PSI - Issue 48

Youcef Cheikhaoui et al. / Procedia Structural Integrity 48 (2023) 81–87 Cheikhaoui et al/ Structural Integrity Procedia 00 (2023) 000 – 000

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Fig. 3. Tributary area theory for wall pillars shown in longitudinal section(Brown and Brady, 1985).

If we define an overall extraction ratio r TAT : = é = ( 0 + ) 2 − 2 ( 0 + ) 2 1− = 2 ( 0 + ) 2

(3) (4) (5) (6)

The average load σ p at the center of the pillar can be written as follows: = 1− 1 2.4. Probability of survival of a pillar

Formula 7, allows us to introduce the probabilistic distribution of defects and the probability of activation of these defects, i.e. the probability of failure. If there is a series of defects, when one of these defects (the discontinuities) is activated, failure occurs. The relationship proposed as an analytical approach to estimate the strength of a pillar such as Rp = σ f the stress to be applied for at least one of the defects is activated (the failure) (Cheikhaoui et al., 2021), so: = [( ⁄ℎ) . − )/(− )] ™‹–Š = ( 0 − ) ( ℎ ⁄ ) (7) where: 0 : Specimen volume (m 3 ). : Pillar volume (m 3 ). ℎ ⁄ : Sample slenderness. ⁄ℎ : Pillar slenderness. : Weibull's dispersion parameter. : The applied stress (MPa). As the exploitation of the mine is done by rooms and pillars, the stress (MPa) applied to the center of the pillar is related to the extraction ratio τ and (Mp>Pa) the vertical stress related to the upper ground load. 3. Case study 3.1. Determination of pillar strength The strength formula of a zinc pillar (Chaabet El Hamra case) found using Weibull parameters is written as follows(Cheikhaoui et al., 2021): ( ) = 103.61 (1/ ) 0.42 ( /ℎ) 5.15 −0.42 ሺ ƒሻ (8) The formula for the strength of a zinc pillar based on the probability of survival P S = 97% means 3% risk of failure, is written as follows: (0.97) = 23.60 ( ⁄ℎ ) 5.15 −0.42 ሺ ƒሻ (9)