Issue 50

G. Belokas, Frattura ed Integrità Strutturale, 50 (2019) 354-369; DOI: 10.3221/IGF-ESIS.50.30 358     2 1 2 2 1 1 var( | ) 1 2            n i i b n n i i i i x y SE n x x x x  (14)     2 2 2 1 2 1 1 1 1 1 2                   n i n n i a b i i n i i i i x SE SE x n n n x x  (15) S TATISTICAL MEASURES FOR THE DIRECT SHEAR TEST – TWO VARIABLES MODEL he direct shear test gives a direct determination of c and tan φ of the linear Mohr – Coulomb criterion (Eq.(5)), with σ n = x and τ = y . For this occasion: a) the best estimates of the mean are c m = a m and (tan φ ) m = b m given by Eqs.(9,8), respectively (with x i = σ ni and y i = τ i being the data measurements) and b) the standards error estimators SE (tanφ) = SE b and SE c = SE a are given by Εqs.(14,15). Concerning the confidence intervals of the linear regression coefficients estimators, the standard method relies on the normality assumption, which is justified if either: a) the errors in the regression are normally distributed (this leads to a t-statistic) or b) the number of observations n is sufficiently large (in this case the estimator is approximately normally distributed). Applying a statistical t-test, the linear regression random variables follow a student’s t-distribution with n -2 degrees of freedom ( dof ), i.e. Tc = ( c m – c )/ SE c ~ t n-2 , T (tanφ) =[(tan φ ) m –tan φ )/ SE (tanφ) ~ t n-2 , where c and tan φ represent the true mean values (or population mean values). The t-test includes the assumptions that the sample is representative of a specific soil unit, the observations are independent, while the variation of x = σ n depends only on the uncertainty of the laboratory measurement. The σ n variation has a negligible influence on the total uncertainty. Ignoring the influence of sampling disturbance and spatial variability, the characteristic values of cohesion, c k , and angle of shearing resistance, (tan φ ) k are given by Eqs.(16,17) and their standard errors by Eqs.(14,15), respectively. The best estimates and their standard errors can be used for probabilistic analyses, by incorporating the standard errors as quantitative measures of the corresponding uncertainties ( u c = SE tanφ = S d,tanφ / n 0.5 and u tanφ = SE tanφ = S d,tanφ / n 0.5 ). , 2    k m p n c c c t SE (16)     , 2 (tan ) tan tan    p n k m t SE    (17) Eqs.(16,17) may be used to give the characteristic Mohr – Coulomb failure envelope: τ= c k + σ n (tan φ ) k . Alternatively, an estimate of the characteristic failure envelope (see also [1,11]) can be obtained by incorporating the shear stress estimates for a specific probability (given by Eq.(18)). The resulting from Eq.(18) failure envelope is non-linear and needs to be ap- proximated by a linear regression to get the characteristic values of the Mohr – Coulomb failure criterion parameters.         2 2 2 2 1 1 1 1 tan 2                               n n m n n j n m j i i x τ c t n n x x     (18) Table 1 shows a fictional example for the application of the abovementioned relationships for a direct shear test. Typically, three specimens are obtained from each sample, which sample corresponds to a specific depth and location. Applying Eqs. (8,9) we obtain the best estimates of strength parameters and their corresponding uncertainties respectively: c m =23.95 kPa, (tan φ ) m =0.37950 (i.e. φ m =20.8 ο ) SE c = u c =5.3 kPa and SE tanφ = u tanφ =0.01561. Applying Eqs.(16,17) for a probability p =5% we get the following characteristic values: c k1 =14.5 kPa and (tan φ ) k1 =0.35186 (see Fig.3, characteristic 1). Applying a linear regression on the characteristic envelope derived from Eq.(18) we get the following characteristic values: c k2 =16.50 kPa and (tan φ ) k2 = 0.38500 (see Fig.3, characteristic 2), which is less conservative than the characteristic 1. T