ICOSSAR 2005, G. Augusti, G.I. Schuëller, M. Ciampoli (eds)
© 2005 Millpress, Rotterdam, ISBN 90 5966 040 4
Reliability analysis of allowable pressure of strip footing in spatially
varying cohesionless soil
Satyanarayana Murthy Dasaka
Research Scholar, Department of Civil Engineering, Indian Institute of Science, Bangalore, India
Rajaparthy Seshagiri Rao
Research Associate, Department of Civil Engineering, Indian Institute of Science, Bangalore, India
G L Sivakumar Babu
Associate Professor, Department of Civil Engineering, Indian Institute of Science, Bangalore, India
Keywords: bearing capacity, shallow foundations, reliability, uncertainty, spatial variability
ABSTRACT: This paper investigates the probabilistic analysis of bearing capacity of strip footing resting on
cohesionless soil deposit. In this study, the cone tip resistance of soil (qc) alone was taken as uncertain parameter. Random field theory was used to analyse the variability of qc. All the three possible phases of uncertainty (inherent, measurement and transformation) are considered in the analysis. The results obtained from
the analysis states that the transformation model plays a major role in the evaluation of variability of design
parameter. The effect of spatial variability and averaging distance on the standard deviation of design parameter is also studied. The probability that an allowable bearing pressure arrived from the deterministic analysis
using an appropriate factor of safety would fall below that defined from probability density function of bearing capacity was evaluated, and the corresponding reliability index is calculated. The study shows that the
conventional deterministic analysis using factors of safety three overestimates the allowable bearing pressure.
For a given target reliability index, the study shows that the simple and traditional probabilistic methodology,
which assumes infinite scale of fluctuation and neglects the spatial averaging, results in lower allowable pressures than that obtained from advanced probabilistic analysis.
1 INTRODUCTION
It is recognized that the in-situ soils are highly variable and the nature of variability of soil significantly
influences the design decisions. Vanmarcke (1977)
indicated in his classical paper on “probabilistic
modeling of soil profiles” that even a homogeneous
soil shows a significant variability in its engineering
behaviour. The variability is attributed to three major sources of uncertainty in geotechnical property
evaluation. The first source is the natural heterogeneity or the in-situ or inherent variability of the soil,
and is due to variation in mineral composition, stress
history, etc. The second source is insitu measurement, which can be attributed mainly to statistical
uncertainty and measurement error. However, this
uncertainty can be reduced at the expense of additional testing. The third source is transformation uncertainty, which occurs when engineering properties
are obtained through correlation with index properties (Phoon and Kulhawy, 1999a).
Many researchers applied probabilistic methods
to geotechnical problems (Vanmarcke, 1977;
Keaveny et al., 1989; Fenton, 1999; Jaksa et al.,
1999; Phoon and Kulhawy, 1999a&b; Duncan,
2000; Sivakumar Babu and Mukesh, 2004). Probabilistic analysis of shallow foundations has received good attention in the recent past (Cherubini,
2000; Griffiths and Fenton, 2001; Fenton and Griffiths, 2003).
The objective of this study is to present the characterization of soil variability and examine the effect
of soil variability on the allowable pressure of a strip
footing placed on a cohesionless soil. Cone penetration data from one of the National Geotechnical Experimentation Sites (NGES) are considered in the
analysis. The soil spatial variability is evaluated using the one-dimensional random field modelling and
the influence of variability is examined using reliability analysis.
2 PROBLEM AND SITE DESCRIPTION
Allowable pressure on a strip footing of width 1.2 m
resting on the surface of sand layer is analysed for
shear failure criterion using probabilistic approach.
All the three major sources of uncertainty, viz., in-
909
herent variability, measurement uncertainty, and
transformation uncertainty, are considered in the
analysis. Effect of spatial variability and averaging
distance on the bearing capacity is also studied.
Cone penetration data from National Geotechnical Experimentation Sites (NGES) Program funded
by the National Science Foundation (NSF) and the
Federal Highway Administration (FHWA) are used
for analysis in the present study. The data used in the
analysis pertain to Texas A & M University riverside-sand site, college station, and described as
CPT27. The soil exploration was carried out upto a
depth of 30 m below the ground level, and penetration results were reported at regular intervals of 2
cm. The ground water table is located at greater
depths. Figures 1 and 2 show the vertical soil profile
of the site and cone tip resistance profile used in the
present analysis.
0
Silty sand
cohesionless soil and cone tip resistance (qc). Since
the authors could not obtain the data for which the
transformation equation has been fitted, an alternative procedure has been adopted in this paper to obtain the bearing capacity. In this, the angle of internal friction (φ) is first calculated from cone tip
resistance using Equation (2). In Equation (2), pa and
σ vo are atmospheric pressure and effective overbur-
den pressure respectively. From the evaluated φ and
using the Equations (3 & 4) proposed by Meyerhof
the bearing capacity is evaluated.
Q u = 28 − 0.0052 (300 − q c )1.5
� q /p
φ = 17.6 + 11.0 log10 � c a
�
� σ vo / p a
(1)
�
�
�
�
�
φ� �
�
N γ = �e π tan φ tan 2 � 45 + � − 1� tan(1.4φ)
2� �
�
�
Q u = 0.5 γ B N γ
(2)
(3)
(4)
-4.0
Clean sand
-8.0
Silty sand
Usually a factor of safety of 3 (Cherubini, 2000)
is used to obtain the allowable bearing pressure from
ultimate bearing pressure to implicitly account for
all possible uncertainties arising right from site exploration to design stage.
-13.5
Very hard clay
Figure 1. Soil profile at the Texas A & M River
Side-Sand site (CPT 27)
0
The first step in reliability analysis is to decide on
the specific performance criterion and the relevant
load and resistance parameters, called the basic variables, Xi, and the functional relationship among
them. Mathematically, this relationship is described
as
1
Z = g(X1, X2, ….., Xn)
2
The limit state function for shear failure criterion
(Cherubini, 2000) is
Cone tip resistance (qc), kPa
0
Depth below ground surface, m
4 LIMIT STATE AND RELIABILITY
ANALYSIS
2000
4000
6000
8000 10000 12000 14000 16000
3
Z = g(q c ) = R − S
4
Figure 2. Typical vertical cone tip resistance profile
at Texas A & M River side-sand Site (CPT27)
3 DETERMINISTIC ANALYSIS
To the authors knowledge there exists only a single
transformation model, shown in Equation (1) proposed by Schmertmann (1978), to relate ultimate
bearing pressure (Qu) of shallow strip foundation on
910
(5)
(6)
where qc is cone tip resistance, R is the variable representing ultimate bearing pressure, and load S is the
deterministic parameter representing the allowable
pressure of the footing, and the probability that a
footing carries the allowable bearing pressure (S) is
evaluated in terms of reliability index (β).
The failure surface or the limit state of interest is
defined as Z = 0. This is the boundary between the
© 2005 Millpress, Rotterdam, ISBN 90 5966 040 4
safe and unsafe regions in the design parameter
space, and it also represents a state beyond which a
structure can no longer perform its intended function, for which it is designed. Hence failure occurs
when Z < 0. Therefore, the probability of failure, pf,
is given by the integral
A classical way of describing random functions is
through the autocorrelation function, ρ(∆z). It is the
coefficient of correlation between values of a random function at separation of ∆z. The statistically
homogeneous data are used to evaluate the autocorrelation function.
p f = � f X (x )dx
The degree of spatial correlation can be expressed
through an autocovariance function:
g ( x )≤ 0
(7)
where fX is the joint probability density function of
the random variables {X i } .
In this paper, reliability analysis is carried out using simple and advanced approaches. In simple approach, the effects of spatial variability of cone tip
resistance and spatial averaging distance are altogether neglected, and the variability of cone tip resistance is expressed in terms of first two moments
(mean and variance). In advanced analysis, along
with first two moments, the effect of spatial variability of cone tip resistance is also taken into account.
4.1 Test for statistical homogeneity
Stationarity is one of the important prerequisites for
statistical treatment of soil data (Jaksa et al., 1999).
It means that the trend in the data should be identified and removed from the raw data before conducting variability studies. The calculation of variance of
the data is likely to be biased with non-stationary
(statistically non-homogeneous) data. For a set of
data to be statistically homogeneous or stationary,
the mean and variance should be constant with depth
and the correlation of residuals at two different
depths is a function only of their separation distance,
rather than their absolute positions. There are many
techniques available in the literature for checking the
stationarity of the data. They are broadly classified
as parametric and non-parametric. Kendall’s τ test
(Daniel, 1990) is one of the most widely used nonparametric tests, and the same is used in the present
analysis.
4.2 Variance reduction function
Variance reduction function is derived in terms of
scale of fluctuation, δ, and spatial averaging distance, L, the distance over which the geotechnical
properties are averaged in conventional determistic
approach.
4.2.1 Autocorrelation function
Almost every engineering property in general, and
soil property in particular exhibits the spatial correlation. It means that the value of the soil property
tends to be similar for adjacent soil elements.
[
]
c(r ) = E (x i − z i )(x j − z j )
(8)
Where r is the vector of separation distance between
point i and j, E[.] is the expectation operator, xi is the
data taken at location i, and zi is the value of the
trend at location i. The autocorrelation function is
defined as
ρ(r)= c(r)/c(0)
(9)
where c(0) is the autocovariance function at zero
separation distance. For the limited data autocovariance function can be described in terms of moment
estimators given by Equation (10).
∧
1 nr
c(r ) =
(10)
� [(x i − z i )(x i + r − z i + r )]
n r − 1 i =1
where nr is the number of all pairs of data having a
separation distance r.
Theoretical functions are fitted to the experimental autocorrelation data, and the best fit is chosen
based on least square error approach. The corresponding autocorrelation distance, the distance
within which the soil property exhibits relatively
strong correlation (Jaksa et al., 1999), and scale of
fluctuation, δ, a parameter introduced by Vanmarcke (1977) are evaluated from the analytical expression of the best theoretical fit. The correlation
distance obtained from theoretical autocorrelation
function for triangular fit is shown in Equation (11).
� ∆z
�1 −
ρ ∆z = �
a
�0
�
for ∆z ≤ a
for ∆z ≥ a
(11)
Where ‘a’ is autocorrelation distance. The scale of
fluctuation is related to autocorrelation distance as
shown in Equation (12).
∞
δ = 2 � ρ r dr
0
(12)
4.2.2 Spatial average
The averaged value of a soil property contributing to
stability along the failure surface does produce more
effect than an extreme value of that property at a
Proceedings ICOSSAR 2005, Safety and Reliability of Engineering Systems and Structures
911
single point on the failure envelope. Hence, effect of
spatial averaging should also be taken into account
while assessing the variance of soil property. A more
useful way to deal with the spatial variability within
a statistically homogeneous soil mass is through spatial averages such as the average of soil property
over a depth interval of L (Vanmarcke, 1977). In
foundation analysis, this length, L, could be taken
equal to depth of zone of influence, i.e., 2B in case
of square footing and 4B for strip footing
(Cherubini, 2000; Lee and Salgado, 2002).
As the ratio of δ/L increases, variance reduction
factor increases. The point standard deviation (or
variance) of cone tip resistance is reduced by multiplying it with the corresponding reduction factor (ΓL
or ΓL2) to include both spatial variability and spatial
averaging effect of cone tip resistance. The point
variance of soil property can be reduced by proper
evaluation of variance reduction function.
Vanmarcke (1983) suggested the following relation for variance reduction factors considering the
scale of fluctuation and spatial averaging length for
a triangular fit.
L
�
L≤δ
�1 − 3δ ,
Γ =�
δ
δ
� (1 − ), L ≥ δ
3L
�L
2
L
(13)
(14)
2
2
� ∂T �
� ∂T �
� ∂T �
2
2
2
SD ξ2d ≈ �
� SD w + � � SD e + � � SD ε (15)
w
e
∂
∂
� ∂ε �
� �
�
�
2
2
� ∂T �
� ∂T �
� ∂T � 2
2
2
2
SDξ2a ≈ �
� SDε (16)
� SDe + �
� Γ (L ) SDw + �
� ∂ε �
� ∂e �
� ∂w �
in which T, t, w, e, and ε refers to the transformation
model, deterministic trend function, inherent soil
variability, measurement error, and transformation
uncertainty respectively.
912
The results of probabilistic analysis of bearing
capacity are presented in the following sections.
The cone tip resistance data are verified for stationarity condition using Kendall’s τ test in which
the value of τ needs to be close to zero to satisfy the
stationarity condition. The Kendall’s τ for the raw
data is 0.72. Detrending process is applied to remove
the trend in the cone tip resistance data and keep the
data fairly stationary. As complete removal of trend
is not practically feasible, a second order polynomial
trend is removed from the experimental data. The
Kendall’s test for the detrended data produced a
τ value of -0.04, which is close to zero. Hence, the
residual data are used for further statistical analysis.
Depth below ground surface, m
m ξd ≈ T(t ,0)
2
From the mean value of cone tip resistance within
the depth zone of influence (the depth till the failure
wedges reach below the base of footing � 2B), the
mean angle of internal friction (φ) is obtained from
Equation (2) as 41.28o. The ultimate bearing pressure, Qu using Equation (3 & 4) corresponding to
mean angle of internal friction is 1668 kPa. Moreover, a factor of safety of 3 on ultimate bearing pressure results in an allowable bearing pressure of 556
kPa.
-5000
0
Cone tip resistance (qc), kPa
From the earlier studies (Kulhawy and Mayne,
1990; Phoon and Kulhawy, 1999b) it is clear that the
design property is a function of inherent soil variability, measurement error, and transformation uncertainty, and can be obtained by combining the individual uncertainties in a consistent manner using
the second-moment probabilistic approach. The
mean, point variance and spatial average variance of
design property are approximated as shown in Equations (14-16) respectively.
2
5 RESULTS AND DISCUSSION
0.5
1
1.5
2
2.5
0
5000
10000
15000
Experimental data
Detrended data
2nd order polynomial trend
20000
Figure 3. Experimental data, trend and detrended cone
tip resistance data used in the analysis
Experimental autocorrelation function is obtained
from the stationary cone tip resistance data within
the zone of influence using Equation (10). From
least square error approach it is observed that triangular function given by Equation (11) best fits the
experimental autocorrelation function. Subsequently
the autocorrelation distance obtained from the parameter of the best fit is calculated to be 0.36 m. The
variance reduction factor for cone tip resistance data
evaluated from Equation (13) is 0.144. Table 1
© 2005 Millpress, Rotterdam, ISBN 90 5966 040 4
shows the statistical parameters such as Kendall‘s τ
values for experimental data as well as detrended
data, mean of trend, standard deviation, theoretical
best fit to autocorrelation function, autocorrelation
distance of cone tip resistance data, averaging
length, and reduction factor used in the first phase of
the reliability analysis. The point coefficient of
variation of inherent variability, which is evaluated
from stationary cone tip resistance data is 24%. The
above value is obtained by neglecting the effect of
autocorrelation and spatial averaging of cone tip resistance on the coefficient of variation. However,
consideration of the combined effect of spatial correlation and spatial averaging reduces the inherent
variability to 9%.
Table 1. Statistical parameters of cone tip resistance
Parameter
Kendall’s τ for experimental data
2nd order polynomial trend in the data
(z=depth of tip measurement from top)
Kendall’s τ for detrended data
Mean of trend, t (kPa)
Standard deviation of the residuals off
the trend, SDw (kPa)
coefficient of variation of inherent variability, CoVqc
Theoretical best fit to experimental
autocorrelation function
�
�
∆z
�
for ∆z ≤ 0.36�
�1 −
�ρ
= � 0.36
∆
z
�
�
��0
for ∆z ≥ 0.36�
�
Autocorrelation distance in vertical distance, m
Standard deviation reduction factor, Γvqc
Averaging distance, L (m)
Standard deviation spatial average, SDwa
(kPa)
Coefficient of variation of spatial average, CoVqca
Value
0.72
qc=927.98z2+
4023.6z
-0.04 �0
6617
1555
24%
1
0.36
0.38
2.4
591
9%
Experimental
autocorrelation
function
Autocorrelation (ρ∆z)
0.6
Theoretical best fit
(Triangular)
2
R =0.93
0.4
0.2
0
-0.2 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
-0.4
-0.6
-0.8
-1
The coefficient of variation of measurement uncertainty (CoVe) for qc in sands produced by electric
cone penetration test is reported to be between 5%
and 15% (Phoon and Kulhawy, 1999a). The standard
deviation of transformation uncertainty (SDε) for
Equation (2) is reported as 2.8° (Kulhawy and
Mayne, 1990). Using Equations (2 & 14-15), the
mean angle of internal friction (mφ) is 41.7° and coefficients of variation of angle of internal friction
(CoVφd) are 7.28% and 7.4% corresponding to 5 and
15% of CoVe. Taking into consideration the benefit
of autocorrelation and spatial averaging, the coefficients of variation (CoVφa) from Equation (16) are
6.82% and 7%. For the transformation model shown
in Equation (2), inherent variability and measurement uncertainty of cone tip resistance do not show
a significant effect on the uncertainty of angle of internal friction. This important observation has been
verified by Kulhawy and Phoon (1999b).
In the second phase of analysis, the uncertainty in
bearing capacity is evaluated from that of angle of
internal friction obtained from first phase. The standard deviation of transformation model given by
Equation (3) is estimated to be 0.5. From Equations
(3-4 & 14), the mean value of Qu is obtained as 1819
kPa. Similarly, from Equations (3-4 & 15-16) the
standard deviations of Qu without and with spatial
correlation and spatial averaging are 1263 kPa and
1105 kPa, which result in CoVQu of 70% and 61%
respectively. A lower scale of uncertainty in bearing
capacity could be seen in the later case. However, in
the second phase of analysis, inherent variability is
critical than the transformation model uncertainty.
Table 2 shows the results obtained from both the
phases of analysis. It is seen from the analysis that
the transformation model has been identified as the
primary factor influencing the degree of variability
of design parameter.
triangular
1.2
0.8
From Equation (13) it is shown that the variance
reduction factor is directly proportional to δ/L ratio
and the effect of autocorrelation and spatial averaging reduces the inherent variability of cone tip resistance and hence the total uncertainty of design parameters and leaves behind an economical design.
Separation distance or Lag (∆z), m
Figure 4. Experimental and theoretical autocorrelation
functions for the stationary cone tip resistance data
The reliability index (β) is calculated for the limit
state function shown in Equation (6), where R is
variable ultimate bearing pressure (Qu) represented
by mean ultimate bearing pressure of 1819 kPa and
coefficients of variation of 70% and 61% respectively using simple and advanced probabilistic approach, and S is the allowable bearing pressure obtained by using a factor of safety on deterministic
ultimate bearing pressure (1668 kPa).
Proceedings ICOSSAR 2005, Safety and Reliability of Engineering Systems and Structures
913
To demonstrate the importance of reliability studies in geotechnical engineering and in particular to
simplify the design procedure a log-normal distribution has been assumed for the variability of the bearing capacity, Qu. Nonetheless a more simplified
normal distribution has been discarded since the
bearing capacity can never take negative values.
The reliability indices obtained using simple and
advanced probabilistic approaches are 1.57 and 1.87
respectively. One could observe the higher reliability
index in advanced probabilistic approach, as it takes
into consideration the beneficial effect of spatial
variability in terms of autocorrelation distance and
spatial averaging. However, Cherubini (2000) suggests that for design of geotechnical structures, a reliability index of three, which corresponds to probability of failure, pf ≈ 10-3 is sufficient.
Table 2. Components and design uncertainty used in the analysis
value of 3, the allowable bearing pressure on the
footing has been reduced to 299 kPa. These allowable pressures correspond to conventional factors of
safety of 7.3 and 5.5 respectively. The results from
the analysis are shown in Table 3 for both simple
and probabilistic analyses. These estimated factors
of safety are higher than the suggested value of 3,
and implies that the deterministic analysis based on
factor of safety of 3 overestimate the allowable bearing pressure.
Table 3. Results of probabilistic analyses of bearing capacity
Method of probabilistic analysis
Simple
Advanced
β corresponding to FS=3
1.57
1.87
β corresponding to FS=4
2.03
2.34
Factor of safety corresponding to target reliability index (βT) of 3
7.3
(=1668/228)
5.5
(=1668/299)
First phase
(Equation 2)
Second phase
(Equation 3)
Mean design property, mξd
41.7°
1819 kPa
Coefficient of variation of
inherent variability, CoVw
24%
7.4%
Coefficient of variation of
spatially averaged inherent variability, CoVwa
9%
7.28%
Coefficient of variation of
measurement error, CoVe
5%-15%
--
Standard deviation of
transformation uncertainty, SDε
2.8°
0.5
6 CONCLUDING REMARKS
Coefficient of variation of
design property, CoVξd
7.28%-7.4%
70%
The following conclusions are made from the above
study.
Coefficient of variation of
spatially averaged design
property CoVξa
6.82%-7%
61%
Hence, allowable bearing pressures on the footing
are back calculated such that they produce required
target reliability index of 3 for the above two approaches. For example, corresponding to 556 kPa,
which is the allowable bearing pressure from shear
consideration using a factor of safety of 3 on ultimate bearing pressure, the reliability index using
simple probabilistic approach is 1.57. To increase
the reliability index to the level of 3, the allowable
bearing pressure on the footing should be reduced to
the level of 228 kPa. If the variance reduction is considered, the reliability index corresponding to the
allowable bearing pressure works out to be 1.87.
Since the reliability index corresponding to allowable bearing pressure with a factor of safety of 3 on
ultimate bearing pressure is less than the suggested
914
Recently Zekkos et al. (2004) also supported this
statement and stated that the conservatism in two
structures designed with same factor of safety would
not be the same, and hence, suggested that the deterministic analysis should always be performed in
alliance with probabilistic treatment of soil properties. The results obtained from the analysis are site
specific and needs further work in this direction.
Removal of second order polynomial trend seems
to be a viable alternative to obtain stationary data, as
the Kendall’s τ calculated from the residual off the
trend approaches zero.
Scale of fluctuation, δ, and spatial averaging
length, L, strongly influence the inherent variability
of soil property, CoVw. The inherent variability decreases with increase in L/δ ratio. The combined effect of spatial variability and spatial averaging reduces the inherent variability.
Transformation model has been identified as the
primary factor influencing the degree of variability
of design parameter. Hence, it plays a major role in
the estimation of degree of variability of design parameter. Depending on the transformation model
chosen the combined effect of all the individual uncertainties (inherent variability, measurement and
© 2005 Millpress, Rotterdam, ISBN 90 5966 040 4
transformation uncertainty) may either be more or
less than their individual uncertainty ranges.
Even though the individual uncertainties in the
first phase of analysis (i.e., inherent variability and
measurement uncertainty of cone tip resistance, and
uncertainty in transformation model) are higher, the
resulting combined variability of design parameter
(angle of internal friction, φ) using Equations (2, 15
& 16) is less.
The back calculated factors of safety corresponding to a target reliability index of 3 are 7.3 and 5.5
respectively for simple and advanced probabilistic
analysis. These factors of safety are generally considered higher than those adopted in routine foundation designs.
The higher values of factors of safety associated
with allowable bearing pressure obtained by probabilistic approach clearly demonstrates the importance of uncertainty studies in geotechnical engineering and strongly demands the need to include
probabilistic framework in geotechnical engineering
design.
ACKNOWLEDGEMENTS
The authors sincerely thank Prof. Phoon, National
University of Singapore, for his guidance in evaluating the transformation uncertainty and Prof. Paul
Mayne, Georgia Tech., USA for providing us the
necessary literature. The authors also thank the two
anonymous reviewers for their critical comments in
the review process.
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915