1 1. INTRODUCTION Undrained shear strength, su , of cohesionless soils is often estimated from back analysis of dams and embankments that underwent various degrees of distress due to undrained loading [1]. Although the existing analytical procedures are based on the assumption of inherently isotropic material behaviour, undrained mechanical response of saturated cohesionless soils is inherently anisotropic [2, 3]. A framework has been proposed recently accounting for inherently anisotropic undrained behaviour of granular soils [4]. In this paper, 28 case histories involving static undrained response of dams and embankments have been back-analyzed using the proposed procedure and to develop a set of correlations between normalized cone tip resistance, qc1, and Standard Penetration Test (SPT) blow count, ( ) N1 60 , and peak undrained shear strength ratio, su σ v′ for various modes of deformation, e.g., plane strain compression and extension, and simple shear. To illustrate the application of the correlations, pre-failure geometries of two embankments that underwent static undrained loading were analyzed using shear strengths obtained from the proposed correlations. 2. ANALYTICAL PROCEDURE FOR INHERENTLY ANISOTROPIC MATERIALS Undrained shear strength of cohesionless soils tends to be higher when a sample is loaded in the direction of deposition than that for loading in any other direction. Such a behaviour is referred to as inherent anisotropy. Inherently anisotropic undrained behaviour of cohesionless soils can be approximated using [4] (su σ v′) (su σ v′)TXC = 0.441× cos 2θ + 0.559 [1] where θ is the angle between the direction of deposition and that of the effective major principal stress. The relationship, presented in Fig. 1, was developed using data from undrained monotonic laboratory tests on undisturbed (frozen) samples [5]. It is apparent from Fig. 1 that su σ v′ at phase transformation in triaxial compression (denoted with subscript “TXC”), is higher than those in simple shear (SS) and triaxial extension (TXE). It ANISOTROPIC UNDRAINED FINITE ELEMENT ANALYSIS OF FLOW FAILURE OF EMBANKMENTS Singh, R. PhD Candidate (QIP), Department of Civil Engineering, Indian Institute of Technology, Kharagpur, WB 721302, INDIA Roy, D. Assistant Professor, Department of Civil Engineering, Indian Institute of Technology, Kharagpur, WB 721302, INDIA ABSTRACT: Undrained shear strength of cohesionless soils is often estimated from back analyses of dams and embankments that have endured static undrained loading with various degrees of distress. The existing procedures for back analysis assume isotropic material behavior. However, saturated cohesionless soils exhibit strong inherent anisotropy during undrained loading. Consequently, the undrained shear strength of granular soils depends on the angle between the major principal direction and the direction of deposition. Using an anisotropic procedure for back analysis, correlations have been developed in this study between normalized standard penetration test (SPT) blow count or cone penetration resistance and anisotropic undrained shear strength for different modes of loading (e.g. compression, simple shear and extension) from back analysis of pre-failure geometries of 28 static flow failure case histories. To illustrate the efficacy of these correlations, finite element analyses were carried out for two earth embankments using the shear strength parameters obtained from the correlations proposed and the computed deformations were compared with the observations. The comparison indicated a reasonable agreement. Keywords: Inherent anisotropy; Back analysis; undrained shear strength; FEM model2 also appears that there is no systematic dependence of su σ v′ on the relative density of the deposit. Fig. 1. Shear strength ratio for undisturbed sand (From Singh et al. 2006) The following procedure was used for back analysis to estimate the peak su σ v′ : • A trial slip surface was first assumed through the zones of lowest penetration resistance in the pre failure configuration. • The undeformed section of the embankment above the assumed slip surface was divided into a number of vertical slices. • The mobilized undrained shear strength was assumed to be governed by the vertical effective stress for the undeformed configuration. • Soils with MPa 6.5 qc1 ≥ or ( ) N1 60 ≥ 12 and those above water table were assigned drained values of friction angle depending on their qc1 or ( ) N1 60 . • The stability analyses were performed using software package [6] and the GLE method assuming the mobilization of su at the base of the slip surface according to Eq. [1] for until obtaining a value of ( ) su σ v′ TXC by trial and error that gave a factor of safety of 1. • The above steps were repeated for other trial surfaces until obtaining the minimum value of ( ) su σ v ′ TXC . 3. CORRELATIONS FOR su σ v′ Pre-failure geometries of 28 embankments (Table 1) that failed due to static undrained loading were back analyzed using the procedure outlined in the preceding section. The results of these back analyses are summarized in Table 2 for anisotropic. Also included are the results obtained from conventional isotropic back analyses for comparison. It appears from the results that the undrained shear strengths from isotropic back analyses are in general similar to the anisotropic undrained shear strengths for simple shear loading. Table 1. Case Histories Embankment qc1 , ( ) N1 60 Reference Aberfan Tip No.4 2.2, 5.2 Lucia ([8]) Aberfan Tip No.7 2.2, 5.2 Lucia ([8]) Asele Embankment 3.8, 7.0 Konard et al. ([9]) Bofokeng Tailings 2.2, 7.4 Lucia ([8]) Calaveras Dam 4.4, 8.0 Hazen ([10]) Fording South Spoil 1.3, 2.1 Dawson et al. ([11]) Copper Tailings Dam 2.1, 6.6 Lucia ([8]) Fort Peck Dam 2.6, 6.5 Casagrande ([12]) Fraser River Delta 2.9, 5.3 Chillarige et al. ([13]) Gypsum Tailings Dam 1.4, 4.7 Lucia ([8]) Greenhills Cougar 7 1.2, 2.4 Dawson et al. ([11]) Helsinki Harbour 2.8, 6.0 Andresen and Bjerrum ([14]) Hoedekenskerke Dyke 4.7, 8.5 Koppejan et al. ([15]) Jamuna Bridge Site 3.2, 7.5 Yoshimine et al. ([16]) Lake Ackerman Road 3.3, 7.0 Hryciw et al. ([17]) Merriespruit Tailings 1.1, 3.2 Fourie et al. ([18]) Mississippi River Bank 3.2, 6.8 Senour and Turnbull ([19]) Nerlerk Slide1 2.6, 6.8 Sladen ([20]) Nerlerk Slide2 1.5, 3.6 Sladen ([20]) Nerlerk Slide3 1.5, 3.6 Sladen ([20]) Wachusett Dam North Dyke 2.3, 4.5 Olson and Stark ([21]) Quintette Marmot 1.3, 2.8 Dawson et al. ([11]) Sullivan Mine 1.8, 3.7 Davies ([22]) Tar Island Dyke 1.2, 3.0 Mittal et al. ([23]) Trondhiem Harbour 2.5, 6.0 Andresen and Bjerrum ([15]) Vlietepolder, Zeeland 2.8, 7.5 Koppejan et al. ([15]) Western US Tailings 0.8, 3.0 Davies et al. ([24]) Wilheminapolder 2.8, 7.5 Koppejan et al. ([15]) The estimates of pre-failure su σ v′ , listed in Table 2, are plotted in Figures 2 and 3 against qc1 …3 Table 2. Pre liquefaction su σ v′ Embankment u v s σ ′ Anisotropic Isotropic TXC SS TXE Aberfan Tip No.4 0.55 0.31 0.07 0.33 Aberfan Tip No.7 0.55 0.31 0.07 0.33 Asele Embankment 0.58 0.30 0.06 0.20 Bofokeng Tailings 0.53 0.28 0.06 0.17 Calaveras Dam 0.64 0.31 0.07 0.28 Fording South Spoil 0.34 0.19 0.04 0.19 Copper Tailings Dam 0.49 0.27 0.06 0.33 Fort Peck Dam 0.56 0.31 0.07 0.14 Fraser River Delta 0.53 0.30 0.06 0.16 Gypsum Tailings Dam 0.40 0.22 0.05 0.20 Greenhills Cougar 7 0.34 0.19 0.04 0.19 Helsinki Harbour 0.58 0.30 0.06 0.24 Hoedekenskerke Dyke 0.70 0.39 0.08 0.25 Jamuna Bridge Site 0.61 0.34 0.07 0.20 Lake Ackerman Road 0.55 0.31 0.07 0.25 Merriespriut Tailings 0.32 0.18 0.04 0.12 Mississippi River Bank 0.52 0.29 0.06 0.27 Nerlerk Slide1 0.54 0.28 0.06 0.16 Nerlerk Slide2 0.33 0.18 0.04 0.16 Nerlerk Slide3 0.32 0.18 0.04 0.14 Wachusett Dam North Dyke 0.49 0.26 0.05 0.25 Quintette Marmot 0.39 0.21 0.05 0.19 Sullivan Mine 0.34 0.19 0.04 0.19 Tar Island Dyke 0.41 0.23 0.05 0.20 Trondhiem Harbour 0.35 0.20 0.04 0.19 Vlietepolder, Zeeland 0.58 0.30 0.06 0.28 Western US Tailings 0.26 0.15 0.03 0.12 Wilheminapolder 0.39 0.22 0.05 0.12 and ( ) N1 60 , respectively. From these data, the following correlations were developed: ( ) su / σ v ′ TXC = 0.272 × qc10.223 [2] ( ) su / σ v ′ TXE = 0.033 × qc10.209 [3] ( ) ( ) su /σ v ′ TXC = 0.123 × N1 600.239 [4] ( ) ( ) su /σ v ′ SS = 0.116 × N1 600.134 [5] ( ) ( ) su /σ v ′ TXE = 0.029 × N1 600.116 [6] The r 2 values for Eqs. [1], [2] and [3] were 0.76, 0.72 and 0.71, respectively, and those for Eqs. [4], [5] and [6] were 0.80, 0.77 and 0.77, respectively. Fig.2 Pre failure su σ v′ - qc1 relationships Fig. 3 Pre failure su σ v′ - ( ) N1 60 relationships4 4. FINITE ELEMENT MODELLING In order to illustrate the efficacy of the proposed correlations, two embankments that underwent static flow failure were analyzed using finite element package [7]. The flow failure case histories are described in the following subsections. 4.1. J-Pit Site, Alberta A full scale experiment was conducted for an 8 m high embankment constructed over a foundation of 10 m thick sand under Phase III of CANLEX Project. This site is situated within the Syncrude premises near Fort McMurray, Alberta. The average normalized SPT blow count was 10 for uppermost 3 m thick while that for the rest of the sand was about 5. To create an undrained condition for failure, the tailings sand was rapidly dumped behind the embankment. This experiment led to the development of a very limited amount of deformation [25]. The cross section of J-Pit Site is shown in Fig. 4. 4.2. Tailings Dam No.1, South America The dam is constructed across a steep – walled valley with bedrock exposed on the valley slopes and alluvium at its base. The starter dam of 15 m height was constructed on silty sand and gravel and included a zone along its upstream face, and along its base, of clean sand and gravel [26]. The impoundment of this dam contains sulfides and is potentially acid generating. The average normalized cone tip resistance was in between 0.6 MPa of liquefiable layer. The cross section of Tailings Dam No.1 is shown in Fig. 5. 4.3. Analysis Procedure Isotropic elastic perfectly plastic (Mohr-Coulomb) material behaviour was used in the finite element computations. Gravity-on analysis was carried out in the first step using drained soil properties to find the major principal directions for each soil element. Elastic soil properties were estimated assuming the Poisson’s ratio to be 0.35 according to [27]. Next, the soil elements below water table with 6.5 MPa qc1 < or ( ) N1 60 < 12 were assigned undrained shear strength depending on the angle between the major principal direction and vertical (i.e., the direction of deposition) according to Eq. [1]. The elastic soil properties for liquefiable soil element were assigned approximately hundred times of drained soil properties and for other soil elements properties used in the first step were left unaltered. The deformations were then computed using Poisson’s ratio of 0.45 and the “large deformation” option of the finite element package. Fig. 4 Cross section of J-Pit Site Fig. 5 Cross section of Tailings Dam No.1 5. RESULTS A framework proposed for undrained analysis of dams or embankments, which considers inherently anisotropic soil behaviour in an approximate manner [5]. In this paper, twenty-eight case histories documenting undrained distress of dams or embankments were back analysed using the proposed framework and based on the results of these back analyses, correlation was developed between the pre liquefaction shear strength ratio, su σ v′ , and normalized penetration resistances, qc1 and (N1)60 . The deformation from finite element analysis using proposed pre failure relationships for anisotropic undrained analyses (Fig. 2 or 3) are shown in Table 3. The results from finite element analyses using proposed anisotropic undrained shear strength are showing agreement with the observed deformations.5 Table 3. Results of finite element analysis Dam or Embankment Vertical Displacement (m) Observed Anisotropic (FEM Analysis) J-Pit Site 0.3 0.37 Tailings Dam No.1 0.0 0.0 REFERENCES [1] Olson, S.M. and Stark, T.D. (2003) Yield strength ratio and liquefaction analysis of slopes and embankments. J. of Geotech. and Geoenviro. Engrg., 129: 727-737. [2] Vaid, Y.P., Chung, E.K.F. and Keurbis, R.H. (1990) Stress path and steady state, Canadian Geotech. J., 27: 1- 7. [3] Yoshimine, M., Ishihara, K. and Vargas, W. (1998) Effects of principal stress direction and intermediate principal stress on undrained shear behavior of sand. Soils and Foundations, 38: 179-188. [4] Singh, R. and Roy, D. (2006) Anisotropic undrained back analysis of embankments, Proc., Int. Conf. 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