On the Estimation of Earthquake Induced Ground Strains from Velocity

Soil dynamic properties can be determined in centr ifuge tests through the shear stress-strain cycles obtained from the accelerations recorded within the soil model. Despite its simplicity, a significant source of uncertainty of the method has been recogn ized to lie in the determination of the strain amplitude. Recently, a method was proposed to compu te the mobilized shear modulus and damping ratio using the nonlinear fit of experiment al ransfer functions with the analytical expression of the amplification function for a visc oelastic layer on a rigid base (Conti & Viggiani, 2012). According to this method, the corresponding shear strain is a function of the particle velocity and of the shear wave velocity. However, t he equation proposed by the Authors strictly holds only for almost mono-frequency signals. This paper addresses the problem of using velocity measurements to compute the shear strains induced b y vertically propagating shear waves in a uniform soil layer. Theoretical results are applied to the interpretation of centrifuge dynamic tests. Introduction The non-linear and hysteretic behaviour of soils is generally described by the shear modulus, G, and the damping ratio, D, and their variation with shear strain level, γ. As amplification phenomena inside a soil layer depend strongly on th e shear stiffness and the damping ratio mobilised during the earthquake, the G(γ) and D(γ) curves are crucial ingredients to determine the seismic response of geotechnical systems in soi l-structure interaction problems. Zhegal et al. (1995) showed that G and D can be determined from in-situ acceleration time histories recorded during real earthquakes. The me thod is based on the evaluation of shear stress-strain cycles obtained from down-hole accele ration time histories recorded at different depths in the soil at instrumented test sites. The same method has been applied to the acceleration time histories recorded in centrifuge models (Zeghal et al., 1998; Brennan et al., 2005). In this case, the greatest source of uncert ainty lies in the determination of the shear strain amplitude (Zeghal et al. 1998) which may be affected by large and not easi ly definable errors. More recently, Conti & Viggiani (2012) suggested to compute the mobilised shear modulus and damping ratio using the nonlinear fit of experiment al transfer functions obtained at different depths in centrifuge models. Moreover, they showed that the free-field shear strain induced into the soil model by a vertically propagating shear wa ve cannot be computed using the theory of wave propagation in homogeneous isotropic elastic m edia, i.e. γ = v/VS (v = particle velocity, VS = shear wave velocity), due to multiple refraction a d reflection of waves generated by boundary conditions. However, the equation proposed by the Authors was derived for the sole Research Assistant, Facoltà di Ingegneria, Universi tà Niccolò Cusano, Roma, Italy, riccardo.conti@unicusano.it case of a uniform undamped soil layer and, also, st rictly holds only for the almost monofrequency signals applied in the centrifuge dynamic tests analysed in their paper. This work addresses the problem of using velocity m easurements to compute the shear strains induced by vertically propagating shear waves in a uniform soil layer. After recalling the main steps of the data reduction procedure proposed by C onti & Viggiani (2012), the exact point wise relation between particle velocity and shear strain is derived through the definition of a suitable transfer function, under the assumption of isotropi c v scoelastic behaviour for the soil. Theoretical results are applied to the interpretati on of centrifuge dynamic tests. Evaluation of Soil Dynamic Properties in Centrifuge Tests According to Conti & Viggiani (2012), the shear mod ulus and the damping ratio mobilised during an earthquake at mid-height of the centrifug e model can be estimated from the amplification function of the soil layer A(f), obtained from the free-field accelerations mea sured at the bottom and close to the surface. The propose d m thod includes three steps: (i) identification of the range of frequencies signific ant for both signals, and calculation of the experimental amplification function within this ran ge; (ii) non-linear interpolation of the experimental curves with the analytical expression of the amplification function of a visco-elastic layer on a rigid base and determination of G and D; (iii) calculation of the mobilised shear strain at mid-height of the sand layer. The procedure for data reduction will be recalled briefly in the following, with reference to one of the test perfor med by the Authors on dry sand (Test CW6). Step 1: Identification of Relevant Frequencies The range of frequencies that are common to signals x(t) and y(t), registered at the bottom and close to the top of the sand layer respectively, ca n be identified by the cross-power spectrum Gxy(f) of the two signals, given by: ) ( ) ( ) ( * f Y f X f Gxy ⋅ = (1) where X(f) is the Fourier spectrum of x(t) and Y*( f) is the complex conjugate of the Fourier spectrum of y(t). In particular, the frequencies that are signific ant for both signals can be identified as those at which the amplitude of the n ormalised cross-power spectrum is greater than a threshold ( TOL = 10÷10). The procedure is illustrated in Figure 1 which s hows, for one of the earthquakes applied in the test, (a,b) the Four ier spectra of the accelerations recorded in free field conditions, (c) the normalised cross-power sp ectrum of the two signals, and (d) the experimental amplification function. In Figure 1(d) , the continuous line is the original amplification function for all frequencies in the F ourier spectra, while the dots represent the subset of the amplification function for those freq u ncies where the normalised cross power spectrum of x and y is larger than the prescribed threshold. Figure 1. Fourier spectrum of the accelerations at (a) the bottom and (b) on top of the sand layer and corresponding (c) normalised cross-power spectr um and (d) amplification function (Conti & Viggiani, 2012 Test CW6, EQ1) Step 2: Computation of Mobilized Shear Modulus and Damping The experimental amplification function A(f) is given by: ( ) ( ) ( ) Y f A f X f = (2) where X(f) and Y(f) are the amplitudes of the Fourier spectra of x(t) and y(t), calculated for the range of frequencies in common to the two signa ls. The analytical expression of the amplification function for a visco-elastic soil lay er on a rigid base, defined at a depth z in the sand layer, is given by: