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Simplified Method for the Lateral, Rotational, and Torsional Static Stiffness of Circular Footings on a Nonhomogeneous Elastic Half-Space Based on a Work-Equivalent Framework

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Although there are many methods for assessing vertical stiffness of footings on the ground, simplified solutions to evaluate lateral, rotational, and torsional static stiffness are much more limited, particularly for nonhomogeneous profiles of shear modulus with depth. This paper addresses the topic by introducing a novel “work-equivalent” framework to develop new simplified design methods for estimating the stiffnesses of footings under multiple degrees-of-freedom loading for general nonhomogeneous soils. Furthermore, this framework provides a unified basis to analyze two existing design methods that have diverging results. 3D finite element analyses were carried out to investigate the soil–footing interaction for a range of continuously varying and multilayered nonhomogeneous soils, and to validate the new design approach.
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Simplified method for the lateral, rotational and torsional static stiffness of
circular footings on a non-homogeneous elastic half-space based on a
work-equivalent framework
Stephen K. Suryasentana1, Ph.D.
Paul W. Mayne2, Ph.D., P.E., M.ASCE
Affiliations
1 Lecturer, Department of Civil and Environmental Engineering, University of Strathclyde, 75
Montrose St, Glasgow G1 1XJ, UK.
2 Professor, School of Civil and Environmental Engineering, Georgia Institute of Technology,
790 Atlantic Dr., Mason Building 2245, Atlanta, GA 30332-0355, USA.
Corresponding author information
Stephen K. Suryasentana
stephen.suryasentana@strath.ac.uk
26 April 2021
2
Abstract
Although there are many methods for assessing the vertical stiffness of footings on the
ground, simplified solutions to evaluate the lateral, rotational, and torsional static stiffness
are much more limited, particularly for non-homogeneous profiles of shear modulus with
depth. This paper addresses the topic by introducing a novel ‘work-equivalent’ framework
to develop new simplified design methods for estimating the stiffnesses of footings under
multiple degrees-of-freedom loading for general non-homogeneous soils. Furthermore,
this framework provides a unified basis to analyze two existing design methods that have
diverging results. Three-dimensional finite element analyses were carried out to
investigate the soil-footing interaction for a range of continuously varying and multi-
layered non-homogeneous soils and to validate the new design approach.
Keywords
Settlement, stiffness, footings, foundations, soil/structure interaction, non-homogeneous
modulus, elasticity
3
NOTATION
depth below ground level
foundation diameter
 normalized depth with respect to foundation diameter
vertical load applied to foundation
lateral load applied to foundation
rotational moment applied to foundation
torsion applied to foundation
vertical displacement of foundation
lateral displacement of foundation
rotation of foundation
torsional displacement of foundation
shear modulus of elastic half-space (soil)
Young’s modulus of elastic half-space (soil) = 2(1+)
Poisson’s ratio of elastic half-space (soil)
displacement influence factor
 foundation rigidity correction factor
average vertical stress at the soil-foundation interface
normalized vertical stress distribution with respect to
factor controlling the rate of increase of the shear modulus with depth
reference shear modulus value
 equivalent constant shear modulus value for a non-homogeneous elastic half-space
elastic strain energy of a half-space
stress-based energy gradient
stress-based weight distribution
strain-based energy gradient
strain-based weight distribution
vertical stiffness of the soil-foundation interaction
lateral stiffness of the soil-foundation interaction
rotational stiffness of the soil-foundation interaction
torsional stiffness of the soil-foundation interaction
 atmospheric pressure
4
1 Introduction
The assessment of the performance of structures under loading depends on an understanding
of the interactions between the soil and the foundation of the structure. For shallow foundations
supporting structures such as wind turbines, transmission towers and offshore platforms, special
concerns must address the evaluation of soil-foundation response under multiple degrees-of-
freedom (DoF) loading, specifically vertical, lateral, rotational and torsional loading. Although
there has been much research in assessing soil-foundation response under multiple DoF
loading, most of them are focused on the ultimate limit response (e.g. Gourvenec and Randolph
2003; Gourvenec 2007; Nouri et al. 2014; Vulpe et al. 2014; Shen et al. 2017; Dunne and Martin
2017; Suryasentana et al. 2020a, b; He and Newson 2020). The assessment of soil-foundation
response at relatively small magnitudes of multiple DoF loading is, however, important for
applications such as structural fatigue analysis and natural frequency analysis.
The initial stiffness of the soil-foundation response at relatively small magnitudes of multiple
DoF loading can be estimated by assuming that the soil response is approximately linearly
elastic at relatively small loads. While there are existing design solutions for representing the
initial stiffness under multiple DoF loading, these solutions typically assume a homogeneous
elastic soil modulus profile where modulus is constant with depth (e.g. Poulos and Davis 1974;
Gazetas 1991), or an idealized non-homogeneous elastic soil modulus profile that conforms to a
specific parametric form (e.g. Doherty and Deeks 2003; Doherty et al. 2005; Efthymiou and
Gazetas 2018). As soils encountered in real life may deviate from the idealized non-
homogeneous profiles, computational methods such as three-dimensional (3D) finite-element
methods (FEM) can be used to obtain more realistic estimates of the initial stiffness for the
foundation. However, 3D FEM is not always practical for routine design purposes in
geotechnical engineering.
Therefore, this paper describes a novel framework for developing new simplified design
methods that can provide quick and approximate values of the initial stiffness of rigid circular
surface foundations under multiple DoF loading in general non-homogeneous (including multi-
5
layered) soils. Although this paper restricts its scope to rigid circular surface foundations as an
exemplar for the framework, simplified design methods can similarly be developed for other
foundation types (different shape, geometry or rigidity), following the procedures described in
the paper.
For this paper,  are defined as the vertical force, lateral force, rotational moment and
torsion that is applied to the center of the foundation base, and  are defined as the
corresponding vertical displacement, lateral displacement, rotation and torsional displacement
of the foundation (see Fig. 1). For homogeneous isotropic linear elastic soil, the reference
analytical solutions for the vertical stiffness  (Boussinesq 1885), lateral stiffness
 (Bycroft 1956), rotational stiffness  (Borowicka 1943), and torsional stiffness
(Reissner and Sagoci 1944) of a rigid circular surface foundation are expressed as
follows (Poulos and Davis 1974; API 2002; Kausel 2010):

(1)


(2)


(3)

(4)
where is the foundation diameter, is the (assumed homogeneous) shear modulus of the
soil, and is the soil Poisson’s ratio. Many natural soil formations exhibit a non-homogeneous
shear modulus profile, however, where the stiffness is represented by a continuously varying
shear modulus with depth; specifically, the initial shear modulus increases with mean
effective stress  in accordance with a power law format:

(5)
where  is the atmospheric pressure, is the reference shear modulus at atmospheric
pressure, and varies from approximately 0.5 for sands (Hardin and Black 1966, Wroth et al.
1979, Kohata et al. 1997, Houlsby et al. 2005) to 1.0 for clays (Hardin and Black 1968, Shibuya
et al. 1997; Yamada et al. 2008).
6
For non-homogeneous linear elastic soil, there is considerable work regarding vertical stiffness
in non-homogeneous ground (e.g. Gibson 1967; Carrier and Christian 1973; Kassir and
Chuaprasert 1974; Boswell and Scott 1975; Vrettos 1991; Selvadurai 1996; Doherty and Deeks
2003). One such design method is that proposed by Mayne and Poulos (1999), for which the
general form is:


(6)
where

(7)
 

(8)

(9)
in which is the Young’s modulus of the soil (which may vary with depth), is the depth below
ground level,  is the normalized depth with respect to the foundation diameter , is the
Boussinesq vertical stress distribution (Boussinesq 1885), is the horizontal stress distribution
for axisymmetric uniform loading (Poulos and Davis 1974) and is the average vertical stress
applied at the soil-foundation interface.  is the rigidity correction factor and equals
for
perfectly rigid foundations and 1 for perfectly flexible foundations. For the special case of
linearly increasing Young’s Modulus for the soil, Eq. 6 simplifies to:

(10)
where is the value of the soil Young’s Modulus directly beneath the foundation base (
and is the displacement influence factor whose values can be obtained from the design
charts in Mayne and Poulos (1999), or in closed-form in Mayne (2019).
Another widely cited design method for estimating the vertical stiffness comes from the field of
contact mechanics, where Gao et al. (1992) proposed the following to represent the settlement
of a rigid cylindrical punch on a non-homogeneous elastic half-space:
7





(11)
where




(12)



(13)
Eq. 11 is a closed-form solution that was derived using a first-order rigorous moduli-perturbation
method, where the reference solution for a homogeneous elastic half-space is used to estimate
the change in settlement in non-homogeneous elastic half-spaces.
For the evaluation of the lateral, rotational, and torsional stiffness of surface foundations on non-
homogeneous elastic soil, most previous research efforts use computational procedures such
as the scaled boundary FEM (e.g. Doherty et al. 2005; Birk and Behnke 2012) to obtain
estimates of these stiffnesses. Semi-analytical approaches based on the Green’s function (e.g.
Andersen and Clausen 2008; Lin et al. 2013) and simplified approaches (Anam and Roësset
2004) have been proposed to estimate the dynamic stiffness of surface foundations on multi-
layered elastic soils. However, there is a lack of simplified design methods that is amenable to
simple spreadsheet calculations, which can estimate the lateral, rotational and torsional static
stiffness of circular surface foundations on soil with general non-homogeneous (including multi-
layered) shear modulus profiles, similar to Eqs. 6 and 11 for the vertical stiffness problem.
Therefore, this paper aims to address this limitation by introducing a novel ‘work-equivalent’
framework that reveals a property of the elastic half-space that stays approximately invariant to
changes to the shear modulus. This framework is then used to develop new simplified design
methods to estimate the stiffness of the foundation on non-homogeneous elastic soil under
multiple DoF loading. This is a timely contribution as there is little guidance in the design codes
(e.g. API 2002) for this common design problem. Furthermore, this paper demonstrates that the
proposed framework provides a common basis to compare Eqs. 6 and 11 and helps shed light
8
on the possible causes for their diverging performance. Moreover, it bears the advantage of
allowing one single implementation to reproduce both design methods.
2 Work-equivalent framework
The work-equivalent framework is a framework that allows any non-homogeneous linear elastic
half-space to be transformed into a work-equivalent homogeneous elastic half-space. In other
words, a non-homogeneous half-space with some arbitrary shear modulus profile can be
converted into a homogeneous half-space with a constant shear modulus, which is defined such
that both half-spaces are ‘work-equivalent’. Under this framework, two linear elastic half-spaces
are ‘work-equivalent’ if it takes the same amount of work to produce the same amount of
displacement on both half-spaces. Conservation of energy then implies that two ‘work-
equivalent’ linear elastic half-spaces have the same amount of elastic strain energy.
To better illustrate the framework, consider the transformation of a non-homogeneous elastic
half-space into its work-equivalent, homogeneous counterpart . The value of may be
determined using two methods, depending on the key assumptions adopted. The following
exposition will describe the first method and its associated assumptions, before describing the
second method.
First method
Consider the elastic strain energy of a half-space:


(14)
where ,  are the stress and strain components of the half-space and the integration is
carried out over the entire volume of the half-space. From linear elasticity theory, it is known
that:
 

(15)
where  is the Kronecker delta.
9
Substituting Eq. 15 into Eq. 14 gives:

 

(16)
As this paper is only concerned with non-homogeneous shear modulus profiles that vary with
depth (i.e. ), Eq. 16 simplifies to:

(17)
where

 

(18)
As is calculated using the stress components and Eq. 17 implies that 
, the
parameter shall be termed the ‘stress-based energy gradient’.
Now, let and  be the elastic strain energy of and  respectively:

(19)

 
(20)
where  is the ‘equivalent shear modulus for .
Since and  are work-equivalent, :




(21)
where

 
(22)
Suppose the following assumption is true.
Assumption 1: is approximately invariant to changes in the shear modulus profile.
10
If Assumption 1 is true, there exists a unique weight distribution (for a fixed ) that can be
used to compute  for any non-homogeneous half-space using Eq. 21. Since is calculated
using the stress-based energy gradient , shall be termed the ‘stress-based weight
distribution’.
Second method
It is also known from linear elasticity theory that:
 


(23)
Substituting Eq. 23 into Eq. 14 and following the same procedure as before produces:
 
(24)
where




(25)
As is calculated using the strain components and Eq. 24 implies that 

, the parameter
shall be termed the ‘strain-based energy gradient’.
Since and  are work-equivalent, their elastic strain energy can be equated to give:

 
  
(26)
where

(27)
Suppose the following assumption is true.
Assumption 2: is approximately invariant to changes in the shear modulus profile.
11
If Assumption 2 is true, there exists a unique weight distribution (for a fixed Poisson’s ratio )
that can be used to compute  for any non-homogeneous half-space using Eq. 26. Since is
calculated using the strain-based energy gradient , is termed the ‘strain-based weight
distribution’.
2.1 Weighted average shear modulus
The work-equivalent framework suggests that if one of the above assumptions is true, there
exists some invariant weight distribution that can convert any non-homogeneous half-spaces
into work-equivalent homogeneous half-spaces using either Eq. 21 or Eq. 26. This would
involve finding the weighted average of the shear modulus after treating the x-y planes of the
half-space as ‘springs in series’ (if invariance is assumed) or ‘springs in parallel’ (if
invariance is assumed).
3 Assessment of assumptions
To assess the validity of the two assumptions, a 3D FEM study was carried out using the
commercial FEM software Abaqus v6.13 (Dassault Systèmes 2014). The 3D FEM model
consists of a rigid circular surface foundation of diameter on non-homogeneous elastic soil
with continuously varying shear modulus profiles (see Fig. 1) of the following form:

(28)
where is the diameter of the foundation, is a factor controlling the rate of increase of the
shear modulus with depth ( represents homogeneous ) and is a reference shear
modulus. Eq. 28 has the same parametric form as that adopted by Doherty et al. (2005).
The soil volume is defined as weightless and isotropic linear elastic. Four continuously varying
shear modulus profiles (and in Eq. 28), and six values of Poisson’s ratio (
) are analysed. Note that although six Poisson’s ratio values are analysed,
the figures in this paper only show the results for the practical values of 
12
(corresponding to drained sandy and undrained clayey materials, respectively) for illustrative
purposes. First-order, fully integrated, linear, brick elements C3D8 (or C3D8H for ) are
assigned to the soil elements. These elements are adequate as comparisons with initial
analyses using their higher-order counterparts (C3D20 or C3D20H) showed insignificant
differences.
The mesh domain is set to  for both width and depth, which is large enough to avoid
boundary effects based on preliminary results. Mesh convergence analyses have been carried
out to determine the mesh fineness. The 3D FEM mesh is shown in Fig. 2. Displacements are
fixed in all directions at the bottom of the mesh domain and in the radial directions on its
periphery. The surface foundation was modelled as a weightless, rigid body, and the loading
reference point RP was set at the center of its base, as shown in Fig. 1. Separation and slip at
the soil-foundation interface was prevented using tie constraints. Vertical, lateral, rotational and
torsional displacements are independently prescribed at the reference point RP to obtain the
vertical, lateral, rotational and torsional stiffness, respectively.
3.1 Assessment of Assumptions 1 and 2
To assess Assumptions 1 and 2, a pair of 3D FEM analysis is investigated for every non-
homogeneous shear modulus profile. Each pair consists of a 3D FEM analysis of the footing on
a non-homogeneous shear modulus profile and another 3D FEM analysis that is identical to the
former, except that the shear modulus profile is now constant with depth and the constant shear
modulus value is set such that both analyses result in the same work done by the footing. These
pair of 3D FEM analyses can then be used to calculate and , following the steps described
in Appendix A.
Figs. 3 and 4 compare the calculated values of and for the different non-homogeneous
shear modulus profiles. It is evident that stays approximately the same, while changes
significantly, for all displacement types. Therefore, the results provide strong support for
Assumption 1, but not for Assumption 2.
13
4 Existing design methods for vertical stiffness
One benefit of the work-equivalent framework is that it provides a common basis to compare
existing (and seemingly disparate) design methods for estimating the vertical stiffness. First,
Eqs. 6 and 11 are reproduced exactly using the work-equivalent framework. For example, Eq. 6
can be reproduced as (see Appendix B):


(29)
where


(30)
 can be manipulated into the form of Eq. 21 as follows:


(31)
where
 
(32)
Similarly, Eq. 11 (for a fixed Poisson’s ratio, ) can be reproduced as (see Appendix C):


(33)
where
  
(34)
 

(35)
It can be observed that Eqs. 29 and 33 have the same form as the reference vertical stiffness
solution for a homogeneous elastic half-space (i.e. Eq. 1), where the constant shear modulus in
Eq. 1 is now replaced by an equivalent, weighted shear modulus (i.e.  or ). Thus, this
suggests that Eqs. 6 and 11 can be viewed as belonging to the same class of weighted shear
modulus design methods, albeit with different ways of applying the weights.
14
The same 3D FEM model described in Section 3 was used to carry out additional FEM analyses
for the vertical stiffness problem, with the only difference being the application of a smooth
constraint at the soil-foundation interface (i.e. soil is free to move horizontally at the interface) in
order to match the assumptions behind Eqs. 6 and 11. Fig. 5 compares the vertical stiffness
estimations of Eqs. 29 and 33 with these 3D FEM results. Note that the numerical integration of
Eq. 31 starts from a small depth ( = 10-5) to avoid a singularity when the shear modulus is
zero at the ground level. It is evident from Fig. 5 that the Mayne and Poulos (1999) estimations
agree well with the 3D FEM results, while the Gao et al. (1992) estimations agree only at low
levels of non-homogeneity (i.e. low values).
To better understand the possible reasons behind the diverging performance of these two
design methods, Fig. 6 compares the weight distributions and calculated from the 3D
FEM results, which shows stronger support for Assumption 1 than for Assumption 2.  and
are also shown in Fig. 6 for comparison, which shows that  agrees well with the
calculated weight distributions . Fig. 6(c), (d) also explains why Eq. 33 estimates increasing
stiffness with increasing in Fig. 5, as there is a larger weighting of the higher shear modulus at
greater depths compared to the true weight distribution, which results in an overestimated
equivalent shear modulus. It can also be observed that the type of constraint at the interface
has little influence on the assumptions assessments, as there is negligible difference between
Fig. 6 and its corresponding results when a fully tied constraint is applied at the interface i.e.
compare Fig. 6(a), (b) with Fig. 3(a), (b), and Fig. 6(c), (d) with Fig. 4(a), (b). In summary, Fig. 6
shows stronger support for the underlying assumptions behind Eq. 29 (i.e. Assumption 1) than
those behind Eq. 33 (i.e. Assumption 2), which possibly explains the diverging performance of
these two design methods.
15
5 New design methods for lateral, rotational and torsional
stiffness
In general, Fig. 3 indicates support for Assumption 1 for all displacement types. Thus,
Assumption 1 is adopted to develop the following new design methods to estimate the lateral,
rotational and torsional stiffness of rigid circular surface foundations on non-homogeneous soils:

 
(36)

(37)


(38)
where  is the constant, equivalent shear modulus that is calculated using Eq. 21 (note that
is different for each stiffness in Eq. 21).
To determine the weight distribution for each stiffness, new parametric equations  are
derived to approximate the invariant weight distributions shown in Fig. 3. For simplicity, these
parametric equations are assumed to have the form of the Weibull distribution,




(39)
where  are the Weibull parameters.
Least squares regression is carried out to identify the optimal Weibull parameter values that
best fit the true weight distributions calculated from the 3D FEM results for each Poisson’s ratio
, under the constraint that . This constraint is applied so that there is zero weight at the
ground level (i.e.  at ), in order to accommodate zero shear modulus at the
ground level in Eq. 21. Consequently, there is some loss of accuracy in the match between the
best-fit  and the weight distributions calculated from the 3D FEM results; however, this is
considered as an acceptable trade-off for the convenience of being able to accommodate zero
shear modulus at the ground level (as is commonly idealised for real world soil profiles).
16
After obtaining the best-fit Weibull parameters for each Poisson’s ratio , the Weibull parameter
was found to vary insignificantly with . Least squares regression is then carried out to fit a
power law-based equation () for the Weibull parameter in terms of . The
resultant equations for the Weibull parameters are listed in Table 1. Fig. 7 shows the fit between
these equations and the best-fit Weibull parameters for each Poisson’s ratio . Fig. 3 also
shows the resultant Weibull-based parametric weight distributions  for estimating the
lateral, rotational and torsional stiffness of the foundation, where it is evident that the parametric
weight distributions capture the salient trends of the true weight distributions calculated from the
3D FEM analyses.
In summary, there are three design methods (Eqs. 36 to 38) for estimating , , and ,
respectively. Each of these design methods has a different Weibull-based weight distribution
 to evaluate . The weight distribution  can be derived by using the relevant
equations for the Weibull parameters (), as shown in Table 1.  can then be calculated
using Eq. 21. For practical purposes, it is sufficient to integrate Eq. 21 to a depth of , instead
of infinite depth, as  

. However, if there is a rock bed at, or if there is limited
ground data up to, a shallower depth of , Eq. 21 should be integrated to this depth and
 should be calculated using the following normalized parametric weight distribution:

 
 
(40)
This will ensure that the 
 is a normalized weight distribution that sums up to 1 (as is the
case for the Weibull distribution). A summary of the workflow for estimating the stiffness using
the new design methods is shown in Fig. 8.
To validate the new design methods, the stiffness values calculated using Eqs. 36 to 38 are
compared with those calculated using the 3D FEM analyses, as shown in Fig. 9. It can be
observed that the estimations of the proposed design methods agree well with the 3D FEM
results, with the maximum deviations being 6.48%, 9.54% and 3.70% for and ,
respectively.
17
6 Assessment of proposed design methods in complex soil
profiles
Although the proposed design methods may be readily applied to any arbitrary non-
homogeneous soils, their reliability for complex (e.g. multi-layered) grounds with shear modulus
profiles that deviate from the idealized form (Eq. 28) have not been validated. Therefore, to
validate this, 11 complex soil profiles were investigated.
The first two soil profiles correspond to multi-layered clay soil profiles that are representative of
realistic ground conditions for offshore wind farm sites (Burd et al. 2020). The first soil profile
(termed ‘BC clayprofile) has soft clay (Bothkennar clay) overlying stiff overconsolidated clay till
(Cowden till). The second soil profile (termed ‘BCB clay profile) is a Bothkennar clay soil matrix
with an interbedded Cowden till layer (see Fig. 10 for the schematic diagrams). These soil
profiles were first investigated to validate the application of a pile design model for layered soil
conditions. The third soil profile (termed ‘EURIPIDES’ profile) corresponds to a sand test site for
the EURIPIDES project (Niazi and Mayne 2010), which investigated the performance of axially-
loaded piles in dense sand. The shear moduli of these three soil profiles are shown in Fig. 11.
The remaining eight soil profiles correspond to three-layered soil profiles similar to those
investigated by Poulos (1979) for his comparisons of solutions for settlement of piles in layered
soil. The Young’s modulus of each soil layer is assumed to be constant and Table 2 lists the
values of the Young’s modulus and Poisson’s ratio for each soil profile (referred to as P1 to P8
in this paper).
Collectively, these 11 soil profiles are highly challenging for existing design methods (e.g.
Doherty et al. 2005), as these design methods typically require an idealized shear modulus form
and it is not straightforward to enforce a ‘best-fit’ of the idealized form (Eq. 28) to these complex
soil profiles. In contrast, these soil profiles do not pose difficulties for the proposed design
methods, since no fitting to an idealized form is required.
18
To validate the proposed design methods, 3D FEM calculations are carried out to estimate the
soil-foundation stiffness using the same FEM model described in Section 3, except for the
different shear modulus profiles. A foundation diameter of 10m is adopted for this numerical
study. Sand and clay soils are assigned a Poisson’s ratio of 0.2 and 0.49, respectively.
The proposed design methods are used to estimate the lateral, rotational and torsional stiffness
of the foundation in these soil profiles. Fig. 12 compares these estimated stiffness values with
their corresponding 3D FEM calculated values, which shows that the proposed method
performs reasonably well in these challenging ground conditions. For the soil profiles that are
more representative of real world ground conditions (i.e. BC clay, BCB clay and EURIPIDES),
the estimated values for and agree very well with the 3D FEM values, with the average
deviations being 1.02% and 2.71% for and , respectively, and the maximum deviations
being 1.44% and 3.88% for and , respectively. The estimations also agree reasonably
well with the 3D FEM calculated values, with the average deviation being 8.62% and the
maximum deviation being 12.23%, which is broadly in line with the maximum deviation of 9.54%
obtained for the calibration cases in Fig. 9 (b). For the P1 to P8 soil profiles, the estimated
values for agree very well with the 3D FEM values, with the average deviation being 3.3%
and the maximum deviation being 5.17%. The estimated values for and agree
reasonably well with the 3D FEM values, with the average deviations being 10.93% and 9.27%
for and , respectively, and the maximum deviations being 17.76% and 17.83% for and
, respectively.
Compared to existing simplified design methods, the proposed design methods are more
versatile as they can be applied to both continuously varying and multi-layered soil profiles
(although it is noted that their accuracies have only been validated for a finite number of soil
profiles due to practical reasons). Compared to 3D FEM analyses, the proposed design
methods are much faster and more computationally efficient. For example, each 3D FEM
analysis in this study took an average of 5 minutes (not including the non-negligible model setup
time) to estimate the static stiffness, while the proposed design methods took less than a
second. This is particularly important for running sensitivity analysis or for the design of large-
19
scale projects such as wind farms, where there is need for a large number of rapid and low-cost
calculations for the optimal sizing of many foundations in variable ground conditions.
Nevertheless, different analysis models are well-suited to meet the requirements at different
design stages. At the preliminary design stage, a simplified model such as those recently
published (e.g. Bordón et al. 2021) or those proposed in this paper would be sufficient to
estimate the foundation stiffness at very low cost; while rigorous but more computationally
intensive models such as 3D FEM would be more appropriate for verification of design at the
final design stages and for modelling very complex, rarely encountered soil profiles or
unconventional distributions of soil-foundation interface pressure.
7 Conclusions
Simplified design methods were derived for evaluating the lateral, rotational and torsional static
stiffness of circular surface foundations on general non-homogeneous (including multi-layered)
elastic soil; they can be implemented numerically using a simple spreadsheet approach. These
design methods were obtained using a novel approach called the ‘work-equivalent’ framework,
which shows that there exists some invariant weight distribution that can be used to convert any
non-homogeneous half-spaces into ‘work-equivalent’ homogeneous half-spaces. 3D FEM
analyses were carried out to validate the assumptions behind this framework and to determine
the weight distributions for each stiffness. Moreover, this framework is used as a common basis
to analyse two existing design methods for estimating the vertical stiffness of the foundation,
which elucidates the plausible reason behind the diverging results of the two design methods.
The proposed design methods have been verified for selected continuously varying shear
modulus profiles that cover the range of practical interest for homogeneous clayey or sandy
grounds. 11 complex, multi-layered shear modulus profiles were also assessed to validate the
proposed design methods in more challenging ground conditions, which showed good results by
the proposed design methods. Further studies are desirable to verify the robustness of the
proposed design methods for a much larger dataset of real world soil profiles.
20
8 Data Availability
Some or all data, models, or code that support the findings of this study are available from the
corresponding author upon reasonable request.
9 Acknowledgements
Parts of the work described here were conducted during the DPhil studies of the first author at
the University of Oxford. The first author would like to thank Professor Byron Byrne, Professor
Harvey Burd and Mr Avi Shonberg for their generous support during the studies, and Ørsted
Wind Power for funding the DPhil studentship.
21
10 Appendix
A: Calculation of and from 3D FEM results
and can be calculated using the following alternative forms of Eqs. 17 and 26:


(A1)


(A2)
where 
 can be approximated from the 3D FEM results through numerical differentiation of the
distribution, noting that  at each depth can be calculated by summing up the elastic strain
energy of all soil elements at that depth (all soil elements at each depth have the same height
as a structured mesh is used).
For each 3D FEM analysis for a non-homogeneous shear modulus profile, there is a
corresponding 3D FEM analysis that is identical to the former, except that its shear modulus is
now constant with depth and its shear modulus value is set such that both analyses result in the
same work done by the footing. The former analysis provides the values of  and  in
Eqs. 22 and 27, while the latter analysis (with the constant shear modulus profiles) provides the
values of 
and 
in Eqs. 22 and 27.
22
B: Derivation of design method equivalent to Mayne and Poulos (1999)
The main principle behind the design method of Mayne and Poulos (1999) is that the vertical
displacement at the center of the foundation base is the integration of the vertical strains
directly beneath it:
 













(B1)
Eq. B1 estimates the stiffness of a flexible, circular surface foundation. For a rigid, circular
surface foundation, a factor of
(Mayne and Poulos 1999) should be applied such that

, which produces Eq. 29.
23
C: Derivation of design method equivalent to Gao et al. (1992)
Assuming that the Poisson’s ratio is constant with depth, the following shows how Eq. 33 can
be derived from Eq. 11, which was proposed by Gao et al. (1992):











 


 


 
(C1)
24
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27
Table 1. Best-fit Weibull parameters (i.e. ) for the parametric weight distributions  (Eq.
39) corresponding to the different stiffness.
Stiffness
Lateral
1.27
0.237 - 0.049
Rotational
1.35
0.17 + 5
Torsional
1.46
0.076
28
Table 2. Young’s modulus profiles for eight three-layered, non-homogeneous soil profiles
analysed in the numerical study.  is the reference Young’s modulus of the soil, is
the soil Poisson’s ratio, is the depth below ground level and is the diameter of the surface
foundation.
Normalised Young’s Modulus, 
Name


P1
0.2
1
2
4
P2
0.2
1
4
2
P3
0.2
2
1
4
P4
0.2
2
4
1
P5
0.49
1
2
4
P6
0.49
1
4
2
P7
0.49
2
1
4
P8
0.49
2
4
1
29
Figure 1 Schematic diagram of a circular surface foundation of diameter bearing on an elastic
half-space with homogeneous and non-homogeneous shear modulus profiles (see Eq. 31).
30
(a)
(b)
Figure 2 (a) 3D FEM mesh for a rigid, circular surface foundation on an elastic half-space. (b)
Enlarged partial view of the foundation
31
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 3 Comparison of under different prescribed displacements: (a), (b) vertical
displacement; (c), (d) lateral displacement; (e), (f) rotation; (g), (h) torsion. Note that
represents the homogeneous elastic half-space case.  are the parametric weight
distributions assumed for the proposed design methods (Eqs. 36 to 38).
32
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 4 Comparison of under different prescribed displacements: (a), (b) vertical
displacement; (c), (d) lateral displacement; (e), (f) rotation; (g), (h) torsion. Note that
represents the homogeneous elastic half-space case.
33
Figure 5 Comparison of the vertical stiffness estimated by the design methods of Mayne and
Poulos (1999) and Gao et al. (1992), normalized by the corresponding 3D FEM results, for the
continuously varying shear modulus profiles.
34
(a)
(b)
(c)
(d)
Figure 6 Comparison of weight distributions and , as calculated from the 3D FEM results
for the vertical stiffness problem, assuming smooth contact between soil and foundation. Note
that  and  correspond to the weight distributions inferred from the Mayne and Poulos
(1999) and Gao et al. (1992) design methods, respectively.
35
(a)
(b)
(c)
(d)
(e)
(f)
Figure 7 Comparison of the best-fit Weibull parameters () for each Poisson’s ratio (shown
as white circle markers) and the fitted equations listed in Table 1 (shown as black solid lines) for
(a), (b) lateral stiffness; (c), (d) rotational stiffness; (e), (f) torsional stiffness.
36
Figure 8 Flow chart showing the steps involved in estimating the stiffness of a foundation on a
site with some arbitrary shear modulus  profile.
37
(a)
(b)
(c)
Figure 9 Comparison of the (a) lateral stiffness; (b) rotational stiffness; (c) torsional stiffness
estimated by the new design methods (Eqs. 36 to 38), normalized by the corresponding 3D
FEM results, for the continuously varying shear modulus profiles, where represents the
homogeneous elastic half-space case. Note that does not vary with .
38
(a)
(b)
Figure 10 Schematic diagram of the two multi-layered soil profiles evaluated in this study.
Similar soil profiles to these were previously investigated in Burd et al. (2020). (a) ‘BC clay
profile comprising of Bothkennar clay overlying Cowden till (b) ‘BCB clay’ profile comprising of a
Bothkennar clay soil matrix with an interbedded Cowden till layer.
39
Figure 11 Comparison of the normalized initial shear modulus of the three complex soil
profiles, where the depth is normalized by the foundation diameter = 10m. ‘BC clay’ and ‘BCB
clay’ correspond to the soil profiles described in Fig. 10, while ‘EURIPIDES’ corresponds to the
soil profile of the EURIPIDES project (according to Niazi and Mayne 2010).
40
Figure 12 Comparison of the normalized lateral, rotational and torsional stiffness estimated by
the simplified solutions (Eqs. 36 to 38) with the corresponding 3D FEM results, for all 11
complex soil profiles. Foundation diameter is 10m. Both axes are in log scale and the dotted
line is a 1:1 line.
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The failure envelope approach is commonly used to assess the capacity of shallow foundations under combined loading, but there is limited published work that compares the performance of various numerical procedures for determining failure envelopes. This paper addresses this issue by carrying out a detailed numerical study to evaluate the accuracy, computational efficiency and resolution of these numerical procedures. The procedures evaluated are the displacement probe test, the load probe test, the swipe test (referred to in this paper as the single swipe test) and a less widely used procedure called the sequential swipe test. Each procedure is used to determine failure envelopes for a circular surface foundation and a circular suction caisson foundation under planar vertical, horizontal and moment (VHM) loading for a linear elastic, perfectly plastic von Mises soil. The calculations use conventional, incremental-iterative finite element analysis (FEA) except for the load probe tests, which are performed using finite element limit analysis (FELA). The results demonstrate that the procedures are similarly accurate, except for the single swipe test, which gives a load path that underpredicts the failure envelope in many of the examples considered. For determining a complete VHM failure envelope, the FEA-based sequential swipe test is shown to be more efficient and provide better resolution than the displacement probe test, while the FELA-based load probe test is found to offer a good balance of efficiency and accuracy.
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