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 athors and IOS Press. This article is pblished online with Open Access by IOS Press and distribted nder the terms of the Creative Commons Attribtion Non-Commercial License. doi: / Effects of Soil Spatial Variability on Bearing Capacity of Shallow Fondations K.E. DARYANI, H. MOHAMAD Faclty of Civil Engineering, Universiti Teknologi Malaysia, Malaysia 371 Abstract. This paper stdies the effect of soil spatial variability on bearing capacity of shallow fondations. The stochastic soil property is the ndrained shear strength considered as a non-gassian random variable. The random variable is simlated sing Local Average Sbdivision Method. A Monte Carlo simlation approach is followed in combination with non-linear finite difference analysis to stdy the realistic behavior of the fondations. It is demonstrated that the inherent spatial variability of soil shear strength can drastically modify the basic form of the failre mechanism in this bearing capacity problem. This behavior of the failre mechanism translates into a sbstantial redction in the ltimate bearing capacity (in an average sense), compared to the corresponding deterministic (homogeneos soil) case. In addition, differential settlements are compted in the stochastic analysis, something impossible in a deterministic analysis of a symmetric problem. A parametric stdy is performed to investigate the effects of varios probabilistic parameters involved in the problem. It is fond that the coefficient of variation and the correlation length of the soil s shear strength (both controlling the amont of loose pockets in the soil mass) are the two most important parameters in variation of the bearing capacity and prodcing sbstantial differential settlements in heterogeneos soils compared to homogeneos soils. Keywords. Soil spatial variability, Stochastic fields, Monte Carlo Simlation, Bearing capacity, Finite Difference method 1. Introdction Fondations of engineering strctres are designed to transfer and distribte their loading to the nderlying soil and/or rock. This design is reqired 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 fondation. Bearing capacity failre occrs as the soil spporting the fondation fails in shear, which may involve either a general, local or pnching shear failre mechanism (Bowles 1988). For these different failre mechanisms, different methods of analysis are sed. Estimation and prediction of the actal bearing capacity of the fondations is the most significant and complicated problems in geotechnical engineering. Conseqently, there is extensive literatre detailing both theoretical and experimental stdies associated with this isse. A list of principal contribtions to the stdy of this sbject may be fond, for example, in Terzaghi (1943), Hansen (1970), Vesic (1974), Meyerhof (1951) and, Chen and McCarron (1991). The focses of these stdies were on the estimation of the ltimate bearing capacity of the fondations nder the combination of vertical, horizontal and moment loading, as well as the effect of fondation shape, soil rigidity, load inclination, tilt of the fondation base, grond srface inclination and the depth of the fondation. Moreover, advanced nmerical methods sch as finite element and finite difference methods are developed recently, which can be sed to simlate the bearing capacity of fondations in complex sitations. However, the above-mentioned methods are based on the assmption of niformity; that is, the properties of the soil are assmed to be homogeneos. In natre, soils are often inherently anisotropic de to the manner in which they are deposited. It has been observed that the performance of fondations is considerably affected by the inherent spatial variability of the soil properties (Fenton and Griffiths 2003). In addition to the inherent variability, there are varios sorces of ncertainties which may arise de to the lack of or information in estimation of ndergrond strength parameters, test and transformation errors, model and analysis assmptions, social and environmental

2 372 K.E. Daryani and H. Mohamad / Effects of Soil Spatial Variability on Bearing Capacity of Shallow Fondations variability (Baecher and Christian 2005). A category of the associated ncertainties is demonstrated in Figre 1. Conventionally, the effects of the ncertainties are considered in the design sing a Factor of safety. However, the vales recommended for the factor of safety are based on experience that may not lead to a safe design (Dncan 2000). Recently, reliability-based design methods are introdced in geotechnical engineering to directly simlate degree of the soil variability sing probabilistic methods, and to estimate the probability of failre of the system. Figre 1. Uncertainty in soil property estimates (Phoon and Klhawy 1999). 2. Random field simlation In a probabilistic analysis the geotechnical parameters, which represent major sorces of ncertainties, are treated as random variables. A random variable is a mathematical fnction 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 vale (or range of vales) cannot be predicted with certainty, bt only with an associated probability, which the possible otcome a particlar experiment in terms of real nmbers. The most important statistical parameters related to the soil variability are the mean vale, the standard of variation, cross-correlation coefficient between parameters, atocorrelation lengths (or scale of flctation), which describes the spatial variability in both horizontal and vertical directions. The possible range of these parameters is reported in the literatre for different soil properties (i.e., Phoon and Klhawy 1996, Lacasse and Nadim 1997, Schneider 1999). From the wide variety of methods developed, the most often sed in the geotechnical applications are (i) Karhnen-Leove Expansion (Spanos and Ghanem 1989), (ii) Local Average Sbdivision (Fenton and Vanmarcke 1990) methods. Varios techniqes of random field simlation have been presented in the literatre. A detailed review of the existing techniqes was presented by Stefano (2009). In addition to these methods, there also exist some other efficient methods for the simlation of random fields, sch as Fast Forier Transformation (FFT), Trning Band Method (TBM), Orthogonal Series Expansion (OSE), and Expansion Optimal linear Estimation (EOLE) methods (Fenton and Griffiths 2008, Phoon 2008). Local Average Sbdivision (LAS) techniqe was developed by Fenton and Vanmarcke (1990) in order to prodce data with specified statistical parameters, which are simltaneosly spatially correlated. As shown schematically in Figre 2, LAS theory follows a recrsive fashion, where a global average is created in stage zero. In stage one, the field is divided into two eqal parts whose local average eqals to the parent global vale. In stage two, two absolte normally distrbed vales 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 vale and so on in this fashion. In this stdy, the soil ndrained shear strength, s, is assmed to be log-normally distribted with mean s, standard deviation s, and spatial correlation length. The lns lognormal distribtion is selected becase it prodces non-negative vales. A log-normally distribted random field is obtained from a normal random filed H lns, having zero mean, nit variance, and spatial correlation length lns throgh the transformation (1)

3 K.E. Daryani and H. Mohamad / Effects of Soil Spatial Variability on Bearing Capacity of Shallow Fondations 373 where is the spatial position at which is desired. The parameters and lns ln s can be obtained from the specified and s sing the s lognormal distribtion transformations (2) (3) strip footing is simlated by an elastic material. The footing elastic properties sed are the Yong s modls E=28 GPa and the Poisson s ratio compared to the soil elastic properties (E=150 Mpa, ). The footing is connected to the soil grids via interface elements sing Colomb law in FLAC, which allows slip and separation and flly soil-strctre interaction. Figre 3 shows a schematic illstration of the model. The correlation between the s at a point x 1 and x 2 is specified by a correlation fnction, lns as (4) where is the absolte distance between the two points. Figre 3. Finite difference model of strip footing Figre 2. Local average random process approach. When the correlation length is small, the field tends to be somewhat rogh. In the limit, when, all points within the field become ncorrelated, which is physically nrealizable. Sch a field is called white noise. Conversely, when the correlation length becomes large ( ), the field becomes smoother. 3. Finite difference analysis of a fondation The deterministic finite difference model in conjnction with Monte Carlo Simlation (MCS) is sed to calclate the ltimate fondation load, the settlement and the fondation slope sing software FLAC 2D V7.0 (Itasca 2005). A conventional elastic-perfectly plastic model based on the Mohr-Colomb failre creation is sed to represent the soil behavior. A For the comptation of the bearing capacity of the footing sbjected to a central vertical load sing 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 introdced by rnning several cycles ntil a steady state of plastic flow is developed in the soil nderneath the footing. At each cycle, the vertical footing load is obtained by sing a FISH (Programming langage in FLAC) fnction that calclates the integral of the normal stress components for all elements in contact with the footing. The vale of the vertical footing load at the plastic steady state is the ltimate footing load. 4. Reliability analysis sing Monte Carlo Simlation A Monte Carlo simlation methodology combining generation of stochastic fields with finite difference analyses was employed. MATLAB (MATLAB 2012) were sed to

4 374 K.E. Daryani and H. Mohamad / Effects of Soil Spatial Variability on Bearing Capacity of Shallow Fondations develop and atomate the stochastic analysis and Monte Carlo Simlations. One hndred samples were sed 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 assmed that the shallow fondations are sally placed on medim to stiff clays, a range of Coefficient of Variation, COV = 10% to 40%, was selected for soil variability beneath the fondation. This was based on finding in the literatre review (Phoon and Klhawy 1996). In addition to COV, a separate correlation strctre with ranges of atocorrelation length h /B=1.25 to 4.0 and v /B=0.25 to 1.0 were considered in the parametric stdy, where B is the width of the fondation, h and v are correlation distances in the horizontal and vertical directions, respectively Reslts of stochastic finite difference model Figre 4(c) shows the reslts of a finite difference analysis with spatially variable soil properties in terms of plastic shear strains. Figre 4(a) shows the point variability of shear strength over the domain of interest. This point variability was mapped to the finite difference mesh sing the LAS method. Figre 4(c) shows how a local shear failre develops below the fondation at a mch lower bearing pressre than the general symmetric failre in Figre 4(b). Sbseqently, increasing the imposed fondation settlement cased another asymmetric general shear failre to develop. It shold be mentioned that both failre srfaces were developed mainly throgh the loose pockets of soil, indicated by darker patches in the figre. (a) (b) (c) Figre 4. A finite difference analysis with spatially variable soil strength (a) realization of ndrained shear strength (b) Contors of plastic shear strain in deterministic model (c) Contors of plastic shear strain in stochastic model 4.2. Monte Carlo Simlation reslts Reslts of a stdy with COV=40% for ndrained shear strength and correlation distances, h /B=1.25 and v /B=0.25 are smmarized in Figre 5(a). This figre presents the reslts of MCS for 100 sample fnctions representing possible realizations by thin lines in terms of normalized pressre vs. normalized settlement. A similar crve reslting from a deterministic analysis is presented by a thick dashed line. The MCS acconting for spatial variability of soil strength, yielded bearing capacity vales that were generally lower than those predicted by the deterministic analysis. Moreover, the MCS reslted in significant rotation of the fondation abot the center. These rotations may become the main criterion for the fondation design. Figre 5(b) shows the normalized pressre verss rotation relations predicted by MCS. Next, the average and 95-percentile of these reslts were calclated at each displacement or fondation slope vale (Figre 6), considering a vale n = as the reference settlement corresponding to ltimate bearing capacity. The average bearing capacity reslting 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 Fondations 375 deterministic analysis. Frthermore, the 95- percentile vale of bearing capacity reslting from MCS exceeds by 38% the bearing capacity reslting from deterministic analysis sing the 95-percentile of soil shear strength. (a) It was observed from literatre (e.g., Popesc, Prevost et al. 1998) that the degree of variability (COV) of soil strength is the most important factor affecting soil behavior. A parametric stdy was performed to investigate its effects on bearing capacity. MCS were performed for five different COV=10%, 20%,, 50% sing 100 sample fnctions for each set of MCS. The reslts are presented on Figre 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 assmed for the shear strength. (b) Figre 5. Comparison of MCS and deterministic analysis reslts: (a) pressre-settlement crves, (b) pressre-rotation crves (no rotation is predicted in the deterministic analysis) Figre 6. Comparison of deterministic and MCS reslts for average and 95-percentile. Figre 7. Inflence of the COV of the soil strength on the bearing capacity. The stdy was frther developed to investigate the effects of correlation distances. The ratio of horizontal correlation distance to fondation width can take a large range of vales. Here a range of 1.25 to 4.0 was investigated. The reslts of MCS for COV=40% are presented in Table 1. The mean bearing capacity ratio, R nbc was defined as the ratio of the mean bearing capacity of heterogeneos soil to that of niform 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. Reslts of MCS for the effects of horizontal correlation distance on predicted bearing capacity. Mean bearing capacity ratio (R nbc) COV of bearing capacity (%)

6 376 K.E. Daryani and H. Mohamad / Effects of Soil Spatial Variability on Bearing Capacity of Shallow Fondations 5. Conclsion The effects of soil heterogeneity on the bearing capacity of strip fondations were examined nder ndrained conditions sing the Monte Carlo techniqe inclding digital generation of non-gassian random fields in conjnction with the finite difference analyses. The following conclsions are derived. Behavior of the soil and soil-strctre systems is strongly affected by the natral spatial variability of soil strength within geologically distinct and niform layers. Increasing soil variability and the amont of loose pockets in the soil mass strongly diminished bearing capacity of the soil and increased differential settlements. The average bearing capacity of heterogeneos soil reslted in consistently lower vales than those predicted assming niform 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 ndrained shear strength of the soil, the variability of the bearing capacity is increased. Acknowledgments The first athor wants to acknowledge the financial spport provided by the Universiti Teknologi Malaysia, UTM (GUP Tier 1) to carry ot the present work. References Baecher, G. B., J. T. Christian (2005). Reliability and statistics in geotechnical engineering, John Wiley & Sons. Bowles, J. E. (1988). Fondation analysis and design. Chen, W.-F. and W. O. McCarron (1991). Bearing capacity of shallow fondations, Springer. Dncan, J. M. (2000). Factors of safety and reliability in geotechnical engineering. Jornal of geotechnical and geoenvironmental engineering 126(4): Fenton, G. A., D. Griffiths (2003). Bearing-capacity s. Canadian geotechnical jornal 40(1): Fenton, G. A., D. V. Griffiths (2008). Risk Assessment in Geotechnical Engineering, Wiley Online Library. Fenton, G. A., E. H. Vanmarcke (1990). Simlation of random fields via local average sbdivision. Jornal of Engineering Mechanics 116(8): Hansen, J. B. (1970). A revised and extended formla for bearing capacity. Itasca, F. (2005). Fast Lagrangian analysis of contina. Itasca Conslting Grop, User s manal, Minneapolis. Lacasse, S., F. Nadim (1997). Uncertainties in characterising soil properties. Pblikasjon-Norges Geotekniske Instittt 201: MATLAB (2012). "The mathworks Inc., Natick, MA." 5: 333. Meyerhof, G. (1951). The ltimate bearing capacity of fodations. Geotechniqe 2(4): Phoon, K.-K. (2008). Reliability-based design in geotechnical engineering: comptations and applications, CRC Press. Phoon, K.-K., F. H. Klhawy (1999). Characterization of geotechnical variability. Canadian geotechnical jornal 36(4): Phoon, K. K., F. H. Klhawy (1996). On qantifying inherent soil variability. Popesc, R., et al. (1998). Characteristic percentile of soil strength for dynamic analyses. Geotechnical Earthqake Engineering and Soil Dynamics III, ASCE. Schneider, H. (1999). Determination of characteristic soil properties Determination des valers characteristiqes. Geotechnical engineering for transportation infrastrctre Amsterdam, Netheredlands, Taylor & Francis. Spanos, P. D., R. Ghanem (1989). Stochastic finite element expansion for random media. Jornal of Engineering Mechanics 115(5): Stefano, G. (2009). The stochastic finite element method: past, present and ftre. Compter Methods in Applied Mechanics and Engineering 198(9): Terzaghi, K. (1943). Theoretical soil mechanics, Wiley New York. Vesic, A. S. (1974). Analysis of ltimate loads of shallow fondations. International Jornal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, Pergamon.