This paper has to be cited as: E. Díaz, Tomás R (2014) Revisiting the effect of foundation
embedment on elastic settlement: A new approach. Computers and Geotechnics 62, 283–
292. The final publication is available at Elsevier via
http://www.sciencedirect.com/science/article/pii/S0266352X14001529
REVISITING THE EFFECT OF FOUNDATION EMBEDMENT ON ELASTIC
SETTLEMENT: A NEW APPROACH
Abstract
The effect of foundation embedment on settlement calculation is a widely researched topic in
which there is no scientific consensus regarding the magnitude of settlement reduction. In this
paper, a non-linear three dimensional Finite Element analysis has been performed with the aim
of evaluating the aforementioned effect. For this purpose, 1800 models were run considering
different variables, such as the depth and dimensions of the foundation and the Young´s
modulus and Poisson’s ratio of the soil. The settlements from models with foundations at
surface level and at depth were then compared and the relationship between them established.
The statistical analysis of this data allowed two new expressions, with a mean maximum error
of 1.80 %, for the embedment influence factor of a foundation to be proposed and these to be
compared with commonly used corrections. The proposed equations were validated by
comparing the settlements calculated with the proposed influence factors and the true
settlements measured in several real foundations. From the comprehensive study of all modelled
cases, an improved approach, when compared to those proposed by other authors, for the
calculation of the true elastic settlements of an embedded foundation is proposed.
Keywords: foundations; soil/structure interaction; settlement; elasticity; finite-element
modelling;
statistical
analysis.
1
1. Introduction
Engineers must consider settlements when designing foundations, in order to avoid excessive,
unacceptable vertical deformations. Therefore, it is important to calculate foundation
settlements as accurately as possible, considering the contact pressure at the base of the
foundation. Foundations are usually buried at depth in order to increase the bearing capacity of
soil, reduce settlements in the foundation or provide protection against frost, desiccation and
erosion. Multiple methods exist for calculating foundation settlements, although due to their
ease of use compared with accuracy, the most commonly used in practice are analytical methods
based on elastic theory and influence factors.
The general expression for calculating the elastic settlement of a foundation with a uniform net
contact pressure distribution (qnet) resting on an elastic, homogeneous and isotropic medium is
[1]:
𝑆 = 𝑞𝑛𝑒𝑡 𝐵
(1−2 )
𝐼
𝐸
[1]
where S is the settlement of the foundation, B is the foundation width, E is the Young´s
modulus of the soil, ν is the Poisson’s ratio of the soil and I is the displacement influence factor.
The displacement influence factor (I) from eq. (1) mainly depends on the foundation shape [1].
However, this factor is also related to the relative foundation-soil stiffness, the existence of
incompressible layers at a certain depth, the embedment of the foundation, the variation of E
with depth (Gibson’s profile), etc. [1]. Therefore, this factor modifies the value of the calculated
settlement according to the conditions of the foundation to provide a more accurate settlement
calculation. Among all displacement influence factors considered by Poulos and David [2],
Milovic [3] and Mayne and Poulos [1], one of the most controversial is that related to the depth
of the foundation (IE).
1
The first proposed method for quantitatively evaluating the effect of foundation embedment on
settlement magnitude was proposed by Fox [4], who studied the relationship between the
settlement of a foundation resting at a certain depth and the settlement of the same foundation at
the surface. To this end a constant  value of 0.5 was considered with a uniformly loaded
flexible foundation, applying Mindlin [5] solution. The relationship between both settlements
(i.e. the displacement influence factor, IE), obtained for different L/B ratios, was plotted in a
graph. The results show that for a given L/B ratio, the greater the embedment depth (D), the
lower the displacement influence factor, which varies from 1.0 (for zero embedment depth) to
0.5 (for an infinite embedment depth). The main drawback of this method is that it is only valid
for  values equal to 0.5. Janbu et al. [6] obtained similar results considering circular and square
foundations, resting at different depths. Nishida [7] studied the influence of the depth of a
circular foundation considering different  values, and concluded that for greater depths, the
obtained displacement influence factors are similar to those proposed by Fox [4] for a  value
equal to 0.5. Schmertmann [8] used cone penetration tests (CPT) to propose an expression
considering the embedment effect, concluding that, as Fox [4] and Janbu et al. [6] had
previously asserted, the displacement influence factor is lesser than 0.5. Burland [9] revised the
IE values proposed by Fox [4] and Janbu et al. [6], by calculating the settlements of circular
foundations by means of the FEM. It was concluded that the effect of the foundation depth is
lesser than that considered by the aforementioned authors (mainly for cases in which the
embedment is high), and proposed a new expression for calculating IE. Butterfield and Banerjee
[10], developed a numerical model based on the Mindlin [5] solution for different depths and a
rigid circular foundation embedded in an elastic half-space, recommending IE values even lower
than those proposed by Fox [4]. Christian and Carrier [11], also performed a detailed analysis of
this problem, concluding that the IE values proposed by Fox [4] and Janbu et al. [6] are
essentially the same and must be reviewed because they provide unsafe values which
underestimate settlements. These authors proposed a chart to consider the existence of a nondeformable layer at a certain depth (depth correction factors, proposed by Giroud [12]) and the
1
effect of the foundation embedment (embedment correction factors proposed by Burland [9]).
However, they do not explicitly consider . Pells and Turner [13], reproduced Burland’s
calculations [9], concluding that his proposed values are inaccurate. They proposed charts, were
derived from FEM analysis, for obtaining IE, considering both rigid and flexible foundations.
Yamaguchi [14], proposed an expression which considers soil deformability, the net contact
pressure of the foundation and the bearing capacity of the soil. Due to the high number of
variables considered by this author the expression is not easy to apply. Bowles [15] advised
against using the formula proposed by Fox [4], as the author considered this approach to give
unsafe values. Christian and Carrier [16] reaffirmed their previous conclusions [11] in response
to the comments made by Bowles [15] about their method. Johnson et al. [17] and Kausel and
Ushijima [18] analysed circular and strip foundations with vertical sides in contact with the
ground. Gazetas et al. [19] applied a parametric boundary element method and proposed an
analytical expression for calculating vertical elastic settlements of rigid arbitrarily shaped
foundations embedded in a homogeneous and elastic half-space with a maximum error of 20%.
Mei and Xu [20] presented an analytical solution for flexible foundations resting on an elastic
half-space, based on the study of the most crucial parameters of Groth and Chapman [21].
Currently, although more reliable models which provide more realistic approximations to the
problem (e.g. [14]) exist, geotechnical engineers often use traditional methods such as those
proposed by Fox [4], Janbu et al. [6] or Burland [9]. For example, Mayne and Poulos [1], who
proposed one of the most complete and widely used formula for the calculation of settlements,
recommend the use of the IE values proposed by Burland [9].
Consequently, there is an evident controversy in the use of this displacement influence factor
and there is no consensus in the magnitude of settlement reduction due to the effect of
foundation embedment. As explained previously, solutions exist in which authors do not
consider  or E, propose algorithms which are difficult to apply, or propose solutions that are
only valid for circular and flexible foundations without considering the shape of the foundation,
etc. Additionally, due to the fact that this is a problem which involves soil-structure interaction,
1
it cannot be satisfactorily solved by using analytical solutions based on the theory of elasticity
[14].
Therefore, the main aim of this investigation is the study of this question, considering all
parameters relevant to the problem (including soil-structure interaction), thereby improving
upon the limitations of the existing models and proposing a new method.
In the present study the relationship between the settlement of a foundation at a given depth
(Semb) and the settlement of the same foundation at surface level (Ssur) is analysed, in order to
obtain the displacement influence factor (IE). For this purpose, different models have been
developed, varying the key parameters for solving the problem using ANSYS+CIVILFEM v.11
software. This software allows the modelling of settlements suffered by a foundation by means
of the FEM.
Additionally, the paper provides analytical expressions for an accurate consideration of the
effect of the foundation depth on the calculated settlement. The study will be focused on the
most common range of embedment/width (D/B) ratios, which in practice vary from 0 to 1.
The paper is organized as follows: section 2 describes the effects of the foundation embedment
on the calculation of settlements, section 3 details the model and the geotechnical and geometric
parameters adopted in the study and examines the most relevant results obtained from the
analysis, section 4 presents the proposed analytical formulae for considering the effect of
embedment on the calculation of settlement, section 5 shows a validation of the proposed
formulae through the analysis of several case studies, and section 6 presents the main
conclusions.
2. Effect of embedment on foundation settlement.
The embedment of a foundation has five different effects on the magnitude of vertical
settlement [14, 19, 20]: The floating foundation effect, the Gibson effect, the trench effect, the
side-wall contact effect and the retaining wall effect. The explanation of these effects is as
follows:
1
-
Reduction of the net pressure (qnet) transmitted by the foundation (floating foundation
effect), that is related to the stress discharge (q0) effect due to excavation (Figure 1b)
which can be calculated as:
𝑞𝑛𝑒𝑡 = 𝑞𝑔𝑟𝑜𝑠𝑠 − 𝑞0 = 𝑞𝑔𝑟𝑜𝑠𝑠 − 𝛾 ∙ 𝐷
[2]
where 𝑞𝑔𝑟𝑜𝑠𝑠 is the gross pressure transmitted to the subsoil by a foundation,  is the
soil unit weight and D is the depth of the foundation base.
-
Deformation modulus usually increases with depth [22, 23] (Figure 1b). Consequently,
foundations at greater depth transmit pressure to a stiffer soil than if they were resting
on the surface, and as such show less settlement in the same loading conditions.
-
The third factor to be considered is the so called trench effect (Figure 1c). This effect
refers to the fact that in a homogeneous half-space, the settlement of a foundation
placed at the bottom of a trench is smaller than that of the same foundation resting on
the ground surface. This effect is illustrated in Figure 1c and is due to the normal ()
and shear () stresses from the overlying soil, which restrict the displacement of the
horizontal plane in contact with the foundation base, thereby reducing the settlements.
In the case of a foundation at surface level (Figure 1a) the deformation of the
aforementioned plane is not influenced by any external stresses, and consequently the
expected settlements are greater. Erden [24] experimentally observed that this effect
takes place even in cases in which non lateral contact between the foundation and the
soil exists, and that it is dependent on the stress level acting over the foundation base
(i.e. the pressure in the surrounding soil at the foundation base level due to soil
overburden, q0).
-
The fourth effect is due to the lateral contact between the foundation and the soil
(sidewall contact effect; Figure 1d). When a foundation’s vertical faces are in contact
with the adjacent soil, a portion of the applied load is transmitted to the ground as shear
stresses () acting along the lateral faces of the foundation. The reduction of the load
1
transmitted by the foundation with sidewall contact compared to the same foundation
without lateral contact also implies a reduction in settlement.
-
The last factor is called the retaining wall effect [20], and is caused by the deformation
of the trench walls towards the excavated area due to being limited by the lateral faces
of the foundation, which has the positive effect of reducing vertical deformations
(Figure 1e).
Figure 1. Illustration of (a) a surface foundation with a stress-free surface (q0=0), (b) reduction
of net pressure (qnet) transmitted to the soil and increase in Young's modulus (E(z)) with depth
(Gibson profile) effects, (c) trench effect, (d) sidewall effect, and (e) retaining wall effect (based
on [14, 19, 20]).
1
The effect of the foundation embedment on settlement magnitude of settlement is due to the
aforementioned causes. Thus, it is difficult to provide a satisfactory explanation of the problem
through the application of numerical methods based on the theory of elasticity [14, 19]. In order
to take into account these five effects, it is necessary to use a tool which is able to consider
nonlinear soil-structure behaviour, solve a three-dimensional problem, include a “Gibson
profile”, take into account the existing natural horizontal and vertical stress level at the
foundation plane before excavation using the command ~TIS from CivilFEM and the
deformation restrictions on foundation planes at depth due to the pressure caused by
overburden.
3. Settlement modelling
The model used in this investigation for the study of the effect of foundation embedment is
described in detail in section 3.1. Additionally, the main results of the modelling are shown in
subsection 3.2.
3.1. Geometry and parameters
A nonlinear three-dimensional model was created using ANSYS+CIVILFEM version 11 [2527] in order to establish a relationship between the settlement from a foundation buried at depth
(Semb) and a surface foundation (Ssur) resting over the same soil (i.e. with the same geotechnical
properties and with a variable E value with depth as shown in Figure 1b and later described in
detail in this section). This was expressed through the IE value, defined as follows:
𝐼𝐸 =
𝑆𝑒𝑚𝑏
𝑆𝑠𝑢𝑟
[3]
The soil and the foundation were meshed using 20-node quadratic solid elements called
SOLID95, with three degrees of freedom per node (i.e. translation in the nodal x, y, and z
directions). These elements were selected because they allow the modelling of soil-foundation
contacts and irregular shapes (e.g. the foundation corners) without a loss of accuracy.
1
The soil-foundation contact was represented by means of a surface-surface contact model using
TARGE170 and CONTA174 elements that accurately represent the reality of the contact [28].
These are elements without thickness or stiffness that are defined by means of the nodes from
both areas (soil and foundation), which are initially at the same position.
TARGE170 elements were employed for the lateral faces and the base of the foundation and
CONTA174 elements for the soil in contact with the foundation, interacting with TARGE170
elements. For modelling the friction between the foundation and the soil, a keyoption for
CONTA174 elements allows the friction coefficient between materials to be considered.
The foundation was considered by ANSYS to be made of a concrete type material in accordance
with Eurocode 2 [29], with a compressive strength of 25 MPa (C25/30 [29]), because this is one
of the most commonly used concretes in practice. A centred vertical load was considered in the
modelling. The foundation was considered to be rigid with a rough interface with the soil, due
to the material being reinforced concrete. Note that this rough interface has been simulated
using a Mohr-Coulomb law in the soil-footing contact. Consequently, horizontal displacement
can only occur when the maximum shear force given by the Mohr-Coulomb law is exceeded.
This shear force is proportional to the normal force and to the friction angle. In this case, a
friction angle equals to 2/3 of the friction angle of the soil has been adopted as recommended by
[30].
From a geometric point of view, different width-length-embedment combinations were
considered and compared with foundations with the same geometry at surface level.
Because the estimation of settlements is performed by applying formulae based on the theory of
elasticity containing E and  only, a surface Young´s modulus of soil has been considered (E0)
and this value has been linearly increased with depth according to the vertical effective stress
level considering an increase rate of 300 kN/m2 per meter of depth. This can be considered a
conservative value for settlement calculation [31, 32].
After the initial stress state is calculated, the foundation is loaded with the net contact pressure.
1
Due to the symmetrical nature of the problem, only half of the model was defined, and
symmetry boundary conditions in the middle plane were considered, which represented a
significant resource optimization.
Regarding the model meshing, mesh refinement was performed in the area affected by the
foundation pressure bulb. For this purpose rectangular prismatic elements were used, with a
variable size depending on the zone of the model, considering 0.1 times the width of the
foundation in the area of the foundation bulb and gradually increasing this value to 0.5 times the
width of the foundation at the model boundaries (Figure 2).
The lower boundary, whose vertical displacements were impeded, was modified as a function of
the dimensions of the foundation, varying from 20 to 25 times the width of the foundation
(Figure 2).
The lateral borders of the model were positioned at a distance of between 6 and 12 times the
foundation width from the edge of the foundation, to avoid modifying the stress distribution
(Figure 2). Horizontal displacements were impeded on these lateral borders.
The validity of the mesh size and the model borders was verified through an optimization study.
In this study, several analyses were performed considering different geometric configurations of
the foundation used in this study, gradually reducing the size of the mesh and increasing the
limits of the model to find the values for which the calculated settlements in consecutive models
were constant.
Once the mesh selection was optimized, the model was verified. For this purpose, the
settlements obtained from the FEM models and from Schleicher´s manual methods [33] were
compared considering a foundation resting at surface level (i.e. not affected by embedment
effects). From the comparison of both sets of data, it may be observed that the results provided
by both methods were very similar and the differences irrelevant, providing a mean error of
4.65% (1.56%). Consequently, it may be concluded that the selected mesh sizes and model
boundaries are valid for the purpose of this research.
A rigid layer at a given depth was not considered in this study. Furthermore, the use of more
advanced models (e.g. plasticity) was avoided because the main aim was to evaluate the effect
1
of the foundation embedment on settlement magnitude, obtained by means of classical elastic
formulae.
The elastic parameters  and E0 were modified in conjunction with the foundation dimensions
(L and B) and the embedment (D) in order to consider all possible situations. Note that due to
convergence problems a  value of 0.49 instead of 0.5 was used, although in the text this value
is referred to as 0.5.
Moreover, due to the similarity between the results obtained from the models for L/B values 5
and 10 and the asymptotic trend of the data series presented in this paper (and in the literature
consulted) for L/B rates greater than 5, IE values were very similar and slightly conservative.
Consequently the value corresponding to the case L/B=5 was adopted as a maximum.
For D/B values lower than 5, IE showed important variations (e.g. Figure 3) and, as a
consequence, most of the models were calculated for values in this interval, in order to provide
accurate IE values. Furthermore, most real world foundations are within this range of D/B
values, which confirms the importance of providing accurate IE values for D/B relationships
lower than 5. In these conditions, 1800 cases were run.
1
Figure 2. Illustration of the model adopted in this study.
3.2. Results
1800 different cases considering most of the situations which are usually encountered in
practice were modelled using the method described in the previous subsection, with the aim of
studying the parameter IE (eq. (3)), i.e. the ratio of settlement in foundation at a given depth
(Semb) and in the same foundation at surface level (Ssur). The cases may be split into three
different groups according to their L/B ratio (1, 2 and 5). In turn, five subgroups were
considered according to the  value (0.1, 0.2, 0.3, 0.4 and 0.5). These cases were run for a set of
D/B ratios (0 for foundations at surface level, 0.1, 0.2, 0.3, 0.4, 0.5, 0.75, 2, 5, and 10).
1
Additionally, all these cases were run for four different values of E0 (2500, 5000, 50000 y
100000 kN/m2). The settlements were calculated at the centre point of the foundation
(maximum settlement).
The results of the modelling for a foundation with an L/B ratio equal to 1 are shown in Figure 3,
which shows the value of IE for different D/B ratios and  values (0.1 to 0.5).
Figure 3. Comparison of the effect of D/B ratio and the embedment influence factor (IE) for
different  values and a constant L/B ratio equal to 1.
1
Figure 4 shows the effect of the dimensions of the foundations (i.e. the L/B ratio) on the value
of IE for different D/B ratios and a constant  equal to 0.1. A similar relationship can be found
for other  coefficients, showing an increase in IE values with .
Figure 4. Comparison of the effect of D/B ratio on IE, for L/B ratios equal to 1, 2 and 5 and a
constant  value equal to 0.1.
This paper considers E as a variable for determining IE, which is novel as it has not been
considered by other authors in the literature consulted. Figure 5 shows the variation of IE against
D/B for different soil E0 values and for a  value equal to 0.1, following the same trend
observed in previous cases.
1
Figure 5. Comparison of the influence of D/B ratio on IE for different E0 values (2.5, 5.0, 50.0
and 100 MPa) and a constant L/B ratio.  is equal to 0.1.
1
Finally, the IE values obtained in this study (Figure 6) were compared with those proposed by
Janbu et al. [6] and Burland [9] for the particular case in which L/B is equal to 1 and  to 0.1.
Note that the values of IE proposed by Burland [9] and Janbu et al. [6] are independent to the
L/B ratio and , respectively.
Figure 6. Comparison of the IE values proposed by Janbu et al. [6] and Burland [9] with those
proposed in this study for L/B equal to 1 and =0.1.
3.3. Results analysis
In the previous sections 1800 FEM models were run to study the effect of different soil
parameters (i.e.  and E0) and foundation geometry (i.e. L and B) on the calculated settlement of
1
a foundation embedded at a depth D. According to these results, for a constant L/B ratio and ,
the relationship between IE and D/B ratio adopts the trend and the shape shown in Figures 3, 5
and 6, which are very similar to those obtained by other researchers such as Burland [9],
Butterfield and Banerjee [10] and Pells and Turner [13] for circular loads, although the IE values
are different. For example, the values suggested by Burland [9] were proposed for circular and
flexible foundations, and although they apparently cannot be directly compared with the results
obtained from this research, there is now a consensus that Burland’s expressions are valid
regardless of the flexibility of the foundation [11].
The IE values proposed in this study were lower than those presented by Burland [9], mainly for
high L/B ratios. The greatest difference was for D/B=10 and L/B=5, although for these cases IE
was practically constant for the different  values.
Regarding the IE values proposed by Fox [4] and Janbu et al. [6], the present study provides IE
values higher than those proposed by the aforementioned authors (Figure 6), mainly for D/B
ratios higher than 1, which even showed differences of up to 50%. These differences are
because these authors did not consider all effects of embedment on foundation settlement
described in section 2. However, there were some exceptions where the IE values presented in
this paper were lower. These cases were: a) situations where D/B ratios were lower than 0.4,
independent of their L/B ratio; and b) cases where D/B varied from 0.4 to 0.8 and L/B=1. A
further reduction in settlements (IE values lower than other formulations) agrees with Christian
and Carrier´s [11] work, who concluded that both methods provide IE values lower than the true
values, overestimating the effect of foundation embedment on calculated settlements. Similarly,
Bowles [15], omitted the use of these methods due to the aforementioned underestimation of
settlement that is less conservative. The present study also reveals this fact, accentuating this
difference for D/B ratios greater than 1.
From the analysis performed a relationship between IE and E not previously described in the
literature was observed, in which IE slightly increases with greater E values.
In the following section an analytical relationship between IE and the variables which were
found to affect it is proposed.
1
4. Proposed formulae
Based on the results shown in the previous sections, two different formulae are proposed for the
determination of IE. The first proposed formula only considers L/B, D/B and  as variables,
similar to the existing formulae and abacuses. However, in the second formula E0 is considered
as an additional variable.
To obtain these analytic formulae, a statistical analysis was performed from POLI-MODEL
software with the experimental data. POLI-MODEL [34, 35] allows the generation of families
of mathematical models with non-linear parameters based on the experimental data of the
intervening variables. The software, which automatically searches among different nonlinear
models within the parameters, is based on the application of the Gauss-Newton algorithm for
obtaining the parameters of nonlinear models. The expression obtained for the calculation of IE
is:
𝐿
2
𝐼𝐸 = 0.7281 + (0.0386 ∙ − 0.0928) + 0.1417 ∙ 𝜈 + (
𝐵
𝐷 2
1
6528
∙( ) )−
𝐵
30482
[4]
𝐷 2
𝐵
−818112∙( ) −137164
giving a squared regression coefficient (R2) of 0.9996 and a mean error of 1.67%.
The formula dependent on E was also fitted to the results obtained from the modelling, giving a
R2 of 0.9998 and a mean error of 1.80% (Figure 7):
𝐼𝐸 = 0.6916 + (
1
13738
𝐿
2
∙ 𝐸0 (𝑀𝑃𝑎)) + (0.0250 ∙ − 0.0719) + 0.1520 ∙ 𝜈 + (
𝐵
1
𝐷
∙ )−
1277 𝐵
379
𝐷
𝐵
−2940∙ −1935
[5]
The range of values for which the formulae are valid are L/B≤5 and 0<D/B≤10. For higher
values it is recommended that the limit values are considered, i.e. the IE values that correspond
to L/B=5 and D/B=10, which provide conservative values. Note that the IE trend is asymptotic,
similar to the curves proposed by others (e.g. [9]), and also giving similar values.
1
Figure 7. Variation of IE against D/B and E0 for different  values according to eq. (5).
5. Case studies
In this section a validation of the proposed values for IE is performed through 94 different case
studies compiled by Burland et al. [36] in which D/B>0. For this purpose, settlements were
calculated using the most commonly used expressions proposed by Schleicher [33] and Mayne
and Poulos [1], considering the foundation at the surface and correcting for the embedment
effect using the IE values proposed by Janbu et al. [6], Mayne and Poulos [1] and the proposed
eqs. (4) and (5). Then, these calculated settlements were compared with those measured in the
field by Burland et al. [36] for foundations buried at depth, and the relative error calculated.
Table 1 shows the most relevant results derived from the analysis performed.
1
The analysis showed that eq. (5) provides a better estimation of the true value of settlements,
with mean relative errors (MRE) between the measured and calculated settlements of up to
9.01% when it is used instead of that proposed by Burland [9] in the Schleicher [33] formula,
and up to 7.84% when Mayne and Poulos [1] formula is used. These differences increase with
the L/B ratio (Table 1). For L/B ratios equal to 1, IE values proposed by Burland [9] agreed very
well with those proposed in the present study, which is reasonable when considering that the
proposed IE values were obtained for circular foundations in which the equivalent square surface
has a L/B ratio near to 1. Note that eq. (4) provided values that agreed better (mean error 1.67%)
than those given by eq. (5) (mean error 1.80%), although the validation shown in this section
was only performed using the best fitting equation (i.e. eq. (5) which provides a R 2 value of
0.9998).
Regarding the IE values proposed by Janbu et al. [6], it may be noted that they present a similar
deviation in the global prediction of the real IE value to those proposed by Burland [9], even
providing better predictions for all considered L/B intervals except for the case in which L/B=1
(Table 1).
The application of the proposed IE values instead of that proposed by Janbu et al. [6] to the
Schleicher [33] and Mayne and Poulos [1] formulae gave mean relative errors in the estimation
of settlements of up to 8.15 and 7.25%, respectively. Note that MRE was calculated relative to
the main prediction method, which is in all cases the one proposed in this paper (eq. (5)).
28 of the 94 cases (i.e. 29.79%) in which the settlement was calculated and corrected using the
Janbu et al. [6] expression showed IE values lower than the true ones, underestimating the
settlements. However, for only 17 of the 94 cases (i.e. 18.09%) in which eq. (5) was applied, the
settlements were underestimated, showing a better agreement.
1
From the global and detailed analysis of this data it may be concluded that the most consistently
accurate formula for calculating IE is the one proposed in this study (eq. 5). However, for L/B
ratios equal to 1 the errors given by Burland’s correction [9]were very low.
6. Conclusions
In this study three-dimensional nonlinear analysis was performed using the FEM to analyse the
effect of foundation embedment on settlement in a rigid foundation resting on an elastic half
space medium. One of the most important developments is the high number of cases considered
(1800) which was much higher than that considered in previous studies. Previous models
proposed by other authors were also improved by considering conditions most commonly
encountered in practice and all plausible variables and effects that determine the influence of the
foundation embedment on the reduction of settlement. The most relevant conclusions are
summarized in the following paragraphs:
1. A satisfactory FEM model was achieved by a previous study of optimization and a
verification process, which showed the validity of the model used.
2. Two new formulae were obtained for calculating IE for all of the cases considered. The first
proposed equation (eq. (4)) is dependent on , L/B and D/B. However the second (eq. (5)) has
an additional dependent variable, E. For both cases the squared regression coefficients were
higher than 0.9996 and the mean relative error provided in the prediction of IE was up to 1.80
%.
3. The proposed IE equation follows a similar trend to that proposed by Burland [9]. However,
IE values derived from eq. (5) provide lower values than those proposed by Burland [9]
probably because these are based on the study of circular flexible foundations (Figure 6). The
difference was greater for larger D/B and L/B ratios. Regarding the values proposed by Fox [4]
and Janbu et al. [6], eq. (5) provided greater values (Figure 6), because as previously mentioned,
these methods considerably overestimate the reduction of the settlement (e.g. [11]). The analysis
performed corroborated this fact, mainly for D/B ratios greater than 1. Nevertheless, for cases
1
with D/B<0.4 independently of L/B, and values for D/B between 0.4 and 0.8 only if L/B=1 (in
both cases independent of ), the IE values presented in this paper were lower.
4. A previously unknown dependence of the embedment influence factor (IE) on E was found by
means of eq. (5) (Figure 7).
5. IE values derived from eq. (5) provided the best estimation of the settlements of real
foundations (Table 1). Consequently, it represents a considerable improvement for the
prediction IE with respect to the most current most commonly used factors, providing a better
alternative to the IE values proposed in literature. However, for L/B ratios equal to 1, the IE
values proposed by Burland [9] gave low errors, which validates their use for foundations with
such characteristics. Furthermore, the IE values proposed by Janbu et al. [6] represent the global
second best alternative for predicting settlements (slightly better than those proposed by
Burland [9]), although they have a higher probability of obtaining lower settlements than the
real values (i.e. they overestimate the value of the settlement reduction).
Acknowledgements
This study was undertaken as part of the UNED “Máster en Teoría y Aplicación Práctica del
Método de los Elementos Finitos y Simulación” and was funded by the Spanish Government
under project TEC2011-28201-C02 and the University of Alicante (VIGROB-157). Terratest
Rosehill supported part of this research. Finally authors are grateful to the research group
“Modelización Matemática de Sistemas” from the University of Alicante and Pedro Robles
Marín (Esfera Consultores S.L.) and Bernardo Robles Marín (IES Ginés Pérez Chirinos) for the
provided support for the fitting of the experimental data.
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