Rudiger Welf Olgert KERSTEN

Student No 7272848

A thesis submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy.

Johannesburg, September 2016


The design of stable pillars in mining is of fundamental importance not only in the Bushveld mines but in the entire mining industry. Wherever mining occurs, pillars will be formed at some stage, and it is essential to predict their behaviour. It is imperative to know whether they would burst, yield or remain stable. Although of major importance, the design of pillars still suffer from major drawbacks and weaknesses that affect the results in a fundamental manner.

Faced with uncertainty, the designs have, for the most part, been conservative, resulting in the loss of millions of tons of ore by being sterilized and unavailable in the future. The reduction of the life of mines with dire consequences for the long-term life of the mining industry.

With the above in mind the thesis investigated the following:

  • The pillar design method currently in use for its strengths and weaknesses
  • Alternative methods in use, actual and suggested.
  • Proposed an improvedd method for calculating the pillar strength and the loading system.
  • Compared the improved method with underground observations.
  • Proposed a simplified pillar strength equation based on the improved method.
  • Presented main conclusions and recommendations.

Examination of the pillar design method currently in use for its strengths and weaknesses:

Pillar design has been based on the empirical equation on the work done by Salamon and Munro (1967) for coal mines and subsequently modified by Hedley and Grant (1972) at the Elliot Lake uranium mine. By changing the exponents relevant to the hard rock mine, the Hedley-Grant equation was created. It has since become known as the Hedley-Grant strength equation for hard rock pillars.

The weaknesses associated with the empirical method are as follows:

  • The method is easy to use but could not be extrapolated beyond the range used for calibration.
  • Very few pillar collapses have occurred in Bushveld mines which could possibly be ascribed to over-conservative design or correct design.
  • Empirical methods are, by nature, observational and no deep fundamental understanding of the variables is required to derive an answer acceptable in general to the problem of stability.

The pillar strength equation is based on the following:


Except for the width and the height of the pillar, the strength factor and the exponents are based on back-analysis of failed pillar areas. In the absence of failed pillar areas, the strength factor is assumed to be a fraction of the uniaxial compressive strength of the rock mass in the pillar; the value varied from 0.3 to 0.8, which is sometimes increased without sound scientific basis when no pillar failure occurred. This type of approach is not only wasteful but could prove to be dangerous.

The stress imposed on the pillar is determined by using the Tributary Area Theory (TAT) including the percentage extraction, depth below surface and the rock mass density.

One major advantage of the method, Hedley-Grant strength equation and the Tributary Area Theory, is its ease of use; few parameters needed to be defined.

Rock masses are difficult to define as an engineering material. At the time of the research, the average uniaxial compressive strength of five samples for an ore body had been deemed sufficient to design a mine. Detailed analysis of strength values showed that the coefficient of variation (COV) for pyroxenite and chromitite was 0.33 nearly falling outside the region of even statistical predictability.

Variability in the actual pillar dimensions, in plan and section, which differ substantially from design dimensions, adds to the uncertainty of the design.

Using the Monte Carlo simulation method, the influence of the variability of the input parameters for a typical bord and pillar chrome mine, using the Hedley-Grant equation and the tributary area theory showed the following results:

Probability of Failure = 23% at a factor of safety of 1.57.

Accepting that the Hedley-Grant equation is correct and that the Tributary Area Theory is a true reflection of the pillar strength and stress, the design safety factor of 1.57 is acceptable but the probability of failure of 23% due to the variation in input parameters raises the question of whether a factor of safety of 1.57 is sufficient?

The above argument presupposed that the values using Tributary Area Theory and the Hedley-Grant equation provided the correct answer. The fact that the predicted failure did not materialise indicated that the input parameters were overly conservative and/or the pillar equation is suspect.

It should also be noted that the probability of failure related to the probability that any given pillar had a factor of safety <1.0. Failure of a panel of pillars would only occur if those pillars happened to be in groups.

Investigate alternative methods in use, actual and suggested:

The deficiencies have been identified by other researchers and alternative approaches have been developed addressing some of these deficiencies. The overall conclusions from a literature survey were as follows:

  • The influence of the stiffness of the loading system was neglected in the pre-failure region but three methods assessed incorporated the strata stiffness concept in the post-failure regime
  • None of the methods considered the interaction between hanging wall and/or footwall and its effect on the pillar strength.
  • Composite pillars were generally treated as a uniform entity.
  • The variability of the mining dimensions and the rock mass properties were dealt with quantitatively in four methods.
  • None proposed a methodology that incorporated a combination of a more realistic strength equation, the system stiffness and the probability approach in design.

Proposal of an improved method for calculating the pillar strength and the loading system.

The proposed alternative approach is based on a semi-analytical strength determination using a two-dimensional mathematical model in conjunction with a failure criterion to calculate the pillar strength.

The selected method is based on the following:-

  • Determining the pillar stress using FLAC2D,
  • Incorporating the modified Hoek-Brown failure criterion.

It is assumed that the pillar/rock mass remained elastic until pillar failure occurred. The interaction between the local mine/pillar stiffness, and an elastic response, determines the loading conditions. The combination of the two concepts, both elastic, was used to determine the factors of safety for the pillar/rock mass.

With further development, the variability of the rock mass properties and mining variations were also incorporated in the proposed methodology.

The FLAC2D/Hoek-Brown model could simulate/incorporate the following conditions:-

  • The pillar strength for a homogeneous pillar.
  • Strength of composite pillars such as found in the chrome mines.
  • Incorporation of the effect of planes of weakness.
  • Use of known geotechnical parameters.
  • Development of a simple equation for a specific set of conditions.
  • Although of great importance, the influence of the hanging and /or footwall properties on pillar strength was not considered in the thesis.

Using the FLAC2D/Hoek-Brown model the following was observed:-

  • The vertical stress was the lowest at the pillar edge at commencement of pillar failure.
  • At the average peak pillar stress, the vertical stress at the core of the pillar generally exceeded the uniaxial compressive strength of the rock.
  • Pillar failure was seen as a progressive process.
  • The volumetric strain increment could be a possible measure of the depth of fracturing in a pillar.

For back analysis, it was obvious that some definitive values had to be used. The required variables were identified as follows:

  • The mean uniaxial compressive strength from available samples for the property was used.
  • The Geological Strength Index was estimated.
  • The mivalue, hence the mr, was based on the widely used RocLab programme.

Pillar loading is a function of the strata stiffness and the areal extent of the mining geometry while the Tributary Area Theory assumes an infinitely mined area with zero stiffness of the overlying rock mass.

This oversimplification leads to overdesign in most practical mining geometries. It is known that the geological losses in the platinum mines varies between 20% and 30% of the mined area resulting in limited mining spans between the “regional pillars” created by the geological losses.

The influence of the geological losses can be simulated using the concept of the load line of the loading system. The amount of convergence in an elastic medium can be calculated for various spans, different Poisson’s ratios and Young’s Moduli.

The research dealt with the pre-failure portion of the pillar design, therefore, the theory of elasticity could be used to the point of pillar failure allowing accurate calculation of rock mass, pillar stresses and deformations.

Figure 1 is a plot of a pillar strength curve intersecting the system loadline. The system obtains equilibrium at the intersection of the two curves. In this example, the system curve is based on the fact that without any support full elastic convergence would occur. The presence of a support medium, that prevents any convergence, is the product of the area and the vertical primitive stress.


Figure 1. Plot showing pillar resistance and system curve

The pillar resistance curve is based on the stress and convergence obtained using FLAC2D and the Hoek-Brown failure criterion, while the system stiffness curve/load line is based on the elastic convergence of a slot in an elastic medium at finite depth. The intersection of the two curves gives the equilibrium condition of a specified geometry.

The combination of the two curves and the intersection point is referred to as the System Pillar Equilibrium Concept, SPEC for short.

The curves in the Figure 1 were obtained by following the process shown in the flow diagram, Figure 2.


Figure 2. Schematic presentation of the steps involved in the FLAC2D/Hoek-Brown and System Pillar Equilibrium/SPEC methodology.

The comparison of the improved method with underground observations:

Two bord and pillar mines in the Bushveld complex were selected for calibration, Two Rivers and Impala Platinum Mines, to test the proposed methodology.

In both cases the extent of pillar failure, as well as the convergence, could be simulated and it was concluded that the method did represent the actual underground conditions better than other current methods.

A propose a simplified pillar strength equation based on the improved method:

One of the main advantage of the Hedley-Grant/Tributary Area Theory is its simplicity in application. In order to achieve a similar ease of modelling, it was attempted to obtain a generic pillar equation that is easy to use.

A simple generic equation was derived, using the Geological Strength Index and the uniaxial compressive strength, based on the detailed back-analysis that could be used for all “normal” situations in the Bushveld mines. It was found that for a specific range of data the equation below can be used:


Where SSand BB are both related to the uniaxial compressive strength and the GSI value.

By employing this equation, it was possible to simplify the calculation procedure as well as for use of the Monte Carlo simulation for sensitivity studies.

Present main conclusions and recommendations:

The weaknesses of the current design procedure have all been addressed and it was concluded that the proposed methodology is an improvement on currently available alternative methods of bord and pillar design.

The volumetric strain increment value for failure initiation lay between 1e-2 to 3e-2 for the fracture zone extent for both pyroxenite and chromitite pillars.

From the volumetric strain increment values, it appeared as if the FLAC2D/Hoek-Brown model in conjunction with the SPEC method required to calculate the pillar stress and convergence, approximated the real underground situation.

The stage has been reached where the methodology can be used to predict most likely failure of pillars at greater depth and alternative pillar mining methods could be modelled. The concept can also be extended to incorporate the energy balance of the system.

A generic equation based on the FLAC2D/Hoek-Brown methodology has been developed for general use in mine design.

Additional research is essential on subsections of the input values to finally establish that the methodology is a representation of underground physical changes.

Rudi KerstenRudi Kersten, rock mechanics consultant, has more than 45 years experience in the mining industry. As consultant to Anglovaal he has been involved in the design and maintenance of both open pit and underground mines. The underground mines included shallow, intermediate, deep and ultradeep operations of archean and Witwatersrand type mines. Also included are manganese, copper, chrome, platinum, andalusite and silica operations. Consultancy work was also done on copper, gold and manganese mines in Brazil, Kazakstan, Uzbekistan, Zambia, Zimbabwe, DRC,United States of America, Australia, United Emirates and Tanzania. Experience covers the evaluation of borehole core to selection of mining method and detail planning and commissioning thereof. He has also been actively involved in designing and installing backfill plants at a number of mines. R Kersten has been an independent mine consultant with South African and overseas clients since 1994 and has been involved in a number of due diligence and resource evaluations. He gained his MSc degree in geology at the University of Pretoria, currently busy with a PhD degree and, throughout his career, he has been involved in the guidance and evaluation of the research activities of the Chamber of Mines research organisation.