Natural Period of Reinforced Concrete Building Frames on Pile Foundation Considering

Seismic Soil Structure Interaction Effects

Author 1

● Nishant Sharma, Research Scholar

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

Assam, India

Author 2

● Kaustubh Dasgupta, Associate Professor

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

Assam, India

Author 3

● Arindam Dey, Associate Professor

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

Assam, India

Full contact details of corresponding author.

Kaustubh Dasgupta, Department of Civil Engineering, Indian Institute of Technology

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

Email: kd@iitg.ac.in

1

NATURAL PERIOD OF REINFORCED CONCRETE BUILDING FRAMES ON

PILE FOUNDATION CONSIDERING SEISMIC SOIL STRUCTURE

INTERACTION EFFECTS

Nishant Sharma, Kaustubh Dasgupta⁎, and Arindam Dey

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

The magnitude of seismic forces induced within a building, during an earthquake, depends on its natural

period of vibration. The traditional approach is to assume the base of the building to be fixed for the estimation

of the natural period and ignore the influence of soil structure interaction (SSI) citing it to be beneficial. However,

for buildings resting on soft soils, the soil-foundation system imparts flexibility at the base of the building and it

is imperative for SSI effects to exist that may prove to be detrimental. Presence of SSI effects modifies the seismic

forces induced within the building, which is dependent on the change in its fixed base natural period. Expressions

for determination of the natural period of a structural system under the influence of SSI (effective natural period)

are available in literature. Although useful and made applicable to all types of foundation systems, these

expressions are developed either using simplified models or are applicable to shallow foundations and may not be

suitable to all types of structural-foundation systems in general. This article investigates the influence of seismic

soil structure interaction on the natural period of RC building frame supported on pile foundations. Detailed finite

element modelling of an exhaustive number of models, encompassing various parameters of the structure, soil

and pile foundation has been carried out in OpenSEES to study the effect of SSI on the natural period of the RC

building frame. The effect of various parameters on the natural period of building frame under SSI is investigated

and the results have been used to develop an ANN architecture model for estimation of the effective natural period

of RC building frame supported by pile foundation. Garson’s algorithm is used to conduct sensitivity analysis for

examining the importance of various parameters that govern the determination of effective natural period. A

predictive relationship for obtaining the effective natural period has been proposed using ANN architecture in the

form of a modification factor that is to be applied on the fixed base natural period, and which depends on various

input parameters of the building frame-pile-soil system. A comparison of the proposed relationship with those

available in literature demonstrates its usefulness and applicability to RC building frame on pile foundations.

⁎ Corresponding author.

Email addresses: nsharmanishant@gmail.com (Nishant Sharma), kd@iitg.ac.in (Kaustubh Dasgupta),

arindam.dey@iitg.ac.in (Arindam Dey)

2

KEYWORDS

Seismic soil structure interaction, RC buildings, Pile foundation, Time history analysis, Natural period, ANN,

Predictive relationship

1. INTRODUCTION

During earthquake shaking, the magnitude of seismic forces induced within the building structure

depends on its natural period of vibration (Tn). This is also reflected in the design codes of various countries. For

example, the seismic design of RC framed buildings, using the Indian seismic code [1], requires the estimation of

the design lateral loads that depends on the natural period (Tn) of the building. The conventional approach is to

assume the base of the building to be fixed for the estimation of the natural period. For buildings resting on rocky

or hard strata, wherein the soil-foundation system imparts high degree of restraint, this assumption holds good.

However, for the buildings resting on softer soils, the soil-foundation system imparts some amount of flexibility

at the base and the building response is affected by the foundation soil medium (this phenomenon is known as

Soil Structure Interaction and is abbreviated as SSI). Under compliant soil-foundation conditions, the natural

period of the building is increased as compared to that obtained by assuming the building to be fixed at the base.

For the same structure, a variation in foundation-soil condition would result in the modification of the natural

period and thus lead to a change in the seismic force levels. Available studies [2, 3] have highlighted the SSI effect

(increase in natural period) to be detrimental towards seismic safety of structures. In this regard, the estimation of

the natural period of the structure under the influence of SSI has been one of the focus areas as far as the subject

matter is concerned. Veletsos and Meek [4] proposed analytical expression for obtaining effective period of a

system with SSI effects (TSSI

). The same expression has been adopted by the seismic codes, ATC-3 [5] and UBC

[6], for structures supported on rigid mat foundations. Gazetas [7] proposed semi-empirical relation for TSSI

that

is an improvement over the work carried out by Veletsos and Meek [4]. Maravas et al. [8] also proposed rigorous

analytical solutions for obtaining TSSI for simple Single Degree of Freedom (SDOF) structures on pile foundation.

Kumar and Prakash [9] proposed semi-empirical relationships for structures founded on pile foundation. Rovithis

et al. [10] introduced the notion of pseudo-natural frequency for SDOF structures supported on single pile

embedded in homogenous viscoelastic soil. More recently, Deng et al. [11] and Medina et al. [12] proposed

regression models for evaluating the effective natural period of shear-type structures (idealized as SDOF systems)

supported on pile group foundation resting on viscoelastic homogenous and inhomogeneous soils, respectively.

Although these studies have focussed on pile group foundations, however the studies are more appropriate for

structures with simple configurations supported on group of piles possessing square grid arrangement connected

3

by a rigid pile cap, thereby behaving as a single unit. The relationships developed in the past incorporate the

following simplifications: (i) idealized SDOF oscillator, (ii) shallow rigid mat foundation or single lumped

foundation and (iii) use of equivalent springs to represent pile foundation stiffness (more useful for pile groups

behaving as a single unit and exhibiting uniform rocking behaviour). The past researches have focused on

obtaining the TSSI based on the simplifying assumptions, primarily for two reasons. Firstly, the evaluation of the

stiffness of the complex foundation systems (viz., spread foundation, soil-pile group foundation, soil-pile raft

foundation, etc.) is a tedious task and that the exact solutions are difficult to obtain. Secondly, the complex

interaction behavior of the individual piles in a group is not well understood and established. In real life situations,

buildings are MDOF systems for which simplified representation of foundation soil system may not be proper.

This is because pile groups under the columns at different locations would behave independently, thereby

exhibiting non-uniform rocking. Therefore, there is a need for developing a relationship for estimation of the

natural period of building under the influence of SSI (TSSI) that could account for the complex soil-foundation

behavior. Artificial Neural Networks (ANNs) have the ability to recognize complex relationships between the

input and output parameters, and have been used widely applied in various fields of civil engineering including

studies involving static and dynamic soil structure interaction. Acharyya and Dey [13, 14] employed ANN to

study and predict bearing capacity of footings on horizontal and sloping grounds. Momeni et al. [15] employed

ANN to evaluate pile bearing capacity, while Das and Basudhar [16] employed the same to evaluate the lateral

load capacity of the pile foundations. Pala et al. [17] demonstrated that ANN could be used for solution of dynamic

SSI problems of buildings.

Multistoried RC framed buildings comprise the most common typology of buildings used for various

purposes. Pile foundations are invariably used as the foundation system wherever the soil is weak or soft. Hence,

it becomes important to assess the natural period of RC building frames supported on pile foundations. The present

article investigates the effect of SSI on the natural period of RC building frame supported by pile foundation.

Detailed finite element modelling, carried out using OpenSEES [18], provides an accurate estimation of effective

natural period (TSSI

) as compared to the simplified models with SDOF system and foundation considered as single

lumped piles or equivalent springs. The effect of SSI on RC building frames has been studied for various

configurations of the superstructure and soil-pile foundation systems. Further, an ANN model has been trained to

develop a relationship for the modification factor (MF) useful in predicting the effective natural period (TSSI

) of

RC building frames supported on pile foundation. Comparison of the proposed relationship with those available

in literature demonstrates its usefulness and applicability to RC building frame on pile foundations.

4

2. NUMERICAL MODELLING AND PARAMETRIC CASE STUDIES

Fig. 1 shows the illustration of the SSI model considered in the present study. The numerical models of

the SSI systems (building, pile foundation and soil medium) were created and analyzed in OpenSEES using the

direct modelling approach. Brief description of the modelling details is provided in the following subsections.

Fig. 1. Representative illustration of the SSI model analyzed for the present study

2.1. Soil Domain

The basic properties of the considered soil types are shown in Table 1. Four types of soil representative

of different categories of site conditions are taken, i.e., loose sand (SS), medium sand (MS), medium dense sand

(MDS) and dense sand (DS). A uniform soil domain depth of 30 m is considered, as per the recommendation of

ATC 40 [19], which is modelled using four-noded quadrilateral elements with four gauss integration points and

bilinear isoparametric formulation. The nonlinear characteristics of the soil is simulated using pressure dependent

constitutive behavior [20] wherein nested yield surface criterion [21] is employed for plastic behavior. The shear

modulus of the soil varies parabolically with depth as per the relationship shown in Table 1 and the value of shear

wave velocity (vs) shown is the average value of the soil layer. The largest size of the soil element at a particular

depth is determined using the relationship prescribed by Kuhlemeyer and Lysmer [22] as shown:

1 0.25

max max0.125 ( )rl f G (1)

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

(typically considered as 15 Hz), Gr is the reference low strain shear modulus of the soil specified at reference

5

mean effective confining pressure of 80 kPa, and ρ is the mass density of the soil. This criterion enforces the mesh

discretization to vary, in the vertical direction, such that the elements possess a fine size at the surface and coarse

at the bedrock level. In the horizontal direction, the mesh size is governed by the geometry of the structure and

pile locations. Finer discretization of the elements was used near the structure and gradually coarsened towards

the boundaries. The mobilization of gravity and inertial loads has been applied with the help of bulk density (Table

1). Similar modelling approach, to simulate nonlinear SSI response of different structures, has been successfully

used by past researchers [23, 24]. For the purpose of the present study, the bedrock is assumed to lie at a depth of

30 m from the ground surface and is not considered for any influence on the dynamic SSI study.

Table 1 Basic properties and constitutive parameters of the soils used in the present study

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

Loose (SS) 1700 29 0.33 193 5.5×104 0.1 0.5 29

Medium (MS) 1900 33 0.33 212 7.5×104 0.1 0.5 27

Medium-Dense (MDS) 2000 37 0.35 234 1.0×105 0.1 0.5 27

Dense (DS) 2100 40 0.35 257 1.3×105 0.1 0.5 27

Note: φ is the friction angle, ν is the Poisson’s ratio, /Gvs is the average shear wave velocity, Gr and γmax is the

reference low strain shear modulus and peak shear strain respectively at reference pressure 'rp = 80 kPa, d is defined by

the relationship dr

ppr

GG )'/'( , 'p is the instantaneous effective confinement, G is the stress dependent shear

modulus and ΦT is the phase transformation angle.

2.2. Building-Foundation System

In the present study, two-dimensional RC building frame supported on pile foundation is considered, as

shown in Fig. 1. The floor-to-floor height and bay width are taken to be 3 m. Fig. 2 shows the details of the various

building configurations considered. Four different storey heights have been considered with 3 stories (9m), 6

stories (18m), 9 stories (27m) and 12 stories (36m). For each height of the building frame, four configurations of

structural width having 3 bays (9 m), 5 bays (15 m), 9 bays (27 m) and 15 bays (45 m) are considered. Each

structural configuration is to be rested on four different soil conditions (viz., SS, MS, MDS and DS) for which the

superstructure design is the same and only the foundation design is modified as per the soil type. Moreover, for a

particular structure-soil combination, three different arrangements of pile foundation are considered. For e.g., a 3

storey 3 bay structure resting on loose sand (SS) is supported by pile foundations under the columns having (a)

single pile, (b) a pile group of two or (c) a pile group of three.

6

Fig. 2. Details of the RC building frame configurations considered in the present study

For the design of the superstructure, gravity loads have been estimated as per the provisions of IS 875:

Part 2 (1987) [25] considering the intended use of the building for residential purpose. The assumed superimposed

dead load and live load are considered as 3 kN/m2. To represent the load of brick walls, 5 kN/m of uniformly

distributed load is considered to act on the beams of the frame. The lateral loading on the superstructure is

considered as per IS 1893: Part 1 (2016) [1]. The structure is assumed located in Zone V as per the seismic zoning

map of India [1] and the design fixed base natural period of the superstructure is obtained using the following

expression:

0.750.075nT H (2)

7

where Tn is the design fixed base natural period and H is the height of the frame building. T

n is utilized to estimate

the design acceleration coefficient and subsequently the base shear (Note: the value of the factors considered are

Zone factor = 0.36; Importance factor = 1 and Response reduction factor = 5). The estimated gravity and seismic

loads at the bottom level of the superstructure are used for the design of foundation under the column members.

The grade of concrete (fck

) and rebar (fy) used for the design of all concrete members in the present study are M30

and Fe500 respectively and the elastic modulus of concrete is obtained as Ec = 5000×(f

ck)

0.5 MPa [26]. The

superstructure is supported on pile foundations and the design of the structure-foundation system has been carried

out with the help of IS 456 (2000) [26], IS 13920 (2016) [27] and IS 2911: Part1/Sec 1 (2010) [28].

Table 2 Details of frame members for various building configurations

Storey

Level

3 Storey 6 Storey 9 Storey 12 Storey

Col Beam Col Beam Col Beam Col Beam

Up to 3 300×300 200×280 350×350 200×350 450×450 250×400 500×500 250×450

3 to 6 - - 350×350 200×350 400×400 250×400 450×450 250×450

6 to 9 - - - - 350×350 250×350 400×400 250×400

9 to 12 - - - - - - 400×400 200×350

Note: All dimensions of frame members are in millimetre (mm)

Table 3 Details of pile groups as used in the present study

No. of

Stories

Loose Soil

SS

Medium soil

MS

Med. dense soil

MDS

Dense soil

DS

Pile

length,

lp (m)

Pile

dia.,

dp (mm)

Pile

length,

lp (m)

Pile

dia.,

dp (mm)

Pile

length,

lp (m)

Pile

dia.,

dp (mm)

Pile

length,

lp (m)

Pile

dia.,

dp (mm)

Pile group of three

3 11.0 300 7.0 300 6.5 250 3.5 250

6 12.5 400 7.5 400 5.5 350 4.5 300

9 15.5 450 10.0 450 6.0 400 5.0 350

12 16.5 500 10.5 500 6.5 450 5.0 400

Pile group of two

3 12.0 350 8.0 350 6.0 300 3.5 300

6 12.0 500 9.5 450 6.0 400 5.0 350

9 15.0 550 12.0 500 7.5 450 5.5 400

12 16.5 600 10.0 600 6.5 450 5.0 400

Single pile

3 15.0 450 9.0 450 6.0 400 4.5 350

6 16.0 600 13.0 550 7.5 500 5.5 450

9 15.5 750 14.0 650 8.5 600 7.0 500

12 18.0 800 16.0 700 12.5 600 7.5 550

The details of the structural member sizes for the frame building are given in Table 2. For a particular

height of the building frame, the dimensions of the frame sections are kept to be the same for different structural

widths. The details of pile foundations are given in Table 3. For piles in a group, the distance between adjacent

8

piles is kept to be three times the diameter of the individual pile, as per the code suggested practice. Elastic beam-

column elements, having two translational DOFs and one rotational DOF, have been used to model the frame and

pile members of the structure-foundation system. Perfectly bonded interface is considered to connect the pile

foundation with the soil nodes. The horizontal inertial forces are simulated in the structure by means of lumping

the mass and loads of the structure at the nodes of the frame and pile members.

2.3. Soil Domain Boundaries

Radiation damping has been accounted for by modelling Lysmer-Kuhlemeyer (L-K) viscous dashpots

[29] at the horizontal and vertical boundaries. L-K boundaries ensure that seismic waves are prevented from being

reflected back into the soil medium. Viscous dashpots along horizontal and vertical directions are assigned on the

vertical boundaries having dashpot coefficients as Cp = ρvpA (where A is the tributary area of the boundary) and

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

(ν) as 0.5{2 (1 ) / (1 2 )}p sv v . At the horizontal base boundary, dashpots in the horizontal direction only have

been employed, having dashpot coefficient as Cs = ρvsA. To simulate the seismic input in the form of vertically

propagating shear waves, equivalent nodal shear forces are applied at the bedrock level. Based on the theory

proposed by Joyner [30], Zhang et al. [31] provided the expression for equivalent nodal shear forces as follows:

( , ) ( , ) 2 ( / )s s t sF x t C u x t C u t x v (3)

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

is the coefficient of dashpot. The first term in Eq. (3) 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 is

assumed to be homogenous, linear elastic, undamped and semi-infinite half-space region. The lateral extent for

each SSI system is fixed based on the study conducted by Sharma et al. [32]. This ensured that the extent of near-

field effects of soil-structure interaction reduces with increasing distance from the pile. Since the soil is modelled

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 finite element modelling of pile embedded in soil domain. The

entire soil domain is considered spatially invariable.

3. METHOD OF ANALYSIS

The fixed base natural period of the building frame is modified by considering the presence of the soil-pile

foundation system. The flexibility of the soil-pile foundation system depends on the confinement action imparted

9

by the soil to the piles. Sandy soils are pressure dependent wherein the shear modulus at any particular depth is a

function of the confining pressure at that depth. For imposition of appropriate confining action to the piles, proper

simulation of static stresses within the soil is essential [24, 31]. On assignment of Lysmer-Kuhlemeyer boundaries,

the confinement action is lost under gravity loading condition. This is because for assignment of L-K boundaries,

the displacement restraints at the soil boundary nodes are to be removed thereby allowing the soil nodes to deform

at the boundary which inhibits the development of proper confinement action. To develop a proper confining state

of stress within the soil, stage wise gravity analysis is carried out prior to conducting dynamic analysis of the

building frames under the influence of SSI. A schematic of the stage wise gravity analysis is shown in Fig. 3. At

each stage, the boundary condition of the SSI model is modified and gradually the boundary conditions required

for dynamic SSI analysis are achieved.

Fig. 3. Procedure of stage-wise static gravity analysis as adopted in the present study

The first stage (stage 1) corresponds to the state wherein the system is analyzed for gravity loading with

soil under elastic condition. The base of the SSI system is restrained both horizontally and vertically, while the

vertical boundaries are restrained only in the horizontal direction. Keeping the soil material model as elastic,

gravity analysis is performed in a single step to achieve equilibrium of the SSI system. In stage 2, the constitutive

model of the soil is made to behave plastically while keeping the boundary conditions to be the same as those at

the end of stage 1. Iteratively, the model is brought to equilibrium and the horizontal and vertical reactions

developed at the boundary restraints are noted. During stage 3, the horizontal restraints of the vertical boundaries

are removed and the corresponding reactions (recorded at the end of stage 2) are applied. Simultaneously,

horizontal and vertical L-K boundary conditions are applied to the vertical boundaries of the SSI model.

Subsequently, the model is iteratively brought into equilibrium under gravity loading. In Stage 4, the horizontal

Stage 1 and 2

Stage 4: Removal of horizontal restraints at base and assignment of L-K boundaries at base

Stage 3: Removal of horizontal restraints and

assignment of dashpots and assignment of L-K

condition at vertical boundaries

10

restraints at the base are removed and the corresponding reactions are applied, along with the assignment of the

basal L-K boundaries in the horizontal direction. Subsequently, the model is iteratively brought into equilibrium.

At this stage when all the boundaries have been incorporated, the SSI model does not have any horizontal

displacement constraint. This renders the stiffness matrix to be singular. Under such circumstance, eigenvalue

analysis does not show appropriate results, as there is a display of rigid body displacements for the first mode.

Additionally, the natural period estimates by eigenvalue analysis are influenced by the size of the soil domain

considered. For e.g., a building frame resting on soil would exhibit a different natural period if the horizontal

extent of the soil domain is changed from 150m to 200m or if the depth of the soil domain is modified from 30m

to 40m. This is so because the stiffness matrix and mass matrix of the SSI model are modified on changing the

extent of soil domain thus producing a different value of the natural period. However, the response of the structural

system under soil-pile flexibility should depend on the confinement action provided by the soil in the vicinity of

the pile foundations. Therefore, to obtain the natural period of the structure under the influence of SSI, time history

analysis is performed.

Once the boundary conditions have been successfully applied, the SSI model is ready to be subjected to

lateral vibrations that are applied as equivalent nodal shear forces at the base of the FE model. The SSI model is

subjected to very low amplitude excitation to restrict the soil material from developing nonlinearities [31]. In

addition, it is to be ensured that the selected motion is capable of exciting the natural frequencies of the structure

located within the SSI system. For this purpose, white noise of very small PGA level (0.005 m/s2) is selected as

the input motion. Figs. 4(a) and 4(b) show the time history and the frequency spectra of the input motion

respectively. The solution of the time history analysis is obtained by solving the equation of motion of the SSI

model shown in Eq. (4):

[ ]{ ( )} [ ]{ ( )} [ ]{ ( )} { ( )} { }vM u t C u t K u t F t F (4)

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

, { }u and { }u represent the nodal acceleration, velocity and displacement respectively; { ( )}F t represents the

input nodal shear force vector; and { }vF 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 considered to be ‘at rest’. The structural response of the SSI system is obtained and the natural

11

period is identified by analyzing it in the frequency domain. The dominant peaks in the Fourier Amplitude Ratio

(FAR) spectrum are indicative of the natural frequencies of the SSI system.

(a) (b)

Fig. 4. Seismic input motion (a) White noise time history (b) Fourier amplitude spectrum

4. RESULTS AND DISCUSSION

4.1. Effective natural period of RC building frame

Prior to analyzing the coupled soil-pile structure system, the structural configurations and the free field

soil system are separately analyzed to check the adequacy of the adopted methodology in predicting their natural

periods. Figs. 5(a) and 5(b) show the FAR (Fourier Amplitude Ratio) spectrum of the fixed-base acceleration

response obtained at the roof level of 6-storey 5-bay building frame and 12-storey 5-bay building frame

respectively. The natural frequencies corresponding to the first three modes have been identified and its

comparison with that obtained from conventional eigenvalue analysis is shown in Figs. 5(c) and 5(d). It can be

seen that a very good match is existent between the results from both the analysis procedures.

Similar methodology is applied to free-field soil domain and the FAR spectrum of the acceleration

response at the surface is obtained corresponding to SS, MS, MDS and DS, as shown in Figs. 6(a)-6(d)

respectively, wherein the natural frequencies of the free-field soil domain have been identified. Theoretical

expressions are available in literature [33] for obtaining the fundamental natural period of a free field soil domain,

as shown in Eq. (5):

4.48 s

s

sH

HT

v (5)

where Ts is the fundamental period of the soil layer, vsH is the shear wave velocity of the soil domain at the depth

Hs (bottom of the soil deposit). The above expression is applicable for soil layers having parabolic variation of

increasing shear modulus with depth [33]. Using the expression, the fundamental natural period of the free-field

soil domain is obtained for the different soil types and are compared with those obtained by the time history

-0.006

-0.003

0.000

0.003

0.006

0 10 20 30 40 50 60

Acc

ele

rati

on

(m

\s2)

Time (s)

0.000

0.002

0.003

0.005

0.006

0 2 4 6 8 10 12 14 16 18 20

Fou

rier

Am

pli

tud

e (

m\s

)

Frequecny (Hz)

Fourier AmplitudeSmoothed Fourier Amplitude

12

analysis of the finite element (FE) model (Fig. 7). As observed, a good agreement is attained between the two

approaches. With these comparisons, it can be said that the adopted methodology for determining the effective

natural period of the SSI system can be convincingly applied for further rigorous analyses.

(a) (b)

(c) (d)

Fig. 5. Fourier amplitude ratio spectrum of the roof-level response and the comparative of natural frequencies

from eigenvalue and finite element (FE) analysis for (a), (c) 6-storey 5-bay building frame (b), (d) 12-storey 5-

bay building frame

(a) (b)

(c) (d)

Fig. 6. Fourier amplitude ratio spectrum of the free field response of soil domain for soil types (a) SS (b) MS (c)

MDS and (d) DS

0.95

2.88

4.94

0

80

160

240

320

400

0 1 2 3 4 5 6

FA

R

Frequency (Hz)

0.70

1.87

3.19

0

40

80

120

160

0 1 2 3 4

FA

R

Frequency (Hz)

0.0

0.3

0.6

0.9

1.2

1 2 3

Nat

ura

l P

erio

d (

s)

Mode

Eigen Analysis

FE Analysis

0.0

0.4

0.8

1.2

1.6

1 2 3

Nat

ura

l P

erio

d (

s)

Mode

Eigen Analysis

FE Analysis

2.04

4.86

7.34

9.33

0

25

50

75

100

125

150

0 2 4 6 8 10

FA

R

Frequency (Hz)

2.23

5.49

8.12

10.19

0

40

80

120

160

200

0 2 4 6 8 10

FA

R

Frequency (Hz)

2.58

6.18

8.96

11.03

0

50

100

150

200

250

0 2 4 6 8 10 12

FA

R

Frequency (Hz)

2.81

6.83

9.76

11.77

0

50

100

150

200

250

0 2 4 6 8 10 12

FA

R

Frequency (Hz)

13

Fig. 7. Comparison of natural periods for free field response of soil domain obtained theoretically and FE

analysis considering different soil types

The SSI system is subjected to the considered white noise motion to induce dynamic excitations and the

acceleration response at the roof level is obtained. The response is analysed in the frequency domain and the

effective natural period (TSSI

) of each RC building frame is obtained corresponding to the different soil types. Fig.

8 shows an example wherein the fundamental natural frequency of a 6-storey 5-bay building frame, supported by

pile foundations, is obtained for the different soil conditions (viz., SS, MS, MDS and DS). The effective natural

period is then estimated from the effective natural frequency. Likewise, the effective natural periods of the various

configurations are obtained and are used for further discussion of results.

(a) (b)

(c) (d)

Fig. 8. Identification of the fundamental natural frequency of 6-storey 5-bay building frame supported by pile

foundation resting on (a) SS (b) MS (c) MDS and (d) DS soils

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

SS MS MDS DS

Nat

ura

l P

erio

d (

s)Soil Type

Theoretical

FE Analysis

0.90

0

20

40

60

80

100

120

0 0.4 0.8 1.2 1.6 2

FA

R

Frequency (Hz)

0.91

0

50

100

150

200

250

300

0 0.4 0.8 1.2 1.6 2

FA

R

Frequency (Hz)

0.92

0

30

60

90

120

150

180

0 0.4 0.8 1.2 1.6 2

FA

R

Frequency (Hz)

0.94

0

20

40

60

80

100

120

0 0.4 0.8 1.2 1.6 2

FA

R

Frequency (Hz)

14

4.2. Parameters influencing SSI effects

This section discusses the parameters that govern the change in the fixed base natural period of the RC

building frame due to SSI effects. The various building frames considered have been analyzed with the base as

fixed to obtain the fixed base natural period (TF). The same frames, along with the various soil-pile configurations,

are analyzed to obtain their respective effective natural periods (TSSI

). The amount of change in the natural period

(from TF to T

SSI) is a measure of the SSI effect on the system and is quantified in terms of the modification factor

(MF) as shown in Eq. (6).

SSI

F

TMF

T (6)

Higher MF indicates greater SSI effect on the natural period of the structure, thus signifying greater influence of

the soil-pile foundation system. A building frame is characterized by its natural period under fixed base condition

(TF). Depending on the structural configuration (height and width), T

F may vary which will in turn influence MF.

Figs. 9(a) and 9(b) show the influence of width of building frame on MF under different soil conditions for short

(3 storey) and tall (12 storey) frames respectively. It can be observed that for short frames (3 storied) increasing

the width from 3 bays to 15 bays increases TF from 0.70s to 0.73s. The variation of MF for narrow (3 bay) frame

under different soil conditions is about 1.10 (SS) to 1.05 (DS) and that for wider (15 bay) frame is about 1.16 (SS)

to 1.06 (DS). For taller frames (12 storied) increasing the width from 3 bays to 15 bays decreases TF from 1.47s

to 1.41s and the variation of MF for narrow (3 bay) frames under different soil condition is about 1.22 (SS) to

1.11 (DS) while that for wider (15 bay) frames is about 1.15 (SS) to 1.05 (DS). Similarly, Figs. 10(a) and 10(b)

show the influence of the height of building frame on MF under different soil conditions for narrow (3 bay) and

wider (15 bay) configurations respectively. For narrow frames (3 bay) on increasing the height from 3 stories to

12 stories, TF increases from 0.70s to 1.47s. The variation of MF for short (3 storey) frames under different soil

condition is about 1.11 (SS) to 1.05 (DS) and that for taller (12 storied) frames is about 1.22 (SS) to 1.10 (DS).

For wider frames (15 bay) increasing the height from 3 stories to 12 stories increases TF from 0.73s to 1.41s. The

variation of MF for short (3 storey) frames under different soil condition is about 1.16 (SS) to 1.06 (DS) while

that for taller (12 storey) frames is about 1.15 (SS) to 1.05 (DS). It can be observed that for any particular structural

configuration, SSI effects are highest (higher MF) for soft soil (SS) and are reduced for relatively stiffer soil

conditions (MS, MDS and DS). Increase in the width of the frame results in an increase or decrease in the fixed

base natural period (TF) depending on the frame height. Increase in the width of the frame causes an addition of

15

mass and stiffness to the structural system. In short frames, being inherently stiff in nature, an increase in width

results in the addition of mass to be dominant, thereby causing the wider frames to be more flexible as compared

to the narrow ones. Taller frames are by nature flexible and an increase in the width results in the addition of

stiffness to be dominant; consequently, the wider frames turn out to be stiffer as compared to the narrow ones.

SSI effects are observed to be marginally greater for frames with extreme configuration (e.g. very short and wide

or very tall and narrow frames). Increasing the height of the frames (corresponding to a fixed width) results in the

system becoming more flexible. Moreover, for narrow frames, the SSI effects increase with greater height;

however, for wider frames SSI effects are comparable corresponding to different heights.

(a) (b)

Fig. 9. Variation of MF with number of bays for (a) 3 storied and (b) 12 storied building frames

(a) (b)

Fig. 10. Variation of MF with number of stories for building frame with (a) 3 bays and (b) 15 bays

Apart from the structural configuration, the soil-foundation characteristics also influence MF.

Corresponding to a particular soil condition, the column members of the RC building frame can be supported by

pile foundations that may consist of a single large diameter pile or multiple piles of smaller diameter. For the same

structure-soil condition, this may result in a variation of MF. Figs. 11(a) and 11(b) respectively show the variation

of MF under different pile foundation configurations of short (3 storied 15 bay) and tall (12 storied 15 bay)

1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

0.70 0.71 0.73 0.73

MF

TF (s)

SS MS MDS DS

3 Bay 5 Bay 9 Bay

15 Bay

1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

1.41 1.42 1.44 1.47

MF

TF (s)

SS MS MDS DS

15 Bay 9 Bay 5 Bay

3 Bay

1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

0.70 1.06 1.19 1.47

MF

TF (s)

SS MS MDS DS

3 Storey 6 Storey

9 Storey

12 Storey

1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

0.73 1.07 1.18 1.41

MF

TF (s)

SS MS MDS DS

3 Storey 6 Storey 9 Storey 12 Storey

16

building frames for various soil conditions. As observed, for the different pile foundation configurations, MF

increases as the properties of the soil get weaker. Moreover, the building frames supported by group of 3 piles

and 2 piles exhibit slight reduction in MF as compared to the building frame supported on single piles. This is

because piles in a group are more effective in arresting rocking behavior, as compared to the single piles, and

impart additional rocking stiffness under the columns due to group action. Hence, MF is relatively larger for

structures supported by single piles as compared to the piles in a group.

(a) (b)

Fig. 11. Variation of MF with the shear wave velocity of soil for (a) 3 storied and (b) 12 storied building frames

For a particular foundation configuration, the type of soil governs the design of foundation. In the present

study, as the soil properties become weaker, the size of the pile foundation increases in diameter (dp) and/or length

(lp) as shown in Table 3. The properties of the pile foundation in conjunction with the soil determine the flexibility

of the soil-pile foundation system. Figs. 12(a) and 12(b) show the variation of MF with average length and

corresponding average diameter of pile for short (3 storey) and tall (12 storey) building frames respectively. It is

observed that MF increases with increase in pile length and diameter. Increase in both pile length and diameter

indicates weak soil conditions that ultimately leads to greater soil-pile foundation flexibility (higher MF).

Although the size of the pile has been segregated in terms of length and diameter, however, the two parameters

are correlated through pile capacity. Therefore, a dimensionless parameter, 0.25( / )( / )H p p p sS l d E E has been

used as a measure of pile foundation flexibility [10], where Ep is the elastic modulus of pile and E

s is the elastic

modulus of soil at a depth corresponding to the diameter of the piles. Stiffer piles possess smaller SH while flexible

piles possess higher SH. Figs. 13(a) and 13(b) show the variation of MF versus average S

H of the pile foundation

for various widths corresponding to short (3 storey) and tall (12 storey) frames respectively. It is observed that for

all structural configurations, the effect of SSI increases with the flexibility of piles (MF increases). Moreover, the

influence of pile foundation flexibility on short (3 storey) frame is greater for wider (15 bay) configuration while

1.00

1.05

1.10

1.15

1.20

1.25

DSMDSMSSS

MF

Soil Type

1 Pile 2 Pile 3 Pile

1.00

1.05

1.10

1.15

1.20

1.25

1.30

DSMDSMSSS

MF

Soil Type

1 Pile 2 Pile 3 Pile

17

that for taller (12 storey) frame is greater for narrow (3 bay) configuration. The results discussed herein are

corresponding to the building frame system wherein the columns are supported by pile group of three. Other

configurations of pile foundation show similar trends and hence are not discussed for the sake of brevity.

(a) (b)

Fig. 12 Variation of MF with pile length for (a) 3 storied and (b) 12 storied building frames

(a) (b)

Fig. 13. Variation of MF with SH for (a) 3 storied and (b) 12 storied building frames with shear wall

5. DEVELOPMENT OF ANN MODEL

Artificial Neural Networks (ANNs) are computing systems that are inspired by the biological neural

network. ANNs have been used widely applied in various fields of civil engineering including studies involving

static and dynamic soil structure interaction. They are capable of learning and modelling nonlinear and complex

relationships between the input and output parameters. ANN is particularly useful for the datasets that do not

follow a particular mathematical pattern (or in other words, datasets that are intricately difficult and complex to

decode). ANNs do not impose any restriction on the input variable (such as the requirement of the input data

conforming to a particular distribution) and its ability to generalize and infer relationships for unseen data is

especially advantageous. Additionally, the mathematical models developed using the ANN are not overly

complex, rather, they can be expressed with the help of multiple sets of simple relationships which are easy to

1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

3.5 6.5 7.0 11.0

MF

Length of pile, lp (m)

3 Bay 5 Bay 9 Bay 15 Bay

dp = 0.25mdp = 0.25m

dp = 0.30m

dp = 0.30m

1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

5.0 6.5 10.5 16.5

MF

Length of pile, lp (m)

3 Bay 5 Bay 9 Bay 15 Bay

dp = 0.40mdp = 0.45m

dp = 0.50m

dp = 0.55m

1.00

1.05

1.10

1.15

1.20

1.25

0 1 2 3 4 5 6 7 8

MF

SH

3 Bay (0.70s)

5 Bay (0.71s)

9 Bay (0.73s)

15Bay (0.73s)

1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

0 1 2 3 4 5 6 7 8

MF

SH

3 Bay (1.47s)

5 Bay (1.44s)

9 Bay (1.42s)

15Bay (1.41s)

18

code. In the present study, an artificial neural network architecture is developed for the prediction of the

modification factor (MF) and further details have been outlined in the following subsections.

5.1. ANN Model Architecture

In the present study, an artificial neural network architecture is developed for the prediction of the

modification factor, MF. The ANN model consists of input, output and hidden layers with various numbers of

nodes in each layer depending on the problem being addressed. Although there have been studies where multiple

hidden layers were used, however it has been noted in many past studies that an ANN model with single layer is

capable of providing good predictive relationships [34-37]. Levenberg-Marquardt learning rule is implemented

for training the neurons in the hidden layer. It is a feed-forward back-propagation algorithm which is the most

widely prevalent technique possessing a suitable prediction capability [36, 38]. The transfer function used in the

input-to-hidden and hidden-to-output layers is a ‘tan sigmoid’ as it can accurately represent the biological behavior

of neurons. The development of the ANN model has been carried out in MATLAB v2016a [39]. The input and

output data are preprocessed by normalizing using Eq. (7):

min

max min

2 ( )1

( )n

X XX

X X

(7)

where, Xn is the normalized value, Xmax and Xmin are the maximum and minimum values of the variable X. This

ensures that the data lies within a range of -1 to 1, Eq. (7), thus ensuring equal weightage to each variable during

the modeling phase.

The input parameters considered are corresponding to the soil, pile foundations and building frame

properties. The governing parameter for soil considered is the shear modulus (Gsoil). The inputs corresponding to

pile foundations are average diameter of individual pile (dp), average length of individual pile (lp) and the number

of piles (np). The structural system can be described using four input parameters namely the effective first mode

modal stiffness (K*), effective first mode modal mass (M*), height (H) and width (W) of the building frame. The

output considered is the modification factor (MF) and has already been discussed in section 4.2. The selection of

optimum number of hidden neurons is essential for ensuring good performance from the adopted model.

Convergence study is conducted to obtain the optimum number of neurons in the hidden layer such that the error

associated with the estimated output is minimum. The quantification of the error is represented as shown:

2

1

( )n

simulated predicted

i

MF MF

MSEn

(8)

19

where, MSE is the mean of the squared error obtained from the dissimilarities in the simulated and predicted

output MFsimulated and MFpredicted respectively and n is the number of data points. Fig. 14 shows that for the present

problem, the MSE of the ANN model is minimum corresponding to 9 neurons in the hidden layer. This results in

an 8-9-1 architecture of the ANN model as shown in Fig. 15.

Fig. 14. Variation of MSE with number of hidden layer neurons

Fig. 15. Architecture of the proposed ANN model

Based on the FE simulations, a dataset comprising 192 input-output combinations are generated which

is provided in Table A1 of Appendix – A. For the development of the ANN model, the complete data set is divided

[40] into samples on which training, validation and testing exercises have been performed. In summary, 134

0.00

0.04

0.08

0.12

0.16

0 2 4 6 8 10 12 14 16 18

MS

E (

MF

)

Number of neurons in hidden layer

20

combinations have been used for training, 29 for validation and 29 for testing. The datasets chosen for each of the

exercises have been chosen randomly from the complete dataset so that the generalization in data is achieved. The

training of the ANN model continually adjusts the weights and biases associated with the neurons through an

iterative feed-forward back-propagation algorithm so that a minimum error between the simulated and predicted

values of the modification factor (MF) is achieved. It is important to stop the training at an optimum stage when

the model is trained just enough to generalize the relationship without overfitting or underfitting of the data. For

this, an early stopping criterion is used [13, 15] in which the error corresponding to training data and testing data

is obtained at each epoch (iteration) and is compared to the preceding epoch. As the number of epoch increases,

the error in the training data continues to decrease; however, for the test data, the error reduces up to a certain

number of iterations, beyond which the error increases. The training is stopped when the error, corresponding to

test data, successively increases for a certain number of successive epochs. Figs. 16(a) and 16(b) show the

comparison of the simulated and the predicted values of the output (MF) during training and testing phases

respectively. The coefficients of efficiency R2 for training and testing are found to be 0.96 and 0.95 respectively.

(a) (b)

(c)

Fig. 16. Performance of ANN model (a) Training phase (b) Testing phase (c) Residual distribution

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

MF

Pre

dic

ted

MFSimulated

R2 = 0.96

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

MF

Pre

dic

ted

MFSimulated

R2 = R2 = 0.95

-0.2

0

0.2

0 50 100 150 200

Resid

ual

Experiment number

21

It can be seen that the model is able to predict the values of the output quite well when exposed to the testing data

set. Residual analysis of the ANN model was carried out by calculating the residuals from the simulated

modification factor and predicted modification factor for the entire data set. The residual is defined in Eq. (9) as

r simulated predictede MF MF (9)

where er is the residual corresponding to a simulated modification factor (MFsimulated) and predicted modification

factor (MFpredicted). The residuals corresponding to each data point are assigned an experiment number (given in

Table A1 in Appendix A) and plotted. Figure 16(c) shows the residual plot of the relationship proposed for the

entire data set. It can be observed that the residuals are scattered randomly about the horizontal line and that its

magnitude is within 20%. Thus, the model can be used for prediction of the modification factor (MF).

5.1.2 Sensitivity Study

For any predictive model, it is important to know the dependency of the output variable towards each

input variable. This can be obtained by ascertaining the relative importance of the input variables involved by

performing a sensitivity analysis. In the present study, Garson’s algorithm [41] has been implemented to study

the sensitivity of various input variables on the modification factor (MF). The algorithm partitions the connection

weights (input-hidden and hidden-output) and utilizes their absolute values to determine the relative importance

of each input variable, as expressed in Eq. (10):

1

1

Relative Importancem

m jn

IH

X Nj k j j

IH HO

k

w

w w

(10)

where, Xm is the mth input variable for which the relative importance is to be obtained, wIH

is the input-hidden

neuron weight, wHO

is the hidden-output neuron weight, N (N = 8) is the total number of input variables, and n (n

= 9) is the total number of neurons in the hidden layer. Table 4 shows the weights of the neurons connecting the

input layer nodes and hidden layer nodes. Table 5 shows the weight between the hidden layer nodes and the output.

Table 6 lists the biases of the input and output neurons after training. The relative importance and ranking of the

various input parameters are given in Table 7.

The attainment of relative importance percentage and the importance rank of individual variables

highlight the fact that the modification factor (MF) is highly dependent on the pile dimensions (dp and lp), followed

by the structural dimension width and height (H and W). The effective stiffness and mass of the structure (M* and

22

K*), followed by soil property Gsoil and the number of piles (np), have relatively lower influence on the value of

MF as compared to the dimensions of the pile foundation and structure.

Table 4 Weights of the neurons connecting the input and hidden layer nodes

Input

Variable

wIH

(input-hidden neuron weights)

N1 N2 N3 N4 N5 N6 N7 N8 N9

X1 GSoil 0.19 -0.18 0.23 -0.78 0.94 -1.10 -0.10 -1.16 0.19

X2 dp 0.37 -1.10 -1.41 -0.01 -0.28 -0.93 0.79 -1.17 0.37

X3 lp -0.02 -0.57 0.14 0.05 -0.20 0.32 2.90 -0.55 -0.02

X4 np -0.27 -1.21 -0.88 0.69 -0.36 0.29 -0.002 -1.01 -0.27

X5 K* 0.56 -1.42 1.37 -0.33 1.47 -0.36 -0.64 0.32 0.56

X6 M* -0.18 0.11 -0.38 0.06 0.63 1.22 0.23 -1.88 -0.18

X7 H -0.30 0.35 0.49 -0.59 1.29 0.77 -1.32 -0.70 -0.30

X8 W -0.82 0.47 -1.26 0.01 -0.59 -0.94 1.36 0.25 -0.82

Table 5 Weights of the neurons connecting the hidden and output layer nodes

Output

TSSI /TF

wHO

(hidden-output neuron weight)

N1 N2 N3 N4 N5 N6 N7 N8 N9

Y 1.88 0.15 0.72 -0.42 -0.58 -3.15 0.82 1.00 1.18

Table 6 Biases of the neurons after training

b0 (Bias of

output

node)

bhj (Bias of hidden layer neurons)

N1 N2 N3 N4 N5 N6 N7 N8 N9

2.07 -2.67 -1.30 -0.50 -0.29 -0.49 -0.56 -1.29 -1.70 -2.59

Table 7 Importance ranking of input parameters based on Garson’s algorithm

Input Variable Outcome of Garson’s Sensitivity Analysis

Relative Importance Relative importance (%) Rank

X1 GSoil 1.04 6.50 7

X2 dp 2.80 17.60 1

X3 lp 2.79 17.56 2

X4 np 0.84 5.26 8

X5 K* 1.56 9.76 6

X6 M* 1.90 11.93 5

X7 H 2.53 15.90 3

X8 W 2.47 15.49 4

5.1.3 ANN Model Equation

The input-output relationship of the developed ANN model can be expressed in the form of a predictive

relationship. Past studies have shown the usefulness of such relationships and various researchers have developed

similar ANN based predictive relationships for different problems [15, 42-43]. Similar approach has been adopted

23

in the present study to form a predictive expression for estimating MF, considering various influencing

parameters, which can be represented by the expression shown in Eq. (11):

0

1 1

n Nj j

n lin HO sig hj IH k

j k

MF f b w f b w X

(11)

where, MFn is the normalized MF varying from -1 to 1, fsig is the sigmoid transfer function, flin is the linear transfer

function, b0 is the bias at output layer, bhj is the bias at the jth neuron of the hidden layer, w

IH is the input-hidden

weight, wHO

is the hidden-output weight, N (N = 8) is the total number of input variables and n (n = 9) is the total

number of neurons in the hidden layer. The developed expression utilizes the weights and biases obtained after

training of the data as shown in Table 4-6. Since the development of the ANN model has been carried out on

normalized data (to ensure equal weightage to all the input parameters), denormalization of the normalized

modification factor (MFn) is necessary. The denormalized value of the modification factor (MF) is obtained using

Eq. (12):

max min min0.5( 1)( )nMF MF MF MF MF (12)

where MFmax = 1.23, MFmin= 1.02, MFn and MF are the normalized and denormalized modification factors,

respectively. Thus Eq. (13) can be used to calculate the modification factor MF using the input variables.

0.105 ( 1) 1.02nMF MF (13)

where,

9

0

1

n j

j

MF b B

(14)

( )j j

j j j

A A

j HO N A A

e eB w

e e

(15)

8

,

1

( )k jj hj k IH X N

k

A b X w

(16)

For the sake of clarity, the expanded form of Aj is shown in the following equation.

1 2 3 4 5

6 7 8

*

, , , , ,

*

, , ,

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

j j j j j

j j j

j hj Soil IH X N p IH X N p IH X N p IH X N IH X N

IH X N IH X N IH X N

A b G w d w l w n w K w

M w H w W w

(17)

In Eq. (14) and Eq. (16), b0 and bhj (j corresponds to the neuron number Nj) can be obtained from Table 6. In Eq.

(15), ( )jHO Nw can be obtained from Table 5. Similarly, in Eq. (16) (or Eq. (17)), ,( )

k jIH X Nw can be obtained from

Table 4.

24

6. COMPARISON WITH PREVIOUS RELATIONSHIPS

As already mentioned, few earlier studies have prescribed relationships for TSSI

, as given in Table 8 in

the form of modification factor (MF), i.e., the ratio of effective natural period of the SSI system (TSSI

) to the fixed

base natural period (TF). The estimates of MF using the proposed ANN predictive relationship is compared with

those available in literature and with that obtained from finite element simulations. Expressions from the past

studies require the estimation of foundation stiffness for which standard relationships have been utilized [33]. The

results shown in this section are corresponding to the building frame system wherein the columns are supported

by a group of three piles. Other configurations of pile foundation show similar trends and hence, are not discussed

for the sake of brevity.

Table 8 TSSI

relationships proposed in past studies

Past Study Expression Remark

Veletsos and

Meek [4] * * 2

1SSI

F x

T K K HMF

T K K

Analytical equation developed for surface

footings. Most widely used and adopted by

seismic code e.g. ATC 3 [5].

Gazetas [7] * * * 2

1SSI

F x x

T K K H K HMF

T K K K

Semi-empirical relation with additional sway

rocking component.

Kumar and

Prakash [9] 1.5

* * 2601SSI

F x

T K K HMF

T H K K

Semi-empirical relationship proposed

specifically for structure on pile foundation

MF = Ratio of fixed effective natural period to the fixed base period of the system

TSSI

= Natural period of the structural system under the influence of SSI

TF = Fixed base natural period of the superstructure

H = Effective height of the superstructure

K* = Effective stiffness of the superstructure under fixed base condition

Kx = Lateral stiffness of the foundation

Kϕ = Rotational stiffness of the foundation

Kxϕ

= Coupled sway rotational stiffness of the foundation

Fig. 17 shows a comparison of MF for a few building frames. Each building frame model is labelled to indicate

the number of stories, bays and type of supporting soil condition. For e.g., a 3 storied 15 bay frame supported on

soft soil (SS) condition is labelled as ‘F3.15_SS’. The estimates of MF using the proposed ANN equation show a

very good and close agreement with those obtained from finite element simulations. This is because the

expressions proposed in the present study has been developed using advanced finite element SSI models,

considering several input parameters which are interrelated by means of a complex network using the theory of

ANN. It can also be observed that the expression given by past researchers provide lower estimates of the MF for

the frames supported on weaker soil conditions and there exists a large difference with respect to the FE simulated

25

values. This is because the expressions proposed by Veletsos and Meek [4] and Gazetas [7] have been developed

and are applicable to structures supported on shallow foundation; however, those expressions have been applied

to structures on pile foundation. The expression of Kumar and Prakash [9] although developed for structures

supported on pile foundation, has been developed using a simplified model and is a modification over the

expression given by Veletsos and Meek [4]. The results presented here highlight the robustness of the ANN based

expressions proposed in the present study.

Fig. 17. Comparison of MF estimates from the proposed relation with past studies

7. CONCLUSIONS

Fundamental natural period of a building is an important characteristic required for seismic analysis or design.

The prevailing trend is to obtain the fixed base natural period (TF) of the buildings and use it for further analysis.

However, the soil-foundation characteristics influence the natural vibrational characteristics and lead to a

modification in the natural period due to SSI effects. In the present study, influence of SSI effects on RC building

frames supported by pile foundations is studied. Detailed finite element modelling has been carried out to obtain

the effective natural period (TSSI

) of several configurations of building frames, under various pile foundation and

soil types. The change in the fixed base natural period under the influence of SSI for the frames, was quantified

in terms of the modification factor (MF) which is the ratio of effective natural period (TSSI

) to the fixed base natural

period (TF) of the building frame. Parametric study was conducted to identify the influence of various input

parameters, of the SSI system, on MF whose higher magnitude indicated greater SSI effects. Subsequently a feed-

F3-5

_S

S

F3-1

5_S

S

F3-5

_D

S

F3-1

5_D

S

F6-5

_S

S

F6-1

5_S

S

F6-5

_D

S

F6-1

5_D

S

F9-5

_S

S

F9-1

5_S

S

F9-5

_D

S

F9-1

5_D

S

F12-5

_S

S

F12-1

5_S

S

F12-5

_D

S

F12-1

5_D

S

1.00

1.05

1.10

1.15

1.20 FE Simulated ATC 3 [5] Gazetas [7]

Kumar and Prakash [9] Present Study

MF

Numerical Specimen

26

forward back-propagation artificial neural network (ANN) model has been developed to form a predictive

relationship for obtaining the modification factor (MF) for the determination of effective natural period (TSSI

) of

reinforced concrete building frames with pile foundations. The main conclusions drawn from the present study

are as follows:

1. Building frames supported by loose soil (SS) exhibited highest SSI effects that reduced for stiffer soil

conditions. A change in the building frame width does not modify the fixed base natural period of the

building frame significantly as compared to the height. Building frames with extreme configurations

such as very short and wide or very tall and narrow frames, showed marginally greater SSI effects.

2. Frames supported on single pile foundation under the columns exhibit greater SSI effects as compared

to those supported on group of two or three piles, as in a group the piles are more effective in arresting

rocking behavior and impart additional rocking stiffness under the columns due to group action. The

flexibility of pile foundation, along with soil, plays a determining role in SSI effects. Building frames on

pile foundations having greater pile foundation flexibility (SH) exhibit greater SSI effects.

3. The developed ANN based model is able to accurately predict the relationship of MF with various input

parameters. Sensitivity analysis using Garson’s algorithm showed that the diameter (dp) and length of

pile (lp), are the most influential input parameters in the determination of the modification factor,

followed by frame height (H) and width (W). Structural characteristics represented by effective modal

stiffness (K*), modal mass (M*), followed by shear modulus of the soil (GSoil

) and number of pile (np),

have relatively lesser importance.

4. The proposed ANN equation is able to provide relatively accurate estimates of MF when compared with

those obtained using the expressions available in literature. This is so because the expressions proposed

by past researchers were originally developed for SDOF on shallow foundation and were either made to

be applicable or modified for structures on pile foundations. The proposed ANN equation is developed

using advanced finite element modelling considering the intricate relationship between the various input

parameters and the output and can be utilized for estimation of (TSSI

) for RC building frame supported

on pile foundations for different soil types.

27

ACKNOWLEDGEMENTS

The support and resources provided by Dept. of Civil Engg., Indian Institute of Technology Guwahati and

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

authors.

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APPENDIX-A

Table A1 Dataset used for training and testing

Data Type Expt. No. G* dp lp np K* M* H W MFsimulated MFpredicted

Training

1 1.000 -1.000 -1.000 -0.636 -0.775 -1.000 -1.000 -1.000 -0.669 -0.695

3 1.000 -1.000 -1.000 0.182 -0.044 -0.787 -1.000 0.000 -0.865 -0.850

5 0.225 -1.000 -0.586 -0.636 -0.775 -1.000 -1.000 -1.000 -0.525 -0.540

7 0.225 -1.000 -0.586 0.182 -0.044 -0.787 -1.000 0.000 -0.573 -0.552

8 0.225 -1.000 -0.586 1.000 0.718 -0.575 -1.000 1.000 -0.420 -0.438

10 -0.414 -0.818 -0.517 -0.364 -0.516 -0.929 -1.000 -0.667 -0.293 -0.393

11 -0.414 -0.818 -0.517 0.182 -0.044 -0.787 -1.000 0.000 -0.420 -0.325

12 -0.414 -0.818 -0.517 1.000 0.718 -0.575 -1.000 1.000 -0.182 -0.234

13 -1.000 -0.818 0.034 -0.636 -0.775 -1.000 -1.000 -1.000 -0.147 -0.127

14 -1.000 -0.818 0.034 -0.364 -0.516 -0.929 -1.000 -0.667 -0.056 -0.121

15 -1.000 -0.818 0.034 0.182 -0.044 -0.787 -1.000 0.000 -0.016 -0.040

16 -1.000 -0.818 0.034 1.000 0.718 -0.575 -1.000 1.000 0.330 0.353

20 1.000 -0.818 -0.862 1.000 0.640 -0.030 -0.333 1.000 -0.782 -0.755

21 0.225 -0.636 -0.724 -0.636 -1.000 -0.890 -0.333 -1.000 -0.583 -0.583

23 0.225 -0.636 -0.724 0.182 -0.090 -0.459 -0.333 0.000 -0.473 -0.406

25 -0.414 -0.455 -0.448 -0.636 -1.000 -0.890 -0.333 -1.000 -0.473 -0.504

26 -0.414 -0.455 -0.448 -0.364 -0.581 -0.747 -0.333 -0.667 -0.796 -0.649

27 -0.414 -0.455 -0.448 0.182 -0.090 -0.459 -0.333 0.000 -0.126 -0.275

28 -0.414 -0.455 -0.448 1.000 0.640 -0.030 -0.333 1.000 -0.223 -0.239

29 -1.000 -0.455 0.241 -0.636 -1.000 -0.890 -0.333 -1.000 -0.131 -0.183

30 -1.000 -0.455 0.241 -0.364 -0.581 -0.747 -0.333 -0.667 -0.690 -0.471

32 -1.000 -0.455 0.241 1.000 0.640 -0.030 -0.333 1.000 0.019 0.070

34 1.000 -0.636 -0.793 -0.364 -0.471 -0.572 0.333 -0.667 -0.451 -0.530

36 1.000 -0.636 -0.793 1.000 1.000 0.495 0.333 1.000 -0.614 -0.575

38 0.225 -0.455 -0.655 -0.364 -0.471 -0.572 0.333 -0.667 -0.324 -0.299

42 -0.414 -0.273 -0.103 -0.364 -0.471 -0.572 0.333 -0.667 -0.193 -0.196

43 -0.414 -0.273 -0.103 0.182 0.125 -0.144 0.333 0.000 -0.255 -0.252

44 -0.414 -0.273 -0.103 1.000 1.000 0.495 0.333 1.000 -0.106 -0.191

45 -1.000 -0.273 0.655 -0.636 -0.757 -0.783 0.333 -1.000 0.190 0.203

46 -1.000 -0.273 0.655 -0.364 -0.471 -0.572 0.333 -0.667 0.078 0.005

48 -1.000 -0.273 0.655 1.000 1.000 0.495 0.333 1.000 0.310 0.337

49 1.000 -0.455 -0.793 -0.636 -0.818 -0.682 1.000 -1.000 -0.170 -0.134

51 1.000 -0.455 -0.793 0.182 0.008 0.160 1.000 0.000 -0.593 -0.615

53 0.225 -0.273 -0.586 -0.636 -0.818 -0.682 1.000 -1.000 -0.003 -0.035

55 0.225 -0.273 -0.586 0.182 0.008 0.160 1.000 0.000 -0.292 -0.293

56 0.225 -0.273 -0.586 1.000 0.834 1.000 1.000 1.000 -0.376 -0.392

58 -0.414 -0.091 -0.034 -0.364 -0.545 -0.403 1.000 -0.667 -0.119 -0.002

61 -1.000 -0.091 0.793 -0.636 -0.818 -0.682 1.000 -1.000 0.918 0.611

62 -1.000 -0.091 0.793 -0.364 -0.545 -0.403 1.000 -0.667 0.045 0.275

63 -1.000 -0.091 0.793 0.182 0.008 0.160 1.000 0.000 0.027 0.082

64 -1.000 -0.091 0.793 1.000 0.834 1.000 1.000 1.000 0.267 0.223

66 1.000 -0.818 -1.000 -0.636 -0.516 -0.929 -1.000 -0.667 -0.665 -0.562

67 1.000 -0.818 -1.000 -0.273 -0.044 -0.787 -1.000 0.000 -0.935 -0.808

68 1.000 -0.818 -1.000 0.273 0.718 -0.575 -1.000 1.000 -0.837 -0.869

69 0.225 -0.818 -0.655 -0.818 -0.775 -1.000 -1.000 -1.000 -0.597 -0.496

70 0.225 -0.818 -0.655 -0.636 -0.516 -0.929 -1.000 -0.667 -0.665 -0.534

71 0.225 -0.818 -0.655 -0.273 -0.044 -0.787 -1.000 0.000 -0.647 -0.740

72 0.225 -0.818 -0.655 0.273 0.718 -0.575 -1.000 1.000 -0.837 -0.769

73 -0.414 -0.636 -0.379 -0.818 -0.775 -1.000 -1.000 -1.000 -0.525 -0.429

74 -0.414 -0.636 -0.379 -0.636 -0.516 -0.929 -1.000 -0.667 -0.293 -0.382

75 -0.414 -0.636 -0.379 -0.273 -0.044 -0.787 -1.000 0.000 -0.497 -0.569

77 -1.000 -0.636 0.172 -0.818 -0.775 -1.000 -1.000 -1.000 -0.147 -0.143

78 -1.000 -0.636 0.172 -0.636 -0.516 -0.929 -1.000 -0.667 -0.215 -0.219

79 -1.000 -0.636 0.172 -0.273 -0.044 -0.787 -1.000 0.000 -0.342 -0.307

82 1.000 -0.636 -0.793 -0.636 -0.581 -0.747 -0.333 -0.667 -0.690 -0.832

83 1.000 -0.636 -0.793 -0.273 -0.090 -0.459 -0.333 0.000 -0.690 -0.818

84 1.000 -0.636 -0.793 0.273 0.640 -0.030 -0.333 1.000 -0.782 -0.864

30

Data Type Expt. No. G* dp lp np K* M* H W MFsimulated MFpredicted

85 0.225 -0.455 -0.655 -0.818 -1.000 -0.890 -0.333 -1.000 -0.690 -0.685

86 0.225 -0.455 -0.655 -0.636 -0.581 -0.747 -0.333 -0.667 -0.690 -0.793

87 0.225 -0.455 -0.655 -0.273 -0.090 -0.459 -0.333 0.000 -0.690 -0.562

90 -0.414 -0.273 -0.172 -0.636 -0.581 -0.747 -0.333 -0.667 -0.473 -0.644

91 -0.414 -0.273 -0.172 -0.273 -0.090 -0.459 -0.333 0.000 -0.473 -0.446

93 -1.000 -0.091 0.172 -0.818 -1.000 -0.890 -0.333 -1.000 -0.473 -0.450

95 -1.000 -0.091 0.172 -0.273 -0.090 -0.459 -0.333 0.000 -0.126 -0.122

96 -1.000 -0.091 0.172 0.273 0.640 -0.030 -0.333 1.000 -0.104 -0.117

97 1.000 -0.455 -0.724 -0.818 -0.757 -0.783 0.333 -1.000 -0.349 -0.314

99 1.000 -0.455 -0.724 -0.273 0.125 -0.144 0.333 0.000 -0.630 -0.540

100 1.000 -0.455 -0.724 0.273 1.000 0.495 0.333 1.000 -0.614 -0.576

101 0.225 -0.273 -0.448 -0.818 -0.757 -0.783 0.333 -1.000 -0.219 -0.212

102 0.225 -0.273 -0.448 -0.636 -0.471 -0.572 0.333 -0.667 -0.324 -0.368

103 0.225 -0.273 -0.448 -0.273 0.125 -0.144 0.333 0.000 -0.383 -0.401

104 0.225 -0.273 -0.448 0.273 1.000 0.495 0.333 1.000 -0.492 -0.444

105 -0.414 -0.091 0.172 -0.818 -0.757 -0.783 0.333 -1.000 -0.086 0.023

106 -0.414 -0.091 0.172 -0.636 -0.471 -0.572 0.333 -0.667 -0.193 -0.155

107 -0.414 -0.091 0.172 -0.273 0.125 -0.144 0.333 0.000 -0.255 -0.215

108 -0.414 -0.091 0.172 0.273 1.000 0.495 0.333 1.000 -0.106 -0.130

109 -1.000 0.091 0.586 -0.818 -0.757 -0.783 0.333 -1.000 0.192 0.165

110 -1.000 0.091 0.586 -0.636 -0.471 -0.572 0.333 -0.667 0.078 0.023

111 -1.000 0.091 0.586 -0.273 0.125 -0.144 0.333 0.000 0.149 0.061

116 1.000 -0.273 -0.655 0.273 0.834 1.000 1.000 1.000 -0.667 -0.665

117 0.225 -0.091 -0.379 -0.818 -0.818 -0.682 1.000 -1.000 -0.170 -0.123

118 0.225 -0.091 -0.379 -0.636 -0.545 -0.403 1.000 -0.667 -0.280 -0.306

119 0.225 -0.091 -0.379 -0.273 0.008 0.160 1.000 0.000 -0.445 -0.507

120 0.225 -0.091 -0.379 0.273 0.834 1.000 1.000 1.000 -0.523 -0.493

121 -0.414 0.273 -0.103 -0.818 -0.818 -0.682 1.000 -1.000 -0.170 -0.122

122 -0.414 0.273 -0.103 -0.636 -0.545 -0.403 1.000 -0.667 -0.119 -0.175

123 -0.414 0.273 -0.103 -0.273 0.008 0.160 1.000 0.000 -0.292 -0.260

125 -1.000 0.273 0.793 -0.818 -0.818 -0.682 1.000 -1.000 0.347 0.377

127 -1.000 0.273 0.793 -0.273 0.008 0.160 1.000 0.000 0.027 -0.102

128 -1.000 0.273 0.793 0.273 0.834 1.000 1.000 1.000 -0.062 0.008

129 1.000 -0.636 -0.862 -1.000 -0.775 -1.000 -1.000 -1.000 -0.068 -0.101

130 1.000 -0.636 -0.862 -0.909 -0.516 -0.929 -1.000 -0.667 -0.056 -0.124

132 1.000 -0.636 -0.862 -0.455 0.718 -0.575 -1.000 1.000 -0.457 -0.437

133 0.225 -0.455 -0.655 -1.000 -0.775 -1.000 -1.000 -1.000 -0.068 -0.182

134 0.225 -0.455 -0.655 -0.909 -0.516 -0.929 -1.000 -0.667 0.027 -0.027

135 0.225 -0.455 -0.655 -0.727 -0.044 -0.787 -1.000 0.000 -0.182 -0.194

136 0.225 -0.455 -0.655 -0.455 0.718 -0.575 -1.000 1.000 -0.213 -0.220

137 -0.414 -0.273 -0.241 -1.000 -0.775 -1.000 -1.000 -1.000 0.093 0.049

138 -0.414 -0.273 -0.241 -0.909 -0.516 -0.929 -1.000 -0.667 0.109 0.239

139 -0.414 -0.273 -0.241 -0.727 -0.044 -0.787 -1.000 0.000 0.155 0.135

140 -0.414 -0.273 -0.241 -0.455 0.718 -0.575 -1.000 1.000 0.130 0.172

141 -1.000 -0.273 0.586 -1.000 -0.775 -1.000 -1.000 -1.000 0.611 0.622

143 -1.000 -0.273 0.586 -0.727 -0.044 -0.787 -1.000 0.000 0.605 0.545

144 -1.000 -0.273 0.586 -0.455 0.718 -0.575 -1.000 1.000 0.592 0.605

145 1.000 -0.273 -0.724 -1.000 -1.000 -0.890 -0.333 -1.000 -0.473 -0.446

146 1.000 -0.273 -0.724 -0.909 -0.581 -0.747 -0.333 -0.667 -0.473 -0.445

148 1.000 -0.273 -0.724 -0.455 0.640 -0.030 -0.333 1.000 -0.566 -0.518

150 0.225 -0.091 -0.448 -0.909 -0.581 -0.747 -0.333 -0.667 -0.473 -0.383

151 0.225 -0.091 -0.448 -0.727 -0.090 -0.459 -0.333 0.000 -0.126 -0.143

154 -0.414 0.091 0.310 -0.909 -0.581 -0.747 -0.333 -0.667 -0.006 -0.037

155 -0.414 0.091 0.310 -0.727 -0.090 -0.459 -0.333 0.000 -0.006 0.017

156 -0.414 0.091 0.310 -0.455 0.640 -0.030 -0.333 1.000 0.144 0.061

157 -1.000 0.273 0.724 -1.000 -1.000 -0.890 -0.333 -1.000 0.118 0.151

159 -1.000 0.273 0.724 -0.727 -0.090 -0.459 -0.333 0.000 0.374 0.389

160 -1.000 0.273 0.724 -0.455 0.640 -0.030 -0.333 1.000 0.403 0.430

162 1.000 -0.091 -0.517 -0.909 -0.471 -0.572 0.333 -0.667 -0.193 -0.099

163 1.000 -0.091 -0.517 -0.727 0.125 -0.144 0.333 0.000 -0.123 -0.158

164 1.000 -0.091 -0.517 -0.455 1.000 0.495 0.333 1.000 -0.237 -0.344

165 0.225 0.273 -0.310 -1.000 -0.757 -0.783 0.333 -1.000 0.217 0.093

167 0.225 0.273 -0.310 -0.727 0.125 -0.144 0.333 0.000 -0.123 -0.055

170 -0.414 0.455 0.448 -0.909 -0.471 -0.572 0.333 -0.667 0.219 0.207

172 -0.414 0.455 0.448 -0.455 1.000 0.495 0.333 1.000 0.310 0.353

173 -1.000 0.818 0.655 -1.000 -0.757 -0.783 0.333 -1.000 0.364 0.397

176 -1.000 0.818 0.655 -0.455 1.000 0.495 0.333 1.000 0.761 0.776

178 1.000 0.091 -0.448 -0.909 -0.545 -0.403 1.000 -0.667 -0.119 -0.156

179 1.000 0.091 -0.448 -0.727 0.008 0.160 1.000 0.000 -0.292 -0.309

31

Data Type Expt. No. G* dp lp np K* M* H W MFsimulated MFpredicted

182 0.225 0.273 0.241 -0.909 -0.545 -0.403 1.000 -0.667 0.045 0.088

183 0.225 0.273 0.241 -0.727 0.008 0.160 1.000 0.000 -0.136 -0.203

186 -0.414 0.636 0.724 -0.909 -0.545 -0.403 1.000 -0.667 0.045 0.192

188 -0.414 0.636 0.724 -0.455 0.834 1.000 1.000 1.000 0.269 0.364

189 -1.000 1.000 1.000 -1.000 -0.818 -0.682 1.000 -1.000 0.530 0.574

190 -1.000 1.000 1.000 -0.909 -0.545 -0.403 1.000 -0.667 0.573 0.367

191 -1.000 1.000 1.000 -0.727 0.008 0.160 1.000 0.000 0.194 0.308

192 -1.000 1.000 1.000 -0.455 0.834 1.000 1.000 1.000 1.000 0.879

Testing

2 1.000 -1.000 -1.000 -0.364 -0.516 -0.929 -1.000 -0.667 -0.593 -0.782

4 1.000 -1.000 -1.000 1.000 0.718 -0.575 -1.000 1.000 -0.573 -0.849

6 0.225 -1.000 -0.586 -0.364 -0.516 -0.929 -1.000 -0.667 -0.369 -0.580

9 -0.414 -0.818 -0.517 -0.636 -0.775 -1.000 -1.000 -1.000 -0.451 -0.525

17 1.000 -0.818 -0.862 -0.636 -1.000 -0.890 -0.333 -1.000 -0.690 -0.605

18 1.000 -0.818 -0.862 -0.364 -0.581 -0.747 -0.333 -0.667 -1.000 -0.889

19 1.000 -0.818 -0.862 0.182 -0.090 -0.459 -0.333 0.000 -0.690 -0.742

22 0.225 -0.636 -0.724 -0.364 -0.581 -0.747 -0.333 -0.667 -0.899 -0.759

24 0.225 -0.636 -0.724 1.000 0.640 -0.030 -0.333 1.000 -0.566 -0.481

31 -1.000 -0.455 0.241 0.182 -0.090 -0.459 -0.333 0.000 -0.006 -0.241

33 1.000 -0.636 -0.793 -0.636 -0.757 -0.783 0.333 -1.000 -0.349 -0.349

35 1.000 -0.636 -0.793 0.182 0.125 -0.144 0.333 0.000 -0.630 -0.509

37 0.225 -0.455 -0.655 -0.636 -0.757 -0.783 0.333 -1.000 -0.219 -0.181

39 0.225 -0.455 -0.655 0.182 0.125 -0.144 0.333 0.000 -0.383 -0.281

40 0.225 -0.455 -0.655 1.000 1.000 0.495 0.333 1.000 -0.366 -0.405

41 -0.414 -0.273 -0.103 -0.636 -0.757 -0.783 0.333 -1.000 -0.086 -0.073

47 -1.000 -0.273 0.655 0.182 0.125 -0.144 0.333 0.000 0.149 0.022

50 1.000 -0.455 -0.793 -0.364 -0.545 -0.403 1.000 -0.667 -0.434 -0.421

52 1.000 -0.455 -0.793 1.000 0.834 1.000 1.000 1.000 -0.667 -0.642

54 0.225 -0.273 -0.586 -0.364 -0.545 -0.403 1.000 -0.667 -0.280 -0.149

57 -0.414 -0.091 -0.034 -0.636 -0.818 -0.682 1.000 -1.000 0.347 0.124

59 -0.414 -0.091 -0.034 0.182 0.008 0.160 1.000 0.000 -0.136 -0.233

60 -0.414 -0.091 -0.034 1.000 0.834 1.000 1.000 1.000 -0.220 -0.149

65 1.000 -0.818 -1.000 -0.818 -0.775 -1.000 -1.000 -1.000 -0.808 -0.490

76 -0.414 -0.636 -0.379 0.273 0.718 -0.575 -1.000 1.000 -0.613 -0.489

80 -1.000 -0.636 0.172 0.273 0.718 -0.575 -1.000 1.000 -0.135 -0.046

81 1.000 -0.636 -0.793 -0.818 -1.000 -0.890 -0.333 -1.000 -0.690 -0.611

88 0.225 -0.455 -0.655 0.273 0.640 -0.030 -0.333 1.000 -0.566 -0.676

89 -0.414 -0.273 -0.172 -0.818 -1.000 -0.890 -0.333 -1.000 -0.561 -0.553

92 -0.414 -0.273 -0.172 0.273 0.640 -0.030 -0.333 1.000 -0.566 -0.430

94 -1.000 -0.091 0.172 -0.636 -0.581 -0.747 -0.333 -0.667 -0.473 -0.483

98 1.000 -0.455 -0.724 -0.636 -0.471 -0.572 0.333 -0.667 -0.451 -0.520

112 -1.000 0.091 0.586 0.273 1.000 0.495 0.333 1.000 0.310 0.192

113 1.000 -0.273 -0.655 -0.818 -0.818 -0.682 1.000 -1.000 -0.170 -0.213

114 1.000 -0.273 -0.655 -0.636 -0.545 -0.403 1.000 -0.667 -0.434 -0.543

115 1.000 -0.273 -0.655 -0.273 0.008 0.160 1.000 0.000 -0.593 -0.705

124 -0.414 0.273 -0.103 0.273 0.834 1.000 1.000 1.000 -0.220 -0.201

126 -1.000 0.273 0.793 -0.636 -0.545 -0.403 1.000 -0.667 0.045 0.087

131 1.000 -0.636 -0.862 -0.727 -0.044 -0.787 -1.000 0.000 -0.182 -0.396

142 -1.000 -0.273 0.586 -0.909 -0.516 -0.929 -1.000 -0.667 0.729 0.648

147 1.000 -0.273 -0.724 -0.727 -0.090 -0.459 -0.333 0.000 -0.473 -0.333

149 0.225 -0.091 -0.448 -1.000 -1.000 -0.890 -0.333 -1.000 -0.473 -0.457

152 0.225 -0.091 -0.448 -0.455 0.640 -0.030 -0.333 1.000 -0.104 -0.276

153 -0.414 0.091 0.310 -1.000 -1.000 -0.890 -0.333 -1.000 -0.006 -0.057

158 -1.000 0.273 0.724 -0.909 -0.581 -0.747 -0.333 -0.667 0.244 0.280

161 1.000 -0.091 -0.517 -1.000 -0.757 -0.783 0.333 -1.000 0.217 0.022

166 0.225 0.273 -0.310 -0.909 -0.471 -0.572 0.333 -0.667 0.078 -0.008

168 0.225 0.273 -0.310 -0.455 1.000 0.495 0.333 1.000 -0.106 -0.080

169 -0.414 0.455 0.448 -1.000 -0.757 -0.783 0.333 -1.000 0.364 0.380

171 -0.414 0.455 0.448 -0.727 0.125 -0.144 0.333 0.000 0.292 0.217

174 -1.000 0.818 0.655 -0.909 -0.471 -0.572 0.333 -0.667 0.364 0.330

175 -1.000 0.818 0.655 -0.727 0.125 -0.144 0.333 0.000 0.437 0.539

177 1.000 0.091 -0.448 -1.000 -0.818 -0.682 1.000 -1.000 -0.004 0.125

180 1.000 0.091 -0.448 -0.455 0.834 1.000 1.000 1.000 -0.220 -0.291

181 0.225 0.273 0.241 -1.000 -0.818 -0.682 1.000 -1.000 0.347 0.473

184 0.225 0.273 0.241 -0.455 0.834 1.000 1.000 1.000 -0.062 -0.078

185 -0.414 0.636 0.724 -1.000 -0.818 -0.682 1.000 -1.000 0.347 0.562