Soil-Structure InteractionJames LewisFSI-ASEN 5519 University of ColoradoApril 27, 2004Overviewz Part I: Introduction– Coupled Problems– Solution Techniquesz Part II:Typical Soil-Structure Problem Formulations– Assumptions– Fields– Time Integration – Element Discretization– Applicationsz Part III: Simple Soil-Structure Problem Setup – Structure Equations – Foundation Equations– Soil Structure Systemz Part IV: Conclusion – Further Difficulties – Direction of the Future z ReferencesPart I: Introductionz Coupled Field Problems– In the age of modern engineering computers have expanded the range and complexity of the problems that can be practically handled. Of particular interest is in the area of multi-physics dynamic interaction problems, using coupled fields. – Numerical solutions of coupled field equations was traditionally achieved with three different approaches:z Field Eliminationz Simultaneous Solution z Partitioned Solution ProcedurePart I: Problematic Solution Techniquesz Field Elimination– By eliminating one of the coupled fields a time integration solution of a single system of increased order is applicable, but maybe more complicated due to the raised order of the resulting single system. z Simulation Solution– Setting up the system of coupled equations into a system of simulation equations to be solved, leads to the loss of the sparseness and solution difficulties.z These solution approaches result in systems that are difficult to implement and solve, and are rendered almost useless in practical applications. z The use of preexisting single field software is not possible with these solution procedures.Part I: Partitioned Solutionz Partitioned Solution Procedure– In this procedure the solution of each field is done separately, making use of preexisting single field software, in a staggered (alternating) procedure where the interaction of fields is accomplished with predicting external forcing quantities, extrapolated from the solution of the previous step. – In this solution procedure the modularity of the separate fields is utilized, as well as preserving the sparseness of the originals systems, simplifying solution procedures for efficient computation, with the possible use of parallel processors.Part II: Typical Soil-Structure Problem Formulationsz Assumptions– Linear elastic material behavior of structure and foundation– Constant contact/No uplift conditionz Fields– Near Field (nf) Structure– Far Field (ff) Soilz Time Integration– Time Domain– Frequency Domain– Cyclic Dynamic/Seismic ExcitationPart II: Typical Soil-Structure Problem Formulationsz Element Discretization– Structurez Finite Elements (FEM)– Soilz Infinite Elements – Elements extend to infinity– Acts as a continuous medium for the propagation of waves– Will not allow a false boundary reflection z Boundary Elements (BEM) for the Soil– For linear problems of homogeneous media– Requires only boundary discretization of considered domain – Acts as a continuous medium for the propagation of waves– Will not allow a false boundary reflectionPart II: Typical Soil-Structure Problem FormulationsPart II: Typical Soil-Structure Problem Formulationsz Applications– Foundations– Bridge Abutments– Driven Piles– Dams– Retaining Walls– Offshore Structure – ConsolidationPart III: Simple Soil-Structure Problem Setup z Seismic excitation of structure embedded in foundation ground. Incident seismic wave is refracted and reflected as it encounters discontinuities in the soil strata. This reflected portion of the wave will be used as the base rock motion.Part III: Simple Soil-Structure Problem Setup (FEM-BEM formulation)Part III: Simple Soil-Structure Problem Setupz Motion Equations of Any Structure– M mass matrix– C damping matrix– K stiffness matrix– d nodal displacement vector– a(t) seismic acceleration()taM1KddCdM −=++&&&[]1111 L=T1Part III: Simple Soil-Structure Problem Setupz Equations of Motion of the Structure in the time domain– btsoil-structure interaction forces– t subscript refers to all the DOF of the structure ()ttttttttta b1MdKdCdM +−=++&&&Part III: Simple Soil-Structure Problem Setupz Equations of Motion of the Structure in the frequency domainz frequency of the excitationz the Fourier Transform of z the Fourier Transform of z the Fourier Transform of ()()()()θθθθθttttttai b1MdKCM +−=++− 2θ()θtd()θtb()θta()ttd()ttb()tatPart III: Simple Soil-Structure Problem Setupz Partition the nodal displacement vector and EOM of the structure in the frequency domain– the displacement vector corresponding to the soil-structure interface– d the displacements of the remaining nodes of the structure()()()=⇒=θθθbtbtdddddd()()()()+=++−θθθθθθbbbbbaib01M00MddKKKKC00CM00M''2bd0b =Part III: Simple Soil-Structure Problem Setupz Equations of Motion of the Soil in the frequency domain– interaction forces which act on the soil, at the soil-structure interface– nodal displacements, at the soil- structure interface, due to interaction forces , – square matrix whose elements are the dynamic stiffnessescorresponding to the DOF of the soil-structure interface. This complex coefficient matrix must be established by analyzing the secondsubstructure, i.e. the foundation ground (soil).()() ()θθθbbbdYb =()θbb()θbd()θbYPart III: Simple Soil-Structure Problem Setupz Equations of Motion of the Soil-structure system in the frequency domainz Conditions for the nodes of the interface of the two substructuresthe compatibility conditionthe equilibrium conditionthe two above conditions provide the following relationshipbbdd =0=+bbbbbbbdYb ×−=Part III: Simple Soil-Structure Problem Setupz Now the EOM for the structure can be expressed as followsz Finally, the complex linear system is solved for all the values of the excitation frequency q for which the matrix has previously been calculated. Thus, the vectors of the displacement are obtained in the frequency domain and a discrete inverse Fourier transform provides the response in the real time domain of the structure in interaction with the foundation ground.bbbbiiKCMGKGKCMG++−==++−=