Static interaction analysis between beam and layered soil using a two-parameter elastic foundation
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The work presents a finite element modeling of beam resting on a two-parameter 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.
KeywordsStatic soil-beam interaction Layered soil Two-parameter elastic foundation model Finite element method Elastic foundation Ballasted layer Soil properties
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 soil-structure interaction researches have been illustrated in the recent art state review (Prakash et al. 2016).
The quantification of the soil-structure 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 two-parameter soil foundation model of beams resting on isotropic or anisotropic elastic half-space (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 half-space 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 soil-structure interaction analysis. Particularly, FEM is well known and widely used in many approaches to study soil-structure interaction behavior and BEM shows many advantages in the modeling field showing a high accord with infinite and semi-infinite 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 soil-structure 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 one-parameter soil foundation model suffers a handicap due to the discontinuity in the supporting medium. To improve the Winkler model, two-parameter and three-parameter soil foundation models have been developed taking into consideration shear deformations of the soil. To overcome the one-parameter 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 two-parameter 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 two-parameter elastic foundation, a finite element procedure was developed (Gan and Kien 2014). Finally, the effect of material non-homogeneity and the two-parameter 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 soil-structure interaction problem has been studied using effectively two-parameter 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, two-dimensional 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.
Finite element formulation
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 structure-borne 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.
A procedure to quantify prismatic beams with perfect adhesion to a homogeneous, linearly elastic and isotropic two-dimensional half-space 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 beam-soil foundation behavior.
The soil properties have a primordial effect on the beam and on the interaction beam-soil foundation.
The deepness of the elastic foundation has a regular effect on the beam-soil foundation and on the beam.
The introduction of a ballast layer (height layer) engenders an influence on the beam-foundation 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 soil-structure interaction due to the impact loads.
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