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By George F. Becker

A Nere record of Emmons's professional career would very inadequately represent the man. That he was eminent mc know, and our successors will realize in due time; but they must depend upon us for knowledge of a singularly ~vholesome, modest, unselfish personality, and of a character that did honor to a profession in which trustworthiness is indispensable. Those members of the Institute who met Emmons are his friends, and I never knew .one of these who was not the better for that friendship. Emmons was born in Boston, Mar. 29, 1841, the son of Xathaniel H. and Elizabeth (Wales) Emmons, and was named Samuel Frankliri after an ancestor who was of the sarne family as Benjamin Franklin. He took the degrcc of Bachelor of Arts at Harvard in 1861, and soon afterwards went abroad to complete his education. From 1862 to 1864 he attended the courses at the fieole Impkriale des Mines at Paris, Elie de Beaumont arid Daubrke being among his professors. The year 1864-1865 he spent at Freiberg under Cotta and other famous teachers; after which he spent another year in traveling through Europe. Like many other renowned geologists, he approached his ultimate profession from its economic side, and was thus from the first imbued with a sense of high responsibility in the promulgation of scientific opinions or conclusions. With hypotheses which were interesting merely because they were ingenious or even plausible, he would have nothing to do. In 1867 he joined the Geological Exploration of the Fortieth Parallel at its organization under Clarence King, serving at first as a volunteer, but soon receiving a regular appointment. This expedition was the first one of purely geological character sent out by the United States government. As Emmons has shown in his admirable presidential address on " The Geology of Government Explorations," its work was founded on a com-

Jan 1, 1912

By Robert B. Johnson

The refraction seismic method has been used as an indirect means of exploring for fluorspar in southern Illinois by the Illinois State Geological Survey. Use of the method has been restricted to those areas where mineralization is associated with faults, a restriction necessitated by the operating principles of the refraction method. The seismic data enable the interpreter to locate fault zones in the subsurface and, under ideal circumstances, to make estimates of dip direction and magnitude and vertical displacement provided the local stratigraphic units are known and are geophysically recognizable. In this instance, refraction seismic exploration has been successful in the surface mapping of well-developed faults, both simple and complex with which fluorspar may or may not be associated. However, less clear results have been obtained where faulting has been less severe and where there are numerous accompanying mineralized joints.

Jan 1, 1964

By A. M. Gaudin, J. H. Brown

In a survey of the effects of heat treatment on the comminution of various rocks, it was found that unusual size distributions often characterize the broken products, and that this is related to a great increase in intergranular fracture1. It is the purpose of the present paper to contribute to an understanding of this effect. In a thesis by one of use, critical consideration was given to many mechanisms which could conceivably be responsible for the increased intergran-ular fracture. It was shown that neither gross chemical changes nor intergranular corrosion are required. Among non-chemical mechanisms the most likely appeared to be (1) Uneven expansion due to thermal gradients, giving spalling (2) In a manner similar to that prevailing in metals, the appearance at high temperature of intergranular weakness. (3) Unequal thermal expansion of adjacent grains. Experiments were designed and conducted to permit a choice among the many alternative mechanisms and in particular among the three listed here. EXPERIMENTAL MATERIALS Tests were made on the materials listed in Table 1. Of each material, only lumps similar in composition and grain size were used. The three marbles listed in Table 1 were analyzed to assure that their chemical compositions did not differ greatly (Table 2). On the basis of the chemical analyses, of differential thermal analyses, and of X-ray powder diffraction patterns, it appeared that the marbles were very similar except for their grain sizes.

Jan 1, 1961

By Ernest M. Spokes

For bituminous and lignite coals the economic picture continued to improve as it has each year since the low of 1961, with production at 480 mil- lion tons in 1964 compared with 1963's 459 million tons. The anthracite industry produced 16.3 million tons in 1964 and 18.3 million tons in 1963. The industry is looking toward a further expansion; among the new developments will be a mining wheel producing 3 million tons of lignite per year in South Dakota. It is expected that electric power generation will continue to provide the market for most of the increase in consumption, with exports second in importance in the expansion. Improvements in methods of mining and transporting coal underground and in strip mines, and the cooperation of carriers in finding ways to reduce transportation costs to distant markets, have made possible these expansions. Research continues to reduce costs of mining, transportation, steam generation of electric power and production of steam for space heating and manufacturing processes, to retain some markets and expand others. Other research seeks to expand markets by developing fuel from coal which will be more attractive than competing fuels, by reducing sulfur and other sources of air pollution. Public relations efforts of the industry are bearing fruit in a greater acceptance of coal as a modern fuel.

Jan 2, 1965

By M. R. TECK, N. Rudd, L. K. Thomas, D. L. Katz

The authors are to be congratulated for presenting the first comprehensive paper on the theory and practice of threshold pressure determination in connection with underground gas storage. It is of particular value since most previous experimental work on this subject has been proprietary to the commercial laboratories engaged in such testing and to their clients. This dlscussion is presented in amplification of the subject paper, not in criticism, and is based upon our experience with approximately 5,000 threshold pressure determinations. Our comments are directed primarily to the experimental procedures as described by Thomas et al. It should be noted that, while clay mineral-rich lithologies are the most common caprock materials, the samples used (Table 1) include low permeability sandstones and carbonates, but do not include shales or other formations in which clav minerals exert a dominant effect. While our experience on samples analogous to those investigated by the authors generally confirms their findings, we have found that even more rigorous experimental conditions must be imposed where clay minerals are the controlling factor. These additional conditions result from the well-known molecular character of the clay minerals themselves in which water is interlaminated with, and bonded to, silicate sheets, thus becoming an essential component of the molecule. In many clay minerals the character of the intersheet bonding, and thus the dimension of the lattice, is strongly affected by variations in the identity and concentration of cations present in the intersilicate water. On a supermolecular scale, adsorbed water films on particle or "flakes" of shale provide much of the cohesive force necessary to maintain mechanical strength. The thickness of these films is variable and inversely proportional to intergranular pressure within the dimensional limits involved in caprock investigation. The thickness of the intersilicate water layers may also be subject to pressure under some conditions. These considerations are reflected in several necessary modifications of the experimental procedures described by the authors. Two of these are, in our judgment and experience, of such importance that they cast real doubt on the validity of data obtained on argillaceous materials without their application. The very careful resaturation procedures proposed by the authors should give good results in connection with the relatively "inert" lithologies used in their experiments. Our experience suggests, however, that they would create more problems than they would eliminate if applied to argillaceous materials. 1. Drying of the sample at the temperature and for the duration indicated by the authors not only will dessicate the sample with consequent shrinkage cracking, parting along bedding planes, and even total disassociation in the case of fissile shales, but it may also alter the character of the clay minerals themselves — in some cases irreversibly. 2. Prolonged evacuation to 2 microns Hg further dessicates the sample and is even more effective in reducing the intersilicate water content. 3. Resaturation with water differing in pH or chemical content from that existing originally in the sample also serves to alter the nature of the sample, particularly its structure and strength. 4. Our experience has been that, even with the thorough precautions proposed by the authors, complete resaturation of very fine grained materials is rarely obtained. Yet our experience confirms the

Jan 1, 1969

By G. Thodos, W. F. Stevens

The point-source function introduced by Horner' us a solution to the general unsteady-state equation for the flow of fluids through porous media has been utilized to calculate pressure profiles for adjacent wells as a function of time. By trial and error, the time at which the pressure waves meet to cause interference can then be calculated. For this calculation, the time of interference is arbitrarily defined as that time when both pressure waves exhibit a pressure decrease of 25 psi ar the same point within the formation. To illustrate the method, an example involving two adjacent wells producing at different rates has been worked out and is presented in detail. INTRODUCTION The initiation of production of a new well is usually accompanied by pressure drawdown measurements in order to establish the extent and characteristics of the formation. The interpretation of such pressure drawdown data is based on mathematical developments involving a single well in an infinite reservoir. If more than one well is present in a reservoir, the possibility of interaction between wells exists. Therefore, care must be taken in interpreting the pressure drawdown data. It becomes important to be able to determine when this interaction, or interference, becomes significant. This paper presents a method for the prediction of the approximate time of interference between two such adjacent wells. The basic differential equation describing the pressure distribution existing during unsteady-state flow of a slightly compressible fluid has been solved by Hor-ner.' for a constant production rate from a circular homogeneous reservoir of infinite extent. The solution utilizes the point-source function to define the pressure at various points within the formation as a function of time. The resulting expression is: q = production rate, STB/D ,;= formation volume factor, dimension-less µ = fluid viscosity, cp k = formation permeability. md H = formation thickness, ft C - fluid compressibility, psi' F = formation porosity, dimensionless R = distance from centerline of well, ft T = production time, hours Ei(—u) = Ei-function of U Eq. 1 may be used to calculate the pressure profiles generated during production from a single well in an infinite reservoir.

By Cyril Wells, Paul E. Busby, Donald P. Hart

EARLY workers in the field have established that the diffusion of nitrogen follows normal diffusion laws. Concentration-penetration data from layer analyses of reasonably pure iron specimens nitrided in an atmosphere of CO-NH, are given in the literature,' and were utilized to calculate diffusion constants for nitrogen. The diffusion of nitrogen in y-iron may be represented approximately by the equation D - 1.07x10-' e-:".'m'"'r The diflusion coefficients, D, given by this relation represent mean values, since carburization occurred simultaneously with nitriding. The rate of diffusion in a-iron has been determined from calculations based on data obtained by measuring the internal friction of samples at each of several frequencies and temperatures.' The expression for the diffusion of nitrogen in a-iron derived from these experiments is D= 1.4x10-1 e-it tudent Increasing carbon content decreases the diffusion of nitrogen in a-iron.3 Oxygen, either dissolved or combined, has been shown' to exert a marked influence on the diffusion of nitrogen from a nitriding atmosphere. D values increase rapidly with increasing amount of dissolved oxide in the metal, but diffusion is completely inhibited if preliminary oxidation has produced the slightest trace of iron oxide on the surface. Diffusion coefficients have also been calculated from permeability and solubility data,' but these values are of questionable accuracy since the boundary conditions assumed may not actually exist. Experimental Method The method selected for determining the diffusion coefficient of nitrogen in a-iron in the present investigation was the so-called fractional saturation method. This method consists of heating the specimen in an atmosphere of H2-NH8 mixtures of known composition for a predetermined time. Following this treatment, the specimen is analyzed for nitrogen; and the diffusion coefficient is calculated from a known relationship of the fractional saturation, time of diffusion, and dimensions of the specimen. This relation is based on a solution of Fick's second law and assumes that the surface of the specimen immediately reaches a given concentration which remains constant during the diffusion anneal. Tabular solutions are available4-5 and provide the relationship between fractional saturation and the quantity \/Dt /a, where D is the diffusion coefficient in cm2 per sec, t is the time of the diffusion anneal in seconds, and a, in the present experiments, is the half-thickness of the slab specimen in cm. The method is adequately discussed elsewhere.9-10 Material The material for this study consisted of 3/32 in. thick (0.24 cm) carbonyl iron plate. The major impurities in this iron are oxygen and carbon, which are readily removed by long time high temperature treatment in an atmosphere of purified hydrogen. In order to reduce the diffusion anneal time for saturation runs, some of this material was rolled to a thickness of 0.015 in. (0.038 cm) prior to the hydrogen treatment. Hydrogen treatments were carried out in a hydrogen-purified Armco iron tube heated in a global

Jan 1, 1957

By Tasuku Fuwa, John Chipman

Equilibrium data on the reactions of gases with carbon and oxygen dissolved in liquid iron are reviewed and correlated. A source of error in oxygen analysis of graphitic samples is exposed. New experimental data on the oxygen content of high -carbon iron in equilibrium with carbon monoxide indicate that the activity coefficient of oxygen is decreased by additions of carbon. The carbon-oxygen product decreases slightly with increasing carbon content. In recent years a number of investigations have been reported dealing with the equilibria between carbon and oxygen in liquid iron and gases containing carbon oxides. The laboratory study of these reactions dates back to 1931 with the publication of a paper by Vacher and Hamilton' on "The Carbon-Oxygen Equilibrium in Liquid Iron". The principal object of this study was to obtain data on the reaction shown by Eq. [3] below; their data and others by vacherZ permit evaluation of the other equilibrium constants as well. Subsequent studies have been reported by Matoba, Phragmen and Kalling and by Marshall and Chipman.5 The equilibria may be expressed by the following three equations: C + CO,= 2CO;Kl = Pc° [1] 0-+CO = CO,;K2=-P co, "0.PCO c + O = CO; K3 = In the equilibrium constants, ac and a0 are taken as equal to [% C] and [% 0] respectively in the infinitely dilute solution. At finite concentrations an activity coefficient is introduced such that ac =fc [%C], and so forth. It is to be noted that these equations represent only two independent relations, the third being derivable from any two. Richardson and Dennis have determined the conditions of equilibrium in reaction [I] at temperatures of 1560" to 1760°C. Their results have been confirmed by Rist and chipman7 who have shown how the data may be extended to cover the entire range of liquid compositions at temperatures from the eutectic up to 1760°C. An approximate confirmation has been reported by Fuwa and chipman.' These investigations also established the activity coefficient of carbon which was shown to increase rather rapidly with increasing concentration. For the very dilute solutions the value of Kl is expressed in the equation: log^i = - 7280/T+ 6.65 This equation leads to a value of Kl = 430 at 1540° C in excellent agreement with the average value of Marshall and Chipman.5 It is shown as line 1 in Fig. 1 together with the experimental observations. In the case of reaction [2] the direct experimental results can be supplemented by observations on the corresponding reaction with hydrogen, namely reaction [4]:

Jan 1, 1961

By Thomas, J. Elmer

WHEN the price of crude oil was advanced on July 26, 1928, with some 4,000,000 bbl. daily of potential production shut in under proration regulations, and with as much more new production shortly available from semi-proved undrilled reserves, students of the oil industry were made to realize that a new order of things now prevailed in petroleum economics. A new interpretation of the law of supply and demand had taken the place of the archaic method of influencing current production by changing crude price levels. Proration had caused the change. The long period of over-production just experienced, the prospects of a potential over-supply far into the future, had set up certain forces which brought about some fundamental changes. Their cause, effect and future influence can best be developed by a detailed study of the history of the period since war-time regulation

Jan 1, 1928

By William P. Shinn

Jan 1, 1881

By Zdzislaw Specht

THIS paper discusses elementary laws pertaining to seismic reflection methods and presents a general and simple criterion for determining the direction of dipping of a reflection horizon from observational data of distance and time. The important development of practical geophysics and the successful results obtained in America, Germany, and other countries, caused introduction of geophysical methods of prospecting in Poland. Of all methods, seismic prospecting has proved most successful in the area of the Carpathian foreland, where new oil reserves are to be expected. Because of the great depth of the sedimentary column in that area, rang-ing sometimes over 2000 meters, it was obvious that only the seismic reflection method could give reliable information regarding the structural features of these strata at such a depth. During the experimental surveys it appeared that the complicated geological conditions of that area do not always allow prediction of the depth and the angle of dipping of reflecting horizons, and for that reason information about the inclination of a given horizon has great importance in the determination of subsurface structure. Therefore the problem arises of obtaining from seismic reflection data reliable information as to the direction of dipping of reflecting strata, without any previous knowledge of the average velocity and depth of geologic horizons under consideration.

Jan 1, 1940

By Kenneth K. Humphreys

INTRODUCTION What is automation? Why automate? Webster's New Collegiate Dictionary defines automation as "the automatically controlled operation of an apparatus, process, or system by mechanical or electronic devices that take the place of human organs of observation, effort, and decision." This definition gives an implied answer to the second question, "Why automate?" Automation, automatic control devices, and process control equipment receive their justification in supplementing or replacing manpower, which results in decreased operating costs. In the past, processes were operated with the minimum amount of instruments needed merely to guide operators. Processes were manually con- trolled by the operator on the basis of simple temperature, pressure, flow, and level indications on a panel. Today the tendency is to design a new plant to operate in a fully automatic manner according to a preset pro- gram utilizing a minimum of manpower. In addition to this direct cost reduction in manual labor, increased equipment capacity results from uniformity of flow and uniform control of process variables. This control eliminates or minimizes surges and permits equipment to be operated at or near its ultimate capacity and, for a given material throughput, permits the use of smaller process equipment. Automatic control, therefore, be it in the field of coal processing or in any industrial field, is simply a problem of economics. Control equipment is justified in terms of its contribution toward increasing productivity, increasing profit margins, or producing a new and salable product at a profit. Instruments and control devices can be divided into two general groups: (1) units which measure and (2) units which both measure and control a process variable.

Jan 1, 1979

By F. F. Skiba, P. J. Root

A mathematical model has been used to investigate crossflow effects in a stratified reservoir during an idealized displacement process. A process in which one incompressible fluid displaces completely another incompressible fluid of the same density and viscosity was considered. The reservoir was assumed to be composed of homogeneous layers which are discrete but interconnected. Approximate five-spot flow geometry was simulated by arranging two pie-shaped cylindrical (two-dimensional) wedges in series. Some of the common production methods rely, for their effectiveness, on the fact that the formation consists of isolated strata. This is particularly true of selective plugging and single-zone production-injection methods. The effectiveness of these techniques for several cases in which adjacent strata are in communication has been evaluated. The usual production-injection schemes for fluid injection processes involve using the same completion interval in both the production and the injection wells. The efficiency of this, as well as alternative productioz-injection procedures has been examined for some cases involving communicating strata. INTRODUCTION The performance of many fluid injection projects is marred by early breakthrough and by production of a high percentage of the displacing phase after breakthrough. This channeling effect, if it occurs in reservoirs which have large permeability variations, is often explained on the basis of a model composed of discrete, isolated strata. Corrective or preventive measures based on this same model are also common. The behavior of stratified models has been the subject of considerable study. An excellent review of the early literature was given by Seba.1 Studies of the depletion performance have since been made for continuous, unrestricted communication between strata2-4 and for production from one permeable bed in continuous but restricted communication with an adjacent permeable bed.5,6 Model studies of displacement processes in stratified systems with crossflow also have been reprted.7-12 Finally, methods which account for crossflow have been developed for computing displacement behavior in stratified systems.13,14 Most methods for controlling or preventing channeling involve some change in the interval open to production and/or injection. Although many variations of the selective completion problem have been studied in regard to the depletion of a reservoir, little attention has been given to the behavior of a displacement process under similar conditions. Sebal used a resistance network model to study the performance of a process in which one incompressible fluid displaces an identical fluid from a linear, stratified reservoir. He investigated the possibility of improving the vertical coverage by limiting the completion interval of the production well. This investigation deals with techniques in which selective completion of the production and/or the injection wells is used in an attempt to control channeling. At one extreme, selective plugging and single-zone isolation (assuming perfect execution) should be successful if the reservoir is actually made up of independent strata. At the other extreme, they can hardly be expected to be successful if the channeling is caused by a mechanism in which stratification effects are negligible even though there are extreme variations in the formation permeability and/or the injectivity profile of the wells is not uniform. By contrast, it is difficult to anticipate the effect that the amount of communication between the beds of a stratified system will have on the efficiency of selective completion techniques. The objective of this study is the quantitative determination of the effect of communication on the behavior of some particular stratified systems which will at least lead to qualitative conclusions.

Jan 1, 1966

By CHARLES R. KETES

I. INTRODUCTORY. INDUSTRIALLY, the most important service that geological science can now render to mining in the Upper Mississippi leadand zinc-fields is to devise some practical scheme whereby the productive ore-belts may be located with greater certainty than they have been in the past. Up to the present time there has been little service of this kind, and in spite of all that has been worked out concerning the geologic features of the region, and notwithstanding all that has been written about these ore-deposits, this particular phase of the subject has been, for some reason or other, the very one that has received the least consideration. The neglect by geologists in this respect has been all but complete. The following notes are suggestive of some practical solutions to the problems presented. As here suggested, the geologic relationships of the ore-bodies were surmised some years ago, though somewhat vaguely perhaps. No opportunity for securing critical data appeared until recent investigations of a private character were undertaken, which bad in view the extensive prospecting of undeveloped areas. Fresh from rather wide experiences of a decade in the principal mining-regions of four continents, and after a lenstrum's residence among the chief mining-camps of the West and of Mexico, under conditions which measured geologic hypotheses in meters and rigidly tested their values in, terms of dollars, the recent return to the Joplin field and a new survey of its ore-conditions were fraught with somewhat unexpected results. Aspects of the ore-deposition were presented of which previously there was no hint. Ten years' absence from the district had tended greatly to clarify former hazy impressions. It enabled many features, then only dimly outlined, to be considered more nearly in their true perspective and in their real relationships to their surroundings. It permitted a great mass of details collected a dozen years before in connection with the work of the Missouri Geological Survey to be viewed under new light and to

Feb 1, 1909

By E. L. Affleck, W. G. Stevenson

In both Canada and the United States, a company making a public offering of securities must comply with regulations which have been passed by Federal, State and/or Provincial Legislatures. These regulations usually require the distribution of an acceptable prospectus which provides essential background information about the issuer, the project and the share offering. when the public offering has been accepted, the unlisted company may then seek a listing of its shares for trading on a Stock Exchange. Once listed, the company will be subject to rules of the Stock Exchange. This paper will deal with the specific information required for inclusion into a prospectus being submitted for a resource company and particularly those parts of concern to mining engineers and geologists.

Jan 1, 1985

By Charles R. Fettke

In connection with the gathering of data for a report on the geology of the Humacao district of southeastern Porto Rico during the summer of 1916, under the auspices of the Scientific Survey of Porto Rico and the Virgin Islands, conducted by the New York Academy of Sciences, the writer had an opportunity to make a brief examination of the magnetite deposits of the eastern portion of the island. Inasmuch as these deposits have attracted some attention as possible future sources of iron ore, and as there is only scanty reference made to them in literature,' a short description based on this reconnaissance survey was deemed to be worthy of record. The magnetite deposits of eastern Porto Rico occur in association with a belt of limestone, trending approximately S 53° E, which begins 11/2 miles east of Juncos and can be traced for about 6 miles toward the coast. Four outcrops of magnetite were observed, which are designated as Nos. 1, 2, 3, and 4, respectively, on the map shown in Fig. 1. Others are probably present, but escaped notice on account of the thick mantle

Jan 1, 1924

Jan 1, 1967

By H. Jehn

Like some other Pt-group metals palladium shows a low but considerable solid solubility of oxygen, which was found already from oxidation studies. The oxygen solid solubility of palladium in equilibrium with gaseous oxygen has been investigated at 900 - 1300 °C in oxygen (665 mbar) and air by gassing and analyzing polycrystalline samples. The 0-concentrations range from 40 to 90 ppm and decrease with rising temperature. The isobaric solubility can be represented by the least-squares fit log c0 = 0.5 log p02- 0.32 + 920/T (c0 in wt. ppm, p02 in mbar, T in K). The thermodynamic data of the exothermic solution reaction are calculated to ?H°= -17.6 kJ?mol-1 and ?S° -76.5 J?mol ? K-1. The behaviour of palladium in an oxygen atmosphere is characterized by the simultaneous reactions of the solution of oxygen in the metal and the formation and evaporation of volatile oxides besides the pure metal evaporation.

Jan 1, 1984

By William P. Shinn

IT seems proper to present in the Transactions of the Institute, from time to time, formal record of the advances made in the arts and sciences to which our organization is devoted-milestones in the highway of progress-from which our successors may measure what they in their turn have gained. The admirable address of the Hon. Abram S. Hewitt, then president elect of the Institute, on "A Century of Mining and Metallurgy in the United States," read at the opening of the meeting held in this hall in June, 1876, forms a point of departure, and the conclusion of the eighth decade of the nineteenth century and the end of the first five years of our country's second century seem to offer a fitting time for again making up our record. The period since 1875 has been marked by a degree of activity in mining and metallurgical industry never before equalled in our history, while the progress made in the science and art has been, if possible, still more noticeable. Whether we consider the mining of iron, ore, and coal, or of the precious metals,-the advances made in smelting the former and in reducing the latter, the extraordinary increase in the production of Bessemer steel, or the science of Thomas and. Gilchrist, which makes the hitherto rejected phosphorus the corner-stone of success,-the past five years are alike memorable for the progress they have shown and to some of the most marked features of this progress I will now oriefly call your attention. It may be of interest in this connection to note the following figures, showing the growth of the population of the United-States during the last three decades :

Jan 1, 1881

By L. Adler

Two basic cases involved in the design of an opening in bedded rock are: 1) where the beds deflect from each other so as to be separated; and 2) where the beds deflect onto their lower neighbor, loading the latter while reducing their own load. Analysis of case 1 is relatively simple, while the interference of the original deflection curves in case 2 complicates analysis of the latter. Figures and formulae are presented for determination of interference loading in this situation. The design of an opening in bedded rock involves two basic alternatives.1'2 In one case the beds deflect away from each other, so as to be separated from their neighbors; in the other case the beds deflect onto their lower neighbor, loading it, while reducing their own load. This is shown in Fig. 1. The first case lends itself to a simple analysis, since the upper and lower boundaries of each beam remain a free surface. In the latter case, however, due to the interference of original deflection curves, the analysis is more complex. Earlier investigators of this interference problem solved it by making the tacit assumption that this type of loading was uniformly distributed along the beam's length. While there is no intuitive justification for this assumption, its use does lead to an approximate solution. Still, a more exact solution does not require this apparently unrealistic restriction. To achieve initial simplicity, the analysis will at first be limited to a two-bedded sequence, as in Fig. 2. Since both beams have identical end conditions and spans, the original deflections of their center lines are members of the same family of curves, as shown in Fig. 3. Therefore, if interference occurs at one point along their center lines, it will occur at every point. For convenience, the midspan point is that usually examined. Deflection at midspan for a uniformly loaded beam regardless of end conditions is: where c is a constant depending on end conditions; w is the body load and equals yh, lb per ft; y is the specific weight of rock, lb per ft'; L is the span, ft; E is Young's Modulus, lb per ft2; I is moment of inertia (for a rectangular section), I = bh3/12 = h3/12, (ft4); b is the width of beam and equals one; and h is the thickness of beam, ft; or 3U > yB an Substituting the appropriate values into Eq. lb, interference between the two beds can be predicted. Should it occur, the additional loads along the lower boundary of the upper bed and the upper boundary of the lower bed must be determined. From the principle of action and reaction, these added loads are known to be equal and opposite, as indicated in Fig. 4. Since the final deflection curves for both beds are identical (Fig. 3), their successive derivatives of y with respect tox must also be identical. That is:

Jan 1, 1961