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Masters Theses
Student Research & Creative Works
1965
Comparison of measured and calculated stiffnesses
for lightweight normal concrete beams
Raman Patel
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Paper 5710.
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COMPARISON OF MEASURED AND CALCULATED STIFFNESSES
FOR LIGHTWEIGHT AND NORMAL CONCRETE BEAMS
BY
RAMAN A. PATEL
A
THESIS
submitted to the faculty of the
UNIVERS ITY OF MISSOURI AT ROLLA
in partial fulfillment of the requirement for the
Degree of
MASTER OF SCIENCE IN CIVIL ENGINEERING
Rolla, Missouri
1965
Approved by
ii
ABSTRACT
This stutiy is a presentation of the finCings involving
experimental measurements of stiffness (EI) for different
percentages of reinforcement steel in normal and lightweight
concrete beams.
The experimental values of stiffness are
compared with calculated values obtained by applying the
standard reinforced concrete theory.
The measured values of
stiffness ob·tainecl by ·the rotation method are compared with
·those obtained by the deflection me·thod.
Of the
~chirty
beams that were tested, fifteen were tested for normal concrete and the remaining fifteen for lightweight concrete.
Each of these beams were tested at different ages, ranging
from seven to twenty--eight days.
The test results indicate that the measured values of
stiffness decrease wi·th an increase in the bending moment.
For noncracked sections the measured values are higher than
the calculated values.
Best agreement between the theoreti-
cal and measured values of EI occurred shortly after the
cracked section was formed.
The best correlation between the
rotation and deflection method for determining stiffness
also occurred in this same region.
iii
ACKNOVV'LEDG.MENT
The author is especially grateful to Dr. Joseph H.
Senne, Jr., Chairman of the Civil Engineering Department,
for his valuable advice and encouragement throughout the
course of this investigation.
The author is also thankful to Professor James E.
Spooner, and the staff of the Civil Engineering Department
for their advice and assistance in preparing this research.
The author would also like to thank Mr. John D. Smith,
the mechanic of the Civil Engineering Department, for his
valuable help in this work.
The author also wishes to express his thanks to
Professor Robert F. Davidson, Chairman of the Engineering
Mechanics Departmen·t, and Professor Rodney A. Schaefer for
their assistance in the use of laboratory equipment.
The author is highly indebted to the St. Louis Slag
Products Company, Inc. for providing the Expanded Blast
Furnace Slag.
In addition, the author wishes to express sincere thanks
to the Computer Science Center for use of their facilities
in making computations.
Finally, the author wishes to express his kind appreciation to graduate students Mr.
and Mr.
v.
v.
R. Shah, Mr. Arvind Desai
T. Patel for helping in this research work.
iv
TABLE OF CONTENTS
Page
ABSTRACT .
.
.
ii
... ... .. ... ..
ILLUSTRATIONS
... ... .
.
ACKNOWLEDGMENT
LIST OF
iii
v
LIST OF TABLES .
• viii
NOMENCLATURE . .
ix
I.
II.
III.
INTRODUCTION .
1
REVIEW OF LITERATURE .
3
EQUIPMENT AND HATERIALS
9
Laboratory Equipment • • • •
Materials
. . • . . . .
Preparation of Specimens .
A.
B.
c.
IV.
TEST PROCEDURE .
c.
D.
v.
VI.
VII.
TEST RESULTS
M~D
13
.
16
Rotation Measurements
• . • •
Deflection Measurements
.
Compression Test for Concrete
Tension Test for Reinforcement
A.
B.
9
11
. • .
. . • • .
. • •
. .
DISCUSSION
33
CONCLUSIONS
RECOMHENDATION .
16
24
26
30
60
...
..... . .
62
BIBLIOGRAPHY .
63
APPENDIX .
65
VITA .
.
.
.
. . . . . . . . . . . . . . . . . . . .
69
v
LIST OF ILLUSTRATIONS
Figure
Page
1
Sieve analysis of fine aggregates . .
12
2
Sieve analysis of coarse aggregates .
12
3
Test specimen loading configuration •
4
Rotation indicator
5
Beam under actual test
6
Crack distribution in normal concrete beams at
28 days .
7
.
.
•
..
17
. . . . . . . . . . .
18
.
19
.
.
.
.
•
•
•
.
.
•
.
•
20
Load values for crack distribution in normal
concrete beams at 28 days . . . .
. . .
21
Crack distribution in lightweight concrete
beams at 28 days
. . . . . .
. . .
22
Load values for crack distribution in lightweight
concrete beams a·t 28 days . . . . . . . . . •
23
Load, Shear, and Bending moment diagrams for
EI computations . . . . . . • . . • . . • . .
25
11
Stress-strain diagrams for lightweight concrete .
27
12
Stress-strain diagrams for normal concrete.
28
13
Ultimate compressive strength versus age
14
Stress-strain curve for steel reinforcement •
15
Measured and calculated values of EI (E by secant
modulus) versus M for normal concrete beams at
8
9
10
17
31
34
Measured and calculated values of EI (E by formula) versus M for normal concrete beams at 7 days.
35
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Measured and calculated values of EI (E by
lOOOf') versus M for normal concrete beams at
c
7 days
18
29
.
7 days
16
.. . .
•
•
.
.
•
.
•
.
•
•
.
•
.
•
.
.
•
.
•
Measured and calculated values of EI (E by secant
modulus) versus M for lightweight concrete beams
at 7 days . . . . . . . . . . . . . • . • . • • .
36
37
vi
LIST OF ILLUSTRATIONS (Continued)
Figure
19
Page
Measured and calculated values of EI (E by formula) versus M for lightweight concrete beams at
7 days .
20
21
22
.
.
.
.
.
.
•
.
.
•
.
•
.
.
.
•
.
•
38
Measured and calculated values of EI (E by
secant modulus) versus M for normal concrete
beams at 21 days • . . • . • . . • • . . • .
39
Measured and calculated values of EI (E by
formula) versus M for normal concrete beams at
21 days
. . . . . . . . . . . . . . . . . .
40
Measured and calculated values of EI (E by
lOOOf~) versus M for normal concrete beams at
21 days
. . . . . . . . . . . . . . . . . . .
41
.
23
Measured and calculated values of EI (E by
secant modulus) versus M for normal concrete beams
42
at 28 days . . . . . . . . . . . . . . . . .
24
Measured and calculated values of EI (E by formula) versus M for normal concrete beams at 28 days
25
Measured and calculated values of EI (E by
lOOOf~) versus M for normal concrete beams at
28 days
26
27
28
29
30
43
................... .
44
Measured and calculated values of EI (E by
secant modulus) versus M for lightweight concrete
beams at 28 days . . . . • . • . • • . . • • . •
45
Measured and calculated values of EI (E by
formula) versus M for lightweight concrete beams
at 2 8 days . . . .
. . . . . . . . . . .
46
comparison between the measured values of EI and
the calculated values of EI by gross section
method for lightweight concrete at 28 days .
48
Comparison between the measured values of EI anu.
the calculated values of EI by gross section
method for normal concrete at 28 days
. . .
49
comparison of measured values of EI by the rotation and the deflection method versus M for
lightweight concrete beam, at 28 days with
3-#3 bars . . . . . . . . . . . . . . . . . .
52
vii
LIST OF ILLUSTRATIONS (Continued)
Figure
31
32
33
34
Page
Comparison of measured values of EI by the rotation and the deflection method versus M for
normal concrete beam, at 28 days with 2-#3 bars.
53
Comparison between the lower measured values and
the theoretical values of EI, considering E by
Secant modulus, Formula method and lOOOf' for
normal concrete beams, at 28 days with 2~#3 and
4-#3 bars
. . • •
• . . . . • • . • • •
54
Comparison between the lower measured values and
the theoretical values of EI, considering E by
Secant modulus, Formula method and lOOOf' for
lightweight concrete beams, at 28 days with
2-#3 and 4-#3 bars.. . . • . .
. . . • • . .
55
Typical cross-section of beam
65
viii
LIST OF TABLES
Table
1
Page
Various recommended methods of determining moment
of inertia . . . . . . . . . . . . . . . . . . . .
2
Concrete mix content
3
Values of modulus of elasticity by different
methods for both normal and lightweight concrete
4
13
0
32
Calculated values of EI by different methods for
normal concrete beams
o
5
6
o
o
o
o
o
o
o
o
o
o
0
58
Calculated values of EI by different methods for
lightweight concrete beams
59
6
Properties of normal concrete beams
67
7
Properties of lightweight concrete beams
68
o
o
o
o
o
o
o
o
ix
NOMENCLATURE
The symbols used are defined where they first occur in
the text and are listed here in alphabetical order for convenience.
A
s
= effective cross section area of reinforcing steel,
2
(in. ) .
b
=
width of test specimen,
d
=
distance from extreme compressive fiber to centroid
(in.).
of tensile reinforcement,
(in.).
c
=
modulus of elasticity of concrete,
s
=
modulus of elasticity of reinforcing steel,
EI
=
stiffness of beam,
f'
=
28-day compressive strength of concrete,
I
=
moment of inertia of a section about the neutral
E
E
c
axis,
k
=
(psi).
(psi).
2
(lb-in. ).
(psi) .
4
(in. ) •
ratio of distance from extreme compressive fiber to
neutral axis to depth d.
L
M
=
=
N.A.=
n
=
span of beam,
(in.).
bending moment due to load P,
(lb-in.). or (kips-in.).
Neutral axis of beam.
ratio of modulus of elasticity (E ) to that of cons
crete (Ec) .
p
=
concentrated load on test specimen,
t
=
depth of test specimen,
w
=
weight of the concrete at time of test,
cp
=
average unit angle change, curvature,
liB
=
deflection under load (at point B),
(lb).
(in.).
(pcf).
(radians/in.).
(in.).
1
I.
INTRODUCTION
The solutions of statically indeterminate structures,
both of normal and lightweight concrete, require the formulation of one or more equations in addition to those of
statics.
These additional equations are based on the be-
havior of the structure under the given loading and support
conditions, and contain the elastic properties of the frame,
or more specifically, the flexural rigidity EI where E is
the modulus of elasticity and I the moment of inertia.
The
term EI, therefore, is a measure of the stiffness, and always appears in equations for determining the elastic deflection of members.
Since E is usually a constant for a
given frame, changes in stiffness are due primarily to variations in I which are influenced by changes in the cross
section of the member (1).
The use of E assumes that
analyses obeying Hook's Law are used.
Reinforced concrete
subject to bending action or compression, in a properly
designed member, should possess adequate stiffness to prevent such deflection or deformation as might impair the
strength and efficiency of the structure or produce cracks
in finishes and in superimposed partitions.
In analyzing
for moments in a frame, the relative stiffness of members
plays a very important role, therefore it is advisable to
be sure that appropriate stiffness values are used in
frame analysis.
Lightweight concrete has been successfully employed in
2
both precast and cast-in-place concrete for many years.
In
recent years more designers are taking advantage of the reduced weight which permits savings in materials for foundations and load carrying members, the superior characteristics
in fire resistance, thermal insulation, and accoustical
properties (2).
This growing interest in lightweight aggre-
gate concrete for structural purposes has created a demand
for reliable design data beyond a knowledge of its weight
and compressive strength (3).
With this in mind, the pres-
ent investigation was undertaken to compare the performance
of measured and calculated stiffnesses for Reinforced Concrete Beams, both lightweight and normal, so that engineers
familiar with standardized structural design procedures
woulc have representative values for use with lightweight
concrete.
3
II.
REVIEW OF LITERATURE
Reinforced concrete subject to bending action or to
compression in a building should possess adequate stiffness
to prevent excessive deflections.
For a given material the
stiffness {EI) is primarily a function of the moment of
inertia {I).
Methods of computing the moment of inertia
suggested in most of the published materials on reinforced
concrete are rather varied.
Reinforced concrete is usually
treated as a homogeneous material by designers when computing moments of inertia.
The ASCE-ACI Joint Committee Report (10)
suggests that
the moment of inertia of flexural members, for purposes of
computing the relative stiffness, may be that of the gross
cross sectional area of the member.
For columns or other
compression members the transformed area of the steel reinforcement should be included.
The CRSI Handbook (9)
recommends that the moments of
inertia can be obtained in one of the following two ways:
(a) For preliminary computations, use the moment of
inertia of the gross outline of the concrete section, omitting the reinforcing steel (MethOQ A, Table 1) •
If the same
method is applied to all the members, this is fairly accurate.
However, in an ordinary continuous tee-beam, there 1s
a tee-section near midspan and a rectangular section at
either end, and a point of inflection which moves under
different loading conditions, so that a high degree of
4
precision is not obtainable.
(b) Some designers, feeling that a heavily reinforced
beam should have more stiffness than one lightly reinforced,
take the moment of inertia of the transformed area
(Method B) .
There is a question as to whether this method
is more precise than A and C.
It can be used only after
considerable preliminary design to establish all data needed.
The British Standard Code of Practice for Reinforced
Concrete (11), in Article 506, mentions that for the purpose
of estimating the stiffness
of
reinforced concrete members,
the moment of inertia may be calculated using the appropriate
modular ratio.
In the case of a beam, the breadth of a com-
pression slab shall be chosen in accordance with Clause
601 (g) , which refers to the allowable flange width.
The
method employed in estimating moments of inertia shall be
the same for all members considered in any one calculation.
The Explanatory Handbook on the B.
s.
Code of Practice
for Reinforced Concrete (11), in Clause 306, recommends that
the moment of inertia, for the purpose of calculating bending
moments in monolithic structures, can be estimated by considering one of the following:
(a) The entire concrete section, ignoring the reinforcement.
(b) The entire section including the reinforcement.
(c) The compression area of the concrete section, combined with the reinforcement on the basis of the
modular ratio.
5
Regardless of the method adopted for the beams, the same
method must be used for the columns.
Both the ACI Code and the Joint Committee Report recommend that in computing the value of I for relative stiffness
of beams, the reinforcement may be neglected; but allowance
shall be made for the effect of the flange in T-shaped sections (14).
From the above discussion there seems to be a general
lack of agreement concerning the best method for determining
the moment of inertia and, further, that one may well take
his choice as to how he calculates the stiffness of various
members of a frame, depending on whose recommendations he
follows (see Table 1).
Thus it becomes necessary for the
designer to know the limitations of the various methods for
the handling of moments of inertia and how the calculated
values compare to the actual values.
There are several methods for calculating Young's
modulus of homogeneous elastic bodies which have been applied, rightly or wrongly, to concrete.
Among them are the
static, flexural resonance and pulse velocity methods.
For
predicting the deflection of structural members, no particular value of Young's modulus is adequate since the deflection is a function of the duration as well as the magnitude
of the load (15).
The modulus of elasticity of the concrete must be determined from the average stress-strain ratio of a relatively
large concrete mass since concrete is a macroscopic
6
heterogeneous mass composed of rather large aggregate particles embedded in a cement paste matrix (16).
Table 1 - Various recommended methods of
determining moment of inertia
Method A
Gross section
Source
Beam
Method B
Method C
Gross sec·tion
with transformed
area of steel
included.
Beam
Col.
Cracked section
with transformed
area of steel
included.
Beam
Col.
British
Standard
Code
X
X
or
X
X
CRSI Design
Handbook
X
X
or
X
X
Taylor,
Thompson
and Smulski
X
X
Joint
Co:rmnittee
Report
X
ACI Code
X
or
X
Col.
X
X
The static modulus of elasticity of both normal and
lightweight structural concrete may be approximately determined by the empirical formula:
E
c
=
2
33(W) 3 /
If'c
Where: Ec = modulus of elasticity of concrete,
(psi).
w = weight of the concrete at time of test, (pcf).
f' = 28-day compressive strength of concrete,(psi).
c
7
The formula is in excellent agreement with other recognized
empirical formulas for normal weight concrete.
The value of the elastic modulus is more depencent on
the weight of the concrete and the method of test used to
determine it than on the compressive strength of the concrete.
The reliability of the proposed formula appears to
be as good as recognized and acceptable empirical formulas
for normal weight concretes (12).
Data pertaining to the elastic properties of the lightweight aggregate concretes are much less extensive than
those covering compressive strengths.
The latter property
seems to have been given much greater attention than the
modulus of elasticity; and yet from the standpoint of deflection, the elastic modulus is most important.
Although
this modulus of lightweight concrete is somewhat lower than
for conventional concrete, the mechanics of designing most
structural members is affected little if any by the higher
value of the modular ratio.
Deflection, however, is a
problem which must be considered (13).
In 1964 R. H. Patel (7) performed a series of tests to
determine the stiffness
o~
lightweight concrete beams.
In
these tests he found that the stiffness obtained from static
flexure tests was less than that obtained using methods A, B
or
c.
His tests were conducted using short beams and load
deflections measurements only, thus limiting the range of
measurements due to the early formation of shear cracks.
8
By using longer beams and measuring rotation in addition to deflection the author has been able to considerably
extend the range of stiffness measurements.
9
III.
A.
EQUIP~·1ENT
AND MATERIALS
Laboratory Equipment
Two types of electrical sieve shakers were used for
screening the coarse and fine aggregate.
The larger shaker, a product of Gibson Screen Company,
Merser, Pennsylvania, has a set of four woven wire screens,
conforming to the ASTM specifications for sieves.
The wire
mesh screens of 1/2", 3/8", #4 and #8, were used for grading
the normal and lightweight coarse aggregates prior to mixing
of the concrete.
The small Rotap shaker, a product of W.
Cleveland, Ohio, which holds six U.
s.
s.
Tyler Company,
standard laboratory
sieves at one time, was used to determine the fine aggregate
gradation.
Different sieve sizes were used, depending on
the type of aggregate.
The concrete mixer used was a stationary, nontilting,
electrically-operated mixer, having a capacity of three cubic
feet.
It is manufactured by Lancaster Iron Works Inc.,
Lancaster, Pennsylvania.
The interior surface was kept very
clean, dry and free from any foreign materials before use.
Two types of testing machines were used.
The first,
a Riehle machine, used for compression test of cylinders,
had two ranges of 60,000 and 300,000 pounds, and was hydraulically operated.
A standard compressometer dial appa-
ratus was used to record the deformation of the concrete
cylinders.
The second, used to determine the tensile
10
strength of reinforcing steel, was a Southwark Compression
and Tensile machine having an attached stress-strain recorder.
This machine had a range of 0 to 20,000 pounds, and was
hydraulically operated.
Both of the above testing machines
are located in the Material Testing Laboratory in the
Engineering Mechanics Department.
A Tinius Olsen Hachine was used for testing beams for
flexural strength.
It is mechanically operated and has a
load capacity ranging from 0 to 10,000 pounds, graduated in
10 pound increments.
It is located in the Structural
Laboratory of Civil Engineering Department.
It is manu-
factured by Tinius Testing Machine Company, Willow Grove,
Pennsylvania.
Two types of forms were used, one for casting beams
and ·the other for cylinders.
The forms were made from 3/4
inch plywood and had inside dimensions of 3.5 x 4.5 x 36.0
inches.
In all cases the depth of the beams were 4.5
inches.
They were kept throughly clean and well oiled be-
fore pouring of the concrete.
The cylinders were stanaard
6 x 12 inches, made of waxed cardboard with a smooth thin
metal bottom.
The air entrainment me-ter consisted of a portable
device made from alkali-resistant lightweight alloy.
It is
known as the Acme Entrained Air Meter and is manufactured
by E.
parts:
w.
Ziwnennan, Chicago, Illinois.
It has three main
upper assembly, bowl and calibration cylinder.
11
B.
Materials
Red Ring 0 Portland Cement, manufactured by the Missouri
Portland Cement Company, was used throughout the experiment.
The fine a9gregate used for the normal concrete was a
river-run sand from the bank of the Meramec River near
Pacific, Missouri, containing mostly chert and quartz.
A
sieve analysis was made using the Rotap machine in accordance
with the "Standard method of test for sieve analysis for fine
and coarse aggregate," ASTM Designation: C 136-61 (8C).
Its
specific gravity was approximate 2.46, the absorption 1.6 per
cent, and fineness modulus 2.83.
The other type of fine
aggregate was an expanded slag, produced and supplied by the
St. Louis Slag Product Co., St. Louis, Missouri.
All parti-
cles were angular and had a porous surface texture.
The
sieve analysis was made in accordance with the standard test
for sieve analysis, ASTM Designation:
C 136-61 (8C).
Both
of the above results are shown in Figure 1.
The coarse aggregates used for normal concrete were
creek grade aggregates from the Big Piney Sand Company,
Waynesville, Missouri, and expanded slag of 3/8 inch maximum
size for lightweight concrete which was obtained from
St. Louis, Missouri.
The standard method of test for "Sieve
analysis of coarse aggregate," ASTM Designation:
C 136-61
(8C), was followed for both and results shown in Figure 2.
The results indicate
the
coarse aggregate meets the grada-
tion requirements of ASTM Designation:
C 33-61 (8B), for
100-
/
80
80
I
I
tTl
I
I
·r-f
I
60
+J
..c:
Il
I
j
I
Q)
i I
i
'!
I
f
;·
~
~
.Q
I
I
I
.
I
_,
60
I
J
tTl
______
II
I
I
I
~
I
·r-1
Ul
Ul
m
I
0...
I
40
Q)
ro
+J
r--- ,I
i
I
!;'
v
: ///
I
0
~
Q)
. I
20
:I
I
1
:--Fine Expanded
I
I
0
0 .___ ---- - -------- ~ -------
#100 #50
~
Q)
0...
i-
/ .
40
tTl
Natural Sand
20
#30
#16
#8
#4
I
_!_ ___ ----
I
I
I
3/8"
/
/)/\
Normal
:
.k~ --s~ag
L~
#8
I
_____ ..._ ____ . [_______ j __
#4
J
3/8" 1/2" 3/4"
Standard square mesh sieves
Standard square mesh sieves
Figure 1 - Sieve analysis of
fine aggregates
Figure 2 - Sieve analysis of
coarse aggregates
13
gravel aggregate, and ASTM Designation:
C 330-60 (8G), for
lightweight aggregate.
The water used in the experiments was unsoftened tap
water from the City of Rolla water distribution system.
The reinforcing steel used was a 3/8 inch intermediate
grade steel, with a yield strength of approximately 56,810
psi, and with a cross sectional area of 0.11 square inches.
C.
Preparation of Specimens
The program included two type of mixes, one for light-
weight concrete and the other for normal concrete, both having a slump of approximately 3 inches.
The compressive
strength of the lightweight and normal concrete was 6,000
psi and 5,000 psi at 28 days, respectively.
The normal con-
crete was designed using the Portland Cement Association's
pamphlet "Design and control of concrete mixtures" (4), and
the lightweight concrete mix was directly taken from
u.
M. R.
Research on "Stiffness of Lightweight reinforced concrete
beams" (7).
The amount of cement, fine aggregate, coarse
aggregate and water used for each batch of concrete mix is
shown in Table 2.
Table 2 - Concrete mix content
Weight in pounds
Ingredient
Normal concrete
Cement
Lightweight concrete
63.0
120.0
Fine aggregate
153.1
145.4
coarse aggregate
173.9
36.0
37.4
60.6
Water
14
Two sets of beams, size 3.5 x 4.5 x 36 inches, were
cast in accordance with the nstandard method of making and
curing concrete compression and flexure test specimens in
the laboratory", ASTM Designation:
C 192-59 (SA).
One set
was for lightweight concrete and the other for normal concrete.
Each batch produced about three cubic feet of
concrete which was sufficient to make five beams and six
standard cylinders.
All of the materials, cement, fine aggregate, coarse
aggregate and water, were weighed on Toledo Scales, accurate to one-fourth of a pound.
For preparing the normal
concrete, air-dry fine aggregate, cement and two-thirds of
the water were first placed in the mixer, after which the
coarse aggregate was added.
This procedure was used for
minimizing the initial bulking of the mixture.
The remain-
ing one-third of the water was added and the concrete mixed
for about two and one-half minutes.
For the lightweight concrete mix (3), the aggregate and
about two-thirds of the mixing water were mixed for about
two minutes; the cement and remaining water were then added
and the mixing continued for three minutes.
s~x
Five beams anG
standard cylinders were cast from each mix.
Slump tests were made on each batch immediately after
mixing and before placing the concrete in the forms.
The
slumps test, conducted in accordance with ASTM Designation:
c 143-58 (8D), gave approximately a three-inch slump in all
cases.
15
Air content tests were made on each batch of concrete.
Air content tests were conducted in accordance with "Standard
method of test for air content of freshly mixed concrete by
the pressure method," ASTM Designation:
c
231-60 (8F).
The unit weight of the fresh concrete was determined
for each batch of concrete.
The test was conducted in ac-
cordance with "Standard method of test for weight per cubic
foot of concrete," ASTM Designation:
C 138-44 (8H).
The beam forms were throughly oiled and the reinforcing
bars were properly positioned with steel rebar chairs.
The
reinforcing bar was clean and free from rust and grease.
The concrete was placed in the forms in two layers, rodded
fifty times per square foot per layer and the top surface
trowelled smooth.
The cylinders were filled in three layers,
each layer being rodded twenty-five times, struck off and
then placed in the curing room at 100 per cent relative
humidity and an average temperature of 72 degrees F.
The
cylinder forms were removed after curing had progressed for
twenty-four hours.
The beams were kept for twenty-four hours outside the
curing room to prevent swelling of the wooden forms due to
moisture.
A period of twenty-four hours was considered to
be the earliest time that the forms could be removed without
damaging the beams.
curing room.
They were marked and placed in the
16
IV.
A.
TEST PROCEDURE
Rotation Measurement
The beams were tested in flexure in accordance with
ASTM Designation:
C 78-59 (8J).
A 10,000 pound Tinius
Olsen testing machine was used to apply a two-point loading
to the beams as shown in Figure 3.
a moderate rate.
The load was applied at
Aluminum plates of 1.5 x 3.5 x 5/16 inch
dimensions, were grouted at the points of load and support
using plaster of paris.
The average unit angle change,
~,
or curvature, near
the center of the beam was measured by a rotation indicator
device (6).
This device consisted of four parallel steel
lever arms of 1.5 x 1.5 x 3/16 inch angles, placed in the
vertical plane symmetrically on either side of the center
line of the beam.
The relative rotation of these arms was
measured by a total of four dial gages, with .0001 inch
divisions, mounted on both faces of the beam.
The two dial
gages mounted above and below the neutral axis, are shown 1n
Figures 4 and 5.
An initial load of about 500 pounds was
placed on the specimen and then released in order to set the
zero reading on the dial gages.
The load was applied gradu-
ally and the corresponding rotation at center noted.
The
crack-propagations were observed with the aid of a magnifying glass and marked with a pencil as shown in Figures 6
through 9.
The curvature,
(~),
of the beam between the arms was
17
2P (Load Applied By Movable
Head of Testing Machine)
~---1-11"
'
7"--1
1.5"
15.~
r- L~o_a_d s_p_r_e_a_d_e_r_s----~------~------~----------------Loading Beam
__
u
Test slpecimen
.~'r-----1-1~
5.5~
l
I
1.5"
~-------16.5'-'------~)~1
Gage
I
~---------18.0-"------~>~,
II--'"'~------
3 6 • 0 II
---'--------------.~to--1
I
I
(syro.)
Figure 3 - Test specimen loading configuration
18
Test Specimen
Test Specimen Rotation
Indicator Angles
~--------
Dial Gages
Section B-B
Test
Specimen
i-------4)-
I~--+-· X
A
Bolts
---l
1- ·1----(
I
..----tch
l
I
-· L
!
I
--·-·1-·-. __I
L
B""----t
y
I
q
~ f==;=::f p-
l
I
I
·-·--"'-
··"""{--
i
I
., A
Bolts
Front View
Section A-A
Figure 4 - Rotation indicator
19
Figure 5 - Beam under actual test .
20
N04 - No reinforcement.
Nl4
-
0.919 % reinforcement , (1-#3 bar} .
N24
-
1.876 % reinforcement, (2 - #3 bars} .
N34
-
3.052 % reinforcement, (3-#3 bars} .
N44
-
3.897 % reinforcement , (4-#3 bars} .
Figure 6 - Crack distribution in normal concrete beams
at 28 days.
21
,------ --- ------i
\
0
I
!
.---.
\
fl.
L _ ____________ _____ } - - -
•
N14 - 0.919 % reinforcement, (1-#3 bar).
N24 - 1.876 % reinforcement,
•
....
o
t
2500
3000
3500
4000
pounds
pounds
pounds
pounds
o
+
x
"
4500
5000
6000
7000
pounds
pounds
pounds
pounds
N34 - 3.052 % reinforcement,
(3-#3 bars).
N44 - 3.897 % reinforcement,
(4-#3 bars).
o
+
x
Scale:
(2-#3 bars).
4500 pounds
5000 pounds
6000 pounds
7000 pounds
• 7500 pounds
~ 8000 pounds
o
1" = 8"
Figure 7 - Load values for crack distribution in
normal concrete beams at 28 days.
22
I r
I
I
3
LOS - No reinforcement.
LlS
-
0 . 892 % reinforcement, (1- #3 bar) .
L25
-
1 . 872 % r einfo r cement ,
L35
-
2 . 869 % reinforcement, (3-#3 bars ) .
L45
(2-#3 bar s) .
3.840 % reinforcement, (4-#3 bars) .
Figure 8 - Crack distribution in lightweight concrete
beams at 28 days .
23
,.--I
I
I!
L-
-
Ll5
f
g
~
-+
Cl
0.892 % reinforcement,
'
(1-#3 bar).
\
L25 - 1.872 % reinforcement, (2-#3 bars).
o
+
a
2500 pounds
3500 pounds
4000 pounds
A
•
5000 pounds
6000 pounds
L35 - 2.869 % reinforcement,
(3-#3 bars).
~I
L45 - 3.840 % reinforcement, (4-#3 bars).
•
o
o
•
Scale:
2500
3500
4500
5000
pounds
pounds
pounds
pounds
o
A
+
x
6000
7000
8000
9000
pounds
pounds
pounds
pounds
1" = 8"
Figure 9 - Load values for crack distribution in
lightweight concrete beams at 28 days.
24
computed by:
Where:
R1 = the change in the bottom dial reading,
R2 = the change in the top dial reading,
X
= horizontal distance between vertical lever
arms,
Y
= vertical distance between dial gages.
Knowing the load and the corresponding curvature and
bending moment, the flexure stiffness (EI) was computed by
the following formula:
EI
= M-4>
Where:
M
=bending moment due to load P, (lb-in.),
4>
= curvature, (radians/in.),
EI =stiffness of the beams,
B.
2
(lb-in. ).
Deflection
Two deflecting dial gages, graduated in 0.001 inch
divisions, were installed symmetrically on both sides under
the load applied on the beam as shown in Figures 3 and 5.
In
order to give a smooth contact for the dial gages, plaster of
paris was applied to the concrete surface at the points of
contact.
A load of 500 pounds was placed on the specimen
and then released in order to set the zero reading of the
dial gages.
The load was applied gradually and corresponding
deflections under the load were noted.
tinued until failure occurred.
The test was con-
25
~
2P (Load)
0
f-
(a)
Load diagram.
(b)
Shear diagram.
(c)
Moment diagram.
c
B
X
l ..
L/3 _J_L/3 --........;...--L/3
IP
1
p
p,_l_ _ _ _ __J
Formula:
Where:
Figure 10
t:J.x
(3LX - 3X 2
= PL/3
6EI
2
L /9)
t:J.B
2
PL
= 18EI (L2 - L /3
L
!:J.B
= 162 EI
6.
= deflection at distance X from support,
X
5PL3
2
/~)
I
I
.
p
= concentrated load on test specimen, (lb)
L
= span of beam, (in.) ,
t:J.B
= deflection at point B, (in.) ,
EI
2
= stiffness of beam, (lb-in. >.
-
1
Load, Shear, and Bending moment diagrams for EI
computations.
26
Knowing the load, the corresponding deflection and the
bending moment the flexural stiffness (EI) was computed by
a standard deflection formula,
EI
=
5 P (L) 3
162
Where:
P
~B
=
concentrated load on test specimen,
L = span of the beam,
~B
C.
=
(lb),
(in.),
deflection at point B,
(in.).
Compression test for concrete
The cylinders of both the lightweight and the normal
concrete were capped with a sulphur compound and tested at
ages of 7, 14, 21, and 28 days, in accordance with the
Standard method of tests for compression strength of molded
concrete cylinders, ASTM Designation:
C 39-61 (8E).
They
were tested in a hydraulically operated Reihle Universal
Testing Machine.
A standard compressometer was used to ob-
tain the stress-strain diagrams as shown in Figure 11 for
lightweight concrete and in Figure 12 for normal concrete.
The static Young's Modulus of Elasticity for each cylinder
and an average static E for each batch, at the ages tested,
was computed.
The static E determined was the secant modulus
at 45 percent of the ultimate compressive strength.
The ul-
timate compressive strength with respect to age is plotted
in Figure 13.
The static modulus of elasticity of both normal and
27
---t----+-----!---1-~·----i
: : : t--l:
.
I
/I
/
I~
!I
----~~ -~-___..--+--LI-tL---~~-------;1
4.0
!
.
U)
U)
ClJ
H
+J
3.0
U)
ClJ
::>
·rl
U)
U)
ClJ
H
0.
8
0
u
2.0
v
7 days
e 14 days
a 28 days
1.0
0
0
1.2
.8
Unit strain, thousandths of
•4
. 1. 6
in./~n.
Figure 11 - Stress-strain diagram for lightweight concrete
2.0
28
5.3r:--------r--------r--------r-------~--------1
1.
1
1
!
5.0~---------~------~~------~--------~--------~
.
·.-1
(f)
~
.
(f)
(f)
Q)
H
3.0
-!-)
(f)
Q)
:>
·r-1
(f)
(f)
Q)
H
~
~
0
u
2.0
()
lt.
0
•
7 days
14 days
21 days
28 d.ays
I
I
.I
I!
1.0
0~------~--------~--------~--------~--------~
1.2
2.0
.8
.4
0
Unit strain, thousandths of in/in.Figure 12 - stress-strain diagrams for norma.l concrete.
29
7.0
·r-i
Ul
I
I
.
-L---
I
5.0
~~-~6
I ----t~---l
~
I , ..
4.0
i
..1'
',:
I
~----~------~--~~--~~------4-------~----~
l /
I ",
.'
I
~----j----J..----
3.0
'1
l/
I,'
I'
2. 0
_/f
1
t
1
I
l
I
I
l
I
I
1
r--
1.0
t
I
I
I
I
1-----+--.JH--~-'
\
I
I
:
I / l
I .
,
I
r·-----+----·
\
I
Normal concrete
Lightweight concrete
'i
!
I
OL-----~------~----~------~-----L------J
0
5
10
20
15
Age in days
25
30
Figure 13 - Ultimate compressive. strength vers.us age
30
lightweight concrete was determined by the empirical formula ( 17)
E
c
= 33
X
w3 / 2
If'.
c
X
Also the static modulus of elasticity for normal concrete
was calculated by the formula (1)
E
c
= lOOOf
I
c
•
The results are shown in Table 3.
D.
Tension test for reinforcement
Tension tests on six test specimens of reinforcing
deformed bars were made.
The stress-strain diagram was
drawn to compute Young's Modulus of Elasticity for the reinforcing steel as shown in Figure 13.
stress was found to be 56,810 psi.
An average yield
An average value of
static modulus of elasticity for the reinforcement was
found to be 28.7 x 10
6
psi.
31
80
~----
. ·-. --· -·-···· ·----:---------r-------r
.
I
.
.
I
I
I
II
l,
'
70 ;-------+-··------"-
60
-d----....----->------:1-----t----~-----+----------,----<
.
i
50
Ul
Ul
I
i
: ____ -!I___ :______ •
-- ----- ____;,,_. _______--.,..-..
~ 40
t---~---
j
1
I
___ i
_1__1
'
'
:
:
:
I
;
-----+----~----4--
'
..jJ
Ul
Q)
....-i
·r-1
I
I
Ul
I
!=:
~
30
1_ _ _.......__ __.__ _- - i - - -
"'---1
---~--
I
I
I
l
20
.
i
~--4------r---~-----~-----~------~---------------~---~
!
I
I
i
1
i
t-l
10
I.
6
=
28.7 X 10 psi
s
f
= 56,810 psi
y
E
T
I
I
!I
1
0
0
.01
.02
.03
.04
.05
.06 . . 07
.08
.09
.1
Unit strain, in./in.
Figure 14 - Stress-strain curve for steel reinforcement.
Table 3 - Values of modulus of elasticity by different methods
for both normal and lightweight concrete.
Type
of
concrete
Ages
in
days
Weight
in pcf
Air content
per cent/volume
Secant
modulus
@ .45f'
-6 c
(10 psi)
By
empirical
formula
( 10-6 psi)
By
lOOOf'
c
(lo-6 psi)
Normal
7
144
1.375
4.594
3.134
3.022
Normal
14
144
1.425
5.687
3.548
4.100
Normal
21
144
1.125
5.843
3.927
4.740
Normal
28
144
1.500
5.960
3.992
4.900
Lightweight
7
118
6.125
3.617
2.991
Lightweight
14
118
6.000
4.091
3.137
Lightweight
28
118
6.500
4.107
3.355
w
N
33
V.
TEST RESULTS AND DISCUSSION
Figure 13, illustrates graphically, the gain in ultimate compressive strength with respect to time in days,
for both the normal as well as lightweight concrete.
The
increase in the ultimate compressive strength for the first
few days was found to be greater than that for the later
days; the slope of the lightweight concrete being steeper
than that of the normal concrete for the first few days.
Figures 15 through 27, illustrate graphs of the EI in
lb-in.
2
versus moment in kips-in. for the twenty five beams
tested; the curves represent the measured values of EI.
The upper values of EI are compared with the values of EI
obtained using the gross section with transformed area of
steel (Method B) •
The calculated values of E were obtained
in three different ways,
(a) Secant modulus (considering
experimental stress-strain curves of concrete) at 0.45f',
c
(b) Formula method and (c)
lOOOf~.
The lower values of EI
have been compared with calculated values of EI for a cracked
section with transformed area of steel (Method C).
In this
case, the different values of E were obtained by using
methods (a),
(b) and (c) for the normal, and (a) and (b)
only for the lightweight concrete.
The four curves in each
of the graphs, obtained by plotting EI versus M, represent
the varying percentages of steel for same age of concrete.
Figures 30 and 31, show typical examples of a comparison
between the measured values of EI by the rotation and the
34
240
200
j=--&-
l--·-1
t---A-180
- - - - - - - - N.A.
~ -0-
~
Method B.
160 ~
0
e'
N
.
c
e
•
""
1-#3 bar.
2-#3 bars.
3- #3 bars.
0
4-#3 bars.
Value of EI from
rotation measurement.
----Calculated value·of EI.
- - The first visible crack.
·r-i
I
..a
,...;
H
IJ;:I
i
100 +-1
II
I
\
80 .-1
I
60
~fl
.
-6---
40
--e---- -----
---
~----
~
'
--A - - - - - -- - -
----------
~
L-o----
20
--
D
.
Method
--o----------N.A.
c.
0 ~------~----~~----~------~-----~------~---~
60
50
0
20
30
40
10
Moment, kips-in.
Figure 15 - Measured and calculated values of EI {E by secant
modulus) versus M for normal concrete beams at
7 days.
35
180
160
---·
140
.
N
l::
- - -f---
Method
fi.
120
·r-1
I
..0
e
•
1-#3 bar.
2-#3 bars.
A
3-#3 bars.
· o 4-#3 bars.
Value of EI from
rotation measurement.
---calculated value of EI •
~ The first visible crack.
r-1
1.0
lo
r-1
100
X
..
80
60
-·---
-€)?--- ----
40
------.-----
.__O __
20
N.A.
Method c.
0
-----
0
10
20
40
30
Moment, kips-in.
.
-·50
-------·
60
Figure 16 - Measured and calculated values of EI (E by formula)
versus M for normal concrete beams .at 7 day~.
(
36
• 1-#3 bar.
11
2-#3 bars.
A 3-#3 bars.
0 4-#3 bars.
-Value of EI from
rotation measurement.
---Calculated value of EI.
---The first visible crack •
180
160
.
1:::
·r-1
I
,.Q
rl
---A-I
I
H
100~
I
r.Ll
.
-. -·-.-."""" ·- N. A.
II
I
80
I
1
r
Method B.
I
!
I
I
I
;
r~~~~~
60'
40
------0---------A----
F----
-----------
__0.-N.A.
20~---
Method C.
I
I
ol0
-------G----
u:z:z:z:
J----~--~--~----~--~~
50
60
20
30
40
10
Moment, kips-in.
Figure 17 - Measured and calculated values of EI (E by lOOOf')
versus M for normal concrete beams at 7 days.
c
37
200:-----r-----r----~-----.-----.----~~~
:--a----i
180 ·-:
----
l
I
--N.A.
!
I
Method B.
r
160'
I
140 (_
.
s::
N
-e----
t
--A---i
·r-l
I
bar.
bars.
bars.
o
bars.
Value of EI from
rotation measurement.
--- Calculated value of EI.
~ The first visible crack.
•
•
•
I
.0
r-1
12oL
~
lo
r-1
1- -G-----
X
100~\~
1-#3
2-#3
3-#3
4-#3
J
soL
I
I
I'
60 l -
_______
_..,.... ____
-----_
1
_....(;).._
!:.I :2: : :::.
40f-
L-o----
----------- --
I
i
2 0 i=----11- - - -
--0---N.A~-~-
•
I
I
l
0
Method
---------
c.
-----L----_J-----~----~----~----~--~
Ll
0
10
20
30
40
50
60
Moment, kips-in.
Figure 18 - Measured and calculated values of EI {E by secant
modulus) versus M for lightweight concrete beams
at 7 days.
(
38
200
180
160
-------140
.
N
• 1-#3 bar.
• 2-#3 bars.
• 3-#3 bars.
0 4-#3 bars.
--Value of EI from
rotation measurement.
---Calculated value of EI.
-- The first visible crack.
~
·r-i
I
..0
r-l
120
1..0
lo
r-l
H
•'
100
N.A.
rx:l
..
Method B.
80
60
:::2-------:.. ::_-_-
40
20
__O __
Method
7-o---'-----
N.A:----:...- --:---
c.
0 ~------~----~~----~~----~------~------~--~
30
40
20
50
10
0
60
Moment, kips-in
Figure 19 - Measured and calculated values of EI (E by formula)
versus M for lightweight concrete beams.at 7 days.
(
39
r-·--·-
320
f-0- ---
265
t-;;: --·-
250
r~...l..'
...
~
~
.
I
J-
~
I
I
__,,....
,--.
-~·---
....
T
---1
_y
.:...
f--
I
1
iI
•
I
I
I
I
! ,\
160 I
:
e
•
1-#3 bar.
2-#3 bars.
A.
3-#3 bars.
0
4-#3 bars.
Value of EI from
rotation measurement .
---Calculated value of EI.
-+-The first visible crack.
: f\
.
~
·r-1
.
I
.
I
I
140
I
\
~~
~
.--l
\
l\
\0
lo
.--l
I
-!- ·
...0
120
II
I
l
: G>
N
.L
I
--
LA,- - -
230 ;
180
T
I
-i
\
~
\
\
100
i
____,I
\
'
I
__ 0 __
\\
I
so
\
\
rL--\ "-,
I
......
zzzzza
a
'.
/
•
----.'
!
~A-A
i
~
-,
irc---- ~~
I
-·-
--€)--.-- - - - - - - :
--ii
-.-,-- -.-- - : ' -
- -·---·--- --··-:~
I
-·--------- --- --,.;.
--l
20 r-l.
..!...
T
I
0
0
:
I
.
L--- \ )·~
40
; ,
I
N.A.
Method C.
\"
kr-·-~
60 t--
tr;=:;
I
10
20
40
30
Moment, kips-in.
50
60
Figure 20 - Measured and calculated values of EI {E by secant
modulus) versus M for normal concrete beams at 21
days.
(
40
200r------.-------.------,-------.------,------~--~
-
-{)--
1aa I
r~
I
I
160
--f----1---
N.A.
Method B.
F\---
.1
.
r~ -A-
I
140
.
N
..0
r-1
~\~ -G1..
\\
120
•
1-#3 bar.
2-#3 bars •
.._ 3-#3 bars.
0
4-#3 bars .
Value of EI from
rotation measurement.
--- Calculated value of EI.
~ The first visible crack.
0
\
a
\\
\\
\\
~
·r-i
I
;'
4
')•. \\
I : \\
i \\\ .
H
100
ril
L~ \\
!
\\\
80 '---
\\ \
I .
60)
\
1-G- --.
40
r"---
\'A~ ....
~ 1-·-~•
__ 0 __
N.A.
- wztzzzzzzztztnllzp•
Method
c.
--e----------
-...a.-- -----------
r-
~-----
--------------
l-..---
-·------------
20-11,
l
o~----~--~----~~----~~----~--~---1
0
10
20
30
40
50
60
Moment, kips-in.
Figure 21 - Measured and calculated values of EI (E by formula)
versus M for normal concrete beams at 21 days.
(
41
i
245 .J..t== -o200
- - - - - - - N.A.
l80
Method B .
.
• 1-#3 bar.
• 2-#3 bars.
A 3-#3 bars.
0 4-#3 bars.
-- Value of EI from
rotation measurement.
---calculated value of EI.
--The first visible crack.
c
·r-1
I
..0
r-i
'Po
r-i
120
H
J:il
..
I
I
100
__o __
80
CIZ?222?2??222
Method
60
40
N.A.
0
c.
-0---------llr-----------
~=~-
------:- __.
....._..._
20_~---.
-··-
-e----------lT
oTL-----L-----L----~----~----~----~~0
10
20
30
40
Moment, kips-in.
50
60
Figure 22 -Measured and calculated values of EI (E by lOOOf')
versus M for normal concrete beams at 21 days. c
42
340
If=-Y--!X
~---
2701
230
200
k---
-- f---
--N .A.
I-A---
180 "·
Method B.
160
•
•
N
1-#3 bar.
2-#3 bars.
A
3-#3 bars.
G
4-#3 bars.
Value of EI from
rotation measurement.
--- Calculated value of EI.
-- The first visible crack.
140
120
100
__ 0 __
80
l'
1
2
2
2 Z I 2
Method
N.A.
?)
c.
---0----------
60
----1!.----·------
-b.---
---·------ ---
40
--·--
20_r
--------
0~----~~----~------~~----~------~------~--~
0
10
20
40
30
Moment, kips-in.
50
60
Figure 23 - Measured and calculated values of EI (E by secant
modulus) versus M for normal concrete beams at
28 days.
43
210
L
t---~-
I
__ j__
18010\
I
r .\
Method B.
r---·\--a--
1
160
'
0---~
\
;
~ .I
140 ::.::.---'
I
I
.
N
~
I
I
•
1-#3 bar.
2-#3 bars.
1t.
3-#3 bars.
o 4-#3 bars.
-Value of EI from
rotation measurement.
---Calculated value of EI.
-- The first visible crack.
---..t.·-
r-l
1
•.
I
120
I.
r-l
·. •
~. \
\\ 'Jil\
I
!
I
I
•
•
I
l-->~\ \
.a
b
\----e-
'
1
•..-I
I
\0
---N .A.
\
\
I
I
100~
j
I
I
J
I
!
aoL
I
I
I
i
I
60~
!
H3----
'
I
I
40 -~--A'-ll---1-e---
20
Method
c.
OL-------~------~------~------+-------~--------~~
·a
10
20
30
40
Moment, kips-in.
50
60
,
Figure 24 - Measured and calculated values of EI (E by formula)
versus M for normal concrete beams at 28 days.
44
270 [
___ ..,
t_--;5-
210 l
180
-+---- -- N .A.
T-----0i \
160
Method B.
L_L4ro--~.
0
$\
\ \
.
N
J:::::
·r-i
I
140
.,
...0
..-1
1.0
I
I
>_
I
'
II
'',
\
o 1-#3 bar.
• 2-#3 bars.
4
3-#3 bars.
O 4- #3 bars .
-- Value of EI from
rotation measurement.
---Calculated value of EI •
-- The first visible crack.
'
0
..-1
120
H
~
I
I
100
80
I
I
L
__ 0·-N.A.
q; zpzz;zz z
zauJtl
Method
c.
60
! -0---
l
---0---------
A---
- - --- - - - - - - - -
I
:--G----
---o----------
--A----------
40 ..___
'--Q---
20
t!
o.T~----~-----~----~--~----~----~~T
0
10
20
30
40
50
60
Moment, .kips-in.
Figure 25 - Measured and calculated values of EI (E by lOOOf~)
versus M for normal concrete beams at 28 days.
45
N
.
~
·r-i
I
•
•
120
..0
r-i
.A.
0
'Po
r-i
100
X
'!
H
"""'+--
~
..
U)
U)
1-#3 bar .
2-#3 bars .
3-#3 bars.
4-#3 bars.
Value of EI from
rotation measurement.
Calculated value of EI.
The first visible crack
I
I
I
I
I
.I
l
I
80
(j)
~
I
!,..._
I
'
4-l
4-l
·r-i
i
I
+l
til
I
60 1f--0----
I
--o------
i
--A-------
f---Lk---
4o
___J
--G-------
F--I
I
__ Q __ N~;-------
r----20
!
Method
c.
O.L-------L---------------'--------~--------L------~---~
0
10
20
30
40
Moment, kips-in.
so
60
Figure 26 - Measured and calculated values of ,EI (E by secant
modulus) versus M for l~ghtweight concrete beams
at 28 days.
(
46
2001
1
180~--~
------- - - -- -- -- -
-- N. A.
I
Method B.
160L
140
.
N
----o---- ------
c
·r-1
I
..a
,...,
120
H
100
~
_'_:_ - - - - ----0-....... - - - - - - - - --
11
1- #3 bar.
e · 2-#3 bars.
J
A. . 3-#3 bars.
0 ."4-#3 bars.
-- Value of EI from
rotation measurement.
---Calculated value of EI.
--The fiirst visible crack.
-~-----~
40
'
20
~·
·----1:1---
--A-------
--e-------- .
.:__0__ :~:~--------I
.l
""
lllli!ill
Method
c.
I
0~------~-------~---------~---------~--------~--------~~
50
60
30
40
0
20
10
Moment; kips-in.
Figure 27 - Measured and calculated values of E~ (E by formula)
versus M for lightweight concrete beams at 28 days.
(
.
47
deflection method.
Figures 28 and 29 show typical examples of a comparison
between the upper measured values of EI and the theoretical
values of EI by. gross section (Method A) .
Figures 32 and 33 show typical examples of a comparison
between the lower measured values of EI and the theoretical
values of EI, considering values of E by (a) Secant modulus,
(b) Formula method and (c)
~ges
and lower
percent~ges
lOOOf~
under the higher percent-
of steel.
The stiffness of the normal as well as the
concrete beams increase as t he age increases.
lightwe~ght
At early ages
initial cracking appeared at low values of moment, whereas
at the
twenty-e~ghth
higher moment values .
normal as well as the
day initial cracking occurred at
This applies equally well to the
lightwe~ght
concrete beams.
In general, it was noted that in all cases the diffe rence between the "high " and "low" values of EI increased as
the percentage of steel decreased.
This differ ence narrows
down in the case of higher percentages of steel .
Thus it
can be said, that the stiffness of a beam for a low percentage of steel woul d vary more with respect to moment
than for a beam with a higher percentage of steel .
In all cases , the stiffness of the beam decreased as
cracking increased.
F~gures
6
thro~gh
9 show t hat , when a
flexure c rack in a beam occurs, it is rapid.
trates relatively deep into the beams.
decrease in the stiffness.
It also pene-
This causes a sudden
48
180 r------,-------,-------.------~------~------~--,
111
G
•
o
I
I
140 L_
0
.
N
c
.,...,
1- #3 bar.
2-#3 bars.
3-#3 bars.
4- #3 bars.
~ The first visible crack
--- Value of EI from
rotation measurement.
---Calculated value of EI
by (E by secant modulus)
-·-·-Calculated values of EI
(E by formula)
160
'·
120
I
..0
r-i
--o-------
1.0
lo
r-i
- -c-------
100
0
-
_ , -
---
-
-
-
· - - -
- - - -- --·-8-·-·-·-·:.
-· -e--· - · -· -·
H
~
-·-A·-·-·-·-
-·--·-·-·-·-
0
40
I
i
I
f
J
1 _ :
20
I
0
~----_i_ _ _ _ _ __ L_ _ _ _ _ _J-------~------~----~--~
0
10
20
30
40
Moment, kips-in.
50
60
Figure 28 - Comparison between the measured values of EI and
the calculated values.of EI by gross section
method for lightweight concrete at 28 days.
49
200
180
o
1-#3 bar
11
2-i*3 bars.
A. 3-#3 bars.
o 4-#4 bars.
~ The first visible crack.
- - Value of EI from
rotation ~easurement.
Calculated value of EI
by (E by secant modulus)·
Calculated values of EI
(E by formula)
-·-· Calculated value of EI
(E by lOOOf~)
160
140
.
N
c
·r-l
I
..Q
~"""-!
120
0
Ia
....-l
H
100
lil
60
40
20
~
I
I
I'
I
0
I
0
10
20
30
40
Moment, kips-in.
50
60
Figure 29 - Comparison between the .measured values of EI and
calculated values of EI by gross section method for
normal concrete at 28 days.
50
A comparison between the lower values of EI obtained by
using net section with transformed area of steel (Method C)
and the measured values of EI seems to indicate a pattern,
such that the differences between the two fall in approximately the same range for different age of beams with the same
percentage of steel.
A comparison between the calculated values of EI (E
calculated by different methods) for gross section with
transformed area of steel, and the measured values of EI in
the uncracked section, indicates a rather erratic pattern.
However, this could be due to experimental error; since it
is difficult to take accurate readings at the lower values
of moments.
Table 4 shows the calculated values of EI obtained by
the gross section (Method A), the gross section with transformed area of steel (Method B) , and the cracked section
with transformed area of steel (Method C) , where the modulus
of elasticity is based on (a) 0.45f',
(b) by formula, and
c
(c)
lOOOf~
for the normal concrete.
Table 5 shows the calculated values of EI by the same
methods as above, where the modulus of elasticity is based
on (a)
0.45f~,
and (b) by formula for the lightweight con-
crete.
The values of EI by the gross section with transformed
area of steel using E as calculated by the formula method,
are more in agreement with the experimental results than
51
using measured values of E taken from experimental stressstrain curves at
0.45f~.
The values of EI by the gross
section (Method A) is less than the values of EI obtained by
the gross section with transformed area of steel (Method B).
With respect to the uncracked values of EI, the method B
is conservative.
The cracked section values of EI obtained
by using net section with transformed area of steel (Method C), compare favorably with those obtained from experimental results.
The results of the tests for the normal
concrete show that, of the three theoretical methods available for determining EI, E computed by
lOOOf~
seems to be
the most satisfactory for higher percentages and by the
secant modulus for the lower percentages of steel.
The
values of EI considering E computed by the secant modulus is
higher, and E by the formula method is lower than E by
for any percentages of steel.
lOOOf~
For the lightweight concrete E
computed by the formula method seems to be the most satisfactory for higher percentages, and the secant modulus for lower
percentages of steel.
In all cases the values of EI consid-
ering E computed by the secant modulus method are higher than
those by the formula method for any percentages of steel.
Figure 25 shows that the maximum values of EI for the
uncracked section apply over a small range of moments especially in beams with small percentages of steel.
Figures 30 and 31 show that, the upper values of EI by
the deflection method are somewhat higher than those obtained
52
180r------r----~r-----~----~-------------
160
(\)._0
\
\
\
\
\
140
®
\
\
.
s::
N
120
·r-1
I
..Q
~
100
\D
I
0
~
><
H
~
..
80
• Rotation Method
• Deflection method
-+-The first flexure crack
~-The first shear crack
--Computed EI using Gross
section with E taken
as the tangent modulus
from the actual stressstrain curve.
(/]
(/]
Q)
s::
4-1
4-1
·r-1
60
~
til
40
20
0 ~------~------~------~------~------~------~
0
10
20
30
40
50
60
Moment, kips-in.
~igure
30 - Comparison of measured values of EI by the rotation and the deflection method versus M for
lightweight concrete beam, at 28 days with
l-i3 bars.
53
200
~
'~
180
''
\
G.
\
\
\
160
0
\
,o
\
.
N
c
·.-i
\
\o
140
I
-•
r-1
\0
120
I
0
r-1
X
H
~
..
Rotation method
Deflection method
The first flexure crack
--<--The first shear crack
---computed EI using Gross
section with E taken as
the tangent modulus from
the actual stress-strain
curve •
A
..a
100
A
Ill
Ill
Q)
c
4-l
4-l
·.-i
80
+J
Cll
60
I
~
l
I
I
__J
I
.)...,
0
T
0
10
30
20
40
Moment, kips-in.
9)
60
Figure 31 - Comparison of measured values of EI by the rotation and the deflection method versus M for
normal concrete beam, at 28 days with 2-#3 bars.
54
8 Q ~-----··· ----
A
\\
I
60
N
~0
.\
--0----
--
..
--a----
f-0·--
~
·r-i
I
..0
·<> --
-- -<>· - -
__
--0----
r-1
~
I
0
r-1
-o-_.,.,_
40
- - u-.- - - -
--+----
~--
H
J:il
-o-Values of EI from
4-#3 bars
I -A-Values of EI from
20
2-13 bars
o , o Calculated values
o,• Calculated values
c / • Calculated values
~The first visible
rotation measurement for
.
r
II
I
rotation measurement for
of EI, E by :~:2cant modulus
of EI, E by Fo~mula method
of EI, E oy lOOOf~ method
crack
I
I
I
0 ~i__________L----------4-----------L----------~--------~
0
10
20
30
Moment, kips-in.
40
50
Figure 32 - Comparison between the lower me&sured values and
the theoretical values of EI, considering E by
Secant modulus, Formula method and lOOOf' for
normal concrete beams,·at 28 days with 2~#3 and
4-#3 bars.
55
~
I
.
N
1::
•r-i
I
--o
--0-----
- -<>- --
--<>------
.0
r-1
1.0
I
0
r-1
X
I
40
- --c--- _]
--~---j
-e-Values of EI from
4-#3 bars
20. -.t.- Values of EI from
2-#3 bars
o , a Calculated values
0 1 + Calcula·ted values
--+---The first visible
rotation measurement for
rotation measurement for
of EI, E by Secant modulus
of EI, E by Formula method
crack
QL---------~--------~----------~--------~----------~
0
10
20
30
40
50
Moment, kips-in.
Figure 33 - Comparison betwee n the lower mea sured values and
the t~ eoretical values of EI, considering E by
Secant modulus, Formula method and lOOOf' for
lightweight concreta beams, at 28 days with 2-#3
and 4-#3 bars.
L
56
by the rotation method but are in reasonable agreement.
It
is believed that any difference is due to variations in fial
gage
readings, since at the early stages of loading very
small readings were recorded.
Calculations in6icated that
for a moment of 2.75 kips-inches an error of 0.001 inch in
deflection produced an error in stiffness of 20 x 10 6 poundssq. inches.
In addition an error of 0.0001 inch in rotation
under the same moment produced an error in stiffness of
7 x 10
6
pounds-sq. inches.
In both methods, at the working
moment of 27.5 kips-inches under the above mentioneu conCitions an error in stiffness was produced equal to 0.5 x 10 6
and 0.05 x 10 6 pounds-sq. inches,respectively.
The values of EI from point A on the graph to the point
of the first flexure crack by both methods are very nearly
identical.
Further, it shoulC be noted that the measured
values of EI by the rotation and deflection methoC after the
working load is reached tend to diverge.
This is due in part
to excessive centerline deflections caused by shear cracks
occurring near the supports.
From the above discussion it can be seen that the
measured values of EI by the rotation and the deflection
method are equivalent except at high values of moment.
In addition Figures 30 and 31 show the calculated values
of EI based on gross section using E calculated as the
tangent modulus from the experimental stress-strain curve.
It can be seen that these curves, while higher, follow the
57
same general shape as the experimental curves for stiffness.
From Figures 15 through 29, it can be concluded that in
all cases EI increases with an increase in the percentages
of steel.
Table 4 - Calculated values of EI by different methods
for normal concrete beams
Age
in
days
Numbers
of
#3 bars
EI b~ formula
(10- lb-in.2)
Method
EI by secant modulus
(lo-6 lb-in.2)
Method
A
B
c
A
B
81.19
324.9
c
EI b* 1000 f'
(lo- lb-in.2)
Method
A
B
79.29
313.2
c
7
0
119.1
476.2
7
1
116.2
241.0
23.0
79.30
150.0
21.0
76.47
143.0
21.0
7
2
120.5
208.9
40.0
82.21
130.2
35.3
79.27
125.0
35.0
7
3
120.5
188.2
53.0
82.21
120.0
46.0
79.27
115.0
45.6
7
4
122.1
180.0
59.0
83.30
116.0
50.4
80.32
111.0
49.7
21
0
143.0
572.0
96.14
384.4
21
1
145.2
318.1
23.9
97.61
194.0
21
2
145.2
266.0
40.9
97.61
21
3
150.7
248.0
56.9
21
4
150.3
232.0
62.8
28
0
153.7
707.3
28
1
166.3
343.0
25.0
28
2
148.1
272.0
41.0
28
3
138.8
202.0
45.9
28
4
153.1
229.0
60.9
116.1
464.0
22.0
117.9
246.3
22.9
161.5
36.0
117.9
204.5
38.6
101.3
153.0
49.0
122.3
192.0
53.1
101.0
144.0
53.9
122.0
181.0
58.5
102.9
473.7
126.4
581.4
111.4
208.1
23.0
136.7
270.0
24.2
99.22
165.8
36.7
121.8
214.0
39.0
92.96
124.0
40.0
114.1
159.0
42.9
142.0
52.0
126.0
181.0
56.2
U1
102.7
co
Table 5 - Calculated values of EI by different methods
for
Age
in
days
Numbers
of
#3 bars
lightwe~ght
concrete beams.
EI by Secant Modulus
( lo-6 lb-in. 2)
Method (A)
Method (B)
7
0
90.69
364.0
7
1
96.12
188.2
7
2
85.25
7
3
7
EI bt formula
(10- lb-in.2)
Method (C)
Method (A)
Method (B)
Method (C)
75.01
303.0
23.0
79.49
150.1
22.0
138.8
33.9
70.50
111.4
31.9
89.15
132.3
46.4
73.75
107.2
43.0
4
93.01
110.0
48.3
76.91
90.4
44.8
21
0
103.35
414.0
84.53
338.0
21
1
90.99
232.7
24.8
74.32
180.8
23.7
21
2
172.7
37.2
83.83
135.0
34.9
21
3
149.6
46.8
76.74
118.0
43.4
21
4
152.0
54.8
86.32
120.1
50.3
102.7
98.37
105.7
60
VI.
1.
CONCLUSIONS
The lower experimental values of EI, compare favor-
ably with those obtained by the cracked section analysis
including the transformed area of steel (Method C) •
The
results of the tests for normal concrete indicate that, of
the three theoretical methods available for determining EI,
E computed by
lOOOf~
seems to be the most satisfactory for
higher percentages and the secant modulus for lower percentages of steel.
For lightweight concrete, E computed by
formula seems to be the most satisfactory method for higher
percentages, and secant modulus for the lower percentages of
steel.
2.
The measured stiffness of EI for lightweight con-
crete is less than that of normal concrete for equal percentage of steel and approximately the same strength.
3.
The values of EI obtained using gross section
analysis with transformed area of steel using E as calculated by the formula method, are in closer agreement with
the experimental results than those obtained by considering
the other values of E, as calculated by the secant modulus
and the lOOOf'c method.
4. The measured values of EI decrease as the bending
moment increases, and the rate of change of EI decreases
noticeably after first cracking occurs.
s.
The values of EI increase as the age increases.
61
6.
The measured values of E increase as the ultimate
compressive strength for the normal as well as the lightweight concrete increase.
7.
The precision of experimental values of EI based
on equal dial_ gage readings up to the working load as obtained by deflection measurements is better than those
obtained by the rotation method.
Near maximum loading EI
as computed by deflection measurements tend to degenerate
due to the formation of shear cracks, therefore the rotation method is superior in this range.
62
VII.
RECOMMENDATION
The author would like to recommend that an additional
investigation be conducted in
determini~g
experimental
values of EI for beams, having different percentages of
tension and compressive steel.
The study should be made for
both normal and lightweight concrete having the same compressive
stre~gth
havior of each.
in order to compare more accurately the be-
63
BIBLIOGRAPHY
1.
Bill G. Eppes, Comparison of measured and calculated
stiffnesses for beams reinforced in tension only.
Journal of the American Concrete Institute,
56: 313-325. 1959-60.
2.
D. W. Lewis, L~ghtweight concrete made with expanded
blast furnace slag. Journal of the American
concrete Institute, 55: 619-633. 1958-59.
3.
J. J. Shideler, Lightweight-aggregate concrete for
structural use. Journal of the American Concrete
Institute, 54: 299-328. 1957-58.
4.
Design and control of concrete mixtures, Tenth edition,
·
Portland Cement Association, Chic~go, Illinois .
5.
Phil M. Fergus9n, Reinforced concrete fundamentals with
emphasis on ultimate stre~gth. ' John Wiley & Sons,
Inc., 1963.
6.
Paul George Hayes, The plastic behavior of beams with
butt welded connections, Thesis, University of
Missouri School of Mines and Metallu!gy, Rolla,
Missouri, 1964.
1.
R. H. Patel, Stiffness of lightweight reinforced concrete
beams, Thesis, University of Missouri at Rolla,
Rolla, Missouri, 1964.
8.
ASTM Standards, Part-4, 1961.
A. Standard method of making and curing concrete
compression and flexure test specimens in the
laboratory. 728-733 .
B. Specification for concrete aggregate (Tentative) .
504-509.
..
.
c. Test for sieve or screen analysis of fine and
coarse aggregates (Tentative) . 585-587 .
D. Test for · slUmp of Portland Cement Concrete.
790-791.
E. Compressive strength of moulded concrete
cylinders. 721-723 .
F. Air content of freshly mixed concrete by the
pressure method. 695-701.
G. Specifications for lightweight-aggregates for
structural concrete ~Tentative). 5~2-529.
H. Standard method of test for weight per cubic
foot of concrete. 689-691.
·
J . Standard method of test for flexural strength
of concrete (using simple beam with thirdpoint loading). · 734-736 .
64
9.
R. c. Reese, CRSI Design Handbook, second edition,
Concrete Reinforci~g Steel Institute, Chicago.
1957.
10.
Report of the joint committee on standard specifications concrete and reinforced concrete. Journal
of the American Concrete Institute.
36: 249-251. 1940.
11.
Scott, w. L . , Glanville w. H., and Thomas E. G.,
Explanatory handbook on the code of practice for
reinforced concrete. Second edition. Concrete
Publications Ltd., London. 1950.
12.
Adrian Pauw, Static modulus of elasticity of concrete
as affected by density. Journal of the American
concrete Institute, 57: 679-687. 1960.
13.
Ralph
14.
Continuity in concrete building frames, fourth edition,
Portland Cement Association, Chicago, Illinois.
1959.
15.
R. E . Philieo , Comparison of results of three methods
for determining Young's modulus of elasticity of
concrete. Journal of the American Concrete
Institute, 51: 461-469. 1954-55.
16.
Teddy J. Hirch, Modulus of elasticity of concrete
affected by elastic moduli of cement paste matrix
and aggregate. Journal of the American Concrete
Institute, 59: 430-434. 1962.
w. Kluge, Structural lightweight-aggregate concrete. Journal of the American Concrete Institute,
53: 383-390. 1956-57.
65
APPENDIX
Sample calculations:
r
b
-Compression area
of concrete
t
•
Transformed area
of steel, nA8 •
Figure 34 - Typical cross-section of beam.
EI based on method (A) :
I
1
= 12
I =
E
=
3
bt ;
=
b
3. 5",
1
I2
( 3. 5) ( 4 • 4) 3 =
3.992 x 10
6
t = 4. 4" (from Table 6) •
4
2 4 • 8 5 in.
psi, (from Table 3).
/
EI = (24.85) (3.992) (10) 6 = 99.22
X
10 6 lb-in~
EI based on method (B) :
3
I=~
kd) 2 + (n-1) (As) (d-kd) 2
12 bt + bt(t2
d
=
(4.40 - 1.05) = 3.35 in.
As = pdb = ( • 0 18 7 6 )
x 10
E = 3.992
= Es/Ec =
(
6
( 3 • 3 5) (3 • 5)
11!11
.
psi, as above.
28.7 X 106
3.992 X 106
= .7 .19.
0 • 22 •
66
Taking the moment of the compression area and that of the
transformed area of
axis (5), (See
reinforci~g
F~gure
steel, about the neutral
32) yields,
(bkd) (kd) = (n • A ) • (d-kd) ,
2
s
and k
=
Substituti~g
2
[2np + (np) ] 1/ 2 - np .
the values of n and p in the above equation
yields,
k
= 0.4016.
Substituting the known numerical values in the equation
I=
2
2
3
~bt
+ (bt) (~- kd) + (n-1) (As) (d-kd)
1
yields, I= 41.56 in .
4
2
and EI = (3.992) (41.56) (10) 6 = (165.85) (10) 6 lb-in.
EI based on method (C) :
substituting the numerical values in the above equation,
4
I= 9.19 in.
and EI
= (3.992) (9.19) (10) 6
= (36. 71) (10)
6
lb-in.
2
67
Table 6 - Properties of normal concrete beams.
?\ge
in
days
Numbers
of
#3 bars
b
(in.)
t
(in.)
d
(in.)
7
p
3.50
4.462
4.462
7
1
3.50
4 . 427
3.377
7
2
3.50
4.480
3.430
7
3
3.50
4 . 480
3.430
7
4
3 . 50
4.500
3 . 325
21
0
3 . 50
4 . 378
4.378
21
1
3.50
4 . 400
3.350
21
2
3.50
4.400
3 . 350
21
3
3.49
4.460
3.410
21
4
3 . 50
4.452
3.277
28
0
3.50
4.668
4.668
28
1
3 . 50
4.470
3.420
28
2
3 . 50
4.400
3.350
28
3
3.48
4.150
3.100
28
4
3.50
4.400
3.225
Note:
b = width of the specimen,
t = depth of the specimen,
d = distance from extreme compressive fiber to
centroid of the tensile reinforcement.
68
Table 7 - Properties of
Age
in
days
Numbers
of
#3 bars
l~ghtwe~ght
concrete beams.
b
(in . )
t
(in.)
d
(in.)
7
0
3.50
4.414
4 . 414
7
1
3.50
4.500
3.450
7
2
3.46
4.340
3.290
7
3
3.44
4.415
3.365
7
4
3.50
4.450
3.275
14
0
3 . 50
4.460
4 . 460
14
1
3.50
4.500
3.450
14
2
3.48
4.434
3.384
14
3
3.50
4.415
3.365
14
4
3.50
4.400
3.225
28
0
3.50
4.421
4.421
28
1
3.50
4.573
3.523
28
2
3.50
4.408
3.358
28
3
3.48
4.354
3.304
28
4
3.50
4.450
3.275
Note:
b = width of the specimen,
t = depth of the specimen,
d = distance from extreme compressive fiber to
centroid of the tensile reinforcement.
69
VITA
Raman A. Patel was born on May 16, 1940 in Saijpur,
Gujarat State, India, son of Mr. and Mrs. Ambalal B. Patel.
He re.c eived his primary and secondary education in the
Local Board School, Saijpur and D. N.
H~ghschool,
Anand,
India and. graduated in June 1957.
In June 1957, he entered Siddharth College of Arts and
Science, Bombay, India and passed Inter Science in June 1959.
In June 1959, he entered Birla Vishvakarma Mahavidyalaya
(Engineeri~g Coll~ge),
Vallabh Vidyanagar, India.
ceived his Bachelor of
Engineeri~g D~gree
i~g
He re-
in Civil Engineer-
from Sardar Vallabhbhai Vidyapeeth (University), Vallabh
Vidyan~gar,
India in June 1963.
After his. graduation from
coll~ge
he joined The PASK
Corporation Private Limited, Engineers and Constructors,
Calcutta, India and worked for one year as a Junior Engineer
in the design office.
He enrolled at the University of Missouri at Rolla,
Rolla in September 1964 to work towards his Master of
Science Degree in Civil
E~gineering.
He is a member of the American Concrete Institute and
ASCE.