Analysis of some key aspects of soil/foundation interaction- Finite elements modelling

This paper is aimed to contribute to the analysis of three important aspects of soil-foundation interaction, which are not clearly investigated, by means of a detailed parametric study based on a finite elements modelling. The first aspect focuses on the effect of the bedrock proximity on the load-settlement behaviour of a continuous or circular shallow foundation. It was found there exists a threshold distance between the foundation base and the top of the bedrock layer beyond which the foundation behaves as in an infinitely deep medium. The second one deals with the behaviour of shallow foundation on a bi-layered soil where the effect of the underlying layer on the bearing capacity as well as on the settlements depends on the distance between the foundation base and the top of the underlying layer, and beyond a threshold value this effect vanishes. The third aspect studied was the interference between two strip footings installed on saturated clay. It was shown within a threshold distance between these foundations, an important modification of the foundation behaviour may occur. Finally, the numerical results were fitted and interpreted which allowed suggesting simple practical formulae for shallow foundations design.


Introduction
These last four decades were marked by an unprecedented scientific interest in the finite element modelling applied to geotechnical engineering. Earliest work was presented by Duncan and Chang (1972) on the use of a hyperbolic stress-strain formulation in a finite element method for computing the settlements of a model strip footing in sand and a circular footing in a saturated undrained clay. Ever since, availability of several non linear constitutive law-based softwares has made possible to explore soil/structure phenomena besides difficult to access by traditional analytical elastic methods [1].
Although the traditional aspect of shallow foundation design based on the analysis of the bearing capacity as well as on the settlement behaviour, it is recognized some aspects related to the soil/foundation interaction are not yet satisfactorily studied, among which the effect of the bedrock proximity on the load-settlement behaviour, the response of shallow foundations on a bi-layered soil, and the interaction between two closely spaced footings.
Based on a non linear 2D finite element modelling, this paper aims at contributing to understand these three aspects of the soil/foundation interaction as well as at proposing simple formulae for practical purposes.
Due to the limited space allowed to this paper, the main results will be briefly presented.

Presentation of the method of analysis
The software CRISP v.5 was used as a computational tool to carry out a detailed parametric study of the loadsettlement behaviour of rigid circular foundation as well as rigid strip footing. CRISP is a specialized 2D programme of elastic plastic finite elements analysis offering many computational options. Soil material is assumed in this study following a perfectly elastic-plastic constitutive law obeying the Drucker-Prager's failure criterion and the soil properties are assumed homogeneous in space.
The soil studied is a purely cohesive saturated clayey soil exhibiting an undrained behaviour and submerged by a ground water table at surface. Three consistencies were studied: soft, medium consistent and hard clay.
The foundation is modelled as a rigid reinforced concrete slab with no interface elements between the foundation and the soil.
The FEM model is a plane strain mesh for a strip footing and an axisymmetric mesh for a circular footing. Mesh size is 20Bx20B, B being the width of the strip or the diameter of the circle [1]. Mesh refinement was considered within a zone of 4Bx4B beneath the foundation base whereas the remaining area of the mesh was automatically generated by the software.

CMSS-2017
The small strain analysis option was considered. Moreover, for nearly incompressible materials like a saturated clay it is possible to avoid an "over-stiff response" of the finite elements by using the technique of integration called the Ḇ method [2].
To model the rigid load-settlement behaviour of the foundation, a series of prescribed displacements up to B/10 was assigned at the nodes of the foundation base.
The bearing capacity q l was conventionally defined as the vertical pressure corresponding to infinite settlement and determined on the basis of the Asaoka's graphical method, whereas the foundation stiffness K v0 which is the initial slope of the load-settlement curve was estimated by the Newton's divided differences technique.

Formulation of the problem
The load-settlement curve parameters K v0 and q l may be formulated according to the following general equation: (1) D, B and H are respectively the foundation embedded length, the width or diameter of foundation and the soil layer thickness beneath the foundation base. E s , C u are respectively the elastic modulus and the undrained shear strength. By using the Vashy-Bukingham's theorem of dimensional analysis, this physical equation is transformed into an equivalent dimensionless equation described by (N-k) dimensionless parameters noted π, k being the number of fundamental units in equation (1). Since N=7 and k=3, the equivalent equation may be written as: is the normalized layer thickness.
The terms π 1 and π 2 should be independent since the first one corresponds to the small displacements whereas the second one corresponds to the large displacements. Consequently, the equation (2) may by uncoupled to the two following equations [3]:

Analysis of the N c factor
In all the cases studied it was found this factor does not depend on the consistency of the clay and slightly increases with D/B. However, sensitive dependency of N c on H/B, as typically illustrated in figure 1, was noticed in both strip and circular foundations. It was shown the existence of a threshold value noted H lim beyond which the foundation behaves as in an infinitely thick layer. The normalized thickness H lim /B was conventionally determined, as shown in figure 1, as the intercept of the straight lines Δ 1 and Δ 2 , which led to the margin of values of 0.7-1.0 depending on D/B for a strip footing with an average value of 0.9, and a margin of 0.67-0.80 with an average value of 0.74 for a circular footing.
By comparison with the Matar-Salençon's method which gives H lim /B equal to 0.79 for a strip footing, this latter fall within the margin found in this study [4].

Analysis of the N c1 factor
It is to be noticed all the cases studied exhibit the same variation of N c with the normalized distance H/B regardless the shape of foundation and the slenderness ratio. Typical curves illustrated in figure 5 show there exists a threshold normalized distance H lim /B beyond which the effect of the lower layer on the bearing capacity vanishes and the foundation behaves therefore as in a infinitely thick mono-layered soil. This threshold value is about 4 for strip footing and 2.5 for circular footings regardless the ratios C 1 /C 2 and D/B. Figure 5 highlights an important effect of the ratio C 1 /C 2 where it can be seen in case of C 1 >C 2 , decrease of H/B results in a decrease in the bearing capacity due to the proximity of the lower layer having a lower undrained shear strength. However, in case of C 1 <C 2 , the bearing capacity increases when H/B decreases.
In case of H< H lim , the effect of proximity of the lower layer should be accounted for. In this regard, the function m as described in equation 8 may be written as follows: q l =f c N c1 C 1 (9) f c is called the influence factor of the lower layer depending on H/B, D/B and C 1 /C 2 . It was estimated by least squares regression technique leading to the following formulation: Values of a and b were found slightly varying with D/B and then tabulated as average values in table 1.
N c1 corresponds to the N c factor of the upper layer considered as infinitely thick and is therefore equal to N c ∞ illustrated in figure 2.

Analysis of the factor I s
Similarly to the effect of C 1 /C 2 on the bearing capacity, the factor I s varies sensitively as function of the stiffness ratio E s1 /E s2 . As shown in figure 6 illustrating a typical variation, in case of E s1 >E s2 any decrease of H/B results in a decrease of I s due to the closeness of a looser lower layer. However, in case of E s1 <E s2 , the factor I s increases when H/B decreases. It was found the existence of a threshold normalized distance H lim /B regardless the ratios D/B, E s1 /E s2 and the shape of foundation, beyond which the foundation settles as in an infinitely thick mono-layered soil. This threshold value is about 6.
The function k described in equation (7) may be rearranged as follows: In case of H<H lim , the results were fitted by simple power functions and the function k may then be formulated as follows: Values of a and b were found slightly varying with D/B and then tabulated as average values in table 2.

Formulation of the problem
As illustrated in figure 7, the general configuration of such a problem leads by dimensional analysis to the following equations: d is the horizontal distance between the vertical axes of two identical strip footings embedded at the same depth D. The footings/soil system is identical to that studied in part 3 except for its dimensions which were fixed by gradual increase until the settlement at foundation base nodes start to stabilize. Due to the vertical symmetry only half a plane was modelled.

Analysis of the N c factor
It was found the consistency of the clay has almost no effect on the N c factor. Average values of N c were plotted versus the normalized distance d/B in figure 8 which clearly shows a threshold distance of 3B beyond which both the strip footings behave separately. In order to formulate the function p described by equation 14, such curves were fitted by the following hyperbolic function using the least squares regression technique:  Table 3 summarized the values of a and b for different values of the slenderness ratio D/B.

Analysis of the factor I s
In contrast with the N c factor, the figure 9 shows the factor I s linearly increases with the normalized distance without exhibiting a threshold value even at distance d of 10B.

Conclusions
Detailed parametric study based on a non linear 2D finite elements modelling was carried out in order to highlight three important aspects of soil/rigid foundation interaction as well as to suggest simple formulae for practical purposes. The first aspect is the effect of the proximity of a bedrock on the strip or circular footing behaviour. It was demonstrated the existence of a threshold depth of the bedrock beyond which the foundation behaves as in a infinitely deep layer.
The second aspect is the behaviour of foundations on a bi-layered purely cohesive soil. The results showed also the existence of a threshold normalized distance H lim /B beyond which the effect of the lower layer on the bearing capacity and on settlement vanishes and the foundation behaves therefore as in a an infinitely thick mono-layered soil.
Finally, study of the behaviour of closely spaced identical strip footings on a mono-layered purely cohesive soil illustrated the consistency of the clay has almost no effect on the N c factor. Moreover, a value of 3B for the distance between the foundations vertical axes is a threshold beyond which both the strip footings behave separately in terms of bearing capacity.