Lyse, I.

English

Lehigh University: Lehigh Preserve

Lehigh University
Lehigh Preserve
Fritz Laboratory Reports Civil and Environmental Engineering
1932
Visualizing concrete economy in terms of strength,
ENR, Vol. 109 (1932), p. 109, Reprint No. 31
(32-6.)
I. Lyse
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Recommended Citation
Lyse, I., "Visualizing concrete economy in terms of strength, ENR, Vol. 109 (1932), p. 109, Reprint No. 31 (32-6.)" (1932). Fritz
Laboratory Reports. Paper 1139.
http://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-reports/1139
Z,ooo 3,000 4,000 5,000 6,000 7000
Strength of Concrete, Lb. per Sq. In.
1\
\; ,-E=i{44600.p, +(p,-0O5/iJ¥), S=-.2510+12CD= 0.82
\~ PI= g12.00 per Ion •P2= 1//.88 per barrel
I'-r--~
o
IPOO
Fig. 2-Relation between -cost and cement
content of concrete mixes for different
prices of aggregate with cement cost
constant.
Fig. I-Relation between cost and cement
content" of concrete mixes for different
prices of cement and with aggregate cost
constant.
2.001-·--f---j--+---t--+---j
Fig. 3-Cost of concrete per unit of strength
0.000 lb. per sq.in.) decreases rapidly with
an increase in concrete strength.
10000,--...,-----,----,r-:-.,--..,...""""-;
12.00r--...,---~---,---,---.---,
1O.001--+---j---j---t---'---:7"'-'--1
;f:5.00
'"~
'"L
<:;4.00
.E
~
~3.00
::l
~~ Z.OO
c.
;<i
31.00
L
8-
1;;
~
...
'" 8.00 I--+---I---j---c;
+~
uS 6.00
~a4.00b"""'+-;;"'''9---j---t---l---l
8-
~
.U
cg:,
in 4.006.ll~:::f-o=~--+----+--+---l
o
u
-tgaoo1--+---1--+, o'~'f--+~"""1
<3
$!
.3 6.00 1--+---:,
cents, where c is given in pounds.
III Fig. 1 the relation between cost
and cement content is given for a price
6f $2 a ton for aggregates and prices
of $1.10, $1.88 and $2.60 per barrel of
cement. In Fig. 2 the price of cement
was $1.88 per barrel, and the prices of
16-~3Dtstri~ by
July 28, 1932 Fr1tz Engineering Leb-~H
these units the cost equation becomes:
P =4460XO. 82 XPI + (P270 . 8Sp,)c.
If the price of aggregate is $2 per
ton or 10 cents per pound, and the price
of cement is $1.88 per barrel, or 50 cents
per pound, the cost of materials for 1
cu.yd. of concrete is:
P =4460XO. 82 XO.lO
+(0.50-0.85 XO.lO)c = 3660+0 .415c
W
27X62.4
1 (_a+_c)
27X62.4 2.653.10
ora = 4460D-0.8Sc
=D+---
P = pIa+p2c= p14460D-PI XO. 85c+p2c
=4460Dpi+(P2-0 . 85p.)c
1 (a c W)
J7 = 27X62.4 ga+~+I:O
1 (a c W)
-+-+-
27X62.4 2.653.10 1.0
D=
Thus the cost of the materials per
cubic yard of concrete varies directly
with the cement content for given unit
prices of aggregate and cement and for
a given consistency of concrete.
The cost of the concrete may also be
expressed in terms of the strength of the
concrete from the following relation:
S-A
S=A+Kc, or c = --,
K
(
S-A)
so thatP=4460D. PI + (P2-0.85pl) K
If the strength-giving qualities of the
materials are known, it is a simple mat-
ter to COll1pute the cost for any given
strength of concrete. It follows that
for concrete containing aggregates of a
given type and gradation and containing
a constant amount of water per cubic
yard of concrete, the cost per unit of
concrete varies with the strength of the
concrete.
The following illustration will make
the significance of this fact more appar-
ent. Assume the use of aggregates re-
quiring 303 lb. of water per cubic yard of
concrete for the desi red consistencv. The
volume of watel' in the concrete then is
303/ (27 X 62.4), Or 18 per cent of the
volume of the fresh. concrete. Since the
density, D, is the difference between
the volume of concrete and the volume
of ,vater, D = 1.0 - 0.18, or 82 per
cent of the volume of the concrete. With
where·
The value 27X62.4 represents the
weight of 1 cu.yd. of water, ga and gc
are specific gravities of aggregates and
cement respectively, and D is density,
or portion of solid materials in the con-
crete.
The cost of the materials is then:
By Inge Lyse
Research Assistant Professor of Engineering
11[aterials, Fritz Engineering Laboratorv.
Lehiu/t University, Bethlehem. Pa.
Straight-line relation between strength and cement-water ratio
permits direct measure of cost in terms of strength-Cost
per unit of strength decreases with an increase in strength
I N PREVIOUS articles the writerhas shown that the strength of con-crete is determined by the concen-
tration of cement particles in the cement
paste (ENR, Nov. 5, 1931, p. 723), and
that the design and control of concrete
mixes are greatly simplified by this re-
lation (ENR, Feb. 18, 1932, p. 248).
Data given in these articles show that,
for concretes containing aggregates of
a given type and gradation, the consist-
ency remains constant regardless of the
richness of the mix if the water con-
tent per unit of concrete remains con-
stant. Thus the straight-line relation-
ship between the strength and the
cement-water ratio of the concrete gen-
erates into a straight-line relation be-
_ tween strength and cement content,
expressed as follows:
S=A +B (c/w) =A + (B/w) c
= A + Kc where S is strength of
concrete, A and B are constants de-
pending upon the materials and condi-
tions of test. and c and ware amounts
of cement and water in a cubic yard of
fresh concrete. If th~ water content per
unit of concrete is held constant, B/w
becomes a new constant, K. Thus for
given materials and conditions of test
the strength of the concrete is propor-
tional to the cement content.
The total cost of the concrete is com-
posed of the cost of the materials plus
the cost of fonnwork, mixing, placing,
curing, etc. Since the costs of form-
work, mixing and placing are prac-
tically the same for concrete mixes of
different richness as long as their con-
sistencies are nearly the same, the
variation in the cost of concretes of
diffel'~nt richness is mainly due to the
difference in the cost of the materials-
aggregates and cement. For 1 cu.yd. of
concrete the cost of the materials may be
expressed by: P = pta + P2C where
P, and P2 are unit prices of aggregates
and cement. and a and c are aggregate
and cement contents per cubic yanl of
concrete. Since the total volume of
properly placed concrete is the sum of
.the absolute volumes of aggregates. ce-
ment and water, the volume of the
freshly placed concrete may be ex-
pressed by: V = Va + Va + V",. The
absolute volumes al'e obtained by divid-
ing the weight of the materials by their
speci fic weights, so that the above re-
lation becomes:
Q Reprinted from Engineering News-Record,
.. Visualizing Concrete Economy
irf Terms of Strength
aggregate were $1.50, $2.00 and $2.50
J'pei" ton.
In structural concrete members, where
the total area is in compression, the
load-carrying capacity of the member is
dii"ectly proportional to the strength of
the concrete. The cost of materials for
1 cu.yd. of concrete per pound of
stt"ength of the concrete is thet"efore a
direct measure of the economv of the
concrete mix. This cost pet: unit of
strength is expressed by the formula:
P 1 [ S-AJE=-=- 4460Dpt+(P2-0.85pl) -'-
S S K
Since for given materials and condi-
tions of construction all units on the
right of the equation are given, the cost
can be computed directly.. The relation
between cost of concrete per unit of
strength and the strength of the con-
crete is shown in Fig. 3 for given ma-
terials and conditions of construction.
It is seen that the cost per unit of
strength decreases rapidly with an in-
crease in strength of the concrete. Thus
for concrete members loaded directly in
compression the more "economical mix is
the one giving the higher strength.
The above relations are valuable aids
for selecting the most economical type
of aggregate and cement for l!se on a
given job. The relations also give in-
formation as to the feasibility of using
ordinary portland or high-eady~strength
cement. Similar relations may also be
-'
found for concrete subjected to flexural
stresses. Thus bv means of the con-
stant water requirement for a given
workability and the straight-line rela-
tion between strength and cement-
water ratio. the most economical mix is
readily determined for all types of plain
and reinforced-concrete members.
In addition to the direct effect of the
strength of the concrete on the economy
of the mix, an additional important eco-
nomic item is the reduction in size of
the member. The benefit of the highet"
strength- concrete is therefore magnified
by the accorilpanied reduction of size of
member for a given load. Fot· columns
the saving in floor space by the use of
high-strength concrete becomes very
important.
Reprinted from Engineering News-Record, July 28, 1932
2000 3po0 4,000 5.000 6iJ00 1fXJO
Stren0th of Concrete, Lb. per SCI_ln.
~
\t ,-E=iL4460 O,PI +(p,-OOSPrJ¥)/ S=-2570+/2CD= 0.82
\ Pt' $12.00 per fon
"'"
P2= 11 1.88 per ba-rre/
.............r--
-
Fig. I-Relation between COSt and cement
content of concrete mixes for different
prices of cement and with aggtegate cost
constant.
2.00 I-----+----\--+----+--+----j
Fig. 2-Relation between cost and cement
content of concrete mixes for different
pri ces of aggregate with cement COSt
constant.
Fig. 3-Cost of concrete per unit of strength
(1.000 lb. per sq.in.) decreases rapidly with
an increase in concrete strength.
10.00,......,.,...--.----,---,---,,...---,--.........,..--;
12.00.---,---,--"T-"--,---,-----,
10.00f---t----\--+----+---'--,7"'~
~5.00
'"t:~
<h4.00
.5
~
8-3.00
::l
~
.. 2.00
l5.
~a1.00
..
l5.
+-2 0
;:; 1900
~~ aoo f-----+----\--_+, o'':..-''r---t--:::iI''''''9
<3
~j 6.00 1--+---,.
., 8.00 f-----+----j--+---:
+~
....
.3 6.00
~a4.00b.~+;;,."9--+---+--+---l
8.
.,
"- .
t; 4.00 1::..=iG::::::F-~::::::j:'---J.----J.--..J------l
o
u
P=4460XO. 82 XO.1O
+(0.50-0.85 XO.IO)c = 3660+0 1415c
cents, wl1ere c is given in pounds. I
In Fig. 1 the relation between cost
and cement content is given for a price
of $2 a ton for aggregates and prices
of $1.10, $1.88 and $2.60 per barrel of
cement. In Fig. 2 the price of cement
was $1.88 per barrel, and the prices of
/ off, ) I
. Distributed bf ......u
rria Enginoorlng~' '
these units the cost equatIOn becomes:
P =4460XO. 82 XPI +(P2-0. 85p,)c.
If the price of aggregate is $2 per
ton or 10 cents per pound, and the price
of cement is $1.88 per barrel, or SO cents
pel" pound, the cost of materials for 1
cu.yd. of concrete is:
w
27X62.4
=D+---
1 (a c W)
r = 27X62.4 ga+~+~
1 (a c W)
-+-+-
.27X62.4 2.65 3.10 1.0
where
P = pla+p2c = p14460D-pl XO. 85c+p2c
= 4460Dpi +(P2--Q . 85p,)c
Thus the cost of the materials per
cubic yard of concrete varies directly
with the cement content for given unit
prices of aggregate and cement and for
a given consistency of concrete.
The cost of the concrete may also be
expressed in terms of the strength of the
concrete from the following relation:
S-A
S=A+Kc, or c = --,
K
so that P=4460D. PI + (P2_0.85PI)(S~A)
If the strength-giving qualities of the
materials are known, it is a simple mat-
ter to compute the cost for any given
strength of concrete. It follows that
fOI" concrete containing aggregates of a
given type and gradation and containing
a constant amount of water per cubic
yard of concrete, the cost per uni t of
concrete varies with the strength of the
concrete.
The following illustration will make
the significance of this fact more appar-
ent. Assume the use of aggregates re-
quiring 303 lb. of water per cubic yard of
concrete for the desired consistency. The
volume of water in the concrete then is
303/ (27 X 62.4). Or 18 per cent of the
volume of the fresh concrete. Since the
density, D, is the diffel"ence between
the volume of concrete and the volume
of water, D = 1.0 - 0.18, or 82 pel"
cent of the volume of the concrete. With
D = 1 (a +--.:_)
27X62.4 ~65 3.10
or a = 4460D ~ O. 85c
The value 27X62.4 represents the
weight of 1 cu.yd. of water, ga and gc
are specific gravities of aggregates and
cement respectively, and D is density,
or portion of solid materials in the con-
crete.
The cost of the materials is then:·
By Inge Lyse
Research Assistant Professor of Engineering
llIaterials. Fritz Engineering Laborator'1l~
Lehigh University. Bethlehem. Pa.
Visualiz~ngConcrete Economy
in· Terms of Strength
Straight-line relation between strength and cement-water ratio
permits direct measure of cost in terms of strength-Cost
.per unit of strength decreases with an increase in strength
I N PREVIOUS articles the writerhas shown that the strength of con-crete is determined by the concen-
tration of cement particles in the cement
paste (ENR. Nov. 5, 1931, p. 723), and
that the design and control of concrete
mixes are greatly simplified by this re-
lation (ENR, Feb. 18, 1932, p. 248).
Data given in these articles showothat,
for concretes containing aggregates of
a given type and gradation, the consist-
ency remains constant regardless of the
richness of the mix if the water con-
tent per unit of concrete remains con-
stant. Thus the straight-line relation-
ship between the strength and the
cement-water ratio of the concrete gen-
erates into a straight-line relation be-
tween strength and cement content,
expressed as follows:
S=A +B (c/w) =A + (B/w) c
= A + Kc where S is strength of
concrete, A and B are constants de-
pending upon the materials and condi-
tions of test, and c and ware amounts
of cement and water in a cubic yard of
fresh concrete. If the water content per
unit of concrete is held constant, B/w
becomes a new constant, K. Thus for
given materials and conditions of test
the strength of the concrete is propor-
tional to the cement content.
The total cost of the concrete is com-
posed of the cost of the materials plus
the cost of formwork, mixing, placing,
curing, etc. Since the costs of form-
work, mixing and placing are prac-
tically the same for concrete mixes of
different richness as long as their con-
sistencies are nearly the same, the
variation in the cost of concretes of
different richness is mainlv due to the
difference in the cost of th~ materials-
aggregates and cement. For 1 cu.yd. of
concrete the cost of the materials may be
expressed by: P = p,a + p2C where
1', and 1'2 are unit prices of aggregates
and cement, and a and c are aggregate
and cement contents per cubic yard of
concrete. Since the total volume of
properly placed concrete is the sum of
the absolute volumes of aggregates. ce-
ment and water, the volume of the
freshly placed concrete may be ex-
pressed by: V = Va + V c + V IV • The
absolute volumes are obtained bv divid-
ing the weight of the materials by their
specific weights, so that the above re-
lation becomes:
aggregate .\~~:e $1.-50, $2.00. and $2.50
per ton. .
In stl'uctural conCl'ete members, where
the total area is in compression, the
load-carrying capacity of the member is
directly proportional to the strength of
the concrete. The cost of materials for
1 cu.yd. of concl:ete per pound of
strength of the concrete is therefore a
direct measure of 'the economv of the
concrete ·mix. This ~ost per' unit of
strength is expressed by the .formula:
P 1 [ S-AJE=-=- 4460Dpl+(P2-0.85pi) --
S S K
Since for given materials and 'condi-
tions of construction all units on the
right of the equation are given, the cost
can be computed directly. The relation
between cost of concrete per uni t of
strength and the strength of the con-
crete is shown in Fig. 3 for given ma-
terials and conditions of construction.
It is seen that the cost per unit of
strength decreases rapidly with an in-
crease in strength of the concrete. Thus
for concrete members loaded directly in
compression the more economical mix is '
the one giving the higher strength.
The above relations are valuable aids
for selecting the most economical type
of aggregate and cement for use on a
given job. The relations also give in-
formation as to the feasibility of using
ordinary portland or high-early-strength
cement. Similar relations may also be
found for concrete subjected to flexural
stresses. Thus by means of the con-
stant water requirement for a given
workability and the straight-line 'rela~
-tion between strength and cen1ent-
water ratio. the most economical mix is
readily det~nllined for all types of plain
and reinforced-concrete members.
In addition to the direct effect of the
strength of the concrete on the eco'noiliy
of the mix, an additi'Onal important eco-p
nomic item is the reduction in size dof
the member. The bem~fit of the higher
strength concrete is -therefol'C' magni fied
by the accompanied reduction' of size- of
member for a given load. For columns
the saving in floor .space by the use of
high-strength concre~~. becomes very
important.
o
.....


Visualizing concrete economy in terms of strength, ENR, Vol. 109 (1932), p. 109, Reprint No. 31 (32-6.)

https://preserve.lehigh.edu/cgi/viewcontent.cgi?article=2138&context=engr-civil-environmental-fritz-lab-reports

Visualizing concrete economy in terms of strength, ENR, Vol. 109 (1932), p. 109, Reprint No. 31 (32-6.)

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