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Evaluation of Undrained BearingCapacities of Bucket Foundations UnderCombined LoadsLe Chi Hung a & Sung-Ryul Kim aa Department of Civil Engineering , Dong-A University , Busan , KoreaAccepted author version posted online: 17 Apr 2013.Publishedonline: 28 Oct 2013.

To cite this article: Le Chi Hung & Sung-Ryul Kim (2014) Evaluation of Undrained Bearing Capacitiesof Bucket Foundations Under Combined Loads, Marine Georesources & Geotechnology, 32:1, 76-92,DOI: 10.1080/1064119X.2012.735346

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Evaluation of Undrained Bearing Capacitiesof Bucket Foundations Under Combined Loads

LE CHI HUNG AND SUNG-RYUL KIM

Department of Civil Engineering, Dong-A University, Busan, Korea

A series of three-dimensional finite element analyses was conducted to investigate theeffects of the embedment depth, the non-homogeneity of clay, and combined loads onthe undrained bearing capacities of bucket foundations. The undrained shear strengthand Young’s modulus of clay were assumed to vary linearly with depth. Meanwhile,the stress-strain response of clay was simulated using the Tresca criterion. The numeri-cal modeling adopted in this study was verified by comparing the calculated capacitieswith those from previous studies. Based on the results of the finite element analyses over1400 cases, new equations were proposed to calculate the vertical, horizontal, andmoment bearing capacities as well as to define the capacity envelopes under generalcombined loads. Comparisons with the capacity envelopes of previous studies showedthat the proposed equations properly predicted the bearing capacities of the bucket foun-dation by considering the effects of the non-homogeneity of clay and embedment depth.

Keywords bucket foundation, clay, combined loads, finite element analyses,undrained bearing capacity, undrained shear strength

Introduction

A bucket foundation is a circular surface foundation with thin skirts around thecircumference. Bucket foundations have been extensively used in offshore facilities,such as platforms, wind turbines, and jacket structures (Tjelta and Haaland, 1993;Bransby and Randolph, 1998; Houlsby et al., 2005; Luke et al., 2005).

The skirt of the bucket foundation is first penetrated into the seabed byself-weight. Further penetration is then conducted by pumping water out of thefoundation, producing a suction pressure inside the foundation. The penetrationstops when the top-plate of the foundation touches with the seabed, and the suctionpressure confines the soil plugged within the skirt.

The bearing capacity of the circular shallow foundations on undrained clay isfundamentally important in many geotechnical problems (Houlsby and Martin,

Received 30 July 2012; accepted 26 September 2012.The present research was supported by a grant from the Korea Institute of Construction

and Transportation Technology Evaluation and Planning (KICTTEP) and from the Ministryof Land, Transport, and Maritime Affairs (MLTM) R&D program (2010 Construction Tech-nology Innovation Program, 10-CTIP-E04), and by Basic Science Research Program throughthe National Research Foundation of Korea (NRF) funded by the Ministry of Education(NRF-2012R1A1A2A10042889).

Address correspondence to Sung-Ryul Kim, Department of Civil Engineering, Dong-AUniversity, 840 Hadan2-dong, Saha-gu, Busan 604-714, Korea. E-mail: sungryul@dau.ac.kr

Marine Georesources & Geotechnology, 32:76–92, 2014Copyright # Taylor & Francis Group, LLCISSN: 1064-119X print=1521-0618 onlineDOI: 10.1080/1064119X.2012.735346

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2003). A number of design codes consider the bucket foundation as a shallowcircular foundation.

Previous researchers conducted two-dimensional (2D) finite element (FE)analyses to study the effects of embedment depth and combined loads on theundrained capacities of the skirted foundation in either non-homogenous clay(Yun and Bransby, 2007a, 2007b; Bransby and Yun, 2009) or homogenous clay(Gourvenec, 2008). The undrained capacity was largely influenced by the embedmentdepth and the magnitude of the vertical load under the general combined loading.However, the 2D FE analysis does not properly consider the effect of skirt embed-ment depth or 3D shape on the bearing capacity. In addition, the clay grounds con-sidered in the previous studies were limited to the very high non-homogeneity of clay.

Although a few studies used 3D FE analyses to study the undrained bearingcapacity of the skirted foundation in non-homogenous clay, the foundation wasmodeled as an equivalent surface shallow foundation without modeling the embed-ment of the foundation (Gourvenec and Randolph, 2003). Previous studies haveshown that the bearing behavior of the foundation was largely affected by theembedment depth (Gourvenec, 2008) and the foundation geometry (Gourvenecand Randolph, 2003). Therefore, modeling the exact 3D shape of the bucket foun-dation is necessary. In addition, the effect of the non-homogeneity of soft clay onthe capacity is particularly important for large foundations such as a bucketfoundation (Houlsby and Martin, 2003).

A series of 3D numerical analyses was performed to analyze the effects of thenon-homogeneity of clay, the embedment depth and the combined loads on theundrained bearing capacities of bucket foundations. Based on the calculated capacitiesfrom FE analyses, new equations were proposed to calculate the vertical, horizontal andmoment capacities and to define the capacity envelope under general combined loads.

Numerical Modeling

Input Properties and FE Modeling

Figure 1 presents the definition of the bucket foundation geometry and the sign con-ventions adopted in this study. The embedment ratio L=D of the bucket foundationwas varied at 0.25, 0.5, 0.75 and 1.0, where L is the skirt length and D is the diameterof the bucket foundation. Preliminary analyses confirmed that the foundation

Figure 1. Foundation geometry, load and displacement conventions and soil conditions(modified from Bransby and Yun, 2009).

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diameter D had no effect on the normalized bearing capacities. Thus D was kept con-stant at 10m for all analyses. The skirt thickness of the foundation was taken ast¼ 25mm, which, in practice, is the common thickness of steel buckets. Thedeformation of foundations in soft clay is difficult to occur. Therefore, thefoundation was modeled as a rigid body.

The undrained shear strength of clay, su, was assumed to vary linearly withdepth (Houlsby and Martin, 2003):

su ¼ sum þ kz ð1Þ

where sum is the undrained shear strength at the ground surface, z is the depth belowthe ground surface, and k is the rate of increase of strength with depth, as shown inFigure 1.

Previous analyses have confirmed that bearing capacity factors are dependent noton the individual parameters of sum or k, but on the normalized parameters of kD=sum(Yun and Bransby, 2007a). Therefore, the non-homogeneity of clay was defined by thenormalized parameter kD=sum. The kD=sum value ranges from zero for homogenousclay to as high as 30 for extremely non-homogenous clay with a low soil shear strengthat the ground surface (Tani and Craig, 1995). In the present study, the kD=sum valuesselected were 0, 2, 4, 6, 10, and 30. The kD=sum, sum, and k values are shown in Table 1.Homogenous clay (kD=sum¼ 0) would be rarely encountered in the in-situ condition.Thus, the kD=sum¼ 0 was investigated for only one directional loading (pure loading)to show the effect of non-homogeneity of clay on the capacity.

The undrained condition of clay during loading can be reasonably analyzedusing total stress analyses (Tani and Craig, 1995). Therefore, the soil was modeledas a linear elastic-perfectly plastic model based on the Tresca failure criterion. Theshear strength was considered as the undrained shear strength of clay su. TheYoung’s modulus Eu of clay was set at 400� su, and the Poisson’s ratio n was fixedat 0.495 to simulate the constant volume response of clay in undrained conditions(Taiebat and Carter, 2000; Yun and Bransby, 2007a, 2007b; Gourvenec, 2008).The effective unit weight of clay was set as c0 ¼ 6 kN=m3.

All FE analyses were conducted using the ABAQUS software (Simulia, 2010).The first-order, eight-node linear brick, reduced integration continuum with a hybridformulation element C3D8RH was used to model the soil.

Figure 2 shows a typical mesh and boundary extensions of the soil domain forthe bucket foundation. By applying symmetric conditions, a half of the entire systemwas modeled. The vertical and horizontal displacements at the bottom boundary and

Table 1. Input properties for undrained shearstrength of clay

kD=sum sum (kPa) k (kPa=m)

0 5 02 6.25 1.254 3.25 1.36 2 1.2

10 1.25 1.2530 0.4 1.2

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the horizontal displacements at the lateral boundaries were constrained. The size ofthe soil elements gradually increased from the bucket foundation to the domainboundary. In addition, the optimum mesh sizes were used to minimize the effectof the mesh size on the results.

The BV and BH indicated in the Figure 2 are the vertical and horizontal bound-ary extents from the skirt tip and the side of the bucket, respectively. BH=D was var-ied from 2 to 6 at a fixed BV=D¼ 2 and vice-versa to investigate the effect of domainsize on the capacity. The bearing capacities gradually decreased as BH=D or BV=Dincreased and became constant at BH=D¼ 4.5 and BV=D¼ 4.5, which were appliedfor subsequent analyses.

The interface between the foundation and the soil was assumed to be rough, and thedetachmentbetween thebucket foundationand the soilwasprevented (YunandBransby,2007a, 2007b; Gourvenec, 2008; Bransby and Yun, 2009; Barari and Ibsen, 2012).

Determination of Bearing Capacities

The loading was applied using the displacement-controlled method, which increasesthe vertical (v), horizontal (h), and rotational (h) displacements at a load referencepoint, RP, as shown in Figure 1 (Bransby and Randolph, 1997; Gourvenec andRandolph, 2003; Yun and Bransby, 2007a, 2007b). This method is known to be moresuitable than the load-controlled method for obtaining the failure loads (Bransbyand Randolph, 1997; Gourvenec and Randolph, 2003). The location of RP wasselected at the base of the foundation, as suggested in previous studies (Bransbyand Randolph, 1998; Cassidy et al., 2004; Yun and Bransby, 2007a; 2007b;Gourvenec, 2008; Bransby and Yun, 2009).

The capacity envelope under combined loads was determined using the probemethod, which increases the displacement at a constant displacement ratio (e.g.,h=v¼ constant). The loading path in the probe method converged at a specific point,which indicated one point on the capacity envelope (Yun and Bransby, 2007a). Thesubsequent points along the capacity envelope were obtained by changing the dis-placement ratio. Figure 3 illustrates examples of the determination of the capacityenvelopes under combined Vertical-Horizontal (VH), Vertical-Moment (VM) andHorizontal-Moment (HM) loads.

The intersection points with the envelope and each axis define the vertical, hori-zontal and moment bearing capacities, which are indicated as Vo, Ho and Mo,respectively. These capacities were carefully determined by controlling the displace-ment increments or by analyzing the load-displacement curves. Vo was determined

Figure 2. Typical mesh for soil and foundation domain.

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by applying the probe method with a small displacement ratio of h=v¼ 1=1000(Taiebat and Carter, 2000). The Ho and Mo were determined using the tangentintersection method (Mansur and Kaufman, 1956), as shown in Figure 4. The methodplots two tangential lines along the initial and later portions of the load-displacementcurve, and the load corresponding to the intersection point of these two lines is takenas the bearing capacity. The capacities determined from the method showed a vari-ation of about 2%–5% according to the drawing deviation of the tangential lines.Therefore, the capacities were properly adjusted by referencing adjacent points inthe capacity envelopes under combined VH (M¼ 0) and VM (H¼ 0) loads.

The capacity envelopes under the general combined VHM loads were obtainedin two steps. First, the vertical loadings of V¼ 0.25Vo, 0.5Vo, 0.75Vo and 0.9Vowere applied to the foundation. In the second step, the probe loading with a constanth=(Dh) ratio was applied to obtain the capacity envelope under HM loads whilekeeping the vertical load constant. A total of 1425 cases were analyzed to definethe capacity envelopes under the general combined VHM loads.

Figure 3. Determination of capacity envelopes via the probe method (L=D¼ 1.0, kD=sum¼ 2).

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The sign conventions for analyzing the capacity were adopted from Gourvenec(2008), as shown in Table 2. The vertical, horizontal and moment bearing capacityfactors are defined as each bearing capacity divided by (cross-sectional area of abucket, A)� (undrained shear strength at a specific depth, suo). suo was consideredas the undrained shear strength at a depth D=4 below the skirt tip level, followingthe suggestion of Byrne and Cassidy (2002).

Validation of FE Analyses

The accuracy of the FE analyses in this study was validated by comparing the FEanalyses results with those of previous studies. Houlsby and Wroth (1983) suggestedthe undrained vertical bearing capacity factors for rough circular surface founda-tions in non-homogenous clay based on the plasticity theory. Yun and Bransby(2007b) reported the undrained vertical bearing capacities of skirted strip founda-tions in non-homogenous soft clay using 100 g centrifuge model tests. Coffmanet al. (2004) conducted a series of 1 g model tests to study the horizontal bearing

Figure 4. Tangent intersection method for determining bearing capacity (L=D¼ 1.0, kD=sum¼ 2).

Table 2. Summary of notation for loads and displacement (modified fromGourvenec, 2008)

Vertical Horizontal Rotational

Load at RP V H MDisplacement at RP v h hBearing capacityfactor

NcV¼Vo=(A.suo) NcH¼Ho=(A.suo) NcM¼Mo=(A.D.suo)

Dimensionless load V=(A.suo) H=(A.suo) M=(A.D.suo)Normalized load V=Vo H=Ho M=Mo

�A¼cross sectional area of the bucket foundation.suo¼ undrained shear strength of clay at depth D=4 below skirt tip level.

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capacities and failure mode of a model bucket foundation in different horizontalloading conditions.

Tables 3 and 4 show the comparisons of the vertical and horizontal bearingcapacities, respectively. The error between the capacities from the FE analyses andprevious studies was only 0.0%�5.5%. The comparisons showed that the capacitiesdetermined from the FE analyses were reliable and could be properly used to analyzethe undrained bearing capacities of the bucket foundation in clay.

Evaluation of Bearing Capacities

From the FE analyses, the effect of the embedment depth and non-homogeneity ofclay on the bearing capacity factors were analyzed. Figures 5, 6, and 7 show thevariations of in the vertical, horizontal, and moment capacity factors according tothe L=D and kD=sum ratios, respectively. The bearing capacity factors increased withthe increase of the L=D ratio because the deeper embedded length of a bucketincreased the resistance against the loadings. In addition, the bearing capacitydecreased with the increase in the kD=sum ratio because the clay with the higherkD=sum has a lower undrained shear strength on the average along the skirt.

Table 3. Comparison of vertical bearing capacities

kD=sum

Vo (kN)

FE resultsHoulsby andWroth (1983) Error (%)

2 3858.3 3735.6 3.24 2312.6 2223.3 3.96 1588.1 1519.0 4.4

10 1174.2 1109.4 5.530 616.7 – –

L=D FE resultsYun and Bransby

(2007b) Error (%)

0.2 6754.9 7108.9 5.00.5 14592.9 14830.9 1.61.0 26447.6 27130.7 2.5

Table 4. Comparison of horizontal bearing capacities

Bucket embeddeddepth (m)

Depth to loadingpoint (m)

Ho (kN)

FEresults

Coffmanet al. (2004)

Error(%)

0.823 0.51 0.31 0.30 3.20.813 0.53 0.34 0.33 2.90.820 0.61 0.33 0.33 0.00.813 0.66 0.28 0.28 0.0

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Based on the FE analyses results, equations that evaluate the vertical,horizontal, and moment bearing capacity factors have been proposed as shown inEquations (2)–(7). The equations were expressed as the function of the L=D ratioand of the non-homogeneity of clay kD=sum.

Vo ¼ NcVAsuo ð2Þ

NcV ¼ 5:2L

D

� �þ 5:1þ 2:7e � kD

sum�0:35ð Þ ð3Þ

Ho ¼ NcHAsuo ð4Þ

NcH ¼ 3:4L

D

� �þ 0:54þ 2e � kD

sum�0:34ð Þ ð5Þ

Mo ¼ NcMADsuo ð6Þ

NcM ¼ ðe �0:38� kDsumð Þ þ 1:22Þ � 0:36e 1:44� L

Dð Þ ð7Þ

Figure 8 shows the effects of embedment depth on the bearing capacity envel-opes under combined VH (M¼ 0), VM (H¼ 0), and HM (V¼ 0) loads at kD=sum¼ 10. The size of the capacity envelopes increased with increasing L=D ratio,as expected. The envelopes under the HM loadings showed a non-symmetrical shapeand a more biased shape with increasing L=D ratio. The non-symmetrical shapeoccurred because the moment capacity varied according to the direction of the hori-zontal loading.

Figure 9 shows the effect of non-homogeneity on the bearing capacity envelopeunder combined loadings at L=D¼ 0.5. The size of the capacity envelopes decreasedwith increasing kD=sum due to the effect of the non-homogeneity of clay, asdiscussed earlier.

Figure 5. Vertical bearing capacity factors with L=D and kD=sum ratios.

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Figure 10 presents the three-dimensional capacity envelopes under generalcombined VHM loading for the foundations with L=D¼ 0.25 and L=D¼ 1.0 atkD=sum¼ 10. The envelopes were made in terms of normalized loads. The combi-nation of VH, VM, and HM generates the load contours, which show athree-dimensional elliptical shape. The three-dimensional capacity envelope can beused to evaluate the safety against any load combination of the vertical, horizontal,and moment loads in design of the bucket foundation (Taiebat and Carter, 2000).

Development of Capacity Envelope Under Combined Loadings

A new equation based on the FE analyses results was proposed to describe thecapacity envelope of the bucket foundations under the general combined VHM loads.

Figure 7. Moment bearing capacity factors with L=D and kD=sum ratios.

Figure 6. Horizontal bearing capacity factors with L=D and kD=sum ratios.

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The new equation was derived by referring to those proposed by otherresearchers, such as Murff (1994), Bransby and Randolph (1998), Taiebat and Carter(2000), Cassidy et al. (2004), Zhang et al. (2012). The new equation presents thecapacity envelope as elliptically shaped with cones. The apexes of the cones changewith changes in the L=D ratio, kD=sum and V=Vo, respectively. The equation can bewritten as follows:

fVHM ¼ M

M0

� �2

�abkH

H0

� �M

M0

� �þ H

H0

� �2

þ V

V0

� �2

�1 ¼ 0 ð8Þ

where Mo, Ho and Vo can be calculated from Equations (2), (4), and (6), respect-ively. a, b, and k are the factors that control the shape and size of the capacityenvelopes.

The a, b, and k factors are the dependent variables of L=D, kD=sum and, V=Vo,respectively. The steps outlined below were applied to evaluate a, b, and k.

Figure 8. Effect of embedment depth on capacity envelopes (kD=sum¼ 10).

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The first step was the normalization of the capacity envelopes under thecombined loadings. In Figure 3, the loads at the intersection points between theenvelopes and coordinate axes are Vo, Ho and Mo. The envelopes can be normalizedby dividing those envelopes by the corresponding Vo, Ho and Mo as shown in

Figure 9. Effect of non-homogeneity of clay on capacity envelopes (L=D¼ 0.5).

Figure 10. Bearing capacity envelopes under general combined VHM load (kD=sum¼ 10).

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Figures 11 and 12. The normalized capacity envelopes under combined VH (M¼ 0)and VM (H¼ 0) loads were unique, which indicates that those capacity envelopescan be evaluated by scaling up Vo, Ho and Mo corresponding to the L=D andkD=sum ratios.

The normalized envelopes under the HM loads were not unique but changedwith the L=D and kD=sum ratios, as indicated in Figures 11(c) and 12(c). Therefore,the second step was to determine a, b and k, which determine the biased shape of thecapacity envelopes under the HM (V¼ 0) loads. H=Ho and M=Mo in Equation (8)were assumed as the independent and dependent variables, respectively. The H=Hovalues obtained from the FE analyses were inputted into Equation (8) and the leastsquare method was applied to determine a, b and k, which yield the M=Mo values ofthe FE analyses. The k factor in this step was assumed as a constant value whichcorresponds to the vertical load of V¼ 0. This procedure was repeated for each ofthe L=D and kD=sum ratios. Those factors were unique and independent of oneanother. The a and b factors were the functions of the L=D and kD=sum ratios,respectively. The final a, b, and k factors in this step are shown in Table 5.

Figure 11. Normalized capacity envelopes for different L=D ratios (kD=sum¼ 10).

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The third step was to evaluate k at the other vertical loads (V=Vo¼ 0.25, 0.5,0.75, 0.9). The a and b values of the first step were remained constant in this step.The same least square method was applied to determine k. The procedure wasrepeated for each vertical load level. In addition, k was dependent only on the ver-tical load level. Table 5 shows the k values according to the vertical load level.

Using the curve fitting technique, the following equations were proposed toevaluate approximately the a, b, and k factors according to the L=D, kD=sum andV=Vo ratios, respectively. The equations are:

a ¼ 0:304L

D

� �0:33

ð9Þ

b ¼ 5:58kD

sum

� ��0:049

� 5:0 ð10Þ

Figure 12. Normalized capacity envelopes for different kD=sum ratios (L=D¼ 0.5).

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k ¼ 0:94þ 0:2V

Vo

� �2:4

ð11Þ

Discussion

The applicability of the proposed method in the present study was evaluated bycomparing the capacity envelopes of the proposed equation with those from theFE results and from previous studies.

Figure 13 shows the comparisons between the capacity envelopes of the FEanalyses and Equation (8). The curves predicted by Equation (8) and FE analysesare designated as ‘‘Proposed’’ and ‘‘FE analyses’’, respectively. The comparison

Table 5. Summary of a, b, and k

Variation of a factors

L=D 0.25 0.5 0.75 1.0a 0.19 0.25 0.28 0.3

Variation of b factors

kD=sum 2 4 6 10 30b 5.41 5.20 5.10 5.0 5.0

Variation of k factors

V=Vo 0 0.25 0.5 0.75 0.9k 0.94 0.95 0.98 1.04 1.1

Figure 13. Comparison of VHM capacity envelopes between the proposed equation and FEanalyses (kD=sum¼ 10).

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showed that the proposed equation could properly predict the capacity of the FEanalyses according to the foundation embedment, non-homogeneity of clay, andloading conditions.

In addition, the size of the capacity envelopes slightly decreased at the verticalload level of V=Vo� 0.5. This result agreed well with the findings of Yun andBransby (2007a), which stated that the decrease in the combined horizontal andmoment bearing capacities for a given vertical load was negligible for a vertical loadless than 40% of the vertical bearing capacity. At V=Vo> 0.5, the size of the capacityenvelopes significantly decreased. This finding is useful in practical respects. Forexample, in designing the bucket foundation, the capacity envelopes under a generalcombined load would be significantly reduced if the vertical load (the weight ofstructures) was larger than 50% of the vertical bearing capacity of the foundation.

The capacity envelope of the proposed equation was compared with those of theprevious equations (Murff (1994); Bransby and Randolph (1998)), as shown inFigure 14. The curves obtained by Murff (1994)’s equation and Bransby andRandolph (1998)’s equation were denoted as ‘‘Murff (1994)’’ and ‘‘Bransby andRandolph (1998)’’, respectively. The foundations with L=D¼ 0.454 and 1.0, as wellas the soil with kD=sum¼ 6, were chosen as compatible with the study of Bransby andRandolph (1998). The Murff’s equation was not suitable in obtaining the VHMcapacity envelopes of the bucket foundation because the method was originally sug-gested for a surface foundation. The curves of ‘‘Bransby and Randolph (1998)’’ and‘‘Proposed’’ showed close agreement at L=D¼ 0.454 and V¼ 0. The differencebetween those became significant at L=D¼ 1 or V¼ 0.75V0. The reason for the dif-ference might be because Bransby and Randolph (1998) assumed the pure slidingcondition, which induces large errors in the high L=D ratio (Gourvenec, 2008).

Conclusions

A series of three-dimensional FE analyses was performed to investigate the effect ofthe embedment depth of the bucket foundation and the non-homogeneity of clay on

Figure 14. Comparison of VHM capacity envelopes between the proposed and previousequations (kD=sum¼ 6).

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the bearing capacities and to define the capacity envelopes under combined loadings.The following conclusions were obtained:

1. The non-homogeneity of clay and the embedment depth of a bucket foundationsignificantly influenced on the bearing capacity of the foundation. The bearingcapacity decreased with both the decrease in the embedment depth and theincrease in the non-homogeneity of clay.

2. The design equations, which evaluate the vertical, horizontal, and moment bear-ing capacities, have been proposed, thereby introducing the bearing capacity fac-tors as the function of the L=D ratio and the non-homogeneity of clay kD=sum.

3. The capacity envelope was developed to define the bearing capacity of a bucketfoundation under combined loadings. The a, b and k factors in the capacity envel-ope were introduced to determine the effects of the embedment depth, thenon-homogeneity of clay, and the vertical loads, respectively. The proposed equa-tion properly considered the effect of the L=D ratio and combined loading on thecapacity compared with previous studies.

References

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