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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
A. Introduction. Three critical aspects of SSI
• Inertia → base shear (V) and moment (M)
• V → relative foundation/free-field displacement (uf)
V
A. Introduction. Three critical aspects of SSI
• Inertia → base shear (V) and moment (M)
• V → relative foundation/free-field displacement (uf)
• M → relative foundation/free-field rotation (θf)
M
A. Introduction. Three critical aspects of SSI
• uf, θf → foundation damping
A. Introduction. Three critical aspects of SSI
• uf, θf → foundation damping
• Radiation damping –foundation acts as wave source
uf
p
s
p
s
p
s
θf
A. Introduction. Three critical aspects of SSI
• uf, θf → foundation damping
• Radiation damping –foundation acts as wave source
• Hysteretic damping in soil
uf
τ
Δ
τ
Δ
Area ∝ hysteretic damping, βs
A. Introduction. Three critical aspects of SSI
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
A. Introduction. Three critical aspects of SSI
2. Kinematic interaction• Incoherent ground
motions → base slab averaging
u1 u2 u3
Sa
T
A. Introduction. Three critical aspects of SSI
2. Kinematic interaction• Incoherent ground
motions → base slab averaging
• Ground motion reductions with depth
Sa
T
u1u2
A. Introduction. Three critical aspects of SSI
3. Foundation deformations
• Loads from superstructure inertia
A. Introduction. Three critical aspects of SSI
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
SSI Can Affect Retrofit Decisions
Fixed-Base
SSI Can Affect Retrofit Decisions
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)
C. Inertial Interaction. Impedance Functions
Static Stiffness (surface foundation)
ux GrKυ−
=2
8
( )3
138
θθ υGrK
−=
uz GrKυ−
=1
4
Circle: Rectangle:
Used in NEHRP Provisions
FEMA-356
C. Inertial Interaction. Impedance Functions
( ) ( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛+=⎟⎟
⎠
⎞⎜⎜⎝
⎛+=
θθθ r
eKKreKK Eu
UEU 21321Circle:
Rectangle:
Static Stiffness (embedment modification)
FEMA-356
C. Inertial Interaction. Impedance Functions
• 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
C. Inertial Interaction. Impedance Functions
• 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
C. Inertial Interaction. Typical Application
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
C. Inertial Interaction. Typical Application
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
Limitations• If distributed shear walls,
must consider coupling of wall rotations
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
Limitations• If distributed shear walls,
must consider coupling of wall rotations
• 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
Limitations• If distributed shear walls,
must consider coupling of wall rotations
• Analysis is conservative for:– High foundation aspect
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=
Limitations• If distributed shear walls,
must consider coupling of wall rotations
• Analysis is conservative for:– High foundation aspect
ratios (a/b > 2)– Deeply embedded
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
Limitations• If distributed shear walls,
must consider coupling of wall rotations
• Analysis is conservative for:– High foundation aspect
ratios (a/b > 2)– Deeply embedded
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.