CC BY-NC-ND
0Procedía Engineering
Volume 143, 2016, Pages 623-634
Advances in Transportation Geotechnics 3 . The 3rd International Conference on Transportation Geotechnics (ICTG 2016)
2D Numerical Analysis of Shallow Foundation Rested Near Slope under Inclined Loading
1* 1f 11 Messaoud Baazouzi Djamel Benmeddour ' Abdelhak Mabrouki 1
Mekki Mellas 1§
'The University of Mohamed kidder, Biskra, Algeria. m.baazouzi@univ-biskra.dz, m_mellas@yahoo.fr, mabrouki_abd@yahoo.fr, benmeddourdj @yahoo. fr
Abstract
The bearing capacity of vertically loaded strip footing near slope in plane strain conditions has always been one of the subjects of major interest in geotechnical engineering for researchers and practical engineers. In the literature few studies deal with inclined loaded strip footings adjacent to a cohesionless slope. This study focuses on the numerical analysis of the bearing capacity for a strip footing near a cohesionless slope, and subjected to a centered inclined load, using the finite difference code Fast Lagrangian Analysis of Continua (FLAC). In this study, several geometrical and mechanical parameters have been considered in order to evaluate the effect of the slope on the bearing capacity. The obtained results are presented in terms of normalised failure envelopes in two-dimensional loading plane (H/Vo, V/Vo). The results show the influence of the load inclination and the position of the footing with respect to the slope on the bearing capacity.
Keywords: bearing capacity, inclined load, strip foundation, slope, ultimate surfaces.
1 Introduction
Establishing the bearing capacity of shallow foundations has long been an important component of geotechnical engineering practice. There are an extensive literature dealing with bearing capacity of foundation during the last century through different methods; experimental investigations, numerical and theoretical analyses. Generally, the bearing capacity of shallow foundations is determined using the equation of (Terzaghi, 1943) [1]. This equation is a superposition of three terms; surface term, cohesion term, and overloading term which depend uniquely on the friction angle of the soil.
* Corresponding author, Ph.D. Candidate, E-mail: messaoud.baazouzi@gmail.com
f Doctor, E-mail: benmeddourdj@yahoo.fr
| Doctor, E-mail: a.mabroukii@yahoo.fr
§ Professor., E-mail: m_mellas@yahoo.fr
Selection and peer-review under responsibility of the Scientific Programme Committee of ICTG 2016 623 © The Authors. Published by Elsevier B.V.
doi:10.1016/j.proeng.2016.06.086
After that, most researchers have focused mainly on a way of investigation of bearing capacity and various correction factors in order to confirm or improve this domain.
Footings sometimes are constructed under inclined loading; this is according to the architectural design or case of factors which affect them. According to the footing failure mechanism of (Meyerhof, 1963), the central shear zone is tilted and the adjacent zones are modified. Also, the bearing capacity would differ from that obtained from the conventional bearing capacity equation. For such cases, particular consideration of the inclination loading factor on the bearing capacity is required.
The effect of the inclination loading on the cohesionless bearing capacity of shallow foundation is represented by the following expression:
Qua = 1 BjNrir + cNJc + qNqiq (1)
Where qu is the ultimate bearing capacity, B: width of foundation, y: the soil density, Ny: bearing capacity factors and iy: inclination factor.
In this study, series of numerical computation, using the finite difference code (FLAC, 2005), are carried out to study the influence of the slope, normalised distance of the footing and load inclination angles on the ultimate bearing capacity. The results of the current study are compared with results available in the literature.
2 Problem Presentation
Various expressions have been proposed for the drained inclination factor iy giving a relatively wide range of results. (Meyerhof, 1963; Hansen, 1970 and Vesic, 1975) have been proposed the following expression for iy respectively:
Where: 8 is load inclination factor and 9: internal friction angle of soil.
ly —
— [l - 0.7tagS\
Where: V and H are the vertical and horizontal component of the load.
Y = - H1 =(1 - tgrf
Both equations of (Hansen, 1970) and (Vesic, 1975) were based mainly on results produced by computations using the method of characteristics (MOC).
The French regulation Fascicule (Fascicule, 1993) has proposed the following expression of inclination factor iT
ly —
Actually, there are a lot of studies, (Gourvenec 2007, Loukidis 2008, Georgiadis 2008, Taiebat and Garter 2010, Terzis 2014, Stergiou 2015, Nguyen 2016, Shen 2016) did not describe the ultimate bearing capacity loading reduced by the correction factors, but with the definition of the ultimate
loading combination in the plan, it is mean replaced by the normalised failure envelope in load plan (V, H)
Expression (1) has been proposed for the footing rested on the horizontal ground surface. However, a very limited work has been done the cohesionless bearing capacity of shallow foundation on the slope under inclined loading.
(Hansen, 1970) proposed an additional factor gy in the equation of bearing capacity (1) to taking account the effect of ground sloping.
qult = " Y
gy = (1 - 0.5tgP)5, for
It should be noted that the solution of (Hansen, 1970) it can be only applicable for a foundation rested in a slope or at the crest of slope and the load is inclined towards the slope. However, often the footings are constructed at distance d/B from the crest of the slope; these cases arise due according to architectural needs. (Meyerhof, 1957) who used the equilibrium methods and proposed design charts which are currently adopted by many designs manual. (Bakir, 1993) who developed empirical equations for the ultimate bearing capacity factors for a footing on a slope based on centrifuge tests.
-d /B < a : iR= 1 - 0.9tgP(2 - tgp)\1 - — I (8)
P I 6 B I
-d / B > a : i p = 1
Where: d/B is normalised footing distance.
The non-symmetry of the ground surface imposed treated distinctly two cases; positive and negative load. Figure 1 shows the particular configuration of a strip foundation established on crest of a slope considering an inclined positive load (S> 0) fig. 1a, and negative load (8 <0) fig.1b.
Figure 1: Problem Geometry
On these sides, (Fascicule, 1993) has proposed the following expressions, when the inclination load is positive S>0, the inclination factor iy is:
= 11 -
8+ p 45
, where : p = 45(1 - )
Also, (Maréchale, 1999) has been proposed the factor iyP which has been evaluated by the simple product the two factor ip and iS:
isp= \is Vp (10)
However, the inclination load is negative S<0, (Fascicule, 1999) has recommended that the factor i5p equal the minimum value of the two following expressions:
p = I1 - £
Actually, there are a lot of studies, (Bransby and Randolph, 1998 [10]; Taiebat and Carter, 2002 [11], Gourvenec, 2007 [12], and Georgiadis, 2009 [13]) did not describe the ultimate bearing capacity loading reduced by the correction factors, but with the definition of the ultimate loading combination in the plan, it is mean replaced by the normalised failure envelope in load plan (V, H).
(Maloum, 2002) [14] has been studied the problem of the cohesionless bearing capacity of a strip footing, with a nonassociated flow rule under central inclined load located at the different distance d/B from the crest of the slope. The results are presented in terms of normalised failure envelopes in vertical (V/V0) and horizontal (H/V0) loading plan, and has been proposed the following expressions:
c1 2 c1 + c 2
Where:
'' .i'P '
v.ln —
, and c2 = -
Where: c1 and c2 are described the shape of the normalised failure envelope loading.
3 Numerical Modeling Procedure
3.1 Mesh Discretization and Boundary Condition
The finite-difference code (FLAC, 2005) was used to estimate the undrained bearing capacity of a strip footing on/or near slopes under conditions of plane strain and subjected to a centered inclined load. FLAC (Fast Lagrangian Analysis of Continua) is a two-dimensional explicit finite-difference program for engineering mechanics computations; it simulates the behaviour of structures built of soil, rock or other materials that undergo plastic flow when their yield limits are reached. Many researchers have used the finite difference code FLAC to study the bearing capacity of strip and circular footings (e.g., Frydman and Burd, 1997).
Evaluated of bearing capacity with FLAC is based on dividing the soil into a number of zones, and applying vertical velocities (displacement-controlled method) onto the zone representing the footing. The importance of the mesh size and the vertical velocity in bearing capacity computations was verified earlier by (Frydman and Burd, 1997). The mesh consists of 147 by 54 zones for width and depth respectively; it has been refined at the region most close to the boundaries of the foundation, under the base and near the crest of the slope. The overall mesh dimensions were selected to ensure
that the zones of plastic shearing and the observed displacement fields were contained within the model boundaries at all times.
The boundary condition for this problem the displacement of the left and right vertical sides is constrained in the horizontal direction and full fixities to the base of the mesh.
3.2 Material Model
The constitutive model used is an elastic-perfectly plastic model following the Mohr-Coulomb failure criterion, it is characterised by a friction angle 9=37.5° (cu=0), dilatancy angle y=10°, the undrained Young's modulus £a=65MPa, Poisson's ratio v=0.33, and unit weight of soil y=16 kN/m3 which has affects to the overall stability of the slope.
As in previous studies (Lee et al, 2005 [16]; and Mabrouki et al, 2010 [17]), it was observed that the values of Young's modulus and the Poisson's ratio affect the evolution of the footing settlement but have no effect in the value of the collapse load. The footing was assumed linear elastic material with concrete Young's modulus of Ec=25GPa and Poisson's ratio v=0.2. The footing is connected to the soil via interface elements. The properties of the interface elements are related to the properties of the adjacent soil elements, is assumed to have a friction angle equal to the soil angle of internal friction and the same dilation angle. It is characterized with the normal stiffness Kn =109 Pa/m and shear stiffness Ks =109 Pa/m, these parameters do not have a major influence on the failure load.
For the simulation of the ultimate load, the probe analysis is the technique which was used (Bransby and Randolph, 1998 and Gourvenec, 2003). This method consist to apply vertical load, smaller than the ultimate vertical loading on the footing, then horizontal velocity is applied, in the horizontal direction, to the nodes situated at the bottom of the footing. An optimal velocity must be chosen in order to reach a value of the ultimate bearing with a reasonable computation time. Several tests were done for chosen the optimal vertical velocity, because the simulation time and speed of analysis is very important part of numerical modeling, also must be not affect the accuracy of results. Finally, the vertical velocity chosen for all analyses is 2x10-7 m/step.
Figure 3: Finite difference mesh and boundary condition for the case: [3=26.6°, H/B=3.5 and d/B=0
4 Results and Discussion 4.1 Horizontal Ground Surface
The results obtained from this case are for associated flow rule 9=y, they will compare to the results obtained with the empirical expressions by (Hansen, 1970; Meyerhof, 1963 and Vesic, 1975).
Figure 3 shows the variation of the inclination factor iy with the inclination of the load 8, on observed the factor iy decreases with increase in the load inclination 8. The results of present study are in good agreement with the results of (Meyerhof, 1963) and (Hansen, 1970). However, the (Vesic, 1975) solutions slightly higher than to those obtained by the finite difference analysis.
tan(S)
Figure 4: Comparison of inclination factor iY as a function of the load inclination S
Recently, the approach of multiplicative factors is being replaced by the ultimate surface (V, H) in load plan (Paolucci et al, 1997; and Sieffert et al, 1998).
Figure 4 shows the ultimate surfaces of normalised failure load in plane (H/Vo, V/Vo) for a foundation established on a ground level surface. The results are compared with those obtained by (Hansen, 1970; and Vesic, 1975). The numerical results are in good agreement with the Hansen's results (Hansen, 1970). Furthermore, the Vesic's results (Vesic, 1975) overestimate the normalised failure loads for height values of normalised horizontal load H/V0.
It should be noted, the horizontal ultimate load Hult of present study is about 0.095Vult obtained for a vertical load between 0.4Vult and 0.5Vult. According to the experimental value (Georgiadis et al, 1988; and Gottardi, 1993) of the horizontal load Hmax is 0.12Vut
Figure 5: Comparisons of normalised failure envelopes for footing rested on horizontal ground surface
4.2 Inclined Loading of a Footing Rested at the Crest of the Slope
The numerical results as shown in the Figure 5 are corresponded to the ultimate normalised load values of (V/V0, H/V0) obtained for each inclination 8 of the load, which are either positive or negative. As seen, the normalised distance d/B it has a very important effect on the normalised loads for inclinations S> -10 Interestingly, the results have shown a significant difference in the normalised loads, which are proving the importance of the position of the foundation near the slope. In contrast, the results are very close for the height negative inclination load. This result can be explained by the failure mechanism occurs towards the horizontal ground surface; and under these conditions the slope has no effect on the normalised loads.
-0.12 -0.08 -0.04 0 0.04 0.0S 0.12
Normalised horizontal failure load (H/H0)
Figure 6: Influence of normalised distance D/B on normalised load curves
For understand the reason for these variations, we will be observing the typical load-displacement horizontal curves which are illustrated in the figure 7. The normalised distance d/B=0, 0.5 and 2, and lower inclination loads S=-5° and -10°, the behaviour is particularly essential, at the beginning of displacement the curves have followed the negative inclination load S=-15° and -20°, until attain a certain value of loading, they have changed their direction of displacement to the positive direction, this phenomenon can be explained by the progression of potential failure mechanism towards the slope, not the horizontal ground surface, as shown in the figure 8; in these cases, it will be interested to present the response of load horizontal versus displacement horizontal for the lower inclination loading 8 as seen in the figure 9. The behaviors are related to the position of the foundation near the slope, for D/B=2 the failure mechanism are occurring in the horizontal ground surface.
Figure 7: load horizontal versus displacement horizontal for: (a) d/B=0.5, (b) d/B=1, (c) d/B=2
Figure8: plasticity zones for d/B=0
Figure 9: load horizontal versus displacement horizontal for S=5
4.3 Normalised Failure Load Envelope
Figure 10 shows the normalised failure load envelope, in two-dimensional (V/Vo) versus (H/Vo) load plan, for a footing at the normalised distance d/B=0, 0.5 and 2, respectively.
The failure envelope it's symmetrical for the foundation located in the horizontal ground surface or located so far from the crest of the slope, but this symmetrical has been disappearing at the presence of the slope.
As seen, the results of present study are in excellent agreement with the results of (Maloum, 2002). However, the Maréchale's solution (Maréchale, 1999) underestimates the normalised loads for negative inclination loads, and (Fascicule, 1993) overestimates the normalised loads for positive inclination loads.
Normalised honsontal failure bad [h]
Figure10: Normalised failure load surfaces for (a) d/B=0, (b) d/B= 0.5 and (c) d/B=2.
5 Conclusion
The finite-difference code FLAC was used to study the influence on the bearing capacity of strip footings on or near slopes under inclined load. Various geometries were considered, the results of the analyses were compared to other available solutions. The displacement confirms that the failure mechanism is made toward the slope for the positive and negative lower inclined load.
It was found that the shape of the vertical versus horizontal load interaction diagram depends the distance of the footing from the slope and the slope angle. The bearing capacity factor Ny increases when the distance of the foundation diverges from the slope, also the slope angle p diminished.
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