Abstract
Simplified procedures, capable of predicting the main effects of soilstructure interaction (SSI), involving minimum computational effort are desired in engineering practice. Besides, userfriendly approaches can be incorporated into guidelines or practical recommended procedures. Nevertheless, approximate solutions need to be carefully validated through rigorous analytical or numerical methods, as well as by the comparison with experimental results or field measurements. In the case of piles, the flexibility of the soilfoundation system plays a fundamental role in structural design and the evaluation of the pile group effects still represents a crucial task in practical problem. In this paper, a Boundary Element approach based on the Stiffness Matrix Method (Kausel in Fundamental solutions in elastodynamics. A compendium, Cambridge University Press, England, 2006) is employed to investigate the key features of SSI, by comparison with the actual field data collected by Stewart et al. (Empirical evaluation of inertial soilstructure interaction effects. Rep. No. PEER98/07, Pacific Earthquake Engineering Research Center, 1998) from instrumented sites and structures.
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1 Introduction
Although various phenomena are involved into a soilstructure interaction (SSI) problem (Kausel 2010), the principal aspects can be summarized in the following three issues (Gazetas 1987):

(1)
the flexibility of the soilfoundation system determines the decreasing of the fundamental frequency of vibration of the structure with respect to the fixedbase condition;

(2)
the vibrational energy of the structure is dissipated into the supporting soil by radiation of waves to infinity and by the hysteretic behavior of the soil;

(3)
the incident seismic waves are filtered by rigid deep foundations and the shaking motion transmitted to the structure is generally modified: the horizontal displacement is reduced with respect to the freefield ground motion; rocking and torsional components may also arise.
From a conceptual point of view, the first two aspects are referred to ‘inertial’ interaction, whereas the last one is indicated as ‘kinematic’. The distinction in two separate types of interaction is very attractive and computationally efficient within the framework of the substructure method (Mylonakis et al. 2006).
Furthermore, owing to the passage of seismic waves into the surrounding soil, piles may suffer important deformations and bending (‘kinematic’) moments in the vicinity of interfaces of alternating soft and stiff soil layers (Gazetas and Mylonakis 1998).
As a matter of fact, among other structures and circumstances, Eurocode 8 (EN 1998–52005) states that SSI analyses shall be performed for structures founded on piles.
In general, it is recognized that SSI effects are most important at short periods of vibration (Stewart et al. 2004), where they tend to reduce the shaking demand on structures, decreasing the design values of the base shear, lateral forces and overturning moments (NEHRP 2015).
Nevertheless, well documented case histories (Mylonakis and Gazetas 2000) show important spectral amplification at somewhat high structural period as a result of resonance of the soft deep soil deposits involved, as well as shaking ground motion directivity.
Detailed numerical models (Chau et al. 2009; Hokmabadi et al. 2014) based on the Finite Element or the Distinct Element Methods permit to analyze in a single step the soilstructure system, taking into account material non linearities, heterogeneous soil profiles, irregular and deformable foundations. Nonetheless, a complete solution is hard to achieve in many practical cases and the results obtained often need to be evaluated with appropriate engineering judgment. Actually, a complete solution is impossible to achieve even with the most advanced and detailed numerical models as the uncertainties for the problem parameters, especially those relevant to the soil properties, even if test results are available, are always present.
The essential features of SSI can be fortunately captured by alternative approaches, such as the Boundary Element Method (Millan and Dominguez 2009; Padron et al. 2011), though working under the superposition law.
In engineering practice, simplified procedures (Dobry 2014) capable of predicting the main results of SSI analyses are very suitable for preliminary studies as these request minimum computational effort. Furthermore, easy to use approaches can be incorporated into guidelines or codes (NEHRP 2020).
Besides the numerous numerical and theoretical procedures available, in recent years experimental methods and laboratory studies broadened (Chau et al. 2009; Hokmabadi et al. 2014, 2015; Durante et al. 2016; Hussien et al. 2016). However, field measurements and recorded data from instrumented structures on piles are rare (Stewart et al. 1999b; Nikolaou et al. 2001; Kim and Stewart 2003).
A welldocumented collection of seismic case history data has been reported by Stewart et al. (1998). On 58 instrumented buildings sited in California, 77 strong motion events occurred from 1975 to 1994 were recorded. The structures were monitored with lateral sensors located at the foundation and roof levels. In 13 of these, vertical sensors at the footings were also arranged to detect base rocking. Freefield accelerographs were also placed in 45 of these sites. The base of the buildings consisted of both shallow and deep foundations. Fourteen of these structures were founded on piles.
The area included in the study was located besides the coast of California, starting from the Humboldt County, on the far North Coast in the north of San Francisco, reaching to Imperial Valley in Southern California.
The physical characteristics of the soils are reported for all the sites in examination (Stewart and Stewart 1997). For 29 soil deposits, the shear wave velocity V_{s} was measured in situ by geophysical tests. In the other cases, V_{s} profiles were calculated using empirical correlations.
The strong motion recordings were used to estimate the main modal structural parameters (frequency of vibration and damping ratio), using appropriate System Identification Analyses (SIA). The principal procedure adopted was based on the derivation of transfer functions, defined in the discrete time domain, between the recorded input motion and the recorded structural response. As known, transfer functions depend on the dynamic properties of the structures, considered with both fixed and flexible base. A second approach made use of transmissibility functions obtained through the ratio of the Fourier transform of the output and input recorded signals, that permit to identify the vibration frequencies at which amplification or deamplification occurs. The basic rules of the two methods are described in details in Stewart et al. (1998).
The actual structure was represented by a generalized singledegreeoffreedom system (SDOF). The evaluation of the natural vibration frequency and the damping ratio, for various cases of base fixity, allowed to characterize the effects of SSI. Stewart et al. (1999a) used the results obtained via SIA to calibrate simplified analytical procedures available in the literature.
Generally, these approaches are not adequate for incorporating pile group effects in both foundation stiffness and damping. In this case, the development of design methods is still hard to be achieved (Maravas et al. 2014).
In the present work, a linear but theoretically robust boundary element approach is employed to investigate the key features of SSI. The case studies collected by Stewart et al. (1998) from instrumented sites and structures are used for comparisons. Although under steadystate conditions, the analyses performed reveal capable to identify the cases in which the SSI effects are important.
2 Outline of the method used
The approach herein addressed is based on the Stiffness Matrix Method originally developed by Kausel and Röesset (1981) to simulate the dynamic response of layered soils. The framework of this approach was previously derived by Cairo et al. (2005), and Cairo and Conte (2006) for the analysis of vertically loaded pile groups under dynamic and static conditions. Recently, Cairo et al. (2020) expanded on the method to account different modes of loading, including seismic waves propagation.
The problem considered consists of a group of vertical piles embedded in a layered soil (Fig. 1). Each layer is modelled as an elastic material of Young’s modulus E_{s}, Poisson’s ratio ν_{s}, mass density ρ_{s}, and damping ratio β_{s}. Each pile is considered to be an elastic cylinder of length L, diameter d, Young’s modulus E_{p}, and mass density ρ_{p}.
The soilpile system is assumed to be under steadystate harmonic oscillations, therefore any timedependent variable is in general expressed as a complex quantity multiplied by the factor \(e^{i\omega t}\), being \(i = \sqrt {  1}\), ω the vibration circular frequency, and t the time. In the following, this factor will be omitted for the sake of brevity, since it is shared by all timedependent variables.
The solution procedure for the analysis of pile groups subjected to an external loading system V_{0}, H_{0}, M_{0} is described first.
The problem can be split into two parts (Fig. 2): the piles subjected to external loads {P} and soilpile interaction forces {P_{p}}; the soil deposit that is subjected to the interaction forces {P_{s}} at the pilesoil interfaces. The piles are subdivided into a finite number of onedimensional elements, and the soil is replaced by a horizontally layered continuum. The nodes are the middle points of the elements and the central end points.
These two systems are first considered separately and then reassembled enforcing equilibrium of the internal pilesoil forces and compatibility of the pile and soil displacements at the pilesoil interfaces. It is worth noting that the difference in the number of degrees of freedom between the piles and the soil has to be accounted for, as pointed out in the following.
Under the above assumptions, the dynamic equilibrium equation for the group is given in matrix form as
where [K_{p}] and [M_{p}] are the assembled stiffness and mass matrices of the pile group, respectively; [K_{s}] is the global stiffness matrix of the layered soil deposit; {u_{p}} is the vector containing the displacements (vertical and horizontal) and rotations of pile nodes; {P} the external pile forces.
Matrices [K_{p}] and [M_{p}] are determined using standard procedures of the structural analysis (Bhatti 2005). The salient feature of the present approach is the evaluation of [K_{s}], which employs the Stiffness Matrix Method (Kausel 2006) as described in the following.
The soil domain is formed by a series of homogeneous and isotropic layers underlain by a homogeneous halfspace. For each frequency ω and wavenumber k, the stiffness matrices of the layers and that of the halfspace are determined and assembled in the stiffness matrix [K] of the complete system. The terms of these matrices are complex and depend on the soil elastic properties and layer thickness. Damping is considered through complex elastic moduli.
The external forces are assumed to be applied in the form of tractions at the layer interfaces and need to be transformed from the space–time domain into the frequencywavenumber domain. Adopting a cylindrical coordinates system, the soilpile interaction forces are considered as ring loads applied to each element of the interfaces and disk loads uniformly distributed on the pile tips (Kausel 1981). The Hankel transforms of these forces in the wavenumber domain are available in closed form (Wolf 1985). Only the horizontal and vertical components of the loading are considered in this work. The rocking component is neglected, since it weakens rapidly with distance and does not contribute to pilesoil interaction. This is consistent with previous studies (Dobry and Gazetas 1988).
The transformed forces {p} with unit amplitude, are applied one at a time on the pile nodes, and the corresponding radial, tangential and vertical displacements \(\left\{ {\overline{u}} \right\}\) are calculated by solving a standard system of equations as
For each harmonic component, the achieved transformed displacements (influence coefficients) at a given depth z are discrete functions of k. The actual displacements are then calculated as a function of the radial distance r from the pile axis and the azimuth θ, using the inverse Hankel transforms.
In the case of vertical force, the following equations have to be used (Kausel 1981; Wolf 1985)
where u_{v}(r) and w_{v}(r) are the radial and vertical displacements, respectively; \(\overline{u}_{v} \left( k \right)\) and \(\overline{w}_{v} \left( k \right)\) are the radial and vertical transformed displacements, respectively, obtained using Eq. (2); J_{0}(kr) and J_{1}(kr) are the Bessel functions of the first kind and order 0 and 1, respectively.
For the horizontal interaction force, the corresponding displacements are given as (Kausel 1981; Wolf 1985)
in which u_{h}(r, θ), v_{h}(r, θ) and w_{h}(r, θ) are the actual radial, tangential and vertical displacements, respectively; \(\overline{u}_{h} \left( k \right)\), \(\overline{v}_{h} \left( k \right)\) and \(\overline{w}_{h} \left( k \right)\) are the corresponding transformed displacements in the wavenumber domain; J_{0}(kr), J_{1}(kr) and J_{2}(kr) are the Bessel functions of the first kind and order 0, 1 and 2, respectively. The horizontal displacements in Cartesian coordinates can be easily computed.
The soil influence matrix of the entire pile group is thus constructed and then inverted to obtain the matrix [K_{s}] of Eq. (1).
Equations (3) and (4) need to be solved by numerical integration, provided that an appropriate interval of discretization Δk and an upper limit of integration k_{max} are adopted. As a matter of fact, the plots of all the transformed displacements \(\overline{u}\left( k \right)\), \(\overline{v}\left( k \right)\), \(\overline{w}\left( k \right)\) exhibit pronounced peaks in the low range of wavenumber \(\left( {k \le k_{lim} } \right)\), and Δk has to be small enough. In the high wavenumber range, these influence functions become sharper and attain rapidly zero, so that larger intervals of Δk may be selected.
According to Kausel (2018), the following limits can be chosen for the entire soil deposit
being \(V_{{s_{{{\text{min}}}} }}\) the minimum shear wave velocity and h_{min} the smallest thickness of the layers.
The pile group response is obtained by Eq. (1) adding the compatibility of displacements and rotations, and the equilibrium of interaction forces and moments between the pile heads and the cap. In this work, the case of a rigid fixedhead freestanding pile group is considered. As known, the foundation impedance matrix can be constructed by successively applying unit vertical and horizontal displacements and rotations to the pile cap.
The kinematic interaction analysis can be carried out in the same framework, considering that the total wave motion can be decoupled into two parts (Banerjee and Butterfield 1981): the freefield wave motion occurring in the soil in the absence of the piles, and the scattered wave motion, consisting of the waves diffracted from the surface of the piles and propagating towards infinity. Equation (1) thus becomes (Cairo et al. 2020)
being {u_{ff}} and {P_{ff}} the vectors containing the displacements and forces in the freefield soil system, respectively.
Prior to solving the complete dynamic equilibrium of the pile group, {u_{ff}} and {P_{ff}} need to be evaluated by a site response analysis performed within the same approach (Kausel 2006). The kinematic response of the group is thus achieved in terms of the translational u_{k0} and rotational θ_{k0} components of the socalled foundation input motion (FIM).
In the final step, the inertial interaction is determined applying the FIM at the base of the structure in which the soilfoundation system is modelled with springs and dashpots (impedance matrix) associated to each mode of vibration.
The structures considered in the next section are simulated by a SDOF system (Fig. 3) with equivalent mass m placed at a distance h (effective height) from the base. The stiffness and damping of the SDOF are associated with the fundamental mode of vibration of the structure built in at its base. The dynamic stiffness of the pile foundation is represented by the horizontal impedance S_{hh}, the rocking impedance S_{rr} and the coupled components S_{hr} = S_{rh}. The foundation has two degrees of freedom consisting of the horizontal displacement u_{0}, and the rocking θ_{0}. The elastic horizontal displacement of the top mass relative to the base mass has amplitude u. The total displacement of the structure is deduced as \(u_{s} = u + u_{0} + h\theta_{0}\).
Ignoring the mass of the pile cap and the mass moment of inertia of the structure and the foundation, the equations of motion of the system can be formulated as (Cairo et al. 2016)
where ω_{fb} is the natural circular frequency of the fixedbase structure, and ζ the damping ratio.
3 Case studies
All the cases examined in the following paragraphs are summarized in Table 1, in which the fundamental parameters used are listed. Further indications are given in the text.
3.1 Case study 1: Imperial County Service Building, El Centro
The first case examined concerns the Imperial County Service Building in El Centro. It was severely damaged during the 1979 Imperial Valley Earthquake (Arnold et al. 1982), subsequently demolished and replaced in 1981. The study refers to the original six story reinforced concrete structure (Shepherd and Plunkett 1983; Kojic et al. 1993), 25.48 m in height, 41.7 m by 26 m in plan (Fig. 4a), founded on 216 Raymond step tapered concrete piles interconnected by tie beams. The piles extended 14 m to 18 m into the soil deposit, which consisted of 120 m deep interbedded silty clays and sands. The shear wave velocity profile based on downhole measurements is shown in Fig. 4b.
Recorded time histories refer to the 1979 Imperial Valley earthquake, characterized by magnitude M_{w} = 6.5 and freefield peak horizontal acceleration 0.24 g. Stewart and Stewart (1997) estimated the fundamental frequency of the soil deposit in the range 0.7–0.8 Hz.
In the present study, the measured shear wave velocity is reduced appropriately in the first 20 m of depth, to take into account the strain level induced by the earthquake motion into the subsoil. This is consistent with the suggestions provided by EC8 and favorably applied by Cairo et al. (2009). In the absence of specific data, the reduction factor of V_{s} is selected as a function of the peak ground acceleration recorded (Table 2).
For the surface softer layer of thickness 4.8 m a reduction factor of 0.51 is adopted, whereas a factor of 0.81 is used for the underlying layer. Similarly, a soil damping ratio β_{s} of 7.6% is assumed.
According to Stewart et al. (1999b), the effective height h of the generalized SDOF can be calculated roughly as 0.7 times the total height of the structure. The equivalent mass m can be obtained from the mass parameter \(\gamma = m/\left( {\rho_{s} A_{b} h} \right) \approx 0.15\) (Stewart et al. 1999a), being A_{b} the area of the actual foundation. All the specific values assumed are listed in Table 1.
Firstly, the kinematic response of the pile group is described in terms of the kinematic factors
being u_{ff} the horizontal freefield motion at the ground surface, and B the width of the base. All the displacements and rotations are complex numbers, so that their amplitudes (absolute values) have to be used in the definition of the kinematic response factors. The trend of I_{u} and I_{θ} as a function of frequency is reproduced in Fig. 5.
At very low frequency, the pile cap follows closely the freefield soil motion (Fig. 5a). With increasing frequency, the horizontal kinematic factor decreases as a consequence of the incompatibility between the wave motion into the soil and the displacement of the rigid pile foundation. At a relatively high frequency, I_{u} becomes to fluctuate. The present result is in agreement with the extensive numerical study provided by Fan et al. (1991).
As known, the seismic interaction among the piles is generally negligible in the frequency range of interest (Gazetas et al. 1992). Therefore, the most important parameters affecting the kinematic response of piles are the relative rigidity E_{p}/E_{s}, heterogeneity of the soil, pilehead fixity conditions. In the case examined, the ratio E_{p}/E_{s1} = 973 is calculated with respect to the surface layer of the deposit and it is strong enough to reduce the horizontal kinematic factor in the intermediatehigh frequency range. Furthermore, the rocking component of the FIM is scarcely significant (Fig. 5b). This can be ascribed to the relative high stiffness contrast (V_{s2}/V_{s1} = 2.3) between the upper and the lower layers around the deep foundation (Cairo and Conte 2014).
In Fig. 6, the harmonic steadystate response (amplification function) of the structural system to the applied foundation input motion is shown as a function of frequency f.
In particular, the amplitude of the total displacement u_{s} of the structure, normalized with the freefield ground motion u_{ff}, is depicted with a red line and compared to the fixedbase displacement u_{fb} (black line). As can be noted, the peak response of the coupled soilstructure system occurs at a smaller resonant frequency (f_{sy} = 1.59 Hz), as it corresponds to a more flexible system. The value provided by Stewart et al. (1999b) is 1.35 Hz. This discrepancy is probably due to a more pronounced nonlinear behavior of the soil. Besides, the calculated natural frequency f_{ff} = 0.82 Hz of the soil deposit is indicated with an asterisk, and it proves to be slightly greater than the range estimated by Stewart and Stewart (1997).
The damping ratio ζ_{sy} of the overall system can be determined from the same amplification function by the halfpower bandwidth method (Seybert 1981). A value of 17.5% is evaluated, which is reduced with respect to the fixedbase damping ζ, and compares favorably with the effective damping (16%) observed by Stewart et al. (1999b).
The normalized elastic displacement u of the structure is also indicated with a blue line. It should be noted that this important response quantity is somewhat smaller than the total structural displacement, and that a significant contribution to the peak response of the structure is likely due to the normalized rocking component hθ_{0} (green line) exhibited by its base. The same conclusion was deduced with the System Identification Analysis processed by Stewart and Stewart (1997), which explained the remarkable peak acceleration (0.56 g) experienced by the structure at the roof level with respect to the ground floor.
Finally, the socalled pseudonatural frequency f_{ps} (Rovithis et al. 2009) of the system is shown (Fig. 6). It corresponds to the frequency in which the ratio of the amplitude u_{s} of the motion of the structure is maximized with respect to the amplitude of the total base motion u_{0t} = u_{0} + u_{k0}. This curve is plotted with a golden line. As can be observed, at this frequency the total base motion u_{0t} is further minimized relative to the freefield u_{ff} (purple line).
A last remark can be drawn applying the freefield ground motion directly at the flexiblebase of the structure instead of FIM (i.e. neglecting kinematic interaction). The comparison reported in Fig. 7 demonstrates that the structural response becomes slightly higher, although the amplification curve broadens and the effective damping increases (18.8%).
3.2 Case study 2: 10story Residential Building, San Jose
The second case study refers to a tenstory reinforced concrete residential building (Fig. 8), sited in San Jose, subjected to the 1989 Loma Prieta earthquake (M_{w} = 6.9). The peak horizontal accelerations recorded at the base and roof of the structure was 0.12 g and 0.36 g, respectively. No freefield accelerographs were available. The structure was rectangular in plan, founded on reinforced concrete piles, 13.2 m long and 300 mm square section. The generalized soil profile of the site is reported in Fig. 8c. The shear wave velocities were evaluated on the basis of empirical correlations. The bedrock was estimated at 450 m. Stewart and Stewart (1997) suggested to adopt a reduction factor of 0.91 to the shear wave velocity within the first 30 m of depth and a soil damping β_{s} = 4.1% to account for the amplitude accelerations induced by the earthquake.
The amplification functions reported in Fig. 9 show important SSI effects on the dynamic response of the structure (red curve), as well as the system identification analysis performed by Stewart et al. (1999b) revealed. The natural frequency f_{sy} of the complete system is equal to 2.09 Hz and the effective damping ζ_{sy} = 8.4%. These values are in good agreement with the observed results (Stewart et al. 1999b): f_{sy} = 2.08 Hz, ζ_{sy} = 6.7%. The variation of ratio hθ_{0}/u_{ff} (green curve) demonstrates the important contribution of base rocking on the total response of the structure and explains the great amplification of the seismic motion at the roof of the building (Stewart and Stewart 1997). The analyses carried out reveal a negligible kinematic interaction, being E_{p}/E_{s1} = 106.
3.3 Case study 3: Ventura 12story Hotel, Ventura
The Ventura hotel is a 12story building, with effective height h = 20.7 m, fixedbase frequency f_{fb} = 1.88 Hz, damping ratio ζ = 5.7%. The foundation (Fig. 10a) consists of reinforced concrete Raymond tapered piles, 20 m long and with diameter d = 300 mm. Beneath the lateral walls, the piles are installed in groups of 4 to 6 at a 3d spacing; in the interior part, they are installed in twopile groups. The heads of all the piles are cast into rigid caps, interconnected between columns by tie beams and a slabongrade floor. The soil profile is reported in Fig. 10b. The shear wave velocities were estimated based on correlations with void ratio and in situ downhole measurements in similar soils near the hotel. Strong motion recordings refer to the 1994 Northridge earthquake with M_{w} = 6.7 and a_{max} = 0.054 g in the freefield site condition.
The calculated harmonic response of this structure (Fig. 10c) reveals a small interaction with the supporting soil (f_{sy} = 1.76 Hz, ζ_{sy} = 6%). The analyses conducted by Stewart et al. (1999b) demonstrated major inertial interaction effects (f_{sy} = 1.41 Hz, ζ_{sy} = 4%) and a significant contribution of base rocking to the response of the structure, which exhibited a peak horizontal acceleration of 0.25 g at the roof level. This discrepancy is likely due to the approximate evaluation of V_{s}, which does not correspond to the actual stiffness of the soils where the foundation of the Ventura Hotel is embedded.
3.4 Case study 4: Vanir Tower, San Bernardino
The Vanir Tower is a 9story structure, 32 m high, sited in the San Bernardino Valley. Each column in the building (Fig. 11a) is founded on a group of four piles connected with a rigid cap. The pile caps are not tied together except for a tie beam around the building perimeter. The piles have length 10 m and diameter 600 mm. Surficial young alluvial soils consist of 6 m of silts and clays, underlain by sands and gavels. At 45 m of depth, older alluvium deposits extend to a depth of 270 m. The shear wave velocities shown in Fig. 11b were determined using correlations with void ratio and SPT blow count.
The structure was subjected to the 1992 Landers earthquake (M_{w} = 7.3). The recorded maximum freefield horizontal acceleration was 0.10 g. According to this amplitude acceleration, soil damping β_{s} is set equal to 3.4% and a 0.925 reduction factor for V_{s} is used (Stewart and Stewart 1997).
The amplification functions displayed in Fig. 11c do not show any SSI effect as confirmed by the empirical findings reported by Stewart et al. (1999b).
3.5 Case study 5: Wadsworth VA Hospital, Los Angeles
The Wadsworth Veteran Hospital, located in West Los Angeles, consists of 6 stories above the ground surface and a twolevel basement, a square base with side of 135 m. The foundation is constituted by 1296 reinforced concrete piles, with L = 15 m and d = 0.8 m, connected by a thick slab. Pile spacing varies from 3 to 4.5 diameter. The shear wave velocity (Fig. 12a) was established using DH tests in nearby sites having similar geological conditions.
The building was subjected to the 1994 Northridge earthquake. The maximum ground acceleration in the freefield was 0.25 g. Sensors located at the basement and roof levels recorded peak accelerations of 0.27 g and 0.36 g, respectively. To account for the strain level induced in the soil by the amplitude of the soil surface motion, the measured shear wave velocity is reduced by a factor of 0.80 within the first 20 m of depth, and soil damping β_{s} = 8% assumed.
The theoretical results obtained show important inertial SSI effects (Fig. 12b). The fundamental frequency of the whole system is considerably reduced (f_{sy} = 0.82 Hz) and effective damping increased (ζ_{sy} = 13.5%) with respect to the fixedbase condition. Base rocking (green line) does not affect the response of the structure, as confirmed by the SIA performed by Stewart and Stewart (1997). The horizontal displacement of the foundation (purple line) appears quite important.
The comparison with the observed effective parameters (f_{sy} = 1 Hz, ζ_{sy} = 9.3%) reported by Stewart et al. (1999b) indicates that SSI is somewhat overestimated. This may be due to the approximate evaluation of the soil stiffness profile.
3.6 Case study 6: Riverside County Office Building, Indio
The Riverside County Office Building is formed by four stories above ground level and a single level basement (Fig. 13a). Each column of the structure is founded on a 2 × 2 pile group (s/d = 3.25). The reinforced concrete tapered piles have length 15 m and diameter 0.4 m. A thin concrete slab is present across the basement.
The foundation is embedded into the level alluvial terrain of the Indio Hills. Soil conditions at greater depth and the presence of bedrock are not known. The shear wave velocity was estimated using correlations with void ratio. The properties of the soil used in the analysis are reported in Fig. 13b.
The accelerometric measurements available refer to the 1992 Landers earthquake. The maximum horizontal acceleration at the freefield was 0.086 g. A minor deamplification (0.076 g) of the motion at the base relative to the freefield was individuated, whereas a significant amplification (0.23 g) at the roof resulted.
The steadystate response of the structure shows a small SSI influence (Fig. 13c). The total damping ζ_{sy} of the soilstructure system becomes 8.7% and the fundamental frequency f_{sy} = 1.40 Hz, which compare well with the empirical results ζ_{sy} = 10.3% and f_{sy} = 1.41 Hz (Stewart et al., 1999b). No kinematic effects are also observed (E_{p}/E_{s1} = 115).
3.7 Case study 7: Pacific Park Plaza Tower, Emeryville
The Pacific Park Plaza Building is a thirtystory reinforced concrete structure, 95.1 m high, sited in Emeryville (San Francisco). The building is Yshaped in plan with three wings connected to the central core (Fig. 14a).
Strong motion instrumentation network was installed to study both the response and the dynamic characteristics of the system (Celebi and Safak 1992; Celebi 2001). The foundation consists of a concrete mat of thickness 1.5 m, supported by 828 prestressed concrete friction piles. The piles have length 30 m, 350 mm square crosssection, embedded in a soil deposit of silty sands and silty clays, which extend to a sedimentary bedrock at 150 m of depth. The site frequency f_{ff} was estimated to be 0.7 Hz.
The recorded data refers to the 1989 Loma Prieta earthquake. The maximum freefield horizontal acceleration was 0.26 g. At the ground floor and the roof of the structure the largest peak accelerations exhibited were 0.21 g and 0.37 g, respectively.
The V_{s} profile is represented in Fig. 14b. A 0.50 reduction factor for shear wave velocity is used for the softer layers, whereas a factor of 0.80 for the stiffer ones, within 20 m depth. Soil damping β_{s} = 8.4% is adopted. The mass density ρ_{s} of the soil varies from 1.73 Mg/m^{3} in the upper layers to 2.04 Mg/m^{3} at greater depth.
The amplification curves obtained are plotted in Fig. 14c. As can be noticed, a slight interaction between the structure and the surrounding soil is highlighted in terms of period lengthening and effective damping. The vibration frequency f_{sy} of the entire system is 0.39 Hz, effective damping ratio ζ_{sy} = 8.5%. The results of SIA were f_{sy} = 0.40 Hz and ζ_{sy} = 13% (Stewart et al. 1999b). The fundamental frequency f_{ff} of the soil is estimated at 0.6 Hz. The kinematic interaction effects turn out to be negligible (E_{p}/E_{s1} = 285).
3.8 Case study 8: Administration Building, Long Beach
The Administration Building in Long Beach is a 7story steel structure founded on 40 groups of 5 reinforced concrete piles located beneath each column (Fig. 15a). The piles, connected by a slab across the ground floor level, have L = 12 m and d = 0.4 m. The soil is composed of a silty sand deposit extending to 30 m of depth. The shear wave velocity (Fig. 15b) was measured by downhole tests.
The accelerometer recordings refer to the 1987 Whittier earthquake (M_{w} = 6), with a maximum freefield acceleration of 0.07 g. A reduction factor of 0.88 is used for V_{s}.
The steadystate response shown in Fig. 15c indicates a reduction of the resonant response, essentially due to the structural distortion, out of radiation damping (ζ_{sy} = 8.9%, f_{sy} = 0.87 Hz). Minor effects (ζ_{sy} = 5.5%, f_{sy} = 0.89 Hz) were observed by Stewart et al. (1999b).
3.9 Case study 9: Avenue of the Stars Building, Los Angeles
The Avenue of the Stars Building in Los Angeles has 19 stories and rectangular plan (Fig. 16a). The structure is supported by 22 m long steel Ibeam piles installed in groups beneath each column. Pile caps are connected by reinforced concrete tie beams. The soil deposit consists of alternating layers of silt and clay, sand, and clayey silt. The shear wave velocity profile (Fig. 16b) was established on the basis of downhole measurements. The 1994 Northridge seismic event was considered in the system identification analysis. The peak ground acceleration in the freefield was 0.28 g. A reduction factor of 0.77 is used for the shear wave velocity of the upper layer, and soil damping β_{s} = 9.2% is chosen. The damping ratio of the equivalent fixedbase structure is not known.
The amplification functions plotted in Fig. 16c demonstrate that the response of the coupled soilstructure system coincides with the response under fixedbase conditions, as pointed out by the empirical findings (Stewart et al. 1999b). This statement is independent of the structural damping assumed.
3.10 Case study 10: 47story Office Building, San Francisco
The seismic response of a 47story Office Building (Fig. 17a) sited in San Francisco, subjected to the 1989 Loma Prieta earthquake, is also analyzed. This steel structure is rectangular in plan (Fig. 17b) and has a reinforced concrete mat foundation supported by 150 steel pipe piles infilled with concrete. The piles are 60 m long and resting on a sandstone bedrock.
No freefield accelerographs were arranged. A maximum acceleration of 0.11 g was measured at the underground basement level, and a peak response of 0.47 g was recorded at the roof level. According to Stewart and Stewart (1997), the shear wave velocity of the soil (Fig. 17c) is reduced by a factor of 0.84 and the damping ratio β_{s} is set equal to 4.3%.
The steadystate analysis carried out (Fig. 17d) clearly shows that the fixedbase condition (black line) disregards the beneficial effect of radiation damping (red line). Moreover, it suggests that base rocking (green line) has a small influence on the structural response. Unfortunately, the study conducted by Stewart et al. (1999b) did not report any data about damping.
3.11 Case study 11: San Bernardino County Services Building, San Bernardino
The San Bernardino County Government Center is a fivestory steel building with an Lshaped plan (Fig. 18a). Groups of 5 piles are placed beneath the columns. The piles have a square section of side 375 mm and length 10 m. The pile caps are connected by grade beams. The surficial alluvial soil deposit consists of gravely sands and silts, and extends to a depth of about 45 m. Soil properties are reported in Fig. 18b. The shear wave velocity was deduced with empirical correlations.
The strong motion data refer to the 1994 Northridge earthquake. The maximum freefield acceleration recorded was 0.04 g.
The results obtained (Fig. 18c) show that considering the effects of SSI reduces considerably the resonant response u_{s} of the structure, entirely due to its elastic distortion u. The fundamental frequency of the system becomes 1.07 Hz, and damping 6.3%. The latter is overestimated with respect to the empirical finding 4.5% obtained by Stewart et al. (1999b), which found out minor inertial interaction effects between the structure and the supporting soil.
3.12 Case study 12: Hollywood Storage Building, Los Angeles
The Hollywood Storage Building is a reinforced concrete structure, 15 stories tall, rectangular in plan (Fig. 19a), founded on concrete piles with length varying from 4 m at the edge to 9 m near the center of the construction. The shear wave velocity profile (Fig. 19b) was evaluated based on downhole measurements. The depth of the bedrock was located at about 120 m.
The structure was subjected to the 1994 Northridge earthquake and a peak freefield acceleration of 0.39 g was measured. The fundamental frequency of the soil deposit was estimated to be around 1 Hz.
According to the amplitude of acceleration at the soil surface, the damping of the soil is assumed to be β_{s} = 10% and the shear wave velocity within the first 20 m of depth is reduced by a factor of 0.78.
The frequency response curves are plotted in Fig. 19c. The vibration frequency f_{sy} of the whole system is equal to 1.25 Hz, and the total damping ζ_{sy} calculated with the halfpower bandwidth method is 10.3%. The corresponding results obtained by Stewart et al. (1999b) with the system identification analysis were 1.25 Hz and 8%, respectively. The calculated fundamental frequency of the soil deposit is f_{ff} = 1.35 Hz. The response of the structure is quite dominated by its flexural displacement u, with a modest contribution of base rocking (green line). No kinematic interaction effects are noticed (E_{p}/E_{s1} = 85).
3.13 Case study 13: 2story Commercial Building, Pomona
A reinforced concrete building with 2 stories and a single level basement, sited in Los Angeles County (Pomona) is considered. Its foundation consists of 168 piles having d = 0.6 m and L = 10 m, installed in groups of 2 and 4 beneath the columns (Fig. 20a).
The soil consists of a sand deposit with occasional gravels and layers of silty fine sand. The shear wave velocity was estimated using correlations with void ratio. A first layer of thickness 6 m having V_{s} = 300 m/s and a 12 m thick lower layer with V_{s} = 450 m/s were characterized. Soil conditions at greater depth are not known.
The strong motion processed referred to the 1990 Upland earthquake (M_{w} = 5.6), which generated in the freefield site condition a peak ground acceleration of 0.21 g. The shear wave velocity of the soil is reduced by a factor of 0.84, and soil damping β_{s} = 6.4% is assumed.
The harmonic steadystate solutions reported in Fig. 20b highlight the inertial interaction effects (red line): the reduction of the fundamental frequency (f_{sy} = 3.42 Hz) and the increase of damping (ζ_{sy} = 6.2%). The observed values obtained by Stewart et al. (1999b) were 3.45 Hz and 9.2%, respectively. In both researches, no significant deamplification of foundation motion relative to the freefield is noticed (E_{p}/E_{s1} = 67). Besides, base motion (purple and green lines) does not affect the structural response.
3.14 Case study 14: 7story Building, Norwalk
The last case examined deals with a rectangular 7story steel building (Fig. 21a) located in Norwalk, subjected to the 1987 Whittier earthquake. It is founded on concrete piles varying in length between 8.6 and 11.6 m. Further data are available in Celebi (1993).
The soil deposit is composed of silty, clayey sands and silts. Sandstone bedrock is located at 150 m of depth. The V_{s} profile is plotted in Fig. 21b. An average reduction factor 0.82 for V_{s} and β_{s} = 7.2% have been assumed according to the recorded peak freefield acceleration of 0.23 g. The results obtained (Fig. 21c) show no SSI effects, as well as the empirical findings substantially confirmed (Stewart et al. 1999b).
3.15 Summary of results
All the results obtained are synthesized in Table 3 along with the observed flexiblebase modal parameters determined from sitespecific analyses.
11 case studies reveal the presence of inertial interaction effects in terms of reduction of the fundamental frequency of vibration f_{sy} and/or ‘effective’ damping ζ_{sy} of the coupled soilstructure system.
The theoretical estimates are reasonably good in many cases. A minor accuracy of the present approach with respect to the empirical findings is noticed in the cases 3 and 5, where shear wave velocity profiles are not given at the specific sites. The same conclusion could be expected for case study 10, but no observed data are available for the actual structure.
Case studies 6, 7 and 13 show an overall underestimation of the damping, probably due to the lack of information about soil conditions at greater depth. An overpredicted damping is calculated in cases 8 and 11, in which the presence of bedrock was not been established.
The modeling of pile foundations proves quite reliable and permits a good prediction of the fundamental frequency of the soilstructure system.
When peak ground acceleration in the freefield condition is available (Table 1), a reduction factor for V_{s} and soil damping β_{s} are adopted as a function of PGA, according to the provisions furnished by EC8 (Table 2). In the absence of freefield measurements (cases 2 and 10), the indications provided by Stewart and Stewart (1997) have been assumed. In so doing, nonlinear effects can be considered approximately captured, although in a very simple manner.
Finally, small kinematic interaction effects are found in case study 1, as a relatively high ratio of the pile modulus to the soil Young’s modulus is matched (Table 1).
4 Conclusions
In this paper, the main aspects of the soilstructure interaction analysis are considered and discussed with the aid of welldocumented case studies. In particular, the influence of pile foundations in the dynamic response of the structure is examined.
Aiming at preserving the clearness and simplicity requested in engineering practice, the study is restricted to the harmonic steadystate solution. The latter permits an easy representation of the peak response of the structure and a fast acquisition of the fundamental parameters featuring the behavior of the soilstructure system under seismic loading. Moreover, use is made of the substructure technique which reveals very attractive for the analysis of large pile groups.
The paper employs a boundary element approach based on the Stiffness Matrix Method, which allows soil heterogeneity to be accounted for in a realistic way. A basic assumption is the validity of the superposition principle, therefore linear soil behavior is mandatory. In order to introduce nonlinear effects, soil stiffness is adequately reduced according to the maximum accelerations recorded at the ground surface in the freefield, and a consistent internal damping is adopted.
In the case examined, the reduced fundamental frequency and the total damping ratio of the soilstructure system are compared with the values observed in the instrumented actual piledstructures. The agreement is found very satisfactory whenever soil stiffness profile was appropriately estimated and nonlinear behavior considered. Finally, the effects of the kinematic interaction prove negligible owing to the small relative pile rigidity and soil layering.
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Acknowledgements
This study was performed in the framework of the research project ReLUIS 20192021, granted by the Italian Civil Protection Department. The author is grateful to Prof. Jonathan P. Stewart who provided all the essential data for writing the present work.
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Cairo, R. A boundary element approach for the evaluation of SSI effects in presence of pile foundations under steadystate conditions. Bull Earthquake Eng 20, 3265–3294 (2022). https://doi.org/10.1007/s10518022013310
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DOI: https://doi.org/10.1007/s10518022013310