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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


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


Lysmer AnalogConstant Lumped Parameters

Richart Hall & Wood(1970)

Gazetas (1983)

Wolf (1988)


Representative Lumped Parameter Values - Square


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


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


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


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




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



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.

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