One-Dimensional Site Response AnalysisWhat do we mean?One-dimensional = Waves propagate in one direction only

One-Dimensional Site Response AnalysisWhat do we mean?One-dimensional = waves propagate in one direction onlyMotion is identical on planes perpendicular to that motionto infinityto infinity

One-Dimensional Site Response AnalysisWhat do we mean?One-dimensional = waves propagate in one direction onlyMotion is identical on planes perpendicular to that motionCant handle refraction so layer boundaries must be perpendicular to direction of wave propagationUsual assumption is vertically-propagating shear (SH) wavesHorizontal surface motion

One-Dimensional Site Response AnalysisWhen are one-dimensional analyses appropriate?StifferwithdepthFocus

One-Dimensional Site Response AnalysisWhen are one-dimensional analyses appropriate?StifferwithdepthHorizontal boundaries waves tend to be refracted toward verticalFocus

One-Dimensional Site Response AnalysisWhen are one-dimensional analyses appropriate?StifferwithdepthNot appropriate here

Retaining structuresTunnelsOne-Dimensional Site Response AnalysisWhen are one-dimensional analyses appropriate?Inclined ground surface and/or non-horizontal boundaries can require use of two-dimensional analysesNot here!

Complex soilconditionsMultiple structuresOne-Dimensional Site Response AnalysisWhen are one-dimensional analyses appropriate?Localized structures may require use of 3-D response analysesNot here!

One-Dimensional Site Response AnalysisHow should ground motions be applied?Not the same!SoilRock

One-Dimensional Site Response AnalysisHow should ground motions be applied?Object motionInput (object) motionIf 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 MethodApproach used in computer programs like SHAKETransfer 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 motionMethods of One-Dimensional Site Response Analysis

Consider the soil deposit shown to the right. Within a given layer, say Layer j, the horizontal displacements will be given byComplex Response Method (Linear analysis)

Defining a*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 simultaneouslyWave amplitudes in Layer jWave amplitudes in Layer j+1So, 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)Propagation of wave energy from one layer to another is controlled by (complex) impedance ratio

Complex Response Method (Linear analysis)Single layer on rigid baseH = 100 ftVs = 500 ft/secx = 10%

Complex Response Method (Linear analysis)Single layer on rigid baseH = 50 ftVs = 1,500 ft/secx = 10%

Complex Response Method (Linear analysis)Single layer on rigid baseH = 100 ftVs = 300 ft/secx = 5%

Complex Response Method (Linear analysis)

Complex Response Method (Linear analysis)

Complex Response Method (Linear analysis)Different sequence of soil layersDifferent transfer functionDifferent response

Complex Response Method (Linear analysis)Another sequence of soil layersDifferent transfer functionDifferent response

Complex Response Method (Linear analysis)Complex response method operates in frequency domainInput motion represented as sum of series of sine wavesSolution for each sine wave obtainedSolutions added together to get total response

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.Equivalent Linear Approach

xAssume some initial strain and use to estimate G and xg(1)g(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 Approach

xg(1)g(1)Use these values to compute responseSoils 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 Approach

xg(1)g(1)Determine peak strain and effective straingeff = Rg gmaxSoils 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 Approachgmaxgeff

xg(1)g(1)g(2)g(2)Select properties based on updated strain levelSoils 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 Approach

xg(1)g(1)g(2)g(2)g(3)g(3)Compute response with new properties and determine resulting effective shear strainSoils 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 Approach

xRepeat until computed effective strains are consistent with assumed effective strainsgeffgeffSoils 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 Approach

Advantages:Can work in frequency domainCompute transfer function at relatively small number of frequencies (compared to doing calculations at all time steps)Increased speed not that significant for 1-D analysesIncreased 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 ApproachThe equivalent linear approach is an approximation. Nonlinear analyses are capable of representing the actual behavior of soils much more accurately.

Nonlinear AnalysisEquation of motion must be integrated in time domainWave equation for visco-elastic mediumzDivide profile into series of layersDivide time into series of time stepst

Nonlinear AnalysisEquation of motion must be integrated in time domainWave equation for visco-elastic mediumzDivide profile into series of layersDivide time into series of time stepstvij = v (z = zi, t = tj)tjzi

Nonlinear AnalysisEquation of motion must be integrated in time domainWave equation for visco-elastic mediumzttjziMore 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 AnalysisEquation of motion must be integrated in time domainWave equation for visco-elastic mediumzttjziMore 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 AnalysisEquation of motion must be integrated in time domainWave equation for visco-elastic mediumzttjziMore 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 AnalysisEquation of motion must be integrated in time domainWave equation for visco-elastic mediumzttjziMore 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 AnalysisEquation of motion must be integrated in time domainWave equation for visco-elastic mediumzttjziMore 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 AnalysisEquation of motion must be integrated in time domainWave equation for visco-elastic mediumzttjziMore 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 BehaviortgtgActualApproximationIn 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 domainCan change properties after each time step to model nonlinearityCan formulate model in terms of effective stressesCan compute pore pressure generationCan compute pore pressure redistribution, dissipation

Avoids spurious resonances (associated with linearity of EL approach)

Can compute permanent strain permanent deformations

Nonlinear ApproachNonlinear 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.

What are people using in practice?Equivalent Linear vs. Nonlinear ApproachesEquivalent linear analysesOne-dimensional 2-D / 3-D Nonlinear analysesOne-dimensional 2-D / 3-D SHAKEQUAD4, FLUSHDESRA, DMODTARA, FLAC, PLAXIS

What are people using in practice?Equivalent Linear vs. Nonlinear ApproachesEquivalent linear analysesOne-dimensional 2-D / 3-D Nonlinear analysesOne-dimensional 2-D / 3-D SHAKEQUAD4, FLUSHDESRATARA

Available CodesSince early 1970s, numerous computer programs developed for site response analysis

Can be categorized according to computational procedure, number of dimensions, and operating system

DimensionsOSEquivalent LinearNonlinear1-DDOSDyneq, Shake91AMPLE, DESRA, DMOD, FLIP, SUMDES, TESSWindowsShakeEdit, ProShake, Shake2000, EERACyberQuake, DeepSoil, NERA, FLAC, DMOD2000

2-D / 3-DDOSFLUSH, QUAD4/QUAD4M, TLUSHDYNAFLOW, TARA-3, FLIP, VERSAT, DYSAC2, LIQCA, OpenSeesWindowsQUAKE/W, SASSI2000FLAC, PLAXIS

Current PracticeInformal survey developed to obtain input on site response modeling approaches actually used in practiceEmailed to 204 peopleAttendees at ICSDEE/ICEGE Berkeley conference (non-academic)Geotechnical EERI members 2003 Roster (non-academic)

SurveyRespondentsWNAENAOverseasPrivatePublicPrivatePublicPrivatePublicNumber of responses3536155

Current PracticeMethod 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 nonlinearOne-dimensional equivalent linear analyses dominate North American practice; nonlinear analyses are more frequently performed overseas

Method of AnalysisWNAENAOverseasPrivate(35)Public(3)Private(6)Public(1)Private(5)Public(5)1-D Equivalent Linear685286502451-D Nonlinear11171204852-D/3-D Equiv. Linear928125602-D/3-D Nonlinear1231252390

Nonlinear BehaviorEquivalent linear vs nonlinear analysis how much difference does it make?30 mVs = 300 m/secVs = 762 m/sec

Nonlinear BehaviorEquivalent linear vs nonlinear analysis how much difference does it make?Topanga motion scaled to 0.05 gWeak motion+stiff soilLow strainsLow degree of nonlinearitySimilar response

Topanga motion scaled to 0.05 gWeak motion+stiff soilLow strainsLow degree of nonlinearitySimilar responseNonlinear BehaviorEquivalent linear vs nonlinear analysis how much difference does it make?

Topanga motion scaled to 0.05 gWeak motion+stiff soilLow strainsLow degree of nonlinearitySimilar responseNonlinear BehaviorEquivalent linear vs nonlinear analysis how much difference does it make?

Topanga motion scaled to 0.05 gWeak motion+stiff soilLow strainsLow degree of nonlinearitySimilar responseNonlinear BehaviorEquivalent linear vs nonlinear analysis how much difference does it make?

Topanga motion scaled to 0.20 gModerate motion+stiff soilNonlinear BehaviorEquivalent linear vs nonlinear analysis how much difference does it make?

Topanga motion scaled to 0.20 gModerate motion+stiff soilRelatively low strainsRelatively low degree of nonlinearitySimilar responseNonlinear BehaviorEquivalent linear vs nonlinear analysis how much difference does it make?Acceleration

Topanga motion scaled to 0.20 gModerate motion+stiff soilRelatively low strainsRelatively low degree of nonlinearitySimilar responseNonlinear BehaviorEquivalent linear vs nonlinear analysis how much difference does it make?

Topanga motion scaled to 0.20 gModerate motion+stiff soilRelatively low strainsRelatively low degree of nonlinearitySimilar responseNonlinear BehaviorEquivalent 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 gStrong motion+stiff soilModerate strainsLow moderate degree of nonlinearityNoticeably different responseNonlinear BehaviorEquivalent linear vs nonlinear analysis how much difference does it make?

Topanga motion scaled to 0.50 gStrong motion+stiff soilModerate strainsLow moderate degree of nonlinearityNoticeably different responseNonlinear BehaviorEquivalent linear vs nonlinear analysis how much difference does it make?

Topanga motion scaled to 1.0 gVery strong motion+stiff soilModerate strainsModerate degree of nonlinearityNoticeably different responseNonlinear BehaviorEquivalent linear vs nonlinear analysis how much difference does it make?

Topanga motion scaled to 0.50 gVery strong motion+stiff soilModerate strainsModerate degree of nonlinearityNoticeably different responseNonlinear BehaviorEquivalent linear vs nonlinear analysis how much difference does it make?

Nonlinear BehaviorEquivalent linear vs nonlinear analysis how much difference does it make?14 mVs = 300 m/secVs = 762 m/sec16 mVs = 100 m/sec

Nonlinear BehaviorEquivalent linear vs nonlinear analysis how much difference does it make?Large strain levels (~6%) near bottom of upper layerEL model converges to low G and high xHigh-frequency components cannot be transmitted through over-softened EL modelNL model: Stiffness stays relatively high except for a few large-amplitude cycles

Nonlinear BehaviorEquivalent linear vs nonlinear analysis how much difference does it make?Large strain levels (~6%) near bottom of upper layerEL model converges to low G and high xHigh-frequency components cannot be transmitted through over-softened EL modelNL model: Stiffness stays relatively high except for a few large-amplitude cyclesMore consistency, but NL model can transmit high-frequency oscillations superimposed on low-frequency cycles too much?

Nonlinear BehaviorEquivalent linear vs nonlinear analysis how much difference does it make?Large strain levels (~6%) near bottom of upper layerEL model converges to low G and high xHigh-frequency components cannot be transmitted through over-softened EL modelNL model: Stiffness stays relatively high except for a few large-amplitude cyclesNL model exhibits stiff behavior following strongest part of record; EL maintains low stiffness, high damping behavior throughout.

Nonlinear BehaviorEquivalent linear vs nonlinear analysis how much difference does it make?Large strain levels (~6%) near bottom of upper layerEL model converges to low G and high xHigh-frequency components cannot be transmitted through over-softened EL modelNL model: Stiffness stays relatively high except for a few large-amplitude cycles

Nonlinear Soil BehaviorSmall cycle superimposed on large cycle (after Assimaki and Kausel, 2002)Low stiffnessHigh stiffnessEquivalent linear model maintains constant stiffness and damping higher stiffness excursions associated with higher frequency oscillations arent seen.

Nonlinear Soil BehaviorSmall cycle superimposed on large cycle (after Assimaki and Kausel, 2002)High dampingLow dampingEquivalent linear model maintains constant stiffness and damping higher stiffness excursions associated with higher frequency oscillations arent seen.

High frequencies are associated with smaller strainsHigh stiffness and low damping are associated with smaller strainsMake stiffness and damping frequency-dependentModified Equivalent Linear ApproachNormalized strain spectra from five motionsNormalized strain spectrum from one motionFrequency (Hz)Frequency (Hz)

Assimaki and Kausel

Modified Equivalent Linear ApproachFrequency-dependent modelConventional modelHigh frequencies oversoftened and overdampedExcellent agreement with nonlinear model

Benchmarking of Nonlinear AnalysesStewart and Kwok

PEER study to determine proper manner in which to use nonlinear analysesWorked 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 AnalysesStewart and Kwok

Considered codes

D-MOD_2 (Matasovic)Enhanced version of D-MOD, which is enhanced version of DESRALumped mass modelRayleigh damping

Damping ratioFrequencyMass-proportionalStiffness-proportionalRayleighBenchmarking of Nonlinear Analyses

D-MOD_2 (Matasovic)Enhanced version of D-MOD, which is enhanced version of DESRALumped mass modelRayleigh dampingNewmark b method for time integrationVariable slice width simulating response of dams, embankments on rock

Benchmarking of Nonlinear AnalysesDecreasing stiffness due to geometry

D-MOD_2 (Matasovic)Enhanced version of D-MOD, which is enhanced version of DESRALumped mass modelRayleigh dampingNewmark b method for time integrationVariable slice width simulating response of dams, embankments on rockCan simulate slip on weak interfacesUses MKZ soil model (modified hyperbola needs Gmax, tmax, a and s)Can soften backbone curve to model cyclic degradation

Benchmarking of Nonlinear Analyses

D-MOD_2 (Matasovic)Enhanced version of D-MOD, which is enhanced version of DESRALumped mass modelRayleigh dampingNewmark b method for time integrationVariable slice width simulating response of dams, embankments on rockCan simulate slip on weak interfacesUses MKZ soil model (modified hyperbola needs Gmax, tmax, a 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)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 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 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 Analyses

Performance

Based on validations against vertical array dataModels produce reasonable resultsSome indication of overdamping at high frequencies, overamplification at site frequencyVariability of predictions due to backbone curves and damping models most pronounced at T

Nonlinear Behavior Effective Stress AnalysesWildlife Superstition Hills recordings

Nonlinear Behavior Effective Stress Analyses

Wildlife Superstition Hills recordings

Nonlinear Behavior Effective Stress AnalysesWildlife Elmore Ranch recordings

Nonlinear Behavior Effective Stress Analyses

Wildlife Superstition Hills recordingsLow frequencyGround surface record???

Site EffectsElmore Ranch record no liquefactionRatio of wavelet amplitudes variation with frequency and timeTime (sec)Frequency (Hz)

Site EffectsElmore Ranch record no liquefactionRatio of wavelet amplitudes variation with frequency and timeTime (sec)Frequency (Hz)

Nonlinear Behavior Effective Stress Analyses

Wildlife Superstition Hills recordings

Nonlinear Behavior Effective Stress Analyses

Wildlife Superstition Hills recordings

One-Dimensional Site Response Analysis SummaryMust be aware of assumptionsUni-directional wave propagation (normal to layer boundaries)Uni-directional particle motion (no surface waves)Particularly useful for profiles with high impedance contrastsEquivalent linear approach works very well for most casesMaterial properties readily availableComputations performed rapidlyNonlinear analyses match equivalent linear when strains are smallNonlinear analyses are preferred when strains are high soft soils and/or strong shakingCan account for shear strength of soilCan handle pore pressure generation some well, some poorlyCan predict permanent deformations for common for 2-D analyses

Thank you

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