An Analytical Solution for Seismic Interaction between Urban River-Canyon Topography and Nearby Buildings

(e interaction between urban river-canyon topography and the river-side building is investigated by using a whole analytic model of a semicircle river-canyon and a shear wall supported by a semicircle rigid foundation embedded in a homogenous halfspace. (e closed-form analytical solution for system response is presented based on the wave function expansion method. (e analysis focuses on the effects of the canyon-building interaction on system response. (e strength of the interaction between the river-canyon topography and the building changes periodically as the distance between the canyon and the structure increases, leading to the interaction having beneficial or harmful effects on the building’s seismic response.(e foundation peak response of the building can be amplified by about 10%, and the peak of the building relative response can be amplified by about 40%. (e distribution of canyon-structure spacing with strong or weak interaction is closely related to the dynamic characteristics of the building and the incident angle of the wave.When designing buildings along the river, the building and canyon should be analyzed as a whole model to determine whether the location of the building is in a position with strong interaction with the river-canyon. (e model in this paper may be useful for obtaining insight into the effects of canyon-structure interaction and interpreting the observed response in buildings and seismic response estimation in general.


Introduction
is paper investigates the interaction between urban rivercanyon and river-side building by using an analytical model. e work in this article is an extension of Trifunac's researches on the seismic response analysis of a single building [1] and a single canyon [2]. e problem studied in this paper involves the seismic interaction between urban local irregular topography and nearby buildings, which is currently termed as site-city interaction [3][4][5].
In the past, the amplification effect of canyon topography on ground motions received widespread attention [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. However, scholars considered this amplification effect of canyon topography on building responses only through studying the ground motion characteristics of canyon sites rather than directly seeing the canyon and the nearby buildings as a whole model to study the interaction. Actually, when a strong earthquake occurs, seismic waves scatter on the canyon boundary and the surface of the building foundation. eir scattering wave fields will couple with each other, forming the complex canyon-structure interaction system. erefore, the ground motion characteristics of the canyon site and the building seismic response are no longer determined only by the seismic response characteristics of the canyon site. e correct results may not be obtained if only the canyon topography and the surrounding soil are taken out for analysis. It is reasonable to analyze the river-canyon, the soil, and the buildings along the river as a whole model. is paper proposes a two-dimensional overall analytical model of the interaction between the canyon topography and the buildings along the river. is model consists of a 2D infinite long shear wall supported by a rigid foundation along a river-canyon embedded in a homogenous half-space, subjected by incident SH wave. e closed-form analytical solution of the canyon-structure interaction system response is obtained. e closed-form analytical solution is further used to reveal the interaction mechanism of the canyon-structure system.
In the next section, the methodology is presented, followed by the verification through comparison with the published results. en, the results in the frequency domain are presented and the anti-plane building response and canyon motion are discussed. Finally, the main findings and the conclusions are summarized.

Analytical Model.
e dynamic interaction model of canyon-structure, as shown in Figure 1, consists of a rivercanyon and a shear wall, infinitely long in the y-direction, standing on a rigid foundation embedded in a half-space. With the points of o 1 and o 2 as the origin, there are two sets of coordinate systems. e rectangular coordinates and the corresponding cylindrical coordinates are drawn. e y-axis of the rectangular coordinate system is perpendicular to the paper (points to the reader). e river-canyon and the rigid foundations are semicircular with the centers at points of o 1 and o 2 , respectively, and the radii are a and b, respectively. e half-space and the shear wall are assumed to be linearly elastic, uniform, and isotropic, and the boundary between the rigid foundation and the half-space is assumed to be tightly connected. e half-space is marked with shear modulus μ, shear wave velocity β, and mass density ρ. Within the unit length along the y-axis, the mass of the rigid foundation is M 0 . e shear wall has shear wave velocity β b , height H, undamped natural frequency ω b , and mass M b per unit length in the y-direction. e distance between the canyon center o 1 and the foundation center o 2 is D. e excitation is an incident plane SH wave with circular frequency ω, incidence angle c measured from the vertical zaxis, and unit amplitude.
To illustrate the canyon deformation, Figure 2 shows 5 observation points on the canyon which are chosen to compute their motions. ese are point 1 4), and point 5 (r 1 � a, θ 1 � − π/2), with point 3 being at the bottom of the canyon.

Governing Equations and Solution.
In the local coordinate system (x 1 -y 1 -z 1 ) of the canyon, the motion in the half-space along the y-axis direction caused by the incident SH wave can be written as where c x and c z represent the phase velocities of waves along the x-and z-axes, respectively (2) e free-field motion consists of the incident wave v i and the reflected wave v r on the surface of half-space.
In the area near the canyon and foundation, the wave field is composed of three parts, which are (A) the free-field motion expressed by equation (3), (B) the scattering field of waves caused by the canyon boundary at r 1 � a, and (C) the scattering field generated by the rigid foundation boundary at r 2 � b. Since (B) and (C) represent the wavefield radiating outward in the half-space, the form of their final solution must be the same. e time factor exp(iωt) is omitted in the following solutions.
All the wave motions v i+r , v R 1 , and v R 2 must satisfy the wave differential equation where v stands for the three wave motions v i+r , v R 1 , and v R 2 , respectively. e three wave motions should satisfy the boundary conditions given by In the local coordinate system (r 1 -o 1 -θ 1 ) of the canyon, the free-field motion v i+r in the half-space along the y-axis caused by the incident SH wave can be expanded into Fourier-Bessel series as Similarly, in the local coordinate system (r 2 -o 2 -θ 2 ), the free-field motion v 1+r can also be expanded into 2 Shock and Vibration Here, J n (x) is the Bessel function of the first kind with argument x and order n, and k � ω/β. e scattering wave motions near the canyon and the two foundations can be written as Here, v R 1 and v R 2 are the scattering wave motions from the canyon and the foundation, respectively, and H (2) n (x) is the Hankel function of the second kind with argument x and order n.
Equations (8)- (11) can be proved to satisfy equation (4) and the boundary condition equation (5). A n , B n , C n , and D n are the unknown complex coefficients that should be determined by the boundary conditions equations (6) and (7).

Shock and Vibration
Δ is the effective input motion of the rigid foundation to the superstructure. Because equations (10) and (11) are expressed in different local cylindrical coordinate systems, equations (10) and (11) should be transformed between any two local cylindrical coordinate systems as shown in Figure 1. Based on Graf's Addition eorem [25,26], equations (10) and (11) can be transformed between any two cylindrical coordinate systems in Figure 1. e details of the derivation of the coordinate transformation formula are omitted.

Shock and Vibration
where, in equation (18e), replacing C with H stands for coordinate transformation for Hankel function. e foundation effective input motion Δ can be determined by the dynamic equilibrium equation of the rigid foundation.
where, f b z , the action per unit length of the shear wall on the foundation, is given by equation (20). f s z is the action of the soil on the foundations, given by equation (21).
By submitting equations (20) and (21) into equation (19), it can be shown that e unknown complex coefficients A n , B n , C n , D n , and Δ are determined by solving the linear equations (12)- (17) and (22) together, and the closed-form analytical solution of the problem can be obtained. Particularly, the relative responses of the shear wall to the foundation can be expressed as To compare with the published results, Δ and Δ b are normalized by the surface displacement amplitude of freefield ground motion as

Impedance Function of the Rigid Foundation.
According to the definition of impedance function, the foundation has unit displacement response (Δ � 1) at this time. e wave field in the half-space only has canyon scattering field v R * 1 and foundation scattering field v R * 2 , but does not include free-field wave motion, e boundary conditions are Shock and Vibration 5 Trifunac 1972 This paper Trifunac 1972 This paper Trifunac 1972 This paper Trifunac 1972 This paper Trifunac 1972 This paper Trifunac 1972 This paper  Shock and Vibration Compared with the derivation in Section 2.2, the unknown coefficients A * n , B * n , C * n , and D * n can be obtained by solving the linear equations (12)∼(17) and (22) but removing the terms that correspond to the free-field. After that, the foundation impedance function K yy can be obtained by integrating the stress on the foundation boundary.
After the normalization by the shear modulus of the halfspace, the foundation impedance function K yy can be further written as follows: where k yy and c yy are the dimensionless stiffness and radiation damping coefficients.

Dimensionless Parameters.
e results in this paper can be seen as functions of the governing dimensionless parameters, which are expressed in terms of the characteristic length of the problem, a, and reference wave velocity, β. e dimensionless frequency η is defined as [27] η where 2a is canyon width and λ is the wavelength of the shear waves in the half-space. Further, based on [28], the dimensionless parameter ε which describes the relative flexibility of the shear wall to surrounding soil is defined as e dimensionless parameter ε has values 0 ≤ ε < ∞. For a given ε, the first-order vibration frequency of the shear wall is at η b1 � 1/(2ε). For a very stiff structure and/or very soft soil, ε � 0 (corresponding to a rigid shear wall added as a rigid mass to the foundation), and for a very flexible structure and/or rigid soil, ε ⟶ ∞.
Other dimensionless parameters used in this paper are the dimensionless distance between the structure and the canyon, D/a, and the ratios, M b /M 0 and M 0 /M S , where M S is the mass of the soil per unit length in the y-direction that has been replaced by the foundation.

Verification.
To show the verification of our solution, we set the canyon radius a equal to the foundation radius b and the dimensionless distance between the canyon and the foundation D/a tends to be infinite, to verify whether our results of the river-side building and the canyon are consistent with the published results that correspond to single building [1] and single canyon [2].   Figures 5-10, respectively, show foundation impedance function, foundation responses, the shear wall's relative response, and the canyon response, at different canyon-structure spacing. For easy comparison with the published results, the responses of single structure [1] and single river-canyon [2] are also shown in each subfigure. e parameters used in this section are as follows. e radius of the canyon is equal to the foundation radius; that is, a � b. e dimensionless distance D/a between the canyon and the river-side building (referred to as "canyonstructure spacing" for short) D/a � 2.5, 3, 4, 5, and 10. e ratios of the foundation mass densities to the half-space mass density are 1/2, which are equivalent to M 0 /M S � 1/2. e mass ratio of the shear wall to the foundation per unit length is M b /M 0 � 1, 2, and 4. e stiffness parameters of the shear walls are ε � 0, 2, and 4. e incident angles of the waves are c � 0°(vertical incident) and 90°(horizontal incident). Figure 5, the stiffness coefficient k yy and damping coefficient c yy of the foundation impedance function are highly related to the   canyon-structure spacing (D/a) and the frequency of the incident wave (η). First, the stiffness coefficient k yy of the foundation impedance function will fluctuate strongly with different canyon-structure spacing and the frequency of the incident wave. For the closer distance between the canyon and the structure (D/a � 2.5, 3, 4, and 5), as the frequency of the incident wave increases, the stiffness coefficient k yy changes more widely. e change in the damping coefficient of the foundation impedance function is similar to the stiffness coefficient k yy . When the frequency of the incident wave is small, the damping coefficient c yy is roughly the same as that of the single rigid foundation [29]. As the frequency of the incident wave increases, the damping coefficient c yy will also fluctuate around the damping coefficient of the single rigid foundation impedance function (D/a ⟶ ∞). However, the amplitude of the fluctuation of the damping coefficient c yy is smaller than that of the stiffness coefficient k yy .

Building Responses for Different Canyon-Structure
Spacing. Figure 6 shows how the foundation response varies with the canyon-structure spacing (D/a) under different incident angles (c). Observing Figure 6, we can see that, under different canyon-structure spacing, the difference between the peak of foundation response and the peak of single foundation response is small. is can be explained as follows. First, when the frequency of the incident wave is small, the wavelength of the wave is longer, resulting in strong wave diffraction. erefore, even if the foundation is close to the canyon, the incident wave cannot feel well the existence of the canyon due to wave diffraction, which leads to a weak interaction between the canyon and the foundation. e foundation vibrates like a single foundation, so the peak of the foundation response has a small difference from that of the single foundation. Secondly, when the distance between the canyon and the structure is large, the system interaction is also naturally weaker. From this, the two factors that determine the strength of the interaction between the canyon and the foundation can be known: the canyon-structure spacing and the wavelength of the incident wave (that is, the frequency of the incident wave). e proof for this is as follows: for example, when η � 0.5, the wavelength of the incident wave is λ � 4a. For canyon-structure spacing D/a � 2.5, 3, 4, and 5, the wavelength is the same order of magnitude and close to the canyon-structure spacing. erefore, foundation response corresponding to these locations under η � 0.5 deviates significantly from that of the single rigid foundation. It can be seen from Figure 7 that the canyon-structure interaction has a great influence on the peak of structure relative response, which is mainly reflected in the case of structural stiffness ε � 2. As shown in Figure 7(a), when ε � 2 and D/a, for different superstructure masses M b /M 0 � 1, 2, and 4, the peaks of the structure relative responses are 10.910, 5.377, and 2.648, respectively, while the corresponding peaks of the single structure relative response are 7.825, 3.857, and 1.893, respectively, which are enlarged by 39.4%, 39.4%, and 39.9%, respectively. Besides, observing the results under different incident angles (c), it can be found that when the wave is incident obliquely (c � 90°), the peak of the structure relative response has a small difference from the corresponding result of the single building. However, when the wave is incident perpendicularly (c � 0°), the difference is large. e influence of the canyon-structure interaction on the structure relative response is mainly reflected in the case of vertical incidence. It can be concluded from this that another factor that determines the strength of the interaction between the canyon and the river-side building is the incident angle of the wave. e influence of the incident angle (c) on the system interaction cannot be ignored.
e deviation magnitude of the foundation response in Figure 6 and the structure relative response in Figure 7 from the single foundation result under the horizontal incidence are significantly smaller than those of the vertical incidence of the wave. is phenomenon can be called the "shielding effect." In other words, when the wave is incident obliquely, the left canyon has the effect of shielding the incident wave reaching the foundation, leading to weak foundation response and structure relative response.

Peak of Structure Relative Response under Different
Canyon-Structure Spacing. Figures 8 and 9, respectively, show the variation of the foundation peak response and the shear wall's peak relative response with the canyon-structure spacing (D/a). As shown in Figures 8 and 9, both the foundation peak response and the shear wall's peak relative response show an attenuation trend as the canyon-structure spacing increases, but the foundation peak response and the shear wall's peak relative response are always greater than those results of single building model [1] ( the gray line in Figures 8 and 9). is phenomenon is still obvious even when the distance between the canyon and the structure is very large (D/a � 40∼50). is illustrates that even when the canyon and the structure are far away, there still is an obvious interaction between them. e peak of the foundation response can be enlarged by approximately 10% (corresponding results of ε � 2 and M 0 /M b � 2 in Figure 8), and the peak of the shear wall's relative response can be amplified by approximately 39.4% (corresponding results of ε � 2 and M 0 /M b � 1 and 2 in Figure 9). e degree to which the color line deviates from the gray line in Figure 9 can be regarded as the degree of interaction between the river-canyon and the building. It can be seen from Figure 9 that the strength of the interaction between the river-canyon topography and the building changes periodically as the distance between the canyon and the structure increases, leading to the interaction having beneficial or harmful effects on the building's seismic response. In many valley-structure intervals, the results of the valley-structure interaction model are greater than the results of single building model [1]. e distribution of valley-structure spacing with strong or weak interaction is closely related to the dynamic characteristics of the building itself and the incident angle of the wave. erefore, when designing buildings along the river, the building and canyon should be modeled and analyzed as a whole to determine whether the location of the building is in a position with strong interaction with the river-canyon. Figure 10 shows the responses of canyon observation points under different incident angles and canyon-structure spacing. When the waves are vertical incident, because point 5 of the canyon is the closest to the building, the response of point 5 is the strongest, especially when the building is close to the river (D/a � 2.5, 3, 4, and 5). When the wave is incident obliquely, since point 1 is on the side of the incoming wave, the response of point 1 becomes the strongest, while the dynamic response at point 5 is less than that at point 1 because the river-canyon shields the wave.

Canyon Response.
Another important phenomenon is that canyon response is affected by the dynamic characteristics of the buildings, especially for small canyon-structure spacing (D/a � ≤5). For example, the response of point 5 becomes zero at the fundamental frequency of the building. is phenomenon is very similar to the foundation response; that is, at the fundamental frequency of the building, the foundation response tends to be zero. As mentioned in Section 3.1, for given ε, the first-order vibration of the structure is at η b1 � 1/(2ε). In Figure 10, since the structural stiffness parameters are ε � 4, the first-order vibration frequency of the building is η b1 � 0.125. It can be seen from Figure 10 that, for D/a � <5 and η � 0.125, the response of point 5 tends to be zero. Even when the buildings are all far from the canyon (D/a � 10), this trend can still be observed.

Conclusions
is paper established a 2D model for the anti-plane dynamic interaction between the urban river-canyon and the river-side building and presented a closed-form analytical solution for the system response based on the wave function expansion method.
e main features of the interaction between the river-canyon and the building along the river are investigated by the analytical solution. e conclusions and findings are as follows: (1) e canyon-structure interaction depends not only on the distances between the canyon and the riverside building but also on the frequency of the incident wave and the incident angle. When the wave is incident obliquely, the canyon on the incoming wave side has the shielding effect on the incident wave. (2) e strength of the interaction between the rivercanyon topography and the building changes periodically as the distance between the canyon and the structure increases, leading to the interaction having beneficial or harmful effects on the building's seismic response. e results in this paper show that the foundation peak response of the building can be amplified by about 10%, and the peak of the building relative response can be amplified by about 40%.
(3) e distribution of valley-structure spacing with strong or weak interaction is closely related to the dynamic characteristics of the building and the incident angle of the wave. When designing buildings along the river, the building and canyon should be modeled and analyzed as a whole to determine whether the location of the building is in a position with strong interaction with the river-canyon. (4) e canyon response of the canyon is closely related to the frequency of the incident wave, the canyonstructure spacing, the incident angle of the wave, and the dynamic characteristics of the building along the river.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that the work is original research that has not been published previously, and not under consideration for publication elsewhere. No conflicts of interest exist in the submission of this paper.

Authors' Contributions
is study was approved by all authors for publication.