It is also known as the multi-step approach. In this approach, the soil and the structure are considered separately, then solved independently and the final seismic response of the structure is calculated by combining these effects using superposition principles (Kramer, 1996). In other words, the substructure approach has the capability to solve inertial and kinematic interactions separately by means of impedance and transfer functions respectively. Since the substructure technique is based on superposition principles, linear assumption of the soil and the structural behaviour is necessary for the substructure approach. Although, this problem has been solved through equivalent-linear and non-linear numerical models developed for the soil domains, such as beam on non-linear Winkler foundation model for shallow foundations (Raychowdhury, 2008) and non-linear spring models for pile foundation (Boulanger et al., 1999). Regardless of the foundation type and the soil model, the non-linear behaviour of the superstructures could be also modelled using simplified constitutive models (Hassani et al., 2018). The solution of SSI problems with the substructure approach is broken down into three main steps as depicted in Figure 1:

1) Evaluation of the Foundation Input Motion (FIM); this includes evaluating the transfer functions to convert the Free-Field Motion (FFM) to FIM that is based on the stiffness and geometry of the foundation and the soil. To evaluate FIM, the structure and foundation are supposed to be massless. The applied seismic demand to the foundation is represented by rotational (θ _FIM ) and translational (U _FIM ) components. The ratio of FIM to FFM is named as a transfer function that expresses the effects of the kinematic interaction only and neglects the inertial interaction.

The direct approach is considered the most rigorous approach to solving SSI problems, especially in the case of complicated structural geometry and non-linear soil modelling. Figure 2 depicts a schematic illustration of the direct approach. Where the soil and the structure are modelled together and analyzed in a single step by solving the governing equations of motion. The soil is represented as a continuum and connected to the foundation with interface elements. Proper boundary conditions must be assigned so that the scattering waves do not reflect when they are passing through it. In this method, the complete system is solved in the time domain with free-field motion as input. Section 4.5 goes into more detail about the modelling approach. The direct approach is a very powerful method to solve a wide range of SSI problems including the linear and non-linear soil model for various foundation types. Although, in engineering practice direct method is rarely used because of its complexity and time-consuming analysis, especially for complex geometry and non-linear problems. The governing equation of motion for a soil-structure system can be written as follows: M u ̈ + C u ̇ + K u = − M u ̈ g (1)

Where [M], [C], and [K] are mass, damping, and stiffness matrices. üg is input ground acceleration. ü, ú, and u are acceleration, velocity, and displacement vectors of the system.

The main computational methods that have been used in SSI studies to discretize the soil domain are: Finite-Element Method (FEM), Finite-Difference Method (FDM), and Boundary-Element Method (BEM).

The FEM is the most well-known computational method in analyzing structural and geotechnical engineering problems. With the availability of powerful computers, numerical methods have been used in a broader range compared to analytical methods. FEM has a very good capability to address the real and complex problem of SSI with significant details, especially in the case of non-linear modelling. However, in the case of complete 3D models and complex structures, it is computationally costly. Nevertheless, FEM is used frequently in the study of soil-structure interaction and has produced some notable achievements in the field of SSI. Both FEM and FDM are discretizing the whole domain into finite numbers of small elements and connect these elements through the nodal points. In FEM the element matrices are combined in order to form a large global stiffness matrix. While this combination is usually not done in the FDM method, because in each time step the finite difference equations are regenerated more efficiently. This made FDM computationally more efficient than FEM. Desai et al. (1973) documented the general principles of FDM and FEM with their application cases.

Following FEM, a new numerical method called the boundary element method was developed. The basic idea of BEM is to discretize the surface of the problem instead of the whole domain and unlike the FEM, the equations of motion of the domain will be formulated in a form of integral equations instead of the differential equation. Furthermore, unlike FEM, BEM does not require the use of special sophisticated non-reflecting boundaries to satisfy the radiation criterion. As a result, the number of degrees of freedom and the computational time will be reduced. In this regard, BEM is more advantageous than FEM. The details on the formulation of BEM and fields of application are well documented (Brebbia et al., 1985). BEM presents an attractive computational framework, especially for problems in unbounded domains. However, in complex engineering problems, BEM is less efficient than FEM because of its difficulty of application in the case of a heterogeneous and anisotropic medium (Wolf and Deeks, 2004). In addition, BEM has limitations to consider non-linear problems (Lou et al., 2011).

Based on the discussion in Sections 3.1 and 3.2, there are several differences between the two approaches, especially in terms of accuracy and efficiency. Since in substructure approaches the semi-infinite soil medium does not need to be discretized, it is computationally more efficient and less time is required to solve the problem compared to the direct approach. Therefore, the substructure approach has a wider range of practical uses than the direct approach and many researchers have adopted this approach for the SSI analysis (Mylonakis et al., 2006; Givens, 2013; Hassani et al., 2018; Mercado et al., 2020; Oz et al., 2020). Moreover, unlike the direct approach, in the substructure approach the effects of inertial and kinematic interactions could be separated and calculated individually. However, a recent study by (Rahmani et al., 2016) questioned the accuracy of the substructure approach and observed an overestimation in top displacement and design base forces. In addition (Mercado et al., 2020), concluded that the substructure approach produces a larger inter-story drift demand in the structure compared to the direct approach. Jahromi et al. (2008), Jabini Asli et al. (2019) also concluded that the substructure approach cannot accurately deal with the material and geometric non-linearity. Therefore, in case of complex and important structures, it is required to use more sophisticated soil models (direct approach) where the soil and structural non-linearity are taken into account. Nevertheless, to draw a better conclusion on the performance of substructure models in various SSI problems in buildings, a more extensive study is required, especially in considering the effects of structural non-linearity.

In this model, the soil medium is idealized and represented by linearly elastic springs, which are distributed independently and discretely in close spaces along the foundation area, as shown in Figure 4. Since the springs are independent, the deformation of the foundation due to the applied load is localized in the loaded region only (Bowles, 1996). The relationship between load and deflection at any point in the soil is given by: p = k w (2)

Where p denotes the applied pressure, k is the coefficient of subgrade reaction and w is the deflection. Due to its simplicity, several studies in the area of soil-structure interaction have been conducted based on the Winkler hypothesis (Popov, 1951; Kramrisch and Rogers, 1961; Vesić, 1961; Daloglu and Vallabhan, 2000).

In a critical review study (Dutta and Roy, 2002), concluded that despite the simplicity and less computational time of this approach, the basic limitation is the consideration of linear stress-strain behaviour of the soil. Furthermore, the determination of the elastic springs used to replace the soil beneath is also a major issue. Because the coefficient of the subgrade is affected not only by the subgrade but also by the dimensions of the loaded area. Since the subgrade reaction is the only parameter in Winkler’s idealization, great care should be taken in the determination of this parameter. To overcome the shortcomings and limitations of this approach, several improved versions of Winkler’s model are available in the literature (Filonenko-Borodich, 1940; Hetenyi, 1946; Pasternak, 1954; Kerr and York, 1965).

Despite the limitations explained above (Dutta and Roy, 2002), recommended using Winkler’s hypothesis for practical purposes at least instead of the fixed-base considerations.

The concept of lumped parameter model represents frequency-dependent soil-structure interaction models of massless foundations resting on or embedded in homogeneous elastic half-space. In this model type, the soil domain is replaced with a set of frequency-dependent springs, dashpots and masses in the direction of degrees of freedom. The system is then could be solved in the frequency or time domains. As per (Wolf, 1994; Ibsen and Liingard, 2006) various versions of physical models based on the lumped parameter concept have been developed. The key differences between these versions are the arrangement and mathematical formulation of foundation impedances. In this section, two commonly used physical models from the lumped parameter family are presented.

Another simplified spring-type model to evaluate SSI effects and consider the non-linearity of the soil medium is the Beam on Non-linear Winkler Foundation (BNWF) model. This model is idealizing the soil domain by numbers of one-dimensional non-linear springs that are distributed along with the interface between soil and foundation (Raychowdhury, 2008). During strong earthquake events, the soil under the foundation is very likely to undergo plastic deformation due to the increase of pressure under the foundation which comes from the rocking and sliding motions of the foundation (Yang, 2016). Consequently, such reduction of contact between the foundation and the soil results in progressive loss of the system stiffness. To consider this phenomenon, the BNWF model provides a set of vertical and horizontal non-linear springs to measure the plastic deformations and energy dissipation in the soil domain. As illustrated in Figure 7 the springs that are distributed vertically along the length of the footing (q-z elements) are capturing rocking, uplift, and settlement. While the horizontal springs (t-x and p-x elements) are placed to capture the sliding and passive resistance of the soil respectively. At the end of the footing, the springs are intensified so that the rotational springs are accounted for appropriately.

The implementation of BNWF into OpenSees is presented by (Raychowdhury and Hutchinson, 2009) for a 2-dimensional structural model. The foundation is modelled as elastic beam-column elements, while the interface is defined by several independent zero-length Winkler springs. The constitutive relation of the (q-z, t-x, and p-x elements) are represented by non-linear backbone curves that were originally developed by (Boulanger et al., 1999) and calibrated by (Harden et al., 2005; Raychowdhury, 2008). The BNWF model is appropriate for every type of surface and embedded foundation. It could be also applied in the case of clay and sand soil deposits regardless of the structural model. Therefore, the model is applicable to a wide range of SSI problems in structural engineering.

The basic limitation of this approach comes from its one-dimensional nature. The spring responds to the loads in the direction of the springs only, any load acting perpendicular to the spring axis does not affect the response of the springs. However, due to the simplicity and abilities of a variety of problems, the BNWF approach is very popular and widely used by many researchers in the past 15 years (Shirzadi et al., 2020; Tahghighi and Mohammadi, 2020; Patro et al., 2021).

PBM is a relatively recent development for solving SSI problems, initiated by (Nova and Montrasio, 1991). The PBM family models aim to make a bridge gap between the BNWF and continuum models by combining elements from both formulations. These model types have the capability of capturing the non-linear response of the soil and rigid foundations. A developed version of PBM is presented by (Gajan and Kutter, 2009) and it is known as Contact Interface Model (CIM). This model uses a single interface element between the soil and rigid foundation to represent flexibility as well as energy dissipation in soil-structure interaction. From a numerical point of view, the macro element is exactly located at the soil-footing interface, which replaces the foundation and the surrounding soil. The notation for the loads and displacements is shown in Figure 8.

Previous macro element models were describing the constitutive relations based on yield surface theory, while CIM is different in the sense that constitutive relations are obtained from tracking the gap geometry and contact interface between soil and foundation. Hence, CIM provides a non-linear constitutive relation between the cyclic loading and displacements of the foundation-soil system when it is subjected to bi-directional cyclic loadings. The CIM is shown to have an advantage over BNWF by having the ability to consider the coupling between the vertical load, shear, and moment capacities (Hutchinson and Stewart, 2008; Gajan et al., 2010). However, the main disadvantage of these model types is their inability to incorporate flexible foundation behavior. In addition, the experimental validation of these models is very limited. Despite the implementation of this model into OpenSees by (Gajan and Kutter, 2009) the practical application of CIM is still very rare in the literature compared to BNWF and cone models.

Micro-element models are a direct approach for solving SSI problems in the time domain by modelling the soil and superstructure together. The soil domain is modelled as a continuum with an infinite number of degrees of freedom. To solve the problem, the whole domain is discretized into smaller sub-domain with finite degrees of freedom for each element. This discretization can be accomplished through the use of various computational techniques, including (FEM, FDM, and BEM) as described in Section 3.2. The micro-element model is capable of solving any complex type of SSI problem with a high degree of accuracy. However, the modelling process and computational time are costly especially for complex geometry and non-linear problems. Nevertheless, with the advances in computer software, the use of FEM numerical methods has been wider used compared to analytical methods. Several researchers adopted this approach to analyze non-linear dynamic SSI problems for different foundation and structural systems (Xu et al., 2016; Van Nguyen et al., 2017; Fatahi et al., 2018; Yeganeh and Fatahi, 2019). In addition, (Bowles, 1996; Dutta and Roy, 2002) highly recommends using micro-element modelling in case of analyzing important structures or when the non-linearity effects in the soil are required. Apart from computational techniques, there are three other issues that need to be considered carefully to obtain more realistic and accurate modelling.

(Abdel Raheem et al., 2015) pointed out that there is a strong connection between structural damage and soil conditions during earthquake events. The study investigated the SSI effects on several typical moment-resisting frame buildings resting on different types of soil with a surface mat foundation. The research showed how seismic demand and structural response of the structure will be altered when considering flexible base conditions. More specifically in the case of soft soil deposits where the lateral displacement and inter-story drift are increasing compared to the fixed-base counterpart. The study concluded that SSI effects are amplified with increasing height of the structure, the lower and higher parts of the structure will be affected more than the middle stories. Similarly, (Ghandil and Behnamfar, 2017) investigated the non-linear response of several short to tall MRFs rested on a soft soil deposit. The study showed that the natural period of the SSI system is increasing compared to the fixed-base counterpart, and these effects are more prominent for taller structures. Contrary to the common belief, this study concluded that in high-rise buildings, the story drifts due to SSI effects were observed to be even larger compared to the fixed-base model, especially in the lower stories of the structure. This happens because of the increase in (P − Δ) effects which comes from the greater relative lateral displacement in the lower stories due to the foundation rocking. However, the base shear and story shear forces are calculated to be less in SSI cases for all heights of buildings. Further, it is found that SSI raises ductility demand in the lower levels of a building while decreasing it in the top stories, with these impacts being more obvious in taller buildings. Similar conclusions were also drawn by (Awchat and Monde, 2021). Regarding SSI effects on deep foundations (Visuvasam and Chandrasekaran, 2019) highlighted that the detrimental consequences of SSI are also existing for pile-raft foundation system structures. The results of the numerical investigations showed that the piles will be subjected to lateral displacement and rocking at the raft foundation level. These effects amplify when the soil type changes from dense to loose sand. It has been shown that spacing between the piles also has significant effects on these deformations. Reduction of pile spacing will reduce soil lateral resistance between the piles. This leads to further lateral displacement and rocking in the piles. As a result, more story drift and lateral displacements are induced in the structure. In addition, (Carbonari et al., 2014) studied the seismic response of reinforced concrete structures supported by monopile foundation systems. Investigations on the building in the case of soft soil deposits showed that SSI has a considerable effect on the seismic response of the building by increasing structural deformation, especially in the early plastic ranges. The study observed that inter-story drifts are also increasing especially in the lower parts of the building, this is possibly due to foundation compliance. As for the pile foundation, it is found that SSI increases pile rotations considerably.

To investigate the influence of SSI on the seismic performance and vulnerability of reinforced concrete buildings, (Tahghighi and Mohammadi, 2020) performed an incremental dynamic analysis on several MRF buildings laying on various soil-type deposits. The study highlighted that SSI reduces base shear for low-rise MRFs while for medium-rise buildings foundation flexibility will cause an increase in the base shear. Further, it is observed that the SSI has a significant role in modifying the fixed-based fragility curves, where considering non-linear SSI might cause significant structural damages. However, the author recommended expanding the studying cases to draw more precise conclusions. In addition, (Arboleda-Monsalve et al., 2020) found that the direct economic loss for buildings is increasing up to 33% when considering SSI in the analysis in comparison to building losses with ignoring SSI. Therefore, the study encouraged researchers and engineers to include SSI in the analysis of seismic performance and losses, especially in high seismicity areas.

As a concern for the differences between equivalent linear and fully non-linear soil models, (Luo et al., 2016) highlighted that the equivalent linear approach is underestimating the ground response acceleration during the site response analysis. Further, it has been shown that seismic soil-pile-structure interaction is significantly affected by soil deformation. A separation of the pile from the surrounding soil has a great effect on the structural response. Therefore, in seismic analysis of a pile-raft foundation system, the fully-nonlinear behaviour of the soil must be considered through a proper constitutive model. Similarly, (Tabatabaiefar et al., 2015) compared the results of an experimental shaking table test result to a numerical result with an equivalent linear soil model for raft foundation. The analysis concluded that the equivalent linear method underestimates the inelastic seismic response of the structure resting on soft soil. Thus, in the inelastic design procedure, an equivalent linear model doe not guarantee the safety of the structure. However, the results from the fully-nonlinear soil model were fitting to the experimental results. Therefore, the fully non-linear soil model is highly recommended to achieve realistic results.

Apart from the moment-resisting frame MRF structural system and regular structures, several researchers tried to investigate the structural irregularity effects on SSI and the influences of the shear wall availability in the system. Patro et al. (2021) compared the SSI effects on the response of a typical eight-story geometric irregular building with the original regular building. Based on the probabilistic seismic demand and fragility curves for the two structures, it has been shown that vertical geometric irregular buildings are more vulnerable to SSI compared to the regular building and the probability of failure of flexible-based structures is higher than the fixed counterpart. In addition, (Shirzadi et al., 2020) investigated several steel structures with mass eccentricity ranging from (0%–30%). The results showed that regardless of eccentricity, seismic soil-structure interaction increases the first story drift ratio up to 30% more than the fix-based. The study concluded that the more eccentric the building, the more sensitive it becomes to SSI.

Regarding shear wall system buildings, (Carbonari et al., 2011) studied the effect of SSI on the response of low-rise buildings on pile foundations with a coupled-wall system. A linear finite element procedure in the frequency domain was conducted. The study showed that the rocking of the wall foundation significantly depends on the soil and it is rapidly increasing with decreasing soil stiffness. Investigation results observed that the seismic base shear of the columns increased while the base shear of the wall decreased when compared to the fix-based system. These findings imply that as the soil softens, shear forces migrate more from the wall to the frame. Carbonari et al. (2012) extended the earlier study for the non-linear behaviour of the structure using the substructure method. The non-linearity of the soil was also taken into account using an equivalent linear approach. According to the findings, SSI increases the overall structure’s lateral deformability. In addition, the rocking of the foundation significantly reduced the base shear at the walls and increased the same in the frame bases. In addition, (Arboleda-Monsalve et al., 2020) pointed out that SSI is less effective for a tall building with a wall-frame system. The availability of shear walls tends to reduce the natural period, inter-story drift, and foundation settlements in comparison to MRF systems. Hence, the earthquake-induced losses are decreased in wall-frame systems. From these studies, it is observed that the response of wall-frame structures is significantly different than MRF buildings which foundation rocking plays a key role in base shear redistribution in the column frame bases. Ignoring SSI in the analysis of wall-frame structures may lead to inadequate design. However, these studies need to be extended considering a more sophisticated model with a full non-linearity of the system.

The structural behaviour of adjacent buildings during pounding impacts has been a subject of discussion for many years and the soil-structure interaction effects were ignored in this regard. However, several studies showed that SSI can increase the lateral deflection of building structures especially when it rested on soft soil deposits. This encouraged researchers to start investigations on the SSI effects on structural pounding for adjacent buildings during earthquakes. Sobhi and Far (2021) conducted a comprehensive review of recent studies of structural pounding with and without the SSI effects. The study emphasized the significance of considering SSI in the calculation of pounding impacts, especially for adjacent structures at close distances. Further, the study recommended that structural engineers consider pounding impact and soil-structure interaction simultaneously to guarantee a safe design for adjacent buildings. Fatahi et al. (2018) assessed the adequacy of separation gaps between adjacent buildings during severe earthquakes. The study considered three adjacent buildings rested on pile foundations with different separation gaps. According to the findings of this research, the minimum gaps to avoid structural pounding between two structures must be 1.75% of the structural height. Nevertheless, this amount could be higher for structures resting on mat foundations. Therefore, the study highly recommends performing similar studies for buildings with mat foundation systems. Similar findings from (Elwardany et al., 2019; Jaradat et al., 2021; Awchat et al., 2022) emphasized the importance of including SSI effects in the calculation for the pounding impacts.

As previously mentioned, the influences of soil-structure interaction on the seismic response of structures are not only functions of the structural system and soil types. Studies have shown that several other factors are playing a key role in determining the SSI effects on building responses. Hokmabadi and Fatahi (2016) investigated the influence of the different types of foundations on the SSI effects of mid-rise moment-resisting concrete buildings. The results demonstrated that type of foundation plays a major role in the seismic response of buildings with SSI. In general, the base shear force of SSI systems was less than for the fixed-based counterpart. However, this reduction ratio was a function of foundation type. The base shear force for the raft foundation was less than for the floated pile and raft-pile types. The structures supported on the raft foundation experienced the most severe rocking angle compared to the piles. Moreover, structures supported by raft-pile foundations experienced around 20% less rocking than floated pile foundations. This is due to the settlement, which was higher than pile-raft. Furthermore, (Bagheri et al., 2018; Ebadi-jamkhaneh, 2019) discovered that there are other factors influencing the performance level of the structures on soft soil deposits such as (area replacement ratio, size and spacing between piles, ground motion characteristics, and aspect ratio of the building). The seismic soil-pile-structure interaction (SSPSI) tends to reduce the base shear force at the foundation level. The redistribution of story forces will be also altered based on different pile sizes, distances, and distributions. Therefore, the study highly recommended changing the mentioned factors to find an optimized force distribution throughout the structure. Nevertheless, further investigation is required to find a relationship between those parameters and the structural response. Similarly, (Van Nguyen et al., 2017) highlighted that the size and type of the piles will influence the seismic response of the building due to the interaction happening between soil and the pile. Based on the findings, increasing the length of the floating pile causes extra lateral deflection and drifts in the superstructure and sometimes it might exceed the life-safety limit. It has also shown that SSPSI alters the distribution of the forces along with the height of the structure. The study recommends that structural engineers should be aware that the reduction of the maximum base shear due to interaction between the pile and the soil is not equal along with the height of the structure. Therefore, it has to be carefully analyzed, otherwise, it could result in an unsafe design. As a concern for shallow foundation systems (Nguyen et al., 2016) investigated the relations between the foundation size on soil-structure interaction effects. According to the findings of the study, a reduction in foundation size brings fewer shear forces compared to a larger foundation. This is because, in a smaller foundation, there is a higher natural period in the structure. Further, it was observed that larger shallow foundations attract more inertial force from the earthquake compared to the small foundation.

Apart from foundation type, other factors (soil profile, soil constitutive model, and ground motion characteristics) have a significant influence on the soil-structure interaction on the seismic response of the building. Yeganeh et al. (2017) highlighted that considering a uniform soil profile to calculate the average shear wave velocity for the whole soil mass is not adequate. Non-uniform soil layers significantly affects on seismic response of the structure. The actual soil profile with different layers is essential in SSI modelling to achieve realistic results. Based on the study findings, the non-uniform soil profile amplifies the ground motion more than the uniform soil profile, especially in the short period ranges. Thus, the results from the uniform soil profile is underestimating the seismic demands of the building in terms of drift and lateral deflections. The study showed the necessity of modelling the soil profile as realistically as possible. However, to improve the findings of this research, further studies are required to be conducted in this area, especially with proper non-linear soil modelling. Furthermore, (Yeganeh and Fatahi, 2019) pointed out that, not only the soil profile but also different constitutive soil models will affect the seismic response of the building and the foundation. The research findings showed a significant difference between the plastic and non-plastic pre-failure soil models in capturing the foundation settlement, rocking, and base shear of the structure. It was observed, the non-plastic soil models are underestimating the total seismic response of the building, as well as the foundation rocking and settlements. The results of this study were for a specific foundation and structural system. Therefore, the findings can be improved for other structural and foundation systems. Another parameter that has been found to be critical in studying soil-structure interaction is the ground motion characteristics. Zhang and Far (2022) studied the effects of far-field and near-field ground motions on a range of hypothetical tall buildings. It has been found that SSI has a remarkable impact on the response of the buildings, however, a significant difference between applying the near-field and far-field earthquake records was observed. For the far-field earthquakes, the increase in lateral deflection and storey drift was more obvious than in the near-field records.

Based on the revision of previous findings and as a general conclusion, SSI has major effects on the seismic response of buildings and there are different factors governing the criticality of these effects. Ignoring SSI effects is not always beneficial, several situations proved that SSI increases the seismic demand of the structure and changes the performance level of the building. Therefore, to guarantee the safety of the buildings, especially in soft soil deposits, the soil-structure interaction effects should not be ignored.