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CHAPTER 3
Findings and Applications
3.2.1.2 Geotechnical Design
3.1 Overview
In an early effort to distinguish different sources of uncer-
This chapter ﬁrst presents the results of a review of current
tainty in geotechnical design, Taylor (1948) proposed the use
LRFD practice in geotechnical design, introduces the basis for
LRFD-based methods for retaining structures, and provides of separate and independent factors of safety for the cohesion
the results of a review of current U.S. practice of soil nailing. and frictional components of soil resistance. However, the con-
Subsequently, the chapter provides discussions of LRFD limit cept of a load factor, which incorporates the uncertainty related
states in the design of SNWs and a synthesis of approaches used to loads, was not used in geotechnical design at that time. All
to calibrate resistance and load factors. Finally, calibrations of uncertainty in geotechnical design was concentrated in the
resistant factors are presented. resistance. The use of both load and resistance factors in geo-
technical engineering was initiated by Brinch-Hansen in Den-
3.2 Review of Current LRFD Practice mark (Brinch-Hansen 1953, 1956, 1966). Later publications
related to the use of LRFD concepts in geotechnical design
3.2.1 Historical Development of LRFD
include Barker et al. (1991) for foundations and retaining
structures, Fellenius (1994) and Meyerhof (1994) for shallow
3.2.1.1 Structural Design
foundations, O’Neill (1995) for deep foundations, Hamilton
The early use of concepts of probability and reliability, as
and Murff (1992) and Tang (1993) for foundations of offshore
used to quantify uncertainties in the design of structures
platforms, Kulhawy and Phoon (1996) for foundations of trans-
(Freudenthal, 1947, 1951; Freudenthal and Gumbel, 1956),
mission towers, Withiam et al. (1991, 1995) and D’Appolonia
set the basis for the subsequent development of the LRFD
(1999) for retaining structures, Allen et al. (2001) and Chen
framework. In the 1970s and 1980s, the development of
(2000a, 2000b) for MSE walls, and Paikowsky et al. (2004,
LRFD methods for structural applications advanced substan-
NCHRP Project 24-17) for deep foundations.
tially when various structural codes started to incorporate
reliability concepts. For example, reliability was used in the
American National Standards Institute code (ANSI) for design 3.2.2 Overview of Uncertainty in Design
loads for buildings (as summarized by Ellingwood et al., 1980; of Structures
Ellingwood and Galambos, 1982; Ellingwood et al., 1982a and
This section provides an overview of common approaches in
1982b). Other design codes incorporating LRFD concepts
dealing with uncertainty in structural design. In the design of
included those for steel construction [American Institute of
structures, a number of uncertainty sources must be addressed.
Steel Construction (AISC), 1994; Galambos and Ravindra,
These sources may include the following:
1978], concrete construction [American Concrete Institute
(ACI), 1995], and offshore platforms [American Petroleum
• Material dimensions and location/extension;
Institute (API), 1989; Moses, 1985, 1986]. International build-
• Material properties, including unit weight/density and
ing codes containing reliability or LRFD methods included the
strength;
National Building Code of Canada (Siu et al., 1975; National
• Long-term material performance;
Research Council of Canada, 1977) and Report 63 developed by
• Possible failure modes;
the United Kingdom’s Construction Industry Research and
• Methods used to analyze loads and evaluate load distribution;
Information Association (CIRIA, 1977).

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• Methods used to predict transient loads; the ultimate capacity of the structural element. Structures have
• Methods used to predict the structural response; and various components that may be subjected to numerous load-
• Potential changes over time associated with the structural ing conditions, possibly involving different potential failure
function. modes. As a result, numerous equations, similar in format
to Equation 3-1, must be considered to achieve a safe design
Besides the sources listed above, in geotechnical design, of each structural component and of the entire system for all
uncertainties also arise from the variability of subsurface con- expected conditions.
ditions, the intrinsic errors made in the estimation of material In Equation 3-1, all uncertainty is concentrated in FS that
appears on only one side of the design equation. FS is typically
properties, and the divergences that occur due to the differences
adopted based on experience, engineering judgment, and com-
between the estimated and actual properties of the structure.
mon practice. It is not usually based on uncertainty quantiﬁca-
The variability of subsurface conditions arises as a result of
tions (i.e., by establishing the probability of failure of a selected
the spatial variability of soil and rock properties. Spatial vari-
failure mode or structural component). Minimum values of FS
ability of soil/rock properties may be caused by differences in
recommended for design of certain structures are selected gen-
geology across a site; in contrast, local variability of soil/rock
erally by agencies with jurisdiction or interest on those struc-
properties commonly results from the inherent heterogeneities
tures. For example, for the design of bridge structures and
of most natural materials. Intrinsic errors in the estimation of
substructures, AASHTO has developed a set of FS values that
material properties (i.e., usually referred to as bias) arise from
is contained in the ASD-based AASHTO Standard Speciﬁca-
(i) sampling methods used to obtain soil/rock specimens
tions (AASHTO, 1996).
[e.g., a standard penetration test (SPT)]; (ii) ﬁeld or labora-
In general, FS values that are selected based on experience
tory testing techniques used to evaluate soil/rock properties
tend to provide safe and reasonably economical designs after
(e.g., SPT blow count or triaxial tests); and (iii) models used
years of practice. However, the selection of new FS values for
to interpret and predict soil/rock properties (e.g., Mohr-
new problems (i.e., use of materials, construction methods, or
Coulomb model). Measurements of soil/rock properties in the
consideration of infrequent loading) may be more challenging
ﬁeld and laboratory produce random errors that are typical of
than simply selecting values based on existing ranges. In deriv-
all measurements. Finally, uncertainty in geotechnical design
ing FS values for new problems, different design practitioners
may also occur due to differences between the assumed or esti-
may select different FS values if only engineering judgment is
mated properties and the actual properties of the constructed
used. The ASD method may occasionally provide inconsistent
structure as a result of differing construction methods or insuf-
levels of safety for structures involving various components
ﬁcient construction quality control and assurance.
with multiple factors of safety (each possibly involving different
probabilities of failure). To overcome some of these limitations
3.2.3 Overview of the ASD Method of the ASD method, the LRFD has been developed.
Uncertainty in engineering design has traditionally been
3.2.4 Overview of the LRFD Method
addressed with factors of safety (FS) in the allowable stress
design (ASD). In the ASD method, allowable “stresses” (or,
3.2.4.1 Objectives and Basic Description
more generally, resistances) of structural components are
of the LRFD Method
obtained by dividing the values of ultimate strengths of those
structural components by FS. The general design condition To address design uncertainty in a more systematic manner
in the ASD method can be expressed as: than in the ASD method, the LRFD method was developed
with the following objectives: (i) to account for uncertainty in
Rn loads and resistances separately with the use of factors for load
ΣQi ≤ Rall = (3-1)
FS and resistance; (ii) to provide reliability-based load and resis-
tance factors based on accepted levels of structural reliability;
where and (iii) to provide consistent levels of safety across a structure
ΣQi = the effect of all combined loads on a given structural when several components are present. This approach is used
component for a given failure mode, in the current AASHTO LRFD Bridge Design Speciﬁcations.
Rall = the allowable stress of that structural component, In the LRFD method, two parameters account for uncer-
Rn = the ultimate or maximum strength of that structural tainty: load factor for load uncertainty and resistance factor for
component, and
material uncertainty. The use of separate parameters is justiﬁed
FS = the factor of safety applied to that ultimate resistance.
because the nature, variability, and hence level of uncertainty
associated with loads are different than the uncertainty related
Allowable stresses represent normal working conditions
to resistance. In principle, the LRFD method can result in more
of a structural element and are therefore selected lower than

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consistent levels of safety across the entire structure because the Nominal resistance is the resistance of an entire structure
(or of one of its components), which is established based on
relationship between the levels of safety of different structural
stresses or deformations or is a specified strength of the
members is accounted for in this method. Resistance and load
materials involved in the structure. In general, nominal resis-
factors are selected using probability-based techniques so that
tances of structural components are derived from the speciﬁed
these factors are related to acceptable levels of structural relia-
materials and dimensions. For example, the speciﬁed tensile
bility, which is equivalent to a tolerable probability of failure.
yield strength of a steel bar is typically a nominal strength.
Unlike the FS, the LRFD-based parameters are calibrated with
However, the nominal resistance of soils and other natural
respect to actual load and resistance data.
materials is obtained differently. The nominal resistance of
Load and resistance factors are related to each other through
soils is derived using suitable field/laboratory methods or
limit states. A limit state is a condition in which the structure as
other acceptable means (e.g., correlations between field test
a whole, or one of its components, has achieved a level of stress,
results and soil strength parameters). The nominal resistance
deformation, or displacement that may affect its performance.
of soils commonly represents an ultimate strength of the soils.
In the LRFD method included in the AASHTO LRFD Bridge
For example, the internal friction angle of granular soils,
Design Speciﬁcations, four types of limit states are deﬁned:
which is routinely estimated from ﬁeld/laboratory tests or cor-
(i) Strength limit states,
relations, is an ultimate strength to be used in establishing the
(ii) Extreme-event limit states,
nominal resistance of soils.
(iii) Service limit states, and
Resistance factors commonly reduce nominal resistances;
(iv) Fatigue limit states.
therefore, they are typically ≤ 1.0. Section 10, Foundations, and
Section 11, Abutments, Piers, and Walls, of the LRFD Bridge
Therefore, the design objectives in the LRFD methodology
are to demonstrate that (i) the available resistance (i.e., for Design Specifications (AASHTO, 2007) present prescribed
strength and extreme-event limit states) is sufﬁcient; (ii) other values of resistance factors for geotechnical design of bridge
structural conditions (e.g., tolerable deformations in service substructure components.
Load factors (γi) are statistically based multipliers that
limit states) are within tolerable limits; and (iii) the structural
performance is adequate for all foreseeable load conditions are used in the LRFD method to account for load variabil-
arising during the design life of the structure. ity sources (e.g., frequency of loads, inaccuracies in load
In general, all of these limit states must be considered in the estimation, and likelihood of simultaneous load occur-
design of structural elements, although not all limit states are rences). While the resistance factors remain the same once
they are selected, different γi are selected for different load
directly applicable for geotechnical design. These limit states
are described in more detail in the following subsections. combinations. For strength limit states, load factors are
typically ≥ 1.0 if the acting load is destabilizing. Conversely,
load factors are ≤ 1.0 if the acting load component tends to
3.2.4.2 Strength Limit States
stabilize the structure. An example of stabilizing loads is the
Strength limit states are those related to the strength (i.e., horizontal force that arises from soil passive pressures that
generally referred to as nominal resistance in the LRFD con- resist the lateral movement of an embedded foundation.
vention, as deﬁned subsequently) and the stability of struc- Guidance for selecting load factors for different load com-
tural components during the design life of the structure. For binations in bridge substructure components are contained
each strength limit state, a design equation can be generically in Table 3.4.1-1, Load Combinations and Load Factors, and
expressed as: Table 3.4.1-2, Load Factors for Permanent Loads, of Sec-
tion 3, Loads and Load Factors, of the LRFD Bridge Design
N
φRn ≥ ∑ γ i ηi Qi Specifications (AASHTO, 2007). The number of load com-
(3-2)
i =1 ponents (N) may vary for different load combinations, as
presented in AASHTO (2007).
Where Factor ηi accounts for redundancy, ductility, and impor-
Rn = the nominal resistance of a given structural component tance of the structure and varies between 0.95 and 1.05. Addi-
for the strength limit state being considered;
tional guidance for the selection of these factors can be found
φ = a non-dimensional resistance factor related to Rn;
in Section 1.3, Design Philosophy, of AASHTO (2007).
Qi = the i-th load type that participates in this limit state;
γi = a non-dimensional load factor associated with Qi;
ηi = a load-modiﬁcation factor; and 3.2.4.3 Extreme-Event Limit States
N = the number of load types considered in the limit state.
Extreme-event limit states are those related to infrequent
These quantities are described in the following paragraphs. but large loads that have return periods exceeding the design

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life of the structure. Extreme–event limit states in bridges and 3.2.4.5 Fatigue Limit States
substructures include loads arising from seismic events, ice
Fatigue limit states are those states in which loads are applied
formation, and vehicle and vessel collision. The same design
repetitively and may affect the performance of a structure,
equation used for strength limit states is commonly used for
while the stress levels are signiﬁcantly below the values used in
extreme-event limit states, although the load factors are dif-
strength limit states. For example, fatigue limit states are appli-
ferent. The load factors that must be considered for different
cable to structures that may be sensitive to fracture as a result
load combinations in extreme-event limit states are contained
of repetitive loads (e.g., vehicular loads and dynamic loads).
in Table 3.4.1-1 of AASHTO (2007).
Additional information on fatigue limit states can be found in
Article 3.6.1.4.1 of AASHTO (2007).
3.2.4.4 Service Limit States
Service limit states are those states related to inadequate 3.2.5 Resistances and Loads
conditions that may arise during normal operation of the as Random Variables
structure but do not cause a collapse. Inadequate conditions
In the LRFD method, loads, Q, and resistances, R, are con-
may include excessive deformation, excessive settlements,
sidered random independent variables with probability density
and cracking. For each service limit state, the following con-
functions fR(R) and fQ(Q) that are usually normal or lognormal
dition must be met:
(as shown in Figure 3-1), mean values Qm and Rm, and standard
SMAX ≤ STOLERABLE (3-3) deviations σQ and σR, respectively. R and Q are commonly
assumed to be probabilistically independent in geotechnical
Where design (Baecher and Christian, 2003). The variability of these
SMAX = the maximum calculated value of a quantity S random variables can be conveniently expressed through co-
(e.g., deﬂection or settlement) expected to occur efﬁcients of variation (COV), which are deﬁned as:
under normal conditions; and
σQ
STOLERABLE = the maximum value of S the structure can sus-
COVQ = (3-4)
tain before its functionality is affected. Qm
The load factors for different load combinations to be con- σR
COVR = (3-5)
sidered in service limit states are contained in Table 3.4.1-1 of Rm
AASHTO (2007).
Importantly, due to reasons that will be presented subse- COVs, which also can be expressed as a percentage, are use-
quently, overall stability, slope stability, and other stability ful as they express uncertainty as a fraction (or percentage) of
states are considered service limit states per AASHTO (2007). the mean values.
For these cases, an equation similar to that of strength limit Nominal values of loads and resistances, Qn and Rn, are
states is used, with the exception that all load factors are deﬁned as:
selected equal to 1.0 to reﬂect the assumption that the struc-
Qm = λ Q Qn (3-6)
ture is under normal conditions.
Qm
Load Effect
probability densities, fQ and fR
Qn Resistance Effect
Rn Rm
γ Qn φ Rn
Load and Resistance, Q and R
Figure 3-1. Probability density functions for load and resistance.

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11
Values of the cumulative distribution of Φ and/or its inverse If the loads are limited to dead and live loads, therefore:
can be obtained from various probability and statistics refer-
γ DC QDC + γ LL QLL
ences (e.g., Baecher and Christian, 2003) or can be computed φ≥ (3-12)
FS (QDC + QLL )
using statistical software.
where subscripts DC and LL refer to permanent and live
3.2.6 Approaches for Calibration of
loads, respectively.
Resistance and Load Factors
One of the objectives mentioned for the LRFD method was Method C: Calibration Using
to provide γ and φ factors that relate to acceptable levels of Pf. Reliability-Based Procedures
This relationship is established through a calibration, which is
In this method, factors are calibrated according to a relia-
performed by ﬁxing one of the factors (usually the load factor)
bility analysis and are based on empirical data (e.g., load-test
and calibrating the other. Therefore, one factor cannot be
data). In addition, a tolerable level of uncertainty is selected.
modiﬁed without modifying the other. Calibrations can be per-
Tolerable levels of uncertainty are expressed through a target
formed using the following methods, each with an increasing
value of the reliability index, βT, which reﬂects an accepted,
level of complexity (Withiam et al., 1998):
low probability of failure for a given structure type and load
• Method A: Calibration using engineering judgment; scenario.
• Method B: Calibration by matching factors to FS in ASD-
This method is more complex than Methods A and B and
based design codes; and requires that adequate and sufficient empirical information
• Method C: Calibration using reliability-based procedures.
be available. Comparative designs help evaluate the factors
obtained in this method and correlate them with factors
A description of each of these methods is presented in the
obtained using other methods. An advantage of this method is
following paragraphs.
that it can provide more explicit insight on the bias of certain
predictive design formulas and can help identify and quantify
Method A: Calibration Using Engineering Judgment
the largest sources of uncertainty arising in design. The method
This method is best suited for situations where a great deal may not be amenable and transparent for engineers unfamil-
of experience is available among a summoned team of design iar with reliability concepts.
professionals (for example, a panel of experts). This method Three different levels of calibration complexity can be
can, in theory, be advantageous because it may incorporate achieved in Method C [Withiam et al. (1998)]—Levels I, II, and
proven design practices that have led to safe and cost-efﬁcient III—each of which is described in the following paragraphs.
projects. This approach may increase the conﬁdence of other
design engineers in certain design procedures. Disadvantages Level I. Level I calibration is referred to as a first-order
of this method include the possibility that the judgment of the second-moment (FOSM) calibration methodology. At this
panel members may be unintentionally biased. level, the random variables R and Q and their mathematical
derivatives used to derive β are only approximated. As dis-
Method B: Calibration by Matching Factors to Safety cussed earlier, R and Q are assumed to be statistically inde-
Factors Contained in Design Codes pendent. The key simpliﬁcation in this method is that only the
ﬁrst-order derivatives of the squared values of R and Q and/or
In this method, resistance factors are calibrated by matching
their derivatives (i.e., known as second moments in probabil-
or calibrating them to FS values used in the ASD format. This
ity) are included, while higher-order terms are disregarded. In
approach is appealing because of its mathematical simplicity,
this method, the reliability index β is expressed as a linear
consistency with earlier design practice, and transparency to
approximation of R and Q around the mean values. An advan-
most practicing engineers. This approach is commonly the ﬁrst
tage of this method is that it can provide approximate, closed-
to be used until load and resistance statistics are available.
form approximations for resistance factors.
However, the approach may not always address all sources of
If the random variables Q and R are normally distributed
uncertainty in an explicit manner.
and statistically independent, the resistance factor can be esti-
In this method, a resistance factor can be calibrated from a
mated as (Withiam et al., 1998):
FS value as follows:
λ R Σγ i Qi
Σγ i Q
φ=
φ≥ (3-13)
(3-11)
FS ΣQi Qm + βT σ 2 + σ Q
2
R
where all variables were deﬁned previously.
where all variables were previously deﬁned.

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13
evaluations that consider iteratively values of the random vari- Step 1: Establish a Limit State Function
ables are conducted and the distance β is recalculated until a
The limit state function is deﬁned as (Allen et al., 2005):
minimum value of β is found. The iteration starts by assum-
ing an initial value for the distance A-B. A disadvantage of this M = R−Q (3-17)
method is that the computational effort can be signiﬁcant for
certain problems and that a signiﬁcant volume of data is nec- where R and Q are random variables representing resistance
and the maximum load, respectively. A design equation repre-
essary to develop the joint probability distribution correctly
senting Equation 3-17 requires that φRRn − γQQmax ≥ 0, where φR
and accurately.
is a resistance factor; Rn is a random variable representing the
nominal resistance, γQ is a load factor, and Qmax is a random
Level III. The Level III calibration represents the highest
variable representing the maximum load. When M = 0, a non-
level of calibration complexity. This level involves formulat-
ing the problem with higher-order derivatives of random random value for Rn can be related by the following relation:
variables. For most geotechnical applications, however, this
γQ
method provides relatively small improvements in the accu- Rn = Qmax (3-18)
φR
racy of calculated load and resistance factors when compared
to those values provided by Level II calibrations. Therefore,
Using the previous equation, the general expression
the additional computational effort demanded by this level of
(Eq. 3-17) for the limit state function, M, can be written as:
analysis generally does not warrant its use.
In this investigation, Method C, Levels I and II, were used.
⎛ γQ ⎞
M = ⎜ Qmax ⎟ − Qmax (3-19)
⎝ φR ⎠
3.2.7 Steps to Perform the Calibration
of Resistance Factors Note that the two terms in Equation 3-19 that contain Qmax
are actually two separate random variables, each with different
To perform the calibration of resistance factors, the follow- statistical parameters and characterization, and each with both
ing steps are taken (Withiam et al., 1997; Allen et al., 2005): non-random and random components. The quantity Qmax as
used in the two terms of Equation 3-19 illustrate that the non-
1. Establish the limit state function (i.e., M = 0) that explic-
random part of the resistance and load random variables can be
itly incorporates load and resistance factors, γ and φ; related. Each of the random variables of Equation 3-19 is gen-
2. Obtain preliminary probability density function (PDF, erated separately in the Monte Carlo simulation. The simula-
usually normal or lognormal), cumulative density func- tions are unaffected if the random variables of Equation 3-19
tions (CDFs), and statistical parameters for random vari- are multiplied or divided by a non-random factor. Therefore,
ables R and Q; to simplify the calculations, both random variables are nor-
3. Select an acceptable probability of failure, Pf, and a corre- malized by the non-random value Qmax, which is equivalent
sponding target reliability index, βT; to adopting Qmax = 1 for the non-random components above
4. Fix load factors in the limit state using statistics or other (Allen et al., 2005).
means;
5. Adjust statistical parameters until there is a best-ﬁt of the
Step 2: Develop PDFs and Statistical Parameters
CDFs with data points;
for R and Q
6. Perform, in a Monte Carlo simulation, the following steps:
a. Estimate an initial, trial value for the resistance factor; In this step, the random variables are assigned a PDF and
b. Generate random numbers and generate values for R their statistical parameters are estimated based on existing
and Q that extrapolate the existing data; and data. The two most common distributions considered in geo-
c. Calculate random values of the limit state function, M; technical design are normal and lognormal.
7. Using graphical methods or other means, obtain the β If the variable Qmax is normally distributed, random values,
value that makes M = 0. Compare the calculated β with Qmax i, of this variable can be generated as:
the target reliability index, βT; modify the resistance fac-
Qmax i = Qmax mean (1 + COVQ z i )
tor and repeat the simulation until the calculated β co- (3-20)
incides with βT. At this point, the ﬁnal, calibrated resistance
where
factor is obtained.
Qmax i = a randomly generated value of the normal vari-
able Qmax;
Each of the previous steps is discussed in the following sub-
Qmax mean = mean of Qmax;
sections.

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14
COVQ = coefﬁcient of variation of the bias of Qmax; In addition, Rn mean = λR Ro, where λR is the normal mean of
zi = standard normal variable, which is the inverse the bias of Rn and Ro is the non-random scaling value deﬁned
Φ−1(uia) of the normal function Φ; and previously.
uia = a random number between 0 and 1 (represent-
ing a random probability of occurrence).
Step 3: Select Target Reliability Index
In addition, Qmax mean = λQ Qo, where λQ is the normal mean
Target reliability indices are selected based on the type of
of the bias of Qmax, and Qo is a non-random scaling value.
structure, importance of structure (i.e., related to conse-
If the variable Qmax is lognormal, random values of this
quences of failure), and the structural redundancy. Structural
variable can be generated as:
redundancy refers to the ability of a structure to transfer loads
to other members if one of its supporting members fails. Tar-
Qmax i = exp ( μ ln Q + σ ln Q z i ) (3-21)
get reliability indices typically range between 2 and 3 for typi-
cal geotechnical design (Barker et al., 1991). Allen et al. (2005)
where
recommend selecting βT close to 2, the lower end of the typical
μln Q = lognormal mean of Qmax and
range, when the structural component is not critical or it is
σln Q = lognormal standard deviation of Qmax.
redundant, and close to 3, the upper end of the range, when the
The above parameters can be obtained from the normal structural component is critical or it is non-redundant.
parameters deﬁned previously as: Zhang et al. (2001) suggested that it is acceptable to assign to
individual structural elements participating in a group a prob-
σ ln Q
2
μ ln Q = ln (Qmax mean ) − ability of failure that is higher than that of the group. Allen
(3-22)
2 et al. (2005) suggested that an individual element of a substruc-
ture can be considered redundant if the reliability index of the
and entire system is signiﬁcantly lower (i.e., 0.5 lower) than that of
individual components. This situation may occur in geotech-
σ ln Q = ln (COVQ + 1) (3-23) nical systems that rely on numerous structural elements (e.g.,
various layers of geosynthetic or steel reinforcement in a retain-
If the resistance is modeled as a lognormal variable, the ﬁrst ing structure or various driven piles in a pile group). Systems
term of Equation 3-19 can be randomly generated as: with various structural elements tend to have greater structural
redundancy and thereby result in a higher overall reliability
γQ
exp ( μ ln R + σ ln R z i )
Rni = (3-24) index than systems with few resisting elements. For example, a
φR
pile group is signiﬁcantly more redundant than a single drilled
pile. This concept will be applied to SNWs, as discussed in the
where
following paragraph.
μln R = lognormal mean of Rn;
Resistance factors for shallow foundations have been cali-
σln R = lognormal standard deviation of Rn;
brated using βT = 3.0 (corresponding to Pf = 0.14%, a relatively
zi = standard normal variable, which is the inverse Φ−1(uib)
low value), as these systems are not highly redundant (Baker
of the normal function Φ; and
et al, 1991). Resistance factors for deep foundations have been
uib = a random number between 0 and 1 (representing a
calibrated for βT = 2.33 (corresponding to Pf = 1%), as driven
random probability of occurrence, and being inde-
piles and drilled shafts are typically installed as part of pile/shaft
pendently generated from uia).
groups (Paikowsky et al., 2004) and thereby carry some struc-
tural redundancy. D’Appolonia (1999) used βT = 2.50 to cali-
The above parameters can be obtained from the normal
parameters for Rn as: brate resistance factors for pullout in geogrids, which is a system
that tends to be redundant as multiple reinforcement layers are
σ ln R
2
installed with a typical vertical spacing of 1 to 1.5 ft. Allen et al.
μ ln R = ln ( Rn mean ) − (3-25)
(2001) adopted βT = 2.33 for the calibration of pullout resis-
2
tance factors in MSE walls.
and
Step 4: Establish Load Factors
σ ln R = ln (COVR + 1) (3-26)
An estimate of the load factor needs to be performed to
evaluate whether the load factors [typically those used in
where
Rn mean = mean of Rn and AASHTO (2007)] are applicable or whether different load
COVR = coefﬁcient of variation of the bias of Rn. factors need to be proposed.

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Allen et al. (2005) present the following equation to estimate Step 7: Compare Computed and Target
Reliability Indices
the load factor when load statistics are available:
γ Q = λ Q (1 + nσ COVQ ) Following a cyclic calculation scheme, computed and tar-
(3-27)
get reliability indices are compared at the end of each Monte
Carlo simulation. The iteration is stopped when the differ-
where
ence between the computed and target reliability indices is
γQ = load factor;
negligible.
λQ = mean of the bias for the load Q;
COVQ = coefficient of variation of the load bias (i.e.,
3.3 Review of Current U.S.
measured-to-predicted ratio for loads); and
nσ = number of standard deviations from the mean of Q. Soil-Nailing Practice
3.3.1 Introduction
This procedure is approximate and is valid for any CDF.
The greater the selected value of nσ is, the lower the probabil-
In this section, the results of a review of current U.S. practice
ity will be that the measured loads exceed the nominal load.
of soil nailing are presented. The results of the review are pre-
Typically, the number of standard deviations of the load bias
sented as descriptions of the most signiﬁcant construction steps
is selected at nσ = 2, which results in a probability of approx-
of SNWs and the main components of an SNW. While this sec-
imately 2% for the factored load values (Allen et al., 2005) to
tion presents a summary of the review, more detailed informa-
exceed the nominal load. This procedure is currently used in tion of construction aspects and SNW elements are contained
the AASHTO LRFD Bridge Design Speciﬁcations and in the in Appendix B. After the main components of a SNW are iden-
Ontario Highway Bridge Design Code (as referenced in Nowak, tiﬁed in this section, a discussion is presented of the limit states
1999; Nowak and Collins, 2000). It is recognized that this pro- to be considered in the design of SNWs based on the LRFD
cedure is based on judgment and not necessarily on a rational method. For each of these limit states, a description of key vari-
procedure (Allen et al., 2005). ables participating in the limit state equation is provided.
Step 5: Best Fit Cumulative Density Functions 3.3.2 Basic Description of Soil Nail Walls
to Data Points
SNWs are earth-retaining structures constructed using
The selected CDFs for load and resistance must be ﬁtted to passive reinforcing elements, referred to as soil nails. The
the data points to assess the adequacy of the selected CDFs and term “passive” is used because soil nails are typically not post-
their statistical parameters. The CDF for loads, which is plot- tensioned. SNWs are constructed using “top-down” methods,
ted as variate, must be compared to the lower tail of the where excavation lifts are created and reinforcing elements
load data point distribution. Conversely, the CDF for resis- are installed after each lift excavation sequence. Soil nails are
tance must be compared to the upper tail of the resistance data installed in each excavation lift to provide lateral support to
point distribution. Finally, both load and resistance approxi- the soil exposed in each excavation level. As each excavation
mations should be plotted side by side and compared. lift is commonly 5 ft deep, nails are installed at a vertical spac-
ing of approximately 5 ft. Soil nails are commonly installed
with a horizontal spacing of 5 ft also.
Step 6: Conduct Monte Carlo Simulation
In U.S. soil-nailing practice, after a lift is excavated, holes
A Monte Carlo simulation is a statistical procedure used to (commonly known as “drill-holes,” regardless of whether
artiﬁcially generate many more values of load and resistance they are drilled or driven) are created on the exposed excava-
than are available from measured data points. Therefore, this tion. Drill-holes are created typically by drilling at an inclina-
technique can be used to extrapolate the data at both ends of tion of approximately 15 degrees from the horizontal; then,
the distribution. soil nails are inserted into the holes, and the annulus between
In a Monte Carlo simulation, random numbers are gener- the drill-hole and nails is ﬁlled with grout. Finally, a facing
ated independently for each of Qmax and Rn, assuming that these layer of reinforced shotcrete is applied over the protruding
variables are statistically independent. New sets of uia and uib nail heads at the face of the excavation. This cycle is repeated
are generated a minimum of 10,000 times to calculate new val- for each subsequent lift of excavation. Appendix B presents
ues for Qmax i and Rn i and to develop complete distributions of detailed information of other aspects of SNW construction,
these random variables. As Qmax and Rn are either normal or including contractor’s qualiﬁcations, information on suitable
lognormal, closed form solutions may be obtained for the methods to store and handle various materials used in SNW
construction, nail installation, grouting, and soil nail testing.
CDFs of the limit state.

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A more detailed description of construction aspects related external structural loads (e.g., weight of superstructure) are
to SNWs is presented in Byrne et al. (1998) and Lazarte applied. Steel bars used in SNWs are threaded and, as men-
et al. (2003). tioned earlier, are not commonly post-tensioned. In some
Soil nails and the facing layer both contribute to excavation cases, however, the upper rows of soil nails are post-tensioned
stability. While soil nails provide support to the soils retained as a means to control and limit the outward movement of the
behind the wall, the facing provides connectivity and struc- wall. Other elements commonly used in connection with the
tural continuity to nails, thus making the SNW act as a unit. soil nail bars are centralizers and bar couplers (not shown in
SNWs have been used successfully in a wide variety of sub- Figure 3-4, see additional descriptions in Appendix B).
surface conditions, including soils and rocks. Although nails
are used in soil and weathered rock, the term “soil nail” will
3.3.3.2 Facing System
be used interchangeably in this document whether the nails
are installed in soil or rock. SNWs can be more advantageous Facing systems typically consist of temporary and perma-
than other top-down retaining systems when the construction nent facing. Temporary facing is applied on the exposed soil
takes place in granular soils exhibiting some cohesion and/or as each lift is excavated to provide temporary stability. Per-
in weathered, soil-like rock. SNWs are generally unsuitable manent facing is applied over the temporary facing to provide
when they are built below the groundwater table. Additional architectural finish and structural continuity. Temporary
information related to favorable and unfavorable subsurface facing most commonly consists of reinforced shotcrete.
conditions for constructing SNWs are presented in Byrne et al. The reinforcement used in the shotcrete usually consists of
(1998) and Lazarte et al. (2003). (i) welded wire mesh (WWM), which is installed over the
In transportation projects, including those involving bridge entire facing; (ii) additional horizontal bars (commonly
substructures, SNWs are routinely used as permanent struc- called “waler bars”) that are placed around nail heads; and
tures having a minimum design service life of 75 years per (iii) additional vertical bearing bars that are also placed
AASHTO (2007). SNWs that are built as temporary structures around nail heads (see bottom of Figure 3-4). Permanent fac-
[i.e., service life up to 36 months per AASHTO (2007)] are rou- ing may consist of cast-in-place (CIP) reinforced concrete,
tinely used in urban settings for shoring up temporary excava- reinforced shotcrete, or precast concrete panels.
tions. However, the use of SNWs as temporary earth-retaining
systems in bridge substructures is uncommon. This document
3.3.3.3 Grout
focuses on SNWs used as permanent structures.
The practice of SNWs varies throughout the United States, Grout used in SNWs may consist of a mixture of neat Port-
particularly in non-public projects. SNW practice differing land cement mortar or ﬁne aggregate, cement, and water. Grout
from that described in this document may include the use of typically covers all the length of the steel bars, transfers tensile
different nail types (e.g., hollow steel bars as opposed to solid stresses from the bars to the surrounding soil, and provides cor-
bars), different nail materials (e.g., synthetic materials instead rosion protection to the bars. Grout is commonly applied in the
of steel), and novel construction procedures. However, none drill-holes under gravity using the tremie method.
of these variations are discussed in this document.
3.3.3.4 Components at the Soil Nail Head
3.3.3 Main Components of Soil Nail Walls
To provide connection between nails and facing at the pro-
truding soil nail heads, connecting components are installed
The main components of SNWs used in typical U.S.
at this location. These components typically consist of nut,
practice are identiﬁed on Figure 3-4. These components are
washers, bearing plate, and headed-studs or anchor bolts.
described in the following paragraphs. Additional informa-
The headed-studs or anchor bolts are attached to the bearing
tion on SNW components is contained in Appendix B. In
plate. Additional descriptions of nail head components are
addition, a more detailed description of typical components
provided in Appendix B.
of SNWs is presented in Byrne et al. (1998) and Lazarte et al.
(2003).
3.3.3.5 Drainage System
3.3.3.1 Steel Bars
A drainage system is typically installed behind the SNW fac-
ing to collect groundwater occurring behind the facing and
Reinforcing soil nails are solid steel bars. The bars develop
to convey it away from the wall. The most commonly used
tensile stresses in response to the outward deformation of
drainage system consists of composite, geosynthetic drainage
soils that are retained in each excavation lift. Soil move-
strips, which are also referred to as geocomposite sheet drains
ment can occur during excavation or after excavation when

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54
Table 3-11. Statistics of bias for nominal bond strength.
Resistance Parameters
Number
Coefficient Log Log
of Points Mean of Standard
of Mean of Standard
Material Distribution
in Bias Deviation
Variation Bias Deviation
Type
Database
λR σR μln σln
N COVR
Sand and
82 Lognormal 1.050 0.25 0.24 0.02 0.24
Sand/Gravel
Clay/
45 Lognormal 1.033 0.05 0.05 0.03 0.05
Fine-Grained
Rock 26 Lognormal 0.920 0.18 0.19 –0.10 0.19
All 153 Lognormal 1.050 0.22 0.21 0.03 0.21
variate on Figure 3-32. The distribution selected to ﬁt the data ulations were conducted to improve initial values presented
was also a lognormal distribution that was adjusted to match previously in this chapter.
the upper tail of the load bias distribution.
The bias calculated for each of these cases is presented
3.7.2 Description of Calibration Process
in Table 3-13. Statistical parameters are summarized in
Table 3-14. These parameters are also used in the calibration. The calibration was performed using the following steps:
3.7 Calibration of Pullout Step 1: Establish a limit state function;
Resistance Factors Step 2: Develop PDFs and statistical parameters for loads
and resistances;
3.7.1 Introduction
Step 3: Select a target reliability index for SNW design;
Step 4: Establish load factors;
This section presents the results of the calibration of pull-
Step 5: Best-ﬁt cumulative density functions to data points;
out resistance factors. The calibration was conducted apply-
Step 6: Conduct Monte Carlo simulation;
ing the calibration framework developed by Allen et al. (2005),
Step 7: Compare computed and target reliability indices; and
which was presented earlier in this chapter. Monte Carlo sim-
0
Data
Approximation
0.2
Normalized Depth, h/H
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4
T
Normalized Nail Force, K γ H S S
a hv
Figure 3-30. Summary of tensile forces measured in instrumented SNWs.

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Table 3-12. Characteristics of monitored soil nail walls.
Feature Case
Swift- Swift- IH-30, IH-30,
Feature Peasmarsh, Guernsey, San Cumberland
Oregon Delta Delta Polyclinic Rockwall, Rockwall, I-78, Allentown
U.K. U.K. Bernardino Gap, 1988
Station 1 Station 2 Section A Section B
Height (m) TBC 5.3 5.6 16.8 11 20 5.2 4.3 7.6 7.9 12.2
Face slope (deg) TBC 0 0 0 20 30 0 0 6 0 3 m bench
55 kN/m 75 kN/m
Back slope (deg) TBC 27 0 0 0 0 5 33 33
surcharge surcharge
Type of facing TBC shotcrete shotcrete shotcrete geogrid geogrid shotcrete shotcrete shotcrete shotcrete concrete panels
Nail length (m) TBC 6.4 5.2 10.7 6–7 10 6.1 6.1 6.7 13.4 6.1–9.2
Nail inclination
TBC 15 15 15 20 20 5 5 12 15 10
(deg)
Nail diameter (mm) TBC NA NA NA NA NA 152 152 203 114 89
Steel diameter (mm) TBC 29 29 36 25 25 19 19 25 29 25–32
Spacing, H x V (m) TBC 1.4 x 1 1.4 x 1 1.8 x 1.8 1.5 x 1.5 1.5 x 1.25 0.75 x 0.75 0.75 x .75 1.5 x 1.5 1.5 x 1.2 1.5 x 1.5

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57
2
11 Instrumented SNWs
(13 Points)
1.5
1
Standard Normal Variable, z
0.5
0
-0.5
-1
-1.5
-2
0.5 0.75 1 1.25 1.5
Bias λQ of Tm
Figure 3-32. Bias for maximum load in soil nails.
Q
Table 3-13. Summary of normalized measured and predicted
maximum nail load.
Normalized Normalized
Bias of
No. Case Measured Predicted
Load
Load, Tm Load, Tp
1 Cumberland Gap, 1988 0.54 1.05 0.51
2 Polyclinic 0.56 0.94 0.59
3 I-78, Allentown 0.68 1.07 0.63
4 Guernsey, U.K. 0.51 0.71 0.72
5 Swift-Delta Station 2 1.11 1.43 0.78
6 Oregon – 3-A 0.81 0.98 0.82
7 Swift-Delta Station 1 0.81 0.97 0.84
8 Peasmarsh, U.K. 0.58 0.65 0.89
9 Oregon – 2-B 1.05 1.10 0.95
10 IH-30, Rockwall, Section B 1.06 0.99 1.01
11 Oregon – 1-A 0.96 0.80 1.11
12 San Bernardino (R) 1.08 0.83 1.20
13 San Bernardino (L) 1.13 0.83 1.36
Table 3-14. Statistics of bias for maximum nail loads.
Load Parameters
Number
Coefficient Log Log
of Points Mean of Standard
of Mean of Standard
Distribution
in Bias Deviation
Variation Bias Deviation
Type
Database
λQ σQ μ ln σ ln
N COVQ
13 Lognormal 0.912 0.290 0.32 -0.140 0.31

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58
and tions. Statistical parameters for maximum loads on a soil nail
were derived previously in this chapter based on the analyses
σ ln = ln (COVQ + 1) of various instrumented walls. These values were summa-
(3-93)
rized in Table 3-14.
where
Tmax mean = mean of the random variable that includes Tmax, Step 3: Select a Target Reliability Index for SNW Design
and
COVQ = coefﬁcient of variation of the bias of the random As discussed earlier in this chapter, the selection of the tar-
get reliability index, βT, is a key factor in a reliability-based
variable that includes Tmax.
design. Because soil nails are installed relatively close to each
If the pullout resistance is modeled as a lognormal variable, other (i.e., vertical and horizontal spacing is typically 5 ft) and
the right-hand side of Equation 3-89 is randomly generated as: the resulting reinforcement density per unit area is relatively
high, SNWs are considered structures with relatively high
γQ
exp ( μ ln R + σ ln R z i )
RPO i = (3-94) structural redundancy. To be consistent with the current
φ PO
practice of selection of a target reliability index for elements
with high structural redundancy, βT = 2.33 (and Pf = 1%) was
where
selected for this study.
RPO i = a randomly generated value of the variable RPO;
γQ = load factor;
φPO = resistance factor for pullout; Step 4: Establish Load Factors
μln R = lognormal mean of RPO;
The expression used to estimate the load factor is as follows:
σln R = lognormal standard deviation of RPO;
zi = an inverse normal function, or Φ−1(uib); and
γ Q = λ Q (1 + nσ COVQ ) (3-97)
uib = a random number between 0 and 1 representing a
probability of occurrence (this number is independ-
where
ent from the number uia deﬁned previously).
γQ = load factor,
λQ = mean of the bias for the load,
The lognormal mean and standard deviation of RPO is
COVQ = coefﬁcient of variation of the measured to pre-
obtained from normal parameters for RPO as:
dicted load ratio, and
σ ln R nσ = number of standard deviations from the mean.
2
μ ln R = ln ( RPO mean ) − (3-95)
2 Using the statistical parameters and nσ = 2, the load factor
can be estimated as:
σ ln R = ln (COVR + 1) (3-96)
γ Q = 0.91(1 + 2 × 0.32 ) = 1.49 ≈ 1.5
where
RPO mean = mean of RPO; and The value γQ = 1.5 best represents the statistics used in
COVR = coefﬁcient of variation of the bias of RPO. AASHTO (2007). However, other load factors can be consid-
ered in the simulation and different resistance factors can be
In addition,
calculated. In this simulation (see Step 6), the following load
RPO mean = λ R Rmax factor values were considered to account for various loading
scenarios of SNWs, γQ = 1.0, 1.35, 1.5, 1.6, and 1.75. Resistance
factors for pullout were calculated for this series of load factors.
where
λR = the normal mean of the bias of RPO, and
Rmax = a non-random scaling factor, similar to the case of Step 5: Best-Fit Cumulative Density Functions
loads. to Data Points
CDFs for loads and resistances were generated via Monte
Step 2: Develop PDFs and Statistical Parameters
Carlo simulations using the statistics for load and resistances.
for R and Q
After the fitting curves were developed for each set of data
Statistical parameters for soil nail pullout resistance were points, they were plotted side by side, as shown in Figures 3-33
developed from the database presented in Appendix C. These through 3-36. The abscissas on these ﬁgures are values of the
values were summarized in Table 3-11 for various soil condi- random variables Tmax and RPO. The ordinates are values of

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59
Figure 3-33. Monte Carlo curve ﬁtting of load and Figure 3-35. Monte Carlo curve ﬁtting of load
resistance—sand. and resistance—rock.
Figure 3-34. Monte Carlo curve ﬁtting of load and
Figure 3-36. Monte Carlo curve ﬁtting of load and
resistance—clay.
resistance—all soil types.

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60
the standard normal variable z. CDFs are shown as essentially
continuous functions on Figures 3-33 through 3-36 (small
markers can be observed at the tails of the CDFs). Data points
for load (13 points) and resistance (varying number for each
soil type) are plotted as circles and diamonds, respectively, in
Figures 3-33 through 3-36. On the left of these ﬁgures, the
generated CDF for loads was compared to the upper tail of
the load data distribution and was veriﬁed to be equal or
greater than all data points. Conversely, on the right of these
ﬁgures, the generated CDF for pullout resistance was com-
pared to the lower tail of the resistance data distribution.
The distribution for pullout resistance was best-fitted
to match the lower tail of the resistance PDF. The curve-
fitting accuracy is unaffected by the upper tail of the resis-
tance CDF because it is the lower tail of the resistance dis-
tribution that controls the calculated reliability factor
(Allen et al., 2005).
Step 6: Conduct Monte Carlo Simulation
The Monte Carlo simulation was conducted to artiﬁcially Figure 3-37. Monte Carlo simulation—sand.
generate additional values of load and pullout resistance than
the ones available from data points and to estimate the prob-
for overall stability as a service limit state (per current AASHTO
ability of failure accurately. For each soil type, random num-
LRFD practice), is also included.
bers were generated independently for the random variables
For the case of λQ = 1.5 (case based on load statistics), the
containing Tmax and RPO. Independent values of the random
range of φPO varies from 0.70 to 0.77. This range is comparable
numbers uia and uib were generated in 10,000 trials to calculate
to the preliminary range varying from 0.63 to 0.70 obtained
new values for Tmax i and RPO i and to develop complete distri-
in Section 3.5.5 for FSPO = 2.0 and QDC/QLL ≥ 2.5.
butions of these two random variables.
Pullout resistance factors were calculated for the range of
γQ listed in Step 4. Figures 3-33 through 3-36 present the curve-
fitting analysis using Monte Carlo for different soils and for
γQ = 1.75. Figures 3-37 through 3-40 present results of the sim-
ulation of the limit function M for different materials and
γQ = 1.75. In all cases, βT = 2.33 and Pf = 1%.
Steps 7 and 8: Compare Computed and Target
Reliability Indices and Iterate, If Necessary
After a few iterations, results converged and the simulation
was stopped when the difference between the computed and
target reliability indices was smaller than 0.5%.
3.7.3 Results
The results of the calibration using Monte Carlo simula-
tions are included in Table 3-15. Various pullout resistance
factors were obtained for the various soil/rock types consid-
ered and for the range λQ = 1.0, 1.35, 1.5, 1.6, and 1.75 to show
the dependency of these factors. This range represents values
that can be commonly used for retaining structures that are
part of bridge substructures. The case of γQ = 1.0, applicable Figure 3-38. Monte Carlo simulation—clay.

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Figure 3-40. Monte Carlo simulation—all soil types.
Figure 3-39. Monte Carlo simulation—rock.
For the case λQ = 1.0, the pullout resistance factors φPO for ous conditions to quantify these differences. As will be seen,
various soils vary between 0.47 and 0.51. This range encom- these differences are small.
passes the value φPO = 0.5, which would be obtained based on The calibrated results also indicate that the reliability in
the ASD-based method as the inverse of a global safety factor design is approximately the same among all selected materi-
FSPO = 2 (see Chapter 4 and Lazarte et al., 2003). als, with soil nails in weathered rock having a slightly lower
Because of the values of the calibrated resistance factors for resistance factor.
pullout, it is expected that a LRFD-based SNW design that Overall, with reference to pullout resistances, the design of
uses this range of resistance factors would not produce signif- SNWs will not be affected signiﬁcantly by use of the LRFD
icant differences in results (i.e., in terms of soil nails, nail bar method in lieu of the ASD method. The same applies for
diameter, etc.) as compared to designs based on the ASD other resistance modes including nail in tension, and facing
method when a safety factor FSPO = 2 is used. Appendix D resistances because the factors associated with these resist-
ances were selected from the ASD practice.
provides detailed comparative designs of SNWs under vari-

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Table 3-15. Summary of calibration of resistance factors for soil nail pullout for various load factors.
Number
Coefficient Log Log λQ
of Points Mean of Standard
of Mean of Standard
Distribution
in Bias Deviation
Material Variation Bias Deviation
Type
Database
1.75 1.60 1.50 1.35 1.00
λR σR μln σln φR = φPO
COVR
N
Sand/Sandy
82 Lognormal 1.05 0.25 0.24 0.02 0.24 0.82 0.75 0.70 0.63 0.47
Gravel
Clay/Fine-
41 Lognormal 1.03 0.05 0.05 0.03 0.05 0.90 0.82 0.77 0.69 0.51
Grained
Rock 26 Lognormal 0.92 0.18 0.19 –0.10 0.19 0.79 0.72 0.68 0.61 0.45
All 149 Lognormal 1.05 0.22 0.21 0.03 0.21 0.85 0.78 0.73 0.66 0.49