Tributary Area Pillar Stress

This is the one relation in the shard with no rock mechanics in it at all, and that is why it is the most trustworthy and the crudest thing here. The argument is pure statics: the overburden has to be carried by something, the openings carry nothing, so each pillar carries the full weight of everything above its own footprint plus everything above the openings around it. For a regular grid of square pillars of width $W_p$ separated by rooms of span $B$, each pillar's tributary area is $(W_p+B)^2$ and its own area is $W_p^2$, and the stress goes up in that ratio.

Written the other way it is the extraction ratio: $\sigma_p = \sigma_v/(1-e)$, where $e$ is the fraction of the ore removed. That form makes the economics visible and brutal. At 50% extraction the pillars carry twice the in-situ stress; at 70% they carry 3.3 times; at 80%, five times; at 90%, ten. Every extra percent of recovery is bought at an accelerating price in pillar load, which is why the last few percent of a resource are the ones that close mines.

$W_p$ is the pillar and $B$ is the opening, and confusing them is the classic mistake. The two symbols sit side by side in the same bracket, they are both widths, and if the numbers are close the error hides completely. It never hides in a safe direction: putting the room span where the pillar width belongs always produces a pillar that looks stronger than it is. Say "pillar width" and "room span" out loud before entering either number.

The method's assumptions are worth checking against your layout every time, because three of them fail routinely. It assumes a WIDE panel, wide compared with its depth, so that no load arches onto the solid abutments at the panel edges. For a narrow panel or a single set of entries a great deal of load does arch, and tributary area badly overestimates the pillar stress — usefully conservative, but it can condemn a workable layout. It assumes a REGULAR grid of identical pillars. A barrier pillar, a panel edge, an irregular remnant left by a fault, or a pillar next to a worked-out area is outside the method entirely and needs numerical modelling. And it returns an AVERAGE. The real stress across a squat pillar is not uniform at all: it peaks near the ribs, where the rock is unconfined, and dips through the confined core — which is exactly why pillars fail by spalling from the outside in rather than by crushing all at once.

One geometric variant to know. This form is written for a square grid. For long rib pillars carrying a panel, the tributary ratio is $(W_p+B)/W_p$ rather than its square, which is a far gentler penalty — the load only concentrates in one direction.