Effect of spatial variability on the bearing capacity of cement-treated ground The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Kasama, Kiyonobu, Andrew J. Whittle, and Kouki Zen. “Effect of Spatial Variability on the Bearing Capacity of Cement-Treated Ground.” Soils and Foundations 52, no. 4 (August 2012): 600–619. © 2012 The Japanese Geotechnical Society As Published http://dx.doi.org/10.1016/j.sandf.2012.07.003 Publisher The Japanese Geotechnical Society Version Final published version Accessed Fri May 27 02:09:30 EDT 2016 Citable Link http://hdl.handle.net/1721.1/91637 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Detailed Terms Soils and Foundations 2012;52(4):600–619 The Japanese Geotechnical Society Soils and Foundations www.sciencedirect.com journal homepage: www.elsevier.com/locate/sandf Effect of spatial variability on the bearing capacity of cement-treated ground Kiyonobu Kasamaa,n, Andrew J. Whittleb, Kouki Zena a Division of Civil and Structural Engineering, Faculty of Engineering, Kyushu University, Fukuoka, Japan Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA b Received 15 September 2011; received in revised form 10 April 2012; accepted 1 May 2012 Available online 7 September 2012 Abstract This paper presents a reliability assessment for the undrained bearing capacity of a surface strip foundation based on the results of a probabilistic study in which the shear strength and unit weight of cement-treated ground are represented as random fields in Monte Carlo simulations of undrained stability using numerical limit analyses. The results show how the bearing capacity is related to the coefficient of variation and correlation length scale in both shear strength and unit weight. Based on the results, the authors propose an overdesign factor, tolerable percentage of defective core specimens, and resistance factors for LRFD ultimate limit state of surface footings on cement-treated ground in order to achieve a target reliability index and probability of failure. The proposed method is illustrated through example calculations based on the spatial variation of unconfined compressive strength measured using a variety of cement-mixing methods from projects in Japan. & 2012 The Japanese Geotechnical Society. Production and hosting by Elsevier B.V. All rights reserved. Keywords: Cement stabilization; Bearing capacity; Limit analysis; Monte Carlo method; Probabilistic methods; Cohesive soils 1. Introduction Cement-mixing techniques such as deep-mixing (DMM; Terashi and Tanaka, 1981) and pre-mixing (Zen et al., 1992) methods are becoming widely established for stabilizing soft soils in applications ranging from the strengthening of weak foundation soils to the mitigation of liquefaction. Although there have been significant advances in the equipment and methods used for cement-mixing, there remains a high degree of spatial variability in the physical and mechanical properties of the treated ground (i.e., unit weight, shear strength, etc.). n Corresponding author. E-mail address: kasama@civil.kyushu-u.ac.jp (K. Kasama). Peer review under responsibility of The Japanese Geotechnical Society. This spatial variability introduces uncertainties in the design of foundations on cement-treated ground. Miyake et al. (1991), Kitazume et al. (2000) and Bouassida and Porbaha (2004) have presented results of physical (centrifuge) model tests to measure the bearing capacity of strip footings founded on arrays of clay-cement columns with area replacement ratios ranging from 18 to 80%. The papers include details of the interpreted failure mechanisms and finite element simulations of the loaddeformation response. Broms (2004) has proposed a semiempirical method for interpreting the bearing capacity based on the unconfined compressive strength of the deep-mixing columns and local shear failure of the adjacent soft clay. A more rigorous theoretical approach is presented by Bouassida and Porbaha (2004) based on upper and lower bound plasticity solutions (based on the earlier yield design theory developed by Bouassida et al., 1995). These analyses treat the cement-treated ground as a homogeneous mass. None of these prior studies has 0038-0806 & 2012 The Japanese Geotechnical Society. Production and hosting by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sandf.2012.07.003 K. Kasama et al. / Soils and Foundations 52 (2012) 600–619 Nomenclture B COVc COVNc COVqu COVg ci cu Fo G Gi K Nc NcDet NcL NcU NcLi NcUi Ncl99% n ne PD Pf P[y] Q QDet QL QU qu R RNc RNc90% RNc95% RNc99% ri width of foundation coefficient of variation of undrained shear strength coefficient of variation of bearing capacity factor coefficient of variation of unconfined compressive strength coefficient of variation of unit weight undrained shear strength of ith element undrained shear strength of cement-treated soil overdesign factor correlated standard Gaussian field with zero mean unit variance local value of standard Gaussian field with zero mean unit variance for ith element; correlation matrix bearing capacity factor bearing capacity factor of analytical Prandtl solution equivalent bearing capacity factor obtained by LB-NLA equivalent bearing capacity factor obtained by UB-NLA bearing capacity factor for ith realization for LB-NLA bearing capacity factor for ith realization for UB-NLA bearing capacity factor with 99% lower confidence level number of Monte Carlo iterations total number of elements in the mesh percent defective probabilities of failure probability the ultimate bearing capacity of cement-treated ground ultimate bearing capacity lower bound estimates of the collapse load upper bound estimates of the collapse load unconfined compressive strength standard Gaussian field with zero mean unit variance mNcU/NcDet, reduced mean bearing capacity ratio reduced bearing capacity ratio with 90% lower confidence level reduced bearing capacity ratio with 95% lower confidence level Ncl99%/NcDet, reduced bearing capacity ratio with 99% lower confidence level standard Gaussian field with zero mean unit variance for ith element S ST SR TPD xi aR bT g gi mc mln c mln Nc mln g mNc mNcL mNcU mqu mg Yln c y yv yh yr yo yln c yln g rij r(xij) sc sln c sln Nc sln g sNc sNcL sNcU squ sg fR F(y) 601 upper triangular forms of matrix K lower triangular forms of matrix K strength ratio of unconfined compressive strength between in situ specimen and laboratory specimen the tolerance of percent defective position vector at center of ith element sensitivity factor variabilities of load component and resistance for a strip foundation target reliability index unit weight unit weight of ith element mean undrained shear strength mean log undrained shear strength mean of log bearing capacity factor mean log unit weight mean of bearing capacity factor mean of bearing capacity factor for LB-NLA through Monte Carlo simulations mean of bearing capacity factor for UB-NLA through Monte Carlo simulations mean unconfined compressive strength mean unit weight yln c/B ¼ yln g/B, normalized correlation length spatial correlation length vertical correlation length horizontal correlation length radial correlation length orthogonal correlation length spatial correlation length for undrained shear strength spatial correlation length for unit weight correlation coefficient between element i and j correlation coefficient between two random values separated by a distance xij ¼ 9xi xj9 standard deviation of undrained shear strength standard deviation of log undrained shear strength standard deviation of log bearing capacity factor standard deviation of log unit weight standard deviation of bearing capacity factor standard deviation of bearing capacity factor for LB-NLA through Monte Carlo simulations standard deviation of bearing capacity factor for UB-NLA through Monte Carlo simulations standard deviation of unconfined compressive strength standard deviation of unit weight resistance factor for the ultimate limit state of a strip foundation on the cement-treated ground Cumulative normal function 602 K. Kasama et al. / Soils and Foundations 52 (2012) 600–619 considered the role of spatial variability of the cementtreated ground on the bearing capacity or ultimate limit state failure modes. Omine et al. (2005) have proposed a model for estimating the shear strength of cement-treated soil that combines a statistical weakest link model (for the brittle ‘soilcrete’) with a bundle model to represent the level of mixing achieved at different scales. Tang et al. (2001) have also evaluated the variation of unconfined compression strength of in situ cement-treated soil due to the change in water content and cement amount. In order to evaluate the effects of the spatial variability of soil parameters on the stability of shallow foundations, Griffiths and Fenton (2001), Griffiths et al. (2002) and Popescu et al. (2005) have analyzed the undrained bearing capacity of strip footings on cohesive soils using displacement-based finite element analyses. In their studies, undrained shear strength is treated as a random field characterized by a lognormal distribution and a spatial correlation length using methods of local area subdivision (LAS; Fenton and Vanmarcke, 1990) and mid-point discretization (Baecher and Christian, 2003). Namikawa and Koseki (2009) investigated the effects of the spatial correlation of the shear strength on the behavior of fullscale, cement-treated columns using similar random finite element analyses. Tokunaga et al. (2009) evaluated the reliability index and safety factor for cement-treated soils with the estimated failure modes based on the results of a series of reliability analyses. This paper presents a reliability assessment for the bearing capacity of a surface strip foundation on cementtreated ground based on the results of a probabilistic study in which the shear strength and unit weight of the cementtreated ground are represented as random fields in Monte Carlo simulations of undrained stability for a surface strip foundation using numerical limit analyses. The numerical limit analysis used in this study offer a convenient method for analyzing undrained stability problems and can readily be adapted to simulate the effects of spatial variability of soil parameters of cement-treated ground and natural soil layers. The originality of the proposed analytical method is to combine the numerical limit analysis with the random field theory, which can offer a more convenient and computationally efficient approach for evaluating effects of variability in soil strength properties in geotechnical stability calculations. The results show how the bearing capacity is related to the coefficient of variation and correlation length scale in the shear strength and unit weight of cement-treated ground. Based on the results of these analyses, we propose a systematic procedure for selecting the overdesign factor, the tolerable percentage of defective core specimens and the resistance factor in LRFD for the bearing capacity of cement-treated ground to obtain a target reliability index. Finally, the proposed method is illustrated using spatial variability data from a range of cement-mixing methods in order to discuss the assessment of stability, design and quality control from the view point of the reliability-based design. 2. Spatial variability of cement-treated ground The main factors influencing the shear strength of the cement-treated ground include the types and amounts of binder/cement (e.g., Clough et al., 1981, Kamon and Katsumi, 1999), physico-chemical properties of the in situ soil (e.g., Chew et al., 2004), curing conditions (e.g., Consoli et al., 2000) and effectiveness of the mixing process (e.g., Larsson, 2001; Omine et al., 1998). Since there are a number of influential factors on shear strength, as mentioned above, the in situ shear strength of cement-treated ground shows a large degree of spatial variability. For example, Table 1 summarizes the data of the unconfined compressive strength from a series of construction projects of cement-treated ground in Japan. In each case, measurements of unconfined compressive strength, qu, were obtained from core samples cured in the field. The mean values of qu range from 100 to 7500 kPa with coefficients of variation, COVqu ¼ squ/mqu ¼ 0.14–0.99. These results are consistent with the findings of a recent review of US deep mixing projects by Navin and Filz (2005) who report COVqu ¼ 0.17–0.67. This level of variability is much higher than that expected for the undrained shear strength of natural clays (e.g., Phoon and Kulhawy, 1999; Matsuo and Asaoka, 1977). Although there is quite extensive data for estimating the coefficient of variation in the unconfined compressive strength, there is much less information available to understand the underlying spatial correlation structure. Table 2 summarizes values of the correlation length, y, (in both the vertical and horizontal directions) reported in five separate studies in the literature. Two of these are based on qu data from installed DMM columns, while two others use cone penetration data in dredged fills. The study by Larsson et al. (2005) uses a miniature penetrometer to evaluate the spatial mixing structure within individual, exhumed lime-cement columns. The results show that the vertical correlation length can range from 0.2 to 4.0 m (this is similar to the range of fluctuation scales quoted for natural clay deposits). Navin and Filz (2005) find that the horizontal correlation length is much larger for wet mix DMM columns (12 m) than for dry-mix (o 3 m), while Larsson et al. (2005) find a radial correlation length, yr o 0.15 m within 0.6 m diameter columns. Overall, these data suggest that the horizontal correlation length for cement-treated ground is much smaller than for natural sedimentary soil layers although some studies have found that the horizontal scale of fluctuation can be an order of magnitude greater than the vertical scale (e.g., James Bay marine clay deposits; DeGroot and Baecher, 1993). In order to examine the difference in shear strength between in situ cement-treatment and laboratory cement– soil mixtures, Fig. 1 summarizes the strength ratio (SR) of qu between in situ specimen obtained from core samples in cement-treated ground (using the pre-mixing method) and laboratory specimens prepared with a similar target cement content. The figure also shows the percentage of defective core Table 1 Project conditions and unconfined compressive strength data for cement-treated soil from construction projects in Japan. Depth (m) Cement type Cement amount (kg/m3) Curing period (day) Water content w (%) Sample number Mean qu (kPa) COVqu Reference NP NP BP NP BP Cement milk 140 150 14 135 150 15 28 28, 49 28 60 28–52 – 95–135 110–150 55–110 100–130 90–100 – 176 222 182 26 29 30 2140 3920 3760 3370 3770 790 0.358 0.353 0.440 0.331 0.485 0.290 CDIT (1999a) Deep mixing method (offshore) 8.0 19.0 9.0 22.0 12.5 36.0 9.0 39.0 1.0 8.0 – 2.0 8.0 0.0 7.0 0.0 10.0 0.0 4.0 1.5 28.2 0.0 36.0 4.0 9.0 0.8 10.0 1.7 9.5 4.0 11.0 0.0 20.5 NP – Special cement NP NP and BP NP BP BP BP BP 74 150 250 – 70–220 100–200 250 150 150 175 105–200 28 28 28 – – 28, 56 28 28 28 28 110–140 – 100–200 – – – – – – – 54 47 493 36 100 116–157 27– 27 54 36 230 2360 3660 2820 1073–1712 754–1248 7321 7155 4127 4578 954–2631 0.480 0.420 0.140 0.423 0.27–0.39 0.26–0.28 0.24 0.3 0.18 0.26 0.12–0.58 CDIT (1999a) Kohinata et al. (1995) Noto et al. (1983) Tamura et al. (1995) Peng and But (2009) Suzuki and Kawamura (2009) Masuda et al. (2009) – – – BP BP BP 7.5% 3% 4% 91 28 28 6.5 9.4 11.8 32 13 25 661 360 120 0.470 0.990 0.750 CDIT (1999b) Premixing method Pneumatic flow mixing 0.0 10.0 – – – – 50 7, 28 28 105 71–180 343 – 296 – 0.37 0.3–0.55 Oota et al. (2009) Tang et al. (2001) Deep mixing method (onshore) Abe et al. (1997) Hioki et al. (2009) K. Kasama et al. / Soils and Foundations 52 (2012) 600–619 Cement mixing method Notes: NP, Portland cement; BP, blast furnace slag cement; CDIT, Costal Development Institute of Technology. 603 K. Kasama et al. / Soils and Foundations 52 (2012) 600–619 604 Table 2 Summary of data on spatial correlation lengths for cement-treated ground. Type of ground Reference strength COV yh (m) yv (m) Reference DMM columns qu – 0.4–4.0 Honjo (1982) Cement-mixed dredged fill Air-transported stabilized dredged fill DMM columns CPT CPT qu 0.21–0.36 (clay) 0.32–0.40 (sand) 0.114–0.194 – 0.34–0.74 (mean: 0.55) 0.5 0.22–0.74 – Tang et al. (2001) Porbaha et al. (1999) Navin and Filz (2005) Lime-cement columns Hand-operated penetrometer 2.0 – 12.0 (wet mixing) o3.0 (dry mixing) yr o 0.15 yo o 0.35 – Larsson et al. (2005) o0.6 Notes: yr, correlation length in radial direction; yo, correlation length in orthogonal direction. Strength ratio SR(=qu in-situ /quo) 100 PDF of SR 0 0.25 0.50 0.25 0.50 0.25 0.50 0.25 0.5 10 1 32.8% 0.1 0.5 1 19.0% PD=2.3% 4.4% 1.5 2 Overdesign factor Fo 2.5 3 Fig. 1. Strength ratio and observed percentage of defective core specimens as function of the overdesign factor used in pre-mixing construction projects. specimens (PD) with qu less than the reference laboratory strength. The data are reported as functions of an ‘overdesign factor’, Fo. The value of Fo is used in a conventional design process to select the cement-mix proportions for achieving a target value of SR for the cement-treated ground while overcoming uncertainties associated with the high variability of the in situ strength (i.e., percent defective). As PD¼ 32.8% at Fo ¼ 1.0 (no overdesign) in Fig. 1, the bearing capacity bearing capacity of the foundation is likely to be less than the target design due to local ground failure and the existence of weak areas of soilcrete. Conventional practice in soil improvement (e.g., Costal Development Institute of Technology, 1999a,b) uses an overdesign factor Fo = 1.5–2.0 (with a corresponding reduction in PD, Fig. 1) in order to guarantee the target bearing capacity even when there is a high degree of spatial variability in cement-treated ground. 3. Random field numerical limit analyses 3.1. Numerical limit analyses The numerical limit analyses (NLA) used in this study are based on 2-D, plane strain linear programming formulations of the upper bound (UB) and lower bound (LB) theorems for rigid, perfectly plastic materials presented by Sloan and Kleeman (1995) and Sloan (1988a). The lower bound analyses (Sloan, 1988a) assume a linear variation of the unknown stresses (sx, sy, txy) within each triangular finite element. The formulation differs from conventional displacement-based finite-element formulations by assigning each node uniquely within an element, such that the unknown stresses are discontinuous along adjacent edges between elements. Statically admissible stress fields are generated by satisfying: (i) a set of linear equality constraints, enforcing static equilibrium with triangular elements and along stress discontinuities between the elements and (ii) inequality constraints that ensure no violation of the linearized material failure criterion. Tresca yield criterion is used to represent both the undrained shear strength of the clay and the cohesive strength of the cement-treated ground. The lower-bound estimate of the collapse load is then obtained through an objective function that maximizes the resultant force, Q, acting on the footing. The linear programming problem is solved efficiently using a steepest edge active set algorithm (Sloan, 1988b). The upper-bound formulation assumes linear variations in the unknown velocities (ux, uy) within each triangular finite element. Nodes are unique to each element and hence, the edges between elements represent planes of velocity discontinuities. Plastic volume change and shear distortion can occur within each element as well as along velocity discontinuities. The kinematic constraints are defined by the compatibility equations and the condition of associated flow (based on an appropriate linearization of the Tresca criterion) within each element and along the velocity discontinuities between elements. The external applied load can be expressed as a function of unknown nodal velocities and plastic multiplier rates. The upperbound on the collapse load can then be formulated as a linear programming problem, which seeks to minimize the external applied load using an active set algorithm (after Sloan and Kleeman, 1995). Recent numerical formulations of upper and lower bound limit analyses for rigid perfectly plastic materials, using finite element discretization and linear or non-linear programming methods, provide a practical, efficient and K. Kasama et al. / Soils and Foundations 52 (2012) 600–619 accurate method for performing geotechnical stability calculations. For example, Ukritchon et al. (1998) proposed a solution to the undrained stability of surface footings on non-homogeneous and layered clay deposits under the combined effects of vertical, horizontal and moment loading to a numerical accuracy of 7 5%. One of the principal advantages of numerical limit analyses is that the true collapse load is always bracketed by results from the upper and lower bound calculations. Moreover, the only parameter used in these NLA is the undrained shear strength (which can vary linearly within a given soil layer). Hence, NLA provides a more convenient method of analyzing stability problems than conventional displacement-based finite element methods which also require the specification of (elastic) stiffness parameters and simulation of the complete non-linear load-deformation response up to collapse. Fig. 2a and b shows typical finite element meshes used in the current UB and LB analyses respectively for surface foundations on cement-treated ground. The model considers a soil layer with depth z/B ¼ 2.0, where B is the width of the surface strip foundation under vertical loading. The dimension of square mesh divided into four quarter elements is 0.125B. Previous studies (e.g., Ukritchon et al., 1998) have used a high element density close to the stress singularities at the edges of the footing in order to achieve more accuracy in lower bound analyses. The current study computes lower bounds using an uniform mesh in order to ensure comparable accuracy in the representation of spatially variable soil properties (i.e., y/B 0 -1 c u / μc SR: 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 -2 0 1 2 3 4 5 x/B y/B 0 -1 c u / μc SR: 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0 1 3 2 4 same element size used in LB and UB analyses). Extension elements are introduced in the LB mesh to ensure that lower-bound conditions are rigorously satisfied in the far field. The soil is underlain by a rigid base, while far-field lateral boundaries of the mesh extend beyond the zone of all potential failure mechanisms. The analyses assume full improvement of soils with cement-mixing around the footing such that the zone of cement-treated ground extends to the boundary. The current simulations also assume that both shear strength and unit soil weight of the cement-treated ground are spatially variable parameters. The sliding resistance at the soil–foundation interface is controlled by the shear strength of the cement-treated ground. Therefore, sliding between soil and foundation occurs when the shear stress on the soil–foundation interface is more than the shear strength of cement-treated ground. The collapse loads for UB and LB meshes as shown in Fig. 2 are represented by an equivalent bearing capacity factors, NcU ¼ QU/(Bcu) and NcL ¼ QL/(Bcu) respectively, where QU and QL are the upper or lower bound estimates of the collapse load. Analyses for uniform clay produce values of NcU ¼ 5.23 and NcL ¼ 5.00, such the analytical Prandtl solution (NcDet ¼ 2þ p) is well bounded (i.e., 5.00rNcDet r5.23) with errors of 72.25%. 3.2. Representation of spatial variability of soil parameter The effects of inherent spatial variability of soil property are represented in the analyses by modeling the undrained shear strength, cu, and unit soil weight, g, of the cementedtreated soil as a homogeneous random field (Vanmarcke, 1984). The undrained shear strength and unit weight are assumed to have an underlying log-normal distributions with mean, mc and mg, and standard deviations, sc and sg, and an isotropic scale of fluctuation (also referred to as the correlation length), yln c and yln g. The current simulations assume that correlation length for unit weight, yln g is similar to that for undrained shear strength,yln c. The use of the log-normal distribution is predicated by the fact that cu and g are always positive quantities. Following Griffiths et al. (2002) the current analyses present results based on assumed values of the ratio of the correlation length to footing width, Yln c ¼ yln c/B, which is referred to as a ‘‘normalized correlation length’’ in this paper. The mean and standard deviation of log cu and log g are readily derived from sc and mc and sg and mg as follows (e.g., Baecher and Christian, 2003): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ slnc ¼ lnð1þ COVc2 Þ; slng ¼ lnð1 þ COVg2 Þ mlnc ¼ lnmc 12 s2lnc ; -2 5 x/B Fig. 2. Typical finite element mesh used in UB and LB numerical limit analyses (Yln c ¼ 1.0, COVc ¼ 0.2). (a) FE mesh for upper bound numerical limit analysis. (b) FE mesh for lower bound numerical limit analysis. 605 mlng ¼ lnmg 12s2lng ð2Þ Spatial variability is incorporated within the numerical limit analyses (both UB and LB meshes) by assigning the undrained shear strength and unit weight corresponding to the ith element ci ¼ expðmlnc þ slnc UGi Þ ð3aÞ gi ¼ expðmlng þ slng UGi Þ K. Kasama et al. / Soils and Foundations 52 (2012) 600–619 ð3bÞ where Gi is a random variable that is linked to the spatial correlation length, yln c. It is assumed that unit weight of ith element, gi is perfectly correlated with ci, based on experimental findings that show strong correlation between undrained shear strength and unit weight of cement-treated soils as presented by Tsuchida et al. (2007) and Kasama et al. (2007). Noted that the auto-correction function is used to realize the spatial variability of the soil properties in both horizontal and vertical directions because the cross correlation between the soil properties in the vertical and horizontal directions has been not fully clarified in prior studies. Values of Gi are obtained using a Cholesky Decomposition technique (CD, e.g., Matthies et al., 1997; Baecher and Christian, 2003; Kasama et al., 2006; Kasama and Whittle, 2011) using an isotropic Markov function which assumes that the correlation decreases exponentially with distance between two points i, j 2xij rðxij Þ ¼ exp ð4Þ ylnc where r is the correlation coefficient between two random values of cu and g at any points separated by a distance xij = 9xi xj9 where xi is the position vector of i (located at the centroid of element i in the finite element mesh). Noted that an exponential autocorrelation function is used to express the covariance structure of cement-treated ground as experimentally shown by Honjo (1982), Navin and Filz (2005) and Larsson et al. (2005) although the influence of autocorrelation function on the variability of the bearing capacity should be clarified in future study. It is emphasized that the coordinate at the centroid of the element is used to represent the spatial variability of soil properties in this study. This coefficient can be used to generate a correlation matrix, K, which represents the correlation coefficient between each of the elements used in the NLA finite element meshes 3 2 1 r12 r1ne 6r 1 r2ne 7 7 6 12 7 K ¼6 ð5Þ 7 6 ^ ^ & 5 4 r1ne r2ne 1 where rij is the correlation coefficient between element i and j, and ne is the total number of elements in the mesh. The matrix K is positive definite and hence, the standard Cholesky Decomposition algorithm can be used to factor the matrix into upper and lower triangular forms, S and ST, respectively: ð6Þ ST S ¼ K The components of ST are specific to a given finite element mesh (for either UB or LB) and selected value of the correlation length, yln c. The vector of correlated random variables, G (i.e., {G1, G2, y, Gne}, where Gi specifies the random component of the undrained shear strength and unit weight in element i, Eq. (3) can then be obtained from the product G ¼ ST R ð7Þ where R is a vector of statistically independent, random numbers {r1, r2, y, rne} with a standard normal distribution (i.e., with zero mean and unit standard deviation). The current implementation implicitly uses the distance between the centroids to define the correlations between undrained shear strengths and unit weights in adjacent elements. This is an approximation of the random field, which involves the integral of the correlation function over the areas of the two elements. Fig. 3 compares the exact correlation function (Eq. (4)) with results using a typical realization obtained using the proposed CD technique for the FE mesh shown in Fig. 2. The data show good agreement with the correlation function for intervals as small as 0.05B, corresponding to the minimum distance between the centroids of adjacent elements. The results suggest that the current mesh can provide an adequate representation for correlation lengths, Yln c Z 0.25. Values of the random variable vector R are re-generated for each realization in a set of Monte Carlo simulations. Fig. 2 illustrates the spatial distribution of shear strength in the cement-treated ground for one example simulation with input parameters cu ¼ 100 kPa, COVc ¼ (sc/mc)¼ 0.2 and Yln c ¼ 1.0. The lighter shaded regions indicate areas of higher shear strength. Based on the literature review of the variability and correlation lengths for cement-treated ground (Tables 1 and 2), a parametric study has been performed using the ranges listed in Table 3. It is noted that input coefficient of variability of undrained shear strength, COVc, ranges from 0.2 to 1.0 while the input coefficient of variability of unit weight, COVg, is fixed at 0.1 because the spatial variability of unit weight of cement-treated ground is less than that of 1.2 : Cholesky Decomposition : Exact Correlation 1 Correlation coefficient 606 0.8 Θ =2.0 lnc 0.6 1.0 0.4 0.5 0.25 0.2 0 Random -0.2 0 0.2 0.4 0.6 distance/B 0.8 1 Fig. 3. Comparison of exact correlation relation and realized correlation obtained by proposed Cholesky Decomposition technique. K. Kasama et al. / Soils and Foundations 52 (2012) 600–619 Table 3 Input parameters for current study. Parameter Selected values Mean undrained shear strength mc COVc of undrained shear strength Mean unit weight mg COVg of unit weight Ratio of horizontal and vertical correlation lengths Normalized correlation length Yln c ¼ yln c/B 100 kPa 0.2, 0.4, 0.6, 0.8, 1.0 15.0 kN/m3 0.1 1.0 Random, 0.25, 0.5, 1.0, 2.0, 4.0 1000 Monte Carlo iteration shear strength reported by Kitazume et al. (2004) and Kitazume and Takahashi (2008). The horizontal correlation length is identical to the vertical correlation length in this study. For each set of parameters, a series of 1000 Monte Carlo simulations have been performed. 4. Stochastic bearing capacity 4.1. Failure mechanism against vertical loading Fig. 4a and b shows the UB failure situations against vertical loading for the initial UB mesh of Fig. 2a. Each figure shows the deformed mesh, vectors of the UB velocity field, zone of plastic shear distortion (dark zones within the velocity field). It can be seen that the computed failure mechanisms are not symmetric (with associated rotation of the footing) and find paths of least resistance, passing through weaker regions of the cement-treated ground with active passive rigid body wedges under the foundation. It can be seen that the failure mechanisms do not extend below the depth z ¼ B (this applies for all of the simulations) which suggests that it is critical to examine the shear strength distribution of the shallow foundation soils (z r B). 4.2. Stochastic bearing capacity factor In order to evaluate the stochastic bearing capacity of cement-treated ground with spatial variability in shear strength and unit weight, the computed bearing capacity factor can then be reported for each realization, i, of the random field, NcUi for UB-NLA and NcLi for LB-NLA. Hence, the mean, mNcU and mNcL, and standard deviation, sNcU and sNcL, of the bearing capacity factor are recorded through each set of Monte Carlo simulations, as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n n 1X 1 X mNcU ¼ ð8aÞ NcUi ; sNcU ¼ ðNcUi mNcU Þ2 ni¼1 n1 i ¼ 1 mNcL n 1X ¼ NcLi ; ni¼1 sNcL sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 1 X ¼ ðNcLi mNcL Þ2 n1 i ¼ 1 ð8bÞ Fig. 5 illustrates one set of results for the case with n ¼ 1000, Yln c ¼ 1.0 and COVc ¼ 0.2, 0.6 and 1.0. The 607 results confirm that the collapse load for any given realization is well bounded by the UB and LB calculations and furthermore the mean and standard deviation of NcU by UB-NLA is always more than those of NcL by LB-NLA for a given COVc and Yln c. Moreover, the mean and standard deviation of Nc both become stable within 1000 simulations and hence, reliable statistical interpretation of the data can be obtained from this set of simulations. Several studies (e.g., Griffiths et al. 2002; Phoon, 2008) have performed to determine an appropriate number of Monte Carlo iteration combining reasonable accuracy of the results in terms manageable computational efforts for a large parametric study. Table 4 summarizes the statistical data for the bearing capacity factor for all combinations of the input parameters. In all cases the results show mNcU 4 mNcL, while difference (mNcU mNcL) increases with increasing COVc and decreasing Yln c. In all cases, except where Yln c is random, the collapse load is bounded within 7 2.5–20% showing acceptable accuracy from the numerical limit analyses. The data also show sNcU 4 sNcL. This latter result may reflect differences in the upper bound and lower bound limit analyses. However, it is notable that the numerical limit analyses generate much smaller coefficients of variation in bearing capacity than were reported by Griffiths et al. (2002) from displacement-based finite element simulations (with LAS representations of the random fields). Fig. 6 shows a 25-bin histogram of the bearing capacity factor from one complete series of Monte Carlo simulations with n ¼ 1000, Yln c ¼ 1.0 and COVc ¼ 0.4 and 0.8. It is seen that most of Nci are less than NcDet, suggesting that weak soil elements have a reducing effect for the bearing capacity of spatially variable ground, as shown in Fig. 4. In order to obtain the distribution function of the bearing capacity factor based on w2 goodness-of-fit tests, Table 4 summarizes w2 statistics for all of the simulations and confirms that normal or log-normal distribution functions can be used to characterize the bearing capacity at a 5% significance level (with acceptance level, w2252122 ½0:05 ¼ 33:92). 4.3. Reduction of bearing capacity factor The role of spatial variability in reducing the expected bearing capacity can be more conveniently seen in Fig. 7, which reports the reduced mean bearing capacity ratio RNc ¼ mNcU/NcDet and mNcL/NcDet, where NcDet is the bearing capacity factor of the analytical Prandtl solution (NcDet ¼ 5.14). Fig. 7a and b is the results of UB-NLA and LB-NLA respectively. There are large reductions in RNc as COVc increases for a given normalized correlation length, Yln c, while the reduction rate of RNc increases with decreasing Yln c. It can be characterized that the expected mean bearing capacity of cement-treated ground for typical coefficients of variation (COVc ¼ 0.4–0.8, c.f. Table 1) is 50–80% of the deterministic value. K. Kasama et al. / Soils and Foundations 52 (2012) 600–619 608 0 y/B y/B 0 -1 -1 SR: 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 -2 0 1 2 3 4 SR: 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 -2 5 0 3 4 x/B Initial mesh and strength distribution 0 1 2 0 1 2 0 1 x/B Initial mesh and strength distribution y/B -1 0 1 2 3 4 (kPa*B): 0.005 0.01 0.015 0.02 0.025 0.03 -2 5 x/B Displacement vector and dissipated energy 3 4 x/B Displacement vector and dissipated energy -1 -1 SR: 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0 1 2 3 4 SR: 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 -2 5 x/B Deformed mesh 3 x/B Deformed mesh -1 0 1 2 3 4 SR: 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 -2 5 x/B Initial mesh and strength distribution 3 4 x/B Initial mesh and strength distribution -1 5 -1 (kPa*B): 0.005 0.01 0.015 0.02 0.025 0.03 0 1 2 3 4 (kPa*B): 0.005 0.01 0.015 0.02 0.025 0.03 -2 5 0 x/B Displacement vector and dissipated energy 0 1 2 3 4 x/B Displacement vector and dissipated energy 5 0 y/B y/B 2 0 y/B y/B 0 -1 -1 SR: 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 -2 5 -1 SR: 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 -2 4 0 y/B y/B 0 -2 5 0 y/B y/B 0 -2 5 -1 (kPa*B): 0.005 0.01 0.015 0.02 0.025 0.03 -2 2 0 0 y/B 1 0 1 2 3 x/B Deformed mesh 4 SR: 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 5 -2 0 1 2 3 x/B Deformed mesh 4 5 Fig. 4. Deformed mesh and plastic failure zone at failure against vertical loading. (a) Yln c ¼ 0.25, COVc ¼ 0.2; (b) Yln c ¼ 0.5, COVc ¼ 0.2; (c) Yln c ¼ 1.0, COVc ¼ 0.2; and (d) Yln c ¼ 2.0, COVc ¼ 0.2. 7 6 COVc =0.2 5 0.6 4 3 1.0 2 : Upper bound limit analysis : Lower bound limit analysis 1 0 609 3 Θ lnc =1.0 Standard Deviation of Bearing Capacity Factor, Nc Mean Bearing Capacity Factor, Nc K. Kasama et al. / Soils and Foundations 52 (2012) 600–619 1 10 100 Number of Simulations Θ lnc =1.0 2.5 : Upper bound limit analysis : Lower bound limit analysis 2 1.5 1.0 1 0.6 0.5 COV c =0.2 0 1000 1 10 100 Number of Simulations 1000 Fig. 5. Summary statistics of bearing capacity factor as functions of the number of Monte Carlo simulations. (a) Mean bearing capacity factor. (b) Standard deviation of bearing capacity factor. Table 4 Bearing capacity factor statistics and goodness of fit results for normal and log-normal distributions. Yln c COVc LB-NLA UB-NLA mNc sNc w2 mlnNc slnNc w2 mNc sNc w2 mlnNc slnNc w2 Random 0.2 0.4 0.6 0.8 1.0 4.23 3.40 2.71 2.16 1.75 0.058 0.090 0.103 0.104 0.099 1.90 2.75 2.63 2.29 2.11 1.44 1.22 0.99 0.77 0.56 0.014 0.027 0.038 0.048 0.057 1.82 3.38 2.93 2.63 2.72 4.78 4.08 3.40 2.83 2.37 0.074 0.126 0.150 0.157 0.159 1.86 2.15 4.06 3.24 3.14 1.56 1.41 1.22 1.04 0.86 0.016 0.031 0.044 0.055 0.067 4.14 3.15 6.58 5.47 5.05 0.25 0.2 0.4 0.6 0.8 1.0 4.44 3.73 3.08 2.55 2.12 0.154 0.245 0.284 0.293 0.285 1.98 1.53 1.88 2.27 1.58 1.49 1.31 1.12 0.93 0.74 0.035 0.065 0.092 0.115 0.134 2.03 3.20 6.30 8.01 8.96 4.81 4.18 3.54 2.98 2.53 0.191 0.332 0.411 0.442 0.447 7.99 8.78 6.71 7.06 2.04 1.57 1.43 1.26 1.08 0.91 0.040 0.079 0.116 0.147 0.175 7.55 7.68 7.72 7.47 4.89 0.5 0.2 0.4 0.6 0.8 1.0 4.54 3.89 3.28 2.76 2.33 0.249 0.396 0.468 0.491 0.487 2.92 1.46 1.81 1.46 1.75 1.51 1.35 1.18 1.00 0.83 0.055 0.102 0.142 0.177 0.206 3.09 4.60 8.14 15.31 18.77 4.80 4.19 3.58 3.05 2.61 0.294 0.511 0.635 0.686 0.692 2.42 4.15 6.50 4.16 4.40 1.57 1.43 1.26 1.09 0.92 0.061 0.121 0.176 0.222 0.261 2.29 3.52 5.04 5.63 8.43 1.0 0.2 0.4 0.6 0.8 1.0 4.63 4.05 3.48 2.99 2.59 0.340 0.547 0.661 0.710 0.721 2.28 3.21 3.10 3.75 4.60 1.53 1.39 1.23 1.07 0.92 0.073 0.134 0.188 0.234 0.273 2.52 6.98 11.95 27.29 31.87 4.82 4.26 3.70 3.21 2.80 0.394 0.684 0.855 0.932 0.951 2.75 2.72 3.43 3.25 4.11 1.57 1.44 1.28 1.13 0.97 0.082 0.160 0.228 0.285 0.331 2.10 2.79 5.28 9.09 13.01 2.0 0.2 0.4 0.6 0.8 1.0 4.72 4.23 3.74 3.31 2.95 0.375 0.618 0.774 0.863 0.905 2.77 1.98 1.92 3.36 3.38 1.55 1.43 1.30 1.16 1.04 0.079 0.145 0.205 0.256 0.300 3.58 11.01 22.53 25.68 21.77 4.86 4.38 3.91 3.48 3.12 0.429 0.757 0.971 1.090 1.148 1.59 2.98 2.68 4.38 4.97 1.58 1.46 1.33 1.20 1.07 0.088 0.171 0.245 0.306 0.356 2.64 3.25 6.00 10.05 16.06 4.0 0.2 0.4 0.6 0.8 1.0 4.81 4.43 4.04 3.68 3.38 0.346 0.586 0.758 0.876 0.952 2.03 2.30 1.92 3.06 3.42 1.57 1.48 1.38 1.28 1.18 0.072 0.132 0.186 0.235 0.276 5.26 16.33 15.20 26.39 18.37 4.94 4.55 4.17 3.83 3.53 0.389 0.700 0.924 1.073 1.168 1.44 1.30 1.83 2.51 4.25 1.59 1.50 1.40 1.30 1.21 0.079 0.153 0.219 0.275 0.323 2.45 5.88 7.71 15.61 28.34 Note: acceptance criterion w2252122 ½0:05r 33:92. K. Kasama et al. / Soils and Foundations 52 (2012) 600–619 610 0.020 Θlnc=1.0 COVc =0.4 NcDet PDF of Nc 0.015 UB-NLA μNc =4.26 COVNc =0.684 LB-NLA μNc=4.05 COVNc=0.547 0.010 0.005 0.000 Θ lnc=1.0 COV c =0.8 UB-NLA μNc=3.21 COVNc=0.932 0.015 PDF of Nc 0.020 LB-NLA μNc=2.99 COVNc =0.710 0.010 0.005 2.3 2.8 3.3 3.7 4.2 4.7 5.2 5.6 Bearing capacity factor Nc 0.000 6.1 NcDet 1 1.7 2.3 3 3.7 4.3 5 5.7 Bearing capacity factor Nc 6.3 100 Reduced bearing capacity ratio, RNc = Nc NcDet Reduced bearing capacity ratio, RNc = Nc NcDet Fig. 6. Histogram of bearing capacity factor. (a) Yln c ¼ 1.0, COVc ¼ 0.4 and (b) Yln c ¼ 1.0, COVc ¼ 0.8. Mean bearing capacity factor 80 60 Θ lnc Random 0.25 0.5 1.0 2.0 4.0 40 20 0 0 0.2 0.4 0.6 0.8 Coefficient of variation of cu, COVc 100 1 Mean bearing capacity factor 80 60 Θ lnc Random 0.25 0.5 1.0 2.0 4.0 40 20 0 0 0.2 0.4 0.6 0.8 Coefficient of variation of cu, COVc 1 100 99% lower confidence bound 80 60 Θ 40 Random 0.25 0.5 1.0 2.0 4.0 20 0 0 R (μ σ ) 0.2 0.4 0.6 0.8 Coefficient of variation of cu, COVc 1 Reduced bearing capacity ratio, Ncl99% RNc 99% = N cDet Reduced bearing capacity ratio, Ncl99% RNc99%= N cDet Fig. 7. Reduced bearing capacity ratio against COVc (mNc). (a) UB-NLA and (b) LB-NLA. 100 99% lower confidence bound 80 60 Θ 40 Random 0.25 0.5 1.0 2.0 4.0 20 0 0 R (μ σ ) 0.2 0.4 0.6 0.8 Coefficient of variation of cu, COVc 1 Fig. 8. Reduced bearing capacity ratio against COVc (99% lower confidence bound of N(c). (a) UB-NLA and (b) LB-NLA. It is well known in most building codes that the nominal value of the material strength (resistance) is usually defined as the 5% or 10% fractile of the material strength. In order to examine nominal values for the bearing capacity, Fig. 8 shows the reduced bearing capacity ratio at the 99% lower confidence level, RNc99% ¼ Ncl99%/NcDet, where Ncl99% is estimated by assuming a log-normal distribution with mNc and sNc (from Table 4). RNc99% shows a large reduction in small range of COVc irrespective of the normalized correlation length, Yln c. In the design of a surface strip foundation, the bearing capacity can be estimated approximately by multiplying NcDet with the characteristic value of soil strength (which is generally assumed to be a conservative value of the strength given by mc a sc) to consider the reduction of bearing capacity due to the spatial variability in shear strength. Fig. 8 shows the reduced bearing capacity ratio for the case where the characteristic soil strength is mc sc (i.e. a ¼ 1), RNca ¼ 1. K. Kasama et al. / Soils and Foundations 52 (2012) 600–619 100 80 60 40 Mean bearing capacity factor 20 0 : UB-NLA : LB-NLA 0 0.5 100 Reduced bearing capacity ratio (%) Reduced bearing capacity ratio (%) COV =0.2 0.4 0.6 0.8 1.0 1 1.5 2 2.5 3 3.5 Normalized correlation length lnc 95% lower confidence bound COV =0.2 80 0.4 60 0.6 0.8 1.0 40 20 : UB-NLA : LB-NLA 0 0 0.5 1 1.5 2 2.5 3 3.5 Normalized correlation length lnc 90% lower confidence bound COV =0.2 80 0.4 60 0.6 0.8 1.0 40 20 0 4 : UB-NLA : LB-NLA 0 0.5 1 1.5 2 2.5 3 3.5 Normalized correlation length lnc 4 100 Reduced bearing capacity ratio (%) Reduced bearing capacity ratio (%) 100 611 : UB-NLA : LB-NLA 80 COV =0.2 60 0.4 40 0.6 0.8 1.0 20 99% lower confidence bound 0 4 0 0.5 1 1.5 2 2.5 3 3.5 Normalized correlation length lnc 4 Fig. 9. Reduced bearing capacity ratio against Yln c. (a) Mean bearing capacity factor, mNc; (b) 90% lower bound of Nc; (c) 95% lower bound of Nc; and (d) 99% lower bound of Nc. 7 7 Target probability of failure,P = 10 6 Overdesign factor Fo Overdesign factor Fo 6 Θ =1.0 5 2.0 0.5 4.0 0.25 4 3 Random 2 1 Target probability of failure,P = 10 2.0 Θ =1.0 0.5 4.0 5 4 0.25 3 Random 2 0 0.2 0.4 0.6 0.8 1 Coefficient of variation of cu, COVc 1.2 1 0 0.2 0.4 0.6 0.8 1 Coefficient of variation of cu, COVc 1.2 Fig. 11. Overdesign factor Fo and COVc. (a) Target probability of failure, Pf ¼ 10 2 and (b) target probability of failure, Pf ¼ 10 3. The results show that for COVc r 0.6, RNc99% r RNca ¼ 1 and hence, a 4 1.0 is needed for a conservative estimate of the nominal bearing capacity. Fig. 9a–d shows the relationships between the reduced mean bearing capacity ratio, RNc, and the 90%, 95% and 99% lower confidence bounds (RNc90%, RNc95%, RNc99%, K. Kasama et al. / Soils and Foundations 52 (2012) 600–619 612 100 10-1 10-2 10-3 COVc =1.0 10-4 0.2 0.4 10-5 1 0.6 2 3 4 Overdesign factor Fo 100 0.4 1 0.6 UB-NLA LB-NLA 0.8 2 3 4 Overdesign factor Fo 100 P[Q < QDet] P[Q < QDet] COVc =1.0 5 Θlnc=1.0 10-1 COVc =1.0 10-3 0.8 10-4 0.2 1 0.4 2 3 4 Overdesign factor Fo 10-2 COVc =1.0 10-3 0.2 10-5 5 0.8 0.6 10-4 UB-NLA LB-NLA 0.6 100 1 UB-NLA LB-NLA 0.4 2 3 4 Overdesign factor Fo 100 Θlnc =2.0 5 Θlnc =4.0 10-1 P [Q<QDet] 10-1 P[Q<QDet] 10-3 10-5 5 Θlnc=0.5 10-2 10-2 COVc =1.0 10-3 0.6 10-4 0.2 10-5 10-2 0.2 10-1 10-5 Θ lnc=0.25 10-4 UB-NLA LB-NLA 0.8 Pf by Baecher & Christian Phoon et al. 10-1 Θlnc=Random P[Q < QDet] P[Q<QDet] 100 Pf by Baecher & Christian Phoon et al. 1 0.8 0.4 2 3 4 Overdesign factor Fo 10-2 COVc =1.0 10-3 0.8 0.6 -4 10 UB-NLA LB-NLA 0.2 5 10-5 1 0.4 2 3 4 Overdesign factor Fo UB-NLA LB-NLA 5 Fig. 10. P[Q o QDet] and overdesign factor Fo. (a) Yln c ¼ random; (b) Yln c ¼ 0.25; (c) Yln c ¼ 0.5; (d) Yln c ¼ 1.0; (e) Yln c ¼ 2.0; and (f) Yln c ¼ 4.0. respectively) of Nc as functions of Yln c and COVc. The results for spatially random shear strength are plotted at Yln c ¼ 0 for simplicity. There is a very small difference between the upper and lower bound solutions for Yln c 4 0.5, and hence the reductions in bearing capacity are well defined in this range of correlation lengths. The widening gap between the solutions for Yln c o 1.0 (i.e., loss of accuracy) reflects the underlying problem of stochastic discretization that requires elements to be smaller than the spatial correlation length (Matthies et al., 1997). Overall, the results show minimum values of RNc90%, RNc95% and RNc99% at Yln c ¼ 1.0 irrespective of COVc while RNc increases with increasing Yln c. Qualitatively similar results have been presented by Griffiths et al. (2002) who also report a local minimum in RNc for Yln c E 0.5, which is not seen in the current numerical limit analyses (Fig. 9a). As a result, the current analyses do not converge to the theoretical limits for the case of spatially random shear strength. K. Kasama et al. / Soils and Foundations 52 (2012) 600–619 35 Target probability of failure,P = 10 30 25 20 COV 0.2 0.4 0.6 0.8 1.0 Allowable percent defective (%) Allowable percent defective (%) 35 15 10 5 0 1 2 3 Normalized correlation length, lnc 4 Target probability of failure,P = 10 30 COV 0.2 0.4 0.6 0.8 1.0 25 20 15 10 5 0 0 613 0 1 2 3 Normalized correlation length, lnc 4 Fig. 12. Allowable percent defective and normalized correlation length. (a) Target probability of failure, Pf ¼ 10 2 and (b) target probability of failure, Pf ¼ 10 3. 5. Reliability assessment 5.1. Overdesign factor and tolerance of percent defective In the conventional design of a surface strip foundation with width, B, on cohesive soil, the ultimate bearing capacity QDet ¼ B mc NcDet, in which the mean undrained shear strength mc and the bearing capacity factor NcDet are based on the simplified failure mechanism obtained for a homogeneous soil mass. If the bearing capacity factor Nc for cement-treated ground is a probabilistic parameter and the mean undrained shear strength mc is augmented (using the overdesign factor, Fo) to account for the increase in mean undrained shear strength due to cement-mixing. The ultimate bearing capacity Q can then be found from Q¼ B Fo mc Nc. In the current calculations, considering the reduction in the bearing capacity of cement-treated ground due to the spatial variability of soil parameter and the increase in mean undrained shear strength due to cement-mixing, the probability that the bearing capacity of cement-treated ground Q is less than QDet can be obtained by assuming that Nc of cementtreated ground is described by a log-normal distribution. The probability P[Q o QDet] is given by lnð½2þ p=Fo ÞmlnNc P½Q o QDet ¼ P½Fo Nc o NcDet ¼ F slnNc ð9Þ where F(y) is the cumulative normal function, mln Nc and sln Nc are mean and standard deviation of ln Nc obtained by following equations using COVNc ¼ sNc/mNc: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð10Þ slnNc ¼ lnð1þ COVN2 c Þ mlnNc ¼ lnmNc 0:5Us2lnNc ð11Þ Fig. 10 summarizes P[Q o QDet] for a given Yln c as a functions of Fo and COVc. The target probabilities of failure considered in LRFD codes for shallow foundations are reported in the range, Pf ¼ 10 2–10 3 (Baecher and Christian, 2003; Phoon et al., 2000). It is noted that the difference of P[QoQDet] between UB-NLA and LB-NLA increases as COVc increases. It can be seen that P[QoQDet] decreases very markedly with increasing Fo irrespective of COVc, but also increases with increasing COVc for a given Fo suggesting that an overdesign factor should be properly determined (depending on COVc) to satisfy the required probability of failure. In addition, P[QoQDet] depends on values of Yln c, however, the probability for a given overdesign factor shows maximum around Yln c=1.0 in current analysis. Fig. 11a and b illustrates the optimized values of the overdesign factor, Fo, for mix designs of cemented-treated grounds (with specified parameters, COVc and Yln c) in order to satisfy the target probability of bearing failure for a surface strip footing with Pf ¼ 10 2 and 10 3. These results use the higher values of Fo from UB and LB computations shown in Fig. 10. It can be seen that the largest values of Fo occur for Yln c ¼ 1.0–2.0 and that Fo always increases with COVc. Typical that cement-treated ground with a large degree of variability (COVc Z 0.6 and Yln c ¼ 1.0) will require Fo 4 3.0 to satisfy Pf ¼ 10 2 while Fo ¼ 1.5–2.5 is appropriate for cement-treated ground with small spatial variability (COVc ¼ 0.2–0.4 similar to naturally deposited soils). The results in Fig. 11 can be re-arranged to focus on the quality of the in situ shear strength of the cement-treated ground. Fig. 12a and b shows the Tolerated Percentage of Defective core specimens TPD needed to achieve Pf ¼ 10 2 and 10 3 as a functions of Yln c and COVc. Noted that TPD is directly related to Fo as shown in Eq. (9). It can be seen that Yln c has much more influence on TPD compared to COVc. The minimum tolerance level, for Pf ¼ 10 2 occurs at Yln c ¼ 0.5 with TPD E 10%, while results for Pf ¼ 10 3 K. Kasama et al. / Soils and Foundations 52 (2012) 600–619 614 1 COVc =0.2 0.8 Resistance factor R Resistance factor R 1 0.4 0.6 0.6 0.8 0.4 1.0 0.2 0 αR = 0.5 Θlnc= Random 1 1.5 2 2.5 3 3.5 Target reliability index T 1.0 0.2 0 4 Resistance factor R Resistance factor R 0.8 0.4 αR = 0.5 Θlnc = 0.25 COVc =0.2 0.4 0.6 0.6 0.4 0.8 1.0 0.2 α R= 0.5 1 1.5 1 1.5 0.8 : UB-NLA : LB-NLA 2 2.5 3 3.5 Target reliability index T 4 COVc =0.2 0.4 0.6 0.6 0.4 0.8 1.0 0.2 αR = 0.5 : UB-NLA : LB-NLA Θlnc = 0.5 2 2.5 3 3.5 Target reliability index T Θlnc = 1.0 0 4 1 1 1.5 : UB-NLA : LB-NLA 2 2.5 3 3.5 Target reliability index T 4 1 0.8 COVc =0.2 Resistance factor R Resistance factor R 0.6 1 0.8 0.4 0.6 0.6 0.4 0.8 1.0 0.2 0 0.4 0.6 : UB-NLA : LB-NLA 1 0 COVc =0.2 0.8 α R= 0.5 1.5 0.4 0.6 0.6 0.8 1.0 0.4 0.2 αR = 0.5 : UB-NLA : LB-NLA Θlnc = 2.0 1 COVc =0.2 0.8 2 2.5 3 3.5 Target reliability index T Θlnc = 4.0 4 0 1 1.5 : UB-NLA : LB-NLA 2 2.5 3 3.5 Target reliability index T 4 Fig. 13. Resistance factor and target reliability index (aR ¼ 0.5). (a) Yln c ¼ random; (b) Yln c ¼ 0.25; (c) Yln c ¼ 0.5; (d) Yln c ¼ 1.0; (e) Yln c ¼ 2.0; and (f) Yln c ¼ 4.0. suggest minimum TPD values in the range 5–6% for Yln c ¼ 0.5–2.0. 5.2. Resistance factor in LRFD In order to extend the numerical results in the current analysis to LRFD code for cement-treated ground, a resistance factor fR for the ultimate limit state of a surface strip foundation on cement-treated ground is calculated assuming that the load component to a strip foundation on the cement-treated ground is modeled by log-normal distribution as follows (JGS, 2006): mNc 1 fR ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expðaR UbT UslnNc Þ N 1þ COVNc cDet ð12Þ where aR is a sensitivity factor to represent the ratio K. Kasama et al. / Soils and Foundations 52 (2012) 600–619 1 COVc =0.2 0.8 0.4 0.6 0.6 0.8 0.4 1.0 0.2 0 α R= 0.8 1.5 2 2.5 3 3.5 Target reliability index T Resistance factor R Resistance factor R 0.8 1.0 0.2 0 4 COVc =0.2 0.6 0.4 0.4 0.6 0.2 0.8 1.0 : UB-NLA : LB-NLA αR = 0.8 Θlnc = 0.5 αR = 0.8 1 1.5 : UB-NLA : LB-NLA 2 2.5 3 3.5 Target reliability index T 4 1 1.5 2 2.5 3 3.5 Target reliability index T 0.8 COVc =0.2 0.6 0.4 0.4 0.6 0.2 αR = 0.8 Θlnc = 1.0 0 4 1 1 1.5 0.8 1.0 : UB-NLA : LB-NLA 2 2.5 3 3.5 Target reliability index T 4 1 0.8 Resistance factor R Resistance factor R 0.6 0.4 1 0.8 COVc =0.2 0.6 0.4 0.4 0.6 0.2 0.8 1.0 0 0.4 0.6 Θlnc = 0.25 1 0 COVc =0.2 0.8 : UB-NLA : LB-NLA Θlnc= Random 1 Resistance factor R Resistance factor R 1 615 αR = 0.8 Θlnc = 2.0 1 1.5 0.8 COVc =0.2 0.6 0.4 0.6 0.4 0.8 1.0 0.2 αR = 0.8 : UB-NLA : LB-NLA 2 2.5 3 3.5 Target reliability index T 4 0 Θlnc = 4.0 1 1.5 : UB-NLA : LB-NLA 2 2.5 3 3.5 Target reliability index T 4 Fig. 14. Resistance factor and target reliability index (aR ¼ 0.8). (a) Yln c ¼ random; (b) Yln c ¼ 0.25; (c) Yln c ¼ 0.5; (d) Yln c ¼ 1.0; (e) Yln c ¼ 2.0; and (f) Yln c ¼ 4.0. between variabilities of the load component and resistance, and bT is a target reliability index. Noted that the nominal bearing capacity (resistance) in Eq. (12) is assumed to be equal to NcDet. Figs. 13 and 14 show the resistance factor fR for aR = 0.5 and 0.8 as functions of the target reliability index for selected values of COVc and Yln c. It is seen that the resistance factor fR for Yln c Z 0.25 decreases with increasing target reliability index bT and the decrease rate of fR increases with increasing Yln c although fR for Yln c = random is almost constant irrespective of COVc. In addition, the difference of UB-NLA and LB-NLA decreases as Yln c increases. JGS (2006) proposed that a resistance factor fR for the ultimate limit state of a strip foundation on a clayey ground with K. Kasama et al. / Soils and Foundations 52 (2012) 600–619 COVc ¼ 0.3 for bT ¼ 1.5–3.0 ranges 0.77–0.62 and 0.67– 0.47 for aR ¼ 0.5 and 0.8 respectively, which is less than those obtained from this study. 6. Case study This section illustrates how spatial variability affects the ultimate limit state design of shallow foundations on cement-treated ground. The numerical limit analyses assume a normalized correlation length Yln c ¼ 0.25, based on empirical data on spatial correlation lengths for a range of ground treatment methods (Table 2) and shallow foundations with widths ranging from 5.0 to 10.0 m. Fig. 15 summarizes COVqu as functions of the mean unconfined compressive strength (mqu) for the cementtreated soils presented in Table 1. Although there is a large scatter in the data associated with different construction methods, the results show that the coefficient of variability generally decreases with the level of the unconfined compressive strength. The behavior can be well Coefficient of Variation of qu, COVqu 1 0.8 DMM(offshore) DMM(onshore) Premixing Method Pneumatic Flow Mixing Method 0.6 COVqu =0.51*exp(-0.0022*μqu )+0.32 0.4 0.2 0 0 1000 2000 3000 4000 5000 6000 7000 Mean unconfined compressive strength qu (kPa) 8000 Reduced bearing capacity ratio, RNc Fig. 15. The relationships between COVqu and mqu. 100 Open: UB-NLA 80 Solid: LB-NLA 60 : Eqn. 13 40 : DMM(offshore) : DMM(onshore) : Premixing Method : Pneumatic Flow Mixing Method 20 0 Θ =0.25 0 1000 2000 3000 4000 5000 6000 7000 8000 Mean unconfined compressive strength qu (kPa) approximated by an exponential decay regression function as follows: COVqu ¼ 0:51Uexpð0:0022Umqu Þþ 0:32 ð13Þ This function suggests that COVqu is approximately constant for mqu 4 2000 kPa. Fig. 16a and b compares the reduced mean bearing capacity ratio (RNc) and 99% lower confidence level (RNc99%) respectively, against mean unconfined compressive strength from numerical limit analyses from Fig. 7 using values of COVqu directly from Table 1 and from the correlation given in Eq. (13). These results confirm that the proposed correlation (Eq. (13)) provides a good approximation for design. There is a large reduction in bearing capacity for mqu o1000 kPa but is well bounded (RNc ¼ 77–86% and RNc99% ¼ 68–73%) for mqu 42000 kPa. This suggests that the design unconfined compressive strength for cement-treated ground should be greater than 2000 kPa to prevent a large reduction in bearing capacity due to the spatial variability. Fig. 17a–d shows similar comparisons for the LRFD resistance factor, fR for aR ¼ 0.5 and 0.8 and bT ¼ 2.0 and 3.0. It can be seen that resistance factor increases with increasing mqu up to 2000 kPa and then remains constant value for mqu 4 2000 kPa irrespective of aR and bT. The resistance factor obtained by Eq. (13) for aR ¼ 0.5 ranges 0.72–0.80 for bT ¼ 2.0–3.0 while 0.68–0.77 for aR ¼ 0.8 which seems to be identical to those for clayey ground proposed by Foye et al. (2006). Finally, in order to guarantee a target bearing capacity for cements-treated ground in terms of probability-based design, Fig. 18 summarizes an overdesign factor Fo for the design shear strength of cement-treated ground against mean unconfined compressive strength mqu. It can be seen that overdesign factor Fo decreases sharply when mqu o 2000 kPa and then remains constant value around 1.5. In addition, the difference of overdesign factor between the target probability of failure, Pf ¼ 10 2 and 10 3 is very small. An overdesign factor Fo ¼ 1.5 can be recommended for the cement-treated ground with mean qu 4 2000 kPa, Reduced bearing capacityratio, RNc99% 616 100 Open: UB-NLA 80 60 Solid: LB-NLA : Eqn. 13 40 : DMM(offshore) : DMM(onshore) : Premixing Method : Pneumatic Flow Mixing Method 20 0 Θ =0.25 0 1000 2000 3000 4000 5000 6000 7000 8000 Mean unconfined compressive strength qu (kPa) Fig. 16. Reduced bearing capacity ratio against mean unconfined compressive strength mqu (Yln c ¼ 0.25). (a) Reduced mean bearing capacity ratio, RNc and (b) 99% lower confidence level, RNc99%. K. Kasama et al. / Soils and Foundations 52 (2012) 600–619 : Eqn. 13 β =2.0, α =0.5, Θ =0.25 0.4 : DMM(offshore) : DMM(onshore) : Premixing Method : Pneumatic Flow Mixing Method 0.2 0 Resistance factor R 1 Resistance factor R 0.6 0 1 Solid: LB-NLA Open: UB-NLA 0.4 : Eqn. 13 0.2 0.6 0.4 : Eqn. 13 0.2 0 1 Solid: LB-NLA Open: UB-NLA 0.6 0.8 0 1000 2000 3000 4000 5000 6000 7000 8000 Mean unconfined compressive strength qu 0.8 Solid: LB-NLA Open: UB-NLA β =2.0, α =0.8, Θ =0.25 Resistance factor R Resistance factor R 1 0 1000 2000 3000 4000 5000 6000 7000 8000 Mean unconfined compressive strength qu 1000 2000 3000 4000 5000 6000 7000 8000 Mean unconfined compressive strength qu Solid: LB-NLA Open: UB-NLA 0.8 0.6 0.4 : Eqn. 13 0.2 β =3.0, α =0.5, Θ =0.25 0 617 β =3.0, α =0.8, Θ =0.25 0 0 1000 2000 3000 4000 5000 6000 7000 8000 Mean unconfined compressive strength qu Fig. 17. Resistance factor against mean unconfined compressive strength mqu (Yln c ¼ 0.25). (a) aR ¼ 0.5, bT ¼ 2.0, Yln c ¼ 0.25; (b) aR ¼ 0.8, bT ¼ 2.0, Yln c ¼ 0.25; (c) aR ¼ 0.5, bT ¼ 3.0, Yln c ¼ 0.25; and (d) aR ¼ 0.8, bT ¼ 3.0, Yln c ¼ 0.25. 4 : DMM(offshore) : DMM(onshore) : Premixing Method : Pneumatic Flow Mixing Method Overdesign factor Fo 3.5 3 Solid: Target probability of failure, P = 10-3 f Open: Target probability of failure,Pf = 10-2 2.5 Θlnc=0.25 2 1.5 1 0 1000 2000 3000 4000 5000 6000 7000 Mean unconfined compressive strength qu 8000 Fig. 18. Overdesign factor Fo against mean unconfined compressive strength mqu (Yln c ¼ 0.25). while higher values are needed for smaller values of mean unconfined compressive strength. 7. Conclusions This paper has presented a reliability assessment for estimating the bearing capacity of a surface strip foundation on cement-treated ground using numerical limit analyses with random field theory and Monte Carlo simulation. Using these results, we propose values of the overdesign factors, tolerable percentage of defective core specimens (TPD) and LRFD resistance factors for the bearing capacity of cement-treated ground in order to obtain a target reliability index and probability of failure. The main conclusions are as follows: (1) The bearing capacity factor of cement-treated ground considering the spatial variability of shear strength and unit soil weight can be characterized by both normal and log-normal distribution functions with 5% significance level. (2) The expected mean bearing capacity of cement-treated ground for a typical coefficient of variation (COVc ¼ 0.4–0.8) is 50–80% of that estimated by assuming an uniform strength ground. It can be characterized that the bearing capacity with a lower confidence level shows the minimum value at normalized correlation length Yln c ¼ 1.0 irrespective of COVc while mean bearing capacity increases with increasing Yln c. (3) Cement-treated ground for COVc Z 0.6 and normalized correlation length Y ¼ 1.0 needs a large overdesign factor Fo over 3.0 although Fo ¼ 1.5–2.5 is 618 K. Kasama et al. / Soils and Foundations 52 (2012) 600–619 appropriate for COVc ¼ 0.2–0.4. The tolerable percentage of defective cores TPD ¼ 10% and 5% are needed to obtain a probabilities of failure Pf ¼ 10 2 and 10 3, respectively, for a simplified quality management in in situ strength of cement-treated ground. (4) For cement-treated ground with mean unconfined compressive strength greater than 2000 kPa, the resistance factor fR ¼ 0.73–0.76 can be used for surface strip foundations (with target reliability indices, bT ¼ 2.0–3.0 and sensitivity factors, aT ¼ 0.5–0.8). An overdesign factor, Fo ¼ 1.5 can be used for the same range of mqu. Acknowledgment A grateful acknowledgment is made to Professors Hidetoshi Ochiai and Noriyuki Yasufuku, Associate Professors Guangqi Chen and Kiyoshi Omine of Kyushu University for their helpful advice and encouragement. References Abe, T., Miyoshi, A., Maeda, T., Fukuzumi, H., 1997. Evaluation for soil improvement of clay mixing consolidation method using dual-way mixing system. In: Proceeding of the 32nd Japan National Conference on JSSMFE, pp. 2353–2354 (in Japanese). Baecher, G.B., Christian, J.T., 2003. Reliability and Statistics in Geotechnical Engineering. Wiley & Sons, New York. Bouassida, M., deBuhan, P., Dormieux, L., 1995. Bearing capacity of a foundation resting on a soil reinforced by a group of columns. Géotechnique 45 (1), 25–34. Bouassida, M., Porbaha, A., 2004. Ultimate bearing capacity of soft clays reinforced by a group of columns—application to a deep mixing technique. Soils and Foundations 44 (3), 91–101. Broms, B.B., 2004. Lime and lime/cement columns. In: Moseley, M.P., Kirsch, K. (Eds.), Ground Improvement, 2nd edition (Chapter 8). Chew, S.H., Kamruzzaman, A.H.M., Lee, F.H., 2004. Physicochemical and engineering behavior of cement treated clays. ASCE Journal of Geotechnical and Geoenvironmental Engineering 130 (7), 696–706. Clough, G.W., Sitar, N., Bachus, R.C., Rad, N.S., 1981. Cemented sands under static loading. ASCE Journal of Geotechnical Engineering 107 (6), 799–817. Consoli, N.C., Rotta, G.V., Prietto, P.D.M., 2000. Influence of curing under stress on the triaxial response of cemented soils. Géotechnique 50 (1), 99–105. Costal Development Institute of Technology, 1999a. Design and Construction Manual on Deep Mixing Method for Off-shore Construction, 136 (in Japanese). Costal Development Institute of Technology, 1999b. Design and Construction Manual on Premixing Method, 144 (in Japanese). DeGroot, D.J., Baecher, G.B., 1993. Estimating autocovariance of in situ soil properties. ASCE Journal of Geotechnical Engineering 129 (1), 147–166. Fenton, G.A., Vanmarcke, E.H., 1990. Simulation of random fields via local average subdivision. Journal of Engineering Mechanics 116 (8), 1733–1749. Foye, K.C., Salgado, R., Scott, B., 2006. Resistance factors for use in shallow foundation LRFD. Journal of Geotechnical and Geoenvironmental Engineering 132 (9), 1208–1218. Griffiths, D.V., Fenton, G.A., 2001. Bearing capacity of spatially random soil: the undrained clay Prandtl problem revisited. Géotechnique 51 (4), 351–359. Griffiths, D.V., Fenton, G.A., Manoharan, N., 2002. Bearing capacity of rough rigid strip footing on cohesive soil: probabilistic study. ASCE Journal of Geotechnical and Geoenvironmental Engineering 128 (9), 743–755. Hioki, Y., Okami, T., Ikeda, K., 2009. Introduction of EX-DJM method (expanded DJM method). In: Proceedings of International Symposium on Deep Mixing & Admixture Stabilization (Deep Mixing 2009), pp. 563–568. Honjo, Y., 1982. A probabilistic approach to evaluate shear strength of heterogeneous stabilized ground by deep mixing method. Soils and Foundations 22 (1), 23–38. Kamon, M., Katsumi, T., 1999. Engineering properties of soil stabilized by ferrum lime and used for the application of road base. Soils and Foundations 39 (1), 31–41. Kasama, K., Whittle, A.J., 2011. Bearing capacity of spatially random cohesive soil using numerical limit analysis. ASCE Journal of Geotechnical and Geoenvironmental Engineering 137 (11), 989–996. Kasama, K., Zen, K., Iwataki, K., 2007. High-strengthening of cement-treated clay by mechanical dehydration. Soils and Foundations 47 (2), 171–184. Kasama, K., Zen, K., Whittle, A.J., 2006. Effects of spatial variability of cement-treated soil on undrained bearing capacity. In: Proceedings of International Conference on Numerical Modeling of Construction Processes in Geotechnical Engineering for Urban Environment, pp. 305–313. Kitazume, M., Hayano, K., Sato, T., Zyoyou, T., 2004. Strength variance and its causes on large-scale artificial ground constructed by pneumatic flow mixing method. Japan Society of Civil Engineers (711/III68), 199–214 (in Japanese). Kitazume, M., Okano, K., Miyajima, S., 2000. Centrifuge model tests on failure envelope of column-type deep mixing method improved ground. Soils and Foundations 40 (4), 43–55. Kitazume, M., Takahashi, H., 2008. Long term property of lime treated marine clay. JSCE Doboku Gakkai Ronbunshuu C 64 (1), 144–156 (in Japanese). Kohinata, T., Nagahashi, H., Sata, M., 1995. Characteristics of strength growth on sandy soil by deep mixing method. In: Proceeding of the 30th Japan National Conference on JSSMFE, pp. 2211–2212 (in Japanese). Larsson, S., 2001. Binder distribution in lime-cement columns. Ground Improvement 5 (3), 111–122. Larsson, S., Stille, H., Olsson, L., 2005. On horizontal variability in limecement columns in deep mixing. Géotechnique 55 (1), 33–44. Masuda, S., Onishi, T., Saito, S., Suzuki, Y., 2009. Deep mixing method using four-shaft mixing machine in building projects. In: Proceedings of International Symposium on Deep Mixing & Admixture Stabilization (Deep Mixing 2009), pp. 553–558. Matsuo, M., Asaoka, A., 1977. Probability models of undrained strength of marine clay layer. Soils and Foundations 17 (3), 51–68. Matthies, H.G., Brenner, C.E., Bucher, C.G., Soares, C.G., 1997. Uncertainties in probabilistic numerical analysis of structures and solids—stochastic finite elements. Structural Safety 19 (3), 283–336. Miyake, M., Akamoto, H., Wada, M., 1991. Deformation characteristics of ground improved by a group of treated soil. Proceedings of Centrifuge 1991, 295–302. Namikawa, T., Koseki, J., 2009. Numerical study of effects of spatial variability on cement-treated column strength. In: Proceedings of International Symposium on Ground Improvement Technologies and Case Histories (ISGI09), pp. 419–424. Navin, M.P., Filz, G.M., 2005. Statistical analysis of strength data from ground improved with DMM columns. In: Proceedings International Conference on Deep Mixing Best Practice and Recent Advances (Deep Mixing’05), CD-ROM. Noto, S., Hita, N., Terashi, M., 1983. On practice and problem on deep soil improvement. JGS Tsuchi-to-Kiso 31 (7), 73–80 (in Japanese). Omine, K., Ochiai, H., Yoshida, N., 1998. Estimation of in situ strength of cement-treated soils based on a two-phase mixture model. Soils and Foundations 38 (4), 17–29. Omine, K., Ochiai, H., Yasufuku, N., 2005. Evaluation of scale effect on strength of cement-treated soils based on a probabilistic failure model. Soils and Foundations 45 (3), 125–134. Oota, M., Mitarai, Y., Iba, H., 2009. Outline of pneumatic flow mixing method and application for artificial island reclamation work. K. Kasama et al. / Soils and Foundations 52 (2012) 600–619 In: Proceedings of International Symposium on Deep Mixing & Admixture Stabilization (Deep Mixing 2009), pp. 569–574. Peng, S.T.G., But, T.V.M., 2009. Soil improvement works for the design and development of SP-PSA international port company ltd project in Bariavung Tau, Vietnam. In: Proceedings of International Symposium on Deep Mixing & Admixture Stabilization (Deep Mixing 2009), pp. 263–268. Phoon, K.K., 2008. Numerical recipes for reliability analysis—a primer. In: Phoon, K.K. (Ed.), Reliability-based Design in Geotechnical Engineering: Computations and Applications. Taylor & Francis (Chapter 1). Phoon, K.K., Kulhawy, F.H., 1999. Characterization of geotechnical variability. Canadian Geotechnical Journal 36, 612–624. Phoon, K.K., Kulhawy, F.H., Grigoriu, M.D., 2000. Reliability-based design for transmission line structure foundations. Computers and Geotechnics 26, 169–185. Popescu, R., Deodatis, G., Nobahar, A., 2005. Effects of soil heterogeneity on bearing capacity. Probabilistic Engineering Mechanics 20 (4), 324–341. Porbaha, A., Tsuchida, T., Kishida, T., 1999. Technology of airtransported stabilized dredged fill. Part 2: quality assessment. Ground Improvement 3, 59–66. Sloan, S.W., 1988a. Lower bound limit analysis using finite elements and linear programming. International Journal for Numerical and Analytical Methods in Geomechanics 12 (1), 61–77. Sloan, S.W., 1988b. A steepest edge active set algorithm for solving sparse linear programming problems. International Journal for Numerical and Analytical Methods in Geomechanics 12 (12), 2671–2685. Sloan, S.W., Kleeman, P.W., 1995. Upper bound limit analysis using discontinuous velocity fields. Computer Methods in Applied Mechanics and Engineering 127, 293–314. 619 Suzuki, K., Kawamura, S., 2009. Compression strength of cement stabilized clay—two cases in the south of Vietnam. In: Proceedings of International Symposium on Deep Mixing & Admixture Stabilization (Deep Mixing 2009), pp. 449–452. Tamura M., Hibino, S., Mizoguchi, E., 1995. Quality control of improved ground considering consumer’s risk (Part3: case study). In: Proceeding of the 30th Japan National Conference on JSSMFE, pp. 2353–2556 (in Japanese). Tang, Y.X., Miyazaki, Y., Tsuchida, T., 2001. Practices of reused dredgings by cement treatment. Soils and Foundations 41 (5), 129–143. Terashi, M., Tanaka, H., 1981. Ground improved by deep mixing method. In: Proceedings of 10th ICSMFE, vol. 3, Stockholm, pp. 777–780. The Japanese Geotechnical Society, 2006. Principles for Foundation Designs Grounded on a Performed-based Design Concept (JGS 4001-2004) (in Japanese). Tokunaga, S., Kitazume, M., Nagao, T., Kitade, K., Muraki, T., 2009. Parametric studies on reliability-based design of deep mixing method. In: Proceedings of International Symposium on Deep Mixing & Admixture Stabilization (Deep Mixing 2009), pp. 369–374. Tsuchida, T., Tang, Y.X., Watabe, Y., 2007. Mechanical properties of lightweight treated soil cured in water pressure. Soils and Foundations 47 (4), 731–748. Ukritchon, B., Whittle, A.J., Sloan, S.W., 1998. Undrained limit analyses for combined loading of strip footing on clay. ASCE Journal of Geotechnical and Geoenvironmental Engineering 124 (3), 265–276. Zen, K., Yamazaki, H., Yoshizawa, H., Mori, K., 1992. Development of premixing method against liquefaction. In: Proceedings of 9th Asian Regional Conference, vol. 1, SMFE, pp. 461–464. Vanmarcke, E.H., 1984. Random Fields: Analysis and Synthesis. MIT press, Cambridge, MA.