Dynamic Impedance Functions of a Square Foundation Estimated with an Equivalent Linear Approach
Abstract
This paper presents the role of the intensity of an earthquake on the estimation of dynamic impedance functions. For various seismic shaking we will have different nonlinear behavior of the soil, which may have significant effects on the amplitude and shape of the dynamic stiffness and damping coefficient. However, under the assumption of linear elastic behavior of the soil, loss of soil stiffness under strong shaking is not taken into account in the dynamic impedance functions. The dynamic impedance functions are estimated using an equivalent linear process and compared relative to the linear elastic case. The vibrations originate from the rigid foundation embedded in the soil layer, which are subjected to harmonic loads of translation, rocking, and torsion. The dynamic responses of the rigid surface foundation are solved from the wave equations by taking into account their interaction. The solution is formulated using the frequency domain boundary element method (BEM), in conjunction with KauselPeek Green’s function for a layered stratum and the thin layer method (TLM) to account for the interaction between the soilfoundation. A parametric analysis is performed for surface foundation in a semiinfinite soil limited by bedrock and subjected to three earthquake records.
1 Introduction
The analysis of the behavior of foundations under dynamic loads has grown considerably over the past four decades. The stringent security requirements imposed on design of certain types of structures have played a particularly important role in the development of analytical and numerical methods. The key step in studying the dynamic response of foundations is the determination of the relationship between forces and displacements. This relationship is expressed using impedance functions (dynamic stiffness) or the compliance functions (dynamic flexibility). The consideration of the soilstructure interaction in the analysis of the dynamic behavior of foundations allows taking realistically into account the influence of soil on its vibration. Several methods have been proposed in the literature to solve the soilstructure interaction problem. To simplify the problem, linearanalysis techniques have been developed. One of the most commonly used approaches is the substructures method that allows the problem to be analyzed in two parts Kausel et al., Aubrey et al. and Pecker. In this approach, the dynamic responses of superstructure and of the substructure are examined separately. The analysis of foundations systems can be reduced to the study of the dynamic stiffness at the soilfoundations interface (known as impedance function). Although a solution of a soilstructure interaction problem in most cases involves a straightforward application of any of the wellestablished soilstructure interaction methods, a relatively small number of 3D investigations have appeared in the related literature. This is probably due to the substantial computational effort required by the Finite Element Method (FEM) and the Boundary Element Method (BEM). Furthermore, there is a noticeable absence of simplified discrete models, which is due, perhaps, to lack of rigorous results that could be used for the verification and calibration of such models. The complexities of the shapes of foundations, of the loadings, and of the soil conditions have discouraged, in general, the development of analytical solutions.
In the nonlinear case, the behavior of dynamically loaded foundations and the soil layers were investigated to a lesser degree (Borja and Wu 1994). For this, we simulate numerically the dynamic response of a massless square foundation resting on a semiinfinite soil considering equivalentlinear soil behavior described through straincompatible shear modulus and damping coefficients for a suite of earthquake records. The method is based on subdivision of the soil mass under the foundation into a number of horizontal layers of different shear modulus and damping ratio like thin layer method (TLM) but compatible with the level of strain imposed by an earthquake motion or a dynamic load. Indeed, the semiinfinite soil is replaced by a layered profile with straincompatible properties within each layer.
In this study, the dynamic responses of the rigid surface foundation are solved from the wave equations by taking into account their interaction. The solution is formulated using the frequency domain boundary element method (BEM), in conjunction with KauselPeek Green’s function for a layered stratum and the thin layer method (TLM) to account for the interaction between the soilfoundation (Sbartai and Boumekik 2008), Sbartai (2016) and Messioud et al. (2016). For a given set of applied loads, characteristic strains are determined in each soil layer and the analysis is repeated in an iterative manner until convergence in material properties is achieved. For limited space, a parametric analysis is performed only for square foundation resting on a semiinfinite soil excited by a suite of three recorded earthquakes motions. The dimensionless results are provided for the variation of foundation stiffness and damping with frequency and excitation level in vertical, horizontal, rocking, and torsional modes.
1.1 Calculation Model
There are two main methods dealing the soilstructure interaction analysis: direct method, and of the substructure method. In the direct method, the response of the soil and structure is determined simultaneously by analysing the idealized soil–structure system in a single step. In substructure method, the soilstructure interaction problem is divided into two sets of simpler problems, which are solved independently, and the results are then superposed to obtain the response of the structure. The basic step in the substructure approach is to determine the force–displacement characteristics of the soil. This relationship may be in the form of an impedance (stiffness) function, or, inversely, a compliance (flexibility) function. By definition, the impedance K(6, 6) of the system is the relation between the load P(6, 1) and the response U(6, 1). Generally, the load, the impedance and the response are complex quantities. The relationship between impedance, displacement and applied load is given by
In the particular case of a symmetric surface foundation (i.e., rectangular, square), it is possible to uncouple the impedance matrix along the two principal axes (x and y) to reduce its dimension (i.e., \( K_{51} = K_{15} = K_{42} = K_{24} = \,0 \)).
Where \( G_{ij} \), represents Green’s functions tensor and \( t_{i} \) is the unknown surface traction.
The equivalent linear analysis has been used to described the nonlinear behavior of the soil through straincompatible shear modulus and damping coefficients for a suite of earthquake records. Assuming that the substratum and the interfaces between the different layers of soil are essentially horizontal, we can consider each layer as linear elastic and develop a model with lumped mass in order to analyze the nonlinear dynamic response of the soil deposit during an earthquake. Thus, Kryloff and Bogoliuboff (1943) and Bogoliuboff and Mitropolsky (1961) proposed the use of both an equivalent linear spring constant, and an equivalent damping ratio for a singledegreeoffreedom system having nonlinear characteristics. Indeed, (Seed and Idriss, 1969) have proposed the use of an equivalent linear scheme wherein the shear modulus and damping are modeled using a linear spring and a dashpot, respectively. The parameters of the spring and the dashpot are calculated based on the secant shear modulus and damping ratio for a given level of shear strain. Shear modulus and damping ratio values are iteratively calculated based on the computed strain. For earthquake input motion, (Seed and Idriss 2012) suggested that the properties must be calculated for a strain equal to 2/3 of maximum strain level in a given layer.
\( \left[ {Fs} \right] \, = \, \{ {\text{G}}^{mn} \}_{ij} \) represents the flexibility matrix of the discretized domain, which includes the terms of Green’s functions; {u} represents the harmonic displacements; and {t} represents the surface tractions.
{D} denotes the displacements of the foundation; and [R] represents the transformation matrix of size (N ×1).
\( k\left( {a_{o} } \right) \) denoting the dimensionless spring coefficient; c denoting the dimensionless damping coefficient and β denoting the constant hysteretic damping coefficient.
2 Results
The response of square rigid foundation resting on a semiinfinite viscoelastic soil over rigid bedrock is considered. The geometry and discretization are shown in Figs. 1 and 2. Due to the space limitations only the vertical, horizontal, rocking and torsion impedances functions of the massless surface foundation are considered according dynamic equivalentlinear soil behavior described through straincompatible shear moduli and damping coefficients for a the effect of excitation amplitude of ground motion and the dimensionless frequency. For this, three seismic records (BoumerdesAzzazga0.1 g, Elcentro0.2 g, BoumerdesKedara0.3 g) are used in this study. Such an application represents a general study that enables the analysis of the influence of different parameters, which we will present in another article later.
In this application, the considered foundation of dimension \( {\text{B}}\,{ = }\,{\text{B}}_{\text{x}} / {\text{B}}_{\text{y}} \,{ = }\, 1 \) is subjected to unit forces \( {\text{P}}_{\text{x}} \,{ = }\,{\text{P}}_{\text{z}} \,{ = }\, 1 \) and unit moments \( {\text{M}}_{\text{x}} \,{ = }\,{\text{M}}_{\text{z}} \,{ = }\, 1 \) for different dimensionless frequencies a_{0}(= ωb/C_{s}). It is noted that ω is the circular frequency and C_{s} is the shear (S)wave velocities. The soil is discretized into 36 quadrilateral constant elements on the soilfoundation interfaces for lowerfrequency case, and it is discretized into 64 quadrilateral constant elements on the soilfoundation interfaces for the higherfrequency case.
Figures 3(a) and (b) show the dimensionless real part \( K_{v} /Kv_{linear  static\,} \) and dimensionless imaginary part \( C_{v} /Kv_{linear  static} \) of the vertical impedance as functions of the dimensionless frequency a _{ o } and three Earthquake records. Figure 3(a) show that the vertical dynamic stiffness coefficient naturally decreases with increasing excitation level. We remarked also of the negative values for the higher excitation level from the cutoff frequency a_{o} = 3 and 2 for the second (0.2 g) and third earthquake (0.3 g) record respectively. However in Fig. 3(b), the vertical damping coefficient increases with increasing excitation level. In the equivalentlinear case, the magnitude of the imaginary part of the impedance increases from the linear case with increasing excitation amplitude. The higher the excitation amplitude, the larger the shift of the curve to higher values.
3 Conclusions
 1.The nonlinear dynamic response of the foundation depends on more parameters than in the linear case:

shear modulus reduction and damping increase curves; and

excitation amplitude and frequency content.

 2.
The dynamic stiffness coefficients decrease with increasing excitation amplitude.
 3.
The dynamic stiffness coefficients become negative for translational modes from specific frequency a_{o}(= 2–5). The same case was observed for rotational modes but with a very low values and a shift of the cutoff frequency towards high frequencies a_{o}(= 5.5–6). It may create a detachment of the soil foundation.
 4.
The damping coefficients are fairly dependent on acceleration maximal of the earthquake. It increases from the linear case with increasing level of shear strain, as expected due to the increase in hysteretic soil material damping.
Notes
Acknowledgments
The authors are grateful to the Ministry of Higher Education and Scientific Research of Algeria for supporting this work.
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