Issue 50

G. Belokas, Frattura ed Integrità Strutturale, 50 (2019) 354-369; DOI: 10.3221/IGF-ESIS.50.30 356 The use of the corrected variance ( s X 2 =var( X ), Eq.(2)) instead the uncorrected sample variance ( σ X 2 ), implies that there is little 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 usually small, despite that population may follow a normal distribution. Therefore, the direct use of the statistical methods may not be applicable, because they may lead to non-representative values for the soil mass (see [2]). In order to account for this difficulty on the statistical error, direct values (e.g. [2,5]) or semi-empirical methods (e.g. [5, 6]) have been proposed for the estimation of the variation coefficient V . The V coefficient can also be used to include other types of errors, such as the error uncertainty due to ground spatial variability, the measurement error and the trans- formation uncertainty of the empirical equations application ([6]). However, a good knowledge of the statistical background can provide a better understanding on the selection of the cautious estimate (see [5,7]), or even to apply Bayesian statistical methods (e.g. [2,8]), not only in cases of a small sample but also of complex uncertainties. Concerning the characteristic values used in deterministic analysis, due to the different types of uncertainties involved (e.g. inherent soil variability, sampling disturbance, Fig.1), engineering judgement is also recommended for their selection (e.g. [9]). Such judgement should evaluate the relative importance of the following uncertainties (see also [6]):  Sampling quality (sampling type and sample condition).  The extent of the in-situ and laboratory investigation (the number and spatial distribution of samples and in-situ and laboratory tests).  The quality of the laboratory tests (accreditation and uncertainty of the laboratory measurements)  The spatial variability of the parameters and samples distribution with respect to the extent of the considered mechanism of geotechnical soil model. In addition, the following should also be taken into account: a) existing experience and data on similar soil units (including their uncertainty) and b) the failure mechanism with respect to the geotechnical profile (e.g. generalized vs local failure, short term vs long term conditions, small – large strains). For a single variable model (e.g. undrained strength, S u ) and a small sample size ( n ) of laboratory data with unknown standard deviation of the population (the usual case for geotechnical engineering) and assuming a normally distributed population, the resulting estimated distribution follows the Student t-distribution. The estimated characteristic value ( X k ), which corresponds to a probability P( X k < μ )=1- p =1- α /2 (i.e. in a certain percentage of the cases the expected value of the true mean, μ , is greater than X k ), is then given by Eq.(3), in which SE X is the sample standard error given by Eq.(4). , 1 , 1 ,       k m p n X m p n d X X X t SE X t S n (3) ,  X d X SE S n (4) where t p,n-1 is the n -1 degrees of freedom student distribution confidence parameter for the one-sided 1- p lower confidence limit of the true mean ( μ ). From Eqs.(1,3) we get the coefficient k = t p,n-1 / n 0.5 . Schneider [10] proposed the approximate re- lationship Χ k = X m –0.5 S d,X , which corresponds to p=5% and n =14. As already mentioned, Eqs.(3,4) correspond to the single variable model (e.g. strength determined from the unconfined compression test, in which q u =2 S u , where S u undrained shear strength). The Mohr – Coulomb failure criterion used in limit state analyses, is a two variables linear model that includes two constants (cohesion, c , and angle of shearing resistance, φ ) and two variables. The for the cases of: a) the direct shear test is given by Eq.(5), in which σ n is the imposed normal stress (the nonrandom or independent variable) and τ the measured – observed shear stress (the random or dependent variable) and b) the typical triaxial test is given by Eq.(6), in which σ 3 is the imposed cell pressure – radial stress (the non- random or independent variable) and σ 1 the measured – observed vertical stress (the random or dependent variable). τ = c + σ n tan φ = a+b·σ n (5) σ 1 =2 c ·cos φ /(1–sin φ )+ σ 3 (1 + sin φ )/(1–sin φ ) =a+b·σ 3 (6) With regards to the experiments, if x i is the imposed value and y i is the corresponding observed value of a sample size n , the linear model is given by Eq.(7), where ε i is the error and a and b the model constants. Eq.(7) gives a best estimate of the mean of y for a given x . The measurement error ε i is the deviation of the y i from its best (deterministic) estimate y = a + x i b (Fig.2). In simple linear regression, the best estimate of the constants may be derived from minimizing the error square sum. y i = a + x i b + ε i (7)