Static interaction analysis between beam and layered soil using a twoparameter elastic foundation
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Abstract
The work presents a finite element modeling of beam resting on a twoparameter layered soil. The behavior of the soil continuum and the beam are assumed to be linear and homogeneous isotropic. Using the strain energy expressions, the shear strain of the beam element and the soil foundation are taken together in the analysis. In this approach, the stiffness matrix of each component is elaborated and integrated in the finite element analysis. First, various examples are elaborated to show the effectiveness of the proposed approach and the ability of the numerical program developed for this concern. Second, the analysis is widened to study the influence of the soil properties on the interface continuum and on the beam responses. Third, a parametric study is carried out to highlight the effect of the position of springs at the interface continuum, the properties of soil, the deepest of the soil foundation and the ballast layer on the response of the interface and the beam itself. Moreover, shear deformations are presented to show the crucial influence on the beam, on the structure and on the interface behaviors. Obtained results show pertinent results corresponding to the interface continuum and the beam responses.
Keywords
Static soilbeam interaction Layered soil Twoparameter elastic foundation model Finite element method Elastic foundation Ballasted layer Soil propertiesIntroduction
The concept of beams on elastic soil foundation has been widely used in different fields of engineering, such as strip foundation, railroads tracks, building, dams and airport runway. The soil mechanic exhibits a very complex behavior of foundations due to the heterogeneity, physical composition, and presence of imperfections and pores of soils. Concepts, status and various analysis methods of the soilstructure interaction researches have been illustrated in the recent art state review (Prakash et al. 2016).
The quantification of the soilstructure interaction is a challenge until the present moment. The modeling of the contact between the structure and the soil foundation is a primordial task of this analysis. Analytical solutions are restricted compared to numerical studies using twoparameter soil foundation model of beams resting on isotropic or anisotropic elastic halfspace (Johnson 1985; Kachanov et al. 2003). In numerical domain, various models have been largely developed for modeling the soil foundation, which are classified into three categories: (1) continuum models, (2) mixed models and (3) spring models.
In continuum mechanic concept, the medium is defined by a continuously distributed matter through the halfspace that the constitutive law can be described by a linear elastic isotropic behavior (Irgens 1980). The solution for a simplified continuum using the finite element idealization was developed by Reissner (1967). The continuum can be analyzed with many numerical methods, such as the finite element method (FEM), the boundary element method (BEM) or combined methods between FEM and BEM, which are suitable to the soilstructure interaction analysis. Particularly, FEM is well known and widely used in many approaches to study soilstructure interaction behavior and BEM shows many advantages in the modeling field showing a high accord with infinite and semiinfinite spaces (Bolteus 1984; Tezzon et al. 2015).
Due to the complexity of the interaction problems, analytical solutions are rarely used and desired alternatives would be a numerical approach (Dinev 2012; Hassan and Doha 2015; Ai and Cai 2016). The FEM is still popular method in this domain (Bourgeois et al. 2012; Su and Li 2013) but it has disadvantages compared to BEM and spectral element method (SEM) (Omolofe 2013; Mokhatari et al. 2016) using absorbent frontier modeling. The FEM requires the discretization of the domain to high number of finite elements but the problem can be solved with BEM where only the boundary of the domains involved can be discretized (Padron et al. 2011; Ai and Cheng 2013; Ribeiro and Paiva 2014).
Beam using Winkler foundation model (1867) is used in various practical problems. In this approach, the foundation flexibility is considered as a set of continuous springs and has employed to model soilstructure interaction (Kim and Yang 2010; Chore et al. 2010; Sapountzakis and Kanpitsis 2011; Raychowdhury 2011; Limkatanyu et al. 2012). Springs introduced provide a resistance in the vertical direction only confining the deformation of the soil foundation. Evidently, the oneparameter soil foundation model suffers a handicap due to the discontinuity in the supporting medium. To improve the Winkler model, twoparameter and threeparameter soil foundation models have been developed taking into consideration shear deformations of the soil. To overcome the oneparameter foundation model deficiencies, many researchers proposed various foundation models to describe rigorously soil response insuring the interconnection between vertical springs.
In the first approach, the interaction between springs has been established by an elastic tensioned membrane (Filonenko 1968) and the second one uses beam elements or a plate to interact between them (Hetényi 1966). In this case, the tensioned membrane is quantified by a shear parameter. Third, to model the mutually effect between springs, Kerr (1964) integrated a shear layer dividing the soil medium of foundation to two different layers.
The twoparameter soil foundation has shown considerable developments during the last decade. A new finite element formulation was developed to study the shallow and the raft foundation response eliminating the limits of Winkler model (Mullapudi 2010). In the way to study large deflections of functionally graded beam resting on twoparameter elastic foundation, a finite element procedure was developed (Gan and Kien 2014). Finally, the effect of material nonhomogeneity and the twoparameter elastic foundation were used to quantify the response of simply supported beams. The foundation medium behavior is assumed to be linear elastic, homogeneous and isotropic with two parameters describing the reaction of the elastic foundation on the beam (Avcar 2016).
In this analysis, the soilstructure interaction problem has been studied using effectively twoparameter model of the layered soil. Both the beam and the substrate are described by means of FEM integrating shear deformations. However, to ensure vanishing displacement at the frontier of the substrate, mesh has to be extended far away from the loaded region. To improve the computational efficiency, twodimensional finite elements are refined in the loaded area. The modeling uses plane stress state for the shear beam in adhesive contact with plane strain state of the soil foundation. Adding, a parametric study is elaborated to show the influence of (1) the horizontal behavior of the soil foundation, (2) mechanical properties of the soil foundation and (3) ballasted layer.
Physical modeling
Finite element formulation
Numerical study
Validation of the program
Really, the calibration of horizontal stiffness plays a primordial contribution in this research theme. In this study, the same stiffness values of horizontal and vertical springs are considered due to the continuity of the medium.
Analysis of the beam on a rigid base
Influence of soil properties on the interaction response
Influence of a ballast layer
Nowadays, requirements for strength and stability of railway tracks are increased that is due to train speed and axle load (Petriaev et al. 2017; Sayeed and Shahin 2018). The ballast need to be elaborated for many roles that are: (1) the isolation of structureborne noise on railway lines in populated areas; (2) the minimization of vibration effects; (3) the stability of railroads and load distribution layer; and (4) provides longitudinal and lateral track support to resist imposed loading from vehicles and thermal rail stress.
Conclusions
A procedure to quantify prismatic beams with perfect adhesion to a homogeneous, linearly elastic and isotropic twodimensional halfspace is proposed. Based on the strain energy expressions, shear deformations of the beam element under plane stress state and of plane strain soil foundation are formulated and employed in the analysis.

There is a notable influence of the laterally interaction on the beamsoil foundation behavior.

The soil properties have a primordial effect on the beam and on the interaction beamsoil foundation.

The deepness of the elastic foundation has a regular effect on the beamsoil foundation and on the beam.

The introduction of a ballast layer (height layer) engenders an influence on the beamfoundation interaction in the longitudinal direction.

The finite element formulation was established independently of the beam boundary conditions. This approach can be easily used for other boundary conditions of beams.

The approach can be considered as issue to nonlinear analysis and vibration analysis of soilstructure interaction due to the impact loads.
Notes
References
 Ai ZH, Cai JB (2016) Static interaction analysis between a Timoshenko beam and layered sols by analytical layer element/boundary element method coupling. Appl Mathem Model 40:9485–9499CrossRefGoogle Scholar
 Ai ZY, Cheng YC (2013) Analysis of vertically loaded piles in multilayered transversely isotropic soils by BEM. Eng Anal Bound Elem 37:327–335MathSciNetCrossRefGoogle Scholar
 Avcar M (2016) Effect of material nonhomogeneity and twoparameter elastic foundation on fundamental frequency parameters of Timoshenko beam. Acta Phys Pol A 130:375–378CrossRefGoogle Scholar
 Binesh SM (2012) Analysis of beam on elastic foundation using the radial point interpolation method. Cient Iran A 9(3):403–409CrossRefGoogle Scholar
 Bolteus L (1984) Soilstructure interaction a study based on numerical methods. Ph. D. theisis. Division of structural design Chalmers University of technology, Goteborg, Sweden, 1984, p 165Google Scholar
 Bourgeois E, de Buhan P, Hassen G (2012) Settlement analysis of piledraft foundations by means of a multiphase model accounting for soilpile interactions. Comput Geotech 46:26–38CrossRefGoogle Scholar
 Cen S, CHen XM, Fu XR (2009) Quadrilateral membrane element with analytical element stiffness matrices formulated by the new quadrilateral area coordinate method. Int J Meth Eng 77:1172–1200MathSciNetCrossRefGoogle Scholar
 Chore HS, Ingle RK, Sawant VA (2010) Building framepile foundationsoil interaction analysis: a parametric study. Interact Multiscale Mech 3(1):55–79CrossRefGoogle Scholar
 Dinev D (2012) Analytical solution of beam on elastic foundation by singularity functions. Eng Mech 19:381–392Google Scholar
 Filonenko B (1968) Theory of elasticity. Mir, Moscow, p 388Google Scholar
 Gan BS, Kien ND (2014) Large deflection analysis of functionally gradedbeams resting on a twoparameter elastic foundation. J Asian Archit Build Eng 13(3):649–656CrossRefGoogle Scholar
 Hassan MT, Doha EH (2015) Recursive differentiation method: application to the analysis of beams on twoparameter foundations. J Theor Appl Mech 55(1):15–26Google Scholar
 Hetényi M (1966) Beams and plates on elastic foundations and related problems. Appl Mech Rev 19:95–102Google Scholar
 Irgens F (1980) Continuum mechanics. Springer, Berlin, Heidelberg, p 649Google Scholar
 Johnson KL (1985) Contact mechanics. Cambridge University Press, CambridgeCrossRefGoogle Scholar
 Kachanov ML, Shafiro B, Tsukrov I (2003) Handbook of elasticity solutions. Kluwer Academic Publishers, Dordrecht, p 324CrossRefGoogle Scholar
 Kerr AD (1964) Elastic and viscoelastic foundation models. J Appl Mech 31(3):491–498CrossRefGoogle Scholar
 Kim SM, Yang S (2010) Moving twoaxle high frequency harmonic loads on axially loaded pavement systems. K J Civ Eng 14(4):513–526CrossRefGoogle Scholar
 Limkatanyu S, Kwon M, Prachasaree W, Chaiviriyawong P (2012) Contactinterface fibersection element: shallow foundation modeling. Geomech Eng 4(3):173–190CrossRefGoogle Scholar
 Logan DL (2012) A first course in the finite element method. CL Engineering, Stamford, p 937Google Scholar
 Mokhatari A, Sarvestan V, Mirdamadi HR (2016) Spectrally formulated finite element for vibration analysis of an EulerBernoulli beam on Pasternak foundation. J Theor Appl Vib Acoust 2(2):119–132Google Scholar
 Mullapudi TRS (2010) Nonlinear finite element formulation of the soil structure interaction through twoparameter foundation model. Master thesis, 2010, MissouriRolla University, p 46Google Scholar
 Omolofe B (2013) Deflection profile analysis of beams on twoparameter elastic subgrade. Latin Am J Solids Struct 10:263–282CrossRefGoogle Scholar
 Padron LA, Aznarez JJ, Maeso O (2011) 3D boundary elementfinite element method for the dynamic analysis of piled buildings. Eng Anal Bound Elem 35:465–477CrossRefGoogle Scholar
 Petriaev A, Konon A, Solovyov V (2017) Performance of ballast layer reinforced with geosynthetics in terms of heavy axle load operation. Proc Eng 189:654–659CrossRefGoogle Scholar
 Prakash MY, Ghugal YM, Wankhade RL (2016) Study on soilstructure interaction: a review. Int J Eng Res 6(3):737–741Google Scholar
 Raychowdhury P (2011) Seismic response of lowrise steel moment resisting frame buildings incorporating nonlinear soil structure interaction (SSI). Eng Struct 33(3):958–967MathSciNetCrossRefGoogle Scholar
 Reissner E (1967) Note on the formulation of the problem of the plate on an elastic foundation. Acta Mech 4(1):88–91CrossRefGoogle Scholar
 Ribeiro DB, Paiva JB (2014) Mixed FEMBEM formulations applied to soilstructure interaction problems. In: Proceedings of the World Congress on Engineering, London, 2–4 July 2014. Lecture Notes in Engineering and Computer Science, 2014, pp 1178–1183Google Scholar
 Sapountzakis EJ, Kanpitsis AE (2011) Nonlinear analysis of shear deformable beamcolumns partially supported on tensionless threeparameter foundation. Arch Appl Mech 81(12):1833–1851CrossRefGoogle Scholar
 Sayeed M, Shahin M (2018) Design of ballast railway track foundation using numerical modeling. Part II: applications. Can Geotech J 55(3):369–396CrossRefGoogle Scholar
 Shen HS (2012) A novel technique for nonlinear analysis of beams on twoparameter elastic foundation. International Journal of Structural Stability Dynamics 11(6):999–1014MathSciNetCrossRefGoogle Scholar
 Su D, Li JH (2013) Threedimensional finite element study of a single pile response to multidirectional lateral loadings incorporating the simplified statedependent dilatancy model. Comput Geotech 50:129–142CrossRefGoogle Scholar
 Tezzon E, Tullini N, Minghini F (2015) Static analysis of shear flexible beams and frames in adhesive contact with an isotropic elastic halfplane using a coupled FEBIE model. Eng Struct 104:32–50CrossRefGoogle Scholar
 Zienkiewicz OC, Taylor RL, Zhu JZ (2005) The finite element method: Its basis and fundamentals. Elsevier, Boston, p 733Google Scholar
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