A Comparative Study on the Interpretation Procedure of Field Tests for Measuring the Dynamic Impedance of a Surface Footing

ABSTRACT The paper analyses in detail and compares different interpretation procedures of snap-back and forced-vibration tests on a full-scale prototype founded on soft-soil, in order to assess their effectiveness for measuring the stiffness and damping of a shallow foundation under dynamic loads. Both properties were back-calculated through three alternative methods, i.e. through impedance functions computed in the frequency domain, or by interpreting force-displacement loops in terms of peak-to-peak amplitudes or of phase-shift. The damping was further calculated from the logarithmic decrement of free-vibration records. The resulting foundation stiffness and damping were observed to vary with the number of cycles and with the load frequency and amplitude. The comparison among the interpretation techniques revealed that the peak-to-peak approach fails when damping is high, because it neglects the delay between force and displacement.


Introduction
The main inertial effects of the dynamic soil-foundation structure (SFS) interaction consist in the elongation of the natural period, which depends on the swaying and rocking stiffness of the foundation, the generation of the radiation damping, accounting for the energy dissipated by the wave scattering from the foundation, and the generation of the hysteretic damping associated to the nonlinear behavior of the soil beneath the foundation. The importance of the above effects increases with the foundation displacement and rotation (Karatzetzou and Pitilakis 2018). The period elongation and the wave scattering occur even when the foundation is close to its static equilibrium position and increase when the foundation motion induces significant shear straining in the soil. The hysteretic damping represents the energy loss due to the nonlinear soil response, thus becoming increasingly significant, if not dominant, at large foundation displacements (Gazetas 2015). The quantification of such effects is based on the accurate evaluation of the complex and frequency-dependent soilfoundation impedance, which integrates the stiffness and the damping ratio mobilized in the soil under the oscillating footing.
The advances in dynamic monitoring and signal processing techniques increasingly raised the interest of researchers in the experimental identification of the dynamic response and in the derivation of the foundation dynamic stiffness and damping ratio.
In previous experimental studies, the abovementioned were derived from data of on-site records during low-amplitude forced vibration tests on real structures or physical prototypes (Amendola et al. 2021b;Crouse et al. 1990;de Barros and Luco 1995;Luco, Trifunac, and Wong 1988;Tileylioglu, Stewart, and Nigbor 2011;Wong, Trifunac, and Luco 1988), as well as from field tests and centrifuge or shaking table tests under force amplitudes able to mobilize the failure of the foundation soil (Faccioli, Paolucci, and Vivero 2001;Fattah, Al-Mosawi, and Al-Ameri 2017;Gajan and Kutter 2008;Negro et al. 2000;Sharma and Deng 2019, 2020a, 2020bShirato et al. 2008). In most of these studies, the interpretation of experimental data differs according to the mobilized foundation motion. In detail, the foundation stiffness and damping ratio are obtained in the frequency domain from the real and the imaginary parts of the impedance functions when the SFS system is loaded by ambient noise or low-amplitude harmonic forces (Amendola et al. 2021b;Crouse et al. 1990;de Barros and Luco 1995;Luco, Trifunac, and Wong 1988;Tileylioglu, Stewart, and Nigbor 2011). Such approach is more rigorous, but it is based on the hypothesis of a linear soilfoundation response. For this reason, it is generally adopted only under very low foundation motions. Conversely, the calculation in the time domain from the moment -rotation loops is preferred when the applied forces are high enough to mobilize soil non-linearity, which means that stiffness and damping ratio evolves during the test. Such approach has been widely applied only to study the soil failure under rocking foundations (Faccioli, Paolucci, and Vivero 2001;Gajan and Kutter 2008;Hakhamaneshi and Kutter 2016;Negro et al. 2000;Sharma and Deng 2019, 2020a, 2020bShirato et al. 2008), while there is a complete lack of data referred to swaying foundations or to combined (rocking and swaying) low foundation motions. The fundamental limit of such a strict separation between the different interpretation procedures is the impossibility to compare the foundation response from very low to very high foundation motions, leading to a very partial comprehension of the effects of SFS interaction. Moreover, evidence of non-linearity in the soil-foundation response has been noticed even under low force amplitudes (Amendola et al. 2021a;Star et al. 2019), suggesting that such a distinction can be overcome.
This paper aims to highlight similarities and differences in the outcomes of different approaches in time and frequency domains to interpret records of forced vibration and snap-back tests under low force amplitudes. The experimental data sets were gathered in the framework of the research project "Seismic Impedance for Soil-structure Interaction From On-site tests, SISIFO" funded by the HORIZON2020 supported program SERA (Seismology and Earthquake Engineering Research Infrastructure Alliance for Europe). The tested facility is the real-scale structure EuroProteas founded on a soft silty-clayey sand at a test site in the North of Thessaloniki, Greece. In a previous paper by the same authors (Amendola et al. 2021b) the data sets were interpreted following a frequency-domain approach and results in terms of impedance functions were successfully compared with analytical relationships well-established in the literature. In this study, the interpretations obtained by different approaches will be discussed in detail, with the purpose to highlight their advantages and drawbacks when measuring the stiffness and damping components from both the rocking and the swaying motions. In particular, the most reliable interpretation technique in the time domain of experimental foundation stiffness or damping derived under small soil strain is investigated, in order to allow a future comparison among the experimental results in the whole range of movements expected under weak-to strong-motion earthquakes, leading to a deeper and complete characterization of the foundation response.

Main Features of the Facility
The prototype structure EuroProteas was built in the Euroseistest experimental array (http://euro seisdb.civil.auth.gr) located in the Mygdonian Valley in Northern Greece. Figure 1a shows the structure consisting of a simple steel frame supported by a square reinforced concrete slab with dimensions 3.0 m × 3.0 m × 0.4 m and overtopped by two similar reinforced concrete slabs (Fig. 1b). The steel frame consists of four squared hollow steel columns (QHS 150 mm × 150 mm × 10 mm) clamped on the foundation through steel bolts. L-shaped cross-braces (100 mm × 100 mm × 10 mm) rigidly connect the steel columns across all the sides of the structure. The foundation can be assumed as 'truly shallow,' since the soil around the concrete slab was intentionally removed before executing the tests reported in the following. More details on the structural features are discussed by  and Amendola et al. (2021a).
The soil profile (Fig. 2a) consists of a series of three soft silty layers in the first 4.6 m immediately below the foundation, resting on a layer of silty sand with a local presence of gravel down to a depth of 22 m. The physical and mechanical properties of the layered soil were measured in numerous geophysical and geotechnical surveys reported in earlier studies (Aristotle University of Thessaloniki, 2012; Raptakis et al. 1970;Pitilakis et al. 1999). In particular, the shear wave velocity, V S , of the uppermost 5 m was estimated as equal to 150 m/s from a down-hole test and to 100 m/s from the interpretation of surface geophysical tests, and as varying between 200 m/s and 250 m/s in the  (a) Soil stratigraphy and associated V S ranges; (b) variation of normalized shear modulus, G/G 0 , and damping ratio, D, versus shear strain, γ, obtained from laboratory tests (markers as adapted from ) and interpolated with sigmoid curves (solid lines).
deeper layers until 25 m. Resonant column tests were performed on representative soil specimens ) taken before the installation of the prototype, to measure the variation of normalized shear modulus, G/G 0 , and equivalent damping ratio, D, with shear strain, γ. Figure 2b shows the results referred to a sample taken at a depth between 2.7 m and 3.0 m, from a borehole close to the foundation (Fig. 1b), and tested at effective stresses 15 kPa, 30 kPa and 60 kPa. The difference among the trends obtained at increasing consolidation stress is negligible, hence all the data points were fitted through the sigmoid curves drawn in Fig. 2b, in order to define a unique non-linear and hysteretic soil behavior in the subsequent analysis of the prototype tests. As a matter of fact, such values of the consolidation stress are representative of the mean overburden stress acting in the shallowest two layers, i.e. those mostly affected by the foundation motion according to the influence depth computed by Amendola et al. (2021b) following the suggestions by Gazetas (1983) and Stewart et al. (2003).
The interpretation in the frequency domain of environmental noise records allowed for identifying the fundamental frequency of the tested configuration of the prototype, ranging between 3.2 Hz and 3.6 Hz (Amendola et al. 2021b). A fixed-base 3D model of the EuroProteas prototype was generated through the SAP2000 finite element code, in order to identify the frequency of the fixed-base system. The configuration of the structural model reproduced the prototype geometry as represented in Fig. 1. The column and the steel frame braces were modelled with elastic beam elements, with their QHS and double L steel sections, respectively, and characterized by a Young's modulus equal to 200 GPa. The base and top concrete slabs were modelled by shell elements with a Young's modulus equal to 30 GPa. The results of a modal analysis led to a fixed-base frequency value equal to 9.13 Hz (Amendola et al. 2021a;, thus significantly higher than the frequency range experimentally derived.
Such a difference highlights the strong influence of the soil-structure interaction on the dynamic response of the facility.

Instrumentation Layout
The dense 3D instrumentation layout shown in Fig. 1 was arranged to record the structure and the foundation response. Five triaxial accelerometers (CMG-5TD and CMG-5TCDE, Guralp Systems Ltd) were mounted on the roof, three along the axis parallel to the direction of shaking (in-plane) and two at the opposite corners of the slab to capture possible out-of-plane motion. Additionally, four triaxial accelerometers were mounted in pairs at the opposite edges of the foundation slab along the direction of shaking, to ensure and validate the proper recording of the foundation translational and rotational response. All the instruments were connected to external global positioning system (GPS) antennas, and their sampling frequency was set to 200 Hz. They were oriented along the positive x-direction of loading at an angle of 30 degrees with the magnetic North. However, for sake of simplicity, the x and the y component of each record were conventionally tagged as North and East, respectively, as shown in Fig. 1.

Performed Field Tests
The dynamic response of the instrumented facility was investigated under snap-back tests and forced vibration tests. Snap-back tests reproduce the soil-foundation response after strong motion, while forced vibration conditions are more representative of the response during strong motion.
During the snap-back tests, the structure was displaced along approximately the NS direction, applying pull-out forces to the roof slab, which was connected to a wire rope anchored from the other side to a counterweight embedded in the soil. The tension was applied by a pulling hoist attached to the counterweight. Once the desired level of force was reached, the wire rope was cut, and the structure underwent free vibration. The test was repeated five times by increasing the pulling force according to the values reported in Table 1. The force amplitudes were selected to check the soil-foundation response at the initial development of nonlinear effects.
For the forced-vibration tests, the MK-500 U (ANCO Engineers Inc) eccentric mass vibrator system was installed at the geometrical center on the prototype roof. The following equation controls the generated harmonic load: where E is the mass eccentricity, t is the time, and f the value of the loading frequency. The test was repeated by changing the frequency of the excitation and the shaker rotating mass, as reported in Table 2. Each applied frequency was locked for a 60s window and then incremented by steps of 1 Hz in the range 1 Hz -10 Hz. The frequency step was reduced to 0.5 Hz around the first vibration mode previously identified from the noise tests. As stated in the Introduction, the experimental campaign focused on investigating the dynamic foundation stiffness and damping in pre-failure conditions. Thus, an excessive increase of the foundation motion induced by the frequency-dependent force amplitude was prevented by stopping the experiments at 8 Hz during the Forced-C and -D tests.

Solution of the Dynamic Equilibrium of the System
The geometric features and the concentration of the masses on the roof allow the dynamic bending response of the facility, u s , to be relatively assimilated to that of a SDOF system endowed with two more degrees of freedom, i.e. the swaying, u f , and rocking, θ f , motions of the foundation.
On this configuration as shown in Fig. 3, the force F s applied on the roof in the forced vibration tests is transmitted at the foundation level as a shear force V B and an overturning moment M B . The null embedment of the foundation makes the rotation induced by the base shear force negligible, so that θ f is mobilized only by the moment M B , and u f is activated only by V B . The rotational degree of freedom of the roof slab is neglected in this work since it has a negligible effect on the foundation overturning moment. The equations of the dynamic equilibrium of the foundation swaying and rocking are expressed as follows: where € u Cf and € u Cs are the horizontal accelerations at the gravity centers of the foundation and roof slabs respectively, while K � xx and K � yy are the unknown dynamic soil-foundation impedances associated with the swaying and rocking motions. They control the "soil reactions" accounting for the stiffness Experiment [Hz] [kN] Forced-A 1-2-3-3.5-4-4.5-5-6-7-8-9-10 0.07-7.30 Forced-B 1-2-2.5-3-3.5-4-5-6-7-8-9-10 0.15-15.51 Forced-C 1-2-2.5-3-3.5-4-5-6-7-8 0.27-17.51 Forced-D 1-2-2.5-3-3.5-4-5-6-7-8 0. 45-28.58 and damping exhibited by the soil-footing system during the motion. The right-hand sides of Eq. (2) and (3) represent the base shear force, V B , and the overturning moment, M B, respectively. They result from the difference between the applied loads (F s in Eq. (2) and F s h in Eq. (3)) and the inertia forces generated by the structural mass, m s , the foundation mass, m f , and its mass moment of inertia, I f . The notations h, h f and h F refer to the height of the top of the structure, to the thickness of the foundation slab and to the height of the gravity center with respect to the bottom of the foundation, respectively. The steel frame mass was considered negligible, so m f = 9 Mg for the foundation and m s = 18 Mg for the jointed roof slabs, while I f = 6.78 Mgm 2 . The accelerations of the roof and foundation centers, € u Cs and € u Cf , were firstly assumed equal to those recorded by sensors 2.4 N and 1.4 N on the surface of the slabs and then calculated by subtracting h f € θ f and h f € θ f =2, respectively. It was verified that the difference in the base shear force, V B , and overturning moment, M B , obtained under the two hypotheses is generally negligible, and reaches 10% only at the highest frequencies. Hence, the motions of the roof and foundation centers could be reasonably assumed equal to those recorded on the top of the slabs.
The rocking acceleration € θ f resulted from the difference between the vertical records 1.3 Z and 1.4 Z divided by their distance (see Fig. 1). Figure 4 reports the evolution of the different acceleration components recorded along the loading direction by sensor 2.4 N, i.e. the contribution of the foundation rocking and swaying motions (h F € θ f and € u f , respectively) and that of the structural bending (€ u s ) during the forced tests. In all the tests, the measured acceleration shows first a sharp amplification, thereafter a slight decrease and then a further increase, following the stepwise increase of the frequency-dependent forces (thick red line, right y-axis scale). The foundation rocking, θ f , was computed by double integration of € θ f , whereas the foundation swaying, u f , was calculated by the displacement obtained by integrating the acceleration recorded by sensor 1.4 N minus that induced by the foundation rocking, i.e. h f θ f . For all the recordings, a time window lasting 20 s was isolated after the steady-state condition was attained in each frequency step. The selected signal window was filtered through a 4th-order band-pass Butterworth filter in the range of the corresponding exciting frequency ±1 Hz. The effectiveness of such treatment of the experimental data was checked by simulating some of the forced-vibration tests through the aforementioned SAP2000 model, equipped at its base with elastic springs, the stiffness of which was calibrated according to the experimental impedance functions. The comparison in terms of displacements was found very satisfying.
In lack of the recording of F S , the phase angle, φ, between the applied force and the response was assumed equal to that characterizing the force-displacement time shift of a simple SDOF oscillator: where the overall damping ξ = 5% and the natural frequency f n were inferred from the interpretation of noise and snap-back tests. The dependency of phase angle on the frequency of the loading force was considered by updating f in Eq. (4) according to the frequency of the test under interpretation, as reported in Table 2. The value of f n was set equal to 3.4 Hz, 3.3 Hz, 3.2 Hz, and 3 Hz for the experimental tests Forced-A, B, C, and D, respectively, as detailed in Amendola et al. (2021b).
The phase angle was added to the sine-wave argument in the force expression (Eq. (1)) to be introduced with the recorded motions in the system's dynamic equilibrium, considering the acceleration recorded on the roof as baseline for the timing, consistently with the SDOF approximation.

Calculation of the Foundation Damping and Stiffness
The stiffness and damping associated with the foundation swaying and rocking modes were evaluated by applying different interpretation procedures: (1) Loop-based (LB) approach, i.e. from the interpretation of the loops described by the time histories of base shear force-displacement, V B -u f , or from the base overturning momentrotation, M B -θ f . The procedure is strictly related to the well-known definition of the equivalent stiffness and damping parameters widely used for synthesizing non-linear and non-reversible cyclic soil behavior (e.g. Hardin (1978)) in equivalent linear analyses. Hence, the equivalent stiffness of the soil-foundation system should be taken as the average slope of the loop taken between its end tips and the equivalent damping ratio, β xx or β yy , should be calculated through the classical formula reported by basic textbooks of structural (e.g. Chopra (2003)) and soil dynamics (e.g. Kramer (1996)): where W Dxx is the energy dissipated in a closed loop (grey area in Fig. 5a) and W Sxx is the 'equivalent elastic energy' stored by a linear elastic system experiencing the same peak forcedisplacement conditions (triangular hatched area in Fig. 5a). The value of β xx at resonance is demonstrated to be coincident with the critical damping ratio of a visco-elastic system (see for instance Chopra (2003); Kramer (1996)), while away from resonance an equivalent viscous damping ratio, β vis xx , can be computed as follows: where f n is the natural frequency of the undamped structure and f is that of the applied force. Rigorously, the experimental frequency is affected by the structural damping ratio; nevertheless, for low damping values it can be fairly assumed equal to that of an undamped structure. When the damping ratio is computed from the moment-rotation loop, W Dxx and W Sxx are replaced by W Dyy and W Syy to obtain β yy and β vis yy . The above definitions can lead to ambiguous interpretations depending on the amount of the phase shift between force (or moment) and displacement (or rotation), inducing a noncontemporary attainment of their peak values, as displayed by the loops in Fig. 5b,c. This does not reflect on the evaluation of the numerator, W Dxx (dissipated energy), in Eq. (5) but it does affect that of its denominator, W Sxx , and of the equivalent stiffness itself. Therefore, in this study two different versions of this procedure were followed: Version a): the foundation secant stiffness for the swaying motion is obtained as the peak-topeak shear force divided by the peak-to-peak displacement (see Fig. 5b): Following this approach, the secant stiffness coincides with the major axis of the ellipse described by the force-displacement loops. From above, it follows that the equivalent elastic energy can be derived as 1/8 of the product of the peak-to-peak amplitudes of forces and displacements, i.e. as follows: The where V B (u fmax ) and V B (u fmin ) are the shear force amplitudes at the same time instants of the maximum and minimum displacements, respectively. The secant stiffness evaluated according to Eq. (9), depicted in Fig. 5c by a continuous black line, is expected to underestimate that computed with Eq. (7) increasingly with the delay between the peak force and the peak displacement, hence with the amount of damping. On the other hand, the damping is expected to be overestimated with respect to the previous interpretation, being the expression of the equivalent elastic energy characterized by the same peak-to-peak displacement multiplied by a lower force amplitude: Again substituting the shear forces, V B (u f max ) and V B (u f min ), and displacement, u f max and u f min , with M B (θ f-max ) and M B (θ f min ), θ f max and θ f min allows calculating rocking foundation stiffness K yy and elastic energy W Syy . (2) Phase shift (PS) approach, i.e. by taking rigorously into account the above-mentioned phase shift between the force acting on the foundation and its associated displacement. In particular, the foundation swaying stiffness can be evaluated from the following formula (the derivation of which is detailed in Appendix): while the foundation damping ratio, β vis xx , can be straightforwardly computed by applying the following formula (the derivation of which is again detailed in Appendix): In the above equation, Ψ xx is the phase shift between the maximum base shear force V B max and the maximum displacement acting on the foundation. Such phase angle can be calculated in the time or frequency domain. The latter approach was adopted in this study, by computing for each cycle the phase shift Ψ xx between the base shear force and displacement as the difference of the phase angles of their respective Fourier transforms, obtained using the Matlab TM 'angle' function. The term K xx switches into K yy ,β vis xx into β vis yy ,Ψ xx into Ψ yy and u fmax into θ fmax when referring to the rocking motion.
(3) Impedance-based (IB) approach, i.e. from the frequency-dependent complex expression of the impedance functions defined by the dynamic equilibrium Eq. (2) and (3) and computed following the Swaying-Rocking Foundation (SRF) assumption described in detail by Amendola et al. (2021b).
The swaying stiffness was assumed to be equal to the real part of the relevant impedance function as follows: while from the imaginary part of the swaying impedance: it was possible to compute the viscous damping ratio relevant to the swaying component: Equation (15) corresponds to the 'energy loss parameter,' analogous to the viscous damping ratio, as can be easily demonstrated (Maravas, Mylonakis, and Karabalis 2014) by exploiting the analogy between the real and the imaginary parts of the impedance functions with the stiffness, K xx , and damping, c xx , of a single degree of freedom system (Richart, Hall, and Woods 1970). The terms K xx *, β vis xx and c xx switch into K yy *, β vis yy and c yy when referring to the rocking motion. Amendola et al. (2021b) used the outcomes of such approach to check the reliability of closed-form solutions for the impedance functions, even considering the increase of the soil initial stiffness produced by the structural weight in the calibration of the analytical formulas (see Amendola et al. 2021a).
The Impedance-Based approach is conceptually consistent with the Phase-Shift approach. Indeed, it was verified in this study that IB leads to the same results as PS when applied to each force-displacement (or moment-rotation) cycle. In the following, however, comparisons are reported between the PS approach applied to each cycle and the IB approach applied to the complete recorded steady-state response as typically adopted in the literature (NIST, 2012).
(4) Log-Decrement (LD) approach, i.e. by applying the formula of the logarithmic decrement to the peak displacements, u fmax i and u fmax i+1 , of any two subsequent cycles recorded during the free vibration tests: where u f max i and u f max i+1 are obtained as the half peak-to-peak value in each cycle.
Theoretically, β vis xx should not vary with the selected i th cycle when the response of a linear visco-elastic system is analyzed; otherwise, it generally decreases from the beginning of the oscillation until the rest.
As an example, Fig. 6 shows the time history of the foundation displacement recorded during the Snap-back A test (a) and the evolution with the number of cycles of the maxima displacements (b). The amplitude of u f max obviously reduces with the number of cycles. Even the difference between two consecutive values in Fig. 6b tends to decrease, highlighting an evolution with the number of cycles of the β vis xx -values. The same trend was observed from the application of the logarithmic decrement to the acceleration time histories recorded during these experimental tests by Amendola et al. (2021b) and from previous experiments on the same facility by . Once again, the formula can be equally applied to the rocking motion to evaluate β vis yy by substituting u f-max i and u f max-i+1 with θ f max i and θ f max i+1 . Tables 3 and 4 list the approaches applied to the data of the performed tests. The extremely low values of the imaginary part of the impedances back-figured from the snap-back tests (see Amendola et al. 2021b) make the 'Impedance approach' inapplicable to such data. Conversely, the 'Log-decrement approach' is significant only with reference to the snap-back tests.  Table 3. Approaches applied to derive the foundation stiffness.

Evolution of the Loop Shape with the Number of Cycles
Each frequency step of the forced vibration tests was interpreted by picking 20s long time-windows of the signals recorded in the steady-state condition by accelerometers 2.4 N, 1.4 N, 1.3Z, and 1.4Z. The selected time histories were then processed through an ad-hoc Matlab™ routine to compute the sheardisplacement V B -u f and the moment-rotation M B -θ f loops, consistently with the equilibrium Eq. (2) and (3).
Each cycle was individually processed to analyze the variability during the loading history of the foundation motion and the derived mechanical parameters. In most cases, an evolution with the number of cycles of the loop shape and peak forces was observed, while the peak displacement amplitudes remained constant. Figure 7 shows the V B -u f loops of two indicative tests, i.e. Forced-B and Forced-C at f = 8 Hz, displayed into four time windows to better emphasize their variation. In the latter case, the first and last cycle relevant to each of the four time windows are highlighted in black.
In the time window 1.5-5 s, the peak force and displacement amplitudes of the Forced-B test are practically simultaneous, while a delay between them is observed in the larger loop of the Forced-C. Moving from the first to the fourth time window, loops derived from the Forced-B broaden without altering their initial shape. Conversely, cycles of the Forced-C enlarge and modify their shape with time; in particular, the maximum displacement u f max keeps constant, while the peak shear force, V B max , increases from V B max ≈7 kN at t = 1.5 s to V B max ≈12 kN at t = 20 s.  The trend observed for the Forced-B test is repeated in the other tests, except at high frequencies, where the initial loops are wider, and their behavior is more similar to that observed for the Forced-C test. Such differences reflect a dependence on the number of cycles of the mechanical parameters derived from the loops, as discussed in the following.

Analysis of the Foundation Motion
The response of the foundation was firstly investigated by analyzing its peak motion amplitudes. Figure 8 shows the maximum displacement induced by the swaying and rocking motion in each cycle of the forced vibration tests. Data are organized into groups characterized by the frequency indicated on the top of the graph, while the different shapes of the markers are associated with the four Forced-A, -B, -C, -D tests reported at the bottom of the plot. The black and white symbols highlight values corresponding to the initial and final cycles, respectively. Finally, the applied force amplitude, F S (which at a given frequency increases from test A to D), is shown through the continuous grey lines referred to the right vertical axis.
The displacements induced on the top of the foundation slab by rocking (h f θ f max ) are comparable with those induced by the swaying, u f max , justifying the adoption of SRF hypothesis above mentioned. As a matter of fact, according to Gavras et al. (2020) and Gajan and Kutter (2009), the foundation rocking can be assumed as predominant in a structure with a slenderness ratio, h/B > 1. Conversely, for squatter structures (h/B≪1) swaying prevails over rocking. For the case at hand, h/B is equal to 1.67, hence the contributions of both swaying and rocking are expected to be significant.
The initial and the final values are practically coincident in any case, except for the Forced-A at 3 Hz, where both motion amplitudes apparently rise with the number of cycles. Being this frequency lower than the resonance natural value at this load amplitude (3.4 Hz, as obtained in the noise tests reported by Amendola et al. (2021b)), the cumulative effects of cycles may have led to a progressive reduction of stiffness of the system (as it will be discussed more in detail later, with reference to Fig. 10) hence to a related increase of displacement and rotation; these latter at the end of the series of cycles result in a bit higher than those measured at 3.5 Hz. Forced-B and -C clearly show stable peak motion amplitudes at 3 Hz, while it is interesting to note that the peak values for Forced-D test are attained during the series of cycles at 2.5 Hz, highlighting the most significant effects of the system's non-linear response at the highest force amplitudes.
Finally, both the peaks of displacement and rotation occur at the same frequency because the foundation exhibits a mode-coupled response, as already observed from previous experimental data by Gavras et al. (2020) and Gajan and Kutter (2009).

Analysis of the Non-Linear Soil Response
To investigate the role of the non-linear soil response on the foundation motions an approximate peak shear strain, γ eff max , was estimated as suggested by Star et al. (2019), i.e. from the ratio between the peak horizontal velocity recorded on the foundation, _ u f max , and the mean shear wave velocity, V S , of the soil volume interacting with the foundation, assumed equal to 100 m/s. The peak shear strains mobilized during the Forced-A and -B -C -D tests are reported in Fig. 9. The strains associated to the peak velocity of the initial and final cycle are plotted only for the Forced-A at 3 Hz, for which the effects of cyclic degradation were already shown as significant in Fig. 8. In the other cases, the initial and final strains are almost coincident. It is interesting to note that, for all the tests, the mobilized strain attains its maximum value around resonance (i.e. either at 2.5 or 3 Hz), but shows also a subsequent increasing trend at higher frequencies, due to the progressive increase of load amplitude. The highest peak shear strain is reached at 2.5 Hz during Forced-D test, where γ eff max is 0.009% as shown in Fig. 9.
The normalized shear modulus and damping ratio mobilized in the soil during the different tests were estimated from the shear strain plotted in Fig. 9 through the sigmoid functions fitting the results of the resonant column test in Fig. 3b. The results are reported in Fig. 10, showing evidence of nonlinearity in all tests. The stiffness reduction is generally less than 15%, except in a few cases, the most significant of which corresponds to a G/G 0 decrease down to about 75% for the Forced-D test. The latter significant decrease motivates the attainment of peak displacement and rotation amplitudes at a frequency as low as 2.5 Hz, already noticed describing Fig. 8. Moreover, the reduction of G/G 0 from  the initial to the final cycle is evident only at 3 Hz in Forced A-test, consistently with the progressive increase of foundation motion previously discussed with reference to Fig. 8.
The soil damping ratio (Fig. 10b) increases with the level of loading and when the degree of nonlinearity is higher, largely overcomes the value of 5% commonly adopted for structures. Such a result highlights that soil damping is not negligible even under low force amplitudes.

Forced Vibration Tests
Figures 11 and 12 respectively show the swaying and rocking stiffness resulting from the Loop Version a) and b) approaches applied to the forced vibration tests. Data are again organized in the same way as in the previous Figs. 8-10. Whatever the interpretation method, below 6 Hz both stiffness components appear steady with the number of cycles and reduce from Forced-A to Forced-D test, i.e. with increasing force amplitude (grey line, right axis scale) and strain level mobilized in the soil (see Fig. 9).
At frequencies higher than 6 Hz, the overall trend appears less clear, with both swaying and rocking stiffness much more scattered around higher mean values. This is particularly apparent for the values obtained by LB Version a) approach (Figs. 11a and 12a). For instance, in the Forced-C test, there is an abnormal growth of the swaying stiffness from the initial to the final cycle of the loading step at f = 8 Hz (Fig. 11a). On the other hand, the stiffness resulting from the Forced-B test at the same frequency appears quite steady with the number of cycles. Such a different trend is clearly shown in Fig. 7, which compares the evolution of the force-displacement loops relevant to these two tests. The progressive counter-clockwise drift of the loop shape during the Forced-C test is characterized by a visible increase of the peak shear force, while the displacement amplitude appears stable, which turns into an increase of the equivalent linear stiffness calculated with the approach LB Version a) (Eq. (7)). An abrupt growth of both swaying and rocking stiffness with the same significance occurs at f = 7 Hz in the Forced-D test and at f = 10 Hz in Forced-B test (Figs. 11a and 12a); meaning that the value of frequency which triggers such a kind of 'loop instability' decreases with the load amplitude.
When the LB Version b) approach is applied (see Figs. 11b and 12b), the decrease of the secant stiffness with the increasing loading force is more apparent for both swaying and rocking components at every frequency, except for the highest values. Due to the steadiness of the peak displacement value, the values calculated by this method for frequencies higher than 6 Hz are much more stable with the number of cycles. As expected from their different definitions (see Fig. 5 and Eq. (7) and (9)), the stiffness components computed by LB Version b) approach are apparently lower with respect to those resulting from LB Version a) method. Figure 13 compares the frequency-dependent dynamic stiffness associated to the swaying (left) and rocking (right) foundation motions, as calculated by applying the four different interpretation approaches listed in Table 3. The comparison is limited to the data of the forced vibration test A, for which the results appeared overall less affected by non-linear effects and variability with cyclic load accumulation. The mean values computed for each frequency step are plotted with symbols connected through solid lines and the variations with the number of cycles are shown through grey shadings. The results obtained following the impedance approach are displayed in black and herein represent the benchmark to check the reliability of the alternative interpretation techniques. As stated before, these experimental impedance functions were validated against different analytical formulations in a previous work by the same authors (Amendola et al., 2021b). Only the mean value is meaningful for the Impedance approach, which is applied to the whole steady-state response of the SFS system. The values predicted by the different approaches are very close each other below 6 Hz. At higher frequencies, the agreement is much more satisfying when the shear force (or moment) synchronous to the maximum displacement (or rotation) is adopted for defining the equivalent elastic stiffness (i.e. LB Version b) approach in Fig. 13b rather than computing it as a peak-to-peak amplitude force/ displacement (or moment/rotation) ratio (i.e. LB Version a) approach, Fig. 13a). Figure 13c compares the real parts of the swaying and rocking impedances with the frequencydependent stiffness computed from the Phase-shift (PS) approach, which shows a scatter (grey shaded area) much lower than that resulting from the LB procedures. The mean values are very similar to those resulting from the LB Version b) approach.
The comparisons in Fig. 13 suggest that the stiffness derived by the LB Version b) and PS approaches closely approximate the real parts of the swaying experimental impedances and slightly overestimate those pertaining to the rocking motion. The predictions of both components progressively worsen with frequency using the LB Version a) approach, being the peak amplitudes of force and moment increasingly out-of-phase with respect to the peak displacement and rotation. It can also be noted that, regardless of the interpretation technique, in correspondence of the resonant frequency both swaying and rocking stiffness exhibit a minimum, consistently with the maximum reduction estimated for the normalized shear modulus around the same frequency value (compare Fig. 10a).
By substituting Eq. (7) into Eq. (8) or Eq. (9) into Eq. (10), it is clear that W SS makes the calculation of the β-values to be dependent on how the foundation stiffness is defined, i.e. by considering either one of the Versions a) or b) of the approach LB. As a matter of fact, Fig. 14 shows (with the same graphical representation as above) the viscous damping ratio calculated for the swaying and rocking motions by applying the different interpretation approaches to data of forced vibration test A.
A fair agreement between β vis xx or β vis yy from LB Version a) and IB approach is achieved until f = 6 Hz, while at higher frequencies for both swaying and rocking modes (Fig. 14a) β vis xx (β vis yy ) values from IB generally exceed the corresponding ones computed by employing the LB Version a) approach. The comparison at high frequencies is significantly improved when comparing the IB viscous damping with the mean β vis xx (β vis yy ) values retrieved from the LB Version b) interpretation techniques (Fig. 14b), especially for the swaying mode.
Finally, Fig. 14c reports the comparison between the IB β vis xx (β vis yy ) and the mean β vis xx (β vis yy ) values computed applying the PS approach. Just like as observed for the stiffness, the trends of the mean PS β vis xx (β vis yy ) are very similar to those in Fig. 14b, again with a better match for the swaying mode with respect to that for the rocking mode, particularly at f > 6 Hz.
Independently of the followed approach, it can be remarked that: • the damping values associated with the swaying and rocking motions are comparable, consistently with the experimental findings by Gajan and Kutter (2009) on a slightly slender shear wall; • there is a significant increase of energy dissipation with frequency for both rocking and swaying modes, highlighting that in this case, the main source of foundation damping is viscous rather than hysteretic; • at a given frequency, the minimum and maximum values of the damping ratio generally coincide with those pertaining to the initial and final cycles of the steady-state response, highlighting that degradation occurs and an increasing amount of energy is progressively dissipated during the test. The results of the other tests (here not reported for sake of brevity, see 'Supplemental materials') show that the increase of the shaker mass from the forced-test A to D significantly affects the stiffness but poorly influences the damping, except around the resonance at 3 Hz, where the damping increases with the shaking energy level. As a final comment on the interpretation of the forced vibration tests, it is once more recalled that the Impedance (IB) approach is expected to be the most straightforward, being the evaluations of stiffness and damping easily distinguished by separating the real from the imaginary parts of the impedance function. Even being conceptually consistent, the PS approach catches instead the evolution with the number of cycles thus it may lead to slightly different results. The Loop approaches require accurate individuation of each single cycle and yield ambiguous results depending on the definition of secant stiffness. In fact, compared to the IB approach, the LB Version a) approach returns the worst predictions, significantly overestimating the stiffness and underestimating the damping at high frequencies. Conversely, the overall agreement between the mean stiffness and damping estimated by LB Version b) and PS approaches with the IB values reciprocally validates their reliability. Figure 15 compares the secant stiffness derived from the snap-back tests computed by Amendola et al. (2021b) as real part of the impedance function (triangles) versus its variation with the number of cycles obtained by the LB Version a) approach in this study (circles). Negligible differences were observed between the results of the approaches LB Version a) and b), being the damping ratio low and the peak force (or moment) and peak displacement (or rotation) almost synchronous. The stiffness is systematically observed to increase from the initial (blue dot) to the final cycle (blue circle), consistently with the progressive reduction of the foundation motion after applying the pulling force. The mean value of the stiffness mobilized during the free-vibration cycles (black circle) results very similar to that identified by the impedance-based interpretation approach (triangle), this latter being applied to the whole length of the free-vibration record and therefore leading to a resulting unique value. The latter can be consequently interpreted as representative of the mean mobilized stiffness. The comparison between the real part of the impedance and the mean secant stiffness is satisfying for all the tests. Moreover, they both tend to decrease from snap-back test A to E, i.e. by increasing the pulling force. Figure 16 compares the foundation damping ratio computed by applying the LB Version a) approach (circles) and Log-decrement (LD) method (triangles) to snap-back test data. As from Eq. (6), β xx = β vis xx and β yy = β vis yy at the natural vibration frequency. Values obtained from the area of the loops, β xx and β yy , are slightly higher and more dispersed with reference to those derived through the  logarithmic decrement approach. Due to the non-linear soil response, both approaches clearly highlight an overall decrease of the damping ratio with the number of cycles, with the highest values in correspondence to the first cycle (blue symbols). The same trend is not evident when comparing data for different tests most probably due to the small difference in loading amplitude which distinguishes the snap-back tests. Independently of the approach, the data scatter tends to reduce from the Snapback test A to E, i.e. with increasing pull-out force amplitude.

Summary of Major Findings and Concluding Remarks
The results of snap-back and forced vibration tests executed on a prototype of a simple structure founded on soft soil were processed by using different interpretation approaches with the purpose to assess their reliability to experimentally measure the dynamic foundation stiffness and damping ratio. In particular, the values obtained through the impedance functions (IB approach) computed in the frequency domain were compared with those back-calculated from the time-domain analysis of the foundation forcedisplacement loops, either neglecting (LB Version a)) or considering (LB Version b)) the phase delay between force and displacement. The force-response foundation phase delay was directly exploited to derive the stiffness and damping ratio in the Phase-shift approach (PS). Hence, the conclusion referred to the LB Version b) approach can be extended to the Phase-shift approach, because they led to the same results due to their conceptual consistency.
In the interpretation of the results of the snap-back tests, the match in terms of stiffness between the frequency and time domain approaches is satisfying due to the short delay between the peak force and displacement. Higher relative differences were found between the damping ratio obtained from the hysteretic loops and that calculated by applying the logarithmic decrement method.
The agreement among the different approaches was found to be more or less satisfying, depending on the frequency of the exciting force during the forced vibration tests. In particular, at higher frequencies the damping increases leading to a significant phase difference between force and displacement. Consequently, the LB Version a) approach, which neglects such phase shift, overestimates the stiffness and underestimates the damping ratio.
Whatever the interpretation criterion, effects of soil nonlinearity were recognized to produce a reduction of the swaying and rocking stiffness and a not negligible foundation damping ratio even under low loading forces. Moreover, the damping increases at high frequency values, due to the combined effect of the viscous dissipation capacity and non-linear hysteretic soil response.
In conclusion, the use of the Phase-Shift and LB Version b) approaches are a valid alternative to the Impedance approach for the interpretation of forced vibration tests with harmonic loading, even when forces and displacements are significantly non-synchronous. By repeating the interpretation for each cycle through simple routines, both above approaches permit an accurate analysis of the evolution of stiffness and damping during the tests due to non-linearity and cyclic degradation. Conversely, more challenging time-frequency analyses, performed by applying frequency domain transforms on subsequent windows of the time records, would be necessary to obtain the same evolution through the Impedance approach.
The comparison among the different approaches reported in this paper is expected to guide the interpretation of tests under low-to medium-amplitude dynamic loads, inducing strain levels in the soil high enough to induce appreciable non-linearity, but still well away from failure conditions. The considerations summarized above are therefore worth of being checked against the interpretation of existing and future experimental tests under motions leading soil closer to failure, to further investigate the reliability of the different interpretation procedures.

Notation
The following symbols are used in this paper: B Foundation width c xx Foundation swaying damping coefficient Phase angle between shear base force and foundation horizontal displacement Ψ yy Phase angle between overturning moment and foundation rotation ω Vibration angular frequency ω n Fundamental angular frequency