Failure Mechanism and Bearing Capacity of Footings Buriedat Various Depths in Spatially Random Soil

Jinhui Li1; Yinghui Tian2; and Mark Jason Cassidy3

Abstract: The objective of this paper is to demonstrate how the spatial variability of random soil affects the failuremechanism and the ultimatebearing capacity of foundations buried at various depths. A nonlinear finite-element analysis combinedwith random field theory is employed toexplore the vertical capacity of foundations embedded at different depths in random soil. Different possibilities of shear failures resulting fromspatial patterns of soil are demonstrated and are used to explain the significant discrepancy between the bearing capacity of the random soil andthat of uniform soil. The effect of the spatial pattern of the soil on the development of shear planes is also investigated, with the coefficients ofvariation for the bearing capacity demonstrated to be closely related to the shear plane length. The results of the statistical variation in the bearingcapacity are provided for different embedment depths, and these are also reported as the failure probability of the footing compared with theestablished uniform soil bearing capacity. Safety factors are proposed for foundations at different levels of failure probability. This study pro-vides a thorough understanding of the failure mechanisms of footings in random soil, especially where structures can penetrate deeply into soil.DOI: 10.1061/(ASCE)GT.1943-5606.0001219. © 2014 American Society of Civil Engineers.

Author keywords: Bearing capacity; Random field; Failure mechanism; Buried foundation; Statistical analysis; Cohesive soils; Offshoreengineering.

Introduction

The vertical bearing capacity of a shallow foundation is a classicalgeotechnical problem.When a foundation is buried deeply in soil, itsfailure mode differs markedly from the surface footing and ischaracterized by a mechanismwhere soil is free to flow around fromunder the footing to the top. In this case, the failure mechanism nolonger extends to the soil surface, as happens for a shallow foun-dation, and becomes fully localized around the foundation (O’Neillet al. 2003). Themechanism of soil failure transforms from a generalshear failure for a surface footing (Craig 2004) to a localized failurefor a buried footing (Hu et al. 1999;Wang and Carter 2002; Hossainand Randolph 2009, 2010; Zhang et al. 2011, 2012, 2014), with theultimate bearing capacity for a deeply buried foundation demon-strated to be considerably larger than that of a shallow foundation(Merifield et al. 2001; Zhang et al. 2012). One example whereunderstanding of the change in bearing capacity at different em-bedment depths is required is the offshore application of spudcanfoundations. These large footings of mobile drilling platformsregularly penetrate and bury into the seabed to depths of three

diameters (Endley et al. 1981; Menzies and Roper 2008; Menziesand Lopez 2011).

The majority of the previous studies on the bearing capacityproblem of buried footings have been limited to homogeneous soilor uniform soil of strength increasing linearly with depth. Thesehave shown that, in uniform soil, a symmetrical logarithmic spiralfailure plane for a surface footing and a symmetrical rotational scoopmechanism for a fully buried foundation are optimal (Merifield et al.2001; Craig 2004). However, under more realistic conditions, soilproperties spatially vary because of a combination of geologic,environmental, and physical-chemical factors (Chiasson et al. 1995;Dasaka and Zhang 2012; Ching and Phoon 2013b; Lloret-Cabotet al. 2014). This spatial variation of soil results in a reduction of thebearing capacity, because the failure plane becomes asymmetricand tends to follow the weakest path (Fenton and Griffiths 2003;Popescu et al. 2005; Cho and Park 2010; Ahmed and Soubra 2012).The influence of soil spatial variation has been also found in slopesthrough the development of different (shallow or deep) failureplanes (Huang et al. 2013; Li et al. 2014). Taking soil spatialvariability into account will lead to a rational and economicaldesign of foundations and geotechnical structures (Salgado andKim 2014; Fan et al. 2014).

The bearing capacity of a footing can be overestimated withoutaccounting for the inherent random heterogeneity of soil. Researchon the bearing capacity of a surface footing on spatially varyingsoil has been conducted over the past 3 decades (Griffiths andFenton 2001; Popescu et al. 2005; Kasama and Whittle 2011; Al-Bittar and Soubra 2013). Nobahar and Popescu (2000) and Grif-fiths et al. (2002) observed that the mean bearing capacity candecrease by 20–30% for random soil compared with the corre-sponding bearing capacity of homogeneous soil with the samemean soil properties. Cassidy et al. (2013) observed that the meanbearing capacity for pure vertical, horizontal, and moment loadsdecreased with increasing variability of the soil strength. Theresults of such probabilistic studies quantify the bearing capacityreduction. However, few studies have revealed how the spatial

1Lecturer, Centre for Offshore Foundation Systems, Univ. of WesternAustralia, Perth, WA 6009, Australia. E-mail: lisa.li@uwa.edu.au

2Research Assistant Professor, Centre for Offshore Foundation Systems,Univ. of Western Australia, Perth, WA 6009, Australia (correspondingauthor). E-mail: yinghui.tian@uwa.edu.au

3Australian Research Council Laureate Fellow and Lloyd’s RegisterFoundation Chair of Offshore Foundations, Centre for Offshore FoundationSystems, Univ. of Western Australia, Perth, WA 6009, Australia. E-mail:mark.cassidy@uwa.edu.au

Note. This manuscript was submitted on January 10, 2014; approved onSeptember 17, 2014; published online on October 20, 2014. Discussionperiod open until March 20, 2015; separate discussions must be submittedfor individual papers. This paper is part of the Journal of Geotechnical andGeoenvironmental Engineering, © ASCE, ISSN 1090-0241/04014099(11)/$25.00.

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variability of random soil affects the failure mechanism andconsequently the bearing capacity. To the authors’ knowledge,there are no studies on the change in the bearing capacity factor forburied footings in spatially random soil.

The objective of this research is to investigate the failuremechanisms and the bearing capacity for footings buried at var-ious depths in random soil. A nonlinear finite-element analysiscombined with a Monte Carlo simulation is employed to achievethe objective. The statistical properties of the bearing capacity areinvestigated and reinterpreted according to the correspondingfailure mechanisms. Finally, the safety factors are proposed fordifferent levels of failure probabilities. This research provides animproved understanding of the failure mechanisms of buriedfootings in random soil.

Methodology

Probabilistic Characteristics of Spatially Random Soil

All soil properties in situ can vary vertically and horizontally. Thespatial variation can be characterized by a trend (representing themean value) and a residual variation (reflecting the variability aboutthe trend). The residuals off the trend are statistically correlated toone another in space and are often a function of their separationdistance. The correlation of residuals at two locations that areseparated by distance d is called the autocorrelation function. Theintegral of the autocorrelation function results in the scale offluctuation. The correlation for the properties at two locations withinthe scale offluctuation is strong.Details on the physicalmeaning andcharacterization of these parameters can be found in Phoon andKulhawy (1999) and Baecher and Christian (2003). In this study, therandomness of the undrained shear strength su is considered andmodeled as a log-normally distributed randomfieldwith ameanvaluems, SD ss, and scale of fluctuation us. The Young’s modulus E isalso a random field because it is considered as perfectly correlatedto the undrained shear strength su with a ratio E=su 5 500 (Hu andRandolph 1998).

The statistical properties of the undrained shear strength arepresented in Table 1. Whereas the mean and SD are familiarconcepts to most engineers, the scale of fluctuation requires moreexamination. Phoon and Kulhawy (1999) conducted an extensiveliterature review and observed that the horizontal scale of fluc-tuation uh is on the order of 40–60 m (with a mean value from thereported literature of 50.7 m). The vertical scale of fluctuation uvis in the range of 2–6m (with a mean value of 3.8 m). In this study,a square exponential model (Baecher and Christian 2003) is usedto describe the autocorrelation of the undrained shear strength.The scale of fluctuation values is taken as 50.7 and 3.8 m in thehorizontal and vertical directions, respectively. The coefficient ofvariation (COV) of the undrained shear strength (i.e., ss=ms) isassumed to be 0.3 in the following simulations, because it isreported to be in the range of 0–0.5 (Uzielli et al. 2005). Randomfields of undrained shear strength su have been generated using

the local average subdivision algorithm (Fenton and Vanmarcke1990).

Random Finite-Element Modeling

The two-dimensional plane-strain condition has been simulated usingthe nonlinear finite-element software ABAQUS 6.11. A footing (withwidth B and height h) embedded in a soil of depth D is illustrated inFig. 1. The soil is modeled with a linear-elastic perfectly plasticconstitutive law. The elastic response is defined by the Young’smodulus andPoisson’s ratio.ThePoisson’s ratio is set as 0.49 tomodelthe undrained conditions of no volume change as well as to ensurenumerical stability (Taiebat and Carter 2000). Soil failure is definedaccording to theTresca criterion,with themaximumshear stress in anyplane limited to the undrained shear strength (su).

The foundation width B is 20 m, and the height h is 4 m. Thesedimensions are typical values for a spudcan foundation today(Cassidy et al. 2009). The embedment depth, D, is taken as 0, 0:5B,1B, 2B, 3B, and 4B, respectively. For a foundation deeper than 4B, itsfailure is contained in a localized area around the foundation that isnot influenced by the embedment depth any more (Rowe and Davis1982; Randolph et al. 2004; Zhang et al. 2011). The foundation isconsidered to be rigid and wished in place. The soil-foundationinterface is fully bonded, which is reasonable to represent undrainedsoil behavior (Gourvenec and Randolph 2003). In the numericalmodeling, the foundation is displaced at the foundation referencepoint in the vertical direction until an ultimate failure load is attained.

The soil domain has a width of 6:4B and a height of 6B, which islarge enough to ensure there are no obvious boundary effects. Thesoil domain is discretized into many zones onto which the randomfield is to be mapped. Hence, the zone size has to be carefullyexamined to avoid excessive spatial averaging in the finite-elementmodeling. Ching and Phoon (2013a) have investigated the effect ofzone sizes and observed that the zone size should be 0.13–0.18 timesthe scale offluctuationwhen the squared exponential autocorrelationis adopted. In this case, the zone sizes are set to be 2.0 m in thehorizontal direction and 0.5 m in the vertical direction, which canensure both a prescribed accuracy and an acceptable computational

Table 1. Summary of Undrained Shear Strength Properties

Soil property Symbol Property value Units

Mean ms 10.0 kPaSD ss 3.0 kPaHorizontal scale of fluctuation uh 50.7 mVertical scale of fluctuation uv 3.8 m Fig. 1. (Color) Model and boundary conditions of a buried footing

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time. The soil properties vary from zone to zone to reflect the spatialvariability of the soil. For the majority of the soil domain, a zone isrepresented by one finite element. However, in a region of size 3B by2B close to the strip footing (as bounded by the bold lines in Fig. 1),a zone is further discretized into four finite elements of 0:53 0:5 m.These smaller elements, eachwith the samematerial properties, havebeen proved to be helpful in improving the numerical accuracy of thesimulation (Cassidy et al. 2013).

Monte Carlo Simulation

For each foundation at a specific embedment depth, Monte Carlosimulations have been performed for n realizations of the soil ran-dom field and the subsequent finite-element simulations of thebearing capacity. The error in the estimate of themean shear strengthdecreases as the number of simulations n increases. The number of

simulations n that estimate themean shear strength to within an errorof e with confidence (12a) is (Fenton and Griffiths 2008)

n��za=2ss

e

�2(1)

where za=2 5 value of the standard normal variant with a cumulativeprobability level (12a=2). If amaximum error e of 0:1ss is allowedon the mean value of the shear strength (i.e., maximum error of 3%on the mean value) with confidence of 95%, then the requirednumber of simulations is 385. In this study, 400 simulations havebeen performed for each foundation at a specific embedment depth.Even with the same statistical properties, each realization can lead toa quite different spatial pattern of the shear strength in the soil. Thisspatial variation can, in turn, result in various failure mechanismsand bearing capacities of the foundations.

Fig. 2. (Color) (a)Variation of bearing capacity factorwith normalized displacement for a surface footing; (b) distribution of the bearing capacity factorfor a surface footing

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Fig. 3. (Color) Development of strain for a surface footing with bearing capacity factor of 6.34 at normalized displacements (d=B) of (a) 0.009; (b) 0.0125;(c) 0.025; (d) 0.0415; (e) 0.052; (f) 0.06; blue regions indicate weaker soil, red regions indicate stronger soil, and gray regions indicate strain development

Fig. 4. (Color) Development of strain for a footing embedded at 3B depth (Nc 5 8:99) at normalized displacements (d=B) of (a) 0.031; (b) 0.036;(c) 0.043; (d) 0.06

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Results and Discussion

The computed bearing capacity factor for each realization, Nci, canbe calculated using

Nci ¼ qfims

(2)

where qfi 5 bearing capacity computed for the ith realization. Themean value of the undrained shear strength ms is maintained ata constant value of 10 kPa, which is the undrained shear strength ofthe uniform soil in the deterministic analysis.

For each analysis, the bearing capacity increases with increasingapplied displacement until plateauing at the failure value (whichtended to be at a displacement of approximately 6% of the footingwidth). The relationship between the bearing capacity factor and thenormalized displacement for a surface footing is demonstrated inFig. 2(a). The 400 realization results from the Monte Carlo simu-lation are compared with both the corresponding deterministicanalysis consisting of uniform soil and the closed-form solution(e.g., Prandtl’s solution). The bearing capacity factor for the de-terministic finite-element analysis using uniform soil strength is5.23, which is 1.7% higher than the Prandtl solution of 5.14. Thereason for this small difference is attributable to the mesh used, whichis to balance the computation time (15min for each buried-foundationcase) and accuracy. The mean value, cumulative distribution, and

probability distribution for the 400 realizations are shown in Fig. 2(b).A majority of the bearing capacity factors obtained from the MonteCarlo simulations are smaller than that of the deterministic case usinguniform soil. This trend indicates that the spatial variability of a soilis prone to decrease the bearing capacity of a foundation, a resultconsistent with the research performed by, among others, Griffithsand Fenton (2001) and Kasama and Whittle (2011).

Failure Mechanism of Foundations in Random Soil

The manner in which the failure plane is formed is discussed in thissection. Fig. 3 shows a failure plane development of a particularrealization of the random field, where the blue regions indicateweaker soil and the red regions indicate stronger soil. The shearstrain contours (i.e., the regions in gray) at different normalizeddisplacements are superimposed on the random field. At the initialstage [as shown in Fig. 3(a)], the largest strain appears at the twobottom corners of the foundation. Then, the shear plane developsdownward until a weak layer of soil beneath the foundation isencountered [as depicted in Figs. 3(b and c)]. The shear planedevelops further along theweak soil layer and is then restricted by anoverlying stronger layer [see Path I in Fig. 3(d)]. In addition,a relatively large shear strain in a weak soil layer at a shallower depthemerges [see Path II in Fig. 3(d)]. The shear plane extends furtheralong Path II through a relatively weak soil region and touchesthe soil surface. Finally, a complete failure plane forms and is

Fig. 5. Different shear planes for different realizations of surface foundations with bearing capacity factors Nc of (a) 5.23 for uniform soils; (b) 2.85;(c) 3.94; (d) 4.98; (e) 5.93; (f) 6.34

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accompanied by several other shear planes in weaker soil. Thefailure plane for the uniform soil is also superimposed in Fig. 3(f),which is the Prandtl failure mechanism. Compared with the uniformcase, the shear plane of this random soil is deeper, mobilizeda greater volume of soil, and in turn, led to a greater ultimate bearingcapacity (6.34). It is interesting that shear planes in random soil candevelop several paths inweak soil instead of a single shear path in thedeterministic analysis. The failure plane seems to find a shear paththat costs the least energy to extend from the corner of the foundationto the soil surface. Therefore, not only theweakest soils but also theirdistribution within the spatial pattern of the random soil will de-termine the failure plane and thus the bearing capacity.

Fig. 4 illustrates the development of the shear strain for a buriedfoundation of 3B depth in a realization of random soil. The largeststrainsfirst occur in theweakest soil surrounding the foundation. The

shear plane is then developed from the regionswith the largest strainsin a circularmanner to form a backflowmechanism,where soil flowsback over the upper surface of the foundation. The failure planefinally passes through a strong soil region to form a localized failureinstead of strictly following the weakest soil path. The failure planefor this footing in uniform soil is also superimposed in Fig. 4(d),which exhibits a symmetrical pattern.

The shear plane of different realizations can be markedly dif-ferent from one another because of different spatial patterns ofrandom soil. In uniform soil, the shear plane is symmetrical fora surface footing [asobserved inFig. 5(a)]. In random soil, Figs. 5(b–f)selectively show the failure planes for the surface footings withbearing capacity factors of 2.85, 3.94, 4.98, 5.93, and 6.34, whichcover the full range of bearing capacities. Fig. 5(b) demonstrates thatthe failure plane is restricted to a shallow and narrow area, which

Fig. 6. Different shear planes for different realizations of embedded foundations with bearing capacity factors Nc of (a) 11.86 for uniform soils;(b) 8.87; (c) 9.48; (d) 10.07; (e) 11.46; (f) 12.23

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mobilizes a small area of soil and leads to a small resistance of thesoil and thus a small bearing capacity factor of 2.85. As the failureplane becomes deeper, with more soil mobilized, the bearing ca-pacity factor increases, as demonstrated in Figs. 5(c–f). Fig. 5(e)shows a comparable failure plane to that in Fig. 5(d), whereas thebearing capacity factor is 16% larger than that of Fig. 5(d). A closeexamination reveals that the shear strength of the soil along the shearplane in Fig. 5(e) is generally larger than that in Fig. 5(d). Therefore,both the size of the shear plane and the soil strength along the shearplane, with both of them depending on the spatial pattern of therandom field, determine the bearing capacity.

The failure planes of the embedded foundation at 3B depth areselected to show different realizations with bearing capacity factorsfrom 8.87 to 12.23 [Figs. 6(b–f)]. The failure plane for the footing inuniform soil is also shown in Fig. 6(a) as a comparison. Generally,the bearing capacity increases upon enlarging the shear plane. Theshear plane in random soil exhibits an unsymmetrical characteristicthat is different from that of the uniform soil case. Interestingly, thefigure suggests that more than one failure plane may be formed,which is different from the common assumption of one failure planein previous studies (Fenton and Griffiths 2003). The shear plane in

Fig. 6(e) is larger than that in Fig. 6(a), whereas the bearing capacityfactor of the former (i.e., 11.46) is smaller than that of the latter(i.e., 11.86). The reason lies in that the shear plane follows theweakest path in random soil, which has smaller shear strengths alongthe shear plane than uniform soil.

The failure plane in Fig. 6(f) appears to be quite close to theboundary of the soil. Hence, a simulation with a larger boundary(i.e., 12:8B3 6B) has been performed to investigate the boundaryeffect. The shear plane for this simulation is shown in Fig. 7,which issimilar to that in Fig. 6(f). The bearing capacity for this analysis is12.40, which is only 1% larger than that of Fig. 6(f). From this, theboundary size used in the simulations [especially in Fig. 6(f)] istherefore considered appropriate.

Bearing Capacity Results for Foundations at DifferentEmbedment Depths

Based on a Monte Carlo simulation for each foundation at a specificembedment depth, the computed bearing capacity factors fromEq. (2) are plotted in the form of a histogram, as presented in Fig. 8.A normal distribution is used to fit the histogram (Fig. 8). It is

Fig. 7. Shear plane with large boundary for the case in Fig. 6(f)

Fig. 8. (Color) Histogram and estimated normal distribution for foundations embedded at different depths

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acknowledged that the use of a normal distribution here has thepossibility of a negative bearing capacity factor. However, witha mean value larger than 4.71 and a SD smaller than 0.74 dem-onstrated in Fig. 8, the probability that the bearing capacity factoris negative is less than 10216. This is considered to be irrelevantfor engineering purposes. A x2 goodness-of-fit test (Ang and Tang2007) is performed to verify the assumed normal distribution.Table 2 summarizes the mean value and x2-statistics for the foun-dations at six different depths. The x2-value is in the range of 11–37,

which indicates an acceptable probability distribution in most cases(Ang and Tang 2007). This normal distribution of the bearing ca-pacity factor has also been reported for a surface footing by Kasamaand Whittle (2011).

The average bearing capacity factors are 4.71, 6.71, 7.59, 9.90,10.84, and 11.25 for the foundations embedded at depths of 0, 0:5B,1B, 2B, 3B, and 4B, respectively [Fig. 9(a)]. The average bearingcapacity factors increase with increasing embedment depth be-cause of the transition from a general failure at a shallow depth toa full-flow failure at a deep depth. The SDs of the bearing capacityfactors are spread in a narrow range of 0.47–0.74 for the foundationsat different depths. The COVs of the bearing capacity factor are0.119, 0.075, 0.062, 0.050, 0.055, and 0.060 for the foundationsburied at depths of 0, 0:5B, 1B, 2B, 3B, and 4B, respectively. Themagnitude of the COV is closely related to the volume of soilmobilized and again to the length of the shear plane, L. The shearplane is characterized by the elements with the largest shear strain.The summation of the length values of the elements with the largestshear strain will give the length of the shear plane. The length of theshear plane for a foundation buried at a certain depth in a random soilis shown to have the same order as that in a uniform soil. The lengthof the shear plane in uniform soil is then investigated and normalizedby the foundation width. The change of the shear plane length ratio,L=B, with the embedment depth is plotted in Fig. 9(b). The resultsindicate that when the shear plane length is larger, the COV issmaller. The reason for this finding is that a larger shear plane willpass through more soil elements, which results in a larger spatialaveraging of the random soil and a smaller COV. The COV cansignificantly affect the failure probability and, in turn, the factor ofsafety of a foundation, as discussed in the “Probability of Failure forFoundations at Different Embedment Depths” section.

Probability of Failure for Foundations at DifferentEmbedment Depths

The failure of a foundation can be defined by the ultimate bearingcapacity or the displacement (Rowe and Davis 1982). In this paper,the authors focus on the bearing capacity. According to Griffiths andFenton (2001), the failure can be defined as the bearing capacities ofthe foundation being less than the corresponding deterministicvalues based on the uniform soil strength. The failure can then bedefined as the normalized bearing capacity factor of a foundation,Nc=Nc,det, being less than 1. The probability of failure can be easilydetermined from the cumulative probabilities of the normalizedbearing capacity for the foundations (as illustrated in Fig. 10). Thefailure probabilities are 82.3, 87.3, 91.7, 95.3, 95.5, and 93.3% forthe foundations buried at 0, 0:5B, 1B, 2B, 3B, and 4B depths, re-spectively. The results demonstrate that the deeply embedded foun-dation generally has a larger probability of failure.

The probability of failure is extremely large when the ultimatebearing capacity obtained from the uniform soil case is used to de-fine the failure criteria. In reality, the allowable load is often obtainedby applying a factor of safety, FS. The probability that the bearingcapacity is less than a targeted level of applied load can be de-termined by considering the factor of safety. The probability offailure can be defined as the bearing capacity being less than thenominated load (i.e., Nc,det=FS). For a normally distributed bearingcapacity factor, the probability of failure can then be calculated using

p�Nc,Nc,det

�FS

� ¼ F

�Nc,det

�FS

�2mNc

sNc

" #(3)

whereF5 cumulative normal function; mNcand sNc 5mean value

and SD of the bearing capacity factor Nc; and Nc,det 5 ultimate

Fig. 9. Properties for foundations at different depths: (a) mean and SDof the bearing capacity factors; (b) COVs of the bearing capacity factors

Table 2. Bearing Capacity Factor Statistics and the Normal DistributionResults

Embedment depth Deterministic result (Nc,det) Mean (mNc) x2

0 5.23 4.71 20.10:5B 7.28 6.71 32.41B 8.24 7.59 37.32B 10.74 9.90 11.33B 11.86 10.84 22.04B 12.36 11.25 15.3

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bearing capacity factor of the deterministic analysis based on uni-form soil.

The probabilities of failure for the foundations at different factorsof safety can be calculated using Eq. (3) with the parameters inTable 2. When the factor of safety is 1.2, the failure probabilities are26.5, 9.9, 6.2, 2.9, 5.5, and 9.9% for the foundations embedded at 0,0:5B, 1B, 2B, 3B, and 4B depths, respectively (Fig. 11). A relativelylarger probability of failure for the foundation buried at the 4Bdepth results from the relatively larger COV of the bearing capacityfactor. As the target probabilities for bearing failure are generally inthe range of 1022 to 1023, a factor of safety of 1.2 would not beacceptable for design. The probability of failure decreases markedlywith an increasing safety factor. For the surface footing, the failure

probability decreases from 82.3 to 1.4% when the factor of safetyincreases from1.0 to 1.5. For a foundation buried at deeper depth, thefailure probabilities are all smaller than 1024 when the safety factorreaches 1.5. The failure probability is essentially reduced to nearly

Fig. 10. (Color) Cumulative probability of the normalized bearing capacity for foundations at different embedment depths

Fig. 11. (Color) Variation of factor of safety for foundations buried at different depths

Table 3. Factor of Safety at Different Levels of Probability of Failure

Embedment depth

Probability of failure 0 0:5B 1B 2B 3B 4B

1022 1.53 1.31 1.27 1.23 1.25 1.291023 1.75 1.41 1.34 1.28 1.32 1.381024 1.99 1.5 1.41 1.33 1.38 1.45

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zero by increasing the factor of safety to 2.0 for all of the foundations.Table 3 summarizes the factor of safety for foundations at differentdepths, which predicts failure probabilities of 1022, 1023, and 1024.The partial safety factor that accounts for the uncertainty in shearstrength for buried anchors specified by Det Norske Veritas (DNV)(2012) is 1.3. This factor of safety indicates a failure probability of0.1%, which is at a reasonable level. Note that the failure probabilityis also dependent on the degree of variation and the scale of fluc-tuation of the random soil, which requires further study.

Summary and Conclusions

This paper has demonstrated how the spatial pattern of random soildominates the development of a failure plane and the ultimatebearing capacity for foundations buried at different depths. Differentpossibilities of shear planes resulting from different spatial patternsof soil are demonstrated, which can explain the significant dis-crepancy between the bearing capacity of random soil and that ofuniform soil. The following conclusions can be drawn.1. A shear plane commences at the weakest soil surrounding the

foundation and extends along the weak soil path. Several shearplanes can be formed in random soil instead of a single shear pathdefined by the logarithmic spiral or circular shape in uniform soil.

2. Generally, the bearing capacity increases upon enlarging theshear planes, which are often unsymmetrical in random soil. Inaddition, the shear strength values along the shear planeimpose a significant effect on the bearing capacity.

3. The average bearing capacity factors increase with an in-creasing embedment depth of the foundation because of thetransition from a general failure at shallow depth to a full-flowfailure at deep depth. The COV of the bearing capacity factor,however, is closely related to the length of the shear plane.

4. When a safety factor of 1 is adopted, the probability of failureof the foundations is extremely large and will increase withincreasing embedment depth. The probability of failuredecreases markedly with an increasing safety factor for allof the foundations. The failure probability is essentially re-duced to nearly zero by increasing the factor of safety to 2.0 forall of the foundations at the given level of variation, althoughdifferent factors of safety for defined probabilities of failure areprovided (again, only relevant for the spatial variation param-eters calculated).

Acknowledgments

This researchwas undertakenwith support from theAustralia-ChinaNatural Gas Technology Partnership Fund and the Lloyd’s RegisterFoundation. Lloyd’s Register Foundation, a U.K.-registered charityand sole shareholder of Lloyd’s Register Group, invests in science,engineering, and technology for public benefit worldwide. Thisstudy comprises part of the activities of the Centre for OffshoreFoundation Systems (COFS), currently supported as a node of theAustralian Research Council Centre of Excellence for GeotechnicalScience and Engineering.

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