Optimum Lateral Extent of Soil Domain for Dynamic SSI ... Corner/2020... · 2 30 response of buildings [6] and (e) substructure method [7]. Although computationally reliable and efficient,

Optimum Lateral Extent of Soil Domain for Dynamic SSI Analysis of RC Framed

Buildings on Pile Foundations

Author 1

Nishant Sharma, Research Scholar

Department of Civil Engineering, Indian Institute of Technology Guwahati, Guwahati,

Assam, India

Author 2

Kaustubh Dasgupta, Assistant Professor


Assam, India

Author 3

Arindam Dey, Assistant Professor


Assam, India

Full contact details of corresponding author.

Arindam Dey, Department of Civil Engineering, Indian Institute of Technology

Guwahati, Guwahati, Assam, India, Pin-781039.

Email: [email protected]

mailto:[email protected]

1

Optimum Lateral Extent of Soil Domain for Dynamic SSI 1

Analysis of RC Framed Buildings on Pile Foundations 2

3

Nishant Sharmaa, Kaustubh Dasguptab and Arindam Deyb* 4

a Research Scholar, Department of Civil Engineering, Indian Institute of Technology Guwahati, Guwahati, Assam, India 5 b Assistant Professor, Department of Civil Engineering, Indian Institute of Technology Guwahati, Guwahati, Assam, India 6 *Corresponding author. E-mail: [email protected] 7

ABSTRACT 8

This article describes a novel approach for deciding optimal horizontal extent of soil domain to be used for finite 9

element based numerical dynamic Soil Structure Interaction (SSI) studies. Soil Structure Interaction model for a 10

twelve storied building frame, supported on pile foundation-soil system, is developed in the finite element based 11

software framework, OpenSEES. Three different structure-foundation configurations are analyzed under 12

different ground motion characteristics. Lateral extent of soil domain, along with the soil properties, were varied 13

exhaustively for a particular structural configuration. Based on the reduction in the variation of acceleration 14

response at different locations in the SSI system (quantified by normalized root mean square error, NRMSE), 15

the optimum lateral extent of the soil domain is prescribed for various structural widths, soil types and peak 16

ground acceleration (PGA) levels of ground motion. Compared to the past studies, error estimation analysis 17

shows that the relationships prescribed in the present study are credible and are more inclusive of the various 18

factors that influence soil structure interaction. These relationships can be readily applied for deciding upon the 19

lateral extent of the soil domain for conducting precise SSI analysis with reduced computational time. 20

Keywords: Soil Structure Interaction, Optimum lateral extent of soil domain length, Multi-storeyed framed 21

building, Pile foundation, OpenSEES, L-K boundaries, Dynamic analysis. 22

1.0 Introduction 23

Soil-structure interaction (SSI) incorporates mutual dynamic interaction between the structure and the 24

supporting foundation medium. When considering finite soil domain, the surrounding foundation soil medium 25

influences the structural response. In the past, many researchers have adopted various numerical approaches for 26

investigating the behaviour of different categories of buildings with SSI, namely (a) frequency domain method 27

for general buildings[1-3], (b) discrete force method for analysis of shear wall buildings [4], (c) domain 28

reduction method [5] , (d) lumped spring approach for incorporating flexibility of foundation soil on the 29

2

response of buildings [6] and (e) substructure method [7]. Although computationally reliable and efficient, these 30

methods are mostly limited to studies on linear or equivalent linear response of soil-structure-foundation system. 31

Moreover, owing to the computational expense associated with the rigorous techniques, these procedures were 32

most widely used for performing SSI studies. Apart from these methods, SBFEM (Scaled Boundary Finite 33

Element Method) provides a powerful technique for modelling of the unbounded domain [8-12]. The method, 34

although powerful, has not obtained widespread popularity until date. This is due to the lack of commercially 35

available software that implement the method and lacking the provision of a library of materials capable of 36

representing nonlinear soil behaviour. With the advancement in computational tools, reliable numerical models 37

for soil, and with the increased number of studies highlighting the necessity of nonlinear SSI research, there has 38

been a paradigm shift towards adopting Finite Element methods for performing SSI studies. The finite element 39

tools have been largely incorporated in many of the commercial software such as OpenSEES, PLAXIS, 40

GeoStudio, and others. Advanced design codes such as the ‘Standard Specification for Concrete Structures, 41

Design (JSCE, 2007)’ [13] recommend incorporating foundation-soil system along with the structure for 42

evaluating the effects of SSI. The direct method of analysis for performing a single-step dynamic analysis, using 43

finite elements (FE), involves the numerical modelling of the foundation soil as continuum along with the 44

supported structure [14]. The advantage of that method is that a time domain approach is followed, and hence 45

the nonlinearity of the soil or structure can be directly incorporated [15]. Hence, considerable research studies 46

are oriented towards performing numerical nonlinear SSI analysis by modelling a finite extent of soil continuum 47

accompanied by appropriate boundary conditions at the lateral extents. 48

For conducting any numerical SSI study, deciding upon the lateral extent of the soil domain (also 49

referred to as soil domain length in this article) is very crucial. Theoretically, considering an infinite lateral 50

extent (as large as possible) of soil domain would be ideal as it produces a response free from the boundary 51

effects, while accounting for the radiation damping as well. However, such choice leads to heavy computational 52

expense and is not practically feasible. To overcome this difficulty, it is a common practice to model the SSI 53

system with a finite length of soil domain, using radiation boundaries at the lateral extents, for obtaining the 54

dynamic response. However, even with the incorporation of these far-off boundaries, the estimation of model 55

response may be inappropriate if the length of the domain is not sufficient. Therefore, it is important to 56

justifiably define the lateral extent of the soil domain to be used for dynamic SSI analysis that would ensure 57

computational efficiency without loss in accuracy of the system response. Standard domain definition available 58

for static cases cannot be used for dynamic or seismic loading as the extent is guided by several factors such as 59

3

the type of soil, interaction mechanism, nonlinear characteristics of the soil or structure, and intensity of 60

shaking, to name a few. Although there have been a few recommendations by past researchers on the soil 61

domain length to be used for dynamic SSI studies, they were based on simple SSI configurations (such as 62

shallow foundation, disc foundation, or single degree of freedom, SDOF, system) and linear soil characteristics 63

without providing due attention to the foundation parameters, except for the width of the foundation. Ghosh and 64

Wilson [16] recommended a horizontal extent of soil as 4W (W is the base width of the foundation); Roesset and 65

Ettorne [17] recommended a domain length of 5W for soils with high internal damping and 10W - 20W for soils 66

with low internal damping. Wolf [18] suggested for an increment in the lateral extent of the soil domain by 67

placing the artificial boundary further away from the structure to improve the accuracy. Table 1 shows the value 68

of normalized lateral extent of soil domain (Ω) used in the more recent studies as a function of structural or 69

foundation width (W). 70

It is worth emphasising that while selecting the lateral extent of soil domain, potentially influencing factors 71

such as the soil type, intensity of shaking, and structural configuration should be taken into consideration. 72

Modern day applications rely extensively on the explicit modelling of the SSI system with detail or with least 73

possible idealizations. A simplistic approach to determine the horizontal extents of the soil domain, based on 74

stress or displacement contours, is helpful only for static cases. For dynamic cases, the stress contours change at 75

every time step of the applied earthquake motion. If an analysis involves various factors (already mentioned), 76

then the determination of appropriate soil domain length becomes a cumbersome task for each case. Hence, it is 77

extremely important to ascertain the extent of the soil domain, on a case-to-case basis, with the help of an 78

exhaustive set of Finite Element (FE) simulations and provide recommendations, which would be of first-hand 79

help to the analysts. Therefore, the intention of the current article is to outline a novel approach for arriving at 80

the horizontal extent of soil domain to be used for DSSI analysis. Recommendations from the study, in the form 81

of relationships and guidelines, would facilitate the decision-making on soil domain length for various 82

representative situations. The recommendations also eliminate the need to make several numerical trials 83

conventionally required to arrive at the decision about the optimum length of the soil domain to be used for any 84

future DSSI analysis. To the best of the authors’ knowledge, such a study has not been reported in the literature 85

so far. 86

2.0 Modelling and Input 87

2.1 Description of the FE model 88

4

Three dimensional (3D) FE modelling and analysis are being adopted for conducting studies related to SSI 89

problems. This is because a 3D model provides an advantage of accurate representation of the real problem and 90

is useful for studying local effects. Nonetheless, 3D modelling is quite intriguing in terms of mesh generation, 91

convergence to a solution, and requires a high performance computation facility [30]. Furthermore, for problems 92

that have symmetry in geometry and loading, or wherein the shaking considered is in a particular direction 93

without resulting in any out–of-plane deformation/stress, or wherein the focus is on studying the global rather 94

than the local effects, a two-dimensional modelling and analysis can provide a fairly good insight into the 3D 95

problem and can be used to draw important conclusions. In the recent past, several SSI studies consisting of 3D 96

problem have been successfully conducted using 2D modelling/analysis [31-35] and it has been quite successful 97

in elucidating the essential response characteristics of even experimentally tested 3D building models [36]. The 98

intention of the present study is to investigate the global effects of boundaries on the response of the building 99

structure. Since the geometry of the problem considered is symmetric and out–of-plane shaking is not 100

considered, therefore, in the present study, 2D FE models, consisting of building frame supported by pile 101

foundation and surrounding soil medium, are developed in the finite element based software framework 102

OpenSEES [37]. The representative illustration of the FE model is shown in Fig. 1. Three different structural 103

widths are considered, and each of the structure-foundation configurations, subjected to three different ground 104

motion characteristics with varying Peak Ground Acceleration (PGA), is analysed. The horizontal extent of soil 105

domain, along with other soil properties, is varied exhaustively for a particular structural configuration. The 106

details of the structural and soil modelling, extent of soil domain used and ground motion input are given in the 107

following subsections. 108

2.1.1 Modelling of soil domain 109

The soil domain considered is of uniform depth, which is modelled using four noded quadrilateral elements with 110

four gauss integration points and bilinear isoparametric formulation. Reference low strain shear modulus of the 111

soil is considered to determine the largest size of the elements, based on the prescribed relationship by 112

Kuhlemeyer and Lysmer [38], as shown in Eq. 1. 113

25.011 )(125.0 rmaxmax Gfl (1) 114

where, lmax is the maximum size of the soil elements, fmax is the value of maximum frequency of input motion, 115

typically considered as 15 Hz. Gr is the reference low strain shear modulus of the soil specified at a reference 116

mean effective confining pressure of 80 kPa, and ρ is the mass density of the soil. 117

5

Three types of cohesionless soil, representing loose, medium and medium-dense sand, have been 118

considered in the study with the basic properties shown in Table 2. Pressure dependent constitutive behaviour 119

[39] is used for simulation of the nonlinear characteristics of the soil wherein plastic behaviour follows the 120

nested yield surface criteria [40]. The chosen constitutive model can simulate response characteristics dependent 121

on the instantaneous confining pressure, and is capable of modelling typical characteristics of cohesionless soil 122

such as dilatancy [41] (based on the non-associative flow rules) and liquefaction. For the purpose of the present 123

study, the bedrock is assumed to lie at a depth of 30 meters from the surface of ground level. The value of shear 124

wave velocity shown in the table is the average value of the entire layer. The actual variation of shear wave 125

velocity increases as the shear modulus increases with depth following the relationship shown in Table 2. The 126

water table was considered present at the bedrock level and is considered not to pose any influence on the 127

dynamic SSI studies. 128

INSERT Fig. 1 Representative illustration of the SSI model considered for the present study. 129

130

2.1.2 Modelling of the structure-foundation system 131

In the present study, the structure considered is a two dimensional RC building frame supported on pile 132

foundations. A representative illustration of the typical plan and elevation of the building considered is shown in 133

Fig. 1. The typical floor-to-floor storey height considered is 3 m and the bay width is considered 3 m. For 134

estimating the sizes of the frame members, the out-of-plane width and height of the structure are considered 135

constant as 15 m (5 bays) and 36 m, respectively, for the entire study (Fig. 1). Three different configurations 136

have been considered for the structure, by varying the number of bays in the direction of the in-plane structural 137

width (W), having width as 15 m (5 bays), 27 m (9 bays) and 45 m (15 bays). As per the guidelines specified in 138

IS 456 [42], it is mandatory to provide an expansion joint at every 45 m. For larger structural widths, this 139

expansion joint separates the two portions of the structure and the Structure-Soil-Structure-Interaction (SSSI) 140

response of the two portions are separately analysed. Hence, the width of the structure in the present study is 141

limited to 45 m, which encompasses the conventional widths of the individually standing structures. The 142

structure is assumed located in the Seismic Zone V as per the seismic zoning map of India [43]. It is supported 143

on pile foundations and has been designed with the help of relevant Indian standards [42, 34] after considering 144

gravity and lateral loading as per [43], [45] and [46]. The loading on the structure, apart from the self-weight, is 145

shown in Fig. 1. The size of the column is 500 mm × 500 mm until the sixth storey, above which the dimensions 146

are reduced by 100 mm along both the directions. The width of the beam is 250 mm and the overall depth is 400 147

6

mm for all storey levels. The size of the square grade beam is taken as 400 mm × 400 mm. The pile foundations 148

have been designed in accordance with the three different soil conditions considered in the study. Distance 149

between adjacent piles in a group is kept to be three times the diameter of the individual pile. The stiffness of 150

the pile group is estimated, and a single equivalent pile possessing the same stiffness as that of the pile group is 151

modelled beneath the columns, to account for the out-of-plane representation provided by the pile group. This 152

methodology has been successfully applied in the past [31]. The details of the pile groups and equivalent piles 153

are shown in Table 3. 154

Elastic beam-column elements, having two translational and one rotational DOF, have been used to model 155

the frame and pile members of the structure-foundation system. The structure-foundation system is positioned in 156

the centre of the soil domain. The discretization of the pile has been carried out ensuring connectivity between 157

the pile nodes and the adjacent soil nodes. The modulus of elasticity of the concrete material is taken to be 25 158

GPa. The horizontal inertial forces are simulated in the structure by means of lumped mass of the structure and 159

the additional loads at the corresponding frame and pile nodes. 160

2.1.3 Modelling of horizontal and vertical boundaries 161

To accurately model the effect of radiation damping, Lysmer-Kuhlemeyer (L-K) viscous dashpots [47] have 162

been assigned at the vertical and horizontal boundaries. These boundaries prevent the reflection of the seismic 163

waves back into the soil medium after being incident on the far-off boundaries. Moreover, the adopted 164

boundaries aid in the truncation of the soil domain to a finite extent. On the vertical boundaries, L-K dashpots 165

are assigned along both the horizontal and vertical directions, having dashpot coefficients as Cp = ρvpA and Cs 166

= ρvsA, respectively. Since the motion is applied at the base of the SSI system and is essentially a shear wave 167

propagating in the vertical direction, only horizontal dashpots are attached at the horizontal boundary having 168

coefficient Cs = ρvsA. The primary wave velocity (vp) is obtained from shear wave velocity and Poisson’s ratio 169

(ν) as 0.52 (1 ) / (1 2 )p sv v . Once the L-K boundaries have been assigned, seismic input is applied in the 170

form of equivalent nodal shear forces at the bedrock level. Based on the theory proposed by Joyner [48], Zhang 171

et al. [31] provided the expression for equivalent nodal shear forces as, 172

( , ) ( , ) 2 ( / )s s t sF x t C u x t C u t x v (2) 173

where, ( / )t su t x v is the velocity of incident motion, ( , )u x t is the velocity of the soil particle motion, and sC 174

is the coefficient of dashpot. The first term in Eq. 2 is the force generated by dashpot, while the second term is 175

7

the applied equivalent nodal force that is proportional to the velocity of the incident motion. The bedrock mass 176

(not modelled herein) is assumed homogenous, linear elastic, undamped and semi-infinite half-space region [31, 177

33]. In addition, the extent of near-field effects of soil-structure interaction reduces with increasing distance 178

from pile. Since the soil is modelled with the aid of ‘pressure dependent constitutive behaviour’ having damping 179

characteristics, the interaction effects result in the development maximum shear strain at the pile-soil interface 180

with an outward decreasing gradient. This aspect is automatically taken care through finite element analysis of 181

pile embedded in soil domain. 182

2.2 Seismic Input motions 183

Based on the categorization provided in Uniform Building Code [49] three ground motions (one near field 184

motion M1, two far field motions M2 and M3), belonging to different seismic events, have been selected for 185

conducting the study, the corresponding accelerograms being shown in Fig. 2. M1 is recorded during the 1995 186

Kobe earthquake at station KJMA, M2 is recorded during the 1980 Mammoth Lakes earthquake at station Long 187

Valley Dam and M3 is recorded during the 1994 Northridge earthquake at station Malibu-Point. The input 188

strong motions have been taken from the PEER ground motion database [50]. It can be observed that the 189

motions are characteristically different, in particular reference to their PGA and the duration of motion. It is 190

assumed that the motions have been recorded at the rock outcrop level, and hence, the same motions are used as 191

input, without any scaling down by 50%, as recommended in [15]. The analyses are performed for the 192

significant duration of each motion [51], which is calculated based on the Arias Intensity [52]. Arias Intensity (193

AI ) is a measure of the intensity of the shaking ground motion and is obtained as, 194

2

0

( )2

T

A iI u t dtg

(3) 195

where, iu is the acceleration of the incident motion, and g is the acceleration due to gravity. The significant 196

duration is the time duration corresponding to 5% AI - 95% AI . The Arias Intensity, as well as the significant 197

duration, of the selected motions, are also shown in Figs. 2b, 2d and 2f. 198

INSERT Fig. 2 Seismic input motions (a) accelerogram of M1, (b) Arias intensity of M1, (c) accelerogram of 199

M2, (d) Arias intensity of M2, (e) accelerogram of M3 and (f) Arias intensity of M3. 200

3 Analysis 201

3.1 Gravity analysis and validation 202

8

A rigorous dynamic analysis is preceded by a stage-wise static gravity analysis which has been outlined in [31] 203

and adopted in [33] with some modification. The analysis steps are suitably modified and listed as follows: 204

Stage 1: Elastic gravity analysis 205

The SSI model, consisting of the soil-pile-structure system, is designed with a base restrained in both the 206

horizontal and vertical directions. The vertical boundaries are restrained only in the horizontal direction and are 207

kept free in the vertical direction. The material model of soil is considered as elastic and the gravity loads of the 208

structure and soil are applied. Single step analysis is performed to achieve the equilibrium. 209

Stage 2: Plastic gravity analysis 210

The material constitutive model is changed to plastic and the SSI system is brought into equilibrium through 211

multiple iteration steps. Once the equilibrium is achieved, the reactions in the horizontal and vertical boundaries 212

are recorded. 213

Stage 3: Assigning L-K boundary condition on vertical boundaries of the SSI model 214

The restraint of the vertical boundaries along the horizontal direction in the SSI model is removed and the 215

reactions obtained in the previous stage are applied. The model is brought into equilibrium through iterations. 216

Subsequently, L-K boundaries are assigned in the horizontal as well as the vertical direction. 217

Stage 4: Assigning L-K boundary condition at the horizontal boundary of the SSI model 218

The restraint of the base boundary of the SSI model along the horizontal direction is removed and the reactions 219

obtained in Stage 2 analysis are applied. The model is brought into equilibrium iteratively and the L-K 220

boundaries are assigned in the horizontal direction. 221

Stage 5: Dynamic analysis 222

Once the boundary conditions have been successfully applied, the SSI model is subjected to seismic excitation 223

that is applied as equivalent nodal shear forces at the base of the FE model. The equation of motion of the SSI 224

model is shown in Eq. 4, and is expressed as. 225

[ ] ( ) [ ] ( ) [ ] ( ) ( ) vM u t C u t K u t F t F (4) 226

where [ ]M , [ ]C and [ ]K are the global mass, damping and stiffness matrices of the SSI system respectively; 227

u , u and u represent the nodal acceleration, velocity and displacements respectively; ( )F t represents 228

9

the input nodal shear force vector as shown in Eq. 2 and vF is the force vector assigned at the viscous 229

boundaries during the staged gravity analysis. The time-step integration scheme adopted is Newmark- β method 230

considering constant variation of acceleration over the time step that renders the scheme as unconditionally 231

stable, and the initial condition of the SSI model is considered to be ‘at rest’. As reported in literature [48] and 232

also observed in the preliminary analysis of the present study, material nonlinearity in soil results in the 233

development of high frequency oscillations, which is attributed to the possible excitation of the high frequency 234

modes of the soil system. Incorporating numerical damping into the system using the HHT-α method of analysis 235

[48], instead of the traditionally used Newmark-β method, may be helpful. However, the spurious oscillations 236

may not necessarily be arrested only by using numerical damping. In such cases, Rayleigh damping may be 237

used. Hence, in the present study, 2% Rayleigh damping has been incorporated into the SSI system, estimated 238

by considering the first two natural frequencies of the entire system [15, 54]. Based on the existing literature, the 239

outcome of the incorporation of boundary conditions and execution of the staged analysis is validated with the 240

results obtained by using a sine wavelet excitation to the model. The total acceleration response in the central 241

region, across the depth of the soil domain is obtained and compared with that in the existing literature [55] (Fig. 242

3). Close agreement of the results indicates successful application of the different stages of the analyses 243

described above and the numerical model incorporating SSI can be used for further rigorous dynamic analysis.244

245

INSERT Fig. 3 Validation of the numerical model. 246

3.2 Analysis cases and response locations 247

For the estimation of an optimum size of soil domain for SSI analysis, it is important to study the effect of soil 248

domain length on response of the SSI system. Instead of estimating the response at just one location 249

(conventionally at the mid-length of the lateral extent), the effect should be ascertained at various locations, 250

including the desired points of interest in the computational domain. Moreover, the response may be different 251

for different widths of the structure included the SSI system, and may vary with the different levels of seismic 252

shaking. Hence, in the present study, three structural-foundation-soil systems have been selected which are 253

different from each other in their in-plane widths (W). The out-of-plane width of these structures is the same as 254

shown in Fig. 1. Each structural configuration is founded on three different types of soil, as described earlier. 255

For a particular configuration, various lengths of soil domain are considered to define the SSI system. A 256

particular SSI system is then subjected to the previously selected three different strong motions. Table 4 shows 257

10

the various soil domain lengths, corresponding to the structural lengths, and motions considered for which the 258

SSI analysis is carried out. The values in the table indicate the time required for completion of the analysis. 259

In the present study, the effect of soil domain length on the response of the SSI system is evaluated by 260

monitoring the acceleration response at various locations in the system. The various locations selected are (i) 261

soil column very near the structure (Loc. A), (ii) soil at pile location (Loc. B), (iii) pile location (Loc. C), (iv) 262

soil column near boundary (Loc. D) and (v) superstructure (Loc. E). For each location, multiple points have 263

been selected for recording the response over the depth or height of the substructure or superstructure, 264

respectively. All the mentioned locations are shown in Fig. 4. 265

INSERT Fig. 4 Details of the response locations chosen for monitoring the acceleration response. 266

4 Results and discussion 267

The following sections discuss in detail results of various analyses, methodology adopted for obtaining the 268

optimum soil domain length and various relationships proposed for practical usage. 269

4.1 Effect of soil domain length on the response of SSI system 270

This section discusses the effect of varying soil domain length on the response of the SSI system. Figs. 5, 6 and 271

7 show the effect of horizontal extent of soil domain on the acceleration response at different locations in the 272

SSI system for structural widths W = 15 m, 27 m and 45 m, respectively. It is observed that there is a noticeable 273

difference, in the response of the SSI system, for the smallest and the largest domain lengths. As the domain 274

length increases, the response gradually approaches to that depicted by the larger domain lengths. Additionally, 275

a minute scrutiny shows that the change in the response is more sensitive for smaller domain lengths and 276

otherwise for larger domain lengths. For example, it can be observed that when the domain length is increased 277

from 33 m to 63 m, the change in the response is significantly large, whereas the same is comparatively lesser 278

when the domain length is increased from 183 m to 753 m. It implies that beyond a particular length of soil 279

domain, there is a marginal change in the response. The observation holds good for the different soil types, 280

motions, and structural widths selected in the present study. Hence, it can be said that for all practical 281

engineering purposes, there exists a particular length of soil domain which would be sufficient enough to 282

consider the obtained results as accurate, and the same is termed as the optimum lateral extent of soil domain 283

(denoted by Ω in this article). Identifying the optimum soil domain length would be helpful in significantly 284

reducing the computational costs incurred for SSI analysis without compromising on the accuracy of the 285

response. In the present article, a new approach is developed for identifying the optimum soil domain length 286

11

considering the influence of several contributory factors. In this regard, a quantitative estimation of the change 287

in the response at various locations in the SSI system has been determined, and the same has been used to define 288

the guidelines. The methodology is described in the next sections. 289

INSERT Fig. 5 Acceleration response for B = 15 m structural width and soil TY-I at (a) Loc. B for M1, (b) Loc. 290

E for M1, (c) Loc. B for M2, (d) Loc. E for M2, (e) Loc. B for M3 and (f) Loc. E for M3. 291





4.2 Quantification of change in response and optimum domain length 296

To quantify the change in the system response due to the increase of soil domain length, a particular foundation-297

structure system is analysed with different horizontal extents of soil domain, as shown in Table 4. Let ‘ itLa ’ be 298

the acceleration response at a particular nodal location ‘i’, at a given time instant ‘t’, and for a particular length 299

of soil domain ‘L’. For example, Fig. 5a shows the acceleration response, itLa for ‘i’ as Loc. B (0 m depth) at 300

different time instants ‘t’ of the ground motion, for various domain lengths ‘L’ (L = 33 m, 45 m, 63 m, 183 m 301

and 753 m). Similarly, the other subplots of Figs. 5, 6 and 7 show likewise. The system response corresponding 302

to the largest domain size can be considered as the most accurate as it is unaffected by the boundaries by virtue 303

of a very large extent of soil domain length considered. For smaller domain lengths, it is observed that there is 304

significant influence of the boundaries on the response. Hence, the response corresponding to the largest domain 305

length is considered as a benchmark of accuracy. It must be pointed out that the largest domain length 306

considered in the study is 50 times the structural width (W), which is quite large to produce a response free from 307

the boundary effects. On the other hand, a particular length of soil domain is said to be sufficient if it is small 308

enough to produce a response within an acceptable margin of tolerance with respect to the benchmark values (309

maxitLa ), and yet maintaining a restriction on the computational cost. The instantaneous absolute difference 310

between the exact response and that obtained from a specific length of soil domain is termed as the error in the 311

response entity, and is expressed as shown in Eq. 5. 312

maxitL itL itLe a a (5) 313

Once the error at all the time instants is obtained, the Root Mean Square Error (RMSE) at a particular 314

location ‘i’ and for a particular length of soil domain ‘L’ is estimated as: 315

12

2

0

T

itL

ttL

e

RMSEN

(6) 316

where, T is the total time duration of the ground motion, itLe is the instantaneous absolute error at a particular 317

time instant ‘t’, location ‘i’ and for a particular soil domain length ‘L’ (evaluated as per Eq. 5), and N is the total 318

number of time samples in the ground motion (N=T/∆t, ∆t is the sampling interval). Subsequently, the RMSE at 319

various locations (Fig. 4) are summed up to produce the cumulative RMSE corresponding to a particular domain 320

length of soil., i.e. L tLRMSE RMSE . 321

Fig. 8 shows the variation of normalized RMSE with the normalized domain length (L/W) for SSI 322

systems with varying structural widths (15 m, 27 m and 45 m) resting on different soil types and subjected to 323

various strong motions (M1, M2 and M3). The normalized RMSE is obtained by normalizing RMSEL by the 324

largest value, i.e., corresponding to the smallest domain length as shown in Eq. 7. 325

LL

Lmin

RMSENRMSE

RMSE (7) 326

It is observed that all the curves have similar trend characteristics: the initial portion of these curves is 327

very steep which gradually takes the shape of horizontal asymptote with an increase in the domain length. The 328

trend suggests that progressively increasing domain lengths have successively lesser influence on the system 329

response. The reduction in the change of the response with increase in normalized domain length is indicative of 330

the reduction in the error of the overall acceleration response in the SSI system. 331

INSERT Fig. 8 Variation of normalized RMSE with normalized domain length for (a) soil TY-I and M1, (b) 332

soil TY-II and M1, (c) soil TY-III and M1, (d) soil TY-I and M2, (e) soil TY-II and M2, (f) soil TY-III and M2 333

(g) soil TY-I and M3 (h) soil TY-II and M3, and (i) soil TY-III, M3. 334

4.3 Optimum soil domain length 335

Based on the qualitative trends as observed in Fig. 8, it can be generalized that there are two distinct zones, one 336

represented by a steep decrement of normalized RMSE with increasing domain length, followed by an 337

asymptotic trend of normalized RMSE. Therefore, the RMSE curves can be justifiably approximated with a 338

bilinear fit, as shown in Fig. 9. The intersection of the two branches provides the optimum magnitude of 339

normalized domain length, beyond which there is very little change in the system response. The normalized 340

domain length corresponding to the optimum point is termed as the optimum normalized length of soil domain 341

13

(Ω). That extent of Ω would be sufficient for an SSI analysis that is not only computationally inexpensive but 342

also helps in estimation of the system response within an adequate tolerance with the benchmark responses. 343

INSERT Fig. 9 Bilinear fit for RMSE for locating optimum point. 344

In order to observe the effect of ground motion intensity (PGA), the various RMSE curves are shown 345

by normalizing with respect to the curve obtained for the motion with the highest PGA level (M1) for various 346

soil domains as shown in Eq. 8. 347

max

LPGA

PGALmin

RMSENRMSE

RMSE (8) 348

Fig. 10 shows the various normalised RMSE curves along with their bilinear idealizations for different 349

soil types, structural widths and ground motions. The details of the idealised bilinear curves along with the 350

estimated optimum normalized domain lengths are provided in Table 6. 351

INSERT Fig. 10 Normalized RMSE curves along with their bilinear idealizations showing the optimum points 352

for (a) W = 15 m and soil TY-I (b) W = 15 m and soil TY-II, (c) W = 15 m and soil TY-III, (d) W = 27 m and 353

soil TY-I, (e) W = 27 m and soil TY-II, (f) W = 27 m and soil TY-III (g) W = 45 m and soil TY-I, (h) W = 45 m 354

and soil TY-II and (i) W = 45 m and soil TY-III. 355

4.4 Relationship of optimum domain length with PGA 356

Once the optimum normalized length of soil domain is obtained, their relationships with PGA for different 357

structural widths and soil types are investigated. Figs. 11a to 11c show the relationship between PGA and the 358

optimum normalized length of soil domain (Ω) for TY-I, TY-II, TY-III soils, corresponding to structural width 359

W = 15 m, 27 m and 45 m respectively. It is observed that, for a particular structural width (W) and PGA, the 360

value of Ω does not vary significantly for different soil types. For the structural width of W = 15 m, the value of 361

Ω corresponding to 0.82g PGA level lies in the range of 8.55-9.95 and that corresponding to 0.22g PGA level 362

lies in the range of 5.02-5.63 for the different soil types; similar feature is noted in all the other figures. 363

Moreover, it is observed that rather than soil type, PGA has more dominant influence on the variation of Ω. For 364

the structural width of W = 15m and TY-I soil, the value of Ω varies in the range of 5.44-9.95 for 0.22g-0.82g 365

PGA levels; similar feature is observed in the other representations as well. Considering negligible influence of 366

the soil type, an average curve is drawn to establish a relationship between PGA level and Ω (Figs 11d to 11f) 367

for structural widths 15 m, 27 m and 45 m, respectively. It is observed that a linear relationship provides a very 368

good fit indicated by sufficiently high R2 value. In all the figures, it is observed that the requirement of domain 369

length increases with the increase in the PGA level. For larger structural widths, the requirement of Ω, at high 370

14

PGA level, is less as compared to that corresponding to smaller structural widths. For e.g., at PGA level of 0.8g, 371

Ω is 8.9 for W = 15 m; however, Ω is 6.1 for the structural width of 45m at the same PGA level. For low PGA 372

levels, Ω is approximately similar. For 0.2g PGA level, for different structural widths, Ω lies between 4.7-5.4. 373

INSERT Fig. 11 Variation of optimum normalized domain length (Ω) with PGA for (a) W = 15 m, (b) W = 27 374

m and (c) W = 45 m for various soil types; variation of linear fitted Ω with PGA for (d) W = 15 m, (e) W = 27 m 375

and (f) W = 45 m. 376

The obtained relationships have been assesed with the help of sensitivity analysis. For such studies, 377

various approaches are available in the literature. Vu-Bac et al. [56] developed a software framework for 378

conducting uncertianity and sensitivity analysis of computationally expensive models. Khader et al. [57] 379

presented a methodology for the stochastic modelling of a problem by incorporating uncertiaintiy in the input 380

and constructing a PCE (Polynomial Chaos Expansion) surrogate model and finally showing its effectiveness in 381

sensitivity analysis. Khader et al. [58] implemented three methodologies for conducting sensitivity analysis in 382

their study, namely, MOAT (Morris One At a Time), PCE-Sobols’ and EFAST (Extended Fourier Amplitude 383

Sensitivity Test). Due to the limited scope of the present study, a simplistic sensitivity analysis is presented in 384

the present article considering two variables, width of the structre (W) and PGA of the input motion, for the 385

different soil types considered. The PGA level is considered to follow normal distribution with mean (μ) as 386

0.49g and standard deviation (σ) as 0.25g, and W is considered to be unifromly distributed between a range of 387

15 m to 45 m. The results of the sensitvity analysis using MOAT and Sobols’ method is shown in Fig. 12a and 388

Fig. 12b respecitvely. From Fig. 12a, it can be observed that for soil type TY-III, Ω shows greater sensitivity to 389

PGA, whereas for soil type TY-I and TY-II, both W and PGA are found influential. The total effect indices ( tS ) 390

and first order indices ( iS ) for the variables considered are shown in Fig. 12b. Based on the sensitivity indices, it 391

can be observed that for TY-I soils, W and PGA are equally sensitive and influential. For TY-II soils, W has 392

more influence compared to PGA, while for TY-III soils, PGA has been found to produce more dominant 393

influence in comparison to W. For any design or analysis of buildings supported by these different types of soils, 394

it is imperative to pay proper attention to the sensitive and contributing parameters. 395

INSERT Fig. 12 Sensitivity analysis (a) MOAT method (b) Sobols’ method. 396

4.5 Engineering application of the findings 397

In the previous section, the relationship between the optimum normalized length of soil domain and PGA of the 398

strong motions is established for individual structural/foundation widths (15 m, 27 m and 45 m). However, for 399

15

engineering purposes, it would be required to develop numerical models to investigate seismic SSI aspect. In 400

this regard, it would be more fruitful to develop relationships to ascertain the optimum normalized length of soil 401

domain required to conduct such studies for various structural or foundation widths. Based on the observations 402

and estimates obtained from the present study, engineering relationships with linear trends are developed for 403

various levels of PGA (Fig. 13), and are expressed in Eqs. 9-11. 404

0.1 11.5 0.82W PGA g (9) 405

0.03 7.5 0.43W PGA g (10) 406

5.5 0.22PGA g (11) 407

In Eqs. 9 and 10, the parameter W is expressed in meter. For intermediate PGA level, linear interpolation can be 408

sought. It is noted that the proposed relationships satisfy all the observations of the analyses, such as: 409

1. Higher the PGA, higher is the value of Ω for a particular structural/foundation width (B), and vice versa. 410

2. For larger structural/foundation widths (W), the optimum Ω is less, and vice-versa. 411

3. For low PGA level, Ω is nearly the same independent of different structural/foundation widths (B). 412

4. In the previous sections, it has already been established that the influence of soil type on Ω is insignificant, 413

and hence this parameter in not included in the expressions. 414

INSERT Fig. 13 (a) Averaged optimum normalized length of soil domain for varying structural width (W) for 415

different PGA levels and (b) proposed relationships of optimum domain length with varying structural width for 416

different PGA levels. 417

The practical applicability of the prescribed relationships is judged based on their performance in comparison 418

to a similar model developed with a very long soil domain (e.g., 50×W, i.e. Ω = 50). Corresponding to the three 419

selected structural widths and PGA levels, various optimum normalized length of soil domain are obtained and 420

dynamic analysis is carried out. Table 7 shows the percentage difference in the estimated peak acceleration 421

response recorded at (i) soil node at the surface very near to the structure, (ii) pile node at the top of the 422

outermost pile, and (iii) structure node at the roof level. It is observed that the maximum error obtained is less 423

than 10%, which is quite acceptable for practical engineering purposes. Moreover, the percentage of the 424

computational time required for performing the analysis is below 5% of the time required for performing exact 425

analysis, which definitely indicates the computational proficiency of the adopted values. 426

In the past, there have been a few recommendations for the horizontal extent of the soil domain to be 427

used for SSI studies. Some researchers have also adopted specific values of horizontal extent of the soil domain 428

16

as suitably required for their purpose without any comprehensive study. Fig. 14 shows a comparison of the 429

prescribed relationships from the present study and the soil domain lengths considered by past researchers. It is 430

observed that the values and the relationships used by past researchers fall within an agreeable zone of the 431

relationships proposed in the current study. However, it is to be noted that the relationships prescribed in the 432

present study are more rigorous since they are inclusive of various factors such as structural width, PGA level 433

and soil type. These relationships are more helpful in deciding about the soil domain length to be considered for 434

soil-structure interaction studies when various factors are to be accounted. It is observed that the relationships 435

proposed by previous researchers are mostly independent of the structural width, which in different cases prove 436

to be computationally inefficient owing to significant underestimation or overestimation of system response and 437

computational effort, respectively. Hence, based on the present findings and relevant comparisons, it can be 438

stated that the prescribed estimates for soil domain length are computationally efficient as well as suitable 439

enough to provide the system response. Thus, these relationships are credible for practical use for future SSI 440

analysis. For any random structural width and PGA of input seismic motion, the optimum normalized length of 441

soil domain length can be arrived at by a suitable linear interpolation. The relationships proposed in the present 442

study, thus, provide a guideline for arriving at the horizontal extent of soil domain to be chosen for SSI analysis. 443

INSERT Fig. 14 Comparion of proposed relationsip of optimum normalized length of soil domain with past 444

studies, for different PGA levels and structural widths. 445

5 Conclusion 446

In the present study, a strategic approach has been outlined to arrive at the optimum normalized length of soil 447

domain to be considered for soil-structure interaction studies. Based on exhaustive finite element simulations, 448

normalized root mean square error (NRMSE) for various domain lengths is obtained for various structural widths 449

and soil types. With the aid of bilinear fit to the NRMSE plots, normalized optimum soil domain lengths have 450

been obtained. Any length beyond the optimum domain length produces insignificant change in the overall 451

response of the SSI system with the percentage difference in the results being less than 10%. Optimum domain 452

lengths for various cases have been obtained, based on which a generalized set of relationships has been 453

prescribed by which the horizontal extent of soil domain is expressed as a function of structural or foundation 454

width and the PGA of the strong motion. It is observed that the domain length to be used for rigorous SSI 455

studies is virtually independent of the soil type. It is concluded that larger extents of soil domain are required for 456

SSI problems considering higher PGA strong motion, and vice versa. SSI problems comprising of smaller 457

structural widths require larger normalized domain lengths (Ω), and vice versa. In comparison to the correlations 458

17

of the domain length provided by earlier researchers, the one prescribed herein proves to be more robust (being 459

inclusive of various factors such as soil type, structural or foundation width and PGA level of input motion), 460

practically feasible and computationally efficient. The developed relationships from the present research can be 461

credibly used as guidelines for numerical modelling in SSI studies. 462

Acknowledgement 463

The support and resources provided by Department of Civil Engineering, Indian Institute of Technology Guwahati 464

and Ministry of Human Resources and Development (MHRD, Govt. of India), is gratefully acknowledged by the 465

authors. 466

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589

21

Table 1 Lateral extent of soil domain used in past studies 590

past studies W (m) H (m) Ω = (L/W) remarks

Lu et al. [19] 14.1 - 10.00 PF/EL/VB/FE

Pala et al. [20] 9.0 25-100 155.00 SF/L/VB/FE

Rayhani and Nagar [21] 5.5 - 5.00 SF/NL/EB/FD

Matinmanesh and Asheghabadi [22] 10.0 50 60.00 SF/L/SD/FE

Tabatabaiefar and Massumi [23] 8.0 - 7.50 SF/L/EB/FE

12.0 - 11.25 SF/L/EB/FE

Tabatabaiefar et al. [24] 12.0 - 5.00 SF/NL/VB/FD

Nateghi and Tabrizi [25] 40.0 - 10.00 SF/L/EB/FE

Saez et al. [26]

6.0 30

-

6.67 SF/NL/O/FE

10.0 7.00 SF/NL/O/FE

Hokmabadi et al. [27] 15.0 30 4.00 SF&PF/NL/VB/FD

Nguyen et al. [28] 15.0 30 9.33 SF/EL/IE/FE

Ghandil and Behnamfar [29] 20.0 45 4.00 PF/EL/VB/FE

note: remarks provide additional information about the study and abbreviations used are given under. general: W is foundation/structural width; H is the depth of soil; Ω is the normalized soil domain length (L/W); L is the length of

horizontal extent of soil domain.

foundation type: SF = shallow foundation; PF = pile foundation. soil property: L = linear; EL = equivalent linear; NL = nonlinear;

boundary: EB = elementary; VB = viscous; SB = spring-dahpot; IE = infinite elements; O = others.

analysis formulation: FE = finite element; FD = finite difference

591

592

22

Table 2 Basic properties and constitutive parameters of the soil 593

type of soil ρ (t/m3) φ () ν vs (m/s) Gr (kPa) γmax d ΦT ()

loose (TY-I) 1.7 29 0.33 180 5.5×104 0.1 0.5 29

medium (TY-II) 1.9 33 0.33 200 7.5×104 0.1 0.5 27

medium-dense (TY-III) 2.0 37 0.35 225 1.0×105 0.1 0.5 27

Note: ρ is the mass density of the soil, φ is the friction angle, ν is the Poisson’s ratio, /Gvs is the shear wave

velocity, Gr and γmax is the reference low strain shear modulus at and peak shear strain respectively at reference pressure

'rp = 80 kPa, d is defined by the relationship d

rpp

rGG )'/'( , 'p is the instantaneous effective confinement, G is the

instantaneous shear modulus and ΦT is the phase transformation angle.

594

23

Table 3 Details of the pile group 595

type of soil dia. of individual pile

(m)

length of individual pile

(m)

no. of plies in a

group

dia. of equivalent

pile (m)

loose (TY-I) 0.50 15 3 1.47

medium (TY-II) 0.45 15 3 1.32

medium-dense (TY-III) 0.40 15 3 1.17

596

24

Table 4 Details of the analysis cases and computational time for various cases. 597 domain

length

(m)

computational time required for analysis (in hours)

motion 1 (M1) motion 2 (M2) motion 3 (M3)

TY-I TY-II TY-III TY-I TY-II TY-III TY-I TY-II TY-III

structural width 15 m

753 12.23 15.35 15.38 8.23 8.51 8.50 44.58 43.63 42.41

681 8.00 11.81 14.56 6.57 8.91 7.63 35.86 33.82 33.65

603 7.86 9.00 11.00 5.13 8.03 7.52 27.57 26.51 26.25

543 5.40 7.02 8.09 5.15 7.27 6.23 22.53 18.02 21.93

483 4.00 5.56 4.91 4.23 6.58 5.42 18.46 17.89 17.66

423 3.28 4.13 3.46 3.15 4.53 3.69 14.96 13.61 14.25

363 2.50 3.06 2.63 2.75 3.64 2.79 12.03 10.78 11.28

303 1.75 `2.23 1.86 1.68 2.65 2.12 9.08 8.08 8.58

243 1.42 1.55 1.47 1.18 1.18 1.18 6.26 5.75 5.95

183 0.83 1.03 0.83 0.78 0.83 0.83 4.07 3.83 3.97

153 0.75 0.75 0.75 0.55 0.55 0.55 3.19 3.07 3.05

123 0.53 0.56 0.54 0.27 0.27 0.27 2.23 2.18 2.16

93 0.46 0.48 0.47 0.23 0.23 0.23 1.58 1.52 1.45

75 0.33 0.35 0.34 0.17 0.17 0.17 1.28 1.12 1.08

63 0.21 0.23 0.22 0.12 0.12 0.12 1.03 0.96 0.87

45 0.14 0.13 0.12 0.06 0.06 0.06 0.61 0.60 0.57

33 0.05 0.05 0.05 0.03 0.03 0.03 0.41 0.41 0.36


1353 34.23 33.78 32.50 32.23 29.155 33.90 282.33 245.17 201.22

1215 29.68 28.96 27.68 24.16 24.56 26.36 218.90 190.46 155.42

1083 24.56 23.93 23.27 20.17 20.39 20.84 166.00 142.59 116.70

975 20.56 19.34 19.92 16.23 17.42 17.79 120.63 107.35 90.45

867 17.26 15.56 15.67 13.56 13.88 14.34 106.20 82.81 66.52

759 10.25 8.37 7.69 8.56 6.03 8.59 78.35 56.18 42.13

651 8.56 5.13 6.31 6.75 4.76 6.32 56.65 38.35 35.52

543 6.58 3.74 4.57 4.49 3.43 4.14 38.56 26.82 24.89

435 4.56 2.68 3.02 2.09 2.25 2.71 24.35 18.35 17.59

327 2.36 1.74 1.82 1.55 1.13 1.64 15.63 11.33 11.46

273 1.56 1.29 1.22 1.16 0.87 1.00 12.36 8.72 8.78

219 1.15 0.64 0.60 0.71 0.67 0.74 10.63 6.17 6.52

165 0.96 0.45 0.43 0.48 0.45 0.48 6.35 3.97 4.73

135 0.65 0.36 0.36 0.34 0.35 0.21 4.35 2.93 3.15

111 0.43 0.29 0.28 0.14 0.16 0.17 2.48 2.19 2.45

81 0.23 0.15 0.20 0.10 0.12 0.12 1.15 1.38 1.53

57 0.10 0.06 0.13 0.06 0.07 0.07 0.15 0.83 0.92


2253 117.5 114.98 371.28 123.18 104.24 96.10 197.26 252.97 207.12

2025 87.66 82.60 244.26 99.5 75.06 71.24 139.37 199.74 147.73

1803 65.44 58.72 145.82 73.30 48.96 51.27 100.56 152.36 109.61

1623 50.72 42.82 129.82 59.83 50.26 60.58 76.64 116.56 80.93

1443 37.34 37.08 95.40 46.89 40.71 24.92 58.35 87.25 61.66

1263 29.73 30.00 49.21 35.26 29.87 24.24 43.48 62.12 48.69

1083 23.80 23.97 49.65 15.63 18.45 19.22 30.15 41.07 35.27

903 16.84 18.17 26.78 10.90 10.04 13.70 19.33 27.14 25.65

723 10.71 11.66 20.00 5.57 7.24 9.20 12.17 16.96 15.21

543 5.78 6.33 10.26 3.16 4.78 3.90 7.72 8.66 7.23

453 2.40 3.63 7.13 2.33 3.58 2.55 5.19 5.03 5.13

363 1.75 2.34 5.23 1.41 1.53 1.99 3.13 4.22 3.28

273 1.24 1.47 3.50 0.89 1.01 1.17 2.53 2.21 2.13

225 0.98 0.89 2.97 0.59 0.80 0.88 1.21 1.14 1.17

183 0.76 0.64 2.33 0.47 0.60 0.59 0.51 0.50 0.48

135 0.51 0.43 2.02 0.30 0.25 0.36 0.30 0.31 0.29

93 0.33 0.26 1.44 0.19 0.16 0.20 0.12 0.18 0.15

Values in the table indicates the analysis duration hours corresponding to various domain lengths considered

TY-I: Soft Soil; TY-II: Med Soil; TY-III: Medium Dense soil

25

Table 6 Details of the bilinear fits of various normalized RMSE curves and the estimated optimum normalized 598 domain length 599

structural

width, W

soil

type

ground

motion 1m

1C

2m

2C 2

1R 22R Ω

TY-I M1 -0.0860 1.0862 -0.0052 0.2819 0.91 0.90 9.95

TY-I M2 -0.0474 0.4663 -0.0044 0.1862 0.97 0.93 6.51

TY-I M3 -0.0221 0.1813 -0.0016 0.0698 0.98 0.96 5.44

TY-II M1 -0.0929 1.1503 -0.0058 0.4058 0.95 0.91 8.55

15 m TY-II M2 -0.0410 0.4015 -0.0034 0.1515 0.97 0.97 6.65

TY-II M3 -0.0205 0.1774 -0.0017 0.0715 0.97 0.93 5.63

TY-III M1 -0.0775 1.1087 -0.0063 0.4815 0.90 0.93 8.81

TY-III M2 -0.0446 0.4056 -0.0037 0.1651 0.97 0.95 5.88

TY-III M3 -0.0258 0.1894 -0.0016 0.0679 0.99 0.90 5.02

TY-I M1 -0.1492 1.235 -0.0054 0.3088 0.92 0.90 6.44

TY-I M2 -0.0522 0.4586 -0.0045 0.1885 0.91 0.90 5.62

TY-I M3 -0.0208 0.1631 -0.0016 0.0758 0.96 0.91 4.55

TY-II M1 -0.1279 1.2095 -0.0054 0.4646 0.91 0.92 6.08

27 m TY-II M2 -0.0416 0.3696 -0.0038 0.1704 0.90 0.94 5.23

TY-II M3 -0.0181 0.1518 -0.0018 0.0736 0.91 0.90 4.80

TY-III M1 -0.1037 1.1712 -0.0062 0.5201 0.93 0.92 6.68

TY-III M2 -0.0362 0.3288 -0.0034 0.1461 0.91 0.85 5.57

TY-III M3 -0.0197 0.1541 -0.0016 0.0651 0.91 0.81 4.92

TY-I M1 -0.1341 1.2002 -0.0044 0.4036 0.90 0.80 6.14

TY-I M2 -0.0695 0.5572 -0.0040 0.1742 0.98 0.75 5.85

TY-I M3 -0.0305 0.2224 -0.0017 0.0769 0.98 0.91 5.19

TY-II M1 -0.1019 1.1841 -0.0036 0.5955 0.96 0.91 5.98

45 m TY-II M2 -0.0475 0.4484 -0.0035 0.2076 0.99 0.71 5.47

TY-II M3 -0.0252 0.1998 -0.0017 0.0732 0.97 0.78 5.38

TY-III M1 -0.1079 1.1702 -0.0063 0.5331 0.85 0.80 6.27

TY-III M2 -0.0454 0.4183 -0.0035 0.1749 0.98 0.81 5.81

TY-III M3 -0.0120 0.1337 -0.0017 0.0771 0.98 0.88 5.49

note: Ω is the optimum value of normalized soil domain length; 21R is the coefficient of determination of the

vertical branch and 22R is the coefficient of determination of the horizontal branch

600

601

26

Table 7 Percentage error in estimate of peak acceleration response 602

structure

width, W

PGA

(g) Ω

Ω×W

(m)

difference in peak acceleration, % computational time#

% soil node pile node structure node

0.82 10.00 150 2 9 5 4.3

15 m 0.43 7.06 106 4 4 7 3.5

0.22 5.53 83 1 2 4 2.0

0.82 8.81 238 5 7 4 3.3

27 m 0.43 6.70 181 4 6 1 2.2

0.22 5.52 149 1 4 6 1.0

0.82 7.00 315 2 4 1 1.0

45 m 0.43 6.15 277 8 6 4 0.9

0.22 5.51 248 9 9 9 0.8 #ratio of the computational time required for analysis with domain length Ω×W to that required with domain length 50W in percentage

603

1

604

Figure 1605

1

606

Figure 2607

1

608

Figure 3609

1

610

Figure 4611

1

612

Figure 5613

1

614

Figure 6615

1

616

Figure 7617

1

618

Figure 8619

1

620

Figure 9621

1

622

Figure 10623

1

624

Figure 11625

1

626

Figure 12627

1

628

Figure 13 629

1

630

Figure 14 631

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Optimum Lateral Extent of Soil Domain for Dynamic SSI ... Corner/2020... · 2 30 response of buildings [6] and (e) substructure method [7]. Although computationally reliable and efficient,