Issue 50

G. Belokas, Frattura ed Integrità Strutturale, 50 (2019) 354-369; DOI: 10.3221/IGF-ESIS.50.30 366 It is obvious that the minimum values of the estimated means SM m =1435.08kPa and SF m =1.639 develop for different critical angle θ cr of the plane of failure. However, the corresponding values of SM p=5% and SF p=5% (calculated in Tables 7 and 8 by applying Eq.(38)) are not the minimum ones, nor are p SM<0 and p SF<1 the maximum ones, as their minimum and maximum values develop for different critical slopes θ cr (presented in Tables 9 and 10). The differences for the problem examined are of the order of 4% for the SM with respect to the min( SM p=5% ) value and of the order of 1 to 2% for the SF with respect to the min( SF p=5% ). Calculations based on min SM for p=5% c m (kPa) (tan φ ) m u c (kPa) u tanφ min( SM p=5% ) (kPa) θ cr ( o ) u SM (kPa) SM m (kPa) V SM p SM<0 66.00 0.58225 15.46 0.03470 554.18 46 546.51 1453.11 2.659 0.39 64.34 0.59415 18.56 0.04526 340.75 46 654.42 1417.49 2.165 1.54 Table 9.1 : Calculated minimum SM for p=5% and the corresponding θ cr , u SM and SM m values. Calculations based on max p for SM<0 c m (kPa) (tan φ ) m u c (kPa) u tanφ max( p SM<0 ) (%) θ cr ( o ) u SM (kPa) SM m (kPa) V SM SM p=5% (kPa) 66.00 0.58225 15.46 0.03470 0.40 45 556.72 1475.12 2.650 559.40 64.34 0.59415 18.56 0.04526 1.54 45 667.07 1440.88 2.160 343.64 Table 9.2 : Calculated maximum p for SM <0 and the corresponding θ cr , u SM and SM m values. Calculations based on min SF for p=5% c m (kPa) (tan φ ) m u c (kPa) u tanφ min( SF p=5% ) (kPa) θ cr ( o ) u SF (kPa) SF m (kPa) V SF p SF<0 66.00 0.58225 15.46 0.03470 1.231 42.5 0.260 1.659 6.383 0.56 64.34 0.59415 18.56 0.04526 1.141 43.5 0.317 1.662 5.243 1.83 Table 10.1 : Calculated minimum SF for p=5% and the corresponding θ cr , u SF and SF m values. Calculations based on max p for SF<0 c m (kPa) (tan φ ) m u c (kPa) u tanφ max( p SF<0 ) (%) θ cr ( o ) u SF (kPa) SF m (kPa) V SF SF p=5% (kPa) 66.00 0.58225 15.46 0.03470 0.62 45 0.287 1.718 5.976 1.245 64.34 0.59415 18.56 0.04526 1.90 45 0.338 1.701 5.035 1.145 Table 10.2 : Calculated maximum p for SF <0 and the corresponding θ cr , u SF and SF m values.. The second set of soil strength parameters lead to more conservative results. The comparison of the results of Tables 9.1 and 9.2 with Table 7 and of Tables 10.1 and 10.2 with Table 8 shows that for the probabilistic analyses it is preferable to determine the critical surface that corresponds to the minimum calculated SM or SF for a probability of exceedance 5% (or 90% confidence level, i.e. min SM p=5% or min SF p=5% ), instead of calculating the critical surface of the minimum mean value SM m or SF m first and then the corresponding SM p=5% or SF p=5% . This happens because there is no linear relationship between SM m (or SF m ) and u SM (or u SF ) for a monotonically increasing or reducing failure plane angle. Concerning the influence of the individual uncertainties Fig.9 shows their influence on the minimum SM and Fig.10 on the probability of having an SM <0. Fig.11 shows their influence on the minimum SF and Fig.12 on the probability of having an SF <1. It is apparent that for this specific problem, the most influential factor on the probabilistic SM or SF is the uncertainty of cohesion. This is important because cohesion generally has a greater uncertainty from the angle of shearing resistance. The dependence SM and SF with the various uncertainties present the same trends. The above results will now be compared with the results from the deterministic analysis, for which the application of Eurocode 7 has been considered and more specifically the design analysis 3 (DA-3). DA-3 has become the national choice