Issue 50

G. Belokas, Frattura ed Integrità Strutturale, 50 (2019) 354-369; DOI: 10.3221/IGF-ESIS.50.30 362 Treating each sample separately we get the Mohr – Coulomb failure envelope constants of Table 4 (sample size n=9), which are represented in Fig.6 by the thin coloured lines, in the classic τ – σ n Mohr – Coulomb diagram. Again the remaining char- acteristic envelopes of Fig.6 are explained next. Alternatively, for the characteristic envelope we can apply a t-student distribution into the y ( x ) estimate (i.e. the predicted σ 1 for given σ 3 , given by Eq.(27)) for a p =5% probability (i.e. t p,n-2 =1.70814) and n -2 dof . In Fig. 5 the characteristic 2 line shows the σ 1 , σ 3 graph of the characteristic envelope from Eq.(27) for a p =5%. This characteristic envelope is non – linear and a linear regression leads to characteristic values a k =190.36 kPa and b k =2.92550 for Eq.(5) and then c k2 =55.6 kPa and (tan φ ) k2 =0.56287 for the Mohr – Coulomb failure criterion (Figs.(5,6), characteristic 2). Taking into account the results for the direct shear test, it is obvious that Eq.(18) or Eq.(27) systematically gives a more optimistic characteristic envelope than the one computed based on mean values and uncertainty calculation of c and tan φ . However, this approach does not take into account the error propagation to calculate the uncertainties. Sample Mean ΓΣ2 (5m) Γ1 (2m) Γ1 (3.5m) ΓΣ13 (17m) Γ1 (18m) Γ1 (6m) ΓΣ2 (9m) ΓΣ13 (24m) Γ3 (12.5m) c (kPa) 64.34 40.99 22.34 21.24 60.41 77.75 29.69 76.83 48.20 201.58 Tan φ 0.59415 0.53191 0.83698 0.64201 0.64099 0.4862 0.64606 0.59134 0.63149 0.34038 φ ( ο ) 30.42 28.01 39.93 32.70 32.66 25.93 32.86 30.60 32.27 18.78 Table 4 : Application of t – test on the values of the observed τ for each σ n . Figure 6 : Characteristic and best estimate Mohr – Coulomb failure envelopes superimposed on the Mohr – Coulomb failure envelopes of each sample.       2 3 3 2 1 3 2 2 1 3 3 1 1 1 2                               n m m n j n j i i a b t n n        (27) An alternative for the characteristic envelope is to compute first the characteristic values of a and b constants by applying a t-student distribution (i.e. a k = a m - t p,n-2 SE a and b k = b m - t p,n-2 SE b ) for p =5% and n -2 dof and then apply Eqs.(20,19) to compute the Mohr – Coulomb characteristic constants c k and (tan φ ) k . This approach, which does not take into account the error propagation, leads to a k =139.86 kPa and b k =2.58981 constants of Eq.(6) (Fig.5, characteristic 3) and to c k3 =43.4 kPa and (tan φ ) k3 =0.493949 constants of the Mohr – Coulomb failure criterion (Fig.6, characteristic 3).