Issue 50

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

Figure 2 : Graphical representation of the regression model.

Assuming that: a) x i variation for each x i

is an accurate observation, b) each x i and d) the uncertainties of the y i

is an independent observation, c) the error ε i

has a constant

observations are equivalent (otherwise weight coefficients are required), the best mean estimators for the linear regression coefficients are given by Eqs.(8,9). Observation x i can be a predetermined imposed loading (e.g. the σ n in direct shear and the σ 3 in the typical triaxial), while observation y i a measured reaction (e.g. the τ in direct shear and the σ 1 in the typical triaxial).          , 1 1 1 1 2 2 , 2 1 , , 1 n n n n i i i i i i d y i i i i m xy n n n d x i x x y y x y y x S Cov x y n b r Var x y S x x                     (8)



i   x

x

 

i

n



1

i



1

1

i

i





(9)

m m a y b x  

, y i ) are the data measurements of the two dimensional sample, , y x are their mean values, S d,x and S d,y are the

In Eqs.(8,9) ( x i

sample standard deviation of x and y measurements (Eqs.(10,11)) and r xy

the Pearson sample correlation coefficient (given

by Eq.(12)). The r xy =1 the correlation is a perfect direct, i.e. increasing). Moreover, an unbiased estimate of the variance of y ( x ) with n-2 degrees of freedom is given by Eq.(13). The standard error estimators SE b and SE a of the b and a regression coefficients are given by Εqs.(14,15), re spectively. Some applications in civil and geotechnical engineering of the two variables linear model have been presented by Baecher & Christian [1], Pohl [8] and Kottegoda & Rosso [11], as for instance the case of a variation with depth. A classic example is the increasing undrained shear strength with depth. The application of this model in the Mohr – Coulomb strength failure criterion has some individual characteristics that will be presented later. is sensitive only to a linear relationship between two variables (| r xy |≤1, when r xy

1

 n



 2

S

 x x



(10)

, d x

i

1  i

n



1

1

 n



 2

S

 y y



(11)

, d y

i

1  i

n



1

        i i i i i i i n x y x y n x x n y   2 2 2   

(12)

r



xy



2

  

     

  

y

i

 n

2

i 

1 2

 n S

ˆ ˆ





i i y a bx     i

,

| var y x S 





(13)

1

i



,

d

n 

2

n





357