PSI - Issue 10

G. Belokas / Procedia Structural Integrity 10 (2018) 120–128 G. Belokas / Structural Integrity Procedia 00 (2018) 000 – 000

126

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the horizontal, θ the angle of the plane of failure to the horizontal and N and T the normal and shear reaction forces on the failure plane. The safety margin is given by Eq.(20). The covariance, ρ R,D , between resistance and action is generally very small. Ignoring the covariance and applying the FORM on the safety margin the standard deviation of the safety is given by Eqs.(21-24), in which u c , u tan φ and u γ are the uncertainties of c , tan( φ ) and γ respectively. Exactly the same methodology is also applicable for the error calculation of the measures calculated from laboratory tests (ISO/IEC Guide 98-3:2008).

Γ

Β

W

Wcos θ

Wsin θ

H

T

N

β θ

ΑΒ = Η /sin θ

Α

Fig. 4. Geometry of planar two dimensional failure.







= c   SM H

sin γH β θ θ φ θ β θ cos tan sin sin sin   

 

1 2 2

(20)



   c u SM SM c u SM u    2 2 2 tan       





2

2

2

 SM u

  

(21)

tan





= sin   SM c H θ

(22)



 



2 tan = 0.5 sin 

  SM

 γH β θ

sin tan β



(23)





 



2 = 0.5 sin

SM γ

 H β θ θ φ θ

cos tan sin sin sin  θ β

 

(24)

A slope geometry of β =60 ο and Η =25m and the material properties from § 4 and § 5 are considered. Table 4 shows for the deterministic analyses, in which the four characteristic strength values have been used, factored by γ Μ =1.25 to give design values ( c d = c k / γ Μ , tan( φ d )=tan( φ k )/ γ Μ ). Minimum safety factor ( FS ) and safety margin ( SM ) and their corresponding critical slip surface ( θ cr ) are presented. The material unit weight has been taken γ soil,m =22 kN/m 3 with uncertainty u γ =2 kN/m 3 . Only characteristic 2 gives an acceptable safety factor ( FS >1.00) and safety margin ( SM >0). The critical failure plane determined by FS and SM do not coincide. The probabilistic approach assumes a normal distribution for SM , which SM is estimated for a probability not greater than 5% (i.e. Eq.(25) with k =1.64485). Best estimate SM m is calculated by applying the best estimates of soil properties (i.e. c m , tan( φ ) m and γ m ). Uncertainty u ( SM ) is calculated by Eqs.(21-24) applying the best estimates and uncertainties of c , tan( φ ) and γ (error propagation considers the standard error of the mean, i.e. u = SE ).

Table 4. Deterministic stability calculation with design parameters. i c ki (kPa) tan φ ki c di (kPa) tan φ di SM i

o )

o )

θ cr,SM,i (

FS

θ cr,FS,i (

i

1 2 3 4

39.60 55.65 43.43 29.82

0.52297 0.56288 0.49395 0.50999

31.38 44.52 34.74 23.86

0.41837 0.45030 0.39516 0.40799

-133.35 431.74 -86.60 -466.99

41 43

0.947 1.174 0.967 0.814

42

40.5

40.5

41

40

43.5