Effect of Spatial Variability on Buried Footings

Effect of Spatial Variability on Buried Footings

Centre for Offshore Foundation Systems, and ARC CoE for Geotechnical Science and Engineering, WA, Australia

Abstract. Natural soil is one of the most variable materials as a result of physical and chemical changes. The variation in soil properties significantly affects the failure mechanism and bearing capacity of a footing. This effect is further complicated by considering the spatial correlation of the soil properties. The purpose of this study is to investigate the influence of the spatial variability of a clay on the bearing capacity of a buried footing. The spatial variation of undrained shear strength of a clay is modelled by random fields. The random fields are generated and mapped into a non-linear finite element analysis. The failure mechanism and bearing capacity of a footing in a spatially random soil can then be revealed. Finally a Monte Carlo simulation is employed to investigate the effect of spatial variability on the buried footing. The random fields with different scales of fluctuation are modelled to reveal the effects of spatial variability. It is found that the mean values of bearing capacity of the foundation are the smallest when the scale of fluctuation is 0.1-1 times of foundation width. The coefficient of variation of the bearing capacity increases with increasing scale of fluctuation. When the scale of fluctuation is on the order of the foundation width the variation in bearing capacity is mainly due to the different shear paths developed in the random soil. When the scale of fluctuation is much larger than the foundation width, the variation in bearing capacity is mainly due to the distinct soil strength in different realizations.

Keywords. footing, bearing capacity, probability, spatial variability

1. Introduction

Soils are among the most variable materials as a result of complex physical, chemical and biological process during earth evolution. A mean value and a coefficient of variation (COV) are often used to describe the variability of soil properties. These statistical parameters, however, are obtained without considering the correlations among the soil properties at different locations. The distribution of soil properties can be very different for soils with the same mean value and COV if the correlations among soils at different locations are considered. Many researchers have reported that the spatial correlation of undrained shear strength affects the bearing capacity of a surface footing on undrained clay (Griffiths and Fenton 2001; Popsecu et al. 2005; Cho and Park 2010; Kassama and Whittle 2011; Cassidy et al. 2013). The mechanism on how the spatial variability affects the bearing capacity, however, is still unclear.

This study will investigate the influence of spatial variability on the bearing capacity and failure mechanism of buried footings. Random field theory is used to generate random fields of undrained shear strength according to different

spatial correlation of the clay. These random fields are mapped into finite elements and used to investigate the bearing capacity of a buried footing.

2. Spatial variability of undrained clay

The undrained shear strength of a clay is considered as a log-normally distributed random variable. The undrained shear strength at a location, su(xi, yi), can be separated into a trend, t(xi, yi) and a residual, r(xi, yi), ) , ( ) , ( ) , ( i i i i i i u x y t x y r x y s  (1)

The trend is characterized by a deterministic equation that represents the mean value. The residual is the variations about the trend. The residuals are characterized statistically as a random variable, usually with zero mean and non-zero variance (Beacher and Christian 2003). The correlation among the residuals at different locations is described by an autocorrelation function with parameter of scale of fluctuation. Within the scale of fluctuation (T) the undrained

© 2015 The authors and IOS Press. This article is published online with Open Access by IOS Press and distributed under the terms of the Creative Commons Attribution Non-Commercial License. doi:10.3233/978-1-61499-580-7-828

shear strength values at different locations are closely related with each other.

Figure 1 shows 4 realizations of random fields of undrained shear strength with Gaussian autocorrelation model. The random fields are generated using the local average subdivision algorithm (Fenton and Vanmarcke 1990). The scale of fluctuation values of Figures 1(a) – 1(d) are 2 m, 20 m, 200 m, and 2000 m, which covers the typical range of scale of fluctuation (Lacasse & Nadim 1996; Phoon and Kulhawy 1999; Uzielli et al. 2005; Cheon and Gilbert 2014; Li et al. 2014; Li et al. 2015b). The mean values of these figures are all 10 kPa and COVs are 0.3. Although the mean value and COV are the same, the distributions of the random fields are very different due to the variation in correlations of the undrained shear strength.

3. Finite Element Model

The non-linear finite element method is used to analyze the bearing capacity problem of a footing (with width B of 20 m and height h of 4 m). This footing is embedded in a depth of 4B (i.e. 80 m) to simulate embedded geotechnical structures such as spudcan in offshore engineering. In each simulation, the foundation is displaced at the foundation reference point in the vertical direction until a failure load is attained.

Soil failure is defined according to the Tresca criterion, with the maximum shear stresses in any plane limited to the undrained shear strength (su). The Young’s modulus E is perfectly correlated to the undrained shear strength with a ratio E/su=500 (Hu and Randolph 1998). The Poisson’s ratio is set as 0.49 to model the undrained conditions of no volume changes as well as to ensure numerical stability.

The soil domain has a width of 11.2B and a height of 8B, which is large enough to ensure there are no obvious boundary effects (as shown in Figure 2). The soil domain is discretized into many small zones of size 2.0 x 0.5 m. A generated random field was then mapped into these small zones. The undrained shear strength varies from zone to zone to reflect the spatial variation of the soil. For the majority of the soil domain, a zone is represented by one finite

element. However, in a region of size 6.6B by 8B close to the footing (as bounded by the bold lines (a) T = 2 m

(b) T = 20 m

(c) T = 200 m

(d) T = 2000 m

Figure 1. Generated random fields (640 x 360 m) with different scale of fluctuations (Red colour indicates strong soil and blue colour indicate weak soil).

in Figure 2), a zone is further discretized into four finite elements of 0.5 x 0.5 m to improve the numerical accuracy of the simulation (Cassidy et al. 2013).

Five scales of fluctuation values, 0.01B, 0.1B, B, 10B, and 100B (i.e. 0.2 m, 2m, 20 m, 200 m and 2000 m) are modeled to investigate the influence of spatial variability. For each scale of fluctuation, 400 realizations of random fields are generated and mapped into the finite element simulations. Hence, 400 simulations are conducted with each of them resulting in a bearing capacity and shear plane.

Figure 2. Finite element model.

4. Results

The bearing capacity factor for each simulation, Nci, can be defined as,

su fi ci q N

P

When the scale of fluctuation is B (i.e. 20 m), 400 bearing capacity factors are obtained from the 400 simulations. The histogram of the bearing capacity factor is plotted in Figure 3. A normal distribution and a lognormal distribution are used to fit the probability distribution of the bearing capacity factor. Visually they both fit the data well. A chi-square test (Ang and Tang 2007) is used to select the best. Results show that a

normal distribution fit is superior to the lognormal distribution fit.

The normal distribution is then used to fit the bearing capacity factor for the cases with scale of fluctuation of 0.01B, 0.1B, B, 10B, and 100B (as shown in Figure 4). The bearing capacity factor for the case with uniform soils (with undrained shear strength of 10 kPa) is 12.36, which is also demonstrated in Figure 4. Results show that the bearing capacity factors of the majority random cases are smaller than the uniform case. The mean for cases with different scales of fluctuation are listed in Table 1.

Figure 3. Histogram and fitted probability distribution for the cases with scale of fluctuation of 20 m.

Figure 4. Fitted normal distribution for the cases with different scales of fluctuation.

The mean bearing capacity factor is the smallest when the scale of fluctuation is 0.1B. As

0 0,2 0,4 0,6 0,8 1 8 10 12 14 16 18 20 22 24 P ro b abil ity de ns ity

Bearing capacity factor T = 0.01B T = 0.1B T = B T= 10B T = 100B in uniform soil Nc =12.36 11.2B 8B B 4B 6.6B h

the scale of fluctuation goes smaller or larger, the mean bearing capacity decreases. The mean bearing capacity factor for the case with scale of fluctuation of 100B is the same as that of the uniform case. This is due to the large value of the scale of fluctuation which results in a more uniform soil. Table 1 also lists the coefficient of variation for the cases with different scales of fluctuation. The COV of the bearing capacity increases with increasing scale of fluctuation. This is consistent with the probability distribution in Figure 4. The variation in bearing capacity factor is caused by the spatial variability of the soils. Figures 5-7 illustrate how the spatial variability influences the bearing capacity.

Table 1. Mean value and COV for the random cases with different scales of fluctuation

Scale of fluctuation Mean COV

0.01B 11.78 0.002

0.1B 11.27 0.014

B 11.39 0.099

10B 12.14 0.314

100B 12.36 0.315

Figure 5 shows the shear paths for the cases with the minimum bearing capacity (Figure 5(a)), average bearing capacity (Figure 5(b)), and maximum bearing capacity (Figure 5(c)) as the scale of fluctuation is 0.1B (i.e. 2 m). The random fields in which the shear paths develop are also shown in Figure 5. The three realizations of random fields are similar. Although the shear paths are unsymmetrical and the patterns are different for different random fields, their sizes are similar. The length of the shear path is closely related with the bearing capacity (Li et al. 2015a). This similar size of shear path results in similar bearing capacity for different cases with the scale of fluctuation of 0.1B. Hence, the COV of the bearing capacity is very small (i.e. 0.002) for random soils with small scale of fluctuation.

Figure 6 shows the random fields and the corresponding shear path for the case with minimum, average and maximum bearing capacity factors as the scale of fluctuation is B (i.e. 20 m). In Figure 6(a), the footing is sitting on soils with small undrained shear strength (blue region). The shear path is developed in the surrounding weak soils. The small undrained shear strength and short shear path jointly lead to

a small bearing capacity in this case (i.e. 8.05). In Figure 6(c), the footing is sitting on strong soils (red region). The shear path tends to avoid the strong soils and develops along the weak soils which are far away from the footing. Hence the shear path is longer when compared with that in

(a) Case with minimum bearing capacity (Nc=10.82)

(b) Case with average bearing capacity (Nc=11.27)

(c) Case with maximum bearing capacity (Nc=11.82)

Figure 5. The Shear path for foundations in soils with scale of fluctuation of 0.1B.

Figure 6(a) and 6(b), which lead to more soil mobilized and a larger bearing capacity (i.e. 15.33). The COV of the bearing capacity is larger due to the distinct pattern in the developed shear path.

Figure 7 shows the shear path development in three random fields which has scale of fluctuation of 10B (i.e. 200 m). The undrained (a) Case with minimum bearing capacity (Nc=8.05)

(b) Case with average bearing capacity (Nc=11.39)

(c) Case with maximum bearing capacity (Nc=15.33)

Figure 6. The Shear path for foundations in soils with scale of fluctuation of B.

(a) Case with minimum bearing capacity (Nc=8.05)

(b) Case with average bearing capacity (Nc=12.14)

(c) Case with maximum bearing capacity (Nc=28.20)

Figure 7. The Shear path for foundations in soils with scale of fluctuation of 10B.

shear strengths are relatively uniform in each of the realizations due to the large scale of fluctuation. The sizes of shear path are similar in the three random fields. The difference in undrained shear strength between different realizations is due to the variations in soils. When the foundation is sitting in a weaker soil

the bearing capacity is smaller than that in a stronger soil. The bearing capacity factors for the 400 realizations range from 5.62 to 28.2. The COV of the bearing capacity is much larger under this circumstance. The large variation in bearing capacity factors are mainly caused by the variation in undrained shear strength.

5. Conclusions

A random finite element method was presented to investigate the effect of spatial variability on buried footings. It is found that the average bearing capacities of the foundation in random soils are smaller than that in uniform soils. This is because the shear path tends to pass the weakest soils which cost the least energy. The mean bearing capacity is the smallest when the scale of fluctuation is about 0.1-1 times of the foundation width. The coefficient of variation of the bearing capacity increases with increasing scale of fluctuation. When the scale of fluctuation is much smaller than the foundation width the resulting shear path is similar in different realizations, which leads to similar bearing capacity factors. When the scale of fluctuation is on the order of the foundation width the variation in bearing capacity is mainly due to the different shear paths developed in the random soil, which, in turn, closely relate to the pattern of the soil. When the scale of fluctuation is much larger than the foundation width, the variation in bearing capacity is mainly due to the distinct soil strength in different realizations. It is recommended that a larger margin of safety should be used if the scale of fluctuation of soils is on the same order of the foundation width.

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