Engineering Geology 204 (2016) 94–107

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Engineering Geology

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Drained residual shear strength at effective normal stresses relevant tosoil slope stability analyses

Hisham T. Eid a,⁎, Khaled H. Rabie a, Dharma Wijewickreme b

a Qatar University, Doha, P.O. Box 2713, Qatarb University of British Columbia, Vancouver, B.C. V6T 1Z4, Canada

⁎ Corresponding author.E-mail addresses: heid@qu.edu.qa (H.T. Eid), Khaled.ra

dharmaw@civil.ubc.ca (D. Wijewickreme).

http://dx.doi.org/10.1016/j.enggeo.2016.02.0030013-7952/© 2016 Elsevier B.V. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 6 May 2015Received in revised form 12 October 2015Accepted 4 February 2016Available online 6 February 2016

An extensive torsional ring shear testing program has been conducted to measure the drained residual shearstrength of soils at the wide range of effective normal stress (10 to 700 kPa) usually mobilized in reactivatedand first-time landslides. Soils, mudstones and shales of different plasticity and gradation were tested in theprogram. The effects of the change in nonlinearity of shear strength envelope over the utilized normal stressranges on slope stability analyses were investigated. Using this data, new empirical residual shear strengthcorrelations were developed as a function of soil index parameters and wide range of effective normal stresses.In essence, the correlations are presented as revised versions of those previously developed for a limited numberof normal stresses utilizing the same soil index parameters. Comparisonsweremadewith a considerable amountof back-calculated shear strength data reported in the literature for reactivated landslides as well as resultspredicted from existing shear strength correlations to verify the increased suitability of the new correlationsfor use in slope stability analyses. A numerical expression was also introduced to express the residual shearstrength correlations for direct incorporation in slope stability software.

© 2016 Elsevier B.V. All rights reserved.

Keywords:ClaysEmpirical correlationFailure envelopeLandslidesNonlinearityResidual strengthShaleShear strengthSlope stabilitySoftware

1. Introduction

Shear strength (τ) is a key input in any stability analysis of soilslopes. The drained residual shear strength (τr) is a crucial parameterin evaluating the stability of slopes that contain a pre-existing shearsurface (Skempton, 1964, 1985). It can be also used along with thefully softened shear strength in determining the factor of safety againstfirst-time sliding in stiff plastic clay slopes (James, 1970; Bishop, 1971;Potts et al., 1997; Stark and Eid, 1997; Mesri and Shahien, 2003).Significant efforts have been reported in the literature for assessingthe residual shear strength through laboratory testing and back analysisof failed case histories. Several empirical correlations have been alsopresented to estimate such strength as a function of soil index parame-ters. Most of these correlations have been summarized in a subsequentsection.

It has been long recognized that the shear strength envelopes ofplastic soils are nonlinear, especially at a low effective normal stress(σ′n b 50 kPa) range (Terzaghi and Peck, 1948; Penman, 1953; Bishopet al., 1965, 1971 and Ponce and Bell, 1971; Charles and Soares, 1984;Atkinson and Farra, 1985; Skempton, 1985; Day and Axten, 1989;

bie@qu.edu.qa (K.H. Rabie),

Maksimovic, 1989). Such low normal stresses are usually relevant inslope stability analyses at locations where the critical slip surface inter-sects the face of the slope or passes through shallow depths or zoneswith high enough pore-water pressures to reduce effective stresses. Inspite of this, parameters derived from laboratory shear tests that havebeen carried out at higher effective normal stresses, at which thecurvature of the shear strength envelope significantly decreases, arecommonly used to represent all zones in slope stability analyses. Evenmost of the existing residual shear strength correlations that incorpo-rate the effect of the normal stress level (e.g., Stark and Eid, 1994;Mesri and Shahien, 2003; Stark and Hussain, 2013) have also beendeveloped based on testing at a limited number of relatively high effec-tive normal stresses. For example, the currently available correlationsdo not efficiently cover certain low and moderate ranges of the averageeffective normal stress (i.e., σ′n b 50 kPa and 100 kPa b σ′n b 400 kPa)that have been mobilized in several reported reactivated and long-term first-time landslides in stiff plastic clays. Fig. 1 illustrates thislimitation for well-documented landslides through English clays(namely; Upper Lias clay, London clay, Oxford clay, Kimmeridge clay,Chalky Boulder clay, Gault clay, Atherfield clay, Etruvia marl, Walton'swood and Jackfield carboniferous mudstone, and Edale shale).

To fill the knowledge gap described above, this paper mainly pre-sents results from a laboratory research program involving residualshear strength tests conducted at effective normal stresses of 10, 25,

Fig. 1. Average normal stresses reported for landslides in English soil formations and thenormal stress ranges not covered by testing utilized to develop the commonly-usedresidual strength correlations that incorporate the effect of normal stress (data pointsfrom Skempton, 1964, 1972, 1977, 1985; James, 1970; Chandler, 1974, 1976, 1977,1982, 1984; Chandler and Skempton, 1974; Bromhead, 1978).

Table 1Soil, mudstone, and shale samples used in ring shear tests.

No. Soil name location LL(%)

PL(%)

CF a

(%) A

1 Fraser-river siltb Vancouver, BC, Canada 21 18 9 0.332 Glacial tillb,d Urbana, IL, USA 24 16 18 0.443 Loessb,d Vicksburg, MS, USA 28 18 10 1.004 Gray siltb Vancouver, BC, Canada 34 17 26 0.655 Bootlegger Cove clayb,d Anchorage, AK, USA 35 18 44 0.396 Duck Creek shalec,d Fulton, IL, USA 37 25 19 0.637 Slide debrisb San Francisco, CA, USA 37 26 28 0.398 Chinle (red) shalec,d Holbrook, AZ, USA 39 20 43 0.449 Slopewash materialb San Luis Dam, CA, USA 42 24 34 0.5310 Colorado shalec,d Montana, MT, USA 46 25 73 0.2911 Panoche mudstoned San Francisco, CA, USA 47 27 41 0.4012 Kaolinite clayb,e Hephzibah, GA, USA 48 26 32 0.6913 Four Fathom shalec,d Durham, England 50 24 33 0.7914 Mancos shaled Price, UT, USA 52 20 63 0.5115 Panoche shaled San Francisco, CA, USA 53 29 50 0.4816 Gulf-bed depositb Doha, Qatar 53 34 18 1.0617 Red Sea white shale Alsokhna, Egypt 55 22 50 0.6618 Comanche shalec,d Proctor Dam, TX, USA 62 32 68 0.4419 Breccia materialb Manta, Ecuador 64 41 25 0.9220 Bearpaw shalec,d Billings, MT, USA 68 24 51 0.8621 Slide debrisd San Francisco, CA, USA 69 22 56 0.8422 Bay Mudb,d San Francisco, CA, USA 76 41 16 2.1923 Patapsco shalec,d Washington D.C., USA 77 25 59 0.8824 Nile depositb Damanhur, Egypt 82 27 58 0.9525 Pierre shalec,d Limon, CO. USA 82 30 42 1.2426 Red Sea gray shale Alsokhna, Egypt 84 27 44 1.3027 Upper Pepper shale Waco, TX, USA 89 29 72 0.8328 Santiago claystoned San Diego, CA, USA 89 44 57 0.7929 Toshka shale Toshka, Egypt 91 30 58 1.0530 Lower Pepper shaled Waco Dam, TX, USA 94 26 77 0.8831 Altamira Bentonitic tuffd Portuguese Bend, CA, USA 98 37 68 0.9032 Brown London clayd Bradwell, England 101 35 66 1.0033 Mokattam yellow shale Cairo, Egypt 103 33 43 1.6334 Cucaracha shalec,d Panama Canal 111 42 63 1.1035 Otay Bentonitic shaled San Diego, CA, USA 112 53 73 0.8136 Denver shalec,d Denver, CO, USA 121 37 67 1.2537 Bearpaw shalec,d Saskatchewan, Canada 128 27 43 2.3538 Mokattam gray shale Cairo, Egypt 134 37 79 1.2339 Pierre shale New Castle, WY, USA 137 32 67 1.5740 Oahe firm shaled Oahe Dam, SD, USA 138 41 78 1.2441 Claggett shalec,d Benton, MT, USA 157 31 71 1.7842 Bentonitic claystone Fayoum, Egypt 164 55 79 1.3843 Taylor shalec,d San Antonio, TX, USA 170 39 72 1.8244 Pierre shalec,d Reliance, SD, USA 184 55 84 1.5445 Oahe bentonitic shaled Oahe Dam, SD, USA 192 47 65 2.2346 Panoche clay gouged San Francisco, CA, USA 219 56 72 2.2647 Midra gray shale Sealine, Qatar 231 74 79 1.9948 Lea Park bentonitic shaled Saskatchewan, Canada 253 48 65 3.1549 Bearpaw shalec,d Fort Peck Dam, MT, USA 288 44 88 2.7750 Bentonitic clayb,e El Hammam, Egypt 293 46 89 2.78

a Quantity of particles b0.002 mm.b Samples not ball-milled.c Index properties from Mesri and Cepeda-Diaz (1986).d Samples reprocessed from those utilized in Eid (1996).e Received in the form of powder passed through sieve # 200.

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200, and 300 kPa on clays, silts, mudstones, and shales with plasticityand gradation varying over a wide range as presented in a table in thesubsequent section. The residual shear strengths of these materials ateffective normal stress of 50, 100, 400, and 700 are also presented.This is intended to: (i) complement the data set developed throughconducting a series of torsional ring shear tests by Eid (1996); and (ii)to consequently revise and update the drained shear strength correla-tions that have been developed entirely (e.g., Stark and Eid, 1994;Mesri and Shahien, 2003) or chiefly (e.g., Stark et al., 2005; Stark andHussain, 2013) based on such a data set. The testing program is limitedto sedimentary fine-grainedmaterials (i.e., materials withmostly plate-like clay particles); as such, the revised correlations presented hereincan be used to predict the drained residual friction angles of suchmaterials. The correlations exclude soils such as carbonate soils(White et al., 2012) and soils composed chiefly of allophane orhalloysite (i.e., soils with non-platy particles such as volcanic ashes)that are unlikely to have particle rearrangement towards somepreferred orientation (Wesley, 1977, 1992, 2003). They also excludemarine soils that contain numerous skeletal remains or foraminifera(Mesri et al., 1975; Najjar et al., 2007). Most of these soils exhibit highfriction angle and small or no difference between the drained peakshear strength and the drained residual shear strength regardless oftheir plasticity (Saldivar and Jardine, 2005).

2. Testing method

A total of 50 clay, silt, mudstone, and shale sampleswere used in tor-sional ring shear testing to measure the drained residual shear strengthat 8 different effective normal stresses (Table 1). To avoid unnecessarytesting repetition, no shear strength tests were conducted on soilswhich were tested by Stark and Eid (1994) at effective normal stressof 50, 100, 400, and 700 kPa using the same testing procedure followedin the present study. As shown in Table 1, the utilized test samples covera wide range of liquid limit (LL), plastic limit (PL), plasticity index (PI),and clay-size fraction (CF). Except for the heavily overconsolidatedclay, mudstone, and shale samples, Atterberg limits and clay-size frac-tions were determined in accordance with the particle disaggregationstandard procedure. Most heavily overconsolidated clays, mudstones,and shales possess diagenetic bonding that results in particle aggrega-tion (induration). This aggregationusually survives the standard samplepreparation procedure and consequently influences the measuredindex parameters and the accuracy of their correlations to the resultsof shear strength testing in which the aggregated particles wouldbe battered (La Gatta, 1970; Townsend and Banks, 1974; Airo'Farullaand La Rosa, 1977; Eid, 2001, 2006). As a result, these materials weredisaggregated by ball-milling of representative air-dried samples until

all particles passed the standard sieve No. 200. The hydrated ball-milled materials were used in determining the index parameters aswell as the shear strengths. This sample preparation procedure wasadopted from that used byMesri and Cepeda-Diaz (1986) to determineliquid limit and clay-size fraction that better infer the clay mineralogyand gradation of shales.

Remolded specimens for ring shear testingwere obtained by addingdistilled water to the air-dried and processed samples until a liquidityindex of 1.5 was obtained. The sample was then allowed to rehydratefor at least one week in a moist room. The ring shear specimen is annu-lar with inside and outside diameters of 7 cm and 10 cm, respectively.Drainage is provided by two bronze porous stones secured on the load-ing platen and the bottom of the specimen container. The specimen isconfined radially by the specimen container which is 5 mm deep. The

Fig. 2. Results of drained ring-shear tests on Toshka shale: (a) multi-stage test formeasuring the residual strength; (b) relationships of the residual stress ratio andresidual strength with the effective normal stress.

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modified Bromhead ring shear apparatus described in Stark and Eid(1993)was utilized formeasuring the residual shear strength. Themod-ified apparatus allows for eliminating the effect of wall friction duringshear which is especially crucial for accurate measurement of lowshear stresses developed when testing soils at effective normal stressesas low as 10 kPa. To insure a drained condition and avoid possible rateeffect, a displacement rate of 0.018 mm/min was used in all tests. Thisrate was estimated using the procedure described by Gibson andHenkel (1954) considering a degree of pore-water pressure dissipationof 0.99 for a normally-consolidated remolded specimen of the mostplastic shale tested in the current investigation.

For each soil, the residual shear strength was determined using aring shear specimen overconsolidated and presheared at 700 kPa. Themultistage shearing process as described in Stark and Eid (1993,1994) and ASTM D6467 (1999c) was followed for shear testing at nor-mal stresses of 10, 25, 50, 100, 200, 300, 400, and 700 kPa. Multistageshear tests were conducted at normal stresses of 10, 25, 200, and 300only for materials tested in Stark and Eid (1994). The overconsolidatedspecimens were used in this ring shear testing program because theyhelp in reducing settlement of the loading platen into the specimencontainer and thus wall friction that may occur during shear and leadto overestimating the shear strength (Stark and Eid, 1993). The use ofoverconsolidated specimens reduces such a settlement by: (1)minimiz-ing soil compressibility and extrusion during reloading and shearing,respectively, and (2) decreasing the shear displacement required toreach a residual condition through concentrating the shear stressesand shear strains on the failure plane by the overconsolidated specimenand the rigid top porous stone, which aids the orientation of clay parti-cles parallel to the direction of shear (Kanji and Wolle, 1977).

The typical stress ratio-displacement relationships yielded from theresidual shear strength testing are presented in Fig. 2a. The stress ratio isdefined as the shear stress divided by the effective normal stress. Thedetermined residual shear strength values as well as the residual stressratios (τr/σ′n) are plotted in Fig. 2b against the corresponding effectivenormal stresses. It can be seen that the residual stress ratios are higherat low effective normal stress levels. This clearly indicates the curvatureof the residual shear strength envelope. Such curvature or nonlinearitysignificantly decreases at effective normal stresses higher than200 kPa. This also shows the importance of having shear strength dataat the low and intermediate to high effective normal stresses consideredin this study to efficiently describe the shape of the drained residualshear strength failure envelopes. Details about the degrees of nonlinear-ity of these envelopes are presented in the subsequent section.

3. Evaluation of test results

Based on the residual testing results yielded from this study, the test-ed soils are sorted into three groups based on their clay-size fraction:less than or equal to 20%, between 25% and 45%, and greater than orequal to 50%. Similar grouping was first introduced by Lupini et al.(1981) to categorize soils that exhibit rolling, translational, and slidingshearing behavior, respectively. This grouping was also adopted bySkempton (1985) for developing his correlation between the residualshear strength friction angle and clay-size fraction. Grouping based onthe clay-size fraction (as an indication of particle size) is effective indeveloping low-scattering correlations between the liquid limit (as anindication of clay mineralogy) and the residual friction angle (Starkand Eid, 1994). For soils with clay-size fractions between 20% and 25%,and 45% and 50%, the friction angle can be estimated by interpolationbetween the clay-size fraction groups.

3.1. Residual shear strength envelope

The ring shear test results were utilized to study the change in non-linearity of the shear strength envelope over the effective normal stressranges occasionally mobilized in slope failures. Fig. 3 was developed to

present the failure envelope degree of nonlinearity over two effectivenormal stress ranges of approximately equal spans. Thefirst range is be-tween 10 and 200 kPa while the second is between 200 and 400 kPa.Friction angle ratios were used to represent the degree of nonlinearityin Fig. 3. Secant residual friction angles at effective normal stresses of10, 200, and 400 kPa [i.e., (ø′r)10, (ø′r)200, and (ø′r)400, respectively]were used to calculate these ratios. The secant residual friction angle isdefined as the angle between the effective normal stress axis and theline formed from the origin to the residual shear strength at a particulareffective normal stress. It can be seen from Fig. 3 that the curvature ofthe residual shear strength failure envelopes, as represented by thefriction angle ratios, is generallymore pronounced for soils withmoder-ate to high plasticity and CF ≥ 50%. Such curvature is significant ateffective normal stresses less than 200 kPa. At the higher effectivenormal stresses considered in this study, the friction ratios are lowand consequently the failure envelope can be assumed as a straightline without significant error (Figs. 2b and 3). A similar conclusionabout the general shape of the residual shear strength envelopeswas made by several researchers (e.g., Hawkins and Privett, 1985;Maksimovic, 1989, 1996 and Tiwari and Marui, 2005). It is also inagreement with residual shear strength testing results presented inthe literature (e.g., La Gatta, 1970; Bishop et al., 1971; Chattopadhyay,1972). Having more curvature at low effective normal stress ranges

Fig. 3. Change in the reduction of secant residual friction angle over different ranges ofeffective normal stress for soils with: (a) CF ≤ 20% and 25% ≤ CF ≤ 45%; (b) CF ≥ 50%.

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can be attributed to orienting most of the clay particles in the directionof shear at such range of stresses. The described general shape of the re-sidual shear strength envelope was considered in presenting thecorresponding correlations introduced in the next subsection.

3.2. Empirical correlations for estimating drained residual shear strength

Several empirical correlations of soil shear strengths with index pa-rameters are available in the literature. The correlations are particularlyuseful for preliminary designs and when soil samples and fundingresources are not readily available for advanced soil testing. Based onthe shape of the residual shear strength failure envelope describedherein and in the literature, having the shear strength at effective nor-mal stresses of 10, 25, 50, and 100 kPa are needed to more accuratelydevelop the potential significantly-curved initial part of the envelope.The approximately straight-line part of the same envelope that startsat effective normal stress of about 200 kPa can be developed by havingthe shear strength data at least at a single higher effective normal stressof those occasionally mobilized in slope failures. Consequently, shearstrengths measured at effective normal stresses of 10, 25, 50, 100, 200,and 400 kPa were utilized in developing the correlations presentedherein. Clearly, the inclusion of data from tests conducted at differenteffective normal stresses between 10 kPa and 50 kPa aswell as between100 kPa and 400 kPa provides an opportunity to improve the correla-tions developed by Stark and Eid (1994),; Mesri and Shahien (2003);Stark et al. (2005), and Stark and Hussain (2013) for estimating the re-sidual friction angles based on shear testing at effective normal stresseslimited to 50, 100, 400, and 700 kPa. For example, drawing a smoothcurve by connecting the data from these four effective normal stresslevels to the origin of τr versusσ′n plot to develop the full shear strengthfailure envelope would lead to underestimating the shear strengths ateffective normal stresses lower than 50 kPa as well as the shearstrengths at effective normal stresses between 100 and 400 kPa.

Fig. 4 shows the residual shear strength relationships developed inthis study as a function of the liquid limit, clay-size fraction, and effec-tive normal stress. Considering these three parameters combined withthe utilization of ball-milled derived index parameters of induratedmaterials seems to have resulted in good relationships with less scatter.It can be seen that the secant residual friction angle of soil decreasesconsiderably and almost linearly up to a liquid limit of 80%, higherthan which the slope of the relation with liquid limit significantlyflattens, especially at high effective normal stresses. Consequently,small errors in measuring the liquid limit of soils with low to mediumplasticity may lead to significant errors in estimated residual shearstrengths. The relationships of Fig. 4 also show that the influence ofthe clay-size fraction, as an indication of particle size and shearingmode, on themeasured residual friction angles decreases with reducingeffective normal stress level. A similar conclusionwasmade by Eid et al.(2015) for residual friction angles of soils at effective normal stresses aslow as 3 kPa. The reduction in the influence of clay-size fraction is clear-er for residual friction angles of soils with clay-size fraction equal to orhigher than 25% as indicated by the nearness of the clay-fraction trend lines for σ′n ≤ 25 kPa (Fig. 4). It should be notedthat the distinguished effect of clay-size fraction on developingdifferent residual shear strength modes, and in turn on the mea-sured friction angles, was presented by several researchers forrelatively higher effective normal stresses [e.g., Lupini et al.(1981) for effective normal stress of 350 kPa and Skempton(1985) for 100 kPa].

To reinterpret slope failures that were presented in the literaturewithout including information on clay-size fraction, Mesri andShahien (2003) re-plotted the residual shear strength data fromStark and Eid (1994) and Eid (1996) against the plasticity index; inessence, plasticity index was considered a parameter that would en-capsulate the effect of both liquid limit and clay-size fraction. A clearrelationship between soil plasticity index and clay-size fraction hasalso been shown by Kenney (1959) and Nelson (1992). Using a sim-ilar line of thinking, the residual shear strength data from this studyfor six different effective normal stresses were also plotted againstthe plasticity index as shown in Fig. 5. Decreasing trends of the se-cant residual friction angle with increasing of plasticity index canbe clearly seen from these results.

The comparison between the data scattering in Figs. 4 and 5 showsthe importance of incorporating the clay-size fraction into the residualshear strength relationships. Similar conclusion was made by Kaya(2010) through a comprehensive sensitivity analysis based on adatabase of residual shear strength. The relationships shown in Fig. 5can still be useful in case of analyzing shallow slopes or mobilizinglow effective normal stresses at which the percentage of clay particleshas a small effect on the residual friction angle. Fig. 5 also helps inanalyzing slopes with difficulties in determining the clay-size fractionas measured by the sedimentation method due to interaction andflocculation between soil particles (Bishop et al., 1971; Mesri andCepeda-Diaz, 1986; Wesley, 2003).

As explained previously, the liquid limit and clay-size fraction ofheavily overconsolidated clays, mudstones, and shales used in thecurrent testing programwere determined using ball-milled samples.The commercial laboratories usually utilize the standard samplepreparation procedures described in ASTM D422 (1999a) andASTM D4318 (1999b) or BS 1377 (1990) for determining Atterberglimits and clay-size fraction. Using these standard-derived indicesmay lead to overestimating the friction angles of heavilyoverconsolidated clays, mudstones, and shales (Eid, 2006). Conse-quently, the standard-derived index parameters should be adjustedbefore estimating the residual friction angles of these indurated ma-terials using the empirical correlations presented in the currentstudy.

Eid (2001, 2006)made a comprehensive study on the effect of parti-cle disaggregation, sample size, and rehydration time on the measured

Fig. 4. Secant residual friction angle relationships with liquid limit and clay-size fraction.

98 H.T. Eid et al. / Engineering Geology 204 (2016) 94–107

classification indices of shales. Liquid limits, plastic limits, and clay-sizefractions derived from the standard sample preparation procedure(LLST, PLST, and CFST) and those resulted from using ball-milled samples(LLBM, PLBM, and CFBM) of the same shales were determined. Datapresented in these studies are plotted to have a relationship betweenLLST and LLBM as shown in Fig. 6a. A similar relationship is presented inFig. 6b for the plasticity indices (i.e., PIST and PIBM). Based on the samedatabase, a relationship between clay-size fraction values derivedfrom the standard sample preparation procedure in terms of the activi-ty, AST and those derived using ball-milled samples is shown in Fig. 6c.The activity is defined as the plasticity index divided by the clay-sizefraction. Results of a similar sample preparation study presented byStark et al. (2005) were also superimposed on Fig. 6a. Because of thelack of availability of the plastic limits, the results of such study werenot represented in parts “b” and “c” of Fig. 6. Fig. 6 clearly shows theeffect of sample preparation procedure on the measured liquid limit,plasticity index and clay-size fraction. For all practical purposes, thenonlinear relationships between the index parameters derived from

the standard and ball-milling sample preparation techniques shown inFig. 6 can be expressed using the following simple equations

LLBM ¼ 1:4LL ST ð1Þ

PIBM ¼ 1:7PIST ð2Þ

CFBM ¼ CFST þ 30A 2ST for AST b 1:0 ð3aÞ

CFBM ¼ CFST þ 30

A 2ST

for AST N 1:0: ð3bÞ

For Red sea gray shale with LLST = 57%, PIST = 32%, CFST = 25%, andAST = 1.28, Eid (2006) has reported values of LLBM = 84%, PIBM = 57%,CFBM= 44%. These ball-milled derived liquid limit, plasticity index, andclay-size fraction can be estimated using Eqs. (1), (2), and (3b) as 80%,54%, and 43%, respectively. It should be noted that data scatteringaround the relationships shown in Fig. 6 can be attributed to the

Fig. 5. Secant residual friction angle relationships with plasticity index.

Fig. 6. Relationships between index parameters of shales, mudstones, andheavily overconsolidated clays disaggregated using the standard and ball-milling procedures for: (a) liquid limit; (b) plasticity index; (c) clay-sizefraction.

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variation in the degree of induration of the tested materials. The in-crease in such scattering for materials with significantly high LLBM andPIBM ranges would not have a considerable effect on predicting the

residual friction angle using the correlations of Figs. 4 and 5 due to theflatness of their trend lines in these ranges.

To facilitate the process of estimating the secant residual frictionangle as a function of liquid limit, clay-size fraction and effective normal

Fig. 7. Drained secant residual friction angle correlation as a function of: (a) liquid limit,clay-size fraction and effective normal stress; (b) plasticity index and effective normalstress.

100 H.T. Eid et al. / Engineering Geology 204 (2016) 94–107

stress based on the test results presented herein, the trend lines ofFig. 4 were re-plotted as shown in Fig. 7a. The equations needed toadjust the index parameters for estimating the friction angles of in-durated materials are also encompassed in the same figure. Thetest data points were not shown in Fig. 7a for clear presentationand better comparison between trend lines of different clay-sizefraction groups and effective normal stress levels. The configurationof Fig. 7a also helps to make the interpolation needed to estimate thedrained friction angle in case of having clay-size fraction or effectivenormal stress different from those shown in the figure. The presenta-tion technique used in Fig. 7a was used to develop residual shearstrength correlation as a function of the plasticity index and effectivenormal stress (Fig. 7b).

The residual shear strength measured in the current study atrelatively low effective normal stresses (i.e., 10 and 25 kPa) andcorrelated to soil index parameters can be utilized in differentgeotechnical engineering applications rather than the slopestability analyses. Such applications include stability assess-ment of earth reinforcement, anchor rods, near-shore pipelinesplaced on seabed, and driven piles supporting shallow-waterplatforms.

4. Suitability of the correlations

The suitability of using the correlations presented in this paper to es-timate soil residual shear strength at effective normal stresses relevantto slope stability analyses is illustrated in this section. This was donethrough checking the back-calculated residual friction angles reportedfor different case histories of reactivated landslides [ø′r]BC againstthose predicted, [ø′r]PR, using these correlations and other residualshear strength correlations available in the literature. Fifty fourlandslides through sedimentary fine-grained materials (i.e., clays,marls, mudstones, and shales) were used for such checking. Thesecase histories (Table 2) were chosen from the large number ofreactivated landslides reported in the literature considering theneed to have the following good reported data/information re-quired for a worthy comparison: (1) a well-defined slip surfacepassing mostly through one material of a type similar to that ofthe materials tested in the current study (i.e., soils with mostlyplate-like clay particles or minerals); (2) measured pore-waterpressures; (3) plasticity limits and clay-size fraction (defined asthe quantity of particles smaller than 0.002 mm); (4) geometricconfigurations and unit weights that would allow accurate calcula-tion of the average effective normal stress acting on the slip sur-face; and (5) drained residual friction angle back-calculated usinga zero cohesion intercept in the effective stress space. For each con-sidered case history, the reported data needed for the comparisonare listed in Table 2.

To predict the residual friction angle for a case history usingeach of the correlations of Fig. 7, a residual shear strength failureenvelope of the concerned material was first developed bydrawing a smooth curve connecting the shear strengths esti-mated at different effective normal stresses utilizing the report-ed liquid limit and clay-size fraction, or plasticity index (notethat the index parameters were adjusted in case of slidingthrough indurated materials). The predicted residual frictionangle was then determined as the inclination angle of a secantof the developed envelope at the reported average effectivenormal stress acting on the slip surface. Comparisonbetween the back-calculated residual friction angles and thosepredicted using the correlations presented in this study isshown in Fig. 8.

To evaluate the performance of the residual shear strength cor-relations presented herein compared with those of similar correla-tions available in the literature, the back-calculated friction anglesreported for the case histories listed in Table 2 were also checkedagainst the friction angles predicted using eighteen of the residualshear strength correlations existing in the literature (Table 3). Theback-calculated friction angles and the corresponding angles pre-dicted using a representative set of these correlations as well asthe correlation of Fig. 7a are shown in Fig. 9 for graphicalcomparison.

In an attempt to quantify statistically the performance of thecorrelations developed in this study (in predicting the residualfriction angles back-calculated from the reactivated landslides)compared with those of the other correlations listed in Table 3,the coefficient of determination (R2) of the regression equationdefined by [ø′r]PR = [ø′r]BC, as well as the mean (μ) standard de-viation (S), and coefficient of variation (COV) of the ratio [ø′r]PR/[ø′r]BC were calculated for each correlation. The correlations de-veloped in this study seem to yield the highest R2, closest μ to1.0, and the lowest S and COV (Table 3) compared to other corre-lations — thus, suggesting the suitability of the correlations pro-posed in the current study in terms of their accuracies andprecisions. This can be attributed to considering the effects ofclay mineralogy, soil texture, sample preparation techniques,and effective normal stress in ranges that are relevant to slopestability analyses.

Table 2Reactivated landslides and reported soil indices and parameters used to calibrate the residual shear strength correlations.

No. Site Stratum LL(%)

PL(%)

CF a

(%)A σn

' b (kPa) [ϕr' ]BCc

(deg.)Reference

1–5 Chiusi della Verna (5 slides),Tuscany, Italy

Clay shale 40d 19d 46d 0. 45d 166–588(av. = 378)

11.7–15.6(av. = 13.2)

Canuti et al. (1994)

6 River Beas Valley, India Siwalik rock 41 25 32 0.5 200 18–20 Henkel and Yudhbir (1966); Skemptonand Hutchinson (1969)

7 Jackfield, Shropshire, UK Carboniferous mudstone 45 20 42 0.6 62.3 17.1 Henkel and Skempton (1955);Skempton (1964)

8 St. Martino Valley, Italy Weathered marl 46d 23d 38d 0.5d 228 17 Bertini et al. (1984)9 Weirton,West Virginia, USA Colluviume 51 26 55 0.46 140 16 D'Appolonia et al. (1967)10 St. Cristoforo, Italy Lugagnano clay 53d 27d 38d 0.7d 115 11.5 Cancelli (1981)11 Mam Tor, Berbyshire, UK Edale shale 53 28 35 0.7 170–210

(av. = 188)14 Skempton et al. (1989)

12–19 Seattle Freeway (8 slides), Seattle, USA Lawton clay 55d 24d 55–60 0.54 85–185(av. = 127)

13.6–16.5(av. = 15.3)

Palladino and Peck (1972)

20 Walton's Wood, Staffordshire, UK Carboniferous mudstone 57 27 70 0.43 54 13.5–15.5 Early and Skempton (1972)21 Lyme Regis, UK Lower Lias clay 57 28 60 0.48 125 11.2 James (1970)22 Bury Hill, Staffordshire, UK Etruria marl 60 27 52 0.64 97 12 Hutchinson et al. (1973); Skempton

(1985)23 Burderop Wood, Swindon, UK Gault clay 64d 29d 47d 0.75d 31 13.3 Skempton (1972, 1985)24 Hodson, Swindon, UK Gault clay 64d 29d 47d 0.75d 42 14 Skempton (1972, 1985)25 Folkestone Warren, slip W4, Kent, UK Gault clay 64d 29d 47d 0.75d 608–749 7.7–9.4 Hutchinson (1969); Hutchinson et al.

(1980); Skempton (1985)26 Barnsdale, Leicestershire, UK Upper Lias clay 64d 28d 52d 0.7d 92 10.4 Chandler (1976, 1982)27 Hambleton, Leicestershire, UK Upper Lias clay 64d 28d 52d 0.7d 107 10.3 Chandler (1976, 1982)28 Uppingham, Leicestershire, UK Upper Lias clay 64d 28d 52d 0.7d 12 16.2 Chandler (1970, 1976)29 Gretton, Northamptonshire, UK Upper Lias clay 64d 28d 52d 0.7d 38 13.5 Chandler et al. (1973); Chandler (1982)30 Daventry, Northamptonshire, UK Upper Lias clay 64d 28d 52d 0.7d 45 13 Biczysko and Starzewski (1977a,

1977b); Chandler (1982)31 Rockingham, Northamptonshire, UK Upper Lias clay 64d 28d 52d 0.7d 34 13.9 Chandler (1971, 1976)32 Weedon, Northamptonshire, UK Upper Lias clay 64d 28d 52d 0.7d 20 14.2 Chandler et al. (1973); Chandler (1982)33 Wansford, Cambs, UK Upper Lias clay 64d 28d 52d 0.7d 120 9.8 Chandler (1979, 1982)34 Wardley, Leicestershire,UK Upper Lias clay 64d 28d 52d 0.7d 18 14.9 Chandler (1982)35 Endcombe, UK Atherfield clay 66d 25d 28d 1.46d 220 13.7 James (1970)36 Patney-Chirton, UK Gault clay 68 25 47d 0.92 33 11.9 James (1970); Skempton (1985)37 Petrofka, Saskatchewan, Canada Bedrock clay 69 27 42 1.0 830 5.7 Eckel et al. (1987)38 Spinney Hill, Saskatchewan, Canada Clay shale 70d 26d N50% NA 183 8.7 Sauer (1984)39 Denholm, Saskatchewan, Canada Bedrock clay 78 31 50 0.94 408 6.7 Christiansen (1983); Sauer (1983);

Sauer and Christiansen (1987)40 Hadleigh, (slip surf. 2&3), Essex, UK Brown London clay 80d 29d 55d 0.9d 35 13.2 Hutchinson and Gostelow (1976)41 Hadleigh, (slip surf. 5), Essex, UK Brown London clay 80d 29d 55d 0.9d 45 13.3 Hutchinson and Gostelow (1976)42 Hadleigh, (slip surf. 8), Essex, UK Brown London clay 80d 29d 55d 0.9d 49 13.5 Hutchinson and Gostelow (1976)43 Hadleigh, (slip surf. 12), Essex, UK Brown London clay 80d 29d 55d 0.9d 67 13.2 Hutchinson and Gostelow (1976)44 Guildford, Surrey, UK Brown London clay 80d 29d 55d 0.9d 69 10.9 Skempton and Petley (1967); Chandler

(1982)45 Beacon Hill, Herne Bay, Kent, UK Brown and Blue London

clay80d 29d 55d 0.9d 108 14.1–14.8 Wise (1957); Hutchinson (1965b);

Bayley (1972); Bromhead (1978)46–47 Queen's Avenue, (1966,1970), Herne

Bay, Kent, UKBrown and Blue Londonclay

80d 29d 55d 0.9d 103–113 12.3–12.5 Hutchinson (1965b); Bromhead (1978)

48–51 Miramar, (1953, 1956, 1966, 1970),Hern Bay, Kent, UK

Brown and Blue Londonclay

80d 29d 55d 0.9d 123–154 10.8–12.3 Hutchinson (1965a, 1965b); Bayley(1972); Bromhead (1972, 1978)

52 San Diego, California, USA Santiago claystone 89f 44f 57f 0.79f 288 7.5 Stark and Eid (1992, 1998)53 Portuguese Bend, California, USA Altamira bentonitic tuff 98f 37f 68f 0.90f 500 6.5 Stark and Eid (1994); Eid (1996)54 Gardiner Dam, Saskatchewan, Canada Bearpaw Shale 128f 27f 43f 2.35f 95 9.8 Jasper and Peters (1979); Mesri and

Cepeda-Diaz (1986); Stark and Eid (1994)

a Quantity of particles b0.002 mm.b Average effective normal stress on the slip surface.c Back-calculated residual friction angle reported for c’ = 0.d Average value.e Not indurated.f Indices for ball-milled materials.

101H.T. Eid et al. / Engineering Geology 204 (2016) 94–107

The work undertaken in this study confirms that considering theeffect of normal stress is important for better prediction of the residualfriction angle as most of the correlations that incorporate the effectivenormal stress yield fair agreement between the prediction and back-calculated angles indicated by having values close to 1.0 for μ, andrelatively low S and COV (Table 3). However, the relative performanceof these correlations would be governed by the levels of effective stress,soil index parameters and their range, method of sample preparationand testing, and number of data points used in presenting each

correlation. For example, in spite of having trend lines that represent ef-fective normal stresses as low as 19.6 kPa, the correlation presented byLambe (1985) as a function of the plasticity index was developed basedon a relatively few number of tests on Amuay soils only. On the otherhand, Stark and Eid (1994) developed their correlation based on a con-siderable number of test results of several soil and shale types. However,they only covered the high level of effective normal stresses relevant toslope stability analyses. By adding more ring shear test results to thosepresented in Stark and Eid (1994), Stark et al. (2005) introduced a

Fig. 8. Comparison between the back-calculated secant residual friction angles and thosepredicted using the correlations developed in this study as a function of: (a) liquid limit,clay-size fraction, and effective normal stress [Fig.7a]; (b) plasticity index and effectivenormal stress [Fig. 7b].

102 H.T. Eid et al. / Engineering Geology 204 (2016) 94–107

revisited residual strength correlation that has the same limitation inaddition to those related to utilizing index parameters of claystonesand shales derived from different sample preparation procedures,miss plotting some of the Stark and Eid (1994) data points (e.g., thoseof Bearpaw shale, Saskatchewan, Canada), and using some resultscollected from commercial sources. Kaya (2009) noted that the revisedcorrelation by Stark et al. (2005) shows a larger data scatter up to LL of100%. This may be the reason for the lower R2 value observed in Table 3for the Stark et al. (2005) correlation compared with that of Stark andEid (1994) in predicting the back-calculated friction angles. The correla-tion of Stark and Hussain (2013) inherited the same limitations but ex-hibits a performance better than that of Stark et al. (2005) correlationdue to extending the covered level of effective normal stress down to50 kPa chiefly based on data presented by Eid (1996).

As it considers the effect of normal stress only, the correlation pre-sented inWhite and Randolph (2007) either overestimates or underes-timates the back-calculated residual friction angles mobilized at low orhigh average effective normal stresses, respectively, of the reactivatedlandslides shown in Table 2. This may interpret having a reasonablevalue of μ, and high values of S and COV for the ratio [ø′r]PR/[ø′r]BC corre-sponding to such correlation (Table 3). As described earlier, by not con-sidering shear strength data at effective normal stress ranges below50 kPa and between 100 kPa and 400 kPa, the other existing residualshear strength correlations that incorporate the effect of normal stressunderestimate the residual shear strength at these effective normalstress ranges and consequently have the tendency to underestimatethe back-calculated residual friction angles (Fig. 9).

5. Expression to account for stress dependency of residual frictionangle

Using residual shear strength correlations shown in Fig. 7, it is possi-ble to propose the following expression that would account for thedependence of the drained residual friction angle on the effectivenormal stress

ϕ0r

� �σ 0

n¼ ϕ0

r

� �Pa

Paσ 0

n

� � mr

ð4Þ

where (ø′r)σ′n is the secant residual friction angle at any effective nor-mal stress σ′n higher than zero, and (ø′r)Pa is the secant residual frictionangle at an effective normal stress equal to the atmospheric pressure “Pa″ (i.e., 100 kPa). The parameter mr is a constant that is practically inde-pendent of σ′n. Values ofmr were determined as the slopes of ln [(ø′r)σ′n/(ø′r)Pa] versus ln [Pa/σ′n] for soils with several liquid limits and clay-size fractions as well as several plasticity indices. An example of suchprocess is shown in Fig. 10. It can be seen that having test results atthe wide range of effective normal stress utilized in this study helpedin better estimation of the values of mr. These values are plotted inFig. 11a to be used in case of utilizing the liquid limit and clay-sizefraction for estimating the secant residual friction angles. In case of uti-lizing the plasticity index for such estimation, Fig. 11b is to be used fordetermining the magnitudes of mr. It should be noted that Eq. (4) hasa form similar to that of the power expressions introduced by Mesriand Shahien (2003) and Lade (2010) for which no continuous relationsbetween the soil index properties and the power parameters were pre-sented. Having the value of ø′r at any effective normal stress relevant toslope stability analyses, in addition to the magnitude of mr, is alsoenough to fully describe the residual strength envelope using Eq. (4).

When (ø′r)Pa is determined using the trend lines of σ′n = 100 kPain the correlations shown in Fig.7, trend lines for other σ′n values ofthese correlations can be reproduced through utilizing Eq. (4). Thereproduced trend lines almost coincide with the original lines devel-oped from lab testing data with maximum deviations limited to ±0.7 .This has been shown in Fig. 12 for σ′n = 700 kPa. Unlike the empiricalcorrelations that are available in graphical form, Eq. (4) can now beeasily incorporated in slope stability software to generate smooth non-linear residual shear strength failure envelopes. Values of (ø′r)Pa, andmr

would be the only inputs needed to fully describe the drained residualshear strength for slope stability analyses. Values of (ø′r)Pa can beprovided through direct laboratory measurements or estimation usingshear strength correlations similar to those given in Fig. 7. The magni-tudes of the corresponding parameter mr can then be provided byusing Fig. 11. Values of (ø′r)Pa, and mr can be also estimated throughthe use of equations that are developed by exact molding of the trendlines of σ′n = 100 kPa in the correlations shown in Fig. 7 and the corre-spondingmr relationships of Fig. 11. In case of utilizing the liquid limit inestimating the drained residual friction angle, the equations can besorted based on the CF groups as.

Table 3Results of statistical analysis and calibration of residual shear strength correlations used in predicting the back-calculated friction angles reported for reactivated landslides considered inthis study.

Correlation reference Source of test data Test device and σn′ used Parameters used forprediction

R2 for[ør′]PR =[ør′]BC

Statisticalparameters for theratio [ør′]PR/[ør′]BC

μ S COV

Skempton (1964)a Skempton (1964) Direct shear, σn′ = 35 to 280kPa

CF –f 1.41 0.53 0.38

Voight (1973) Kenney (1967); Nonveiler (1967); Skemptonand Petley (1967); D'Appolonia et al. (1967);Skempton et al. (1969); Hutchinson (1967,1969); Chandler (1969, 1970); Hamel (1970)

Direct shear, σn′ = 10 to 885kPa

PI –f 1.32 0.48 0.36

Kanji (1974) Bishop et al. (1971); Tulinov and Molokov(1971); Kanji (1970, 1972); Townsend andGilbert (1973)

Direct shear and Ring shear, σn′

=6.9 to 1177 kPaPI –f 0.84 0.27 0.33

Mitchell (1976)a Deere (1974) ―b LL –f 1.28 0.37 0.29Mitchell (1976)a Deere (1974) ―b PI –f 1.23 0.39 0.32Cancelli (1977) Jamiolkowski and Pasqualini (1976) ―b LL –f 1.14 0.33 0.29Lupini et al. (1981) Lupini et al. (1981) Ring shear, σn′ = 177 to 352

kPaPI –f 1.85 0.65 0.35

Lambe (1985)e Lambe (1985) Direct shear, σn′ = 19.6, 48,98, and 196 kPa

PI, and σn′ 0.27 1.10 0.31 0.28

Skempton (1985)a Skempton (1985) Ring shear, σn′ ≃ 100 kPa CF –f 1.06 0.39 0.36Mesri and Cepeda-Diaz(1986)

Kenney (1967); Mesri and Cepeda-Diaz (1986) Direct shear, σn′= 19.6 to 785kPa

LLc –f 0.79 0.26 0.32

Collotta et al. (1989)a Collotta et al. (1989) Direct shear, σn′ = 100 to 700kPa

CALIP= (CF)2 × LL × PI × 10−5 –f 0.96 0.54 0.56

Collotta et al. (1989)a

Collotta et al. (1989)Ring shear, σn′ = 100 to 700kPa CALIP= (CF)2 × LL × PI × 10−5 –f 0.85 0.37 0.44

Stark and Eid (1994)e Stark and Eid (1994) Ring shear, σn′ = 100, 400,and 700 kPa

LLc, CFc, and σn′ 0.23 0.90 0.18 0.20

Mesri and Shahien (2003)a,e Stark and Eid (1994); Eid (1996) Ring shear, σn′ = 50, 100, and400 kPa

PI c, and σn′ –f 0.88 0.18 0.20

Wesley (2003)a Townsend and Gilbert (1973); Lupini et al.(1981); Wesley (1992)

Ring shear, and Direct shear,σn′ = 6.9 to 1177 kPa

Δ Ip = PI – 0.73 (LL-20) –f 1.06 0.55 0.52

Stark et al. (2005)e Stark and Eid (1994); Stark et al. (2005) Ring shear, σn′ = 100, 400,and 700 kPa

LLd, CFd, and σn′ 0.12 0.88 0.21 0.23

White and Randolph (2007) White and Randolph (2007) Ring shear, σn′ = 50 to 300kPa

σn′ 0.21 1.09 0.52 0.48

Stark and Hussain (2013)eEid (1996); Stark et al. (2005); Stark andHussain (2013)

Ring shear, σn′= 50, 100, 400,and 700 kPa

LLd, CFd, and σn′ 0.26 0.89 0.19 0.21

This studye,g Stark and Eid (1994); Eid (1996); This study Ring shear, σn′ = 10, 25, 50,100, 200, and 400 kPa

LLc, CFc, and σn′ 0.65 0.95 0.14 0.15

This studye,g Stark and Eid (1994); Eid (1996); This study Ring shear, σn′ = 10, 25, 50,100, 200, and 400 kPa

PI c, and σn′ 0.40 0.94 0.19 0.20

a Average curve for the correlation upper and lower bounds was used in determining [ør′]PR.b No available information.c Reported index properties of shales, mudstones, and overconsolidated clays were adjusted using the relationships of Fig. 6 or Eqs. (1), (2), and (3) of this study.d Reported index properties of shales, mudstones, and overconsolidated clays were adjusted using the relationships presented in the correlation reference.e [ør′]PR determined as the inclination of a secant of the envelope-developed as a smooth curve connecting shear strengths estimated at σn′ values utilized in the correlation-at the

reported average σn′ acting on the slip surface.f Failure to represent the linear relationship of [ør′]PR = [ør′]BC.g Ranked the best based on their ability to predict the back-calculated friction angles for the considered cases (i.e., having highest R2, closest μ to 1.0, and the lowest S and COV).

103H.T. Eid et al. / Engineering Geology 204 (2016) 94–107

For CF ≤ 20% and 21% ≤ LL ≤ 76%

ϕ0r

� �pa

¼ 39:2−0:344 LLð Þ þ 0:00123 LLð Þ2 ð5aÞ

mr ¼ −0:013þ 8:2� 10−4 LLð Þ−2� 10−6 LLð Þ2 ð5bÞ

For 25% ≤ CF ≤ 45% and 34% ≤ LL ≤ 128%

ϕ0r

� �pa ¼ 30:7−0:123 LLð Þ−0:0016 LLð Þ2 þ 1� 10−5 LLð Þ3 ð6aÞ

mr ¼ −0:0413þ 0:00406 LLð Þ−9:1� 10−5 LLð Þ2 þ 1� 10−6 LLð Þ3−3:74� 10−9 LLð Þ4 ð6bÞ

For CF ≥ 50% and 46% ≤ LL ≤ 120%

ϕ0r

� �pa ¼ 33:23−0:386 LLð Þ þ 0:0015 LLð Þ2 ð7aÞ

mr ¼ −0:103þ 0:00313 LLð Þ þ 1:9� 10−5 LLð Þ2−2:21� 10−7 LLð Þ3ð7bÞ

For CF ≥ 50% and 120% ≤ LL ≤ 293%

ϕ0r

� �pa ¼ 10:79−0:019 LLð Þ ð8aÞ

mr ¼ 0:276−0:0014 LLð Þ þ 5� 10−6 LLð Þ2−8:68� 10−9 LLð Þ3 ð8bÞIn the case of utilizing the plasticity index in estimating the residual

friction angle, (ø′r)Pa, and mr can be predicted using the followingequations.

Fig. 9. Comparison between the back-calculated secant residual friction angles and those predicted using different empirical correlations.

104 H.T. Eid et al. / Engineering Geology 204 (2016) 94–107

Fig. 10. Definition of the parameter mr using trend lines of the residual shear strengthcorrelations developed based on ring shear test results.

105H.T. Eid et al. / Engineering Geology 204 (2016) 94–107

For 3% ≤ PI ≤ 120%.

ϕ0r

� �pa ¼ 33:5−0:657 PIð Þ þ 0:0057 PIð Þ2−1:7� 10−5 PIð Þ3 ð9aÞ

Fig. 11. Values of parameter mr for estimating secant residual friction angle in terms of:(a) liquid limit and clay-size fraction; (b) plasticity index.

Fig. 12. Comparison of secant residual friction angle trend lines developed using differenttechniques as a function of: (a) liquid limit and clay-size fraction; (b) plasticity index.

mr ¼ 0:0119−8:5� 10−4 PIð Þ þ 1:4� 10−4 PIð Þ2−1:9� 10−6 PIð Þ3þ 7:22� 10−9 PIð Þ4 ð9bÞ

For 120% ≤ PI ≤ 247%.

ϕ0r

� �pa ¼ 9:51−0:0179 PIð Þ ð10aÞ

mr ¼ 0:319−0:00236 PIð Þ þ 8:4� 10−6 PIð Þ2−1� 10−8 PIð Þ3 ð10bÞEqs. (5a), (5b) to (10) can be also incorporated, alongwith Eq. (4), in

slope stability software to generate smooth nonlinear residual shearstrength failure envelope. The values of LL, and CF, or PI are the onlyinputs needed to fully describe the drained residual failure envelope.

Several expressions are presented in the literature for estimatingthe residual friction angles as a function of the soil index parameterswithout considering the effective normal stress to yield linear shearstrength envelopes (e.g., Kanji, 1974; Cancelli, 1977; Nelson, 1992;Sridharan and Raghuveer Rao, 2004) or at specific normal stresses todevelop tri-linear or quad-linear shear strength envelopes (e.g., Starkand Hussain, 2013). In conducting analyses using slope stabilitysoftware, utilizing smooth nonlinear envelopes leads to an accuratedetermination of the shear strength that corresponds to the effectivenormal stress acting on the base of each slice of the proposed failurewedge. Errors in determining these strengths may occur when linearenvelopes or envelopes with limited number of linear segments areutilized in the analyses.

6. Conclusions

A study was undertaken to evaluate the results from a comprehen-sive torsional ring shear test programconducted tomeasure the drainedresidual shear strengths of clays, silts,mudstones, and shales at thewide

106 H.T. Eid et al. / Engineering Geology 204 (2016) 94–107

range of effective normal stresses relevant to slope stability analyses. Itwas found that nonlinearity of the residual shear strength envelopes ismost pronounced at effective normal stresses lower than 50 kPa. Foreffective stresses equal or higher than 200 kPa that are occasionallymobilized in slope failures, these envelopes can be approximated asstraight lines. The influence of clay-size fraction, which is the governingfactor in developing different shearing behaviors, on the measuredresidual friction angle decreased with decreasing the effective normalstress level.

Thework led to the development of new residual shear strength cor-relations for sedimentary soils as a function of soil index parameters anda wide range of effective normal stress levels (i.e., 10 kPa to 400 kPa).This is a significant advancement since the currently available correla-tions do not cover the residual strength at effective normal stresseslower than 50 kPa, and also between 100 kPa and 400 kPa. The new cor-relations are suitable for fine-grained materials with mostly plate-likeclay particles. It is noted that, they would underestimate the drainedresidual friction angles of soilswith distinct contents - such as carbonatesoils, and soils that contain allophan, halloysite, or foraminifera- thatare unlikely to have particle rearrangement towards some preferredorientation.

For shales, mudstones, and heavily overconsoildated clays, the liquidlimit and clay-size fraction used in developing the presented correla-tions were derived from a modified sample preparation procedurethat involves ball-milling to disaggregate the particles of such induratedmaterials. To facilitate the use of the empirical shear strength correla-tions introduced herein, a set of new equations were developed –

using previously published data- to convert the index parametersderived from the commonly used standard procedures to result in indi-ces equivalent to those derived from the ball-milled sample preparationprocedure.

The suitability of the new correlations in predicting the residualstrength was verified through comparisons with back-calculated shearstrength data reported in the literature as well as results yielded fromsimilar existing correlations. It was revealed that accounting for theeffect of normal stress is an important consideration in arriving atbetter predictions using such empirical correlations. New simple formu-la was generated to express the residual shear strength correlations de-veloped in this study. The formula should facilitate incorporating thenonlinearity of the residual shear strength envelopes in slope stabilitysoftware.

Acknowledgment

The funding provided by the Qatar National Research Fund (QNRF),Qatar, under Projects No: NPRP 5-488-2-194 and NPRP 08-203-2-064,for a major part of this research work is deeply appreciated. Some ofthe samples used in this study were reprocessed from those previouslytested by the first author and provided by G. Mesri and T. Stark of theUniversity of Illinois at Urbana-Champaign.

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