PSI - Issue 10

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

125

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

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Another alternative for the characteristic envelope is first to compute the characteristic values constants of a and b by applying a t-student distribution (i.e. a k = a m - t p,n-2 SE a and b k = b m - t p,n-2 SE b ) for p =5% and n -2 dof and then apply Eqs.(12,11) to compute the Mohr – Coulomb characteristic constants c k and (tan φ ) k . This approach does not take into account the error propagation, leads to a k =139.86 kPa and b k =2.58981 of Eq.(5) and to c k3 =43.4 kPa and (tan φ ) k3 = 0.493949 of Mohr – Coulomb failure criterion. Finally, a common approach applied in engineering practice for the estimation of all statistical measures of c and tan φ is first to derive the Mohr – Coulomb constants ( c , tan φ ) for each sample separately (see Table 2) and then apply a statistical t-test on each constant independently for n /3-1 degrees of freedom (where n the complete number of speci mens and n /3 the number of specific locations of soil sampling). For the Herakleion marl case the n /3-1=8 degrees of freedom lead to: a) mean values (best estimates): c m =64.34 kPa, (tan φ ) m =0.59415, b) standard deviation: S d,c =55.69 kPa, S d,tanφ =0.13577, c) standard error: SE c =18.56 kPa, SE tanφ =0.04526 and d) characteristic values: c k4 = 29.82 kPa, (tan φ ) k4 =0.5099. Table 3 summarizes the comparison of all Mohr – Coulomb envelopes statistical measures. Note that the mean Mohr – Coulmb strength parameters are practically the same, with a difference on the standard errors. This results in great difference for the characteristic envelope and also influences probabilistic analyses.

Table 3. Mohr – Coulomb failure criterion constants and statistical measures for different approaches. Best estimate c m (kPa) (tanφ) m SE c (kPa)

SE tan( φ)

a m b m from linear regression of all σ 1 , σ 3 data points c m , (tanφ) m from Eq. 12 and 11 SE c and SE tan φ from FORM error propagation (this gives characteristic 1)

66.00

0.58225

15.50

0.03470

c m , (tanφ) m from mean values SE c and SE tan φ from single variable model (this gives characteristic 4)

64.34

0.59415

18.56

0.04526

Characteristic values

SE c (kPa)

SE tan( φ)

c k (kPa)

(tanφ) k

Characteristic 1 (from FORM): c k = c m - t p,n-2 SE c , tan( φ ) k = tan( φ ) m - t p,n-2 SE tan( φ ) Error propagation (FORM) for corresponding SE Characteristic 2: a k , b k from linear regression on σ 1pred , σ 3 data points σ 1 pred = a m + σ 3 b m ± t n-2 SE σ 1 c k , (tanφ) k from Eq. 12 and 11

39.60

0.52297

–

–

55.65

0.56288

–

–

Characteristic 3: a k = a m - t p,n-2 SE a , b k = b m - t p,n-2 SE b c k , (tanφ) k from Eq. 12 and 11

43.43

0.49395

–

–

Characteristic 4: c m and (tanφ) m from mean values c k = c m - t p,n-2 SE c , tan( φ ) k = tan( φ ) m - t p,n-2 SE tan( φ ) Errors from one variable model (Eq. 2) for c and tan φ

29.82

0.50990

–

–

6. Applications of the FORM to a simple planar failure problem

Reliability analysis can be performed with respect to the safety margin SM = R - E ( R is resistance and Ε action, e.g. EC7 ). In FORM application SM is preferable compared to the safety factor FS = R / E , because Ε in the denominator enhances any non-linearity effects, especially in the error propagation. The FORM is applied herein to a planar failure problem (Fig.4), which can be adapted to any kind of failure surface (e.g. Wu (2008)) of limit equilibrium problems (e.g. method of slices). W is the weight of the wedge, H the height of the slope, β the angle of the slope to