PSI - Issue 2_B

Alberto Ramos et al. / Procedia Structural Integrity 2 (2016) 2591–2597 Author name / Structural Integrity Procedia 00 (2016) 000–000

2594

4

where,

(3)

  1

P

P

s A  ,

f A  ,

i

i

The scale factor allows calculating the probability of failure for any area size, and therefore the probability of failure of each finite element at a given stress from the expression:

   

   

  

A

    

  

(4)

ref

    1 exp

;

P



 

, f A i

A



The problem is that although theoretically for the 4-point bending test, the profile of the stress distribution along the selected dimension is commonly considered constant, actually it is not. Being more rigorous, it considers that in many cases the stress varies along the longitudinal direction of the plate and even in the central section between the loading rollers, so it would be an important error taking only the maximum as a representative value. Analyzing the numerical model of the 4-point bending test shown in Fig. 3, it is verified that it cannot be considered uniform stress in the central part of the plate, as there is a significant variation along the plate width. Therefore there will be a big difference between considering the stress distribution of the numerical model and the misnamed maximum stress obtained from the standard, since the formulation gives the stress value on a single point, the center point, being higher in other sample areas. In this case and because of the dimensions of the plate, stresses are greater at the edges and slightly higher under loading rollers than in the center of the specimen, the one calculated with the standard formulation and considered constant over the central area between the loading rollers.

Fig. 3. Stress distribution of the 4-point bending tests.

Taking into account the stress distribution obtained numerically for each specimen, it can be referred to an effective area, A ef , such that under this constant maximum stress, σ max , present the same probability of failure that the complete plate. The effective area may be calculated for Weibull bi-parametric models according to Choi et al. (2000), however, this article proposes a correction for the same method and for a three-parameter Weibull model, so the expression deduced (5) to calculate the characteristic A ef of each of the tests is:       eq ef dA x y A    * , (5)

  

 







A