## 2 Extended Pick Failure Probability Assessment Approach
### Pick Failure Probability during the Shearing-Down Stage
To address this issue, the pick tip failure probability for cutting the \(s\)-th segments in all the revolutions in the shearing-down stage will be estimated first, and then the pick tip failure probability over the whole shearing-down stage is estimated. With a consideration of the variation of rock thickness, the pick tip failure probability for cutting the \(s\)-th segments in all the revolutions over the shearing-down stage is given by  
\[F_{fs}=\sum\
olimits_{j=1}^{n_{bf}}\left[1-\prod\
olimits_{i=1}^{n_{bf}}(1-F_{ fj}\left(h_{fj},\ x_{a}^{t}\right))\right]p\left(h_{fj}\right) \tag{16}\]  
where \(F_{fs}\) is the failure probability of the pick tip for cutting the \(s\)-th segments in all the revolutions in the shearing-down stage, \(n_{bf}\) is the number of mean floor rock thickness values in consideration, \(F_{fj}\left(h_{fj},\ x_{a}^{t}\right)\) is the pick tip failure probability for cutting the \(s\)-th segment in the \(i\)-th drum revolution in the shearing-down stage when the mean floor rock thickness is \(h_{fj}\) (mm), \(p\left(h_{fj}\right)\) is the likelihood (%) that the mean floor rock thickness is \(h_{fj}\) (mm), and \(n_{df}\) is the number of drum revolutions in the shearing-down stage:  
\[n_{df}=\text{ceil}\left(\frac{h_{t}-D_{d}}{v_{n}}\right) \tag{17}\]  
Then, the failure probability of the pick tip over the shearing-down stage, \(F_{2}^{t}\left(x_{a}^{t}\right)\), is  
\[F_{2}^{t}\left(x_{a}^{t}\right)=1-\prod\
olimits_{s=1}^{n_{sf}}(1-F_{fs}) \tag{18}\]  
where \(n_{sf}\) is the number of segments in a revolution in the shearing-down stage:  
\[n_{sf}=\text{ceil}\left(\frac{360}{\alpha_{sf}}\right) \tag{19}\]  
In Equation (19), \(\alpha_{sf}\) is the central angle of a segment in the shearing-down stage (in this study, all drum revolutions are equally segmented). From Equations (16) and (18), it can be seen that to estimate the failure probability of the pick tip in the shearing-down stage, \(F_{fj}\left(h_{fj},x_{a}^{t}\right)\) must be calculated first. To calculate \(F_{fj}\left(h_{fj},\ x_{a}^{t}\right)\), it is necessary to determine whether an individual segment in the shearing-down stage is hard rock. To address this issue, the mid-point method [26] applied in [11] is applied again. If the midpoint of a segment is within the floor hard rock, the whole segment is regarded as the same hard rock. Based on this concept, the pick tip failure probability for cutting the \(s\)-th segment in the \(i\)-th drum revolution with the rock thickness of \(h_{fj}\) is given by  
\[F_{fs}\Big{(}h_{fj},\ x_{a}^{t}\Big{)}=\ \left\{ \begin{array}{c}P_{is}\Big{(}x_{a}^{t }\Big{)},\ \text{when}\ h_{fj}>D_{is}\ \text{and}\ \alpha_{fci}\geq\alpha_{is}\\ 0,\ \text{others}\end{array} \right. \tag{20}\]  
where, \(P_{is}\Big{(}x_{a}^{t}\Big{)}\) is the probability that actual bending force is greater than the allowable bending force \(x_{a}^{t}\) of the pick tip, and \(\alpha_{is}\) is the angular position of the \(s\)-th segment in the \(i\)-th drum revolution (degree). In Equation (20), \(D_{is}\) is the distance of the \(s\)-th segment in the \(i\)-th drum revolution from the roadway floor:  
\[D_{is}=h_{t}-v_{n}\Big{(}n_{di}-1+\frac{\alpha_{is}}{360}\Big{)}-\frac{D_{d}}{ 2}\Big{[}1+\ \sin(\frac{\pi\alpha_{is}}{180})\Big{]} \tag{21}\]  
The computation based on Equations (20) and (21) can often be simplified, because in the real world, the chord of a drum engaging in rock cutting cannot exceed its tip-to-tip diameter which is usually about 1000 mm. The rock thickness normally does not change dramatically over this short distance. In this case, it is reasonable to assume that once drum starts to cut rock in a revolution, it will continue to cut the rock over the rest of its cutting sector in the same revolution. This means that although rock thickness changes randomly, its mean value over a drum revolution remains constant.