Effects of Soil Spatial Variability on Bearing Capacity of Shallow Foundations

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(1)Geotechnical Safety and Risk V T. Schweckendiek et al. (Eds.) © 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:10.3233/978-1-61499-580-7-371. 371. Effects of Soil Spatial Variability on Bearing Capacity of Shallow Foundations K.E. DARYANI, H. MOHAMAD Faculty of Civil Engineering, Universiti Teknologi Malaysia, Malaysia. Abstract. This paper studies the effect of soil spatial variability on bearing capacity of shallow foundations. The stochastic soil property is the undrained shear strength considered as a non-Gaussian random variable. The random variable is simulated using Local Average Subdivision Method. A Monte Carlo simulation approach is followed in combination with non-linear finite difference analysis to study the realistic behaviour of the foundations. It is demonstrated that the inherent spatial variability of soil shear strength can drastically modify the basic form of the failure mechanism in this bearing capacity problem. This behavior of the failure mechanism translates into a substantial reduction in the ultimate bearing capacity (in an average sense), compared to the corresponding deterministic (homogeneous soil) case. In addition, differential settlements are computed in the stochastic analysis, something impossible in a deterministic analysis of a symmetric problem. A parametric study is performed to investigate the effects of various probabilistic parameters involved in the problem. It is found that the coefficient of variation and the correlation length of the soil’s shear strength (both controlling the amount of loose pockets in the soil mass) are the two most important parameters in variation of the bearing capacity and producing substantial differential settlements in heterogeneous soils compared to homogeneous soils. Keywords. Soil spatial variability, Stochastic fields, Monte Carlo Simulation, Bearing capacity, Finite Difference method. 1. Introduction Foundations of engineering structures are designed to transfer and distribute their loading to the underlying soil and/or rock. This design is required to satisfy two main design criteria, namely bearing capacity (i.e. strength) and the total and differential settlements (i.e. serviceability) considering economic feasibility of the foundation. Bearing capacity failure occurs as the soil supporting the foundation fails in shear, which may involve either a general, local or punching shear failure mechanism (Bowles 1988). For these different failure mechanisms, different methods of analysis are used. Estimation and prediction of the actual bearing capacity of the foundations is the most significant and complicated problems in geotechnical engineering. Consequently, there is extensive literature detailing both theoretical and experimental studies associated with this issue. A list of principal contributions to the study of this subject may be found, for example, in Terzaghi (1943), Hansen (1970), Vesic (1974), Meyerhof (1951) and, Chen and McCarron (1991). The focuses of these studies were on the estimation of. the ultimate bearing capacity of the foundations under the combination of vertical, horizontal and moment loading, as well as the effect of foundation shape, soil rigidity, load inclination, tilt of the foundation base, ground surface inclination and the depth of the foundation. Moreover, advanced numerical methods such as finite element and finite difference methods are developed recently, which can be used to simulate the bearing capacity of foundations in complex situations. However, the above-mentioned methods are based on the assumption of uniformity; that is, the properties of the soil are assumed to be homogeneous. In nature, soils are often inherently anisotropic due to the manner in which they are deposited. It has been observed that the performance of foundations is considerably affected by the inherent spatial variability of the soil properties (Fenton and Griffiths 2003). In addition to the inherent variability, there are various sources of uncertainties which may arise due to the lack of our information in estimation of underground strength parameters, test and transformation errors, model and analysis assumptions, social and environmental.

(2) 372. K.E. Daryani and H. Mohamad / Effects of Soil Spatial Variability on Bearing Capacity of Shallow Foundations. variability (Baecher and Christian 2005). A category of the associated uncertainties is demonstrated in Figure 1. Conventionally, the effects of the uncertainties are considered in the design using a ‘Factor of safety’. However, the values recommended for the factor of safety are based on experience that may not lead to a safe design (Duncan 2000). Recently, reliability-based design methods are introduced in geotechnical engineering to directly simulate degree of the soil variability using probabilistic methods, and to estimate the ‘probability of failure’ of the system.. Figure 1. Uncertainty in soil property estimates (Phoon and Kulhawy 1999).. 2. Random field simulation In a probabilistic analysis the geotechnical parameters, which represent major sources of uncertainties, are treated as random variables. A random variable is a mathematical function defined on a sample space that assigns a probability to each possible event within the sample space. In practical terms, it is a variable for which the precise value (or range of values) cannot be predicted with certainty, but only with an associated probability, which the possible outcome a particular experiment in terms of real numbers. The most important statistical parameters related to the soil variability are the mean value, the standard of variation, cross-correlation coefficient between parameters, autocorrelation lengths (or scale of fluctuation), which describes the spatial variability in both horizontal and vertical directions. The possible range of these. parameters is reported in the literature for different soil properties (i.e., Phoon and Kulhawy 1996, Lacasse and Nadim 1997, Schneider 1999). From the wide variety of methods developed, the most often used in the geotechnical applications are (i) Karhunen-Leove Expansion (Spanos and Ghanem 1989), (ii) Local Average Subdivision (Fenton and Vanmarcke 1990) methods. Various techniques of random field simulation have been presented in the literature. A detailed review of the existing techniques was presented by Stefanou (2009). In addition to these methods, there also exist some other efficient methods for the simulation of random fields, such as Fast Fourier Transformation (FFT), Turning Band Method (TBM), Orthogonal Series Expansion (OSE), and Expansion Optimal linear Estimation (EOLE) methods (Fenton and Griffiths 2008, Phoon 2008). Local Average Subdivision (LAS) technique was developed by Fenton and Vanmarcke (1990) in order to produce data with specified statistical parameters, which are simultaneously spatially correlated. As shown schematically in Figure 2, LAS theory follows a recursive fashion, where a global average is created in stage zero. In stage one, the field is divided into two equal parts whose local average equals to the parent global value. In stage two, two absolute normally disturbed values are generated whose means and variances are selected so as to satisfy three criteria: (a) they show the correct variance according to local averaging theory; (b) they are properly correlated with one another; (c) they average to the parent value and so on in this fashion. In this study, the soil undrained shear strength, su, is assumed to be log-normally distributed with mean su, standard deviation su, and spatial correlation length lnsu. The lognormal distribution is selected because it produces non-negative values. A log-normally distributed random field is obtained from a normal random filed Hlnsu, having zero mean, unit variance, and spatial correlation length lnsu through the transformation (1).

(3) K.E. Daryani and H. Mohamad / Effects of Soil Spatial Variability on Bearing Capacity of Shallow Foundations. where is the spatial position at which is desired. The parameters lnsu and lnsu can be obtained from the specified su and su using the lognormal distribution transformations (2). (3). 373. strip footing is simulated by an elastic material. The footing elastic properties used are the Young’s modulus E=28 GPa and the Poisson’s compared to the soil elastic ratio ). The footing is properties (E=150 Mpa, connected to the soil grids via interface elements using Coulomb law in FLAC, which allows slip and separation and fully soil-structure interaction. Figure 3 shows a schematic illustration of the model.. The correlation between the su at a point x1 and x2 is specified by a correlation function, lnsu as (4). where. is the absolute distance. between the two points. Figure 3. Finite difference model of strip footing. Figure 2. Local average random process approach.. When the correlation length is small, the field tends to be somewhat ‘rough’. In the limit, , all points within the field become when uncorrelated, which is physically unrealizable. Such a field is called ‘white noise’. Conversely, when the correlation length becomes large ), the field becomes smoother. (. 3. Finite difference analysis of a foundation The deterministic finite difference model in conjunction with Monte Carlo Simulation (MCS) is used to calculate the ultimate foundation load, the settlement and the foundation slope using software FLAC 2D V7.0 (Itasca 2005). A conventional elastic-perfectly plastic model based on the Mohr-Coulomb failure creation is used to represent the soil behaviour. A. For the computation of the bearing capacity of the footing subjected to a central vertical load using FLAC, the following method is adopted: an optimal controlled downward vertical velocity of m/timestep (i.e., displacement per timestep) is applied to the central node of the footing. Damping of the system is introduced by running several cycles until a steady state of plastic flow is developed in the soil underneath the footing. At each cycle, the vertical footing load is obtained by using a FISH (Programming language in FLAC) function that calculates the integral of the normal stress components for all elements in contact with the footing. The value of the vertical footing load at the plastic steady state is the ultimate footing load.. 4. Reliability analysis using Monte Carlo Simulation A Monte Carlo simulation methodology combining generation of stochastic fields with finite difference analyses was employed. MATLAB (MATLAB 2012) were used to.

(4) 374. K.E. Daryani and H. Mohamad / Effects of Soil Spatial Variability on Bearing Capacity of Shallow Foundations. develop and automate the stochastic analysis and Monte Carlo Simulations. One hundred samples were used for each analysed case. The effects of sample size on the predicted mean and COV was investigated. The predicted mean and variance remain practically constant for sample sizes larger than 100. It is assumed that the shallow foundations are usually placed on medium to stiff clays, a range of Coefficient of Variation, COV = 10% to 40%, was selected for soil variability beneath the foundation. This was based on finding in the literature review (Phoon and Kulhawy 1996). In addition to COV, a separate correlation structure with ranges of autocorrelation length h/B=1.25 to 4.0 and v/B=0.25 to 1.0 were considered in the parametric study, where B is the width of the foundation, h and v are correlation distances in the horizontal and vertical directions, respectively. 4.1. Results of stochastic finite difference model Figure 4(c) shows the results of a finite difference analysis with spatially variable soil properties in terms of plastic shear strains. Figure 4(a) shows the point variability of shear strength over the domain of interest. This point variability was mapped to the finite difference mesh using the LAS method. Figure 4(c) shows how a local shear failure develops below the foundation at a much lower bearing pressure than the general symmetric failure in Figure 4(b). Subsequently, increasing the imposed foundation settlement caused another asymmetric general shear failure to develop. It should be mentioned that both failure surfaces were developed mainly through the loose pockets of soil, indicated by darker patches in the figure. (a). (b). (c). Figure 4. A finite difference analysis with spatially variable soil strength (a) realization of undrained shear strength (b) Contours of plastic shear strain in deterministic model (c) Contours of plastic shear strain in stochastic model. 4.2. Monte Carlo Simulation results Results of a study with COV=40% for undrained shear strength and correlation distances, h/B=1.25 and v/B=0.25 are summarized in Figure 5(a). This figure presents the results of MCS for 100 sample functions representing possible realizations by thin lines in terms of normalized pressure vs. normalized settlement. A similar curve resulting from a deterministic analysis is presented by a thick dashed line. The MCS accounting for spatial variability of soil strength, yielded bearing capacity values that were generally lower than those predicted by the deterministic analysis. Moreover, the MCS resulted in significant rotation of the foundation about the center. These rotations may become the main criterion for the foundation design. Figure 5(b) shows the normalized pressure versus rotation relations predicted by MCS. Next, the average and 95-percentile of these results were calculated at each displacement or foundation slope value (Figure 6), considering a value n=0.0125 as the reference settlement corresponding to ultimate bearing capacity. The average bearing capacity resulting from MCS was 25% lower than that predicted by the.

(5) K.E. Daryani and H. Mohamad / Effects of Soil Spatial Variability on Bearing Capacity of Shallow Foundations. deterministic analysis. Furthermore, the 95percentile value of bearing capacity resulting from MCS exceeds by 38% the bearing capacity resulting from deterministic analysis using the 95-percentile of soil shear strength. (a). 375. It was observed from literature (e.g., Popescu, Prevost et al. 1998) that the degree of variability (COV) of soil strength is the most important factor affecting soil behaviour. A parametric study was performed to investigate its effects on bearing capacity. MCS were performed for five different COV=10%, 20%, …, 50% using 100 sample functions for each set of MCS. The results are presented on Figure 7. As the COV of shear strength increases, the average predicted bearing capacity decreases. This again emphasizes the effects of loose pockets in the soil mass. Also, it was observed in all cases that the scatter in the predicted bearing capacity was significantly lower than those assumed for the shear strength.. (b). Figure 7. Influence of the COV of the soil strength on the bearing capacity. Figure 5. Comparison of MCS and deterministic analysis results: (a) pressure-settlement curves, (b) pressure-rotation curves (no rotation is predicted in the deterministic analysis). The study was further developed to investigate the effects of correlation distances. The ratio of horizontal correlation distance to foundation width can take a large range of values. Here a range of 1.25 to 4.0 was investigated. The results of MCS for COV=40% are presented in Table 1. The mean bearing capacity ratio, RnBC was defined as the ratio of the mean bearing capacity of heterogeneous soil to that of uniform soil having the same average shear strength. It can be seen in Table 1 that both the mean and COV of the predicted bearing capacity increases with increasing horizontal correlation distance. Table 1. Results of MCS for the effects of horizontal correlation distance on predicted bearing capacity.. Figure 6. Comparison of deterministic and MCS results for average and 95-percentile.. 1.25 2.0 4.0 . Mean bearing capacity ratio (RnBC). COV of bearing capacity (%). 0.81 0.82 0.84 1.0. 13.1 14.9 17.3 40.

(6) 376. K.E. Daryani and H. Mohamad / Effects of Soil Spatial Variability on Bearing Capacity of Shallow Foundations. 5. Conclusion The effects of soil heterogeneity on the bearing capacity of strip foundations were examined under undrained conditions using the Monte Carlo technique including digital generation of non-Gaussian random fields in conjunction with the finite difference analyses. The following conclusions are derived. Behaviour of the soil and soil-structure systems is strongly affected by the natural spatial variability of soil strength within geologically distinct and uniform layers. Increasing soil variability and the amount of loose pockets in the soil mass strongly diminished bearing capacity of the soil and increased differential settlements. The average bearing capacity of heterogeneous soil resulted in consistently lower values than those predicted assuming uniform soil strength. Moreover, the predicted bearing capacity had a lower variability than that of the soil shear strength. Also, the horizontal correlation distance of soil shear strength affects the variability of bearing capacity. By increasing the horizontal correlation length of the undrained shear strength of the soil, the variability of the bearing capacity is increased.. Acknowledgments. The first author wants to acknowledge the financial support provided by the Universiti Teknologi Malaysia, UTM (GUP Tier 1) to carry out the present work.. References Baecher, G. B., J. T. Christian (2005). Reliability and statistics in geotechnical engineering, John Wiley & Sons. Bowles, J. E. (1988). Foundation analysis and design.. Chen, W.-F. and W. O. McCarron (1991). Bearing capacity of shallow foundations, Springer. Duncan, J. M. (2000). Factors of safety and reliability in geotechnical engineering. Journal of geotechnical and geoenvironmental engineering 126(4): 307-316. Fenton, G. A., D. Griffiths (2003). Bearing-capacity

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