The reduced-order model consists of a set of n buildings (e.g., a model of a city blocks), distributed over a shared soil stratum, as shown in Figure 1. A known ground displacement field x g and y g is applied in the x - and y - direction respectively, at all foundations. Each superstructure can be modelled as low-order model with translational DOF x i and y i , and their sway-flexural lateral stiffness k b x i and k b y i . The wave passage effects, coherence effects, and spatially heterogeneous ground displacements are neglected in the presented work. We only aim to model the important rotational interactions between the buildings.

Each building’s foundation has two orthogonal auto-rotational springs k x i and k y i and various interbuilding-rotational springs defined as a vector κ i j = κ x i j , κ y i j , κ x y i j , κ y x i j (i.e., the coupling between building ″ i ″ and building ″ j ″ ), where κ x i j is the interbuilding-rotational stiffness coefficients in the x -direction, κ y i j is the interbuilding-rotational stiffness coefficients in the y -direction, and κ x y i j , κ y x i j are the cross-coupled terms. Note that the springs k x i , k y i and κ i j represent the complete dynamic behavior soil stratum, where only rotational DOFs at the foundation level θ x i , θ y j are retained. The system geometry and nomenclature used here are described in Figure 2. The kinematic interaction is not considered in the present work, i.e., the transfer function between foundation input motion and free-field motion is one.

The potential energy of the complete system is given by Eq. 1 and is calculated by the sum of: (i) the internal work due to auto-rotational springs, and (ii) the internal work due to inter-rotational springs κ x i j , κ y i j and the cross-coupled term κ x y i j and κ y x i j . The kinetic energy of the complete system is defined by Eq. 2, where each term corresponds to: (i) translational kinetic energy due to sway and foundation rotation of each building’s mass, (ii) the rotational energy of each soil/foundation mass. Note that, in the particular case where the foundation is square, the term collapse to κ x i j = κ x j i , κ y i j = κ y j i , and κ x y i j = κ y x i j . U = 1 2 ∑ i = 1 n k x i θ x i 2 + k y i θ y i 2 + 1 2 ∑ i = 1 n − 1 ∑ j = i + 1 n ( κ x i j θ x j − θ x i 2 + κ y i j θ y j − θ y i 2 + κ x y i j θ y j − θ x i 2 + κ y x i j θ x j − θ y i 2 ) (1) T E = 1 2 ∑ i = 1 n ( m b i x ˙ i + x ˙ g − h i θ y i 2 + m b i y ˙ i + y ˙ g − h i θ x i 2 + m b i y ˙ i − h i θ x i 2 + m s i r i 2 θ x i 2 + θ y i 2 ) (2) where h i is the height, m b i is the total lumped modal mass of building i and m s i is the foundation/soil mass underneath building i . k b i is the modal building lateral stiffness and r i is the soil/foundation mass radius of gyration, and n is the number of buildings. Hence, the Euler-Lagrange equations of motion describing the dynamics of the discretised system of Figure 3 can be derived in the standard way by calculus and is written in matrix form, as follows, M x ¨ + C x ˙ + Kx = p 1 x ¨ g + p 2 y ¨ g (3) where the mass matrix M (Eqs 4 and 7), the damping matrix C , the stiffness matrix K (Eq. 8), the force vectors p 1 , p 2 (Eqs 5 and 9) and the DOFs vector x of the complete system of n buildings are stated as follows, M = M ^ 1 ⋯ 0 ⋯ 0 ⋮ ⋱ ⋮ ⋱ ⋮ 0 ⋯ M ^ i ⋯ 0 ⋮ ⋱ ⋮ ⋱ ⋮ 0 ⋯ 0 ⋯ M ^ n , K = K ^ 1 ⋯ κ ^ 1 i ⋯ κ ^ 1 n ⋮ ⋱ ⋮ ⋱ ⋮ κ ^ i 1 ⋯ K ^ i ⋯ κ ^ i n ⋮ ⋱ ⋮ ⋱ ⋮ κ ^ n 1 ⋯ κ ^ n i ⋯ K ^ n (4) p = p ^ 1 ⋯ p ^ i ⋯ p ^ n T , x = x ^ 1 ⋯ x ^ i ⋯ x ^ n T , x ^ i = x i y i θ y i θ x i T (5) C = M ∑ j = 1 4 n 2 ξ j ω j ϕ j T M ϕ j ϕ j ϕ j T M (6)

The system’s linear viscous damping matrix C assumes that each natural mode is damped at ξ j = 0.05 of the critical damping, ϕ j is the modal eigenvector of the mode j , ω j are the natural frequencies of the system. Thus, the Caughey orthogonal damping matrix C can be calculated as (Clough and Penzien, 1993), by Eq. 6. Note that the damping matrix refers to the coupled system. The fundamental frequencies ω j of the coupled system do not change very much compared to the uncoupled system (Vicencio and Alexander, 2018b), with a maximum of 9% variation in the natural frequencies. Hence, the damping matrix does not vary substantially between the SSSI and SSI systems.

The global mass matrix M corresponds to a diagonal block matrix, where each different blocks M ^ i represent the mass matrix for each building i . M ^ i = m b i 0 − m b i h i 0 0 m b i 0 − m b i h i − m b i h i 0 m b i h i 2 + m s i r i 2 0 0 − m b i h i 0 m b i h i 2 + m s i r i 2 (7)

The global stiffness matrix K includes the interaction effects between the buildings (SSSI). Therefore, the diagonal block terms K ^ i (of the global stiffness matrix K ) correspond to the stiffness matrix for each building i , including the additional stiffening effect of the adjacent footings. The off-diagonal block terms κ ^ i j represent the interaction between the buildings i and j . Note that all the interaction between buildings are considered. K ^ i = k b i 0 0 0 0 k b i 0 0 0 0 k x i + ∑ j = 1 n κ x i j + κ x y i j 0 0 0 0 k y i + ∑ j = 1 n κ y i j + κ x y i j , κ ^ 1 i = 0 0 0 0 0 0 0 0 0 0 − κ x 1 i − κ x y 1 i 0 0 − κ x y 1 i − κ y 1 i (8)

Finally, the global excitation vectors p 1 , p 2 are assembled by the block vector p ^ i of each building i in both directions. p ^ i x = − m b i 0 − m b i h i 0 T , p ^ i y = 0 − m b i 0 − m b i h i T (9)

In this paper, the dynamic properties of each building “ i ”, which is required in the Eq. 3, are deduced according to the following assumptions,

• The fundamental natural period of the structure on a rigid foundation (i.e., with no foundation/soil rotation) can be defined as T x i = T y i = T i = c 0 h i 3 / 4 . This empirical form is adopted by the approximate empirical relationship proposed by the Euro Code 8 (British Standards Institution, 1996). In this equation the height of the building h i is taken in meters, and the factor equal to c 0 = 0.075 , corresponding for reinforced concrete moment resisting frames.

We are interested in evaluating the change in the response between the coupled (structure-soil-structure interaction SSSI) and the uncoupled (soil-structure interaction, SSI) system. Initially the SSSI solution of Eq. 2 is calculated through time-history analysis. Then, the SSI response of each building is evaluated, where the dynamic analysis is evaluated by setting the inter-rotational springs κ x i j , κ y i j and the cross-coupled term κ x y i j and κ y x i j equal to 0. As a measure of change in the response between SSSI and SSI, we will employ the displacement U i (horizontal sway + rocking, Eq. 10) and the total acceleration A i (horizontal sway + ground + rocking, Eq. 10) for the top of the building i (in the x-direction), denoted by, U i = x i − h i θ y i , A i = x ¨ i + x ¨ g − h i θ ¨ y i (10)

In addition, it is valuable to characterize the change in total power caused by the multiple SSSI among the buildings. So, the percentage change in total power χ ¨ i for the building i , when using the uncoupled SSI analyses rather than coupled SSSI analyses is expressed in terms of the total power spectral densities E P S D A i , χ i = 100 E P S D U i S S S I E P S D U i S S I − 1 , χ ¨ i = 100 E P S D A i S S S I E P S D A i S S I − 1 (11) where E P S D U i and E P S D A i is based on the average of the square Fourier Transform of all data points of the response acceleration time-series U i and A i . The change of power χ i and χ ¨ i would be zero if there is no difference in overall response power between SSSI and SSI analyses. Using the Eq. 11 as a comparative metric, delivers a statistical estimate of magnitude that is more robust than employing a single peak of the function (displacement or acceleration).

To determine the effect of SSSI on the system described in Eq. 3 we consider fifty records that are taken from 25 events that occurred between 1971 and 2007 (see Figure 4), from the Pacific Earthquake Engineering Research (PEER-NGA) West database. Nine of records occurred in California (namely, San Fernando, Imperial Valley, Coalinga, Morgan Hill, Hector Mine, Whittier Narrows, Superstition Hills, Loma Prieta, Northridge and Parkfield), and six of them are taken from different places around the world (namely, Kocaeli (Turkey), Chi-Chi (Taiwan), Duzce (Turkey), Irpinia (Italy), Kobe and Chuetsu-oki (Japan)). Each record has two horizontal components. Event magnitudes range from M w = 6.5 to M w = 7.6 with an average magnitude of M w = 7.0 . Values of their peak ground accelerations (PGAs) vary from 0.21 g to 0.82 g with a mean value of 0.43 g. All ground motions were recorded on weak soils, which correspond to sites of an average shear wave velocity of less than 180 m/s, i.e., loose sand. Figure 4 displays the elastic response spectrum for all the records and their mean.

To evaluate the effect of cluster interaction between the buildings, three types of building cluster with the same building heigh are considered, as shown in Figure 5 (Layout 1: 3 × 3 city blocks of nine equispaced identical buildings, Layout 2: 4 × 4 city blocks of sixteen equispaced identical buildings, Layout 3: 5 × 5 city blocks of twenty-five equispaced identical buildings). The fundamental natural period of the structure on a rigid foundation (i.e., with no foundation/soil rotation) is T x i = T y i = T i = 0.5 s . The buildings have the same square plan area, with a height to width ratio equal to h i / b = 1.5 . The centre-to-centre interbuilding distances are equispaced at ∆ x i = ∆ y i = 1.2 b . The system is subjected to all earthquake events (fifty in total) in both directions simultaneously (East-West and North-South).

Figure 6 displays the variation of change in power for the displacement on top of the buildings, corresponding to the configuration Layout 1. The results are shown as the mean for Far-Field (FF), Near-Field Without Pulse (NFWP), and Near-Field Pulse Like (NFPL), in order to evaluate the changes depending on the different types of ground motion. The maximum increase in total power response occurred at the centre of the cluster (building 5), with a maximum of χ 5 = 12.9 % , corresponding to the Far-Field records. In addition, there is a reduction in the response for some buildings, with a maximum for building 7 ( χ 7 = − 21.2 % ) to the Near-Field Without Pulse records.

Figure 7 shows the variation of change in power for the acceleration on top of the buildings. Similar to the displacement, the maximum increase in total power response occurred at the centre of the cluster (building 5), with a maximum of χ ¨ 5 = 22.6 % , corresponding to the Far-Field records. In addition, there is a reduction in the response for some buildings, with a maximum for building 7 ( χ ¨ 7 = − 21.2 % ,).

Figures 8A, B show the uncoupled SSI (blue line) and coupled SSSI (red line) response for the top of building 1 and 5, and Figures 8C, D depict the corresponding power spectral density for the total acceleration, considering the Supertition Hill earthquake. Comparing the seismic response, it is clear that building 1 is affected by the surrounding buildings, where the change in power is χ ¨ 1 = − 23.8 % . On the other hand, the building 5 (at the center of the city block), there is an amplification of χ ¨ 5 = 16.9 % .

As we can observe in the previous figure, the amplification/reduction of change in power (displacement and acceleration) between the three different earthquake-type events (FF, NFPL, and NFWP) are similar and follow equivalent trends for maximum values. Therefore, from now on we are plotting the mean of all earthquakes. Figures 9, 10 display the variation of change in power for the displacement χ i % and acceleration χ ¨ i % on top of the buildings, respectively.

The maximum increase in total power response occurred at the centre of the cluster, with a maximum of χ ¨ 6 = 6.79 % for the acceleration. In the case of the displacement, there is a reduction in the response of χ 6 = − 4.25 % .

Figure 11 shows the variation of change in power for the acceleration χ ¨ i % on top of the buildings, corresponding to the configuration Layout 3 (5 × 5 city blocks of twenty-five equispaced identical buildings and the same height). In this case, there is an amplification of χ ¨ 13 = 24 % at the center of the city block, and a reduction of χ ¨ 13 = − 29 % at the corner of the city block. These results are consistent with the transfer functions shown by (Isbiliroglu et al., 2015), where the change in response is calculated in different clusters with variable number of buildings and spacing. As mentioned in the same work, the SCI effects increase as the number of structures increases and the separation between buildings decreases. This is why, in this paper we did not consider larger interbuilding separations (the SCI effects decreases).

Here we evaluate the effect of cluster interaction between buildings with different height. As before, three types of building cluster are considered (Layout 1: 3 × 3 city blocks of nine equispaced identical buildings, Layout 2: 4 × 4 city blocks of sixteen equispaced identical buildings, Layout 3: 5 × 5 city blocks of twenty-five equispaced identical buildings). The fundamental natural period of the structure on a rigid foundation (i.e., with no foundation/soil rotation) covers a range of T x i = T y i = 0.2 s − 1.0 s . All the buildings have the same square plan area, therefore the centre-to-centre interbuilding distances are equispaced at ∆ x i = ∆ y i = 1.2 b . The system is subjected to all earthquake events (fifty in total) in both directions simultaneously (East-West and North-South).

Figure 12 shows the variation of change in power for the acceleration χ ¨ i % on top of the buildings, corresponding to the configuration Layout 2 (4 × 4 city blocks of sixteen equispaced identical buildings), where the four central buildings are taller. In this case, due to the differences in height, there is an amplification at the edges of the city blocks, with a maximum of χ ¨ 5 = 20.1 % . This highlights the complexity of the interaction, and the need to evaluate the interaction for each particular case, especially when there are important differences between the heights of the buildings.

Finally, Figure 13 depicted the variation of change in power for the acceleration χ ¨ i % on top of the buildings, corresponding to the configuration Layout 3 (5 × 5 city blocks of twenty-five equispaced identical buildings and different height). As the previous discussed, there is a transfer of energy between the taller buildings, to the shorter buildings ( χ ¨ 7 = 15.6 % ). These results highlight the relevance of studying the seismic interactions between the buildings and consider the site-city interaction effects, especially in highly dense urban areas. Future studies should consider different buildings configurations, and nonlinear effects. In the same way, data from instrumented buildings are required in order to validate this news numerical simulations.