NUMERICAL STUDY OF THE UNDRAINED BEARING CAPACITY OF

STRIP FOOTING ON SLOPE

M. BAAZOUZI, M. MELLAS, A. MABROUKI; D. BENMEDDOUR Civil engineering laboratory, University of Biskra, BP 145 Biskra 07000, Algeria Abstract: The bearing capacity of shallow foundation near slope has always been one of the subjects of major interest in geotechnical engineering for researchers and practical engineers. This study focuses on the numerical analysis of the undrained bearing capacity for a strip footing near a slope, and subjected to a centered vertical load, using the explicit finite difference code FLAC (Fast Lagrangian Analysis of Continua). Theoretical and experimental studies confirm that, when a strip footing is near a slope, the bearing capacity must be assessed using reduction coefficients. In this study, several geometrical and mechanical parameters have been considered in order to evaluate the effect of the slope on the undrained bearing capacity. The numerical values have been compared with those available in the literature. The results show the influence on the undrained bearing capacity of the location of the footing with respect to the slope. Keywords: bearing capacity; strip footing; vertical loads; finite difference analysis; slope.

INTRODUCTION The prediction of the bearing capacity of a shallow foundation is a very important problem in Geotechnical Engineering. There is 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 Terzaghi (1943). This equation is a superposition of three terms; surface term, cohesion term, overloading term which depend uniquely on the friction angle of the soil. In the horizontal ground surface case and vertical load, the undrained bearing capacity is restricted on the cohesion term and it is represented by the following expression: q=Cu Nc (1) Where Nc is the undrained bearing capacity factor (=π+2), Cu is the undrained shear strength of the soil. However, there are many circumstances where foundations must be constructed on or near a slope. Due to land limitation and some other specific reason, such as bridge abutments, towers footings of electrical transmission lines and buildings that are located next to a ravine (Shield et al. 1990). In these cases the bearing capacity may be significantly reduced depending on the location of the footing with respect to the slope, slope height, and soil type. However, the difficulty in solving a footing on slope problem is that the ultimate bearing capacity is affected by either the local foundation failure or the global slope failure which has damaging consequences on the environment in general (Azzouz et al. 1983). Many empirical equations have been developed from correlation between in situ data and/or laboratory data in order to understand this phenomenon. Hansen (1961) proposed the following expression for the undrained bearing capacity factor Nc based on a method of characteristics:

c

25.14 1

2N

(2)

Where: β is slope angle. Vesic (1975) incorporated Hansen’s and included the soil weight term of the undrained bearing capacity equation:

c

25.14 1 sin

2 u

BN

C

(3)

Where: γ is the soil unit weight, and Cu undrained shear strength. Hansen (1961) and Vesic (1975) solutions are applicable only for a foundations established at the crest of a slope. Moreover, they did not study the influence of the slope height, and Hansen didn’t take account the effect of soil properties. Practically, the foundations are created at distance λ from the crest of a slope. Meyerhof (1957) is proposed design charts which are currently adopted by many designs manual. Georgiadis (2010) based on the finite element results; design charts, equations, and a design procedure are proposed for the calculation of the undrained bearing capacity factor Nc:

For λ < λ0: 0

0 00

5.14 - 1 1-2

c c cN N N

(4)

For λ ≥ λ0: Nc = 5.14 (5) Where: λ normalized footing distance, and λ0 is critical normalized footing distance. Many analytical and numerical studies in bearing capacity have conducted via different approach, including the method of stress characteristics which studied by Giroud and Tran-Vo-Nhiem (1971) and Graham et al (1987). Kusakabe et al (1981) has been researched about bearing capacity of clays on slope that are being given a continuous load using upper bond theorem, he is the first to introduce the concept of the soil strength ratio Cu/γB, Buhan and Garnier (1998) has been also used yield design theory to evaluate the ultimate bearing capacity of foundation on slope. Saran et al. (1989) who introduced an analytical solution to obtain the bearing capacity of foundation adjacent to slopes using both limit equilibrium and limit analysis approaches and Yung et al (2007) presented design charts of bearing capacity on slope using an energy dissipation method of plasticity. The limit equilibrium method has also been applied by several investigators to provide solutions and design charts for footing on slope, Azzouz and Baligh (1983) who studied three dimensional of strip and square footings on clay slopes, they used circular arc methods. Bowles (1997) who developed an expression which takes into account the effect of the distance of he footing from the slope, Narita and Yamaguchi (1990) presented three dimensional footing located on top of a slope has been used log-spiral analysis. Castelli and Motta (2008) who used a method of slices to determine the bearing capacity of footing located on top of a slope. In present time numerical analysis such as finite element difference and finite element method become a powerful technique in geotechnical problems to analyze complex behavior of stress and strain due to external loading. The finite difference, Shiau and Watson (2008) studies 3D bearing capacity of smooth footing on homogeneous clay (φ = 0). The finite element, upper bound plasticity and stress field methods were used by Georgiadis (2009) who is investigate the influence of load inclination on the undrained bearing capacity of strip footing on or near slopes. Georgiadis (2010) as mentioned earlier, based on the finite element results; the results of analysis were presented in the form of design charts, and a design procedure was proposed for the calculation of the undrained bearing capacity factor Nc. Also, Shiau et al (2011) proposed design charts based on averaged lower bound and upper bound results based on the finite element limit analysis method for rough and smooth footings placed on purely cohesive slopes. Nguyen et al (2012) and Mofidi et al (2014) Each one of them used upper bound method based on edge-based smoothed finite element method and finite element formulation of lower bound method to investigate the bearing capacity of rough and smooth footing rest

on/or the slope. In addition, presented design charts for both purely cohesive and cohesive-frictional soils. Furthermore, some experimental works on the bearing capacity of foundation near slope have been conducted by Shields et al. (1977) and Bauer et al. (1981) were used full-scale model tests. Also, Gemperline (1988) who developed empirical equations for the ultimate bearing capacity factors for a footing on a slope based on centrifuge tests. In this study, series of numerical computation using the finite difference code FLAC (2D) (2005) are carried out to evaluate the influence of the various parameter that affect undrained bearing capacity of strip footings on or near slopes. The results of the analyses are compared to available methods. PROBLEM DEFINITION The geometry of the analyzed problem is shown in Fig 1. In the current study, we consider a rigid strip footing of width B=2m located on a homogeneous clay soil which makes a various angle β° with the horizontal and slope height H/B at a several distance λ/B measured from the edge of the foundation to the crest of the slope. The soil is modeled as a Tresca material using the Mohr-Coulomb elastic-perfectly plastic constitutive model with a shear strength Cu (φ=0), the undrained Young’s modulus Eu=30MPa, Poisson’s ratio v= 0.49 and unit weight of soil γ=20KN/m3 which has affected (original: affects to) the overall stability of the slope. The footing is assumed to be smooth and rigid. The undrained bearing depends on a wide range of dimensionless ratios, namely the slope angle β, Cu/γB, λB, H/B. Furthermore, these parameters are affecting the failure mode of slope, which can to take place according to either of two modes: the first two failure modes fig.2 (a-b) which is governed by either the local foundation failure, where the shear surface does not intersect the slope and can be referred to as “bearing capacity” failure modes. However, the second is overall slope failure where the critical shear surface extends beyond the crest and therefore; involves part of the slope as shown on fig.2c.

Fig.1. problem geometry

Fig.2. Different typical failure modes for footing/slope problem: bearing capacity failure (a,b) and overall slope failure(c).

NUMERICAL MODELING PROCEDURE 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 vertical 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 isolated footings (e.g., Frydman and Burd, 1997). Because of the nonsymmetrical problem, the entire model is considered in the computations. Fig 3 shows a typical finite-difference mesh, used for the case of footing located at the crest of slope (λB=0). For evaluated 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 177 by 71 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. The soil is modeled as a Tresca material using the Mohr-Coulomb elastic-perfectly plastic constitutive model with a shear strength Cu(φ=0), the undrained Young’s modulus Eu=30MPa, Poisson’s ratio v=0.49 and unit weight of soil γ=20KN/m3 which has affects to the overall stability of the slope. The footing was assumed linear elastic material with concrete Young’s modulus of Ec=25 GPA and Poisson’s ratio v=0.4. The properties of the interface elements are related to the properties of the adjacent soil elements, for the smooth case, the cohesion is set as C=0, these

ensures that the shear stress underneath the footing is zero and the displacement of nodes in the horizontal direction is free, while for the rough case C=Cu, it is characterized with normal stiffness Kn =109 Pa/m and shear stiffness Ks =109 Pa/m. Several tests were done for chosen the optimal vertical velocity, because the time and rapidity 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 2×10-7 m/step.

Fig.3. Finite difference mesh and boundary condition for the case: β=45°, H/B=3 and λB=0 RESULTS AND DISCUSSIONS BEARING CAPACITY The loading of the rigid strip footing is simulated by applying equal vertical velocities at the area representing the footing. The progressive displacement of the footing induced by the vertical velocity applied at the footing nodes is accompanied by the increase of the pressure in the soil. Finally, the pressure under the footing stabilizes for a value that indicates a limit load (Fig. 4). The vertical bearing capacity qu=5.21Cu, was predicted in the present study for horizontal ground surface, representing an overestimation of less than 1.36% from the analytical solution of qu=5.l4Cu. It means that numerical prediction is in excellent agreement with a solution of Prandtl (1921). Figure 4 shows the load-displacement curves for rough footing at distance λB=1 from the slope, (i.e., the curves of the vertical load V versus the vertical displacement). The comparison is made for three values of slope inclination, it can be seen that the limit load V decreases substantially with increasing β.

0

100

200

300

400

500

600

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Displacement Uy

Lo

ad

V (

KN

/m)

β=15°

β=30°

β=45°

Fig.4. load displacement curves for: β=15°, 30° and 45°, H/B=3m, and λ=1.

INFLUENCE OF THE INTERFACE The influence of the footing interface is investigated through finite-difference analysis results for four normalized footing distance λB= 0, 1, 2 and 3, and two slope angles β=60° and 90°. Three ratios are considered Cu/γB = 1, 3 and 5, as shown in the table 1 and 2. Note the values of ratio p/γB differ depending on whether smooth or rough conditions are assumed, the footing roughness increases the bearing capacity slightly compared a smooth interface footing-soil. It can be seen that the results of Shiau et al (2011) are lower than the results of present study for the footing located at the crest of slope (β=60° and λB=0) with rough interface. However, the other cases are slightly higher. The reason for these differences are shown in the figure 5(a, b) which are showed the deformed shaped for smooth and rough footing rest on the top of slope (β=60° and λB=0) respectively. Table 1: Bearing capacity factor Nc for smooth (s) and rough (r) footings for β=60°.

p/γB

λ= 0 λ= 1 λ= 2 λ= 3

Rough Smooth Rough Smooth Rough Smooth Rough Smooth

Present study 2.74 2.20 2.97 2.84 3.38 3.27 3.97 3.82 Cu/γB=1

Shiau et al.2011 2.63 2.63 3.19 3.18 3.66 3.66 4.20 4.20

Present study 10.38 8.88 12.53 11.75 14.02 13.00 15.66 14.76 Cu/γB=3

Shiau et al.2011 9.08 8.95 12.46 12.46 14.54 14.40 15.96 15.31

Present study 15.10 18.97 21.15 23.63 Cu/γB=5

Shiau et al.2011 15.44 15.16 21.27 21.26 24.81 24.80 27.00 26.68

Table 2: Bearing capacity factor Nc for smooth (s) and rough (r) footings for β=90°.

p/γB

λ= 0 λ= 1 λ= 2 λ= 3

Rough Smooth Rough Smooth Rough Smooth Rough Smooth

Present study - - - 2.05 - 2.25 - 3.13 Cu/γB=1

Shiau et al.2011 1.32 1.32 1.20 1.20 1.46 1.46 1.99 1.99

Present study 6.99 4.73 8.98 7.69 11.07 9.94 13.73 Cu/γB=3

Shiau et al.2011 5.50 5.50 9.01 9.01 10.88 10.85 12.78 12.72

Present study 12.20 9.5 15.74 12.87 19.07 15.94 22.5 Cu/γB=5

Shiau et al.2011 9.50 9.50 16.17 16.12 19.65 19.64 22.74 22.73

Fig.5. Deformed of shape for smooth (a) and rough (b) footing respectively (β=60°, λB=0, Cu/γB=1)

INFLUENCE OF THE RATIO Cu/γB: In this section, investigated the effect of the ratio Cu/γB on the dimensionless bearing capacity p/γB for smooth footing rest on different slopes angle β=30°,60° and 90°, five normalised distance λB=0, 1, 2, 3 and 4. As seen in the figure 6(a-e), the dimensionless bearing capacity p/γB increases with the increase of ratio Cu/γB and reduces with the increasing of slope inclination β. Furthermore, when the normalised footing distance λB is faraway from the top of the slope it can be seen that the results of three slopes are converging between them. Interestingly, the dimensionless bearing capacity in primarily increases no linear for the low ratio Cu/γB, and to become linear from the ratio Cu/γB>1, this phenomenon indicates that the type of mode which had occurred. The non linear curve indicated the overall slope failure mode and the linear curve shows the bearing capacity failure mode; for further clarification, the figure 7 shows the velocity contours for various values of Cu/γB.

(a) (b)

0

10

20

30

40

50

0 2 4 6 8 10

Cu/γB

p/γ

B

Present study, β=30°Present study, β=60°Present study, β=90°Shiau et al. (2011) β=30°Shiau et al. (2011) β=60°Shiau et al. (2011) β=90°

0

10

20

30

40

50

0 2 4 6 8 10Cu/γB

p/γ

B

Present study, β=30°

Present study, β=60°

Present study, β=90°

Shiau et a.l (2011), β=30°

Shiau et a.l (2011), β=60°

Shiau et a.l (2011), β=90°

0

10

20

30

40

50

0 2 4 6 8 10

Cu/γB

p/γ

B

Present study, β=30°Present study, β=60°Present study, β=90°

Shiau et al. (2011), β=30°Shiau et al. (2011), β=60°Shiau et al. (2011), β=90°

Cu/γB=0.7

0

10

20

30

40

50

60

0 2 4 6 8 10Cu/γB

p/γ

B

Present study, β=30°Present study, β=60°Present study, β=90°Shiau et al. (2011), β=30°Shiau et al. (2011), β=60°Shiau et al. (2011), β=90°

0

10

20

30

40

50

0 2 4 6 8 10

Cu/γB

p/γ

B

Present study, β=30°

Present study, β=60°

Present study, β=90°

Shiau et al. (2011), β=30°

Shiau et al.(2011), β=60°

Shiau et al. (2011), β=90°

Fig.6. Variation of dimensionless bearing capacity p/γB with Cu/γB for λB=0, 1, 2, 3 and 4

respectively

Fig.7. Velocity contours for decreasing values of Cu/γB.

INFLUENCE OF THE SLOPE INCLINATION β

Cu/γB=0.6

Figure 8 shows the variation of the calculated values of the undrained bearing capacity factor for a rough footing with the slope angle β for ratios cu/γB=1. The results presented in this study are compared to the solutions of Hansen (1961), Vesic (1975), Bowles (1996), Kusakabe et al. (1981) and Georgiadis (2010) for the case of a footing at the crest of a slope. The present study confirms a linear decrease of the undrained bearing capacity factor with the increase of the slope angle. It is seen that the results of present study is in good agreement with upper bound solutions that are investigated by Kusakabe (1981) and Georgiadis (2010).however, Vesic solution underestimates the value of Nc to 13%. Hansen solution is slightly overestimated the value of undrained bearing capacity factor, it should be noted that Hansen’s equation does not take account to the effect of Cu/γB ratio.

1.5

2

2.5

3

3.5

4

4.5

5

5.5

0 10 20 30 40 50 60 70

β°

Nc

Hansen (1961)Vesic(1975)Kusakabe (1981)Bowles (1996)Georgiadis (2010)Present study

Fig.8. Variation of Nc with slopes angle β for footing at crest of the slope λB=0, Cu/γB=1 and H/B=3

(Figure modified from Georgiadis, 2010). INFLUENCE OF SLOPE HEIGHT Figure 9 shows the variation of bearing capacity factor Nc with various normalized slope height H/B for β=30° and β=45°, Cu/γB=1, 2.5 and 5 will the normalised distance of footing λB=0. As seen in the figure 8, it can be distinguished a three failures modes. The first mode decreases linearly until reaching a value where it becomes constant, these modes represent a transition from the bearing capacity failure mode on the horizontal ground to the bearing capacity failures mode on slope and start the second mode of failure which takes remains constants at the critical value which represents the bearing capacity failure on slope as shown in the figure 2a. However, the third failure mode which is begun when the slope height attains the critical value then decreases (Original: decrease) linearly, this mode represented the overall slope failure. To make the difference between these failure modes, it is common to calculate the difference using the number stability of slope N equation which has been proposed by Taylor (1937) who use the friction circle method.

γ

cN

HFs

Where N = stability number and Fs= factor of safety for stability of the slope.

For β=30° and β=45°, φ=0° the number stability of slope N=0.18, so the critical height H/B=5.55, 13.9 and 27.8 for Cu/γB=1, 2.5 and 5 respectively, which is close to the present study.

0

1

2

3

4

5

6

0 5 10 15 20 25 30

H/B

Nc

Cu/γB=1, β=30°

Cu/γB=2.5, β=30°

Cu/γB=5, β=30°

Cu/γB=1, β=45°

Cu/γB=2.5, β=45°

Cu/γB=5, β=45°

Fig.9. Variation of Nc with slope height H/B for λ= 0

INFLUENCE OF NORMALISED FOOTING DISTANCE Figure 10 shows the variation of Nc with λ. for slope angles 30° and 45°. As seen, Meyerhof’s solution overestimates the value of bearing capacity factor Nc. For all values of β, the increase in footing capacity tends to stop at certain values of λ. From the present study Nc reaches a constant value of 5.25 approximately at λ=1.5 for β=30°. For β=45° the limite value of Nc is reached at λ=2.5. The finite difference results are in good agreement with the finite element results solution proposed by Georgiadis (2010).

3

3.5

4

4.5

5

5.5

0 0.5 1 1.5 2 2.5

λ/B

Nc

Meyerhof (1957)

Kusakabe (1981)

Georgiadis (2010)

Présente étude

Fig.10. Variation of Nc with normalised footing distance λ/B, β=30°, 45° and cu/ γB=2.5

(Figure modified from Georgiadis, 2010).

Failure mode (a) Failure mode (c)

Β=30°

Β=45°

CONCLUSIONS The finite-difference code FLAC was used to study the influence on the undrained bearing capacity of strip footings on or near slopes under vertical load. Various geometries and soil properties were considered, the results of the analyses were compared to other available solutions. It was found that the bearing capacity factor Nc depends on the slope height, the distance of the footing from the slope and the slope angle. It is worthwhile noting that most of the available methods for the evaluation of the undrained bearing capacity factor Nc do not take account of these parameters. The increase of the distance of the footing from the slope increases the value of bearing capacity factor Nc, also, at high values of H/B, overall slope failure is observed. The footing capacity decreases as the slope angle β is increased. REFERENCES

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