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


What do we mean?

One-dimensional = waves propagate in one direction only

Motion is identical on planes perpendicular to that motion

to infinityto infinity


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


When are one-dimensional analyses appropriate?

Stifferwith

depth

Focus



Stifferwith

depth

Horizontal boundaries – waves tend to be refracted

toward vertical

Decreasing stiffness causes refraction of waves

to increasingly vertical path

Focus



Stifferwith

depth

Not appropriate

here

Retaining structuresRetaining structures

Dams andembankments

Dams andembankments

TunnelsTunnels



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



Localized structures may require use of 3-D response analyses

Localized structures may require use of 3-D response analyses

Not here!Not here!


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


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.


Propagation of wave energy from one layer to another is controlled by

(complex) impedance ratio


Single layer on rigid base

H = 100 ft

Vs = 500 ft/sec

= 10%


H = 100 ft

Vs = 500 ft/sec

= 10%



H = 50 ft

Vs = 1,500 ft/sec

= 10%


H = 50 ft

Vs = 1,500 ft/sec

= 10%



H = 100 ft

Vs = 300 ft/sec

= 5%


H = 100 ft

Vs = 300 ft/sec

= 5%


Different sequence of soil layers

Different transfer function

Different response


Another sequence of soil layers

Different transfer function

Different response


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)





)log( eff)log( eff

max/GG

(1)(1)

Use these values to compute responseUse these values to compute response





(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





(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





)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





)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





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.

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


2 3

2 2u u

z t z t



z

Divide profile into series of

layers

Divide time into series of time steps t

vij = v (z = zi, t = tj)

tj

zi


2 3

2 2u u

z t z t



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.


2 3

2 2u u

z t z t



z

ttj

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.


2 3

2 2u u

z t z t



z

ttj

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.

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


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



Topanga motion scaled to 0.05 g

Weak motion+

stiff soil

Low strains

Low degree of nonlinearity

Similar response


Weak motion+

stiff soil

Low strains

Low degree of nonlinearity

Similar response




Moderate motion+

stiff soil

Relatively low strains

Relatively low degree of nonlinearity

Similar response




Moderate motion+

stiff soil



Similar response


Equivalent linear vs nonlinear analysis – how much difference does it make?Acceleration

Velocity


Moderate motion+

stiff soil



Similar response



Equivalent linear overpredicts nonlinear response at certain frequencies – “spurious resonances”

Stress-strain response becoming more complicated

– more variable stiffness and less “elliptical” shape


Moderate motion+

stiff soil



Similar response



Stiffness starting to vary more significantly over

course of ground motion


Strong motion+

stiff soil

Moderate strains

Low – moderate degree of nonlinearity

Noticeably different response



Acceleration


Strong motion+

stiff soil

Moderate strains

Low – moderate degree of nonlinearity





Very strong motion+

stiff soil

Moderate strains

Moderate degree of nonlinearity



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


Very strong motion+

stiff soil

Moderate strains

Moderate degree of nonlinearity






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



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.








Acceleration

More consistency, but NL model can transmit high-frequency oscillations superimposed on low-frequency cycles – too much?








Acceleration

NL model exhibits stiff behavior following strongest part of record; EL maintains low stiffness, high damping behavior throughout.

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


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



Rayleigh dampingNewmark method for time integration

Variable slice width – simulating response of dams, embankments on rock


Decreasing stiffness due to geometry



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




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)


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


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


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).


Nonlinear Behavior – Effective Stress AnalysesNonlinear Behavior – Effective Stress Analyses

Wildlife – Superstition Hills recordings

Nonlinear BehaviorNonlinear Behavior – Effective Stress Analyses– Effective Stress Analyses



Wildlife – Elmore Ranch 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)


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

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