THE GENERAL EARTHWORK TABLE. (At the end of the book.) This Table, with the help of the Auxiliary Earthwork Tables, Nos. 1 and 2, on the same sheet, possesses the advantage of being general for all varieties of slopes and bottom-widths in common use, as well as for decimal parts of feet in the depths. It may also, with a very trifling preliminary calculation, be made to extend to every variety of bottom-width and ratio of slope that can occur, if even the slopes of the two sides differ in the same cutting; and with the help of a table of square roots it will apply, with all attainable mathematical accuracy, to cuttings where the surface of the ground is uneven. The investigation of the method of forming the Tables and using them, will be given at the end of these Problems. The contents in the general Table, and those in Table No. 2., are calculated to the nearest unit for one chain in length, and checked by differences; the sideslopes being assumed to be extended till they intersect. The auxiliary Table, No. 1., gives the depths of the intersection of the side-slopes below the balance-line, and the corresponding number of cubic yards to be deducted from the contents for each chain in length. PROBLEM VII. M CASE I.-To find the Contents of Cuttings by the general Earthwork Table, and the Auxiliary Table, No. I., at the end of the Book. Let A B bd c C be a cutting, A B = a b = bottom-width on the formation level, M M' and m m' the perpendicular depths at the middle of the two ends of the cutting; A C, B D, a c, b d the side-slopes, which, being prolonged two and two, will meet at the points N and n; also M M' and m m', being prolonged, will meet at the same points. The distance M' N = = m'n in feet, and decimals is given in the Auxiliary Earthwork Table, No. 1., for all bottom-widths a m A and ratios of slopes in common use, at which distance a line must be ruled on the section, parallel to the balance-line, or at the same distance +2 feet from the line of the rails, in which latter case the balanceline need not be drawn. From the line thus ruled, the depths of the cutting must be measured to adapt them to the General Earthwork Table; or a mark might be made on the vertical scale with Indian ink (which is easily washed off) at the same distance, which mark might then be applied to the line of the rails in measuring off the depths. For measuring the depths of embankments, the line must be ruled at the same distance 2 feet above the line of the rails. When the several quantities of a cutting or embankment have been taken from the Table, and their sum multiplied by the ratio of the slopes, the cubic yards to be deducted for each chain in length, for the particular bottom width and ratio of slopes, must be taken from Table No. 1. and multiplied by the whole length of the cutting, and the product, being subtracted from the result obtained from the General Table, will give the content of the cutting in cubic yards. The method of using the Tables will best appear from the following examples: Ex. 1. Let the several depths of a railway cutting to the intersection of the slopes, at the end of every chain, be as in the annexed table, the bottom width 30 feet, and the ratio of the slopes 1 to 1; required the content of the cutting in cubic yards. = Dist. in Qts. per Table. 0 10 1.00 1003 2.00 32 2276 3.00 33 2582 5.00 33.50 6.00 For slope 1 to 1.. 13751 to 1.. 6875.5 1 to 1..20626.5 NOTE. 4.00 By the Table No. 1. the depth to be added to the depths of the cutting, for bottom width 30 ft. and ratio of slopes 1 to 1, is 10 ft., therefore the line from which the depths in the annexed table are measured is 10+2 12 ft. below the line of the rails. The corresponding number of cubic yards to be subtracted is carried to two places of decimals, or, if the nearest whole number had been taken, the quantity would have been in excess or defect by several cubic yards, when the cutting is of a considerable length. Subtract cubic yds. whole distance, and the final result divided by 66, as in the annexed example. See Demonstration at the end of these Problems. CASE II. To find the Contents of Cuttings by the Tables, when the Depths are given in Feet and Decimals of Feet. RULE. Let any two succeeding depths be denoted by a and b, and let the decimal parts of the depths be respectively denoted by a' and b'; find the quantity corresponding to a and b from the General Table, as in the former case; then, 2a+b 10 26+a, 10 or its nearest whole number, and the decimal a' will show the number of cubic yards to be added in Table No. 2., and or its nearest whole number, and the decimal b' will show the cubic yards to be added in the same Table. Ex. 1. Let the depths to the intersection of the slopes be 61.6, and 39.4 feet, their distance 1 chain, the bottom width 36 feet, and the ratio of slopes 2 to 1; required the content of the cutting in cubic yards. Then the depths a and b per General Table give 2a+b=16.1, or its nearest whole No. 16 and (a') .6 per 10 6210 78 26+a 10 =13.9, or its nearest whole No. 14 and (b) .4 per 46 For slopes 1 to 1 6334 2 For slopes 2 to 1 12668 By Table No. 1., for bottom width 36 feet, and ratio of slopes 2 to 1, No. of cubic yds. to be deducted 396 decimal b' with NOTE 1. Care must be taken to use the decimal a' with in finding the quantities in Table No. 2., as a mistake might easily be made in this matter, which would lead to an erroneous result. 2. If any two succeeding depths be nearly equal, and the sum of their decimal parts be together equal to 1 foot, or nearly so, by adding 1 foot to the lesser depth, and rejecting the decimal in the larger depth, and using the depths thus altered as whole numbers, a result sufficiently correct will be obtained; as in the following example. Ex. 2. Let the depths of a cutting be 50.29 and 48.7 feet, their distance 1 chain, the bottom width 33 feet, and the ratio of the slopes 1 to 1; required the content. By adding the decimal .29, in the larger depth, to 48.7, the depths may be called 50 and 49, for which, by the General Table, the content is From which deduct, from Table No. 1. NOTE 1. The results of all the examples in the two preceding cases only differ by a very small fraction of a cubic yard from the true contents obtained by calculation from formula (1.), page 411. The method of finding the contents to decimals, or tenths of a foot in the depths having been particularly discussed, on account of its utility in finding the contents for actual contract-work from the working drawings; in which, as great accuracy is required, the contents should be found to two places of decimals, or to ths of a foot in the depths, as in Case II. of the following Problem. 2. When one or both of the given depths exceed the limits of the table, find the content corresponding to half the two depths, and four times the result will be the content required. PROBLEM VIII. CASE I. To find the Content of a Cutting between two Cross Sections the Areas of which, to the Intersection of the Slopes, the Length, the Bottom Width, and the Ratio of the Slopes, being given. RULE. Find the square roots of the given areas either by a table of square roots*, or by actual extraction; with these roots, as depths, proceed to find the content from the General Table, as in Prob. VII., from which deduct the quantity corresponding to the given bottom width and ratio of slopes from Table No. 1., and multiply the remainder by the length for the content. NOTE. - If the length be given in feet, multiply the content found by the above rule by the feet, and divide by 66 for the content. Ex. 1. Let the areas of the two ends of a cutting be 5141 and 1444 square feet, the bottom width 30 feet, the length 1.60 chains, and the ratio of the slopes 1 to 1; required the content of the cutting in cubic yards. = 1444; and since the bottom-width A B = a b, and the ratio of the slopes are given, the solidity of the prism, the ends of which are ABN a bn, is given in Table No. 1., and is to be deducted from the content found by the General Table, as in Prob. VII. The square roots of 5141 and 1444 are 71.7 and 38 7483 By Table No. 1., for bottom width 30, and slopes 1 to 1 366.67 |