THEORY AND PRACTICE OF PROPORTIONING CONCRETE.

American engineers proportion concrete mixtures by measure, thus a 1-3-5 concrete is one composed of 1 volume of cement, 3 volumes of sand and 5 volumes of aggregate. In Continental Europe concrete is commonly proportioned by weight and there have been prominent advocates of this practice among American engineers. It is not evident how such a change in prevailing American practice would be of practical advantage. Aside from the fact that it is seldom convenient to weigh the ingredients of each batch, sand, stone and gravel are by no means constant in specific gravity, so that the greater exactness of proportioning by weight is not apparent. In this volume only incidental attention is given to gravimetric methods of proportioning concrete.

VOIDS.—Both the sand and the aggregates employed for concrete contain voids. The amount of this void space depends upon a number of conditions. As the task of proportioning concrete consists in so proportioning the several materials that all void spaces are filled with finer material the conditions influencing the proportion of voids in sand and aggregates must be known.

Voids in Sand.—The two conditions exerting the greatest influence on the proportion of voids in sand are the presence of moisture and the size of the grains of which the sand is composed.

Table I.—Showing Effect of Additions of Different Percentages of Moisture on Volume of Sand.

Per cent of water in sand	0	0.5	1	2	3	5	10
	Lbs.	Lbs.	Lbs.	Lbs.	Lbs.	Lbs.	Lbs.
Weight per cu. yd. of fine sand and water	3,457	2,206	2,085	2,044	2,037	2,035	2,133
Weight per cu. yd. of coarse sand and water	2,551	2,466	2,380	2,122	2,058	2,070	2,200

The volume of sand is greatly affected by the presence of varying percentages of moisture in the sand. A dry loose sand that has 45 per cent. voids if mixed with 5 per cent. by weight of water will swell, unless tamped, to such an extent that its voids may be 57 per cent. The same sand if saturated with water until it becomes a thin paste may show only 37½ per cent. voids after the sand has settled. Table I shows the results of tests made by Feret, the French experimenter. Two kinds of sand were used, a very fine sand and a coarse sand. They were measured in a box that held 2 cu. ft. and was 8 ins. deep, the sand being shoveled into the box but not tamped or shaken. After measuring and weighing the dry sand 0.5 per cent. by weight of water was added and the sand was mixed and shoveled back into the box again and then weighed. These operations were repeated with varying percentages of water up to 10 per cent. It will be noted that the weight of mixed water and sand is given; to ascertain the exact weight of dry sand in any mixture, divide the weight given in the table by 100 per cent. plus the given tabular per cent.; thus the weight of dry, fine sand in a 5 per cent. mixture is 2,035 ÷ 1.5 = 1,98 lbs. per cu. yd. The voids in the dry sand were 45 per cent. and in the sand with 5 per cent. moisture they were 56.7 per cent. Pouring water onto loose, dry sand compacts it. By mixing fine sand and water to a thin paste and allowing it to settle, it was found that the sand occupied 11 per cent. less space than when measured dry. The voids in fine sand, having a specific gravity of 2.65, were determined by measurement in a quart measure and found to be as follows:

Sand not packed, per cent. voids	44½
Sand shaken to refusal, per cent. voids	35
Sand saturated with water, per cent. voids	37½

Another series of tests made by Mr. H. P. Boardman, using Chicago sand having 34 to 40 per cent. voids, showed the following results:

Water added, per cent.	2	4	6	8	10
Resulting per cent. increase	17.6	22	19.5	16.6	15.6

Mr. Wm. B. Fuller found by tests that a dry sand, having 34 per cent. voids, shrunk 9.6 per cent. in volume upon thorough tamping until it had 27 per cent. voids. The same sand moistened with 6 per cent. water and loose had 44 per cent. voids, which was reduced to 31 per cent. by ramming. The same sand saturated with water had 33 per cent. voids and by thorough ramming its volume was reduced 8½ per cent. until the sand had only 26¼ per cent. voids. Further experiments might be quoted and will be found recorded in several general treatises on concrete, but these are enough to demonstrate conclusively that any theory of the quantity of cement in mortar to be correct must take into account the effect of moisture on the voids in sand.

The effect of the size and the shape of the component grains on the amount of voids in sand is considerable. Feret's experiments are conclusive on these points, and they alone will be followed here. Taking for convenience three sizes of sand Feret mixed them in all the varying proportions possible with a total of 10 parts; there were 66 mixtures. The sizes used were: Large (L), sand composed of grains passing a sieve of 5 meshes per linear inch and retained on a sieve of 15 meshes per linear inch; medium (M), sand passing a sieve of 15 meshes and retained on a sieve of 50 meshes per linear inch, and fine (F), sand passing a 50-mesh sieve. With a dry sand whose grains have a specific gravity of 2.65, the weight of a cubic yard of either the fine, or the medium, or the large size, was 2,190 lbs., which is equivalent to 51 per cent. voids. The greatest weight of mixture, 2,840 lbs. per cu. yd., was an L₆M₀F₄ mixture, that is, one composed of six parts large, no parts medium and 4 parts fine; this mixture was the densest of the 66 mixtures made, having 36 per cent. voids. It will be noted that the common opinion that the densest mixture is obtained by a mixture of gradually increasing sizes of grains is incorrect; there must be enough difference in the size of the grains to provide voids so large that the smaller grains will enter them and not wedge the larger grains apart. Turning now to the shape of the grains, the tests showed that rounded grains give less voids than angular grains. Using sand having a composition of L₅M₃F₂ Feret got the following results:

	—Per cent. Voids—
Kind of Grains.	Shaken.	Unshaken.
Natural sand, rounded grains	25.6	35.9
Crushed quartzite, angular grains	27.4	42.1
Crushed shells, flat grains	31.8	44.3
Residue of quartzite, flat grains	34.6	47.5

The sand was shaken until no further settlement occurred. It is plain from these data on the effect of size and shape of grains on voids why it is that discrepancies exist in the published data on voids in dry sand. An idea of the wide variation in the granulometric composition of different sands is given by Table II. Table III shows the voids as determined for sands from different localities in the United States.

Table II.—Showing Granulometric Compositions of Different Sands.

Held by a Sieve.	A	B	C	E
No. 10	35.3%
No. 20	32.1	12.8%	4.2%	11%
No. 30	14.6	49.0	12.5	14
No. 40	...	...	44.4	...
No. 50	9.6	29.3	...	53
No. 100	4.9	5.7	...	...
No. 200	2.0	2.3	...	...
	——	——	——	——
Voids	33%	39%	41.7%	31%

Note.—A, is a "fine gravel" (containing 8% clay) used at Philadelphia. B, Delaware River sand. C, St. Mary's River sand. D, Green River, Ky., sand, "clean and sharp."

Table III.—Showing Measured Voids in Sand from Different Localities.

Locality.	Authority.	Percent Voids.	Remarks.
Ohio River	W. M. Hall	31	Washed
Sandusky, O.	C. E. Sherman	40	Lake
Franklin Co., O.	C. E. Sherman	40	Bank
Sandusky Bay, O.	S. B. Newberry	32.3	......
St. Louis, Mo.	H. H. Henby	34.3	Miss. River
Sault Ste. Marie	H. von Schon	41.7	River
Chicago, Ill.	H. P. Broadman	34 to 40	......
Philadelphia, Pa	39	Del. River
Mass. Coast	31 to 34	......
Boston, Mass	Geo. Kimball	33	Clean
Cow Bay, L. I.	Myron S. Falk	40½	......
Little Falls, N. J.	W. B. Fuller	45.6	......
Canton, Ill.	G. W. Chandler	30	Clean

Voids in Broken Stone and Gravel.—The percentage of voids in broken stone varies with the nature of the stone: whether it is broken by hand or by crushers; with the kind of crusher used, and upon whether it is screened or crusher-run product. The voids in broken stone seldom exceed 52 per cent. even when the fragments are of uniform size and the stone is shoveled loose into the measuring box. The following records of actual determinations of voids in broken stone cover a sufficiently wide range of conditions to show about the limits of variation.

The following are results of tests made by Mr. A. N. Johnson, State Engineer of Illinois, to determine the variation in voids in crushed stone due to variation in size and to method of loading into the measuring box. The percentage of voids was determined by weighing the amount of water added to fill the box:

Size.	Method of Loading.	Per cent. of Voids.
3 in.	20-ft. drop	41.8
3 in.	15-ft drop	46.8
3 in.	15-ft. drop	47.2
3 in.	Shovels	48.7
1½ in.	20-ft. drop	42.5
1½ in.	15-ft. drop	46.8
1½ in.	15-ft. drop	46.8
1½ in.	Shovels	50.5
¾ in.	20-ft. drop	39.4
¾ in.	15-ft. drop	42.7
¾ in.	15-ft. drop	41.5
¾ in.	15-ft. drop	41.8
¾ in.	Shovels	45.2
¾ in.	Shovels	44.6
⅜ in.	Shovels	41.0
⅜ in.	Shovels	40.6
⅜ in.	Shovels	41.0

The table shows clearly the effect on voids of compacting the stone by dropping it; it also shows for the ¾-in. and the ⅜-in. stone loaded by shovels how uniformly the percentages of voids run for stone of one size only. Dropping the stone 20 ft. reduced the voids some 12 to 15 per cent. as compared with shoveling.

Table IV.—Showing Determined Percentages of Voids in Broken Stone from Various Common Rocks.

Authority.	Percent Voids.	Remarks.
Sabin	49.0	Limestone, crusher run after screening out ⅛-in. and under.
"	44.0	Limsetone (1 part screenings mixed with 6 parts broken stone).
Wm. M. Black	46.5	Screened and washed, 2-ins. and under.
J. J. R. Croes	47.5	Gneiss, after screening out ¼-in. and under.
S. B. Newberry	47.0	Chiefly about egg size.
H. P. Broadman	39 to 42	Chicago limestone, crusher run.
"	48 to 52	" " screened into sizes.
Wm. M. Hall	48.0	Green River limestone, 2½-ins. and smaller dust screened out.
"	50.0	Hudson River trap, 2½-ins. and smaller, dust screened out.
Wm. B. Fuller	47.6	New Jersey trap, crusher run, 1/6 to 2.1 in.
Geo. A. Kimball	49.5	Roxbury conglomerate, ½ to 2½ ins.
Myron S. Falk	48.0	Limestone, ½ to 3 ins.
W. H. Henby	43.0	" 2-in size.
"	46.0	" 1½-in size
Feret	53.4	Stone, 1.6 to 2.4 ins.
"	51.7	" 0.8 to 1.6 in.
"	52.1	" 0.4 to 0.8 in.
A. W. Dow	45.3	Bluestone, 89% being 1½ to 2½ ins.
"	45.3	" 90% being 1/6 to 1½ in.
Taylor and Thompson	54.5	Trap, hard, 1 to 2½ ins.
"	54.5	" " ½ to 1 in.
"	45.0	" " 0 to 2½ in.
"	51.2	" soft, ¾ to 2 ins.
G. W. Chandler	40.0	Canton, Ill.
Emile Low	39.0	Buffalo limestone, crusher run, dust in.
C. M. Saville	46.0	Crushed cobblestone, screened into sizes.

Table V.—Showing Percentages of Voids in Gravel and Broken Stone of Different Granulometric Compositions.

				———Per cent Voids in———
Passing a ring of	2.4"	1.6"	0.8"	Round	Broken
Held by a ring	1.6"	0.8"	0.4"	Pebbles.	Stone.
Parts	1	0	0	40.0	53.4
"	0	1	0	38.8	51.7
"	0	0	1	41.7	52.1
"	1	1	0	35.8	50.5
"	1	0	1	35.6	47.1
"	0	1	1	37.9	40.5
"	1	1	1	35.5	47.8
"	4	1	1	34.5	49.2
"	1	4	1	36.6	49.4
"	1	1	4	38.1	48.6
"	8	0	2	34.1	....

Table IV gives the voids in broken stone as determined by various engineers; it requires no explanation. Table V, taken from Feret's tests, shows the effect of changes in granulometric composition on the amount of voids in both broken stone and gravel. Considering the column giving voids in stone it is to be noted first how nearly equal the voids are for stone of uniform size whatever that size be. As was the case with sand a mixture of coarse and fine particles gives the fewest voids; for stone an L₁M₀F₁ mixture and for gravel an L₈M₀F₂ mixture. Tamping reduces the voids in broken stone. Mr. Geo. W. Rafter gives the voids in clean, hand-broken limestone passing a 2½-in. ring as 43 per cent. after being lightly shaken and 37½ per cent. after being rammed. Generally speaking heavy ramming will reduce the voids in loose stone about 20 per cent.

It is rare that gravel has less than 30 per cent. or more than 45 per cent. voids. If the pebbles vary considerably in size so that the small fit in between the large, the voids may be as low as 30 per cent. but if the pebbles are tolerably uniform in size the voids will approach 45 per cent. Table V shows the effect of granulometric composition on the voids in gravel as determined by Feret. Mr. H. Von Schon gives the following granulometric analysis of a gravel having 34.1 per cent. voids:

Retained on 1-in. ring, per cent.	10.70
Retained on ⅜-in. ring, per cent.	23.65
Retained on No. 4 sieve, per cent.	8.70
Retained on No. 10 sieve, per cent.	17.14
Retained on No. 20 sieve, per cent.	21.76
Retained on No. 30 sieve, per cent.	6.49
Retained on No. 40 sieve, per cent.	5.96
Passed a No. 40 sieve, per cent.	5.59
Passed a 1½-in ring, per cent.	100.00

As mixtures of broken stone and gravel are often used the following determinations of voids in such mixtures are given. The following determinations were made by Mr. Wm. M. Hall for mixtures of blue limestone and Ohio River washed gravel:

Per cent. Stone.		Per cent. Gravel.	Per cent. Voids in Mix
100	with	0	48
80	"	20	44
70	"	30	41
60	"	40	38½
50	"	50	36
0	"	100	35

The dust was screened from the stone all of which passed a 2½-in. ring; the gravel all passed a 1½-in. screen. Using the same sizes of gravel and Hudson River trap rock, the results were:

Per cent. Trap.		Per cent. Gravel.	Per cent. Voids in Mix.
100	with	0	50
60	"	40	38½
50	"	50	36
0	"	100	35

The weight of a cubic foot of loose gravel or stone is not an accurate index of the percentage of voids unless the specific gravity is known. Pure quartz weighs 165 lbs., per cu. ft., hence broken quartz having 40 per cent. voids weighs 165 × .60 = 99 lbs. per cu. ft. Few gravels are entirely quartz, and many contain stone having a greater specific gravity like some traps or a less specific gravity like some shales and sandstone. Tables VI and VII give the specific gravities of common stones and minerals and Table VIII gives the weights corresponding to different percentages of voids for different specific gravities.

Table VI.—Specific Gravity of Stone. (Condensed from Merrill's "Stones for Building.")

Trap,	Boston, Mass.	2.78
"	Duluth, Minn.	2.8 to 3.0
"	Jersey City, N. J.	3.03
"	Staten Island, N. Y.	2.86
Gneiss,	Madison Ave., N. Y.	2.92
Granite,	New London, Conn.	2.66
"	Greenwich, Conn.	2.84
"	Vinalhaven, Me.	2.66
"	Quincy, Mass.	2.66
"	Barre, Vt.	2.65
Limestone,	Joliet, Ill.	2.56
"	Quincy, Ill.	2.51 to 2.57
Limestone, (oolitic)	Bedford, Ind.	2.25 to 2.45
"	Marquette, Mich.	2.34
"	Glens Falls, N.Y.	2.70
"	Lake Champlain, N. Y.	2.75
Sandstone,	Portland, Conn.	2.64
"	Haverstraw, N. Y.	2.13
"	Medina, N. Y.	2.41
"	Potsdam, N. Y.	2.60
"	(grit) Berea, O.	2.12

Table VII.—Specific Gravity of Common Minerals and Rocks.

Apatite	2.92-3.25
Basalt	3.01
Calcite, CaCO₃	2.5-2.73
Cassiterite, SnO₂	6.4-7.1
Cerrusite, PbCO₃	6.46-6.48
Chalcopyrite, CuFeS₂	4.1-4.3
Coal, anthracite	1.3-1.84
Coal, bituminous	1.2-1.5
Diabase	2.6-3.03
Diorite	2.92
Dolomite, CaMg (CO₃)²	2.8-2.9
Felspar	2.44-2.78
Felsite	2.65
Galena, Pbs	7.25-7.77
Garnet	3.15-4.31
Gneiss	2.62-2.92
Granite	2.55-2.86
Gypsum	2.3-3.28
Halite (salt) NaCl	2.1-2.56
Hematite, Fe₂O₃	4.5-5.3
Hornblende	3.05-3.47
Limonite, Fe₃O₄ (OH)⁶	3.6-4.0
Limestone	2.35-2.87
Magnetite, Fe₃O₄	4.9-5.2
Marble	2.08-2.85
Mica	2.75-3.1
Mica Schist	2.5-2.9
Olivine	3.33-3.5
Porphyry	2.5-2.6
Pyrite, FeS₂	4.83-5.2
Quartz, SiO₂	2.5-2.8
Quartzite	2.6-2.7
Sandstone	2.0-2.78
" Medina	2.4
" Ohio	2.2
" Slaty	1.82
Shale	2.4-2.8
Slate	2.5-2.8
Sphalerite, ZnS	3.9-4.2
Stibnite, Sb₂S₃	4.5-4.6
Syenite	2.27-2.65
Talc	2.56-2.8
Trap	2.6-3.0

Table VIII.—Showing Weight of Stone with Different Percentages of Voids for Different Specific Gravities.

			Weight in Lbs. per cu. yd. when Voids are
Specific Gravity.	Weight in Lbs. per cu. ft.	Weight in Lbs. per cu. yd.	30%	35%	40%	45%	50%
1.0	62.355	1,684	1,178	1,094	1,010	926	842
2.0	124.7	3,367	2,357	2,187	2,020	1,852	1,684
2.1	130.9	3,536	2,475	2,298	2,121	1,945	1,768
2.2	137.2	3,704	2,593	2,408	2,222	2,037	1,852
2.3	143.4	3,872	2,711	2,517	2,323	2,130	1,936
2.4	149.7	4,041	2,828	2,626	2,424	2,222	2,020
2.5	155.9	4,209	2,946	2,736	2,525	2,315	2,105
2.6	162.1	4,377	3,064	2,845	2,626	2,408	2,189
2.7	168.4	4,546	3,182	2,955	2,727	2,500	2,273
2.8	174.6	4,714	3,300	3,064	2,828	2,593	2,357
2.9	180.9	4,882	3,418	3,174	2,929	2,685	2,441
3.0	187.1	5,051	3,536	3,283	3,030	2,778	2,526
3.1	193.3	5,219	3,653	3,392	3,131	2,871	2,609
3.2	199.5	5,388	3,771	3,502	3,232	2,963	2,694
3.3	205.8	5,556	3,889	3,611	3,333	3,056	2,778
3.4	212.0	5,724	4,007	3,721	3,434	3,148	2,862
3.5	218.3	5,893	4,125	3,830	3,535	3,241	2,947

In buying broken stone by the cubic yard it should be remembered that hauling in a wagon compacts the stone by shaking it down and reduces the volume. Table IX shows the results of tests made by the Illinois Highway Commission to determine the settlement of crushed stone in wagon loads for different lengths of haul. The road over which the tests were made was a macadam road, not particularly smooth, but might be considered as an average road surface. The wagon used was one with a dump bottom supported by chains, which were drawn as tight as possible, so as to reduce the sag to a minimum. It will be noticed that about 50 per cent. of the settlement occurs within the first 100 ft., and 75 per cent. of the settlement in the first 200 ft. Almost all of the settlement occurs during the first half mile, as the tests showed practically no additional settlement for distances beyond. Some of the wagons were loaded from the ground with shovels, others were loaded from bins, the stone having a 15-ft. drop, which compacted the stone a little more than where loaded with shovels, so that there was somewhat less settlement. But at the end of a half mile the density was practically the same, whatever the method of loading. The density at the beginning and at the end of the haul can be compared by the weight of a given volume of crushed stone. For convenience, the weight of a cubic yard of the material at the beginning of the haul and at the end was computed from the known contents of a wagon.

Table IX.—Showing Settlement of Broken Stone due to Different Lengths of Haul on Ordinarily Good Road in Wagons.

		Per cent Settlement for Hauling.									Weight per Cu. Yd. in Lbs.
Size.	Method of Loading.	100'	200'	300'	400'	500'	600'	700'	½ Mile	1 Mile	At start.	At finish.
Screenings	15 ft. drop	....	...	....	....	....	....	....	11.5	11.5	2,518	2,840
Screenings	15 ft. drop	....	...	....	....	....	....	....	12.6	12.6	2,518	2,886
Screenings	15 ft. drop	7.3	8.3	8.9	9.2	9.5	10.1	10.1	11.2	....	2,450	2,770
Screenings	15 ft. drop	5.0	9.6	10.2	10.2	10.4	10.4	10.4	12.4	....	,425	2,780
1½ inch	15 ft. drop	...	....	....	....	....	....	....	11.5	11.5 [C]	2,305	2,600
1½ inch	15 ft. drop	5.3	6.2	7.1	7.7	7.9	8.0	8.3	9.2	....	2,380	2,625
1½ inch	15 ft. drop	2.6	3.7	4.9	5.3	5.3	5.3	5.4	5.4	....	2,450	2,600
1½ inch	Shovels	3.5	4.1	4.8	5.3	5.3	5.7	6.5	7.25	....	2,270	2,445
1½ inch	Shovels	...	....	....	....	....	....	....	12.6	12.6	2,305	2,642
3 inch	15 ft. drop	...	....	....	....	....	....	....	10.1	10.1	2,376	2,638
3 inch	15 ft. drop	3.5	4.2	4.5	4.8	5.0	5.0	5.0	6.0	....	2,360	2,505
3 inch	15 ft. drop	0.5	2.5	2.5	4.1	4.3	4.3	4.3	4.9	....	2,470	2,595
3 inch	Shovels	...	....	....	....	....	....	....	12.6	12.6	2,270	2,601
3 inch	Shovels	5.0	5.6	6.5	6.5	6.8	6.8	6.8	7.1	....	2,335	2,510

[C] —Same per cent of settlement for two-mile haul.

THEORY OF THE QUANTITY OF CEMENT IN MORTAR AND CONCRETE.—All sand contains a large percentage of voids; in 1 cu. ft. of loose sand there is 0.3 to 0.5 cu. ft. of voids, that is, 30 to 50 per cent. of the sand is voids. In making mortar the cement is mixed with the sand and the flour-like particles of the cement fit in between the grains of sand occupying a part or all of the voids. The amount of cement required in a mortar will naturally depend upon the amount of voids in the particular sand with which it is mixed and since a correct estimate of the number of barrels of cement per cubic yard of mortar is very important, and since it is not always possible to make actual mixtures before bidding, rules based on various theories have been formulated for determining these quantities. In this volume the rule based on the theory outlined by one of the authors in 1901 will be followed. The following is a discussion of the authors' theory:

When loose sand is mixed with water, its volume or bulk is increased; subsequent jarring will decrease its volume, but still leave a net gain of about 10 per cent.; that is, 1 cu. ft. of dry sand becomes about 1.1 cu. ft. of damp sand. Not only does this increase in the volume of the sand occur, but, instead of increasing the voids that can be filled with cement, there is an absolute loss in the volume of available voids. This is due to the space occupied by the water necessary to bring the sand to the consistency of mortar; furthermore, there is seldom a perfect mixture of the sand and cement in practice, thus reducing the available voids. It is safe to call this reduction in available voids about 10 per cent.

When loose, dry Portland cement is wetted, it shrinks about 15 per cent, in volume, behaving differently from the sand, but it never shrinks back to quite as small a volume as it occupies when packed tightly in a barrel. Since barrels of different brands vary widely in size, the careful engineer or contractor will test any brand he intends using in large quantities, in order to ascertain exactly how much cement paste can be made. He will find a range of from 3.2 cu. ft. to 3.8 cu. ft. per barrel of Portland cement. Obviously the larger barrel may be cheaper though its price is higher. Specifications often state the number of cubic feet that will be allowed per barrel in mixing the concrete ingredients, so that any rule or formula to be of practical value must contain a factor to allow for the specified size of the barrel, and another factor to allow for the actual number of cubic feet of paste that a barrel will yield—the two being usually quite different.

The deduction of a rational, practical formula for computing the quantity of cement required for a given mixture will now be given, based upon the facts above outlined.

Let	p	=	number of cu. ft. cement paste per bbl., as determined by actual test.
	n	=	number of cu. ft. of cement per bbl., as specified in the specifications.
	s	=	parts of sand (by volume) to one part of cement, as specified.
	g	=	parts of gravel or broken stone (by volume) to one part of cement, as specified.
	v	=	percentage of voids in the dry sand, as determined by test.
	V	=	percentage of voids in the gravel or stone, as determined by test.

Then, in a mortar of 1 part cement to s parts sand, we have:

n s	=	cu.	ft.	of	dry sand to 1 bbl. of cement.
n s v	=	"	"	"	voids in the dry sand.
0.9 n s v	=	"	"	"	available voids in the wet sand.
1.1 n s	=	"	"	"	wet sand.
p - 0.9 n s v	=	"	"	"	cement paste in excess of the voids.

Therefore:

1.1 n s + (p - 0.9 n s v) = cu. ft. of mortar per bbl.

Therefore:

		27		27
N	=	————————————	=	——————————
		1.1 n s + (p - 0.9 n s v)		p + n s (1.1 - 0.9 v)

N being the number of barrels of cement per cu. yd. of mortar.

When the mortar is made so lean that there is not enough cement paste to fill the voids in the sand, the formula becomes:

		27
N	=	———
		1.1 n s

A similar line of reasoning will give us a rational formula for determining the quantity of cement in concrete; but there is one point of difference between sand and gravel (or broken stone), namely, that the gravel does not swell materially in volume when mixed with water. However, a certain amount of water is required to wet the surface of the pebbles, and this water reduces the available voids, that is, the voids that can be filled by the mortar. With this in mind, the following deduction is clear, using the nomenclature and symbols above given:

ng	=	cu.	ft.	of	dry gravel (or stone).
ng V	=	"	"	"	voids in dry gravel.
0.9 ng V	=	"	"	"	"available voids" in the wet gravel.
p + n s (1.1 - 0.9 v) - 0.9 ng V	=	excess	of	mortar	over the available voids in the wet gravel.
ng + p + n s (1.1 - 0.9 v) - 0.9 ng V	=	cu.	ft.	of	concrete from 1 bbl. cement.

		27
N	=	———————————————————————
		p + n s (1.1 - 0.9v) + ng (1 - 0.9 V)

N being the number of barrels of cement required to make 1 cu. yd. of concrete.

This formula is rational and perfectly general. Other experimenters may find it desirable to use constants slightly different from the 1.1 and the 0.9, for fine sands swell more than coarse sands, and hold more water.

The reader must bear in mind that when the voids in the sand exceed the cement paste, and when the available voids in the gravel (or stone) exceed the mortar, the formula becomes:

	27
N	= ———
	ng

These formulas give the amounts of cement in mortars and concretes compacted in place. Tables X to XIII are based upon the foregoing theory, and will be found to check satisfactorily with actual tests.

In using these tables remember that the proportion of cement to sand is by volume, and not by weight. If the specifications state that a barrel of cement shall be considered to hold 4 cu. ft., for example, and that the mortar shall be 1 part cement to 2 parts sand, then 2 barrel of cement is mixed with 8 cu. ft. of sand, regardless of what is the actual size of the barrel, and regardless of how much cement paste can be made with a barrel of cement. If the specifications fail to state what the size of a barrel will be, then the contractor is left to guess.

Table X.—Barrels of Portland Cement per Cubic Yard of Mortar.

(Voids in sand being 35%, and 1 bbl. cement yielding 3.65 cu. ft. of cement paste.)

Proportion of Cement to Sand	1 to 1	1 to 1½	1 to 2	1 to 2½	1 to 3	1 to 4
	Bbls.	Bbls.	Bbls.	Bbls.	Bbls.	Bbls.
Barrel specified to be 3.5 cu. ft.	4.22	3.49	2.97	2.57	2.28	1.76
" " " 3.8 "	4.09	3.33	2.81	2.45	2.16	1.62
" " " 4.0 "	4.00	3.24	2.73	2.36	2.08	1.54
" " " 4.4 "	3.81	3.07	2.57	2.27	2.00	1.40
Cu. yds. sand per cu. yd. mortar	0.6	0.7	0.8	0.9	1.0	1.0

Table XI.—Barrels of Portland Cement per Cubic Yard of Mortar.

(Voids in sand being 45%, and 1 bbl. cement yielding 3.4 cu. ft. of cement paste.)

Proportion of Cement to Sand	1 to 1	1 to 1½	1 to 2	1 to 2½	1 to 3	1 to 4
	Bbls.	Bbls.	Bbls.	Bbls.	Bbls.	Bbls.
Barrel specified to be 3.5 cu. ft.	4.62	3.80	3.25	2.84	2.35	1.76
" " " 3.8 "	4.32	3.61	3.10	2.72	2.16	1.62
" " " 4.0 "	4.19	3.46	3.00	2.64	2.05	1.54
" " " 4.4 "	3.94	3.34	2.90	2.57	1.86	1.40
Cu. yds. sand per cu. yd. mortar	0.6	0.8	0.9	1.0	1.0	1.0

If the specifications call for proportions by weight, assume a Portland barrel to contain 380 lbs. of cement, and test the actual weight of a cubic foot of the sand to be used. Sand varies extremely in weight, due both to the variation in the per cent. of voids, and to the variation in the kind of minerals of which the sand is composed. A quartz sand having 35 per cent. voids weighs 107 lbs. per cu. ft.; but a quartz sand having 45 per cent. voids weighs only 91 lbs. per cu. ft. If the weight of the sand must be guessed at, assume 100 lbs. per cu. ft. If the specifications require a mixture of 1 cement to 2 of sand by weight, we will have 380 lbs. (or 1 bbl.) of cement mixed with 2 × 380, or 760 lbs. of sand; and if the sand weighs 90 lbs. per cu. ft., we shall have 760 ÷ 90, or 8.44 cu. ft. of sand to every barrel of cement. In order to use the tables above given, we may specify our own size of barrel; let us say 4 cu. ft.; then 8.44 ÷ 4 gives 2.11 parts of sand by volume to 1 part of cement. Without material error we may call this a 1 to 2 mortar, and use the tables, remembering that our barrel is now "specified to be" 4 cu. ft. If we have a brand of cement that yields 3.4 cu. ft. of paste per bbl., and sand having 45 per cent. voids, we find that approximately 3 bbls. of cement per cu. yd. of mortar will be required.

Table XII.—Ingredients in 1 Cubic Yard of Concrete.

(Sand voids, 40%; stone voids, 45%; Portland cement barrel yielding 3.65 cu. ft. paste. Barrel specified to be 3.8 cu. ft.)

	1:2:4	1:2:5	1:2:6	1:2½:5	1:2½:6	1:3:4
Bbls. cement per cu. yd. concr't	1.46	1.30	1.18	1.13	1.00	1.25
Cu. yds. sand " "	0.41	0.36	0.33	0.40	0.35	0.53
Cu. yds. stone " "	0.82	0.90	1.00	0.80	0.84	0.71
Proportions by Volume.	1:3:5	1:3:6	1:3:7	1:4:7	1:4:8	1:4:9
Bbls. cement per cu. yd. concr't	1.13	1.05	0.96	0.82	0.77	0.73
Cu. yds. sand " "	0.48	0.44	0.40	0.46	0.43	0.41
Cu. yds. stone " "	0.80	0.88	0.93	0.80	0.86	0.92

Note.—This table is to be used where cement is measured packed in the barrel, for the ordinary barrel holds 3.8 cu. ft.

It should be evident from the foregoing discussions that no table can be made, and no rule can be formulated that will yield accurate results unless the brand of cement is tested and the percentage of voids in the sand determined. This being so the sensible plan is to use the tables merely as a rough guide, and, where the quantity of cement to be used is very large, to make a few batches of mortar using the available brands of cement and sand in the proportions specified. Ten dollars spent in this way may save a thousand, even on a comparatively small job, by showing what cement and sand to select.

It will be seen that Tables XII and XIII can be condensed into the following rule:

Add together the number of parts and divide this sum into ten, the quotient will be approximately the number of barrels of cement per cubic yard.

Table XIII.—Ingredients in 1 Cubic Yard of Concrete.

(Sand voids, 40%; stone voids, 45%; Portland cement barrel yielding 3.65 cu. ft. of paste. Barrel specified to be 4.4 cu. ft.)

Proportions by Volume.	1:2:4	1:2:5	1:2:6	1:2½:5	1:2½:6	1:3:4
Bbls. cement per cu. yd. concr't	1.30	1.16	1.00	1.07	0.96	1.08
Cu. yds. sand " "	0.42	0.38	0.33	0.44	0.40	0.53
Cu. yds. stone " "	0.84	0.95	1.00	0.88	0.95	0.71
Proportions by Volume.	1:3:5	1:3:6	1:3:7	1:4:7	1:4:8	1:4:9
Bbls. cement per cu. yd. concr't	0.96	0.90	0.82	0.75	0.68	0.64
Cu. yds. sand " "	0.47	0.44	0.40	0.49	0.44	0.42
Cu. yds. stone " "	0.78	0.88	0.93	0.86	0.88	0.95

Note.—This table is to be used when the cement is measured loose, after dumping it into a box, for under such conditions a barrel of cement yields 4.4 cu. ft. of loose cement.

Thus for a 1:2:5 concrete, the sum of the parts is 1 + 2 + 5, which is 8; then 10 ÷ 8 is 1.25 bbls., which is approximately equal to the 1.30 bbls. given in the table. Neither is this rule nor are the tables applicable if a different size of cement barrel is specified, or if the voids in the sand or stone differ materially from 40 per cent. to 45 per cent. respectively. There are such innumerable combinations of varying voids, and varying sizes of barrel, that the authors do not deem it worth while to give other tables. The following amounts of cement per cubic yard of mortar were determined by test:

Authority	Neat.	1 to 1	1 to 2	1 to 3	1 to 4	1 to 5	1 to 6	1 to 7	1 to 8
	Bbls.	Bbls.	Bbls.	Bbls.	Bbls.	Bbls.	Bbls.	Bbls.	Bbls.
Sabin	7.40	4.17	2.84	2.06	1.62	1.33	1.14	....	....
W. B. Fuller	8.02	4.58	3.09	2.30	1.80	1.48	1.23	1.11	1.00
H. P. Boardman.	7.40	4.50	3.18	2.35	....	....	....	....	....

The proportions were by barrels of cement to barrels of sand, and Sabin called a 380-lb. barrel 3.65 cu. ft., whereas Fuller called a 380-lb. barrel 3.80 cu. ft.; and Boardman called a 380-lb. barrel 3.5 cu. ft. Sabin used a sand having 38 per cent. voids; Fuller used a sand having 45 per cent. voids; and Boardman used a sand having 38 per cent. voids. It will be seen that the cement used by Sabin yielded 3.65 cu. ft. of cement paste per bbl. (i. e. 27 ÷ 7.4), whereas the (Atlas) cement used by Fuller yielded 3.4 cu. ft. of cement paste per bbl. Sabin found that a barrel of cement measured 4.37 cu. ft. when dumped and measured loose. Mr. Boardman states a barrel (380 lbs., net) of Lehigh Portland cement yields 3.65 cu. ft. of cement paste; and that a barrel (265 lbs., net) of Louisville natural cement yields 3.0 cu. ft. of cement paste.

Mr. J. J. R. Croes, M. Am. Soc. C. E., states that 1 bbl. of Rosendale cement and 2 bbl. of sand (8 cu. ft.) make 9.7 cu. ft. of mortar, the extreme variations from this average being 7 per cent.

Frequently concrete is made by mixing one volume of cement with a given number of volumes of pit gravel; no sand being used other than the sand that is found naturally mixed with the gravel. In such cases the cement rarely increases the bulk of the gravel, hence Table XIV will give the approximate amount of cement, assuming 1 cu. yd. of gravel per cubic yard of concrete.

Table XIV.—Showing Barrels of Cement per Cubic Yard of Various Mixtures of Cement and Pit Gravel.

Spc. Vol. of bbl. cu. ft.	Barrels of Cement per Cubic Yard of Concrete for Mixtures of
Spc. Vol. of bbl. cu. ft.	1-5	1-6	1-7	1-8	1-9	1-10	1-12
3.8	1.41	1.18	1.01	0.874	0.789	0.71	0.59
4.4	1.25	1.02	0.875	0.766	0.681	0.61	0.51

PERCENTAGE OF WATER IN CONCRETE.—Tests show that dry mixtures when carefully deposited and well tamped produce the stronger concrete. This superiority of dry mixtures it must be observed presupposes careful deposition and thorough tamping, and these are tasks which are difficult to have accomplished properly in actual construction work and which, if accomplished properly, require time and labor. Wet mixtures readily flow into the corners and angles of the forms and between and around the reinforcing bars with only a small amount of puddling and slicing and are, therefore, nearly always used because of the time and labor saved in depositing and tamping. The following rule by which to determine the percentage of water by weight for any given mixture of mortar for wet concrete will be found satisfactory:

Multiply the parts of sand by 8, add 24 to the product, and divide the total by the sum of the parts of sand and cement.

For example if the percentage of water is required for a 1-3 mortar:

(3 × 8) + 24
——————	=	12.
4

Hence the water should be 12 per cent. of the combined weight of cement and sand. For a 1-1 mortar the rule gives 16 per cent.; for a 1-2 mortar it gives 13½ per cent., and for a 1-6 mortar it gives 10.3 per cent.

To calculate the amount of water per cubic yard of 1-3-6 concrete for example the procedure would be as follows: By the above rule a 1-3 mortar requires

(3 × 8) + 24
——————	=	12 per cent. water.
4

A 1-3-6 concrete, according to Table XII, contains 1.05 bbls. cement and 0.44 cu. yd. sand. Cement weighs 380 lbs. per barrel, hence 1.05 bbls. would weigh 380 × 1.05 = 399 lbs. Sand weighs 2,700 lbs. per cu. yd., hence 0.44 cu. yd. of sand would weigh 2,700 × 0.44 = 1,188 lbs. The combined weight of the cement and sand would thus be 399 + 1,188 = 1,587 lbs. and 12 per cent. of 1.587 lbs. is 190 lbs. of water. Water weighs 8.355 lbs. per gallon, hence 190 × 8.355 = 23 gallons of water per cubic yard of 1-3-6 concrete.

METHODS OF MEASURING AND WEIGHING.—The cement, sand and aggregate for concrete mixtures are usually measured by hand, the measuring being done either in the charging buckets or in the barrows or other receptacles used to handle the material to the charging buckets. The process is simple in either case when once the units of measurement are definitely stated. This is not always the case. Some engineers require the contractor to measure the sand and stone in the same sized barrel that the cement comes in, in which case 1 part of sand or aggregate usually means 3.5 cu. ft. Other engineers permit both heads of the barrel to be knocked out for convenience in measuring the sand and stone, in which case a barrel means 3.75 cu. ft. Still other engineers permit the cement to be measured loose in a box, then a barrel usually means from 4 to 4.5 cu. ft. Cement is shipped either in barrels or in bags and the engineer should specify definitely the volume at which he will allow the original package to be counted, and also, if cement barrels are to be used in measuring the sand and stone, he should specify what a "barrel" is to be. When the concrete is to be mixed by hand the better practice is to measure the sand and stone in bottomless boxes of the general type shown by Fig. 10 and of known volume, and then specify that a bag of cement shall be called 1 cu. ft., 0.6 cu. ft., or such other fraction of a cubic foot as the engineer may choose. The contractor then has a definite basis on which to estimate the quantity of cement required for any specified mixture. The same is true if the measuring of the sand and stone be done in barrows or in the charging bucket. The volume of the bag or barrel of cement being specified the contractor has a definite and simple problem to solve in measuring his materials.

Fig. 10.—Bottomless Box for Measuring Materials in Proportioning Concrete.

To avoid uncertainty and labor in measuring the cement, sand and stone or gravel various automatic measuring devices have been designed. A continuous mixer with automatic measuring and charging mechanism is described in Chapter XIV. Figure 11 shows the Trump automatic measuring device. It consists of a series of revolving cylinders, each opening onto a "table," which revolves with the cylinders, and of a set of fixed "knives," which, as the "tables" revolve, scrape off portions of the material discharged from each cylinder onto its "table." The illustration shows a set of two cylinders; for concrete work a third cylinder is added. The three tables are set one above the other, each with its storage cylinder, and being attached to the same spindle all revolve together. For each table there is a knife with its own adjusting mechanism. These knives may be adjusted at will to vary the percentage of material scraped off.

Fig. 11.—Sketch Showing Trump Automatic Measuring Device for Materials in Proportioning Concrete.

Automatic measuring devices are most used in connection with continuous mixers, but they may be easily adapted to batch mixers if desired. One point to be observed is that all of these automatic devices measure the cement loose and this must be allowed for in proportioning the mixture.