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Bearing capacity of strip footings on layered sands
Computers and Geotechnics 114 (2019) 103101
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Research Paper
Bearing capacity of strip footings on layered sands ⁎
T
Seyednima Salimi Eshkevari , Andrew J. Abbo, George Kouretzis Priority Research Centre for Geotechnical Science and Engineering, School of Engineering, University of Newcastle, Callaghan, NSW 2308, Australia
A R T I C LE I N FO
A B S T R A C T
Keywords: Bearing capacity Strip footings Layered sands
In this paper we investigate the bearing capacity of strip footings resting on the top of a relatively thin layer of dense sand, which overlays a weaker sand layer. We use the Finite Element Limit Analysis method to calculate the collapse load and identify the geometry of the failure mechanism, as well as how this geometry changes with varying problem parameters. Results are benchmarked against published experimental data and methods used in practice for the design of working platforms for tracked plant, based on the punching shear model. The observed discrepancies are attributed to the development of a transitional mechanism when the bottom layer is rather dense, and the increase in the shear strength at the interface of the two layers is not abrupt. To cover this gap, we propose a new factor to calculate the bearing capacity in cases where a transitional mechanism will govern failure of the footing.
1. Introduction Since Terzaghi [1] published his theory on estimating the bearing capacity of shallow foundations on homogeneous soil, a number of researchers have investigated the bearing capacity of footings on stratified natural soils, or on a layer of selected coarse-grained fill placed on top of weak subgrade to increase the capacity of the foundation. However, estimating the bearing capacity of shallow foundations on layered soils remains a challenging problem as, owing to its complexity, deriving closed-form plasticity solutions is cumbersome. Some simplifying assumptions are required to develop easy-to-use expressions for use in practice, similar to the ones originally proposed by Terzaghi for homogeneous soil. The most common scenario examined in the literature is that of footings on dense sand over soft clay, which is relevant to the design of working platforms in soft soil sites. Meyerhof [2] first used a semi-empirical approach to calculate the bearing capacity of strip and circular footings on dense sand over soft clay, while he also considered the case of loose sand over stiff clay. The pioneer punching shear model proposed by Meyerhof was later refined in a number of experimental, numerical and analytical studies, including the works of Hanna and Meyerhof [3], Griffiths [4], Das and Dallo [5], Michalowski and Shi [6], Kenny and Andrawes [7], Burd and Frydman [8], Okamura et al. [9], Shiau et al. [10], Qin and Huang [11] and, more recently, by Salimi et al. [12]. The case of shallow footings on layered sand profiles has attracted much less attention in the literature, despite its importance for the design of working platforms for tracked piling rigs and cranes on sites
⁎
where relatively loose sand deposits are encountered near the ground surface. Current practice for the design of working platforms on loose sandy deposits is based on the method described in the Building Research Establishment BRE470-2004 guidelines [13]. The BRE470 method is a simplified version of the semi-empirical method proposed by Hanna [14], which in turn is based on Meyerhof’s punching shear model. Other researchers have approached this problem analytically, using limit equilibrium and limit analysis techniques based on specific, simplified failure mechanisms e.g. Ghazavi and Eghbali [15] derived bearing capacity factors for footings on two-layered granular soils, using an extension of the failure mechanism initially proposed by Richards et al. [16]. Analytical upper bound solutions have also been obtained for the more general case of footings on two-layered cohesivefrictional soils by e.g. Purushothamaraj et al. [17], Florkievicz [18] and Huang and Qin [19]. More recently, Khatri et al. [20] used the numerical Finite Element Limit Analysis (FELA) method to calculate upper and lower bounds of the collapse load, using conic optimisation [21–24]. Experimental results to benchmark the abovementioned solutions are sparse, however Hanna [14], Das and Munoz [25] and Kumar et al. [26] report findings from small-scale 1 g tests on footings resting on layered sand, for different problem geometries. Extrapolation of these test results to full-scale footings must be done with caution, as scale effects may lead to overestimating the collapse load, due to the dependency of sand mechanical characteristics to the level of confining stress [27,28]. Nevertheless, results of such model tests are invaluable, as they can provide insight on the developing failure mechanisms. This study was motivated by the observation that neither the Hanna
Corresponding author at: The University of Newcastle, School of Engineering, University Drive, Callaghan, NSW 2308, Australia. E-mail address: [email protected] (S. Salimi Eshkevari).
https://doi.org/10.1016/j.compgeo.2019.103101 Received 14 January 2019; Received in revised form 10 May 2019; Accepted 14 May 2019 0266-352X/ © 2019 Elsevier Ltd. All rights reserved.
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homogeneous loose and dense sand layer, embedded at depth H + D or D, respectively:
[14] nor the BRE470 [13] method, used widely in the design of working platforms for tracked plant, compare well with more refined numerical simulations for a certain range of material parameters, encountered in practice by the authors. To identify the reason behind the observed discrepancies, we performed a series of analyses considering different problem geometries and sand parameters, using the FELA technique implemented in an in-house code developed in the University of Newcastle [29–32]. Apart from rigorous lower and upper bounds of the collapse load, we were able to gain insight on the developing failure mechanisms, and how the mechanism diverges from Meyerhof’s and Hanna’s punching shear assumption as the relative strength of the sand layers and the problem geometry changes. To support our findings, we compare FELA results against the experimental results of Meyerhof and Hanna [33] and Das and Munoz [25], and predictions of the methods proposed by Hanna [14] and embraced by the BRE470. We conclude by proposing a practical method to estimate the bearing capacity of shallow strip footings in cases where Hanna’s method yields un-conservative estimates.
(4)
KP tanδ − γ1 H ≤ qt B
(5)
A key parameter in Eq. (5) is the mobilised friction angle δ, which is not constant but varies along the thickness of the top layer [2]. Hanna [14] used experimental results to determine a parabolic distribution for δ, which is a function of the relative shear strength of the two layers. To simplify the estimation of the bearing capacity qu, Hanna followed Meyerhof’s approach and used the coefficient of punching shear resistance Ks:
K S tan ϕ1 = KP tan δ
(6)
to eliminate δ and Κp from Eq. (5). Ks is provided from a chart (Fig. 2) as function of the friction angle of the top (φ1) and of the bottom (φ2) layer, and the expression providing the bearing capacity of the strip footing is simplified as:
qu = qb + γH 2
Ks tan ϕ1 − γ1 H ≤ qt B
(7)
The method embraced by the BRE470-2004 guidelines is based on the punching shear model, described above, however a number of conservative assumptions are introduced viz. – The embedment depth D is taken as zero and the surcharge term γ1ΗΝq2 is ignored in the calculation of the contribution of the bottom sand layer to the resistance, qb. – The mobilised friction angle δ is taken equal to δ = 2/3φ and accordingly the term Kptanδ is calculated using Coulomb’s passive earth pressure coefficient. – BRE470-2004 adopts the formulas for the bearing capacity factor Nγ proposed by Vesic [35], which are slightly higher than those suggested by Meyerhof [34].
(1)
in which, H and D is the thickness of the top sand layer and the embedment depth of the footing, respectively. In addition, Kp is the mobilised coefficient of lateral earth pressure and γ1 is the unit weight of the top layer. The contribution of the bottom layer is taken equal to the ultimate bearing capacity of an imaginary footing on homogeneous sand, with width equal to that of the actual footing. As a result, the bearing capacity of a strip footing on layered sand can be estimated as:
2 (PP sin δ ) − γ1 H ≤ qt B
qt = 0.5γ1 B Nγ1 + γ1 DNq1
qu = qb + γ1 H 2
We begin the presentation by providing an outline of Hanna’s [14] and of the BRE470 [13] methods for estimating the bearing capacity of footings on layered sand, in order to identify the key simplifications introduced in these methods. As mentioned earlier, Hanna [14] extended the semi-empirical punching shear model of Meyerhof [2] to the case of shallow footings on dense sand overlying loose sand. Hanna considered the failure mechanism depicted in Fig. 1, where general shear failure occurs within the bottom sand layer, as the footing punches an inverted frustum of top sand into the bottom layer. The bearing capacity of the footing results as the sum of the ultimate shear resistance mobilised in the top and bottom sand layers. More specifically, the contribution of the top layer to the resistance is associated with the passive thrust Pp acting on the (assumed for simplicity) vertical failure planes shown in Fig. 1, which depends on the mobilised friction angle on the failure planes, δ. The passive thrust can be estimated with the following expression:
qu = qb +
(3)
The inequality in Eq. (2) suggests that the bearing capacity cannot exceed the bearing capacity of a footing resting on the surface of homogenous dense sand, therefore there is a critical thickness Hcr, beyond which the properties of the bottom layer do not affect the capacity of the footing, and failure develops entirely within the surficial layer. The bearing capacity factors Nq and Nγ used in Eqs. (3) and (4), are calculated according to Meyerhof [34]. Combining Eqs. (1) and (2) results in the following expression:
2. Existing methods for estimating the bearing capacity of footings on layered sands
2D ⎞ KP / cos δ PP = 0.5γ1 H 2 ⎛1 + H⎠ ⎝
qb = 0.5γ2 B Nγ 2 + γ1 (H + D) Nq2
(2)
in which qb and qt is the bearing capacity of a strip footing on a
Fig. 1. Assumed failure mechanism for dense sand over loose sand in Hanna [14].
Fig. 2. Determination of the coefficient of punching shear resistance Ks (modified after Hanna [14]). 2
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bearing capacity of foundations requires assuming that plastic flow in sand is associated. Considering an associated flow rule in conjunction with the Mohr-Coulomb yield criterion (used in this study) implies that the dilation angle of sand is equal to its internal friction angle at failure, which is not realistic for granular materials. While this simplification affects the estimated value of the collapse load, the question of to what extent the flow rule influences the limit loads has not been yet accurately answered. Davis [44] correlated the error introduced in the estimation of the collapse load due to considering associated flow with the degree of kinematic constraint of the problem. Drescher and Detournay [45] proposed to use the effective strength parameters defined in Eqs. (9) and (10) below and associated flow to estimate upper and lower bounds of the collapse load of foundations on geomaterials that exhibit coaxial non-associated flow.
As a result of the above, bearing capacity values, estimated via the BRE470 method are consistently lower than the corresponding values resulting from Eq. (8), for top layer thickness less than the critical thickness. The abovementioned methods may be used to obtain quick estimates of the bearing capacity. However, they are based on the assumption that the failure mode does not depend on the relative strength of the two layers. Intuitively one would expect that there is a transitional failure mode between the punching mechanism and general failure within the surficial layer, which will govern the response for intermediate values of the relative strength of the two layers. This is investigated in the following sections of this paper. 3. Computational method
tan ϕ∗ =
3.1. Background Limit analysis [36] is an efficient technique for estimating rigorous upper and lower bounds of the collapse load of foundations. Application of this technique was initially limited to relatively simple problems, since complicated geometries result in complex equations that have to be solved analytically. More recently, the use of the finite element method to solve the equations of limit theorems has extended the field of application of this technique to realistic geometries and heterogeneous soil stratigraphies [37]. This computational technique called Finite Element Limit Analysis (FELA) method can provide upper and lower bound estimates of the collapse load that bracket its true value, with the difference between the upper and lower bound providing a direct measure of the error in estimating the true collapse load. Taking the “exact” solution to be equal to the average QM of the upper bound Qu and of the lower bound QL estimates of the collapse load QM=(QuQL)/2, the relative error is defined here as [38]:
c∗ =
(cos ψ cos ϕ) × tan ϕ (1 − sin ψ sin ϕ)
(cos ψ cos ϕ) ×c (1 − sin ψ sin ϕ)
(9)
(10)
where φ is the friction angle of soil, c is its effective cohesion and ψ is the peak dilation angle while φ* = ψ* and c* are the effective friction angle, dilation angle and cohesion, respectively. However, as discussed by Krabbenhoft et al. [46], using the above effective strength parameters results in significant underestimation of the bearing capacity of shallow foundations. As discussed later, FELA results obtained in the following while assuming that the dilation angle is equal to the plane strain friction angle of sand agree fairly well with published experimental results. 3.2. FELA model As mentioned above the two sand layers are modelled as associated Mohr-Coulomb materials. The footing is resting on the top of the surface layer (embedment depth is equal to zero in all simulations, as it will usually be the case for working platforms) and is modelled with continuum elements. This allows simulating a rough surface strip footing, corresponding to the tracked plant, by setting the cohesion of the footing elements equal to an arbitrarily high value and their friction angle equal to the friction angle of the top sand layer. To contain the relative error RE to values less than 5% we employ four (4) adaptive remeshing iterations per analysis, with initial and target mesh sizes equal to 20,000 and 30,000 elements respectively [32]. These values, as well as the required dimensions of the domain to encapsulate the failure surface, resulted from a sensitivity analysis, not presented here for brevity. Fig. 3 depicts the evolution of an indicative finite element mesh (comprising only 1000 elements) over four adaptive re-meshing iterations. Note that we present a mesh with a lower number of elements than that used in the actual simulations to improve the clarity of the presentation. The finer mesh generated in areas with high power dissipation intensity allows minimising the relative error in the estimated
(QM − QL ) (Qu − QM ) (Qu − QL ) RE (%) = × 100 = × 100 = × 100 QM QM (Qu + QL ) (8) In this study we will estimate the bearing capacity of strip footings on layered sand using the in-house FELA code, developed in the University of Newcastle on the basis of the formulation originally presented by Sloan [29], and later refined by Lyamin and Sloan [30] and Krabbenhoft et al. [23,31]. This method has recently been employed by a number of researchers for the estimation of the collapse load of shallow foundations, and can be used together with advanced models for describing the shear strength of geomaterials [e.g. 39–43]. In addition, to minimise the relative error in the estimation of the bearing capacity, we employ the adaptive re-meshing technique developed by Lyamin et al. [32]. More specifically, the number of elements and of adaptive remeshing iterations in each analysis is refined so as to achieve RE of less than 5%. Employing the limit analysis theorems for the estimation of the
Fig. 3. Indicative initial (left) and final (after four adaptive re-meshing iterations, right) finite element mesh. 3
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4. Validation and discussion
Table 1 Problem parameters considered in the parametric analysis. Friction angle of top layer, φ1 Friction angle of bottom layer, φ2 Unit weight of top layer, γ1 Unit weight of bottom layer, γ2 Improvement thickness-over-width ratio, H/B
4.1. Error in the estimation of collapse loads
40–50° 27.5–35° 18 kN/m3 15 kN/m3 0.2–2.0
The use of adaptive re-meshing led to upper and lower bound collapse loads being close in all simulations. Fig. 4 depicts upper and lower bound bearing capacities calculated in a series of analyses where the thickness of the top layer H and the friction angle of the bottom layer φ2 were increased while keeping the rest of the problem parameters constant. Notice that the relative error RE increases as the shear strength of the bottom sand layer increases, still it remains lower that the target value of 5%. As a result, the average bearing capacity from the upper bound and lower bound simulations qult = (qupper + qlower)/2 is considered in the following as representative of the true collapse load.
limit loads. Finally, the boundary conditions comprise fixed horizontal and vertical displacements at the base and fixed horizontal displacements at the lateral boundaries of the domain. Each model was solved twice to obtain both the lower and upper limit load, from which the average bearing capacity was estimated, following Salgado et al. [38]. Apart from rigorous estimates of collapse loads, the kinematically admissible velocity field obtained from upper bound analyses provides insight on the governing failure mechanism for each analysis. The mechanism is identified from plots of nodal velocity vectors and power dissipation intensity contours. Areas of high power dissipation intensity represent planes of shear failure and plots of velocity vectors computed at each node depict the collapse mechanism [47]. Following an investigation on the error in the estimated collapse loads, the numerical technique is benchmarked against the published experimental results of Meyerhof and Hanna [33] and of Das and Munoz [25] and predictions of the punching shear models of Hanna [14] and BRE470-2004 [13]. Accordingly, a range of material parameters and problem geometries were analysed parametrically, to cover cases beyond the thin very dense sand-over-loose sand profile for which the punching shear model was proposed. The range of parameters considered in the analysis is listed in Table 1, and covers most cases encountered in practice during the design of working platforms for tracked plant. Note that the unit weight of the sand layers was kept constant during the parametric analysis; the effect of this assumption on the governing failure mechanism is discussed later. In addition, H/B values outside the standard range reported in Table 1 were considered in selected cases, in order to estimate the critical thickness of the top layer i.e. the thickness where failure takes place exclusively in the top sand layer.
4.2. Comparison with experimental results and published solutions The aim of the comparison that follows is two-fold: To validate the computational methodology, but also to identify the conditions under which existing semi-empirical solutions diverge from numerical results and later determine the reasons behind the observed discrepancies. As mentioned earlier the scaled physical model test results reported in Meyerhof and Hanna [33] and Das and Munoz [25] will be used in this section. Meyerhof and Hanna [33] loaded up to failure a 0.05 m-wide strip footing, resting on the surface of two layers of sand, with the top and bottom layers featuring plane-strain friction angles φ1 = 47.7° and φ2 = 34°, respectively. The dry density of the top and bottom layer was γ1 = 16.3 kN/m3 and γ2 = 13.8 kN/m3, respectively. The relative thickness of the top layer varied from H/B = 0 up to H/B = 5. The comparison of FELA results against the experimental data and the corresponding results obtained by applying Hanna’s and the BRE470 method is presented in Fig. 5. In addition, Fig. 6 presents a comparison of the same solutions against the bearing capacities measured by Das and Munoz [25], who tested a 0.1016 m-wide strip footing resting on the surface of a sand layer with φ1 = 43°, underlain by a looser sand layer with φ2 = 36°. Results plotted in Fig. 5 suggest that FELA and Hanna’s method provide very similar bearing capacity values and agree well with the experimental data at least for tests where H/B ≤ 3. BRE470 estimations
Fig. 4. Lower and upper bounds of the bearing capacity from a selected series of FELA analyses. 4
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Fig. 5. Comparison between FELA average bearing capacities, semi-empirical solutions [13,14] and the experimental results of Meyerhof and Hanna [33].
Fig. 6. Comparison between FELA average bearing capacities, semi-empirical solutions [13,14] and the experimental results of Das and Munoz [25].
non-linear variation of sand friction angle (but also of the dilation angle) with mean stress level is rather trivial. Similar conclusions are drawn from Fig. 6, at least for the FELA analyses. It is worth mentioning that numerical analyses capture well the critical thickness (where the bearing capacity does not further increase as H increases) observed in both experimental datasets, which increases as the friction angle of the top layer increases. Notice however that Hanna’s method overestimates the bearing capacity measured in these experiments, where the shear strength of the two sand layers is comparable. On the other hand, the BRE470 method results in lower capacity values, and consistently underestimates the measured bearing capacity. In order to shed some light on the observed discrepancies, we plot in Fig. 7 the failure mechanisms obtained from three characteristic simulations of the abovementioned experiments, corresponding to different relative strength of the two sand layers qb/qt and top layer
are consistently lower, and in better agreement with experimental data for this range of material parameters. For thickness of the top sand layer larger than the critical thickness, where failure takes place entirely within the surficial layer, all methods underestimate the capacity. This is attributed to the small scale of experiments (footing width 0.05–0.1016 m), and the fact that all methods consider an average, constant sand friction angle with depth. In reality the friction angle will be higher near the surface, due to the lower confining stress, while the variation of the friction angle with mean stress level is more pronounced in dense sands [28]. Thus when the top sand layer is relatively thin (but thicker than the critical thickness owing to the relatively small width of the footing), considering a constant friction angle in the sand layer will result in conservative predictions of the collapse load. On the other hand, when H/B < 3 the contribution of the top dense layer to the bearing capacity of the footing is relatively low, hence the effect of
5
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Fig. 7. Velocity magnitudes depicting the failure mode observed in three characteristic analyses: (a) Punching shear mechanism (Analysis corresponding to Meyerhof and Hanna [33] experiment for H/B = 1). (b) Transitional failure mechanism (Analysis corresponding to Das and Munoz [25] experiment for H/B = 1.5). (c) General shear failure mechanism (Analysis corresponding to Das and Munoz [25] experiment for H/B = 5 > Hcr/B).
Fig. 8. Simplified punching shear (a) and transitional failure (b) mechanisms.
6
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failure mechanisms described by Vesic [48] for footings on uniform soils, which consider the effect of soil compressibility. The above suggest that the punching shear mechanism cannot describe all possible failure modes, and as a result Hanna’s method overpredicts the bearing capacity measured in Das and Munoz [25] experiments. A new model applicable in cases where a transitional failure mechanism will develop is described in the following section.
Table 2 Constants a, b, c (Eq. (11)). φ1 (deg)
a
b
c
50 47.5 45 42.5 40
17.11 16.35 16.08 18.12 18.34
−7.49 −6.98 −7.9 −8.61 −8.79
26.65 28.47 31.11 34.60 38.15
5. Transitional bearing capacity mechanism thickness H/B. Notice that three different mechanisms develop:
5.1. Conditions for development of transitional failure mechanism
– Punching shear mechanism (Fig. 7a), that occurs where the top layer has significantly higher shear strength compared to the bottom layer, and the thickness of the former is less than the critical thickness. This mechanism is compatible with the mechanism considered by Hanna [14]. – General failure mechanism (Fig. 7c), that occurs when the thickness of the top layer is larger than the critical thickness, and – An intermediate mechanism (Fig. 7b) called transitional failure mechanism in the following, that is observed where the relative strength of the two layers qb/qt is higher than that required for punching to develop, while the thickness of top sand layer is not sufficient to fully accommodate the failure mechanism.
A series of 400 parametric FELA analyses were performed, considering the range of parameters listed in Table 1. The aim of these analyses was to identify under which conditions the mechanism changes from punching shear (Fig. 8a) to transitional (Fig. 8b), using power dissipation intensity and velocity contours to infer the geometry of the failure mechanism from each analysis. As depicted in Figs. 7 and 8, we can use the inclination of the shear planes in the top layer θ to describe an idealised wedge, formed below the strip footing. Inclination θ, calculated directly from power dissipation intensity contour plots, is a function of the relative strength of the sand layers and of H/B. Coevaluation of all parametric analyses resulted in the following fitting expression for θ:
Note that the fact that the failure mechanism shown Fig. 7c appears to be non-symmetric is merely an artefact of the averaging method used to create velocity contours. Also, we must clarify that the described mechanisms are not related to the general, local and punching shear
H θ = aln(tan (ϕ2)) + b ln ⎛ ⎞ + c (θ in deg) ⎝B⎠
(11)
in which, a, b and c are constants depending on the friction angle of the top layer φ1 and are given in Table 2. Whether a transitional
Fig. 9. Failure mechanisms for different qb/qt and H/B combinations. 7
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Fig. 10. Results of additional analyses performed to confirm that the geometry of the failure mechanism is independent of the width of the footing when H/B is constant, and independent of the unit weight of sand layers when γ1/γ2 is constant: (a) Standard analysis, (b) effect of footing width, (c) effect of unit weight. Results for φ1 = 45° and φ2 = 30°.
Fig. 11. Bearing capacity factor Nγ* for φ1 = 40°. 8
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Fig. 12. Bearing capacity factor Nγ* for φ1 = 42.5°.
Fig. 13. Bearing capacity factor Nγ* for φ1 = 45°.
Note that Fig. 9 suggests that the geometry of the failure mechanism is independent of (i) the width of the footing when Η/Β is constant; (ii) the actual value of the unit weight of sand layers when the ratio γ1/γ2 is constant. Additional simulations were performed to confirm that, some indicative results of which are presented in Fig. 10. Velocity vectors and power dissipation intensity contours shown in Fig. 10 confirm that angle θ is independent of the abovementioned parameters.
mechanism will develop or not depends on the value of the equivalent width B′ = B − 2Htanθ (Fig. 8a), which is calculated using θ from Eq. (11). If B′ < 0 then a transitional failure mechanism will develop. Note however that while B́ > 0 is a necessary condition for a punching shear failure mechanism, it is not sufficient. As discussed earlier, punching shear occurs when there is an abrupt change in shear strength of the two layers. B́ may be greater than zero even for relatively high qb/qt, if the thickness of the top layer is not sufficient to fully accommodate the elastic wedge below the footing. Such failure mechanisms, that feature a geometry similar to that of transitional mechanisms yet correspond to cases where B́ > 0, are referred to as “shallow transitional” mechanisms. The problem parameters corresponding to each observed mechanism are grouped in Fig. 9, which provides an overview under which conditions the shallow and the standard transitional mechanisms govern the bearing capacity of the footing. Therefore Fig. 9 can be used to identify the dominant mode of failure.
5.2. Bearing capacity factor for transitional failure mechanism A new bearing capacity factor Nγ* can be now defined to estimate the bearing capacity in cases where a (standard or shallow) transitional mechanism is likely to govern collapse, according to Fig. 9 above. Nγ* is back-calculated, using results of the parametric FELA investigation and considering the standard bearing capacity equation for surface strip footings on sands: 9
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Fig. 14. Bearing capacity factor Nγ* for φ1 = 47.5°.
Fig. 15. Bearing capacity factor Nγ* for φ1 = 50°.
qult = 0.5γ1 BNγ ∗ ≤ qt
(12)
Note that for the cases where a punching shear mechanism will develop according to Fig. 9 Nγ* is not applicable, hence the curves in Figs. 11–15 cover specific H/B ranges. The method proposed by Hanna [14] can be used for estimating the bearing capacity when a punching shear mechanism is identified via Fig. 9 to govern the collapse load.
The upper bound qt is the bearing capacity of a footing with the same geometry, resting on the surface of a uniform layer with the properties of the top dense sand, and applies when the thickness of top layer exceeds the critical thickness. In other words, the upper bound to the bearing capacity factor Nγ* is the conventional bearing capacity factor Nγ. Results of the parametric study suggest that Nγ* is a function of φ1, φ2, Η/Β (as the geometry of the failure mechanism) but also γ1/ γ2. As discussed earlier, the assumption of constant γ1/γ2 = 1.2 is a reasonable approximation for practical cases. This assumption was adopted while preparing Figs. 11–15, which can be used to estimate Nγ* and accordingly the bearing capacity from Eq. (12) in cases where a transitional mechanism will develop according to Fig. 9. Figs. 11–15 simplify the estimation of the bearing capacity, as calculating angle θ is not required for determining Nγ*.
6. Concluding remarks We have shown that apart from the punching shear mechanism (proposed by Hanna [14] and embraced by BRE470 [13]), a transitional failure mechanism may govern the bearing capacity of strip footings on layered sands, in cases where the thickness of the top layer is not sufficient for general shear failure to be contained within it. This transitional mechanism develops when there is not an abrupt increase in the shear strength at the interface of the two sand layers, and when the thickness of the top layer (normalised against the width of the footing) 10
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is rather small. We have also shown that using bearing capacity models based on the punching shear mechanism under these conditions leads to overestimating the collapse load of the footing. To cover this gap, we have identified under which conditions a transitional mechanism will be critical (Fig. 9) and propose a new bearing capacity formula (Eq. (12) and Figs. 11–15) to calculate the collapse load in these cases. This formula applies for surface rough strip footings, hence it is appropriate for the design of working platforms for tracked plant. Results of this numerical study also suggest that the method proposed by Hanna [14] provides accurate estimates of the bearing capacity, when a punching shear mechanism develops. On the other hand, the BRE470 method provides conservative results, which diverge from the FELA-calculated bearing capacities as the relative strength of the two layers qb/qt increases. In cases where a transitional mechanism is expected to govern the response, use of the proposed method will result in working platform thickness about 50% higher compared to the thickness calculated from Hanna’s method, but still about 70% lower than the thickness calculated via BRE470.
dense sand layer over loose sand strata. Int J Geomech 2017;17(10):06017018. [21] Makrodimopoulos A, Martin CM. Lower bound limit analysis of cohesive frictional materials using second-order cone programming. Int J Numer Methods Eng 2006;66(4):604–34. [22] Makrodimopoulos A, Martin CM. Upper bound limit analysis using simplex strain elements and second-order cone programming. Int J Numer Anal Methods Geomech 2007;31(6):835–65. [23] Krabbenhoft K, Lyamin AV, Sloan SW. Formulation and solution of some plasticity problems as conic programs. Int J Solids Struct 2007;45(5):1533–49. [24] Krabbenhoft K, Lyamin AV, Sloan SW. Three-dimensional Mohr-Coulomb limit analysis using semidefinite programming. Int J Numer Methods Eng 2008;24(11):1107–19. [25] Das BM, Munoz RF. Bearing capacity of eccentrically loaded continuous foundations on layered sand. Transp Res Rec 1984:28–31. [26] Kumar A, Ohri ML, Bansal RK. Bearing capacity tests of strip footings on reinforced layered soil. Geotech Geol Eng 2007;25(2):139–50. [27] Zhu F, Clark JI, Phillips R. Scale effect of strip and circular footings resting on dense sand. J Geotech Geoenv Eng 2001;127(7):613–21. [28] Cerato AB, Lutenegger AJ. Scale effects of shallow foundation bearing capacity on granular material. J Geotech Geoenv Eng 2007;133(10):1192–202. [29] Sloan SW. Lower bound limit analysis using finite elements and linear programming. Int J Numer Anal Methods Geomech 1988;12(1):61–77. [30] Lyamin AV, Sloan SW. Upper bound limit analysis using linear finite elements and non-linear programming. Int J Numer Anal Methods Geomech 2002;26(2):181–216. [31] Krabbenhoft K, Lyamin AV, Hjiaj M, Sloan SW. A new discontinuous upper bound limit analysis formulation. Int J Numer Methods Eng 2005;63(7):1069–88. [32] Lyamin AV, Krabbenhoft K, Sloan SW. Adaptive limit analysis using deviatoric fields. Proc. 6th int. conf. adaptive modeling and simulation. 2013. p. 448–55. [33] Meyerhof GG, Hanna AM. Ultimate bearing capacity of foundations on layered soils under inclined loads. Can Geotech J 1978;15(4):565–72. [34] Meyerhof GG. Influence of roughness of base and groundwater conditions on ultimate bearing capacity of foundations. Geotechnique 1955;5(3):229–42. [35] Vesic AS. Bearing capacity of shallow foundations. In: Fang WA, editor. Foundation engineering handbook. New York: Van Nostrand Reinhold; 1975. p. 121–47. [36] Drucker DC, Prager W, Greenberg HJ. Extended limit design theorems for continuous media. Quart J Appl Math 1952;9:381–9. [37] Sloan SW. Geotechnical stability analysis. Geotechnique 2013;63(7):531–72. [38] Salgado R, Lyamin AV, Sloan SW, Yu HS. Two- and three-dimensional bearing capacity of foundations in clay. Geotechnique 2004;54:297–306. [39] Ukritchon B, Keawsawasvong S. Lower bound limit analysis of an anisotropic undrained strength criterion using second-order cone programming. Int J Numer Anal Methods Geomech 2018;42:1016–33. [40] Ukritchon B, Keawsawasvong S. Three-dimensional lower bound finite element limit analysis of an anisotropic undrained strength criterion using second-order cone programming. Comput Geotech 2019;106:327–44. [41] Ukritchon B, Keawsawasvong S. Three-dimensional lower bound finite element limit analysis of Hoek-Brown material using semidefinite programming. Comput Geotech 2018;104:248–70. [42] Ukritchon B, Keawsawasvong S. Bearing capacity of shallow foundations in clay with linear increase in strength and adhesion factor. Mar Georesour Geotechnol 2018;36(4):438–51. [43] Ukritchon B, Keawsawasvong S. Unsafe error in conventional shape factor for shallow circular foundations in normally consolidated clays. J Geotech Geoenv Eng 2017;143(6):02817001. [44] Davis EH. Theories of plasticity and failure of soil masses. Selected topics. New York: Elsevier; 1968. [45] Drescher A, Detournay E. Limit load in translational failure mechanisms for associative and non-associative materials. Geotechnique 1993;43(3):443–56. [46] Krabbenhoft 1 K, Karim MR, Lyamin AV, Sloan SW. Associated computational plasticity schemes for nonassociated frictional materials. Int J Numer Meth Eng 2012;90:1089–117. [47] Wilson DW, Abbo AJ, Salimi Eshkevari S, Sloan SW. Graphical interpretation of upper bound finite element limit analysis results. 15th int. conf. international association for computer methods and recent advances in geomechanics, IACMAG. 2017. [48] Vesic AS. Analysis of Ultimate Loads of Shallow Foundations. ASCE J Soil Mech Found Div 1973;99(SM1):45–73.
References [1] Terzaghi K. Theoretical soil mechanics. London: Chapman and Hall; 1943. [2] Meyerhof GG. Ultimate bearing capacity of footings on sand layer overlying clay. Can Geotech J 1974;11(2):223–9. [3] Hanna AM, Meyerhof GG. Design charts for ultimate bearing capacity of foundations on sand overlying soft clay. Can Geotech J 1980;17(2):300–3. [4] Griffiths DV. Computation of bearing capacity on layered soils. Numerical methods in geomechanics. Proc., 4th int. conf. vol. 1. 1982. p. 163–70. [5] Das BM, Dallo KF. Bearing capacity of shallow foundations on a strong sand layer underlain by soft clay. Civ Eng Pract Des Eng 1984;3(5):417–38. [6] Michalowski RL, Shi L. Bearing capacity of footings over 2-layer foundation soils. J Geotech Eng 1995;21(5):421–8. [7] Kenny MJ, Andrawes KZ. The bearing capacity of footings on a sand layer overlying soft clay. Geotechnique 1979;47(2):339–45. [8] Burd HJ, Frydman S. Bearing capacity of plane-strain footings on layered soils. Can Geotech J 1997;34(2):241–53. [9] Okamura M, Takemura J, Kimura T. Bearing capacity predictions of sand overlying clay based on limit equilibrium methods. Soils Found 1998;38(1):181–94. [10] Shiau JS, Lyamin AV, Sloan SW. Bearing capacity of a sand layer on clay by finite element limit analysis. Can Geotech J 2003;40(5):900–15. [11] Qin HL, Huang MS. Upper-bound method for calculating bearing capacity of strip footings on two-layer soils. Yantu Gongcheng Xuebao/Chinese J Geotech Eng 2008;30(4):611–6. [12] Salimi Eshkevari S, Abbo AJ, Kouretzis G. Bearing capacity of strip footings on sand over clay. Can Geotech J 2018. https://doi.org/10.1139/cgj-2017-0489. [13] Working platforms for tracked plant. Good practice guide to the design, installation, maintenance and repair of ground-supported working platforms. BRE470 (Building research Establishment); 2004. [14] Hanna AM. Foundations on strong sand overlying weak sand. J Geotech Eng 1981;107(7):915–27. [15] Ghazavi M, Eghbali AH. A simple limit equilibrium approach for calculation of ultimate bearing capacity of shallow foundations on two-layered granular soils. Geotech Geolog Eng 2008;26(5):535–42. [16] Richards M. Seismic bearing capacity and settlements of foundations. J Geotech Eng 1993;119(4):662–74. [17] Purushothamaraj P, Ramiah BK, Venkatakrishna Rao KN. Bearing capacity of strip footings in two layered cohesive-friction soils. Can Geotech J 1974;11(1):32–45. [18] Florkiewicz A. Upper bound to bearing capacity of layered soils. Can Geotech J 1989;26(4):730–6. [19] Huang M, Qin HL. Upper-bound multi-rigid-block solutions for bearing capacity of two-layered soils. Comput Geotech 2009;36(3):525–9. [20] Khatri V, Kumar J, Akhtar S. Bearing capacity of foundations with inclusion of
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Research Paper
Bearing capacity of strip footings on layered sands ⁎
T
Seyednima Salimi Eshkevari , Andrew J. Abbo, George Kouretzis Priority Research Centre for Geotechnical Science and Engineering, School of Engineering, University of Newcastle, Callaghan, NSW 2308, Australia
A R T I C LE I N FO
A B S T R A C T
Keywords: Bearing capacity Strip footings Layered sands
In this paper we investigate the bearing capacity of strip footings resting on the top of a relatively thin layer of dense sand, which overlays a weaker sand layer. We use the Finite Element Limit Analysis method to calculate the collapse load and identify the geometry of the failure mechanism, as well as how this geometry changes with varying problem parameters. Results are benchmarked against published experimental data and methods used in practice for the design of working platforms for tracked plant, based on the punching shear model. The observed discrepancies are attributed to the development of a transitional mechanism when the bottom layer is rather dense, and the increase in the shear strength at the interface of the two layers is not abrupt. To cover this gap, we propose a new factor to calculate the bearing capacity in cases where a transitional mechanism will govern failure of the footing.
1. Introduction Since Terzaghi [1] published his theory on estimating the bearing capacity of shallow foundations on homogeneous soil, a number of researchers have investigated the bearing capacity of footings on stratified natural soils, or on a layer of selected coarse-grained fill placed on top of weak subgrade to increase the capacity of the foundation. However, estimating the bearing capacity of shallow foundations on layered soils remains a challenging problem as, owing to its complexity, deriving closed-form plasticity solutions is cumbersome. Some simplifying assumptions are required to develop easy-to-use expressions for use in practice, similar to the ones originally proposed by Terzaghi for homogeneous soil. The most common scenario examined in the literature is that of footings on dense sand over soft clay, which is relevant to the design of working platforms in soft soil sites. Meyerhof [2] first used a semi-empirical approach to calculate the bearing capacity of strip and circular footings on dense sand over soft clay, while he also considered the case of loose sand over stiff clay. The pioneer punching shear model proposed by Meyerhof was later refined in a number of experimental, numerical and analytical studies, including the works of Hanna and Meyerhof [3], Griffiths [4], Das and Dallo [5], Michalowski and Shi [6], Kenny and Andrawes [7], Burd and Frydman [8], Okamura et al. [9], Shiau et al. [10], Qin and Huang [11] and, more recently, by Salimi et al. [12]. The case of shallow footings on layered sand profiles has attracted much less attention in the literature, despite its importance for the design of working platforms for tracked piling rigs and cranes on sites
⁎
where relatively loose sand deposits are encountered near the ground surface. Current practice for the design of working platforms on loose sandy deposits is based on the method described in the Building Research Establishment BRE470-2004 guidelines [13]. The BRE470 method is a simplified version of the semi-empirical method proposed by Hanna [14], which in turn is based on Meyerhof’s punching shear model. Other researchers have approached this problem analytically, using limit equilibrium and limit analysis techniques based on specific, simplified failure mechanisms e.g. Ghazavi and Eghbali [15] derived bearing capacity factors for footings on two-layered granular soils, using an extension of the failure mechanism initially proposed by Richards et al. [16]. Analytical upper bound solutions have also been obtained for the more general case of footings on two-layered cohesivefrictional soils by e.g. Purushothamaraj et al. [17], Florkievicz [18] and Huang and Qin [19]. More recently, Khatri et al. [20] used the numerical Finite Element Limit Analysis (FELA) method to calculate upper and lower bounds of the collapse load, using conic optimisation [21–24]. Experimental results to benchmark the abovementioned solutions are sparse, however Hanna [14], Das and Munoz [25] and Kumar et al. [26] report findings from small-scale 1 g tests on footings resting on layered sand, for different problem geometries. Extrapolation of these test results to full-scale footings must be done with caution, as scale effects may lead to overestimating the collapse load, due to the dependency of sand mechanical characteristics to the level of confining stress [27,28]. Nevertheless, results of such model tests are invaluable, as they can provide insight on the developing failure mechanisms. This study was motivated by the observation that neither the Hanna
Corresponding author at: The University of Newcastle, School of Engineering, University Drive, Callaghan, NSW 2308, Australia. E-mail address: [email protected] (S. Salimi Eshkevari).
https://doi.org/10.1016/j.compgeo.2019.103101 Received 14 January 2019; Received in revised form 10 May 2019; Accepted 14 May 2019 0266-352X/ © 2019 Elsevier Ltd. All rights reserved.
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homogeneous loose and dense sand layer, embedded at depth H + D or D, respectively:
[14] nor the BRE470 [13] method, used widely in the design of working platforms for tracked plant, compare well with more refined numerical simulations for a certain range of material parameters, encountered in practice by the authors. To identify the reason behind the observed discrepancies, we performed a series of analyses considering different problem geometries and sand parameters, using the FELA technique implemented in an in-house code developed in the University of Newcastle [29–32]. Apart from rigorous lower and upper bounds of the collapse load, we were able to gain insight on the developing failure mechanisms, and how the mechanism diverges from Meyerhof’s and Hanna’s punching shear assumption as the relative strength of the sand layers and the problem geometry changes. To support our findings, we compare FELA results against the experimental results of Meyerhof and Hanna [33] and Das and Munoz [25], and predictions of the methods proposed by Hanna [14] and embraced by the BRE470. We conclude by proposing a practical method to estimate the bearing capacity of shallow strip footings in cases where Hanna’s method yields un-conservative estimates.
(4)
KP tanδ − γ1 H ≤ qt B
(5)
A key parameter in Eq. (5) is the mobilised friction angle δ, which is not constant but varies along the thickness of the top layer [2]. Hanna [14] used experimental results to determine a parabolic distribution for δ, which is a function of the relative shear strength of the two layers. To simplify the estimation of the bearing capacity qu, Hanna followed Meyerhof’s approach and used the coefficient of punching shear resistance Ks:
K S tan ϕ1 = KP tan δ
(6)
to eliminate δ and Κp from Eq. (5). Ks is provided from a chart (Fig. 2) as function of the friction angle of the top (φ1) and of the bottom (φ2) layer, and the expression providing the bearing capacity of the strip footing is simplified as:
qu = qb + γH 2
Ks tan ϕ1 − γ1 H ≤ qt B
(7)
The method embraced by the BRE470-2004 guidelines is based on the punching shear model, described above, however a number of conservative assumptions are introduced viz. – The embedment depth D is taken as zero and the surcharge term γ1ΗΝq2 is ignored in the calculation of the contribution of the bottom sand layer to the resistance, qb. – The mobilised friction angle δ is taken equal to δ = 2/3φ and accordingly the term Kptanδ is calculated using Coulomb’s passive earth pressure coefficient. – BRE470-2004 adopts the formulas for the bearing capacity factor Nγ proposed by Vesic [35], which are slightly higher than those suggested by Meyerhof [34].
(1)
in which, H and D is the thickness of the top sand layer and the embedment depth of the footing, respectively. In addition, Kp is the mobilised coefficient of lateral earth pressure and γ1 is the unit weight of the top layer. The contribution of the bottom layer is taken equal to the ultimate bearing capacity of an imaginary footing on homogeneous sand, with width equal to that of the actual footing. As a result, the bearing capacity of a strip footing on layered sand can be estimated as:
2 (PP sin δ ) − γ1 H ≤ qt B
qt = 0.5γ1 B Nγ1 + γ1 DNq1
qu = qb + γ1 H 2
We begin the presentation by providing an outline of Hanna’s [14] and of the BRE470 [13] methods for estimating the bearing capacity of footings on layered sand, in order to identify the key simplifications introduced in these methods. As mentioned earlier, Hanna [14] extended the semi-empirical punching shear model of Meyerhof [2] to the case of shallow footings on dense sand overlying loose sand. Hanna considered the failure mechanism depicted in Fig. 1, where general shear failure occurs within the bottom sand layer, as the footing punches an inverted frustum of top sand into the bottom layer. The bearing capacity of the footing results as the sum of the ultimate shear resistance mobilised in the top and bottom sand layers. More specifically, the contribution of the top layer to the resistance is associated with the passive thrust Pp acting on the (assumed for simplicity) vertical failure planes shown in Fig. 1, which depends on the mobilised friction angle on the failure planes, δ. The passive thrust can be estimated with the following expression:
qu = qb +
(3)
The inequality in Eq. (2) suggests that the bearing capacity cannot exceed the bearing capacity of a footing resting on the surface of homogenous dense sand, therefore there is a critical thickness Hcr, beyond which the properties of the bottom layer do not affect the capacity of the footing, and failure develops entirely within the surficial layer. The bearing capacity factors Nq and Nγ used in Eqs. (3) and (4), are calculated according to Meyerhof [34]. Combining Eqs. (1) and (2) results in the following expression:
2. Existing methods for estimating the bearing capacity of footings on layered sands
2D ⎞ KP / cos δ PP = 0.5γ1 H 2 ⎛1 + H⎠ ⎝
qb = 0.5γ2 B Nγ 2 + γ1 (H + D) Nq2
(2)
in which qb and qt is the bearing capacity of a strip footing on a
Fig. 1. Assumed failure mechanism for dense sand over loose sand in Hanna [14].
Fig. 2. Determination of the coefficient of punching shear resistance Ks (modified after Hanna [14]). 2
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bearing capacity of foundations requires assuming that plastic flow in sand is associated. Considering an associated flow rule in conjunction with the Mohr-Coulomb yield criterion (used in this study) implies that the dilation angle of sand is equal to its internal friction angle at failure, which is not realistic for granular materials. While this simplification affects the estimated value of the collapse load, the question of to what extent the flow rule influences the limit loads has not been yet accurately answered. Davis [44] correlated the error introduced in the estimation of the collapse load due to considering associated flow with the degree of kinematic constraint of the problem. Drescher and Detournay [45] proposed to use the effective strength parameters defined in Eqs. (9) and (10) below and associated flow to estimate upper and lower bounds of the collapse load of foundations on geomaterials that exhibit coaxial non-associated flow.
As a result of the above, bearing capacity values, estimated via the BRE470 method are consistently lower than the corresponding values resulting from Eq. (8), for top layer thickness less than the critical thickness. The abovementioned methods may be used to obtain quick estimates of the bearing capacity. However, they are based on the assumption that the failure mode does not depend on the relative strength of the two layers. Intuitively one would expect that there is a transitional failure mode between the punching mechanism and general failure within the surficial layer, which will govern the response for intermediate values of the relative strength of the two layers. This is investigated in the following sections of this paper. 3. Computational method
tan ϕ∗ =
3.1. Background Limit analysis [36] is an efficient technique for estimating rigorous upper and lower bounds of the collapse load of foundations. Application of this technique was initially limited to relatively simple problems, since complicated geometries result in complex equations that have to be solved analytically. More recently, the use of the finite element method to solve the equations of limit theorems has extended the field of application of this technique to realistic geometries and heterogeneous soil stratigraphies [37]. This computational technique called Finite Element Limit Analysis (FELA) method can provide upper and lower bound estimates of the collapse load that bracket its true value, with the difference between the upper and lower bound providing a direct measure of the error in estimating the true collapse load. Taking the “exact” solution to be equal to the average QM of the upper bound Qu and of the lower bound QL estimates of the collapse load QM=(QuQL)/2, the relative error is defined here as [38]:
c∗ =
(cos ψ cos ϕ) × tan ϕ (1 − sin ψ sin ϕ)
(cos ψ cos ϕ) ×c (1 − sin ψ sin ϕ)
(9)
(10)
where φ is the friction angle of soil, c is its effective cohesion and ψ is the peak dilation angle while φ* = ψ* and c* are the effective friction angle, dilation angle and cohesion, respectively. However, as discussed by Krabbenhoft et al. [46], using the above effective strength parameters results in significant underestimation of the bearing capacity of shallow foundations. As discussed later, FELA results obtained in the following while assuming that the dilation angle is equal to the plane strain friction angle of sand agree fairly well with published experimental results. 3.2. FELA model As mentioned above the two sand layers are modelled as associated Mohr-Coulomb materials. The footing is resting on the top of the surface layer (embedment depth is equal to zero in all simulations, as it will usually be the case for working platforms) and is modelled with continuum elements. This allows simulating a rough surface strip footing, corresponding to the tracked plant, by setting the cohesion of the footing elements equal to an arbitrarily high value and their friction angle equal to the friction angle of the top sand layer. To contain the relative error RE to values less than 5% we employ four (4) adaptive remeshing iterations per analysis, with initial and target mesh sizes equal to 20,000 and 30,000 elements respectively [32]. These values, as well as the required dimensions of the domain to encapsulate the failure surface, resulted from a sensitivity analysis, not presented here for brevity. Fig. 3 depicts the evolution of an indicative finite element mesh (comprising only 1000 elements) over four adaptive re-meshing iterations. Note that we present a mesh with a lower number of elements than that used in the actual simulations to improve the clarity of the presentation. The finer mesh generated in areas with high power dissipation intensity allows minimising the relative error in the estimated
(QM − QL ) (Qu − QM ) (Qu − QL ) RE (%) = × 100 = × 100 = × 100 QM QM (Qu + QL ) (8) In this study we will estimate the bearing capacity of strip footings on layered sand using the in-house FELA code, developed in the University of Newcastle on the basis of the formulation originally presented by Sloan [29], and later refined by Lyamin and Sloan [30] and Krabbenhoft et al. [23,31]. This method has recently been employed by a number of researchers for the estimation of the collapse load of shallow foundations, and can be used together with advanced models for describing the shear strength of geomaterials [e.g. 39–43]. In addition, to minimise the relative error in the estimation of the bearing capacity, we employ the adaptive re-meshing technique developed by Lyamin et al. [32]. More specifically, the number of elements and of adaptive remeshing iterations in each analysis is refined so as to achieve RE of less than 5%. Employing the limit analysis theorems for the estimation of the
Fig. 3. Indicative initial (left) and final (after four adaptive re-meshing iterations, right) finite element mesh. 3
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4. Validation and discussion
Table 1 Problem parameters considered in the parametric analysis. Friction angle of top layer, φ1 Friction angle of bottom layer, φ2 Unit weight of top layer, γ1 Unit weight of bottom layer, γ2 Improvement thickness-over-width ratio, H/B
4.1. Error in the estimation of collapse loads
40–50° 27.5–35° 18 kN/m3 15 kN/m3 0.2–2.0
The use of adaptive re-meshing led to upper and lower bound collapse loads being close in all simulations. Fig. 4 depicts upper and lower bound bearing capacities calculated in a series of analyses where the thickness of the top layer H and the friction angle of the bottom layer φ2 were increased while keeping the rest of the problem parameters constant. Notice that the relative error RE increases as the shear strength of the bottom sand layer increases, still it remains lower that the target value of 5%. As a result, the average bearing capacity from the upper bound and lower bound simulations qult = (qupper + qlower)/2 is considered in the following as representative of the true collapse load.
limit loads. Finally, the boundary conditions comprise fixed horizontal and vertical displacements at the base and fixed horizontal displacements at the lateral boundaries of the domain. Each model was solved twice to obtain both the lower and upper limit load, from which the average bearing capacity was estimated, following Salgado et al. [38]. Apart from rigorous estimates of collapse loads, the kinematically admissible velocity field obtained from upper bound analyses provides insight on the governing failure mechanism for each analysis. The mechanism is identified from plots of nodal velocity vectors and power dissipation intensity contours. Areas of high power dissipation intensity represent planes of shear failure and plots of velocity vectors computed at each node depict the collapse mechanism [47]. Following an investigation on the error in the estimated collapse loads, the numerical technique is benchmarked against the published experimental results of Meyerhof and Hanna [33] and of Das and Munoz [25] and predictions of the punching shear models of Hanna [14] and BRE470-2004 [13]. Accordingly, a range of material parameters and problem geometries were analysed parametrically, to cover cases beyond the thin very dense sand-over-loose sand profile for which the punching shear model was proposed. The range of parameters considered in the analysis is listed in Table 1, and covers most cases encountered in practice during the design of working platforms for tracked plant. Note that the unit weight of the sand layers was kept constant during the parametric analysis; the effect of this assumption on the governing failure mechanism is discussed later. In addition, H/B values outside the standard range reported in Table 1 were considered in selected cases, in order to estimate the critical thickness of the top layer i.e. the thickness where failure takes place exclusively in the top sand layer.
4.2. Comparison with experimental results and published solutions The aim of the comparison that follows is two-fold: To validate the computational methodology, but also to identify the conditions under which existing semi-empirical solutions diverge from numerical results and later determine the reasons behind the observed discrepancies. As mentioned earlier the scaled physical model test results reported in Meyerhof and Hanna [33] and Das and Munoz [25] will be used in this section. Meyerhof and Hanna [33] loaded up to failure a 0.05 m-wide strip footing, resting on the surface of two layers of sand, with the top and bottom layers featuring plane-strain friction angles φ1 = 47.7° and φ2 = 34°, respectively. The dry density of the top and bottom layer was γ1 = 16.3 kN/m3 and γ2 = 13.8 kN/m3, respectively. The relative thickness of the top layer varied from H/B = 0 up to H/B = 5. The comparison of FELA results against the experimental data and the corresponding results obtained by applying Hanna’s and the BRE470 method is presented in Fig. 5. In addition, Fig. 6 presents a comparison of the same solutions against the bearing capacities measured by Das and Munoz [25], who tested a 0.1016 m-wide strip footing resting on the surface of a sand layer with φ1 = 43°, underlain by a looser sand layer with φ2 = 36°. Results plotted in Fig. 5 suggest that FELA and Hanna’s method provide very similar bearing capacity values and agree well with the experimental data at least for tests where H/B ≤ 3. BRE470 estimations
Fig. 4. Lower and upper bounds of the bearing capacity from a selected series of FELA analyses. 4
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Fig. 5. Comparison between FELA average bearing capacities, semi-empirical solutions [13,14] and the experimental results of Meyerhof and Hanna [33].
Fig. 6. Comparison between FELA average bearing capacities, semi-empirical solutions [13,14] and the experimental results of Das and Munoz [25].
non-linear variation of sand friction angle (but also of the dilation angle) with mean stress level is rather trivial. Similar conclusions are drawn from Fig. 6, at least for the FELA analyses. It is worth mentioning that numerical analyses capture well the critical thickness (where the bearing capacity does not further increase as H increases) observed in both experimental datasets, which increases as the friction angle of the top layer increases. Notice however that Hanna’s method overestimates the bearing capacity measured in these experiments, where the shear strength of the two sand layers is comparable. On the other hand, the BRE470 method results in lower capacity values, and consistently underestimates the measured bearing capacity. In order to shed some light on the observed discrepancies, we plot in Fig. 7 the failure mechanisms obtained from three characteristic simulations of the abovementioned experiments, corresponding to different relative strength of the two sand layers qb/qt and top layer
are consistently lower, and in better agreement with experimental data for this range of material parameters. For thickness of the top sand layer larger than the critical thickness, where failure takes place entirely within the surficial layer, all methods underestimate the capacity. This is attributed to the small scale of experiments (footing width 0.05–0.1016 m), and the fact that all methods consider an average, constant sand friction angle with depth. In reality the friction angle will be higher near the surface, due to the lower confining stress, while the variation of the friction angle with mean stress level is more pronounced in dense sands [28]. Thus when the top sand layer is relatively thin (but thicker than the critical thickness owing to the relatively small width of the footing), considering a constant friction angle in the sand layer will result in conservative predictions of the collapse load. On the other hand, when H/B < 3 the contribution of the top dense layer to the bearing capacity of the footing is relatively low, hence the effect of
5
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Fig. 7. Velocity magnitudes depicting the failure mode observed in three characteristic analyses: (a) Punching shear mechanism (Analysis corresponding to Meyerhof and Hanna [33] experiment for H/B = 1). (b) Transitional failure mechanism (Analysis corresponding to Das and Munoz [25] experiment for H/B = 1.5). (c) General shear failure mechanism (Analysis corresponding to Das and Munoz [25] experiment for H/B = 5 > Hcr/B).
Fig. 8. Simplified punching shear (a) and transitional failure (b) mechanisms.
6
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failure mechanisms described by Vesic [48] for footings on uniform soils, which consider the effect of soil compressibility. The above suggest that the punching shear mechanism cannot describe all possible failure modes, and as a result Hanna’s method overpredicts the bearing capacity measured in Das and Munoz [25] experiments. A new model applicable in cases where a transitional failure mechanism will develop is described in the following section.
Table 2 Constants a, b, c (Eq. (11)). φ1 (deg)
a
b
c
50 47.5 45 42.5 40
17.11 16.35 16.08 18.12 18.34
−7.49 −6.98 −7.9 −8.61 −8.79
26.65 28.47 31.11 34.60 38.15
5. Transitional bearing capacity mechanism thickness H/B. Notice that three different mechanisms develop:
5.1. Conditions for development of transitional failure mechanism
– Punching shear mechanism (Fig. 7a), that occurs where the top layer has significantly higher shear strength compared to the bottom layer, and the thickness of the former is less than the critical thickness. This mechanism is compatible with the mechanism considered by Hanna [14]. – General failure mechanism (Fig. 7c), that occurs when the thickness of the top layer is larger than the critical thickness, and – An intermediate mechanism (Fig. 7b) called transitional failure mechanism in the following, that is observed where the relative strength of the two layers qb/qt is higher than that required for punching to develop, while the thickness of top sand layer is not sufficient to fully accommodate the failure mechanism.
A series of 400 parametric FELA analyses were performed, considering the range of parameters listed in Table 1. The aim of these analyses was to identify under which conditions the mechanism changes from punching shear (Fig. 8a) to transitional (Fig. 8b), using power dissipation intensity and velocity contours to infer the geometry of the failure mechanism from each analysis. As depicted in Figs. 7 and 8, we can use the inclination of the shear planes in the top layer θ to describe an idealised wedge, formed below the strip footing. Inclination θ, calculated directly from power dissipation intensity contour plots, is a function of the relative strength of the sand layers and of H/B. Coevaluation of all parametric analyses resulted in the following fitting expression for θ:
Note that the fact that the failure mechanism shown Fig. 7c appears to be non-symmetric is merely an artefact of the averaging method used to create velocity contours. Also, we must clarify that the described mechanisms are not related to the general, local and punching shear
H θ = aln(tan (ϕ2)) + b ln ⎛ ⎞ + c (θ in deg) ⎝B⎠
(11)
in which, a, b and c are constants depending on the friction angle of the top layer φ1 and are given in Table 2. Whether a transitional
Fig. 9. Failure mechanisms for different qb/qt and H/B combinations. 7
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Fig. 10. Results of additional analyses performed to confirm that the geometry of the failure mechanism is independent of the width of the footing when H/B is constant, and independent of the unit weight of sand layers when γ1/γ2 is constant: (a) Standard analysis, (b) effect of footing width, (c) effect of unit weight. Results for φ1 = 45° and φ2 = 30°.
Fig. 11. Bearing capacity factor Nγ* for φ1 = 40°. 8
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Fig. 12. Bearing capacity factor Nγ* for φ1 = 42.5°.
Fig. 13. Bearing capacity factor Nγ* for φ1 = 45°.
Note that Fig. 9 suggests that the geometry of the failure mechanism is independent of (i) the width of the footing when Η/Β is constant; (ii) the actual value of the unit weight of sand layers when the ratio γ1/γ2 is constant. Additional simulations were performed to confirm that, some indicative results of which are presented in Fig. 10. Velocity vectors and power dissipation intensity contours shown in Fig. 10 confirm that angle θ is independent of the abovementioned parameters.
mechanism will develop or not depends on the value of the equivalent width B′ = B − 2Htanθ (Fig. 8a), which is calculated using θ from Eq. (11). If B′ < 0 then a transitional failure mechanism will develop. Note however that while B́ > 0 is a necessary condition for a punching shear failure mechanism, it is not sufficient. As discussed earlier, punching shear occurs when there is an abrupt change in shear strength of the two layers. B́ may be greater than zero even for relatively high qb/qt, if the thickness of the top layer is not sufficient to fully accommodate the elastic wedge below the footing. Such failure mechanisms, that feature a geometry similar to that of transitional mechanisms yet correspond to cases where B́ > 0, are referred to as “shallow transitional” mechanisms. The problem parameters corresponding to each observed mechanism are grouped in Fig. 9, which provides an overview under which conditions the shallow and the standard transitional mechanisms govern the bearing capacity of the footing. Therefore Fig. 9 can be used to identify the dominant mode of failure.
5.2. Bearing capacity factor for transitional failure mechanism A new bearing capacity factor Nγ* can be now defined to estimate the bearing capacity in cases where a (standard or shallow) transitional mechanism is likely to govern collapse, according to Fig. 9 above. Nγ* is back-calculated, using results of the parametric FELA investigation and considering the standard bearing capacity equation for surface strip footings on sands: 9
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Fig. 14. Bearing capacity factor Nγ* for φ1 = 47.5°.
Fig. 15. Bearing capacity factor Nγ* for φ1 = 50°.
qult = 0.5γ1 BNγ ∗ ≤ qt
(12)
Note that for the cases where a punching shear mechanism will develop according to Fig. 9 Nγ* is not applicable, hence the curves in Figs. 11–15 cover specific H/B ranges. The method proposed by Hanna [14] can be used for estimating the bearing capacity when a punching shear mechanism is identified via Fig. 9 to govern the collapse load.
The upper bound qt is the bearing capacity of a footing with the same geometry, resting on the surface of a uniform layer with the properties of the top dense sand, and applies when the thickness of top layer exceeds the critical thickness. In other words, the upper bound to the bearing capacity factor Nγ* is the conventional bearing capacity factor Nγ. Results of the parametric study suggest that Nγ* is a function of φ1, φ2, Η/Β (as the geometry of the failure mechanism) but also γ1/ γ2. As discussed earlier, the assumption of constant γ1/γ2 = 1.2 is a reasonable approximation for practical cases. This assumption was adopted while preparing Figs. 11–15, which can be used to estimate Nγ* and accordingly the bearing capacity from Eq. (12) in cases where a transitional mechanism will develop according to Fig. 9. Figs. 11–15 simplify the estimation of the bearing capacity, as calculating angle θ is not required for determining Nγ*.
6. Concluding remarks We have shown that apart from the punching shear mechanism (proposed by Hanna [14] and embraced by BRE470 [13]), a transitional failure mechanism may govern the bearing capacity of strip footings on layered sands, in cases where the thickness of the top layer is not sufficient for general shear failure to be contained within it. This transitional mechanism develops when there is not an abrupt increase in the shear strength at the interface of the two sand layers, and when the thickness of the top layer (normalised against the width of the footing) 10
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is rather small. We have also shown that using bearing capacity models based on the punching shear mechanism under these conditions leads to overestimating the collapse load of the footing. To cover this gap, we have identified under which conditions a transitional mechanism will be critical (Fig. 9) and propose a new bearing capacity formula (Eq. (12) and Figs. 11–15) to calculate the collapse load in these cases. This formula applies for surface rough strip footings, hence it is appropriate for the design of working platforms for tracked plant. Results of this numerical study also suggest that the method proposed by Hanna [14] provides accurate estimates of the bearing capacity, when a punching shear mechanism develops. On the other hand, the BRE470 method provides conservative results, which diverge from the FELA-calculated bearing capacities as the relative strength of the two layers qb/qt increases. In cases where a transitional mechanism is expected to govern the response, use of the proposed method will result in working platform thickness about 50% higher compared to the thickness calculated from Hanna’s method, but still about 70% lower than the thickness calculated via BRE470.
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