitudes – variation with frequency and time
Ratio of wavelet amplitudes – variation with frequency and time
Time (sec)
Fre
qu
ency
(H
z)
Site EffectsSite Effects
Elmore Ranch record – no liquefaction
Ratio of wavelet amplitudes – variation with frequency and time
Ratio of wavelet amplitudes – variation with frequency and time
Time (sec)
Fre
qu
ency
(H
z)
Nonlinear BehaviorNonlinear Behavior – Effective Stress Analyses– Effective Stress Analyses
Wildlife – Superstition Hills recordings
Nonlinear BehaviorNonlinear Behavior – Effective Stress Analyses– Effective Stress Analyses
Wildlife – Superstition Hills recordings
One-Dimensional Site Response AnalysisOne-Dimensional Site Response Analysis
Summary
Must be aware of assumptions
Uni-directional 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
Thank youThank you