A Hossain

• M S I Sarker

• M K R Khan

F A Khan

• M Kamruzzaman

M M Rahman

M S I Sarker

Structural, magnetic, and electrical properties of sol-gel derived cobalt ferrite nanoparticles

This work reports the synthesis and studies of semi-soft ferrimagnetic CoFe 2 O 4 nanoparticles using sol-gel method. The X-ray diffraction patterns confirm the formation of cubic spinel CoFe 2 O 4 nanoparticles. The average crystallite size was found from XRD data is about 30 nm. The high resolution transmission electron microscopy analysis shows that nanoparticles are highly crystalline. The magnetic properties reveal that the particles are ferrimagnetically ordered soft magnetic materials with coercive field of 620 Oe and saturation magnetization M s = 60 emug -1 . The higher value of saturation magnetization is due to the ordered single-domains magnetic nanoparticles and lower magnitude of coercivity is attributed to the decrease of interparticle interactions and magneto-elastic anisotropy. The value of remanence ratio R < 0.5 is responsible for magnetostatic interactions of the particles. The nanoparticles possess low values of dielectric constant which decreased with increasing frequency. The low dielectric constant makes the nanoparticles as a promising candidate for high frequency magnetic devices.

Introduction

Magnetic metals and metal oxides are very important in materials science and engineering due to their wide range of application and outstanding properties. Soft magnetic materials constitute an important class of engineering materials [START_REF] Fish | Proc. IEEE[END_REF][START_REF] Abraham | [END_REF] as they can be easily magnetized and demagnetized under a small external field. Their magnetic properties are important for applications involving power generation and distribution, actuator, magnetic shielding, data storage, and microwave communication.

Co based alloy nanoparticles (NPs) as well as its bulk counterpart such as FeCo, CoNi, etc., are important class of materials owing to their interesting properties like low coercivity and high saturation magnetization [START_REF] Spaldin | Magnetic Materials[END_REF]. Moreover, CoFe 2 O 4 NPs has great importance due to its unique physical properties such as high Curie temperature, moderate saturation magnetization, large magneto crystalline anisotropy, high magneto-strictive coefficient, excellent chemical stability and mechanical hardness [START_REF] Rajath | [END_REF]. In addition to that, CoFe 2 O 4 NPs shows high electrical resistance and low dielectric constant. For these properties CoFe 2 O 4 is technologically important and suitable for high density magnetic recording media [5], ferrofluid technology, biomedicine, magnetic resonance imaging, biosensors, magnetic hyperthermia based therapy, data storage, magnetic refrigerators and microwave devices [6].

Dielectric ferrites have a high conduction grains separated by poorly conducting grain boundaries [7,8]. These grain and grain boundaries affect on conductivity due to the hopping of ions with applied external electric field. The above mentioned properties of spinel CoFe 2 O 4 NPs are affected by composition and microstructure due to the octahedral and tetrahedral occupancy of cations Fe 3+ and Co 2+ in the lattice sites. Moreover, the size and shape of the NPs also affects to magnetic properties. Therefore, the control of the size and shape of the NPs are quite important.

Different chemical routes have been used to synthesis the magnetic NPs such as sol-gel method [9], micro emulsion [10], co-precipitation method [11], polyol reduction method [12] etc. Among these methods, sol-gel is an excellent chemical route for obtaining fine nanocrystalline and homogeneous powder because it is simple, environment friendly, cost effective and low temperature operation.

Despite several researches on CoFe 2 O 4, but the systematic studies are rarely reported. In this research, we report a simple process to synthesis CoFe 2 O 4 NPs using sol-gel method in which ferric (III) chloride and cobalt (II) chloride were used as a precursors and citric acid was used as a capping agent. The crystal structure, surface morphologies, as well as the magnetic and electrical properties of the as synthesized NPs were investigated.

Experimental procedure

The experimental sample was prepared by conventional sol-gel method. Stoichiometric amounts of 0.596 g CoCl 2 • 6H 2 O (97%: Merck), 0.408 g FeCl 3 (98%: Merck) dissolved in 100 ml distilled water and a few drops of concentrated nitric acid were also added to prevent hydrolysis. Citric acid and a few ml of dimethylamine were subsequently added to the solution, which was then magnetically stirred at 70 °C for 2 h and evaporated at 100 °C to form a gel. The gel mixture was dried at 150 °C for 12 h and the resulting powder was decomposed in air and heated in a furnace at 250 °C for 3 h. Finally, the powder samples were ground thoroughly using an agate mortar pestle and heat treated further at 600 °C for 4 h inside a furnace at ambient temperature.

X-ray diffraction (XRD) analysis was performed using Rigaku Smart Lab X-ray diffractometer, operating at a voltage of 45 kV and current of 150 mA with Cu-k α (λ = 1.54184 Å) radiation to identify the phases and to determine lattice parameter. The morphological studies were performed by field emission scanning electron microscopy (FE-SEM) with accelerating voltage 5 keV (Philips XL30 FEG) and with a high resolution transmission electron microscopy (HRTEM) with 200 keV (Philips FEG TEM CM200). The thermal analysis was carried out from thermogravimetric analysis (TGA) and differential thermal analysis (DTA) in a controlled atmosphere. The TGA and DTA were performed (Pyris Series STA-8000 Perkin Elmer) on synthesized NPs from 30 to 900 °C to obtain its phase transition temperature, thermal stability and crystalline formation by observing weight loss and heat flow with the increasing of temperature. During TGA and DTA analysis, sample was heated in a tubular electrical furnace under a constant flow of nitrogen gas. The molecular bond in the NPs and the presence of functional group with associated molecular vibration were determined with wavenumber in the range of 225-4000 cm -1 with a high resolution Fourier transform infrared radiation (FTIR) spectrometer (Perkin Elmer 100). The resolution of FTIR spectroscopy was 1 cm -1 (KBr was used with synthesized NPs as annealing materials in FTIR investigation). The magnetic measurements were carried out using a vibrating sample magnetometer (VSM) at room temperature (using BRUKER type magnet serial no.

811336 and PAR Lock-in Amplifier 5210). The ac electrical measurements were done in a Wayne Kerr 6500B impedance analyzer with a frequency range 100 Hz-10 MHz at different temperatures. A disk shaped sample having 14 mm in diameter and 3 mm in thickness was prepared to determine electrical properties. The surfaces of the disk were polished roughly and then coated with silver paste on both sides for the measurements.

Results and discussion

Structural characterization

Figure 1 shows the XRD patterns of the as synthesized powder sample annealed in air at 600 °C. The observed peaks (220), ( 311), (222) (400), ( 422), ( 511), ( 440) and ( 533) can be indexed from the Joint Committee Powder Diffraction Standard (JCPDS) card for spinel CoFe 2 O 4 cubic structure (No. 22-1086) [11]. Besides, the peaks around 44° and 65° corresponding to the crystal planes ( 110) and ( 200) indicate a small amount of FeCo alloy formation in the composition [START_REF] Rajath | [END_REF]11]. The space group of CoFe 2 O 4 is Fd-3m and the calculated lattice parameter is found to be 8.403 Å [11]. XRD pattern also confirms no other unreacted species or impure phase is detected in the sample. The average crystallite size was determined from XRD peak broadening using the Debye-Scherrer formula [13]. D = K ∕ cos , where, λ is the wavelength of the X-ray radiation, K α is a constant (0.94), β is the full width at half maximum in radian and θ is Fig. 1 X-ray diffraction patterns of the as synthesized materials annealed at 600 °C the diffracting angle. The average crystallite size was found to be 30 nm. In the spinel structure, the cations Co 2+ and Fe 3+ may occupy two different sites, i.e., octahedral and tetrahedral sites within the lattice structure. The actual cation distribution is characterized by degree of inversion, which is the fraction of divalent cations occupied in octahedral sites [14]. The divalent Co atoms shared both octahedral and tetrahedral sites in CoFe 2 O 4 NPs. Therefore, in the microstructural state of the as synthesized spinel CoFe 2 O 4 , it is assumed Co atoms may occupy the same tetrahedral and octahedral sites.

Surface morphologies

Figure 2 shows the scanning electron microscopy (SEM), transmission electron microscopy (TEM) and high resolution TEM images of the as synthesized cobalt ferrite NPs. It is seen from the SEM micrograph in Fig. 2a that the synthesized particles are formed as octahedron and tetrahedron nanometer in size with some micrometer phases. The microstructure of synthesized NPs reveals the crystallites are perfectly ordered with porous structure. The porosity was determined using the empirical formula:

P = 1 -( m ∕ x ) ,
where, ρ m and ρ x are measured density and X-ray density of the NPs, respectively [10]. The porosity of the NPs is about 17.5%. It can be seen from TEM image in Fig. 2b that the sizes of the particle is in the range of 30-40 nm, which corresponds XRD data. The high resolution TEM images shown in Fig. 2c,d indicate the NPs are highly ordered and crystalline. Furthermore, the particles are assumed to be single domain.

Thermal analysis

The thermal stability CoFe 2 O 4 nanomaterials was investigated by TG-DTA from room temperature to 900 °C with increasing temperature 20 °C/min. Figure 3 shows the relative weight variation during the heating of the samples in air. It is seen from Fig. 3 that the first weight loss of 5% is associated to an endothermic peak and is related to water loss, which occurs in two steps, dehydration and then loss of adsorbed water until 250 °C [15]. The second weight loss between 20 to 60% is related to a high exothermic peak as well as decarbonation of the samples [15,16]. The decarbonation has lasted up to the temperatures 500 °C (Fig. 3a). Above that temperature (around 600 °C) the samples has presumably entered in an ordered state. The differential thermal analysis (DTA) for the synthesized nanomaterials in Fig. 3b has shown some exothermic peaks around 273 and 340 °C temperatures, which is attributed to the formation

FTIR spectra analysis

The surface chemical properties of the NPs were investigated using FTIR spectroscopy with high resolution of 1 cm -1 . Figure 4 shows the FTIR spectra of the CoFe 2 O 4 NPs observed in the range from 225 to 4000 cm -1 . The vibrational spectra of the absorption band were 385.56 and 487.15 cm -1 due to vibration of metal oxide such as Fe-O and Co-O which is consistent with ferrite for the octahedral site [17]. The intense peak at 588.48 cm -1 in the FTIR spectrum was attributed to the Co-O stretching vibration indicates tetrahedral site in the spinel structure [16]. Thus, the cations of the NPs randomly occupied in octahedral and tetrahedral sites [10,[18][19][20]. The other peaks such as 1637.83 and 1618.46 cm -1 were represented the N-H bending vibration and 1384.48 cm -1 was due to bending vibration of C-H group. Moreover, the peaks 1105.15 and 878.02 cm -1 were shown in Fig. 4 for C-N, N-H stretching vibration [17]. The peak 3413.45 cm -1 was due to stretching vibration of O-H group, which indicate the presence of moisture in the molecule.

Magnetic properties

It is important to study the magnetic properties of CoFe 2 O 4 NPs due to the mixed valence state of Co. The anisotropic behavior of Co 2+ ions is responsible to change the magnetic properties of the synthesized CoFe 2 O 4 NPs. Figure 5 shows the magnetization, M as a function of magnetic field, H at room temperature in the range of -7 to 7 kOe of the as synthesized spinel CoFe 2 O 4 NPs. The NPs behaves as a soft ferrimagnet with coercivity, H c = 620 Oe and saturation magnetization, M s = 60 emug -1 at room temperature. It is observed that the saturation magnetization of CoFe 2 O 4 is higher than the values obtained by modified wet chemical process and hydrothermal process [22,23]. The higher saturation magnetization reported in this study is attributed to the perfectly ordered single-domains magnetic NPs [START_REF] Spaldin | Magnetic Materials[END_REF]. On the other hand, the lower coercivity is found due to the decrease of interparticle interactions and magneto-elastic anisotropy [22]. The retentivity of the magnetic nanomaterials is equal to M r = 13.5 emug -1 as shown in Fig. 5 and the remanence ratio (R = M r /M s ) of the NPs is 0.22. The 4 FTIR spectra of the as synthesized NPs Fig. 5 The magnetic hysteresis loop of the cobalt ferrite NPs measured at room temperature existence or absence of the different types of inter grain group exchanges is determined by the amount of R that varies from 0 to 1 [24]. The variation of remanence ratio with the particle size can be explained based on domain structure, crystal size and anisotropy of the crystal [25]. More explicitly, the magnetic properties of ferrite NPs depend on the preparation method. For example, it is reported that 16 nm cobalt ferrite NPs made by co-precipitation [25] give the value of remanence ratio 0.27 at room temperature, whereas 13-nm cobalt ferrite NPs prepared by the polymeric precursor method [24] give the value of remanence ratio 0.5 for similar conditions. Moreover, 39 nm particles made by thermal method found the remanence ratio 0.407 [27]. It has been reported that for particles having R < 0.5 interact by magnetostatic interaction, while R = 0.5 is for randomly oriented noninteracting particles that undergo coherent rotations [START_REF] Wang | Materials Systems and Applications I[END_REF][START_REF] Stoner | [END_REF]. Therefore, in this research the value of R lower than 0.5 is attributed to particles that interact by magnetostatic interactions.

Electrical properties

AC conductivity

To understand the conduction mechanism, the variation of AC conductivity with frequency at different temperatures is shown in Fig. 6. It is seen that at room temperature the AC conductivity increases with the increase of frequency. This can be explained using Koop's dispersion relations based on the Maxwell-Wagner model [8] considering inhomogeneous double layer dielectric structure. The grains with high conductivity are formed during the ferrite preparation. These grains are separated by poorly conducting grain boundaries. As the frequency of applied field increases, conducting grains more active and the hopping between the Fe 2+ and the Fe 3+ ions enhances which causes an increase in the mobility of charge carriers, resulting the increase of conductivity. Moreover, it is noticed that increasing the temperature the conductivity is also increased. This increase in conductivity with temperature is assumed due to the increase of drift mobility of the charge carriers thereby enhancing the charge hopping. However, at higher frequencies conductivity is less temperature dependent due to the hopping ions could not follow the applied frequency and the conductivity decreases at higher frequencies. In recent past, studies of the electrical conduction of a series of ferrites Co 1 -x Fe 2 + x O 4 were performed and reported the conductivity of ferrites resulting from hopping of holes between Co 2+ and Co 3+ ions and hopping of electrons between Fe 2+ and Fe 3+ ions [7]. Thus, the predominant conduction mechanism in spinel cobalt ferrite can be summarized as follows: Co 2+ + Fe 3+ ↔ Co 3+ + Fe 2+ .

Dielectric properties

The dielectric properties of materials are affected by synthesis technique, temperature, the percentage of composition, density, crystal structure, porosity and heat treatment etc [30,31]. The dielectric constant (ε′), ( � = Cd∕ o A , where, C is the capaci- tance, d is the thickness of the disk, A is the cross-sectional area and ε o is the permittivity of the free space) and loss factor (tan δ) as a function of frequency of the as synthesized NPs annealed at 600 °C are illustrated in Fig. 7. In an AC field, the dielectric constant is a complex quantity and we have measured the real part of this quantity as a function of frequency. It is observed (Fig. 7a) that at room temperature the value of ε′ is higher at low frequencies and the dielectric constant decreases rapidly with increasing the frequency. Moreover, the dielectric constant possesses low values at temperatures 318 K, 328 K, 337 K and 354 K (inset), which is also decreased with the increasing of frequency. This is due to change of cationic states as mentioned in Sect. 3.6.1 and interfacial polarization which can be explained by Maxwell-Wagner model [8]. In this model, the dielectric materials are assumed to be composed of conducting grains separated by poorly conducting grain boundaries. The poorly conducing grain boundaries have been found to be effective at lower frequencies while at higher frequencies conducting grains is fairly effective. Moreover, the mechanism of polarization in polycrystalline ferrite can be explained through the hopping of electrons between ions of the same element in different oxidation states at tetrahedral site [32]. When a field is applied, the electron flow in the direction of the field within the grain through hopping and pile up at the grain boundaries to produce polarization. If the frequency of the field increases, the electron cannot follow the applied field, i.e., electron changes their direction as well as decreases the Fig. 6 The variation of AC conductivity as a function of frequency at different temperatures probability of electrons to reach the grain boundaries. Thus, the polarization and dielectric constant decreases [12].

The dissipation factor ( tan = �� ∕ � , where, ε′ is the real part of the permittivity and ε′′ is the imaginary part of the permittivity) is illustrated in Fig. 7b. It is seen that the loss tangent decreases with the increasing of frequency and in the high frequency region it is nearly zero. The low values of loss tangent indicate the low eddy loss of current [12,20].

Conclusion

The cobalt ferrite nanoparticles have been synthesized using citrate assisted sol gel route. The 17.5% porosity, which we found in synthesized NPs plays an important role in technological applications, e.g., water and chemical filtration related to bio-medical applications, materials science, etc. The phase transition temperature and thermal stability over a wide range of temperature have been investigated through the TGA and DTA curves. Magnetization measurement reveals that the synthesized NPs are soft ferrimagnetic with low coercivity and high saturation magnetization of single domain particles. The AC conductivity of the NPs confirms the mobility increases with frequency resulting to the hopping of electrons. The low dielectric constant and dissipation factor are found in the synthesized CoFe 2 O 4 NPs. This low dielectric constant and low dissipation factor of the NPs indicate the low eddy loss in the system, which makes the CoFe 2 O 4 NPs as a promising tool for high frequency applications.

Fig. 2 a

 2 Fig. 2 a SEM microstructure, b TEM image, and c, d are HRTEM images of the synthesized NPs annealed at 600 °C

Fig. 3 a

 3 Fig. 3 a TGA (solid line) and b DTA (dashed line) curves of the as synthesized CoFe 2 O 4 nanomaterials

Fig. 7

 7 Fig. 7 Variation of a dielectric constant (inset for better clarity of data points) and b dissipation factor with frequency at different temperatures

Acknowledgements

The authors would like to acknowledge the Center of Super-Diamond and Advanced Films (COSDAF), City University of Hong Kong, Hong Kong SAR, P. R. China to provide the facility for measuring XRD, SEM and HRTEM measurements. This work was supported by the UGC grant no. A-843-5/52/UGC/Science-10/2015. The authors are thankful to the International Science Programs (ISP), Uppsala University, Sweden and Bangladesh University of Engineering and Technology (BUET) for technical support related to magnetic and electrical properties studies.


Zheng Li
email: zheng.li@ifsttar.fr

Sandra Escoffier
email: sandra.escoffier@ifsttar.fr

Panagiotis Kotronis
email: panagiotis.kotronis@ec-nantes.fr

Study on the stiffness degradation and damping of pile foundations under dynamic loadings

This paper presents a comprehensive study on the different behaviour of batter and vertical pile foundations in terms of stiffness degradation and damping properties under dynamic loadings. Firstly, dynamic centrifuge tests were carried out and the variations of translational/rotational stiffness and the associated damping properties were identified from a series of hysteretic loops. Results show that for the rocking behaviour, the presence of batter piles causes a small decrease of the rotational stiffness and a great increase of the rotational damping; for the horizontal translation behaviour, batter pile foundation has much higher horizontal stiffness and translational damping compared to the vertical foundation. Then, a set of stiffness degradation and damping curves for the translational and rotational behaviour of batter and vertical pile foundations are proposed. Finally, the proposed stiffness degradation and damping curves were validated numerically by using equivalent linear approach. The numerical validation shows a good agreement with the experimental results.

Introduction

Soil-Structure Interaction as a very important issue affecting the seismic responses of structures, has attracted much attentions from the engineering community. Many experimental results [START_REF] Negro | Large scale soil-structure interaction experiments on sand under cyclic loading[END_REF][START_REF] Shirato | Large-scale experiments on nonlinear behavior of shallow foundations subjected to strong earthquakes[END_REF][START_REF] Drosos | Soilfoundation-structure interaction with mobilization of bearing capacity: experimental study on sand[END_REF][START_REF] Deng | Seismic design of rocking shallow foundations: displacement-based methodology[END_REF] of foundation response and nonlinear soil-structure interaction under strong seismic loadings have shown the effects of SSI of foundation system. Ahmadi et al. [START_REF] Ahmadi | Importance of soil material damping in seismic responses of soil-MDOF structure systems[END_REF] pointed out that not only the flexibility, but also the energy dissipation of the foundation will greatly influence the inertial response of soil-structure interaction. For a rational and integrated displacementbased seismic design, SSI effects cannot be ignored [START_REF] Pecker | The role of non-linear dynamic soil-foundation interaction on the seismic response of structures, Keynote Lecture[END_REF][START_REF] Pecker | The role of non-linear dynamic soil-foundation interaction on the seismic response of structures[END_REF].

For the SSI analysis, many approaches have been developed. Macroelement modeling, which simplifies the dynamic interaction between soil and foundation by integrating the nonlinearities into a single plastic computational unit, has attracted considerable attention [START_REF] Cremer | Cyclic macro-element for soil-structure interaction: material and geometrical non-linearities[END_REF][START_REF] Grange | A macro-element to simulate dynamic soil-structure interaction[END_REF][START_REF] Figini | A macro-element model for non-linear soilshallow foundation-structure interaction under seismic loads: theoretical development and experimental validation on large scale tests[END_REF][START_REF] Li | A hypoplastic macroelement for single vertical piles in sand subject to three-dimensional loading conditions[END_REF]. At the same time, using spring-type models to simulate the dynamic response of soil-structure systems is popular in design practice because of their ease of use and clear physical meaning. Lu et al. [START_REF] Lu | A simplified Nonlinear Sway-Rocking model for evaluation of seismic response of structures on shallow foundations[END_REF] developed a simplified Nonlinear Sway-Rocking (NSR) model that is capable of simulating the load-displacement response of mat foundations subjected to seismic excitations. Anastasopoulos and Kontoroupi [START_REF] Anastasopoulos | Simplified approximate method for analysis of rocking systems accounting for soil inelasticity and foundation uplifting[END_REF] proposed a nonlinear rotational spring without requiring iterations. Monotonic and cyclic pushover analyses were utilized to define the moment-rotation, damping-rotation and settlement-rotation relations. The proposed method is straightforward and efficient in predicting the rotation and settlement of shallow foundation on inelastic soil. Besides the above mentioned methods, for the analysis of SSI for shallow foundation, the Beam-on-Nonlinear Winkler Foundation (BNWF) model is also widely used for its of simplicity [START_REF] Naggar | Analytical moment -rotation curves for rigid foundations based on a Winkler model[END_REF].

In recent years, more and more researchers carried out SSI analyses from the point of view of stiffness degradation and damping properties. Tileylioglu et al. [START_REF] Tileylioglu | Dynamic stiffness and damping of a shallow foundation from forced vibration of a field test structure[END_REF] studied the transnational and rotational behavior of shallow foundation by large-scale model test. It is found that the damping has more pronounced energy dissipation for translational than rotational vibration modes, and increases of damping with frequency highlights the importance of radiation damping. Anastasopoulos et al. [START_REF] Anastasopoulos | Rocking response of SDOF systems on shallow improved sand: an experimental study[END_REF] studied the effectiveness of shallow soil improvement for the heavily-loaded system by rocking stiffness degradation of shallow foundation. Kokkali et al. [START_REF] Kokkali | Centrifuge modeling of rocking foundations on improved soil[END_REF] performed centrifuge tests to study the effect of shallow soil improvement on the reduction of rocking-induced settlement of shallow foundaion. Empirical equations were also proposed for shallow foundations by many researchers. For example, Gajan et al. [START_REF] Gajan | Centrifuge modeling of load-deformation behavior of rocking shallow foundations[END_REF] proposed a equation for the rotational stiffness degradation of shallow foundation. Gazetas et al. [START_REF] Gazetas | Nonlinear rocking stiffness of foundations[END_REF] proposed formulas and charts in a general dimensionless format which allow easy computation of the nonlinear effective rotational stiffness of foundations of any (reasonable) shape. The proposed formulas are suitable for the practice of design. Paolucci et al. [START_REF] Paolucci | Seismic behaviour of shallow foundations: large scale experiments vs. numerical modelling and implications for performance based design[END_REF][START_REF] Paolucci | Accounting for non-linear dynamic soilstructure interaction in the displacement-based seismic design[END_REF][START_REF] Paolucci | Introducing dynamic nonlinear soil-foundationstructure interaction effects in displacement-based seismic design[END_REF] proposed equivalent damping curve that related to the rotational stiffness degradation curves. Studies in the similar direction are available in the literature [START_REF] Gazetas | Analysis of machine foundation vibrations: state of the art[END_REF][START_REF] Mergos | Rocking isolation of a typical bridge pier on spread foundation[END_REF][START_REF] Chen | Seismic isolation of viaduct piers by means of a rocking mechanism[END_REF][START_REF] Sullivan | Soil-foundation-structure interaction considerations for performancebased design of RC wall structures on shallow foundations[END_REF][START_REF] Kourkoulis | Dimensional analysis of SDOF systems rocking on inelastic soil[END_REF][START_REF] Gelagoti | Rocking isolation of low-rise frame structures founded on isolated footings[END_REF][START_REF] Deng | Seismic design of rocking shallow foundations: displacement-based methodology[END_REF]. Numerical studies concerning the stiffness degradation and damping of shallow foundaitons were also conducted by researchers. Taeseri et al. [START_REF] Taeseri | Non-linear rocking stiffness of embedded foundations in sand[END_REF] studied numerically the the degradation of the rocking stiffness of shallow foundation. Adamidis et al. [START_REF] Adamidis | Equivalent-linear stiffness and damping in rocking of circular and strip foundations[END_REF] and Pitilakis [START_REF] Pitilakis | Nonlinear soil-foundation-structure interaction aiming in performance based design[END_REF][START_REF] Pitilakis | Equivalent linear soil-foundation-structure interaction and performance based design[END_REF] implemented the stiffness degradation and damping curves. Equivalent linear substructure approximation was used to analyse the rocking oscillation of shallow foundation.

The existing studies mainly focused on shallow foundations. However, currently, studies for stiffness degradation and energy dissipation properties for deep foundations are very rare. To this end, this paper presents a comprehensive study of the stiffness degradation and damping for both batter and vertical pile foundations. The outline of this study is as following. The different behaviour in terms of stiffness degradation and damping properties of batter and vertical pile foundations are highlighted by a series of centrifuge tests. Then, based on the experimental results, stiffness degradation and damping curves are proposed for both batter and vertical pile foundations. Finally, numerical validation using equivalent linear approach with an iterative process is exploited to validated the proposed curves. The comparison of numerical and experimental results shows good agreement.

Centrifuge tests

Experimental campaign was carried out to investigate the rocking and lateral translation behavior of batter and vertical foundations. In this section, the preparation of centrifuge model is discussed while the experimental program will be introduced hereafter.

Experimental set-up

In the centrifuge tests, Fontainebleau sand (NE34) was prepared by air pluviation technique (with the relative density about 80%). The material properties of Fontainebleau sand are listed in Table 1.

Fig. 2 shows the friction (or floating) piles and pile caps used in the experiments. The distances of piles to the boundaries of container are also indicated. Equivalent Shear Beam (ESB) container which has the inner dimensions 800 × 340 × 416 mm was used in the tests. Vertical shear rods were attached to the bottom of the container to provide complimentary shear stresses [START_REF] Li | Centrifuge modeling of batter pile foundations under earthquake excitation[END_REF][START_REF] Li | Centrifuge modeling of batter pile foundations under sinusoidal dynamic excitation[END_REF]. The model piles were made of aluminum alloy and had a hollow section. The distance between the pile cap and the ground surface is 28 mm at model scale (1.12 m at prototype scale). This design was to avoid the additional soil-cap interaction [START_REF] Li | Centrifuge modeling of batter pile foundations under sinusoidal dynamic excitation[END_REF][START_REF] Li | Centrifuge modeling of batter pile foundations under earthquake excitation[END_REF]. The spacing of the piles was 4 times the diameter of the piles (4D) for both configurations. Table 2 gives the characteristics of the piles and the pile cap in model and prototype scales.

The following pile installation method was adopted in tests. A temporary sustaining system was used to maintain the pile group during pluviation, composed of an aluminum plate, 2 two steel rods and thin plastic wires. For both batter and vertical piles the same sustaining system was used, an example for positioning batter piles is shown in Fig. 1. The paper plate was temporarily used to verify the inclination of the piles. The thin aluminium plate (see in Fig. 1) was used to maintain precisely the inclination angle of °15 . The two piles were suspended by two thin steel rods in the container that are temporarily attached. Due to the falling of sand, the exact positions of piles were influenced. In order to avoid this, two thin plastic wires were attached on the tips of the two piles to provide additional supports. The sand was filled by air pluviation from the bottom up to the level of the ground surface. When the sand surface was about 5 cm higher than the tips of the piles, the wires were cut to free the pile tips. After the sand surface arrived at the prescribed level, the pluviation stopped and the steel rods were removed. Then, the pile cap was installed on to the pile head by tightening the bolts with a dynamic torque screw driver. Finally, the superstructure was installed on to the top of the pile cap. It should be pointed out that the adopted procedure of pluviation could minimize the "shadowing effect"1 in the zone between the two piles. However, the disturb of the homogeneity and density of sand underneath the batter piles was inevitable. For the details of installation of foundation models refers to Li et al. [START_REF] Li | Centrifuge modeling of batter pile foundations under earthquake excitation[END_REF][START_REF] Li | Centrifuge modeling of batter pile foundations under sinusoidal dynamic excitation[END_REF].

Three types of single degree of freedom superstructures were used in the tests, i.e. short, medium-tall and tall superstructures, Fig. 3(a frequency; the same top mass weight; and the same total weight of the whole foundation-superstructure system. With the increase of height of the top mass, the inertial loading generated by these three superstructures varies from a horizontal force dominated to a overturning moment dominated loading case. In this study, the soil column response frequency was estimated between 3.5 and 3.7 Hz. In order to highlight the influence of the superstructure and to distinguish it from the influence of the soil column, the frequency of the superstructure was designed to be far away from that of the soil column [START_REF] Li | Centrifuge modeling of batter pile foundations under earthquake excitation[END_REF][START_REF] Li | Centrifuge modeling of batter pile foundations under sinusoidal dynamic excitation[END_REF]. By considering the working frequency range of the shaker for sine input, the fixed base frequency of superstructure was designed to be 2.0 Hz. The pile foundation with a tall superstructure after installation is shown in Fig. 3(b). Since the column supporting the building cannot be regarded as weightless, according to Harris and Piersol [START_REF] Harris | Harris' shock and vibration handbook[END_REF], approximately 23% of the weight of the column should be included in the mass at the top while the rest 77% contributes to the weight of the base. Then foundation superstructure system can be idealized as a lumped mass and column system, see Fig. 3(c). The main properties of the idealized foundation-superstructure system are summarized in Table 3.

Experimental program

The centrifuge tests in this study were in the framework of a project -experimental study of the performance of batter piles under dynamic 3. In order to adapt the capacity of centrifuge shaker [START_REF] Chazelas | Original technologies for proven performances for the new LCPC earthquake simulator[END_REF], the earthquake records were attenuated. The attenuation refers to the reduction of the signal strength and is represented in decibels (dB) -calculated as 20 times the logarithm (base 10) of the output divided by the input.

excitations. In the first phase of this project, foundations (batter and vertical) with short and medium tall superstructures were subjected to real earthquake excitations, see Table 4 [33]; In the second phase, a taller superstructure was designed to highlight the influence of different superstructures. In order to reduce the experimental parameters, both short and taller superstructures were subjected to simple sinusoidal signals (2.0 Hz and 3.5 Hz, see Tables 5 and6) [START_REF] Li | Centrifuge modeling of batter pile foundations under sinusoidal dynamic excitation[END_REF]. All the loading cases are summarized in Table 7.

Definitions and assumptions

This section presents several definitions and assumptions concerning the interpretation of the experimental results and the numerical modelling in the following Sections 4 and 5.

Calculation of rotation and lateral displacement of foundation

In the experiments, the responses of the foundation and the superstructure are recorded by sets of accelerometers2 as shown in Fig. 4. The vertical accelerations of the pile cap are recorded by the sensors CH12 and CH13, by which the rotation of the foundation can be determined. The horizontal accelerations of the foundation and the superstructure are measured by the sensors CH10 and CH11, respectively. Sensor CH09 is far from the pile foundation and it was used to capture the farfield acceleration near the soil surface. Then the displacement of foundation and far-field soil surface can be calculated by integrating two time the accelerations in time domain. Then the relative translation of the foundation with respect to the soil surface can be calculated by the difference between CH09 and CH10 which corresponding to the displacements of the foundation and the far-field soil surface.

Overturning moment and total horizontal force

The overturning moment M and total horizontal force F acting at the foundation are calculated from the inertia forces from superstructure and the foundation using Eqs. ( 1) and ( 2). The overturning moments and horizontal forces are calculated only from the measured horizontal accelerations. Due to the small rotation of the foundation, the overturning moment caused by rotational inertia which is found less than 3% of the total overturning moment and thus was neglected. For tall superstructure, the additional overturning moment caused by P effect was also ignored.

= + M m u H m u H ¨ẗ t t b b b (1) = + F m u m u ¨ẗ t b b (2)
where m t and m b are the masses, H t and H b the C.G. and u ¨t and u ¨b the measured accelerations at the top mass and the foundation, respectively.

Pseudo foundation stiffness and damping

When subjected to dynamic loadings, the inertial forces acting on the foundation f I are balanced by the structural resisting forces f S and damping forces f D . The equilibrium of forces at the C.G. of foundation can be written as Eq. ( 3). The matrix form is expressed by Eq. ( 4).

= + f f f I S D (3) = + F M k k k k u c c c c u h hr hr r b h hr hr r b (4)
where u b and are the lateral displacement and rotation of foundation, respectively; F, the total horizontal force and M, the overturning moment and acting at the center of foundation, see Eqs. ( 2) and (1); k k , h r and k hr are the stiffness coefficients; c c , h r and c hr are the damping coefficients. Currently, in the dynamic tests and with the measured test data, it is impossible to determine directly these coefficients i.e. k k k c c , , , , h r hr h r and c hr , especially the coupling terms k hr and c hr . Al- ternatively, a uncoupled relationship could be established by assuming the coupling between foundation lateral displacement and rotation are included in some uncoupled coefficients. Based on this assumption, the coupled relationship between rotation and lateral translation of foundation in Eq. ( 4) can be decoupled and written as Eq. ( 5). Since these uncoupled terms are not the true stiffness and damping coefficients, therefore in this paper, these new coefficients are called pseudo coefficients marked with • , i.e. pseudo lateral stiffness coefficients k h and k r and pseudo rotational stiffness coefficients c h and c r . In this study, these uncoupled pseudo relations facilitate the modeling of pile foundation by simple uncoupled nonlinear springs. However, these terms may not be applicable for the rigorous evaluation of the response of pile foundation. The determination of these coefficients will be presented hereafter.

= + F M k k u c c u 0 0 0 0 ḧ r b h r b (5)
For the damping of the foundation, normally, two kinds of damping are involved in the dynamic response of foundation system i.e. radiation damping and material damping. Radiation or geometric damping is a fundamental concept in modern methods of analysis of dynamic soilstructure interaction. Whereas a foundation element moves against the surrounding soil, stress waves originate at the contact sufface and spread outward. These waves carry away some of the energy tranmitted by the foundation into the soil [START_REF] Gazetas | Simple radiation damping model for piles and footings[END_REF][START_REF] Çelebi | Comparison of damping in buildings under low-amplitude and strong motions[END_REF][START_REF] Çelebi | Response of olive view hospital to Northridge and Whittier earthquakes[END_REF][START_REF] Çelebi | Radiation damping observed from seismic responses of buildings[END_REF][START_REF] Benaroya | Mechanical vibration: analysis, uncertainties, and control[END_REF][START_REF] Kramer | Geotechnical earthquake engineering[END_REF]. Material damping is caused by inelastic behavior of the soil supporting the foundation which depends on the level of strain induced in the soil [START_REF] Kramer | Geotechnical earthquake engineering[END_REF]. Due to the presence of the container, the stress waves were reflected back by the side walls. The energy of these waves is enclosed in the soil container which cannot be spread outward. Therefore, in this study, only material damping was considered.

Equivalent rotational, lateral stiffness and associated damping ratio

With the overturning moment (or horizontal force) and rotation (or translation) hysteresis loops, equivalent stiffness (rotational and transnational) and the associated damping ratio can be determined. A schematic representation is shown in Fig. 5, where k r,max and k r are the initial rotational and equivalent rotational stiffness, respectively; similarly, k h,max and k h are the initial translational and equivalent trans- lational stiffness, respectively.

In the dynamic tests, quiet often, the loops are in irregular shapes, shown in Fig. 6. It is difficult to determine the apexes of the enclosed loop so as to the equivalent stiffness. Alternatively, the stiffness could be obtained with Eq. ( 6) [START_REF] Brennan | Evaluation of shear modulus and damping in dynamic centrifuge tests[END_REF].

= = k k , F F u u M M h max( ) min( ) max( ) min( ) r max( ) min( ) max( ) min( ) b b (6)
where u b and F are the horizontal translation (or rotation) and force (or moment), respectively. The geometrical interpretation of the equivalent stiffness is shown as the line between A and B, see in Fig. 6. For an arbitrary or even irregular close loop, points A and B are often outside the loops. The damping ratio could be calculated by Eq. ( 7):

= D W W 1 4 ( 7
)
where W is the energy dissipated in a cycle and W is the stored energy measured at the peak of the same cycle, respectively. Then the damping coefficients c h and c r can be calculated with the damping ratios. In addition, a method [START_REF] Li | Using centrifuge tests data to identify the dynamic soil properties: application to Fontainebleau sand[END_REF] to reduce the dispersion of equivalent stiffness and damping ratio from irregular loops was adopted. This method refers to collecting hysteresis loops in the same level (small range or short window) of rotation (translation) and then an average loop is found.

For the whole range of rotation (translation), with the moving of this short window, less dispersed results can be obtained.

Stiffness degradation and damping of pile foundations under dynamic loading

In the dynamic centrifuge test, the soil-pile-superstructure system is loaded at the base which is close to the real loading case of ground motion. Both kinematic and inertial interaction can be included in dynamic centrifuge tests. Under dynamic loading, pile foundations move horizontally combined with rocking movements. The rocking and translational behavior of the foundations can be significantly influenced by the presence of batter piles. This section presents the different rocking and translational behavior of batter and vertical pile foundations under dynamic loadings.

Rotational behavior of pile foundations

First of all, the batter and vertical pile foundations under sine 2.0 Hz excitations are analyzed. The results are presented in terms of hysteretic loops of foundation-superstructure system under different excitation intensities, see Fig. 7. For example, Fig. 7(a) presents the results of batter pile foundation with a short superstructure under 2.0 Hz excitations with different peak acceleration level (from 0.05 to 0.4 g, Table 5). The degradation of rotational stiffness and damping for different foundation configurations are shown in Fig. 8. From the results, it is clear that the batter pile foundation has larger energy dissipation than the vertical pile group, Fig. 8 (b) and (d). However, the presence of batter piles does not influence so significantly the rotational stiffness degradation curve, see in Fig. 8 (a) and (c). Due to the dispersion of the data points, it is difficult to distinguish, from the current experimental data, the differences caused by the height of superstructure. The trends for stiffness degradation and damping are almost identical for batter (or vertical) pile foundation with short and tall superstructures.

After the analyses of the results from foundation-superstructure system under sine 3.5 Hz and real earthquake excitations (Tables 4 and6), it is observed that the rotational stiffness degradation and damping ratio curves follow almost the same trends in all tests, (see in Fig. 9). The results for rocking behavior of foundation-superstructure system under sine 3.5 Hz and real earthquake excitations are summarized in A. The good agreement between all the tests indicates that the presence of a superstructure does not influence the mechanical behavior of the foundations. The presence of batter piles has limited effect on the rotational stiffness degradation behavior, however it significantly influences the energy dissipation properties of the pile foundation system. An empirical equation is proposed to fit the data points of rotational stiffness of batter and vertical pile foundations in the following form:

= + k k 1 r r,max (8)
where k r is the secant rotational stiffness of the pile foundation; k r,max the maximum rotational stiffness; and and constants. k r,max is es- timated empirically as 2.35× 10 6 kN • m/rads for batter pile foundation and 2.50× 10 6 kN • m/rads for the vertical, respectively. With these two values, curves calculated by Eq. ( 8) fit well with the experimental results. These initial values for rotational stiffness are also verified by FEM models 3 . From the FEM model, the values of k r,max are found 2.33× 10 6 kN • m/rads for batter pile foundation and 2.61× 10 6 kN • m/ rads for the vertical, which are very close to the empirically estimated values. For the numerical determination of the initial stiffness, the FEM model utilized advanced soil constitutive law with soil stiffness in small-strain range. It means alternatively a simpler way of estimation of these values by linearly elastic model could be adopted.

For both batter and vertical configurations and are 5.0× 10 3 and 1.4, respectively. The maximum rotational stiffness for the batter pile foundation is lower than that of the vertical pile foundation due to the presence of batter piles. The fitting curves together with all the data points are plotted in Fig. 10.

As for the damping properties, the damping ratio curves for batter and vertical foundation are linked with their rotational stiffness degradation curves. The proposed form for the fitting equation is the following:

= + D D m k k n k k 1 r r,max r r,max 1.1 r r,max (9)
where D r is the rotational damping ratio; D r,max is the maximum rota- tional damping ratio; m and n are fitting parameters. D r,max is estimated around 60% for batter pile foundation and 25% for vertical pile foundation, respectively. For both cases m and n are 0.63 and 1.5, respectively. Nonlinear FEM simulations were also carried out to estimate the maximum damping ratios. Results show the maximum damping ratios are about 40% and 20% for batter and vertical pile groups. Different from the good estimation of initial stiffness, the nonlinear FEM simulation cannot well predict the maximum damping ratio. To this end, in this study, the empirically estimated damping ratios are used to have the best fitting curves with the experimental data. The relationships between rotational stiffness and damping ratio for the batter and vertical foundations together with the fitting curves are plotted in Fig. 11.

The main constants for Eqs. ( 8) and ( 9) are summarized in Table 8. Finally the damping ratio curves for batter and vertical pile foundations are obtained and plotted in Fig. 12. It should be acknowledged that the values of D r,max and parameters m and n are em- pirically determined to ensure that Eq. ( 9) fits well with the experimental data in its range. The accuracy cannot be guaranteed if the fitting curves are out of the range of the data points i.e. rotation < = 10 4 and > = 10 2 , where the curves are marked by dashed lines in Fig. 12. The comparison of stiffness degradation and damping between batter and vertical pile foundations is summarized in Fig. 13. It can be seen that the rotational stiffness is not significantly influenced by the presence of the batter piles. However, batter pile foundation has larger energy dissipation capacity.

Lateral behaviour of pile foundations

Similar as the rotational behavior of foundation-superstructure system, the horizontal behavior of foundation is analyzed using hysteretic loops (results for horizontal translation and force hysteretic loops are summarized in Appendix B).

Plotting the results from all different dynamic loadings together, see in Fig. 14, it is obvious that they have very similar tendencies. This confirms the former conclusion that the behavior of foundations is not influenced by the superstructures. Following the same procedure in the previous sections, the horizontal translation stiffness degradation are fitted by empirical equations:

= + k k u 1 h h,max (10)
where k h is pseudo horizontal translational stiffness; k h,max is the maximum pseudo horizontal translation stiffness; u is the horizontal translation;

and are fitting parameters. The maximum pseudo horizontal stiffnesses are estimated as approximately 2.3× 10 5 kN/m and 0.75× 10 5 kN/m for batter and vertical pile foundation, respec- tively. These initial values are also verified by FEM models. From the FEM model, the values of k h,max are found 2.4× 10 5 kN/m for batter pile foundation and 0.97× 10 5 kN/m for the vertical, which agree well with the empirically estimated values. and for both cases are 200 and 1.05 respectively. For the damping ratio, a similar equation is proposed in function of the degradation of translational stiffness:

= + D D m k k n k k 1 h h,max h h,max 2 h h,max (11)
where D h is the damping ratio associated with the translation of the pile foundation; D h,max is the maximum damping ratio; m and n are two fitting parameters. D h,max is estimated around 60% for batter pile foundation and 35% for vertical pile foundation. m and n for both cases are 0.88 and 1.8 respectively. Nonlinear FEM simulations found the maximum damping ratios were about 35% and 25% for batter and vertical pile groups. These values cannot fit well with the experimental data. In this study, the empirically estimated damping ratios are used to have the best fitting curves with the experimental data. Figs. [START_REF] Tileylioglu | Dynamic stiffness and damping of a shallow foundation from forced vibration of a field test structure[END_REF][START_REF] Anastasopoulos | Rocking response of SDOF systems on shallow improved sand: an experimental study[END_REF][START_REF] Kokkali | Centrifuge modeling of rocking foundations on improved soil[END_REF] show the comparisons of the fitting curves with the all the data points.

As it was explained before, with empirical parameters, Eq. ( 11) fits well within the range of experimental data. For other ranges, accuracy cannot be guaranteed and marked by dashed curves, see Fig. 17. Table 9 summarized the parameters for the empirical equation for horizontal translational behavior of foundations. Finally, by comparison, it can be concluded from Fig. 18 that batter piles greatly increase the horizontal stiffness of foundation system by around 3.0 times; while batter pile foundation also have greater damping ratio for horizontal behavior.

Summary

In this study, a comprehensive experimental study of the performance of batter and vertical pile foundations is carried out. For the rocking behavior, the presence of the batter piles increases the rotational damping ratio without losing so much rotational stiffness; for the horizontal translation behavior. Batter piles have a more important horizontal stiffness than the vertical and energy dissipation ability. Due to the inclinations, batter piles provides substantial horizontal resistance to the foundation. When subjected to horizontal loading which causes rotation and translation of foundation, batter piles may have much more complicated nonlinear soil-pile interaction in both axial and lateral directions. This is in accordance with the numerical results from 3D nonlinear FEM simulations of Li et al. [START_REF] Li | A hypoplastic macroelement formulation for single batter piles in sand[END_REF]. It is found that under cyclic horizontal loading, batter pile has more important axial reaction force and energy loss due to plastic deformation. The current results from a relative large number of tests show that the behavior of the pile foundation system is not influenced by the type of superstructures.

Modelling of nonlinear behaviour of pile foundations

Based on the proposed stiffness degradation and damping curves, Equivalent Linear Approach (ELA) was used to validate the feasibility of using these curves to analyse the nonlinear response of pile foundations. The main numerical approach and validations will be presented hereafter.

The dynamic equilibrium of the system is shown in Fig. 19. The foundation system can be idealized as springs and dampers which control the translational and rotational behaviour. The dynamic equilibrium of this simplified foundation-superstructure system can be described by the following set of equations, Eq. ( 12 of foundation,

= c D k J D •2 ; r r r t
h , pseudo horizontal translational damping ratio of foundation; D r , pseudo rotational damping ratio of foundation; J t , moment of inertia of foundation; k is the lateral stiffness of the column. In the governing equation of the system Eq. ( 12), the damping in the column is ignored.

In order to implement the foundation stiffness degradation and damping curves, Equivalent Linear Approach (ELA) was adopted. This method was initially proposed by Schnabel et al. [START_REF] Schnabel | SHAKE: a computer program for earthquake response analysis on horizontally layered sites[END_REF] for analyzing free-field responses of ground under seismic actions. In this study, in stead of using the recursion formulas in frequency domain, step-by-step integration in time domain was adopted (Newmark-Beta method integration with constant average acceleration). For evaluating the effective displacement or rotation, the same factor i.e. 0.65 (in the original equivalent linear approach) is taken. In other words, the effective displacement or rotation of foundation is taken as 65% of their peak values. In the iterative procedure, the initial values of stiffness and damping ratios are taken the values listed in Tables 8 and9. Maximum damping ratios were taken as the initial values to get a fast convergence for the iteration process. Then, the numerical model is validated and compared with the experimental results. For the dynamic inputs to the numerical model, the far-field accelerations measured at the ground surface were taken as the base excitation signals, see CH09 in Fig. 4. Fig. 20 shows the comparison of model performance with experimental results. The simulated case is batter pile foundation with a short superstructure under 2.0 Hz excitation. A very good agreement is found between the numerical and experimental results. After several iterations, the stiffness and damping are stabilized as their final values, see Fig. 21(a). An example of iterative process of rotational stiffness is shown in Fig. 21(b). Other simulation cases are shown in Figs. 22232425. Equivalent linear analysis requires a number of iterations to obtain more accurate results. Although enough accuracy with differences of less than 5 to 10% are usually achieved in 3 5 iterations [START_REF] Schnabel | SHAKE: a computer program for earthquake response analysis on horizontally layered sites[END_REF], in this study, in order to further ensure the convergence of the iterative process, a large number of iterations was adopted. This large number is also recommended by similar code implementations (DeepSoil [START_REF] Hashash | DEEPSOIL 6.1, User Manual, Tech. rep[END_REF] recommends 15 as the number of iterations, for example).

From the results, it is proved that for batter (or vertical) foundation with short (or medium-tall and tall) superstructure, the numerical model has very good performance in reproducing the dynamic response of foundations under various dynamic loadings.

Numerical validation shows a good agreement with the experimental results. However, some limitation of the numerical simulations should be acknowledged. First of all, that equivalent linear approach that adopted in this study is not able to take account of the strong strain dependence to secant modulus and damping ratio. For large deformation under strong ground motions, using constant secant stiffness and damping ratio may not yield good results. Secondly, due to the limitation of accelerometers in the centrifuge tests, the residual displacement or rotation cannot be obtained which is also the case for equivalent linear approach. In this study, the good agreement of the comparisons only refers to dynamic displacements and rotations. However, despite these limitations, the proposed model may be utilized for preliminary evaluation of nonlinear SSI of pile foundations.

Conclusions

In this paper, the study of the performance of batter and vertical foundation in terms of translational/rotational stiffness and the damping properties is carried out. Empirical equations are also proposed to estimate the stiffness degradation and damping for foundations under dynamic loadings. It is shown that for the rocking behaviour, the presence of batter piles causes a small decrease of the the rotational stiffness and a great increase of the rotational damping; for the horizontal translation behaviour, batter pile foundation has much higher horizontal stiffness and translational damping compared to the vertical foundation. Finally, the proposed stiffness degradation and damping curves are implemented and integrated with equivalent linear approach. The numerical validation shows a good agreement with the experimental results. The feasibility of analysing the nonlinear response of pile foundations using simple stiffness degradation and damping curves is convinced. This work provides not only the insights to the stiffness degradation and energy dissipation properties of batter and vertical pile foundations, but also a simple and reliable numerical modelling approach for the design practice concerning the seismic response of deep foundations.

A.2. Rocking behavior of batter and vertical foundations under real earthquake excitations

See Figs. 28 and29.

Fig. 1 .

 1 Fig. 1. Positioning the batter piles in the ESB container.

Fig. 2 .

 2 Fig. 2. Dimensions of pile foundations and ESB container (in prototype scale), refers to Li et al. [33,34].

Fig. 3 .

 3 Fig. 3. (a) Superstructures; (b) Centrifuge model; (c) Idealized lumped mass model, refers to Li et al. [33,34].

Fig. 4 .

 4 Fig. 4. Illustration of the response of the foundation and superstructure, after Li et al. [33,34].

Fig. 5 .

 5 Fig. 5. (a) Equivalent lateral stiffness and damping ratio; (b) Equivalent rotational stiffness and damping ratio.

Fig. 6 .

 6 Fig. 6. A hysteretic loop extracted from experimental data.

Fig. 7 .

 7 Fig. 7. Rotation-moment hysteretic loops, 2.0 Hz sine input: (a) Batter pile foundation with short superstructure (IS, 5.12 m); (b) Vertical pile foundation with short superstructure (VS, 5.12 m); (c) Batter pile foundation with tall superstructure (ITT,14.16 m) and (d) Vertical pile foundation with tall superstructure (VTT, 14.16 m).

Fig. 8 .

 8 Fig. 8. Rotational stiffness degradation curve of batter and vertical pile foundations, 2.0 Hz input. (a) Rotational stiffness degradation curve of batter pile foundations with short (IS) and tall (ITT) superstructures and (b) Damping ratio; (c) Rotational stiffness degradation curve of vertical pile foundations with short (VS) and tall (VTT) superstructure and (d) Damping ratio.

Fig. 9 .

 9 Fig. 9. Rotational stiffness degradation curves from all input signals (sine 2.0 Hz, 3.5 Hz and real earthquakes): Batter pile foundation with (a) short (IS), medium-tall (IT) and tall (ITT) superstructures and (b) damping ratio; (c) Vertical pile foundation with short (VS), medium-tall (VT) and tall (VTT) superstructures and (d) damping ratio.

Fig. 10 .

 10 Fig. 10. Rotational stiffness degradation fitting curves: (a) Batter pile foundation and (b) Vertical pile foundation.

Fig. 11 .

 11 Fig. 11. Damping ratio in function of normalized rotational stiffness: (a) Batter pile foundation and (b) Vertical pile foundation.

Fig. 12 .

 12 Fig. 12. Damping ratio in function of rotation of foundation: (a) Batter pile foundation and (b) Vertical pile foundation.

Fig. 13 .

 13 Fig. 13. Comparison of rocking behaviour of batter and vertical foundations: (a) degradation of rotational stiffness and (b) damping ratio.

Fig. 14 .

 14 Fig. 14. Translational stiffness degradation curves for all input signals (sine 2.0 Hz, 3.5 Hz and real earthquakes): (a) Batter pile foundation with short and tall superstructures and (b) damping ratio; (c) Vertical pile foundation with short and tall superstructure and (d) damping ratio.

Fig. 15 .Fig. 16 .

 1516 Fig. 15. Fitted horizontal translation stiffness degradation curves: (a) Batter pile foundation and (b) Vertical pile foundation.

Fig. 17 .

 17 Fig. 17. Damping ratio in function of horizontal translation of foundation: (a) Batter pile foundation and (b) vertical pile foundation.

Fig. 18 .

 18 Fig. 18. Comparison of translation behavior of batter pile foundation and vertical foundation: (a) degradation of rotational stiffness and (b) damping ratio.

Fig. 19 .

 19 Fig. 19. Computational model for nonlinear dynamic soil-structure interaction analysis.

Fig. 21 .

 21 Fig. 21. Simulation of batter pile foundation with short superstructure (5.12 m, sine 2 Hz input). (a) Evolution history of stiffness and damping ratio from their initial values; (b) An example of iterations of k r .

Fig. 20 .

 20 Fig. 20. Simulation of batter pile foundation with short superstructure (5.12 m, sine 2 Hz input). Comparison of displacement history of top mass.

Fig. 23 .

 23 Fig. 23. Simulation of batter pile foundation with tall superstructure (14.16 m, sine 2.0 Hz input). Comparison of displacement history of top mass.

Fig. 24 .

 24 Fig. 24. Simulation of vertical pile foundation with short superstructure (5.12 m, Northridge input) (a) Comparison of displacement history of top mass (b) Evolution history of stiffness and damping ratio from their initial values.

Fig. 25 .

 25 Fig. 25. Simulation of vertical pile foundation with tall superstructure (14.16 m, sine 2.0 Hz input). Comparison of displacement history of top mass.

Fig. 22 .

 22 Fig. 22. Simulation of batter pile foundation with medium-tall superstructure (8.48 m, Kobe input). Comparison of displacement history of top mass.

Fig. 27 .

 27 Fig. 27. Rotational stiffness degradation of batter and vertical pile foundations, 3.5 Hz input. (a) Rotational stiffness degradation of batter pile foundations with short (IS) and tall (ITT) superstructure (b) Damping ratio; (c) Rotational stiffness degradation of vertical pile foundations with short (VS) and tall superstructures (VTT) (d) Damping ratio.

Fig. 28 .

 28 Fig. 28. Rotation-moment hysteretic loops, real earthquake input (a) Batter pile foundation with short superstructure (5.12 m); (b) Vertical pile foundation with short superstructure (5.12 m); (c) Batter pile group with medium-tall superstructure (8.48 m) and (d) Vertical pile foundation with medium-tall superstructure (8.48 m).

Fig. 31 .

 31 Fig. 31. Horizontal stiffness degradation of batter and vertical pile foundations, 2.0 Hz input: (a) Horizontal stiffness degradation of batter pile foundations with short (IS) and tall (ITT) superstructures (b) Damping ratio (c) Horizontal stiffness degradation of vertical pile foundations with short (VS) and tall superstructures (VTT) (d) Damping ratio.

Fig. 32 .

 32 Fig. 32. Horizontal translation-force hysteretic loops, 3.5 Hz sine input (a) Batter pile group with short superstructure (5.12 m) (b) Vertical pile group with short superstructure (5.12 m) (c) Batter pile foundation with tall superstructure (14.16 m) and (d) Vertical pile group with tall superstructure (14.16 m).

Fig. 33 .

 33 Fig. 33. Horizontal stiffness degradation of batter and vertical pile foundations, 3.5 Hz input: (a) Rotational stiffness degradation of batter pile foundation with short (IS) and tall (ITT) superstructures (b) Damping ratio (c) Rotational stiffness degradation of vertical pile foundations with short (VS) and tall (VTT) superstructures (d) Damping ratio.

Fig. 34 .

 34 Fig. 34. Horizontal translation-force hysteretic loops: (a) Batter pile foundation with short superstructure (5.12 m); (b) Vertical pile foundation with short superstructure (5.12 m); (c) Batter pile foundation with medium-tall superstructure (8.48 m) and (d) Vertical pile foundation with medium-tall superstructure (8.48 m).

Fig. 35 .

 35 Fig. 35. Horizontal stiffness degradation of batter and vertical pile foundations, real earthquake input (a) Batter pile foundations with short (IS) and tall (ITT) superstructures and (b) damping ratio; (d) Vertical pile foundations with short (VS) and tall (VTT) superstructures and (d) damping ratio.

  ). These superstructures were designed to have the same fixed base

	Nomenclature				u s	Horizontal translation of top mass due to the deformation
						of the column
	List of symbols				H	Length of the column
						k h	Pseudo horizontal translational stiffness of foundation
	m t	Lumped mass of superstructure		k r	Pseudo rotational stiffness of foundation
	m b	Lumped mass of foundation		c h	Pseudo horizontal translational damping coefficient of
	u t	Horizontal translation of top mass		foundation
	u b	Horizontal translation of foundation		c r	Pseudo rotational damping coefficient of foundation
		Rotation of foundation			D h	Pseudo horizontal translational damping ratio of founda-
	H t	Height of top mass			tion.
	H b	Height of center of foundation		D r	Pseudo rotational damping ratio of foundation
	F	Horizontal force
	M	Overturning moment
	Table 1
	Fontainebleau sand NE34: Material properties.
	Soil		e min	emax	e	D r (%)
	Fontainebleau NE34	0.51	0.866	0.58	80%

Note: e:void ratio; D r : relative density.

Table 2

 2 Characteristics of the piles and of the pile cap, refers to Li et al.[START_REF] Li | Centrifuge modeling of batter pile foundations under sinusoidal dynamic excitation[END_REF][START_REF] Li | Centrifuge modeling of batter pile foundations under earthquake excitation[END_REF].

	Item	Model scale 40 g	Prototype scale 1 g
	Pile-embedded depth	326.5 mm	13.06 m
	Pile-external diameter	18 mm	0.72 m
	Pile-internal diameter	15 mm	0.6 m
	Pile-bending stiffness	197 N • m 2	505 MN • m 2
	Pile-yield strength	245 MPa	4245 MPa
	Pile cap-dimensions	140 mm × 40 mm × 40 mm	5.6 m × 1.6 m × 1.6 m
	Cap-weight	5.5 N	331.24 kN

Table 3

 3 Main characteristics of the pile foundations with three types of superstructures.

		Top	Found.	C.G. of Top	C.G. of	Column
		mass mt	mass mb	mass Ht	found. Hb	stiffness
	Short	51.2	64.0 tonne	5.12 m	1.92 m	8090 kN/m
	Medium-tall	tonne		8.48 m
	Tall			14.16 m

Note: 1. All values are presented in prototype scale; 2. C.G.: center of gravity.

Table 4

 4 Applied sequences of the real earthquake signals.

	NO.	Events	Excitations	Attenuation (dB)	PGA	Arias
					(m/s 2 )	Intensity (m/
						s)
	Test 1 1, 2, 3	Martinique Jara	-1	0.9	0.2
	Test 2 4, 5, 6	Northridge	-20	1.35	0.2
	Test 3 7, 8, 9	Northridge	-9	4.79	2.43
	Test 4 10, 11, 12 Kobe	-4	3.83	3.51
	Test 5 13, 14, 15 Martinique Jara	-1	0.9	0.2
	Test 6 16, 17,18	Northridge	-20	1.35	0.2

Note: 1. Earthquake records are: Martinique earthquake (Jara station EW); Northridge earthquake (Tarzana station 090) and Kobe earthquake (DAI8-G, N43W) records. 2. Values of PGA (peak ground acceleration) and Arias Intensity are in prototype.

Table 5

 5 Applied sequence of sinusoidal signals, 2.0 Hz (prototype scale).

	NO.	Events	Frequency (Hz)	PGA (g)	Arias Intensity (m/s)
	Test 1	1, 2	2.0	0.05	0.34
	Test 2	3, 4	2.0	0.1	1.35
	Test 3	5, 6	2.0	0.2	5.37
	Test 4	7, 8	2.0	0.4	21.38

Table 6

 6 Applied sequence of sinusoidal signals, 3.5 Hz (prototype scale).

	NO.	Excitations	Frequency (Hz)	PGA (g)	Arias Intensity (m/s)
	Test 1	1, 2	3.5	0.05	0.35
	Test 2	3, 4	3.5	0.1	1.39
	Test 3	5, 6	3.5	0.2	5.54
	Test 4	7, 8	3.5	0.4	22.06

Table 7

 7 Loading cases for different configurations of foundation-superstructure systems.

	Configurations	Real	Sine 2.0 Hz Sine 3.5 Hz
		earthquakes
	Batter & vertical found. with short	✓	✓	✓
	Batter & vertical found. with medium-tall	✓	-	-
	Batter & vertical found. with tall	-	✓	✓

Note: Frequencies are in prototype.

Table 8

 8 Main constants for rotational stiffness degradation and damping curves of batter and vertical pile foundation.

		k r,max	D r,max m	n
	Batter pile found.	2.35× 10 6 kN • m/rads 5.0× 10 3 1.4 60%	0.63 1.5
	Vertical pile found. 2.50× 10 6 kN • m/rads	25%

Table 9

 9 Main constants for horizontal stiffness degradation and damping curves of batter and vertical pile foundation.

		k h,max			D h,max	m	n
	Batter pile found.	2.3× 10 5 kN/m	200	1.05	60%	0.88	1.8
	Vertical pile found.	0.75× 10 5 kN/m			35%

shadowing effect: during sand pluviation the presence of certain objects influence the falling of sand into the container and thus disturbs the homogeneity of the sand deposit.

The type of accelerometers used in the tests is Brüel & Kjaer 4517, which is a piezoelectric shear type accelerometer with a frequency range from 1 to 20 kHz.

FEM models of pie foundations were carried out using advanced hypoplastic constitutive model[START_REF] Von Wolffersdorff | A hypoplastic relation for granular materials with a predefined limit state surface[END_REF][START_REF] Gudehus | The soilmodels.info project[END_REF] and the numerical modeling technique proposed by Li et al.[START_REF] Li | A hypoplastic macroelement for single vertical piles in sand subject to three-dimensional loading conditions[END_REF] was adopted.

Acknowledgments

Financial support from National Natural Science Foundation of China (NSFC, Grant No. 51708009) is highly acknowledged. The support of IFSTTAR (Institut français des sciences et technologies des transports, de l'aménagement et des réseaux) and of the Région Pays de la Loire is gratefully appreciated. The authors would like also to thank the valuable support and help of the technical staff of the IFSTTAR centrifuge team. The first author would acknowledge special thanks to Ms. Xuejiao Tang for her help.

Declaration of Competing Interest

The authors declared that there is no conflict of interest.