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EQUIVALENT-LINEAR DYNAMIC IMPEDANCE FUNCTIONS OF SURFACE FOUNDATIONS Dimitris Pitilakis1, Arezou Moderessi – Farahmand – Razavi2 and Didier Clouteau3 1
Lecturer, Aristotle University of Thessaloniki, Department of Civil Engineering, 54124, Thessaloniki, Greece;
[email protected] 2 Professor, Ecole Centrale Paris, Grande Voie des Vignes, 92295 Chatenay-Malabry, France;
[email protected] 3 Professor, Ecole Centrale Paris, Grande Voie des Vignes, 92295 Chatenay-Malabry, France;
[email protected]
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ABSTRACT
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An approximate linearization method using the familiar concept of G-γ and D-γ curves is presented
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for determining the dynamic impedance (stiffness and damping) coefficients of rigid surface
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footings accounting for nonlinear soil behavior. The method is based on subdivision of the soil mass
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under the footing into a number of horizontal layers of different shear modulus and damping ratio,
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compatible with the level of strain imposed by an earthquake motion or a dynamic load. In this way,
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the original homogeneous or inhomogeneous soil profile is replaced by a layered profile with strain-
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compatible properties within each layer, which do not vary in the horizontal sense. The system is
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solved in the frequency domain by a rigorous boundary-element formulation accounting for the
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radiation condition at infinity. For a given set of applied loads, characteristic strains are determined
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in each soil layer and the analysis is repeated in an iterative manner until convergence in material
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properties is achieved. Both kinematic and inertial interaction can be modelled simultaneously by
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the method, which thus encompasses primary and secondary material nonlinearity in a single step.
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Results are presented for a circular footing resting on: (i) a halfspace made of clay of different
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plasticity index and (ii) a halfspace made of sand of different density, excited by a suite of recorded
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earthquake motions. Dimensionless graphs are provided for the variation of foundation stiffness and
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damping with frequency and excitation level in vertical, swaying, rocking and torsional oscillations.
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Keywords: soil-foundation-structure interaction; soil-footing compliance; stiffness; damping;
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earthquake engineering
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INTRODUCTION
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Following the seminal publications by Barkan (1960) and Richart, Hall and Woods (1970), the
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dynamic response of coupled soil-foundation-structure systems has been the aim of numerous
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studies for over four decades. The problem is typically decomposed into two subtasks, namely
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kinematic and inertial interaction, the solution of which provides the effective excitation and the
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response of the superstructure respectively. Solving the inertial interaction problem requires
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determining the dynamic impedance functions (i.e., stiffness and damping) of the foundation, which
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constitutes the single most important subtask in such analyses. Detailed reviews of the subject can
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be found in Gazetas (1983, 1991), Pais and Kausel (1988), Mylonakis et al. (2006) and a recent
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ATC report (NEHRP 2011). Applications on earthquake structural response are discussed by
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Mylonakis and Gazetas (2000).
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In current engineering practice, soil-foundation-structure interaction analyses are typically
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performed assuming linearly elastic or viscoelastic material behavior. In some recent guidelines,
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soil nonlinearity is taken into account using two approximate methods. In the first one, soil behavior
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is modeled using strain-compatible shear and damping moduli considering a single characteristic
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strain for the whole soil profile, as described in Stewart et al. (2003) and the current NEHRP
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Provisions (BSSC 2004). In the second method, which is adopted in ATC-40 and FEMA-440
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guidelines, soil reaction to foundation movement is modeled by uniaxial elasto-plastic springs and
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dashpots distributed along the soil-foundation interface (NEHRP 2011). The latter approach is
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attractive because of its simplicity and straightforward treatment of nonlinearity. Yet, it does not
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take into account either the continuity of soil medium, or the interplay between plastic deformations
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in different directions (plastic flow rule).
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Field tests and rigorous analyses pioneered by Barkan (1948) and Luco and Westman (1971),
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respectively, have demonstrated the frequency dependence of foundation stiffness and damping
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under harmonic loading. More recent forced-vibration field tests on large scale model structures (de
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Barros and Luco 1995, Tileylioglu et al. 2011) confirmed the frequency dependence of these
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parameters. With reference to kinematic interaction and using recorded data, Kim and Stewart
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(2003) demonstrated how foundation response is affected by slab averaging and wave incoherence
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effects. These studies have shown that in the elastic regime, available solutions can, with minor
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adjustments, model dynamic soil-foundation interaction (SFI) effects in a realistic manner.
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In the inelastic regime, however, the behavior of dynamically loaded foundations and the
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underlying soil has been investigated to a lesser degree and no established analysis methods are
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presently available (Borja and Wu 1994, NEHRP 2011). This is in contrast to the dynamics of the
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superstructure, for which a plethora of nonlinear analysis procedures are available in the context of
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finite-element modeling (Chopra 2011). Recently Gajan et al. (2010) reviewed two numerical tools
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for nonlinear SFI analysis, notably a Beam-on-Nonlinear-Winkler-Foundation and a contact
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interface model, implemented in general-purpose finite-element platform OpenSees (PEER 2008).
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Both tools were shown to be capable of capturing a wealth of nonlinear effects, including material
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and geometric nonlinearities, as well as hysteretic energy dissipation. Notwithstanding the
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theoretical significance and practical appeal of these tools, they are all rather complex and difficult
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to be implemented by non-experts.
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Contrary to numerical models, simple approximate solutions incorporating soil material
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nonlinearity in foundation impedance functions have not yet been developed. Available information
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is limited to purely numerical (Borja and Wu 1994, Chatterjee and Basu 2008, Lopez-Caballero and
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Modaressi-Farahmand-Razavi 2008) and experimental data (Star 2011). These studies have
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demonstrated an anticipated increase in foundation response with increasing soil material
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nonlinearity. An important finding (Borja and Wu 1994) is that amplification of foundation motion
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may occur at high frequencies for constant level of applied load. This behavior can be attributed to
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stress-induced inhomogeneity (which acts as a reflector) in the soil mass, and will be discussed in
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the ensuing.
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It is worth mentioning that available foundation impedance functions lack, at both low and
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high strain level, adequate validation against experimental data, due to the difficulties in carrying
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out such experiments and interpreting results. Nevertheless, all equivalent-linear procedures
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(originally developed for free-field response) were advanced before experimental data became
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available to verify them.
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With reference to surface footings, we investigate the effect of soil material nonlinearity on
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dynamic impedance functions for earthquake excitation. To this end, we simulate numerically the
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dynamic response of a massless circular footing on the surface of a layered half-space considering
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equivalent-linear soil behavior described through strain-compatible shear moduli and damping
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coefficients for a suite of earthquake records. Kinematic interaction is inherent in the analyses,
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while inertial interaction corresponds to the kinematically-induced strain level that is, for an
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infinitesimal additional load imposed at the footing. We develop dimensionless charts for the real
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and imaginary parts of the dynamic impedance functions, as affected by soil characteristics,
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excitation amplitude and frequency content. These results are meant to be used in the context of
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preliminary assessment of nonlinear soil material behavior on foundation response.
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NUMERICAL MODELING
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The dynamic impedance S of a rigid foundation along any degree of freedom can be expressed in
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the familiar complex form as:
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S=K+iωC
(1)
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where the real part K stands for dynamic stiffness and is represented by a spring. The imaginary part
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(ωC) is referred to as loss stiffness, ω being the cyclic excitation frequency; C is typically
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represented by a dashpot coefficient accounting for the combined effect of radiation and material
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damping in the soil medium. The first component (radiation damping) corresponds to energy
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dissipation due to waves emanating from the soil-foundation interface in perfectly elastic soil. The
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second part (hysteretic damping) is associated with energy loss due to hysteretic action in the soil
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material. Eq. (1) can be cast in the alternative form (Gazetas 1983):
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S = K [k(α0, ) + i α0 c(α0, ) ] (1+2 i )
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(2)
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where now dynamic stiffness is expressed in terms of a static part, K, times a dynamic modifier, k;
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the radiation dashpot coefficient is similarly expressed in terms of static stiffness and the product of
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a dimensionless frequency, α0 (= ωr/Vs, where r = characteristic footing dimension) times a
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dynamic modifier, c. In the above equation,
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Poisson’s ratio and i = the imaginary unit. Linearity is implicit in Eq. (2), which therefore does not
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account for nonlinear.
denotes the hysteretic material damping ratio,
the
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MISS3D-EqL (Pitilakis and Clouteau 2010, Pitilakis 2006) is a newly-developed boundary-
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element numerical code based on the linear MISS3D software developed in Ecole Centrale Paris
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(Clouteau and Aubry 2003), which is employed herein to calculate the dynamic foundation
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impedance functions by the equivalent-linear approach. MISS3D employs a direct three-
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dimensional boundary-element formulation in the frequency domain for linearly elastic or
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viscoelastic soil, accounting for layering and radiation condition at infinity.
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MISS3D-EqL incorporates a novel equivalent-linear procedure for the analysis of the soil
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domain. To this end, the soil under the footing is subdivided into a number of (real or fictitious)
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horizontal layers of infinite extent having different shear moduli and hysteretic damping ratios. In
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the realm of this approach, a layered profile with strain-compatible properties within each layer
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replaces the original homogeneous or inhomogeneous soil. The system is solved in the frequency
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domain by the rigorous numerical procedure employed in MISS3D. For any given set of applied
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loads, a characteristic (―effective‖) shear strain is computed from the displacement field for each
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soil layer and the dynamic shear modulus and damping ratio of the layer are adjusted based on
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pertinent G-γ and D-γ relationships. The analysis is repeated in an iterative manner until
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convergence in shear strain in all layers is met (threshold is set to 3%). The input motion at bedrock
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level is modified in every iteration to account for the downward propagating waves generated (after
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reflection) at the soil-foundation interface and the free soil surface. The method can incorporate any
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combination of earthquake loads, external loads applied directly on the foundation or the soil
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surface, as well as loads transmitted onto the foundation by an oscillating superstructure. Contrary
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to conventional equivalent-linear codes developed solely for soil response such as SHAKE and
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EERA, in the present solution we take into account -simultaneously- both kinematic and inertial
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stress fields in the soil resulting from structural, footing and soil response. The accuracy of the
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numerical procedure has been validated in Pitilakis et al. (2008) and Pitilakis and Clouteau (2010)
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using experimental data from shaking table and centrifuge tests.
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To determine the characteristic strain in each soil layer, a number of control points are
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considered at the mid-depth of each soil layer in the region under the footing, where the octahedral
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deviatoric strain is computed as function of time through an FFT algorithm. Maximum octahedral
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deviatoric strain among all control points in the layer defines the characteristic layer strain in the
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particular iteration. The associated effective strain-compatible shear and damping moduli are
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applied over the whole soil layer (i.e., both under the footing and in the area not covered by the
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footing). Accordingly, vertical inhomogeneity and lateral homogeneity in the soil profile are
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retained, which greatly simplifies the numerical analysis and the evaluation of the results. This
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represents an essential advantage of the particular approach over more rigorous schemes employing
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different soil moduli in all directions (i.e., on a point-to-point basis). Whereas the above
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simplification may naturally lead to some ―softening‖ of the soil at large horizontal distances from
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the footing, this violation was found to be insignificant from a practical viewpoint. This finding
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conforms to analyses of experimental data reported by Dunn, Hiltunen and Woods (2009) on a pair
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of spread footings in a Houston site (Mylonakis 2012, personal communication).
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An aspect of importance relates to initial stresses (due to vertical static loads carried by the
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footing), which are known to affect nonlinear problems. The increase in soil stiffness under the
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foundation due to overburden is taken into account in a straightforward way following the approach
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proposed by Chu et al. (2008) which considers shear stresses associated with base shear and base
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rocking. More discussion on this issue is provided later on.
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PARAMETRIC ANALYSES
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The problem considered in this work consists of a rigid circular footing of diameter 2r = d resting
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on the surface of a homogeneous halfspace (Fig. 1). Soil properties are described by shear modulus
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G, mass density ρ (leading to shear wave propagation velocity Vs=G/ρ), Poisson’s ratio
and
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material damping . Two types of fine-grained soil with plasticity index (PI) 0 (representing silt)
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and 30 (representing clay of medium plasticity), and two types of coarse-grained soil corresponding
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to the upper and lower modulus reduction curves of Seed et al. (1986) for sands are considered (Fig.
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2). Dependence of shear modulus and hysteretic damping ratio on level of strain for the fine-grained
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materials is described according to the established curves by Vucetic and Dobry (1991) (Fig. 2).
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Low-strain hysteretic damping
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velocities (100m/s, 180m/s, 250m/s, 350m/s, 500m/s) are considered for the soil, thus classifying
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the site to category types B, C and D according to EC8. The soil profiles in all analyses have a unit
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weight of 2Mg/m3 and a Poisson’s ratio of 1/3. Finally, a single footing radius r = 5m is selected,
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which does not restrict generality as the governing parameter is dimensionless frequency a0 = ω
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r/Vs , and not r itself.
is set equal to 2% for all material types. Five different shear wave
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Different mesh configurations of the soil-footing interface were tested for accuracy and
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computational efficiency. The results presented below were obtained with 334 quadrilateral and 4
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triangular elements for the soil-footing contact interface (Fig. 1), with an average size of 5% the
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footing diameter.
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As explained in the foregoing, the soil profile is discretized in horizontal layers to establish
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characteristic shear strains at different depths. The selected discretization employs five layers of
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thickness equal to 10%, 20%, 40% 80% and 150% of the footing diameter, respectively. The above
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mesh was found to provide almost identical results at lower computational cost to a finer mesh of
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fifteen layers of constant thickness equal to 20% of the footing diameter (Fig. 3).
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Five earthquake events (Friuli 1976, Vrancea 1977, Umbria 1984, Aegion 1995, Kozani 1995)
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were selected to cover a significant range of magnitudes (6.0-7.2), distances from source (25-95km)
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and predominant frequencies (0.8-10Hz), as shown in Fig. 4. All records were scaled to five
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different PGA levels (0.01g, 0.10g, 0.20g, 0.30g, 0.50g) to trigger different levels of nonlinearity in
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the soil. The analysis parameters are summarized in Table 1.
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RESULTS
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All vibrational modes of the circular footing (horizontal, vertical, rocking and torsion) are
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examined. The real part of the dynamic impedance in Eq. (2) is denoted by Re(Si), whereas the
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imaginary part is denoted by Im(Si), subscript i being h, v, r, or t corresponding, respectively, to the
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aforementioned oscillation modes. Both real and imaginary parts are normalized with respect to
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linear static stiffness Ki,LINEAR,STATIC for the corresponding mode i, calculated as zero-frequency
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stiffness. For the real part of dynamic impedance, Re(S), the familiar dimensionless frequency
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parameter α0 = ωr/Vs is expressed in the form α0 = ωr/Vs30, the last parameter being the average
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shear wave propagation velocity over the upper 30m, so that information from depths outside the
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conventional static ―stress bulbs‖ under the footing is incorporated. In addition, the dimensionless
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frequency parameter is enhanced by a correction factor Vs30,LIN /Vs30,EQL corresponding to the initial
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and current deformation state, to account for material nonlinearity. Accordingly, the dimensionless
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frequency factor takes the form:
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0e = 0
Vs30 ,LIN Vs30 ,EQL
=
r Vs30 ,LIN
Vs 30 Vs30 ,EQL
(3)
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It is noted that Vs30 and Vs30,LIN do not coincide, as the former stands for the shear wave velocity
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at first assigned to the soil profile, whereas the latter is the shear wave velocity after the first
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iteration, incorporating the downward propagating component of the wave field created by wave
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reflection and diffraction at the soil surface and the soil-foundation interface.
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Even though in a true nonlinear dynamic analysis soil shear wave velocity varies with time, the
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above approximation of linear and equivalent-linear shear wave velocity provides the means for a
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rational approximation of soil material behavior. This approach goes beyond the approximate 8
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method for correcting soil moduli due to structural overburden effects adopted by Kim and Stewart
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(2003).
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Note that attention should be paid in interpreting the results, as the equivalent-linear shear
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wave velocity (after convergence) differs from one soil profile to the other, depending on the
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material properties and the dynamic characteristics of the input signal. From a practical point of
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view, multiplying the dimensionless frequency by Vs30,LIN /Vs30,EQL will shift the abscissa to higher
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values. This shift will be counteracted by the shift in fundamental natural frequency of the medium
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to lower values due to soil softening. As will be shown in the ensuing, the two actions balance each
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other, making the period of peak footing compliance (minimum footing stiffness) almost invariant
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to excitation level.
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Contrary to dynamic stiffness, the imaginary part of dynamic impedance is plotted in the form
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Im(Si)/Ki,LINEAR,STATIC against α0 = ωr/Vs30 without the aforementioned correction in the abscissa.
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The imaginary part is plotted in its integrity, encompassing both material and radiation damping, to
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avoid divergence at low frequencies. For simplicity, the imaginary part of the dynamic impedance
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will be referred hereafter to as ―dashpot coefficient‖.
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An issue that should be addressed is the judicious estimation of the soil shear wave velocity
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beneath the foundation. In the current NEHRP Provisions, an effective depth equal to 75% of the
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characteristic dimension of the foundation (i.e. radius for circular footings) is chosen, for which the
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corresponding shear wave velocity is estimated. In this way, the non-uniformity of the profile and
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the shear modulus reduction with increasing strain are taken into account in an approximate yet
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realistic manner (Stewart et al. 2003). Nevertheless, in the present study the shear wave velocity
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Vs30 of the upper 30m of the soil is adopted. Despite its aim at characterizing site effects (i.e., no
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soil-foundation-structure interaction effects) and, consequently, its difficulty to represent soil
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stiffness close to the surface (Kokusho and Sato 2008), the particular parameter was selected to
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ensure compatibility with EC8 and IBC. It should be noted, however, that the results we present
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below should be interpreted with the understanding that Vs30 is an imperfect (yet convenient) index.
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Evidently, engineering judgment is required to address individual cases depending on the
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circumstances.
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Effect of excitation amplitude of ground motion
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Results for a loose silt profile (Vs30=180m/s, PI0) subjected to the 1995 Aegion, Greece earthquake
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record are shown in Fig. 5 for the four vibrational modes at hand. The dynamic stiffness (Fig. 5(a)
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to 5(d)) and dashpot (Fig. 5(e) to 5(h)) coefficients are plotted for six intensity levels, including
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linear case.
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The horizontal dynamic stiffness coefficient naturally decreases with increasing excitation level
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(Fig. 5(a)). The undulations in the curves are due to resonance phenomena (corresponding to wave
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reflections) in the soil. Evidently, the initially homogeneous halfspace behaves in an inelastic way
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and stress-induced interfaces are formed between layers. Consequently, waves emitted from the
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vibrating foundation tend to be reflected back towards the source, creating undulations in the
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frequency response curves. The result of this complex wave pattern is revealed by an increase in
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foundation motion (decrease in stiffness) in the frequency range close to the fundamental frequency
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of the inelastic medium. Nevertheless, the lack of sharp peaks in Fig. 5(a) implies no significant
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impedance contrasts between consecutive layers.
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Interestingly, contrary to the linear case, in the lower frequency range the magnitude of
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dynamic stiffness increases with increasing frequency. It attains a peak and then starts to decrease at
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higher frequencies, as in the linear case. This contradictory behavior in low frequencies –
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dominated by the response of the soil profile to low frequency earthquake components – is caused
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by inversion (decrease) of shear wave velocity with depth. For earthquake amplitude of 0.30g, the
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equivalent-linear shear wave velocity is calculated to decrease with depth. Thus, the low frequency
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response is dominated by pulses propagating at significantly lower velocities than in the initial
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linear case, resulting in stiffness coefficients significantly lower in amplitude. On the other hand, in
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the high frequency range, the footing response is dominated by surface waves. These waves
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propagate at higher velocities, with shorter wavelengths than the low frequency deeper body waves, 10
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leading to an increase in the dynamic stiffness of the footing.
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The imaginary part of the horizontal dynamic impedance function, plotted in Fig. 5(e), stands
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for the combined effect of radiation (viscous) and material (hysteretic) damping. The non-zero
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value of dashpot coefficient at very low frequency denotes the presence of hysteretic dissipation
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(assumed 2% at small strains). In the linear case, the imaginary part of the impedance increases at a
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nearly constant rate with frequency, implying that radiation damping is practically frequency
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independent. In the equivalent-linear case, the magnitude of the imaginary part of the impedance
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increases from the linear case with increasing excitation amplitude. The higher the excitation
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amplitude, the larger the shift of the curve to higher values. This increase is attributed primarily to
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hysteretic material damping of the soil, which increases with increasing excitation level. For an
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earthquake amplitude of 0.30g, hysteretic damping increases up to 17% in the deeper soil layers.
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Interesting to note is that radiation viscous damping in the horizontal mode increases at the same
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rate with increasing frequency, as in the linear case. This explains why the equivalent dashpot
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coefficients are simply offset from the linear case, apparently unaffected by nonlinearity.
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In the low frequency range, however, hysteretic damping in equivalent-linear analyses does not
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deviate from the linear case. This stems from the fact that at low frequencies no surface waves are
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created. Instead, the response of the soil is dominated by longer-wavelength pulses, which create
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resonant phenomena deeper in the soil. As the halfspace is discretized into progressively thinner soil
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layers close to the surface, the increase in material hysteretic damping is associated primarily with
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the uppermost soil layers. Thereby, the longer wavelengths in the deeper soil layers do not affect
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significantly the equivalent-linear soil properties, as they induce a minor increase in hysteretic
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damping relative to the linear case.
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The effect of soil softening due to nonlinear behavior is more pronounced in the vertical mode
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(Fig. 5(b)). This is anticipated because of the deeper zone of influence of vertical normal stresses
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associated with vertical loading. The magnitude of stiffness coefficient decreases with excitation
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amplitude, reaching 80% decrease for peak ground acceleration of 0.50g. Furthermore, the
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equivalent-linear soil becomes stratified, as evident from the peaks and valleys, in dynamic stiffness
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for values of α0e less than 1. The frequencies associated with valleys correspond to resonant
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frequencies of the soil, as discussed in the foregoing.
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Concerning the vertical dashpot coefficient (Fig. 5(f)), the increase in hysteretic damping with
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increasing excitation amplitude is apparent. The low frequency range is dominated by the response
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of deeper soil layers, which do not influence to an appreciable extent the overall dashpot
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coefficient. The frequency-dependent radiation damping coefficient in the vertical mode is shown to
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be slightly affected by the increase in excitation amplitude, increasing with frequency at a slightly
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higher rate than in the linear case.
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Similar trends are observed in the rocking (Fig. 5(c,g)) and torsional (Fig. 5(d,h)) modes.
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Specifically, stiffness decreases relative to the linear case with increasing excitation amplitude, and
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tends to become frequency independent. On the other hand, hysteretic damping increases with
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excitation amplitude, while radiation damping is relatively unaffected. Radiation damping for the
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nonlinear cases increases with frequency at approximately the same rate as in the linear one. The
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small (compared to the swaying modes) dashpot coefficients for the rocking and torsional modes
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hold for the nonlinear cases as well. They are at least 50% lower than those for the translation
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modes, a trend which is known to result from wave interference effects (Mylonakis et al. 2006).
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Effect of initial soil shear wave velocity
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Considering a stiffer silt with an initial shear wave propagation velocity of the soil Vs30=350m/s, the
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dynamic response of the footing resting on the stiffer soil is depicted in Fig. 6 for the four
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oscillation modes.
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The stiffer soil medium forces the dynamic stiffness coefficients to decrease to a lesser extent
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relative to the softer soil halfspace in the previous figure (Fig. 5). This is anticipated, given that the
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softer soil develops larger deformations and, thereby, nonlinearities are naturally more pronounced.
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With reference the stiffer soil, the horizontal stiffness coefficient increases monotonically with
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increasing frequency – in contrast with the linear case – due to the constant increase in strain12
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compatible shear wave velocity with depth. Some minor undulations in the results are observed,
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which can be attributed to stress-induced soil inhomogeneity and the discretization of the soil
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profile. In the vertical mode, the peaks and valleys are again more pronounced than in the horizontal
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mode, suggesting stronger resonance phenomena due to the deeper zone of influence (―stress
316
bulbs‖) in the particular mode. These undulations tend to become progressively flatter with
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increasing level of seismic load, because of the increasing hysteretic damping in the soil. Likewise,
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rocking and torsional dynamic stiffness coefficients decrease less relative to softer soil. On the other
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hand, their frequency dependence is weaker in the whole frequency range.
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The dashpot coefficients for the horizontal and vertical modes resemble the ones of the softer
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soil, both in magnitude and frequency dependence. On the contrary, in the stiffer soil the equivalent
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dashpot coefficients for rocking and torsion are less than half of the corresponding ones in softer
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soil. In the particular case of the torsional mode, the dashpot coefficient is practically independent
324
of forcing frequency.
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Effect of frequency content of ground motion
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It is well known that identical soil profiles subjected to different earthquake motions can respond
327
differently, depending on the amplitude and frequency content of the input motion. For the purposes
328
of this analysis, the soft silt halfspace examined above (PI0, Vs30=180m/s) is subjected to the scaled
329
San Rocco, 1976 Friuli, Italy earthquake record. The resulting horizontal, vertical, rocking and
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torsional dynamic coefficients are depicted in Fig. 7.
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In horizontal mode (Fig. 7(a,e)), footing stiffness naturally decreases with increasing excitation
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amplitude. Nevertheless, contrary to Fig. 5(a,e), for 0.50g earthquake amplitude the ordinates of
333
stiffness coefficient plot higher than for excitation amplitude of 0.30g. This can be explained by
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inspection of the equivalent-linear shear wave velocity profile. For earthquake amplitude of 0.30g,
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the equivalent-linear shear wave velocity at depth 0.5m, 2m, and 5m is calculated at 173m/s,
336
154m/s and 128m/s respectively. Remarkably, for 0.50g earthquake amplitude, equivalent-linear
337
shear wave velocity at the same depths is higher (176m/s, 162m/s and 140m/s respectively). This 13
338
indicates that stiffness at higher frequency range (dominated by the response of the upper soil
339
layers) will be larger for the amplitude of 0.50g. This counter intuitive phenomenon can be
340
explained in view of the higher frequency content of the particular signal, which excites different
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resonant frequencies of the soil profile. The uppermost soil layers seem to be excited less strongly
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by the 0.50g record. On the other hand, radiation damping seems not to be significantly affected by
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the frequency content of the ground motion.
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In the vertical mode (Fig. 7(b,f)), similar trends are observed for the stiffness. Whereas for
345
amplitudes up to 0.30g stiffness decreases with increasing amplitude, for 0.50g amplitude the
346
stiffness coefficient seems to decrease by a lesser amount than it does for lower input amplitudes.
347
Besides, steeper valleys and flatter peaks appear in the low frequency range, as compared to the
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case of the 1995 Aegion earthquake record (Fig. 5(b,f)). The sharp drop in stiffness for 0.50g
349
amplitude at dimensionless frequency α0e 0.5 can be attributed to the rapid increase in radiation
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damping at that frequency range.
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In the rocking and the torsional modes (Fig. 7(c,g) and Fig. 7(d,h) respectively), dynamic
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stiffness resembles that of horizontal and vertical modes. The results corresponding to the highest
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considered amplitude for this record (0.50g) tend to decrease less with frequency than in Fig. 5. In
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contrast, the dynamic stiffness coefficients of the rocking and torsional modes do not experience
355
any resonance phenomena in the examined frequency range, decreasing monotonically with
356
frequency. The dashpot coefficients increase with increasing amplitude of input motion, due to
357
progressively higher hysteretic material damping in the soil, and increase with increasing frequency
358
at a rate similar to the linear elastic case. As expected, smaller values of radiation damping are
359
exhibited in the rocking and torsional modes compared to the translational ones.
360
Effect of soil material type
361
For given foundation geometry, soil profile, initial shear wave velocity, and earthquake excitation
362
and intensity, the dynamic response of the footing might change in the light of an equivalent-linear
363
soil analysis, depending on the shear modulus reduction and damping increase curves that are 14
364
assigned to the soil. The dynamic behavior of an identical soil-footing configuration to that
365
presented in Fig. 5, but with the soil material comprising of sand (upper bound shear modulus)
366
instead of silt, is presented in Fig. 8.
367
In horizontal mode (Fig. 8(a,e)) the dynamic stiffness coefficient exhibits the same trend as in
368
the case of loose silt with PI0 (Fig. 5(a,e)) but attains higher values. In addition, stiffness increases
369
with frequency, most likely because of shear wave velocity inversion at the deeper layers.
370
Obviously, the higher shear modulus of the sand makes the soil-footing system stiffer for the same
371
level of excitation, as witnessed from comparing the shear modulus reduction curves in Fig. 2.
372
Besides, dashpot coefficients for silt and sand are quite similar, suggesting that wave propagation in
373
the medium is not sensitive to material type.
374
In the vertical mode (Fig. 8(b,f)), stiffness is higher than for the silty PI0 soil (Fig. 5(b,f)) while
375
same undulations are noted for dimensionless frequencies less than 0.5. At higher frequencies, the
376
dynamic stiffness coefficient for sand is practically frequency independent. As in the horizontal
377
mode, the vertical dashpot coefficient is not sensitive to material type.
378
For the rocking (Fig. 8(c,g)) and torsional (Fig. 8(d,h)) modes, trends for both dynamic
379
stiffness and dashpot coefficients resemble those for silt with PI0. For the same level of loading,
380
however, the shear modulus of the sand is higher than the modulus of the silt, so strain levels are
381
lower and the dynamic stiffness coefficients attain higher values.
382
Averaging dynamic excitation
383
From the above observations, it becomes clear that the dynamic response of a footing under strong
384
earthquake motion may strongly depend on the stress-strain properties of the soil material, the
385
initial shear wave velocity and the amplitude and frequency content of the input motion. The soil
386
stress-strain characteristics seem to affect mainly the amplitude of dynamic stiffness coefficients,
387
while the excitation frequency characteristics seem to affect its shape.
388
To eliminate the influence of excitation characteristics, it appears desirable to develop a set of
389
mean curves encompassing different ground motions with varying amplitude and frequency content, 15
390
for the dynamic stiffness and dashpot coefficient for each soil type. Since shear wave velocity
391
profiles depend on the characteristics of earthquake motions, averaging of dimensionless parameters
392
α0e and α0 should be performed as well. Based on analytical results of this study, standard deviation
393
of the mean curve does not exceed a mere 10%. For sake of simplicity and in the interest of space,
394
only mean values are reported here.
395
For a homogeneous halfspace consisting of silty soil material with PI0 and initial shear wave
396
velocities 180m/s and 350m/s, subjected to the five earthquake records of Table 1, the mean
397
dynamic stiffness coefficients are shown in Fig. 9. The mean curves exhibit patterns similar to the
398
ones discussed in the foregoing.
399
For the softer soil (Vs30=180m/s) and for earthquake amplitudes larger than 0.20g, there is more
400
than 20% decrease in stiffness over the linear case. The mean vertical dynamic stiffness coefficient
401
(Fig. 9(b)) drops over the linear case with increasing excitation amplitude. For earthquake
402
amplitudes larger than 0.20g, a decrease from the linear case of over than 40% is attested.
403
Moreover, the curves show clear peaks in dimensionless frequencies lower than 0.5, indicative of
404
resonant phenomena in the soil. For dimensionless frequencies higher than 0.5, vertical stiffness
405
coefficients decrease with frequency in more or less the same way as in the linear case. Rocking
406
(Fig. 9(c)) and torsional (Fig. 9(d)) stiffness coefficients decrease with increasing level of ground
407
motion in an essentially frequency independent manner.
408
For stiffer soil (Vs30=350m/s) the deviation from the linear case is generally smaller, as evident
409
in
410
CONCLUSIONS
411
An equivalent-linear method was presented for an approximate – yet reasonable – assessment of the
412
dynamic impedance functions of surface footings accounting for material nonlinearity in the soil
413
halfspace. The main findings of the study are summarized as follows:
414
Fig.
9(e,f,g,h).
The
individual
patterns
are
as
discussed
in
the
foregoing.
1. The dynamic response of the footing depends on more parameters than in the linear case.
16
415
Specifically the complexity of the linear problem is augmented by the influence of the: (i)
416
initial shear wave velocity of the soil profile; (ii) shear modulus reduction and damping
417
increase curves; (iii) excitation amplitude and frequency content.
418
2. The dynamic stiffness coefficient is found to decrease monotonically with increasing
419
excitation amplitude and decreasing initial shear wave velocity. On the other hand, no clear
420
conclusions can be drawn for the frequency dependence of the parameters. The coefficient
421
may increase, decrease, or remain constant with frequency.
422
3. In the equivalent-linear soil profile, distinct soil layers of different stiffness are formed (a
423
behavior often referred to as stress-induced inhomogeneity) and, consequently, resonant
424
frequencies may appear, giving rise to undulations in dynamic stiffness coefficients with
425
frequency, even in homogeneous halfspace. These undulations are pronounced in the vertical
426
and horizontal modes, yet may not appear in rocking and torsion. The larger fluctuations for
427
the vertical mode are attributed to the deeper zone of influence (―stress-bulb‖) induced by
428
vertical loading.
429
4. Depending on the individual characteristics of the input ground motion, different behaviors
430
may be observed for the stiffness coefficients, resulting from resonances of the exciting
431
signal with the soil. These differences are generally filtered out when averaging results from
432
different earthquake ground motions, to produce a smoother stiffness function of frequency.
433
5. The dashpot coefficient is found to be fairly dependent on excitation amplitude. It increases
434
from the linear case with increasing level of strain, as expected due to the increase in
435
hysteretic soil material damping. Also, smooth undulations are observed near the stress-
436
induced resonant frequencies.
437
6. The dashpot coefficient is much larger in the translational modes than in the rotational ones,
438
as in the linear case. In the rotational modes, however, the dependence of radiation
439
coefficient on soil type (clay or sand) is higher than in the translational modes.
440
The proposed dimensionless charts for foundation impedances are sought to be used for a first
17
441
assessment of whether nonlinear soil behavior is a considerable factor in system response, without
442
resource to sophisticated computer platforms for nonlinear dynamic analysis of continua.
443
ACKNOWLEDGEMENTS
444
This study was performed in the framework of the European research project ―New Methods
445
for Mitigation of Seismic Rick of Existing Foundation‖ (acronym NEMISREF, EC contract No
446
G1RD-CT-2002-00702, EC project No GRD1-2001-40457). The first author would like to
447
acknowledge Professor George Mylonakis for fruitful discussions on the topic throughout the last
448
year. The authors would also like to thank Fernando Lopez-Caballero for all the helpful discussions
449
over the course of this research.
450
REFERENCES
451
Barkan, D. D. (1948). Dynamics of Beds and Foundations [in Russian], Stroivoenmorizdat,
452
Moscow.
453
Barkan, D. D. (1960). Dynamics of bases and foundations. McGraw-Hill, New York, N.Y.
454
Borja, R. I. and Wu, W. (1994). Vibration of foundations on incompressible soils with no elastic
455
regions. Journal of Geotechnical Engineering Division - ASCE, 120(9), 1570-1592.
456
Building Seismic Safety Council (BSSC) (2004). NEHRP Recommended Provisions for Seismic
457
Regulations for New Buildings and Other Structures, Federal Emergency Management
458
Agency, Washington, D.C.
459 460 461 462
Chatterjee, P. and Basu, B. (2008). Some analytical results on lateral dynamic stiffness for footings supported on hysteretic soil medium. Soil Dynamics and Earthquake Engineering, 28, 36-43. Chopra, A. K. (2011). Dynamics of Structures: Theory and Applications to Earthquake Engineering, 4th Edition, Prentice Hall.
463
Chu, D. B., Stewart, J. P., Boulanger, R. W. and Lin, P. S. (2008). Cyclic softening of low-plasticity
464
clay and its effect on seismic foundation performance. Journal of Geotechnical and
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465 466 467
Geoenvironmental Engineering, 134(11), 1595-1608. Clouteau, D. and Aubry, D. (2003). Computational soil-structure interaction. Boundary element methods for soil-structure interaction (pp. 61-126). Kluwer Academic Publishers.
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de Barros, F. C.P. and Luco, J. E. (1995). Identification of foundation impedance functions and soil
469
properties from vibration tests of the Hualien containment model. Soil Dynamics and
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Earthquake Engineering, 14(94), 229-248.
471
Dunn, P. W., Hiltunen, D. R. and Woods, R. D. (2009). In Situ Determination of Dynamic
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Impedance Functions of Shallow Foundations. Proceedings of the XVII International
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Conference on Soil Mechanics and Geotechnical Engineering, Alexandria, Egypt.
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Gajan, S., Raychowdhury, P., Hutchinson, T. C., Kutter, B. L. and Stewart, J. P. (2010). Application
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and validation of practical tools for nonlinear soil-foundation interaction analysis. Earthquake
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Spectra, 26(1), 111-129.
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Gazetas, G. (1983). Analysis of machine foundation vibrations: State of the art. Soil Dynamics and Earthquake Engineering, 2(1), 2-42. Gazetas, G. (1991). Formulas and charts for impedances of surface and embedded foundations. Journal of Geotechnical Engineering Division - ASCE, 117(9), 1363-1381.
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Kim, S. and Stewart, J. P. (2003). Kinematic soil-structure interaction from strong motion
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recordings. Journal of Geotechnical and Geoenvironmental Engineering, 129, 323-335.
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Kokusho, T. and Sato, K. (2008). Surface-to-base amplification evaluated from KiK-net vertical
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array strong motion records. Soil Dynamics and Earthquake Engineering, 28, 707-716.
485
Lopez-Caballero, F. and Modaressi Farahmand-Razavi, A. (2008). Numerical simulation of
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liquefaction effects on seismic SSI. Soil Dynamics and Earthquake Engineering, 28(2), 85-98.
487
Luco, J. E. and Westman, R. A. (1971). Dynamic response of circular footings. Journal of the
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Engineering Mechanics Division, ASCE, 97, 1381-1395. Mylonakis, G. and Gazetas, G. (2000). Soil-Structure Interaction: Beneficial or Detrimental? Journal of Earthquake Engineering, 4(3), 377-401.
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Mylonakis, G., Nikolaou, S. and Gazetas, G. (2006). Footings under seismic loading: Analysis and
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design issues with emphasis on bridge foundations. Soil Dynamics and Earthquake
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Engineering, 26(9), 824-853.
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NEHRP Consultants Joint Venture (2011). Soil-Foundation-Structure Interaction for Building Structures. Report No. NIST GCR 11-917-14. Pais, A. and Kausel, E. (1988). Approximate formulas for dynamic stiffnesses of rigid foundations. Soil Dynamics and Earthquake Engineering, 7(4), 213-227.
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PEER (2008). Open system for earthquake engineering simulation (OpenSees)—development
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platform by the Pacific Earthquake Engineering Research Center (PEER), http://
500
opensees.berkeley.edu.
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Pitilakis, D. (2006). Soil-structure interaction modeling using equivalent linear soil behavior in the substructure method. Ph.D. Thesis, Ecole Centrale Paris, France.
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Pitilakis, D., Dietz, M, Wood, D.M., Clouteau, D. and Modaressi, A. (2008). Numerical simulation
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of dynamic soil-structure interaction in shaking table testing. Soil Dynamics and Earthquake
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Engineering, 28(6), 453-467.
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Pitilakis, D. and Clouteau, D. (2010). Equivalent-linear substructure approximation of soil-
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foundation-structure interaction: Model presentation and validation. Bulletin of Earthquake
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Engineering, 8(2), 257-282.
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Richart, F. E., Hall, J. R. and Woods, R. D. (1970). Vibrations of soils and foundations. PrenticeHall, Englewood Cliffs, N.J.
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Seed, H. B., Wong, R., Idriss, I. M. and Tokimatsu, K. (1986). Moduli and damping factors for
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dynamic analyses of cohesionless soil. Journal of Geotechnical Engineering - ASCE, 112(11),
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1016-1032.
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Star, L. (2011). Seismic Vulnerability of Structures: Demand Characteristics and Field Testing to
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Evaluate Soil-Structure Interaction Effects. Ph.D. Thesis, Department of Civil Engineering,
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Stewart, J. P., Kim, S., Bielak, J., Dobry, R. and Power, M. (2003). Revisions to soil structure interaction procedures in NEHRP design provisions. Earthquake Spectra, 19, 677-696.
519
Tileylioglou, S., Stewart, J. P. and Nigbor, R. L. (2011). Dynamic stiffness and damping of a
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shallow foundation from forced vibration of a field test structure. Journal of Geotechnical and
521
Geoenvironmental Engineering, 137(4), 344-353.
522 523
Vucetic, M. and Dobry, R. (1991). Effect of soil plasticity on cyclic response. Journal of Geotechnical Engineering Division - ASCE, 117(1), 89-107.
524
21
525
List of Tables
526
Table 1. Summary of the input (column-wise) used in all parametric analyses. Footing
Soil
Material
Vs
Earthquake record
PGA
Circular
ρ=2Mg/m3
Loose Silt (PI0)
100m/s
Friuli 1976
0.01g
Rigid
=1/3
Clay (PI30)
180m/s
Vrancea 1977
0.10g
Massless
=2%
Sand lower bound
250m/s
Umbria 1984
0.20g
Sand upper bound
350m/s
Aegion 1995
0.30g
500m/s
Kozani 1995
0.50g
d=10m
22
List of Figures Fig. 1. a) Rigid massless circular footing and b) homogeneous halfspace
Fig. 2. Shear modulus reduction and damping increase curves for all four materials
Fig. 3. Comparison between various horizontal layering mesh configurations for the swaying (a, b) and rocking modes (c, d), and dynamic stiffness (a, c) and damping (b, d) coefficients. Impedance functions for the chosen configuration are almost identical to the ones for the very fine configuration.
Fig. 4. a) Acceleration time histories and b) Fourier spectra of the five earthquake records.
Fig. 5. Dynamic stiffness coefficient (a, b, c, d) and dashpot coefficient (e, f, g, h) for footing resting on a halfspace soil profile, composed of silt with PI0, with initial soil shear wave velocity Vs30=180m/s, for the horizontal (a, e), vertical (b, f), rocking (c, g) and torsional (d, h) vibration modes, when subjected to the upand down-scaled Aegion, 1995 Aegion, Greece earthquake record.
Fig. 6. Dynamic stiffness coefficient (a, b, c, d) and dashpot coefficient (e, f, g, h) for footing resting on a halfspace soil profile, compose of silt with PI0, with initial soil shear wave velocity Vs30=350m/s, for the horizontal (a, e), vertical (b, f), rocking (c, g) and torsional (d, h) vibration modes, when subjected to the upand down-scaled Aegion, 1995 Aegion, Greece earthquake record.
Fig. 7. Dynamic stiffness coefficient (a, b, c, d) and dashpot coefficient (e, f, g, h) for footing resting on a halfspace soil profile, composed of silt with PI0, with initial soil shear wave velocity Vs30=180m/s, for the horizontal (a, e), vertical (b, f), rocking (c, g) and torsional (d, h) vibration modes, when subjected to the upand down-scaled San Rocco, 1976 Friuli, Italy earthquake record.
Fig. 8. Dynamic stiffness coefficient (a, b, c, d) and dashpot coefficient (e, f, g, h) for footing resting on a halfspace soil profile, composed of upper bound sandy soil, with initial shear wave velocity Vs30=180m/s, for the horizontal (a, e), vertical (b, f), rocking (c, g) and torsional (d, h) vibration modes, when subjected to the up- and down-scaled Aegion, 1995 Aegion, Greece earthquake record.
23
Fig. 9. Dynamic stiffness coefficient for footing resting on a halfspace soil profile, composed of silt with PI0, with initial shear wave velocity Vs30=180m/s (a, b, c, d) and Vs30=350m/s (e, f, g, h), for the horizontal (a, e), vertical (b, f), rocking (c, g) and torsional (d, h) vibration modes. The response is averaged concerning the earthquake records used in the sets of parametric analyses.
24
Fig1a Click here to download Figure: fig1a.eps
Fig1b Click here to download Figure: fig1b.eps
Fig2 Click here to download Figure: fig2.pdf
Fig3a Click here to download Figure: fig3a.eps
Swaying Re(Sh) / Klinear static
1 Chosen Fine Coarse
0.8 0.6 0.4 0.2 0
0
1
2 α0e
3
Fig3b Click here to download Figure: fig3b.eps
Swaying Im(Sh) / Klinear static
1 Chosen Fine Coarse
0.8 0.6 0.4 0.2 0
0
1
2 α0
3
Fig3c Click here to download Figure: fig3c.eps
Rocking Re(Sr) / Klinear static
1 Chosen Fine Coarse
0.8 0.6 0.4 0.2 0
0
1
2 α0e
3
Fig3d Click here to download Figure: fig3d.eps
Rocking 1 Im(Sr) / Klinear static
Chosen Fine Coarse
0.8 0.6 0.4 0.2 0
0
1
2 α0
3
Fig4a Click here to download Figure: fig4a.eps
Acceleration records
2
Acc. (m/s )
5
0
−5 0
5 Time (s)
10
Fig4b Click here to download Figure: fig4b.eps
Frequency range covered 3 2.5
Ampl.
2
Aigio95 Vrancea77 Friuli76 Kozani95 Umbria84
1.5 1 0.5 0 0 1 2 3 4 5 6 7 8 9 10 Frequency (Hz)
Fig5a Click here to download Figure: fig5a.eps
Swaying Re(Sh) / Klinear static
1 0.8 0.6 lin 0.01g 0.10g 0.20g 0.30g 0.50g
0.4 0.2 0
0
1
2 α 0e
3
Fig5b Click here to download Figure: fig5b.eps
Vertical Re(Sv) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
1
2 α 0e
3
Fig5c Click here to download Figure: fig5c.eps
Rocking Re(Sr) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
1
2 α 0e
3
Fig5d Click here to download Figure: fig5d.eps
Torsion Re(St) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
1
2 α 0e
3
Fig5e Click here to download Figure: fig5e.eps
Im(Sh) / Klinear static
1 0.8 0.6 0.4
lin 0.01g 0.10g 0.20g 0.30g 0.50g
0.2 0
0
0.5
1 α0
1.5
2
Fig5f Click here to download Figure: fig5f.eps
Im(Sv) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
0.5
1 α0
1.5
2
Fig5g Click here to download Figure: fig5g.eps
Im(Sr) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
0.5
1 α0
1.5
2
Fig5h Click here to download Figure: fig5h.eps
Im(St) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
0.5
1 α0
1.5
2
Fig6a Click here to download Figure: fig6a.eps
Swaying Re(Sh) / Klinear static
1 0.8 0.6 lin 0.01g 0.10g 0.20g 0.30g 0.50g
0.4 0.2 0
0
1
2 α 0e
3
Fig6b Click here to download Figure: fig6b.eps
Vertical Re(Sv) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
1
2 α 0e
3
Fig6c Click here to download Figure: fig6c.eps
Rocking Re(Sr) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
1
2 α 0e
3
Fig6d Click here to download Figure: fig6d.eps
Torsion Re(St) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
1
2 α 0e
3
Fig6e Click here to download Figure: fig6e.eps
Im(Sh) / Klinear static
1 0.8 0.6 0.4
lin 0.01g 0.10g 0.20g 0.30g 0.50g
0.2 0
0
0.5
1 α0
1.5
2
Fig6f Click here to download Figure: fig6f.eps
Im(Sv) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
0.5
1 α0
1.5
2
Fig6g Click here to download Figure: fig6g.eps
Im(Sr) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
0.5
1 α0
1.5
2
Fig6h Click here to download Figure: fig6h.eps
Im(St) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
0.5
1 α0
1.5
2
Fig7a Click here to download Figure: fig7a.eps
Swaying Re(Sh) / Klinear static
1 0.8 0.6 lin 0.01g 0.10g 0.20g 0.30g 0.50g
0.4 0.2 0
0
1
2 α 0e
3
Fig7b Click here to download Figure: fig7b.eps
Vertical Re(Sv) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
1
2 α 0e
3
Fig7c Click here to download Figure: fig7c.eps
Rocking Re(Sr) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
1
2 α 0e
3
Fig7d Click here to download Figure: fig7d.eps
Torsion Re(St) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
1
2 α 0e
3
Fig7e Click here to download Figure: fig7e.eps
Im(Sh) / Klinear static
1 0.8 0.6 0.4
lin 0.01g 0.10g 0.20g 0.30g 0.50g
0.2 0
0
0.5
1 α0
1.5
2
Fig7f Click here to download Figure: fig7f.eps
Im(Sv) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
0.5
1 α0
1.5
2
Fig7g Click here to download Figure: fig7g.eps
Im(Sr) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
0.5
1 α0
1.5
2
Fig7h Click here to download Figure: fig7h.eps
Im(St) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
0.5
1 α0
1.5
2
Fig8a Click here to download Figure: fig8a.eps
Swaying Re(Sh) / Klinear static
1 0.8 0.6 lin 0.01g 0.10g 0.20g 0.30g 0.50g
0.4 0.2 0
0
1
2 α 0e
3
Fig8b Click here to download Figure: fig8b.eps
Vertical Re(Sv) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
1
2 α 0e
3
Fig8c Click here to download Figure: fig8c.eps
Rocking Re(Sr) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
1
2 α 0e
3
Fig8d Click here to download Figure: fig8d.eps
Torsion Re(St) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
1
2 α 0e
3
Fig8e Click here to download Figure: fig8e.eps
Im(Sh) / Klinear static
1 0.8 0.6 0.4
lin 0.01g 0.10g 0.20g 0.30g 0.50g
0.2 0
0
0.5
1 α0
1.5
2
Fig8f Click here to download Figure: fig8f.eps
Im(Sv) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
0.5
1 α0
1.5
2
Fig8g Click here to download Figure: fig8g.eps
Im(Sr) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
0.5
1 α0
1.5
2
Fig8h Click here to download Figure: fig8h.eps
Im(St) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
0.5
1 α0
1.5
2
Fig9a Click here to download Figure: fig9a.eps
Re(Sh) / Klinear static
1 0.8 0.6 0.4
lin 0.01g 0.10g 0.20g 0.30g 0.50g
0.2 0
0
2 α0e
4
Fig9b Click here to download Figure: fig9b.eps
Re(Sv) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
2 α0e
4
Fig9c Click here to download Figure: fig9c.eps
Re(Sr) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
2 α0e
4
Fig9d Click here to download Figure: fig9d.eps
Re(St) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
2 α0e
4
Fig9e Click here to download Figure: fig9e.eps
Re(Sh) / Klinear static
1 0.8 0.6 0.4
lin 0.01g 0.10g 0.20g 0.30g 0.50g
0.2 0
0
0.5
1 α0e
1.5
Fig9f Click here to download Figure: fig9f.eps
Re(Sv) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
0.5
1 α0e
1.5
Fig9g Click here to download Figure: fig9g.eps
Re(Sr) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
0.5
1 α0e
1.5
Fig9h Click here to download Figure: fig9h.eps
Re(St) / Klinear static
1 0.8 0.6 0.4 0.2 0
0
0.5
1 α0e
1.5
