numerical investigation of the behavior of strip …

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NUMERICAL INVESTIGATION OF THE BEHAVIOR OF STRIP FOOTINGS UNDER LARGE ECCENTRIC LOADS

Sadok BENMEBAREK1, Ahlem GUETARI1*, Mohamed Saddek REMADNA1, Naima BENMEBAREK1

Address

1 Dept. of Civil Engineering and Hydraulics, MNISSI Laboratory, University of Biskra, Algeria

* Corresponding author: [email protected]

Abstract

In this paper, numerical computations using the FLAC code are carried out to investigate of the behavior the bearing capacity of strip footings embedded in sandy soil and under large eccen-tric vertical load conditions. The study focuses on an evaluation of the non-dimensional reduction factor (RF), which is the ratio between the average bearing capacity of an eccentric footing and its bearing capacity when subjected to a central load. The results indicate a significant decrease in RF when the eccentricity increases and the curve of the decrease is parabolic, even in large eccentricity cases. Based on the numerical results obtained, a new formula of the RF is proposed, and the reasons for the discrepan-cy between the RF proposed by several authors are identified. The values provided by the proposed formula are in close agreement with some experimental results available in the literature.

Key words

● Bearing capacity, ● Reduction factor, ● Numerical modelling, ● Large eccentricity.

1 INTRODUCTION

The bearing capacity of shallow foundations is one of the most important subjects in the geotechnical field. Various theoretical and experimental studies have been published on this subject. Terzaghi (1943) proposed a formula for the bearing capacity of a strip footing subjected to a centered vertical load. This formula was based on an approximate solution, which had used superposition, and the solution of (Prandtl, 1920). According to (Terzaghi, 1943) and (Prandtl, 1920), different theories are usable for the computation of the ultimate bear-ing capacity of shallow foundations subjected to central vertical loads (Meyerhof, 1951; Hjiaj et al., 2005); eccentric vertical loads (Prakash and Saran, 1971; Okamura et al., 2002; Nawghare et al., 2010; Krab-benhoft et al., 2012); and eccentric inclined loads (Meyerhof, 1953, 1963; Hansen 1961; Purkayastha and Char 1977; Saran and Agrawal 1991; Loukidis et al., 2008; Yamamoto and Hira 2009; Patra et al., (2012, 2016); Ornek 2014; Krabbenhoft et al., 2013; Ganesh et al., 2016).

Shallow foundations are often subjected to eccentric and/or in-clined loads, where the problem becomes more complicated because of detachment at the soil-foundation interface. The most commonly

applied equations for the bearing capacity of foundations subjected to eccentric loads are based on the work carried out by (Meyerhof, 1953) and (Hansen, 1961). Meyerhof (1953) proposed a semi-em-pirical method to estimate the ultimate bearing capacity of a shallow foundation subjected to eccentric loads; this method is called the ef-fective width, B’ = B-2e, where B is the foundation’s width, and e is the eccentricity of the load. Prakash and Saran (1971) provided a mathematical formulation to obtain the ultimate bearing capacity of a strip foundation subjected to a vertical eccentric load. Purkayastha and Char (1977) conducted stability analyses of an eccentrically load-ed strip footing supported by sandy soil using the method of slices. They introduced a non-dimensional reduction factor (RF) to obtain the bearing capacity of an eccentrically loaded footing from the val-ue of a similar centrally loaded footing. Saran and Agarwal (1991) carried out a limit-equilibrium analysis for calculating the bearing capacity of rigid strip footings with a rough base under eccentric in-clined loading conditions. Okamura et al., (2002) carried out centrif-ugal model tests to study the effect of shape and size on the bearing capacity of rectangular footings under vertical eccentric loading on sand. Loukidis et al., (2008) used the finite element method to study the bearing capacity of a shallow foundation on sand subjected to

Vol. 27, 2019, No. 3, 29 – 36

DOI: 10.2478/sjce-2019-0019

Slovak Journal of Civil Engineering

© 2019 The Author(s). This is an open access article licensed under the creative commons attribution-noncommercial-noderivs license (http://creativecommons.Org/licenses/by-nc-nd/3.0/).


NUMERICAL INVESTIGATION OF THE BEHAVIOR OF STRIP FOOTINGS UNDER LARGE ECCENTRIC...30

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eccentric and inclined loads; these authors demonstrated that for an eccentricity e (0 < e ≤ 1/3B), the results were very close to the ap-proach of Meyerhof’s effective width. Yamamoto and Hira (2009) used the finite element method to calculate the bearing capacity of shallow foundations under eccentric loads. Based on their numerical results, these authors clearly demonstrated that the bearing capacity calculated using Meyerhof’s equation, which is expressed in terms of an effective width B−2e, is not accurate and tends to overestimate the bearing capacity, particularly when the eccentricity ratio is large (e/B ≥ 1/3). Krabbenhoft et al., (2012) also used the finite element method to evaluate the bearing capacity of eccentrically loaded foot-ings in cohesionless soil for the eccentricities in the interval 0 to 0.5B with varying surcharges, without using an approximate superposition of the contributions from the self-weight of the soil and the surcharge. They demonstrated, contrary to past authors (Yamamoto and Hira, 2009), that in a case of no surcharge, the results correspond greatly with the results obtained by the effective width approach. Based on the work of (Purkayastha and Char, 1977), Patra et al. (2012) carried out experimental tests for a strip footing subjected to an eccentric and inclined load embedded in sandy soil. Ganesh et al., (2016) per-formed a regression analysis of the values of the bearing capacity taken from laboratory model tests reported in the literature by differ-ent researchers; they proposed an empirical RF for determining the ultimate bearing capacity of shallow strip foundations supported on sand under eccentric and/or inclined loads.

Several reduction factors have been proposed by many authors. The average bearing capacities obtained from these factors are largely different, especially in the case of a strip footing subjected to ver-tical and eccentric loading on sandy soil. Nevertheless, despite all this research, there is still some ambiguity about the behaviour of the bearing capacity with an increase in the depth of a footing sub-jected to vertical and eccentrically loading conditions. In the present work, numerical simulations using the finite difference method and the Fast Lagrangian Analysis of Continua 2007 (FLAC) code were carried out to study the behaviour of the bearing capacity of a shallow strip footing subjected to vertical centric and large eccentric loading conditions.

Moreover, the aim of the present study is to first make a contribu-tion related to the evaluation of the RF characterising large eccentric footings and then to thoroughly check the scientific reasons for the discrepancy between the RF proposed by several authors. This pa-per hopes to facilitate future research by leading the way and provid-ing a clear and detailed explanation of the subject.

2 OVERVIEw OF PREVIOUS wORk

Following the generalised formula of (Terzaghi, 1943), the bear-ing capacity of a strip footing with a width B, embedded in the soil at a depth Df, and subjected to a vertical central load, is generally expressed as: (1)

Where qu is the ultimate bearing capacity per unit area of the foundation; c is the soil cohesion; γ is the soil’s unit weight; and B is the width of the footing; Nγ, Nc and Nq are the bearing capacity factors for a shallow foundation, depending only on the soil friction; φ, dγ, dc and dq are the depth factors.

In addition to vertical loads, foundations are often subjected to eccentric loads; for example, in the design of foundations for light-weight buildings, masts, pylons, steel chimneys, and wind turbines, eccentricities will invariably occur. For these eccentric load cases, Meyerhof (1953) proposed a formula for the bearing capacity called the effective width method. This formula is derived from the Terzaghi

formula; it considers a reduced width B’ of the footing instead of the actual width B. Where B’ is the effective width of the foundation, which is equal to the width of the foundation minus twice the distance of the eccentricity, B’ = B - 2e, while e is the load eccentricity as de-picted in Fig. 1 below.

The principle, therefore, of the concept of the Meyerhof method is to substitute the actual width B with an effective width of B’ = B - 2e. The effective ultimate bearing capacity q’u is thus applied to this effective width. In order to allow comparisons between the present work and the effective width method, it is necessary to first establish the average ultimate bearing capacity quavr applied to the entire width B of the footing. This average ultimate bearing capacity is the ratio of the ultimate vertical load Qu to the width B, which can be deduced from Eq. (1), as modified by Meyerhof’s concept. Qu can be written for a cohesionless soil (c = 0), as:

(2)

The average ultimate bearing capacity per unit of the actual area of a strip foundation can then be written as:

(3)

The rearrangement of Eq. 3 gives:

(4)

According to various authors, including (Prakash and Saran, 1971; Purkayastha and Char, 1977; Patra et al., 2012; Ganesh et al., 2016), the RF is expressed as:

(5)

Where = the ultimate vertical load per unit length of a footing with an embedment ratio (Df /B) and eccentric-ity e; = the ultimate vertical load per unit length of a footing with an embedment ratio (Df /B) and eccentricity (e = 0);

= the average vertical ultimate bearing capacity of a strip footing at a given depth with a load and at an eccentricity e with respect to the central vertical axis of the footing; and = the ultimate vertical bearing capacity of a footing with centric load-ing (e = 0) at the same depth Df.

Following the concept of RF, Purkayastha and Char (1977) car-ried out stability analyses of eccentrically loaded strip foundations supported by frictional soil using the slice method. They proposed the following RF:

(6)

According to the authors, the parameters b and c are independent of the soil friction angle φ and depend only on Df /B. When Df /B varies from the value zero to one, b varies from 1.862 to 1.820, and c varies from 0.730 to 0.888.

Based on the work of (Gottardi and Butterfield, 1993) and on the results obtained from a numerical analysis, Loukidis et al. (2008) pro-posed the following mathematical formulation of the RF:

(7)

Patra et al., (2012) carried out experimental tests on a strip foot-ing subjected to eccentric and inclined loads and embedded in sandy soil. Based on the work of (Purkayastha and Char, 1977), they pro-posed the following empirical relationship for RF:

(8)


NUMERICAL INVESTIGATION OF THE BEHAVIOR OF STRIP FOOTINGS UNDER LARGE ECCENTRIC... 31

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Eq. 8 shows that, for a strip footing subjected to vertical and ec-centric loadings, the Df /B ratio has no effect on the average ultimate bearing capacity. Moreover, for the case where Df /B=0, with vertical and eccentric loading conditions, the RF proposed by (Patra et al., 2012) clearly overestimates the average ultimate bearing capacity, as will be explained in detail below.

Ganesh et al., (2016) proposed an empirical formula for the RF. According to the authors, this was developed after analysing a series of laboratory model test results available in the literature. The empir-ical RF is expressed as:

(9)

where n = 2 for (9a)

and for for (9b)

3 NUMERICAL MODELLING PROCEDURE

This work deals with a numerical study of the bearing capacity of a rigid, rough strip footing of width B, embedded at a depth Df in a sandy soil and subjected to an eccentric vertical load. The soil is assumed to be elastic and perfectly plastic following the Mohr–Cou-lomb failure criterion with an associated flow rule. The finite differ-ence code FLAC (2007) is used. This code is an efficient tool in the analyses of problems of the bearing capacity of different foundation systems. This efficiency has clearly been demonstrated in analyses conducted by (Benmebarek et al., 2012; Massih and Soubra, 2007; Michalowski and Dawson, 2002).

The physical and mechanical characteristics used in the present study are: the shear modulus G = 10 MPa, the elastic bulk modulus K = 20 MPa, the soil unit weight γ = 20 kN/m3, and a series of four values of the angle of the soil’s internal friction φ = 25-40° with an increment of 5°. For each φ value, three values of the embedded ratio (Df /B = 0, 0.5 and 1) are considered, and for each couple (φ, Df /B), four values of the eccentricity ratio (e/B = 0, 0.1, 0.2, 0.3 and 0.4) are considered. The rigid footing was also discretised as depicted in Fig. 2. It is assumed to be elastic material with parameters considered to be comparatively very high, with the soil support, namely a shear modulus G = 1.65 GPa and a bulk modulus K = 1.85 GPa. To simulate the soil-foundation contact, interface elements defined by Coulomb’s shear-strength criterion encoded in FLAC were placed between the soil and the footing laterally and at the base of the foundation. The interfaces were defined with a friction equal to the soil friction and a null cohesion. These values taken for the interface are based on the FLAC manual recommendations.

The domain study boundaries were located at a height H = 5B and a width L = 28B respectively. The domain study was chosen to be very large in order to minimize any boundary effects.

The bottom boundary was assumed to be fixed in both directions, and the vertical boundaries were restrained in a horizontal direction, as shown in Fig. 1. The numerical model is discretised into elements of different dimensions. The mesh is refined near the edges of the foundation because these points present singularities caused by the abrupt change in the direction of the displacement in their vicinity, as mentioned in (Loukidis et al., 2008) among other authors. Fig. 2 shows the mesh used for this analysis.

The loading of the strip footing was simulated by imposing a ver-tical downward velocity to the upper node of the foundation situated at a distance e from the footing’s centreline equal to the eccentricity

Fig. 2 Mesh used in the FLAC simulations.

Fig. 1 Geometry and boundary conditions of the numerical model with Df /B=0.



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of the vertical load. Five values were used for e/B in this study. These values were 0 to 0.40, with an increment of 0.10 (see Fig. 1). In order to develop an acceptable analysis scheme for the later computations, preliminary simulations were carried out by testing the size of the domain, the grid, the boundary conditions, and the magnitude of the downward applied velocity.

The modelling procedure used two steps. In the first one, the geo-static stresses were computed assuming the soil to be elastic. At this stage, some stepping is required to bring the model to equilibrium. In the second step, a downward velocity of 10-7 m/step was applied, until a steady plastic flow was achieved (i.e., until a constant pressure was realised). A large number of steps are necessary for each simulation to reach the limit load.

4 RESULTS AND DISCUSSIONS

4.1 Vertical central loading

Some preliminary analyses were carried out in order to validate the results. These included the calculation of the bearing capacity fac-tor Nγ of a rough rigid strip footing of a width B = 1.0 m resting on a cohesionless soil of Mohr Coulomb and subjected to a central vertical load for the values of a friction angle equal to 25°, 30°, 35° and 40°. In all the tests, the soil unit weight was taken as γ = 20 kN/m3. The results obtained using the FLAC software are given in (Tab. 1).

Tab. 1 also presents a comparison of the present values of Nγ (the central vertical load) with other authors. As can be seen, the FLAC Nγ values are in perfectly good agreement with those obtained in the literature, see (Hjiaj et al., 2005; Krabbenhoft et al., 2012; Meyer-hof, 1963; Hansen, 1961; Loukidis et al., 2008; Yamamoto and Hira, 2009;).

These preliminary results have confirmed the suitable choices of the mesh and the velocity of the loading used for the numerical mod-elling. The simulations related to the eccentric loading could then be undertaken.

4.2 Embedded strip footing subjected to vertical eccentric loading

The values of the bearing capacity for Df /B = 0, 0.5, 1 and e/B = 0, 0.1, 0.2, 0.3 and 0.4 are listed in (Tab. 2). It can be seen that the increase in e/B brings about a decrease in the average bearing ca-pacity. Table 2 also shows an increase in the bearing capacity values with the increase in Df /B. The values of the RF reported in (Tab. 2) are calculated from the numerical values of the bearing capacity by using Eq. 5.

Fig. 3 illustrates the variations of the non-dimensional RF with the variations of e/B taking the values 0 to 0.15 and for three values of the embedment of the foundation (Df /B = 0, 0.5 and 1). Moreover,

Fig. 3 shows a comparison of the variations of RF obtained by the present numerical analysis with the experimental values of RF report-ed by (Patra et al., 2012).

The results are presented as a non-dimensional factor expressing the ratio of two mean pressures; the first pressure is due to the ulti-mate eccentric load and the second to the ultimate central load.

This RF factor permits avoiding and neutralising scale effects when comparing the numerical models with the experimental pro-totypes. It can be seen in Fig. 3 that the finite difference code FLAC values match very closely with the experimental results reported by (Patra et al., 2012).

According to (Meyerhof, 1953), the decrease in the average bear-ing capacity with the increase in the e/B could be explained by the

Tab. 1 Comparison of the values of Nγ with other authors

ϕ° FLAC* Hjiaj et al., (2005)

Krabbenhoft et al., (2012)

Meyerhof, (1963) Hansen, (1961) Loukidis et al.,

(2008)Yamamoto and

Hira, (2009)

25 6.83 6.74 6.35 6.77 / / /

30 15.58 15.24 14.27 15.67 15 15.37 16.68

35 36.55 35.65 32.7 37.15 33.92 36.09 37.76

40 91.19 88.39 / 93.7 79.6 89 94.36

*Present study

Fig. 3 Comparison of RF, obtained from the present numerical analysis with those obtained from the experimental tests reported by (Patra et al., 2012), for different e/B values. (a) Df /B = 0, (b) Df /B = 0.5 and (c) Df /B = 1.

(a)

(b)

(c)



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decrease in the contact pressure beneath the footing. Furthermore, when the limit state is reached, the foundation tilts towards the side of the eccentricity. The contact stress distribution for an eccentrically loaded footing is typically assumed to be linear in design. Howev-er, it should be understood that this is a simplification of the actu-al distribution, which is not even approximately linear (Meyerhof, 1953; Loukidis et al., 2008; Yamamoto and Hira, 2009). Indeed Fig. 4 shows the vertical stress distribution beneath a rough strip footing when subjected to a vertical centric loading (e/B = 0) and the stress distribution for eccentric conditions e/B = 0.1, 0.2, 0.3 and 0.4 as obtained by the FLAC code in the case of a cohesionless soil with an angle of friction φ = 30° and the footing resting on the surface of the soil (Df /B = 0).

In the case of the vertical centric load conditions, the vertical stress distribution is symmetrical with respect to the center of the footing, and the maximum value of the distribution is obtained at the center of the footing. On the other hand, for eccentric loading conditions, the vertical stress distribution is not symmetrical, and the contact surface becomes smaller as the e/B increases. This is in good agreement with the loss of contact between the footing and the soil at the back edge of the footing. Furthermore, the maximum normal stress occurs almost at the point of the application of the eccentric load.

The normal stress distribution contact beneath the footing as de-picted by Fig. 4, subjected to vertical centric and eccentric loading conditions, is clearly not linear at all, but its shape typically changes in size with the increase in e/B. This is in good agreement with the experimental observations of (Meyerhof, 1953) and with (Yamamoto and Hira, 2009), who obtained similar shapes of the normal stress distribution; some zigzags affecting the stress curves were present, but they were most likely due, according to (Yamamoto and Hira, 2009), to the effect of the discretization of the continuum in the finite element method, the interpolation from the integration points of the

elements, and the average value for the nodes constituting several elements.

The zone mobilized in the limit state of the rupture of the soil beneath the footing is represented by contours of the maximum shear strain increment in Fig. 5. The figure shows some cases of the col-lapse mechanism as obtained from the present analyses for φ = 40° and for e/B = 0, 0.2 and for Df /B = 0, 1. Many similar plots are not presented for lack of space. The FLAC finite difference code provides a mechanism, which is in close agreement with that of the rigorous solutions (Okamura et al., 2002). One could see that the size of the mechanism becomes smaller as e/B becomes larger, which means that the lowest average bearing capacities correspond to the highest eccentrities. However, for a given e/B, the size of the mechanism becomes greater with the increase in Df /B.

Tab. 2 The values of the bearing capacity

ϕ° Df /B

Numerical average bearing capacity qu (kN/m2) Numerical RF

Variation of e/B Variation of e/B

0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4

30

0.00 155.8 101.9 58.27 26.79 7.64 1.00 0.654 0.374 0.172 0.049

0.25 322.0 237.7 155.2 83.85 26.27 1.00 0.738 0.482 0.260 0.082

0.50 463.5 355.4 244.6 143.4 50.58 1.00 0.767 0.528 0.309 0.109

0.75 591.4 470.5 333.5 206.5 96.01 1.00 0.796 0.564 0.349 0.162

1.00 740.9 603.5 443.0 293.1 156.0 1.00 0.815 0.598 0.396 0.211

35

0.00 365.5 243.6 138.6 63.7 17.9 1.00 0.666 0.379 0.174 0.049

0.25 674.4 493.5 319.7 175.0 61.86 1.00 0.732 0.474 0.259 0.092

0.50 931.7 716.1 492.8 292.7 112.7 1.00 0.769 0.529 0.314 0.121

0.75 1160 912.9 644.4 398.3 181.5 1.00 0.787 0.556 0.343 0.156

1.00 1429 1162 841.8 550.4 298.0 1.00 0.813 0.589 0.385 0.209

40

0.00 911.9 608.1 349.8 162.4 45.6 1.00 0.667 0.384 0.178 0.050

0.25 1533 1122 717.2 390.8 139.7 1.00 0.732 0.468 0.255 0.091

0.50 2025 1537 1047 620.2 263.8 1.00 0.759 0.517 0.306 0.130

0.75 2471 1931 1338 814.3 379.0 1.00 0.781 0.541 0.330 0.153

1.00 2953 2354 1667 1075 564.8 1.00 0.797 0.565 0.364 0.191

Fig. 4 Distribution of the normal stress contact below the footing under centric and eccentric vertical loadings, for φ =30° Df /B=0. X, horizontal distance from the footing’s centerline; σn, normal stress.



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As can be seen in Fig. 5, and following the terminology used by (Bransby and Randolph, 1998), the collapse mechanism consists of two parts; i.e., the first part, a ‘wedge’ part is on the side upon which the load is applied, and the second part, a ‘scoop’ (rotational part), is located on the other side. The ‘wedge’ has two components, i.e., a passive wedge and a fan region. Plastic shearing occurs inside both the fan and the passive wedge. On the other hand, the plastic shearing in the ‘scoop’ part tends to occur along the single convex shear band bounding the rotating ‘scoop’, with plastic deformations developing inside the ‘scoop’.

Fig. 6 shows a comparison between the present values of the RF obtained from the FLAC simulations for a friction angle φ = 40° and the different values of Df /B. As could be expected, the RF increases with the embedment of the footing in the same manner as revealed by the experimental results reported by (Patra et al., 2012), which also means an increase in the bearing capacity with the increase in Df /B.

Furthermore, Fig. 6 raises some issues: the RF proposed by (Pur-kayastha and Char, 1977) shows some agreement with the RFs of other methods for small eccentricities (e/B ≤ 0.15); it tends to be lin-ear but then overestimates the value of RF with the increase in the e/B; this is clearly observed in the case of (Df /B=0) (see Fig. 6a). Indeed, the value of the error is greater and more easily observed in the case of (Df /B=0). Regarding the method suggested by (Patra et

al., 2012), it invariably gives the same results regardless of the depth of the footing for an eccentric vertical loading. This contrasts with the experimental results presented by the same authors (Patra et al., 2012), where it was obvious from the experimental data that the RF increased with the depth of the footing.

This same issue was raised by (Ganesh et al., 2016); according to these authors, this error might be due to the fact that the effect of Df /B was not taken into consideration for computing the RF in the presence of eccentric loads.

The RF of (Patra et al., 2012) is clearly linear for all the cases of the embedded ratio Df /B, which makes it markedly overestimated in the case of (Df /B=0) and tends to be in some agreement with the other methods as much as the depth of the footing increases.

As the concept of RF proposed by (Patra et al., 2012) is based on the effective width concept proposed by (Meyerhof, 1953), and in the case of a strip footing rests on a free surface, the evaluated bearing capacity that used this suggestion is not an average.

The RF proposed by (Ganesh et al., 2016) is generally in good agreement with that reported by authors such as (Meyerhof, 1953; Hansen, 1961; Loukidis et al., 2008). For the case of a strip foot-ing resting on a free surface and under eccentric vertical load con-ditions, the suggestion of (Ganesh et al., 2016) for the evaluation of the RF coincides perfectly with that proposed by (Meyerhof, 1953), as shown in Fig. 6a, where the concept behind this suggestion is the same as that of Meyerhof’s method [(B’/B)2]. However, in the case of (Df /B ≠ 0), the empirical method suggested by (Ganesh et al., 2016) is totally different from Meyerhof’s method. It is based on a power (n), Eq. 9b, where (n) is a non-linear function of the embedded ratio

Fig. 5 Distribution of maximum shear strain with displacement fields for φ= 40°. Df /B = 0, 1. and e/B = 0, 0.2.

Fig. 6 Comparison of the present RF with those obtained from the literature for different e/B values for φ= 40° and for (a) Df /B = 0, (b) Df /B = 0.5, and (c) Df /B =1.

(a) Df /B=0, e/B=0

(a)

(b)

(c)

(b) Df /B= 0, e/B = 0.2

(c) Df /B= 1, e/B = 0

(d) Df /B= 1, e/B = 0.2



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Df /B, and, as shown in Figs. 6b and 6c, the method gives reasonable values for the RF, but tends to provide RF values slightly higher than Meyerhof’s results.

In addition, in this case (Df /B ≠ 0), Ganesh’s concept was to pro-vide an expression of RF that had to take into account both the eccen-tricity and the embedment in the soil in the same formula. This idea, besides being very interesting, is aimed at substituting the exponent equal to 2 in the initial formula of Meyerhof to an exponent equal to an expression n, i.e., a function of the embedded ratio Df /B, Eq. 9b. However, it might be noted that the exponent n in Eq. 9b should converge to the value of 2 when Df /B converges towards the value zero to be consistent with Eq. 9a, but, from the point of view of math-ematical rigor, this convergence does not occur. To overcome that problem, it could be proposed that the condition in Eq. 9a might be replaced by (0 ≤ n ≤ 0.025) and in Eq. 9b by the condition (n > 0.025).

Following the above ideas, one can conclude that it has become clear that a design with an exponent depending on the variables of the problem (embedment, eccentricity, etc.), in the concept of the RF needs to be quantified in our present numerical study. To do this a greater understanding of the behaviour of the bearing capacity with the increase in e/B is needed.

According to (Meyerhof, 1953), the bearing capacity decreases approximately parabolically, with the increase in e/B. This may be limited to cases where the strip footing is resting on a free surface or where the amount of the embedment is small. The decrease in the RF becomes approximately linear with the increase in Df /B, as is shown in Fig. 7. In other words, in the case of a strip footing resting on a free surface, the behavior of the bearing capacity with the increase in e/B is parabolic and tends to be linear as much as the depth of the footing increases, whatever the large value of e/B is. That means that increasing the depth of the footing has a positive effect on the bearing capacity of a strip footing under vertical eccentric loads.

One can see that, in order to build a concept that provides con-sistent and appropriate values of RF, the varying of the behaviour of the bearing capacity with both the increase in e/B and Df /B should be taken into consideration.

Following this concept and based on a review of published stud-ies related to the estimation of the ultimate bearing capacity of shal-low strip foundations subjected to eccentric vertical loading, and also mostly based on the present numerical analysis, we will try to propose an analytical formula for calculating the RF in this paper. It should be pointed out that the concept of RF proposed by almost all the au-thors is based on the effective width concept proposed by (Meyerhof, 1953). Thus, it can initially be assumed that:

(10)

where the factor k is a function of Df /B. In order to find the suitable mathematical expression of k, the scientific workplace software 5.5 has been used. Several numerical analyses were carried out on a rigid strip footing subjected to an eccentric vertical load. Three values of the soil friction φ = 30°, 35° and 40° were retained for the numerical computations; for each friction value, five values of the Df /B ratio 0, 0.25, 0.5, 0.75 and 1, and five values of the e/B ratio 0, 0.1, 0.2, 0.3 and 0.4 were considered. Furthermore, using a nonlinear least-squares method, the expression of k as shown in Eq. 11 was deter-mined by minimizing the mean square error between the proposed model and the numerical data. The variation of the factor k with Df / B is shown in Fig. 8.

(11)

It has been determined that the value of k only depends on Df /B. This confirms the previous work of (Purkayastha and Char, 1977; Ganesh et al., 2016). The RF can then be expressed as:

(12)

This proposed analytical equation of the RF is represented in Fig. 6 for the value of φ = 40°; as can be seen, the proposed RF is efficient for calculating the ultimate bearing capacity of shallow strip founda-tions embedded in cohesionless soils subjected to vertical eccentric loads. The RF computed from the present numerical analysis com-pares well with the generally used theoretical, numerical and experi-mental results existing in the literature.

5 CONCLUSIONS

The FLAC finite difference code was used in this paper to evalu-ate the bearing capacity and the RF for rough, rigid strip footings sub-jected to large eccentric vertical loads and embedded in cohesionless Mohr-Coulomb associated soils. Based on the numerical analyses and the results presented in this paper and within the range of the param-eters tested, we can reach the following conclusions:

1 - The distribution obtained from the normal contact stresses confirms that the decrease in the bearing capacity with the in-crease in the eccentricity is due to the loss of contact between the footing and the soil at the back edge of the footing and is in good agreement with Meyerhof’s effective footing width;

2 - It has been determined that the RF is independent of the angle of friction and depends only on e/B and Df /B as observed by (Purkayastha and Char, 1977);

Fig. 7 Values of RF obtained from the present numerical analysis for Df /B = 0, 0.5, 1 and, e/B are variable from 0 to 0.4.

Fig. 8 Variation of the factor k with Df / B.



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Vol. 27, 2019, No. 3, 29 – 36

3 - The empirical relationship suggested by (Patra et al., 2012) overpredicts with a maximum error in the case of a strip foot-ing resting on a free surface (Df /B= 0) under eccentric loading and tends to be in some agreement with the other methods as much as the depth of the footing increases. This error is due to the fact that in the formula proposed by (Patra et al., 2012), the bearing capacity is not an average;

4 - The variation of RF with the increase in the eccentricity ratio is parabolic and tends to be linear as much as the depth of the

footing increases. In other ways, increasing the depth of the footing has a positive effect on the bearing capacity of a strip footing under vertical eccentric loads;

5 - Based on the present finite difference analysis results, a new RF formula was proposed, which compares very well with the experimental results reported by (Ganesh et al., 2016) by per-forming a regression analysis of the laboratory model’s test results reported by different researchers.

numerical investigation of the behavior of strip …

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