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By Rossen A. Halatchev

This study introduces a new probabilistic approach to seismic stability analysis of embankments and slopes. The approach is based on Sarma's solution, which is modified through presentation of the seismic force by its horizontal and vertical components; i.e. the force may have an arbitrary inclination. The probabilistic modelling includes Monte Carlo simulation of the shear strength parameters of the soil mass, which are assumed to have a normal distribution. The probability of failure is determined based on the seismic co-efficient, which is treated as a random quantity. The probabilistic approach is illustrated through a solution of a numerical example from an opencast coal mine.

Jan 1, 1992

By D. S. Young

The random variable transformation technique was introduced to make probabilistic structural analysis simple for such structures as slopes, dams, foundations, or retaining walls, where two shear strength parameters are required to analyze their stability by deterministic engineering methods. The Gaussian transformation was performed by using the Hermite orthogonal polynomial function. This transforms the experimental histogram into the standard normal distribution, whose mean is zero with a variance of one. Then, the transformed standard Gaussian variable was used for the bivariate normal probability analysis on the structure. The Gaussian transformation technique and the Hermite polynomial functions were chosen because the bivariate joint normal distribution is well-known and easily available for engineers. The binormal probability analysis was made on the probability region of a certain confidence level by superposing the critical line on the same probability region that divides the region into two zones: stable and unstable zones. The critical line was determined from the back calculation of the deterministic engineering method of the structural design. Finally, the probability of structural failure at a desired confidence level was calculated by simply integrating the volume of the binormal density surface above the unstable zone in the probability region. A simple case of slope analysis is introduced to illustrate this probabilistic approach, and its effectiveness and simplicity is demonstrated.

Jan 1, 1987

By Neil T. Russo, Peter M. Kandaris, Jogi S. Gadok

"Assessment of subsurface data for foundation design in support of long high voltage electric transmission lines is best accomplished using a probabilistic strategy since only a small portion of structure locations are typically sampled. Probabilistic approaches provide a tool to evaluate site uncertainty in a systematic and rational manner. This paper provides a case history of a risk-based design approach for a recently completed 191-mile long 69/161/345kV steel monopole transmission line traversing Central Iowa. The project owner made available limited subsurface information at project initiation, requiring the design team to obtain additional data within a constrained budget for 1067 large diameter, laterally-loaded deep drilled shaft reinforced concrete piers. The team commissioned a geologic desktop study to reduce uncertainty in subsurface evaluations and aid in selecting additional sample sites. In total, investigation programs tested approximately 15 percent of structure sites. The paper details the probabilistic approach used to estimate foundation geotechnical design parameters, with information from the various studies categorizing strata both laterally and with depth. Low-bound parameters were estimated via a statistical approach at a predetermined reliability level. Significant savings resulted from this approach when compared to conservative deterministic processes. Project QA/QC logged subsurface soil conditions during foundation excavation, then compared as-found conditions to the conditions assumed for design. The team redesigned foundations if either a non-conservative variance in soil conditions was observed or if bedrock elevation differed significantly from design assumptions. As a result, only 22 out of 1067 foundations required modification during construction (2% variance from initial design) with negligible impact on the construction budget and schedule.INTRODUCTIONFoundation design for long high voltage transmission lines presents a unique challenge in that the assessment of subsurface conditions typically requires substantial data that may or may not be obtainable. Investigations are usually performed well in advance of construction, often with limited access to structure sites and typically in remote areas. Probability and risk-based investigation and design strategies offer an opportunity to systematically estimate subsurface profiles and geotechnical design properties with limited data. Risk can be further reduced with dedicated field inspection of foundation construction along with good field and design team communication during construction. This paper presents a case history of probabilistic methods used for the investigation and design of electric transmission line foundations."

Jan 1, 2017

By C. O. Swanson

CALCULATING the average grade of a gold deposit is a problem that presents certain unique difficulties. Particularly troublesome is the fact that the average of the samples taken from a deposit is likely to be higher than the grade of the ore that is eventually mined, even when allowance is made for dilution and other factors resulting from the mining operations. It would seem that this situation is due to the presence in the deposits of structural peculiarities which have not been properly taken into account, since estimating the average grade of the broken ore produced from the deposits does not present the same difficulty. It is therefore evident that structural considerations are of prime importance in estimating the grade of a deposit. This fact is clearly realized and taken into account in cases where the boundaries of the orebodies and the variations within them are easily recognized. But in many gold deposits this condition is not satisfied, a circumstance which can be regarded as the main source of difficulty. Lacking definite information concerning the structure, we are apt to use certain procedures which are either empirical or based on general probabilities.

Jan 1, 1945

In the stability assessment of open pit walls, geotechnical engineers have to deal with uncertainty related to structural geology, rock mechanical parameters, groundwater and so on. This uncertainty will immediately and significantly impact on wall stability assessment and therefore. influence mineral recovery and ultimately the financial return of open pit mining. However, the conventional factor of safety applied to measure the stability of slopes and the relevant analytical computer packages cannot quantitatively consider the variability or uncertainty of the input data. As a supplement, therefore, geotechnical engineers have begun to apply probability to the stability assessment of open pit walls over the last two decades. To-date unfortunately, no powerful computer package is available to geotechnical engineers to conduct the probability analysis of slope stability. This paper presents a computer package of probability analysis for rock and soil slope stability. In this computer package, the strength parameters of rock and/or soil material such as cohesion, friction angle and unit weight of the different rock and/or soil type domain in a complex structural geology environment can be considered as the basic input random variables. Groundwater level can also be taken into account as a random variable in this package. Two `rigorous' limit equilibrium solutions, the Morgenstern/Price method and the Sarma method, can be used to calculate the factor of safety during Monte Carlo Simulation applied in this program. Five types of probability distribution are included in this package for the user to simulate basic input random variables. Two realistic cases from BHP Iron Ore are used to illustrate this package.

Jan 1, 1997

By C. W. Eggert

"The idea for this talk came to me last year when I was attending the Canadian Gold Metallurgists Meeting. I talked about it to a friend of mine who is an authority on mineral dressing, particularly the theoretical end of the business. He remarked that this was an excellent technical note, why don't you write it up. This is what I have for presentation this morning. I hope in presenting this type of idea in this way it is in keeping with the tradition of the Canadian Gold Metallurgists which was to function on a not too sophisticated level and where operators could get together and trade ideas in an informal way.My basic thesis is that it should be possible to evaluate screen analyses by the application of the standard statistical probability theory. In any statistical analysis the sample is the first consideration. If we look at existing methods of size evaluation we find it is usually based on the fact that a given particle will pass one square aperature and not another. One system of standardizing how these aperatures will be set up is based on Rittenger' s work with the ratio of hole sizes proportional to the square root of 2. Keeping in mind the limitation we have accepted this as our method of 11 sizing"" in attempting to apply standard statistical methods to screen analysis."

Jan 1, 1968

By W. E. Roscoe

The probability of making an exploration discovery in Canada can be calculated from Derry's (1970) data on exploration expenditures and number of discoveries over the last nineteen years. Considering an average exploration "try" to be $30,000, the probability of a discovery with one try has decreased tenfold from about 0.01 in 1951 to 0.001 in 1969. A plot of these probabilities can be extrapolated to the future, giving a 1969-71 three-year average probability of 0.00096 with 90 per cent confidence limits of 0.00042 and 0.00153. Using this probability, various calculations can be made, such as (1) the probability of making at least one discovery for any given exploration expenditure, (2) the probability of ultimate ruin or survival of the exploration investment process and (3) the expected value of an exploration investment. There is a critical minimum amount per discovery that must be re-invested in exploration, no matter how large the original investment is, for exploration to survive at all in the long run. Currently, this critical minimum is $31,200,000 with 90 per cent confidence limits of $19,600,000 and $71,400,000.

Jan 1, 1971

By A. F. Silva, R. Dockendorff, J. M. Girao Sotomayor, V. F. Navarro Torres, C. Castro

This paper shows the importance of performing probabilistic analyses in open pits, especially for mine planning, which can lead to more efficient ore extraction and meeting the acceptability criteria for safety in mine slopes. Three-dimensional stability analyses were performed to evaluate the future geometry of a large open pit for iron ore extraction in Brazil. The strength parameters of the lithologies were calibrated using ruptures in the pit walls. After determining the factors of safety (FoSs) from the calibrated parameters, probabilistic analyses were performed using the total range of values of each parameter under different field conditions to verify the reliability of the initial analysis. In this sense, it was possible to plot the probability of failure (PoF) and the FoS on the graph of slope height versus slope interramp angle (IRA) for the future pit in each lithology. IRA recommendations are made for two scenarios: (1) the best scenario: dry without ubiquitous joints and (2) the worst scenario: the water table at 10 m depth with ubiquitous joints in the most unfavourable direction. The results show that probabilistic evaluation is an important tool for establishing alert mechanisms in slopes that can be termed stable.

Jul 2, 2022

By John G. Kovac

A computer simulation has been developed to estimate the chances of a miner making a successful escape while wearing a SCSR. The model takes into account: (1) training in the use of SCSRs, (2) apparatus integrity, and (3) the oxygen cost of a mine escape. This Bureau of Mines report examines survival odds for a prototypical escape, and illustrates how these odds change when SCSR training is improved.

Jan 1, 2009

By John G. Kovac

A computer simulation has been developed by the US. Bureau of Mines to estimate the chances of a miner making a successful escape while wearing a self-contained self-rescuer (SCSR). The model takes into account (1) training in the use of SCSR's, (2) apparatus integrity, and (3) oxygen cost of a mine escape. This report examines survival odds for a prototypical escape, and illustrates how these odds change when SCSR training is improved.

Jan 1, 2009

By Calvin L. Stevenson

The probability of a particle passing through a maze is predictable and consistent. This principle may be utilized to effect a particle size separation of materials by presenting a physical maze to a stream of material that will allow the fine particles to pass, while deflecting the coarse particles. A maze is effective even though all apertures encountered are many times larger than the particles being separated. By designing a machine that allows the probability parameters to be controlled and varied without changing physical dimensions, probability sizing becomes a useful tool for the separation of granular materials.

Jan 1, 1967

By C. W. Hoffman

A fairly recent development in the field of particle separation is now under investigation to determine its value to the mining industry. The method employed is probability sizing through a vibrating network of decreasing screen openings. While probability sizing as such is not new, the possibility of using it for coarse, highly abrasive ores at high tonnage rates has only recently appeared to merit practical consideration. The advantages to be gained by the use of probability sizing vs. conventional screening methods are: reduced space requirements for a high capacity unit, fewer moving parts, more versatile product sizing selection, reduced maintenance requirements and lower power consumption.

Jan 1, 1968

By Randal J. Barnes

In, any open pit mine the ability to make accurate production estimations for the shovel-truck system is an important asset. This is true in an operating mine wishing to estimate production under a set of system modifications (i.e. more trucks, fewer loaders, change in roads, etc.), as well as in the pre- mining planning stages of a property. Many different methods have been proposed for solving this problem; some of the methods rely heavily on "rules of thumb" while others are highly mathematical and require access to a high speed digital computer. Analysis of shovel-truck systems with the use of simulation models is a widely accepted technique within the industry. Through the use of such models it is possible to account for much of the inherent variability of the system and thus produce meaningful results. However, simulation studies are expensive. Formulation and setup of the actual simulation model requires a significant commitment of time and energy. In addition, the cost of the necessary computer runs is often of considerable magnitude.

Jan 1, 1979

INTRODUCTION TO PROBABILITY THEORY The mathematical theory of probability provides the foundation for the interpretations and inferences that can be made from statistical techniques. Probability theory is a broad field of study in itself, and can only be touched on here in order to provide the necessary back- ground for a more complete understanding of statistical analysis. Probability is concerned with the outcome of random events. It had its beginnings in the theory of games of chance which were of great interest to European mathematicians during the last half of the 17th century. The outcome of a random process or experiment is defined as an event. The collection of all mathematical points associated with each event is called the sample space for an experiment. Consider a random experiment having a finite number, n, of possible outcomes, some of which have a characteristic A. What is the probability that A will result from a single

Jan 1, 1980

By G. Pan

The production data of many mines have not reconciled geostatistical estimations based on exploration and development drilling. The discrepancies in recovered metals are sometimes as large as 30%. A number of factors are responsible for such unpleasant outcomes. A major cause, however, is the lack of geological controls in the implementation of geostatistical techniques. Geological modeling establishes geological constraints and mineralization/ore envelopes. Geological constraints are usually established through drill hole interpretation. Mineralization and ore envelopes have been delineated manually based on assay grades. The issues associated with manual methods are: •Time requirements . •Inconsistent dilution. •Uncontrollable implementation variance. Although the time issue may not be a big concern for some companies, the other two have been responsible for many undesired estimation errors. A new geostatistical approach described in this study is designed on the basis of indicator and ordinary kriging. The central component of the method is the procedure for automatic delineation of mineralization and ore envelopes. The method, referred to as probability assigned constrained kriging (PACK), is demonstrated on a Carlin-type gold deposit. The estimation results are compared to blast-hole based production panels, showing that the estimated tonnage is about 2.1 % higher than the actual production, and the average grade is about 2.4% lower.

Jan 1, 1995

By B Mohanty, G Yamin

A number of parameters, aside from blast design and pre-existing rock conditions, can change the performance of the explosives system in actual blasting practice. Sympathetic pressures caused by detonation of preceding holes have been identified as a major contributor to explosive system malfunctions, leading to uneven blasting performance. These effects can occur both within a hole or across several holes. These can consist of high transient pressures related to the shock phase of the explosion or the relatively longer or quasi-static pressures associated with the explosion gas expansion phase. This study presents the findings from an on- going investigation on measurement and analysis of these pressures between donor and receptor holes in a granitic outcrop for representative inter-hole distances and borehole conditions, employing single cartridges of emulsion explosive in 45 mm diameter holes. The results, despite wide scatter in the measured values, show the very short duration pressures to range up to 7 MPa and arriving within a few milliseconds after detonation, whereas the much lower amplitude pressures arrive after tens of milliseconds. Both have the potential to seriously affect the performance of an explosives system.

Jan 1, 2005

By John Reilly

Costs of complex/underground projects have been consistently underestimated and delivery on time and on budget has been problematic. This has been attributed, to inadequate cost estimates and unidentified and/or insufficiently managed risk. Use of probabilistic cost estimates like the Washington State Department of Transportation?s CEVP® process (developed by the author and others) has produced more realistic cost estimates, explicitly identified and quantified risk (as input to risk management plans) and early awareness of key project issues. This paper updates previous work (NAT 2004), adds implementation detail for probalistic cost estimating as used by WSDOT, FHWA and other agencies and will address key issues in risk management, as related to probalistic cost estimates.

Jan 1, 2008

By T. T. Read

A SHORT time ago, one of the large steel companies courteously furnished the author with detailed records of the operations of a considerable number of iron blast furnaces over a period of two months. These will be made the basis of a subsequent study, but in beginning this study one of -the first points requiring consideration was the probable error of the records, for what otherwise might be regarded as significant differences of performance under different conditions were perhaps really due to errors of observation, or failure to observe and record variations. If it is assumed, in blast-furnace records and calculations, that known quantities or factors are constant, when as a matter of fact they are variable, the effects of these variations may erroneously be ascribed to some other cause. This is quite clearly expressed in the following quotation from E. Buckingham:1 Theory always operates on an ideally simplified picture of reality because real phenomena are unmanageably complicated. The results obtained are not exactly true for any, real phenomenon, though they may be for an ideal one; and the approxi¬mation with which a theoretical equation, however obtained, represents the actual facts, always depends on the approximation in essentials between the ideal picture and its real prototype. The purpose of dimensional reasoning is to find out how some quantity which is involved in the phenomenon under consideration is related to certain others; or to find the relation connecting two or more quantities which vary, or may vary, simulta-neously during the course of the phenomenon. Since we know that we must have a complete equation to start with, we begin by thinking the matter over, to see whether the quantities we have in mind are the only ones involved. Usually it is evident that they are not; so we next make a list of all the quantities we can think of which might under any circumstances be important. Upon considering this list, which is often a long one, it is usually evident that under the actual circumstances a number of these quantities may safely be ignored; so we cross them off the list and thus pass from our most general conception of the phenomenon to an ideal simpler one, in which these quantities are not involved at all.

Jan 3, 1925

By John E. Kinnison

The Mission mine, located in southern Arizona near Tucson, is a recently developed open pit which produces 15,000 tons per day of copper ore. The mine lies on a wide and gently sloping bajada sweeping northeasterly from the Sierrita mountains. It derives its name from the nearby Mission San Xavier del Bac, built circa 1700. The ore body is everywhere covered by about 200 feet of alluvium, as are the adjacent Pima and Palo Verde mines. The geology of the Mission deposit is known principally through the study of diamond drill holes spaced 150 to 300 feet apart. The open pit is in an infant stage and has not yet revealed much of the deposit, although information obtained there by the operating staff, and from a few thousand feet of exploratory underground workings, has added significantly to the general fund of knowledge. Permission to publish was granted by the American Smelting and Refining Company. I am indebted to Kenyon Richard and J. H. Courtright for criticism of this paper and for their direction during the several years in which I studied the Mission deposit. Acknowledgment is given to consultant in petrography, Robert L. DuBois of the University of Arizona. The only previous publication pertaining directly to the geology of file Mission deposit appeared in 1959 (Richard and Courtright), and may not be everywhere readily available. Cooper (1960) has published a short paper on the district

Jan 1, 1963

By Dorricott M. G, McAllister J

This paper is concerned with the application of a probabilistic approach to the risk assessment of mining projects.In current practice, sensitivity analysis is the most commonly used deterministic approach of accounting for the uncertainties involved in mining project evaluation, while Monte Carlo simulation is considered to be the best probabilistic method available for the same purpose. The methodology of the approach proposed in this paper is as simple as that of sensitivity analysis. However, the results obtained can be more comprehensive than those obtained from Monte Carlo simulation. Various important statistical features, such as correlation and skewness, of the basic random parameters can be easily taken into account by the proposed approach. Example studies are demonstrated. Lotus 123 was employed for the associated probabilistic calculations.

Jan 1, 1992