a practical iterative procedure to estimate

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A practical iterative procedure to estimate

seismic-induced deformations of shallow

rectangular structures

Antonio Bobet, Gabriel Fernandez, Hongbin Huo, and Julio Ramirez

Abstract: An iterative procedure is proposed to estimate seismic-induced distortions of cut-and-cover rectangular struc-

tures. The procedure is based on an existing analytical solution for deep rectangular structures subjected to far-field shear

stress which assumes elastic behavior of the soil and structure, tied contact at the soilstructure interface, and static load-

ing. The new proposed procedure builds on the analytical solution and approximates dynamic response with a pseudo-

static analysis and incorporates soil-stiffness degradation through an iterative scheme where the soil shear modulus is

changed in each iteration based on the shear strain of the soil obtained in the previous iteration. The presence of the

ground surface and slip at the soilstructure interface are neglected in the method proposed, but their effects are shown to

be small and have compensating results when soil nonlinearity is introduced. Predictions obtained from the analytical solu-

tion have been verified by a series of numerical tests, which include the response of the Daikai subway station during the

1995 Kobe earthquake in Japan and the Los Angeles Civic Center subway station subjected to the 1994 Northridge earth-quake in California. The relative errors in terms of deformation between analytical and numerical results are smaller than

15%. The procedure results in stresses on the structure that compare well with those obtained with the numerical method

when there is no slip between soil and structure. If slip is allowed, the analytical solution overpredicts tensile normal

stresses and underpredicts compressive normal stresses.

Key words: seismic design, soilstructure interaction, underground structures, relative stiffness, shear modulus degradation,

cut-and-cover.

Resume : On propose une procedure iterative pour estimer les distorsions induites par des seismes dans des coupes et des

structures rectangulaires quelles soutiennent. La procedure est basee sur une solution analytique existante pour des structu-

res rectangulaires profondes soumises a une contrainte de cisaillement etendue qui suppose un comportement elastique du

sol et de la structure, un contact serre a linterface sol structure, et un chargement statique. La nouvelle proce dure propo-

see est basee sur la solution analytique et la reponse dynamique approximative avec une analyse pseudo-statique, et incor-

pore une degradation de la rigidite du sol au cours du processus diteration ou le module de cisaillement du sol est change

a chaque iteration en partant de la deformation en cisaillement du sol obtenue au cours de literation precedente. Dans lamethode proposee, on neglige la presence de la surface du sol et du glissement a linterface sol structure, mais il est

montre que leurs effets sont faibles et ont des re sultats compensatoires lorsquon introduit la non linearite du sol. Les pre-

dictions obtenues par la solution analytique ont ete verifiees par une serie dessais numeriques qui comprenaient la reponse

de la station de Daikai durant le seisme de Kobe en 1995, et la station du Centre Civique de Los Angeles durant le se isme

de Northridge en 1994. Les erreurs relatives en termes de de formation entre les resultats analytiques et numeriques sont

plus petites que 15 %. La procedure resulte en des contraintes sur la structure qui se comparent bien avec celles obtenues

avec la methode numerique lorsquil ny a pas de glissement entre le sol et la structure. Si on permet un glissement, la so-

lution analytique surestime la prediction des contraintes normales de traction et sous estime les predictions de contraintes

normales en compression.

Mots-cles : conception sesmique, interaction sol structure, structures souterraines, rigidite relative, degradation du mo-

dule de cisaillement, coupe et couvert.

[Traduit par la Redaction]

Introduction

As an integral part of the infrastructure of modern society,underground structures are used for a wide range of applica-

tions, including storage, sewage, water conveyance, andtransportation systems such as subways and railways. Thesupport of underground facilities in seismic zones must bedesigned for static overburden loads and for additional de-

Received 14 November 2006. Accepted 12 February 2008. Published on the NRC Research Press Web site at cgj.nrc.ca on 2 July 2008.

A. Bobet1 and J. Ramirez. School of Civil Engineering, Purdue University, West Lafayette, IN 47907, USA.G. Fernandez. School of Civil Engineering, University of Illinois at Urbana, Urbana, IL 61801, USA.H. Huo. Earth Systems, 79-811B Country Club Drive, Indio, CA 92203, USA.

1Corresponding author (e-mail: [email protected]).

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formations imposed by earthquakes. Seismic-induced defor-mations can be produced by ground failure or shaking.Ground failure includes several types of ground instability,such as direct shear displacements of active faults intersect-ing the structure, landslides, liquefaction of the surroundingground, and tectonic uplift and subsidence. Ground shakingrefers to vibrations produced by propagating waves through

the ground. Shear waves are the most damaging to under-ground structures because they produce racking or ovaliza-tion of the structure (Merritt et al. 1985; Hashash et al. 2001).

There are two basic approaches in present seismic designof underground structures. One approach is to carry out dy-namic, nonlinear soilstructure interaction analysis using nu-merical methods. The input motions in these analyses aretime histories emulating design response spectra and are ap-plied to the boundaries of a soil island to represent verti-cally propagating shear waves. The second approachassumes that the seismic ground motions induce a pseudo-static loading on the structure. This approach allows for thedevelopment of analytical relationships to evaluate the mag-nitude of seismic-induced strains in underground structures

(Timoshenko and Goodier 1970; Peck et al. 1972; Einsteinand Schwartz 1979; Hendron and Fernandez 1983; Merrittet al. 1985; Wang 1993; Penzien and Wu 1998; Penzien2000; Hashash et al. 2001). Such relationships generally ap-ply to deep tunnels in relatively stiff ground. Cut-and-coversubway structures are generally rectangular shallow struc-tures often placed on soft soils, which exhibit significantchanges of stiffness with deformation.

A closed-form solution for rectangular tunnels has beendeveloped by Huo et al. (2006) that addresses some of theshortcomings of previous solutions. The development of thesolution is based on complex variable theory and conformalmapping techniques and relies on the following assumptions:(i) a deep rectangular structure inside an infinite medium;

(ii) plane-strain conditions in any transverse section perpen-dicular to the longitudinal axis of the structure; ( iii) homo-geneous and isotropic ground; (iv) elastic response ofstructure and surrounding ground; (v) pseudo-static analysis.The solution provides the normalized deformation of a rec-

tangular structure in an infinite elastic medium subjected toa far-field shear stress (Fig. 1). Because of the symmetry ofthe problem, the stress distribution around the structure is asdepicted in Fig. 2a. The shear stress (ti) is constant alongthe perimeter of the structure, and the normal stress has alinear distribution along the four sides of the rectangle, with

a maximum magnitude pi1 along the side of length a and pi2

along the side of height b. Because of moment equilibrium,pi1a

2pi2b2. A positive sign of the normal and shear stresses

indicates a distribution, as shown in Fig. 2a.

The structure deformations are given by

1stru

ff 1 2s Npi2 Mp

i2

iLG

b

where Dstru/Dff is the ratio of the structure racking deforma-tion over the free-field ground deformation; ns is thePoissons ratio of the structure; G is the shear modulus ofthe ground; b is the height of the structure; M, N, and L areparameters that depend on the magnitude of the stresses ap-plied at the perimeter of the structure and on the stiffness of

the ground and the structure; i is the deformation of thefree-standing structure due to a unit shear stress ti; and pi

2

is the deformation of the free-standing structure due to thelinear normal stress distribution (Fig. 2a) with pi2 1 and

pi1 b=a2. Alternatively, when the magnitudes of ti and

pi2 are known and for plane strain, the structure deformationDstru, which is the difference between the displacements ofthe top and bottom of the structure (see Fig. 2b), can be es-timated as

2 stru 1 2s

ii pi2pi2

The values of i and pi2

can be obtained analyticallyor numerically from structural analysis. For the case of a

rectangular structure with width a and height b with an inte-rior central column, with member stiffnesses: (EI)w (lateralwalls), (EI)s (bottom slabs), and (EI)c (central column), thedeformations are as follows (Huo et al. 2006):

3

i 1

24b4

2

EIc

EIs

1

EIw

24

35

2EIs

3

2EIs

2

EIw

24

35

2

EIc

1

EIw

3

EIs

pi2

1

4b4

1

45EIc

EIs 1

EIw24 35

480EIs

7

2EIs 5

EIw24 35

1

3EIc

1

6EIw

2EIs

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where = a/b is the aspect ratio of the structure (Fig. 1). Ifthere is no central column, (EI)c = 0, and eq. [3] simplifiesto

4

i 1

24b4

EIs

1

EIw

24

35

pi2

1

60b4

EIs

1

EIw

24

35

The parameters M and N in eq. [1] depend on thePoissons ratio of the ground (n) and the aspect ratio of thestructure () (Huo et al. 2006) and are given in Fig. 3. Theparameter L is defined as

5 L 1 2s Npi2

aGF1

1 2s Mpi2 i aGF2

where F1 and F2 can be found from Fig. 4 and are also afunction of n and .

The stresses along the perimeter of the structure are

6

pi1 pi2

2

pi2 Mi N

G

bff

i LG

bff

The structure deformation can be obtained directly fromeq. [1] and the stresses from eq. [6], or alternatively thestresses can be computed from eq. [6] and the deformationsfrom eq. [2]. Huo et al. (2006) provided extensive compari-sons between predictions obtained with the analytical solu-tion and results from numerical simulations using finiteelement methods (FEMs). The comparisons showed that theanalytical solution provides results with differences typicallywithin a 2%8% range with respect to numerical methods.Huo et al. (2006) also showed that the normalized structuredisplacement is independent of the absolute size of thestructure; in other words, the normalized displacement de-

pends on = a/b, the ratio of length to height of the struc-ture, which is a measure of the shape of the structure. Theyalso showed that the normalized displacement depends onthe relative stiffness between the structure and the surround-ing ground, and not on the absolute stiffness of the structureor the ground.

There are fundamental difficulties for the application of

the analytical solution to practical problems. First, the solu-tion was developed for deep rectangular underground struc-tures, whereas cut-and-cover structures have relativelyshallow burial depths. Second, the analytical solution wasbased on a pseudo-static approach, whereas earthquakes in-duced dynamic loading. Third, the ground was consideredelastic while soils exhibit elastoplastic hysteretic behavior.Lastly, derivation of the solution was done with the assump-tion of a tied interface between ground and structurewhereas in reality slip may occur. All these difficulties areinvestigated and evaluated in this paper, and an approach isoffered to overcome the difficulties. A practical method isproposed where the analytical solution is combined with aniterative scheme to account for the ground shear modulus

degradation with cyclic loading (i.e., equivalent linearmethod). The new method is validated using comparisonswith numerical simulations of the seismic responses of theDaikai subway station in Kobe, Japan, and the Civic Centersubway station in Los Angeles, Calif.

Description of the method

The first step in the method is to obtain the seismic-induced free-field ground deformations at elevationscorresponding to the top and bottom of the structure. Thesefree-field ground deformations can be estimated as Dff =gffb = tffb/G, where Dff is the displacement difference be-tween the top and bottom of the structure; G is the ground

shear modulus; b is the height of the structure; and tff andgff are the far-field ground shear stress and shear strain, re-spectively. The shear modulus, shear stress, and shear strainare those computed at the center of the structure.

The far-field shear strain can be obtained from ground re-sponse analyses. The computer program SHAKE (Schnabelet al. 1972), is widely employed to estimate the free-fieldsoil deformations for various types of ground and for anygiven ground motion. The normalized structure deformationcan thus be obtained from eq. [1], given the structure shearand normal compliances i and pi

2, the aspect ratio , the

ground shear modulus G, and the Poissons ratio n. The ap-propriate shear modulus G is the result of the iterative proc-ess discussed later in the paper.

The analytical formulation is based on the assumption thatboth the ground and the structure behave elastically. Thismay be acceptable for a well-designed structure in a rela-tively low seismic area, where no damage during seismicmotions is anticipated and thus the reduction of stiffness ofthe soil and structure is moderate. Under higher seismic mo-tions, this assumption may not be appropriate. Therefore, inthis paper, the change of soil stiffness with structure defor-mations is introduced in the formulation by an iterativeprocess, where the shear modulus of the soil input into theequations is adjusted depending on the ground deformationsobtained in the previous iteration (equivalent linear method).

Fig. 1. Rectangular structure in an infinite medium.

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The process ends when the shear modulus used in the lastiteration corresponds to the ground deformations (Bobet etal. 2006). If the structure moves into its nonlinear regime,

the degradation of the structure stiffness may be significantand thus the constant-stiffness assumption for the structuremay introduce large errors (a methodology similar to thatproposed for the soil could also be used). Figure 5 showsthe flow chart for the iterative procedure, which is summar-ized in the following steps:

(1) Obtain the free-field ground deformation Dff from theshear strain at the center of the structure. The free-fieldground deformation can be estimated from ground re-sponse analyses, e.g., SHAKE.

(2) Compute the structure compliances i and pi2

and theaspect ratio . This can be done from closed-form solu-

tions such as those of eqs. [3] and [4] or from structuralanalysis.

(3) Compute parameters M, N, and L. Use the value of the

ground shear modulus obtained in the previous iteration(step 6). An initial estimate of the ground shear modulusis needed to start the iteration process.

(4) Find the structure distortion Dstru from eq. [1], given Dfffrom step 1.

(5) Determine the shear strain g of the ground close to thestructure. The soil deformations around the structure areconstrained by the presence of the structure. Huo et al.(2005) showed that there is a volume of ground at-tached to the structure that moves with the structureand determines the structure response. Thus, g = Dstru/b.

(6) Update the ground shear modulus on the basis of the

Fig. 2. Seismic-induced structure (a) loading and (b) deformations.

Fig. 3. Parameters M and N.


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shear strain estimated. The shear modulus is obtainedfrom the Gg curve of the soil (e.g., from resonant col-umn tests; Seed et al. 1986), given the magnitude of theshear strain found in step 5.

(7) Evaluate the difference between the soil strains com-puted in the current and previous iterations. If the differ-ences are small (e.g., error 3 < 1%), convergence has

been reached and the iteration process is completed. Thestructure deformation is that obtained in step 4. If thedifferences are larger than the maximum error allowed,a new iteration is attempted with the updated shear mod-ulus of the soil. In the cases presented in this paper, con-vergence has always been found after a reasonablenumber of iterations.

(8) Compute normal and shear stresses along the perimeterof the structure from eq. [6] using the final, updated,shear modulus of the ground.

The procedure assumes that the soil around the structureis homogeneous. A gradual change of soil properties withdepth can still be effectively captured with the model be-cause (i) free-field deformations are computed with actual

soil properties, and (ii) changes of soil stiffness around thestructure are averaged using the stiffness of the soil locatedat mid-depth of the structure as reference. If significant dif-ferences in soil properties are encountered at the site (e.g., amuch softer or stiffer soil layer, the fill material between the

structure and the original soil is much softer or stiffer thanthe in situ soil, or there are elements that locally affect thesoil response such as the remains of temporary support dur-ing construction), the method proposed may not give accu-rate predictions. However, if soil properties can bebracketed within a reasonable range of values, the solutioncan provide a quick estimate of expected results, which

then can be used as a preliminary design or as a first stepfor more complex (and expensive) numerical methods.

Verification of the method

The analytical solution has been verified with four cases:three different cross sections of the Daikai subway station inKobe, Japan, subjected to the 1995 Kobe earthquake and theCivic Center subway station of the Los Angeles, California,Metro Red Line subjected to the 1994 Northridge earth-quake. The responses of the two stations were calculated us-ing a full dynamic numerical analysis with hystereticelastoplastic soil model (Huo et al. 2005) and the analytical

method proposed. The soil model used in the numericalanalysis included an increase in damping and a reduction ofthe shear modulus of the ground with an increase in strain.Details of the model can be found in Huo et al. (2005). Inessence, stresses and strains are related by

7 toct toct;c Gmaxgoct goct;c1

1 jgoct goct;cj=n goct;y

" #

Fig. 4. Parameters F1

and F2.

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where toct and goct are the octahedral shear stress and strain,respectively; toct,c and goct,c are the octahedral shear stressand shear strain at the last load reversal, respectively; Gmaxis the initial tangent shear modulus and has a magnitudeGmax = ty/gy in which ty and gy are the reference stressand strain, respectively; and n is the scale factor (n = 1 forinitial loading; n = 2 for unloading and reloading).

The tangent shear modulus can be obtained from eq. [7]as

8 G @ oct

@ oct Gmax

1

1 joct oct;cj=noct;y2

The model has been incorporated into the finite elementcode ABAQUS (Hibbitt, Karlsson and Sorensen, Inc. 2002).

The following sections provide a comparison of structuredistortions obtained from the dynamic numerical method

with a nonlinear soil model and with those from the newmethod proposed. Comparisons are provided for three dis-tinct sections of the Daikai subway station and the Los An-geles Civic Center subway station.

Daikai subway station

The Daikai subway station, a cut-and-cover rectangular

structure with central columns, constructed in the subwaysystem in Kobe, Japan, collapsed during the HyogokenNambu (Kobe) earthquake of 17 January 1995. Three differ-ent cross sections can be distinguished in the station(Fig. 6): (i) section 1, the wide (platform) section that col-lapsed during the earthquake; (ii) section 2, which includesthe running tunnels between stations; and (iii) section 3,which corresponds to the two-story access structure.

Section 1 (Fig. 6a) was a rectangular, reinforced concretebox structure, 7.17 m high and 17.0 m wide, with a soilcover of 4.8 m above the top of the structure (Shawky andMaekawa 1996). The central columns had a rectangularcross section of 0.4 m 1.0 m with an axial spacing of3.5 m in the longitudinal direction. The thickness of the lat-

eral wall was 0.7 m, and the top and bottom slabs were 0.8and 0.85 m thick, respectively. The running tunnel (Fig. 6b)was also a rectangular, reinforced concrete box structure,6.36 m high and 9.0 m wide, with a 5.2 m soil cover. Thecentral columns of the running tunnel had cross sections of0.4 m 0.6 m with axial spacing of 2.5 m in the longitudi-nal direction. The thickness of the lateral walls and the topslabs was 0.4 m and the bottom slab 0.44 m. The access sta-tion, section 3, had two levels (Fig. 6c). The section was10.12 m high and 26.0 m wide, with an average soil coverof 1.9 m. The central columns were identical to those of sec-tion 1. The lateral walls were 0.5 m thick, the top slab was0.75 m thick, of the bottom slab was 0.85 m thick at thecenter and 0.55 m at the sides, and the middle slab was0.35 m thick.

At the location of the station, the soil profile consisted ofmanmade fill close to the surface underlain by a sequence ofHolocene clay and sand layers and Pleistocene sand, clay,and gravel layers. The thickness of the fill ranged from 1 to2 m and was underlain by Holocene clay with a thickness ofabout 11.5 m. The clay was followed by a Holocene sandto a depth of about 58 m below the surface. Layers of dif-ferent thickness of Pleistocene sand and clay were encoun-tered between depths of 5 and 17.5 m; a Pleistocene gravelat least several metres thick was found below the abovelayers. The standard penetration test (SPT) values increasedwith an increase in depth. For the Holocene deposits, the

blow counts generally ranged between 10 and 20, and thelower Pleistocene soil deposits showed larger values, partic-ularly at the center of the station, and increased to 50 ormore on the Pleistocene clay and gravel at depths below1517 m. The groundwater table was located at a depth of68 m below the surface. Further details can be found inHuo et al. (2005).

Extensive modeling of each of the sections and the sur-rounding soil was carried out by Huo et al. (2005) andParra-Montesinos et al. (2006) to investigate the load trans-fer mechanisms between the structures and the soil and toidentify the causes for the observed different behavior ofthe sections during the earthquake. In essence, each of the

Fig. 5. Iterative procedure for computing structure deformations.


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three sections of the Daikai subway station was placed in-side a large discretization of the soil medium, with the lat-eral boundaries far enough from the structure to minimizetheir influence on the structure. The actual acceleration re-corded at Port Island was input at the bottom of the discreti-zation (Fig. 7). The soil properties used for the model wereobtained from shear wave velocity and SPT measurements

at the site (Shawky and Maekawa 1996; Huo et al. 2005).The small-strain shear modulus of the soil increased fromGmax = 80 MPa at the surface to 200 MPa at 58 m belowthe surface, which was the limit of the discretization of themodel. The Poissons ratio was taken as 0.35, and the totalunit weight of the soil was 19.6 kN/m3 (Shawky andMaekawa 1996). The structure, with the central columns,was assumed to have elastic behavior throughout the entireanalysis or at least prior to failure (this is an acceptable as-sumption, even for section 1, since it exhibited a brittle fail-ure, as shown in Parra-Montesinos et al. 2006). The concreteof the structure was modeled as an elastic material, with aunit weight of 25 kN/m3, Poissons ratio of 0.15, andYoungs modulus of 24 GPa for the frame (Chuto Industrial

Inc. 1997) and 7 GPa for the columns. The Youngs modu-lus of the central column of the station was obtained by per-forming a three-dimensional (3D) finite element method(FEM) analysis of the structure with the actual dimensionsand spacing of the central columns and matching the stiff-ness of the 3D structure with that of a two-dimentional(2D) structure where the column was assumed to be a con-tinuous wall; hence, the actual spacing of the column wastaken into consideration with the reduced stiffness (7 GPa isapproximately the ratio of 24 GPa, the stiffness of theframe, to 3.5 m, the spacing of the columns of section 1;the procedure was also applied to sections 2 and 3). It wasassumed that a frictional interface existed between the struc-ture and the surrounding ground which was supposed to fol-

low the Coulomb friction law with a coefficient of friction mof 0.4. Special contact elements were placed between thesoil and the structure, which allowed for detachment of thesoil from the structure in tension and enforced the Coulombfriction law in compression. Further details about the discre-tization and simulations are found in Huo et al. (2005). Fig-ure 8 shows the predicted central column distortions forsection 1 (results after maximum distortion are theoretical,since the central column should have collapsed beyond thispoint). Analogous simulations were conducted for sections 2and 3. Table 1 shows the maximum central column distor-tion for each section.

For the analytical solution, the dimensions of the structure

used are those in Fig. 6. It has been found that the analyticalsolution provides the best results, compared with those ofthe numerical model, when the dimensions of the structureused in the equations correspond to those of the interioropening (Huo et al. 2006). This behavior is the result of theconstraint that the thickness of the structure slabs imposeson the ground behind, and so for practical purposes the ef-fective structure dimensions are those of the interior open-ing. Thus, a and b were measured from wall to wall andcorrespond to the interior dimensions of the structure, ratherthan to the distance between the axes of the horizontal andvertical members. The aspect ratios = a/b were 2.82 forsection 1, 1.48 for section 2, and 2.78 for section 3. The

elastic properties of the structure elements are the same asthose used for the numerical analyis, with the Youngs mod-ulus of the central column in each section reduced accord-ingly to approximate the 3D response. The structuredeformations under a unit shear and unit normal stress, iand pi

2, were obtained numerically from structural analysis

and were i 0:6 and pi2

0:092 for section 1,

i 0:76 and pi2 0:23 for section 2, and i 0:62and pi

2 0:08 for section 3. The small-strain ground shear

modulus at the center of each structure was equal to Gmax =97 MPa for sections 1 and 2 and 94 MPa for section 3. Thefree-field soil deformations induced by the Kobe earthquakewere obtained from ground response analysis using the pro-gram SHAKE 91 (Idriss and Sun 1992), given actual soilproperties and input accelerations. The free-field ground de-formations obtained from the shear strain of the soil at thecenter of each cross section were 14, 12, and 19 cm for sec-tions 1, 2, and 3, respectively.

The structure deformations for each section were obtainedfollowing the proposed iterative method (Fig. 5). The proce-

dure followed to obtain the solution for section 1 is includedin Appendix A as an example; the same process was re-peated for sections 2 and 3. The shear modulus reduction ofthe soil with an increase in shear strain was taken fromFig. 9, which also shows the increase of damping with anincrease in shear strain (data from Seed et al. 1986; Vuceticand Dobry 1991) and the results of soil stiffness and damp-ing predicted by the soil model from eqs. [7] and [8]. Thecomparison between the model and experiments is good.The final horizontal structure (column) deformations of sec-tions 1, 2, and 3 were 3.8, 3.5, and 2.1 cm, respectively. Theanalytical results compare well with the numerical results,namely deformations of 4.0, 3.1, and 2.2 cm, respectively(Table 1). The differences between the analytical and nu-

merical results are 5%, 12.9%, and 4.5%, respectively, wellwithin the degree of accuracy expected given the uncertain-ties involved in seismic analysis.

It is important to point out that the Dstru/Dff ratios ob-tained (e.g., 3.8/14 = 0.27 for section 1 and 0.29 and 0.11for sections 2 and 3, respectively) result in structural defor-mations significantly lower than those in the free-field.Thus, the proposed method is a considerable improvementof current analytical methods, which suggest that under-ground structures behave as flexible members that followfree-field deformations.

Los Angeles Civic Center Metro Red Line station

The Metro Red Line is the first modern heavy-rail subwayin Los Angeles. The downtown part of the Red Line takespassengers under downtown Los Angeles, from Union sta-tion to Civic Center station. The total length of the MetroRed Line is 29 km. Construction started in September 1986,and the downtown part of the Metro Red Line was openedto service on 30 January 1993. The 6.8 magnitude North-ridge earthquake in 1994 caused no significant damage tothe existing subway tunnels and stations.

The Civic Center station is a reinforced concrete structurebuilt by cut-and-cover. The transverse cross section is a rec-tangular box with a central column, with dimensions some-what variable along the longitudinal axis of the station.


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Figure 10 shows a slightly idealized cross section of the

west part of the station. The structure at this location is19.3 m wide 18.5 m high, with an overburden of 3.17 m.The thickness of the top and bottom slabs is 1.0 and 1.5 m,respectively. The thickness of the lateral walls is 1.2 m. Thetop one third of the central column has a 0.9 m 0.9 msquare cross section, and the bottom two thirds has a rectan-gular cross section with 0.9 m in the transverse direction and1.4 m in the longitudinal direction. For simplicity, the crosssection of the column is modeled as 0.9 m 1.4 m in thefollowing analyses. The longitudinal axial spacing of thecolumns is 9.0 m. There are two intermediate slabs in thestructure that divide the station into three floors, withthicknesses of 0.56 and 0.5 m, respectively, as shown in

Fig. 10. The concrete of the structure had a unit weight of22.6 kN/m3. A Youngs modulus Es = 30000 MPa wasused for the lateral walls and slabs and Es = 4700 MPa forthe central column to account for the spacing of the col-umns. The Poissons ratio of the structure was taken as 0.15

(Southern California Rapid Transit District 1986).Soil exploration consisting of borings and SPT tests was

performed for the design of the station (Southern CaliforniaRapid Transit District 1986). The site at the Civic center sta-tion was composed of a weathered silty claystone, known asthe Fernando Formation. In the west portion of the station, ashallow layer of alluvial silty clay was found. The silty claylayer extended from the surface down to 7 m depth on topof the claystone layer, the Fernando Formation. The clay-stone layer can be divided into two sublayers: a shallowersoft silty claystone down to 16 m and a deeper stiff siltyclaystone down to the maximum depth of the borings,36.6 m (Fig. 10). Based on SPT blowcounts, the maximumsmall-strain shear modulus of the soil layers was taken as40, 95, and 177 MPa for the top alluvium silty clay and thesoft and stiff claystones of the Fernando Formation, respec-tively. The Poissons ratio of the soil was taken as 0.35.

The strong motions from the Northridge earthquake on17 January 1994 were used in the analyses. The ground mo-tions recorded at the Glendale Las Palmas earthquake sta-tion, approximately 3 km north of the Civic Center station,were used in both the numerical and analytical simulations.Figure 11 shows the recorded horizontal accelerations,which shows a peak value of 0.36g. In the models, the ac-celerations were arbitrarily imposed at the bottom of thesoil exploration, which was at 36.6 m below the surface.

Fig. 7. Daikai subway station: input accelerations.

Fig. 8. Daikai subway station, section 1: column distortion.

Table 1. Comparison of predicted column distortions from nu-

merical and analytical models.

StationNumerical(cm)

Analytical(cm)

Error(%)

Daikai, section 1 4.0 3.8 5.0



Los Angeles Civic Center 6.8 6.6 2.9

Fig. 9. Shear modulus reduction and damping increase with an in-

crease in strain.

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Modeling and discretization of the Civic Center stationwere similar to those of the Daikai station. The cross sectionof the Civic Center station was placed at the center of a soilisland 36.6 m deep, with free lateral boundaries far enoughfrom the station such that their presence had no effect on thestructure motions. The earthquake accelerations (Fig. 11)were input at the bottom of the discretization. The structurewas modeled as elastic and with the dimensions shown inFig. 10. The soil behavior was approximated with the soilmodel described previously, with properties obtained fromsoil exploration. The structureground interface was mod-eled, similar to that of the Daikai station, as frictional withm = 0.4, using special interface elements. Figure 12 showsthe center column deformationtime history. The maximumcolumn distortion is 6.8 cm (Table 1), which corresponds toa column drift of 0.39%, much smaller than that of section 1of the Daikai station (0.8%). This column drift is below thedistortion required for yielding of the columns and is consis-tent with the observation that no damage occurred in the sta-tion during the Northridge earthquake.

The first step for the analytical solution was to obtain thefree-field deformations at the center of the structure. Thiswas done with SHAKE 91 and with soil properties and inputaccelerations exactly the same as those used for the numeri-cal simulation. The free-field shear strain produced a struc-

Fig. 10. Los Angeles Civic Center subway station. All dimensions are given in millimetres.

Fig. 11. Los Angeles Civic Center subway station: input accelera-

tions.


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ture deformation of 21 cm, measured from axis to axis be-tween the top and bottom slabs. The aspect ratio of the sta-tion was = 1.05, and the structure compliances due to unitshear and normal loads were i 1:35 and pi

2 0:36, re-

spectively, which were obtained numerically. The maximumshear modulus Gmax of the ground at the Civic Center stationsite, taken at the middle height of the structure, was

95 MPa. The structure deformations were obtained follow-ing the iterative procedure discussed earlier, with shearmodulus degradation with shear strain taken from Fig. 9.The end result was a column distortion of 6.6 cm (Table 1)or a column drift of 0.38%.

The difference between the analytical and numerical re-sults for the Los Angeles Civic Center station is 2.9%. Thedifferences for sections 1, 2, and 3 of the Daikai station are2.5%, 12.9%, and 4.5% (Table 1), respectively. Such differ-ences are within the degree of accuracy expected in this typeof problem, given the uncertainties involved in the analysis.

Dynamic, depth, and interface effects

The method proposed assumes that the dynamic amplifi-cation of stresses associated with a stress wave impingingon the structure is negligible (i.e., a pseudo-static analysis isappropriate). This assumption is correct if the wavelength ofpeak velocities is at least eight times greater than the widthof the opening (Hendron and Fernandez 1983). Under theseconditions, the free-field stress gradient across the openingis relatively small, and the seismic loading can be consid-ered a pseudo-static load. In most underground openings,the quasi-static conditions are usually satisfied, and thus thepseudo-static approach followed in the proposed methoddoes not introduce significant errors.

The analytical solution was developed with the assump-tion that the underground structure was far from the surfaceand there was no slip between the ground and the structureat the interface. However, the depth of cut-and-cover struc-tures is usually small compared with the characteristic sizeof the structure, and slip at the ground-surface contact ispossible. The two factors, depth and interface, have smalleffects on the structure deformations; when the nonlinear be-havior of the soil is included in the analysis, there are com-pensating results that minimize the effects of the twofactors.

Wang (1993) showed that the depth of an undergroundstructure has relatively minor effects compared with otherfactors such as free-field ground movements and relativestiffness. The normalized structure deformation Dstru/Dff

slowly decreases as burial depth decreases, with a maximumreduction with respect to a deep structure of about 15%20%. The authors expanded the analyses done by Wang(1993) and conducted a series of linear elastic numericalanalyses on rectangular structures with aspect ratios rangingfrom = 1 to 3, with different relative stiffnesses at increas-ing depths. Even though it was found that the influence ofdepth on structure deformations depended on the shape ofthe structure (Huo 2005), the results essentially confirmedprevious observations (Wang 1993). For overburden (dis-tance from the surface to the top of the structure) to struc-ture height ratios greater than about 1.5, the deformationsare those of a deep structure. Thus, the analytical solution,

which considers a deep tunnel, would initially provide aconservative estimate of the structure deformations.

To investigate the significance of friction at the groundstructure interface, a series of FEMs were carried out withABAQUS (Hibbitt, Karlsson and Sorensen, Inc. 2002). Arectangular structure in an infinite medium was considered(Fig. 1). The boundaries were placed at about 1020 timesthe dimensions of the structure such that the presence of theboundaries had no influence on the response of the structure.The ground was assumed isotropic, homogeneous, and elas-tic. Special interface elements were placed between thestructure and the surrounding ground such that slip occuredwhen the applied shear stress was larger than the shearstrength, which was modeled following the Coulomb frictionlaw. A small constant normal stress with magnitude 120 kPawas first applied to the boundaries of the discretization. Thiswas done to prevent the soil from detaching from the struc-ture (i.e., the normal stresses at the interface were compres-sive). The seismic-induced deformations were modeled byapplying a constant shear stress at the far boundaries of thediscretization with a magnitude of 50 kPa. Different inter-faces were evaluated, with coefficients of friction rangingfrom 0 (full slip) to very large (no slip, tied interface). Co-

hesion was always taken as zero. The properties used in theanalysis were as follows: Es = 24 000 MPa and ns = 0.15 forthe structure, and G = 80 MPa and n = 0.35 for the ground.The thickness of the slabs and walls of the structure was0.5 m. Figure 13 shows the effects of the frictional interface.The structure distortions, normalized with respect to the dis-tortions obtained with a tied interface (the assumption madewith the analytical solution), were plotted as a function ofthe interface friction and for three different structure aspectratios, i.e., = 1 (square, 4 m 4 m), 2 (4 m 8 m), and 3(4 m 12 m). The structure deformations increase with anincrease in the friction coefficient. The structure deformationswith an interface m > 1.2 (friction angle of 508) converge on

Fig. 12. Los Angeles Civic Center subway station: column distor-

tion.

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those with a tied interface. The structure deformations with atied interface and assuming elastic soil behavior are 12%, 9%,and 7% greater than those with m = 0.4 for = 1, 2, and 3,respectively. The structure deformations are the smallestwith m = 0.2 and are about 84%88% of the case with a tiedinterface. In Fig. 13, results with 0 < m < 0.2 have beenshaded because openings between the ground and the struc-ture appear in the numerical model owing to the small valuesof the friction coefficient. The results indicate that an inter-face with a tied contact yields larger structure deformationsthan those with a frictional interface. Hence, predictionswith the analytical solution would initially lie on the safe

side. A similar observation was made for circular tunnelsby Peck et al. (1972) who indicated that it is conservativeto assume no slip between the ground and the tunnel linerduring a seismic event.

When the elastoplastic soil behavior is included in the dy-namic numerical analyses, the structure deformations areslightly larger with a frictional interface than with a tiedinterface. The nonlinear numerical analyses for each of thethree sections of the Daikai station and the Los AngelesCivic Center station have been repeated exactly as wasdone originally with a frictional interface but now using atied interface. Table 2 shows a comparison between struc-ture distortions with a frictional interface and those with a

tied interface. A tied interface decreases structure deforma-tions about 5%10% with respect to a frictional interfacewhen the nonlinear soil behavior model is used. This is incontrast with what happens with an elastic soil model wherethe structure deformations increase 10%15% with a tied in-terface. The reason for the difference is the interplay thatexists between the constraint that the structure imposes onthe soil and the soil stiffness degradation with an increasein strain. With an elastic model, the soil stiffness does notchange. Since shear stresses from the soil are better trans-ferred to the structure with a tied interface, the structure de-formations are larger than those with a frictional interface.However, with a strain-softening soil model, the stiffness of

the soil is reduced with soil deformations. A frictional inter-face has two opposing effects: (i) it limits the magnitude ofthe shear stress that can be transmitted to the structure; and(ii) it allows larger deformations to develop within the soilsurrounding the structure, since slip can occur at the inter-face. As the soil deforms more, the stiffness of the soil isreduced, which in turn induces further deformations in the

soil with the imposed earthquake motions. This is consistentwith the numerical predictions that indicate a reduction ofthe soil shear modulus G/Gmax to 78% around the structurewith a frictional interface compared with a reduction to85% with a tied interface. Thus, the softer soil will deformmore, but the capacity of the soil to transmit deformations tothe structure in shear is reduced. The end result is that thesoil stiffness degradation phenomenon prevails, and the finaloutcome from the interrelated structure and soil behavior islarger structure deformations with a frictional interface.

The effects and interplay among depth, friction, and shearmodulus degradation tend to reduce the differences betweenthe analytical and numerical results. First, neglecting deptheffects in the analytical solution would result in overesti-

mating the structure deformation by 15%20%. Second, theanalytical solution (tied interface) with elastic soil behavioralso results in overestimating the structure deformation by10%15%. On the other hand, the elastoplastic hystereticsoil model shows that a frictional interface, which is the ex-pected behavior, coupled with shear modulus degradation re-sults in an increase of the structure deformation by about5%10%. Hence, the net effect of the three factors resultsin the analytical solution (tied interface) giving predictionswithin 5%15% of those of the numerical model, which iswithin the error range shown in Table 1.

Interface stresses

Once the structure deformations are obtained with theiterative procedure proposed, the stresses at the interface be-tween the structure and the soil can be computed witheq. [6] using the soil shear modulus found in the last itera-tion. As with eq. [1], the inside dimensions of the structureare used for the computations. The magnitudes of the nor-mal and shear stresses obtained with the analytical solutionand the numerical method are examined at the peak re-sponse of the structure (at about 4 s for the Daikai stationand about 10 s for the Los Angeles Civic Center station).The following cases are investigated: tied interface and fric-tional interface with m = 0.4.

Table 3 shows the magnitude of the normal and shear

stresses applied to the structure obtained with the numericaland analytical methods for the Daikai and Los AngelesCivic Center stations, with the assumption of a tied inter-face. Figure 2a provides the shape of the stress distributionaround the structure. The differences are acceptable, giventhe uncertainties involved in this type of analysis, and arewithin a 15%20% difference range. The normal stresses re-ported in Table 3 are due only to the dynamic-induced de-formations. In other words, the initial normal stresses due togeostatic conditions need to be added to find the final nor-mal stress. Although the similarities between the numericaland analytical results are not unexpected because, after all,structure distortions are not significantly affected by the in-

Fig. 13. Effect of friction coefficient on normalized structure de-

formation.


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terface (Table 2), the results are not realistic because tensiondevelops at the tied interface. In reality, the soil cannot sus-tain tension, and the shear stresses at the soilstructure con-tact are limited by the frictional strength of the interface.

Figure 14 shows the stress distribution around section 1 ofthe Daikai station with a frictional interface, where the soilcan detach from the structure and slip. There are importantdifferences with the analytical solution: (i) the stress distri-bution is no longer antisymmetric, and (ii) there is no ten-sion at the interface (the total normal stresses should beobtained from those shown in Fig. 14a plus geostaticstresses). Both effects are the result of allowing slip and pre-venting tension along the interface. The shear stresses are nolonger constant, since their magnitude is limited to the totalnormal stress times the coefficient of friction, m = 0.4. As aconsequence, the analytical solution does not predict wellthe actual stress distribution around the station. It is interest-ing to note that a frictional interface redistributes the normalstresses, with a clear shift to compression, as tension is notpossible; and the shear stress distribution is no longer con-stant but proportional to the normal stresses in those areaswhere there is slip. This redistribution of stresses does notsubstantially change the structure deformations (Table 2).

This is to some extent an expected result, since deformationsare mostly controlled by the relative stiffness between thesoil and structure while interface and depth effects have rel-atively minor importance. Similar conclusions were obtainedfor the Los Angeles Civic Center station.

Even though the analytical solution is unable to approxi-mate stresses with a frictional interface, these stresses canbe obtained employing numerical methods (e.g., finite ele-ment method with frictional interface elements at the soilstructure interface) using as input parameter results from theanalytical solution. In this case, the far-field shear stressesthat are needed for the simulation can be obtained from theanalytical procedure proposed, tff = Ggff, where G is the

soil shear modulus obtained at the last iteration, and gff isthe far-field shear strain of the soil from SHAKE; the stiff-ness of the soil that needs to be used in the numerical modelis the magnitude of the shear modulus resulting from the lastiteration. As an alternative, and for predimensioning of thestructural elements, shear forces and bending moments canbe estimated, given the stresses resulting from the analyticalsolution, or by following the recommendation by Wang(1993), where the loads and moments are the result of ahorizontal load applied to the top slab of the structure, witha magnitude such that the load produces the structure defor-mations given by the iterative method.

Summary and conclusions

An iterative scheme is presented in this paper to estimatedeformations of cut-and-cover rectangular structures duringearthquakes. The method is based on an existing analyticalsolution for deep rectangular structures in an elastic medium(Huo et al. 2006) and on previous work on the failure of theDaikai subway station in Japan during the 1995 Kobe earth-quake (Huo et al. 2005). The previous research confirmedthat the relative stiffness between the soil and the structurewas the parameter that controlled structure deformationsduring an earthquake and that the presence of a structurestiffer than the surrounding soil significantly constrained thedeformations of the soil adjacent to the structure.

In this paper, the existing analytical solution has beenmodified by including an iterative procedure where theground shear modulus is reduced based on the deformationsof the structure (e.g., equivalent linear method). The itera-tion scheme is completed when the ground stiffness used asinput in the analytical solution matches the stiffness that theground should have with the deformations predicted by thesolution. In the scheme, the deformations of the ground aretaken as those of the structure, since the groundstructure

Table 2. Predicted column distortions from the numerical model for frictional and

tied interfaces.

StationFrictional interface(m = 0.4; cm)

Tied interface(cm)

Difference(%)




Los Angeles Civic Center 6.8 6.0 11.8

Table 3. Comparison of interface stresses from elastoplastic dynamic numerical and analytical meth-

ods for the Daikai and Los Angeles Civic Center stations with a tied interface.

StationStress(kPa) Numerical Analytical Error (%)

Daikai, section 1 Normal pi1 205; pi2 1088 p

i1 164; p

i2 1287 19.9; 15.4

Daikai, section 1 Shear ti = 335 ti = 334 0.2


i1 190; p

i2 401 10.8; 4.1

Daikai, section 2 Shear ti = 290 ti = 264 9.0


i1 150; p

i2 1149 10.3; 1.7

Daikai, section 3 Shear t

i

= 261 t

i

= 298 14.2Los Angeles Civic Center Normal pi1 140; pi2 154 p

i1 160; p

i2 176 14.3; 14.3

Los Angeles Civic Center Shear ti = 181 ti = 193 6.6

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interface is assumed to be tied, and the ground adjacent tothe structure controls the deformations of the structure.

Four cases were used for verification of the proposed sol-ution: the three sections of the Daikai subway station (sec-tions 1, 2, and 3) and the Los Angeles Civic Center subway

station. The four cases were analyzed with a dynamic nu-merical method with an elastoplastic hysteretic soil modelusing actual ground motions recorded during the Kobeearthquake of 1995 for the Daikai station and the Northridgeearthquake of 1994 for the Los Angeles Civic Center sta-tion. The same four cases were also analyzed using the ana-lytical solution with the iterative scheme, with free-fieldground deformations obtained at mid-depth of the structure,given the actual ground motions and ground properties ateach of the sites. Comparisons between the numerical andanalytical results show that the deformations are well pre-dicted by the analytical method, with differences within5%15%.

The fact that the analytical solution matches the numeri-cal results is attributed to three factors that produce compen-sating effects: depth, interface, and nonlinear soil behavior.Depth and interface effects are not considered in the anal-ytical solution; disregarding these two factors should pro-duce predictions that overestimate the results by about15%20%. As the ground around the structure is less con-strained with a frictional interface, its shear modulus degra-dation increases, and thus the analytical solution shouldunderpredict structure displacements. Numerical simulationsindicate that the errors are of the order of 10%. Hence, thecombination of the three factors, namely depth, interface,and shear modulus degradation, results in the analytical sol-

ution giving predictions that should be accurate within anerror of 5%15%.

The analytical solution provides the magnitude and distri-bution of normal and shear stresses along the soilstructureinterface reasonably well when there is a tied contact. Atied interface imposes the same deformations of the soil andstructure at the contact; for a small overburden, tension may

be produced at some areas along the interface, as the initialnormal stress may be too small compared with the tensilenormal stresses induced by the shear-induced loading. Foran elastic material or a stiff soil, this behavior may still bepossible, but it may not be reasonable for soft soils and inparticular for sandy soils. A frictional interface where slipis allowed produces a deviation of the normal stress distribu-tion compared with that assumed in the analytical solution,with a significant increase of compression relative to ten-sion. The end result is that, when a frictional interface is in-troduced, the analytical solution, which assumes a tiedinterface, overpredicts tension and underpredicts compres-sion. Independently of interface characteristics, the anal-ytical solution predicts quite well the structure deformations

because the primary factor that controls deformations is therelative stiffness between the structure and the ground,whereas the interface has secondary effects. Although theproposed method may be used for predimensioning of thestructural elements, accurate prediction of structure stressesrequires a numerical model.

Given the level of accuracy provided by the analyticalsolution and the number of cases investigated for its verifi-cation, it can be concluded that the proposed solution is agood approximation for a preliminary seismic design of cut-and-cover rectangular structures, in particular where defor-mations need to be estimated. Because the method does notinclude nonlinear deformations of the structure and is devel-oped with the assumption of a structure more rigid than thesurrounding ground, it should only be applied to structuresthat are not expected to experience significant damage. Itmay not be appropriate for assessment of deformations orrisk for existing structures that are underdesigned or forproperly designed structures under seismic ground motionthat significantly exceeds design levels. In both these cases,response involves highly nonlinear structural or soil behav-ior that could well exceed the ability of the proposed ap-proach. However, the level of deformation estimated withthe approach can be used to assess if nonlinear structural be-havior will occur and thus if a more sophisticated modelwill be needed.

The proposed method, taking into consideration the level

of uncertainty that still exists for predicting the response ofground and structure during cycles of loading, should beviewed as a first approximation and not as a replacementfor detailed dynamic numerical models. The analytical solu-tion indicates that, in addition to seismic motion input data,the relative stiffness between the structure and the degradedsoil is the key parameter that determines the soilstructureresponse behavior. The analytical solution also provides aquantitative assessment of the influence of the relative stiff-ness on soilstructure behavior and allows one to determinewhen efforts should be made to accurately determine thesoil and structure stiffnesses. Even though the four casesanalyzed in this paper provide confidence in the solution de-

Fig. 14. (a) Normal stresses and (b) shear stresses (both in kPa)

alonga interface (m = 0.4) of section 1 of the Daikai subway station.


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veloped, additional verifications should be made with otherstructures and earthquake input motions with a sufficientlylarge and diverse number of cases.

Because of the uncertainty associated with estimation ofseismic-induced deformations imposed on undergroundstructures, their design must include a significant level ofductility such that brittle failures are avoided. This is partic-

ularly important in cut-and-cover structures that have to re-sist significant bending moments, and thus their behavior isnot that different from that of above-ground structures. Col-lapse of an underground structure may be caused, similar tothat of above-ground structures, in critical elements wherethe moment capacity and (or) shear capacity is exhausted.This is quite different from deep circular tunnels where thesupport works mostly in compression. Formation of plastichinges in the support during earthquake-induced motionsmay not necessarily cause the collapse of the structure, asthe compression capacity of the liner may not be signifi-cantly reduced. This provides a level of redundancy that isnot available in shallow rectangular structures.

Acknowledgements

The research has been supported in part by the US Na-tional Science Foundation, Structural Systems and HazardsMitigation Program, under grant CMS-0000136. This sup-port is gratefully acknowledged.

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Hibbitt, Karlsson and Sorensen, Inc. 2002. ABAQUS/explicit users

manual, version 6.3. Hibbitt, Karlsson and Sorensen, Inc., Paw-

tucket, R.I.

Bobet, A., Fernandez, G., Ramirez, J., and Huo, H. 2006. A practi-

cal method for assessment of seismic-induced deformations of

underground structures [CD-ROM]. In GeoCongress 2006: Pro-ceedings of Geotechnical Engineering in the Information Tech-

nology Age, Atlanta, Ga., 26 February 1 March 2006. Edited

by D.J. DeGroot, J.T. DeLong, J.D. Frost, and L.G. Baise.

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Chuto Industrial Inc. 1997. Recovery record of the Daikai station

after Kobe earthquake. Chuto Industrial Inc., Japan. [In Japanese.]

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tunnel supports. Journal of the Geotechnical Engineering Divi-

sion, ASCE, 105(GT4): 499518.

Hashash, Y.M.A., Hook, J.J., Schmidt, B., and Yao, J.I. 2001. Seis-

mic design and analysis of underground structures. Tunnelling

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Hendron, A.J., and Fernandez, G. 1983. Dynamic and static design

considerations for underground chambers. In Seismic design ofembankments and caverns. Edited by T.R. Howard. ASCE, New

York. pp. 157197.

Huo,H. 2005. Seismic design and analysis of rectangular underground

structures. Ph.D. thesis, Purdue University, West Lafayette, Ind.

Huo, H., Bobet, A., Fernandez, G., and Ramrez, J. 2005. Load

transfer mechanisms between underground structure and sur-

rounding ground: evaluation of the failure of the Daikai station.

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131:12(1522).

Huo, H., Bobet, A., Fernandez, G., and Ramrez, J. 2006. Analyti-

cal solution for deep rectangular underground structures sub-

jected to far-field shear stresses. Tunnelling and Underground

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design of underground structures. In Proceedings of the 1985

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Mining Engineers of theMining, Metallurgical and Petroleum

Engineeers, New York. Vol. 1, pp. 104131.

Parra-Montesinos, G.J., Bobet, A., and Ramirez, J. 2006. Evalua-

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kai subway station during Kobe earthquake. American Concrete

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Appendix A

Calculations of deformations of section 1 of the Daikaisubway station during the Kobe earthquake are included asan example. Calculation steps follow the procedure depictedin Fig. 5.

(1) Free-field ground deformations are computed from the

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shear strain at the mid-depth of the structure using thecomputer code SHAKE 91. Dff = 14 cm.

(2) Structure compliances are obtained from numerical ana-lysis of the cross section of the structure by imposingunit shear and normal loads. Thus, i 0:6 andpi

2 0:092. = 15.6/5.52 = 2.82.

(3) Soil and structure Poissons ratios are n = 0.35 and ns =0.15, respectively. From Fig. 3, M = 3.941 and N =0.145. From Fig. 4 and eq. [5], L = 0.65. It is assumed

in the first iteration that G = Gmax = 97 MPa. Note thatL depends on G and changes with iterations.

(4) From eq. [1], Dstru/Dff = 2.881. Thus, Dstru = 40.34 cm.(5) g1 = Dstru/b = 0.073.(6) A shear modulus G1 = 0.16 MPa is obtained from the Gg

curve of Fig. 9, given the magnitude of the shear strain g1.

(7) A new iteration is attempted with G = G1 = 0.16 MPa.New iterations are performed with updated values of theshear modulus until the error is small. All iterations areincluded in Table A1. The final structure deformation is3.8 cm.

The number of iterations could be significantly reducedwith a better estimate of the final shear modulus in the firstiteration. In the example, G = Gmax was taken as the startingvalue. For strong ground motions, it is unlikely that the soil

deformations are going to be that small. In general, asmaller number of iterations may be attained with the initialguess G = 0.5Gmax.

Table A1. Iterations for section 1 of the Daikai station.

Iteration gi Gi (MPa) Dstru/Dfree-field Dstru (cm) gi+1 Gi+1 (MPa) |gi+1 - gi|/max(gi+1, gi) (%)

1st not applicable 97.00 2.8811 40.34 7.31 102 0.16 1002nd 7.31 102 0.16 0.0222 0.31 5.64 104 30.00 993rd 5.64 104 30.00 1.9636 27.49 4.98 102 0.30 994th 4.98 102 0.30 0.0413 0.58 1.05 103 4.60 985th 1.05 103 4.60 0.5720 8.01 1.45 102 0.85 936th 1.45 102 0.85 0.1164 1.63 2.95 103 2.90 807th 2.95 103 2.90 0.3642 5.10 9.24 103 1.08 688th 9.24 103 1.08 0.1456 2.04 3.69 103 2.70 609th 3.69 103 2.70 0.3584 5.02 9.09 103 1.17 5910th 9.09 103 1.17 0.1568 2.20 3.98 103 2.50 5611th 3.98 103 2.50 0.3350 4.69 8.50 103 1.40 5312th 8.15 103 1.40 0.1876 2.63 4.76 103 2.30 4213th 4.76 103 2.30 0.3053 4.27 7.74 103 1.80 3914th 7.74 103 1.80 0.2390 3.35 6.06 103 2.15 2215th 6.06 103 2.15 0.2854 4.00 7.24 103 1.90 1616th 7.24 103 1.90 0.2522 3.53 6.40 103 2.10 1217th 6.40 103 2.10 0.2788 3.90 7.07 103 2.00 1018th 7.07 103 2.00 0.2655 3.72 6.73 103 2.05 519th 6.73

103 2.05 0.2721 3.81 6.90

103 2.03 2

20th 6.90 103 2.03 0.2695 3.77 6.83 103 2.04 121st 6.83 103 2.04 0.2708 3.79 6.87 103 2.04 0


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a practical iterative procedure to estimate

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