Soil-Structure Interaction ECIV 724A Fall 2004
Jan 15, 2016
Soil-Structure Interaction
ECIV 724A Fall 2004
SSI – Problem Definition
Earthquake AnalysisStructures supported by rigid foundationsEarthquakes=>Specified motion of base
RigidBaseAnalysis
Tall Buildings Acceptable• Light & Flexible• Firm Foundations• Methods focus on
modeling of structure• Displacements wrt fixed
base• Finite Element Methods
Nuclear Power Plants Wrong Assumption• Massive & Stiff• Soft Soils
• Interaction with supporting soils becomes important
SSI – Problem Definition
Machine Foundation
Parameters•Local Soil Conditions•Peak Acceleration•Frequency Content of
Motion•Proximity to Fault•Travel Path etc
Inertial InteractionInertial forces in structure are transmitted to flexible soil
Kinematic InteractionStiffer foundation cannot conform to the distortions of soil
TOTAL=INERTIAL + KINEMATIC
Seismic Excitation
SSI Effects
0.00E+00
2.50E-05
5.00E-05
7.50E-05
1.00E-04
1.25E-04
1.50E-04
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
Am
plitu
de in
ft.
SSI - Proposed BE-FE
SSI - Spring DashpotModelProposed BE-FE, StiffSoilFixed Base Analysis
Posin( t)
Half Space
2b
H
SSI Effects
-1.0E-05
-5.0E-06
0.0E+00
5.0E-06
1.0E-05
1.5E-05
2.0E-05
2.5E-05
0 25 50 75 100 125 150 175 200 225 250
Time x 1.08x10-4 (sec)
Ho
rizo
nta
l Am
pli
tud
e
U1 SSI - Relative
U1 Fixed Base
U2 SSI - Relative
U2 Fixed Base
P(t)
Half Space
m
m
Cross Interaction Effects
1. Moment is applied
2. Waves Propagate…
3. …Reach Receiver…
4. …and life goes on…
SSI Effects
Alter the Natural Frequency of the Structure
Add Damping
Through the Soil Interaction Effects
Traveling Wave Effects
Methods of Analysis
Objective:
Given the earthquake ground motions that would occur on the surface of the ground in the absence of the structure (control or design motions), find the dynamic response of the structure.
Methods of Analysis
Methods
IdealizedComplete
Direct MultiStep
Complete Interaction Analysis
• Account for the variation of soil properties with depth.• Consider the material nonlinear behavior of the soil• Consider the 3-D nature of the problem• Consider the nature of the wave propagation which
produced the ground motion• Consider possible interaction with adjacent structures.
High Degree of ComplexityHigh Degree of Complexity
Idealized Interaction Analysis
Idealization
Horizontal LayersSimplified Wave Mechanismsetc
Idealized Interaction AnalysisPreliminary description of free field motionbefore any structure has been built
The definition of the motion itselfthe control motion in terms of response spectra, acceleration records etc
The location of the control motionfree surface, soil-rock interface
The generation mechanism at the control point vertically or obliquely incident SH or SV waves, Rayleigh waves, etc.
Idealized AnalysisIdealized Interaction Analysis Idealized Interaction Analysis
Tools: FEM, BEM, FDE, Analytical solutionsTools: FEM, BEM, FDE, Analytical solutions
Direct MethodsDirect MethodsEvaluation of Dynamic Evaluation of Dynamic Response in a Single Response in a Single StepStep
MultiStep MethodsMultiStep MethodsEvaluation of Dynamic Response Evaluation of Dynamic Response in Several Stepsin Several Steps
SUPERPOSITIONSUPERPOSITION
• Two-StepTwo-Step Kinematic+Inertia InteractionKinematic+Inertia Interaction
• Three-StepThree-Step Rigid FoundationsRigid Foundations Lumped Parameter ModelsLumped Parameter Models
• SubstructureSubstructure Division to SubsystemsDivision to Subsystems Equilibrium & CompatibilityEquilibrium & Compatibility
True Nonlinear True Nonlinear SolutionsSolutions
Finite Element Method (FEM)
tttt fKuuCuM Governing EquationGoverning Equation
• Modal AnalysisModal Analysis• Direct IntegrationDirect Integration• Fourier Analysis - Complex ResponseFourier Analysis - Complex Response
Solution TechniquesSolution Techniques
FEM Solution Techniques
Selection CriteriaSelection Criteria Cost and FeasibilityCost and FeasibilityParamount ConsiderationParamount Consideration Accuracy Accuracy
DifferencesDifferences
- Handling of Damping- Handling of Damping- Ability to Handle High Frequency- Ability to Handle High Frequency Components of Motion Components of Motion
FEM - Modal Analysis
Damping is neglected during early stagesDamping is neglected during early stages
Actual displacements are dampedActual displacements are damped
Damping is considered in arbitrary mannerDamping is considered in arbitrary manner
Structural Dynamics: First few modes need to be evaluated Structural Dynamics: First few modes need to be evaluated (<20)(<20)
SSI: Acceleration response spectra over a large frequencySSI: Acceleration response spectra over a large frequency range and large number of modes need to be considered range and large number of modes need to be considered (>150)(>150)
Not recommended for Direct SSI - Stiff Massive Structure SoftNot recommended for Direct SSI - Stiff Massive Structure Soft Soil Soil
OK for SubstructureOK for Substructure
FEM - Direct Integration
Time Marching SchemesNewmark’s Methods, Wilson Methods, Bathe and WilsonCubic Inertia Method
Small Time Step for Accuracy Stability and Convergence Choice of Damping Matrix
Frequency Dependent Damping Ratio - filters out high frequency components
Proportional Damping
Good Choice if True Dynamic Nonlinear Analysis is feasible
FEM - Complex Response
Fourier Transformation - Transfer Functions
Transfer Functions Independent of External Excitation
Control of Accuracy Efficient Only Linear or Pseudo non-linear analysis
FEM - Geometric Modeling
FEM Modeling
Matrix MassMixed5
1
Matrix MassConsistent8
1
Matrix MassLumped8
1
max
s
s
s
h
Max Element Size Governed by Highest frequency Max Element Size Governed by Highest frequency which must be transmitted correctly within the which must be transmitted correctly within the
element element
FEM Modeling of Infinite Space
FEM Modeling of Infinite Space
Modeling Introduces Artificial Boundaries that Modeling Introduces Artificial Boundaries that Reflect WavesReflect Waves
FEM Modeling of Infinite Soil
Absorbing Boundaries Viscous Boundary Variable Depth Method Damping proportional to Wave Velocities
Radiating Boundaries (Hyperelements) Satisfy Boundary Conditions at Infinity Eigenvalue Analysis Frequency Domain Analysis
SSI – FEM Methods
FEM
Advantages• Non-Linear Analysis• Well Established
Shortcomings• Finite Domains• Volume
Discretizations
Boundary Element Methods
jjiijiji ufucucc ,22,
22
21
Governing Equation
Small Displacement Field
Homogeneous Isotropic Elastic
Boundary Element Method
GOVERNING EQUATION
BOUNDARY INTEGRAL EQUATION
Dynamic Reciprocal Theorem
Indirect DIRECT
Transform Domain TIME DOMAIN
Dirac- Step Impulse B-SPLINE
System of Algebraic Equations Time Marching Scheme
Boundary Element Method
BOUNDARY INTEGRAL EQUATION
B-SPLINE FUNDAMENTAL SOLUTIONS
SPATIAL DISCRETIZATION TEMPORAL DISCRETIZATION
BOUNDARY INTEGRAL EQUATION IN A DISCRETE FORM
TIME MARCHING SCHEME & B-SPLINE IMPULSE RESPONSE
RESPONSE TO ARBITRARY EXCITATION
1
1
2N
n
nNnN fBu NN HFf
BEM – Methods
BEM
Advantages• Infinite Media• Surface Discretization
Shortcomings• Non-symmetric
matrices• Not Efficient for
Nonlinear
SSI Methods
Combined BEM-FEM eliminate disadvantages of each method
and retain advantages
ApproachApproach• FEM ApproachFEM Approach• BEM ApproachBEM Approach• Staggered SolutionsStaggered Solutions
Governing Equations
jjiijiji ufucucc ,22,
22
21
tt
tt
fKu
uCuM
FEM MethodTime Marching Scheme
tttt fKuuCuM
NN fDu
Governing EquationGoverning Equation
Discrete Form in TimeDiscrete Form in Time
FEM-BEM CouplingStaggered Solutions
Can be Solved in a Staggered Approach...Can be Solved in a Staggered Approach...
NNBEM
NBEM HFfu
NFEM
NFEM fDu
BEMBEM
FEMFEM
FEM-BEM CouplingStaggered Solutions
Compatibility of DisplacementsCompatibility of Displacements at Interfaceat Interface
BEMBEMSolverSolver
FEMFEMSolverSolver
Equilibrium of ForcesEquilibrium of Forcesat Interfaceat Interface
ExternalExternalExcitationExcitation
ExternalExternalExcitationExcitation
intFEMuint
BEMu
intFEMfint
BEMf
At Every Time Step...At Every Time Step...
FEM-BEM CouplingAdvantages
Independent Solutions for BEM and FEM
Independent Time Step Selection Smaller Systems of Equations BEM System of Reduced Size In the Absence of Incidence
Displacement Field in Soil, BEM does not require Solution.
Lumped Parameter Models for SSI
P(t) m
Half Space
P(t) m
Spring-Dashpot ModelStick Model
Lumped Parameter Foundation Models
Reissner (1936) Analytic Solutions to Vertical Vibration of Circular Footing Due to Harmonic Excitation
Assumptions:Elastic ½-spaceMaterial G,v,Uniform Vertical Pressure
Formed Basis of Almost All Analytical Studies
Lumped Parameter Foundation Models
Quinlan and SungAssumed Different Pressure Distributions
Richart & WhitmanEffects of Poisson’
Bycroft (1956)Displacement Functions
Hsieh K and C in terms of Soil and Foundation Parameters
Lumped Parameter Foundation Models
Lysmer AnalogConstant Lumped Parameters
Richart Hall & Wood(1970)
Gazetas (1983)
Wolf (1988)
Lumped Parameter Foundation Models
Representative Lumped Parameter Values - Square
Lumped Parameter Foundation Models
Mode K C B D
Vertical (z)
1
4 oGr
Gro
2
1
4.3
34
1
or
m
zB
425.0
Sliding
(x)
2
8 oGr
Gro
2
2
6.4
38
2
or
m
xB
288.0
Rocking
()
13
8 3oGr
B
Gro 11
8.0 4
58
13
or
I
BB1
15.0
Torsional
()
3
6 3oGr
B
GB
21
4
5or
I
B21
5.0
Representative Lumped Parameter Values Circular
Lumped Parameter Foundation Models
Stehmeyer and Rizos (2003)
Properties k, and c are known to be frequency () dependent
The Real System Equivalent SDOF System
n
n
mc
MK
2
Lumped Parameter Foundation Models
Horizontal Displacement with Horizontal Impulse Applied
-0.002
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
0.018
0.020
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00
Time
Dis
pla
ce
me
nt
Discrete BEM Solution
Simplified Closed Form Solution
2b
B(t)
Half Space
y
z
x
B(t
)
n = 3.3
= 0.975
SSI Effects
0.00E+00
2.50E-05
5.00E-05
7.50E-05
1.00E-04
1.25E-04
1.50E-04
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
Am
plitu
de in
ft.
SSI - Proposed BE-FE
SSI - Spring DashpotModelProposed BE-FE, StiffSoilFixed Base Analysis
Posin( t)
Half Space
2b
H
SSI Effects
-1.0E-05
-5.0E-06
0.0E+00
5.0E-06
1.0E-05
1.5E-05
2.0E-05
2.5E-05
0 25 50 75 100 125 150 175 200 225 250
Time x 1.08x10-4 (sec)
Ho
rizo
nta
l Am
pli
tud
e
U1 SSI - Relative
U1 Fixed Base
U2 SSI - Relative
U2 Fixed Base
P(t)
Half Space
m
m
SSI Effects
Based on the Simplified Lumped Parameter Models it can be shown that
k
k
k
k
T
T h
h
2
1~
P(t) m
Longer Period of Foundation-Structure System
SSI Effects – Cross Interaction
Receiver FoundationReceiver FoundationSource FoundationSource Foundation
SSI Effects – Cross Interaction
0.0E+00
5.0E-11
1.0E-10
1.5E-10
2.0E-10
2.5E-10
0 0.5 1 1.5 2 2.5 3 3.5 4
Dimensionless Frequency ao
Ho
rizo
nta
l Am
pli
tud
e
1
Source M=10Receiver M=10Source M=5Receiver M=5Source M=1Receiver M=1
Receiver FoundationReceiver FoundationSource FoundationSource Foundation
Receiver FoundationReceiver FoundationSource FoundationSource Foundation
SSI Effects – Cross Interaction
0.0E+00
5.0E-11
1.0E-10
1.5E-10
2.0E-10
2.5E-10
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
Dimensionless Frequency ao
Ho
rizo
nta
l Am
plit
ud
e
d/a=0.25
Source Foundation
d/a=1.00
d/a=2.00
d/a=3.00
d/a=0.25
Receiver Foundation
d/a=1.00
d/a=2.00
d/a=3.00
Receiver FoundationReceiver FoundationSource FoundationSource Foundation
Receiver FoundationReceiver FoundationSource FoundationSource Foundation
Traveling Wave Effects
After Betti et al.
Traveling Wave Effects
After Betti et al.
Traveling Wave Effects
After Betti et al.
Traveling Wave Effects
After Betti et al.
SH-Waves
After Betti et al.
P-Waves
After Betti et al.
SV-Waves
After Betti et al.
Rayleigh Waves
After Betti et al.
Traveling Wave Effects Inertia Effects were Not Important but yet
SSI significantly affects the response
Asynchronous Motion Excite Antisymmetric Vibration Modes
SSI effects cannot be ignored
After Betti et al.