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pillars within five times the square root of the depth of cover of the excavation front. Designing with a stability factor of at least 1.5 is recommended when the depth of cover is greater than 1,000 ft. Probabilistic Coal Pillar Study Two mine geometries were assumed to test the probabilistic procedure involved with underground coal pillar design. Both geometries were six-entry panels with 20-ft entries. One sce- nario had 40- x 40-ft pillars, the other had 60- x 60-ft pillars. T h e r e s u l t s f o r t h e s e d e s i g n s w e r e c a l c u l a t e d f o r m u l t i p l e depths. A mean value was assigned to each required input parameter, and a reasonable standard deviation for each value was estimated. E a c h i n p u t p a r a m e t e r w a s a s s u m e d t o b e n o r m a l l y d i s - tributed. Except for the pillar dimensions and the depth of cover, the distribution of all parameters remained the same for both scenarios. The compressive strength of coal is notori- ously difficult to test in a laboratory. Coal samples degrade quickly once removed from confinement, and coal exhibits a rather extreme size effect. The in situ strength of coal is gener- ally accepted to be 900 psi, but this value has been known to vary. In situ coal strength has been determined to fall between 780 and 1,070 psi. Therefore, the mean value for in situ coal strength was assumed to be 900 psi with a standard deviation of 72.5 psi. This range results in the typical range of in situ c o a l s t r e n g t h r e p o r t e d b y M a r k a n d B a r t o n t o b e a p p r o x i - mately ±2 standard deviations. The room-and-pillar layout was assumed to consist of 40-ft square pillars on 60-ft centers for the first scenario and 60-ft square pillars on 80-ft centers for the second scenario. Both of these geometries result in an entry and crosscut width of 20 ft. The pillar width was given these mean values with a standard deviation of 0.5 ft. The entry width was assumed to be the center-to-center spacing minus the pillar width to keep the mine geometry consistent. The mining height was given an arbitrary mean of 6 ft and standard deviation of 0.25 ft. All of the input parameters required for this simple study are listed in Table 1. The only parameter changed between the first and second runs is the pillar size. Depth was varied to compare the probabilities of failure from the probabilistic study to the stability factors from ARMPS. The stability factors were calculated for the mine geometries under development loading only. Monte Carlo simulations were performed for each of the two runs to determine distributions for the likely factors of safety resulting from these synthetic data sets with the stress and strength equations discussed above. Random numbers were generated by MATLAB for each of the input parameters, with t h e e x c e p t i o n o f e n t r y w i d t h , w h i c h i s a f u n c t i o n o f p i l l a r width, and the factor of safety was determined. Each Monte Carlo simulation consisted of 1 million iterations for each of the two runs. Results The results from the Monte Carlo simulations were compared to the stability factors calculated by ARMPS and the determin- istic safety factors for a variety of depths. The deterministic fac- t o r o f s a f e t y r e s u l t s w e r e f o u n d b y i n p u t t i n g o n l y t h e distribution means into the stress and strength equations. For the first scenario, the depth of cover was varied from 400 ft to 1,400 ft. The depth of cover was varied from 400 ft to 2,400 ft for the second scenario. Plotted in Figure 4 are the probability of failure from the stochastic analysis, the safety factor from the deterministic analysis, and the stability factor from ARMPS for varying depths of cover for 40 ft x 40 ft pillars. There is relative- ly good agreement between all three pillar design methods. The minimum stability factor recommended for a "satisfactory" pil- lar system is 1.5 for depths of cover greater than 1,000 ft. The d e p t h o f c o v e r c o r r e s p o n d i n g t o a s t a b i l i t y f a c t o r o f 1 . 5 i s approximately 1,100 ft. At this same depth, the safety factor resulting from the deterministic method is approximately 1. The probability of failure at 1,100 ft is approximately 0.6, which means 60% of pillars subjected to this loading scheme are expected to fail. This probability of failure is more conservative than the other results. The distribution of safety factors is nor- mally distributed and centered close to one, as shown in Figure 5. The largest safety factor calculated was approximately 1.6 and the smallest value was approximately 0.6. This range is due to the standard deviations assigned to the input parameters. m i n e d e s i g n c o n t i n u e d 32 www.coalage.com March 2014 Table 1: List of statistical parameters for each input value. All distributions were assumed to be normal. Figure 4: ARMPS stability factor, deterministic safety factor and the probability of failure for the first scenario (40-ft x 40-ft pillars) plotted versus depth of cover. CA_pg30-33_V2_CA_pg46-47 3/12/14 8:44 AM Page 32