Engineering Structures 71 (2014) 212–221

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Engineering Structures

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Foundation and overall structure designs of continuous spread footings along with soil spatial variability and geological anomaly

S. Imanzadeh, A. Denis, A. Marache ⇑

Université de Bordeaux, CNRS, UMR 5295-I2M, GCE Department, Bât. B18, Allée Geoffroy St Hilaire, CS 50023, 33615 Pessac Cedex, France

a r t i c l e i n f o

Article history:Received 27 June 2013Revised 21 February 2014Accepted 8 April 2014Available online 4 May 2014

Keywords:Continuous spread footing Differential settlementFoundation and overall structure designs Geological anomalyLow stiffness zone Spatial variability Finite element methodSoil–foundation interaction Geostatistics

a b s t r a c t

Spatial variability of soil properties and geological anomaly can be very important in the case of low weight buildings with continuous spread footings inducing differential settlements which can have harmful consequences on the structure. They are also the major source of uncertainty in the choice of the soil design parameters. In this study, the design of continuous spread footings is performed with two approaches: the first approach with a foundation design using a one-dimensional finite element modeling and the second approach with an overall structure design using a three-dimensional finite element modeling. These approaches are compared for two cases: the first case dealing with the spatial variability of soil modulus and the second case with the spatial variability of soil modulus coupled with the presence of a geological anomaly (low stiffness zone of soil). Spatial variability of soil modulus is modeled by geostatistical methods using data from a real construction site. The values of the maximum settlements, maximum differential settlements and maximum bending moments obtained from the both approaches for the first case are nearly close together where the latter values for the second case are sig-nificantly greater than the first case. These results show that in the case of the presence of a geological anomaly on the construction site, the overall structure design appears the more appropriate approach compared to the foundation design in the design of continuous spread footings.

2014 Elsevier Ltd. All rights reserved.

1. Introduction

Soil exhibits spatial heterogeneities resulting from the history of its deposition and aggregation processes, which occur in differ-ent physical and chemical environments. This inherent or natural variability can be also accompanied by a geological anomaly. A geological anomaly is any inclusion that is of different properties from that normally expected in a design soil profile. This anomaly may include weak pockets or lenses of clay in a sand layer, cavities or boulders in soils. The presence of these unfavorable materials could lead to unsatisfactory foundation and overall structure performance.

The natural variability accompanied by a geological anomaly can be very important in the case of superficial geotechnical works inducing differential settlements, which can have harmful conse-quences on the structure. For example in low weight buildings with continuous spread footings, damage can range from sticking doors and hairline plaster cracks to complete destruction. Kumar et al. studied

the sources of these natural variability and the

⇑ Corresponding author. Tel.: +33 5 40 00 88 27; fax: +33 5 40 00 31

13. E-mail addresses: s.imanzadeh@i2m.u-bordeaux1.fr (S.

Imanzadeh), alain. denis@u-bordeaux.fr (A. Denis), antoine.marache@u-bordeaux.fr (A. Marache).

http://dx.doi.org/10.1016/j.engstruct.2014.04.019 0141-0296/ 2014 Elsevier Ltd. All rights reserved.

presence of a geological anomaly in foundation design parameters [1]. Raychowdhury et al. studied the shallow foundation response variability due to parameter uncertainty [2].

In foundation design, a low weight building is simply modeled with a one-dimensional modeling of a continuous spread footing with a loading [3]. However, in overall structure design, this low weight building is modeled with a two or three dimensional modeling of its continuous spread footings along with building elements such as columns, beams, walls and slabs [4].

In these conventional designs and dimensioning computations, continuous spread footings are often designed on the basis of the deterministic approaches where natural variability of soil and uncertainty related to imperfect knowledge of the presence of a geological anomaly of soil, in their longitudinal directions are usu-ally not considered. These effects and the soil–shallow foundation interaction along the longitudinal direction of continuous spread footings need to be taken into account and studied in order to perform an accurate analysis leading to correct designs.

In this research work, two approaches are used for the design of continuous spread footings: the first approach with a foundation design using a one-dimensional finite element modeling (1D) and the second approach with an overall structure design using a three-dimensional finite element modeling (3D). These approaches

S. Imanzadeh et al. / Engineering Structures 71 (2014) 212–221 213

are compared for two cases: the first case, taking into account the spatial variability of soil

modulus (Es) and the second case, taking into consideration the spatial variability of soil modulus accompa-nied by the presence of a geological anomaly as a lens of clayey soil of weak mechanical properties. Through these both approaches and cases, the soil–shallow foundation interaction along the longi-tudinal direction of continuous spread footings is then studied in order to better understand the

influence of the spatial variability of Es and a geological anomaly on the maximum settlement, max-imum differential settlement and maximum bending moment.

In order to achieve this goal, geological conditions of the studied construction site and available data from the geophysical and geo-technical investigations are presented. Thereafter, the appropriate geostatistical methods (collocated ordinary cokriging and condi-tional simulations [5,6]) are used to model the spatial variability of Young’s soil

modulus (Es) on a construction site. This spatial var-iability are then used through the finite element modeling of the Winkler soil–foundation interaction model in the longitudinal direction [7–14] along with and without the presence of a geolog-ical anomaly for both geotechnical and structural designs of continuous spread footings. From these numerical models, the maximum settlements, maximum differential settlements, maximum bending moments and their uncertainties are obtained in order to perform a statistical analysis that describes the longitu-dinal behavior of continuous spread footings in 1D and 3D models. Finally, a comparison between the obtained results from founda-tion and overall structure designs is done to study firstly, the influ-ence of the spatial variability of soil modulus and secondly, the influence of this spatial variability coupled with the presence of a geological anomaly on the behavior of continuous spread footings.

2. Soil–shallow foundation interaction model

In the conventional calculations of the shallow foundations design, the behavior is only studied in a cross section to represent the transverse behavior of the foundation elements. In the case of a continuous spread footing and particularly when a differential set-tlement may appear, the longitudinal behavior of spread footing should be taken into consideration.

In the past, many researchers have worked on the soil–structure interaction which is referred to as beams and plates on elastic foundations. Most of the previous work began with Winkler’s well known model with one parameter [15], which was originally developed for the analysis of railroad tracks. This model is expressed by the following

equation (Eq. (1)):

pðxÞ ¼ ks _ b _ wðxÞ

where ks is the coefficient of subgrade reaction, w(x) is the deflec-tion, b is a width of the foundation and p(x) is the reactive pressure of the foundation. Winkler’s idealization considers the soil as being a system of identical but mutually independent, closely spaced, dis-crete, linearly elastic springs. According to this idealization, defor-mation of foundation due to applied load is confined to loaded regions only. Furthermore this model cannot transmit the shear stresses which are derived from the lack of spring coupling [16,17]. Vlassov and Leontiev [18], recognizing the difficulty to determine values of

ks for soils, postulated a two-parameter model. The continuity in this model is characterized by the consideration of the shear layer. Kerr [19] attempted to make Winkler’s model more realistic by assuming some forms of interaction among the spring elements that represent the soil continuum even though it requires more parameters (three-parameter mathematical model).

Winkler’s model, due to its simplicity, has been extensively used to solve many soil–foundation interaction problems and has given satisfactory results for many practical problems. Further-

more, this model seems, from a practical point of view, to be appropriate for lightweight structures such as a low weight building.

The differential equation governing the deflection, w(x), of a homogeneous elastic bending beam with constant bending stiffness resting on Winkler’s model and subjected to a vertical continuous load, q(x), can be written as [20]:

Ec _ I

d4wðxÞþ ks _ b _ wðxÞ ¼ qðxÞ4

dxwhere Ec _ I is the constant bending stiffness of the beam (Ec and I are respectively Young’s modulus of concrete and the moment of inertia of the cross section of the foundation). When the deflection w(x) is known, the bending moment and shear force can be determined.

Numerous expressions or semi-empirical models are available to determine the soil

reaction modulus (ks) as a function of the studied applications [9,21–24]. The Vesic semi-empirical model (Vesic [25]), commonly used in the design of continuous spread footings, is considered in this study in order to obtain a value of the soil reaction modulus (Eq. (3)).

ks0:65

:12 12Esb

3

:

Es3

¼ b sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

_

m2

Ec h s

where Es is the Young’s soil modulus, ms the

Poisson’s ratio of soil, b, h and Ec are respectively width, height and Young’s modulus of a continuous spread footing.

Soil reaction modulus (ks) is not an intrinsic parameter of soil. The calculation of this modulus is a function of soil parameters (Es,

ms), the parameters related to the geometry of the continuous spread footing (b, h) and a mechanical property of the continuous spread footing (Ec) (Eq. (3)).

Induced reactions of the whole structure in the Winkler model (be it a single foundation beam or an overall superstructure) deduced on the basis of a certain distribution of a subgrade coeffi-cient if applied in the opposite sense on the supporting soil mass with a given directly determined geotechnical property as the

Young modulus (Es) and the Poisson ratio (ms) of soil, cannot ensure that the very same settlements which have been assumed for the subgrade system will be developed also on the soil surface.

When the structure rigidity is significant, using the Winkler model is valid. This has been pointed out earlier in the well known study carried out by Stavridis et al. for the two dimensional analy-sis of the concrete tunnel frame [16].

3. Finite element models for foundation and overall structure designs of continuous spread footings

The influence of the soil spatial variability

on a spread footing, using a finite element model, was studied by Cassidy et al. [26]. The finite element method has been largely used in numerous studies to model the soil–structure interaction: Denis et al. studied soil–shallow foundation interactions [27], Dubost et al. [7] and Niandou et al. [8] analyzed soil–pile interaction, Elachachi et al. [9–11], Buco et al. [12–14] studied soil–buried pipe interactions.

In this section the finite element models for foundation and overall structure designs of the considered continuous spread foot-ings in this study are presented. We take a low weight building with four continuous spread footings with lengths of 10 m and 6 m along with concrete columns (cross section: 20 _ 20 cm2), beams (cross section: 20 _ 20 cm2) and floor slab (thickness: 15 cm). Continuous spread footings, for low weight buildings with relatively lightly loaded walls, consist of concrete strips with a rectangular cross section, placed under masonry walls. We take the common dimensions of a spread footing for a low weight

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buildings: a width of 0.5 m and a height of 0.3

m. Young’s modulus of the foundation (Ec) and Poisson‘s ratio of soil are respectively equal to 20 GPa and 0.3 with a uniform loading of 30 kN per running meter.

In foundation design (1D), this low weight building is modeled with a continuous spread footing (for example for a spread footing of 6 m) as a beam resting on an elastic soil with a uniform loading of 30 kN per running meter as illustrated in Fig. 1. Finite element modeling of this spread footing has 12 elements and 13 nodes.

However, in overall structure design (3D), this low weight building is modeled with four continuous spread footings along with concrete columns, beams and floor slab in order to make it close to reality. Finite element modeling of these spread footings has 64 elements and 64 nodes ( Fig. 2).

The computations are performed with the CASTEM software [28] using the Winkler model.

In the following the geological conditions of the studied construction site and available data from the geophysical and geotechnical investigations are briefly presented.

4. Presentation of the studied site and available data

The study site with a surface area of

25,000 m2 is located to the south of the city of Pessac in France. In view of its large area, it was decided to implement a preliminary VLF-R type of geophysical survey campaign in order to qualify the homogeneity of the site, and ascertain zones most suitable for construction, before proceed-ing with borehole and pressure meter soundings.

The reconnaissance of a site of large surface area using an R (Resistivity) mode VLF (Very Low Frequency) technique can provide an interesting alternative to conventional methods such as the direct current electrical method [29,30]. The measurement points of soil resistivity obtained from this technique are distrib-uted over uniform profiles with mean spacing of 10 m between measurement points. The profiles are aligned in the east–west direction, over a length of approximately 190 m. The full reconnaissance campaign included a total of 272 soil resistivity measurement points ( Fig. 3).

The reconnaissance soundings (6 auger boreholes to a depth of 8 m) and twelve pressuremeter tests (depth of boreholes ranging between 1 and 7 m) enabled the surface formations detected using the VLF-R technique to be confirmed ( Fig. 3).

Deformation modulus (EPMT) and limit

pressures (PL) are obtained for each sounding from the pressuremeter test. The pressuremeter soundings made to a maximum

depth of 12 m, including a test measurement every meter, confirmed the sandy-clayey character of the soil with, for some soundings, the presence of sand, which occurs as embedded lenses rather than continuous seams or layers at depths greater than 7 m. Pressuremeter sounding allows the detection of a lens of clayey soil of weak mechanical properties at the depth of 2– 4 m. The average values found for the limit pressure and the

Fig. 1. Schematic view of the finite element modeling for foundation design of spread footing of 6 m with free ends as boundary conditions (one dimensional model (1D)).

Fig. 2. Schematic view of the finite element modeling for overall structure design of spread footings (a three dimensional model (3D)).

deformation modulus for each sounding are provided in Table 1. The ratio between the mean deformation modulus and the mean limit pressure ranged between 5 and 8.1, thus confirming the sandy-clayey character of the soils encountered [31].

5. Geostatistical workflows

Geostatistics are well suited for estimating the geotechnical parameters in heterogeneous soils [27]. Compared to classical statistics, geostatistical methods take into account the spatial variability of the target parameter, in order to provide realistic spatial estimates together with a quantification of the associated uncertainty.

We choose two appropriate geostatistical methods (multivari-ate geostatistical estimation following by conditional simulations) to model the spatial variability of soil modulus (Es) on the con-struction site using ISATIS software. For doing this, we need to perform a variographical analysis of available data. In the following sections, we explain the theoretical parts for a variographical anal-ysis and the two considered geostatistical methods together with the application of these methods using real data in order to model the spatial variability of soil modulus. This modulus can be assessed from pressuremeter modulus, using common relation: EPMT = a _ Es where a is structural or rheological coefficient [31].

5.1. Variographical analysis

Most geostatistical methods rely on a variogram model, which quantifies the spatial structure of the target parameter [5]. Firstly, the experimental variogram must be calculated. It describes how the spatial variability between data values evolves with the dis-tance between the data. The

experimental variogram c(h) is defined by:

cðhÞ ¼1NðhÞ

ðZðxi þ hÞ _

ZðxiÞÞ 22NðhÞ i¼1

X

where Z(x) denotes the target variable measured at location x and N(h) corresponds to the number of pairs of points separated by a distance of h.

During the variographical analysis, it is important to compute the experimental variogram in various directions in order to iden-tify directions of anisotropy if such anisotropy exists.

The experimental variogram is not sufficient for a geostatistical estimation or simulation and an analytical mathematical function has to be fitted to the experimental variogram: the variogram model. A variogram model is usually defined by some parameters: the type of basic mathematical model, its sill and its range for a stationary model. Usually, the variability between points increases

S. Imanzadeh et al. / Engineering Structures 71 (2014) 212–221 215

Fig. 3. Localization of the VLF point measurements, wells and pressuremeter tests in the studied area.

Table 1Average values for the deformation modulus EPMT and pressure limit PL.

FP1 FP2 FP3 FP4 FP5 FP6 FP7 FP8 FP9 FP10 FP11 FP12

PMT (MPa) 3.71 3.21 5.1 3.55 4.41 6.5 3.88 4.38 4.46 3.58 3.65 4.19E

L (MPa) 0.73 0.64 0.8 0.5 0.81 1.05 0.69 0.87 0.88 0.7 0.45 0.7P

PMT =

L 5.08 5.02 6.38 7.1 5.44 6.19 5.62 5.03 5.06 5.11 8.11 7E P

and becomes stabilized at a given variogram value called the sill. The range (or correlation length) is the distance at which the variogram reaches the sill value. The variogram will be used for the geostatistical estimation and conditional simulations that are explained in the ensuing sections.

5.2. Multivariate geostatistical estimation (collocated cokriging)

When we have a variable of interest or main variable known in few points and an auxiliary variable known in a lot of points in the domain, in this case, a multivariate geostatistical estimation called collocated ordinary cokriging is particularly appropriate [6]. The collocated ordinary cokriging estimator is (Eq. (5), [6]):

nð5ÞZ_ðx0Þ ¼ w0 Sðx0Þ þ ðwz

a ZðxaÞ þ wsa SðxaÞÞ

a¼1

Xwhere x0 is the point where the main variable (Z) is to be estimated, Z ⁄(x0) is the estimated value of the main variable at point x0, w0 is the weight assigned to the value of the auxiliary variable (S) at point x0, a is an index

numbering the samples from 1 to n, Z(xa) are the values of the main

variable at point xa, S(xa) are the values of the auxiliary variable at point xa,

waz are the weights assigned to the values of the main variable (Z) at point

xa and was are the weights assigned to the values of the auxiliary variable

at point xa.

This multivariate technique requires the computation and fitting of a variogram model that contains simple variograms for each variable and a cross variogram measuring the spatial correla-tion between both variables.

In order to use this method, a correlation must exist between the variable of interest (soil modulus in this case) and the auxiliary variable (soil resistivity). In a clayey-sandy soil with no important change in water content, the more the proportion of sand there is, the more there are deformation modulus and resistivity. Thus, it is assumed that deformation modulus is indirectly linked to soil resistivity derived from geophysical methods [3,27,32]. A linear relationship with a correlation of 0.6 was obtained between the deformation modulus and resistivity values.

The dataset in this study is constituted of a few set of pressure-

meter boreholes with the deformation modulus EPMT values (12 data of interest) and a lot of soil resistivity values (272 auxiliary data) [27]. For the collocated cokriging a bivariate variogram model is required. Nevertheless, because the small number of soil modulus values, this bivariate model is tedious to establish. For this reason, a bundled version of the collocated cokriging is used which takes into account only the variogram model of the auxiliary variable (soil resistivity) with the circular neighborhood equal to 50 m, along with an exponential

component model with a sill equal to 85 (X m)2 associated with a range equal to 37 m (Eq. (6)), which represents the isotropic variogram for the dataset,

216 S. Imanzadeh et al. / Engineering Structures 71 (2014) 212–221

which is also represented by a red line in Fig. 4. The bivariate model is deduced from this model and from the coefficient of correlation and variance ratio between both variables.

h 85: 1

_

exp _3h ; h : lag m

37cð Þ ¼ _ _ __ ð ÞThe collocated cokriging map of soil modulus (Es) (mean and its coefficient of variation equal to 8.3 MPa and 0.18 respectively) is displayed in Fig. 5. This estimation is done over a regular grid with a mesh of 10 m _ 10 m. The spatial soil modulus distribution illus-trates the heterogeneity of the surface layer formations on this site.

This estimation method gives a smoothed image of the reality, thus underestimates the proportion of extreme values. The final aim of this work is to study the effects of the spatial variability of soil modulus on the foundation and overall structure designs of continuous spread footings. Thus it is important to be able to evaluate a geotechnical parameter value and its uncertainty at a location. In our case, extreme low values especially can be very important in the longitudinal behavior of continuous spread footings.

In order to satisfy these remarks, we have chosen to use the conditional simulation method that is explained in the following section.

5.3. Conditional simulations

Conditional simulations are useful to obtain realistic pictures of spatial variability. There are many conditional simulation methods that can be used. We have chosen to use the turning bands method [5] which enables the construction of simulations in space from simulations on lines. The turning bands method was first used by Chentsov [33] in the special case of Brownian random functions. The general principle of the method appears as a remark in Matérn [34], but its development for simulations is due to Matheron [35]. Chilès and Delfiner [5] gave a fairly complete description of this method.

Because modulus values will be input data

of finite element modeling (see Section 3)

1000 simulations will be realized on a regular

grid with a mesh of 0.5 _ 0.5 m2 to obtain more

values of

Fig. 5. Collocated ordinary cokriging estimate for soil modulus.

Es beneath a structure. Starting from estimation results on a 10 _ 10 m2 mesh (Section 5.2) and computing new results on a 0.5 _ 0.5 m2 mesh is authorized by the application of the three perpendicular theorem [5]. Finally, results will be analyzed in terms of cumulative distribution function by post-processing simulations results.

In the following, the influences of the spatial variability of soil modulus and geological anomaly of soil on the maximum settle-ment, maximum differential settlement and maximum bending moment of continuous spread footings using foundation and overall structure designs are studied.

6. Foundation and overall structure designs of continuous spread footings

The location of the considered low weight building along with the four spread footings on the construction site is depicted in Fig. 6. The soil parameters, the load, concrete columns, beams, floor slab, the mechanical property and the geometrical dimensions of these spread footings are identical to those previously presented in this paper (see Section 3). For this location of spread footings, the values of Young’s soil modulus are obtained at each node of a

Fig. 4. Experimental (black line) and modeled (red line) isotropic variograms of thesoil resistivity. (For interpretation of the references to color in this figure legend, the Fig. 6. Four spread footings of a low weight building with lengths of 10 m and 6 mreader is referred to the web version of this article.) on the construction site.

S. Imanzadeh et al. / Engineering Structures 71 (2014) 212–221 217

grid (every 0.5 m) from the 1000 conditional

simulations results. From these values of Es,

the values of subgrade reaction modulus (ks) for the Vesic model are obtained by Eq. (3),

representing 64,000 values of ks. These values of subgrade reaction modulus are then introduced in the finite element model to obtain the max-imum settlement, maximum differential settlement and maximum bending moment for each spread footing of the building for each simulation. Then the 1000 results for the maximum settlement, maximum differential settlement and maximum bending moment for each continuous spread footing can be transformed in the form of cumulative distribution function.

Fig. 7 depicts the data distribution of Es for the four spread footings taking into account the

spatial variability of Es on the con-struction site. For the soil modulus probability between 1% and 99%, the interval of soil modulus values for the spread footings (1), (2), (3) and (4) are respectively [7.2,8.26], [7.25,9.2], [7.5,9.37] and [7.3,8.36] MPa.

The results of the foundation (1D) and overall structure (3D) designs will be presented for the spread footing (2) using two cases: the first case dealing with the spatial variability of

Es and the second case with this spatial variability and a geological anom-aly of soil where assumed to be a lens of clayey soil of weak mechanical properties under spread footing.

6.1. Taking into consideration the spatial variability of soil modulus in foundation and overall structure designs of continuous spread footings

The longitudinal variation of soil properties below the spread footings induces differential settlement and bending moment that cannot be predicted when assuming soil homogeneity.

In order to explain the behavior of the continuous spread foot-ing (2) with both foundation and overall structure designs in the

presence of the spatial variability of Es for the considered location, Fig. 8 depicts the bending moment along the lengths of the spread footings for one of the 1000 simulations. For the foundation design of the spread footing (2) the maximum value of the bending moment is smaller than those obtained from the overall structure design. For the foundation design of the spread footing (2), the high values of the bending moments, as expected, are not located at the two ends of this spread footing where for the overall structure design of the spread footings (1), (2), (4) the high values of the bending moments, as expected, are found at their two ends. The high values of the bending moments for the spread footing (3) are found at its two ends and in its middle.

In the overall structure design of the spread

footing (2) the high values of the bending moments at its two ends are due to the

Fig. 7. Cumulative distribution function of soil modulus for the four spread footings taking into account the spatial

variability of Es in the case of the study site.

Fig. 8. Bending moment along the spread footings of the low weight building for both foundation and overall structure designs in the presence of the spatial variability

of Es for one simulation along with an overall structure design with a simple deterministic study.

influence of its orthogonal spread footings (spread footings (1) and (3)), the spread footing (4) and more generally to the global structure rigidity ( Fig. 6) whereas in the foundation design there is only the spread footing (2) with free ends as boundary condi-tions ( Fig. 1).

Furthermore, an overall structure design with a simple deterministic study (the mean values of the Es equal to 7.5, 7.9, 8.3, 7.7 MPa ( Fig. 7) respectively for the spread footings 1, 2, 3 and 4) was carried out. For the latter, the values of the bending moments (ranging from _0.71 to 0.74 kN m) are smaller than those ones obtained from an overall structure design (ranging from _3.2 to 3.6 kN m) with taking into account the spatial variability of Es ( Fig. 8).

These high negative bending moments for the overall structure design are found to be about 3.5 MPa which can generate theoret-ically cracks on the upper parts of the spread footings (for the cross sectional area: 0.5 _ 0.3 m2). These negative bending moments can impose the installation of longitudinal steel reinforcements in the upper part of the foundation footings [27].

The 1000 results obtained for the maximum settlement, maximum differential settlement and maximum bending moment for each of these two designs, are transformed in the form of cumu-lative distribution function ( Fig. 9).

For the foundation design of this continuous spread footing the intervals of the maximum settlement, maximum differential set-tlement and maximum bending moment ranging between 1% and 99% of cumulative probability are respectively [7.6, 8.4] mm, [0.26, 3] mm and [0.14, 2.21] kN m. The latter intervals for the overall structure design of this continuous spread footing are respectively [7.3, 8.1] mm, [0.2, 2.1] mm and [0.56, 3.86] kN m ( Fig. 9). From these results we obtain for each design

the means (E[D], E[Dd], E[M]), variances

(Var[D], Var[Dd], Var[M]) and then the

coefficients of variation (CV[D], CV[Dd], CV[M]) of the maxi-mum settlement (D), maximum

differential settlement (Dd) and maximum bending moment (M) ( Tables 2 and 3).

6.2. Taking into consideration the spatial variability of soil modulus and the presence of a geological anomaly in foundation and overall structure designs of continuous spread footings

Only one geological anomaly (a lens of clayey soil with low stiff-ness) has been encountered during site investigation but we hypothesize that other lenses can exist on the construction site. The reason for taking this hypothesis is that the soil resistivity measurement points are distributed over uniform profiles with

218 S. Imanzadeh et al. / Engineering Structures 71 (2014) 212–221

Fig. 9. Cumulative distribution function of the (a) maximum settlement, (b) maximum differential settlement and (c) maximum bending moment for the Vesic model

taking into account the spatial variability of Es for the continuous spread footing (2).

mean spacing of 10 m between measurement points and a low stiffness zones of soil such as weak pockets or lenses of clay in a sand layer with a length of 2 m and a depth of 2–4 m is difficult to detect by geophysical and geotechnical survey campaign. In the following, for the same spatial variability of soil modulus (Section 6.1) and for the continuous spread footing (2) we consider that there is a low stiffness zone of clayey soil. For this zone the values of soil modulus and soil reaction modulus are respectively equal to 1.64 MPa and 1.5 MN m_3 with a length of 2 m under the continuous spread footing and not detected during digging out the soil for placing the concrete ( Fig. 10).

Fig. 11 shows for both designs the bending moment along the lengths of the spread footings for the same simulation as in Fig. 8 with considering a low stiffness zone of 2 m in the middle of spread footing (2). For this considered simulation, for the founda-tion design of the spread footing (2) the maximum value of the bending moment is greater than that one obtained from the overall structure design at the same position of the maximum bending moment.

The 1000 results obtained for the maximum settlement, maximum differential settlement and maximum bending moment for each of foundation and overall structure designs of this contin-uous spread footing are transformed in the form of cumulative dis-tribution function ( Fig. 12).

For the foundation design of this continuous spread footing the intervals of the maximum settlement, maximum differential settlement and maximum bending moment ranging between 1% and 99% of cumulative probability are respectively [11.12, 12.44] mm, [3.49, 5.42] mm and [26.7, 28.52] kN m. The latter intervals for the overall structure design of this continuous spread footing are respectively [8.8, 9.8] mm, [0.86, 2.29] mm and [17, 18.9] kN m ( Fig. 12).

From these results we obtain for each design the means (E[DA], - E[DdA], E[MA]), variances (Var[DA], Var[DdA], Var[MA]) and then the coefficients of variation (CV[DA], CV[DdA], CV[MA]) of the maximum settlement (DA), maximum differential settlement (DdA) and maximum bending moment (MA) ( Tables 4 and 5).

6.3. Comparison and discussion

We compare the results presented for the considered continu-ous spread footing in the two previous sections (Sections 6.1 and 6.2) in order to show the influence of the spatial variability of soil modulus and the presence of a geological anomaly on the estima-tions of the maximum settlement, maximum differential

settle-ment, maximum bending moment and their associated uncertainties.

The values of the maximum settlements and maximum differ-ential settlements obtained for the continuous spread footing (2)

taking into account only the spatial variability of soil modulus, from the overall structure design are softly smaller than those obtained from the foundation design ( Fig. 9a and b). Their associ-ated uncertainties quantified by the coefficient of variation are also

Table 2Statistical parameters of the maximum settlement, maximum differential settlement and maximum bending moment for the

foundation design of continuous spread footing (2) with taking into account the spatial variability of Es.

Statistical parameters (foundation design, 1D)

Maximum settlement (D) Maximum differential settlement (Dd) Maximum bending moment (M)

E[D] (mm) 8.006 E[Dd] (mm) 1.472 E[M] (kN m) 0.7540Var[D] (mm)2

0.0204 Var[Dd] (mm)20.2692 Var[M] (kN m)2

0.1827

CV[D] 0.0179 CV[Dd] 0.3524 CV[M] 0.5670

S. Imanzadeh et al. / Engineering Structures 71 (2014) 212–221 219

Table 3Statistical parameters of the maximum settlement, maximum differential settlement and maximum bending moment for the overall structure design of continuous spread footing (2)

with taking into account the spatial variability of Es.

Statistical parameters (overall structure design, 3D)

Maximum settlement (D) Maximum differential settlement (Dd) Maximum bending moment (M)

E[D] (mm) 7.750 E[Dd] (mm) 1.094 E[M] (kN m) 1.876Var[D] (mm)2

0.0332 Var[Dd] (mm)20.1271 Var[M] (kN m)2

0.4562

CV[D] 0.0235 CV[Dd] 0.3257 CV[M] 0.3601

Fig. 10. Schematic view of the finite element modeling of the spatial variability of soil modulus and the presence of a geological anomaly for the continuous spread footing (2).

Fig. 11. Bending moment along the spread footings of the low weight building for both

foundation and overall structure designs in the presence of the spatial variability of Es

and a geological anomaly for one simulation.

close together ( Tables 2 and 3). In this case, a one dimensional finite element modeling (foundation system) is sufficiently ade-quate for the design of spread footings. It should be noted that this result is valid for the identical charges on each of the spread footings.

However, there is a difference between the values of maximum bending moments obtained from these both designs. The obtained values of the maximum bending moments from 1D modeling (foun-dation system) are smaller than those obtained from 3D modeling (overall structure system, Fig. 9c) and then the same interpretation for the mean of the maximum bending moments ( Tables 2 and 3). Note that, the values of the maximum bending moments remain low compared to the value of the maximum elastic bending moment equal to 37.5 kN m for such structural elements. The uncertainty of the maximum bending moment from 1D modeling is greater than that one obtained from 3D modeling ( Tables 2 and 3). This shows that, although the mean value of the maximum bending moment obtained from the overall structure design is 2.5 times greater than that one obtained from the foundation design, the uncertainty of the maximum bending moment for the overall structure design is less than the same uncertainty for the geotechnical design.

For the overall structure design, all of the spread footings are affected by almost the same spatial variability of soil modulus

Fig. 12. Cumulative distribution function of the (a) maximum settlement, (b) maximum differential settlement and (c) maximum bending moment for the Vesic model taking into account the spatial variability of soil modulus and a geological anomaly for the continuous spread footing (2).

220 S. Imanzadeh et al. / Engineering Structures 71 (2014) 212–221

Table 4Statistical parameters of the maximum settlement, maximum differential settlement and maximum bending moment for the foundation design of continuous spread footing (2) with taking into account the spatial variability of soil modulus and the presence of a geological anomaly.

Statistical parameters (foundation design, 1D)

Maximum settlement (DA) Maximum differential settlement (DdA) Maximum bending moment (MA)

E[DA] (mm) 11.783 E[DdA] (mm) 4.332 E[MA] (kN m) 27.557Var[DA] (mm)2

0.0826 Var[DdA] (mm)20.1485 Var[MA] (kN m)2

0.1364

CV[DA] 0.0244 CV[DdA] 0.0890 CV[MA] 0.0134

Table 5Statistical parameters of the maximum settlement, maximum differential settlement and maximum bending moment for the overall structure design of continuous spread footing (2) with taking into account the spatial variability of soil modulus and the presence of a geological anomaly.

Statistical parameters (overall structure design, 3D)

Maximum settlement (DA) Maximum differential settlement (DdA) Maximum bending moment (MA)

E[DA] (mm) 9.281 E[DdA] (mm) 1.421 E[MA] (kN m) 17.702Var[DA] (mm)2

0.0480 Var[DdA] (mm)20.0768 Var[MA] (kN m)2

0.1127

CV[DA] 0.0236 CV[DdA] 0,1950 CV[MA] 0.0190

(isotropic spatial variability). The influences of continuous spread footings (1), (3), (4) in the overall structure design, and more generally, its global structure rigidity ( Fig. 6) are the reasons for differences between the resulting spread footing (2) of this design and those of foundation design.

In this case, we can conclude that the foundation design is a significantly simplified approach compared to the overall structure design for an estimation of bending moment when spatial variabil-ity of soil modulus is considered.

The values of the maximum settlements, maximum differential settlements and maximum bending moments obtained from the foundation and overall structure designs, taking into account both the spatial variability of soil modulus and the presence of a geolog-ical anomaly, are very different from each other ( Fig. 12). The latter values obtained from foundation design are significantly greater than those obtained from overall structure design but their associ-ated uncertainties obtained from the both designs are almost the same ( Tables 4 and 5). The foundation design in this case, appears more on safety side respect to the overall structure design and the maximum angular distortion of the spread footing (2) (1/600) remains smaller than the limit angular distortion defined for a low weight building (1/500 [36]).

These results show that taking into consideration a geological anomaly in addition, increases significantly, for the both founda-tion and overall structure designs, the values of maximum settle-ments, maximum differential settlements and maximum bending moments ( Figs. 9 and 12). It also decreases the uncertainties on the estimations of the maximum differential settlement and max-imum bending moment compared to those obtained when only dealing with the

spatial variability (CV[Dd], CV[M], CV[DdA], CV[MA], Tables 2–5).

In fact, in the overall structure design, the presence of a geolog-ical anomaly leads to an anisotropic spatial variability of soil modulus on the beneath the spread footings which leads to signif-icant impacts on their settlements and their bending moments. This is illustrated for the bending moment as previously presented in Fig. 11 only for one simulation. The values of the bending moment for the spread footings (1), (3) and (4) are between _4.3 to +5.7 kN m ( Fig. 11), which are identical to some extent to those obtained without taking into account the geological anomaly ( Fig. 8). However, we observe a considerably different distribution of these values along their lengths which is due to an effect of load redistribution in the structure [21]. The values of the bending moment for the spread footing (2) are smaller than those obtained from the foundation design ( Fig. 11). The load redistribution effect

reduces the value of the bending moment to _17.4 kN m but it cannot completely absorb it. The overall structure design in this case, appears closer to reality than a foundation design in which the effect of this load redistribution cannot be considered.

7. Conclusions

The main aim of this study was to design the continuous spread footings, for low weight buildings with relatively lightly loaded walls (using the Winkler soil–foundation interaction model), from two approaches: the first approach with a foundation design using a one-dimensional finite element modeling (1D) and the second approach with an overall structure design using a three-dimen-sional finite element modeling (3D). These approaches were com-pared for two different cases: the first case dealing with the spatial variability of

Young‘s soil modulus (Es) and the second case with the spatial

variability of Es coupled with the presence of a geolog-ical anomaly as a lens of clayey soil of weak mechanical properties.

The values of the maximum settlements, maximum differential settlements and their associated uncertainties obtained from the both foundation and overall structure designs of the continuous spread

footing taking into account only the spatial variability of Es (first case) are nearly close together. In this case, the foundation design of a continuous spread footing is sufficiently adequate for the estimations of the maximum settlements, maximum differen-tial settlements and their associated uncertainties.

For the considered first case, the obtained value of the maximum bending moment for the overall structure design is greater than the maximum bending moment value for the founda-tion design. Then, in this case the overall structure design of a con-tinuous spread footing is appropriate for the estimations of the maximum bending moments and their associated uncertainties.

Considering the spatial variability of soil modulus and a geolog-ical anomaly as a lens of clayey soil of weak mechanical properties (second case where a geological anomaly is presented in the mid-dle of a spread footing) increase significantly, for both foundation and overall structure designs, the values of maximum settlements, maximum differential settlements and maximum bending moments but they decrease the uncertainties on the estimations of the maximum differential settlement and maximum bending moment. Then in this second case, the results obtained from the overall structure design appear the more appropriate approach compared to the foundation design due to the load redistribution

S. Imanzadeh et al. / Engineering Structures 71 (2014) 212–221 221

effects in the structure which cannot be considered in the founda-tion design.

Finally, the results obtained here show for continuous spread footings, the importance of the longitudinal behavior of these structures when the spatial variability and a geological anomaly of soil properties can be present. In the case of the spatial variabil-ity of soil properties on a construction site, the foundation design (1D) can be performed only in this case to estimate the settlement and differential settlement but in the other cases, and particularly in the case of a construction site with a strong probability of the issue of a geological anomaly (zone of low stiffness) the overall structure design (3D) appears the more appropriate approach in the design of continuous spread footings.

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