Roof beams are modeled with a cracked beam column finite element incorporating subcritical propagation of edge cracks and compressive strength deterioration by subcritical propagation of wing cracks. For limestone beams strength is reduced to 80% of short-term strength in 2 hours, 50% in 1 year and 30% in 100,000 years. Stand-up time may be related to a factor of safety (F.S.) computed as short-term compressive strength of the rock divided by maximum compressive stress in the beam following relief of tension by cracking. F.S. of 2.5 gives stand-up time of decades or centuries, whereas F.S. of 3.3 confers effective immortality. The full benefits of roof support are realized if installation occurs within 5% of the stand-up time and before compressive stress exceeds 99% of its final value. If mean and standard deviation of short-term strength are known, probability of failure can be calculated for any loading duration.
Mine roof beams fail by a combination of tensile crack propagation, and compressive failure. At the low temperature and zero or very small confining stress of roof beams, both failure modes result from propagation of cracks governed by linear elastic fracture mechanics. Discrete edge cracks can propagate from the top or bottom of the beam in response to tension. Propagation may be effectively instantaneous at high stress (mode I stress intensity KI equal to critical stress intensity KIC), but it also proceeds more slowly in a subcritical state (KIKIC), giving rise to time dependent propagation. Compressive failure is also modeled as a fracture mechanics process, with time dependence resulting from subcritical propagation of wing cracks from initial microcracks. I first describe the characteristics of the finite element code used to model the beams. The methods of analyzing both tensile and compressive failure are then described, followed by the results of the analysis.
Behavior of individual beams is studied using a new time-dependent version (FEBCT) of the cracked beam-column finite element program FEBRF (Tharp 1987, 1994). A crack in a beam column introduces a local compliance that affects deflection and stability. The cracked finite element has rectangular cross section with two nodes, zero length, and a single edge crack in either the upper or lower surface. Because it includes the three degrees of freedom commonly associated with nodes in beam- column elements, it is easily alternated with standard beam-column elements.
Values of c and n were calculated for linear portions of the log (velocity) vs log (Kj) curves interpreted by Atkinson (1984) to correspond to region 1, i.e. control by reaction rate at the crack tip. Because this region corresponds to the lowest stress intensities, it may be extrapolated to lower stress intensities than have been investigated experimentally. Parameters for Falerans micrite (Henry et al., 1977) are used in this paper except where indicated. Crack propagation velocity for limestone is adjusted to an underground temperature of 130C using H measured by Henry (1978) for St. Pons marble.
