Issue 50

G. Belokas, Frattura ed Integrità Strutturale, 50 (2019) 354-369; DOI: 10.3221/IGF-ESIS.50.30 360 0 50 100 150 200 250 0 50 100 150 200 250 300 350 400 450 500 550 Shear stress τ (kPa) Normal stress σ n ' (kPa) Experimental Data Best estimate Characteristic 1 Best estimate (alternative) Characteristic 4 (alternative) Figure 4 : Best estimates and corresponding characteristic envelopes comparison. S TATISTICAL MEASURES FOR THE TYPICAL TRIAXIAL COMPRESSION TEST BY FORM n the typical triaxial compression test the Mohr – Coulomb failure criterion parameters are determined indirectly. For this test, a constant horizontal radial stress (the cell pressure) σ r = σ c = σ 3 is applied to a cylindrical specimen, while the reaction of the axial stress is measured (Δ σ a , σ 1 = σ 3 +Δ σ a ). We can consider that the cell pressure is an accurate ob- servation (i.e. the non – random). The Mohr – Coulomb failure criterion in terms of principal stresses is given by Eq.(6), in which the statistical measures of constants a and b can be determined directly from the two variables models, while the best estimates of the mean for c and tan φ constants are calculated indirectly by the transformation of Eqs.(19,20), respectively. sin φ =( b –1)/(1+ b )  φ =asin[( b –1)/(1+ b )] and tan φ =tan{asin[( b –1)/(1+ b )}] (19) c = a (1 – sin φ )/(2cos φ ) (20) A way to calculate the uncertainties of c and tan φ is to apply the FORM , which method makes use of the second moment statistics (the mean and the standard deviation) of the random variables and assumes a linearized form of their perform- ance function (e.g. z =g( X 1 ,…, X n )) at the mean values of the random variables and independency between all variables. Truncating at the linear terms the Taylor expansion of the performance function about the mean, it is possible to obtain the first order approximation of the variance ( σz 2 ) of the true mean ( μ z ) of z . Assuming uncorrelated non – random variables X 1 , …, X n , the approximation of the variance is given by Eq.(21), an equation commonly used to estimate the uncertainties by error propagation for laboratory tests results.   2 2 2 2 , 1 1 var( )                         i n n z i d z X i i i i g g X SE SE X X (21) The sample variance var( X i )= s Xi 2 =( S d,Xi ) 2 of a X i variable (where X i is tan φ or c ) relates to the standard error SE Xi by Eq.(4), while SE Xi is a quantitative measure of the corresponding uncertainty u Xi (i.e. u Xi = SE Xi ). The variances s tanφ 2 and s c 2 of the Mohr – Coulomb constants may be now calculated by applying Eq.(21) and considering Eqs.(19,20) as the perform- ance functions (i.e. c =g( a , φ ) and tan φ =g( b ) or φ =g( b )):   2 2        b SE SE b   ,    2 2 tan tan           b SE SE b (22)     2 2 2 2                    c a c c SE SE SE a (23) I