Site Response

8/10/2019 Site Response

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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 onlyMotion is identical on planes perpendicular to that motion

to infinityto infinity




What do we mean?

One-dimensional = waves propagate in one direction onlyMotion 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 !"# waves

Horizontal input motion

Horizontal surface motion




When are one-dimensional analyses appropriate?

!tiffer

with

depth

Focus





!tiffer

with

depth

Horizontal boundaries –waves tend to be refracted

toward vertical

Decreasing stiffness

causes refraction of waves

to increasingly vertical

pat

Focus





!tiffer with

depth

!otappropriate

ere



$etaining structures

%ams and

emban&ments

'unnels



(nclined ground surface and)or non-

hori*ontal boundaries can re+uire use

of two-dimensional analyses

!ot ere"!ot ere"



Comple, soilconditions

%ams in

narrow

canyons

Multiple

structures



ocali*ed structures may re+uire

use of .-% response analyses

!ot ere"!ot ere"




"ow should ground motions be applied?

#ncoming motion

u i

Roc$outcropping

motion

2u i

%edroc$ motion

u i + u r

Free surface

motion

u s

!ot te same"!ot te same"

Soil

Rock




"ow should ground motions be applied?

Ob&ect motion

Free surface

motion

u s

(nput ob/ect# motion

(f recorded at roc& outcrop0 apply as

outcrop motion program will remove

free surface effect#1 2edroc& should

be modeled as an elastic half-space1

(f recorded in boring0 apply as within-

profile motion recording does notinclude free surface effect#1 2edroc&

should be modeled as rigid1



Comple, $esponse Method

3pproach used in computer programs li&e !"345

'ransfer function is used with input motion to compute surface motion

convolution#

6or layered profiles0 transfer function is 7built8 layer-by-layer to go from input

motion to surface motion

Amplification

De-amplification

'etods of One-Dimensional Site Response Analysis'etods of One-Dimensional Site Response Analysis

Single

elastic

layer



(ayer j )*

(ayer j

* 9

9

9

9

9

9

ξ

ξ

ξ

ξ

ξ

ξ

ρ

ρ

ρ

ρ

ρ

ρ

:

; < :

;

/ < :

/

=

/ < :

:

=

/

;

:

=

/

/ < :

;

; < :

:

=

/

/ < :

;

; < :

:

=

/

/ < :

;

; < :*

h

h

h

h

h

*

*

*

*Consider the soil deposit shown to the

right1 Within a given layer0 say ayer j 0 the hori*ontal displacements will be

given by

( ) ( ) j j j i k z

j ik z i t u z t A e B e e j j j j,

* *= +

− ω

3t the boundary between layer j and layer j <:0 compatibility of displacements

re+uires that

j j j i k h j i k h A B A e B e j j j j+ + −+ = +1 1* *

Continuity of shear stresses re+uires that

j j j j

j j

ik h j j

ik h A BG k

G k

A e B e s j s j+ ++ +

−− = −

1 1

1 1

* *

* *

* *

+omple, Response 'etod (inear analysis.+omple, Response 'etod (inear analysis.

Amplitudes of upward- anddownward-traveling waves in (ayer j

5+uilibrium satisfied

;o slip



%efining α> / as the comple, impedance ratio at the boundary between layers

j and j <:0 the wave amplitudes for layer j <: can be obtained from the

amplitudes of layer j by solving the previous two e+uations simultaneously

( ) ( ) j j j i k h

j j i k h A A e B e j j j j+

−= + + −11

2 1

1

2 1* ** *

α α

( ) ( ) j j j

i k h j j

i k h B A e B e j j j j

+

−= − + +11

2 1

1

2 1* ** *

α α

Wave amplitudes in ayer j

Wave amplitudes in ayer j <:

!o0 if we can go from ayer j to ayer j <:0 we can go from j <: to j <0 etc1

'his means we can apply this relationship recursively and e,press the amplitudes in

any layer as functions of the amplitudes in any other layer1 We can therefore 7build8

a transfer function by repeated application of the above e+uations1


ropagation of wave

energy from one layer to

another is controlled by

comple,# impedance ratio




!ingle layer on rigid base

H = :@@ ft

V s = A@@ ft)sec

ξ = :@B





H = A@ ft

V s = :0A@@ ft)sec

ξ = :@B





H = :@@ ft

V s = .@@ ft)sec

ξ = AB










%ifferent se+uence of soil layers

%ifferent transfer function

%ifferent response




3nother se+uence of soil layers

%ifferent transfer function

%ifferent response




Comple, response method operates in fre+uency domain

(nput motion represented as sum of series of sine waves!olution for each sine wave obtained

!olutions added together to get total response

rinciple of

superposition

τ

γ

inear

system

Can we capture important effects of nonlinearity with linear model?



!oils e,hibit nonlinear0 inelastic behavior under cyclic loading conditions

!tiffness decreases and damping increases as cyclic strain amplitude increases

'he nonlinear0 inelastic stress-strain behavior of cyclically loaded soils can be

appro,imated by e+uivalent linear properties1

)log( eff γ )log( eff γ

ξ

5+uivalent shear modulus5+uivalent shear modulus 5+uivalent damping ratio5+uivalent damping ratio

max/ GG

/0uivalent (inear Approac/0uivalent (inear Approac




ξmax

/ GG

3ssume some initial strain and use to estimate 9 and ξ 3ssume some initial strain and use to estimate 9 and ξ

*. *.









ξmax

/ GG

γ :# γ :#

Use these values to compute responseUse these values to compute response






γ t #

t




ξmax

/ GG

γ :# γ :#

%etermine pea& strain and effective strain

γ eff

= $γ

γ ma,

%etermine pea& strain and effective strain

γ eff = $γ γ ma,






γ t #

t

γ ma,γ eff




ξmax

/ GG

γ :# γ :#γ #

γ #

!elect properties based on updated strain level!elect properties based on updated strain level









ξmax

/ GG

γ :# γ :#γ #

γ #γ .# γ .#

Compute response with new properties and determine

resulting effective shear strain

Compute response with new properties and determine

resulting effective shear strain









ξmax

/ GG

$epeat until computed effective strains are

consistent with assumed effective strains

$epeat until computed effective strains are

consistent with assumed effective strains

γ eff γ eff








3dvantages

Can wor& in fre+uency domainCompute transfer function at relatively small number of fre+uencies

compared to doing calculations at all time steps#

(ncreased speed not that significant for :-% analyses

(ncreased speed can be significant for -%0 .-% analyses

5+uivalent linear properties readily available for many soils D familiarity

breeds comfort)confidence

Can ma&e first-order appro,imation to effects of nonlinearity and inelasticity

within framewor& of a linear model


'he e+uivalent linear approach is an appro,imation1 ;onlinear analyses are

capable of representing the actual behavior of soils much more accurately1

E often0 a very good oneF



!onlinear Analysis!onlinear Analysis

2 3

2 2

u u

z t z t

∂τ ∂ ∂= ρ + η∂ ∂ ∂ ∂

5+uation of motion must be integrated in time domain

Wave e+uation for

visco-elastic medium

*

%ivide profile

into series of

layers

%ivide time into series of time stepst




2 3

2 2

u u

z t z t

∂τ ∂ ∂= ρ + η∂ ∂ ∂ ∂


Wave e+uation for


*

%ivide profile

into series of

layers

%ivide time into series of time stepst

v i/ = v z = z i0 t = t /#

t /

z i




2 3

2 2

u u

z t z t

∂τ ∂ ∂= ρ + η∂ ∂ ∂ ∂


Wave e+uation for


*

t t /

z i

, 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 steps0 but basic process

involves using wave e+uation to

predict conditions at time j <: from

conditions at time j for all layers in

profile1




2 3

2 2

u u

z t z t

∂τ ∂ ∂= ρ + η∂ ∂ ∂ ∂


Wave e+uation for


*

t t /

z i





profile1

Can change material properties

for use in ne,t time step1

Changing stiffness based on

strain level0 strain history0 etc1 can

allow prediction of nonlinear0

inelastic response1




2 3

2 2

u u

z t z t

∂τ ∂ ∂= ρ + η∂ ∂ ∂ ∂


Wave e+uation for


*

t t /

z i





profile1






inelastic response1




2 3

2 2

u u

z t z t

∂τ ∂ ∂= ρ + η∂ ∂ ∂ ∂


Wave e+uation for


*

t t /

z i





profile1






inelastic response1




2 3

2 2

u u

z t z t

∂τ ∂ ∂= ρ + η∂ ∂ ∂ ∂


Wave e+uation for


*

t t /

z i





profile1






inelastic response1




2 3

2 2

u u

z t z t

∂τ ∂ ∂= ρ + η∂ ∂ ∂ ∂


Wave e+uation for


*

t t /

z i





profile1






inelastic response1

rocedure steps through time from

beginning of earth+ua&e to end1

Step through time



!onlinear %eavior !onlinear %eavior

τ

γ

τ

γ

Continuous inear

segments

3ctual 3ppro,imation

(n a nonlinear analysis0 we appro,imate the continuous actual

stress-strain behavior with an incrementally-linear model1 'he

finer our computational interval0 the better the appro,imation1



3dvantages

Wor& in time domainCan 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 redistribution0 dissipation

3voids spurious resonances associated with linearity of 5 approach#

Can compute permanent strain permanent deformations

!onlinear Approac!onlinear Approac

i+uefaction

;onlinear analyses can produce results that are consistent with e+uivalent

linear analyses when strains are small to moderate0 and more accurate

results when strains are large1

'hey can also do important things that e+uivalent linear analyses can’t0 such

as compute pore pressures and permanent deformations1



What are people using in practice?

/0uivalent (inear vs1 !onlinear Approaces/0uivalent (inear vs1 !onlinear Approaces

5+uivalent linear analysesOne-dimensional D

-% ) .-% D

;onlinear analyses

One-dimensional D

-% ) .-% D

!"345

GU3%H0 6U!"

%5!$30 %MO%

'3$30 63C0 3I(!



What are people using in practice?

/0uivalent (inear vs1 !onlinear Approaces/0uivalent (inear vs1 !onlinear Approaces

5+uivalent linear analysesOne-dimensional D

-% ) .-% D

;onlinear analyses

One-dimensional D

-% ) .-% D

!"345

GU3%H0 6U!"

%5!$3

'3$3



Dimensions OS /0uivalent (inear !onlinear

:-%%O! %yne+0 !ha&eJ: 3M50 %5!$30 %MO%0

6(0 !UM%5!0 '5!!

Windows !ha&e5dit0 ro!ha&e0!ha&e@@@0 55$3

CyberGua&e0 %eep!oil0;5$30 63C0 %MO%@@@

-% ) .-% %O!

6U!"0

GU3%H)GU3%HM0'U!"

%K;36OW0 '3$3-.0 6(0

L5$!3'0 %K!3C0 (GC30Open!ees

Windows GU345)W0 !3!!(@@@ 63C0 3I(!

Available +odes

!ince early :J@s0 numerous computer programs developed for site

response analysis

Can be categori*ed according to computational procedure0 number of

dimensions0 and operating system





+urrent 2ractice

Method of 3nalysis

Method of

3nalysis

W;3 5;3 Overseas

rivate.A#

ublic.#

rivateN#

ublic:#

rivateA#

ublicA#

:-% 5+uivalent inear N A N A@ H A

:-% ;onlinear :: : : @ H A

-%).-% 5+uiv1 inear J : A N @

-%).-% ;onlinear : . : A . J@

Of the total number of site response analyses you perform, indicate theapproximate percentages that fall within each of the following categories:

[ ] a. One-dimensional equivalent linear

[ ] b. One-dimensional nonlinear

[ ] c. wo- or three-dimensional equivalent linear

[ ] d. wo- or three-dimensional nonlinear

One-dimensional e+uivalent linear analyses dominate ;orth 3merican

practiceP nonlinear analyses are more fre+uently performed overseas






5+uivalent linear vs nonlinear

analysis D how much difference

does it ma&e?

'opanga motion scaled to @1@A g

Wea& motion<stiff soil

ow strains

ow degree of nonlinearity

!imilar response







ow strains


!imilar response




does it ma&e?





ow strains


!imilar response




does it ma&e?



'opanga motion scaled to @1@ g

Moderate motion<stiff soil

$elatively low strains

$elatively low degree of nonlinearity

!imilar response




does it ma&e?







'opanga motion scaled to @1@ g

Moderate motion<stiff soil

$elatively low strains

$elatively low degree of nonlinearity

!imilar response




does it ma&e?

Stiffness starting to &ar#more significantl# o&er

course of ground motion



'opanga motion scaled to @1A@ g

!trong motion<stiff soil

Moderate strains

ow D moderate degree of nonlinearity

;oticeably different response




does it ma&e? "cceleration




!trong motion<stiff soil

Moderate strains

ow D moderate degree of nonlinearity





does it ma&e?

! li % i



'opanga motion scaled to :1@ g

Lery strong motion<stiff soil

Moderate strains

Moderate degree of nonlinearity





does it ma&e? "cceleration

Substantial softening by/( metod causes

underprediction of initial

portion of record(inearity inerent in /( metod

causes overprediction response in

strongest portion of record

Softening by /( metod

causes underprediction

! li % i




Lery strong motion<stiff soil

Moderate strains

Moderate degree of nonlinearity





does it ma&e?

! li % i! li % i




5+uivalent linear vs nonlinear analysis D how much difference does it ma&e?

14 m V s = .@@ m)sec

V s = N m)sec

16 m V s = :@@ m)sec

uH 0t #

u@0t #

: m

:A m

J m

! li % i! li % i






does it ma&e?

arge strain levels QNB# near

bottom of upper layer

5 model converges to low 9

and high ξ

"igh-fre+uency componentscannot be transmitted through

over-softened 5 model

!( model? Stiffness stays relatively ig

e,cept for a few large-amplitude cycles

"cceleration

5 model predicts very soft behavior at beginning of earth+ua&e0

before any large strains have developed1

! li % i! li % i






does it ma&e?




and high ξ





"cceleration

More consistency0 but ; model can transmit high-fre+uencyoscillations superimposed on low-fre+uency cycles D too much?



! li % i!onlinear %eavior






does it ma&e?




and high ξ





!onlinear Soil %ea ior!onlinear Soil %eavior



T i m e

!onlinear Soil %eavior !onlinear Soil %eavior

!mall cycle

superimposed on large

cycle after 3ssima&i and4ausel0 @@#

(ow stiffness

Hig stiffness

/0uivalent linear modelmaintains constant

stiffness and damping –

iger stiffness

e,cursions associated

wit iger fre0uency

oscillations aren@t seen1

!onlinear Soil %eavior!onlinear Soil %eavior



T i m e

!onlinear Soil %eavior !onlinear Soil %eavior

!mall cycle

superimposed on large

cycle after 3ssima&i and4ausel0 @@#

Hig damping

(ow damping

/0uivalent linear modelmaintains constant

stiffness and damping –

iger stiffness

e,cursions associated

wit iger fre0uency

oscillations aren@t seen1



'odified /0uivalent (inear Approac'odified /0uivalent (inear Approac



3ssima&i and 4ausel

'odified /0uivalent (inear Approac'odified /0uivalent (inear Approac

Fre0uency-dependent model +onventional model

"igh fre+uenciesoversoftened and

overdamped

5,cellent agreement

with nonlinear model

%encmar$ing of !onlinear Analyses%encmar$ing of !onlinear Analyses




!tewart and 4wo&

55$ study to determine proper manner in which to use nonlinear analyses

Wor&ed with five e,isting

nonlinear codesP hired

developers to run their codes

and comment on results

5stablished advisory

committee to oversee

analyses and assist with

interpretation

Met regularly with advisory

committee and developers





!tewart and 4wo&

Considered codes




%-MO%R Matasovic#

5nhanced version of %-MO%0 which is enhanced version of %5!$3

umped mass model

$ayleigh damping

% a m p i n g

r a t i o

6re+uency

Mass-proportional

!tiffness-proportionalRayleig





%-MO%R Matasovic#


umped mass model

$ayleigh damping

;ewmar& β method for time integration

Lariable slice width D simulating response of dams0 emban&ments on roc&


Decreasing stiffnessdue to geometry




%-MO%R Matasovic#


umped mass model

$ayleigh damping



Can simulate slip on wea& interfaces

Uses M4S soil model modified hyperbola D needs Gma,0 τma,0 α and s#

Can soften bac&bone curve to model cyclic degradation





%-MO%R Matasovic#


umped mass model

$ayleigh damping



Can simulate slip on wea& interfaces

Uses M4S soil model modified hyperbola D needs Gma,0 τma,0 α and s#

Can soften bac&bone curve to model cyclic degradationUses Masing rules for unloading-reloading behavior

;eed input parameters for

M4S bac&bone curve H#

Cyclic degradation . for clay0 H for sand#

ore pressure generation H for clay0 H for sand#

ore pressure redistribution)dissipation at least #

$ayleigh damping coefficients #

2asic layer properties density0 shear wave velocity0 half-space properties#





%55!O( "ashash#

!imilar to %MO%- lumped mass0 derives from %5!$3-#

More advanced $ayleigh damping scheme lower fre+uency dependence#

'5!! y&e#

6inite difference wave propagation analysis not lumped mass#

Cundall-y&e hypothesis for loading-unloading behavior

!imilar bac&bone curve to %MO%- and %55!O(

(nviscid sort of# low-strain damping scheme

Open!ees Kang0 5lgamal#

6inite element model :%0 %0 .% capabilities#

Multi-surface plasticity model von Mises yield surface0 &inematic

hardening0 non-associative flow rule#

6ull $ayleigh damping

!UM%5!

6inite element model

2ounding surface plasticity model ade-li&e yield surface0 &inematic

hardening0 non-associative flow rule#

!implified $ayleigh damping







erformance

2ased on validations against vertical array data

T Models produce reasonable results

T !ome indication of overdamping at high fre+uencies0 overamplification at

site fre+uency

T Lariability of predictions due to bac&bone curves and damping models

most pronounced at T @1A sec and is significant only for relatively thic&

profiles1 Model-to-model variability most pronounced at low periods1

T ;onlinearity modeled well up to levels for which ade+uate data is

available generally up to about @1g#1 %ata for stronger sha&ing being

sought centrifuge tests0 recent ;igaata earth+ua&e#1

T %MO%-0 %55!O(0 and Open!ees generally produced similar

amplification factors and spectral shapesP '5!! produced differentresponse at high fre+uencies different damping formulation#0 !UM%5!

results were significantly different than all others for deep sites probably

due to simplified $ayleigh damping#1


!onlinear %eavior – /ffective Stress Analyses!onlinear %eavior – /ffective Stress Analyses



!onlinear %eavior /ffective Stress Analyses!onlinear %eavior /ffective Stress Analyses

Wildlife D !uperstition "ills recordings

!onlinear %eavior!onlinear %eavior – /ffective Stress Analyses – /ffective Stress Analyses



!onlinear %eavior o ea e a o /ffective Stress Analysesect e St ess a yses


!onlinear %eavior!onlinear %eavior – /ffective Stress Analyses – /ffective Stress Analyses



!onlinear %eavior /ffective Stress Analysesy

Wildlife D 5lmore $anch recordings

!onlinear %eavior !onlinear %eavior – /ffective Stress Analyses – /ffective Stress Analyses



yy


ow

fre+uency

"igh

fre+uency

9round surface record

???

Site /ffectsSite /ffects



5lmore $anch record D no li+uefaction

$atio of waveletamplitudes D variation

with fre+uency and time

ime sec.

F r e 0 u e n c y - H z .

Site /ffectsSite /ffects



5lmore $anch record D no li+uefaction

$atio of waveletamplitudes D variation

with fre+uency and time

ime sec.

F r e 0 u e n c y - H z .




yy





yy






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Site Response