Issue 50

G. Belokas, Frattura ed Integrità Strutturale, 50 (2019) 354-369; DOI: 10.3221/IGF-ESIS.50.30 365   2 sin 1 1 = cos tan sin 2 sin          β θ SM γH θ θ β  (32)     2 sin 1 1 = cos tan sin sin 2 sin           β θ SM H θ φ θ γ θ β (33)   2 2 2 2 2 2 tan tan c SF SF SF u SF u u u c                                (34)   2 1 = sin sin sin SF c H β θ θ β     (35) 1 = tan tan SF     (36)   2 2 1 = - sin sin sin SF c γ γ H β θ θ β         = 2c / γ sin cos tan sin sin sin sin SF H β θ θ φ β H β θ θ β            (37) A slope geometry example of β =60 ο and Η =25m and the material properties from Tables 5 and 6 are considered to exhibit the safety probability calculation. A deterministic calculation of the best estimate of the mean of SM m and SF m from Eqs.(28,29) respectively is computed first, by the best estimates of the mean of the soil properties (i.e. c m , tan( φ ) m and γ m ). Then a normal distribution for SM and SF is applied, in order to estimate their value for a probability not greater than 5% (i.e., Eq.(38) with k =1.64485). The uncertainties u ( SM ) and u ( SF ) are calculated by Eqs.(30,34) applying the best estimates of the mean and the corresponding uncertainties of c , tan( φ ) and γ (error propagation considers the standard error of the mean, i.e. u = SE ). The values of SM and SF for probability p =5% and the corresponding probability p for SM <0 and SF <0, all calculated for the critical plane θ cr that gives the minimum SM m , are presented in Tables 7 and 8. The two sets of statistical measures from Table 5 for soil strength properties have been used. ( )    m SM SM k u SM ,  ( ) m SF SF k u SF    (38) c m (kPa) (tan φ ) m u c (kPa) u tanφ SM m = min( SM m ) (kPa) θ cr ( o ) u SM (kPa) SM p=5% = SM m - ku SM (kPa) V SM p (SM<0) (%) 66.00 0.58225 15.46 0.03470 1435.08 48 527.49 567.43 2.721 0.33 64.34 0.59415 18.56 0.04526 1396.92 48 631.85 357.62 2.211 1.35 Table 7 : Calculations of SM for p=5% and p for SM <0 based on the minimum best estimate of the SM m and the corresponding θ cr . c m (kPa) (tan φ ) m u c (kPa) u tanφ SF m = min( SF m ) (kPa) θ cr ( o ) u SF (kPa) SF p=5% = SF m - ku SF (kPa) V SM p (SF<1) (%) 66.00 0.58225 15.46 0.03470 1.639 40 0.241 1.243 6.800 0.40 64.34 0.59415 18.56 0.04526 1.6300 40 0.284 1.163 5.740 1.33 Table 8 : Calculations of SF for p=5% and p for SF <1 based on the minimum best estimate of the SF m and the corresponding θ cr .