PSI - Issue 10

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

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Concerning the variation coefficient V , various values (e.g. Duncan (2000); Orr & Breysse (2008)) and some semi empirical methods for its estimation (e.g. Duncan (2000); Schneider and Fitze (2013)) have been proposed. Apart from the statistical error, the coefficient V can also take into account the error-uncertainty due to ground spatial variability, the measurement error and the transformation uncertainty of the empirical equations application (see Schneider and Fitze (2013)). A good knowledge of the statistical background can provide a better understanding of the cautious estimate (see Duncan (2000); Fellin (2005)) concept, or even to apply Bayesian statistical methods (e.g. Orr and Breysse (2008); Pohl (2011)) not only in cases of a small sample but also of complex uncertainties. Due to uncer tainties involved (e.g. inherent soil variability, sampling disturbance, Fig.1), we cannot ignore that engineering judge ment is also important for the selection of the characteristic value (e.g. Frank et al. (2004); Schneider and Fitze (2013)). For a small sample size ( n ) of laboratory data with unknown standard deviation of the population and assuming a normally distributed population, a Student t-distribution should be considered. The estimated characteristic value ( X k ), which corresponds to a probability P ( X k < μ )=1- p =1- α/2 , is given by Eq.(3). SE X = S d,X / n 0.5 is the sample standard error and t p,n-1 is the n -1 degrees of freedom student distribution confidence parameter for the one-sided 1- p lower con fidence limit of the true mean ( μ ). From Eqs.(1,3) we get the coefficient k = t p,n-1 / n 0.5 . Schneider (1997) proposed the approximate relationship Χ k = X m – 0.5 S d,X , which corresponds to p =5% and n =14. A common laboratory test for obtaining the Mohr – Coulomb failure criterion parameters is the typical triaxial compression test, in which the specimen is cylindrical. During this test, a constant horizontal radial stress (the cell pressure) is applied σ r = σ c = σ 3 , while the reaction of the axial stress is measured ( Δ σ a , σ a = σ 1 + Δ σ a ). Therefore, we may consider that the cell pressure is an accurate observation (i.e. the non – random or independent variable). The Mohr – Coulomb failure criterion in terms of principal stresses is given by Eq.(4), or in our case by Eq.(5), where a =2 c cos φ /(1 – sin φ ) and b =(1 + sin φ )/(1 – sin φ ) are the constants of the linear relationship between σ 1 and σ 3 . σ 1 – σ 3 = ( σ 1 + σ 3 )sin φ + 2 c · cos φ (4) σ 1 = σ 3 (1 + sin φ )/(1 – sin φ ) + 2 c · cos φ /(1 – sin φ ) = b · σ 3 + a (5) By expressing σ 1i = b·σ 3i + a + ε i (with y i = σ 1i , x i = σ 3i , Fig.2 ), where ε i =y i – a m – x i b m , Eqs. (6-9) are used to determine the best estimates of constants a , b and their uncertainties u ( a )= SE a , u ( b )= SE b , respectively. In Eqs.(6,7) , y x are the mean values of σ 1 , σ 3 measurements, respectively, S d,x , S d,y the sample standard deviation of x , y (Eq.(2)) and r xy the Pearson sample correlation coefficient (Eq.(10)). r xy is sensitive only to a linear relationship between two variables. It is | r xy | ≤1 and for r xy =1 there is a perfect direct (increasing) correlation. Applications in civil and geotechnical engineering of the two variables linear model have been presented by Baecher and Christian (2003), Kottegoda and Rosso (2008) and Pohl (2011), mainly for increasing undrained shear strength with depth. The estimates of the Mohr – Coulomb constants are given by Eqs.(11,12) respectively, which in our case represent the transformation model mentioned in Fig.1.   , ,  m xy d x d y b r S S ,   m m a y b x (6), (7)   2 2 1 1 1 2               n n b i i i i SE x x n , 2 1 1    n a b i i SE SE x n (8), (9)     2 2 2 2                            xy i i i i i i i i r n x y x y n x x n y y (10) sin φ = ( b – 1)/(1 + b )  φ = asin[( b – 1)/(1 + b )] and tan φ = tan{asin[( b – 1)/(1 + b )]} (11) c = a (1 – sin φ )/(2cos φ ) (12) , 1  , 1 ,  m p n d X   X X t   SE X t k m p n X S n (3) 3. Statistical measures for the typical triaxial compression test