SSI – Problem Definition Earthquake Analysis Structures supported by rigid foundations Earthquakes=>Specified motion of base Rigid Base Analysis 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 Interaction Inertial forces in structure are transmitted to flexible soil Kinematic Interaction Stiffer foundation cannot conform to the distortions of soil TOTAL=INERTIAL + KINEMATIC Seismic Excitation
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 DirectMultiStep
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 Complexity
Idealized Interaction Analysis Preliminary description of free field motion before any structure has been built The definition of the motion itself the control motion in terms of response spectra, acceleration records etc The location of the control motion free surface, soil-rock interface The generation mechanism at the control point vertically or obliquely incident SH or SV waves, Rayleigh waves, etc.
Idealized Analysis Idealized Interaction Analysis Tools: FEM, BEM, FDE, Analytical solutions Direct Methods Evaluation of Dynamic Response in a Single Step MultiStep Methods Evaluation of Dynamic Response in Several Steps SUPERPOSITION Two-Step Two-Step Kinematic+Inertia Interaction Kinematic+Inertia Interaction Three-Step Three-Step Rigid Foundations Rigid Foundations Lumped Parameter Models Lumped Parameter Models Substructure Substructure Division to Subsystems Division to Subsystems Equilibrium & Compatibility Equilibrium & Compatibility True Nonlinear Solutions
Finite Element Method (FEM) Governing Equation Modal Analysis Modal Analysis Direct Integration Direct Integration Fourier Analysis - Complex Response Fourier Analysis - Complex Response Solution Techniques
FEM Solution Techniques Selection Criteria Cost and Feasibility Paramount Consideration Accuracy Differences - Handling of Damping - Ability to Handle High Frequency Components of Motion
FEM - Modal Analysis mDamping is neglected during early stages m Actual displacements are damped m Damping is considered in arbitrary manner m Structural Dynamics: First few modes need to be evaluated (<20) m SSI: Acceleration response spectra over a large frequency range and large number of modes need to be considered (>150) m Not recommended for Direct SSI - Stiff Massive Structure Soft Soil m OK for Substructure
FEM - Direct Integration Time Marching Schemes Newmark’s Methods, Wilson Methods, Bathe and Wilson Cubic 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 Method Time Marching Scheme Governing Equation Discrete Form in Time
FEM-BEM Coupling Staggered Solutions Can be Solved in a Staggered Approach... BEM FEM
FEM-BEM Coupling Staggered Solutions Compatibility of Displacements at Interface at Interface BEMSolverFEMSolver Equilibrium of Forces at Interface ExternalExcitationExternalExcitation At Every Time Step...
FEM-BEM Coupling Advantages 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 Foundation Models Reissner (1936) Analytic Solutions to Vertical Vibration of Circular Footing Due to Harmonic Excitation Assumptions: Elastic ½-space Material G,v, Uniform Vertical Pressure Formed Basis of Almost All Analytical Studies
Lumped Parameter Foundation Models Quinlan and Sung Assumed Different Pressure Distributions Richart & Whitman Effects of Poisson’ Bycroft (1956) Displacement Functions Hsieh K and C in terms of Soil and Foundation Parameters
Lumped Parameter Foundation Models Lysmer Analog Constant Lumped Parameters Richart Hall & Wood(1970) Gazetas (1983) Wolf (1988)
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.