Evaluation of the strength of slender pillars

Pillars with width-to-height ratios of less than 1.0 are frequently created in underground hard rock mines. The strength of slender pillars can be estimated using empirically developed equations. However, the equations can provide variable results when the width-to-height ratios approach 0.5. This paper investigates some of the issues affecting pillar strength at low width-to-height ratios in hard brittle rock. The investigation includes an evaluation of empirical pillar strength data presented in the literature and observations of pillar performance in underground limestone mines in the eastern United States, supplemented by numerical modeling in which failure processes and sensitivity of slender pillars to variations in rock mass properties are evaluated. The results showed that the strength of slender pillars is more variable than that of wider pillars. The numerical model results demonstrated the increasing role of brittle rock failure in slender pillar strength. The absence of confinement in slender pillars can result in a fully brittle failure process, while wider pillars fail in a combined brittle and shearing mode. The onset of spalling in slender pillars occurs at or near the ultimate strength, while this is not the case for wider pillars. Slender pillars are shown to be more sensitive to the presence of discontinuities than wider pillars, which can partly explain the increased variability of slender pillar strength. Two examples are presented that illustrate failure initiation by brittle spalling and the sensitivity of slender pillars to the presence of discontinuities. Introduction The National Institute for Occupational Safety and Health's (NIOSH's) Pittsburgh Research Laboratory has embarked on a project to develop pillar design guidelines for underground limestone mines. A survey of mining methods and pillar and room dimensions in 70 underground limestone mines (Iannacchione, 1999) showed that the room-and-pillar method was used in 69 of the 70 mines surveyed. The average depth of cover was 80 m (260 ft), varying between 7 m (23 ft) and 610 m (2,000 ft). Pillars were typically square in plan view, but rectangular or rib pillars are also used. During initial development, the average pillar width-toheight (w:h) ratio was 1.73 but was reduced to 0.92 after bench mining of the floor. The minimum and maximum w:h ratio observed in the study was 0.4 and 3.13, respectively. Nine cases of pillar failure were identified, with all of them at width-toheight ratios of less than or equal to 1.5. Because floor benching is conducted in more than half of the limestone mines, pillars that were previously stable could become unstable when their widthto-height ratio is reduced during benching. NIOSH has, therefore, initially focused the project on the strength of slender pillars. The design of stable pillars requires that both the strength and loading of the pillars be known. In addition, an appropriate safety factor should be selected to ensure that the variability and uncertainty of the pillar strength and loading is accounted for. In the case of regular arrays of flat lying pillars, the load can be estimated by the tributary area method (Salamon and Munro, 1967), or if the layout is more complex., estimates of average pillar loading can be obtained from numerical models (Brady and Brown, 1985). Pillar strength can be estimated from empirical equations that have been developed by observing both failed and stable pillar configurations. The pioneering work in this field was carried out for coal mine pillar design (Salamon and Munro, 1967; Bieniawski and van Heerden, 1975). Several empirically based pillar strength equations have since been developed for hard rock mines (Hedley and Grant, 1972; von Kimmelman et al. (1984); Lunder and Pakalnis, 1997). Analytical methods to estimate pillar strength have been developed, such as Wilson's confined core model (Wilson, 1972) and a similar model by Barron (1986). Although these methods have assisted in understanding pillar failure mechanics, they have not found wide acceptance as design tools in the mining industry. More recently, numerical models have found increasing use in pillar design (Mark, 1999). For example, Hoek and Brown (1980) used the results of elastic models to estimate the strength of pillars in various rock-mass classes. Martin and