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By S. H. Williston

IT not often that bore hole surveys can be checked by actual. civil engineering methods. A recent Arizona survey was checked by normal surveying methods and the comparison of the results should be of value to both oil and mining men. During the summer of 1948 the Phelps-Dodge Corporation, at its Copper Queen property near Bis-bee, Ariz., drilled a 1245 ft, 8 in. diarn, churn drillhole in a mineralized area and cased part of it, intending to use it to transfer mill tailings for stope fill. The hole, as frequently occurs, was not straight, and, in endeavoring to locate the bottom in the underground workings, they found no evidence of the hole at the underground coordinates directly below the surface location. The noise of the drilling tools was reasonably clear, but the direction of sound was uncertain. Preliminary tests with available equipment were not successful in locating the bottom of the hole. Because of the mineralized character of the area and the fact there was some casing in the hole, any magnetic method of well surveying would give results of doubtful value. Sperry-Sun gyroscopic well-surveying instruments were finally used to locate the bottom of the hole. These instruments consist of a gyroscope to determine azimuth and either a pendulum or bubble inclino-meter. A multiple shot camera photographs both instruments on a single film and superimposes the photograph of a watch. Coordination of depth with time at the surface makes it possible to select the corresponding picture for any depth. After making several runs of the empty instrument housing from the top of the hole to the bottom to make sure there were no obstructions in the hole, three surveys on wire line were completed during the afternoon. The three surveys, in which readings were taken at different points in the hole on each survey, were computed and gave the following locations of the bottom of the hole in relation to the surface collar: survey No. 1—24.92 N, 30.30 W; survey No. 2—24.24 N, 31.11 W; survey No. 3—26.54 N, 27.72 W. Then the data from the three surveys were combined into a single set of calculations which gave a location for the bottom of the hole: combined surveys—24.27 N, 30.16 W. (Fig. 1.) Immediately upon the determination of the coordinates at the bottom of the hole, a drift on the 1300 ft level was started toward the indicated loca- tion some 38 ft northwest of the coordinates of the surface location, The bottom of the hole was located within the drift round in which it was expected, and the transit survey run to the actual location of the hole indicated N 27.18, W 29.71. This shows a discrepancy between the well survey and the transit survey of 0.45 ft in the westerly direction and 2.91 ft in the northerly direction. All surveys, both gyroscopic and transit, fell well within the width of an ordinary drift. While this is satisfactory for almost any and all mining requirements, a theoretical examination was made as to reasons for the discrepancy. A study of the course of the hole indicates that considerable right turn or spiral existed, and in all probability the surveying instrument was pulled out of alignment while traversing the turn by approximately 0.05 ft at the top and another 0.05 ft in the opposite direction at the bottom of the instrument. If such an allowance were to be made, the survey calculations would almost exactly correspond with those determined by transit. This sort of discrepancy would be minimized by the use of stabilizing guides. It is unfortunate that physical laws probably effectively prevent the use of gyroscopic instruments in EX and AX diamond drill holes. The directive power of a gyroscope falls off inversely at some rate between the third power and the sixth power of the diameter. Present instruments can be run in casing 53/4 in. ID or over and might be adapted to somewhat smaller diameters, but the difficulty of reducing these diameters to 11/4 in. or 2 in. is almost insurmountable at the present time. Acknowledgment The author wishes to express his appreciation to the Phelps-Dodge Corporation for permission to publish this article, and to the Operating and Engineering Departments for their cooperation on the survey; also to Donald Hering, of the Sperry-Sun Well Surveying Co., who actually made the survey and calculated the results.

Jan 1, 1951

By S. H. Williston

IT not often that bore hole surveys can be checked by actual. civil engineering methods. A recent Arizona survey was checked by normal surveying methods and the comparison of the results should be of value to both oil and mining men. During the summer of 1948 the Phelps-Dodge Corporation, at its Copper Queen property near Bis-bee, Ariz., drilled a 1245 ft, 8 in. diarn, churn drillhole in a mineralized area and cased part of it, intending to use it to transfer mill tailings for stope fill. The hole, as frequently occurs, was not straight, and, in endeavoring to locate the bottom in the underground workings, they found no evidence of the hole at the underground coordinates directly below the surface location. The noise of the drilling tools was reasonably clear, but the direction of sound was uncertain. Preliminary tests with available equipment were not successful in locating the bottom of the hole. Because of the mineralized character of the area and the fact there was some casing in the hole, any magnetic method of well surveying would give results of doubtful value. Sperry-Sun gyroscopic well-surveying instruments were finally used to locate the bottom of the hole. These instruments consist of a gyroscope to determine azimuth and either a pendulum or bubble inclino-meter. A multiple shot camera photographs both instruments on a single film and superimposes the photograph of a watch. Coordination of depth with time at the surface makes it possible to select the corresponding picture for any depth. After making several runs of the empty instrument housing from the top of the hole to the bottom to make sure there were no obstructions in the hole, three surveys on wire line were completed during the afternoon. The three surveys, in which readings were taken at different points in the hole on each survey, were computed and gave the following locations of the bottom of the hole in relation to the surface collar: survey No. 1—24.92 N, 30.30 W; survey No. 2—24.24 N, 31.11 W; survey No. 3—26.54 N, 27.72 W. Then the data from the three surveys were combined into a single set of calculations which gave a location for the bottom of the hole: combined surveys—24.27 N, 30.16 W. (Fig. 1.) Immediately upon the determination of the coordinates at the bottom of the hole, a drift on the 1300 ft level was started toward the indicated loca- tion some 38 ft northwest of the coordinates of the surface location, The bottom of the hole was located within the drift round in which it was expected, and the transit survey run to the actual location of the hole indicated N 27.18, W 29.71. This shows a discrepancy between the well survey and the transit survey of 0.45 ft in the westerly direction and 2.91 ft in the northerly direction. All surveys, both gyroscopic and transit, fell well within the width of an ordinary drift. While this is satisfactory for almost any and all mining requirements, a theoretical examination was made as to reasons for the discrepancy. A study of the course of the hole indicates that considerable right turn or spiral existed, and in all probability the surveying instrument was pulled out of alignment while traversing the turn by approximately 0.05 ft at the top and another 0.05 ft in the opposite direction at the bottom of the instrument. If such an allowance were to be made, the survey calculations would almost exactly correspond with those determined by transit. This sort of discrepancy would be minimized by the use of stabilizing guides. It is unfortunate that physical laws probably effectively prevent the use of gyroscopic instruments in EX and AX diamond drill holes. The directive power of a gyroscope falls off inversely at some rate between the third power and the sixth power of the diameter. Present instruments can be run in casing 53/4 in. ID or over and might be adapted to somewhat smaller diameters, but the difficulty of reducing these diameters to 11/4 in. or 2 in. is almost insurmountable at the present time. Acknowledgment The author wishes to express his appreciation to the Phelps-Dodge Corporation for permission to publish this article, and to the Operating and Engineering Departments for their cooperation on the survey; also to Donald Hering, of the Sperry-Sun Well Surveying Co., who actually made the survey and calculated the results.

Jan 1, 1951

Organization Place Date 1919 National Drainage Congress St. Louis, Mo. Nov. 11-13 American Mining Congress St. Louis, Mo. Nov. 1.7-21 Canadian Mining Institute Vancouver, B. C. Nov.. 26-28 American Society of Mechanical Engineers:.... New York, N. Y. Dec. 2-5 American Institute of Chemical Engineers Savannah, Ga. Dec. 3-6 Geological Society of America...; Boston, Mass. Dec. 29-31. American Society of Heating and Ventilating 1920 Engineers New York, N. Y. Jan. Mining and Metallurgical Society of America ... New York, N. Y. Jan. 13. American Institute of Min. and Met. Engineers... New York, N. Y. Feb. 16-19 American Ceramic Society Philadelphia, Pa. Feb. 23-26

Jan 11, 1919

By Francis B. Foley

In examining steel under the microscope, one is constantly confronted with structures that are difficult to interpret. Recently, in a collection of samples for exhibition purposes, the writer found a piece of open-hearth runner metal, 2 in. (50 mm.) in diameter, that had been fractured for use as a sample of cast-steel structure. On polishing and etching a section for the purpose of photographing a field to typify the microstructure of steel in the "as cast" condition, a variety of structures was found (Fig. 1) so arranged as to arouse interest at once. For a distance of about 0.065 in. (1.6 mm.) from the skin of the runner, normal ingot structure was found (Fig. 2); this was followed by a zone, about 0.215 in. (5.5 mm.) wide, of considerably lower carbon content in which occurred a most pronounced Widmannstattian structure (Fig. 3); from this point on to the center of the runner, the metal was again of normal ingot structure (Fig. 4). Representative samples taken from the metal in the three zones showed the following analysis: Carbon Nickel Chromium Per Cent. Per Cent. PeR Cent. Metal from skin........................... 0.34 1.26 0.63 Widmannstattian zone.................... 0.21 1.18 0.63 Metal from center......................... 0.34 1.32 0.67 Such a condition might result from decarburization followed by recarburization of the skin. This section came from one of the branch runners to the bottom of the mold, not from the main runner, and it is difficult to conceive of decarburization occurring in this portion of a runner. If the metal were stripped while quite hot, some oxidation would take place but the recarburization is unaccounted for. It could not occur by segregation after solidification—the tendency then is for diffusion and homogeneity. The following seems to fit the conditions and is offered as an explanation. The outside layer of normal structure 0.065 in. (1.6 mm.) thick was chilled on the walls of the runner brick when the molten metal first entered the runner. As the molten metal flows by, the outside of the stream

Jan 1, 1920

J. E. Stead, Middlesborough, England (communication to the author): Prof. Howe's valuable paper on cast-iron brings forward most prominently the correct explanation of the part played by combined carbon in pearlite and cementite, in determining the strength and hardness of cast-iron. On a previous occasioil I have shown that castings made by melting a white Cleveland iron and glazed iron, one containing 1.5 and the other from 4 to 5 per cent. of silicon, and each about 3 per cent. of carbon, were stronger than those made of ordinary foundry-iron; the difference in the final castings being a differ-

Jan 1, 1902

By B. M. Larsen, K. Heindlhofer

Jan 1, 1945

By W. S. Morris

PETROLEUM'S importance in World War II can perhaps be better realized by the recitation of a few facts and figures: Gasoline needs in this war are already eighty times greater than in the last war. It takes nearly three pounds of gasoline to deliver one pound of bombs, filled with petroleum explosives, on an enemy objective. Every American soldier overseas requires an average of more than fifty gallons of petroleum products per week, with the oil used by Navy and supply ships and America's Allies boosting the consumption of oil per soldier to an even higher figure. Every oil well in the United States now must produce each day an average of four barrels of oil for war and combat jobs. More than fifty million gallons of gasoline, fuel oils, lubricants, and other petroleum products are now going directly to the fighting forces every day.

Jan 1, 1945

By Joseph Jensen, J. B. Stevens

The history of the normal oil field is supposed to show an oil graph starting high in flush production, descending more or less steeply into the curve of settled production and dropping gradually to the vanishing point of economical operation. Together with this, somewhere in the early history of the field, there appears from a vanishing point a graph showing the production of water. During the period of settled production this graph rises higher and, toward the close, unless extensive work is done in the line of correction, rises far above the oil graph to the point at which the cost of operation terminates profitable pumping. Fields that start out with a high relative content of water are generally very short-lived. Fields that never develop a water problem are exceedingly rare. The McKittrick oil field, Kern County, California, with 840 acres of proved ground, 300 wells and a present recovery per acre of 94,598 bbl. from 1898 to Aug. 1, 1930, presents a study which is an apparent reversal of the outlined normal history. Though the history of the field now covers over 30 years of operation, it is still too early and the conditions too unique to project the graph of production to completion. The present conditions and directions of change are most interesting, in that the curve of oil production is remarkably flat and the curve of water invasion, after reaching an appalling height, has shown a definite reversal and a continued decline for 10 years. Early drilling in the McKittrick field was in or prior to 1898. Development continued steadily to 1918 when the present limitations of the field were determined. Shut-offs, according to accepted practice of the time, were made by driving the shoe into a shale stratum or into a tamped-in bridge of sand, rosin and clay. Records of early days speak of three waters: (1) surface water which was fresh; (2) an intermediate sulfur water, encountered below the top oil or tar sand; (3) bottom salt water. In different portions of the field any one of these waters let into a well by faulty completion might require immediate correction to obtain satisfactory production. In many wells,

Jan 1, 1931

By J. Gurland

THE microstructure of sintered carbides consists of particles of metal carbides, such as WC and TiC, embedded in a metallic binder which is usually a cobalt—or nickel-rich solid solution. One of the important factors influencing the strength of the alloys is the degree of dispersion of the carbide particles in the binder matrix. While it is generally accepted

Jan 1, 1960

By R. V. Norris

1. INTRODUCTION. THE anthracite coal trade, with a shipment averaging about 70,000,000 tons per year, differs essentially from other coal business, in the fact that the larger sizes, comprising about 65 per cent. of the total, are used almost exclusively for domestic purposes, principally during the winter months; and that for proper combustion close-sizing is imperative, so that eight sizes are made; broken, egg, stove, and nut, known as prepared sizes, and pea, buckwheat, rice, and barley, known as steam sizes. These sizes are made in the regular course of preparation, and but little variation in the natural percentage of each is practicable. Thus it is necessary either to work the collieries intermittently, or to dispose of all the varying sizes in their fixed proportions. Unfortunately, the market does not at all times absorb the coal in proper proportions, and rather than interfere with the regular operation of the mines it has been found economical to provide storage for the excess. While the cellars of the consumers are the great storage-plants of the country, this capacity is not under the control of the trade, except so far as reduced prices during the spring and summer months may tempt the use of this reserve. The same remark applies but with less force to the retail yards, which,, though usually small, store in the aggregate a large amount of anthracite. While every effort is made to utilize to the utmost the individual storage-capacity of the country, there still remains the necessity for taking care of the irregularities of the market by the construction of storage-plants under the direct control of the producers. Such plants are usually the property of the railroads, and are situated at points most convenient from a traffic stand-point.

Jun 1, 1911

By E. J. Eisenach, W. E. Jones

SAFETY and industrial hygiene have always been recognized as highly important in company policy, and the co-operative support of the company officials and entire plant personnel has contributed largely to the present success of the Company's program in this field. Growth has necessarily been slow, and dependent, at times, on trial and error but the program in its present form combines the regulations, working methods, and equipment that have proved their value under actual operating conditions. Every effort is made to keep the program on a practical basis. Continuous experimental work is carried on in conjunction with the health and safety program, with resultant periodic changes for improvement. As a proof of accomplishment, in 1945 the frequency and severity rates for accidents were the lowest in the history of the operation. A full-time safety engineer was employed in 1933 and an assistant safety engineer in 1936. During the following year a stenographer was employed in the office and a safety inspector was added for the mine. During the gradual expansion of operations that was climaxed by

Jan 1, 1946

By Thomas T. Read

IN reviewing the field of mineral industry education last year reference was made to recent assertions, mostly emanating from sources not in a position to know the facts, that mining engineers as a class are disappearing. Attention might also then have been called to a statement that appeared in Engineering News, August 11, 1892 (p. 130), ". . . the great absolute decline in number of mining graduates, which last amounts almost to a complete abandonment of mining engineering as a distinct branch of study." Evidently we outrank opera stars in the number of farewell appearances we are able to make; 43

Jan 1, 1936

By N. V. Hansell

l. INTRODUCTION. THE last few years have shown an increasing interest in the subject of beneficiating iron-ores -in all iron-producing countries. In the United States, this movement has been slower than in certain parts of Europe, for the obvious reason that the abundance and relative cheapness of the Lake Superior iron ores has hindered the development of enterprises for the preparation of low-grade ores. Gradually, however, conditions have changed. The world's consumption of iron is increasing enormously. Being in 1900 about 40,000,000 tons, in 1910 ii was already 60,000000; and there is no reason to doubt that for the near future the increase in iron-consumption will continue at the same rapid rate, especially with, the development of China, certain parts of Africa, and other regions which have hardly been touched as yet by modern industrialism. It is. therefore, not surprising that apprehension is felt that the known available iron-ore resources of the world are gradually being depleted. Signs of this feeling are the recent investigation, under the direction of the U. S. Geological Survey, of the iron-ore resources of the United States, and the estimate presented to the last International Geological Congress, at Stockholm, Sweden, of the iron-ore resources of the world. These investigations have emphasized the fact that the known high-grade ore-deposits are limited in extent, and will, at a not distant future, be exhausted, compelling the iron industry to depend for raw material on the enormous deposits of low-grade ores which are distributed over almost all the world. The discovery of new deposits of high-grade ore, in countries not yet thoroughly explored, will not materially change this prospect. Of course, the part which such new deposits will play in the future cannot be forecast. The difficulty and cost of transportation will probably exclude many of them from use at the present centers of the iron industry.

May 1, 1912

By G. J. Wilson

Difficulties experienced in correlating vertical and lateral locations of subsurface features that are encountered in directional wells prompted critical review of the tangential method of computing directional surveys. Inherent discrepancies were indicated to cause small incremental but significantly large cumulative errors in computed vertical depth and horizontal departure. A radius of curvature method, developed for planning and analyzing directional wells, was studied as a means of more accurately calculating survey data. Errors incurred in the tangential method were not present in the radius of curvature method. Conflicting assumptions in calculating dog-leg severity by the tangential method were resolved by a new procedure that was derived from the radius of curvature method. Results of the investigation showed the radius of curvature method to be an improved method for computing directional surveys. Introduction Many improvements in directional drilling tools and techniques have been made in the past 10 years,' but the method of computing survey data has remained virtually unchanged. While use of computers greatly aids the process of calculation' and eliminates many human errors, manual and computer calculations are still made with the assumption that the wellbore consists of straight-line segments. This paper points out discrepancies in the method and presents an improved method in which the wellbore is assumed to be curved. Tangential Method The method presently used to calculate vertical depth, horizontal departure and direction coordinates with measured values of well depth, inclination angle and direction angle is termed the tangential method. This term stems from performing calculations with a series of straight-line segments, with each segment assumed to be tangent to the wellbore at a corresponding survey point. A segment is defined by inclination and direction angles measured at only the lower or deeper end. Consequently, a discrepancy or error is introduced for each segment. Errors are not of extreme magnitude, but cumulatively are significant due to the need to define as accurately as possible the vertical and lateral location of subsurface features such as pay tops, fluid contacts and fault cuts encountered in expensive directional well bores. In Fig. 1, the curved-line segment ab represents a portion of the actual borehole as viewed in the vertical plane. Points a and b are survey stations and +, is the inclination angle at point b. The curved-line segment ab is assumed to be the straight-line segment arb of a length equal to ab and tangent at point b to the curved-line segment. Lengths a'c' and c'b are equal to a'b cos +, and a'b sin +,, respectively. Point a already has been fixed by calcillations performed with survey data obtained at point a, and calculated lengths m'c' and c'b are projected as lengths from point a, rather than from point a'. Point b1 is thus fixed and assumed to represent the actual point b. Calculated length ac" is shorter than actual length ac, and calculated length c"bl is longer than actual length cb. Since b1 is the point from which the next deeper segment will be projected, errors are cumulative. Cumulative error continues to increase as long as the inclination angle continues to increase. When the inclination angle decreases (Fig. 2), calculated length ac" is longer than actual length ac, and calculated length c"bl is shorter than actual length cb. These errors also are cumulative, but differences between calculated and actual lengths are opposite in mathematical sign to differences demonstrated in

Jan 1, 1969

By J. F. Olk, E. D. Spaulding, R. E. Thurmond, G. A. Komadina, R. W. Hernlund, J. A. Journeay

THE Pima pit is a 1700x1400-St oval, the long T axis parallel to the strike of the orebody. The north side of the pit is carried as a final pit slope that coincides with the footwall of the orebody. The south side and east and west ends of the pit are working slopes continually being stripped back toward the final slopes. An inclined roadway extending from the natural ground surface on a 5 pct grade down to the north- east corner provides access to the pit. This road enters the pit 130 ft below the natural surface and continues as a pit ramp on a 5 pct grade to the 3150-ft bench (roughly the base of the alluvial cover). At this point the ramp system is steepened to a 12 pct grade and continues to the pit bottom. The 5 pct grade is maintained in the alluvial section to facilitate scraper haulage out of the pit. In addition to this main access ramp, temporary working ramps in the alluvial areas allow shorter routes to dumps. These are left on top of working benches and do not change the overall working slopes. Below the base of the alluvium, haulage is by truck down to the skip loading point or up from the bottom to the skip loading point.

Jan 4, 1958

By H. M. Howe

THIS paper presents an experimental verification of some of the predictions made by one of us' concerning 'the influence of certain conditions of casting upon the size and position of the pipe', and the position of the segregate, in steel ingots. The predictions which we have been able to verify are the following A. That the pipe is shortened and the segregate raised 1. By slow casting; 2. By casting with the large end up instead of down; 3. By retarding the cooling of the top, e.g., by means of a sinking-head. B. That the pipe is shortened by slow cooling. C. That the pipe and segregate lie in the last freezing-part. Our procedure was to cast ingots of wax (commercially pure stearic acid) containing a little bright green copper oleate (usually 1.5 per cent.) under varying conditions; to saw each ingot open along a longitudinal plane passing through its axis ; and to examine the longitudinal section thus laid bare. In order to make the segregate or enriched part more distinct in color from the impoverished part, a very little red cerasine was added. An experiment which will be described later showed that the cerasine itself does not segregate markedly, if at all. Hence we may assume that the strong contrasts between the green and red shown in so many of our ingots are due, not to segregation of cerasine, but to that of the green oleate, in the wax which in itself is uniformly reddened with cerasine.

Jul 1, 1907

Jan 1, 1908

By Lucien Eaton

THE use of filling in mines is less common in the United States than it is in Europe, where in some places it is required by law. In most cases the filling is placed by hand, and the material used for this purpose is the waste from development and that sorted from the ore in the stopes. When these sources provide an insufficient supply, additional rock must be quarried on surface or mined underground, or some other material :st be used. In different parts of the world sand, clay, gravel, culm, nulated slag, mill tailings, and even ashes, have been used as filling material in place of or in conjunction with rock. Of all the materials used for filling the best is probably clean sand. Gravel, finely crushed rock, granulated slag and coarse mill tailings are almost as good, and the methods used for placing and confining sand will apply equally well to any of these. The advantages of sand are its comparatively slight compressibility and the ease, economy and convenience with which it may be placed. On account of the voids, which necessarily occur in fills made of coarse waste, such fills will compress 10 to 25 per cent before they develop their full strength, whereas clean sand seldom compresses more than 5 per cent. Sand is especially desirable when practically complete filling is required, and sand derived from mill tailings, especially when not thoroughly cleaned, has been remarkably effective in sealing off and extinguishing mine fires. When natural sand is available on surface, or when it can be obtained from the mill tailings or tailings dump, the cost of using this material for fill will probably be considerably less than that of rock. In this paper it is not proposed to discuss all methods of filling, butt to describe some of the methods used for placing sand fill and to make a brief comparison of their merits.

Jan 1, 1932

By Vanderlei Beisiegel, Arthur L. Bernardelli, Arthur W. Ruff, John H. Tremaine

Estimated reserves of high-grade iron ore at Serra dos Carajás total 16 billion metric tons and have an average grade of 66.7 per cent iron, 2.2 per cent combined silica and alumina, and 0.05 per cent phosphorous on a dry basis. Hematite is predominant and no significant sulfur on other deleterious elements are present. Serra dos Carajás is located in northern Brazil, about 550 kilometers southwest of Belém, Pará. It is six degrees south of the equator between the Xingu and Tocantins Rivers. Both rivers flow northward toward the Amazon River. The greater part of the area is covered by thick jungle, and the most practical means of access is by air.

Jan 1, 1974

By L. Adler

The proper transfer of roof loads from props and bolts to ribs and pillars can result in appreciable savings. The author shows how to plan such load reduction in underground mines. For openings in bedded rocks, analyzed by simple beam theory, it has been shown that roof loads can be shifted from one support to another.&apos; This transfer is effected by controlling the relative deflection of adjacent supports. If this action is used to reduce loads in the props or bolts and increase them in the ribs or pillars, it could be productive of economies averaging 30 pct. Such savings can be realized by: 1) using smaller artifical supports such as props and bolts, and 2) increasing loads in the rib to induce eventual failure for longwall caving. Since the rib is relatively load-insensitive, it provides a natural load sink. However, caution must be used in applying such permissive deflections, for while the supports are being handled properly, the stresses in the roof itself are very sensitive to such manipulations.1 An underground opening overlain by a relatively thin immediate roof, and a thick main roof, is usually laid out as shown in Fig. 1. The total open span (T) between pillars or ribs is calculated on the basis of the main roof, and the secondary spans (L) on the basis of the immediate roof, so that a complete accounting of the existing loads is made. For this purpose the immediate roof is treated as a uniformly loaded continous beam, which is an indeterminate structure as shown in Fig. 2(A). The term systematic supports places the following two restrictions on the continuous beam: 1) Spans between all supports are equal: uniformity of spans. 2) Loads on all supports (excepting ribs) are equal: uniformity of supports. The usual mode of analysis of such a structure by beam theory is to divide it up into a series of simply supported beams with end moments as shown in Fig. 2(B)2 The problem now is to transfer a maximum roof load to the rib from the span immediately adjacent to it by permitting its support to deflect, and yet not fail the roof. Once this has been done, the load can be equalized on all other supports by permitting further degrees of relative deflection. It is necessary, therefore, to determine an expression for the constant load on each support, and the required deflections, in terms of properties of the roof. From Fig. 1: T/L0 = No and T/L = N [1] Where: T = allowable span of main roof, ft. (Calculated from S = Mc/l) S is allowable modulus of rupture, psi. M is maximum moment, lb-ft (in this case wT2/12). c is distance from neutral axis to extreme fiber, ft. I is moment of inertia with respect to the neutral axis, ft4. L,, = allowable span of immediate roof, ft (calculated as was T). L = actual span of immediate roof, ft. N0 and N are pure numbers where No < N and N is the integer just higher than Nu. For example: If T =100 ft and L0 - 15 ft 100 N0 =100/15= 6.667 . . .N= 7 L = T/N = 100/7 = 14.28 ft Consequently N = No + n where n is a positive fraction i.e.: 0< n < 1 Since from Eq. 1, T = NL = N0L0 Lo /L = N/No = N/N-n

Jan 1, 1961