2SSI – Problem Definition Earthquake AnalysisStructures supported by rigid foundationsEarthquakes=>Specified motion of baseRigidBaseAnalysisTall BuildingsAcceptableLight & FlexibleFirm FoundationsMethods focus on modeling of structureDisplacements wrt fixed baseFinite Element MethodsNuclear Power PlantsWrong AssumptionMassive & StiffSoft SoilsInteraction with supporting soils becomes important
3SSI – Problem Definition Machine FoundationSeismic ExcitationParametersLocal Soil ConditionsPeak AccelerationFrequency Content of MotionProximity to FaultTravel Path etcInertial InteractionInertial forces in structure are transmitted to flexible soilKinematic InteractionStiffer foundation cannot conform to the distortions of soilTOTAL=INERTIAL + KINEMATIC
6Cross Interaction Effects 3. …Reach Receiver…1. Moment is applied2. Waves Propagate…4. …and life goes on…
7SSI Effects Alter the Natural Frequency of the Structure Add Damping Through the Soil Interaction EffectsTraveling Wave Effects
8Methods 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.
9Methods of AnalysisMethodsCompleteIdealizedDirectMultiStep
10Complete Interaction Analysis High Degree of ComplexityAccount for the variation of soil properties with depth.Consider the material nonlinear behavior of the soilConsider the 3-D nature of the problemConsider the nature of the wave propagation which produced the ground motionConsider possible interaction with adjacent structures.
12Idealized Interaction Analysis Preliminary description of free field motionbefore any structure has been builtThe definition of the motion itselfthe control motion in terms of response spectra, acceleration records etcThe location of the control motionfree surface, soil-rock interfaceThe generation mechanism at the control point vertically or obliquely incident SH or SV waves, Rayleigh waves, etc.
13Idealized Analysis MultiStep Methods Direct Methods Idealized Interaction AnalysisTools: FEM, BEM, FDE, Analytical solutionsMultiStep MethodsEvaluation of Dynamic Response in Several StepsSUPERPOSITIONTwo-StepKinematic+Inertia InteractionThree-StepRigid FoundationsLumped Parameter ModelsSubstructureDivision to SubsystemsEquilibrium & CompatibilityDirect MethodsEvaluation of Dynamic Response in a Single StepTrue Nonlinear Solutions
15FEM Solution Techniques Selection Criteria Cost and FeasibilityParamount Consideration AccuracyDifferences- Handling of Damping- Ability to Handle High Frequency Components of Motion
16FEM - Modal Analysis Damping is neglected during early stages Actual displacements are dampedDamping is considered in arbitrary mannerStructural Dynamics: First few modes need to be evaluated (<20)SSI: Acceleration response spectra over a large frequency range and large number of modes need to be considered (>150)Not recommended for Direct SSI - Stiff Massive Structure Soft SoilOK for Substructure
17FEM - Direct Integration Time Marching SchemesNewmark’s Methods, WilsonJ Methods, Bathe and WilsonCubic Inertia MethodSmall Time Step for AccuracyStability and ConvergenceChoice of Damping MatrixFrequency Dependent Damping Ratio - filters out high frequency componentsProportional DampingGood Choice if True Dynamic Nonlinear Analysis is feasible
18FEM - Complex Response Fourier Transformation - Transfer Functions Transfer Functions Independent of External ExcitationControl of AccuracyEfficientOnly Linear or Pseudo non-linear analysis
28BEM – Methods BEM Advantages Infinite Media Surface Discretization ShortcomingsNon-symmetric matricesNot Efficient for Nonlinear
29eliminate disadvantages of each method and retain advantages SSI MethodsCombined BEM-FEMeliminate disadvantages of each method and retain advantagesApproachFEM ApproachBEM ApproachStaggered Solutions
31FEM Method Time Marching Scheme Governing EquationDiscrete Form in Time
32FEM-BEM Coupling Staggered Solutions Can be Solved in a Staggered Approach...
33FEM-BEM Coupling Staggered Solutions Compatibility of Displacementsat InterfaceBEMSolverFEMEquilibrium of ForcesExternalExcitationAt Every Time Step...
34FEM-BEM Coupling Advantages Independent Solutions for BEM and FEMIndependent Time Step SelectionSmaller Systems of EquationsBEM System of Reduced SizeIn the Absence of Incidence Displacement Field in Soil, BEM does not require Solution.
36Lumped Parameter Foundation Models Reissner (1936) Analytic Solutions to Vertical Vibration of Circular Footing Due to Harmonic ExcitationAssumptions:Elastic ½-spaceMaterial G,v,rUniform Vertical PressureFormed Basis of Almost All Analytical Studies
37Lumped Parameter Foundation Models Quinlan and SungAssumed Different Pressure DistributionsRichart & WhitmanEffects of Poisson’Bycroft (1956)Displacement FunctionsHsiehK and C in terms of Soil and Foundation Parameters
38Lumped Parameter Foundation Models Lysmer AnalogConstant Lumped ParametersRichart Hall & Wood(1970)Gazetas (1983)Wolf (1988)
39Lumped Parameter Foundation Models Representative Lumped Parameter Values - Square
40Lumped Parameter Foundation Models Representative Lumped Parameter Values Circular
41Lumped Parameter Foundation Models Stehmeyer and Rizos (2003)The Real SystemEquivalent SDOF SystemProperties k, and c are known to be frequency (w) dependent
42Lumped Parameter Foundation Models wn = 3.3x = 0.975
57Traveling Wave Effects Inertia Effects were Not Important but yet SSI significantly affects the responseAsynchronous Motion Excite Antisymmetric Vibration ModesSSI effects cannot be ignoredAfter Betti et al.