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Overview of Soil-Structure Interaction Principles

Jonathan P. StewartUniversity of California, Los Angeles

Overview

A. IntroductionB. General methods of analysisC. Inertial interactionD. Kinematic interaction

A. Introduction

• Structure• Foundation• Underlying soil/rock

Response dictated by interactions between:

System analysis evaluates response given free-field motion, ug

No SSI when___________ SSI effect =______________

A. Introduction. Three critical aspects of SSI

• Inertia → base shear (V) and moment (M) F=ma

V

M


• Inertia → base shear (V) and moment (M)

• V → relative foundation/free-field displacement (uf)

V


• Inertia → base shear (V) and moment (M)

• V → relative foundation/free-field displacement (uf)

• M → relative foundation/free-field rotation (θf)

M


• uf, θf → foundation damping



• Radiation damping –foundation acts as wave source

uf

p

s

p

s

p

s

θf



• Radiation damping –foundation acts as wave source

• Hysteretic damping in soil

uf

τ

Δ

τ

Δ

Area ∝ hysteretic damping, βs


1. Inertial soil structure interaction• Inertia from vibration of structure and

foundation• Causes foundation translation and rotation

(uf and θf)• Directly affects system flexibility and mode

shapes• Introduces foundation damping


2. Kinematic interaction• Incoherent ground

motions → base slab averaging

u1 u2 u3

Sa

T


2. Kinematic interaction• Incoherent ground

motions → base slab averaging

• Ground motion reductions with depth

Sa

T

u1u2


3. Foundation deformations

• Loads from superstructure inertia


3. Foundation deformations

• Loads from superstructure inertia

• Deformations applied by soil

Nikolaou et al. (2001)

Beyond scope of current presentation

B. General Methods of Analysis

• Direct approach– Full modeling of soil,

foundation, structure– Propagate waves

through system

Beyond scope of current presentation

B. General Methods of Analysis

• Direct approach• Substructure

approach

Focus of this seminar

C. Inertial Interaction

• Springs used to represent soil-foundation interaction

• Complex-valued– Real part represents

stiffness– Imaginary part related

to damping

Combination of real and complex parts comprises

“Impedance function”

C. Inertial Interaction

• Springs used to represent soil-foundation interaction

• Complex-valued• If rigid foundation,

simplifies to:– 3 springs for 2D

system– 6 springs for 3D

system),(),( 00 υωυ aciakk jjj +=

xkzk

θk

C. Inertial Interaction.Effects on System Behavior

• Concepts of period lengthening and foundation damping– System period

– System dampingθkhk

kk

TT fixed

x

fixed2

1~

++=

( )30 ~ TTi

fβββ +=

Foundation damping factor

kxkθ

θ

ug uf hθ u

m

hK*fixed, c

C. Inertial Interaction.Effects on System Behavior

Hysteretic soil damping

1 1.5 2Period Lengthening, T/T

0

10

20

30

Foun

datio

n D

ampi

ng, β

f(%)

e/ru = 0PGA > 0.2gPGA < 0.1g

h/rθ = 0.5

1.0

2.0

∼

0.0 0.1 0.2 0.3 0.4

h/(vs ×T)

0

4

8

12

16

20

β = 0.1β = 0

~ζ 0(%)h/r = 1

h/r = 2

h/r = 4

βf

• Force-based procedure

• SSI affects design spectral ordinate

• Usually not considered for design of new buildings

C. Inertial Interaction.Effects on Base Shear

T

S a

S a

~S a

~S a

T ~T

Flexible-base period, damping ratio(includes SSI effects)

Fixed-base period, damping ratio(neglects SSI effects)

0 1 2Period (s)

0.1

0.2

0.3

0.4

0.5

0.6

Spec

tral A

ccel

erat

ion

(g)

~T

(a)

βi

β0

T, β0 =

T, βi =

∼ ∼

• Initial seismic demand– Should be drawn for

foundation motion, not free-field

– Spectral ordinates should reflect system damping ratio

• Pushover curve– Soil springs in

pushover analysis

C. Inertial Interaction.Effects on Displacement-Based

Pushover Analysis

Initial seismic demand (free-field)Reduced seismic demand (SFSI effects)

Reduced seismic demand(SFSI + extra str. damping)Pushover curve

Performance point

Sa

Sd

Are these effects important?• YES, especially for

short-period structures

• Field data shows:– Foundation damping

ratios up to ~ 10-20%– Period lengthening up

to ~ 1.5– Foundation/ff Sa’s at

low period as low as ~0.5

SSI Can Affect Retrofit Decisions


Fixed-Base


Flexible-Base

C. Inertial Interaction. Impedance Functions

),(),( 00 υωυ aciakk jjj +=

j = u (translation, x or z)θ (rocking)

uuu Kk α=S

uuuu V

rKc β=

θθθ α Kk =SVrK

c θθθθ β=

a0 = ωr/Vs ν = Poisson’s ratio

πfu Ar =

4 4 πθ fIr =

Two aspects of impedance function analysis: 1) Static stiffness (e.g., Kx)2) Dynamic modifiers (e.g., αx, βx)


Static Stiffness (surface foundation)

ux GrKυ−

=2

8

( )3

138

θθ υGrK

−=

uz GrKυ−

=1

4

Circle: Rectangle:

Used in NEHRP Provisions

FEMA-356


( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+=⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

θθθ r

eKKreKK Eu

UEU 21321Circle:

Rectangle:

Static Stiffness (embedment modification)

FEMA-356


• Issue: What is the effective Vs for a non-uniform profile?

• Vs increase with depth– Increases foundation

stiffness– Impedes radiation damping

at large λ (low f) relative to halfspace

Dealing with nonuniform profilesVS

Depth


• For stiffness, use

– Ze = 0.75ru or 0.75rθ–

• For damping, use (Vs)0

Dealing with nonuniform profilesVS

Depth

ttZ

V es =

( )∑ Δ=

is

i

Vztt

Δzi

C. Inertial Interaction. Impedance FunctionsDealing with nonuniform profiles

0 1 2

a0 = ωr/Vs0

0.0

0.5

1.0

β u

0 1 2

a0 = ωr/Vs0

0.00

0.15

0.30

β θ

TRANSLATION ROCKING

Half., β=0.1Half., β=0

α=0.025

α=0.

23n=0.5

n=1

α=0.02

5

α=0.23

Half., β=0.1

Half., β=0

0 2 4 6 8G(z)/G0

6

4

2

0z/

r

α=0.23

α=0.025

0 2 4 6 8G(z)/G0

6

4

2

0

z/r

n=12/31/2

z

G(z) υ, ρ

2r

after Gazetas, 1991

BIAS

1. Evaluate foundation radii••• Analysis of If must consider shear wall

configuration and potential rotational coupling between walls

2. Evaluate foundation embedment, e3. Evaluate effective height of structure, h4. Initial fixed base damping, βi (usually 5%)

πfu Ar =

4 4 πθ fIr =

C. Inertial Interaction. Typical Application

5. Evaluate T/T using structure-specific model :

• Fixed-base period T

∼

Displacement

Forc

e

1

k


5. Evaluate T/T using structure-specific model :

• Fixed-base period T• Flexible-base period

T• Calculate ratio T/T• Ductility correction:

∼

∼

Displacement

Forc

e

1keff

∼

5.02

1~

1~

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+=

TT

TT

s

f

eff

eff

μμ


6. Evaluate foundation damping βf based on Teff/Teff, h/rθ and e/ru∼

1 1.5 2Period Lengthening, T/T

0

10

20

30

Foun

datio

n D

ampi

ng, β

f(%)

e/ru = 0PGA > 0.2gPGA < 0.1g

h/rθ = 0.5

1.0

2.0

∼ 1 1.5 2Period Lengthening, T/T

0

10

20

30

Foun

datio

n D

ampi

ng, β

f (%

)

e/ru = 0.5PGA > 0.2 gPGA < 0.1g

h/rθ = 0.5

1.0

2.0

∼C. Inertial Interaction. Typical

application

7. Evaluate flexible-base damping ratio, β0

8. Evaluate the effect on spectral ordinates of the change in damping from βi to β0

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

3

021 )100ln(~C

CCSS aaβ Eq. 3-7 and 3-8 of FEMA440

(assumes βi = 0.05)

( )30 ~effeff

if

TT

βββ +=

C. Inertial Interaction. Typical application

Limitations• If distributed shear walls,

must consider coupling of wall rotations

160’-0”

100’

-0”

Plan

Elevation @ wall Section @ wall

Roof

2nd

1st

20’-0”

Footing 26’L x 3’B x 1.5’t

3’D

10’-0”typical

8” R/C wall – 20’Ltypical



h uug uf

m

k,c h

kukθ

θ

θ

~TT

kk

khku

= + +12

θ

• Evaluate k• Evaluate ku• Derive kθ• Derive rθ from kθ



• Analysis is conservative for:– High foundation aspect

ratios (a/b > 2)0.0 0.5 1.0 1.5

0.0

0.2

0.4

0.6

0.8

1.0

c rx =

cθ,

x(β=0

)/(ρV

LaI x)

0.0 0.5 1.0 1.50.0

0.2

0.4

0.6

0.8

1.0

c ry =

cθ,

y(β=0

)/(ρV

LaI y)

(a) rocking around x-axis

(b) rocking around y-axis

L/B > 10

L/B = 5

range for L/B = 1 - 2and circles

L/B = 4-5

L/B = 3L/B = 2

range for L/B = 1and circles

Footing

2L

2B

L/B → ∞

y

x

( )V

VLa

s=−

3 41.

π υ

a BVS

0 =ω

Modified from: Dobry and Gazetas, 1986




ratios (a/b > 2)– Deeply embedded

foundations (e/ru > 0.5)

a0

βu βθ

0 2 4 60

1

2

3

0 2 4 60

1

2

3

e/r = 1

1/2

0

1

1/2

e/r = 1

1

1/2

0

**

a rVS0 = ω

Modified from: Apsel and Luco, 1987

s

uuuu V

rKc β=sVrKc θθ

θθ β=





foundations (e/ru > 0.5)

• Analysis unconservativefor:– nonuniform profiles, a0<1

0 1 2

a0 = ωr/Vs0

0.0

0.5

1.0

β u

0 1 2

a0 = ωr/Vs0

0.00

0.15

0.30

β θ

TRANSLATION ROCKING

Half., β=0.1Half., β=0

α=0.025

α=0.

23n=0.5

n=1

α=0.02

5

α=0.23

Half., β=0.1

Half., β=0

0 2 4 6 8G(z)/G0

6

4

2

0

z/r

α=0.23

α=0.025

0 2 4 6 8G(z)/G0

6

4

2

0

z/r

n=12/31/2

z

G(z) υ, ρ

2r

BIAS

after Gazetas, 1991

0 2 4 6 8G(z)/G0

6

4

2

0

z/r

α=0.23

α=0.025

0 2 4 6 8G(z)/G0

6

4

2

0

z/r

n=12/31/2

z

G(z) υ, ρ

2r





foundations (e/ru > 0.5)• Analysis unconservative

for:– nonuniform profiles, a0<1– large impedance contrast at

depth; Vs2 ≥ 2 × Vs1

a

DS

vs1

vs2

ρ1

ρ2

D. Kinematic Interaction

• Contributions from: – Base-slab averaging– Foundation

embedment

after Veletsos et al., 1997

D. Kinematic Interaction.Base Slab Averaging

• Existing theoretical models– User-specified

incoherence parameter, κ

– Rigid foundation, soil is uniform halfspace

• Result is foundation / free-field transfer function, not RRS 2

22

,0 sin~

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

ev

rs

e

bb

Vb

a ακω

after Veletsos and Prasad, 1989; Veletsos et al., 1997

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

Tran

sfer

Fun

ctio

n A

mpl

itude

Disk

a/b=1

a/b=1/4, 4

αv = 0

0a~

Calibration against field data

κ = -0.037 + 7.4E-04 Vs (m/s)

0 200 400 600Vs (m/s)

0.00

0.20

0.40

0.60

κa

Surface foundationsShallowly emb.

σ = 0.55

90% confidence intervals

0

1

2

3

4

Ampl

itude

(|H

3|)

1

κ = 0.11

0 5 10 15 20 25Frequency (Hz)

Kim and Stewart, 2003

Calibration against field data

κ = -0.037 + 7.4E-04 Vs (m/s)

0 200 400 600Vs (m/s)

0.00

0.20

0.40

0.60

κa

Surface foundationsShallowly emb.

σ = 0.55

90% confidence intervals

Kim and Stewart, 2003

222

,0 sin~

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

ev

rs

e

bb

Vb

a ακω

rs

eo V

ba,2

~ κω=

2

1

2

1

,0 222

~nnb

VnVnb

Vb

a e

s

se

rs

e ωωκω=≈=

Site Class 0.1 0.4 0.8A 1.00 1.00 1.00B 1.00 0.97 0.95C 0.97 0.87 0.77D 0.95 0.71 0.32E 0.77 0.22 *F * * *

Peak Ground Acceleration, PGA (g)

Note: Use straight line interpolation for intermediate values of PGA* = should be estimated from site-specific analysis

Shear Wave Velocity Reduction Factor, n2

D. Kinematic Interaction.Embedment Effects

Elsabee and Morray (1977) and Day (1978):• Evaluated transfer functions for vertically incident,

coherent waves• Developed simple model

,

0 2 4 6a0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

rθFI

M/u

g

0 2 4 6a0=ωr/Vs

0.0

0.2

0.4

0.6

0.8

1.0

1.2

u FIM

/ug

ApproximationHalfspaceFinite soil layer

e/r = 0.5

Translation Rocking

D. Kinematic Interaction.Embedment Effects

Elsabee and Morray (1977) and Day (1978):• Evaluated transfer functions for vertically incident,

coherent waves• Developed simple model

,

0 2 4 6a0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

rθFI

M/u

g

0 2 4 6a0=ωr/Vs

0.0

0.2

0.4

0.6

0.8

1.0

1.2

u FIM

/ug

ApproximationHalfspaceFinite soil layer

e/r = 1

Translation Rocking

D. Kinematic Interaction.Transfer Function to RRS

Veletsos and Prasad (1989):• Evaluated RRS at 2%

damping for conditions where transfer function amplitude (TFA) known

• Result:– RRS ≈ TFA for T > 0.2 s– RRS ≈ TFA @ T = 0.2 s for

T < 0.2 s

• Result valid for free-field spectrum shown to right Power spectral density of ff motion

Source: Veletsos and Prasad (1989)

D. Kinematic Interaction.Transfer Function to RRS

0.01 0.1 1 10 Period (s)

0

0.2

0.4

0.6

0.8

1

1.2

Tran

fer F

unct

ion

Am

plitu

de, R

RS

CAP_fn (Tm = 0.51s)Transfer FunctionRRS, 2% dampingRRS, 5%RRS, 10%RRS, 20%

2 4 6 8 10 12Time (s)

-0.4

-0.2

0

0.2

0.4

Acc

eler

atio

n (g

)

RecordedFiltered

0.01 0.1 1 10 Period (s)

0

0.2

0.4

0.6

0.8

1

1.2

Tran

fer F

unct

ion

Am

plitu

de, R

RS

NWH_fn (Tm = 0.70s)Transfer FunctionRRS, 2% dampingRRS, 5%RRS, 10%RRS, 20%

2 4 6 8 10 12Time (s)

-0.8

-0.4

0

0.4

0.8

Acc

eler

atio

n (g

)

RecordedFiltered

Procedure for KI

• Evaluate effective foundation size, be = √ab

• Evaluate embedment depth, e

e

a

b

Procedure for KI

• Evaluate RRS from base slab averaging, RRSbsa

0 0.2 0.4 0.6 0.8 1 1.2Period, T (s)

0.4

0.5

0.6

0.7

0.8

0.9

1

Foun

datio

n/Fr

ee-F

ield

RR

SSimplified Model

be = 65 ftbe = 130 ftbe = 200 ftbe = 330 ft

Procedure for KI

• Evaluate RRS from embedment: RRSe

• RRS = RRSbsa× RRSe

0 0.4 0.8 1.2 1.6 2Period, T (s)

0

0.2

0.4

0.6

0.8

1

1.2

Foun

datio

n/Fr

ee-F

ield

RR

SSite Classes C and D

e = 10 fte = 20 fte = 30 ft

C

D

Limitations of KI Procedure

• Neglect KI effects for soft clay sites (NEHRP E)• Firm rock sites (i.e., NEHRP A and B):

– Neglect embedment effects– Based slab averaging model conservative (over-

estimates RRS)• Base slab averaging model not applicable for

– Flexible foundations (non-interconnected)– Pile-supported foundations with slab-soil gap

ReferencesApsel, R.J. and Luco, J.E. (1987). “Impedance functions for foundations embedded in a layered

medium: an integral equation approach,” J. Earthquake Engrg. Struct. Dynamics, 15(2), 213-231.Day, S.M. (1978). “Seismic response of embedded foundations,” Proc. ASCE Convention, Chicago,

IL, October, Preprint No. 3450.Dobry, R. and Gazetas, G (1986). “Dynamic response of arbitrarily shaped foundations,” J. Geotech.

Engrg., ASCE, 112(2), 109-135.Elsabee, F. and Morray, J.P. (1977). “Dynamic behavior of embedded foundations,” Rpt. No. R77-33,

Dept. of Civil Engrg., MIT, Cambridge, Mass.FEMA-356: Prestandard and commentary for the seismic rehabilitation of buildings, Federal

Emergency Management Agency, Washington, D.C., 2000. FEMA-440: Improvement of Nonlinear Static Seismic Analysis Procedures, Department of Homeland

Security, Federal Emergency Management Agency, June, 2005. Gazetas, G. (1991). Chapter 15: Foundation Vibrations, Foundation Engineering Handbook, H.-Y.

Fang, ed., 2nd Edition, Chapman and Hall, New York, NY.Kim, S. and Stewart, J.P. (2003)."Kinematic soil-structure interaction from strong motion recordings,"J.

Geotech.. & Geoenv. Engrg., ASCE, 129 (4), 323-335.Nikolaou, S., Mylonakis, G., Gazetas, G., and Tazoh, T. (2001). “Kinematic pile bending during

earthquakes: analysis and field measurements,” Geotechnique, 51(5), 425-440. Veletsos, A.S. and Verbic, B. (1973). “Vibration of viscoelastic foundations,” J. Earthquake Engrg.

Struct. Dynamics, 2(1), 87-102.Veletsos, A.S., Prasad, A.M., and Wu, W.H. (1997). “Transfer functions for rigid rectangular

foundations,” J. Earthquake Engrg. Struct. Dynamics, 26 (1), 5-17.Veletsos, A.S. and Prasad, A.M. (1989). “Seismic interaction of structures and soils: stochastic

approach,” J. Struct. Engrg., ASCE, 115(4), 935-956.

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