PSI - Issue 10

G. Belokas / Procedia Structural Integrity 10 (2018) 120–128

121

G. Belokas / Structural Integrity Procedia 00 (2018) 000 – 000

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presents an alternative for geotechnical engineering limit states analyses (FORM, e.g. Baecher & Christian (2003); Orr & Breysse (2008)), a method for the estimation of error propagation, such as the uncertainty of the experimental results from laboratory tests (e.g. GUM:1995, ISO/IEC Guide 98-3:2008), and applies directly to cases where a closed form of analytical solution exists. Under the framework of EC7, the reliability analysis (uncertainty calculation) for a limit equilibrium problem can be performed with respect to the safety margin (SM). The safety margin is expected to have a value of SM ≥ 0.0 for a certain level of confidence, which 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 internal and exter nal loads (permanent and mobile) and the strength constants, as well as the spatial variability and uncertainty of the model (see Fig.1 from Kulhawy (1992)). One of the most critical components on the overall uncertainty is strength uncertainty both due to spatial variability and model uncertainty.

IN SITU or LAB MEASUREMENT

TRANSFORMATION MODEL

ESTIMATED SOIL PROPERTY

SOIL

Inherent soil variability

Statistical uncertainty

Model uncertainty

Data scatter

Sampling disturbance

Measurement error

Fig. 1. Factors affecting the property uncertainty (Kulhawy (1992), modified).

Limit equilibrium methods of analyses require an estimation of Mohr – Coulomb strength parameters, cohesion ( c ) and angle of shearing resistance ( φ ). Their best estimates and their corresponding variations (or uncertainties) may be calculated either directly from 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 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. A deterministic analysis according to EC7 (ΕΝ -1997-1) makes use of the characteristic values of actions ( F k ) and of the soil parameters ( X k ). Specifically, Χ k is a cautious estimate of the mean value (i.e. the best estimate of the mean, Χ m ) (see ΕΝ -1997- 1, 2.4.5.2 §2) , which has to be representative for 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). Therefore, for a sample size, n , X k corresponds to a worse value governing the occurrence of the soil parameter with a calculated prob ability not greater than 5% (i.e. 90% confidence interval, ΕΝ -1997- 1, 2.4.5.2 §11 ), given by Eq.(1) ( k is the confidence level coefficient for a given probability distribution, S d,X the sample standard deviation and V the variation coefficient):   , , 1 /         k m d X m d X m X X k S X k V V S X (1) 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. The use of the corrected variance ( s X 2 =var( X ), Eq.2) instead of the uncorrected sample variance ( σ X 2 ), implies that there is not great confidence that σ X 2 is a close estimate of the population variance, σ 2 . Had it been σ X 2  σ 2 , the sample would closely follow a normal distribution, which is not the case in geotechnical engineering investigations as sample size is often very small. However, the population may follow a normal distribution.     2 2 2 , 1 1 var( ) 1        n X d X i m i X s S X X n (2) 2. Characteristic values of soil properties