Rock Mechanics - Application of Extreme Value Statistics to Test Data

In general, many problems relating to the exploitation of mineral deposits are probabilistic in nature. This derives from the fact that the geologic universe is inherently random. Probability theory and statistics have been found useful for forecasting the behavior of natural events that occur in the geologic universe. The objective of this paper is to illustrate the application of the theory of extremes to this fore-casting problem. For example, it is customary for design purposes to determine the rupture strength of geologic materials. The theory of extremes is exceedingly useful in describing that portion of the frequency distribution of rupture strength which contains the least strengths. Parameters describing the distribution of the least strengths are more important to the designer of mining excavations than parameters describing the total distribution. The basic principles of the theory of extremes will be detailed and illustrated. Any person required to work in the laboratory of nature is aware that uncertainty is a salient feature of all mining enterprises. A mining engineer required to plan the most efficient, practicable, profitable, and safe mine finds himself face to face with numerous ill-understood and often unquantifiable states of nature. Basic information necessary for adequate planning is often lacking or derived from incomplete tests on samples or experience of doubtful validity. The planning procedure usually takes the form of determining a feasible layout with the intent of determining an optimal layout when and if the necessary details and information become available. The crux of the entire procedure is the choosing of numbers to put into the operational and structural models which encompass the plan. Many times these numbers must be assigned qualitatively from past experiences and are called the "most probable ones." At other times, load records, performance records and material tests provide a basis for extrapolation. In any event, the numbers are chosen from a distribution or set of all numbers. Since each number in the distribution represents a possible state, the choice of any particular value is based upon a decision rule. To illustrate, consider the design of an underground structure or the design of a rock slope. The initial step is the formulation of the various possible structural actions which result from the geometry of the layout. For a given structural model various intensities of behavior are possible depending upon the load, deformation, and material characteristic spec-trums, respectively. Of particular interest to mining people is the failure behavior or condition, i.e., when there is a complete collapse of structural resistance by either structural instability or fracture. A necessary feature of the analysis is the "rupture strength" of the material. Information on the rupture strength is derived from testing either in situ or in the laboratory and the usual outcome is a variation in the test results. The methodology used to overcome this variation is to construct a frequency distribution of rupture strengths, and then determine a measure of central tendency and variability. The main idea involved is that the central tendency number will be used in the failure calculations and the measure of dispersion will be used to estimate the probability of failure. In particular if the distribution of rupture strength is normal, the mean rupture strength is the central tendency number and the standard deviation of the rupture strength is the measure of variability. Suppose the mean value of rupture strength is 1000 psi and the standard deviation is 200 psi. Insertion of 1000 psi into the failure calculation produces results that are unsafe, hence a common decision rule is to reduce the mean value by a "factor of ignorance" so that the failure calculation will produce a "safe result." If two is chosen as a factor of ignorance, this means the value inserted in the calculation is 500 psi or 2.5 times the standard deviation. The next step is to determine the percentage chance that failure will occur from a design created on this basis. Tables on the normal distribution function show that this percentage chance is 0.621% or approximately 7 times out of 1000. In practice, however, the situation is more complicated than represented by the foregoing illustration. The laboratory or field testing program usually constitutes a pathetically small sample of the geologic universe of interest and not enough testing is carried out to determine the exact form of the distribution of the test results. The normal, Cauchy and Student's T distributions are strikingly similar, and it becomes a matter of mathematical convenience to assume the normal law for phenomena which follow other laws.