Optimum lateral extent of soil domain for dynamic SSI analysis of RC framed buildings on pile foundations

Abstract

This article describes a novel approach for deciding optimal horizontal extent of soil domain to be used for finite element based numerical dynamic soil structure interaction (SSI) studies. SSI model for a 12 storied building frame, supported on pile foundation-soil system, is developed in the finite element based software framework, OpenSEES. Three different structure-foundation configurations are analyzed under different ground motion characteristics. Lateral extent of soil domain, along with the soil properties, were varied exhaustively for a particular structural configuration. Based on the reduction in the variation of acceleration response at different locations in the SSI system (quantified by normalized root mean square error, NRMSE), the optimum lateral extent of the soil domain is prescribed for various structural widths, soil types and peak ground acceleration levels of ground motion. Compared to the past studies, error estimation analysis shows that the relationships prescribed in the present study are credible and more inclusive of the various factors that influence SSI. These relationships can be readily applied for deciding upon the lateral extent of the soil domain for conducting precise SSI analysis with reduced computational time.

Keywords

soil structure interaction / optimum lateral extent of soil domain length / multi-storyed framed building / pile foundation / OpenSEES / L-K boundaries / dynamic analysis

Introduction

Soil structure interaction (SSI) incorporates mutual dynamic interaction between the structure and the supporting foundation medium. When considering finite soil domain, the surrounding foundation soil medium influences the structural response. In the past, many researchers have adopted various numerical approaches for investigating the behavior of different categories of buildings with SSI, namely: frequency domain method for general buildings [¹–³], discrete force method for analysis of shear wall buildings [⁴], domain reduction method [⁵], lumped spring approach for incorporating flexibility of foundation soil on the response of buildings [⁶], and substructure method [⁷]. Although computationally reliable and efficient, these methods are mostly limited to studies on linear or equivalent linear response of soil-structure-foundation system. Moreover, owing to the computational expense associated with the rigorous techniques, these procedures were most widely used for performing SSI studies. Apart from these methods, Scaled Boundary Finite Element Method (SBFEM) provides a powerful technique for modeling of the unbounded domain [⁸–¹²]. The method, although powerful, has not obtained widespread popularity until date. This is due to the lack of commercially available software that implement the method and lacking the provision of a library of materials capable of representing nonlinear soil behavior. With the advancement in computational tools, reliable numerical models for soil, and with the increased number of studies highlighting the necessity of nonlinear SSI research, there has been a paradigm shift toward adopting FE methods for performing SSI studies. The FE tools have been largely incorporated in many of the commercial software such as OpenSEES, PLAXIS, GeoStudio, and others. Advanced design codes such as the ‘Standard Specification for Concrete Structures, Design, [¹³] recommend incorporating foundation-soil system along with the structure for evaluating the effects of SSI. The direct method of analysis for performing a single-step dynamic analysis, using FE, involves the numerical modeling of the foundation soil as continuum along with the supported structure [¹⁴]. The advantage of the method is that a time domain approach is followed, and hence the nonlinearity of the soil or structure can be directly incorporated [¹⁵]. Hence, considerable research studies are oriented toward performing numerical nonlinear SSI analysis by modeling a finite extent of soil continuum accompanied by appropriate boundary conditions.

For conducting any numerical SSI study, deciding upon the lateral extent of the soil domain (also referred to as soil domain length in this article) is very crucial. Theoretically, considering an infinite lateral extent (as large as possible) of soil domain would be ideal as it produces a response free from the boundary effects, while accounting for the radiation damping as well. However, such choice leads to heavy computational expense and is not practically feasible. To overcome this difficulty, it is a common practice to model the SSI system with a finite length of soil domain, using radiation boundaries at the lateral extents, for obtaining the dynamic response. However, even with the incorporation of these far-off boundaries, the estimation of model response may be inappropriate if the length of the domain is not sufficient. Therefore, it is important to justifiably define the lateral extent of the soil domain to be used for dynamic SSI analysis that would ensure computational efficiency without loss in accuracy of the system response. Standard domain definition available for static cases cannot be used for dynamic or seismic loading as the extent is guided by several factors such as the type of soil, interaction mechanism, nonlinear characteristics of the soil or structure, and intensity of shaking, to name a few. Although there have been a few recommendations by past researchers on the soil domain length to be used for dynamic SSI studies, they were based on simple SSI configurations (such as shallow foundation, disc foundation, or single degree of freedom, system) and linear soil characteristics without providing due attention to the foundation parameters, except for the width of the foundation. Ghosh and Wilson [¹⁶] recommended a horizontal extent of soil as 4W (W is the base width of the foundation); Roesset and Ettouney [¹⁷] recommended a domain length of 5W for soils with high internal damping and 10W–20W for soils with low internal damping. Wolf [¹⁸] suggested for an increment in the lateral extent of the soil domain by placing the artificial boundary further away from the structure to improve the accuracy. Table 1 shows the value of normalized lateral extent of soil domain (Ω) used in the more recent studies as a function of structural or foundation width (W).

It is worth emphasizing that while selecting the lateral extent of soil domain, potentially influencing factors such as the soil type, intensity of shaking, and structural configuration should be taken into consideration. Modern day applications rely extensively on the explicit modeling of the SSI system with detail or with least possible idealizations. A simplistic approach to determine the horizontal extents of the soil domain, based on stress or displacement contours, is helpful only for static cases. For dynamic cases, the stress contours change at every time step of the applied earthquake motion. If an analysis involves various factors (already mentioned), then the determination of appropriate soil domain length becomes a cumbersome task for each case. Hence, it is extremely important to ascertain the extent of the soil domain, on a case-to-case basis, with the help of an exhaustive set of FE simulations and provide recommendations, which would be of first-hand help to the analysts. Therefore, the intention of the current article is to outline a novel approach for arriving at the horizontal extent of soil domain to be used for DSSI analysis. Recommendations from the study, in the form of relationships and guidelines, would facilitate the decision-making on soil domain length for various representative situations. The recommendations also eliminate the need to make several numerical trials conventionally required to arrive at the decision about the optimum length of the soil domain to be used for any future DSSI analysis. To the best of the authors’ knowledge, such a study has not been reported in the literature so far.

Modeling and input

Description of the FE model

Three dimensional (3D) FE modeling and analysis are being adopted for conducting studies related to SSI problems. This is because a 3D model provides an advantage of accurate representation of the real problem and is useful for studying local effects. Nonetheless, 3D modeling is quite intriguing in terms of mesh generation, convergence to a solution, and requires a high performance computation facility [³⁰]. Furthermore, for problems that have symmetry in geometry and loading, or wherein the shaking considered is in a particular direction without resulting in any out-of-plane deformation/stress, or wherein the focus is on studying the global rather than the local effects, a two-dimensional modeling and analysis can provide a fairly good insight into the 3D problem and can be used to draw important conclusions. In the recent past, several SSI studies consisting of 3D problem have been successfully conducted using 2D modeling/analysis [³¹–³⁵] and it has been quite successful in elucidating the essential response characteristics of even experimentally tested 3D building models [³⁶]. The intention of the present study is to investigate the global effects of boundaries on the response of the building structure. Since the geometry of the problem considered is symmetric and out-of-plane shaking is not considered, therefore, in the present study, 2D FE models, consisting of building frame supported by pile foundation and surrounding soil medium, are developed in the FE based software framework OpenSEES [³⁷]. The representative illustration of the FE model is shown in Fig. 1. Three different structural widths are considered, and each of the structure-foundation configurations, subjected to three different ground motion characteristics with varying peak ground acceleration (PGA), is analyzed. The horizontal extent of soil domain, along with other soil properties, is varied exhaustively for a particular structural configuration. The details of the structural and soil modeling, extent of soil domain used and ground motion input are given in the following subsections.

Modeling of soil domain

The soil domain considered is of uniform depth, which is modeled using four noded quadrilateral elements with four gauss integration points and bilinear isoparametric formulation. Reference low strain shear modulus of the soil is considered to determine the largest size of the elements, based on the prescribed relationship by Kuhlemeyer and Lysmer [³⁸], as shown in Eq. (1).

(1)l_{\max }=\mathrm{0.125}{f}_{\max }^{-1}{\left(G_{\mathrm{r}}{\rho }^{-\mathrm{1}}\right)}^{\mathrm{0.25}},

where l_max is the maximum size of the soil elements, f_max is the value of maximum frequency of input motion, typically considered as 15 Hz. G_r is the reference low strain shear modulus of the soil specified at a reference mean effective confining pressure of 80 kPa, and r is the mass density of the soil.

Three types of cohesionless soil, representing loose, medium, and medium-dense sand, have been considered in the study with the basic properties shown in Table 2. Pressure dependent constitutive behavior [³⁹] is used for simulation of the nonlinear characteristics of the soil wherein plastic behavior follows the nested yield surface criteria [⁴⁰]. The chosen constitutive model can simulate response characteristics dependent on the instantaneous confining pressure, and is capable of modeling typical characteristics of cohesionless soil such as dilatancy [⁴¹] (based on the non-associative flow rules) and liquefaction. For the purpose of the present study, the bedrock is assumed to lie at a depth of 30 m from the surface of ground level. The value of shear wave velocity shown in the table is the average value of the entire layer. The actual variation of shear wave velocity increases as the shear modulus increases with depth following the relationship shown in Table 2. The water table was considered present at the bedrock level and does not pose any influence on the dynamic SSI studies.

Modeling of the structure-foundation system

In the present study, the structure considered is a two dimensional RC building frame supported on pile foundations. A representative illustration of the typical elevation of the building considered as shown in Fig. 1. The typical floor-to-floor story height considered is 3 m and the bay width as 3 m. For estimating the sizes of the frame members, the out-of-plane width and height of the structure are considered constant as 15 m (5 bays) and 36 m, respectively, for the entire study (Fig. 1). Three different configurations have been considered for the structure, by varying the number of bays in the direction of the in-plane structural width (W), having width as 15 m (5 bays), 27 m (9 bays), and 45 m (15 bays). As per the guidelines specified in IS 456 [⁴²], it is mandatory to provide an expansion joint at every 45 m. For larger structural widths, this expansion joint separates the two portions of the structure and the Structure-Soil-Structure-Interaction (SSSI) response of the two portions are separately analyzed. Hence, the width of the structure in the present study is limited to 45 m, which encompasses the conventional widths of the individually standing structures. The structure is assumed to be located in the seismic zone V as per the seismic zoning map of India [⁴³]. It is supported on pile foundations and has been designed with the help of relevant Indian standards [⁴²–⁴⁴] after considering gravity and lateral loading as per [⁴⁵,⁴⁶]. The loading on the structure, apart from the self-weight, is shown in Fig. 1. The size of the column is 500 mm × 500 mm until the 6th story, above which the dimensions are reduced by 100 mm along both the directions. The width of the beam is 250 mm and the overall depth is 400 mm for all story levels. The size of the square grade beam is taken as 400 mm × 400 mm. The pile foundations have been designed in accordance with the three different soil conditions considered in the study. Distance between adjacent piles in a group is kept to be three times the diameter of the individual pile. The stiffness of the pile group is estimated, and a single equivalent pile possessing the same stiffness as that of the pile group is modeled beneath the columns, to account for the out-of-plane representation provided by the pile group. This methodology has been successfully applied in the past [³¹]. The details of the pile groups and equivalent piles are shown in Table 3.

Elastic beam-column elements, having two translational and one rotational DOFs, have been used to model the frame and pile members of the structure-foundation system. The structure-foundation system is positioned in the center of the soil domain. The discretization of the pile has been carried out ensuring connectivity between the pile nodes and the adjacent soil nodes. The modulus of elasticity of the concrete material is taken to be 25 GPa. The horizontal inertial forces are simulated in the structure by means of lumped mass of the structure and the additional loads at the corresponding frame and pile nodes.

Modeling of horizontal and vertical boundaries

To accurately model the effect of radiation damping, Lysmer-Kuhlemeyer (L-K) viscous dashpots [⁴⁷] have been assigned at the vertical and horizontal boundaries. These boundaries prevent the reflection of the seismic waves back into the soil medium after being incident on the far-off boundaries. Moreover, the adopted boundaries aid in the truncation of the soil domain to a finite extent. On the vertical boundaries, L-K dashpots are assigned along both the horizontal and vertical directions, having dashpot coefficients as C_p = rv_pA and C_s= rv_sA (A=tributary area), respectively. Since the motion is applied at the base of the SSI system and is essentially a shear wave propagating in the vertical direction, only horizontal dashpots are attached at the horizontal boundary having coefficient C_s = rv_sA. The primary wave velocity (v_p) is obtained from shear wave velocity and Poisson’s ratio (ν) as v_{\mathrm{p}}\hspace{0.17em}=\hspace{0.17em}{\left(\mathrm{2}{v}_{\mathrm{s}}\right(\mathrm{1}-v)/}^{}{(\mathrm{1}-\mathrm{2}v))}^{\mathrm{0.5}}. Once the L-K boundaries have been assigned, seismic input is applied in the form of equivalent nodal shear forces at the bedrock level. Based on the theory proposed by Joyner [⁴⁸], Zhang et al. [³¹] provided the expression for equivalent nodal shear forces as,

(2)F(x,t)\hspace{0.17em}=\hspace{0.17em}-C_{\mathrm{s}}\dot{u}(x,t)+\mathrm{2}{C}_{\mathrm{s}}{\dot{u}}_{\mathrm{t}}(t+x/v_{\mathrm{s}}),

where {\dot{u}}_{\mathrm{t}}(t+x/v_{\mathrm{s}}) is the velocity of incident motion, \dot{u}(x,t)\hspace{0.17em} is the velocity of the soil particle motion, and C_{\mathrm{s}} is the coefficient of dashpot. The first term in Eq. (2) is the force generated by dashpot, while the second term is the applied equivalent nodal force that is proportional to the velocity of the incident motion. The bedrock mass (not modeled herein) is assumed homogenous, linear elastic, undamped and semi-infinite half-space region [³¹,³³]. In addition, the extent of near-field effects of SSI reduces with increasing distance from pile. Since the soil is modeled with the aid of ‘pressure dependent constitutive behavior’ having damping characteristics, the interaction effects result in the development of maximum shear strain at the pile-soil interface with an outward decreasing gradient. This aspect is automatically taken care of through FE analysis of pile embedded in soil domain.

Seismic Input motions

Based on the categorization provided in Uniform Building Code [⁴⁹] three ground motions (one near field motion M1, two far field motions M2 and M3), belonging to different seismic events, have been selected for conducting the study, the corresponding accelerograms being shown in Fig. 2. M1 is recorded during the 1995 Kobe earthquake at station KJMA, M2 is recorded during the 1980 Mammoth Lakes earthquake at station Long Valley Dam, and M3 is recorded during the 1994 Northridge earthquake at station Malibu-Point. The input strong motions have been taken from the PEER ground motion database. It can be observed that the motions are characteristically different, in particular reference to their PGA and the duration of motion. It is assumed that the motions have been recorded at the rock outcrop level, and hence, the same motions are used as input, without any scaling down by 50%, as recommended in Ref. [¹⁵]. The analyses are performed for the significant duration of each motion [⁵⁰], which is calculated based on the Arias Intensity [⁵¹]. Arias Intensity (I_{\mathrm{A}}) is a measure of the intensity of the shaking ground motion and is obtained as,

(3)I_{\mathrm{A}}={\displaystyle \frac{\mathrm{\pi }}{\mathrm{2}g}}{\displaystyle \underset{\mathrm{0}}{\overset{\mathrm{T}}{\int }}{\ddot{u}}_i^2\left(t\right)\mathrm{d}t},

where {\ddot{u}}_iis the acceleration of the incident motion, and g is the acceleration due to gravity. The significant duration is the time duration corresponding to 5%I_{\mathrm{A}}–95%I_{\mathrm{A}}. The Arias Intensity, as well as the significant duration, of the selected motions, are also shown in Figs. 2(b), 2(d), and 2(f).

Analysis

Gravity analysis and validation

A rigorous dynamic analysis is preceded by a stage-wise static gravity analysis which has been outlined in Ref. [³¹] and adopted in Ref. [³³] with some modification. The analysis steps are suitably modified and listed as follows:

Stage 1: Elastic gravity analysis

The SSI model, consisting of the soil-pile-structure system, is analyzed with a base restrained in both the horizontal and vertical directions. The vertical boundaries are restrained only in the horizontal direction and are kept free in the vertical direction. The material model of soil is considered as elastic and the gravity loads of the structure and soil are applied. Single step analysis is performed to achieve the equilibrium.

Stage2:Plastic gravity analysis

The material constitutive model is changed to plastic and the SSI system is brought into equilibrium through multiple iteration steps. Once the equilibrium is achieved, the reactions in the horizontal and vertical boundaries are recorded.

Stage3:Assigning L-K boundary condition on vertical boundaries of the SSI model

The restraint at the vertical boundaries along the horizontal direction in the SSI model is removed and the reactions obtained in the previous stage are applied. The model is brought into equilibrium through iterations. Subsequently, L-K boundaries are assigned in the horizontal as well as the vertical direction.

Stage4:Assigning L-K boundary condition at the horizontal boundary of the SSI model

The restraint at the base boundary of the SSI model along the horizontal direction is removed and the reactions obtained in Stage 2 analysis are applied. The model is brought into equilibrium iteratively and the L-K boundaries are assigned in the horizontal direction.

Stage5:Dynamic analysis

Once the boundary conditions have been successfully applied, the SSI model is subjected to seismic excitation that is applied as equivalent nodal shear forces at the base of the FE model. The equation of motion of the SSI model is shown in Eq. (4), and is expressed as:

(4)\mathit{M}\ddot{\mathit{u}}\left(t\right)+\mathit{C}\dot{\mathit{u}}\left(t\right)+\mathit{K}\mathit{u}\left(t\right)=\mathit{F}\left(t\right)+{\mathit{F}}_{\mathrm{v}},

where \mathit{M}, \mathit{C}, and \mathit{K}are the global mass, damping, and stiffness matrices of the SSI system, respectively; \ddot{\mathit{u}}, \dot{\mathit{u}}, and \mathit{u} represent the nodal acceleration, velocity, and displacements, respectively; \mathit{F}\left(t\right)represents the input nodal shear force vector as shown in Eq. (2) and {\mathit{F}}_{\mathrm{v}} is the force vector assigned at the viscous boundaries during the staged gravity analysis. The time-step integration scheme adopted is Newmark-β method considering constant variation of acceleration over the time step that renders the scheme as unconditionally stable, and the initial condition of the SSI model is considered to be ‘at rest’. As reported in Ref. [⁴⁸] and also observed in the preliminary analysis of the present study, material nonlinearity in soil results in the development of high frequency oscillations, which is attributed to the possible excitation of the high frequency modes of the soil system. Incorporating numerical damping into the system using the HHT-α method of analysis [⁵²], instead of the traditionally used Newmark-β method, may be helpful. However, the spurious oscillations may not necessarily be arrested only by using numerical damping. In such cases, Rayleigh damping may be used. Hence, in the present study, 2% Rayleigh damping has been incorporated into the SSI system, estimated by considering the first two natural frequencies of the entire system [¹⁵,⁵³]. Based on the existing literature, the outcome of the incorporation of boundary conditions and execution of the staged analysis is validated with the results obtained by using a sine wavelet excitation to the model. The total acceleration response in the central region, across the depth of the soil domain is obtained and compared with that in the existing Ref. [⁵⁴] (Fig. 3). Close agreement of the results indicates successful application of the different stages of the analyses described above and the numerical model incorporating SSI can be used for further rigorous dynamic analysis.

Analysis cases and response locations

For the estimation of an optimum size of soil domain for SSI analysis, it is important to study the effect of soil domain length on response of the SSI system. Instead of estimating the response at just one location (conventionally at the mid-length of the lateral extent), the effect should be ascertained at various locations, including the desired points of interest in the computational domain. Moreover, the response may be different for different widths of the structure included the SSI system, and may vary with the different levels of seismic shaking. Hence, in the present study, three structural-foundation-soil systems have been selected which are different from each other in their in-plane widths (W). The out-of-plane width of these structures is the same. Each structural configuration is founded on three different types of soil, as described earlier. For a particular configuration, various lengths of soil domain are considered to define the SSI system. A particular SSI system is then subjected to the previously selected three different strong motions. Table 4 shows the various soil domain lengths, corresponding to the structural lengths, and motions considered for which the SSI analysis is carried out. The values in the table indicate the time required for completion of the analysis.

In the present study, the effect of soil domain length on the response of the SSI system is evaluated by monitoring the acceleration response at various locations in the system. The various locations selected are soil column very near the structure (Loc. A), soil at pile location (Loc. B), pile location (Loc. C), soil column near boundary (Loc. D), and superstructure (Loc. E). For each location, multiple points have been selected for recording the response over the depth or height of the substructure or superstructure, respectively. All the mentioned locations are shown in Fig. 4.

Results and discussion

The following sections discuss in detail results of various analyses, methodology adopted for obtaining the optimum soil domain length and various relationships proposed for practical usage.

Effect of soil domain length on the response of SSI system

This section discusses the effect of varying soil domain length on the response of the SSI system. Figures 5, 6, and 7 show the effect of horizontal extent of soil domain on the acceleration response at different locations in the SSI system for structural widths W = 15, 27, and 45 m, respectively. It is observed that there is a noticeable difference, in the response of the SSI system, for the smallest and the largest domain lengths. As the domain length increases, the response gradually approaches to that depicted by the larger domain lengths. Additionally, a minute scrutiny shows that the change in the response is more sensitive for smaller domain lengths and otherwise for larger domain lengths. For example, it can be observed that when the domain length is increased from 33 to 63 m, the change in the response is significantly large, whereas the same is comparatively lesser when the domain length is increased from 183 to 753 m. It implies that beyond a particular length of soil domain, there is a marginal change in the response. The observation holds good for the different soil types, motions, and structural widths selected in the present study. Hence, it can be said that for all practical engineering purposes, there exists a particular length of soil domain which would be sufficient enough to consider the obtained results as accurate, and the same is termed as the optimum lateral extent of soil domain (denoted by Ω in this article). Identifying the optimum soil domain length would be helpful in significantly reducing the computational costs incurred for SSI analysis without compromising on the accuracy of the response. In the present article, a new approach is developed for identifying the optimum soil domain length considering the influence of several contributory factors. In this regard, a quantitative estimation of the change in the response at various locations in the SSI system has been determined, and the same has been used to define the guidelines. The methodology is described in the next sections.

Quantification of change in response and optimum domain length

To quantify the change in the system response due to the increase of soil domain length, a particular foundation-structure system is analyzed with different horizontal extents of soil domain, as shown in Table 4. Let ‘a_{itL}’ be the acceleration response at a particular nodal location ‘i’, at a given time instant ‘t’, and for a particular length of soil domain ‘L’. For example, Fig. 5(a) shows the acceleration response, a_{itL}for ‘i’ as Loc. B (0 m depth) at different time instants ‘t’ of the ground motion, for various domain lengths ‘L’ (L = 33, 45, 63, 183, and 753 m). Similarly, the other subplots of Figs. 5, 6, and 7 show likewise. The system response corresponding to the largest domain size can be considered as the most accurate as it is unaffected by the boundaries by virtue of a very large extent of soil domain length considered. For smaller domain lengths, it is observed that there is significant influence of the boundaries on the response. Hence, the response corresponding to the largest domain length is considered as a benchmark of accuracy. It must be pointed out that the largest domain length considered in the study is 50 times the structural width (W), which is quite large to produce a response free from the boundary effects. On the other hand, a particular length of soil domain is said to be sufficient if it is small enough to produce a response within an acceptable margin of tolerance with respect to the benchmark values (a_{it{L}_{\max }}), and yet maintaining a restriction on the computational cost. The instantaneous absolute difference between the exact response and that obtained from a specific length of soil domain is termed as the error in the response entity, and is expressed as shown in Eq. (5):

(5)e_{itL}\hspace{0.17em}=\hspace{0.17em}{a}_{it{L}_{\max }}\hspace{0.17em}-\hspace{0.17em}{a}_{itL}.

Once the error at all the time instants is obtained, the Root Mean Square Error (RMSE) at a particular location ‘i’ and for a particular length of soil domain ‘L’ is estimated as:

(6)RMS{E}_{iL}\hspace{0.17em}=\hspace{0.17em}\sqrt{{\displaystyle \frac{{\displaystyle \mathrm{\sum }_{t=\mathrm{0}}^{\mathrm{T}}{e}_{itL}^{\mathrm{2}}}}{N}}},

where T is the total time duration of the ground motion, e_{itL}is the instantaneous absolute error at a particular time instant ‘t’, location ‘i’ and for a particular soil domain length ‘L’ (evaluated as per Eq. (5)), and N is the total number of time samples in the ground motion (N = T/∆t, ∆t is the sampling interval). Subsequently, the RMSE at various locations (Fig. 4) are summed up to produce the cumulative RMSE corresponding to a particular domain length of soil, i.e., RMS{E}_L\hspace{0.17em}={\displaystyle \sum RMS{E}_{iL}}.

Figure 8 shows the variation of normalized RMSE with the normalized domain length (L/W) for SSI systems with varying structural widths (15, 27, and 45 m) resting on different soil types and subjected to various strong motions (M1, M2, and M3). The normalized RMSE is obtained by normalizing RMSE_L by the largest value, i.e., corresponding to the smallest domain length as shown in Eq. (7).

(7)NRMS{E}_L={\displaystyle \frac{RMS{E}_L}{RMS{E}_{L_{\min }}}}.

It is observed that all the curves have similar trend characteristics: the initial portion of these curves is very steep which gradually takes the shape of horizontal asymptote with an increase in the domain length. The trend suggests that progressively increasing domain lengths have successively lesser influence on the system response. The reduction in the change of the response with increase in normalized domain length is indicative of the reduction in the error of the overall acceleration response in the SSI system.

Optimum soil domain length

Based on the qualitative trends as observed in Fig. 8, it can be generalized that there are two distinct zones, one represented by a steep decrement of normalized RMSE with increasing domain length, followed by an asymptotic trend of normalized RMSE. Therefore, the RMSE curves can be justifiably approximated with a bilinear fit, as shown in Fig. 9. The intersection of the two branches provides the optimum magnitude of normalized domain length, beyond which there is very little change in the system response. The normalized domain length corresponding to the optimum point is termed as the optimum normalized length of soil domain (Ω). That extent of Ω would be sufficient for an SSI analysis that is not only computationally inexpensive but also helps in estimation of the system response within an adequate tolerance with the benchmark responses.

To observe the effect of ground motion intensity (PGA), the various RMSE curves are shown by normalizing with respect to the curve obtained for the motion with the highest PGA level (M1) for various soil domains as shown in Eq. (8).

(8)NRMS{E}_{\mathrm{PGA}}={\displaystyle \frac{RMS{E}_L}{RMS{E}_{L_{\min }}|PG{A}_{\max }}}.

Figure 10 shows the various normalized RMSE curves along with their bilinear idealizations for different soil types, structural widths and ground motions. The details of the idealised bilinear curves along with the estimated optimum normalized domain lengths are provided in Table 5.

Relationship of optimum domain length with PGA

Once the optimum normalized length of soil domain is obtained, their relationships with PGA for different structural widths and soil types are investigated. Figures 11(a) to 11(c) show the relationship between PGA and the optimum normalized length of soil domain (Ω) for TY-I, TY-II, TY-III soils, corresponding to structural widths W = 15, 27, and 45 m, respectively. It is observed that, for a particular structural width (W) and PGA, the value of Ω does not vary significantly for different soil types. For the structural width of W = 15 m, the value of Ω corresponding to 0.82g PGA level lies in the range of 8.55-9.95 and that corresponding to 0.22g PGA level lies in the range of 5.02-5.63 for the different soil types; similar feature is noted in all the other figures. Moreover, it is observed that rather than soil type, PGA has more dominant influence on the variation of Ω. For the structural width of W = 15 m and TY-I soil, the value of Ω varies in the range of 5.44-9.95 for 0.22g-0.82g PGA levels; similar feature is observed in the other representations as well. Considering negligible influence of the soil type, an average curve is drawn to establish a relationship between PGA level and Ω (Figs. 11(d)-11(f)) for structural widths 15, 27, and 45 m, respectively. It is observed that a linear relationship provides a very good fit indicated by sufficiently high R² value. In all the figures, it is observed that the requirement of domain length increases with the increase in the PGA level. For larger structural widths, the requirement of Ω, at high PGA level, is less as compared to that corresponding to smaller structural widths, e.g., at PGA level of 0.8g, Ω is 8.9 for W = 15 m; however, Ω is 6.1 for the structural width of 45 m at the same PGA level. For low PGA levels, Ω is approximately similar. For 0.2g PGA level, for different structural widths, Ω lies between 4.7 and 5.4.

The obtained relationships have been assessed with the help of sensitivity analysis. For such studies, various approaches are available in the literature. Vu-Bac et al. [⁵⁵] developed a software framework for conducting uncertainty and sensitivity analysis of computationally expensive models. Hamdia et al. [⁵⁶] presented a methodology for the stochastic modeling of a problem by incorporating uncertainty in the input and constructing a Polynomial Chaos Expansion (PCE) surrogate model and finally showing its effectiveness in sensitivity analysis. Hamdia et al. [⁵⁷] implemented three methodologies for conducting sensitivity analysis in their study, namely, Morris One At a Time (MOAT), PCE-Sobols’ and EFAST (Extended Fourier Amplitude Sensitivity Test). Due to the limited scope of the present study, a simplistic sensitivity analysis is presented in the present article considering two variables, width of the structure (W) and PGA of the input motion, for the different soil types considered. The PGA level is considered to follow normal distribution with mean (m) as 0.49g and standard deviation (s) as 0.25g, and W is considered to be uniformly distributed between a range of 15 to 45 m. The results of the sensitivity analysis using MOAT and Sobols’ method is shown in Figs. 12(a) and 12(b), respectively. From Fig. 12(a), it can be observed that for soil type TY-III, Ω shows greater sensitivity to PGA, whereas for soil type TY-I and TY-II, both W and PGA are found influential. The total effect indices (S_{\mathrm{t}}) and first order indices (S_{\mathrm{i}}) for the variables considered are shown in Fig. 12(b). Based on the sensitivity indices, it can be observed that for TY-I soils, W and PGA are equally sensitive and influential. For TY-II soils, W has more influence as compared to PGA, while for TY-III soils, PGA has been found to produce more dominant influence in comparison to W. For any design or analysis of buildings supported by these different types of soils, it is imperative to pay proper attention to the sensitive and contributing parameters.

Engineering application of the findings

In the previous section, the relationship between the optimum normalized length of soil domain and PGA of the strong motions is established for individual structural/foundation widths (15, 27, and 45 m). However, for engineering purposes, it would be required to develop numerical models to investigate seismic SSI aspect. In this regard, it would be more fruitful to develop relationships to ascertain the optimum normalized length of soil domain required to conduct such studies for various structural or foundation widths. Based on the observations and estimates obtained from the present study, engineering relationships with linear trends are developed for various levels of PGA (Fig. 13), and are expressed in Eqs. (9)–(11):

(9)\mathrm{\Omega }=-\mathrm{0.1}W+\mathrm{11.5},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\vee PGA\hspace{0.17em}=\mathrm{0.82}g.

(10)\mathrm{\Omega }=-\mathrm{0.03}W+\mathrm{7.5},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\vee PGA=\mathrm{0.43}g.

(11)\mathrm{\Omega }=\mathrm{5.5},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\vee PGA=\mathrm{0.22}g.

In Eqs. (9) and (10), the parameter W is expressed in meter. For intermediate PGA level, linear interpolation can be sought. It is noted that the proposed relationships satisfy all the observations of the analyses, such as:

1) Higher the PGA, higher is the value of Ω for a particular structural/foundation width (B), and vice versa.

2) For larger structural/foundation widths (W), the optimum Ω is less, and vice-versa.

3) For low PGA level, Ω is nearly the same independent of different structural/foundation widths (B).

4) In the previous sections, it has already been established that the influence of soil type on Ω is insignificant, and hence this parameter in not included in the expressions.

The practical applicability of the prescribed relationships is judged based on their performance in comparison to a similar model developed with a very long soil domain (e.g., 50 × W, i.e., W = 50). Corresponding to the three selected structural widths and PGA levels, various optimum normalized lengths of soil domain are obtained and dynamic analysis is carried out. Table 6 shows the percentage difference in the estimated peak acceleration response recorded at soil node at the surface very near to the structure, pile node at the top of the outermost pile, and structure node at the roof level. It is observed that the maximum error obtained is less than 10%, which is quite acceptable for practical engineering purposes. Moreover, the percentage of the computational time required for performing the analysis is below 5% of the time required for performing exact analysis, which definitely indicates the computational proficiency of the adopted values.

In the past, there have been a few recommendations for the horizontal extent of the soil domain to be used for SSI studies. Some researchers have also adopted specific values of horizontal extent of the soil domain as suitably required for their purpose without any comprehensive study. Figure 14 shows a comparison of the prescribed relationships from the present study and the soil domain lengths considered by past researchers. It is observed that the values and the relationships used by past researchers fall within an agreeable zone of the relationships proposed in the current study. However, it is to be noted that the relationships prescribed in the present study are more rigorous since they are inclusive of various factors such as structural width, PGA level and soil type. These relationships are more helpful in deciding about the soil domain length to be considered for SSI studies when various factors are to be accounted. It is observed that the relationships proposed by previous researchers are mostly independent of the structural width, which in different cases prove to be computationally inefficient owing to significant underestimation or overestimation of system response and computational effort, respectively. Hence, based on the present findings and relevant comparisons, it can be stated that the prescribed estimates for soil domain length are computationally efficient as well as suitable enough to provide the system response. Thus, these relationships are credible for practical use for future SSI analysis. For any random structural width and PGA of input seismic motion, the optimum normalized length of soil domain length can be arrived at by a suitable linear interpolation. The relationships proposed in the present study, thus, provide a guideline for arriving at the horizontal extent of soil domain to be chosen for SSI analysis.

Conclusions

In the present study, a strategic approach has been outlined to arrive at the optimum normalized length of soil domain to be considered for SSI studies. Based on exhaustive FE simulations, normalized root mean square error (NRMSE) for various domain lengths is obtained for various structural widths and soil types. With the aid of bilinear fit to the NRMSE plots, normalized optimum soil domain lengths have been obtained. Any length beyond the optimum domain length produces insignificant change in the overall response of the SSI system with the percentage difference in the results being less than 10%. Optimum domain lengths for various cases have been obtained, based on which a generalized set of relationships has been prescribed by which the horizontal extent of soil domain is expressed as a function of structural or foundation width and the PGA of the strong motion. It is observed that the domain length to be used for rigorous SSI studies is virtually independent of the soil type. It is concluded that larger extents of soil domain are required for SSI problems considering higher PGA strong motion, and vice versa. SSI problems comprising of smaller structural widths require larger normalized domain lengths (Ω), and vice versa. In comparison to the correlations of the domain length provided by earlier researchers, the one prescribed herein proves to be more robust (being inclusive of various factors such as soil type, structural or foundation width and PGA level of input motion), practically feasible and computationally efficient. The developed relationships from the present research can be credibly used as guidelines for numerical modeling in SSI studies.

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