A still better concrete would have resulted with the use of a coarse sand having a curve similar to the line OMN, since then to make the combination of lines OMN and DBKLA pass through C, coarse sand less broken stone should be used than with fine sand, which is as should have been expected.
When the sieve-analysis curves for two materials overlap or extend past each other in the diagram corresponding to Fig. 12, or when more than two materials are to be used, the problem is more difficult; and the reader is referred to pages 197-205 of Taylor and Thompson's Concrete Plain and Reinforced (1905 edition) for detailed explanations. However, the following example, taken from pages 207 and 208 of that book, will give a fair idea of the method of solution, and will also show the value of sieve-analysis curves in proportioning concrete.
"Given a medium sand and three sizes of crushed stone, as shown in Fig. 13, to find what percentage of each will best combine to make the strongest and most impermeable concrete." The parabola passing through the zero point and the point at which curve No. 4 reaches 100 per cent is shown in Fig. 13.
"We see at once that the percentage of No. 4 stone required is (To be sure, about 8 per cent of No.4 is overlapped by No. 3, but this is so slight it need not here he considered.) "Let us determine sand curve No. 1 at 0.10 diameter ordinate, since it can be seen by inspection that the portion Oh of curve No. 1 very nearly fits the parabola, and that grains smaller than 0.10 diameter must be supplied wholly from this curve, while the larger grains represented by portion hG are found also in No. 2 curve.
Accordingly, we have the percentage "A part of No. 3 curve, that portion extending from D to 1 is overlapped by nearly the whole of No. 2 curve. We can see, however, that No. 3 curve alone must supply 14 per cent of the material in the parabola (that portion extending from e to k). This leaves 100— (36 +23 +14) = 27 per cent of . the mixture to be furnished by the overlapping portions of No. 3 and No. 2 in such ratio as best fits the parabola.
"From a study of the two curves, we find by inspection and trial plottings that most of the material required would be better supplied by No. 2 curve, since it contains stone corresponding very well to the needs of that part of the parabola extending from / to e. Let us consider 23 per cent as the proper amount of the final mixture to be furnished by No. 2 curve, which would leave 14+4=18 per vent as the total portion which must be supplied by No. 3 curve.
"Now, on any of the ordinates, we can locate points through which a curve may be drawn which represents a mixture of the given sand and stone in the proportions just found, for example: "These percentages are plotted on the diagram as small circles. The same points would have been obtained if we had begun at the left of the diagram and calculated the percentages passing the sieve." These points lie quite close to the theoretical curve, and hence we may assume that about the best concrete that can be made of the given materials will consist of 23 per cent of the sand, 23 per cent of the finest stone (No. 2), 18 per cent of medium stone (No. 3), and 36 per cent of the coarsest stone (No. 4).
This method affords a means of determining t'.e best
proportions in which to mix the fine and the coarse aggregate, and also shows how the aggregate may be improved by adding or sub tracting some particular size. Sieve analyses can be made from time to time as the work progresses to see whether or not the sizes of the aggregate have changed; and if sizes have changed, the pro portions can be varied to secure the most economical and the densest concrete. In a work of any magnitude the greater labor required in determining the proportions by sieve-analysis curves is likely to be justified by the better quality, or the less cost, of the concrete; and the extra labor required to make sieve analyses during the prog ress of the work will be worth all it costs because of the better control of the proportions of the concrete.
To secure the maximum benefit of this method of proportioning, it is necessary to screen the aggregate to several sizes and then com bine them in the proportions indicated by the sieve-analysis curves. As to whether or not the increased cost of screening and proportion ing would be justified by the saving of cement, depends upon the magnitude of the work and other conditions. The following example illustrates the possibilities: "The ordinary mixture for water-tight concrete is about 1 : 2 : 4i, which requires 1.37 barrels of cement per cubic yard of concrete. By carefully grading the materials by methods of sieve analysis the writer [Fuller] has obtained water-tight work with a mixture of about 1 : 3 : 7, which requires only 1.01 barrels of cement per cubic yard of concrete. This saving of 0.36 barrel is equivalent, with portland cement at $1.60 per barrel, to $0.58 per cubic yard of con crete. The added cost of labor for proportioning and mixing the concrete because of the use of five grades of aggregate instead of two, was about $0.15 per cubic yard, thus effecting a net saving of $0.43 per cubic yard."* Modification of the Parabolic Curve. The principle that the best combination of sizes is that corresponding to the ordinates of a parabola, was deduced from a series of experiments made by Mr. Fuller at Little Falls, N. J., in 1901; but experiments made by Mr. Fuller at Jerome Park Reservoir, New York City, in 1904-05 t seem to show that the parabola does not give quite enough coarse sand and fine stone, and that the ideal sieve-analysis curve for this material consists of an ellipse for the sand and a tangent thereto for the stone. The exact curve starts upon and is tangent to the vertical zero axis of percentages at 7 per cent—that is, at least 7 per cent of the aggre gate plus cement is finer than the No. 200 sieve—and runs as an ellipse to a point on a vertical ordinate whose value represents a size about one tenth of the diameter of the maximum fragment of the aggregate, and thence by a tangent to the 100 per cent point on the ordinate of the maximum diameter. For the exact data for the curves for the materials experimented on at the Jerome Park Reser voir, see Transactions American Society of Civil Engineers, Vol. lix, page 90. The form of the best curve for any material is nearly the same for all sizes of stone; that is, the curves for a 1-inch, a 1-inch, or a 2f-inch maximum stone are nearly the same except the horizontal scale.