Issue 50

G. Belokas, Frattura ed Integrità Strutturale, 50 (2019) 354-369; DOI: 10.3221/IGF-ESIS.50.30 355 applied to the uncertainty estimation of the experimental results of laboratory tests (e.g. GUM: 1995, see [3]) and can be applied directly wherever a closed form of analytical solution exists. Under the framework of EC7 , the reliability analysis (uncertainty calculation) for a limit equilibrium problem can be per- formed with respect to the safety margin ( SM ), which is expected to have a value of SM ≥0.0 for a certain level of con- fidence. That requires the knowledge of the uncertainty of the parameters that affect the value of SM . These parameters normally include the value and the uncertainty of the external and internal loads (permanent and mobile) and of the strength constants, as well as the spatial variability and uncertainty of the model (see Fig.1 based on Kulhawy [4]). One of the most critical components on the overall SM uncertainty is the strength uncertainty both due to spatial variability and model uncertainty. On the other hand, a deterministic analysis requires the best estimate of the loading conditions and the materials properties, which correspond to a specific level of confidence. Inherent soil  variability SOIL IN SITU or LAB  MEASUREMENT TRANSFORMATION  MODEL ESTIMATED SOIL  PROPERTY Data scatter Statistical  uncertainty Measurement  error Model  uncertainty Sampling  disturbance Figure 1 : Factors affecting the property uncertainty (based on Kulhawy [4]). Therefore, for the Mohr – Coulomb strength parameters, cohesion ( c ) and angle of shearing resistance ( φ ), which are used in limit state analysis, an estimation of their mean value and their corresponding variation (or uncertainty) is required. These measures may be calculated either by direct application of statistical methods (e.g. for the direct shear test) or by an error propagation method (e.g. FORM for the typical triaxial test). The present work explores on the application of the FORM for the statistical evaluation of the strength parameters and for the slope stability analytical solution of a wedge failure mechanism. Issues with respect to the design and characteristic strength are also discussed, as well as the capability to apply the FORM into a general limit equilibrium slope stability problem. S TATISTICAL MEASURES OF SOIL PROPERTIES deterministic analysis requires the knowledge of the best estimates of its individual components, i.e. loading conditions and material properties, for a specific confidence level. These correspond to the characteristic values of actions ( F k ) and of soil parameters ( X k ) defined in EC7 (ΕΝ-1997-1). Focusing on soil parameters, for any specific parameter, X , that affects the development of the limit state condition, its characteristic value Χ k is defined as a cautious estimate of the mean value, (i.e. of the best estimate) of the mean, Χ m (see ΕΝ-1997-1). The selection of this characteristic value has to be representative of the volume involved in the considered failure mechanism and it can depend on the type of the failure mechanism (e.g. local vs generalized failure). When the sample size, n , is large enough to apply statistical methods, the characteristic value corresponds to a worse value governing the occurrence of the soil parameter with a calculated probability not greater than 5% (it is 90% confidence interval, see ΕΝ-1997-1), which for a single variable model is given by Eq.(1).   , , 1 /         k m d X k d X m X X k S X k V V S X (1) where k is the confidence level coefficient for a given probability distribution, S d,X is the sample standard deviation and V the variation coefficient. For a specific sample with unknown standard deviation, the S d,X is the corrected standard deviation, which is related to the corrected – unbiased sample variance ( s 2 ) according to Eq.(2).     2 2 2 , 1 1 var( ) 1        n X d X i m i X s S X X n (2) A