ssifibo: a matlab toolbox for soil-structure interaction...

SSIFiBo: a Matlab toolbox for soil-structure interaction problems with finite andboundary elements

P. Galvın, A. RomeroEscuela Tecnica Superior de Ingenieros, Universidad de Sevilla, Camino de los Descubrimientos, 41092 Sevilla, Spain

email: [email protected], [email protected]

ABSTRACT: Following theSSIFiBo toolbox is presented.SSIFiBo is a numerical model developed inMATLAB to study soil-structure interaction problems. The model is based on a three dimensional boundary element-finite element coupled formulation intime domain. This model allows computing structural forced-vibrations, as well as seismic responses. Two numerical examples aresolved with the proposed technique: ground-borne isolation with open and filled trench, and the seismic response of a tall chimney.

KEY WORDS: MATLAB toolbox, soil-structure interaction, seismic incident wave field, BEM-FEM model, transient analysis.

1 INTRODUCTION

Soil-Structure Interaction (SSI) is a field of interest thatinvolvesstructural analysis considering flexibility and damping due tothe soil. Induced vibrations by rotatory machines, dynamiceffects due to railway traffic, seismic problems, and dynamicbehaviour of foundation systems, are examples where SSI isan important issue. The effects of SSI are important andcannot be neglected in this type of problems [1]. Kausel [2]presented an exhaustive review of the main developments in thistopic. Recently, Clouteau et al. [3] also reviewed the proposednumerical models to study the dynamic behaviour of structureson elastic media.

Numerical models based on the Boundary Element Method(BEM) and the Finite Element Method (FEM) allow to studySSI problems rigorously. The BEM [4] is especially suitedfor the analysis of wave propagation in soils. Sommerfeldradiation condition [5] is satisfied implicitly and the semi-infinite character of soils is well considered. The FEM is veryuseful to analyse the dynamic behaviour of structures taking intoaccount nonlinear effects [6].

This communication presents a numerical tool to study soil-structure interaction problems. TheSSIFiBo (Soil StructureInteraction with Finite and Boundary elements) is a set ofMATLAB1 functions based on a fully coupled 3D BEM-FEMmodel formulated in time domain. The numerical modelis suitable for studying general dynamic problems of soil-structure interaction. The boundary element formulation allowsto represent seismic load sources, describing the incidentwavefield and the scattered waves. BEM-FEM coupling is performeddirectly. Once the interaction problem is solved, the solution atany internal soil point can be obtained. The toolbox is illustratedwith two problems: soil scattering waves due to open and filledtrenches, and the seismic response of a tall chimney.

1www.mathworks.es

2 NUMERICAL MODEL

TheSSIFiBo toolbox offers a set of function based on a three-dimensional BEM-FEM model formulated in time domain.SSI analysis are carried out by domain decomposition in twosubdomains represented by the BEM and FEM. Soil behaviouris represented by the BEM, while the structures are modelledwith the FEM. The wave propagation problem deals to thedecomposition of the total wave field in two terms: the incidentwave field and the scattered wave field.

The BEM is based on a time marching procedure to obtain thetime variation of the boundary unknowns; i.e. displacementsand tractions. Piecewise constant time interpolation functionsare used for tractions and piecewise linear functions fordisplacements. The fundamental solution for displacementandtraction are evaluated analytically, and nine node rectangularquadratic elements are used for spatial discretization. Explicitexpressions of the fundamental solution for displacementsandtractions due to an impulse point load in a three dimensionalelastic full-space can be seen in Reference [7]. An approachbased on the idea of using a linear combination of equations forseveral time steps in order to advance one step is used to ensurethat the stepping procedure is stable in time. The algorithmefficiency is enhanced using the truncation technique previouslypresented in Reference [8]. After boundary unknowns aresolved, the scattered wave field at any internal point is computedby means of the integral representation of Somigliana identity.

Coupling of BEM and FEM equations is carried out byimposing equilibrium and compatibility conditions at the soil-structure interface. Both systems of equations are assembledinto a single global system, together with the equilibrium andcompatibility equations [10].

3 SSIFIBO TOOLBOX

SSIFiBo is implemented in aMATLAB package of functions.The toolbox includes a complete set of subroutines for the BEMin time domain, including the computation of the boundaryelement influence matrices and the time marching procedure.The FEM module does not include any preprocessor. Instead,

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014Porto, Portugal, 30 June - 2 July 2014

A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)ISSN: 2311-9020; ISBN: 978-972-752-165-4

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a gateway forANSYS2 allows to import directly the structuremodel.

The source code is structured in three parts. Firstly,boundary element matrices over all time steps and the FEMdynamic stiffness matrix are computed. After, BEM-FEM nodalconnectivity at the soil-structure interface is checked and theglobal system of equation is assembled. Finally, the solutionfor all time steps is assessed.

The BEM matrices are derived from an element subdivisionprocedure. The spatial integration is done only in those partsof the elements under the effects of the fundamental solutionwaves, according to the causality condition of each term of thefundamental solution. The element subdivision integration isdone with a Gauss quadrature using four integration points whenthe collocation point does not belong to the subdivision. A peaksingularity appears at the fundamental solution if the collocationpoint belongs to the element subdivision. In this case, thesingularity is avoided with a new subdivision. An alternativeprocedure was tried out with an adaptive quadrature inMATLABusing a double integral over a planar region containing thesingular integrand.quad2 MATLAB command allows to solvethis kind of problem with a numerical integration over a mappedregion. However, the refined mapping necessary to provide highaccuracy leads to an excessive computational cost.

The finite element dynamic stiffness matrix is computedfollowing an implicit time integration GN22 Newmark schema[6] without any difficulty.

The solution for all time steps is addressed from the solutionof the global BEM-FEM system of equations. Previously, theLU system factorisation is computed since the global assembledmatrix remains the same for all time step. The right-hand-side account for the boundary conditions at the current timestep and the influence of the previous time steps. The systemfactorisation and the time solution are carried out using theSuperLU library [11], [12]. That library allows to reducethe computational effort regarding with thelinsolve ormldivide (back slash)MATLAB commands for solving linearsystem of equations.

Several enhancements are taking into account to improvethe numerical model performance. BEM and FEM algorithmsrequire large computing resources (CPU and memory storage).The package takes advantage of parallel computing usingmulticore processors and computer clusters throughParallelComputing Toolbox. The workload related with thecomputation of the BEM matrices and the influence of theprevious time steps is distributed among the available processorswith high time performance. Currently, the maximum numberof parallel process for a local cluster is 12. Also, C/C++ codesfor some of BEM subroutines were generated using theMATLABCoder to enhance the toolbox capability.

TheMATLAB toolbox implementation facilitates its use eitherin academic and engineering environments.MATLAB softwareis common used in most of engineering schools by students andresearchers. The package modularity makes possible simpleand efficient implementation of new enhancements. Therefore,

2www.ansys.com

it could be a powerful tool for researching. A trial version isavailable for further researches3.

4 NUMERICAL EXAMPLES

The SSIFiBo functioning is illustrated with two numericalexamples. Firstly, the effects of an attenuation barrier inwave propagation on soil are shown. Numerical results werecompared with experimental measures presented by Klein et al.[13]. After, the seismic response of a tall chimney is studied.This structure was previously analysed by Luco [14] with asimplified beam model.

In these problems, the structures were modelled with FEMand the soil was represented as a homogeneous half-space withBEM. The surrounding soil around the structures was extendeda distance enough to avoid truncation effects [4]. Frequencyresponses were computed from time histories applying the FastFourier Transform (FFT).

4.1 Vibration isolation by open and filled trenches

The BEM-FEM methodology is applied to examine theeffectiveness of opened and filled trenches to mitigate wavesgenerated by impulsive loads. Numerical results were comparedwith experimental tests done by Klein et al. [13]. Soil propertiesat the tests site were: P-wave propagation velocitycp = 250m/s,S-wave propagation velocitycs = 100m/s, and densityρ =1750kg/m3.

The influence of the filling material trench is also shown.Filling material properties were: Young’s modulusE = 0.26×106N/m2, Poisson’s rationν = 0.3, and densityρ = 100kg/m3.The BEM-FEM model presented in Figure 1.(b) was used tostudy this problem. The filling material was represented withsolid finite elements. The time step was set as∆t = 0.005sin order to avoid stability problems with the proposed spatialdiscretization [8].

Listing 1 shows the input file for this example. Input dataincludes soil properties, the structural damping and the time stepsettings.SSIFiBo calling reads soil and trench discretizationsfrom a job database, as well the load time history applied at theground excitation point. This example does not account neithersoil nor structural damping, according with the assumptionmade in the Reference [13]. Also, the trench filled materialdamping was neglected in order to simplify the problem.

% Job t i t l ejob= ' f i l led trench '

% Soil propertiescp=250 % P−wave propagation velocity [m/ s ]cs=100 % S−wave propagation velocity [m/ s ]ro=1750 % Density [kg/mˆ3]damp=0 % Damping

% Structural damping (C=alpha*M+beta*K)alpha=0 % [sˆ−1]beta=0 % [s ]

% Time step sett ingnstep=400 % Number of time stepsat=5e-3 % Time interval [ s ]

3personal.us.es/pedrogalvin/ssifibo.en.html

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

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0.6 0.6 0.6

2.0

p(t)

3.0

z

x

(a)

z

yx

(b)

Figure 1. (a) Geometry of a filled trench and (b) BEM-FEM discretization of one quarter model.

% Call SSIFiBossifibo(job,cp,cs,ro,damp,alpha,beta,nstep,at)

Listing 1. Input file to callSSIFiBo for the solution of soilisolation by a filled trench.

Figure 2 shows wave field displacements due to a Heavisideload pH(t) = p0H(t), where p0 = 1N. Results are plottedin dimensionless formuH

o = uH(t)πGr/p0, where G is theelasticity shear modulus andr is the distance from the source tothe observation points. The effect of the open trench is clearlyobserved in the soil response (Figures 2.(b,e)). An effectivenessloss of soil displacements is reached for points after the trench.Rayleigh waves are mitigated for a soil depth close to the trenchdepth, as can be seen in Figure 2.(e). Similar effects occursforfilled trench (Figures 2.(c,f)), without a reduction of the trencheffectiveness.

Figure 3 shows the amplitude reduction factor (AR) computedas the relation between the vertical displacements for openor filled trench, and the initial configuration. The amplitudereduction factor was estimated at different soil points alonga straight line defined byy = 0, z = 0 and x ≥ 0 (Figure1). Experimental test were carried out for a harmonic load at20Hz. Experimental and numerical results match quite well forpoints near to the trench. For larger distance the differences aremore important probably due to the soil at the test site was notperfectly homogeneous [13]. Results with filled trench are quitesimilar, and it allows an easier construction of the trench.

0 5 10 150

0.5

1

Distance [m]

AR

[−]

Figure 3. Experimental (grey circles) and numerical (blacklines) amplitude reduction factor for a harmonic load at20Hz computed for open trench (solid line) and filledtrench (dashed line) at different distances from the trench.

4.2 Seismic effects on a tall chimney

Second example concerns with the analysis of the seismicresponse of a tall chimney studied previously by Luco [14].Seismic responses due to horizontal SH incident wave field arestudied. Numerical result is compared with the presented byLuco. After, induced effects by El Centro (1940) accelerogramare analysed.

The Caletones chimney is a 152.4m tall reinforced concretestructure (Figure 4.(a)). The external diameters of the structureat the base and at the top are 16.3m and 6.6m, respectively.The thickness of the reinforced concrete wall varies linearlyfrom 0.61m at the base to 0.165m at the top. A stainlesssteel shell with a thickness 6.35× 10−3m and height 4.2mwas placed atop the chimney. Concrete material properties are:Young’s modulusE = 30×109N/m2, Poisson’s ratioν = 0.25,and densityρ = 2500kg/m3. The structure rests over a soilwith cp = 400m/s,cs = 200m/s, and density 1800kg/m3. Thefoundation has a total height of 6.2m and the equivalent basediameter is 26.8m. Structural damping ratio is considered 2%(α = 0.17s−1, β = 3.74×10−4s).

Figure 4.(b) shows discretization of the soil-structure system.Shell and solid finite elements were used to represent thechimney and the foundation, respectively. The time step∆t =0.015s was chosen to compute accurate results for maximumrange frequency of 25Hz, enough to represent the dynamicbehaviour of the structure. Resonance frequencies vary in therange between 0.71Hz and 16.31Hz [14]. A similar input file toListing 1 was used.

Figure 5 shows the dynamic behaviour due to a horizontal SHincident wave field, with unitary amplitude. Numerical resultcomputed with the proposed method and those presented byLuco for the horizontal response at the foundation are compared.The agreement between both results is good, especially atlow frequencies. Several differences are found for highermodes, probably due to the characteristics of the models.Luco modelled the structure as variable section beam, and soilbehaviour was considered using a compliance matrix.

Next, the dynamic behaviour of the structure due to El Centro(1940) seismic accelerogram is evaluated. This seismic wascharacterized by energy distribution over a range frequencyup to 10Hz. The energy was concentrated mainly in the N-S component. Figure 6 shows the response at the top ofthe foundation in N-S direction. The accelerogram is alsorepresented.


707

(a) (b) (c)

(d) (e) (f)

-0.73 -0.48 -0.23 0.02 0.4Figure 2. Dimensionless soil displacements at (a-c) the surface and (d-f) the planey= 0 in different cases: (a,d) without trench,

(b,e) opened trench and (c,f) filled trench.

4.2

148.2

6.2

∅= 6.6∅= 6.6

∅= 26.8∅= 16.3

SH

(a) (b)

Figure 4. (a) Caletones chimney and (b) BEM-FEM model.

Time history presents similar evolution than the accelero-gram, with high amplifications for the top of the structure.Maximum level was reached at time 2.58s from the beginning.Figure 7 shows the chimney response at this time. Deformedshape practically corresponds with the first bending mode shapeat 0.52Hz. Maximum displacement occurs at the top with arelative base amplitude of 0.38m.

Frequency content exhibits maximum levels around the peakspresented in Figure 5. The seismic wave field is widelyamplified around 1.95Hz, matching with the main frequencycontent of accelerograms and the natural frequencies of thesoil-structure system.

This example has shown how the SSIFiBo can be used toassess the seismic effects on structures. General seismic fieldscan be included in the transient methodology by computing theincident waves at any point of the soil’s boundary. The incidentwave field is read during theSSIFiBo calling. This procedureallows considering non-plane waves or irregular soil geometry.

5 CONCLUSIONS

This work presented theSSIFiBo toolbox for MATLAB.SSIFiBo is a package of functions to study general Soil-Structure Interaction problems with Finite and Boundaryelements. The numerical model is based on the three


708

0 10 20 30 40 50−5

0

5

Time [s]

Acc

eler

atio

n [m

/s2 ]

(a)

0 2 4 6 8 100

2

4

6

Frequency [Hz]

Acc

eler

atio

n [m

/s2 /H

z]

(b)

Figure 6. (a) Time history and (b) frequency content of acceleration at N-S direction at the top of foundation (black line) inducedby El Centro (1940) accelerogram (grey line).

0 2 4 6 8 1010

−2

10−1

100

101

102

Frequency [Hz]

Acc

eler

atio

n [m

/s2 /H

z]

Figure 5. Frequency content of the horizontal accelerationdue to vertical SH-waves atop of the foundation: proposedsolution (black solid line) and Luco’s result (grey circles).

0.310.230.140.06

-0.07

Figure 7. Horizontal displacement at N-S direction inducedbyEl Centro (1940) accelerogram, at timet = 2.58s.

dimensional BEM-FEM formulation in time domain. Structuresare modelled with finite elements. The soil is representedwith boundary elements considering the radiation conditionimplicitly. The toolbox is available for a trial version.

In this communication, the model was validated by compari-son of two numerical examples with experimental or numericalresults presented previously by other authors. All the exampleswere aimed with the characterisation of the SSI effects. Theexamples illustrated the importance of SSI influence on wavepropagation barriers and the seismic behaviour of structures.

ACKNOWLEDGMENTS

This research is financed by the Ministry of Science andInnovation (Ministerio de Innovacion y Ciencia) of Spain underthe research project BIA2010-14843. The financial support isgratefully acknowledged. The authors also wish to acknowledgethe support provided by the Andalusian Scientific ComputingCentre (CICA).

REFERENCES

[1] J.P. Wolf, Dynamic soil-structure interaction, 1985.[2] E. Kausel, Early history of soil-structure interaction, Soil Dynamics and

Earthquake Engineering 30 (9) (2010) 822–832.[3] D. Clouteau, R. Cottereau, G. Lombaert, Dynamics of structures coupled

with elastic media - A review of numerical models and methods, Journalof Sound and Vibration 332 (10) (2013) 2415–2436.

[4] J. Domınguez, Boundary elements in dynamics, Computational Mechan-ics Publications and Elsevier Aplied Science, Southampton, 1993.

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[6] O.C. Zienkiewicz, The finite element method, 3rd Edition, McGraw-Hill,1986.

[7] P. Galvın, J. Domınguez, Analysis of ground motion dueto moving surfaceloads induced by high-speed trains, Engineering Analysis with BoundaryElements 31 (11) (2007) 931–941.

[8] A. Romero, P. Galvın, J. Domınguez, 3D non-linear timedomainFEM-BEM approach to soil-structure interaction problems,EngineeringAnalysis with Boundary Elements 37 (3) (2013) 501–512.

[9] D.D. Barkan, Dynamics of Bases and Foundations, McGraw-Hill, NewYork, 1962.

[10] M.J. Prabucki, O. von Estorff, Dynamic response in the time domain bycoupled boundary and finite elements, Computational Mechanics 6 (1)(1990) 35–46.

[11] X. Li, J. Demmel, J. Gilbert, iL. Grigori, M. Shao, I. Yamazaki, SuperLUUsers’ Guide, Tech. Rep. LBNL-44289, Lawrence Berkeley NationalLaboratory,http://crd.lbl.gov/ xiaoye/SuperLU/. Last up-date: August 2011 (September 1999).

[12] J. W. Demmel, S. C. Eisenstat, J. R. Gilbert, X. S. Li, J. W. H. Liu, Asupernodal approach to sparse partial pivoting, SIAM J. Matrix Analysisand Applications 20 (3) (1999) 720–755.

[13] R. Klein, H. Antes, D. Le Houedec, Efficient 3D modelling of vibrationisolation by open trenchs, Computers & Structures 64 (1-4) (1997) 809–817.

[14] E. Luco, Soil-Structure Interaction effects on the seismic response of tallchimney, Soil Dynamics and Earthquake Engineering 5 (3) (1986) 170–177.


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[15] R.W. Clough, J. Penzien, Dynamic of Structures, McGraw-Hill, NewYork, 1975.


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