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The second Monte Carlo analysis was performed assuming 60-ft x 60-ft pillars. The depths for this geometry were varied from 400 ft to 2,400 ft. The resulting ARMPS stability factors, deterministic safety factors, and probabilities of failure are plotted versus depth in Figure 6. The synthetic study resulted in more conservative values than ARMPS for the second scenario. The ARMPS stability factor approaches 1.5 for a depth of 2,400 ft. At 2,400-ft deep, the deter- ministic safety factor is approximately 0.8 and the probability of failure is 0.992. Discussion The results from the synthetic study and the results from the ARMPS varied somewhat. The difference between the results of the two methods increases with increasing depth of cover. This is most likely because the tributary area method of pillar stress determina- tion was used for the synthetic study. The tributary area method leads to a conservative estimate of the stress on a pillar by as much as 40%. Before the probabilistic approach to coal pillar design could become widely applied, standards would need to be created. Geotechnical engineering involves too much variation to adhere to the Six Sigma standard. An acceptable probability of failure thresh- old would have to be decided upon. For example, if a probability of failure of 30% were deemed an appropriate design threshold, this would correspond to a stability factor of approximately 1.6 for the first scenario and 1.8 for the sec- ond scenario. A more realistic approach for estimating coal pillar stress should be used that would result in a more reasonable value that could be consistent with industry-wide standards. The scope of the probabilistic approach could be increased by performing a similar calculation for an entire panel rather than just one pillar. This could be done with a Gaussian field representing the spatial distribution of coal strengths, entry height, etc. While the Monte Carlo for the simple, single-pillar example has a rather quick computation time, a panel-sized probabilistic method could take a significant amount of time to perform. Conclusions Uncertainty quantification is necessary in all engineering prac- tices. This is especially true when regarding geotechnical engineer- ing. Probabilistic analyses for engineering design are becoming more prevalent in many fields where a high degree of uncertainty is present but remain mostly absent in coal pillar design. Performing a probabilistic study rather than a deterministic one is a relatively easy way to quantify the uncertainty in an engineer- ing design. A sensitivity analysis requires actively varying the inputs a slight degree to determine how the output is affected. This process can be much more time consuming, more user intensive, and less comprehensive than a probabilistic analysis like the one performed here. A probabilistic approach to coal pillar design has potential to be a viable alternative to traditional deterministic methods. Accounting for variability or uncertainty in the inputs and outputs of engineer- ing design calculations provides more meaningful results. Designing to arbitrary deterministic thresholds indirectly admits the fault in the deterministic approach. The simple synthetic study performed involved comparing the results of a simple factor of safety determination to the stability fac- tor calculated by ARMPS. Both the probability of failure and a deterministic factor of safety were determined using the Bieniawski equation for coal pillar strength and the tributary area assumption for coal pillar stress. For 40-ft square pillars in a six-entry panel under development loading, ARMPS resulted in a stability factor of 1.5 at approximately 1,100 ft of cover. This same scenario in the synthetic study resulted in a probability of failure of approximately 60%. The results of the synthetic study agreed somewhat with the output from ARMPS, but were more conservative than ARMPS. The conservative nature of the synthetic study was found to increase with depth. Ben Fahrman is a graduate student and Erik Westman is an associate professor with Virginia Tech's Mining & Minerals Engineering Department. They presented this paper at the Ground Control Conference, which took place in Morgantown, W.Va., dur- ing August 2013. m i n e d e s i g n c o n t i n u e d March 2014 www.coalage.com 33 Figure 5: Factor of histogram for the second mine geometry (40-ft x 40-ft pillars) at a depth of 1,100 ft. Figure 6: ARMPS stability factor, deterministic safety factor and the probability of fail- ure for the second scenario (60-ft x 60-ft pillars) plotted versus depth of cover. CA_pg30-33_V2_CA_pg46-47 3/12/14 8:44 AM Page 33