Pillar Strength in Schistose Rock

Pillar Strength Estimates for Foliated and Inclined Pillars in Schistose Material

L.J. Lorig
Itasca Consulting Group, Inc., Minneapolis, MN, USA
A. Cabrera
Itasca S.A., Santiago, Chile


Pillar strength curves used worldwide are not particularly helpful in the case of foliated/inclined orebodies, as the data they are based on is mostly composed of brittle isotropic rocks.  Many room-and-pillar mining operations face the need to design their pillars in schistose rocks within inclined orebodies and therefore it is necessary to make numerical simulations to assess the feasibility of the designs.  This article refers to specific studies performed in such an environment and is aimed at providing a reference for the effect of foliation and pillar inclination on pillar strength.


Analysis of a room-and-pillar mining method often relies on empirical estimates of ultimate pillar strength. Unfortunately, empirical pillar strength curves normally used are not particularly helpful in the case of foliated/inclined orebodies due to the following reasons:

  • The actual data those curves are based on does not include real failures with width to height (W/H) ratios greater than about 1.5 even though actual pillars can have W/H ratios of 2 or more.
  • The pillar design curves do not explicitly account for inclined orebodies, square pillars or weak hangingwall and/or footwall contacts, and
  • The empirical database of pillar behavior does not include foliated rocks.

Consequently, numerical models provide an alternative that can be used to gain additional insight into the effect of pillar W/H ratio on pillar strength. There is a problem with this approach, however, in that the pillar ultimate strength and/or transition to pillar hardening behavior are largely dependent on and sensitive to post-peak material response, including softening rate, residual strength and brittle-to-ductile transition, which are still uncertain and among the least known parameters in rock mechanics.

What is observed in laboratory tests is that, under zero to low confinement, the material response is brittle softening. When confinement is increased, besides increasing peak strength, the post-peak response changes from brittle softening to plastic, and eventually to strain hardening. That is not the case in continuum-based numerical models. The increase in confinement increases the peak strength, but does not change the post-peak response. If we specify softening response, it is independent of confinement. Thus, the continuum models are conservative (but also unrealistic), because they don’t have a brittle-to-ductile transition. Instead, the material post-peak response is independent of confinement. There is uncertainty because we do not have the data about this transition for schistose materials, and therefore we cannot determine if the model represents this transition correctly.

The rock behavior in pillars is simulated in this study using a strain-softening material model in which the rock mass cohesion and tension change from peak to residual strength over a defined amount of plastic shear strain. The residual cohesive and tensile strengths are assumed to be zero. Pillars are loaded by applying a constant velocity to the hangingwall side of the model.  In subsequent sections of this article we first validate the numerical tool for reasonably known geometries and brittle rock masses and then we address the effect of changes to these assumptions, as discussed above.



In order to validate the numerical modeling tool and to adjust the post-peak parameters needed for the analysis, a three-dimensional model has been constructed to study an individual vertical pillar without anisotropy. In consequence, a 1/8th pillar model has been used simulating this symmetry in the three axes, as shown in Figure 1. The model was parametric and it allowed representation of W/H ratios ranging from 0.3 to 3.5, in 0.5 intervals. Height (H) was fixed at 1.3 m. Figure 1 shows the models used to simulate W/H = 1.0 (left) and W/H = 2.0 (right).

Figure 1. Models used to analyze individual vertical pillars with W/H = 1 (left) and W/H = 2 (right).

The rock behavior in pillars is simulated using a strain-softening constitutive relation in which the rock mass cohesion and tension goes from peak to residual strength over a defined amount of plastic shear strain. The material is homogeneous in the whole model, with the properties shown in Table 1. Zone size for all models has been the same (0.065 m cubic zones) in order to represent the post-peak behavior using the same criterion in all the models.

Table 1. Geomechanical properties used in modeling for peak condition and post-peak (residual) condition for pillar schist.


At a critical strain of 0.4%, analyses show the expected behavior, i.e. brittle type failure in pillars having low W/H ratios (smaller than or equal to 1.0), and softening – hardening type failure in pillars having high W/H ratios (above 1.0). It is observed that slim pillars only fail inside the pillar, while squat pillars fail inside the pillar as well as in the hangingwall sector near the pillar. All of this is presented in Figure 2, with the representation of shear strain contours in the post-peak condition for all the pillars analyzed, as well as a pillar loading versus displacement history.

By plotting all the pillar load histories together (Figure 3), we can notice the difference in behavior of slim pillars (brittle failure) versus squat pillars (plastic failure). Figure 4 shows the relation between pillar strength and uniaxial compressive rock strength for different W/H ratios. This graphic also shows the relationship obtained by Lunder & Pakalnis (1997), based on real cases. We can observe how the analyzed pillar is reasonably adjusted for the range of W/H between 2 and 3.

Figure 2. Contours of shear strain at peak load for each model analyzed. Load vs. axial strain is superimposed.
Figure 3. Relationship between pillar strength and pillar axial strain for vertical pillars.
Figure 4. Relationship between vertical pillar strength and W/H ratio, together with the relationship proposed by Lunder & Pakalnis (1997).


This section presents the models built for the case of inclined pillars in anisotropic material. It includes an inclined vein geometry with a 20° dip and vein parallel anisotropy (also at 20° dip). This analysis is aimed at finding the ultimate individual pillar strength.

In addition to the changes described above, a realistic feature regarding degradation because of blasting damage was included. All analyses considered that this damage corresponds to 0.5 meter on each side of the pillar. This skin of blast damaged material is assumed to have residual strength, i.e. a Hoek-Brown damage factor D = 1.


A 4 m wide pillar was analyzed, modeling only half of it, using the available symmetry condition. The geometry is shown in Figure 5. Boundary conditions are based on rollers in side boundaries and fixity in the bottom boundary. Boundaries are deemed to be far enough to avoid its influence on the results of the pillar system behavior. However, these vertical boundaries could be a limitation in the model because the convergence in real pillars will tend to not be constrained vertically but will tend to occur normal to the extracted ore. The other choice is to use periodic boundaries. Both boundary condition assumptions assume infinite arrays of pillars, which never occurs.

The same zone size used in the vertical pillar analysis was used (i.e. 0.065 m) because this zone size is associated to the calibrated post-peak behavior (in the strain-softening constitutive model).

Figure 5. Model representing the 4 m wide square pillar.


One material has been considered in the model, corresponding to a schist, with the properties described in Table 1. Anisotropy was assigned using c = 407 kPa and f = 51.5°, empirical values for discontinuities in hard schist from Jimenez Salas (1975). Figure 6 summarizes the parameters used in the analysis. As indicated previously, the first half meter of the pillar was assumed to be damaged (D = 1) and have residual strength.

Figure 6. Summary of geometry and properties used in inclined pillar model.


Results of strength analyses of inclined pillar with anisotropy resulted in a lower strength compared to vertical isotropic pillars. Figure 7 shows strength versus vertical strain curves for the pillar analyzed. Maximum strength for the pillar analyzed resulted in about 45 MPa.

Figure 7. Pillar strength versus pillar vertical strain for inclined pillar under analysis.

The reduction in strength with respect to vertical and isotropic pillars is about 10%, which seems reasonable for a W/H ratio equal to 1.8. Figure 8 shows the strength difference between both pillar geometries (vertical and inclined).

Figure 8. Pillar strength versus the W/H ratio for vertical and inclined pillars, also shown the empirical relation ratio proposed by Lunder & Pakalnis (1997).

Figure 9 shows peak and post-peak behaviors for the pillar analyzed. Spalling and punching in peak situations without causing a pillar failure are observed. Thus, the pillar failure is developed from degraded conditions of the pillar system, caused by the punching mechanism, resulting in a shear failure.

Figure 9. Shear strains for the analyzed model in a status prior to peak, (left) and in the post-peak status (right). Load history applied on the inclined pillar is superimposed.


The analyses confirm the expected reduction in strength triggered by inclination of pillars and also by anisotropy in shear strength, caused for example by schistosity.  The reduction observed was approximately 10% for a pillar with W/H ratio of 1.8. In practical terms, there are considerable uncertainties associated with estimating the foliation strength. Obviously, lower foliation strength would lead to even lower pillar strengths. Lower strengths also result from steeper inclinations (i.e. greater than 20°).

The biggest limitation of the modeling shown here is the assumption of vertical lateral boundaries with horizontal displacement constraint (i.e., rollers). In most real pillars in inclined orebodies displacement is not constrained to the vertical, but will tend to occur normal to the extracted ore. This deformation induces shear in the pillar and will reduce pillar strength more than shown here. For inclined orebodies, an infinite array of pillars is simulated by attaching lateral boundaries on each of the up- and down-dip sides to each other and thereby forming periodic boundaries. In another study without anisotropy we found that periodic boundaries produced a 35% reduction (for 15° orebody inclination) with respect to zero normal displacement (i.e. roller boundaries) on the lateral model boundaries. In all likelihood, the true pillar strength is bounded by the strengths given by these two boundary assumptions. For critical applications neither boundary condition should be used, but rather the boundaries should be extended to include several pillars located within panels.   


Itasca Consulting Group, Inc. 2012. FLAC3D – Fast Lagrangian Analysis of Continua in 3 Dimensions, Version 5.0.  Minneapolis: Itasca.

Jiménez Salas, J.A., et al. 1975. Geotecnia y Cimientos. Ed. Rueda. Madrid

Lunder, P.J. & Pakalnis, R. 1997. Determining the strength of hard rock mine pillars. Bull. Can. Inst. Min. Metall., V.90, pp. 51-55.

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