Influence of the soil properties on the seismic response of structures
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Abstract
The objective of this study is to model the interaction between a concrete wall and a soil under seismic loading using the finite element method. The stiffness matrix of the soil is integrated to that of the structure to formulate the stiffness matrix of the entire system. To simulate the dynamic of soil–structure interaction, a numerical program was developed for this concern. So to resolve governed dynamic equations, the central difference method is used to compute displacement, velocity, and acceleration fields of soil, interface medium, and concrete wall nodes. The significance of soil foundation–structure interaction over fixedbase structure analysis showed that the integration of soil and foundation produces considerable changes in the seismic response. Obtained results using the soil–structure model and impedance functions are compared to those of fixedbase structure. The purpose is to calibrate the effects of soil properties and the soil–structure interaction on the seismic response of the structure and on the interaction medium.
Keywords
Soil–structure interaction Soil properties Central difference method Impedance functions Concrete wall Structure Seismic response Finite element methodIntroduction
In earthquake engineering, the dynamic soil–structure interaction is a primordial task that has attracted a great interest of researchers and engineers. This phenomenon is taken into account to improve the seismic response of structures and evaluate their vulnerability (Sharma 2017). When a structure on soil is subjected to seismic loading, foundation oscillates depending on the supporting and surrounding soil, the foundation, and inertia of the superstructure. So the dynamic response of structures built on soft soil may significantly differ from the fixedbase structures (Kwon and Elnashai 2017). This fact is principally due to the dissipated energy of flexibly supported structures.
The literature offers the direct method, the substructure method or the hybrid method to model soil–structure interaction. In the direct method, the structure and the soil region around it are modeled together. So the geometry, the properties of the soil, and the behavior of the medium, and the structure could be considered in a unique step (Gullu and Pala 2014). When the substructure method is employed, the soil supporting foundation is described by impedance functions (Chatzigogos et al. 2007; Mahmoudpour et al. 2011). At last, the hybrid method considers the macroelement concept of the soil–structure interaction (Pecker 2010; Lu et al. 2016). This approach combines the soil halfspace, the foundation, and the interface between the soil and the structure (Figini et al. 2012).
Methods analyzing the soil–structure interactions are regrouped into two categories: (1) analytical methods and (2) numerical methods. There was a considerable lack in high power computing machine, analytical methods were popular and can only be used to solve simple problems. With the known innovation in computer science and numerical methods, many simulations are mostly used to study the soil–structure interaction. Thus, modeling of the interaction problems is elaborated based on the dimensional concept of problems under static or dynamic loading. Moreover, various computational methods such as the finite difference method (Dolicanin et al. 2010; Challamel et al. 2015), the finite element method (Zienkiewicz and Taylor 2013; Smith et al. 2013), and the boundary element method (Fedeliński 2004; Gernot et al. 2008) have been employed to analyze the interaction problems.
On the other hand, procedures of structural design under seismic loading based on the performance objective have been developed over the last two decades (Ghobarah 2001). The procedure of performancebased earthquake engineering can quantify the probabilistic future seismic performance of buildings by a combination of structural capacity and seismic domain (Zarein and Krawinkler 2009). Really, the structural performance under seismic loading is neatly affected by the soil–structure interaction and its surrounding. In this case, a simplified approach is presented to study the effect of the soil–structure interaction on the nonlinear seismic response of reinforced concrete structure (Mekki et al. 2014). Later, an approximate approach is formulated to analyze the soil–structure interaction and to evaluate the relative importance of its effects on structural performance (Mekki et al. 2016). In the same way, the assessment of the seismic performance considering the soil–structure interaction is studied to quantify the contribution of the soil Poisson’ ratio, density of soil, shear wave velocity, soil dumping, and structure dumping on the lateral response of structure (Zoutat et al. 2016). Moreover Moghaddasi et al. (2012) have already defined the correlation between soil, structure, and interaction effects on the structural response where the effect of soil properties, structure characteristics, and their interaction on structural response have been analyzed.
This contribution is a prolongation of our developed investigation to analyze the effect of mechanical properties of soil on concrete wall responses under static loadings (Bourouaiah et al. 2017). In this way, the finite element model of soil–wall structure is integrated in the developed numerical program to study the seismic performance of structures considering the soil–structure interaction. The soil and the concrete wall were modeled using eightnode elements with two degrees of freedom for each node. The dimensional problem is justified by accuracy and satisfaction of results obtained by various studies using twodimensional analysis for simple and regularly problems (Wood 2004).
Modeling of the soil–structure interaction
The resistance of ground to forces generate by different movements may transmit additional forces to the superstructure. This transmission of forces between the soil and the adjacent structures continues until the equilibrium of the soil–structure system is achieved. The energy transfer mechanism from the soil to the building during seismic loading is critical for design and conception of a structure. In this case, the behavior of the structure and the soil medium is greatly different, which can engender a relative response between them. Therefore, the approach to model rigorously the soil–structure interaction can be selected by providing accurate analytical or experimental results.
Various researches using the finite element method have been elaborated to investigate the soil–structure interaction problems. In this case, models can be regrouped in (1) modeling of the soil–structure interaction using nodetonode fulfilling compatibility conditions. This kind is quite often in practical applications (Langen and Vermeer 1991; Day and Potts 1994). These models are inspired from the relationship between relative displacements and stresses of common nodes (Fig. 1a). (2) In other cases, the behavior of the interface may be modeled by conventional finite element meshing conveying suitable mechanical properties of the contact media. They are used to predict the effect of the soil–structure interaction with interface elements and constitutive law. This approach is well established and largely used in finite element codes. In these analysis, the failure can occur in the nearest stress point for weak mechanical properties of elements (Viladkar et al. 1994; Skejic 2012; Barros et al. 2017; Sharma 2017) (Fig. 1b).
Finally, the zerothickness interface elements or thinlayer elements are used to model the behavior of the discontinuity of joints in rock mechanics (Viladkar et al. 1994; Bouzid et al. 2004; Skejic 2012; Barros et al. 2017; Sharma 2017). These elements are used with respect to the finite element formulation, which can be triangular finite elements of 6–15 nodes or quadrilateral elements with 4, 8, and 9 nodes (Fig. 1c).
In this work, the modeling of the soil–structure interaction uses the direct method involving the soil and the concrete wall components in the same phase. Also, the nodetonode contact elements for modeling the behavior of the interface are employed for their suitability with that of the selected numerical method. Then the concrete wall is modeled by twodimensional plane stress quadrilateral finite elements; even soil elements are modeled using plane strain elements.
Numerical formulation
The differential operator \( \left[ \partial \right] \) is \( \left[ {\begin{array}{*{20}c} {\frac{\partial }{\partial \xi }} & 0 \\ 0 & {\frac{\partial }{\partial \eta }} \\ {\frac{\partial }{\partial \eta }} & {\frac{\partial }{\partial \xi }} \\ \end{array} } \right]. \)
\( \left J \right \) is the determinant of the Jacobian matrix, t and \( \rho \) is the thickness of the element and the density of the used material, respectively.
In the same way, the formulation of the stiffness matrix can be developed by substituting Eqs. (5) and (7a, 7b) into Eq. (8a). To analyze planed problems, two cases can be distinguished as:
Numerical dynamic analysis
The numerical solution can be established using the central difference method to compute the solution of the dynamic response of the structural system for its stability for real dynamic time and its implementation in the developed numerical program (Wu et al. 2009; Grobeholz et al. 2015).
Central difference method
Technique of the resolution
Description of studied cases and modeling
Geometrical and mechanical properties of soil and concrete wall structure
Material properties  Geometrical dimensions L_{x}× L_{y}× t (m)  Shear modulus (G) (kN/m^{2})  Poisson’s ratio ν  Density, ρ (kN/m^{3}) 

Soil (S1)  17 × 10 × 8  19.23 10^{5}  0.45  15 
Soil (S2)  17 × 10 × 8  35.71 10^{5}  0.40  18 
Soil (S3)  17 × 10 × 8  74.07 10^{5}  0.35  20 
Soil (S4)  17 × 10 × 8  107.14 10^{5}  0.30  22 
Concrete  3 × 4 × 0.3  125 10^{5}  0.20  23 
In this model, the soil medium and the concrete wall were discretized into quadrilateral finite elements (Fig. 4b). To satisfy boundary conditions, the located soil nodes at lateral sides were fixed against horizontal displacement while nodes of longitudinal side boundary were restrained in vertical and longitudinal displacements.
Results and discussions
Effect of soil properties on the interface media
Obtained results show that soil proprieties have a great influence on the behavior of the interface medium and they are proportional to the soil nature. The peaks of oscillations are very important for soft soil and decrease as well as the soil becomes very stiff (Fig. 6). The seismic force is applied in the horizontal direction, so horizontal displacements are very significant compared to the vertical ones (Fig. 6a, b). Moreover, when the soil varies from S2 to S4, the difference between vertical displacements is not remarkable (Fig. 6b), but horizontal displacements are underlined (Fig. 6a).
Effect of soil properties on the concrete wall
Effect of soil properties on horizontal displacements
Soil used  Very soft soil (S1)  Soft soil (S2)  Firm soil (S3)  Stiff soil (S4) 

Peak displacement (m)  0.078  0.041  0.018  0.01 
Displacement ratio  1.00  1.90  4.34  7.80 
It is emphasized here that the design of the concrete wall with respect to the horizontal behavior must be taken into consideration.
Influence of the soil–structure interaction
Modeling of the soil foundation
Soil stiffness under rectangular foundation
Nature of soil  Very soft soil  Soft soil  Firm soil  Stiff soil 

Density, \( \rho \) (KN/m^{3})  15  18  20  22 
Poisson ratio, \( \nu \)  0.45  0.40  0.35  0.30 
a/b = 2  
\( K_{h} \) (10^{6} kN/m)  1.38  5.36  22.12  97.33 
\( K_{v} \) (10^{6} kN/m)  2.03  7.46  28.44  125.16 
Influence of translational springs
Thus, the impedance functions substituting the soil of foundation can predict highly the dynamic soil–structure interaction. Consequently, spring stiffens taken as average values between stiffness of concrete and the soil predict well the seismic loading.
Effect of soil–structure interaction on concrete wall
Spring rigidity effect on the concrete wall
Conclusions

The flexibility of the soil foundation reduces, notably, the seismic response of the concrete wall and its neglecting leads to larger values of displacements of concrete wall nodes.

Displacements of concrete wall nodes are important relative to the interaction medium nodes when the foundation is so rocking.

Damage of the concrete wall base is highly affected by the soil–structure interaction.

The developed numerical program is an open tool to be improved for nonlinear or elastoplastic analyses.

Formulated analytical model using central difference method of the soil can reproduce considerably the soil–structure interaction.

Horizontal and vertical behaviors of the interface medium and the wall concrete depend directly on the soil properties. When feeble properties of the soil are used, horizontal and vertical displacements of the concrete wall nodes are largely observed and the horizontal displacements are more pronounced compared to the vertical displacements.
Notes
References
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