Methods for Earthwork Computations

An excerpt from the INTRODUCTION: Purpose of Chapter I. - It is generally admitted that, as a rule, the use of the so-called prismoidal formula leads to a value of a volume more accurate than that obtained by any other approximate method in common use, but the labor involved is so great that it is customary to use a less exact method, usually the method of average end areas. The calculation of the areas of the two ends, and the determination of the dimensions and then of the area of the mid-section, with the subsequent combination of these to find the volume, indeed form a lengthy process, and the purpose of Chapter I is to explain a method by which the labor is greatly reduced. Rule 1 of that chapter formulates a statement by which the terms involved in the computation may be written directly from the notes, without any intermediate steps and without drawing any figure, the symbolized form of the rule enabling the computer to apply it without difficulty; and Arts. 24 and 25 explain in full its application to the complicated solid shown in Fig. 17. Other rules in the same chapter provide simpler methods for special cases, but Rule 1 is the fundamental rule, applicable to the general solid. The reader may test the general method, without preliminary preparation, by studying Arts. 16, 17, 18, 24 and 25, remembering that the problem was selected, not to demonstrate the ease with which the method can be used in some special case, but to explain the way in which the terms are formed in a solid that is by no means simple. 2. Chapter II contains a number of rules by which the computer can form in a systematic manner, directly from the notes, the terms that occur in the determination of a volume by the average end area method, Rule 14 being the general rule, applicable to any polygonal cross-section. 3. Chapter III determines, in certain cases, the correction to be subtracted algebraically from the volume computed by the average end area method in order to find the volume as it would be given by the prismoidal formula, the quantities being taken directly from the notes. 4. Chapter IV treats of the determination of the volume, by the average end area method, when the transverse slope of the surface is measured; and the resulting formulas are so systematized in Art. 124 that the volumes in side-hill work as well as in thorough work may be readily computed. 5. Chapter V is devoted to the consideration of the correction for curvature in railroad work. 6. Chapter VI describes the Crockett Volume Slide Rule, and explains the method of applying it to the determination of volumes. 7. Approximate Results of Earthwork Determination. - For two reasons the computation of the volume of earthwork is, at its best, merely approximate: First. - The measurements, upon which the computation depends, are approximate and liable to error. Thus, when the measurements are made to the nearest tenth of a foot, any measurement is liable to an error of 0.05 foot; and the error, in the volume of a parallelopiped whose base is 14 X 100 feet and whose altitude may be in error by 0.05 foot, might be 70 cubic feet or 2.6 cubic yards. Second. - The formulas are based upon the assumption that the surfaces are either plane surfaces or hyperbolic paraboloids, - or upon yet cruder assumptions, - and in practice this cannot be other than approximately true....