Empirical Model for Estimation of the Residual

Strength of Liquefied Soil

Steven L. Kramer, M.ASCE 1; and Chwen-Huan Wang 2
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Abstract: An empirical model for the estimation of residual strength is proposed. The proposed model predicts residual strength as a nonlinear
function of both penetration resistance and initial effective stress. The model is consistent with steady-state concepts and behavior observed in
laboratory tests, but was calibrated on the basis of residual strengths back-calculated from flow-side case histories. The back-calculated
strengths considered inertial effects, hydroplaning, uncertainties, the conditions under which flow sides are known not to have occurred
in past earthquakes, and the relative quality of each case history. The proposed model predicts residual strengths that are generally between
those predicted by direct approaches and those predicted by normalized strength approaches. The proposed model also allows estimation of the
probability distribution of residual strength. DOI: 10.1061/(ASCE)GT.1943-5606.0001317. © 2015 American Society of Civil Engineers.

Introduction Castro (1969) investigated the response of triaxial specimens of

loose to medium dense sands under stress-controlled, monotonic,
Liquefaction resulting from static or dynamic loading can cause triaxial loading conditions and observed three primary types of
large permanent deformations of soil masses. In cases of earthquake- behavior. Very loose specimens were observed to reach a peak
induced liquefaction, permanent deformations most often occur shearing resistance at a low strain level after which the shearing
primarily during the period of ground shaking; these deformations resistance dropped to a constant, lower value at large strains. Dense
are driven by high dynamic shear stresses and occur through the specimens were observed to dilate and reach high shearing resis-
mechanism of cyclic mobility. In some cases, however, the liquefied tances that were still increasing when the capacity of the loading
soil is weak enough that static shear stresses can drive permanent equipment was reached. Specimens of intermediate density were
deformations, which can become very large as in cases of flow sides. observed to mobilize a peak shearing resistance at low strain levels
Because they can be so devastating, evaluation of the potential for followed by a reduction in shearing resistance, followed in turn
flow side development is an important part of geotechnical earth- by an increase in shearing resistance. Eventually, all specimens
quake engineering practice. Evaluation of that potential depends reached a constant shearing resistance at large strain levels; Castro
critically on the shear strength that can be mobilized by the liquefied and Poulos (1977) and Poulos (1981) defined the state in which a
soil in the field, i.e., the residual strength of the soil. soil is shearing with constant shearing resistance, constant effective
This paper presents reviews of the physical processes that lead stress, constant volume (i.e., constant void ratio), and constant
to liquefaction-induced permanent deformations, previous proce- strain rate as the steady state of deformation, and postulated that the
dures for estimation of residual strength, and issues involved in steady state is a unique function of void ratio. Thus, the steady-
the interpretation of case histories. It also develops a steady (critical) state effective stress and steady-state strength, Ssu , depend only
state-based framework that illustrates the effects of characteristics on the density of the soil. The steady state and critical state differ
such as effective stress on the steady state strength. It then presents in the steady-state requirement of a constant strain rate and in the
the results of back-analyses that account for uncertainties inherent in unproven steady-state postulate of a flow structure that develops
the flow-side case histories and presents an empirical model that under stress-controlled loading conditions (Casagrande 1976).
predicts behavior consistent with both flow-side and lateral spread- For the purposes of this paper, the steady state and critical state
ing case histories, with the steady (critical) state framework, and that are considered to be equivalent; because of its long use in geotech-
allows probabilistic estimation of residual strength. nical earthquake engineering, the term steady state will be used in
the remainder of this paper.
Yoshimine and Ishihara (1998) presented a framework that
Shear Strength Behavior of Liquefied Soil clearly describes the relationship between the quasi-steady state
(QSS) and ultimate steady state (USS) at different soil densities.
The excess porewater pressures generated during liquefaction The QSS occurs at the local minimum effective stress (and shearing
generally reduce the stiffness and strength of coarse-grained soils. resistance) between contractive and dilative behavior at intermedi-
ate strain levels, and the USS is mobilized at very high strain levels.
1
Professor, Dept. of Civil and Environmental Engineering, Univ. of Fig. 1 illustrates this framework schematically, and also shows the
Washington, Seattle, WA 98195-2700 (corresponding author). E-mail: relative position of the initial consolidation line (ICL) as typically
kramer@u.washington.edu observed in laboratory tests on liquefiable soils; the ICL is known
2
Asia Branch Manager, Geopier UK Limited, 5 F, No. 10, SingYun St., to not be unique for sands. The ultimate steady-state line (USSL)
Lane 33, Neihu District, Taiwan City 114, Taiwan (ROC). E-mail:
is steeper than the ICL and crosses the ICL at a relatively low
chwang@geopier.com
Note. This manuscript was submitted on August 9, 2014; approved on effective stress level. If an element of soil is on the ICL at a lower
February 18, 2015; published online on May 6, 2015. Discussion period open initial effective stress (Case A), the sample will initially contract but
until October 6, 2015; separate discussions must be submitted for individual then begin to dilate after reaching the quasi-steady state line
papers. This paper is part of the Journal of Geotechnical and Geoenviron- (QSSL); the sample will continue to dilate until it reaches the
mental Engineering, © ASCE, ISSN 1090-0241/04015038(15)/$25.00. USSL. Since there is no local minimum in the shearing resistance,

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Fig. 1. Stress-strain and stress path behavior for elements of soil consolidated to three different initial states (adapted from Yoshimine and Ishihara
1998, soils and foundation, The Japanese Geotechnical Society, with permission)

Yoshimine and Ishihara (1998) refer to this as phase transformation 0

ψ ¼ ψ1 þ ðλss − λc Þ log σvo ð1Þ
behavior. An element of soil consolidated to a somewhat higher
0
initial effective stress (Case B), and hence lower void ratio, would When the vertical effective stress is σvo , the corresponding USS
exhibit similar behavior, but would exhibit a peak shearing resis- effective stress is given by
tance at low strain followed by a local minimum shearing resistance
σss0 ¼ σvo
0
10−ψ=λss ð2Þ
at the QSSL, followed then by dilation to the USSL. An element of
soil at higher initial effective stress (Case C) would show strain so the USS strength can be written as
softening following a low-strain peak shearing resistance but no
0 0
subsequent dilation due to coincidence of the QSS and USS lines Ssu ¼ σvo 10−½ψ1 −ðλss −λc Þ log σvo =λss ð3Þ
at high effective stress levels. Because the ICL is flatter than the
USSL, the soil becomes more contractive as the effective confining Eq. (3) indicates that the USS strength is proportional to vertical
pressure during consolidation increases. effective stress only if the USS line is parallel to the consolidation
These aspects of liquefiable (i.e., liquefaction-susceptible) soil curve (i.e., if λss ¼ λc ). The partial derivative of Eq. (3) with
respect to σvo0
behavior illustrate one of the most critical issues in the practical
estimation of residual strength—the definition of residual strength.
∂Ssu λc −½ψ1 −ðλss −λc Þ log σvo0 =λss
0 ¼ 10 ð4Þ
In many laboratory investigations (e.g., Sladen et al. 1985; Vaid and
Chern 1985; Mohammad and Dobry 1986), QSS and USS data ∂σvo λss
have been combined. In other investigations (e.g., Verdugo and can be seen to decrease monotonically with increasing σvo 0
Ishihara 1996; Yamamuro and Lade 1998; Yoshimine and Ishihara when λss > λc .
1998), QSS and USS strengths are treated separately. From a prac- For λss > λc, Eq. (4) indicates that the partial derivative,
tical standpoint, the relative significance of QSS and USS strengths 0 0
∂Ssu =∂σvo , decreases with increasing σvo , so the USS strength
depends on what level of deformation is considered to correspond of a soil at a particular SPT resistance should increase at a decreas-
to failure. If deformations consistent with shear strains on the order ing rate with increasing initial effective stress as shown in Fig. 2(b).
of 5–10% are unacceptable, then the QSS strength may be most Olson and Stark (2003) summarized laboratory data for a number
appropriate; if larger deformations are of greater concern, then of sands that showed the USS line slope, λss , to be an average of
the USS may govern. In this paper, residual strength will be inter- about 15% greater than the one-dimensional consolidation curve
preted as the shearing resistance that can be mobilized at very large slope, λc , with the difference greater for clean sands than for silty
strain levels. Under ideal conditions (constant volume, effective sands. While the relatively restrictive conditions of the ultimate
stress, shearing resistance, and velocity), this definition of residual steady state of deformation are, as discussed subsequently, rarely
strength would therefore correspond to USS conditions. It should met in actual flow sides, the type of behavior illustrated qualita-
be recognized, however, that strains associated with the QSS can tively in Fig. 2 provide a useful conceptual indication of how the
also lead to unacceptably large deformations. strength of a liquefied soil should be expected to vary with initial
effective stress.

Framework for Ultimate Steady-State Strength

Estimation of Residual Strength
Useful insight into the manner in which different factors can be
expected to affect ultimate steady-state (USS) strength can be ob- Estimation of residual strength is a critical part of evaluating
tained from fundamental ultimate steady-state concepts. Assuming flow-side potential. Over the years, two primary approaches to
the one-dimensional consolidation curve and USS line for a given residual strength estimation have developed. Experimental ap-
soil are both linear over the stress range of interest in e- log σv0 space proaches based on the type of laboratory tests first used to identify
[Fig. 2(a)] with respective intercepts (at σv0 ¼ 1 kPa) of ec1 and steady-state behavior were developed in the 1970s and 1980s.
ess1 , and slopes of λc and λss , the state parameter will clearly vary Case history-based approaches were developed in the 1980s and
with σv0 if λc and λss are not equal, i.e. subsequently.

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(a) (b)

Fig. 2. Conceptual illustration of ultimate steady-state strength: (a) relationship between one-dimensional normal consolidation line, ultimate steady-
state line, and state parameter; (b) resulting trend of ultimate steady-state strength with vertical effective stress

Laboratory Testing-Based Approach show shearing resistances continuing to change at shear strains of
30% or more. The rapid softening that occurs upon triggering of
Numerous laboratory investigations of the USS behavior of sands
liquefaction generally prevents testing equipment from maintaining
have been undertaken (Olson and Stark 2003). Early experimental
stress-controlled loading through the mobilization of the USS.
investigations formed the basis for current understanding of the
Although the steady-state concept postulates that USS strength
behavior of liquefiable soils, and led to the development of a labo-
depends only on soil density (e.g., Poulos 1981; Been et al. 1991;
ratory testing-based procedure for evaluation of USS strength.
Ishihara 1993), a number of experimental investigations (e.g., Miura
Procedures and Toki 1982; Hanzawa 1980; Kuerbis et al. 1988; Vaid et al.
Poulos et al. (1985) proposed a method for evaluation of USS 1990; Riemer and Seed 1997; Yoshimine and Ishihara 1998) have
strength based on laboratory testing of carefully retrieved undis- concluded that USS strength is influenced by stress path. Given that
turbed specimens and testing of reconstituted specimens of the flow failures often involve multiple stress paths, estimation of the in
same soil. The procedure involves determining the in situ void ratio situ USS strength from tests performed using one particular stress
from one or more undisturbed specimens, measuring the USS path may be inaccurate.
strength of undisturbed specimens consolidated to confining pres- Discussion
sures sufficiently high to ensure contractive behavior, determining Although conceptually useful, experimentally based procedures for
the slope of the USS line by testing specimens reconstituted at dif- measurement of the residual strength mobilized in the field suffer
ferent void ratios, and correcting the measured USS strengths of the from a number of practical limitations that have tempered their use
undisturbed specimens by assuming the USS line of the undis- in geotechnical engineering practice. While some of these limita-
turbed specimens is parallel to that of the reconstituted specimens. tions could potentially be reduced by using, for example, frozen
Sadrekarimi (2013) used a framework similar to that described samples and multiple stress path testing, a number of them still re-
in Eqs. (1)–(4) to express the normalized USS strength ratio as a main. Even more significantly, however, it must be recognized that
function of the state parameter, soil compressibility, and steady- the USS conditions that can exist under idealized conditions in
state friction angle. The relationship showed good agreement with laboratory tests usually do not exist in actual flow sides. As a flow
laboratory data for a number of sands with different relative com- failure develops in the field, drainage can occur, leading to changes
pressibilities, i.e., ratios of ψ to λss . The use of this procedure for in effective stresses, volume, and density; strain rates are variable as
determination of USS strength, therefore, requires knowledge or the failing soil accelerates and then comes to rest; stresses and
estimation of the state parameter and of the in situ compressibility stress paths are different; and different soils can be mixed together
of the soil. during flow. All of these factors tend to limit the direct applicability
of laboratory-measured shear strengths to actual flow sides.
Experimental Issues
Experimental investigations have involved different soils prepared
in different manners and tested at different rates using different Case History-Based Approach
types of equipment. In a number of instances, however, they have Even before all of the issues associated with the laboratory
produced conflicting and/or inconsistent results, which have iden- testing-based approach were identified and explored, the notion
tified important issues in the problem of estimating the strength of of using back-calculated strengths from flow failure case histories
liquefied soil. was proposed (Seed 1987). In the case history–based approach,
The effects of sampling disturbance can be pronounced back-calculated strengths are correlated to the penetration resis-
for loose, saturated sands. Even with disturbance corrections tance of the liquefied soil. Since actual flow failures frequently
(e.g., Poulos et al. 1985), the flat slopes of typical USS lines involve drainage, pore pressure (or void) redistribution, mixing,
can lead to significant uncertainty in estimated USS strength variable strain rates, and other factors that violate the assumptions
(Kramer 1989). Laboratory strength tests also may not be able to of the USS of deformation, back-calculated strengths are frequently
reach the very large strains associated with the USS while main- referred to as residual strengths. Two basic approaches, termed here
taining reasonable uniformity of stresses and strains; many labora- the direct approach and the normalized strength approach, have
tory tests (e.g., Verdugo and Ishihara 1996; Yamamuro and Lade been proposed. The primary versions of both approaches are de-
1998; Yoshimine and Ishihara 1998; Yamamuro and Covert 2001) scribed in the following sections.

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 Direct Approach uncertainty, and the limited number of case studies back-analyzed
In the direct approach to case history-based residual strength to date.”
estimation, the residual strength is considered to be a function Idriss (1998) reinterpreted the case history database of Seed and
of penetration resistance alone. To the extent that penetration resis- Harder (1990) and presented a graphical model for estimation of
tance can be considered directly related to soil density, this ap- residual strength (Fig. 3). The Idriss model removed a good deal
proach is consistent with classical USS concepts, which hold that of the ambiguity in determining a representative SPT resistance in
USS strength is a unique function of soil density. It implicitly previous models by explicitly correlating residual strength to the
assumes that any effect of initial effective stress on the residual median SPT resistance in the liquefied zone. Fig. 3 shows that
strength of an element of soil is entirely accounted for by its mea- the Idriss (1998) curve is of similar shape to the curves of Seed
sured ðN 1 Þ60 value. and Harder (1990) and is closer to the Seed and Harder lower bound
Seed (1987) correlated the apparent residual strength from 12 curve than to the upper bound curve. While the Idriss (1998) curve
case histories involving liquefaction with substantial soil deforma- was expressed graphically, it was consistent with a residual strength
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tions to standard penetration test (SPT) resistance. Only three of the (in atm) computed as
case histories (Calaveras Dam, Fort Peck Dam, and Lower San
Fernando Dam) were from case histories with clean sand SPT re- Sr ≈ 0.0236 exp½ðN 1 Þ60  ð6Þ
sistances greater than 10. The residual strength was back-calculated
by varying the residual strength in a zone considered to have lique- Wride et al. (1999) reviewed the case histories used by Seed
fied until limit equilibrium procedures produced a factor of safety (1987), Seed and Harder (1990), and Stark and Mesri (1992), and
of unity. The postfailure slope geometry was analyzed for most of related their interpretations of the most reliable undrained strength
the case histories, but the prefailure geometry was analyzed for to the minimum SPT resistance, which was described as “a reason-
some. The back-calculated residual strength was plotted against a able lower bound ðN 1 Þ60in situ .” Wride et al. (1999) noted the diffi-
representative clean sand SPT resistance, ðN 1 Þ60−cs , which was culty of expressing the conditions in a particular flow-side case
defined using history by a single undrained strength and a single SPT resistance.
Gutierrez and Eddy (2011) back-calculated residual strengths
ðN 1 Þ60−cs ¼ ðN 1 Þ60 þ ΔðN 1 Þ60 ð5Þ from postfailure geometries of 38 flow-side case histories, a num-
ber of which have been identified by others as lateral spreads. The
back-calculation did not consider inertial effects, and residual
where ΔðN 1 Þ60 is a fines correction (Fig. 3). The procedures by strengths were correlated to minimum SPT resistances with no fines
which the representative SPT resistance or the fines correction were correction. These analyses established the relationship
obtained were not explicitly described, but the fines correction was
the same as that recommended for evaluation of liquefaction poten- Sr ¼ 0.87ðN 1 Þ60 þ 0.1ðN 1 Þ260 ð7Þ
tial. The results of the back-calculation analyses fell within a band
that showed a trend of increasing residual strength with increasing Gutierrez and Eddy (2011) also characterized uncertainties in
SPT resistance, as shown in Fig. 3. residual strengths using first-order reliability methods. Uncertainty
Seed and Harder (1990) expanded (to 17 case histories) and re- in the residual strength was characterized by a beta distribution and
interpreted the case history database and developed an updated re- only shown graphically; the 16th percentile curve is about 60% of
lationship for residual strength estimation. Back-calculated residual the 50th percentile curve, which would correspond to a σln Sr value
strengths were correlated to a representative clean sand SPT resis- of 0.51 if Sr was a lognormally distributed quantity.
tance, with the fines correction identical to that of Seed (1987). The
variation of residual strength with clean sand SPT resistance is Normalized Strength Procedures
shown graphically in Fig. 3. Seed and Harder (1990) recommended Castro (1987) observed that the density of a given soil increased
use of “the lower-bound, or near lower-bound relationship” be- with increasing initial effective stress and that residual strength
tween residual strength and SPT resistance “owing to scatter and increased with increasing density, and concluded that the USS
strength should be related to initial effective stress. As indicated
previously, the ratio of USS strength to effective stress should
be constant for a given soil if the USS line and consolidation curve
are parallel (λss ¼ λc ).
Castro (1987), Castro and Troncoso (1989), and Castro (1991)
investigated several tailings dams in South America and reported
ratios of USS strength to initial major principal effective stress,
0 , ranging from 0.12 to 0.19.
Sus =σ1c
Stark and Mesri (1992) developed a database of 20 case histor-
ies for which the ratios of residual strength to initial vertical effec-
tive stress, hereafter referred to as the normalized residual strength
0 , were computed. These stress ratios were found to
ratio, Sr =σvo
correlate better to a representative clean sand SPT resistance when
a different fines correction (Table 1) than that of Seed (1987) was
used. Noting that the strength of a liquefied soil at a given vertical
effective stress should increase with factors (e.g., gradation, particle
angularity, particle roughness) that also increase SPT resistance,
Fig. 3. Variation of residual strength with equivalent clean sand SPT 0 to representative clean sand
Stark and Mesri (1992) related Sr =σvo
resistance using direct approach residual strength models; upper and
SPT resistance. Supplementing the case history database with in-
lower bound curves are shown for Seed (1987) and Seed and Harder
terpreted laboratory test results, Stark and Mesri (1992) proposed
(1990) models
that the normalized residual strength ratio could be estimated as

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 Table 1. Back-Calculated Residual Strengths and Pertinent Characteristics for Flow-Slide Case Histories
Case history Weighting SPT Fines Initial vertical Residual
type Case history factor resistance content (%) effective stress (atm) strength (atm)
Primary Calaveras Dam 0.55 7.9 34 3.237 0.311
Fort Peck Dam 0.85 11.7 54 3.528 0.322
Hashiro-Gata 0.55 4.4 15 0.188 0.030
Lake Ackerman 1.00 4.8 0 0.396 0.045
Lower San Fernando Dam 1.00 12.6 25 1.672 0.240
Route 272 0.70 6.6 33 0.493 0.061
Shibeca-Cho 0.70 3.7 20 0.495 0.099
Uetsu 0.55 3.0 0 0.433 0.021
Wachusett Dam 1.00 7.3 8 1.209 0.164
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Secondary Asele road 0.20 8.9 32 0.310 0.060

Chonan 0.74 5.1 18 0.930 0.085
El Cobre Tailings Dam 0.60 1.2 95 4.608 0.092
Helsinki Harbor 0.39 6.1 0 0.417 0.025
Hokkaido 0.31 1.1 50 1.603 0.119
Kawagishi-cho 0.50 4.3 2 0.646 0.058
Koda Numa 0.44 3.05 40 0.504 0.023
La Marquesa D/S 0.72 8.2 20 0.874 0.163
La Marquesa U/S 0.76 4.1 30 0.795 0.087
La Palma Dam 0.80 2.9 15 0.745 0.092
Lake Merced 0.39 6.1 0 0.620 0.066
Metoki road 0.39 2.1 0 1.246 0.055
Mochi-Koshi 1 0.34 4 73 0.826 0.075
Mochi-Koshi 2 0.67 5.2 73 1.365 0.111
Nalbland 0.51 4 30 0.607 0.066
Nerlerk Berm 0.41 10.6 10 0.682 0.085
Sheffield Dam 0.37 5 40 1.099 0.047
Snow River 0.50 7 20 0.984 0.024
Solfatara 0.42 5.1 0 0.578 0.036
Soviet Tajik 0.22 7.6 15 1.948 0.158
Tar Island 0.32 8.2 13 2.966 0.172
Zeeland 0.39 7.7 11 2.225 0.107

Sr effective stress, Idriss and Boulanger (2007) produced relationships

0 ¼ 0.0055ðN 1 Þ60−cs
σvo
ð8Þ
for normalized residual strength ratio that can be expressed as

  
Sr ðN 1 Þ60−cs ðN 1 Þ60−cs − 16 3
with a different fines correction procedure than that introduced by
0 ¼ exp þ − 3.0
Seed (1987). σvo 16 21.2

 
Olson and Stark (2002) developed a case history database of 33 ðN 1 Þ60−cs
flow-side case histories and performed detailed residual strength × 1 þ β exp − 6.6 ≤ tan ϕ 0 ð10Þ
2.4
back-calculation analyses on a number of them. These analyses ac-
counted for inertial effects and variable initial vertical effective where β ¼ 0 or 1 when void redistribution effects are significant or
stresses within the liquefied zone of each case history. Other case insignificant, respectively.
histories were not investigated or documented in sufficient detail to The residual strengths predicted by the normalized strength
warrant detailed analyses and were analyzed using a simple, infinite models of Stark and Mesri (1992), Olson and Stark (2002), and
slope approach. Olson and Stark (2002) proposed that the normal- Idriss and Boulanger (2007) are shown for three effective stress
ized residual strength ratio could be estimated as levels in Fig. 4. Due to the implicit assumption of proportionality,
the residual strengths at low effective stress levels are quite low
Sr over a broad range of SPT resistances. The linear increase in
0 ¼ 0.03 þ 0.0075ðN 1 Þ60
σvo
ð9Þ
residual strength with increasing effective confining pressure is
inconsistent with the USS framework for typical liquefiable soils.
for ðN 1 Þ60 ≤ 12 and found that no fines correction was necessary
with this relationship. Case History Issues
Idriss and Boulanger (2007) considered a subset of 18 flow sides Because flow sides do not occur frequently, relatively few case his-
from the case history database compiled by Olson and Stark tories are available. Since many occur in remote and undeveloped
(2002). This group included 10 of the 12 cases considered by Seed locations, information on prefailure geometric and material proper-
(1987), 13 of the 17 cases considered by Seed and Harder (1990), ties is frequently unavailable or poorly known. A number of flow
and 18 of the 33 cases considered by Olson and Stark (2002). The sides have been investigated following failure, but such investiga-
18 case histories were divided into three groups based on adequacy tions are frequently limited to a small number of borings in areas
of in situ measurements (i.e., SPT and/or CPT) and geometric in- near, rather than on, the failure itself, and to limited measurements
formation. Normalizing the average back-calculated strengths of the postfailure geometry. Although generally analyzed assum-
(from as many as available) computed by Seed (1987), Seed ing sliding on distinct failure surfaces, some case histories have
and Harder (1990), and Olson and Stark (2002) by initial vertical involved slumping and distributed deformations. Measured SPT

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Fig. 4. Residual strengths predicted by normalized strength models

of Stark and Mesri (S-M), Olson and Stark (O-S), and Idriss and
Boulanger (I-B) for three initial vertical effective stress levels

resistances, for example, are nonexistent, sparse, or obtained after

failure for many of the case histories. Values for Calaveras Dam, for
example, have been assigned based on estimated relative densities,
and others have been based on a single measurement. Lack of in-
formation leads to increased uncertainty in the parameters that go
into a residual strength back-analysis. This uncertainty, which is
larger for some parameters than others and larger for some case
histories than others, can lead to bias and/or uncertainty in the
back-calculated residual strength. The uncertain parameters typi-
cally include soil density or penetration resistance, strengths of Fig. 5. Sliding block analog to the flow-side problem illustrating
nonliquefied soils, surface and subsurface slope geometry, failure inertial effects
surface geometry, phreatic surface location, hydroplaning, mixing,
and three-dimensional effects.
The different case history-based approaches to residual strength back-analysis based on the final geometry (Point C) would under-
estimation have used different measures of SPT resistance. Several estimate the resistance (ϕapp ¼ θC ). The correct geometry at which
(e.g., Seed 1987; Seed and Harder 1990; Stark and Mesri 1992; to back-analyze the resistance would be that at Point B, where the
Idriss and Boulanger 2007) used a representative SPT resistance inertial force (i.e., the acceleration of the block) is zero and for
that was not explicitly or uniformly defined but appears to be tar- which ϕapp ¼ θB ϕ. Methods for consideration of inertial effects
geted toward the looser soils involved in a flow side and determined have been reported by Davis et al. (1988) and Olson et al. (2000).
with considerable engineering judgment on a case-by-case basis. These methods use a sliding block analog to account for the dy-
Wride et al. (1999), however, interpreted representative SPT resis- namics of the failure mass during flow. The sliding block is as-
tances as being closer to the average SPT resistance, and proposed sumed to have a weight equal to that of the failure mass and to
the use of minimum SPT resistance for residual strength prediction. slide from the prefailure center of gravity to the postfailure center
Olson and Stark (2002) took representative SPT values as mean of gravity along a path parallel to the failure surface. This requires
values. The model of Idriss (1998) is explicitly based on median that the shape of the failure surface be known or estimated; the
SPT resistance. Because different users can consider different SPT sensitivity of the back-analyzed residual strength to the shape of
resistances to be representative, and those values can be inconsis- the failure surface has not been established.
tent with those used in the interpretation of case histories, this im- The importance of pore pressure (and void ratio) redistribution
precise definition of SPT resistance can contribute to uncertainty in in the development of flow sides is becoming increasingly recog-
residual strength prediction beyond that associated with the number nized (Whitman 1985; Boulanger and Truman 1996; Kokusho
and consistency of SPT measurements. 1999; Kulasingam et al. 2004; Boulanger and Idriss 2011). When
Back-analyses of flow-side case histories involving large or high pore pressures are generated within or beneath a slope,
rapid deformations can also be complicated by inertial effects. hydraulic gradients cause flow of the porewater. If this flow is
The situation is analogous to the sliding block shown in Fig. 5. impeded, as may occur at boundaries between liquefiable soils
At Point A, the shearing resistance between the block and the curv- and layers of silt or clay that may cap or exist within them, the
ing surface beneath it is insufficient to resist the gravity-induced effective confining pressure decreases and the void ratio increases
driving force, so the block begins to slide. The block accelerates (due to rebound of the soil skeleton). Because the residual strength
until it reaches Point B at which the driving and resisting forces is sensitive to void ratio, the residual strength may drop signifi-
are equal and equilibrium could be achieved under static condi- cantly due to this local rebound of the soil skeleton. Many of
tions. At Point B, however, the block has some velocity and its the soils involved in the case history database are silty sands de-
momentum causes it to continue sliding until it stops sliding at posited in water; as such, some degree of segregation or banding of
Point C. Back-analysis of the apparent sliding resistance based on coarser and finer particles existed prior to failure. In such cases, the
the initial geometry (block at Point A) would overestimate the ac- back-calculated strengths would be expected to include the effects
tual resistance (the apparent friction angle would be ϕapp ¼ θA ), and of void redistribution so that further correction for those effects

© ASCE 04015038-6 J. Geotech. Geoenviron. Eng.

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 would not be necessary (Seed 1987). This phenomenon compli- back-calculate average residual strengths that reflected the effects
cates interpretation of case histories because the measured SPT re- of uncertainties in the inputs to a back-analysis. The alternative
sistances do not reflect the local loosening that produces the back-analysis procedure attempted to characterize and account for
mobilized residual strength. Because the back-calculated strength uncertainties in the various flow slide case histories, correct for
corresponds to a looser condition than that inferred by the meas- inertial effects for those case histories for which they were impor-
urement of SPT resistance, the error can introduce a potential bias tant and data was available, and evaluate the quality of each case
into the relationship between residual strength and SPT resistance. history.
The extent to which subsurface density and permeability can be
characterized sufficiently to determine whether void redistribution
Identification and Characterization of Uncertain
potential is or is not significant at a particular site is unclear.
Variables
The residual strength of layered soils can also be affected by the
mixing of constituent materials during flow. Naesgaard and Byrne The stability of slopes, like most other problems of geotechnical
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(2005) showed that the large-strain strength of a mixture of finer and structural engineering, is a function of geometry and material
and coarser soils can have a lower residual strength than that of properties. Back-analysis of shear strength from a slope failure case
either constituent material at the same void ratio. To the extent that history, therefore, requires characterization of the geometry and
layered soils exist in many of the case histories, this effect will be material properties of the slope. Geometric variables include those
reflected in back-calculated residual strengths. that describe the external geometry of the slope, the geometry of
A number of the case histories involved flow sides into bodies of individual layers of soil/rock within the slope, the geometry of any
water. When such slides occur at high velocities, water in front of phreatic surface within the slope, and the position of the failure
the slide may become trapped under the leading edge of the flow, surface used in the back-analysis. Material property variables in-
resulting in hydroplaning (Horne and Joyner 1965; Mohrig et al. clude measures of density and available shearing resistance.
1998). When hydroplaning occurs, the length of the failure surface
is effectively shortened, which can lead to underestimation of back- Prefailure and Postfailure Geometry
calculated residual strength when not accounted for. Although hy- Uncertainties in slope coordinates were related to the manner in
droplaning may occur at displacement levels greater than those at which the prefailure and postfailure geometries were measured
which residual strength is mobilized, it can affect final geometries and documented. For some case histories, the geometries were sur-
that are used in common inertial correction procedures. veyed with accurate instruments and reported in clear drawings; in
Models using the normalized strength approach predict other cases they were measured more crudely and documented
extremely low residual strengths at low initial vertical effective in less detail. Standard deviations of external geometry points typ-
stresses (i.e., at shallow depths), regardless of SPT resistance. This ically varied from 8 to 15 cm in the horizontal direction and from 30
aspect of the model suggests that flow sliding should be expected to 60 cm in the vertical direction. For internal boundaries, standard
in moderately dense to dense soils subjected to very strong ground deviations ranged from 30 cm to 1.2 m in the horizontal and vertical
shaking; while such soils may develop displacements due to lateral directions, respectively. Water table elevations were assumed to
spreading, the absence of flow slides in such events calls this aspect vary with standard deviations ranging from 10 to 30 cm at the slope
of the normalized model into question. surface and from 30 cm to 1.0 m within the slope.

Discussion Failure Surface Geometry

Estimation of residual strength is an extremely difficult problem. Previous procedures for correction of inertial effects have been
Laboratory studies show that the physical phenomena are sensitive based on interpolating between the residual strengths back-
to many factors that cannot be well characterized in available flow analyzed from the prefailure and postfailure geometries. Unlike
slide case histories. The preceding sections have briefly described a the rigid block in the sliding block model, however, the geometry
number of these issues that affect estimation of residual strength by of a failing slope changes and often does so in a complex manner
back-analysis of flow slide case histories. Because the effects of that may affect the driving stress after inertial forces have been re-
these issues are embedded in the back-calculated residual strength moved. A new procedure was developed to improve the accuracy
values, residual strength should be recognized as a system response with which inertia-free driving stresses could be estimated. The
parameter rather than a soil property. Previous efforts at residual procedure allows estimation of an inertia-corrected geometry for
strength estimation have recognized and dealt with some of these selected case histories using the following steps:
issues explicitly, but have either not addressed or made assumptions 1. Obtain the most accurate available cross-sections for the pre-
as to the potential effects of others. failure and postfailure conditions.
At this time, two basic approaches have been taken to the 2. Identify, using those cross sections, both the prefailure and
residual strength estimation problem. The direct approach, which postfailure positions of as many specific reference points on
assumes residual strength to vary only with corrected SPT resis- the slope as possible.
tance, predicts relatively high strengths at shallow depths and 3. Sketch, considering soil mechanics, kinematics, and all avail-
relatively low strength at large depths. The normalized residual able descriptions of the failure, the anticipated paths taken by
strength approach, which considers residual strength to be affected the points identified in Step 2 in moving from their prefailure
by corrected SPT resistance and to be directly proportional to ef- to postfailure positions.
fective confining pressure, predicts very low residual strength at 4. Using sliding block analyses similar to those reported by
shallow depths and relatively high strengths at large depths. Olson and Stark (2002), estimate the zero-inertia factor, ZIF,
for the case history. The zero-inertia factor is defined as the
fraction of total displacement corresponding to the zero-inertia
Back-Analysis of Case Histories position; with reference to the schematic illustration of Fig. 5,
ZIF ¼ lAB =lAC .
In an attempt to consider the important issues described in the pre- 5. Apply the ZIF to each of the kinematic paths developed in
vious section, an alternative approach to the back-analysis of flow Step 3 to estimate the locations of the various reference points
slide case histories was developed. The goal of this approach was to at the zero-inertia state.

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 6. Construct the zero-inertia slope geometry to be consistent with Once the zero-inertia geometry was obtained, the position of
the positions of the points identified in Step 5 and with the the failure surface was characterized by a best-estimate geometry
volumes of the various soil zones (i.e., maintaining constant that was then altered by a half-sine function with a wavelength of
volume). twice the length of the failure. The amplitude of this function was
An example of this procedure is shown for the case of Wachusett assigned based on the nature of the apparent failure mechanism
Dam in Fig. 6. The procedure is laborious and approximate but and on the quality of the information with which that failure mecha-
does allow back-analysis of residual strength from a reasonable nism could be deduced. The half-sine function used for the well-
geometry rather than interpolating between residual strength values documented, 30.5-m-high Lower San Fernando Dam case history,
(those based on prefailure and postfailure geometries) that are for example, had a standard deviation of 80 cm (2.6% of the height
known to be incorrect. of the slope), while the corresponding value for the 4.6-m-high
This procedure, while applied objectively and consistently, must Hachiro-Gata roadway embankment, which failed with a slumping
be recognized as uncertain. The zero-inertia geometry depends on mechanism that produced no clear failure surface, was 1.1 m (24%
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the ZIF, which was estimated from the sliding block model of of the height of the slope). The nature of the failure mechanism also
Olson (2001). Uncertainty in the zero-inertia geometry was taken affected the accuracy with which the position of the top of the fail-
as a function of the level of documentation of the individual flow ure surface could be determined; the horizontal position of this
slide and of the degree to which the failure mechanism was con- point was taken to vary over a range related to the length of the
sistent with the simplified sliding block model used in estimation of failure surface and the accuracy with which the available informa-
the ZIF. A coefficient of variation ranging from 0.1 to 0.3 was as- tion constrained that location; the range varied from 0 to 6% of the
signed to the ZIF; lower values were used for block-type failures on length of the failure surface. Case histories with broadly distributed
well-defined failure surfaces and higher values for slumping fail- failure zones, slumping mechanisms, and/or retrogressive deforma-
ures or failures with highly distributed zones of shear strain pro- tion mechanisms were assigned higher levels of uncertainty in this
ducing poorly defined failure surfaces. geometric parameter.

Fig. 6. Estimation of zero-inertia geometry for Wachusett Dam: (a) identification of reference points on prefailure and postfailure geometries;
(b) estimated geometry under zero-inertia conditions

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 The effects of hydroplaning were accounted for by an approxi- Analyses of Case Histories
mate procedure. Experimental studies have shown that hydroplan-
The flow-slide case history database of Olson (2001) has been
ing tends to occur when the densimetric Froude number exceeds
used for the development of several residual strength models,
values of 0.3–0.5 (Mohrig et al. 1998). In the laboratory, water has and it forms the basis for the model described in this paper.
been observed to penetrate beneath a hydroplaning debris flow by The database was divided into primary and secondary case histor-
a distance of up to 10 times the thickness of the flow. For case ies. The primary group consisted of nine case histories for which
histories in which materials flowed into water, the distance to inertial effects were significant and sufficient data for detailed
which hydroplaning extended back from the head of the slide back-analysis was available; the remainder were classified as sec-
was estimated based on an estimated Froude number (Wang ondary case histories. Monte Carlo simulations of the primary case
2003) and accounted for in the residual strength back-calculation. histories were used to account for the effects of uncertainties in
The fraction of available residual strength on the hydroplaned por- geometric and material parameters on back-calculated residual
tion of the failure surface ranged from 0.5 (for silty sands and fi-
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strength. The secondary case histories were interpreted using a

ner-grained soils) to 0.9 (for coarse-grained soils). This procedure procedure that relied in part upon previous residual strength
was recognized as approximate, and was therefore randomized in back-analyses.
the Monte Carlo analyses, but was applied consistently to all pri-
mary case histories. Primary Case Histories
The primary case histories were back-analyzed using Monte Carlo
Material Variables simulations that randomized initial and final geometry, failure
The primary material variables that affect back-analyzed residual surface position, water table position, and SPT resistance. Each
strengths are unit weight and shear strength. In nearly all flow slide uncertain parameter was characterized by a probability distribution
case histories, failure mechanisms involve nonliquefied as well as and the spatial variability of SPT resistance was also modeled.
liquefied soils. In the back-analyses described in this paper, the For each primary case history, iterative limit-equilibrium stability
properties of nonliquefied soils were assigned coefficients of varia- back-analyses were used to compute the residual strength along
tion equal to those recommended by Phoon and Kulhawy (1999) the failure surface for each of a total of 50,000 Monte Carlo real-
unless case-specific information suggested that other values were izations. The fact that spatial variability of SPT resistance was
more appropriate. modeled meant that the SPT resistance varied over the length of
Residual strengths have been most commonly correlated to the failure surface, which required consideration of variable
SPT resistance, which reflects soil density and other pertinent residual strength over the failure surface. This aspect of the back-
characteristics of the soil. Estimation of the uncertainty in SPT analysis procedure was handled by assuming a relationship of the
resistance was complicated by the widely varying amounts of mea- form
sured SPT data in the various case histories. A procedure developed
to allow consistent and objective estimation of the uncertainty in
SPT resistance computed the standard deviation of SPT resistance Sr ¼ 0.0236 exp½kðN 1 Þ60  ð11Þ
as the weighted average of standard deviations given by three
different procedures. The procedure, while approximate, was ap- where Sr is in atm [this relationship is of the same form as that
plied uniformly and consistently to the case histories for which implied by Idriss (1998) and is similar to the Idriss curve shown in
back-analyses were performed. It assigned very high uncertainties Fig. 3 when k ¼ 0.16]. The value of k was then back-analyzed for
to cases where SPT resistances were not measured or were only each Monte Carlo realization, and used with Eq. (11) to determine
measured at one or two locations, and assigned the statistically the residual strength at all points along the particular failure sur-
calculated standard deviation to cases where a significant number face for that realization. Those strengths and the spatially variable
(30 or more) of SPT resistances were measured. In between, the ðN 1 Þ60 values were then averaged over the length of the liquefied
standard deviation was computed as a fraction of the range (maxi- portion of the failure surface. The average residual strengths given
mum minus minimum) of measured SPT values using the approach by this procedure were insensitive to the coefficient (0.0236) used
of Burlington and May (1970). In order to prevent negative pen- in Eq. (11).
etration resistances for cases where σSPT was large, the SPT resis-
tances were modeled as being lognormally distributed with Secondary Case Histories
logarithmic mean, λ ¼ μln SPT , and logarithmic standard deviation, The secondary case histories have also been analyzed by previous
ζ ¼ σln SPT . investigators but at a lower level of detail than the primary case
Although sufficient data to characterize it on a case-specific histories. These case histories generally involved smaller slopes
basis did not exist, spatial variability of SPT resistance was for which lower levels of investigation/documentation precluded
modeled with two-dimensional, anisotropic random fields based the type of detailed back-analyses performed for the primary case
on the method of Yamazaki and Shinozuka (1988). In recognition histories. In many cases, simple infinite slope analyses were used
of the fact that soil properties vary more rapidly in the vertical to back-analyze residual strength. It should be noted that back-
direction, the spatial variability was defined by scales of fluc- calculating residual strengths based on a noncritical failure surface
tuation of 2.4 and 12 m in the vertical and horizontal directions, will tend to produce lower values than would be obtained using the
respectively. actual, critical surface. For these cases, the mean residual strength
was taken as the mean of the residual strength values obtained in
Discussion previous investigations. Given the relatively simple back-analysis
The uncertainties assigned to various input variables were, due to the procedures used in the previous investigations, and the generally
wide variation in investigation/documentation of available flow slide good agreement between the residual strength values obtained in
case histories, necessarily based on a combination of data and those investigations, repetition of the same types of analyses was
judgment. Systematic procedures for assigning uncertainty values not considered necessary. The mean SPT resistances for the secon-
were developed (Wang 2003) and applied consistently to all case dary case histories were obtained by the same procedure used for
histories. the primary case histories.

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 Table 2. Values of σln Sr for Different Combinations of SPT Resistance and paper. It is an unfortunate fact that insufficient data is currently
Initial Vertical Effective Stress available to resolve all of those issues. In the absence of such data,
0 it is important to develop a model that is consistent with current
Initial vertical effective stress, σvo (atm)
SPT resistance
understanding of the basic mechanics of liquefiable soils and flow
ðN 1 Þ60 0.01 0.1 0.25 0.5 1.0 2.0 4.0
slides, and also with the reliable empirical data that is currently
0 0.563 0.434 0.397 0.383 0.386 0.410 0.454 available.
2 0.570 0.432 0.390 0.371 0.370 0.389 0.431 A new model for estimating the residual strength of liquefied
4 0.581 0.438 0.391 0.367 0.361 0.375 0.414 soil was developed. The model is of a flexible form consistent with
6 0.597 0.449 0.399 0.371 0.361 0.370 0.404
expected soil behavior and capable of representing either direct
8 0.619 0.468 0.415 0.383 0.367 0.371 0.400
10 0.643 0.491 0.436 0.402 0.382 0.381 0.404 or normalized strength behavior. It was calibrated against back-
12 0.671 0.519 0.463 0.427 0.404 0.397 0.415 calculated residual strengths from case histories with consideration
14 0.702 0.550 0.493 0.457 0.431 0.421 0.432 of the uncertainties in, and quality of, those case histories. The cal-
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16 0.736 0.585 0.528 0.491 0.463 0.449 0.455 ibration process also considered conditions under which flow slides
have been observed not to occur. The details of the model develop-
ment process are described in the following sections.

Back-Analysis Results
Form of Predictive Equation
Each primary case history back-analysis produced a distribution of
50,000 average ðN 1 Þ60 and Sr values, each representing an individ- Multiple functional forms of equations relating residual strength
ual Monte Carlo realization of the slope failure. The mean values of (direct approach) and residual strength ratio (normalized strength
the SPT resistances and residual strengths, along with mean fines approach) to SPT resistance were investigated. The former used
contents and initial vertical effective stresses, are given for each a relationship of the direct form
case history in Table 2. Sr ¼ a1 exp½a2 ðN 1 Þa603  ð12Þ
Direct comparison of the back-calculated residual strengths
for the primary case histories with those obtained by previous in- and the latter a normalized relationship
vestigators is complicated by fundamental differences in the mod-
eling techniques. The consideration of spatial variability in the Sr b3
0 ¼ b1 þ b2 ðN 1 Þ60 ð13Þ
back-analyses performed in this investigation results in average σvo
residual strength values that are greater than the value that would
both of which are capable of producing models very similar to
be obtained using the average SPT resistance with no spatial vari-
those shown in Figs. 3 and 4. Calibration of both models using
ability; this effect increases with increasing residual strength and
nonlinear least-squares regression produced model coefficients that
slope height. Fig. 7 shows the residual strengths from the primary
fit the data well but had residuals that were heteroscedastic and/or
case histories that have been evaluated by other investigators. The
exhibited trends with respect to the predictor variables. The accu-
agreement is generally good with the average residual strengths
racy of the normalized strength model was somewhat better than
from the current back-analyses falling near the high ends of the
that of the direct model, and was found not to be improved by
ranges obtained by others. This behavior results from consideration
the addition of fines content terms.
of spatial variability and hydroplaning, and from differences in the
These preliminary results motivated the development of a model
way inertial effects were calculated.
of different form that has elements of both the direct and normal-
ized strength modeling approaches. The search for an appropriate
model form was guided by the following basic criteria:
Development of an Empirical Model 1. Residual strength should increase with increasing SPT
resistance.
The development of a model for evaluation of residual strength
2. Residual strength should increase with increasing initial ver-
must consider the many complex issues discussed earlier in this
tical effective stress.
3. The sensitivity of residual strength to SPT resistance should
increase with increasing SPT resistance.
4. The residual strength need not vary in direct proportion to in-
itial vertical effective stress.
5. Residual strengths at low initial vertical effective stresses
should be consistent with lateral spreading activity observed
in actual earthquakes.
6. The model should allow probabilistic characterization of resi-
dual strength.
A number of potential predictive models satisfying as many of
these criteria as possible were investigated using nonlinear least-
squares regression. The results of these analyses showed that a
model of the basic form
0 Þ θ4 g
Sr ¼ θ1 expfθ2 ½ðN 1 Þ60  þ θ3 ðσvo ð14Þ

could potentially satisfy all of the criteria and produce a better fit to
Fig. 7. Comparison of back-calculated primary case history residual
the case history data than either the direct or normalized residual
strengths with residual strengths computed by others (LSFD = Lower
strength models. This basic equation was modified to include a
San Fernando Dam; FPD = Fort Peck Dam)
fines content term and an indicator variable for void redistribution

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 potential (set to 1 or 0 based on the presence or absence of potential Model Calibration
flow-impeding soil layers) in various stages of the model calibra-
The predictive model of Eq. (14) was calibrated using the maxi-
tion process.
mum likelihood estimation assuming that residual strengths are log-
normally distributed. The assumption of lognormality is common
Case History Quality for strength parameters and is particularly appropriate for highly
uncertain parameters such as residual strength due to its inherent
As previously discussed, the case histories included small and large elimination of negative parameter values. The likelihood func-
slopes whose failures caused relatively small to very large losses. tion, which describes the likelihood of observing the case history
Some of the case histories were investigated and documented in data given the parameters assigned to the proposed model, can be
great detail, while others were investigated quickly and/or incom- written as
pletely documented. As a result, the extent to which each case his-
Yn
   
tory should be allowed to influence the proposed residual strength 1 1 ln Sr;i − λ 2
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model was judged to be variable. Lðλ; ζjSr;i Þ ¼ pffiffiffiffiffiffi exp − ð16Þ

i¼1 2πζSr;i 2 ζ
A procedure for rating the relative quality of the flow-slide case
histories was developed. The procedure rated each case history with where λ ¼ μln Sr , ζ ¼ σln Sr ; and Sr;i = residual strength for the
respect to three factors: (1) the level of detail with which each was 0 to sim-
ith case history. Letting R ¼ Sr , N ¼ ðN 1 Þ60 , and S ¼ σvo
investigated; (2) the number of independent residual strength eval- plify some of the succeeding equations
uations used to compute the average residual strength for the sec-
ondary case histories; and (3) the level of agreement between the λi ¼ ln θ1 þ θ2 ½N i þ θ3 ðSi Þθ4  ð17Þ
secondary case history residual strengths obtained by previous
investigators. The total case history weighting factor for each case Using a first-order, second-moment approximation, the contri-
history (Table 2) was then computed as the product of the three butions of uncertainties in the input parameters to the total uncer-
component weighting factors. tainty can be expressed as
The procedures used to develop these weighting factors were
necessarily subjective, but they were judged to be reasonable and ζ 2i ≈ θ22 σ2N;i þ θ22 θ23 σ2N;i þ Si2θ4 −2 σ2S;i þ θ5 ð18Þ
were applied consistently to all of the case histories. Idriss and
Boulanger (2007) divided the case history database into four groups Maximizing the natural logarithm of the likelihood function
on the basis of the adequacy of in situ measurements and com- yielded a set of coefficients θ1 − θ5 that provided a very good fit
pleteness of geometric data. The mean values of the weighting to the data, but implied residual strengths at low effective stress
factors for the case histories in Idriss and Boulanger’s Groups levels that were lower than the minimum strengths implied by the
1–3 and the remaining case histories (implied as being excessively Youd et al. (2002) lateral spreading database. The residual strength
poor) were 0.81, 0.64, 0.50, and 0.41, respectively. These values at low effective stress was determined to be most sensitive to the
indicate an overall consistency between the weighting factors origi- stress exponent, θ4 . A series of parametric analyses indicated that
nally reported by Wang (2003) and the independently determined values of θ4 ≥ 0.1 would produce residual strengths at low effective
adequacy/completeness groups of Idriss and Boulanger (2007). stresses that were consistent with the upper range of minimum
Given the paucity of available flow-slide case histories, it was con- strengths inferred from the lateral spread case history database. The
sidered preferable to use all of the case histories with appropriate maximum likelihood regression was then repeated with the con-
weighting factors than to eliminate any group entirely. straint that θ4 ≥ 0.1, which yielded θ1 ¼ 0.000215, θ2 ¼ 0.109,
θ3 ¼ 49.35, θ4 ¼ 0.10, and θ5 ¼ 0.121. Analyses of modified
models indicated that neither the addition of fines content or void
Influence of Lateral Spreading redistribution indicator parameters improved the predictive capabil-
Normalized residual strength models can predict extremely ities of the model.
low residual strengths at low initial vertical effective stress levels, The calibrated predictive model can then be written as
i.e., at shallow depths. While some of the flow sides in the case
history database included liquefiable soils at low initial vertical ef- ln Sr ¼ −8.444 þ 0.109 N þ 5.379 S0.1 ð19Þ
fective stresses, there is considerable case history data for lateral
spreads at low initial vertical effective stress levels. Given that subject to the constraint that the residual strength not exceed the
residual strength is taken to correspond to large-strain, flow con- drained strength (which it may at shallow depths). Values of the
ditions in this paper, the shear stresses driving these lateral spreads median residual strength, Sr ¼ expðln Sr Þ, are shown in Fig. 8(a).
were generally too low to cause flow sliding and can therefore be The fit to the case history data is shown in Fig. 8(b).
interpreted as a lower bound to residual strength. The lateral The variance of can be expressed as
spreading case history database of Youd et al. (2002) was used σ2ln Sr ¼ 1.627 þ 0.00073N 2 þ 0.0194N − 0.027NS0.1
to estimate a minimum driving shear stress for ground slope case
histories as − 3.099S0.1 þ1.621S0.2 þ 0.00073σ2N þ 4.935S−1.8 σ2S
ð20Þ
−1
τ min ¼ γ sat T 10 sin½tan ðS=100Þ ð15Þ
Fig. 9 illustrates the uncertainty in residual strength from the
proposed model. Because the flow-side case history data is so scat-
where γ sat = saturated unit weight assuming e ¼ 0.7 (approxi- tered, the terms related to the uncertainty in N and S contribute
mately 50% relative density for typical uniform sands), T 10 is the relatively little to the overall uncertainty in Sr , and can therefore
cumulative thickness of soils with SPT resistances less than or be neglected for practical purposes. With that simplification, the
equal to 10 bpf, and S is the ground slope in percent. This equation values of σln Sr for various combinations of N and S are listed in
implicitly assumes a water table at the ground surface and that all Table 2. The uncertainty can be seen to vary with both ðN 1 Þ60 , and
0
layers comprising T 10 are contiguous from the surface. σvo and to be lower where case history data is more plentiful.

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 Sr ðPÞ ¼ exp½ln Sr þ Φ−1 ðPÞσln Sr  ð21Þ
where Φ−1 ðPÞ is the inverse standard normal variate for P.

Comparison with Other Residual Strength Models

Direct comparisons of residual strength models are complicated by
differences in SPT value definition (i.e., representative, median,
mean values, etc.), differences in case history databases, differences
in back-calculated strengths, and differences in the intents of the
models. Most models to date have been intended for deterministic
use and therefore include an inherent conservatism that tends to
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underpredict back-calculated strengths. The model described in this

paper is a probabilistic model and hence seeks to be accurate rather
than conservative. Conservatism, in a deterministic sense, can be
achieved with the proposed model through the use of a low percen-
tile strength value.
As indicated by the exponential nature of the proposed model,
its variation of residual strength with SPT resistance [Fig. 8(a)] is
consistent with that assumed in the direct models of Seed (1987),
Seed and Harder (1990), and Idriss (1998). In contrast with those
models, however, the proposed model also makes use of initial
vertical effective stress, although not in the linear manner used
by Stark and Mesri (1992), Olson and Stark (2002), and Idriss and
Boulanger (2007). Figs. 10(a–c) illustrate the variation of residual
strength with initial vertical effective stress predicted by the Seed–
Harder, Idriss–Boulanger, and proposed models for ðN 1 Þ60 ¼ 5, 10,
and 15, respectively. The proposed model predicts residual
strengths that are higher than normalized residual strength model
at low initial vertical effective stresses and lower at high initial ver-
tical effective stresses.
The higher residual strength at low effective stress levels results
from both the flow case history data and consideration of the mini-
mum strengths implied by shallow lateral spreading case histories.
Fig. 8. (a) Median residual strengths predicted by hybrid model; Fig. 10(d) shows the combinations of minimum shear stress and
(b) comparison of predicted residual strengths with those back- initial vertical effective stress from the lateral spreading ground
calculated from flow-side case histories slope case histories. The average SPT resistances for each of the
lateral spreading case histories cannot be determined from the def-
inition of T 10 , but are obviously less than 10; therefore, a residual
The proposed model can be used to estimate percentile strength strength model for a soil with ðN 1 Þ60 ¼ 10 should predict residual
values, i.e., values of residual strength with specific probabilities of strengths greater than these values at low initial effective stress lev-
nonexceedance. For a percentile value, P, the corresponding per- els. Fig. 10(d) also shows back-calculated residual strengths for the
centile strength is given by flow-side case histories at low initial vertical effective stress levels.

1.2
N = 10 NN==10
10
S = 1 atm SS==11atm
atm
1.0 COVS = 0 COVNS==00
COV

COVN = 0 COVS = 0
0.8 COVN = 0.1 COVS = 0.01
COVN = 0.2 COVS = 0.05
fSr(Sr)

0.6 COVN = 0.5 COVS = 0.10

0.4

0.2

0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6
(a) Residual Strength, Sr (atm) (b) Residual Strength, Sr (atm)

Fig. 9. Probability density functions for residual strength: (a) different uncertainties in SPT resistance; (b) different uncertainties in initial vertical
effective stress

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J. Geotech. Geoenviron. Eng., 2015, 141(9): 04015038

Downloaded from ascelibrary.org by University of Washington Libraries on 09/06/16. Copyright ASCE. For personal use only; all rights reserved.

Fig. 10. Predicted variation of residual strength with initial vertical effective stress for (a) ðN 1 Þ60 ¼ 5; (b) ðN 1 Þ60 ¼ 10; (c) ðN 1 Þ60 ¼ 15 bpf
(Idriss–Boulanger curves shown with and without void redistribution; (d) for ðN 1 Þ60 ¼ 5 and 10 with minimum strengths implied by lateral spreading
case histories and back-calculated strengths from flow-side case histories at low initial vertical effective stress levels; Seed–Harder values taken at
lower-third point; shallow soil friction angle of 25° used for drained strength constraint

These case histories have ðN 1 Þ60 values ranging from 2.9 to 10.6 and ultimate steady-state line slope. While the theoretical frame-
with an average of 5.2 and standard deviation of 2.2. It should work cannot be expected to accurately predict strengths that are
therefore be expected that the predicted strengths for ðN 1 Þ60 ¼ 5 mobilized under field conditions that violate its basic assumptions,
should be consistent with the strengths from those case histories. the fact that the proposed model shows trends that are consistent
Fig. 10(d) confirms that the proposed model predictions exceed the with those implied by the theoretical framework help support its
lateral spreading data (gray circles) for ðN 1 Þ60 ¼ 10 and are con- applicability. The direct and normalized strength approaches are
sistent with the flow-side data (black circles) for ðN 1 Þ60 ¼ 5. not consistent with the theoretical framework given available ex-
perimental data for typical liquefiable soils.
Several aspects of the proposed model are particularly note-
Discussion worthy. First, the model allows probabilistic characterization of
The model described in this paper presents an alternative to residual residual strength. As such, it does not attempt to be conservative
strength models currently used in engineering practice. It shares as most deterministic models do. Second, the model does not make
some characteristics with the direct residual strength models and use of a fines correction; the addition of fines content-related terms
some with normalized strength models; it was formulated in a to the predictive model did not result in improved residual strength
way that could have produced results consistent with either if the predictions. This characteristic was also observed by Olson and
data had indicated that either was most appropriate. The residual Stark (2002). Third, the use of a void redistribution potential indi-
strengths back-calculated from the flow-side case history database, cator term that would distinguish between cases where void redis-
supplemented by relevant data from a lateral spreading case history tribution would and would not be expected also did not result in
database, however, provided better support for the proposed model. improved predictions. This observation could be taken to indicate,
The manner in which the residual strength predicted by the as some have suggested, that void redistribution effects occur to
proposed model varies with SPT resistance and effective stress is some extent in virtually all flow-side case histories; the accuracy
consistent with that predicted by a theoretical steady-state frame- of that interpretation, however, cannot be proven with available
work that considers the relationship between soil compressibility data. Finally, the proposed model predicts residual strengths at low

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J. Geotech. Geoenviron. Eng., 2015, 141(9): 04015038

 initial vertical effective stresses that are significantly higher than Acknowledgments
those predicted by the normalized strength models and similar
(slightly higher at low SPT resistance but significantly lower at high The research described in this paper was supported by the
SPT resistances) to those predicted by the direct models. The higher Washington State Department of Transportation and the Pacific
strengths predicted by the proposed model are consistent with the Earthquake Engineering Research Center; the support of Tony
flow-side case histories at low initial vertical effective stresses and Allen and Kim Willoughby is gratefully acknowledged. A portion
are also greater than the shear stresses at which lateral spreading, of the work was completed while the senior writer was on sab-
rather than flow sliding, has been observed in the field. Normalized batical leave at the International Centre for Geohazards at the
residual strength models tend to predict shallow flow sliding under Norwegian Geotechnical Institute. The authors would also like to
conditions where lateral spreading has been observed; while lat- acknowledge the residual strength data made available by
eral spreading can cause damaging deformations, it is a different Dr. Scott Olson and the lateral spreading data made available to
mechanism than that considered in this paper. all researchers by Dr. T.L. Youd. The writers are also grateful to
Downloaded from ascelibrary.org by University of Washington Libraries on 09/06/16. Copyright ASCE. For personal use only; all rights reserved.

Ross Boulanger, Liam Finn, and Tom Shantz for their constructive
comments on a draft of this paper.
Conclusions

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