PSI - Issue 10

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

123

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

4

σ 1

y = a + xb = a + x tan( β )

ε 3

y 3

y 1 y 2

ε 2

ε 1

β

a

σ 3

x 1

x 2

x 3

Fig. 2. Graphical representation of the regression model for the typical triaxial test.

4. Application of the first order reliability method to triaxial test data

The FORM method makes use of the second moment statistics (the mean and the standard deviation) of the random variables and assumes a linearized form of their performance function (e.g. z =g( X 1 , …, X n )) at the mean values of the random variables and independency between all variables. Truncating at the linear terms the Taylor expansion of the performance function about the mean, it is possible to obtain the first order approximation of the variance ( σ z 2 ) of the true mean ( μ z ) of z . Assuming uncorrelated non- andom variables X 1 , …, X n , the approximation of the variance is given by Eq.(13), an equation commonly used to estimate the uncertainties by error propagation for laboratory tests results.

n

n

  2

 2











2

2

var( ) g X X SE

g X SE

 z

  

   

(13)

i

i

, d z

i

X

i

i

i

1

1





The sample variance var( X i )= s Xi i variable (where X i is tan φ or c ) relates to the standard error SE Xi , while SE Xi is a quantitative measure of the corresponding uncertainty u Xi (i.e. u Xi = SE Xi ). The variances s tanφ 2 and s c 2 of the Mohr – Coulomb constants may be now calculated by applying Eq.(13) and considering Eqs(11,12) as the performance functions (i.e. c =g( a , φ ) and tan φ =g( b ) or φ =g( b )):     2 2      b SE b SE ,     2 2 tan tan      b SE b SE (14) d,Xi ) 2 of a X 2 =( S

   2 2  SE

     2 2 c a SE c a SE c      



(15)

where:

2 2 ( 1) 1 ( 1) / ( 1) )     b b 2 2

n ta 

 2 1 1 ( 1) (     b 2 2







,





(16)

3/2

b

b b







 

3

b

b

1

)





,

2

   sin c

(   a

cos

   c a

2 sin cos ) ( / 

1 / 2 ) (

) 



(

1 –

) 

(17), (18)

Laboratory results from the Herakleion marl (Table 1, with 27 specimens and 9 samples) are considered. Treating each specimen separately (i.e., sample size n =27), the best estimates of constants a and b of Eq.(5) are determined from a linear regression (i.e., Eqs.(6,7)), which gives a m =229.6 kPa and b m =3.02553. Applying these values into Eqs.(12,11) we get c m =66.0 kPa and (tan φ ) m =0.58225 (best estimate, Fig.3). Fig.3 also shows all test data. The FORM (Eqs.( 15,14,9,8)) gives SE c = u c =15,5 kPa and SE tanφ = u tan φ =0.0347. A t-student distribution for n -2 dof (i.e., Eqs.(1,3)) to account for error propagation leads to c k1 =39.60 kPa and (tan φ ) k1 =0.52297 (Fig.3, characteristic 1). Treating each sample separately we get the Mohr – Coulomb failure envelope constants of Table 2 (sample size n =9). The remaining characteristic envelopes of Fig.3 are explained in the next section.