Evaluation of Factors of Safety against Basal Heave for DeepExcavations in Soft Clay Using the Finite-Element Method

Tuan-Nghia Do1; Chang-Yu Ou, M.ASCE2; and Aswin Lim3

Abstract: This study investigates basal heave stability of deep excavations in soft clay using FEM with reduced shear strength. Three failureand one excessive deformation case histories are utilized. Three methods of estimating factors of safety against basal heave are employed, in-cluding the convergence criterion method involving numerical convergence, the angle method of quantifying the kick-out phenomenon of thewall, and the intersection method of examining the behavior of nodal displacements. Results show that the convergence criterion method andthe angle method overestimated the factors of safety (FS) against basal heave. The intersection method, on the other hand, resulted in goodagreement with the field observations because the calculated factors of safety were close to unity. The intersection method was also verifiedby a parametric study wherein its factors of safety increased linearly with the undrained shear strength of soil. DOI: 10.1061/(ASCE)GT.1943-5606.0000940. © 2013 American Society of Civil Engineers.

CE Database subject headings: Excavation; Finite element method; Clays; Safety; Differential settlement.

Author keywords: Deep excavations; Basal heave stability analysis; Finite-element method.

Introduction

Conventional methods, including Terzaghi’s method, Bjerrum andEide’s method, and the slip circle method, are commonly used forevaluation of basal heave stability of excavations in soft clay. Inthese methods, a failure surface of soil is first assumed, and thefactor of safety (FS) against basal heave is then calculated as theratio of the resistant force to the driving force. Although thiscalculation procedure is convenient for practical application, theactual failure surface may not be the same as that assumed in theconventional methods. In addition, these methods do not take intoaccount the stiffness and embedment depth of the wall in thecalculation, which should have significant effects on the stability ofexcavations, especially as supported by relatively stiff structuraldiaphragm walls.

The FEM with reduced shear strength proposed by Zienkiewiczet al. (1975) has been applied to slope stability problems by manyresearchers, such as Tan and Donald (1985) and Griffiths and Lane(1999). Goh (1990) and Faheemet al. (2003), on the other hand, usedthis method to analyze the stability of excavations in soft clay. Inprinciple, the stability of an excavation at the final stage is studied byreducing the soil strength successively until convergence of nu-merical solutions no longer remains. The factor of safety is thendefined as the ratio of the original soil strength to the reduced soil

strength when numerical solutions diverge. This approach simplyutilizes the convergence criterion to specify convergence of nu-merical solutions as the strength reduction procedure is performedwithout making any further assumptions. The FEM has manyadvantages over the conventional method in stability analysis. Thefailure surface in the FEM is formed automatically among the failurepoints in soil, which are subject to the shear stress greater than orequal to the shear strength of soil, while this surface needs to beassumed in the conventional method. The FEM is also capable ofmodeling effects of the retaining system, such as the excavation walland horizontal struts. Therefore, the FEM is a promising alternativeapproach for stability analysis of deep excavations in clay. However,few researchers have applied this method to analyze stability of casehistories. The verification of the FEM remains to be resolved.

This study uses three failure and one excessive deformation deepexcavations in soft clay to evaluate the capability of the FEM withreduced shear strength for stability analysis. Three methods areadopted, including the convergence criterion method, the anglemethod, and the intersection method.

FEMs with Reduced Shear Strength

Large Deformation Analysis

The updated mesh finite-element (FE) analysis, an advanced op-tion in PLAXIS, was used for the current study to analyze largeexcavation-induced deformations of the soil and wall. In this ap-proach, the updated Lagarangian formulation, the so-called relativedescription, is adopted to describe the deformation of the FE meshin which a current mesh configuration at a time t, is treated as thereference mesh configuration for the description of the past andfuture mesh configurations. The reference mesh configurationtherefore varies with the variable time t and requires updating thedeformed FE mesh consecutively as the calculation proceeds. Theconventional FE analysis, commonly used for small deformationproblems, on the other hand, utilizes the referential description toindicate deformation in which the reference mesh configuration is

1Ph.D. Student, Dept. of Construction Engineering, National TaiwanUniv. of Science and Technology (Taiwan Tech), Taipei 106, Taiwan(corresponding author). E-mail: D10005806@mail.ntust.edu.tw

2Professor, Dept. of Construction Engineering,National TaiwanUniv. ofScience and Technology (Taiwan Tech), Taipei 106, Taiwan. E-mail:ou@mail.ntust.edu.tw

3Lecturer, Dept. of Civil Engineering, Parahyangan Catholic Univ.,Bandung, 40291, Indonesia.

Note. This manuscript was submitted on August 7, 2012; approved onApril 8, 2013; published online on April 10, 2013. Discussion period openuntil May 1, 2014; separate discussions must be submitted for individualpapers. This paper is part of the Journal of Geotechnical and Geoenvir-onmental Engineering, Vol. 139, No. 12, December 1, 2013. ©ASCE,ISSN 1090-0241/2013/12-2125–2135/$25.00.

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taken arbitrarily at the particular time t5 0. It can be recognized thatthe referential description is a special case of the relative descriptionas the reference time t is chosen at 0.When the FEmesh experienceslarge deformations, the remaining arbitrary reference mesh con-figuration at time t5 0 will cause an inaccurate description of thefuture mesh configurations; this limitation can be circumvented byregularly updating the reference mesh configuration after each de-formation increment.Also,with the application of theHill stress rate toconstitutive laws, the updated mesh FE analysis is capable of con-sidering the change in Cartesian stress components caused by thesignificant rotation of elements as the FE mesh deforms. Therefore,this analysis takes into account the effects of large deformation moresufficiently than the conventional FE analysis and provides an ade-quate means of estimating stability of the case histories in this study.Based on the large deformation analysis, three FEMs with reducedshear strength are presented subsequently to determine factorsof safetyagainst the basal heave of the excavations.

Convergence Criterion Method

The strength reduction procedure was performed in the convergencecriterion method based on the following assumptions: (1) a re-duction in soil strength parameters [cohesion (c) and friction angle(f)] does not impact other soil properties, e.g., Young’s modulusand Poisson’s ratio; and (2) both c and tanf are equally important inproviding shear strength for the soil. Therefore, the same strengthreduction ratio (SR) can be applied to both c and tanf. Denoting theoriginal and reduced strength parameters by the subscripts originaland input, respectively

cinput ¼coriginalSR

(1)

finput ¼ tan21�tanforiginal

SR

�(2)

Failure is definedwhen no convergence of numerical solutions in thecalculation occurs, and the corresponding SR value is treated as thefactor of safety of excavations. The standard setting of the conver-gence criteria inPLAXIS adopted in this study is that the global erroris lower than the tolerated error of 0.01 and the number of inaccurate

soil (interface) points is lower than 3 plus 10% of the number ofplastic soil (interface) points.

Intersection Method

Fig. 1(a) shows the typical nodal displacement versus SR curve. Atthe beginning, the nodal displacement increases slightly but thendevelops rapidly for large SR values. Such behaviors are denoted asthe lower and upper parts of the curve in Fig. 1(a), respectively. Asextended, these two parts will intersect each other at an intersectionpoint, marking the onset of rapid development of the nodal dis-placement with any further increase in the strength reduction ratio.The intersectionmethod, hence, treats the corresponding SRvalue asthe factor of safety of excavations.

Angle Method

When an excavation approaches the basal heave failure, a kick-outphenomenon of the excavation wall is often observed. The soil inthe back of the wall tends to move downward and forward to theexcavation zone, causing the embedded part of the wall to move andbend toward the excavation zone and the maximum horizontal dis-placement occurs at its toe, as shown in Fig. 1(b). The angle methodtends to quantify this phenomenon. If indicating the deformations ofthe wall above and below the final excavation depth by anglesa1 anda2, respectively

a1 ¼ tan21�

He

DHe2D0

�(3)

a2 ¼ tan21�

Hp

DHeþHp2DHe

�(4)

where He 5 excavation depth; Hp 5 penetration depth of thewall; and D0, DHe, and DHe1Hp 5 horizontal deformations of thewall at the ground surface, final excavation level, and wall toe,respectively.

As shown in Fig. 1(b), the kick-out phenomenon of wall is said tooccur whena1 .a2. The SR value corresponding to the onset of thisphenomenon is defined as the factor of safety of excavations.

Fig. 1. Analyses of FE result: (a) nodal displacement versus SR curve; (b) kick-out phenomenon

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Fig. 2. Plan of excavation and layout of internal struts for Taipei Rebar Broadway case (Pio-Go Hsieh, Chang-Yu Ou, Hui-Tzu Liu, “Basal heaveanalysis of excavations with consideration of anisotropic undrained strength of clay”, Canadian Geotechnical Journal, Vol. 45, No. 6, pp. 788–799,© 2008 Canadian Science Publishing or its licensors. Reproduced with permission)

Fig. 3.Subsoil profile, construction sequence, and undrained shear strength of soil for Taipei RebarBroadway case (Pio-GoHsieh, Chang-YuOu,Hui-Tzu Liu, “Basal heave analysis of excavations with consideration of anisotropic undrained strength of clay”, Canadian Geotechnical Journal, Vol. 45,No. 6, pp. 788–799, © 2008 Canadian Science Publishing or its licensors. Reproduced with permission)

Fig. 4. Finite-element model of Taipei Rebar Broadway case used for analysis

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Case Studies

Taipei Rebar Broadway Case

Ground Conditions and Construction SequenceThe Taipei Rebar Broadway case was a 100:08-3 25:8-m, in-ternally braced excavation with a maximum excavation depth of13.45 m, as shown in Fig. 2 (Hsieh et al. 2008). Fig. 3 plots the sitestratigraphy, comprising four alternating layers of clayey and sandysoils. The first layer is a 4.5-m-thick silty sand (SM) with a standardpenetration test (SPT) N value of 14 blows=m. Below this layer,4.2-m-thick silty clay (ML-CL) exists. The third layer is SM, rangingfrom ground surface level (GL) 28:7m to GL 210:7m, with aSPT2N value of 10 blows=m.The next layer is 34-m-thick clay (CL)underlain by hard gravel. It is this clay layer that has a predominant

effect on the excavation behavior in the case. The liquid limit andplastic index for this layer are in the range of 10–20 and 30–45,respectively. The groundwater table is 2.8 m below the groundsurface. Fig. 3 shows the undrained shear strength obtained fromtriaxial K0-consolidated undrained compression (CK0U-AC) tests,extension (CK0U-AE) tests, and direct simple shear (DSS) tests, inwhich soil samples were taken at 200 m far from the constructionsite.

A 70-cm-thick and 24-m-deep diaphragmwall was used to retainthe excavation. In addition, four levels of struts at GL21:0,2 3:55,26:85, and 210:15m were adopted to support the wall, spacingfrom 4.1 to 5.8 m along the longitudinal direction of the excavation,as shown in Fig. 2.

Fig. 3 shows the construction sequence for this case. When ex-cavation reached the final depth (GL 213:45m), the basal heavefailure occurred in the construction site, causing collapse of thebraced system and a 132-3 40-m subsided area.

FE AnalysisTomodel the variation of soil properties with depth, soil layers weredivided into many sublayers, each with a maximum thickness of3 m. The Mohr-Coulomb model was adopted to simulate stress-strain behavior of the soil. This is because the yield or failure be-havior of the soil is of interest in this study, and the Mohr-Coulombmodel is capable of providing an acceptable approximation of thefailure surface of soil (Yong and McKyes 1971). This soil modelrequiresfive input parameters: Young’smodulus (E), Poisson’s ratio(y), friction angle (f), cohesion (c), and dilatancy angle (c). Clayeysoils were modeled with undrained materials, at which yu 5 0:495and fu 5 0� The undrained shear strength (su) was taken as anaverage result of CK0U-AC, CK0U-AE, and DSS tests following thesuggestion by Ladd and Foott (1974) and multiplied by a correctionfactor (mt) for strain rate effect, which was summarized by Kulhawyand Mayne (1990) as follows:

mt ¼�12 0:1 log10

�tf =tlaboratory

��(5)

where tf 5 time to get failure of soil in the field; and tlaboratory 5 timeto get failure of soil in the laboratory.

Fig. 5. Wall displacement and soil heave at final excavation stage forTaipei Rebar Broadway case during strength reduction procedure

Fig. 6. Displacement versus SR for Taipei Rebar Broadway case

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It was assumed that tf was 2 weeks for all failure cases as a roughestimate, and tlaboratory was 5 h for the consolidated undrained testwith a strain rate of 1%=h, which was considered as the standardreference test. Hence, mt 5 0:817. The undrained Young’s modulus(Eu) was determined using the empirical ratioEu=su 5 450 for Taipeiclay (Aswin 2010). Sandy soils, on the other hand, were simulatedwith drained materials, in which the effective parameters were as-sumed to be y95 0:3, f95 30�, c95 0, and c95 0�. The effectiveYoung’s modulus (E9) was calculated based on the method sug-gested by Ou and Lai (1994) as follows:

E9 ¼ 2bPa

�Pa

P

�rV2

s ð1þ nÞ (6)

where r 5 density of the sand; Vs 5 shear wave velocity equal to93:11ðN11Þ0:33 (Wu 1990); N 5 standard penetration resistance;Pa 5 atmospheric pressure; P 5 mean effective stress; and b5 reduction factor equal to 0.5 for most excavation problems.

The plate element and anchor elementwere used for the diaphragmwalls and struts, respectively, in which the linear-elastic responsewas assumed. In the analysis, the stiffness (EI) of the diaphragm wallwas estimated from the compressive strength of the concrete ( fc9)

according to ACI 318-11 (American Concrete Institute 2011) andreduced by 20%, considering the possibility of bending moment–induced crack in thewall and the relatively lowquality of concreting inwater. The axial stiffness of the struts was reduced by 40% due to thefact that the struts in the field were installed by splicing H steelstogether, not easily lined up totally straight, and bending phenomenoncaused by heave of the central posts, which all reduced the axialstiffness of the struts (Ou 2006).

The interaction between the wall and soils was simulated withinterface elements whose behaviors follow the Mohr-Coulombmodel. The strength parameters (ci and fi) of interfaces were cal-culated from those of corresponding soils (csoil and fsoil) with theapplication of strength reduction factor (Rinter) as follows:

ci ¼ Rinter csoil (7)

tanfi ¼ Rinter tanfsoil (8)

where the Rinter values were roughly taken at 0.67 for both Taipeisilty clay and sand.

Fig. 4 shows the two-dimensional FE model used for analysis,assuming that the deformations of the soil and wall are under plainstrain conditions. Only half of the excavation was modeled becauseof its symmetric characteristics. The left boundary was the cen-terline of the excavation; the right boundary was located at a dis-tance of 70 m from the centerline beyond the excavation influencezone; and the bottom boundary was at GL244:7mwhere the hardgravel existed. The boundary conditions were imposed on themodel, with the restraint of horizontal displacements on the leftand right boundaries, and the restraint of both vertical and hori-zontal displacements on the bottom boundary. Soils, walls, andstruts related to the excavation were described by 15-node tri-angular elements, two-node spring elements, and plate elements,respectively. Surcharge caused by the presence of heavy constructionequipment was assumed to be 10 kPa.

Fig. 5 shows the wall displacement and soil heave at the finalexcavation stage (GL 213:45m) for different SR values. The wallexhibits firstly a deep inward movement for the SR values ranging

Fig. 7. Plan of excavation and layout of internal struts for Taipei Shi-Pai case (Pio-Go Hsieh, Chang-Yu Ou, Hui-Tzu Liu, “Basal heaveanalysis of excavations with consideration of anisotropic undrainedstrength of clay”, Canadian Geotechnical Journal, Vol. 45, No. 6, pp.788–799, © 2008 Canadian Science Publishing or its licensors.Reproduced with permission)

Fig. 8. Subsoil profile, construction sequence, and undrained shear strength of soil for Taipei Shi-Pai case (Pio-GoHsieh, Chang-YuOu, Hui-Tzu Liu,“Basal heave analysis of excavations with consideration of anisotropic undrained strength of clay”, Canadian Geotechnical Journal, Vol. 45, No. 6, pp.788–799, © 2008 Canadian Science Publishing or its licensors. Reproduced with permission)

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from 1.0 to 1.04 and then a kick-out movement for the SR valuesranging from 1.05 to 1.75, but the maximum horizontal displace-ment occurs at its toe for all the SR values. The maximum soil heaveat the excavation bottom, on the other hand, always appears at thecenter of the excavation.

The convergence of numerical solutions no longer remainswhen the strength reduction ratio is increased up to 1.76 so thisvalue is defined as the factor of safety by the convergence criterionmethod. The wall also exhibits a kick-out movement when thestrength reduction ratio is greater than or equal to 1.05. The anglemethod hence judges this SR value as the factor of safety for thecase.

The maximum wall displacement, the displacement of the walltoe, and the maximum soil heave are plotted against the strengthreduction ratio in Fig. 6. It is clear that all intersection points of the

plotted curves have the same corresponding SR value of 1.0. Whenthe strength reduction ratio is greater than this value, a rapid de-velopment of the displacements can be observed, as shown in Fig. 6.This value is therefore treated as the factor of safety by the in-tersection method.

To compare the effects of the reduction in su and the reduction inbothE and su on the displacements of nodes near excavations used inthe intersectionmethod and the angle method, an additional analysiswas performed. The results show that there is almost no difference inthe nodal displacements between the two reductionways. The reasonmay be that these displacements are mainly caused by the plasticresponse governed by su rather than the elastic response based on Eof the soil. Therefore, the reduction in su is still adopted in thestability analyses of the remaining cases.

Taipei Shi-Pai Case

Figs. 7 and 8 show the subsoil conditions, construction sequence,and bracing system for the Taipei Shi-Pai case (Hsieh et al. 2008).The main soil layer at this site is clay, ranging from GL 28:5 to2 41m, with a liquid limit and plastic index of 45 and 25, re-spectively. The basal heave failure occurred at the final excavationstage (GL 29:3 m), with the collapse of the bracing system andadjacent buildings.

The simulation of the soils, bracing system, and constructionsequence herein were the same as those presented in the TaipeiRebarBroadway case. Fig. 9 shows the deformations of the wall andsoil at the final excavation stage obtained from numerical analysisfor different SR values from 1.00 to 2.45. The factors of safetyjudged by the convergence criterion method (the maximum SRvalue) and the angle method (the SR value corresponding to theonset of the kick-out phenomenon of the wall) are 2.46 and 1.00,respectively.

Fig. 10 shows the relationships between the displacementscalculated from the analysis, the maximum wall displacement, thedisplacement of the wall toe, and the maximum soil heave, andthe strength reduction ratio. A pronouncing intersection point,corresponding to the SR value of 1.00, can be observed at themaximum wall displacement versus SR curve. Each of the tworemaining curves, on the other hand, exhibits two intersection points

Fig. 9. Wall displacement and soil heave at final excavation stage forTaipei Shi-Pai case during strength reduction procedure

Fig. 10. Displacement versus SR for Taipei Shi-Pai case

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at the SR values of 1.00 and 1.75. The factor of safety of 1.00 isjudged for this case by the intersection method. In general, when thenodal displacement versus SR curve exhibits more than one in-tersection point, it is recommended that the intersection point firstappearing on the curve when the SR value increases be treated as thefactor of safety for excavations because this point marks the onset ofthe rapid development of the nodal displacement.

Taipei Electricity Distribution Center Case

The Taipei Electricity Distribution Center case was an L-shapedexcavation near a hillside, which includes two 26-3 73-m rectan-gular sections supported by a sheet-pile wall (Wang and Su 1996).Fig. 11 shows the subsoil conditions, construction sequence, andbracing system for this case. The fourth clay layer has themain effecton excavation-induced deformations in this case. This layer rangesfrom GL29:4 to 2 30m and has an average water content of 40%,which is very close to its liquid limit. After the third strut level wasinstalled, excavation was carried out to the final depth (GL29:2m)at a corner of the construction site. The first damage of the strutsystem occurred at this corner and then caused an overall collapse ofthe construction site in a few minutes.

Analysis was performed in a way similar to that in the TaipeiRebar Broadway case. Fig. 12 plots the analysis results, walldeformation, and soil heave at the final excavation stage for thedifferent SR values from 1.00 to 1.99. The convergence criterionmethod and angle method provide factors of safety of 2.00 and1.26, respectively. As shown in Fig. 13, all the curves, relatingthe maximum wall displacement, displacement of the wall toe,and maximum soil heave to the strength reduction ratio, exhibitthe definite intersection points corresponding to the same SRvalue of 1.06, which can be treated as the factor of safety for thiscase.

Oslo Case

The Oslo case was approximately a 60-3 140-m rectangular ex-cavation, as shown in Fig. 14 (Aas 1984). The ground surface at thesite generally inclines from the north (Gunnerus Street) to the south,with a 1- to 1.5-m difference in elevation. Prior to excavation, the top1.5–2 m of fill was removed, and then two rows of sheet-pile walls

Fig. 11.Subsoil profile, construction sequence, and undrained shear strength of soil for Taipei ElectricityDistributionCenter case (adapted fromWangand Su 1996)

Fig. 12.Wall displacement and soil heave at final excavation stage forTaipei Electricity Distribution Center case during strength reductionprocedure

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were installed, as denoted by the bold lines in Fig. 14, dividing theconstruction site into the peripheral and central zones. The peripheralzone was first excavated. Excavation began at sections No. 4, 14, and19 and went further in two opposite directions, as indicated by thearrows in Fig. 14. Fig. 15 shows the subsoil conditions, constructionsequence, and bracing system at section No. 19 (shaded area inFig. 14),whichwas analyzed in this study.Mostof the soils are soft-to-medium clays except 2–3 m of top fill. The fifth layer is the maindeposit with a liquid limit and plastic index ranging from 40 to 50 and20 to 30, respectively.

The ratio Eu=su was assumed to be 200 for Oslo clay in thisanalysis (Bjerrum andEide 1956), and the friction angle of the topfillwas 35�. The Rinter value of interfaces between the sheet-pile walland soil was determined based on sheet-pile pulling tests by theNorwegian Geotechnical Institute (NGI) (1962) at the site near thisexcavation. It was reported that the average adhesion of Oslo clay to

the pile was 6 kN=m2 after 100 days from the time of pile driving,which was assumed to be equal to the period of excavation at sectionNo. 19 since the construction of the wall. As shown in Fig. 15, theaverage undrained shear strength of the soil with the consideration ofthe strain rate effect (the black diamond line) was approximately25 kN=m2 from GL 25 to 2 15 m. Therefore, Rinter was found at0.24 for interfaces between the sheet-pile wall and Oslo clay. In thecase of the top fill, Rinter was assumed to be 0.5.

Analysis was performed in a way similar to that in the TaipeiRebar Broadway case. Fig. 16 shows the calculated wall displace-ment and soil heave at the final excavation stage when the strengthreduction ratio increases from 1.0 to 1.16. A significant asymmetrycan be observed in the shape of the deformations due to the unevenground surface and surcharge by Gunnerus Street, i.e., the wall nextto Gunnerus Street, namely the outer wall, exhibits a deep inwardmovement, while the inner wall shows a kick-outmovement, and thesoil heave reaches themaximummagnitude that is closer to the innerwall than the outer one. For this case, the factor of safety judged bythe convergence criterion method is 1.17. The angle method, on theother hand, provides inconsistent estimates, which are 1.16 and 1.00for the outer and inner walls, respectively. As shown in Fig. 17, thedeformations of the soil and walls, i.e., the maximum displacementof the outer wall, the maximum displacement of the inner wall, thedisplacement of the outer wall toe, the displacement of the inner walltoe, and the maximum soil heave, obtained from the numericalanalysis exhibit a remarkable increase in magnitude when thestrength reduction ratio is greater than 1.01, which is treated as thefactor of safety for the case.

Based on the results of the previous stability analyses, it can beobserved that among the three numerical methods, the intersectionmethod gives the most appropriate estimate of factors of safetyagainst basal heave. Further validation of the intersection methodwill be performed in the following section.

Validation of FE Intersection Method

For equilibriummethods of stability analysis, the factor of safety canbe defined as the ratio of the soil strength to the shear stress requiredfor equilibrium (Duncan and Wright 1980). Therefore, the factor ofsafety should increase linearlywith the shear strength of the soil. The

Fig. 13. Displacement versus SR for Taipei Electricity Distribution Center case

Fig. 14. Site plan showing system of sheet-pile walls, excavationsections, and instrumentation installations for Oslo case (Aas 1984, withpermission from the International Conference on Case Histories inGeotechnical Engineering)

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Fig. 15. Subsoil profile, construction sequence, and undrained shear strength of soil for Oslo case

Fig. 16. Wall displacement and soil heave at final excavation stage for Oslo case during strength reduction procedure

Fig. 17. Displacement versus SR for Oslo case

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intersection method will be reliable if its results also exhibit suchbehavior.

A hypothetical excavation used for validation had the same di-mensions, construction sequence, and retaining system as the TaipeiRebar Broadway case, but the subsoil was homogeneous clay, asshown in Fig. 18. The numerical analyses were performed fordifferent normalized shear strength values (su=sv9) of 0.25, 0.35,0.50, 0.60, and 0.75.

Fig. 18 plots the FS calculated by the intersection method againstthe normalized soil shear strength. It is observed that the FS valuescalculated by the method increase linearly with the su=sv9 ratio. Sucha linearly increasing trend matches the basic definition of the factorof safety. The FEMwith the intersectionmethod is further confirmedto result in a reliable estimate of the factor of safety against basalheave.

Discussion

Table 1 summarizes the results of stability analyses using the FEMsand the conventional methods (Terzaghi’s method, Bjerrum andEide’s method, and the slip circle method) for four case histories.

The convergence criterionmethod provides the highest FS valuesamong the FEMs. For the Taipei Rebar Broadway case, the TaipeiShi-Pai case, and the Taipei Electricity Distribution Center case, the

results of this method are much greater than unity despite the factthat they are failure cases. However, for the Oslo case, this methodgives a FS value of 1.17, which is rather in good agreement with thefield observation.

The angle method cannot provide a consistent FS value for theOslo case. The reason is that the outer and inner walls do not exhibitthe kick-out phenomenon simultaneously because of asymmetriccharacteristics of the excavation, as shown in Fig. 16. The FS valuescalculated by this method are 1.05 and 1.26 for the Taipei RebarBroadway case and the Taipei Electricity Distribution Center case,respectively, and therefore overestimate the basal heave stability ofthese two cases. For the Taipei Shi-Pai case, the FS value is 1.00 andagrees well with the field observation.

The FS values calculated by the intersection method for all casehistories ranging from 1.00 to 1.06 are generally close to unity. Itsestimates are consistent with the field observations because threeTaipei cases once experienced the basal heave failure, and the Oslocasewas reported to have excessivewall deformations. Estimating thefactor of safety against basal heave based on the rapid development ofthe deformations of the soil and wall is hence a rational approach.

As summarized in Table 1, the FS values calculated by threeconventional methods give conservative estimates of stability of theexcavations because the values are generally less than or equal tounity. Terzaghi’s method, Bjerrum and Eide’s method, and the slipcircle method provide the highest, medium, and lowest FS values,respectively.

Conclusion

The following conclusions can be drawn on the basis of the workpresented herein:1. The convergence criterion method provides the highest FS

values among the FEMs and generally overestimates basalheave stability of the case histories. The reasonmay be that thedivergence of the numerical solutions often occurs with thelarge deformations of the soil and wall and a number of plasticpoints in the soil; hence, it does not indicate the start but the fulldevelopment of the excavation failure.

2. The angle method does not give a consistent FS value for theOslo case because in this case, the inner wall exhibits a kick-out phenomenon but the outerwall does not during the strengthreduction procedure. Also, this method slightly overestimatesthe basal heave stability of the Taipei Rebar Broadway caseand the Taipei Electricity Distribution Center case but predictsthe basal heave failure for the Taipei Shi-Pai case well.

3. Results from the intersection method are generally in goodagreement with the field observations. The method has alsobeen verified. It is therefore the most appropriate FEM forestimating basal heave stability of deep excavations in softclay.

Fig. 18. Factor of safety obtained by intersection method versusnormalized undrained shear strength of soil for parametric study

Table 1. Calculated Factors of Safety against Basal Heave for Case Histories

FEM Conventional method

Type offailureCase history

Convergencecriterion method

Intersectionmethod

Anglemethod

Terzaghi’smethod

Bjerrum andEide’s method

Slip circlemethod

Taipei Rebar Broadway case 1.76 1.00 1.05 1.09 0.95 0.81 Total failureTaipei Shi-Pai case 2.46 1.00 1.00 0.90 0.79 0.73 Total failureTaipei ElectricityDistribution Center case

2.00 1.06 1.26 1.17 1.04 0.83 Total failure

Oslo case 1.17 1.01 1.16a and 1.00b 0.86 0.83 0.59 Near failureaFactor of safety for the outer wall.bFactor of safety for the inner wall.

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