PSI - Issue 10

G. Belokas / Procedia Structural Integrity 10 (2018) 120–128

127

G. Belokas / Structural Integrity Procedia 00 (2018) 000 – 000

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( )    m SM SM k u SM

(25)

In § 4 the Mohr – Coulmb mean values are given, which are c m =66.00kPa, tan( φ ) m =0.58225 and the uncertainties u c =15.46 kPa, u tan( φ ) =0.03470 (applying FORM , see Table 3). The resulting safety margin is SM m =1435.08 kPa, which develops at a failure plane angle of θ cr =48 o , with an uncertainty of u SM =527.49 kPa. These correspond to SM = 567.43 kPa. The SM m and u SM give a 0.33% probability of having a SM <0, according to normal distribution. The minimum safety factor for the c m and tan( φ ) m is FS m =1.639 with θ cr =40 o . However, analyzing all possible failure planes we find that the critical SM (i.e. the minimum for 5% probability) corresponds to a different failure plane angle than that of the minimum SM m and a lower SM . The minimum safety margin for 5% probability is SM =554.18 kPa at a critical plane of θ cr =46 o , with an uncertainty u SM =546.51 kPa and a 0.39% probability of having a SM <0. Yet, the maximum probability of having a SM <0 is 0.40% for a plane of θ cr =45 o , corresponding to a SM =559.40 not the minimum one. A recalculation incorporating the properties from paragraph 5 ( c m =64.34 kPa, tan( φ ) m =0.59415 and u c =18.56 kPa, u tan( φ ) =0.04526) gives slightly different results. The best estimate for safety margin is SM m =1396.92kPa at a failure plane angle of θ cr =48 o , with an uncertainty of u SM =631.85kPa. These correspond to a safety margin of SM =357.62 SM =654.42 kPa and a 1.52% probability of having a SM <0. Again the maximum probability of having a SM <0 is 1.54% for a plane of θ cr =45 o , which corresponds to a SM =343.54 not the minimum one. These results are more conservative than the previous probabilistic analysis.Therefore, for the probabilistic analyses it is preferable to check that the maximum probability of SM <0 is less than 5% and not to consider either one of the minimum safety margin best estimate or the minimum safety margin for 5% probability. This happens because there is no linear relationship between SM m and u SM for a monotonically increasing or reducing failure plane angle. Fig.5 shows the influence of each uncertainty coefficient to the minimum SM and Fig.6 to the probability of having an SM <0. For this specific problem, the most influential factor is the uncertainty of cohesion, which is generally greater than the rest of the uncertainties. kPa and a 1.35% probability of having a SM <0, according to normal distribution. The minimum safety margin for 5% probability is SM =340.57kPa for a θ cr =46 o , with an uncertainty u

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Safety Margin, SM (kPa)

Safety Margin, SM (kPa)

Safety Margin, SM (kPa)

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Uncertainty u c (kPa) ο ) Fig. 5. Influence of each uncertainty coefficient on the value of SM = SM m - ku SM . Uncertainty u φ ( ο ) Uncertainty u γ (

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Probability of SM <0

Probability of SM <0

Probability of SM <0

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Uncertainty u c (kPa) ο ) Fig. 6. Influence of each uncertainty coefficient on the probability of SM <0. Uncertainty u φ ( ο ) Uncertainty u γ (