Pseudo-natural SSI frequency of coupled soil-pilestructure

Pseudo-natural SSI frequency of coupled soil-pilestructure systems E.N. Rovithis Institute of Engineering Seismology and Earthquake Engineering (ITSAK), Thessaloniki, Greece K.D. Pitilakis Department of Civil Engineering, Aristotle University of Thessaloniki, 544, Thessaloniki, Greece G.E. Mylonakis Department of Civil Engineering, University of Patras, 65, Rion, Patra, Greece ABSTRACT Within the identification of the prevailing frequencies in a coupled soil-pile-structure system, a pseudo-natural SSI frequency is defined as the frequency where pile-head motion is minimized with respect to both free-field and structural motion. This frequency is determined analytically and the proposed solution is compared to numerical results from finite-element analyses for a single-degree-of-freedom structure supported on a single end-bearing pile. The role of foundation vibration modes on pseudo natural-to-effective natural SSI frequency ratio and effective damping of the coupled system is explored, as affected by salient model parameters such as relative stiffness and inertia between structure and soil, foundation flexibility and pile slenderness. It is observed that for stiff massive structures founded on a stiff and/or short pile, pseudo-natural frequency may deviate substantially from the conventional effective natural frequency, providing a possible design frequency associated with lower levels of pile radiation damping. Conversely, when a flexible tall structure is supported on a flexible pile for which foundation rocking prevails, pseudo-natural frequency and conventional effective natural frequency nearly coincide regardless of the superstructure mass, allowing prediction of the actual flexible-base vibrational characteristics of a SSI system even in the absence of free-field recordings. Keywords: Soil-pile-structure interaction, coupled system, seismic response, finite element analysis, pseudo-natural SSI frequency INTRODUCTION Soil-pile-structure interaction possesses an important role in controlling the vibrational characteristics of the structure and the effective seismic motion imposed at the foundation of pilesupported structures on soft soil. Several numerical and analytical solutions have been proposed for quantifying the effect of soil compliance near the pile-head in terms of effective natural frequency and damping ratio of the complete system [-3]. System identification techniques have also been adopted to derive flexible-base parameters of structures in the absence of base rocking or free-field recordings [4]. Within the identification of the characteristic frequencies that dominate pile-head response in a coupled soil-pile-structure system Rovithis et al. [5] introduced the notion of a pseudo-natural SSI frequency (f pssi ) as the frequency where pilehead motion is minimized with respect to both superstructure and free-field motion. Based on a Corresponding Author. Institute of Engineering Seismology and Earthquake Engineering (ITSAK), Ag. Georgiou 5, Patriarchika Pylaias, P.O.Box 53, GR-55 Finikas, Thessaloniki, Greece, rovithis@itsak.gr

frequency domain analysis of a 3D finite-element model the above study showed that pseudonatural frequency is always higher than the conventional effective natural frequency (f SSI ) especially for stiff structures founded on stiff and/or short end bearing piles. In the herein reported study the problem is treated analytically leading to an exact solution for the pseudo-natural SSI frequency encompassing frequency-dependent pile impedance functions and higher-order terms of effective damping, as opposed to common approximations [, 6]. Partial base fixity conditions as defined by Stewart and Fenves [4] are implemented for a single-degree-of-freedom structure supported on a single end-bearing pile. Rigorous numerical data are employed to validate the proposed analytical solution. Pseudo-natural-to-effective natural frequency ratios and effective damping of the coupled system are investigated in connection to pile vibration modes. The effect of salient SSI factors such as relative stiffness between structure and soil, foundation flexibility, pile slenderness and superstructure mass is explored in dimensionless form, allowing identification of coupled systems where pseudo-natural frequency is a prevailing parameter of SSI response. DOMINANT FREQUENCIES OF COUPLED SPSI SYSTEMS The examined soil-pile-structure (SPSI) system comprises of a single end-bearing pile supporting a SDOF structure founded on a homogeneous viscoelastic soil layer over rigid rock. The effect of governing SSI factors was investigated by means of dimensionless parameters. The wave parameter (/σ) was adopted as a normalized fundamental frequency of the superstructure []: f fixed H V s () where f fixed, H and V s stand for the naturalfrequency of the fixed-base structure, the height of the structure and the shear wave velocity of the soil, respectively. The selected values of (/σ) concern both stiff and flexible superstructures, varying between.4 and.4. Pile slenderness ratio (L p /D p ) and soil-pile stiffness contrast (E p /E s ) were examined through the dimensionless pile flexibility parameter [7]:.5 S L D E E () H p p p s The above parameter was considered in the range of.8 to 8.5 representing both stiff and/or short piles (S H < 5) and long piles characterized by higher values of S H.. Superstructure mass (m ) effect was investigated by implementing structure-to-soil relative mass parameter []: m H s r (3) where ρ s is the soil density and r is the foundation halfwidth. For most conventional building structures γ =.-. [8]. In this study, γ varies between and referring to both light and massive structures. The coupled system was analyzed in the frequency domain using a rigorous three-dimensional finite-element model under harmonic horizontal displacement of unit amplitude specified at the base of the soil profile [5]. Typical results of superstructure to free-field (U s /U ff ), superstructure to pile-head (U s /U p ) and Amplification 5 5 No Fixed-base SSI (ffixed) (No SSI) Us/UffU s Us/UpU s p Up/UffU p ff f fixed f SSI f pssi 3 4 5 6 7 8 9 f (Hz) Figure. Numerically derived amplification ratios U s /U ff, U s /U p and U p /U ff corresponding to a coupled system with (/σ)=.4 and S H =3.56.

pile-head to free-field (U p /U ff ) amplification ratios obtained numerically are plotted in Fig. for (/σ), (S H ) and (γ) at.4, 3.56 and.7, respectively. Evidently, pile-head to free field response ratio is dominated by two discrete frequencies: a lower frequency where pile motion is amplified and a higher one where a significant deamplification is observed. This frequency variation of U p /U ff is in agreement with transient response analyses of pile-supported structures using real earthquake motions [9] as well as analytical studies of the complete soil-pilestructure system under harmonic excitation []. These dominant frequencies were clearly identified by means of both numerical and experimental data [5]. It was found that pile-head motion is amplified at the effective natural frequency f SSI, whereas significant de-amplification occurs at the pseudo-natural frequency f pssi relative to both superstructure s and free-field motion. Thereby, the pseudo-natural frequency possesses a double role in affecting the seismic response of a SSI system: First, it defines the frequency where the translational motion of the structure is maximized relative to the translational motion of the pile-head. Second, it is the frequency where the pile head motion is minimized with respect to the free field surface motion. 3 ANALYTICAL SOLUTION FOR f pssi The proposed analytical solution is based on the definition of the pseudo-natural frequency as the frequency where the ratios U s /U p and U p /U ff are maximized and minimized respectively, denoting a foundation rocking-related SSI response. For this reason a fictitious soil-pile-structure system with the pile-head restrained against translational motion is adopted, representing partial base fixity conditions [4]. This reduced SSI model accounts solely for structural deflection and rocking of the foundation (Fig.). Accordingly, the lateral displacement of the superstructure (U s ) relative to free-field motion (U ff ) is given by U U H (4) s p H Uff Ug Us θph U θp K Up= m Figure. Deflection of a fictitious soil-pile-structure system with the pile-head restrained against translational motion. where U and θ p H represent the structural deflection and the rigid-body displacement due to pile-head rocking, respectively. The reference system is transferred from the pile-head to a certain depth e where the resultant soil reaction is applied to address the issue of coupled swayingrocking stiffness of the foundation []. Eq.(4) suggests that the dynamic impedance of the fictitious SSI system may be expressed as a set of complex springs attached in series: ˆ * K K ( H e) * * K (5) * * where ˆK, K and K * represent the corresponding complex-valued impedances defined by * the general function K K( i ) in which K is the real part of the impedance function and ζ represents a damping factor analogous to the viscous damping coefficient of a simple oscillator. * Substituting K in Eq. (5) for each available degree of freedom leads after some straightforward algebra to the solution for the pseudo-natural frequency and the corresponding damping ratio of the coupled system: pssi ˆ ˆ 4 4 4 4 (6)

ˆ str str str. 4.. 4 4 4 (7) In the above expressions, K m and K m( H e) denote the uncoupled natural frequencies under constraint oscillations of the superstructure and pile rocking, respectively, while ˆ pssi Km reflects the pseudonatural frequency related to the effective stiffness ( ˆK ) of the reduced SSI model. ˆ, and represent the corresponding damping ratios. Note that neglecting higher-order damping terms, the above equations duly reduce to the solution of Jennings and Bielak [] and Stewart and Fenves [4]. Since only rocking is allowed at the pile-head of the modified SSI system, the real part K θ of pile rocking impedance and the associated damping ratio ζ θ introduced in Eqs. (6)- (7) are given by [] 8 p p x p x 34 K 4E I k m c cos( ) 4 4 (8) tan( ) (9) 4 where E p I p stands for the flexural stiffness of the pile and c x Arc tan kx m p () In the above expression (k x +iωc x ) represent the dynamic impedance of the Winkler medium [7, ] and ω is the cyclic excitation frequency. 3. Comparison with FE results A comparison of the analytical solution (Eq. (6)) to the pseudo-natural frequency computed numerically for the examined coupled systems fpssi/ffixed.4..8.6.4. FE solution Analytical Solution (Eq. 6) S H =.78 S H =4.3 S H =8.46.5.3.45 /σ Figure 3. Ratios of the pseudo-natural frequency of the coupled system to the fixed-base natural frequency of the structure against parameter (/σ) for three values of S H is shown in Fig. 3 in terms of f pssi /f fixed ratios plotted against the wave parameter (/σ). Each curve in Fig, 3 corresponds to a different value of pile parameter S H. It is evident that the analytical and the FE solutions provide nearly identical results, establishing further the rocking-related nature of the pseudo-natural frequency. 4 EFFECT OF SALIENT SSI FACTORS ON f / f RATIOS AND DAMPING pssi SSI The role of salient SSI parameters is investigated in connection to the foundation vibration modes. For the purpose of this investigation, f pssi /f SSI ratios of the coupled system were evaluated as affected by wave parameter (/σ), pile flexibility parameter (S H ) and inertia parameter (γ). Fig.4a shows f pssi /f SSI ratios plotted against (/σ) for different values of (γ). In this graph, S H parameter was set at.78 representing a stiff and/or short end bearing pile. Ratios of the effective natural frequency to the fixed-base natural frequency of the structure (f SSI /f fixed ) are also shown (Fig.4b) as a measure of SSI effects. Corresponding results for a flexible foundation (S H =8.46) are given in Figs. 5a and 5b, respectively. It is observed that the deviation between f SSI and f pssi becomes notable for high values of (/σ) and (γ) and low values of S H, corresponding to stiff massive structures founded on stiff and/or short end-bearing piles (Fig.4a). This should be

fpssi/fssi fssi/ffixed.4..8.6.4. a S H =.78.5.3.45 /σ. b.8 Series γ =.5.6 Series. Series3...4 Series4.3. Series5.6 Series6 S H =.78.5.3.45 /σ Figure 4. (a) f pssi /f SSI ratios against mass parameter (γ) (b) f SSI / f fixed ratios against mass parameter (γ). In all plots S H =.78 attributed to a strong contribution of translational pile motion to the total deflection of the system. Indeed, low S H values denote substantially lower translational stiffness than corresponding rocking stiffness of the pile or, conversely, pronounced translational compliance with respect to the rocking one. In this case, knowledge of pseudonatural frequency (Eq.6) may be of importance in design when the predominant frequency of excitation is close to f pssi. Naturally, modest SSI effects are anticipated with decreasing values of (/σ) and/or (γ), leading to f SSI /f fixed ratios close to unity (Fig.4b). On the contrary, for large values of pile parameter S H (flexible and/or long piles), strong SSI effects are mobilized due to foundation rocking that dominates the response, leading to f SSI /f fixed ratios lower than (Fig. 5b). In this case, f pssi /f SSI ratios are close to unity regardless of both (/σ) and (γ) (Fig. 5a), which implies that pseudo-natural and effective natural frequencies practically coincide. This suggests that the actual flexible-base vibrational characteristics of a SSI fpssi/fssi fssi/ffixed.4..8.6.4. a S H =8.46.5.3.45 /σ. b.8 Series γ =.5.6 Series..4 Series3.. Series4.3. Series5.6 Series6 S H =8.46.5.3.45 /σ Figure 5. (a) f pssi /f SSI ratios against mass parameter (γ) (b) f SSI / f fixed ratios against mass parameter (γ). In all plots S H =.78 system with prevailing rocking motion (e.g., a tall structure on a flexible foundation) may be sufficiently predicted by standard identification procedures even when free field recordings are not available [4]. Effective damping of the SSI system was also examined, as function of relative inertia between soil and structure. For this reason, the damping ratio in Eq.7, of the reduced SSI model, is compared to the effective damping of the complete system [], which takes into account both translational and rotational pile motion. The corresponding damping ratios are plotted in Fig. 6a and 6b, respectively, against excitation frequency f, for the selected values of (γ). The results refer to S H =.78 and (/σ)=.4 where deviation between the vibrational characteristics of the two systems is substantial. When the complete system is examined allowing mobilization of both translational and rotational pile modes, higher radiation damping associated with lateral deflection of pile is produced leading to significant effective SSI

ζ ˆζ.4...8.6.4. 4 6 8 f (Hz).4. γ Series =.5 Series.3 Series3. Series4.6. Series5. Series6.8.6.4. b 4 6 8 f (Hz) Figure 6. Effective damping of the complete (a) and the reduced (b) SSI system against excitation frequency. Results correspond to S H =.78, /σ=.4. damping. On the other hand, when translational pile-head motion is restrained radiation damping is reduced resulting in lower overall damping (Fig. 6b). The above behaviour becomes more pronounced with increasing superstructure mass due to stronger SSI effects. Thereby, the response of the coupled soil-pile-structure system at the pseudo-natural frequency is accompanied by lower levels of damping that should be considered in the context of an analysis or design procedure. 5 CONCLUSIONS A pseudo-natural SSI frequency (f pssi ) where pile-head motion is minimized with respect to free field and structural motion is analytically determined (Eq.6). Fundamental frequencies and effective damping of the coupled SPSI system were explored, as affected by salient SSI factors. For stiff massive structures founded on stiff and/or short end-bearing piles, f pssi may deviate a substantially from the conventional effective SSI frequency (f SSI ) providing a possible design frequency associated with lower levels of pile radiation damping. For coupled systems with flexible and/or long piles, f pssi and f SSI practically coincide, allowing estimation of the actual flexiblebase parameters even in the absence of free-field recordings. REFERENCES [] A,S. Veletsos and J.W. Meek, Dynamic behavior of building foundation systems, Earthquake Engineering and Structural Dynamics 3 (974), -38. [] A. Maravas, G. Mylonakis and D. Karabalis, Dynamic characteristics of simple structures on piles and footings. In: Proceedings of the 4th International Conference on Earthquake Geotechnical Engineering, Thessaloniki, Greece, 7, paper No. 67. [3] G. Gazetas, Soil dynamics and earthquake engineering Case studies, Simeon Publications, Athens (In Greek), 996. [4] J.P. Stewart and G.L. Fenves, System identification for evaluating soil-structure interaction effects in buildings from strong motion recordings, Earthquake Engineering and Structural Dynamics, 7 (998), 869-885. [5] E.N Rovithis, K.D. Pitilakis and G.E. Mylonakis, Seismic analysis of coupled soil-pile-structure systems leading to the definition of a pseudo-natural SSI frequency, Soil Dynamics and Earthquake Engineering, 6 (9), 5-5. [6] J.P. Wolf, Seismic Soil-Structure Interaction, Prentice Hall, 985 [7] R. Dobry, E. Vicente, M.J. O Rourke and J.M. Roesset, Horizontal Stiffness and Damping of Single Piles, Journal of the Geotechnical Engineering Division ASCE, 8 (98), 439-459. [8] J.P. Stewart, G.L. Fenves and R.B. Seed, Seismic soilstructure interaction in buildings. I: Analytical methods, Journal of Geotechnical and Geoenviromental Engineering, 5 (999), 6-37. [9] T. Ohta, S. Uchiyama, M. Niwa and K. Ueno, Earthquake response characteristics of structure with pile foundation on soft subsoil layer and its simulation analysis. Proceedings of the 7th World Conference on Earthquake Engineering. Istanbul, Turkey, 98, Vol.3 [] G. Mylonakis, A. Nikolaou and G. Gazetas, Soil-Pile- Bridge seismic interaction: Kinematic and Inertial effects. Part I: Soft Soil, Earthquake Engineering and Structural Dynamics, 6 (997), 337-359. [] P.C. Jennings and J. Bielak,. Dynamics of building-soil interaction. Bulleting of the Seismological Society of America, 63 (973), 9-48. [] G. Gazetas and R. Dobry, Horizontal response of piles in layered soils, Journal of Geotechnical Engineering, (984), -4.