In the Engineering and Mining Journal there appeared an article by Fred T. Greene, describing a method of measuring stopes by the use of strings, a clinometer and a tape. I had occasion to use a somewhat similar method, for the purpose of accurately ascertaining the volume of a very irregular drift connecting two mines. This drift was an important factor in a lawsuit in which it was necessary to determine accurately the volume originally occupied by the material extracted from the drift.
While it is unnecessary to give all details of the controversy, some explanation of the conditions is required for a clear understanding of the following description. This drift was started by the “ K.” Co. from its workings, in about 7 ft. of ore, and headed toward the “ G.” property, the plan being to extract all the ore between foot- and hanging-walls, but to take out nothing but ore. As the “ G.” workings were approached, the ore-body became very irregular and much thinner. It was, however, very closely followed, and where the drift broke through into the “ G.” workings, the hole was only 11 inches high by 14 inches wide, or just about the size of a manhole in a boiler.. Thus, this connection-drift tapered sharply; and the very irregular foot-wall sloped toward the “ G.” workings, and entered them at a place where the stope was only about 3 ft. high, and the ground had already commenced to cave. It may be added that the ore-body was a nearly flat deposit of lead carbonate in limestone, the ore and gangue being quite soft.
I was called upon by the “ G.” people to make the survey required to show the relation of this drift to their own workings. The opening into their own stope being so small, the roof being so low, and the drift itself being so irregular, it was quite impossible to get any satisfactory results with the transit in the ordinary way, because it was impossible to set up the instrument in any position outside the hole so that a new station could be located within, high enough to permit setting up the instrument. Upon consideration, I devised a method which gave us the volume very accurately and tied this drift to our other workings. What was sought was a series of parallel cross-sections, at a known distance apart, from which we could calculate the volume, as in railroad earth-work.
About 7 ft. in front of the 11 by 14 in. hole, and about square with it, stood a 12 by 12 in. post. A horseshoe nail was put in each of the two edges facing the hole. Two nails were now driven at A and B, Fig. 1, into the roof of the drift as far inside as was feasible, and strings were stretched between them (aA and bB, Fig. 1).
Then working up from the nearest station, a new transit station, I, was located as far away as possible, yet in such a position that all the string outside the hole was visible. From this station two points on each string (a and a1, b and b1) were now carefully located and marked by means of fine piano-wire wound around the string; a and b were placed close up to the nails in such a manner that a line connecting them would make about the same angle with each of the strings; a1 and b1 were put as far along the strings as they were visible from the instrument.
Sighting at a, the distance Ia was measured and the vertical angle was read. The angle aIa1 was next turned off and read, after which the distance Ia1 was measured, and the vertical angle was read. Thus we had two sides and the included angle of the triangle; however, as a check, the distance aa1 was measured. A similar operation was gone through with bIb1. In this manner the plane of the two strings was accurately located. The next step was to measure the distances aA and bB. This done, as many parallel cross-sections as were desired could be obtained with the aid of plumb-bobs, nails, fish-line and a tape. The locations of these cross-sections were selected in an arbitrary manner, according to the irregularities of the drift, so as to obtain the precise volume.
The modus operandi was as follows : Let us assume that a section was desired at, say, a5 b5, Fig. 1. The distance from a to a5 was measured along the string aA, say 16.3 ft.; the same distance, 16.3 ft. was laid off on bB from b, and each point was marked with a lead-pencil on the string, or by a string tied around the point. Nails were then driven in the roof, directly over these points, in such a manner that plumb-lines suspended from them just touched the strings aA and bB at the marks, a5 and b5. By this means, two points in the roof (c and d, Fig. 2) and two points in the floor (e and f) were fixed, by measuring the distances a5 c, a5 e, b5 d, and b5 f. It was found convenient in most cases to mark the points in the floor with nails.
Points g and i were next located by stretching a string across the drift, so that it just touched at a5 and b5; and then the distances a5 i and b5 g were measured. It will be seen that g and i are points in the plane of the strings Aa, Bb. Point h was next located by making an offset of, say, 0.9 ft. from a5 toward i, and measuring vertically to the roof. Point k was fixed by measuring, say, 0.8 ft. down both from a5 and b5 and measuring across the plane of these two points, which is parallel to the plane of the two main strings.
As many points as are desired may be located in a similar manner, the main idea to be kept in view being that all measurements should be made vertically, or in a plane parallel to the plane of the two main strings.
A sketch of each separate section is made in the note-book and all the measurements are marked thereon; the sketch being labelled at the top of the page with the distance from the initial points a and b.
Having secured as many sections as are desired, the office-work to be done consists of a series of very simple calculations. From the work at the instrument, after the proper calculations have been made, the plan and elevation of the plane of the strings can be drawn, and from these the horizontal distances between the different sections can be scaled off; or, knowing the angle of inclination of the two strings, we can calculate the distances horizontally. Each cross-section can be laid off accurately with triangle and T-square, and the area can be calculated in the ordinary way.
Having calculated the areas of all sections, and knowing the horizontal distance between each two, the volume of each prism can be obtained by the formula:
Volume equals one-half the sum of the area of the two bases, multiplied by the altitude—the bases being any two adjacent sections, and the altitude the horizontal distance between them. The sum of all the volumes gives the total volume, up to the line across the two points A and B.
From our previous work we have the location of these points A and B; to obtain the remaining volume, we set up our transit under either A or B and proceed to make cross-sections from our instrument in the usual way.
It should be added that the two sketches shown herewith are hypothetical, as I have no drawings of the real workings at hand.
At the beginning of his paper, Mr. Herzig refers to an article of mine in the Engineering and Mining Journal. I would like to add that the method described in that article was afterwards used to find the amount of ground “ in place,” broken down by each successive blast, with the object of getting data upon which a fair scale of contract-prices could be based. To attain this end, the method described in the article was used with very slight modifications. The work had to be done so as not to stop the machine-drills or interfere in any way with the men employed in the stope. Each machine had to be visited as soon after blasting as the mass broken down could be cleared away; and as there were
nine workings (one with three machines, three with two, and five with one), the amount of time available for measurements in each place was necessarily small. Fig. 1 is a plan, and Fig. 2 a vertical cross-section, of one of the working-places where one machine was at work, breaking the ground down from the back of the stope, which was floored as shown in Fig. 2. A wooden plug pierced longitudinally with a hole 3/8 in. in diameter was put in the hanging-wall at a convenient point, A, and a rod AB, of 3/8-in. round steel, with one end bent to an eye, was inserted in this plug. From B, a string BX (Fig. 1), knotted at every two feet, was stretched horizontally, parallel to the hanging-wall. At B, and at each knot (k1, k2, k3, etc.), cross-sections, at right-angles to BX, were taken in vertical planes at 1-1, 2-2, etc. (Fig. 2), by stretching the tape from B, or from the appropriate knot, to points of change of contour, such as c, d, e, f, g, h, i and k ; and determining the angle from the horizontal by means of a very light pendulum-clinometer, clamped
to the tape. The tape-distance and vertical angle to each point, when plotted, gave a very accurate cross-section at each knot. A reference line, L M, Fig. 2, having been chosen, the areas of the several cross-sections were ascertained with a planimeter. These gave the ends of a series of prismoids, from which the cubic volumes could be readily figured. In the cross-section shown in Fig. 2, the contours were first taken on January 5th, to get an outline, c, d, e, f, g, h, i, k, from which to start. The next contour, n, o, p, r, s, t, v, was taken January 8th, and the increase in area of each cross-section furnished the basis for calculating the volume broken down.
An accurate record was kept of the time occupied in drilling, blasting and delays, as well as an account of all supplies, powder, etc., consumed at each working-place. From this information there was calculated a scale of prices, for each character of working-place, covering every condition likely to be encountered at that place, and to affect the fair operation of the contract-system. In the stope shown in my article cited by Mr. Herzig, we had a very good check on the accuracy of the method. That particular stope, according to the office-records, produced 11,134 tons; and our cross-sectioning showed 10,867 tons; an error of 2.15 per cent. In this case, however, the accuracy of the method depended more on the care taken in determining the specific gravity of the ore than on the survey itself. If the ore had been homogeneous throughout, there would have been little difficulty under this head; but a great many experiments had to be conducted, with pieces of ore as large as possible, in order to obtain the average number of cub. ft. per ton of ore. The number obtained in this case was 8.85, corresponding to a specific gravity of 4.05, while from the analysis of the ore the specific gravity obtained was 3.97, indicating that the factor should be 9.04 cub. ft. per ton. The mean of these two factors, 8.96, was used in obtaining the above result, showing only about 2 per cent, of error.