This article applies two modifications to the elastic beam approach to represent failures and cavities produced above shallow stopes of hard rock mines: using the compression and tension modulus of elasticity of the rock instead of a single value, producing different levels of imposed tension and compression, and a strata loading and failure mechanism that models the Fayol beam stacking observations and the stabilizing cavity formation normally seen in the field. A relationship has also been derived to calculate the total cavity height normal to stratification in order to verify the potential for shallow hard rock (metal) mine failure to surface. Case studies are presented to verify the application of this model and identify realistic strata behaviour. The failure model provides predicted cavity height values similar to actual failures. When loading of the lowest stratum from detached strata is in effect, the thinner strata have a maximum calculated cavity height about 1/4 the original stope span, 1/6 the original stope span in the case of thicker strata. This model also serves to identify beam action versus fully supported plate action.


Stratification of rock introduces distinct behaviour of rock masses, leading to parting and failure of layers. The evaluation of performance of such conditions requires distinct procedures other than those applied in unstratified, but otherwise jointed rock masses.

Metal mine rock masses can contain extensive,parallel discontinuities, such as gneissic fabric, shear planes or even sedimentary bedding joints. These dominant rock structures extend continuously, effectively separating the rock mass into strata, often of similar thickness, typically from a few to tens of centimeters. In the case of metamorphic, fault-displaced and some sedimentary terrains occurring at mine sites, cross-jointing may be random or poorly developed.

In the context of near-surface stope failure research, a new approach to predict the potential failure extent of strata was established in order to arrive at the extent of the failure cavity for comparison to possible breakthrough to surface where impact consequence to infrastructure and the safety of the population must be assessed.


he failure model developed here is based on a simple 2D elastic beam analysis using strata of identical thickness and material properties but uses realistic material behaviour and complex loading conditions.

2.1 Beam with Different Materials

Elastic beam theory (where stratum thickness is less than 1/4 roof span) and plane strain conditions can be used to quantify the imposed stresses to a stratum under consideration in two-dimension situations (the structure divided into 1 m wide beams). The rock stratum can be considered as an elastic beam structure with both ends cantilevered.

The induced stress considering a plain strain analysis is:

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where M = moment applied to the beam by the load, y = distance from neutral axis (half thickness) to point of reference (t/2), I = moment of inertia of the beam cross-section (bt3/12, b = 1).

Examination of single stratum tensile failure (no loading by other detached strata from above) follows conventional solid mechanics consideration for a beam of unit width of rock.

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