Issue 50

G. Belokas, Frattura ed Integrità Strutturale, 50 (2019) 354-369; DOI: 10.3221/IGF-ESIS.50.30





sin

 β θ

1 1 = tan sin 2   SM θ 

  



2

cos

γH

θ

(32)

sin

β







 

sin

 β θ

1 1 = sin 2  

  

SM





2

cos tan sin 

H

θ φ θ

(33)

sin

γ θ

β





2

2

2

SF

SF

SF

  

  

  

  







  

 

 

2 c   u

2

2

u SF

u

u

(34)







tan





tan

c









 SF c H β θ θ β     (35) 1 = tan tan SF      2 1 = sin sin sin

(36)

2 SF c γ  2 = -

1







sin γ H β θ θ β  sin sin











= 2c / γ sin  

cos tan sin H β θ θ φ β H β θ θ β     sin sin sin

SF

 

(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 , 

(38)

( ) m SF SF k u SF   

SM m = min( SM m ) (kPa)

SM p=5% =

c m (kPa) (tan φ ) m

u c (kPa)

θ cr ( o )

u SM (kPa)

p (SM<0) (%)

u tanφ

SM m

- ku SM

V SM

(kPa)

66.00 0.58225 15.46 0.03470 64.34 0.59415 18.56 0.04526

1435.08 1396.92

48 527.49 567.43 48 631.85 357.62

2.721 2.211

0.33 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 .

SF m = min( SF m ) (kPa)

SF p=5% = SF m - ku SF (kPa)

c m (kPa) (tan φ ) m

u c (kPa)

θ cr ( o ) 40 40

u SF (kPa) 0.241 0.284

p (SF<1) (%)

u tanφ

V SM

66.00 0.58225 15.46 0.03470 64.34 0.59415 18.56 0.04526

1.639

1.243 1.163

6.800 5.740

0.40 1.33

1.6300

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 .

365