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Using a probabilistic approach for calculating factors of safe- ty can easily be visualized in a few different ways. Figure 1 shows possible distributions of stress and strength for an exam- ple probabilistic analysis. The probability of failure in the example component or structure is the area confined in the overlap where the stress exceeds the strength. Figure 2 shows a second possible visualization of results from a probabilistic analysis. With the factor of safety output plotted as a single dis- tribution, the probability of failure is simply the area under the curve to the left of a factor of safety equal to one. This is a sim- plified version of possible results from stochastic analyses, but a more complex analysis will likely take the same general form. The probabilistic approach has some inherent advantages over the more widely used deterministic approach. Two of these advantages are: • I n p u t p a r a m e t e r s r e p r e s e n t e d b y s i n g l e v a l u e s s u g g e s t knowledge of the exact value. Input parameters represented by distributions show uncertainty or variability in the data, which makes them a more realistic representation. • Probability of failure is more meaningful than safety factor in many instances. Designing to an acceptable probability of failure is a better engineering practice than setting an arbi- trary factor of safety threshold. These are two of the more important advantages of the proba- bilistic method over the deterministic method, but there are additional, less obvious advantages. There is a smaller chance of user error when a probabilistic analysis is used. This is inherent to the method, because the user is not forced to determine a single value for the input parameters. Fundamentally accounting for any uncertainty in the calculation allows the user to be more cer- tain of inputs. Furthermore, the process can be quicker and easi- er to implement. After performing a deterministic design calculation, a sensitivity analysis it typically required to deal with uncertainty. A probabilistic analysis removes the need for a sensi- tivity analysis because the uncertainty is handled inherently. Probabilistic analyses for engineering design have been used in many industries to determine probability of failure. The nat- ural variability of the ground makes geotechnical design a unique setting where a probabilistic approach should be used m o r e w i d e l y . A t r a n s i t i o n t o w a r d u s i n g t h e p r o b a b i l i s t i c approach for engineering design of geologic structures is taking place for rock wedge failure (Park and West, 2000), slope stabil- ity, and room-and-pillar failure in oil shale mining and under- ground uranium mines, among others. This transition to using a probabilistic approach for geotechnical engineering design has been slow, controversial, and mostly neglected by the room-and-pillar coal mining industry. Coal Pillar Design Like many other structural design problems, coal pillar design can be expressed as a factor of safety. Estimating stress on a coal pillar can be relatively straightforward, but the method for determining coal pillar strength is not as well established. There are many equations for coal pillar strength, and any of them can be appropriate in a given set of conditions. The stress supported by a pillar in a room-and-pillar mine can be assumed to be a function of the tributary area and the pillar area, shown in Figure 3, as well as the overburden stress. It is assumed that each pillar supports the volume of overbur- den in the column above the tributary area of that pillar. For square pillars with a consistent, rectangular pattern, the equa- tion for pillar stress, σ p , becomes Where σz, the vertical stress, is the product of ϒ , the unit weight of the overburden, and z, the overburden thickness. As shown in Figure 3, W p is the pillar width and Wo is the width of the opening. Determination of coal pillar strength is much less straightforward. Pillar strength is commonly estimated using the Bieniawski equation: Where S i is the compressive strength of coal, and W/H is the coal pillar width to height ratio. Like most equations for coal pillar strength, the Bieniawski equation is empirical. This equa- tion was chosen because its ubiquitous use for calculating the stress of square coal pillars. The Analysis of Retreat Mining Pillar Stability (ARMPS) pro- gram created by the National Institute for Occupational Safety and Health (NIOSH) is often used in addition to or instead of a factor of safety calculation. The ARMPS program reports a "sta- bility factor," which is similar to a deterministic safety factor. The stability factor is the ratio of the load bearing capacity of the pillars within the active mining zone (strength) to the total l o a d a p p l i e d t o t h e p i l l a r s w i t h i n t h e a c t i v e m i n i n g z o n e (stress). The active mining zone is defined to include all of the m i n e d e s i g n c o n t i n u e d March 2014 www.coalage.com 31 Figure 3: Tributary area in a room-and-pillar mine (Zipf, 2001). CA_pg30-33_V2_CA_pg46-47 3/12/14 8:44 AM Page 31