Issue 50

G. Belokas, Frattura ed Integrità Strutturale, 50 (2019) 354-369; DOI: 10.3221/IGF-ESIS.50.30 368 Figure 12 : Influence of each uncertainty coefficient on the probability of SF <1. and their corresponding critical slip surface ( θ cr ) are presented. The material unit weight has been taken γ soil,m =22kN/m3 with uncertainty u γ =2kN/m 3 . According to the calculations, only characteristic 2 gives acceptable safety margin ( SM >0) and safety factor ( SF >1.00) and. The critical failure plane determined by SM and SF do not coincide. Characteristic values 1 and 4 were derived from the statistical measures used in the above analyses. It is obvious that probabilistic and deterministic analyses can give the opposite result. In our case the deterministic analysis was more conservative, which has to do with the selection of the partial safety factors. i c ki (kPa) tan φ ki c di (kPa) tan φ di SM i θ cr,SM,i ( o ) FS i θ cr,FS,i ( o ) 1 39.60 0.52297 31.38 0.41837 -133.35 41 0.947 42 2 55.65 0.56288 44.52 0.45030 431.74 43 1.174 40.5 3 43.43 0.49395 34.74 0.39516 -86.60 40.5 0.967 41 4 29.82 0.50999 23.86 0.40799 -466.99 40 0.814 43.5 Table 11 : Deterministic stability calculation with design parameters.. C ONCLUSIONS Reliability methods can be incorporated for the determination of soil strength statistical measures and for probabilistic limit equilibrium analyses. Regarding soil strength, the determination of the uncertainties of the Mohr – Coulomb failure criterion constants c and tan φ from results of the typical triaxial test can be calculated through an error propagation technique, such as the FORM , applied to an appropriate performance function. This function relates the measured ( σ 1 ) and applied ( σ 3 ) quantities to the material constants ( c , φ ) of the M-C failure criterion by a nonlinear relation- ship. The FORM was applied to triaxial data and the results from different approaches for calculating the best estimate of the mean, the uncertainties and the characteristic values of the M-C constants were compared. For the approaches con- sidered, the direct application of the FORM gave a lower uncertainty, since the consideration of each single specimen as independent increased the sample size. The uncertainties together with the corresponding best estimates were also used to estimate the characteristic failure envelope. Regarding the characteristic failure, envelope four different approaches were presented with different results. The FORM gave the more reliable results. In any case, engineering judgement on the results is necessary. The results were applied to a simple planar failure problem, which allows for a direct comparison of the various statistical measures of the SM and SF , since a single failure surface is considered. It was observed that the maximum probability of having a SM <0 or SF <1 does not correspond to the minimum best estimate of the SM or SF . This is influenced by the non – linear relationship of SM and u SM (or SF and u SF ) function and needs further exploring. The sensitivity analysis showed that the only influential uncertainty was that of cohesion, which generally happens to have the greater variability with regards to other soil constants. A sensitivity analysis is recommended in probabilistic analyses in order to determine the influence of each uncertainty separately. R