Numerical Methods in Soil Mechanics BEM FEM soil structure

Numerical Methods in Soil Mechanics BEM FEM soil structure The chapter reviews aspects of the soil–structure interface behavior at the element level and the numerical integration of the corresponding interface constitutive models. The design of structures subjected to soil–structure interaction and to contact with friction should be tackled using soil–structure interface constitutive equations. These laws differ from the laws for soils because of three main features: the size of the relative displacements and of relative rotations between grains, the high level of dilatancy and contraction under shearing, and the presence of an intense degradation effect resulting from localization in the pattern of a shear band. The elastoplastic interface constitutive equations are easy to use but do not modelize all these effects. The incrementally non-linear interface constitutive equations are versatile for modeling all these interdependent phenomena. In addition, applications to piles under tension loading are presented to illustrate the results of these procedures.

Applications of Staggered BEM-FEM Solutions to Soil-Structure Interaction D.C Rizos, M ASCE and Z.Y Wang University of Nebraska Lincoln, Lincoln NE 68588 drizos@unlinfo.unl.edu Abstract: The present work introduces a direct time domain BEM-FEM formulation for 3-D SoilStructure Interaction analysis The proposed BEM that is based on the B-Spline family of fundamental solutions computes the dynamic response of the soil domain through a superposition of the characteristic B-Spline impulse responses Standard direct integration FEM procedures are used to compute the dynamic response of the structure A staggered solution process is proposed for the coupling of the two methods The proposed methodology is applied on the problem of the dynamic through-the-soil interaction of massive foundations, and the examples presented in this work demonstrate the accuracy and efficiency of the method Introduction The idea of coupling the FEM and the BEM finds its origins in the work of McDonal and Wexley in the beginning of 1970’s on the microwave theory The first organized formulation was presented by Zienkiewicz and his coworkers (1977) for analysis of solids Coupled BEM-FEM procedures are of three general types The first one is a Boundary Element (BE) approach, which considers the Finite Element (FE) subdomains as equivalent BE subregions by transforming the force-displacement relations of the FEM to “BEM-like” traction-displacement relations The second approach is the FE, in which, the BE equations are considered as a special case of the FE procedures Staggered BEM-FEM solutions have been implemented in fluid-structure interaction analysis The coupling of the FEM with the BEM for wave propagation and soil-structure interaction problems follows similar procedures The solutions are obtained in either a direct time domain, or a frequency (transform) domain approach Most of the coupled FEM-BEM solutions reported in the literature are in the frequency domain and adopt the FE or BE approach One can mention the work of Bielak et al (1984), Gaitanaros and Karabalis (1986), Aubry and Clouteau (1992), and Chuhan at al (1993), among others Only a few publications have dealt with the time domain BEM-FEM techniques for SSI and wave propagation problems Karabalis and Beskos (1985) and Spyrakos and Beskos (1986) have reported on 2-D and 3-D flexible foundations following the BE approach Fukui (1987) and Estorff and Kausel (1989,1990) reported on more generally applicable coupling formulations for 2-D scattering of anti-plane waves and 2-D plane-strain This work employs the B-Spline BEM formulation for 3-D wave propagation and SSI in elastic media reported by Rizos (1993) and Rizos and Karabalis (1994,1998) A staggered solution algorithm for the coupling with standard FEM processes in the direct time domain is introduced BEM Formulation The direct time domain BEM employed in this work is developed based on a special case of the Stokes fundamental solutions for the infinite elastodynamic space that assumes that the time variation of the body forces in the domain is defined by the B-Spline polynomials The derived B-Spline fundamental solutions accommodate virtually any order of parametric representation of the time dependent variables without excessive computational effort and implicitly satisfy the continuity conditions of the Stokes fundamental solutions A detailed formulation and integration of the B-Spline fundamental solutions and of the associated BEM can be found in the work of Rizos (1993), and Rizos and Karabalis (1994, 1998) Under the assumption of a small displacement field in a linear isotropic and homogeneous space, the Navier-Cauchy equations of motion can be expressed in the Love integral identity form as, { [ ] } cij (ỵ )u i (ỵ , t ) = ∫ U ijB x, t ; ỵ t (n )i (x, t ) − TijB [x, t ; î ui (x, t )] dS (1) S where S is the bounding surface of the elastodynamic domain, ỵ, and x represent the receiver and source points, respectively, and the tensor cij is known as the “jump” term that depends on the geometric characteristics of the domain in the neighborhood of the receiver point The tensors ui (x, t ) , and t (n )i (x, t ) are the displacement and traction fields of the actual elastodynamic state, and the tensors U ijB , and TijB are the B-Spline fundamental solutions of the infinite elastodynamic space (Rizos and Karabalis 1998) Appropriate spatial and temporal discretization schemes along with a transformation of tractions to forces are applied on equation (1) and the system of algebraic equations is derived as ~ N N T*u N = G *L−1f soil + RN = Gf soil + RN (2) where T* and U* are coefficient matrices derived on the basis of the B-Spline fundamental solutions and depend only on the first and/or second time steps, L is the traction-force transformation matrix obtained on a virtual displacement approach and vector RN represents the influence of the past time steps on the current step N and is always known Equation (2) can be solved in a time marching scheme to obtain the B-Spline impulse response of the elastodynamic system The B-Spline impulse response of all degrees of freedom due to unit force excitations can be collected in a matrix form that represents the time dependent flexibility matrix, BR(t), of the elastodynamic system The response u(t) to an arbitrary excitation f(t) can be calculated by an appropriate superposition of the B-Spline impulse responses as j u(t ) = ∑ BR(t − t i )f (τ i ), i =1 [ ] t ∈ t1 , t j and τ i = ti + ti +1 + L + t i +k −1 , k −1 k >1 (3) where k is the order of the B-Spline fundamental solutions Matrices BR are independent of the actual external excitations and need to be computed only once for each soil region FEM Formulation The FEM system of equations for the structural model is also solved in a time marching scheme using Newmark’s algorithm The FEM equations relate incremental forces, ˜f iFE , to displacements ˜u iFE on discrete nodes in the FE model at time interval i, and are presented symbolically as, ˆ ˜u FE = ˜fˆ FE K i i (4) ˆ is the dynamic stiffness matrix and ^ indicates quantities related to the where K Newmark’s process BEM-FEM Coupling At the interface of the FE and BE models, the compatibility conditions of the displacement vector, u, and force vector, f, need to be satisfied at all time instances tj, BE (t j ) = u intFE (t j ) u int f intBE (t j ) = −f intFE (t j ) (5) The coupling of the two models is achieved through a staggered solution approach according to which the solution of one method serves as initial conditions to the other at every time step, as depicted in Figure In view of equation (3) the BEM solver evaluates displacements by a superposition of the B-Spline impulse responses without solving any system of equations The FEM solver, however, needs to solve for the unknown forces at the BE-FE interface It is apparent that this coupling scheme increases the efficiency of the solution by reducing the computing time The proposed scheme has shown superior accuracy and stability for the examples examined so far fFE f intFE (t j ) = −f intBE (t j ) fBE FEM BEM Solver Structure Solver Soil uBE uFE FE (t j ) = u intBE (t j ) u int uBE Figure Staggered Solution Scheme Numerical Examples Analysis of Rigid Surface Massive Foundations This example examines the through-the-soil interaction of two adjacent massive rigid foundations The “structural component” of this soil-structure system pertains only to inertia forces, which are not known a priori, and demonstrates the compatibility of the BE solver with the proposed solution scheme The footings are square of side 2b=5 and their weights correspond to mass ratio M=10 The constants of the surrounding soil medium are Poisson’s ratio ν = 1/3, mass density ρ=10.368 lb.sec2/ft4 and modulus of elasticity E=2.5898x109 lb/ft2 The surface of the half space is modeled by node quadrilateral elements and each footing covers an area of 4x4 elements The rigid conditions are implemented according to the rigid surface element introduced by Rizos (1999) The frequency domain solutions are due to harmonic forcing functions of unit magnitude applied on one foundation The B-Spline impulse response matrices of the foundation system are obtained only once Subsequently, for each excitation, the solution is obtained in the time domain through the procedure outline above and the maximum amplitude of the steady state is defined This approach is very efficient since the BE solver reduces to a mere superposition of pre-computed quantities, as implied by Equation (3) Figure shows the maximum amplitude of response of the excited and the unloaded foundations as a function of the dimensionless frequency In this example the footings are spaced at a distance d/b=0.25 apart The results are compared to the ones reported by Huang (1993) and the accuracy is evident Other modes of vibration as well as distance ratios have shown the same accuracy 1.2E-10 Excited - Proposed Work Unloaded-Proposed Work 1.0E-10 Excited - Huang Unloaded - Huang Posin(ωt) Amplitude 8.0E-11 2b d Half Space 6.0E-11 4.0E-11 2.0E-11 0.0E+00 0.5 1.5 2.5 3.5 Dimensionless Frequency ao Figure Through-the-Soil interaction of Massive Rigid Foundations Hollow Rigid Surface Foundation This example examines a square footing of size 2a=5 ft with a centered square hole of size 2d for which d/a=0.75 The mass of the foundation varies so that the mass ratio M takes the values of M=1,3,5 and 10 The footing rests on the elastic half-space described in the previous example A series of harmonic excitations of various frequencies are applied at its center in order to define the amplitude of the steady state response The amplitudes of the vertical mode of vibration are shown in Figure as function of the dimensionless frequency for all mass ratios considered A comparison with a solution reported by Huang (1993) for mass ratio M=3 is also shown All modes of vibration have been examined, as well as a number of d/a ratios The proposed method compared favorably and always converged for the considered frequency range 1.2E-10 M=10 M=5 M=3 M=1 Huang M=3 M=10 Posin(ωt) 1.0E-10 Amplitude (ft) M=5 2d 2a 8.0E-11 6.0E-11 Half Space M=3 4.0E-11 M=1 2.0E-11 0.0E+00 Dimensionless Frequency a0 Figure Vertical Response of Hollow Foundation Conclusions A methodology is developed for the efficient coupling of the Finite Element with the Boundary Element Method for 3-D wave propagation and Soil-Structure Interaction Analysis in the direct time domain The method uses the newly developed B-Spline BEM along with standard FEM processes The coupling is obtained through a staggered scheme, which satisfies the compatibility and equilibrium conditions at the interface boundary between the BEM and FEM domains This article presented the first attempt to implement the method and the problem of analysis of massive foundations was selected In such problems, kinematic and inertial interaction effects are present Although the FEM domain does not contain elastic or damping forces of a real structure, this class of problems verifies the suitability of the B-Spline BEM method to such staggered solution schemes It has been shown that the proposed methodology is accurate and stable for the class of problems considered, and is very efficient The present formulations will be further developed to account for elastic and damping forces of a structure References D Aubry and D Clouteau (1992): A subdomain approach to soil-structure interaction J Bielak, R.C MacCamy and D.S McGhee (1984): On the coupling of finite element and boundary integral methods In S.K Datta (Ed.): Earthquake Source Modeling, Ground Motion, and Structural Response, AMD-Vol 60, pp 115-132, ASME New York Z Chuhan, J Feng and W Guanglun (1993): A method of FE-BE-IBE coupling for seismic interaction of arch dam-canyons In M Tanaka et al (eds): Boundary Element Methods Elsevier O von Estorff (1990): Soil-structure interaction analysis by a combination of boundary and finite elments In: S.A Savidis (ed.): Earthquake Resistant Constructions and Design Balkem (Rotterdam) O von Estorff, and E Kausel (1989): Coupling of boundary and finite elements for soil-structure interaction problems, Earthquake Engineering and Structural Dynamics, 18, 1065-1075 T Fukui (1987): Time marching BE-FE method in 2-D elastodynamic problem International Conference BEM IX Stuttgart A.P Gaitanaros and D.L Karabalis (1986): 3-D flexible embedded machine foundations by BEM and FEM In: D.L Karabalis (ed.): Recent Applications in Computational Mechanics ASCE (New York) C-F Huang (1993): Dynamic Soil-Foundation and Foundation-Soil-Foundation Interaction in 3-D Ph.D Dissertation, University of South Carolina, Columbia SC D.L Karabalis and D.E Beskos (1985): Dynamic response of 3-D flexible foundations by BEM and FEM In D.L Karabalis (ed.): Recent Applications in Computational Mechanics ASCE (New York) D.C Rizos (1999): A Rigid Surface Element for the B-Spline direct time domain BEM Computational Mechanics (To appear) D.C Rizos (1993): An Advanced Time Domain Boundary Element Method For General 3-D Elastodynamic Problems Ph.D Dissertation, University of South Carolina, Columbia South Carolina D.C Rizos, and D.L Karabalis (1998): A time domain BEM for 3-D elastodynamic analysis using the B-Spline fundamental solutions, Computational Mechanics, 22, No 1, pp 108-115 D.C Rizos, and D.L Karabalis (1994): An advanced direct time domain BEM formulation for general 3-D elastodynamic problems Computational Mechanics 15, 249-269 C.C Spyrakos and D.E Beskos (1986): Dynamic response of flexible strip foundations by boundary and finite elements Soil Dynamics and Earthquake Engineering, 5, 84-96 O.C Zienkiewicz, D.W Kelly, and p Bettess (1977): The coupling of the Finite Element Method and Boundary Solution procedures, International Journal for Numerical methods in Engineering, 11, 355-375  ... by reducing the computing time The proposed scheme has shown superior accuracy and stability for the examples examined so far fFE f intFE (t j ) = −f intBE (t j ) fBE FEM BEM Solver Structure. .. damping forces of a structure References D Aubry and D Clouteau (1992): A subdomain approach to soil- structure interaction J Bielak, R.C MacCamy and D.S McGhee (1984): On the coupling of finite... FE-BE-IBE coupling for seismic interaction of arch dam-canyons In M Tanaka et al (eds): Boundary Element Methods Elsevier O von Estorff (1990): Soil- structure interaction analysis by a combination of