Seismic Performance of Modal Transfer Stations on Soft Clays

Abstract

In densely populated urban zones, seismic performance evaluation of strategic infrastructure during seismic events has become more challenging because the distance between surface and underground structures has been shortened to optimize the urban environment functionality. This is even more important in transit transfer stations, which usually comprise tunnels, bridges, and buildings, in which wave propagation interference is exacerbated. This paper explores the seismic interactions between on-ground and underground structures in soft-soil environments, focusing on a typical urban modal transfer station in Mexico City. The study is conducted through comprehensive parametric analyses using 3D numerical simulations in FLAC3D (v.6.0), considering both intraplate and interplate earthquakes, to assess the effect of differences in their frequency content, duration, and intensity. Multiple scenarios are considered in the numerical study, and the relative distances among the structures are varied to investigate both detrimental and beneficial interaction effects, and to identify the zone of influence where this interaction leads to ground motion variability. The study’s findings established the key variables in the interaction between underground and on-ground structures, providing valuable insights into the seismic design and retrofitting of urban infrastructure in densely populated areas.

1. Introduction

Densely populated urban areas with increasing service demands require the development of proper strategies to enhance the resilience of urban infrastructure and lifeline systems to natural hazards [1,2,3,4]. To achieve this in earthquake-prone regions, a proper assessment of the seismic demand expected to occur in the infrastructure systems is required. This need has led to a growing interest in the research community for assessing the vulnerability of transportation systems, following a holistic approach, where each component affects the behavior of the whole system [3,4]. Regarding seismic performance, important interaction effects have been observed to occur in structures that are closer to each other, considerably modifying their seismic demands. Thus, more robust methodologies and analysis models should be implemented to capture such effects, considering the structure proximity of each element of the transportation system [5,6,7].

Multiple research groups have studied the interaction between surface and underground structures, implementing several methodologies. These include advanced 3D numerical modeling [8,9,10], experimental small-scale testing using shaking tables [5,11], or centrifuge testing [12], as well as large-scale research involving real instrumented structures [13]. Overall, these studies have provided significant contributions, establishing that structural proximity, soil characteristics, input motion frequency content, fundamental vibration modes of on-ground structures, foundation type, tunnel diameter, and burial depth are the key variables that define the cases where interaction is beneficial or detrimental for the structural seismic performance [14]. Regarding the interaction among surface structures, significant advances have been made through recent studies [15,16,17,18]. This effect has been called site–city or structure–soil–structure interaction (SSSI). In the particular case of the Mexico City Basin (MCB), some anomalous beating cycles have been identified in the recorded seismic “free-field motions” of recent earthquakes. Some researchers have associated this signal pollution with SSSI effects [19,20,21], which are exacerbated by the very low soil shear stiffness and damping of the shallow clay layer typically found in the MCB, and the large impedance ratio of such a layer with respect to the underlying stiff soils. Furthermore, building-soil deposit resonance and the seismic motion frequency content in which the energy of the excitation is concentrated also play a major role in the SSSI. Under those characteristics, wave fields radiating from nearby buildings are trapped in shallower layers and are then propagated as surface waves, affecting neighboring structures. Considering both phenomena, there is a complex interaction between the incident, reflected, and diffracted seismic waves by underground structures, and those generated by the swaying of the surface structures during earthquakes, as Figure 1 depicts.

Although such contributions have provided important insights, they considered idealized scenarios, such as plane–strain conditions, one-degree-of-freedom oscillator representations of surface structures, and harmonic time history excitations, and, more importantly, they neglected the interaction effects among on- and underground structures. These limitations can potentially lead to expensive and, in some cases, unsafe structure designs. In the worst-case scenario, interaction effects can increase the seismic demand of the structures in terms of internal forces, story drifts, or bridge decks’ relative displacements, reaching or exceeding a particular damage level, which can be directly associated with potential economic losses, and reductions in seismic resilience.

This paper presents a parametric study, using numerical three-dimensional finite-difference models, on the seismic building–tunnel–bridge interaction effects commonly encountered in urban environments. The study focuses on the impact of the frequency content, duration, and intensity of the excitation, the distance between the building and the bridge, and the tunnel position relative to each structure on the seismic performance of the system. The case study location is the lacustrine soil deposits of the MCB, where the aforementioned SSSI effects have been observed.

As depicted in Figure 2, this paper starts by describing the idealized tunnel–bridge–building system considered in the parametric study. This includes a detailed description of the typology, structural features, subsoil conditions, geotechnical properties, and seismic environment. Then, the numerical model used to simulate the free field, and the seismic tunnel–building–bridge interaction is described. This involves the criteria for the finite-difference size selection, boundary conditions, and the constitutive models used in the simulation, as well as the type of finite-difference elements representing the tunnel lining, the building, the bridge, and their foundations. Subsequently, the interaction analysis methodology is explained, along with the structural demand parameters used to conduct the structure performance assessment. Finally, the main findings, conclusions, and future research recommendations are presented.

2. Definition of the Infrastructure Assets

The seismic tunnel–soil–building–bridge interaction in soft clays was studied considering a common typology usually encountered in Mexico City, as depicted in Figure 3. Based on this schema, the numerical models were developed. The tunnel diameter, D, was set to 10 m; the building height, Hb, and length, L, of the square foundation were 21 m and 20 m, respectively; and the urban bridge height, Ho, was assumed to be 12 m. The building was supported by a 6m deep, 20 m side square box foundation, and the bridge foundation was composed of a 6 m deep, 12.5 m side square box foundation with 9 m long friction piles. The depth of the tunnel was kept constant and equal to two times the tunnel diameter (i.e., 20 m). The distance between the building and the bridge, called “interaction longitude” here, was considered to be from 10 to 30 m, which corresponds to 1 to 3 times the tunnel diameter. For each interaction longitude, the tunnel position was varied as follows: case (A) corresponds to the tunnel underneath the bridge, case (B) to the tunnel between the building and the bridge, and case (C) to the tunnel underneath the building (Figure 3).

Table 1. Tunnel–building–urban bridge cases considered.

Interaction Longitude (Diameters)	Position of the Tunnel for the Tunnel–Building–Bridge Scenarios			Position of the Tunnel for the Tunnel–Bridge Scenarios			Position of the Tunnel for the Tunnel–Building Scenarios			Building–Bridge Without Tunnel
	A	B	C	A	B	C	A	B	C
1	TBU_A1	TBU_B1	TBU_C1	TU_A1	TU_B1	TU_C1	TB_A1	TB_B1	TB_C1	BU_1
2	TBU_A2	TBU_B2	TBU_C2	TU_A2	TU_B2	TU_C2	TB_A2	TB_B2	TB_C2	BU_2
3	TBU_A3	TBU_B3	TBU_C3	TU_A3	TU_B3	TU_C3	TB_A3	TB_B3	TB_C3	BU_3

compiles the cases considered for the parametric study.

2.1. Soil Profile and Dynamic Properties

The studied site is located in the so-called Zone IIIb in Mexico City, where high-plasticity clays are found (i.e., PI > 200). The dynamic properties and seismic response of the site have already been calibrated by several authors [23,24] studying the seismic effects of past strong earthquakes. To continue from the previous research presented by Mayoral and Mosqueda [22], it was deemed appropriate to use this geotechnical profile to study the seismic interaction effects discussed above (Figure 3). Figure 4 summarizes the soil stratigraphy of the site studied, some index properties, and results of the Cone Penetration Test (CPT). Figure 4 also shows the shear wave velocity distribution, which was obtained in previous research [23] with various techniques including suspension logging and down-hole.

The key factors that contributed to the large seismic wave amplification observed in past earthquakes in the MCB are the very low shear stiffness (Figure 4), and the quasi-linear elastic range and very low damping exhibited by the soft clay for shear strain as high as 0.1%. These last two characteristics have been associated with very high plasticity (PI > 200), in accordance with the relation found by Darendeli [27]. To represent this behavior, González and Romo’s model [28] was used to estimate the normalized modulus degradation and damping curves for clays (Figure 5). Through a series of resonant column and cyclic triaxial tests, the authors related the model parameters to the plasticity index (PI) of the high-plasticity clays. For non-plastic materials such as silty sands, the curves proposed by Seed and Idriss [29] were used to characterize their behavior during seismic loading. Figure 5 depicts the normalized modulus degradation and damping curves used in the numerical analyses. It should be mentioned that similar curves have been used in 1D wave propagation analyses to reproduce the measured response during past earthquakes in the MCB, and good agreement was observed when comparing predictions with measured responses [23].

2.2. Building Features

The building has seven floors, 3 m high each, a square footprint of 20 by 20 m, and a compensated box-like foundation, 6 m deep. This structure typology experienced major damage during the 2017 Mexico City earthquake [30]. Following the method proposed and validated by Romo and Bárcena [31], the structure was simplified to a shear beam comprising solid elements, with equivalent stiffness, $k_i$ and mass, $m_i$, for each story, $i$. The dimensions of the equivalent shear beam are the same as those of the building considered. The structural period can be estimated with Expression (1). Table 2 presents the properties of the buildings considered in the analysis.

$$Te=4\sum \sqrt{{\displaystyle \frac{m_i}{k_i}}},$$

(1)

• $m_i=$ mass of each floor.
• $k_i=$ stiffness of each floor.

Table 2. Building structural fundamental period.

Stories	Basements Levels	Estimation for Stiff Buildings ${\mathit{T}}_{\mathit{e}}=\mathit{S}\mathit{t}\mathit{o}\mathit{r}\mathit{i}\mathit{e}\mathit{s}\times 0.1$ (s)	Estimation for Flexible Buildings ${\mathit{T}}_{\mathit{e}}=\mathit{S}\mathit{t}\mathit{o}\mathit{r}\mathit{i}\mathit{e}\mathit{s}\times 0.2$ (s)	Te Calculated Using Expression 1 (s)	Height (m)
7	2	0.7	1.4	1.01	21.0

Table 2. Building structural fundamental period.

Stories	Basements Levels	Estimation for Stiff Buildings ${\mathit{T}}_{\mathit{e}}=\mathit{S}\mathit{t}\mathit{o}\mathit{r}\mathit{i}\mathit{e}\mathit{s}\times 0.1$ (s)	Estimation for Flexible Buildings ${\mathit{T}}_{\mathit{e}}=\mathit{S}\mathit{t}\mathit{o}\mathit{r}\mathit{i}\mathit{e}\mathit{s}\times 0.2$ (s)	Te Calculated Using Expression 1 (s)	Height (m)
7	2	0.7	1.4	1.01	21.0

2.3. Tunnel Description

The tunnel has a circular cross-section with an external 10 m diameter. The tunnel has a 0.4 m lining of reinforced concrete with a compression strength of $f’c$ = 45 MPa, at 28 days, and rebar reinforcement with a yield stress of $fy$ = 420 MPa, distributed according to Figure 6. More details about the considerations adopted for modeling the tunnel lining can be found in [32].

2.4. Urban Bridge Description

The bridge superstructure comprises a 9 m wide hollow beam with a trapezoidal section supporting the upper deck. This beam is, in turn, supported by a 12 m high precast column with a hollow rectangular section which is structurally tied to the upper deck beam. The column is connected to box-type foundations with a 12.5-by-12.5 m square footprint, 6 m deep, structurally tied to nine 0.5-by-0.5 m square friction piles 12.5 m long (Figure 7). Foundation elements exhibit an unconfined compression strength of 30 MPa.

Table 3. Fundamental period of the bridge onrigid base.

Direction	Structural Period (s)
Transversal	0.45
Longitudinal	0.60

depicts the fixed-base period of the bridge, which was computed with the numerical model assuming free vibration conditions in each direction. More details about the characteristics of the urban bridge can be found in [33,34].

3. Seismic Environment Characterization

The seismic tunnel–soil–building–bridge interaction effects were studied considering both interplate and intraplate events to assess the impact of frequency content, duration, and seismic intensity in the system interaction. Mexico City’s seismic hazard is controlled by two main earthquake sources: interplate events, which originate between the Cocos and North American plates along the Mexican subduction zone, and intraplate earthquakes, which occur within the subducted Cocos plate. It can be highlighted that the two most destructive earthquakes that have occurred in Mexico City originated from these seismogenic sources. Recently published papers described the observed damage patterns that occurred during both earthquakes and the interaction effects typically observed in more detail [30,35,36,37,38].

For this purpose, the ground motions recorded at station CUP5 during the earthquakes in September 1985 (interface event) and September 2017 (intraplate earthquake), which is located at a rock outcrop, were used as seed ground motion. The local building code [39] requires an intensity level related to a return period of 250 years to design most structures (i.e., base design earthquake); thus, the acceleration histories were adjusted to the uniform hazard spectra (UHS) developed for this return period for each seismogenic source. The selected time history, also called seed ground motion, was modified by applying the method developed by Lilhanand and Tseng [40], as modified by Alatik and Abrahamson [41], to match the target UHS. As can be seen in Figure 8, the response spectrum calculated from the above methodology reasonably matches the target spectrum. The seed ground motions’ characteristics are compiled in

Table 4. Earthquakes considered in the analysis.

Seismogenic Zone	Earthquake Name	Year	Mw, Moment Magnitude	PGA, Peak Ground Acceleration (g)	TD (s)	Frequency Content (Hz)	Arias Intensity (cm/s)
Intraplate	Puebla, Mexico, CU17	2017	7.1	0.059	29.6	0.23 to 3.5	12.7
Interface	Michoacan, Mexico, CU85	1985	8.1	0.033	49.7	0.38 to 1.36	15.5

Note: T_D is the significant duration of the ground motion defined as the difference between T-95 and T-5, which are the times where 95% and 5% of Arias intensity is reached, respectively.

.

Site Response Analyses

Initially, the computer code SHAKE-91 [42] was used to conduct one-dimensional equivalent linear site response analyses and deconvolve the motion to the base of the soil column. Afterwards, the site response analysis in the time domain was carried out with the software FLAC3D v.6.0 [43]. The deconvolved ground motion was applied at the base of the model. The compliant-base approach was employed. Thus, seismic motion was applied as a shear stress history. To validate the site response analysis model, the records of the seismic station SCT of the 2017 earthquake were used. Mayoral and Mosqueda [22] compared the response spectrum calculated with the numerical model with the response spectrum of the SCT ground motion record. Good agreement was observed between the computed and measured responses. Following this validation, the same dynamic properties were considered for the 3D model.

4. Numerical Model

The seismic tunnel–building–bridge interaction was established through numerical models developed with the software FLAC3D v.6.0. FLAC3D is designed to simulate the mechanical behavior of three-dimensional continua under static or dynamic loading conditions, employing an explicit finite-difference scheme. The software is driven by strains, laws of motion, and constitutive equations, resulting in partial differential equations for specific geometries and properties. These equations, which are solved by the program, relate mechanical (stress) and kinematic (strain rate velocity) variables, allowing them to simulate complex geotechnical systems under specific boundary and initial conditions. The main advantage of FLAC3D regarding handling dynamic problems, such as seismic wave propagation, is that it incorporates inertial effects, which makes it well suited to simulate soil–structure interactions.

The model has 166,263 solid elements and 239,548 nodes, representing the building and the soil. The minimum thickness of the elements was 2 m based on the recommendation provided by Lysmer and Kuhlemeyer [44] to adequately simulate wave transmission through the mesh. Similar element sizes have been employed in previous research, and good agreement has been found between FLAC3D and SHAKE models [8,22]. Wave reflection at the model base was avoided by using quiet boundary conditions, with the Lysmer and Kuhlemeyer [44] formulation. For the horizontal faces of the model, free-field boundaries available in FLAC3D were used to avoid energy reflection. Thus, the main grid’s lateral boundaries are coupled to a free-field grid with the same properties as the lateral faces using viscous dashpots to simulate a quiet boundary, and the unbalanced forces from the free-field grid are applied to the main grid’s boundary in such a way that plane waves traveling upward suffer no distortion at the boundary because the free-field grid provides similar forces to those in an infinite model [43].

The Mohr–Coulomb failure criterion was used to represent the stress–strain soil behavior, and the structure elements were simulated with an elastic relationship. The bridge superstructure was simulated with beam elements, and equivalent density and stiffness to account for inertial effects in the full-scale bridge, properly accounting for the rocking generated by the mass distribution [33]. The walls and slabs of the box-type foundation were simulated with shell elements and the concrete piles with pile elements, which are basically beam elements coupled with normal and tangential springs that are defined as a function of the surrounding soil parameters. The pile element parameter was defined following the procedure described by Yeganeh et al. [45], which allows the pile–soil to slip during ground shaking. Therefore, when a pile is loaded vertically or horizontally, a relative displacement between the pile and the host medium can occur. The lining of the tunnel and the box foundation of the bridge were simulated by SHELL elements.

Figure 9a shows the numerical model considered for the seismic tunnel–building–bridge interaction analyses, with the assumed boundary conditions, and Figure 9b shows the corresponding structural elements.

During the dynamic simulation, the soil elements develop hysteretic closed loops of load–unload cycles; nevertheless, the Mohr–Coulomb model is not capable of reproducing that behavior. Therefore, to simulate the soil nonlinearities, the hysteretic model Sig3, available in FLAC3D, was implemented. This model allows the simulation of stiffness degradation and generation of damping during cyclic stress–strain loops. Although it is not a constitutive model, it can be used to provide proper shear stiffness and damping according to the equivalent linear approach. It should be stated that, with this model, stress depends only on the deformation and not on the number of cycles, and the corresponding damping is given directly by the hysteresis loop during cyclic loading [43]. The Sig3 model is defined according to Equation (2):

$${\displaystyle \frac{G}{G_{max}}}={\displaystyle \frac{a}{1+exp\left(-\frac{L-x_0}{b}\right)}}$$

(2)

• $G/Gmax=$ normalized secant modulus.
• $L=$ logarithmic strain defined as ${\log }_{10}\left(\gamma \right)$.
• $a$, $b$, and $x_0$ = fitting curve parameters used by the sig3 model, to be matched with the modulus degradation curves.

Table 5. Dynamic properties of soil elements.

Strata	Mohr–Coulomb Properties			Elastic Properties		Sig3 Hysteretic Model			Estimated Level of Nonlinearity
	γ (kN/m3)	c (kPa)	φ (°)	Gmax (GPa)	ν	a	B	x0	γ (%)	G/Gmax	λ (%)
Sandy silt	16.0	30	35	12.0	0.30	1.014	−0.480	−0.600	0.01–0.15	0.4–0.9	9–21
Very soft clay	12.0	40	0	5.6	0.45	1.000	−0.460	0.320	0.13–0.69	0.8–0.95	3–4
Silty sand with stiff clay	17.0	15	27	126.0	0.30	1.014	−0.480	−1.250	0.02–0.04	0.7–0.8	11–14
Soft clay	14.0	80	0	34.7	0.45	1.000	−0.490	−0.020	0.01–0.11	0.85–1.0	2–3
Silty sand	19.0	15	40	390 to 613	0.28	1.014	−0.550	−1.500	0.01–0.02	0.8–0.9	5–6

summarizes the Mohr–Coulomb parameters of each material, as well as the parameters of the hysteretic model Sig3. These parameters were defined by a fitting process using the least square method to reproduce the stiffness degradation and damping curves of Figure 5. The fitting is driven by the modulus shear stiffness degradation, $G/Gmax$, curves, considering that in a nonlinear response history analysis, the soil stiffness is a function of the evolving shear strains developed during seismic loading, which diverges from the steady-state response used to obtain the experimental damping curves.

The coupling springs of the pile elements were computed with the following equations, which simulate the shear and normal interface behavior at the pile–soil contact, according to Yeganeh et al. [45]. Table 6 shows the calculated pile–soil spring values.

$$c{s}_c=\alpha C_{soil} p$$

(3)

$$c{s}_f=\alpha {\phi }_{soil}$$

(4)

$$c{s}_k=\left[10 Max {\displaystyle \frac{\left(K+\frac{4}{3}G\right)}{\Delta Z_{min}}}\right]p$$

(5)

$$c{n}_c=9C_{soil}{D}_{pile}$$

(6)

$$c{n}_f\approx {\mathsf{\phi }}_{soil}$$

(7)

$$c{n}_t=\left({\displaystyle \frac{C_{soil}}{\tan \left({\phi }_{soil}\right)}}\right)D_{pile}$$

(8)

$$c{n}_k=\left[10 Max {\displaystyle \frac{\left(K+\frac{4}{3}G\right)}{\Delta Z_{min}}}\right]D_{pile}$$

(9)

• $\alpha$ = the value of this parameter depends on the condition of the pile surface. For the rough surface (like the cast-in situ pile), it is suggested to be equal to 1 [43].
• $c_{soil}$ and ${\phi }_{soil}$ = cohesion and internal friction angle of the soil adjacent to the pile.
• $P$ = exposed perimeter of the pile.
• $K$ and $G$ = bulk and shear moduli of the soil adjacent to the pile.
• $\Delta Z_{min}$ = smallest width of an adjoining zone in the normal direction.
• $D_{pile}$ = pile diameter.
• $c{s}_c$ and $c{s}_f$ = cohesion and frictional strength components of the shear springs.
• $c{n}_c$, $c{n}_f$ and $c{n}_f$ = cohesion, frictional, and tensional strength components of the normal springs.
• $c{s}_k$ and $c{n}_k$ = stiffness of the shear and normal springs.

Table 6. Coupling spring properties of the pile elements.

Soil	$\mathit{c}{\mathit{s}}_{\mathit{c}}$ (kN/m)	$\mathit{c}{\mathit{s}}_{\mathit{f}}$ (°)	$\mathit{c}{\mathit{s}}_{\mathit{k}}$ (MPa)	$\mathit{c}{\mathit{n}}_{\mathit{c}}$ (kN/m)	$\mathit{c}{\mathit{n}}_{\mathit{f}}$ (°)	$\mathit{c}{\mathit{n}}_{\mathit{k}}$ (MPa)
Very soft clay	60	0.1	1077	180	0.1	270

Table 6. Coupling spring properties of the pile elements.

Soil	$\mathit{c}{\mathit{s}}_{\mathit{c}}$ (kN/m)	$\mathit{c}{\mathit{s}}_{\mathit{f}}$ (°)	$\mathit{c}{\mathit{s}}_{\mathit{k}}$ (MPa)	$\mathit{c}{\mathit{n}}_{\mathit{c}}$ (kN/m)	$\mathit{c}{\mathit{n}}_{\mathit{f}}$ (°)	$\mathit{c}{\mathit{n}}_{\mathit{k}}$ (MPa)
Very soft clay	60	0.1	1077	180	0.1	270

5. Seismic Tunnel–Bridge–Building Interaction

The deconvolved ground motions calculated in the free-field response analyses were employed as input in the dynamic analyses of the tunnel–soil–bridge–building model, using the same compliant-base approach. For both CU17 and CU85 input motions, the shear stress histories were applied in the two orthogonal directions “x” (i.e., transversal) and “y” (i.e., longitudinal) at the model’s base, considering 100% in each direction.

To study the seismic interaction effect on the tunnel–bridge–building system, the results were presented in terms of the “Interaction factor for the urban bridge” (Tuf; Figure 10 and Figure 11), which is the transfer function computed between the upper deck of the bridge and its foundation without the presence of the adjacent structures, TF U-F, divided by the transfer function between the upper deck of the bridge and its foundation with the presence of the adjacent structures in the different scenarios, TF U-F_T. Similarly, the “Interaction factor for the building” (Tbf; Figure 12 and Figure 13) was also defined. These results are presented for all the scenarios considered in Figure 3 and Table 1, for the transversal and longitudinal components. Tuf and Tbf allow us to clearly show the effect of the amplification potential and energy distribution within the frequency content of interest. Values of Tuf and Tbf above one correspond to beneficial interactions (i.e., the building and bridge amplification potential decreases due to the adjacent structures), whereas a value below one means a detrimental interaction (i.e., the building and bridge amplification potential increases due to the adjacent structures). These relationships based on the transfer functions allow us to assess the amplification potential and energy distribution for each frequency. In all cases, the relevant frequency is indicated, depending on the fundamental period of the structure and direction analyzed, equal to Tb = 0.45 and 0.6 s (2.2 and 1.6 Hz) for the bridge in the transverse and longitudinal directions, respectively, and Te = 1.01 s (0.99 Hz) for the building in both directions. The characteristic elastic site period of the soil Tpe $\approx$ 2.0 s (0.5 Hz) is also within this range.

First, it can be clearly appreciated in all cases (Figure 10, Figure 11, Figure 12 and Figure 13) that the interaction among structures is greater for high frequencies (>1 Hz) and that is almost negligible for low frequencies (<1 Hz). Also, the interaction is larger for the CU17 synthetic ground motion, which concentrates its energy on higher frequencies, than for the CU85 ground motion. This can be associated partially with the dispersion and wave-scattering effect of the short waves, which corresponds to the high-frequency seismic motion component. Wave scattering can occur if there is an important heterogeneity within a uniform soil profile. Although this does not affect the total amount of energy carried by seismic motion, at the limits of heterogeneities, scattering tends to disorganize a wave field through reflection and refraction in such a way that the amplitudes of waves, especially at higher frequencies, are dispersed over different wavelengths [46]. Therefore, in systems formed by several vibration structures (bridges and buildings) located above a soil medium with heterogeneities (such as tunnels), it is most probable that their seismic response would be affected at high frequencies [47].

Figure 10 and Figure 11 show the interaction factor for the bridge; it is clear that for frequencies around 1 and 2 Hz, the amplification potential is affected by the presence of the building, while for higher frequencies, the presence of the tunnel increases the amplification potential. Also, the interaction among the three structures decreases in the frequency range of interest when the tunnel moves away from the bridge (case TBU_C). This can be clearly appreciated by comparing the black line (“BU”, which represents the case of the urban bridge and building without the tunnel) with the red lines (“TBU”, which represent the cases with the three structures in their different positions). This can be explained by considering that in almost all the cases studied, these lines are very similar within the relevant frequency range. This suggests that the presence of the building has a greater impact on the seismic response of the bridge within these frequencies; nevertheless, as the frequency increases, the red lines tend to be more similar to the blue lines (“TU”, which represents the case of the urban bridge with the tunnel in different positions), which suggests that the tunnel has a greater impact at higher frequencies. Similar effects can be observed on the building response (Figure 12 and Figure 13), and in addition, there is a clear amplification potential (i.e., detrimental interaction) in both directions as the tunnel gets closer to the building underneath its foundation, as shown by all the discontinuous lines (i.e., cases B and C).

The above results agree with the relative maximum displacements of the bridge and the building (Figure 14). In order to evaluate the seismic demand on the structures in more detail, the relative displacement between the base and the top of the structures was monitored during the seismic analyses for all the scenarios. Figure 14 depicts the numerical results for each case and the comparison with the independent structure case for each seismic motion. A detrimental interaction can be clearly observed on the maximum relative displacement of the upper deck and the building roof (Figure 14), computed as the maximum absolute value of the instantaneous relative displacement between the center of the upper deck bridge and the building roof and the base of their box foundations. In general, the maximum relative displacement is greater in the transverse (x) direction than in the longitudinal (y) direction. For subduction earthquakes, the maximum relative displacement is observed in the transverse direction when the tunnel is under the bridge. This amplification increases when the distance between the bridge and the building reduces (i.e., one diameter, 1D, in comparison with three diameters, 3D). The amplification in relative displacements is higher for the building in cases TB_C1 and TB_C2, which corresponds to the tunnel beneath the building. In addition, the differences between the relative displacements with normal and subduction earthquakes are more evident in the building, with greater values for CU85 observed, which can be associated with its frequency content which is closer to that of the structure. Thus, for the bridge performance, the tunnel induces up to 20% of amplification potential when it is located beneath the bridge; nevertheless, such an effect is exceeded by the presence of the building that generates an increase in the column drift up to 60% when it is located at a 1D distance.

Similar findings have been obtained in previous research. Based on the shaking table experiments carried out by Wang et al. [11], the seismic response of tunnel–soil–surface structures depends largely on the earthquake type, having less impact at lower frequencies. Furthermore, multiple numerical analyses of several authors [14,48,49] have demonstrated that the influence of the underground structure on the aboveground seismic response decreases gradually as the horizontal distance between them grows, being detrimental to the aboveground structure in most cases.

Figure 15a,b present the calculated peak acceleration at the urban bridge deck and its foundation, and Figure 16a,b at the building roof and its foundation. From the accelerations computed at the bridge deck in the transverse direction, it can be clearly noted that the most critical scenario is when the building is one diameter away. Thus, the amplification is almost 50% in comparison to the independent structure scenario. This effect decreases as the building moves away from the bridge. Furthermore, it can be highlighted that the tunnel effect is almost negligible when the building is one diameter away from the bridge, because the computed accelerations for TBU and BU cases are similar. Nevertheless, for the TU cases (i.e., when the building is not present), there is a sudden reduction in the acceleration, which continuously decreases as the tunnel moves away from the bridge. Thus, it can be concluded that the structure that affects the seismic response of the bridge the most is the building. Regarding the performance of the building, the maximum detrimental interaction observed can be associated with the tunnel, which leads to an amplification of the maximum acceleration computed at the building roof in the TB_B2 case (Figure 16a). In addition, the same effect is noticed for the scenario considered at the building foundation level (Figure 16b), which suggests that additional energy comes from the soil below. This can be related to the wave diffraction and reflection effect that occurs at the tunnel lining, causing a larger impact on the building. On the other hand, it seems that the presence of the bridge leads to a reduction in the computed acceleration TBU_B2 (i.e., beneficial interaction), which is the same scenario as previously discussed, but with the presence of the bridge.

In previous research [50,51], it has been established through analytical and numerical models that when the natural frequencies of two aboveground structures are different, there is an energy transfer between them, causing a beneficial interaction for the structure with a smaller frequency, and detrimental for the other one. In addition, this trend is exacerbated when the masses of the structures are different, being significantly detrimental for such structures with small masses and higher natural frequency. Therefore, this can explain why the building amplifies the seismic response of the bridge, while the building’s response is decreased. According to the above definition, the bridge structure is the most affected because its natural frequency (2.22 Hz) is higher than that of the building (0.99 Hz) and it is almost five times more lightweight than the building.

The maximum accelerations were also computed at the ground surface on the space between the two aboveground structures, called “near field” here. Figure 17 shows the comparison of the values obtained in each scenario with the free-field case (the scenario without structures). Interestingly, the maximum acceleration for the transverse direction appears between the tunnel and the building, but without the bridge, in the TB_C1 and TB_C2 cases. In these scenarios, the building and the tunnel seem to contribute to the generation of surface waves, associated with the building rocking back and forth and the wave scattering by the presence of the tunnel, leading to ground motion amplification in this zone. On the other hand, in the longitudinal direction, there is a decrease in accelerations when the tunnel is just below the point of analysis. This can be associated with the shadow effect observed by other researchers [52], defined as a zone above the tunnel where the energy is blocked by its presence, which is generated by the wave-scattering effect mentioned before.

In addition to the detrimental effects on the bridge and the building due to the presence of the tunnel, the bending moments acting on the piles also increase due to the presence of the tunnel. Figure 18 depicts the location of each pile relative to the tunnel–bridge–building system, and Figure 19 and Figure 20 depict the maximum bending moments in the cross-section of each bridge pile at the instant of maximum horizontal displacement of the foundation for CU17 and CU85 seismic motions. These figures compare the results for cases TBU_A, TBU_B, TBU_C, TU_A, TU_B, TU_C, and BU, considering interaction distances of 1D, 2D, and 3D (where D is the tunnel diameter). The results show that the presence of the tunnel in certain positions (cases TBU_B, TBU_C, TU_B, and TU_C) leads to an increase in the bending moments compared to the case without the presence of the tunnel (i.e., BU cases). This effect is more significant at the middle length of the pile and more noticeable for the CU17 seismic motion, which has a higher frequency content. On the other hand, when the tunnel is located directly beneath the bridge (cases TBU_A and TU_A), there is a reduction in bending moments for both seismic motions. Interestingly, the variations in bending moments are almost negligible for CU85 motion, which suggests that the frequency of the excitation plays a critical role in the distribution of internal shear forces and bending moments along the piles.

6. Conclusions

The results gathered from this numerical study provide valuable insights, and lead to design and retrofitting recommendations for earthquake-resistant structures in densely populated areas, where underground (i.e., tunnels) and on-ground structures (i.e., bridges and buildings) are often close to each other. In comparison with previous studies, the current research introduces a robust methodology, employing 3D numerical simulations, considering a suite of motions with varying frequency content, seismic intensity, and duration, to evaluate the seismic response of bridge–building–tunnel systems during intraplate and interplate events. In addition, the study introduces the concept of interaction factor as a function of the frequency range in which the energy of the excitation is focused as a structural demand parameter. This provides a new metric for assessing the interaction effects in complex infrastructure systems.

The main findings revealed that seismic interactions among the structures in soft-soil conditions are highly frequency-dependent. Thus, remarkable differences were observed between the seismic responses generated by intraplate and interplate events. For all the scenarios studied, interaction effects were strongest at high frequencies (>1 Hz), with wave scattering and dispersion as the main contributors to the interaction mechanism, particularly when the tunnel was located beneath the bridge or the building. Focusing on the bridge performance, it can be observed that the tunnel induces an amplification potential up to 20% when is located beneath the bridge; nevertheless, such an effect is exceeded by the presence of the building that generates an increase in the bridge column drift up to 60% when it is located one diameter away. The seismic demand on the pile foundation is also affected by the tunnel’s proximity, leading to an increase in the shear and bending moment at the pile as the tunnel gets closer to the bridge. Regarding the seismic performance of the building, the presence of the tunnel extensively affects the seismic demand, with an increase of up to 10% in relative displacements and greater accelerations observed at the roof level. Such effects were most notable when the tunnel was beneath the building, and for the interplate earthquakes, which exhibited a frequency content with energy concentrated around the fundamental period of the structure (Tf = 1.0 s (1.0 Hz)) and the characteristic elastic site period of the soil. On the other hand, minor interaction effects were observed in the presence of the bridge. In fact, the change in the seismic demand was less than 6% in comparison to the case of the building alone. This was associated with the difference in mass of both structures, with the building having five times the mass of bridge. Another key factor is the actual distance between each component.

Particularly, for the case studied, such interaction effects tend to increase when the system is subjected to high-frequency seismic ground motion (larger than or equal to 1 Hz) and when the structures are closer to each other. Thus, such key variables should be considered in densely populated areas for the analysis of earthquake-resistant structures, when similar conditions are presented, particularly in soft soils.

Although this paper partially fills the gap in information regarding the seismic interaction among on-ground and underground structures, further research is needed as this problem will become increasingly common in future infrastructure projects in large, densely populated cities. While numerical modeling can provide a robust methodology for exploring a wide range of scenarios, physical experimental validation will always be required to enhance the predictive capabilities of the numerical models. To deal with these shortcomings, instrumentation of actual structures must be conducted. Furthermore, mitigation strategies for detrimental interaction, based on isolation devices or frequency-dependent barriers using innovative materials, are another subject that is demanding attention.