The Possibility of Detrimental Effects on Soil–Structure Interaction in Seismic Design Due to a Shift in System Frequency

Abstract

Soil–structure interaction (SSI) leads to a modification in the dynamic properties of structure, but due to the complexity of analysis, it is traditionally assumed in seismic designs that the structure is fixed-supported on the ground, which brings about potential risks to the seismic performances of structure. The study works on the possibility of SSI having detrimental effects by comparing the dynamic responses of the SSI system to a fixed-base structure, and presents charts for an evaluation of the system frequency of SSI for the purpose of engineering practice. In order to reveal the physical nature, the SSI model is reduced to its simplest form, consisting of a SDOF oscillator, a three-dimensional rectangular foundation, and a multi-layered half-space. The energy dissipation in the soil is achieved by foundation impedances and the substructure method. Previously, the foundation impedances are usually acquired by two-dimensional or axisymmetric three-dimensional models in uniform half-space to avoid the high cost of the more realistic, fully 3D models, while a high-precision indirect boundary element method is employed, combined with the non-singular Green’s functions of distributed loads to calculate the foundation impedances. Although SSI dampens the peak amplitude of structure response in the frequency domain, case studies on four buildings’ responses to 42 earthquakes in the time history show a possibility of 15–20% that SSI amplifies the dynamic responses of structures, such as the maximum and the mean values in the time history, depending on the properties of the structures and the site, as well as the frequency component of incident waves.

1. Introduction

It is assumed in coded seismic designs that structures are fixed-supported at their foundations. In reality, soil experiences obvious flexibility during dynamic excitations, which is responsible for the mechanism of soil–structure interaction (SSI). Since the soil usually acts as a cushion under the structure foundation to decrease the structure responses, SSI was first considered beneficial for the seismic performances of structures several decades ago. Nevertheless, scholars realized, especially after the 2000s, that SSI may sometimes amplify the structure responses compared with a fixed-base assumption, or have detrimental effects on the seismic performances of structures, and that ignoring this mechanism may result in potential risks to seismic designs [1].

In recent years, progress on seismological numerical simulations added more proof regarding this recognition. It was shown, using a probabilistic model based on damage and repair cost, that buildings on very soft soil are extremely likely to incur smaller losses due to SSI, but SSI has an adverse effect on the buildings on moderately soft soil, with a probability up to 0.4 [2]. It was concluded using a statistical study on SSI that the underlying soil may either increase or decrease the structural response depending on the particular site–structure–earthquake scenario [3]. An analysis using damage spectra on SSI reveals that SSI increases the damage index before a threshold period, and the conventional fixed-base model underestimates the damages of buildings with periods less than the threshold [4]. A study using fragility curves shows that neglecting SSI results in nonconservative predicted damage levels of structural and non-structural elements [5].

Although the analysis of SSI is necessary for structural seismic designs, due to its complexity and cost, it currently only applies to a few very important infrastructures, such as nuclear power plants [6], huge dams [7] or long bridges [8,9]. Actually, it was the development of the nuclear power plants in the 1970s that promoted the extensive research on SSI. The analysis of SSI applies to a very small number of civil structures used only for some special purposes; for example, the influence of adjacent topography on buildings [10], the influence of a poroelastic site on deep piled foundations [11,12], and so on. For most civil structures, however, little attention is paid to seismic designs involving SSI. The conventional fixed-base assumption is usually adopted for convenience, and it is only possible to very roughly determine the potentially detrimental effects of SSI by the shear wave velocity, or the period, of the site. Th method, although widely used among civil engineers, is not accurate and cannot be parametrically quantified. For example, many studies using numerical simulations [13,14] and earthquake damage investigations [1,15] have revealed that structures with some of the same features in a certain local area exhibit much more serious damage than other structures without those features.

Nevertheless, there is an ongoing effort to integrate SSI analysis into the framework of coded seismic designs in engineering practice. In ASCE/SCI-7 [16] and NEHRP (FEMA P-1050-1/2015) [17], there are coded provisions in recommended chapters in all versions after the 1990s (v1997, v2000, v2003, v2009, v2015) to involve SSI analysis, which is based on the substructure method and foundation impedance functions. In CECS 160 (general rules for the performance-based seismic design of buildings) [18], similar provisions are also recommended, and the formulas on the evaluation of structure modes are improved. A research report by the National Institute of Standards and Technology (NIST) provides modified coefficient curves, considering SSI based on the concept of traditional coefficient curves in seismic designs [19]. However, these methods are rarely adopted in practice for the following two reasons: The analysis involves the calculation of foundation impedance functions, which are too complex for engineering designs. In addition, the analysis considers only the beneficial effects of SSI and reduces base shear force by at most 30%, which is unsafe or lack of safety reverse.

The foundation of a structure is usually comprised of various shapes, and it forms a group of multi-scatters in infinite half-space, which creates a complex scattering field under seismic excitations. In the past few decades, intensive effort has been put into its numerical simulations, such as raising the incident wave to a more shallow layer to reduce the range of the near-field [20], setting incident waves inside the artificial boundary to reduce the data storage on free-field motion and seismic loads [21], or carrying out a large-scale calculation, from earthquake sources to structure sites, through multi-scale simulation [22], and so on. However, up to now, the numerical simulations on SSI are still very costly and not widely available among civil engineers [23].

This paper investigates the possibility of SSI having detrimental effects using case studies on four buildings whose structural data, site data and earthquake records are available. The scattering field of the half-space is modeled by foundation impedances, which are obtained by a high-precision indirect boundary element method that we proposed in our previous research [24]. Under the earthquake excitations, kinematic interaction and inertial interaction are coupled for SSI, so some studies focus on the former in a half-space with a foundation [25,26], or on the latter with a structure in a uniform half-space [27]. The SSI model, considering the two types of interactions, consists of an oscillator, a 3D rectangular foundation, and a multi-layered half-space. It is shown in the study that the shift in peak frequency between SSI system and fixed-base oscillator leads to the different seismic performances between them, so the paper also proposes a corresponding chart of peak frequency, or system frequency, between SSI and the fixed-base oscillator, to shed some light on SSI analysis in engineering seismic designs. The target parameters in the chart consist of the foundation embedment, soil-layer thickness, length-to-width ratio, and so on, to cater a variety of scenarios in coded seismic designs.

2. Methodology

2.1. Model

The model of the soil–structure interaction system (SSI) consists of a superstructure, a three-dimensional rectangular foundation, and a horizontally multi-layered half-space, as shown in Figure 1. The superstructure is simplified to a single-degree-of-freedom (SDOF) oscillator, and its dynamic properties are characterized by mass M_b, spring coefficient k_b, damping coefficient c_b and height H, so the fixed-base frequency is ${\omega }_{\mathrm{b}}=\sqrt{k_{\mathrm{b}}/M_{\mathrm{b}}}$ and the damping ratio is ${\xi }_{\mathrm{b}}=c_{\mathrm{b}}/2M_{\mathrm{b}}{\omega }_{\mathrm{b}}$. The foundation is represented by a rigid rectangular foundation of mass M₀, width a, length b and embedment c. The half-space consists of a soil layer of thickness D on a bedrock, and each sub-layer, as well as the bedrock, is assumed to be elastic and isotropic. The material properties of the jth sub-layer are characterized by shear-wave velocity β_j, Poisson’s ratio ν_j, mass density ρ_j and damping ratio ξ_j (j = 1, 2, 3, …), and those of the bedrock are characterized by shear-wave velocity β_R, Poisson’s ratio ν_R, mass density ρ_R and damping ratio ξ_R. The SSI system is excited by a train of harmonic SV waves coming from the bedrock, with motion in the XOZ-plane. The circular frequency of the excitation is ω, and the incident angle is θ, measured horizontally.

The deformation of the structure is represented by the relative horizontal translation u_b with respect to the z’-axis, which is fixed on the foundation, and the foundation motion is represented by the generalized vector $U={\{u,\phi \}}^{\mathrm{T}}{e}^{\mathrm{i}\omega t}$, with u of the horizontal translation and φ of the clockwise rotation about the reference point (0, 0, c), and e^iωt is the harmonic time factor, omitted hereafter. The vertical displacement, as well as the torsional motion of the SSI system, are not considered in our problem.

2.2. Dynamic Equilibrium

The dynamic equilibrium equations of the SSI are as follows [28,29,30]:

$$M\ddot{V}+C\dot{V}+KV=-M^{\prime }{\ddot{U}}_{*}$$

(1a)

where

$$M=\left[\begin{array}{ccc}{M}_{\mathrm{b}}& M_{\mathrm{b}}& M_{\mathrm{b}}\\ M_{\mathrm{b}}& M_0+M_{\mathrm{b}}& S_0+M_{\mathrm{b}}\\ M_{\mathrm{b}}& S_0+M_{\mathrm{b}}& I_0+I_{\mathrm{b}}\end{array}\right],C=\left[\begin{array}{ccc}{c}_{\mathrm{b}}& 0& 0\\ 0& c_{\mathrm{hh}}& c_{\mathrm{hm}}\\ 0& c_{\mathrm{mh}}& c_{\mathrm{mm}}\end{array}\right],K=\left[\begin{array}{ccc}{k}_{\mathrm{b}}& 0& 0\\ 0& k_{\mathrm{hh}}& k_{\mathrm{hm}}\\ 0& k_{\mathrm{mh}}& k_{\mathrm{mm}}\end{array}\right],M^{\prime }=\left[\begin{array}{cc}{M}_{\mathrm{b}}& M_{\mathrm{b}}\\ M_0+M_{\mathrm{b}}& S_0+M_{\mathrm{b}}\\ S_0+M_{\mathrm{b}}& I_0+M_{\mathrm{b}}\end{array}\right],V=\left\{\begin{array}{c}{u}_{\mathrm{b}}\\ u_0\\ {\phi }_0{H}^{\prime }/a\end{array}\right\}$$

(1b)

The symbol S₀ is the area moment of the foundation with respect to the axis z = c, I₀ and I_b are the inertia moments of the foundation and the superstructure, respectively, and we have I_b = M_bH’ for an oscillator here, for simplicity. The parameter H’ = H + c is the structure height from the structure top to the foundation bottom. Vector $U_{*}={\{u_{*},{\phi }_{*}H\}}^{\mathrm{T}}$ is the foundation input motion, corresponding to the dynamic response of the excavation where the foundation is placed, and $U_0={\{u_0,{\phi }_{0}H\}}^{\mathrm{T}}$ is the relative displacement of the foundation with respect to the foundation input motion [28,29,30], so we have $U=U_{*}+U_0$. The physical meaning of $U$, $U_{*}$ and $U_0$ in SSI is illustrated in Figure 2.

Foundation impedances are complex-valued and are shown by

$$K_0(\omega )=\left[\begin{array}{cc}{K}_{\mathrm{hh}}(\omega )& K_{\mathrm{hm}}(\omega )\\ K_{\mathrm{mh}}(\omega )& K_{\mathrm{mm}}(\omega )\end{array}\right]=\left[\begin{array}{cc}{k}_{\mathrm{hh}}(\omega )& k_{\mathrm{hm}}(\omega )\\ k_{\mathrm{mh}}(\omega )& k_{\mathrm{mm}}(\omega )\end{array}\right]+\mathrm{i}\omega \left[\begin{array}{cc}{c}_{\mathrm{hh}}(\omega )& c_{\mathrm{hm}}(\omega )\\ c_{\mathrm{mh}}(\omega )& c_{\mathrm{mm}}(\omega )\end{array}\right]$$

(2)

The real and the imaginary parts act as the spring and dashpot coefficients of the site, with the elements K_hh of horizontal impedance, K_hm = K_mh of coupled horizontal-rocking impedance and K_mm of rocking impedance. Foundation impedances are obtained using a high-precision indirect boundary element method (IBEM), combined with the non-singular Green’s functions of distributed loads presented in [24]. The foundation impedances in Equation (1) involve both translation and rotation, and the elements are already normalized by H’/a to be unified to the same units.

The boundary element method (BEM) needs element meshing only at boundaries, and it does not need artificial boundaries like the Finite Element Method (FEM) because it uses Green’s functions to automatically satisfy the radiation conditions at infinity. The calculation on Green’s functions is carried out in the frequency domain by Fourier transformation, and its logic is similar to the “displacement method” in the Structural Mechanics textbook. The Green’s functions correspond to the displacement or traction responses of some point in the half-space, subjected to some unit-distributed load, then the foundation impedances are obtained by a series of fictitious distributed loads acting on the foundation-soil boundary. The value of the fictitious loads, as well as the foundation impedances, are all found by the continuous conditions on the boundary.

The dimensionless parameters necessary for the following analysis are:

(1)

The dimensionless mass of the foundation and the structure are M₀/M_s and M_b/M₀, respectively, where M_s is the mass of soil replaced by the foundation.

(2)

The dimensionless frequency of incident wave is defined as $\eta ={\lambda }_1/2a=\omega a/\pi {\beta }_1$, where ${\lambda }_1$ is the shear wavelength of the first sublayer. Correspondingly, the dimensionless fixed-base frequency is ${\eta }_{\mathrm{b}}={\omega }_{\mathrm{b}}a/\pi {\beta }_1$, and a general range of η_b in engineering practice is 0–0.4 [28,29,30], and the dimensionless system frequency is ${\tilde{\eta }}_{\mathrm{b}}={\tilde{\omega }}_{\mathrm{b}}a/\pi {\beta }_1$.

(3)

The dimensionless size of the foundation and the dimensionless thickness of the soil layer are a/a, b/a, c/a and D/a, respectively.

(4)

The dimensionless mass density and the shear wave velocity of the soil layer are ρ_R/ρ_j and β_R/β_j, respectively.

3. SSI System versus Fixed-Base Oscillator

3.1. Dynamic Properties of SSI

The simplest layered half-space, a single soil layer on bedrock, is taken for the analysis of the dynamic properties of the SSI system. The following parameters of the site are used: ρ_R/ρ_j = 1, ν_R = ν_j = 1/3, ξ_R = ξ_j = 0.02, β_R/β_j= 2, D/c = 2, (j = 1, 2, 3, …). The foundation embedment is c/a = 0.5, and the foundation mass is M₀/M_s = 0.2. Meanwhile, for the structure, we have a damping ratio ξ_b = 0.02, a height H/a = 2, and mass M_b/M₀ = 10 following the recommended relationship M_b/M₀ = 4.9H/a in [31]. The length-to-width ratio of the foundation varies, as b/a = 1/4, 1/2, 1, 2 and 4.

Figure 3a shows the spectrum amplitude of structure response $\left|u_{\mathrm{b}}\right|$ for a fixed-base oscillator and SSI systems with a varying b/a ratio. The x-axis is the dimensionless incident frequency η = 0–0.3, which covers the predominant frequency domain of energy in earthquakes. Three structure frequency values are concerned: η_b = 0.1, 0.15, 0.2. For a fixed-base oscillator, a rigid connection between the structure and the ground is assumed without any energy dissipation through the soil, while for an SSI system, the soil behaves like a cushion under the structure and adds additional flexibility to the system, so the system frequency of the SSI is obviously lower than the resonant frequency of the corresponding fixed-base oscillator. Further, as the length-to-width ratio b/a decreases, the peak has a higher frequency and a higher amplitude, approaching the fixed-base case.

Figure 3b shows the dynamic properties (system frequency, system damping, peak amplitude) of the structure response $\left|u_{\mathrm{b}}\right|$, as well as the varying b/a ratios. The x-axis is the dimensionless structure frequency η_b = 0–3. The system frequency ${\tilde{\eta }}_{\mathrm{b}}$ corresponds to the peak frequency of $\left|u_{\mathrm{b}}\right|$, or the eigen-value of Equation (1) [28,29,30], and here we use the ratio ${\tilde{\eta }}_{\mathrm{b}}/{\eta }_{\mathrm{b}}$ to indicate a shift in system frequency with respect to the structure frequency ${\eta }_{\mathrm{b}}$. Although the spectrum shape is similar between the fixed-base oscillator and the SSI system with various length-to-width ratios, for the values of b/a and η_b concerned in the figure, the system frequency shifts up to 50% from the structure frequency. For example, it shifts by 12% for a framed structure of width 2a = 40 m, length 2b = 40 m and fixed-base frequency ${\omega }_{\mathrm{b}}\hspace{0.17em}=\hspace{0.17em}6.28\hspace{0.17em}\mathrm{Hz}$ on a moderate rigid site (β₁ = 300 m/s), and it shifts by 16% for the same structure on a moderate soft site (β₁ = 250 m/s). In fact, the shift in peak frequency between the SSI system and the fixed-base oscillator leads to a completely different correspondence with respect to the frequency of excitations, which is further investigated in Figure 4.

The second sub-figure of Figure 3b describes the system damping ${\tilde{\xi }}_{\mathrm{b}}$ which is obtained by the half-power method from $\left|u_{\mathrm{b}}\right|$, and we use the ratio ${\tilde{\xi }}_{\mathrm{b}}/{\xi }_{\mathrm{b}}$ to show a comparison between the SSI system and the fixed-base oscillator. The ratio ${\tilde{\xi }}_{\mathrm{b}}/{\xi }_{\mathrm{b}}\hspace{0.17em}>\hspace{0.17em}1$ indicates that the system damping is obviously larger than the structure damping, while the third sub-figure is the peak amplitude $\left|u_{\mathrm{b}}^{\max }\right|$ between the SSI system and the fixed-base oscillator. Correspondingly, the peak amplitude in the frequency domain of the SSI system is smaller than the fixed-base oscillator, which gives $\left|u_{\text{b\_SSI}}^{\max }/u_{\text{b\_fixed-base}}^{\max }\right|\hspace{0.17em}<\hspace{0.17em}1$.

3.2. Shift in System Frequency

Figure 4 shows a spectrum correspondence between the structure response $\left|u_{\mathrm{b}}\right|$(the line style follows Figure 3) and the incident wave. Due to the shift in peak frequency, the SSI system demonstrates different seismic performances compared to the fixed-base oscillator. When the predominant frequency of the incident wave matches the peak frequency of the SSI system (acceleration record 1 in Figure 4), the dynamic responses of the SSI system are probably larger than the fixed-base oscillator, which probably has detrimental SSI effects on the seismic performances of a structure, so the seismic designs on the assumption of a fixed-base oscillator are probably unsafe. Whereas, for a case in which the predominant frequency matches that of the fixed-base oscillator (acceleration record 2 in Figure 4), the responses of the former are probably smaller than the latter, which probably has beneficial SSI effects, so the seismic designs on the assumption are probably too conservative. Certainly, for a case in which the predominant frequency of the incident wave is not obvious, it also probably results in beneficial SSI effects, since the peak amplitude of the SSI system is always smaller than that of the corresponding fixed-base oscillator. In summary, a smaller peak amplitude of SSI system in the frequency domain does not necessarily mean that the dynamic responses of SSI in the time domain is always smaller than the fixed-base structure. SSI effects are sometimes detrimental to the seismic performances of structures, depending on the dynamic properties of both the structure itself and the soil.

Figure 5 presents the ratio ${\tilde{\eta }}_{\mathrm{b}}/{\eta }_{\mathrm{b}}$ (system frequency/fixed-base frequency) as that which is shown in Figure 3b, but for different structure and site parameters. The structure mass is for three values M_b/M₀ = 5, 10, 15, and two kinds of half-space are concerned, β_R/β_j = 1 and 2, with the other parameters the same as Figure 3. β_R/β_j = 1 corresponds to a uniform half-space. The shift in system frequency in uniform half-space is larger than that in layered half-space, since the bedrock adds rigidity to the site. Also, the shift in system frequency for a high structure is larger than that for a short structure, since a high structure is more flexible according to our parameters.

4. Detrimental Effects of SSI

4.1. Data on Structures and Sites

A further study on the time domain was performed on four buildings in California, USA, to investigate the possibility of SSI having detrimental effects. The parameters were selected from the published literature: the Hollywood Storage Building (California Strong Motion Instrumentation Program (CSMIP) station No. 24236) [32], the Millikan Library (National Strong Motion Project (NSMP) No. 5407) [33], a building in Sherman Oaks (CSMIP station No. 24322) and a building in Walnut Creek (CSMIP station No. 58364) [19]. These buildings have been used for strong-motion observations for decades, and their structural and site data are available, as well as earthquake records, which is helpful to establish proper SSI models. The details of the four buildings and the stratifications of the site can be found in our previous paper [34], and we list key information on the buildings in

Table 1. Buildings for case studies.

Parameters		Hollywood Storage Building	Millikan Library	Sherman Oaks Building	Walnut Creek Building
a (short side)		7.77 m	11.65 m	18.15 m	15.9 m
b (long side)		33.15 m	12.55 m	28.8 m	22.6 m
c		8.74 m	4.30 m	16.10 m	4.20 m
H		45.6 m	43.9 m	50.0 m	39.2 m
He		26.3 m	25.3 m	28.9 m	22.6 m
M0 (kg)		1.38 × 107 kg	1.43 × 106 kg	4.57 × 107 kg	1.43 × 106 kg
Mb		1.17 × 107 kg	1.07 × 107 kg	1.66 × 107 kg	1.07 × 107 kg
Mb/M0		0.85	7.48	0.24	7.48
${\xi }_{\mathrm{b}}$		2%	2%	2%	2%
NS	b/a	4.27	0.93	1.59	1.42
	c/a	1.12	0.34	0.89	0.26
	He/a	3.38	2.17	1.59	1.42
	${\omega }_{\mathrm{b}}a/\pi {\beta }_1$	0.070	0.168	1.10	0.14
	${\omega }_{\mathrm{b}}/2\pi$	0.83 Hz	2.16 Hz	1.54 Hz	1.52 Hz
	${\tilde{\omega }}_{\mathrm{b}}/2\pi$	0.78 Hz	1.86 Hz	1.38 Hz	1.40 Hz
	${\tilde{\omega }}_{\mathrm{b}}/{\omega }_{\mathrm{b}}({\tilde{\eta }}_{\mathrm{b}}/{\eta }_{\mathrm{b}})$	0.93	0.84	0.90	0.92
EW	a/b	0.23	1.08	0.63	0.70
	c/b	0.26	0.37	0.56	0.186
	He/b	0.79	2.02	1.00	1.00
	${\omega }_{\mathrm{b}}a/\pi {\beta }_1$	0.72	0.11	0.61	0.28
	${\omega }_{\mathrm{b}}/2\pi$	2.00 Hz	1.26 Hz	1.69 Hz	2.10 Hz
	${\tilde{\omega }}_{\mathrm{b}}/2\pi$	1.78 Hz	1.20 Hz	1.55 Hz	1.88 Hz
	${\tilde{\omega }}_{\mathrm{b}}/{\omega }_{\mathrm{b}}({\tilde{\eta }}_{\mathrm{b}}/{\eta }_{\mathrm{b}})$	0.89	0.94	0.92	0.89

and on the sites in

Table 2. Site parameters for case studies.

	S-Wave Velocity (m/s)	Thickness (m)	Mass Density (kg/m3)	Damping Ratio
Hollywood Storage Building
1	184.7	15.2	2050.4	2%
2	362.7	15.2	2082.4	2%
3	554.7	30.5	2082.4	2%
4	624.8	41.5	2082.4	2%
Bedrock	1045.5	∞	2082.4	2%
Millikan Library
1	298.7	5.49	1846.9	2%
2	387.1	4.26	1846.9	2%
3	454.2	3.66	1846.9	2%
4	487.7	6.71	1846.9	2%
5	609.6	82.29	1846.9	2%
6	762.0	16.16	1846.9	2%
Bedrock	944.8	∞	1846.9	2%
Sherman Oaks Building
1	160	2.7	2040.8	2%
2	205	6.7	2040.8	2%
3	260	13.7	2040.8	2%
4	330	23.8	2040.8	2%
Bedrock	514	∞	2040.8	2%
Walnut Creek Building
1	336	5.0	1846.9	2%
Bedrock	434	∞	1846.9	2%

, which are used to establish the SSI model. The influence of SSI is probably not taken into account in the seismic designs of the buildings. The fixed-base frequency ω_b/2π for the four buildings ranges from 0.83 Hz to 2.16 Hz, and the shear-wave velocity of the soil at the top of the site ranges from 184.7 m/s (soft soil) to 336 m/s (moderate rigid soil). A rough formula $H_e=H/\sqrt{3}$ suggested in [35] is used as the effective height of the structures, since the structures are simplified to the equivalent SDOF oscillator.

4.2. Case Studies

Figure 6a,b are the dynamic responses of the Hollywood Storage Building for an assumed SSI system and an assumed fixed-base structure, respectively. The excitation, as shown at the top of the figure, is the acceleration record obtained at the ground floor of the building in the NS direction (a component of the horizontal direction) during the Northridge earthquake in 1994, and it is normalized to an acceleration of a peak value of 0.1 g. The record can be found on the USGS website https://strongmotioncenter.org/ (accessed on 28 January 2019). It is also assumed that the excitation is a train of SV waves coming from the bedrock with an incident angle θ = 90°. The acceleration responses of both the SSI system and the fixed-base structure are shown in the time history ${\ddot{u}}_{\mathrm{b}}(t)$ (left column) and the response spectrum of the time history $RS({\ddot{u}}_{\mathrm{b}})$ (right column).

The responses in the time history in Figure 6 were gathered according to the following steps: First, the time history of an earthquake acceleration was converted into the frequency domain at 4096 + 1 equally spaced points from 0 to 50 Hz using Fourier transformation to acquire the spectrum of earthquake acceleration. The corresponding values on the spectrum of the structure response u_b were calculated at these frequency points as well. Thereafter, the spectrum of the structure response was multiplied by the spectrum of the earthquake acceleration at the corresponding frequency points to produce the amplified spectrum of the structure. Finally, the response in the frequency domain was converted to the time domain using inverse fast Fourier transformation.

In the time history graph (left column), for the time window 0–40 s, the value denoted by an asterisk “*” is the maximum value during the time history, and the value presented in the bottom right corner is the mean value. In the response spectrum graph, the value denoted by an asterisk shows the same. It can be noticed that the maximum value of the SSI system (0.33 g) is 57% larger than that of the fixed-base oscillator (0.21 g), and the mean value of the SSI system (4.33 × 10⁻² g) is 48% larger than that of the fixed-base oscillator (2.92 × 10⁻² g). Also, the maximum value in the response spectrum of the SSI system (0.96 g) is 25% larger than that of the fixed-base oscillator (0.80 g). We list these three values (the maximum and the mean values in the time history, and the maximum value in the response spectrum) for the four buildings to 42 earthquake events in

Table 3. Dynamic responses of structures in the time domain of target buildings in the XOZ-plane.

Earthquake	Year	$\mathbf{Time}\mathbf{History}{\ddot{\mathit{u}}}_{\mathbf{b}}\mathbf{(}\mathit{t}\mathbf{)}$				$\mathit{R}\mathit{S}\mathbf{(}{\ddot{\mathit{u}}}_{\mathbf{b}}\mathbf{)}$
		Maximum (g)		Mean (g) ×10−2		Maximum (g)
		SSI System	Fixed-Base	SSI System	Fixed-Base	SSI System	Fixed-Base
Hollywood Storage Building (NS)
Kern County	1952	0.43	0.28	7.72	5.25	1.67	1.40
San Fernando	1971	0.37	0.15	5.52	3.12	1.17	0.85
Whittier Narrow	1987	0.32	0.15	4.92	4.34	0.97	0.90
Northridge	1994	0.33	0.21	4.33	2.92	0.96	0.80
Chino Hills	2008	0.41	0.18	7.24	2.75	1.73	0.68
Encino	2014	0.27	0.11	1.49	0.61	0.57	0.23
Millikan Library (NS)
Lytle Creek	1970	0.41	0.33	11.40	9.83	3.44	2.56
San Fernando	1971	0.13	0.21	3.41	5.61	0.57	1.10
Whittier Narrow	1987	0.16	0.37	3.12	7.10	0.90	2.66
Yorba Linda	2002	0.13	0.26	2.91	5.76	0.69	1.96
San Simeon	2003	0.12	0.13	3.45	3.49	0.87	1.03
Sherman Oaks Building (NS)
Lander	1992	0.21	0.28	5.64	6.55	1.30	1.66
Whittier Narrow	1987	0.13	0.15	1.38	1.52	0.34	0.33
Northridge	1994	0.20	0.22	2.43	3.70	0.78	1.51
Chatsworth	2007	0.095	0.102	1.00	1.04	0.33	0.47
Chino Hills	2008	0.20	0.16	2.44	2.57	0.73	1.01
Encino	2014	0.061	0.12	0.053	0.06	0.18	0.33
Walnut Creek Building (EW)
Livermore	1980	0.33	0.34	4.94	6.20	1.32	1.48
Livermore Aftershock	1980	0.27	0.39	4.33	7.84	1.50	2.37
Loma Prieta	1989	0.24	0.33	4.54	5.83	1.19	1.68
Alamo	2008	0.10	0.12	1.25	1.77	0.30	0.45

(XOZ plane) and

Table 4. Dynamic responses of structures in the time domain of target buildings in the YOZ-plane.

Earthquake	Year	$\mathbf{Time}\mathbf{History}{\ddot{\mathit{u}}}_{\mathbf{b}}\mathbf{(}\mathit{t}\mathbf{)}$				$\mathit{R}\mathit{S}\mathbf{(}{\ddot{\mathit{u}}}_{\mathbf{b}}\mathbf{)}$
		Maximum (g)		Mean (g) ×10−2		Maximum (g)
		SSI System	Fixed-Base	SSI System	Fixed-Base	SSI System	Fixed-Base
Hollywood Storage Building (EW)
Kern County	1952	0.44	0.32	7.84	7.02	2.39	2.50
San Fernando	1971	0.38	0.19	4.01	3.17	1.45	1.36
Whittier Narrow	1987	0.49	0.53	8.60	15.42	2.09	4.69
Northridge	1994	0.45	0.32	6.60	4.15	3.11	1.95
Chino Hills	2008	0.35	0.41	5.52	6.45	1.94	3.14
Encino	2014	0.19	0.25	2.88	3.67	0.88	1.48
Millikan Library (EW)
Lytle Creek	1970	0.39	0.49	9.47	15.44	2.37	3.77
San Fernando	1971	0.20	0.19	6.65	7.74	1.35	1.53
Whittier Narrow	1987	0.27	0.34	5.53	6.36	1.30	1.80
Yorba Linda	2002	0.107	0.105	2.62	3.54	0.56	0.70
San Simeon	2003	0.44	0.55	11.74	14.90	3.13	4.19
Sherman Oaks Building (EW)
Lander	1992	0.19	0.27	5.20	8.10	1.20	1.99
Whittier Narrow	1987	0.20	0.26	2.95	5.50	0.87	1.64
Northridge	1994	0.187	0.190	2.91	3.11	1.13	1.15
Chatsworth	2007	0.055	0.106	0.060	0.087	0.19	0.43
Chino Hills	2008	0.20	0.24	3.58	5.18	1.46	1.63
Encino	2014	0.061	0.12	0.053	0.06	0.18	0.33
Walnut Creek Building (NS)
Livermore	1980	0.28	0.37	6.25	8.17	2.07	2.42
Livermore Aftershock	1980	0.14	0.17	2.75	3.27	0.85	1.01
Loma Prieta	1989	0.14	0.20	3.09	2.94	0.96	1.33
Alamo	2008	0.13	0.19	1.11	2.15	0.36	0.59

(YOZ plane) for both the SSI system and the fixed-base oscillator. The excitations are the acceleration records from the ground floor of each building for each earthquake event (https://strongmotioncenter.org/, accessed on 28 January 2019).

Figure 7 illustrates the ratio of the three responses (maximum response in the time history, mean response in the time history, and maximum response spectrum of structure acceleration response) of the SSI system to those of the fixed-base oscillator. The x-axis corresponds to each event in Table 3 and Table 4, and the y-axis is the percentage value. The symbol “FB” represents a “fixed-base oscillator”. A value larger than 100% on the y-axis indicates that the dynamic response of the SSI system is larger than that of the corresponding fixed-base oscillator, or that SSI amplifies the dynamic response of the structure and has a detrimental effect on the seismic designs. As shown in the figure, about 15–20% of the 42 selected earthquakes on four buildings’ events had detrimental effects. The SSI mechanism amplifies the structure responses by as much as 1.5 times for a few cases. This justifies the idea that a shift on the spectrum may lead to considerably different responses in the time domain, depending on the frequency component of the incident waves and the parameters of the structure as well as the soil.

4.3. Evaluation of System Frequency

Involving SSI in the seismic designs could result in neutral/negligible, beneficial or detrimental effects on structure safety. Due to the uncertainties of seismic motions, there is little necessity for the seismic analysis on SSI having beneficial effects, while it is necessary to develop a method that determines the detrimental effects to ensure the seismic safety of structures. However, seismic hazard assessments on SSI are costly, even for linear problems. For a seismic engineering analysis, an efficient method that is applicable in coded standard procedures is needed. For an engineering seismic analysis, it needs efficient method that is applicable in coded standard procedures. Since the structure period (frequency) is the most important parameter in the seismic designs, it makes sense to predict on the shift of structure period (frequency) in engineering practice. In the following, we describe a method, using Figure 5 to predict the system frequency of the Millikan Library.

The parameters of the structure are a = 11.65 m, b = 12.55 m, c = 4.3 m, H_e = 29.3 m, M₀ = 1.43 × 10⁶ kg, M_b = 1.07 × 10⁷ kg, so we have b/a = 1.08, c/a = 0.37, H_e/a = 2.52, M_b/M₀ = 7.48. The fifth layer (β₅ = 609.6 m/s) is assumed as bedrock, and we simplify the multi-layered half-space to a uniform half-space by the equivalent shear wave velocity of the site

$${\beta }^{\mathrm{eq}}={\displaystyle \frac{5.49+4.26+3.66+6.71}{\frac{5.49}{298.7}+\frac{4.26}{387.1}+\frac{3.66}{454.2}+\frac{6.71}{487.7}}}=393.0\mathrm{m}/\mathrm{s}$$

(3)

so we have the dimensionless frequency in the NS direction in the equivalent uniform half-space as

$${\eta }_{\mathrm{b}}={\displaystyle \frac{(2.16\mathrm{Hz}\times 2\pi )\times 11.65\mathrm{m}}{393.0\mathrm{m}/\mathrm{s}\times \pi }}=0.128$$

(4)

We find the curve in parameters b/a = 1, H_e/a = 2, M_b/M₀ = 10 in uniform half-space at the x-axis of η_b = 0.128, then we measure the value of y-axis of ${\tilde{\eta }}_{\mathrm{b}}/{\eta }_{\mathrm{b}}\hspace{0.17em}=\hspace{0.17em}0.85$, to evaluate the system frequency ${\tilde{\omega }}_{\mathrm{b}}/2\pi \hspace{0.17em}=\hspace{0.17em}1.836\hspace{0.17em}\mathrm{Hz}$. The evaluation of the system frequency using Figure 5 is very close to the true value 1.84 Hz. While in an EW direction, it gives ${\eta }_{\mathrm{b}}\hspace{0.17em}=\hspace{0.17em}0.075$, and we measure ${\tilde{\eta }}_{\mathrm{b}}/{\eta }_{\mathrm{b}}\hspace{0.17em}=\hspace{0.17em}0.94$ from y-axis, to evaluate the system frequency ${\tilde{\omega }}_{\mathrm{b}}/2\pi \hspace{0.17em}=\hspace{0.17em}1.18\hspace{0.17em}\mathrm{Hz}$, which is also very close to the true value 1.20 Hz.

The parameters of the other three buildings are a little far from the parameter combinations described in Figure 5, so it is difficult to use Figure 5 to predict their system frequency. It is necessary to find a general method capable of covering a variety of structure and site parameters. However, the SSI system is multi-parametric, and dozens of target parameters are necessary to obtain the global function, which maps subdomain parameters to the system frequency. At last, we describe a possible way, based on an artificial neural network, to realize the evaluation of system frequency. The artificial neural network is an algorithm that fits a complex mapping of multiple inputs and outputs. Through appropriate machine learning, this algorithm could theoretically converge to any function.

The system frequency of SSI depends on the properties of the structure itself as well as the underlying soil. The soil that can determine the system frequency is the surface layer across dozens or hundreds of meters, the constructions and the parameters of which are statistically consistent in adjacent areas with the same geological environment. Moreover, the structural parameters follow general principles in structural designs and engineering practices. Therefore, it is feasible to create a neural network mapping the parameters of the two substructures (structure and site) to the SSI system, and the method is applicable to cities whose geological stratum data have been accumulated plentifully over the past decades, in order to obtain a numerical model on seismic hazard assessment involving SSI effects. This will be the focus of our future work.

5. Conclusions

To date, it has been assumed in most seismic designs that the structures are fixed-supported on the foundation, or that the underlying soil is completely rigid. In fact, the mechanism of the soil–structure interaction introduces considerable differences to the seismic performance of a structure. This study investigated the possibility of SSI amplifying the dynamic responses of a structure, or SSI having detrimental effects on the seismic performance of a structure, by comparing the dynamic responses of the SSI system and a fixed-base structure. The SSI model consists of an oscillator, a 3D rectangular foundation, and a multi-layered half-space. The foundation impedance functions are obtained by the high-precision indirect boundary element method from our previous studies.

The soil behaves like a cushion under the structure and adds additional flexibility, so from the viewpoint of the spectrum in the frequency domain, the system frequency of SSI is lower than the resonant frequency of the fixed-base structure, the system damping of SSI is larger than the structure damping, and the system amplitude of SSI is smaller than the peak amplitude of fixed-base structure. However, a smaller peak amplitude of the SSI system in the frequency domain does not necessarily mean that the dynamic responses of SSI in the time domain are always smaller than the fixed-base structure. Case studies on four buildings to 42 earthquakes in the time domain show a 15–20% possibility that SSI would have detrimental effects on the seismic performances of structure, or SSI amplifying the dynamic responses of structure.

This is because there is a shift in peak frequency between the SSI system and the fixed-base structure. When the predominant frequency of the incident wave matches the peak frequency of the SSI system, the dynamic responses of the SSI system are probably larger than the fixed-base oscillator, which probably has detrimental SSI effects on the seismic performance of a structure. The influence of SSI on the seismic performance of a structure depends on the dynamic properties of both the structure itself and the soil.

It should be noted that the possibility of SSI having detrimental effects is based on the four selected buildings as well as the earthquake records obtained from the buildings, so the database is not very plentiful, and the result is not very generalized. More cases are needed in the future to enrich the topic and the conclusion. It should also be noted that both the soil and the structure are likely to experience some degree of nonlinearity during strong earthquake events, and the detrimental effects of SSI under nonlinear conditions are the focus of our future work.