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The Effect of Uncertain Spatial Variability of Soil on Differential Settlernentof Footings
JoshuaW. Nine
Departmentof Civil Engineering
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The Effect of Uncertain Spatial Variability of Soil on
Differential Settlement of Footings
Joshua W. Nine
April 12, 2011
Abstract
While current practice in geotechnical engineering commonly uses an observational approach a method which is founded on engineering judgment learned only from years of experience - to
compensate for the uncertainty caused by the heterogeneous nature of natural soil, a reliability
based approach - similar to that used in other branches of engineering - might be used. In this
study, a finite element approach in tandem with Monte Carlo technique is used to probabilistically estimate the differential settlement under a pair of isolated footings. The soil properties
of primary interest in this study are shear modulus, G, and soil cohesion, c. The above soil
properties along with their associated uncertainties are quantified by analyzing CPT data using
random field technique.
1
Motivation
Uncertainty in soils has long troubled the geotechnical engineer. Even Karl Von Terzaghi, also known
as the father of soil mechanics, explained the importance of minor variation of soil in one of his early
papers [1]. He recommended designers assume the least favorable possibilities in their designs, a
task that is not economically feasible. Terzaghi later suggested a ‘learn-as-you-go’ approach which
was then expanded upon to the ‘observational method’ by Ralph B. Peak [2]. It is more economical
to design for the most probable than the most unfavorable. The observational method allows the
missing information in the design to be filled in during construction. The design is then altered to
accommodate for the new information [3]. However, this method requires that the uncertainty is
acknowledged and accepted by the project managers as well as the community [4].
There are essentially four ways for a geotechnical engineer to deal with uncertainty: ignore it, be
conservative, use the observational method, or quantify the uncertainty [4]. Ignoring the uncertainty
requires the engineer to make unjustified assumptions which lead to a design likely full of error.
While this method is illogical, it can be seen in practice by a considerable number of agencies and
corporations. Being conservative often leads to designs that are expensive, time consuming, and overdesigned. The observational method is far more logical, yet several limitations still exist. The engineer
must be able to contact the project manager if the design needs changed during construction. Also,
1
continuous field measurements and testing can be expensive. To reduce this expense, the engineer
would have to identify where the failure would likely occur, which is impossible [5]. Quantifying
the uncertainty is a relatively new approach based on reliability analysis. This method has been
successfully established in several other branches of engineering; however, is not used in current
practice in geotechnical engineering. This is largely due to fact that geotechnical engineers typically
are not familiar with the concepts of reliability theory and because this method often requires more
data and time than available [6].
The reliability based approach offers several advantages over the observational method. The observational method requires the geotechnical engineer to rely on engineering judgment and the opinion of
experts [4]. Good engineering judgment is learned from experience, not lectures or textbooks. It can
only be improved from years in the industry, studying the reasoning and conclusions behind designs.
A major limitation of the opinion of experts is that their response can widely vary. The opinion of
several experts will likely be better than the opinion of one expert. It should also be noted that
experts tend to underestimate uncertainty and are overly confident with their estimates as shown by
Haynes and Vanmarke [7]. While good engineering judgment should always be applied, a reliability
approach could be used to quantify the uncertainty without an expert’s opinion.
2
Techniques for Estimating Settlement of Shallow Foundations
Settlement is usually calculated using empirical approaches, analytical approaches, or through numerical approaches. Empirical approaches are those based observations and experiments such as the
Burland and Burbidge’s Method [8].
"
#2 0.7 0 1.25 BL
q
B
Se
= α1 α2 α3
L
BR
BR
pa
0.25 + B
(1)
where α1 , α2 , and α3 are correction factors with α2 dependent upon the N60 value, the standard
penetration resistance obtained from the field. An empirical approach gives the advantage of a simple
calculation which can be computed quickly and depends on untransformed data obtained from the
field. This method also has its limitations. It has been shown that empirical methods typically
overestimate settlement [9] which leads to over-designed geostructures, increased cost, and increased
construction time. Also, empirical calculations typically assume the soil to be a homogenous material,
the same throughout, which is clearly inaccurate. This leads to several problems such as cracks and
tilting from differential settlement due to variation in the soil.
Analytical approaches are based upon theory. In geotechnical engineering, analytic settlement
calculations are limited to using a linear elastic model based on Hooke’s law.
1
Se =
Es
Z
H
(∆σz − νs ∆σx − νs ∆σy ) dz
0
2
(2)
In reality, soil is an elastic-plastic material. By assuming a linear elastic model the entire strength of
the soil is not used. Further assumptions are typically made, such as the foundation is perfectly flexible
[10], to simplify calculations even more. Linear elastic assumptions yield quick, easy calculations but
often underestimate settlement.
Modeling soil as an elastic-plastic (nonlinear) material is more realistic. However, the resulting
set of equations is not amenable in closed form solution. Numerical approaches are used in such
situations. In civil engineering, finite element analysis is the most popular numerical technique. In
finite element analysis (FEA), the site being investigated is transformed into a mathematical model.
This model is then discretized into a finite number of elements, each with a predetermined amount
of nodes, called a mesh. To improve accuracy, more elements, or more nodes per element, can be
used but this also increases the number of equations to be solved. The mesh is essentially a piecewise
interpolation of the field with measurements taken at each node. The stiffness matrix of each element
is determined from the internal virtual work while the load vector for each element is determined
from the external virtual work. These describe the behavior of each element. The stiffness matrix
and load vector is determined by assembling the elemental stiffness matrices and load vectors. By
applying the boundary conditions, the system of equations can be solved to determine the behavior
of the structure. In Equation 3, [K] is the stiffness matrix derived from the internal virtual work, {u}
is the displacement at each node, and {P} is the force vector derived from the external virtual work.
[K]{u} = {P}
(3)
In geotechnical engineering, it is typical to assume a pinned connection, which prevents translation
in all directions, along the bottom of the model to simulate rigid bedrock. Also a roller connection,
which prevents translation in one direction, is typically assumed along the sides of the model. Modern
computers with FEA programs require the analyst to input the data which describe the geometry,
loads, boundary conditions, material properties, and mesh density (size and shape of elements). The
software will then generate the matrices and solve the system of equations. The analyst may then
select the information desired to be displayed.
The finite element method is still an approximation of the settlement. The approximation can be
improved by selecting different modeling methods of the soil constitutive behaviors, increasing the
number of elements used, or by increasing the number of nodes per element at the cost of an increase
in calculation time. Besides the advantage of improved accuracy, the finite element method could also
aid in accounting for variation in the field as discussed in the following sections.
3
Uncertainties in Soil Properties
While precise, technical mathematical equations carried to many decimal points will give very exact
solutions, soils are full of variation. The mathematical equation may suggest the soil will behave in
certain way when the true nature of the soil may be completely different. Therefore, calculations can
only be used as an estimation of how the soil will behave. This was explained well by Terzaghi at a
conference address in 1936,
3
“Unfortunately, soils are made by nature and not by man and the products of nature are
always complex. . . As soon as we pass from steel and concrete to earth, the omnipotence
of theory ceases to exist. Natural soil is never uniform. Its properties change from point
to point while our knowledge of its properties are limited to those few spots at which the
samples have been collected. Furthermore, its properties are too complicated for rigorous
theory, and approximate mathematical solutions are difficult for even the most common
problems. In soil mechanics the accuracy of computed results never exceeds that of a
crude estimate, and the principal function of theory consists in teaching us what and how
to observe in the field...” [11]
This non-uniformity described by Terzaghi can be classified as uncertainty. There are three generally
accepted sources of this variation in geotechnical engineering: inherent soil variability, measurement
error [12], and error from using correlation or transformation models to assume soil properties [13].
Terzaghi was referring to inherent soil variability in this statement which results from the geological
formation of the soil and the continuous modification of the soil with time. This type of uncertainty
resembles epistemic uncertainty which can be reduced with additional information such as additional
sampling sites [14]. The analogy between a deck of cards and soil formation has been used to describe
this uncertainty [15]. To predict the arrangement of cards in the deck is nearly impossible and
completely related to luck. However, by examining several cards from the deck, the arrangement
will become clearer. While additional information will likely result in a better understanding of the
geological formation of the soil at a site, it will not eliminate the uncertainty and is expensive. At
some point, the cost of reducing the uncertainty becomes more than it is worth [4].
Measurement error is introduced by human and equipment data collection and the selection of
sample site location. To reduce measurement error, human interference with data collection should
be minimized to restrict the error to the inaccuracy of the equipment. Equipment should be properly
maintained and calibrated, and the operator should be familiar with possible sources of error. One
example is the cone penetration test (CPT) which is a procedure used to classify materials and
measure soil properties vertically below the soil surface. If the cone encounters gravel or cobbles,
the penetration may be obstructed and cause the tip to travel diagonally instead of vertically. Also,
if parts of the cone become rusted or clogged, inaccurate measurements will occur [16]. The data
collection location also induces error if a site does not represent the general soil properties of the
surrounding area. For this reason, a design should not be based on a single sample site.
The error caused by transformation and correlation models is introduced when the field data,
such as CPT cone tip resistance, is used to estimate soil properties by an empirical correlation. The
data is also transformed if the engineer needs to assume a distribution for probabilistic calculations.
For some soil properties, a logarithmic distribution is assumed because it is bounded below by zero.
Another common transformation is raising the data to some power between 0 and 1. This is typically
used if the probability of returning a negative value is negligible [17].
There are several methods to attempt to account for uncertainty, the simplest being to use an
average or weighted average of the soil properties [10]. This has the advantage of being applicable
to empirical and analytical calculations. However, applying a probabilistic approach could prove to
be a more accurate means of dealing with uncertainty, such as the random finite element employing
Monte Carlo simulation. Here, the mean, variance and scale of fluctuation are determined for the
4
soil property. The scale of fluctuation, or correlation length, is essentially the length over which the
data is related. These three statistical parameters are used to generate a random field of the data
which models what the soil behavior of the site might be similar too. The loads are applied to the
generated soil structure, and settlement can be calculated using FEA. Then a new realization of the
random field is generated and settlement is calculated again. This process is known as Monte Carlo
simulation. After a predetermined number of realizations, the results are evaluated probabilistically.
The Monte Carlo simulation allows a deterministic model to resemble a stochastic model. The goal
of this simulation is to represent how the soil variability and measurement error affects the behavior
of the soil continuum.
3.1
Statistical Parameters
The classical sample mean and sample variance are used, calculated as,
n
µx =
n
1X
xi ;
n i=1
σx2 =
1X
(xi − µx )2
n i=1
(4a,b)
A biased variance estimator is used because it yields an effective error variance that is smaller, a
nonnegative definite covariance matrix [18], and because it is popular in time series analysis [19].
The scale of fluctuation, θ, is slightly more difficult to estimate. However, it should first be
noted that the scale of fluctuation is a parameter used in finite scale models. A finite scale model
assumes that the relationship between two points decreases rapidly with distance and is also known as
short-memory [20]. A fractal scale model assumes that two points are highly related, even with large
separation distances. Also, the estimation of the scale of fluctuation is dependent on the distance
over which it is estimated. Soil may vary greatly over just a few meters relative to those few meters
but may vary less when a larger distance is used, such as a thousand kilometers [21].
As the scale of fluctuation approaches zero, θ → 0, the values of any two points become independent
of each other, creating a white noise field. Due to the averaging effect, the results should be nearly
the same as if a deterministic approach was applied using the mean of the value. As the scale of
fluctuation approaches infinity, θ → ∞, the field becomes statistically homogeneous throughout.
Again, the results should be nearly the same as if a deterministic approach was used, and differential
settlement would approach zero [21].
There are several ways to estimate the scale of fluctuation. One method involves calculating the
area under the sample correlation function or fitting a curve to the sample correlation function and
determining the area below the curve. However, this method is typically highly biased [18]. Another
method involves iterating the sample variance function [22], but has not been found advantageous by
other authors [21]. A third method involves the maximum likelihood in the space domain and will be
used in this paper because it is linked to the commonly used 1D Markov model [18].
To determine the scale of fluctuation using the maximum likelihood in the space domain, the data
must be transformed into something that is approximately normally distributed. For the purpose of
this paper, the data will be transformed into the log-normal distribution by taking the logarithm of
5
the data. This creates a distribution that is nonnegative. A scale of fluctuation for each data sounding
is calculated using the following method. The average is then used for future calculations.
First, an initial scale of fluctuation is guessed. The corresponding correlation coefficient matrix,
ρ, is then calculated by Equation 5 where x is the data sounding.
ρij = e−2|xi −xj |/θ
(5)
Then Equations 6a and 6b are solved for r and s respectively. 1 is a vector with all elements equal
to 1.
ρr = x;
ρs = 1
(6a,b)
If r and s are solutions to Equations 6a and 6b, then the estimator for the mean can be determined
from Equation 7.
µ̂x =
1T r
1T s
(7)
The estimator for the variance can be calculated from Equation 8 which is derived from taking the
derivative of the likelihood function and setting it equal to zero.
σ̂x2 =
1
(x − µ̂x 1)T r
n
(8)
The log-likelihood value is then determined from Equation 9 [18].
n
1
L (x|θ) = − ln σ̂x2 − ln|ρ|
2
2
(9)
This is then repeated until the global maximum is found. Guesses for the scale of fluctuation can
simply be selected by stepping through a likely range of values. The scale of fluctuation should be
less than the maximum depth of the data. In this study, guesses were selected by stepping from 0.01
m to the maximum depth of the data by 0.01 m increments.
4
Random Field Modeling of CPT Data
For the purpose of this study, CPT records from a site will be analyzed to determine the statistical
parameters (mean, standard deviation, and scale of fluctuation) of the tip resistance. These statistical
parameters will then be used to generate pseudorandom CPT tip resistance records of the site which
will be transformed into soil mechanical properties. These transformed records are then used in
6
Figure 1: Local and global regression with sample CPT record.
conjunction with a FEA software and Monte Carlo simulation to determine the differential settlement
between two footings. A probability of unacceptable performance is then determined from the results.
4.1
Data Selection
CPT soundings were collected from a site around Berkeley, California provided by the United States
Geological Survey [24]. For the 47 soundings the usable depths ranged from 5 to 30.5 m in 0.05 m
increments. Only the tip resistance readings, qt, are analyzed in this paper as it best represents a
point property [17] and is easily transformed empirically to other soil properties such as the shear
modulus, G, and the cohesion, c.
It is assumed that there are no large voids in the soil; therefore, tip resistance readings should
not be negative or zero. If a sounding began or ended in negative values or zeros, the sounding was
shortened by removing these values. Also, if there was a zero or negative value at some point in the
CPT record, the value was replaced with a linearly interpolated value derived from the measurement
directly before and after that point. Soundings that contained large extended periods of negative or
zero values were rejected for the purpose of this study. Two records were discarded due to this fact.
Figure 1 is a sample CPT sounding from the Berkeley site.
4.2
Statistical Properties
The mean, standard deviation, and scale of fluctuation are needed to generate the soundings. The
mean is computed by determining the global linear regression of the data. The mean is thus broken
7
Figure 2: Average, minimum, and maximum of qt at each depth.
into two parts: a slope and an intercept, which allows for variation with depth. The standard deviation
is calculated by Equation 4b. The standard deviation of the soil for the use of this study refers to the
mean of the standard deviations of each sounding taken individually. The same applies for the scale
of fluctuation. It is expected that tip resistance will increase with depth; however, the Berkeley data
suggests that the tip resistance decreases with depth as seen in Figure 2. This is believed to be caused
by two or more layers of different types of soil. A soil with a lower tip resistance may lie underneath
a soil of a higher tip resistance. For this reason a decreasing linear regression will be used. Future
studies could be conducted using multiple regressions, one for each soil layer.
The data is then log-normally transformed which restricts the generation of negative and zero
readings. The scale of fluctuation is determined from the transformed data as previously described.
The standard deviation and mean of the transformed data is then calculated using Equations 10 and
11, respectively [21].
σln qt = ln 1 + σq2t /µ2qt
0.5
1 2
µln qt = ln (µqt ) − σln
2 qt
8
(10)
(11)
Figure 3: Generated qt sounding with depth
4.3
Generation of Pseudorandom Soundings
Because it is assumed that the scale of fluctuation is greater than zero, each generated sounding
should consist of correlated random numbers with an ultimate depth less than or equal to the ultimate
depth of the usable data. By assuming a log-normal distribution, generating these correlated random
numbers becomes simple. First, the correlation matrix, ρ, is determined from Equation 5. Since ρ is
positive definite in all physically feasible situations, a Cholesky factorization may be used to factor
the correlation matrix into an upper triangular matrix, S, and a lower triangular transpose, ST [25].
ρ = ST S
(12)
A vector, X, of n normally distributed pseudorandom numbers is generated where n is the number
of variables per generated sounding. Due to the limitations of the software used in this study, a one
meter step was used. A vector, Y, of n normally distributed variables related to X by S is then:
Y = ST X
(13)
The values are now correlated according to the scale of fluctuation but are standardized. To obtain
the random variables with the correct mean and standard deviation, the vector Z is computed by
Equation 14. Figure 3 is an example of a generated sounding using the previously described procedure.
zi = eµln qt +σln qt yt
9
(14)
Once the tip resistance soundings were generated, they must be converted into properties used in
settlement calculations: shear modulus, G, cohesion, c, and friction angle, φ0 . Shear modulus, defined
as the ratio of shear stress, τ , to the shear strain, γ, is used in measuring the stiffness of a material.
The shear modulus can be estimated from the tip resistance by Equation 15 [26] where γT is the unit
weight of the soil and g is the gravitational acceleration constant (9.81 m/s2 ). The unit weight was
assumed to be 20 kN/m3 .
G=
γT 2
V ;
g s
where
Vs = 1.75qt0.627
(15)
Cohesion is the force that holds the particles of the soil together and is usually determined by a
direct shear test or a triaxial test. However, it can be estimated from tip resistance using Equation
16 [26]. The total overburden stress, σo ,which varies with depth is dependent on unit weight of the
soil and pore water pressure and NK is a bearing capacity factor. In this study, the bearing capacity
factor was assumed to be 20. This is the suggested value according to Mayne and Kemper [27] for a
mechanical cone.
c=
qt − σo
NK
(16)
The friction angle is essentially the maximum angle before which the soil begins to shear. Like
cohesion, the friction angle is typically determined by a direct shear test or a triaxial test but can also
be estimated with an empirical correlation. Equation 17 [26] was used to correlate tip resistance to
friction angle where σo0 is the effective vertical overburden stress which is the total vertical overburden
stress minus the pore water pressure.
φ0 = arctan [0.1 + 0.38 (qt /σo0 )]
4.4
(17)
Modeling and Monte Carlo Simulation
As previously stated, the maximum depth of the model should be less than or equal to the ultimate
depth of data used. For this reason, the maximum depth of the model used in this study was restricted
to 10 meters. The width of the model was selected to be 30 meters to ensure the boundary of the
model did not affect the results of the settlement calculations. The 10 meter by 30 meter model was
then broken into 300, 1 meter by 1 meter square sections. A generated sounding was then assigned to
each column of square sections. Each section was then divided into 2 triangular elements with each
element consisting of 15 nodes as seen in Figure 4.
Two foot foundations with a six foot center-to-center span were added to the model. A distributed
load of 3000 kN/m was then added to each foundation. Assuming each story of a concrete frame
building induces a 300 kN load on a foundation, the applied load would be approximately equivalent
to a 20 story building. This was believed to cause a significant amount of settlement and would allow
the differential settlement to occasionally reach the maximum value as defined later in this paper.
10
Figure 4: FEA model with 600 triangular elements.
Whereas, a shorter infrastructure, say four or five stories, might not cause enough settlement to reach
the maximum settlement limit, let alone cause enough differential settlement.
Because soil and rock tend to behave in a nonlinear manner under loading, the Mohr-Coulomb
model was selected. This assumes the soil to be elastic perfectly plastic and is a first order approximation. The basic principle of elastoplasticity is that strains are decomposed into an elastic part and
a plastic part as seen in Figure 5 and Equation 18.
= e + p
(18)
To determine if plasticity occurs, a function of stress and strain called the yield function, f, is introduced. Plasticity occurs when f=0. A perfectly plastic model is a continuous model with a yield
surface fully defined by the model parameters [28].
The Mohr-Coulomb model requires five input parameters: shear modulus, cohesion, friction angle,
Poisson’s ratio, and dilatancy angle. Shear modulus, cohesion and friction angle are determined as
previously described. The Poisson’s ratio was assumed to be 0.25, a typical value of stiff sandy or
silty clays. Dilatancy angle is the angle where soil particles tend to expand in volume as they are
sheared. For the purpose of this study, dilatancy angle is assumed to be zero, a typical assumption
for clayey soils.
First, the effect of using randomly generated shear modulus were considered. The friction angle
and cohesion were set as their mean values while the shear modulus was assumed random. One
hundred realizations were conducted and settlement under each foundation was recorded. This was
then repeated by setting the friction angle and shear modulus to their mean values while allowing
cohesion to vary randomly. Finally, friction angle was set to its mean value and both shear modulus
and cohesion were assumed random and 20 realizations were conducted.
As a heuristic method, 20 realizations of a Monte Carlo simulation will provide a mean that is
relatively close to true mean when only one variable is random. One hundred realizations are typically
11
Figure 5: Elastic perfectly plastic model stress-strain curve.
needed to estimate the standard deviation when only one variable is random. However, to estimate
the mean when two variables are random, 2020 or approximately 1.05e26 realizations are required.
For this reasoning, it is assumed that the mean and standard deviation are relatively well estimated
when only shear modulus or only cohesion was random. The realizations where both shear modulus
and cohesion was assumed random should be loosely viewed as estimates as to whether shear modulus
or cohesion mostly controls differential settlement.
5
Probabilistic Simulation of Differential Settlement using
Random Finite Element
As previously stated, the depth used in this study was 10.5 m. From this the overall mean of the
qt data was 6060 kPa with a standard deviation of 7185 kPa (a coefficient of variance of 118%).
After taking the natural logarithm of the data, the coefficient of variation was 10.5%. The mean is a
function of depth therefore the slope was determined to be -1.0e-2 ln(qt )/m with an intercept of 8.3
ln(qt ). The scale of fluctuation, as determined from the maximum likelihood function, was estimated
to be 7.10 m.
Soundings of qt were then generated from the preceding statistics. From the generated soundings,
the overall mean of the data was 6100 kPa, only 0.6% larger than the true data. The standard
deviation was estimated to be 4300 kPa, significantly less than the true data. This is likely due to the
number of generated soundings. For the purpose of this paper, the difference in standard deviation
will be ignored. Further studies could be spent on the number of generations needed to better estimate
the standard deviation. The qt soundings were then converted to the desired soil property.
A differential settlement exceeding D/500 will likely cause serviceability or working limit state failure where D is the center-to-center distance between the two footings [29].Any differential settlement
12
beyond 12 mm for the two foundations with the center-to-center spacing of 6 m will be assumed to
fail. It is expected that using randomizing cohesion will have a lesser effect on differential settlement
than randomizing shear modulus as shear modulus tends to play a greater role in soil stiffness.
5.1
Uncertain Shear Modulus
As previously stated, one hundred realizations were calculated assuming only the shear modulus was
random. The assumed values for unit weight (20 kN/m3 ) and Poisson’s ratio (0.25) were used. The
mean values for friction angle (35◦ ) and cohesion (300 kPa) were also used. The shear modulus was
randomly determined as previously described. From the one hundred realizations, the mean of the
shear modulus was approximately 387,000 kPa with a standard deviation of approximately 700,000
kPa. Therefore, the coefficient of variation is approximately 180%.
Settlement under the footings ranged from 11 mm to 89 mm while differential settlement ranged
from 1 mm to 67 mm. From this the mean of the differential settlement was determined to be
approximately 14 mm, greater than the point of failure, with a standard deviation of 11.7 mm. After
assuming the results to be normally distributed, it is estimated that there is a 56.4% likelihood the
differential settlement will exceed the limit (12 mm). Figure 6 shows the differential settlement with
uncertain shear modulus.
Figure 6: Differential settlement with uncertain shear modulus. All other soil properties are assumed
deterministic.
13
5.2
Uncertain Cohesion
The same procedure used when shear modulus was considered random is used here varying cohesion
instead of shear modulus. The unit weight, Poisson’s ratio, and friction angle remained constant as
before at 20 kN/m3 , 0.25, and 35◦ respectively. The shear modulus was assigned its mean value of
400,000 kPa. From the varying cohesion, a mean of 298 kPa and a standard deviation of 366 kPa
was determined. The coefficient of variation is 123% which is significantly less than that when shear
modulus was allowed to vary but is still extremely large.
Settlement ranged from 11 mm to 21 mm. The differential settlement ranged from 0 mm to 7
mm with a mean of 2.3 mm and a standard deviation of 1.9 mm. This is significantly less than the
failure point. From this, it is estimated that there is a 1.03e-5% probability of failure. Therefore, the
probability of failure lies beyond 5 standard deviations from the mean. As expected, varying cohesion
had a significantly less effect on differential settlement compared to varying shear modulus. Figure 7
shows the assumed normal distribution with differential settlement with uncertain cohesion.
Figure 7: Differential settlement with uncertain cohesion. All other soil properties are assumed
deterministic.
5.3
Uncertain Shear Modulus and Cohesion
The simulation was then repeated with only 20 realizations varying both shear modulus and cohesion.
It is unrealistic to use a Monte Carlo simulation when both soil properties are varied due to the large
14
number of realizations need to determine an accurate probability of soil failure. Thus the purpose
of this set of realizations is to estimate whether the soil will respond similar to when either shear
modulus or cohesion was varied independently or in a new unique manner.
The unit weight, Poisson’s ratio, and friction angle remained constant as in the previous tests.
The shear modulus was assumed random and had a mean of 380,000 kPa and standard deviation of
613,000 kPa. The cohesion was also assumed random with a mean of 297 kPa and standard deviation
of 348 kPa.
Settlement of the two footings ranged from 17 mm to 95 mm while differential settlement ranged
from 3 mm to 60 mm. From the 20 realizations, the mean of the differential settlement was approximately 20.5 mm. Because the mean of the differential settlement is significantly greater than when
cohesion was solely varied, it is believe that further realizations would prove that varying both shear
modulus and cohesion would yield results similar to when strictly shear modulus was varied. Due to
the small amount of realizations computed, it is unclear as to how similar the two tests may be. Also,
for this reason, the probability of failure should not be estimated.
6
Conclusion
Assuming the least favorable conditions in a design application is not economically feasible. It is far
more economical to design for the most probable. The reliability based approach used in this study
is one possible method. Compared to the observational method, engineering judgment and expert
opinion no longer becomes the driving force behind a geodesign.
While empirical and analytical approaches can be used as a quick estimate when assuming soil
parameters to be constant, this should not be used to justify a design. In reality, the soil data
obtained from a site is highly uncertain due to inherent soil invariability and measurement error. A
finite element analysis is one method that can be used in conjunction with Monte Carlo simulation
to quantify the inherent soil invariability. Also, this method is only advantageous when assuming one
variable to be uncertain. When multiple variables are assumed to vary with position, this method is
no longer timely feasible as the required number of Monte Carlo run grows exponentially.
The primary result of this study is that when assuming one variable to be uncertain, it appears that
randomizing the shear modulus of the soil will affect the differential settlement more than assuming
soil cohesion uncertain. This was expected but was proven after 100 realizations of both situations.
When both were assumed uncertain, the data suggests that the outcome will be similar to that of
when only soil shear modulus was assumed uncertain. However, an insufficient number of realizations
were conducted and no further conclusions can be drawn.
Future studies should suggest a more efficient method for dealing with uncertainty using a probabilistic approach. Also other methods should be considered to handle the situation when multiple
soil properties are considered uncertain as they are in naturally occurring soil. A primary factor of
this study is assuming a normal distribution and the Mohr-Coulomb model. Further investigation in
other distributions and stress-strain models may be of interest.
15
7
Appendix I. References
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method in applied soil mechanics. Geotechnique, 19(2), 171-187.
[3] Terzaghi, K., Peck, R. B., & Mesri, G. (1996). Soil mechanics in engineering practice (3rd ed., pp.
34). New York: John Wiley & Sons, Inc.
[4] Christian, J. T. (2004). Geotechnical engineering reliability: How well do we know what we are
doing? Journal of Geotechnical and Geoenvironmental Engineering, 985-1003.
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Engineering Mechanics Division Specialty Conf., University of Waterloo Press, Waterloo, Ont., Canada.
[8] Burland, J. B., & Burbidge, M. C. (1985). Settlement of foundations on sand and gravel. Procedings, Institute of Civil Engineers, Part I, Vol. 7, 1325-1381.
[9] Sivakugan, N., & Johnson, K. (2004). Settlement predictions in granular soils: A probabilistic
approach. Geotechnique, 54(7), 499-502.
[10] Bowles, J. E. (1987). Elastic foundation settlement on sand deposits. Journal of Geotechnical
Engineering, ASCE, 113(8), 846-860.
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Engineers.
[12] Phoon, K. K., & Kulhawy, F. H. (1999). Characterization of geotechnical variability. Can.
Geotech., 36, 612-624.
[13] Phoon, K. K., & Kulhawy, F. H. (1999). Evaluation of geotechnical property variability. Can.
Geotech., 36, 625-639.
[14] Hacking, I. (1975). The emergence of probability. Cambridge, UK: Cambridge University Press.
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[16] (1990). Manual on estimating soil properties for foundation design. Ithaca, NY: Cornell University.
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[19] Priestley, M. B. (1981). Spectral analysis and time series (Vol. 1). New York: Academic Press.
[20] Beran, J. (1994). Statistics for long-memory processes. Monographs on statistics and applied
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random soil. ASCE J. Geotech. & Geoenv. Engrg., 128(5), 381-390.
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[24] United States Geological Survey. (2010, September 2). CPT Data. In Earthquake Hazards Program. Retrieved September 20, 2010, from http://earthquake.usgs.gov/regional/nca/cpt/data/?map=alameda-
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Appendix II. Notation
B = Width of foundation
BR = Reference width
c = Soil cohesion
D = Center-to-center distance between 2 footings
Es = Modulus of elasticity of soil
f = Yield function
G = Shear modulus of soil
g = Acceleration due to gravity
H = Thickness of soil layer
K = Structural stiffness matrix
L = Length of foundation
L = Log-likelihood function
n= Population size
NK = Bearing capacity factor
P = Force vector
pa = Atmospheric pressure
q0 = Applied pressure on foundation
qt = CPT cone-tip resistance
Se = Elastic settlement of foundation
r = vector used in maximum likelihood estimator
S = Cholesky upper triangular matrix
s = vector used in maximum likelihood estimator
u = Displacement vector
Vs = Shear wave velocity
X = Vector of standard normally distributed variables that are statistically independent
x = Vector of observations of sounding
Y = Vector of standard normally distributed variables related to X by S
z = Vector of random variables with correct means and standard deviations
α1 = Burland and Burbidge constant
α2 = Compressibility index
α3 = Burland and Burbidge correction for the depth of influence
γ = Shear strain
γT = Unit weight of soil
∆σx = Stress increase due to the net applies foundation load in x direction
17
∆σy = Stress increase due to the net applies foundation load in y direction
∆σz = Stress increase due to the net applies foundation load in z direction
= Strain
e = Strain due to elasticity
p = Strain due to plasticity
θ = Scale of fluctuation
µx = Sample mean
µ̂x = Estimated mean
νs = Poisson’s ratio of soil
ρ = Correlation coefficient matrix
σo = Total vertical overburden stress
σo0 = Effective vertical overburden stress
σx = Sample standard deviation
σx2 = Sample variance
σ̂x2 = Estimated variance
τ = shear stress
φ0 = Friction angle of soil
1 = Vector with all elements being 1
18