The bearing capacity of spudcan foundations under combined loading in spatially variable soils

Accepted Manuscript The bearing capacity of spudcan foundations under combined loading in spatially variable soils

Li Li, Jinhui Li, Jinsong Huang, Hongjun Liu, Mark J. Cassidy PII: DOI: Reference:

S0013-7952(17)30483-0 doi: 10.1016/j.enggeo.2017.03.022 ENGEO 4535

To appear in:

Engineering Geology

Received date: Revised date: Accepted date:

16 August 2016 21 March 2017 26 March 2017

Please cite this article as: Li Li, Jinhui Li, Jinsong Huang, Hongjun Liu, Mark J. Cassidy , The bearing capacity of spudcan foundations under combined loading in spatially variable soils. The address for the corresponding author was captured as affiliation for all authors. Please check if appropriate. Engeo(2017), doi: 10.1016/j.enggeo.2017.03.022

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ACCEPTED MANUSCRIPT The bearing capacity of spudcan foundations under combined loading in spatially variable soils Li Lia; Jinhui Lia,*; Jinsong Huangb; Hongjun Liua; Mark J. Cassidyc

ABSTRACT: Predicting the bearing capacity of a spudcan foundation under combined vertical (V), horizontal (H) and moment (M) loads is a challenging problem encountered by

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geotechnical engineers. In previous studies the combined VHM capacity was defined for a uniform soil profile, ignoring any variability in soil stratification and properties. In offshore

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conditions, however, both the soil profile and soil properties vary spatially. Therefore, it is of

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interest to account for the spatial variability of soil in the analysis of the bearing capacity of a spudcan. It is shown in this paper how the spatial variability of a clay affects the bearing

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capacity of a deeply buried spudcan foundation under combined loadings. Three-dimensional random fields are generated to model the spatial variability of undrained shear strength of clay and combined with a non-linear finite element analysis to investigate and define the

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VHM failure envelope of a spudcan foundation. Because of the random nature of soils VHM failure envelopes of different probability of occurrence are proposed. Results from this study

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provide guidance to the practical assessment of spudcan foundations in spatially varied soil

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conditions that can be encountered offshore.

Risk.

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Keywords: Spudcan foundation; Bearing capacity; Clay; Failure envelope; Random field;

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Corresponding author. Email address: [email protected] (J. H. Li). Department of Civil and Environmental Engineering, Harbin Institute of Technology Shen Zhen Graduate School, Shenzhen, China b ARC Centre of Excellence for Geotechnical Science and Engineering, University of Newcastle, Newcastle, Australia c Centre for Offshore Foundation Systems and ARC Centre of Excellence for Geotechnical Science and Engineering, The University of Western Australia, Australia a

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ACCEPTED MANUSCRIPT 1. Introduction In the offshore industry, mobile jack-up drilling rigs are a key contributor to explore oil and gas reservoirs in water depths up to around 150m. The drilling rigs are typically supported by individual, conical spudcan foundations that are around 20 m in diameter. During installation and preloading of jack-up platforms, spudcan foundations can penetrate

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deeply into the seabed (up to 3 spudcan diameters) (Menzies and Roper, 2008). Industrial

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guidelines, such as SNAME (2008) and ISO (2012), provide formulas to assess the vertical

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load-penetration curves for jack-ups. Based on bearing capacity theory, these have been calibrated from model experiments, finite-element and limit analysis (e.g., Taiebat et al.,

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2000; Templeton, 2009; Hossain and Randolph, 2010; Zhang et al., 2011). These studies have assumed homogeneous soils or uniform soils of strength linearly increasing with depth.

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However, under more realistic conditions found offshore, the strength of seabed soils often varied spatially as a result of depositional and post-depositional process (Baecher and

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Christian, 2003).

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The influence of spatial variability of soil properties (e.g., undrained shear strength)

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on the bearing capacity of a strip footing and slope stability has been studied extensively (e.g., Fenton and Griffiths, 2008; Popescu et al., 2005; Cassidy et al., 2013; Jiang et al. 2014; Li et al., 2017). It is found that the spatial variability can dramatically reduce the bearing capacity of a foundation (e.g., Popescu et al., 2005; Li et al., 2015, 2016b). The majority of the studies employed two-dimensional random fields and plane-strain conditions (i.e. an infinite foundation with soil properties not varying into the plane). This simplifying assumption could be influential on the bearing capacity of conical-shaped spudcan foundation. Therefore, in this paper, three-dimensional (3D) random fields representing the spatial variation of soil properties in all directions are investigated. Once a jack-up is installed and operational, the spudcan foundations need to be able to 2

ACCEPTED MANUSCRIPT resist the combined vertical (V), horizontal (H) and moment (M) loads due to the environmental loadings (e.g., wind, waves and currents) on the superstructure. However, there currently is no guidance in the effect of the uncertainty in the spatially varied seabed on the combined VHM bearing capacity. The application being addressed in this paper is illustrated in Fig. 1. Combined loading problems have traditionally been addressed by the

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classical bearing capacity approach (Hansen, 1970), which considers the correction with the

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load eccentricity and inclination for the vertical bearing capacity. However, it is more

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accurate to demonstrate the ultimate bearing capacity of the foundation under combined loads written directly in VHM loading envelopes. The load combinations inside the envelope are

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assumed as safe, but the load combinations outside the envelope are deemed to result in failure. The jack-up industry uses this approach when assessing if a jack-up is safe to operate

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at a particular site (SNAME, 2008; ISO, 2012), with the VHM failure envelopes defined through numerical and experimental studies in uniform soils (Martin and Houlsby, 2000;

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Templeton, 2009; Zhang et al., 2013).

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With no advice currently available, this paper provides guidance on how the spatial

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variability of clay soil affects the bearing capacity of a deeply embedded spudcan foundation under combined loads. Randomness of undrained shear strength in three dimensions is accounted for and VHM failure envelopes with different probability of occurrence are proposed. The results can be used in assessing the probability of exceeding an ultimate limit state of failure of a spudcan foundation within the methodology used to assess the safety of jack-ups structure operating in large storms.

2. Spatially varying soil The spatial variability of a soil is often described by a trend and a residual variable. A trend is estimated by fitting well-defined mathematical functions to spatial data points. The 3

ACCEPTED MANUSCRIPT residuals around the trend are spatially correlated to one another in space. This correlation is generally a function of the distance, described by an autocorrelation function and a parameter called scale of fluctuation (see details in Baecher and Christian, 2003). In this study, the undrained shear strength (su) of clay was considered as a random variable and modeled as a log-normally distributed random field with mean value, standard

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deviation, and scale of fluctuation. The autocorrelation function of exponential model was

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employed. The average undrained shear strength was assumed to increase linearly with depth

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at a strength gradient of 1.2 kPa/m. The mean value of the mudline undrained shear strength su0 was assumed as 0.1 kPa with this small value adopted for computational stability (Zhang

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et al., 2011). The coefficient of variation (COV) of undrained shear strength was assumed to be 0.3 (Li et al., 2015). Previous researches (e.g., Tang, 1979; Keaveny et al., 1989; Phoon

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and Kulhawy, 1999; Uzielli et al., 2006; Cheon and Gilbert, 2014) found that the scale of fluctuation of offshore soils in vertical direction ranges from 0.05 m to 14 m. The horizontal

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scale of fluctuation ranges from 7 m to 9000 m, which is much larger than that in the vertical

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direction. The mean horizontal and vertical scales of fluctuation were found to be 50.7 m and

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3.8 m, respectively (Phoon and Kulhawy, 1999). These mean scales of fluctuation were then used in this study. The clay was assumed to be linear-elastic perfectly plastic. The Young’s modulus and Poisson’s ratio were adopted to represent the elastic response. The Young’s modulus E was considered to be perfectly correlated to the undrained shear strength su with a ratio E/su = 500 (Hu and Randolph, 1998). The effective unit weight of the soil was γ= 7kN/m3. The Poisson’s ratio of 0.49 was chosen to simulate the undrained conditions of no volume change and soil failure determined by the Tresca criterion. There are several random field generation methods available (Fenton and Griffiths, 2008; Bari and Shahin, 2015; Jamshidi and Kamyab, 2015). The Karhunen-Loeve expansion method was used because it has analytical solutions for the exponential autocorrelation 4

ACCEPTED MANUSCRIPT function considered in this study. As the Karhunen-Loeve expansion needs to be truncated to a finite number of terms, a significant concern is that the simulated variance will be reduced. In order to control this reduction, the eigenvalues were sorted in descending order and the number of terms decided when the last eigenvalue was at least ten thousand times smaller than the first eigenvalue. A realization of the three-dimensional random field is demonstrated

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in Fig. 2.

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3. Methodology 3.1. Random finite element method

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Random finite element method (RFEM) was performed using the nonlinear finiteelement software ABAQUS to investigate the bearing capacity of a spudcan foundation

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buried in spatially varying soils (Li et al. 2016a). The spudcan was wished in place which meant the installation process was not simulated. The foundation was modelled as a rigid

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body with loads and displacements applied to a reference point (see Fig. 3). The reference

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point was chosen at the center of the section of lowest maximum bearing area following the

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convention of the ISO standard (2012). The soil domain had a diameter of 6D and a height of 6D (108 m), which proved to be large enough to ensure no obvious boundary effects (Zhang et al., 2011, 2012). The horizontal displacements at the sides of the model were constrained and all displacements in the three coordinate directions fixed at the base of the model. The foundation-soil interface was assumed to be fully bonded, which is reasonable to represent undrained soil behavior (Gourvenec and Randolph, 2003; Zhang et al., 2012). Figure 4 shows the finite-element mesh for the spudcan and cylindrical soil model, containing about 48,000 first order hexahedral elements. Soil elements around the spudcan were meshed in a higher density to ensure the accuracy of analyses. However, refining the

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ACCEPTED MANUSCRIPT mesh can cost extra time for finite element analysis. The number of 48,000 elements was chosen to provide the balance between accuracy of results and time cost for analyses.

3.2. Probe test to define the failure envelope One combination of V, H, and M loading was applied to the spudcan foundation by

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translating or rotating the foundation to the ultimate state. The displacement ratio (i.e., the

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ratio between the displacements in designated directions) was fixed according to the applied

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loading combination, which is known as probe test. In a probe test, displacement in 2 directions (e.g., vertical and horizontal displacement in the VH plane) was applied at a

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constant ratio until failure, which indicates the resistance load (V, H, M) paths will ultimately become constant with increasing displacements. Each of this ultimate load combination

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defines one point on the failure envelope. The envelope was constructed by combining sufficient number of the ultimate loading points. In this study each failure envelope was

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defined by 37 independent probe tests. Figure 5 shows the 37 displacement paths for probe

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tests in VH (M = 0), VM (H = 0) and HM (V = 0) planes. The displacement ratios for the 37

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paths are also shown in Fig. 5. The failure envelope was determined by connecting the failure points of the corresponding probe tests in each plane. More details about this numerical methodology with various foundation types were provided in, amongst others, Elkhatib (2006), Gourvenec and Randolph (2003), and Zhang et al. (2011). Monte-Carlo simulations have been performed for 400 realizations of the 3D random fields and the subsequent random finite element analysis. The 400 random fields have a maximum error of 3% of the mean value of the undrained shear strength (Li et al., 2015). 37 probe tests were simulated on each of the 400 realizations of soil conditions, resulting in 14,800 finite element analyses.

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ACCEPTED MANUSCRIPT 3.3. Sign conventions The V, H and M loads as well as the corresponding spudcan movements w, u and β are illustrated in Fig. 3. The sign convention follows Butterfield et al. (1997) and Cassidy et al. (2013). As summarized in Table 1, the uniaxial bearing capacities are denoted by V0, H0 and M0 for pure vertical (H = M = 0 for ultimate pure vertical bearing capacity), horizontal and det

is adopted to identify bearing capacity (V, H or M) results

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moment loadings. Subscript

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derived for the homogeneous soils and subscript ran for the random soils. The bearing capacity

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factors (Ncv, Nch, Ncm) are derived with the corresponding ultimate uniaxial bearing capacity (V0, H0 and M0) normalized by the foundation diameter D and mean value of undrained shear

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strength su, i.e. Ncv = V0/(Asu), Nch = H0/(Asu), Ncm = M0/(ADsu). A is the bearing area of the

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spudcan foundation.

4. Results

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4.1. Deterministic analysis

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A deterministic case with the spudcan foundation embedded in soil with the undrained

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shear strength linearly increasing with depth is first simulated. Figure 6 demonstrates the failure envelopes for the spudcan in VH (M = 0), VM (H = 0) and HM (V = 0) planes. A comparison of the failure envelope obtained from the circular plate footing (thickness-todiameter aspect ratio (T/D) of 0.05 in Zhang et al. (2012)) and a spudcan embedded in 3.5D (Zhang et al., 2011) are also presented in Fig. 6. The envelopes for the circular plate footing are consistent with the envelopes in this study except the horizontal bearing capacity. This is because the inverted conical shape of the spudcan produces a larger horizontal capacity compared to the flat circular footing. The failure envelopes obtained from this research and those in Zhang et al. (2011), which were evaluated for a spudcan, agree well.

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ACCEPTED MANUSCRIPT Table 2 summarizes a comparison of the uniaxial bearing capacity factors for deterministic soil profiles. For example, Martin and Randolph (2001) proposed a theoretical solution of vertical bearing capacity factor of 13.11 for a thin plate with a rough contact surface embedded deeply in uniform normally consolidated soils. The vertical bearing capacity factor obtained from this study is 13.12, which is consistent with the theoretical

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value.

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4.2. Failure of spudcan in spatially varying soil

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In uniform soils a vertically loaded spudcan has a localized failure mechanism as shown in Fig. 7a. The displacement vectors show that the soils at the spudcan edges flow from the

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bottom to the top of the spudcan symmetrically as a result of symmetrical distribution of soil undrained shear strength (see Fig. 7b).

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This symmetry is lost in spatially varying soils. An example of a soil flow mechanism for a spudcan foundation in spatially random soils is shown in Fig. 7c. The displacement

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vectors at the right edge of the spudcan are much larger than that at the left edge. This

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unsymmetrical soil flow is because that the undrained shear strength at the right edge is lower

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than that at the left edge (see Fig. 7d). Therefore, the soil flow mechanism can be altered by the spatial pattern of the varying soil strength. The bearing capacity, and hence the failure envelope in spatially varying soils, is closely related with the failure mechanism and varies from case to case. The percentage of failure envelopes that are smaller than the deterministic case can be investigated by the uniaxial bearing capacity. The histogram of uniaxial bearing capacities normalized by the uniform deterministic capacity (Vran/Vdet,0, Hran/Hdet,0 and Mran/Mdet,0) are shown in Fig. 8. Also presented are the normal distributions and log-normal distributions used to fit the histogram (with parameters shown in Table 3). Both distributions provide a good fit to the probability density function. It is found that the mean value of Vran/Vdet,0, Hran/Hdet,0 and

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ACCEPTED MANUSCRIPT Mran/Mdet,0 are all less than 1, indicating that the mean uniaxial bearing capacity for spudcan in random soils are smaller than the corresponding deterministic case. This result is consistent with those for the shallow and embedded strip footings (Griffiths et al., 2002; Cassidy et al., 2013; Li et al., 2015). It indicates that the spatial variability of soil will, on average, reduce the bearing capacity of a foundation. For the buried spudcan in this study, about 70% of the

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cases in spatially variable soils have smaller vertical bearing capacity than in the uniform

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random cases are smaller than the deterministic results.

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soils. Regarding the bearing capacity in horizontal direction and moment, about 61% of the

Figure 9 shows the failure envelopes for the 400 realizations of the random fields along

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with that for the deterministic case. The shape of the failure envelopes for a spudcan in spatially varied soils changes with the spatial pattern of the soil. A smaller failure envelope

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indicates that less combined loadings can be maintained. Therefore, if a spudcan foundation is designed using the deterministic soil properties without considering the spatial variability

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the designed combined loading may result in failure of the foundation.

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4.3. Probabilistic VHM failure envelope As the failure envelope of the spudcan in spatially varied soils changed significantly with the pattern of the soil, it is appropriate to define the failure envelope with a certain probability of occurrence instead of a single failure envelope from the deterministic analysis. One direct method is to sort all of the failure envelopes from smallest to largest and then determine the failure envelopes with a certain confidence accordingly. However, it is hard to sort the failure envelopes by sizes as they are intersecting and crossing over one another for the random cases (see Fig. 9). Another approach, proposed for surface strip footings subjected to VHM loading (Cassidy et al., 2013), is to rank the 400 ultimate load combinations from the smallest to the largest for each probe test as determined by Eq. (1):

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V   H   M  C   ran    ran    ran  V   H     det,0   det,0   M det,0 

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(1)

where C is the distance from the origin to the point of failure on the ultimate combined bearing capacity envelope. The ultimate bearing capacity in each loading direction of the 37 probe tests was denoted as a dot in the VHM space. Figure 10 selectively shows several

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groups of bearing capacity points with the 1%, 5%, 10%, and 50% smallest values (i.e. 4th,

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20th, 40th, 200th smallest values out of 400 cases) as sorted by Eq. (1). The 50% smallest

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values represent that the bearing capacity in 50% of the random cases (i.e., 200 cases) are smaller than these values. The 50% smallest values are not the same as the bearing capacity

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in the deterministic uniform soils using the mean parameters (as shown in Fig. 10). An envelope connecting these sorted bearing capacity points in each direction reflects a

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probabilistic failure envelope with a certain probability of occurrence. For example, the envelope that connecting the group of the points of 1% smallest values indicates that there is

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only 1% probability that the spudcan will fail if the load combination is within this envelope.

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In another words, there is 99% reliability to ensure that it is safe if the load combination lies

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within this failure envelope. To obtain such a probabilistic envelope, an expression is proposed as:

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 V   H ran   M ran  H ran M ran f   ran    1  0      2e vV  h H h0 H det,0 m0 M det,0  0 det,0   0 det,0   m0 M det,0 

(2)

where v0, h0 and m0 affect the failure envelope size (i.e. v0 is the normalized pure vertical bearing capacity; h0 is the normalized pure horizontal bearing capacity; m0 is the normalized pure moment bearing capacity); e is a fitted parameter which determines the shape of the envelope. Examples of the fitted probabilistic failure envelopes against the original bearing capacity points are presented in Fig. 10. Results show that the proposed probabilistic failure envelope can fit the results very well. Meanwhile, the fitted failure envelope for the 10

ACCEPTED MANUSCRIPT deterministic case is also shown in Fig. 10, of which the best fitting parameter of v0, h0, m0 and e are 0.998, 1.042, 1.006 and 0.125 respectively. The values of v0, h0 and m0 are very close to 1, which indicates Eq. (2) can model the deterministic failure envelope well. Moreover, the failure envelope obtained from the deterministic case is very close to the envelope fitting the 50% smallest values. Therefore, there is 50% probability that the spudcan

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foundation will fail if the average undrained shear strength is adopted and the corresponding

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deterministic failure envelope is used in the design. This result was obtained from the cases

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where the typical COV of the undrained shear strength of clay (i.e., 0.3) was assigned. If the COV of the soil strength changed, this result should be re-evaluated.

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Table 4 summarizes values of v0, h0, m0 and e of the best fitted parameters to establish failure envelopes for a spudcan with different probability of occurrence. The fitted

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parameters of v0, h0 and m0 of the failure envelope are about 0.76 times that for the deterministic results if 95% probability is required to ensure the safety of the spudcan.

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Therefore, with any given confidence level, the parameters determining the probabilistic

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failure envelopes can be achieved by multiplying such a factor accordingly. These can be

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referred as different reliable levels in the offshore engineering design. It will help to provide a reliable guideline to the design of spudcan foundations embedded in heterogeneous soils.

4.4. Verification of the Probabilistic VHM Envelopes It is significant to prove that Eq. (2) can fit the probabilistic failure envelopes of the 400 random cases as well. Figure 11 demonstrates a more comprehensive comparison between the failure envelopes of Eq. (2) and all of the normalized results (i.e. Vran/Vran,0, Hran/Hran,0 and Mran/Mran,0). The results show that the shape of the normalized probabilistic failure envelopes can be fitted well by Eq. (2). Figure 12 presents the failure envelopes with different probability of occurrence in

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ACCEPTED MANUSCRIPT three-dimensions. These failure envelopes are compared with the single envelope obtained from the traditional deterministic analysis. Results show that the deterministic failure envelope is much larger than the envelope within which the spudcan is safe with 90% probability. The failure envelope shrinks with the increase in the confidence of the safety of the spudcan. The results indicate that the failure envelope should be much smaller than that

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obtained from the deterministic analysis if a higher level of confidence on the failure

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envelope is required.

5. Summary and conclusions

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This study investigated the failure mechanism and failure envelopes of a spudcan foundation embedded in spatially variable soils under combined loading. The spatial

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variability of the soil was modelled by three-dimensional random fields. The failure envelopes were investigated by applying combined vertical (V), horizontal (H) and moment

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(M) loads. By expressing the VHM failure envelopes within a probability framework, failure

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envelopes with different probability of occurrence were defined. An analytical expression

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that can describe the probabilistic failure envelopes was proposed. The following conclusions can be drawn:

(1) The spatial variability in soils can reduce the bearing capacity of a spudcan foundation. About 70% of the cases in spatially variable soils have smaller vertical bearing capacity than in the uniform soils. Regarding the bearing capacity in horizontal direction and moment, about 61% of the random cases are smaller than the deterministic results. The soil flow mechanism of failure becomes unsymmetrical and varies with the spatial pattern of the soil strength. (2) There are 50% probability that the failure envelopes for the spudcan in spatially varying soils are smaller than the failure envelopes for the spudcan in the assumed uiform

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ACCEPTED MANUSCRIPT soils with mean shear strength. (3) For the design of the spudcan foundation, with any given confidence level, the parameters determining the failure envelopes of the spudcan can be achieved by multiplying a factor accordingly. This approach can help to provide guideline to the design of spudcan

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foundations considering the spatial variability of soils.

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Acknowledgments

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The authors would like to acknowledge the support of Natural Science Foundation of China (Grant nos: 51379053, 51422905 and 51679060) and Shenzhen Science and

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Technology Commission (Grant No. CXZZ20151117174345411). . The authors would like to thank Dr. Youhu Zhang from NGI for his valuable discussion. This study represents part of

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the activities of the Centre for Offshore Foundation Systems (COFS) in the University of Western Australia and the Australian Research Council Centre of Excellence for

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Geotechnical Science and Engineering.

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deeply buried flat circular footings under general loading. J Geotech Geoenviron Eng 138(3), 385-397.

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Zhang, Y., Bienen, B., Cassidy, M.J., 2013. Development of a combined VHM loading

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apparatus for a geotechnical drum centrifuge. Int J Phys Modell Geotech 13(1), 13–30.

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Table 1 Summary of the sign conventions adopted in this paper. Table 2 Comparison of the uniaxial bearing capacity factors in a deterministic soil profile. Table 3 Fitting parameters of the normal distribution and lognormal distribution for normalized bearing capacity.

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Table 4 Summary of the fitting parameters defining the failure envelopes. List of Figures

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Fig. 1. Illustration of the jack-up platform in heterogeneous soil subjected to combined loading (red: stronger soil; blue: weaker soil).

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Fig. 2. A three-dimensional random field realization for su: (a) cylindrical finite-element model; (b) vertical slice; and (c) horizontal slice.

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Fig. 3. Spudcan geometry, loads and displacements convention. Fig. 4. Geometry and mesh of the finite element model.

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Fig. 5. Ratios of the displacement paths in probe tests.

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Fig. 6. Failure envelopes of spudcan in soils with linearly increasing shear strength for (a) VH (M = 0); (b) VM (H = 0); and (c) HM (V = 0) plane.

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Fig. 7. Comparison of soil flow mechanisms and su distribution: (a) soil flow in uniform soil; (b) su distribution in the uniform soil; (c) soil flow in a spatially varying soil; and (d) su distribution the spatially varying soil. (red indicates stronger soil; blue indicates weaker soil) Fig. 8. Histograms and probability density functions of normalised pure uniaxial capacities: (a) Vran/Vdet,0; (b) Hran/Hdet,0; and (c) Mran/Mdet,0. Fig. 9. Failure envelopes for the 400 random fields and the uniform soil. Fig. 10. Failure envelopes at different probability levels. Fig. 11. Verification of the probabilistic failure envelope. Fig. 12. Comparison of the fitted three-dimensional failure envelopes at different probability of occurrence with the deterministic case.

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Wind

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Horizontal loads

Vertical loads

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Seabed soil

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Fig. 1. Illustration of the jack-up platform in heterogeneous soil subjected to combined loading (red: stronger soil; blue: weaker soil).

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(a)

(c)

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(b)

su (kPa) 150 120 90 60 30 0

Fig 2. A three-dimensional random field realization for su: (a) cylindrical finite-element model; (b) vertical slice; and (c) horizontal slice.

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2.16 m

0.66 m

7.2 m

4.38 m

2.82 m

Reference point

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1.88 m

D=18 D =18 m

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w

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Fig. 3. Spudcan geometry, loads and displacements convention.

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Fig. 4. Geometry and mesh of the finite element model.

Fig. 5. Ratios of the displacement paths in probe tests.

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H/Asu

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Probe test path Envelope Spudcan foundation (Zhang 2011) Circular footing T/D=0.05 (Zhang 2012)

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Fig. 6. Failure envelopes of spudcan in soils with linearly increasing shear strength for (a) VH (M = 0); (b) VM (H = 0); and (c) HM (V = 0) plane.

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(c)

(d)

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Fig. 7. Comparison of soil flow mechanisms and su distribution: (a) soil flow in uniform soil; (b) su distribution in the uniform soil; (c) soil flow in a spatially varying soil; and (d) su distribution the spatially varying soil. (red indicates stronger soil; blue indicates weaker soil)

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0.08 20

Histogram Normal distribution Lognormal distribution

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1 1.2 Vran/Vdet,0

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(c)

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Fig. 8. Histograms and probability density functions of normalised pure uniaxial capacities: (a) Vran/Vdet,0; (b) Hran/Hdet,0; and (c) Mran/Mdet,0.

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Deterministic uniform soil case Random spatially varing soil

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Fig. 9. Failure envelopes for the 400 random fields and the uniform soil.

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Mran/Mdet,0

Vran/Vdet,0

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99% Fitted envelope 1% Smallest capacity points 95% Fitted envelope 5% Smallest capacity points 90% Fitted envelope 10% Smallest capacity points 50% Fitted envelope 50% Smallest capacity points Fitted deterministic envelope

Hran/Hdet,0

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1 Bearing capacity points normalized onto one curve Fit of Eq.(2)

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Fig. 11. Verification of the probabilistic failure envelope.

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Deterministic failure envelope 90% 99%

Fig. 12. Comparison of the fitted three-dimensional failure envelopes at different probability of occurrence with the deterministic case.

ACCEPTED MANUSCRIPT Table 1. Summary of the sign conventions adopted in this paper. Horizontal H u

Rotational M β

Vdet,0

Hdet,0

Mdet,0

Vran,0

Hran,0

Mran,0

Ncv = V0/(Asu) Vran/Vdet,0

Nch = H0/(Asu) Hran/Hdet,0

Ncm = M0/(ADsu) Mran/Mdet,0

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Load Displacement Uniaxial bearing capacity (deterministic case) Uniaxial bearing capacity (random soil) Bearing capacity factor Normalized load

Vertical V w

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Note: A = πD2/4 = bearing area of the spudcan foundation; su is the mean value of soil undrained shear strength at the reference point.

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Table 2. Comparison of the uniaxial bearing capacity factors in a deterministic soil profile. Nch

Ncm

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Hossain and Randolph (2009), numerical analysis 13.10

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Martin and Randolph (2001), theoretical solution (circular footing)

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Zhang et al. (2012), numerical analysis (circular footing)

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3.51

1.63

Zhang et al. (2011), numerical analysis

12.96

4.88

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13.12

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1.65

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Ncv

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This study, numerical analysis

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ACCEPTED MANUSCRIPT Table 3. Fitting parameters of the normal distribution and lognormal distribution for normalized bearing capacity Normalized Probability bearing density Expression Parameters capacity function

 x   2   exp   2  2  2   





 ln x   ln 2   exp   2  2  2  ln2 x ln  

ln

ln

1

f ( x) 

x=Vran/Vdet,0

f ( x) 

Normal distribution



  y   2   exp   2  2  2   

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 ln y   ln 2   exp   2  2  2  ln2 y ln  

ln -

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f ( y) 

Normal distribution

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Lognormal distribution

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f ( z) 

z=Mran/Mdet,0 Lognormal distribution

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0.167

 ln z   ln 2   exp   2 2  2  2  ln z ln  

ln 0.031

ln

0.135

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0.971 0.131

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f ( z) 

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Cases

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Table 4. Summary of the fitting parameters defining the failure envelopes. Probability of occurrence

v0

h0

m0

e

Deterministic case 50%

0.998

1.042

1.006

0.125

1.00

1.00

1.00

0.941

1.002

0.966

0.125

0.94

0.96

0.96

90%

0.811

0.838

0.816

0.125

0.81

0.80

0.81

95%

0.765

0.783

0.776

0.125

0.77

0.75

0.77

99%

0.663

0.706

0.692

0.125

0.66

0.68

0.69

Random soil

v0/v0,det h0/h0,det m0/m0,det

Note: The Subscript 0,det mean fitted parameters of v0, h0 and m0 from the finite element analysis of a deterministic case (i.e. v0,det = 0.998, h0,det = 1.042, m0,det = 1.006).

ACCEPTED MANUSCRIPT Highlights

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

How spatial variability affects the bearing capacity under combined loadings is studied. 3D random fields are combined with a finite element analysis to model spatially varying soils. The failure envelopes with various probability of occurrence are constructed for spudcan in random soils. There are 50% probability the failure envelopes in random soils are smaller than that in uniform soil.

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  