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Simplifiedseismicanalysisofsoil–well–piersystemforbridges

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Soil Dynamics and Earthquake Engineering ] (]]]]) ]]]–]]]

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Soil Dynamics and Earthquake Engineering

0267-72

doi:10.1

n Corr

E-m1 Fo

Pleas(201

journal homepage: www.elsevier.com/locate/soildyn

Simplified seismic analysis of soil–well–pier system for bridges

Goutam Mondal a, Amit Prashant b, Sudhir K. Jain b,n,1

a Department of Civil Engineering, Indian Institute of Technology Kanpur, UP 208016, Indiab Department of Civil Engineering, Indian Institute of Technology Gandhinagar, GJ 382424, India

a r t i c l e i n f o

Article history:

Received 15 February 2011

Received in revised form

21 July 2011

Accepted 8 August 2011

61/$ - see front matter & 2011 Elsevier Ltd. A

016/j.soildyn.2011.08.002

esponding author. Tel.: þ91 7923972574; fa

ail address: [email protected] (S.K. Jain).

rmerly Professor, Indian Institute of Technolo

e cite this article as: Mondal G, et a1), doi:10.1016/j.soildyn.2011.08.002

a b s t r a c t

Seismic analysis of soil–well–pier system was carried out using three different approaches to evaluate

their comparative performance and associated complexities. These approaches were (a) two-dimen-

sional nonlinear (2D-NL), (b) two-dimensional equivalent-linear (2D-EqL), and (c) one-dimensional

spring–dashpot (1D). Soil was modeled as 2D plane-strain elements in the 2D-NL and 2D-EqL

approaches, and as springs and dashpots in the 1D approach. Nonlinear behavior of soil was captured

rigorously in the 2D-NL approach and approximately in the remaining two approaches. Results of the

two approximate analyses (i.e., 2D-EqL and 1D) were compared with those of the 2D-NL analysis with

the objective to assess suitability of approximate analysis for practical purposes. In the 1D approach,

several combinations of Novak’s and Veletsos’ springs were used to come up with a simplified 1D

model using three types of spring–dashpots. The proposed model estimates the displacement and force

resultants relatively better than the other 1D models available in literature.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Well foundations (also known as caisson foundation) arefrequently adopted in the Indian subcontinent and other coun-tries like Japan, USA, Thailand, etc., for the deep foundation ofrailway and highway bridges on rivers. Because of its large crosssection and high rigidity, such foundations are often believed tobe safe foundation systems against earthquake. However, it wasobserved during recent earthquakes that structures supported onsuch foundations also suffered damage during moderate to severeearthquakes mainly because of the large permanent displacementassociated with soil liquefaction. For example, many structuressupported on well foundations sustained severe damage during1995 Kobe earthquake. In India, many such bridges are located inthe high seismic region where moderate to severe earthquakesare expected to occur. Therefore, seismic analysis should beperformed for the design of new bridges and retrofitting of theexisting bridges supported on well foundation.

In practice, well foundation is analyzed by modeling soil as two-dimensional (2D) plane-strain element or one-dimensional (1D)spring–dashpot (also named as 1D model herein). However, in theseanalyses, effect of soil nonlinearity is generally ignored or some-times considered approximately. Out of these two approaches, 1Dapproach is widely used in practical purposes. The main advantage

ll rights reserved.

x: þ91 7923972622.

gy Kanpur, UP 208016, India.

l. Simplified seismic analys

of the 1D approach over the 2D is that the former is very efficientand easy to implement in design offices than the latter. However,most of the 1D spring–dashpot models available in the literature areintended mainly to the flexible pile foundation or shallow founda-tion except for a few of these models [1–3], which were specificallydeveloped for relatively rigid embedded structures like well founda-tion. Japanese code [3] specified a Winkler’s spring model with sixtypes of springs for the stability check of well foundation. However,since damping is not considered (in absence of dashpots), this modeloften overestimates the force and displacement responses of welland pier. Gerolymos and Gazetas [1] proposed 1D model with fourtypes of spring–dashpots considering soil and interface nonlinearity.However, since this model comprehensively considers the issuesrelated to soil–structure interaction, it requires many parametersand therefore it may not be suitable for practical purposes. Varunet al. [2] proposed a simplified model with three types of spring–dashpots for well foundation embedded in linear soil. In thisformulation, material damping in soil was not considered. All thesimplified models (e.g., 1D and 2D-EqL) have some limitations,which stem from the assumptions associated with them. It is ofinterest to see if these models are suitable to analyze well founda-tion embedded in saturated cohesionless soil susceptible to liquefac-tion. For this purpose, three types of approaches were considered:(a) two-dimensional nonlinear (2D-NL), (b) two-dimensionalequivalent-linear (2D-EqL), and (c) one-dimensional spring–dashpot(1D). Results of both the approximate approaches (2D-EqL and 1D)were compared with those of the rigorous one (i.e., 2D-NL) to studythe performance and the computational efficiency of the approximateapproaches. In the 1D approach, various combinations of Novak’s [4]

is of soil–well–pier system for bridges. Soil Dyn Earthquake Eng

www.elsevier.com/locate/soildyn

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dx.doi.org/10.1016/j.soildyn.2011.08.002

mailto:[email protected]



18 m

11 m

11.35m

3.3 m

Pier

Well

1

2 2

5.3 m

50 m

G.L.

Superstructure

Direction of Earthquake Motions

1

Fig. 1. Geometry of the well foundation and piers analyzed in the present study.

Table 1Parameters for constitutive model.

Parameters Layer 1 Layer 2 Layer 3

Depth 0 m–20 m 20 m–50 m 50 m–100 m

Type of soil Medium sand Medium-dense sand Dense sand

Unit weight (t/m3) 1.9 2.0 2.1

Poisson’s ratio 0.33 0.35 0.35

Gra (kN/m2) 7.5�104 1.0�105 1.3�105

fb 331 371 401

gmaxc 0.1 0.1 0.1

fPTd 271 271 271

Contrace 0.07 0.05 0.03

Dilat1f 0.4 0.6 0.8

Dilat2f 2 3 5

Liquefac1g (kN/m2) 10 5 0

Liquefac2h 0.01 0.003 0

Liquefac3i 1 1 0

Note:

a Gr is the reference shear modulus specified at confining pressure of

80 kN/m2.b f is the angle of internal friction.c gmax is the octahedral shear strain at which the maximum shear strength is

reached.d fPT is the phase transformation angle.e Contrac is a non-negative constant defining the rate of shear induced volume

contractionf Dilat1 and Dilat2 are non-negative constants defining the rate of shear-

induced volume increase.g

G. Mondal et al. / Soil Dynamics and Earthquake Engineering ] (]]]]) ]]]–]]]2

and Veletsos’ [5,6] springs were considered and parametric study wasperformed to propose a 1D model with three types of spring–dashpots by simplifying expressions of Novak’s spring coefficients.

Liquefac1 is the pressure below which cyclic mobility takes place.h Liquefac2 is the maximum amount of perfectly plastic shear strain developed

at zero effective confinement during each loading phase.i Liquefac3 is the maximum amount of biased perfectly plastic shear strain

accumulated at each loading phase under biased shear loading conditions.

2. Geometry of the model and earthquake motions

Fig. 1 shows the geometry of the bridge substructure systemand cross-sectional dimensions of its components. The bridgesubstructure system consisted of a typical double-D cellular wellfoundation supporting two hollow circular piers on its top. Theheight of each pier was 13.47 m, and the depth of well foundationwas 50 m. Load on the pier cap due to bridge deck and live loadwas 3000 kN. The soil profile considered in the present studyconsisted of three layers of cohesionless soil with bedrock under-neath. The bedrock was assumed at 100 m depth from the groundsurface. The properties of soil in these layers are shown in Table 1.Such a bridge substructure and soil system is considered to betypical for bridges in alluvial rivers in India and believed to berepresentative of similar such structures. Because of the uncer-tainty associated with the design ground motion, variation ofground motions was preferred over that of geometry of thefoundation and properties of soil layers. Therefore, nine earth-quake motions recorded at different geographical locations wereselected for the seismic analysis (Table 2). These ground motionsrepresent different source mechanisms and epicentral distances.These were recorded at ground level at rock-outcrop as free-fieldmotions during strong earthquakes with magnitude 6.5 andabove. The ground motions with PGA ranges from 0.1 g to 0.3 g,0.3 g to 0.5 g, and 0.5 g to 0.7 g were scaled to PGA values of 0.2 g,0.4 g, and 0.6 g, respectively. These three sets of ground motionswere termed as low, moderate, and severe ground motions, asshown in Table 2. Seismic motion was assumed to be along thedirection of traffic (i.e., in the longitudinal direction).

3. Two-dimensional (2D) nonlinear analysis

Seismic analysis of the soil–well–pier (SWP) system is inherentlya three-dimensional (3D) problem. It is known that 2D modelingunderestimates the dynamic-spring coefficient and overestimatesthe radiation damping [7]. Luco and Hadjian [8] studied thefeasibility of modeling 3D linear soil-structure-interaction problem

Please cite this article as: Mondal G, et al. Simplified seismic analys(2011), doi:10.1016/j.soildyn.2011.08.002

by 2D plane-strain representation. By comparing the response of arigid shallow foundation of circular shape and a strip footing placedon elastic half-space as obtained from the 2D and 3D analyses, theydemonstrated that it was possible to obtain approximately thenatural frequencies of the system, but the radiation dampingassociated with the low-frequency modes was significantly over-estimated, which may underestimate the response parameters.Watanabe and Tochigi [9] simulated the shaking table test onsoil–foundation–structure system using 2D FE model and demon-strated that the 2D FE analysis could represent the main features ofthe system when mass and stiffness of the superstructure andsubstructure were divided by the width of the foundation perpen-dicular to the direction of excitation. Seed and Lysmer [10] com-pared the seismic response of a nuclear power plant by modeling itusing 2D and 3D FE formulations. They showed that the 2D analysisunderestimates the response of superstructure and substructure.

However, all the above studies were for shallow foundationsresting on linear and elastic soil. Therefore, further research isneeded to verify these limitations for deep foundations andinelastic soil. Despite the best effort, a similar study comparingthe response of 2D versus 3D modeling for deep foundation withinelastic soil could not be found in the literature. Moreover,seismic analysis of a 3D FE model of the SWP system with soilnonlinearity is very cumbersome and computationally expensive.Jeremic et al. [11] performed seismic analysis of 3D FE model of apile-supported viaduct considering linear–elastic soil. Gerolymosand Gazetas [12] performed 3D analysis of a small size circularwell foundation (depth 6 m and diameter 3 m) considering soilnonlinearity; however, coarse mesh was used since they did staticand dynamic analyses by applying sinusoidal motion at the top ofthe well. Recently, Elgamal et al. [13] performed seismic analysisof a bridge supported on pile foundation using a ‘‘pilot 3D FE’’model considering nonlinear soil and with coarse mesh and theanalysis took 40 h to complete. Therefore, considering the



Table 2Ground motions considered for the seismic analysis of soil–well–pier system.

Level Earthquake Mwa Station PGA (g) Component Epicentral distance (km) Symbol for ground motion

Low Northridge, 1994 6.7 LA-Griffith Park Observatory, (USGS 141) 0.289 270 25.4 L1

Turkey, 1999 7.1 Lamont 531 0.117 90 27.7 L2

Uttarkashi, 1991 7.0b Bhatwari, India 0.246 355 19.3 L3

Moderate Chi-Chi, 1999 7.6 CWB 99999 CHY041 0.302 90 51.2 M1

Loma Prieta, 1989 6.9 Corralitos (CDMG 57007) 0.479 90 7.2 M2

Northridge, 1994 6.2 UCSB 99999 LA 00 0.388 90 14.4 M3

Severe Chi-Chi, 1999 7.6 CWB 99999 TCU071 0.655 0 15.4 S1

Northridge, 1994 6.7 DWP 77 Rinaldi Receiving Station 0.633 318 10.9 S2

Petrolia, 1992 7.8 CGS-89156 0.685 90 4.5 S3

Note:

a Mw¼Moment magnitude.b Surface wave magnitude (Ms).

100

-75

-50

-25

0

0

Well

Rigid massless outrigger

Pier

Y

XZ

Radiationboundary

1 2

v1

u1

θ1

v2

u2

θ21

v1

u1

2

v2

u23

v3

u3

4v4

u4

25 50 75 100 125 150

Fig. 2. Details of the finite element model of soil–well–pier (SWP) system.

(a) Finite element discretization of the SWP system, (b) Plane-strain element for

soil, (b) Plane-strain element for at the boundaries and (d) Beam-column element

for well and pier.

G. Mondal et al. / Soil Dynamics and Earthquake Engineering ] (]]]]) ]]]–]]] 3

complexity of analyzing rigorous 3D FE model, 2D representationof the foundation soil was used in the present study by assumingplane-strain condition. Further, this 2D model is able to capture,at least qualitatively, the key aspects of the effects of soilnonlinearity (liquefaction) on the overall seismic responsemechanism of the SWP system. One may note that even thoughthis 2D model may have somewhat underestimated the seismicresponse of pier and well, it has captured many important aspectsof the seismic behavior.

Fig. 2 shows the FE discretization of half of the SWP system. Inthe dynamic behavior of a bridge substructure, the superstructurestiffness does not contribute significantly [14], and hence only themass of the superstructure was applied at the pier cap. Hydro-dynamic mass was not considered in the present study. The massof water and sand inside the well foundation was thoughincorporated. In the 2D FE model, soil domain was discretizedusing four-noded, bilinear, isoparametric finite elements with twodegrees of freedom (DOF) at each node. Soil was analyzed under


plane-strain condition. Well foundation and piers were discretizedusing two-noded linear beam-column elements with three DOF ateach node. Massless rigid beam-column elements were added tothe embedded part of the well foundation to account for the widthof the well foundation along the earthquake motion. Interfacenonlinearity (i.e., gapping and sliding at the soil–well interface)was ignored since it does not have significant influence in thedesign force resultants and displacements of well foundation [15].In other words, the soil–well interfaces were assumed to beperfectly-bonded. Viscous boundary proposed by Lysmer andKuhlemeyer [16] was used as radiation boundary at the two verticalboundaries and at the base of the FE model. Ground motion wasapplied at the base of the FE model in the form of equivalent shearforce proportional to velocity of incident wave motion [17]. Seismicanalysis of the SWP system was performed in open source codeOpenSees, the Open System for Earthquake Engineering Simulation,developed specially to simulate the performance of structural andgeotechnical systems subjected to earthquake motions [18].

In two-dimensional nonlinear (2D-NL) analysis, the soilresponse was simulated using elasto-plastic pressure-dependentmulti-yield surface (nested-yield surface) constitutive model(‘‘PressureDependMultiYield’’ model) available in OpenSees[18–20]. In this model, a set of Drucker–Prager nested yieldsurfaces with a common apex and different sizes were used tosimulate nonlinear behavior of undrained cohesionless soil. Thevalues of the parameters required for the constitutive model aretaken from the table given in the user’s manual [19]. Soil dampingwas primarily captured through hysteretic energy dissipation,and therefore, no other damping was considered.

The numerical model in OpenSees was validated withSHAKE2000 [21] to check the adequacy of the mesh size andtime step considered for the seismic analysis, and to verify theeffectiveness of Lysmer–Kuhlemeyer (L–K) radiation boundaries.Since SHAKE2000 can perform only free-field analysis (i.e., onlysoil layers in absence of structure), the FE model in OpenSees wasmodified by excluding the piers and replacing the well foundationby soil elements. Verification was carried out under the north–south component of the horizontal acceleration (PGA¼0.314 g)recorded at El Centro during 1940 Imperial Valley earthquake(Fig. 3a). Effectiveness of the L–K boundary was verified byassuming soil to be linear–elastic and undamped to check theperformance of L–K boundary in the worst scenario. Furtherdetails of the verification can be found elsewhere [22].

Fig. 3b–d shows comparison of the acceleration responsehistories obtained from OpenSees and SHAKE2000 at groundsurface, 25 m depth and 50 m depth, respectively. The accelera-tion response histories obtained from both the proceduresmatched satisfactorily throughout the duration of the motion.However, peak horizontal acceleration (PHA) values were



0 5 10 15 20 25 30-0.8

-0.4

0

0.4

0.8

PGA = 0.319g

Acc

eler

atio

n (g

)

Time (s)

Input Motion

-0.8

-0.4

0

0.4

0.8

Acc

eler

atio

n (g

)

At 0 m

-0.8

-0.4

0

0.4

0.8

Acc

eler

atio

n (g

)

At -20 m

SHAKE

0 5 10-0.8

-0.4

0

0.4

0.8

Time (s)

Acc

eler

atio

n (g

)

At -50 m

OpenSees

0 0.2 0.4 0.6 0.8

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Peak Horizontal Acceleration (g)

Dep

th (m

)

Fig. 3. Comparison of acceleration response of soil obtained from free-field analysis in SHAKE and OpenSees for the El-Centro motion (a) input motion (b)–(d) acceleration

response histories at specific points (e) variation of peak horizontal acceleration with depth.


underestimated in OpenSees by 6%, 1%, and 2%, respectively, atground surface, 25 m depth and 50 m depth. Hence, such smallvariations in the peak responses were considered acceptable.Moreover, the variation of PGA along depth as obtained fromthe two models also matched well (Fig. 3e). Therefore, it wasconcluded that dimensions of the finite elements and time stepconsidered during the seismic analysis were adequate, and theL–K boundaries were effective in eliminating reflection of spur-ious waves.

Analysis was performed in several steps to simulate the initialcondition of the SWP model. In the first step, gravity analysis wasperformed for the self-weight of soil and weight of the embeddedportion of well by assuming the soil as linear–elastic. In this step,the vertical boundaries of soil domain were restrained inhorizontal direction only, and the base of FE model was restrainedin both vertical and horizontal directions to develop the desiredconfining pressure to all the soil elements. In the second step, thesoil constitutive model was switched from linear–elastic toelasto-plastic. The new equilibrium state under gravity wasobtained iteratively. In the third step, self-weight of pier, wellcap, and other gravity loads were applied statically to the non-linear soil model. Reaction forces at the boundary nodes wereobtained at the end of gravity analysis. In the fourth step, all therestraints along the boundary nodes were removed and thereaction forces obtained from the gravity analysis of the SWPsystem were statically applied at the corresponding nodes. Thisstate of the model was assumed to be the initial condition ofthe SWP system for the seismic analysis. In the fifth step, bothhorizontal and vertical radiation dampers were added at thenodes of lateral boundaries and base boundary. These dampers


were zero-length elements with two nodes; one node wasconnected to the boundary node and the other node was fixedin space. Finally, seismic analysis of the SWP system was per-formed by applying horizontal seismic excitation in the form ofeffective nodal forces applied at the base of the computational soildomain.

The six response parameters, namely, the maximum displace-ment at pier top (DPT) and well top (DWT), the maximum shear forcein pier (SFP) and well (SFW), and the maximum bending moment inpier (BMP) and well (BMW) were obtained from the 2D-NL analysisand compared with those obtained from the 2D-EqL and 1Danalyses, as discussed in the subsequent sections to evaluate theperformance of the simpler models (i.e., 2D-EqL and 1D models).

4. Two-dimensional (2D) equivalent-linear analysis

4.1. Method of analysis

In the two-dimensional equivalent-linear approach (2D-EqL),the FE model was similar to the 2D-NL model except the analysisbeing linear in the former. In this approach, the soil nonlinearitywas considered indirectly using the equivalent-linear (or effec-tive) soil properties (shear modulus G and damping z). These soilproperties were obtained from the free-field analysis of soilcolumn (without the foundation structure) subjected to a giveninput earthquake motion at the column base using the commer-cial software SHAKE2000 [21]. SHAKE2000 uses 1D wave propa-gation theory to iteratively calculate the level of maximumstrain in soil and determines the equivalent-linear properties of




soil at any desired depth. The above two steps of the 2D-EqLapproach (i.e., 1D analysis in SHAKE and 2D linear analysis inOpenSees) significantly reduce the computational time from thesingle step 2D-NL analysis.

Since the properties G and z depend upon the shear strain levelin soil, their variations with shear strain are needed during thefree-field analysis in SHAKE. These variations are generallyevaluated from laboratory test. However, in absence of laboratorytest, these can be estimated from empirical relationships availablein the literature [23–26], or from the experimental results of a

0

0.2

0.4

0.6

0.8

1

Shear Strain (%)

G/G

max

p’ = 6 kN/m2 (0m−2m)p’ = 25 kN/m2 (2m−7m)p’ = 80 kN/m2 (7m−20m)p’ = 200 kN/m2 (20m−50m)p’ = 500 kN/m2 (50m−100m)

0

5

10

15

20

Shear Strain (%)

Dam

ping

Rat

io (%

)

p’ = 6 kN/m2 (0m−2m)p’ = 25 kN/m2 (2m−7m)p’ = 80 kN/m2 (7m−20m)p’ = 200 kN/m2 (20m−50m)p’ = 500 kN/m2 (50m−100m)

10−4 10−3 10−2 10−1 100

10−4 10−3 10−2 10−1 100

Fig. 4. Shear–strain dependent (a) shear modulus reduction curves, and

(b) damping curves used in the 2D-EqL in SHAKE [24].

−30

−15

0

15

30L1

−30

−15

0

15

30

Err

or (

%)

L2

−30

−15

0

15

30L3

Respons

DPT DWT SFP

Fig. 5. Bar chart of percentage error in the maximum responses (i.e., maximum displac

well (SFW), and maximum bending moment in pier (BMP) and well (BMW)) in the 2D-


similar type of soil. In the present study, these were obtainedfrom the procedure proposed by Zhang et al. [24]. Fig. 4 shows thevariation of shear modulus (normalized with the maximum shearmodulus Gmax) and damping ratio with shear strain at differentlevels of confining pressure (p0). Ideally, these curves should becalculated for each layer corresponding to p0 of that layer.However, for simplicity, only five sets of G/Gmax and dampingratio curves representing p0 6 kN/m2, 25 kN/m2, 80 kN/m2,200 kN/m2, and 500 kN/m2 were used in this study by followingthe method suggested by Stokoe et al. [27]. The parameter Gmax

(in kN/m2), which is required to estimate G from the G/Gmax curve,can be obtained from Eq. (1):

Gmax ¼ Grp0

p0r

� �d

ð1Þ

where Gr is the shear modulus at reference confining pressure ofpr0 (¼80 kPa), and d (¼0.5) is a positive constant defining the

variation of G as a function of instantaneous effective confiningpressure p0.

In linear time-domain analysis, the stress–strain curve of soilis linear, and therefore, hysteretic energy dissipation does notoccur. However, the energy dissipation in soil was approximatelycaptured through the equivalent-linear damping zeff in soilobtained from the SHAKE analysis. This damping in soil was usedin the form of mass and stiffness proportional Rayleigh damping.Rayleigh damping coefficients were determined by consideringtwo target modes, ith and jth with damping ratio zeff following theconcept proposed by Hudson et al. [28]. Damping in structure wasassumed to be 5% of the critical damping. The ground motions inthis analysis were applied by following the same procedure as inthe 2D-NL analysis.

4.2. Comparison of 2D equivalent-linear and 2D nonlinear analyses

Displacement and force responses at different locations in pierand well foundation from the 2D-EqL analysis were compared withthose from the 2D-NL analysis. The percentage error is shown inFig. 5. It was observed that the error in the maximum absolutedisplacement at pier top (DPT) and well top (DWT) was satisfactorilyestimated by the 2D-EqL analysis with a maximum error within 10%except in S1 motion. In S1 motion, the observed error was 24% and16% in DPT and DWT, respectively. However, the average absolute

M1 S1

M2 S2

e Parameters

M3

SFW BMP BMW

S3

ement at pier top (DPT) and well top (DWT); maximum shear force in pier (SFP) and

EqL analysis as compared to the 2D-NL analysis.



0 25 50 75 100

−50

−45

−40

−35

−30

−25

−20

−15

−10

−5

0

Dep

th (

m)

Shear Force (MN)

S1

2D−NL 2D−Eql 1D−Proposed

0 500 1000 1500

Bending Moment (MN−m)

S1

Fig. 6. Comparison of envelopes of the maximum force resultants (a) shear force and (b) bending moment along the depth of well foundation obtained from

two-dimensional nonlinear (2D-NL) model, two-dimensional equivalent-linear (2D-EqL) model, and 1D proposed model for severe motion S1.

Dis

plac

emen

t tim

e hi

stor

ies

Translational base

Distributed translationsprings (kx, cx)

Distributed rotational springs (k�, c�)

Well

Pier

Superstructure mass

Rotational base springs (kb�, cb�)

.

.

.

.

.

.

.

.

.


error and the associated standard deviation (SD) were 8.9% and 5.7%,respectively, for DPT, and 7.4% and 4.5%, respectively, for DWT. Themaximum error in the force resultants (both shear force and bendingmoment) in both pier and well foundation were within 27% whilethe average absolute error was about 13% with SD of 7%. Themaximum error in all the response parameters considered in thisstudy was limited to 27% while the average absolute error was 11%with SD of 7%. Moreover, envelopes of the maximum shear forcealong the depth of well foundation match considerably (Fig. 6(a)).Similar match was also observed in the maximum bending momentenvelopes of well foundation (Fig. 6(b)). It is worth mentioning thata point in an envelope is obtained by estimating the maximumresponse parameter (e.g., shear force) at that location during theearthquake; thus any two points (i.e., responses) on the envelopesmay not occur at the same instant of time. In general, it can beinferred that equivalent-linear analysis can satisfactorily estimateresponse of well foundation for small to severe earthquakes in spiteof the approximations involved in the analysis.

spring (kbx, cbx)

Fig. 7. 1D model with four types of spring–dashpot connected in parallel. Note: all

the springs are connected with dampers in parallel. However, dampers are not

shown for clarity.

5. One-dimensional (1D) spring–dashpot model

5.1. Method of analysis

In the 1D spring–dashpot model, a model combining Novak’sand Veletsos’ springs was used as a basic model. These twosprings were used in this study since Novak’s springs are widelyacceptable for the deep foundations and Veletsos’ springs areoften used for rigid footings resting on homogeneous half-space.Several combinations of these springs were used for the para-metric study. Details of the models are described below.

Novak et al. [4] proposed the following dynamic impedancesor complex stiffness of unit length of soil when an infinitely longrigid cylinder is embedded in homogeneous soil and subjected totranslational and rotational modes of vibration independentlyassuming plane-strain condition along the length of the cylinder:

Kx ¼ G½Sx1ða0,n,DÞþ iSx2ða0,n,DÞ� ð2Þ

Ky ¼ Gr20 ½Sy1ða0,DsÞþ iSy2ða0,DsÞ� ð3Þ


where Kx and Ky are the dynamic impedance of soil for transla-tional and rotational vibrations, respectively; r0 is the radius ofthe circular cylinder or equivalent radius of the foundation withother cross sections; a0ð¼ r0o=VsÞ denotes the dimensionlessfrequency; o is the circular frequency; Vs represents the shearwave velocity of soil; i¼

ffiffiffiffiffiffiffi�1p

; Sx1 and Sx2 are the real and theimaginary parts, respectively, of the dimensionless complexstiffness for horizontal vibration; similarly, Sy1 and Sy2 are thereal and the imaginary parts, respectively, of the dimensionlesscomplex stiffness for rocking; n is Poisson’s ratio, and D is thematerial damping. The real (kx and ky) and the imaginary parts(cx and cy) of the impedance functions represent the stiffness anddamping coefficients (both material damping of soil and radiationdamping), respectively, of the complex spring (spring and dashpotconnected in parallel). Therefore, the following stiffness and




damping coefficients are obtained from Eqs. (2) and (3):

kx ¼ GSx1ða0,n,DÞ

cx ¼Go Sx2ða0,n,DÞ

)ð4Þ

ky ¼ Gr20Sy1ða0,DsÞ

cy ¼Gr2

0o Sy2ða0,DsÞ

9=; ð5Þ

These coefficients are frequency (o) dependent; however, forsimplicity, the stiffness and damping coefficients were evaluatedcorresponding to the predominant frequency of input motion(i.e., o¼oip).

In the present study, the springs and dashpots with coeffi-cients obtained from Eq. (4) were connected in parallel anddistributed throughout the embedment depth of well shaft asshown in Fig. 7. Similarly, Eq. (5) was used to obtain thecoefficients of distributed rotational springs along the embed-ment depth of well shaft (Fig. 7).

For a well foundation of finite depth embedded in a layeredsoil, the coefficients in Eqs. (4) and (5) were assumed to be valid.In other words, the stiffness and damping coefficients per unitdepth of a particular layer were obtained by assuming that a rigidwell foundation of the same cross section but infinite depth wasembedded in a homogeneous half-space with property of thatlayer. The same assumption has been made by Novak and Aboul-Ella [29] for flexible pile embedded in layered soil.

Unlike pile where the lateral soil reaction is the sole resistingmechanism during earthquake, in well foundation base reactionalso plays an important role along with the lateral reaction as theresisting mechanism. Since, solution of Novak et al. [4] was for theinfinitely long foundation, concept of base springs was not there.Therefore, springs at the well base were introduced in this studyto model the base resistance mechanism of the finite depth ofwell foundation considered in this study. The following expres-sions for the dynamic impedance of rigid circular foundationresting on surface of elastic homogeneous medium proposed byVeletsos and co-authors [5,6] were used as the coefficients of theconcentrated translational and rotational spring–dashpot systemat the well base:

kbx ¼8Gr02�n

cbx ¼8Gr02�n U

0:6a0o

9=; ð6Þ

kby ¼8Gr3

03ð1�nÞ Uk0by

cby ¼8Gr3

03ð1�nÞ U

0:35a30

oð1þa20Þ

9>=>; ð7Þ

Table 3Types of 1D spring–dashpot models analyzed in the present study.

Springs type Kx Cx Ky Cy Kbx Cbx K

1D-1 NV – NV – VS – V

1D-2 NV NV NV NV VS VS V

1D-3 NV 0.1 NV NV 0.1 NV VS 0.1 VS V

1D-4 NV 0.2 NV NV 0.2 NV VS 0.2 VS V

1D-5 NV 0.2 NV NV 0.2 NV VS 0.2 VS –

1D-6 NV 0.2 NV NV 0.2 NV – – V

1D-7 NV 0.2 NV NV 0.2 NV – – –

1D-8 NV 0.2 NV – – VS 0.2 VS V

1D-Proposed PR PR PR PR VS 0.2 VS –

Note: NV¼Novak springs; VS¼Veletsos springs; PR¼Proposed.


where

k0by ¼1�0:2a0 for a0r2:5

0:5 for a042:5

(

The parameter n is the Poisson’s ratio of soil at well base. Theabove four-spring–dashpots model consisted of (a) distributedtranslational spring–dashpots along the well shaft, (b) distributedrotational spring–dashpots along the well shaft, (c) the concen-trated base translational spring–dashpot, and (d) the concen-trated base rotational spring–dashpot. One end of all the springswas connected to the well foundation. The other end of all therotational springs (i.e., springs b and d) was fixed in space whilethat of the translational springs (i.e., springs a and c) wassubjected to earthquake motions in the form of displacementhistory corresponding to that soil layer estimated from theequivalent-linear analysis in SHAKE. In fact, the fixed end of therotational springs should be subjected to dynamic rotation ye(t)(¼due(t)/dz, where due(t) is the differential displacement of thetop and bottom nodes of the layer and dz is the thickness of thelayer) imposed by the horizontal displacement profile. However,for simplicity, small amount of dynamic rotation at the end of therotational springs was neglected in this study.

The spring and dashpot coefficients used in Eqs. (2)–(7) arebased on linear soil. In order to consider soil nonlinearity,approximately, the spring coefficients were estimated using theequivalent-linear properties of soil (G and z) obtained from 1Dwave propagation analysis in SHAKE, as discussed earlier.

5.2. Comparison of 1D and 2D-NL analyses

Table 3 illustrates nine different 1D models obtained bydifferent combinations of the springs and dashpots as discussedin the previous section. In 1D-1, all the four sets of springs wereconsidered and dashpots were neglected while in 1D-2 all thesprings and dashpots were considered. The models 1D-3 and 1D-4were developed from 1D-2 by reducing the damping coefficientsto 10% and 20%, respectively, of that considered in 1D-2. Thereduction in damping is due to the presence of frequency cut-offfor radiation damping when natural period of the SWP system ismore than the fundamental period of the soil strata [12,30,31].In such a case, the radiation damping is significantly small ascompared to its half-space value.

Fig. 8 shows the percentage error in the maximum displace-ment and force resultants in pier and well foundation obtainedfrom these analyses. The percentage error was estimated basedon the 2D-NL analysis. About 75% error was observed in theresponse parameters in model 1D-1, which neglected dashpots.The model 1D-2, which considered full damping, resulted inabout 50% error in the maximum force and displacement

by Cby Remarks

S – No dashpot

S VS All springs and dashpots

S 0.1 VS Dashpot coefficients are 10% of the half-space value

S 0.2 VS Dashpot coefficients are 20% of the half-space value

– No base rotational spring and dashpot

S 0.2 VS No base translational spring and dashpot

– No base springs and dashpots

S 0.2 VS No distributed rotational springs and dashpots

– Proposed distributed springs with no base rotational spring



−500

50100150200 L1 M1 S1

−500

50100150200

Err

or (

%)

L2 M2 S2

1D−1 1D−2 1D−3 1D−4

−500

50100150200 L3

1D−1 1D−2 1D−3 1D−4

Types of Analysis

M3

DPT DWT SFP SFW BMP BMW

1D−1 1D−2 1D−3 1D−4

S3

Fig. 8. Bar chart of percentage error in the maximum responses (i.e., maximum displacement at pier top (DPT) and well top (DWT); maximum shear force in pier (SFP) and

well (SFW), and maximum bending moment in pier (BMP) and well (BMW)) obtained from the analysis using four types of 1D models.

0 25 50 75 100

−50

−45

−40

−35

−30

−25

−20

−15

−10

−5

0

Dep

th (

m)

Shear Force (MN)

S1

2D−NL 1D−2 1D−3 1D−4

0 500 1000 1500


S1

Fig. 9. Comparison of envelopes of the maximum force resultants, (a) shear force and (b) bending moment, along the depth of well foundation obtained from the two-

dimensional nonlinear (2D-NL) model and three one-dimensional spring–dashpot models (1D-1, 1D-2, and 1D-3) for severe motion S1.


responses of pier and well foundation. However, the models1D-3 and 1D-4 with reduced damping indicated about 730%and 720% maximum error, respectively. Envelopes of the max-imum force resultants (shear force and bending moment) of wellobtained from the 1D models were also compared with thoseobtained from the 2D-NL model. Fig. 9 shows such comparison forthe case of S1 motion. The best agreement was observed in case ofmodel 1D-4.

From the above discussion, it was concluded that the coefficientof dashpots should be limited to 20% of that for foundationembedded in half-space when the natural period of the SWP systemis more than the fundamental period of the soil profile, whichis used. Therefore, the model 1D-4 was used as a basis for the


subsequent analysis to study the effect of individual springs on theresponse of well foundation. Since the ground motion was applied inthe horizontal direction only, the distributed horizontal springs weredominant over the other springs. Therefore, the individual signifi-cance of the other three types of spring was studied.

5.3. Effect of base rotational spring–dashpot

The effect of rotational spring–dashpot (kby, cby) 1D-5 at wellbase on the response of well and pier was studied by comparingthe results of models 1D-5 and 1D-4. The model 1D-5 wasprepared by removing the base rotational spring and dashpotfrom the model 1D-4. It was observed that the magnitudes of



1D−5 1D−6 1D−7 1D−8−40

−20

0

20

40

Err

or w

ith r

espe

ct to

1D

−4

(%)

Types of Analysis


Fig. 10. Effect of different spring components on the maximum responses (i.e.,

maximum displacement at pier top (DPT) and well top (DWT); maximum shear

force in pier (SFP) and well (SFW), and maximum bending moment in pier (BMP)

and well (BMW)).

0 10 20 30 40

−50

−40

−30

−20

−10

0

Shear Force (MN)

1D−4

0 200 400 600 800


1D−5

−50

−40

−30

−20

−10

0

1D−6

−50

−40

−30

−20

−10

0

Dep

th (

m)

1D−7

−50

−40

−30

−20

−10

0

1D−8

Fig. 11. Effect of individual spring–dashpots on the maximum shear force and

maximum bending moment envelopes of well foundation: (a–b) base rotational

spring–dashpot (Kby�Cby), (c–d) base translational spring–dashpot (Kbx�Cbx),

(e–f) both translational and rotational base spring–dashpots (Kbx�Cbx, Kby�Cby),

(g–h) rotational distributed spring–dashpots (Ky–Cy).


error in the maximum force and maximum displacementresponses of well foundation and piers were less than 5% inabsence of base rotational spring–dashpot (Fig. 10). Comparisonof shear force envelopes obtained from these two models (1D-4and 1D-5) shows that the envelopes almost overlapped eachother (Fig. 11(a)). Similarly, the maximum bending momentenvelopes of well foundation obtained from these two modelsalso compared well with each other except near the base of thewell foundation where the model 1D-5 estimated slightly lowervalue of the maximum bending moment as compared to model1D-4 (Fig. 11(b)). Therefore, it can be inferred that the baserotational spring and dashpot can be removed without causingsignificant error in the response parameters. Hence, one can use amodel with only three types of spring–dashpots.

5.4. Effect of base translational spring–dashpot

To study the influence of translational spring and dashpot atwell base (kbx, cbx), the maximum displacement and maximumforce responses of model 1D-6 were compared with those of themodel 1D-4. The model 1D-6 was prepared by removing the basetranslational spring and dashpot from model 1D-4. It wasobserved that the displacement responses (i.e., DPT and DWT)were reduced by only 5% and the maximum force resultants inwell foundation and pier were reduced by 10% to 13% in absenceof base translational spring and dashpot (Fig. 10). However, theenvelopes of the maximum shear force were highly underesti-mated near the base of the well foundation (Fig. 11c). Therefore,one should not neglect the base translational spring during theseismic analysis of the SWP system.

5.5. Combined effect of base translational and rotational

spring–dashpots

The combined effect of the base spring–dashpots (both trans-lational and rotational) were studied by analyzing the wellfoundation using only distributed spring–dashpots (kbx, cbx andkby, cby) along the well shaft (in model 1D-7) and comparing theresults with those of the model 1D-4. In absence of base spring–dashpots (both translational and rotational), the maximum valuesof the displacement at pier cap (DPT) and well cap (DWT) werereduced by 8% to 10%. Similarly, magnitudes of the maximumshear force and bending moment were reduced by about 9% inpier and about 15% in well foundation. Moreover, the maximumshear force and bending moment below the central half of wellfoundation were underestimated if base springs were neglected(Fig. 11(e) and (f)). Therefore, the combined effect of the


translational and rotational base spring–dashpots is significantand cannot be neglected.

5.6. Effect of distributed rotational spring–dashpots along well shaft

The effect of distributed rotational spring–dashpots (ky, cy)was studied by analyzing the model 1D-8 and comparing theresults with those of the model 1D-4. The model 1D-8, which wasprepared by excluding the distributed rotational spring–dashpotsfrom model 1D-4, resulted in overestimation of the maximumdisplacement of pier cap and well cap by about 20% and themaximum force resultants in well foundation and pier by about12% (Fig. 10). The envelopes of the maximum shear force in wellfoundation obtained from the analysis using the models 1D-8 and1D-4 vary significantly (Fig. 11(g) and (h)). Therefore, it can be



0

1

2

3

4

5

S x1

0

5

10

15

20

S x2

0 0.5 1 1.5 20

1

2

3

4

5

S θ1

0 0.5 1 1.5 20

1

2

3

4

5

6

a0 = ωr0/Vs

S θ2

ζ = 5% ζ = 10% Proposed

ν = 0.3ν = 0.4

ν = 0.3

ν = 0.4

a0 = ωr0/Vs

Fig. 12. Stiffness (Sx1 and Sy1) and damping parameters (Sx2 and Sy2) for translational and rotational springs proposed by Novak et al. [4] along with the simplified

parameters proposed in the present study.

Distributed translational springs (k , c )

Superstructure mass

Pier


concluded that the distributed rotational springs and dashpotshave significant influence in the response of well foundationand pier.

Dis

plac

emen

t tim

e hi

stor

ies

Translational base spring (kbx, cbx)

x x

Distributedrotationalsprings (k�, c�)

Well

Fig. 13. Proposed 1D model with three types of spring–dashpots. Note: all the

springs are connected with dampers in parallel; however, dampers are not shown

for clarity.

6. Proposed 1D model

In the previous section, it was concluded that when the naturalperiod of the SWP system is more than the fundamental period ofthe soil strata, the coefficient of dashpots should be reduced toabout 20% of that for foundation embedded in half-space. It wasalso observed that the rotational base spring and dashpot may beneglected without causing significant error in the estimation of themaximum displacement and force resultants in well foundationand pier.

Based on the above findings, a spring–dashpot model consistingof (1) distributed translational spring–dashpots, (2) distributedrotational spring–dashpots, and (3) a base translational spring–dashpot is proposed (Fig. 13). The spring and dashpot parametersin Eqs. (4) and (5) have been simplified by some linear approxima-tion of these parameters with dimensionless frequency (a0). Fig. 12shows that the translational spring stiffness parameter Sx1 proposedby Novak et al. [4] depends on the dimensionless frequency (a0) andn. However, in the frequency range of interest (0.5oao1.5), Sx1 canbe assumed to be independent of a0 and n. Hence, Sx1 is crudelyapproximated to be equal to 3.5. However, the translational damp-ing parameter Sx2 almost linearly varies with a0. Neglecting theeffect of n on Sx1, it is proposed that Sx2E10a0. Similarly, therotational spring and damping parameters have been simplified asSy1E3.0�0.75a0 and Sy2E3a0. The proposed simplified expressionsof the Novak’s spring coefficients are plotted in Fig. 12 along withthe original expressions. Summary of the proposed spring–dashpotscoefficients is shown in Eqs. (8)–(10)

kx ¼ 3:5G

cx ¼2Ga0o

)ð8Þ


ky ¼ Gr20ð3:0�0:75a0Þ

cy ¼0:6Gr2

0a0

o

9=; ð9Þ

kbx ¼8Gr02�n

cbx ¼8Gr02�n U

0:12a0o

9=; ð10Þ

Results of analysis of the well foundation using the proposedthree-spring model (model 1D-Proposed) were compared with thoseobtained from the 2D-NL analysis. The values of percentage error inthe response parameters were also compared with those for 2D-EqL



−60

−30

0

30

60

L1 M1 S1

−60

−30

0

30

60

Err

or (

%)

L2 M2 S2

2D−EqL 1D−4 1D−Proposed−60

−30

0

30

60

L3

2D−EqL 1D−4 1D−ProposedTypes of Analysis

M3

DPT

DWT

SFP

SFW

BMP

BMW

2D−EqL 1D−4 1D−Proposed

S3


well (SFW), and maximum bending moment in pier (BMP) and well (BMW)) obtained from the various approaches.


and 1D-4 analyses (Fig. 14). It was observed that the proposed modelsatisfactorily estimates the maximum force and maximum displace-ment responses in pier and well foundation; the maximum percen-tage error in the response parameters was within 30% except for themotion S3. Moreover, the values of the percentage error for theproposed model and the model 1D-4 match well with each other.Therefore, it can be concluded that the effects of simplification ofNovak’s [4] spring and dashpot coefficients, and the base rotationalspring on the maximum force and displacement responses of pier andwell foundation are not significant.

7. Performance of some other existing 1D models

As stated earlier, there are only a few spring–dashpot models,available in the literature, which were developed specifically forthe seismic analysis of well foundation. In the previous discussion,the distributed translational and rotational springs and dashpotsproposed by Novak et al. [4] were used to analyze the SWP systemand a simplified model with three types of springs was proposed.In this section, three other 1D models were studied; one modelwas based on the recommendation of Japanese code [3], and theother two were proposed by Gerolymos and Gazetas [1] and Varunet al. [2]. Fig. 15 illustrates the comparison of spring and dashpotcoefficients of the soil layer considered in this study. It wasobserved that the value of translational spring coefficient kx

specified in JRA [3] was significantly higher in comparison to thoseof the other three models. Values of the coefficients ky, cx, and cyobtained from these models significantly varied from each other. Itis worth noting that the dashpot coefficient of the rotationalsprings (cy) proposed by Gerolymos and Gazetas [1] and Varunet al. [2] attained negative values. The negative values of cy maysometimes cause convergence problem during time history analy-sis. Since these models were developed specifically for frequencydomain analysis, their application in the time domain may not besuitable.


Performance of the above spring–dashpot models was checkedby analyzing the SWP system using these models and comparingresults with those of the 2D-NL analysis. The coefficient cyproposed by Gerolymos and Gazetas [1] and Varun et al. [2]was assumed to be zero when it attained a negative value. It wasobserved that the proposed model performed considerably well incomparison to the other models considered in this study. Sig-nificant error (more than 50%) was observed in the displacementand force response parameters of well foundation and pier formodels proposed by Gerolymos and Gazetas [1] and Varun et al.[2] (Fig. 16). In a few cases, the JRA model, which does notconsider dashpots, also estimated the responses satisfactorily.However, it significantly overestimated (error4100%) the max-imum shear force and maximum bending moment in pier for themotions L2, M1, S1, and S2.

8. Computational efficiency of different approaches

Computational efficiency of the 2D-NL, 2D-EqL, and 1Dapproaches was compared to study the acceptability of theavailable approaches in design offices. It was observed that thecomputational time required for the above three approaches were15 h, 6 h, and 5 min for the L1 motion. Hence, one can imaginethat the 3D SWP model with appropriate model parameters maytake significantly long time for the execution under seismic load-ing. Although there was no significant difference in the estimatedresponse parameters, computational efficiency of the 2D-EqLanalysis was about three times better than that of the 2D-NLanalysis. On the other hand, the 1D spring–dashpot model req-uired negligible computational time as compared to the 2D-NLand 2D-EqL analyses. However, the 1D approach is comparativelyless accurate than the 2D-EqL model. Therefore, one shouldchoose correct approach depending on the importance of theproblem. In general, 1D approach can be chosen for first handanswer and 2D-EqL approach may be chosen for detailed analysis.



0 4 8 12

−50

−40

−30

−20

−10

0

Kx (kN/m) × 106

0 1 2 3 4 5

Cx (kN.s/m) × 104

0 1 2 3

−50

−40

−30

−20

−10

0

Dep

th (

m)

Kθ (kN.m) × 108

Novak et al. [4] JRA [3]

−3 −2 −1 0 1

Cθ (kN.m.s) × 107

Gerolymos and Gazetas [1] Varun et al. [2]

Fig. 15. Comparison of spring and dashpot coefficients of distributed side springs obtained from various 1D models.


9. Summary and conclusions

Seismic response of soil–well–pier (SWP) system wasperformed by three approaches, namely, two-dimensional non-linear (2D-NL), two-dimensional equivalent-linear (2D-EqL), andone-dimensional spring–dashpot (1D) approaches. Out of thesethree approaches, the 1D approach is the simplest one and it iseasy to implement in design offices. Nine earthquake motions withpeak ground acceleration ranging from 0.2 g to 0.6 g were used torepresent the variations in ground motion parameters. Displace-ment and force responses obtained from the 2D-EqL analysis werecompared with those determined by the rigorous 2D-NL analysis.In the 1D approach, various 1D models combining springs pro-posed by Novak et al. [4], and Veletsos and co-authors [5,6] wereprepared and used to analyze the SWP system. Results obtainedfrom these models were compared with the results of 2D-NLmodel. Based on this study, the coefficients of Novak’s springswere further simplified and a three-spring–dashpot model wasproposed. The proposed model consisted of distributed transla-tional and rotational spring–dashpots along the well shaft andconcentrated base translational spring–dashpot. The proposedmodel estimated relatively well the seismic response of the SWPsystem. Performances of some other 1D spring–dashpot models


specifically developed for well foundation were also studied. Basedon the present study, the following conclusions can be made:

1)

is o

The 2D-EqL approach, which is simpler and computationallymore efficient than the 2D-NL approach, can be very useful forthe seismic analysis of well foundation though it considers soilnonlinearity approximately. Such analysis may cause up to30% error in design displacement and force resultants in pierand well foundation. However, computationally, it is aboutthree times more efficient than the 2D-NL analysis.

2)
For routine work in design offices, engineers may use the 1Dspring–dashpot model, which requires negligible computationaltime as compared to the 2D analyses (both 2D-NL and 2D-EqL).In this model, radiation damping should be considered otherwisedisplacement and force responses may be overestimated.
3)
For long span bridges supported on well foundation, thenatural period of the SWP system is generally more than thefundamental period of the soil. In this case, the use of radiationdamping for half-space may lead to underestimation of dis-placement and force resultants in well foundation and pier.In such a situation, it is recommended that the coefficient ofdashpots should be reduced to 20% of that for foundationembedded in half-space. The spring and damping coefficients
f soil–well–pier system for bridges. Soil Dyn Earthquake Eng


−100

0

100

200 L1 M1 S1

−100

0

100

200

Err

or (

%)

L2 M2 S2

Proposed JRA [3] Gerolymosand

Gazetas [1]

Varunet al. [2]


Gazetas [1]

Varunet al. [2]


Gazetas [1]

Varunet al. [2]

−100

0

100

200 L3

Types of Analysis

M3


S3


well (SFW), and maximum bending moment in pier (BMP) and well (BMW)) obtained from the 1D analysis.


P(2

should be estimated based on the equivalent-linear propertiesof soil.

4)
One can use the proposed model, which consists of three typesof spring–dashpots and ignores base rotational spring–dashpotto analyze approximately the well foundation embedded insoil. The proposed model estimated the displacement andforce resultants in pier and well foundation relatively betterthan the available 1D models. In a few cases, the JRA springs[3] also estimated the responses satisfactorily.
In the present study, seismic analysis was performed on atypical well foundation embedded in a layered cohesionless soil.Considering the fact that such foundations may be located indifferent seismic zones and may be subjected to different inten-sities of earthquake motion, the present study focuses on thevariation of ground motions. Further comprehensive research isneeded to study the influence of the geometry of the foundation andproperties of soil layers on the conclusions drawn in this paper.

Seismic analysis of the SWP system is a 3D problem, which hasbeen represented as a 2D plane-strain problem in this paper.Despite the limitations of the 2D model, it is able to capture (atleast qualitatively) the key aspects of the effects of soil non-linearity (liquefaction) on the overall seismic response mechan-ism of the SWP system. More comprehensive research may beneeded in future for carrying out seismic analysis of 3D SWPmodel with nonlinear soil.

Acknowledgements

Authors gratefully acknowledge Poonam and Prabhu Goel

Foundation at Indian Institute of Technology Kanpur for the financialsupport in conducting the present study.

lease cite this article as: Mondal G, et al. Simplified seismic analys011), doi:10.1016/j.soildyn.2011.08.002

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