Issue 50

G. Belokas, Frattura ed Integrità Strutturale, 50 (2019) 354-369; DOI: 10.3221/IGF-ESIS.50.30 364 SE c =18.56kPa, SE tanφ =0.04526 and d) characteristic values: ck 4 =29.82kPa, (tan φ ) k4 =0.5099. The mean value from this alternate approach (characteristic 4, Fig.7) is almost the same with the previously calculated. Fig.7 compares the character istic value from this approach (characteristic 4) with the characteristic 1 from the FORM . The alternate approach is very conservative because it gives a higher SE than the FORM . All the above results are summarized into Tables 5 for the mean values and the corresponding uncertainties and Table 6 for the characteristic values. A PPLICATION OF THE FORM TO A SIMPLE PLANAR FAILURE PROBLEM urther extending the FORM application to engineering problems calculations, a limit state analysis may be performed in terms of either the safety margin SM = R - E ( R is resistance and Ε action as defined in EC7) in terms of the safety factor SF = R / E . More specifically, in FORM applications the SM application is preferable compared to the FS because the actions Ε in the denominator of the SF enhances the non-linearity effects in the error propagation. For the safety margin the reliability index is β = SM m / S d,SM , where SM m is the best estimate of the mean and S d,SM the standard deviation, while by definition it is β SM =1/ V SM , where V SM the variation coefficient. Since resistances and actions describe different types of random variables, they are expected to be uncorrelated, and their covariance, ρ R,D , can be considered zero. Moreover, the variation of permanent loads is generally small compared and should not greatly affect the SM or SF value. The FORM is applied herein to a planar failure problem, which can be adapted to any type of failure surface (e.g. [13]) of limit equilibrium problems (e.g. method of slices). For the planar wedge failure problem considered herein Fig.8 shows the geometry. W is the weight of the wedge, H is the height of the slope, β is the angle of the slope to the horizontal, θ is the angle of the plane of failure with respect to horizontal and N and T are the normal and shear reaction forces on the plane of failure. The safety margin is then determined from the equilibrium equations that lead to Eq.(28). On the other hand the SF is given by Eq.(29).    2 1 2 = c sin cos tan sin sin sin SM H γH β θ θ φ θ β θ         (28)      = 2c / γ sin cos tan sin sin sin sin SF H β θ θ φ β H β θ θ β            (29) H β θ W Wcosθ Wsinθ N T Α Β Γ ΑΒ = Η /sin θ Figure 8 : Geometry of two dimensional planar failure. Applying the FORM on Eq.(28) the standard deviation of the SM is given by Eqs 30 to 33, in which u c , u tanφ and u γ are the uncertainties of c , tan( φ ) and γ respectively. Likewise, applying the FORM on Eq.(29) the standard deviation of the SF is given by Eqs.(34-37).   2 2 2 2 2 2 tan tan                            c SM SM SM u SM u u u c     (30) = sin   SM H c θ (31) F