25 Stochastic Approach to Slope Stability Analysis with In-Situ Data

Transcription

1 Chapter 5-x 5 Stochastic Approach to Slope Stability Analysis with In-Situ Data Authors: Jonathan Nuttall Michael Hicks Marti Lloret-Cabot Motivation Technologies to properly model the influence of soil heterogeneity on geotechnical performance are desired. Traditional deterministic approaches based on single representative property values are to be replaced by an alternative stochastic approach, combining random field theory with finite elements. Main Results A methodology for the stochastic approach has been developed incorporating the spatial variation of material properties, thereby enabling a probabilistic evaluation of the performance of slopes and other geotechnical structures. 1. R = 95% Reliability Reliability P(f) R = 95% P(f) =.5 57

2 5 Stochastic Approach to Slope Stability Analysis with In-Situ Data 5-1 Introduction The inherent spatial variability of geo-material properties influences material behaviour and geo-structural response [Hicks and Onisiphorou, 5]. It also leads to uncertainty in design due to incomplete knowledge regarding in-situ conditions, and thereby leads to the need for statistical definitions of material properties, probabilistic analysis, and global response quantified in terms of reliability and probability of failure [Hicks, 7]. Spatial variability is particularly important due to structural response typically being governed by the soil profile, with failure mechanisms often following the weaker zones within the soil [Griffiths and Fenton, ; 1; 4; Hicks and Samy, a; Hicks and Spencer, 1]. Moreover, the need for adequate representation of soil properties and their variation has been recognized in geotechnical design codes, with the European Union s Eurocode 7 [Eurocode 7, 4] introducing the concept of characteristic values into the design process [Hicks and Nuttall, 1]. Probabilistic modelling of slope stability problems has been carried out since the 197s, albeit mainly through the use of limit equilibrium methods combined with various statistical approaches [El-Ramly et al., ]. These statistical approaches have included Monte Carlo methods, as well as so-called approximate methods such as the First Order Second Moment (FOSM) and Point Estimation methods. Moreover, the representation of soil variability has generally followed two approaches. [Tang et al., 1976; Harr, 1987; Duncan, ] used the point statistics of a material property, implying an infinite spatial correlation of properties; whereas [Vanmarcke, 1977; Mostyn and Soo, 199; El-Ramly et al., ; 3; 5] accounted for spatial variation by reducing the variance of material properties along prescribed failure planes as a function of the correlation distance and failure plane length. However, the Finite Element Method (FEM) has more recently been adopted as a viable alternative to the limit equilibrium method, while random field theory may be used to provide more realistic modelling of the spatial variation of soil properties. For example, Illustration of the statistics of property X: X as a function of depth (left) and probability density function of X (right) [Hicks and Samy, a; c] F.5-1 X θ µ Depth pdf σ σ µ X 58

3 Site Characterization 5- [Paice and Griffiths, 1997; Griffiths and Fenton, ; 4; Hicks and Samy, a; b; c; 4; Hicks and Onisiphorou, 5] modelled failure mechanism development by combining FEM and random field theory, an approach often referred to as the Random Finite Element Method (RFEM) and which forms the basis of this paper. This approach gives rise to the need for an adequate spatial representation of the soil, as well as a suitable method of characterization. In particular, in addition to the standard statistical properties typically measured (i. e. mean, µ, and standard deviation, σ ), a measure of the spatial variation is required. This is often defined by the scale of fluctuation, θ, which is a measure of the distance between zones of similar property value, or, in other words, the distance over which the property is significantly correlated [Vanmarcke, 1983]. It is important that a site is characterized adequately in terms of, in order to represent the spatial variability in both the horizontal and vertical directions, i. e. as given by θ h and θ v respectively. This stochastic approach can be summarized by the following stages [Hicks and Samy, c]: 1) Statistical characterization: The material properties are represented in terms of statistics that may be obtained through site investigation. ) Prediction of spatial variability and analysis: The soil is modelled according to its statistics, giving rise to an infinite number of possible soil profiles and the need for Monte Carlo analysis, e. g. using the Random Finite Element Method (RFEM). 3) Probabilistic definitions of response: The RFEM results provide the probability of failure and/or reliability; thereby providing the basis for a risk analysis. This paper discusses these three steps in further detail, from the initial in-situ measurements through to the final risk analysis. 5- Site Characterization An important step in the stochastic approach is to characterize the material properties at a site in terms of their point and spatial statistics. F.5-1 illustrates this process. F.5-1 (left) shows the variation with depth of a property X in a soil layer. In a typical deterministic analysis X would be represented by a single characteristic value, such as the mean or lower bound. However stochastic analysis uses all the data, by expressing them in the form of a probability density function, or pdf, characterized by the mean and standard deviation of the property value, µ and σ respectively, as shown in F.5-1 (right). A third statistical parameter, the scale of fluctuation, θ, defining the degree of spatial correlation is required as indicated in F.5-1 (left). As θ increases, the degree of spatial correlation increases and the distribution of the property values becomes more uniform. There are various methods in literature for estimating the vertical scale of fluctuation, θ v, although there is little research on how to estimate the horizontal scale of fluctuation, θ h. However a number of studies have demonstrated that θ h is an important consideration in geotechnical modelling [Hicks and Samy, a; b; c; 4; Hicks and Onisiphorou, 5; Hicks and Spencer, 1; Nuttall, 11]. 59

4 5 Stochastic Approach to Slope Stability Analysis with In-Situ Data Cone Penetration Tests (CPT) may be used to estimate the statistics of the site. Due to the high resolution of the CPT in the vertical direction, θ v may be easily estimated. However the low resolution of data in the horizontal plane, due to the limited number of CPTs that may be conducted across a site, means that θ h is more difficult to estimate Statistical Evaluation [Wickremesinghe and Campanella, 1993; Wong, 4; Lloret-Cabot et al., 13] discuss methods for estimating the scale of fluctuation, θ. The approach outlined in this section was used by [Lloret-Cabot et al., 13] to study the soil variability of artificial sand islands constructed in the Canadian Beaufort Sea in the 197s and 198s; more specifically using data from the Tarsuit P-45 island, which had been the focus of several previous studies [Hicks and Smith, 1988; Wong, 4; Lloret-Cabot et al., 1]. In this approach it is assumed that the in-situ data are statistically homogeneous (or stationary), that is: a) the data have a constant mean and constant standard deviation throughout the profile; b) the data have an autocorrelation function independent of location and dependent only on the separation, or lag distance, τ, i. e. the correlation between property values at two locations is only a function of the distance that separates them. Condition a) is achieved by de-trending the data, as illustrated in F.5- for CPT tip resistance data. F.5- (above) shows the original tip resistance data, q c, and the underlying Example of a CPT profile: original data (above) and de-trended data (below) [Lloret-Cabot et al., 13] F.5-16 Tip resistance q c [MPa] trend t(z) = mz+b Trend removed tip q c * = q c (mz + b) [MPa] Depth [m] µ =

5 Modelling Spatial Variability 5-3 depth trend, t(z), whereas F.5- (below) shows the same data with the trend removed. This approach has provided a useful approximation in a number of previous studies [Hicks and Onisiphorou, 5; Uzielli et al., 5; Lloret et al., 1]. A constant standard deviation, σ, and condition b) are likely to be present if the data are taken from the same soil layer, as approximately uniform fluctuations are likely to occur in layers of the same soil type [Phoon and Kulhawy, 1999]. However, the data can be de-trended if they exhibit a depth trend in the standard deviation, by using a similiar procedure to that followed for the mean. The next step in this approach is to estimate the scale of fluctuation, θ, by fitting a theoretical correlation function, ρ( τ) (E.5-1), to the experimental correlation function, ˆ( ρτ) (E.5-), as illustrated in F.5-3. A number of correlation models exist [Fenton and Griffiths, 8]; however the exponential correlation model is given by: τ ρτ ( ) = exp E.5-1 θ where θ can represent the scale of fluctuation in either the horizontal or vertical directions and τ is the lag distance. The experimental correlation function is given by: n + j ρ τ = 1 ˆ 1 ( j ) ( µ + µ σ ˆ)( ˆ) i i j ˆ ( n j) X X i= 1 E.5- where µˆ and σˆ are the estimated mean and standard deviation respectively, taken from the in-situ CPT data, and the interval τ is the spacing of consecutive data [Fenton and Griffiths, 8]. F.5-3 illustrates the difficulty of estimating the horizontal scale of fluctuation, θ h ; that is, in the horizontal direction there are often relatively few data points available over the site, as indicated by the circles in F.5-3 (below). 5-3 Modelling Spatial Variability Having statistically characterized a soil layer based on data obtained at discrete locations, it is possible to generate random field predictions of the spatial variability across the entire site Random Field Generation Random fields are generated using random field theory. The present strategy generates random fields using the Local Average Subdivision (LAS) method [Vanmarcke, 1983; Fenton, 199; Fenton and Vanmarcke, 199]; a D algorithm was first developed [Hicks and Samy, a; 4] and subsequently a 3D algorithm [Spencer, 7; Nuttall, 11]. The algorithm is broken down into the following basic steps [Hicks and Spencer, 1]: 531

6 5 Stochastic Approach to Slope Stability Analysis with In-Situ Data Example of estimating the scale of fluctuation θ : vertical direction, θ v (above) and horizontal direction, θ h (below) [Lloret-Cabot et al., 13] F.5-3 Correlation function ρ ( τ ) Correlation function ρ ( τ ) Optimised θ v =.4 m theoretical ρ( τ ) estimated ρ( τ ) Vertical lag [m] Optimised θ h = 1.67 m τ v theoretical ρ( τ ) estimated ρ( τ ) Horizontal lag [m] τ h 1) LAS [Vanmarcke, 1983; Fenton, 199; Fenton and Vanmarcke, 199] generates a square (D) or cubic (3D) isotropic standard normal (Gaussian) random field of size D. The field is generated by uniformly subdividing the domain into square or cubic cells of dimension d, maintaining the mean value of the subdivided cells through upward averaging, with each cell value spatially correlated with its neighbours based upon an exponential Markov covariance function [Fenton and Vanmarcke, 199]. For D this is given by: τ τ βτ τ = σ 1 ( + 1, ) exp E.5-3 θ θ 1 and for 3D: τ τ τ βτ τ τ σ 1 3 ( = 1,, 3) exp + E.5-4 θ θ θ 1 3 where β is the covariance, τ is the lag distance and subscripts 1 3 denote the vertical and the two lateral directions respectively. However, due to the overall statistics being poorly preserved in the anisotropic process, the random field is initially gener- 53

7 Modelling Spatial Variability 5-3 ated with a constant scale of fluctuation, i. e. θ = θ1= θ = θ3, where θ is taken to be the largest scale of fluctuation over the domain; in geotechnical practice this usually corresponds to the horizontal plane, i. e. θ = θ h, due to the method of deposition for most geo-materials. ) An anisotropic field is generated by squashing the isotropic field; i. e. preserving θ h in the horizontal plane while compressing θ in the vertical direction to become θ v. The compressing process averages several layers of cells into a single layer, the number of these cells being equal to the degree of anisotropy of the heterogeneity, ξ = θh / θ. v F.5-4 illustrates the effect of increasing ξ on a single post-processed random field. 3) The anisotropic field is transformed from a standard normal (Gaussian) distribution to a normal, or other, distribution, by using a suitable transformation, i. e. for the normal conversion: X( x) = µ ( z) + σ( z) Z( x ) E.5-5 T where x = ( xyz) defines the centroid of the random field cell and Z(x) is the local average for a random field cell at location x. µ and σ are taken from the site statistics and re-incorporate their respective depth trends, removed during the statistical characterization, into the random field. A random field has now been generated that exhibits statistical properties that are consistent to those of the in-situ conditions. An infinite number of possible random fields exist for a given set of statistics. The aim is to analyze the geotechnical problem for a sufficient number of random fields, so as to give a statistically representative solution. One such method is the Random Finite Element Method (RFEM) [Griffiths and Fenton, 4; Hicks and Onisiphorou, 5; Nuttall, 11]. Note that the range of random fields may be reduced by conditioning the random fields to the field measurements (e. g. CPT data) [Lloret-Cabot et al., 1] Random Finite Element Method (RFEM) Although the Random Finite Element Method (RFEM) has been around in various guises since the 198s, the technique was developed by [Fenton and Griffiths, 1993a; 1993b] in the 199s. It involves mapping a random field onto a finite element mesh and subsequent analysis of the problem by finite elements. The mapping of the properties in- Illustration of the effect of increasing anisotropy for a single random field F.5-4 Increasing ξ 533

8 5 Stochastic Approach to Slope Stability Analysis with In-Situ Data volves assigning each random field cell value to a finite element, or sampling point within the element, thus mapping the spatial variation of the properties to the deterministic FE analysis. Due to the range of possible random fields, the analysis is carried out within a Monte Carlo framework, where, in each realization, the random field is mapped onto the FE domain, a deterministic FE analysis is undertaken, and the required measure of performance is recorded. This repetitive process continues until the performance measure has statistically converged to within an acceptable tolerance. The results are typically presented in the form of a performance pdf or cumulative distribution function (cdf), which expresses the probability of occurrence of a structure response, e. g. in terms of reliability, or probability of failure, P(f), as shown in F Risk Risk may be defined as (probability of failure) (consequence of failure). Although RFEM provides the probability of failure of a given structure, until recently little research had been carried out to quantify the consequence. The consequence of a given failure can be far reaching and is often a matter of opinion or conjecture, going beyond the mere financial cost of a failure. Therefore it is difficult to quantify consequence in a simple manner or cost estimation. It could be argued that the ultimate cost, or consequence, of a failure is proportional to the size of the failure; for example, the consequence of a landslide, or liquefaction slide, may be considered to be proportional to the volume of the failure, whereas the consequence of a dike or dam failure may be proportional to the depth of the failure within the structure. Hence: Consequence Failure (volume, length, etc.) E.5-6 and therefore: Risk P(f) Failure (volume, length, etc.) E.5-7 Illustrations of performance from an RFEM analysis F.5-5 Total area = pdf R = 95% cdf Reliability P(f) Reliability R = 95% P(f) =.5 Structure response Structure response 534

9 Case Study 5-5 This approach to risk makes it possible to estimate a value for the relative risk associated with a particular structure and failure using RFEM. During each realization, not only is the structure analyzed for global response (or failure), but also the corresponding consequence of the response is measured; i. e. the volume, depth and/or length of the failed zone are determined, which can be used to estimate risk according to E Case Study To explore this methodology, a case study is presented using RFEM. F.5-6 shows the geometry and mesh for a 45 slope founded on a 3 m layer of soil. The slope and foundation are considered to be from the same layer. The problem is modelled using eight node quadratic finite elements with Gaussian integration [Smith and Griffiths, 4]. The boundary conditions are that the left and right sides of the domain are allowed to move vertically, whereas the base is fixed. In this analysis the soil is modelled as an elastic, perfectly plastic Von Mises material, characterized by a spatially varying undrained shear strength, c u, defined at the sampling points. The spatial statistics for the soil layer have been defined to illustrate the application, instead of using actual site data, although in reality they would be taken from insitu data using the procedure set out in section 5-. Therefore, random fields of c u have Mesh geometry for D slope stability analysis F m 5 m 1 m 3 m 5 m A realization of the c u random field F

10 5 Stochastic Approach to Slope Stability Analysis with In-Situ Data been generated using the LAS method assuming a normal property distribution, with µ = 4 kpa, σ = 8 kpa, θ v = 1 m and θ h = 1 m (i. e. ξ = 1). For each realization of the random field (F.5-7), the factor of safety of the slope is calculated, by repeatedly analyzing the slope for increasing gravity loading until failure occurs [Nuttall, 11]. From these results the probability of failure P(f) of the slope is calculated as a function of the factor of safety (FOS), using a Monte Carlo simulation, in this case with 35 realizations (F.5-8). F.5-8 shows the probability of failure of the slope against FOS. It also indicates the value of FOS computed for a deterministic analysis of the same slope based only on c u = µ = 4 kpa (i. e. without the influence of heterogeneity). This graph clearly shows that the spatial variability has a significant impact on the failure of the slope. In comparing the homogeneous and stochastic analyses, it is seen that, for the slope based only on µ, the factor of safety corresponds to a probability of failure of P(f) 81.5 % when spatial variation is incorporated. Furthermore, the circles in F.5-8 indicate the failure volumes of the slope, which can be combined with the reliability results to give the associated risk of designing the slope to a specific factor of safety, using E Conclusions This paper has demonstrated that traditional deterministic slope stability analyses (i. e. ignoring heterogeneity) provide results that can overestimate the strength of a slope. An alternative stochastic approach has been presented; this starts with the characterization of a soil layer using in-situ data, then representing the soil profile using a random field generated using Local Average Subdivision (LAS), and finally conducting a reliability analysis using the Random Finite Element Method (RFEM). The inclusion of failure volume Probability of failure and failure volume versus factor of safety (FOS) F P(f) Volume Deterministic P(f) [%] / Vol [%] FOS 536

11 References 5 and consequence, to the analysis, has led to an evaluation of risk that can be incorporated into the design process. References Duncan, J. M.,. Factors of Safety and Reliability in Geotechnical Engineering. J Geotech Eng, 16(4): El-Ramly, H., Morgenstern, N. R. and Cruden, D. M.,. Probabilistic Slope Stability Analysis for Practice. Can Geotech J, 39(3): El-Ramly, H., Morgenstern, N. R. and Cruden, D. M., 3. Probabilistic Stability Analysis of a Tailings Dyke on Pre-Sheared Clay-Shale. Can Geotech J, 4(1):19 8. El-Ramly, H., Morgenstern, N. R. and Cruden, D. M., 5. Probabilistic Assessment of Stability of a Cut Slope in Residual Soil. Géotechnique, 55(1): Eurocode 7, 4. Eurocode 7: Geotechnical Design. Part 1: General Rules. EN European Commitee for Standardisation (CEN). Fenton, G. A., 199. Simulation and Analysis of Random Fields. PhD Thesis, Princeton University, USA. Fenton, G. A. and Griffiths, D. V., 1993a. Seepage Beneath Water Retaining Structures Founded on Spatially Random Soil. Géotechnique, 43(6): Fenton, G. A. and Griffiths, D. V., 1993b. Statistics of Block Conductivity through a Simple Bounded Stochastic Medium. Water Resources Research, 9(6): Fenton, G. A. and Griffiths, D. V., 8. Risk Assessment in Geotechnical Engineering. John Wiley & Sons, New Jersey, USA. Fenton, G. A. and Vanmarcke, E. H., 199. Simulation of Random Fields via Local Average Subdivision. J Eng Mech, 116(8): Griffiths, D. V. and Fenton, G. A.,. Influence of Soil Strength Spatial Variability on the Stability of an Undrained Clay Slope by Finite Elements. In Slope Stability, Proceedings of Sessions of Geo-Denver, ASCE, Griffiths, D. V. and Fenton, G. A., 1. Bearing Capacity of Spatially Random Soil: the Undrained Clay Prandtl Problem Revisited. Géotechnique, 51(4): Griffiths, D. V. and Fenton, G. A., 4. Probabilistic Slope Stability Analysis by Finite Elements. J Geotech Geoenviron Eng, 13(5): Harr, M. E., Reliability-Based Design in Civil Engineering. McGraw-Hill. Hicks, M. A., 7. Risk and Variability in Geotechnical Engineering. Ed., Thomas Telford. Hicks, M. A. and Nuttall, J. D., 1. Influence of Soil Heterogeneity on Geotechnical Performance and Uncertainty: A Stochastic View on EC7. Proceedings 1 th International Probabilistic Workshop, Universität Stuttgart, Stuttgart, Hicks, M. A. and Onisiphorou, C., 5. Stochastic Evaluation of Static Liquefaction in a Predominantly Dilative Sand Fill. Géotechnique 55(): Hicks, M. A. and Samy, K., a. Influence of Heterogeneity on Undrained Clay Slope Stability. Quart J Eng Geol Hydrogeol, 35(1): Hicks, M. A. and Samy, K., b. Influence of Anisotropic Spatial Variability on Slope Reliability. Proceedings 8 th International Symposium on Numerical Models in Geomechanics, Rome,

12 5 Stochastic Approach to Slope Stability Analysis with In-Situ Data Hicks, M. A. and Samy, K., c. Reliability-based Characteristic Values: a Stochastic Approach to Eurocode 7. Ground Engineering, 35(1):3 34. Hicks, M. A. and Samy, K., 4. Stochastic Evaluation of Heterogeneous Slope Stability. Ital Geotech J, 38(): Hicks, M. A. and Smith, I. M., Class A Prediction of Artic Caisson Performance. Géotechnique, 38(4): Hicks, M. A. and Spencer, W. A., 1. Influence of Heterogeneity on the Reliability and Failure of a Long 3D Slope. Computers and Geotechnics, 37(7 8): Lloret, M., Hicks, M. A. and Wong, S. Y., 1. Soil Characterisation of an Artificial Island Accounting for Soil Heterogeneity. In GeoCongress 1, San Francisco, Lloret-Cabot, M., Hicks, M. A. and van den Eijnden, A. P., 1. Investigation of the Reduction in Uncertainty due to Soil Variability when Conditioning a Random Field using Kriging. Géotechnique letters, (3): Lloret-Cabot, M., Hicks, M. A. and Nuttall, J. D., 13. Investigating the Scales of Fluctuation of an Artificial Sand Island. Proceedings International Conference on Installation Effects in Geotechnical Engineering, Rotterdam, the Netherlands, Mostyn, G. R. and Soo, S. W., 199. The Effect of Autocorrelation on the Probability of Failure of Slopes. Proceedings 6 th Australia New Zealand Conference on Geomechanics: Geotechnical Risk, New Zealand, Nuttall, J. D., 11. Parallel Implementation and Application of the Random Finite Element Method. PhD Thesis, University of Manchester, UK. Paice, G. M. and Griffiths, D. V., Reliability of an Undrained Clay Slope Formed from Spatially Random Soil. Proceedings 9 th International Conference on Computer Methods and Advances in Geomechanics, Wuhan, Phoon, K.-K. and Kulhawy, F. H., Characterization of Geotechnical Variability. Can Geotech J, 36(4): Smith, I. M. and Griffiths, D. V., 4. Programming the Finite Element Method. 4 th ed., Wiley & Sons. Spencer, W. A., 7. Parallel Stochastic and Finite Element Modelling of Clay Slope Stability in 3D. PhD Thesis, University of Manchester, UK. Tang, W. H., Yucemen, M. S. and Ang, A. H. S., Probability-Based Short-Term Design of Slopes. Can Geotech J, 13(3):1 15. Uzielli, M., Vannucchi, G. and Phoon, K.-K., 5. Random Field Characterisation of Stress- Normalised Cone Penetration Testing Parameters. Géotechnique, 55(1):3. Vanmarcke, E. H., Reliability of Earth Slopes. J Geotech Eng Div, 13(11): Vanmarcke, E. H., Random Fields: Analysis and Synthesis. Cambridge, Massachusetts: The MIT Press. Wickremesinghe, D. S. and Campanella, R. G., Scale of Fluctuation as a Descriptor of Soil Variability. Proceedings Conference on Probablistic Methods in Geotechnical Engineering, Canberra, Wong, S. Y., 4. Stochastic Characterisation and Reliability of Saturated Soils. PhD Thesis, University of Manchester, UK. 538