Center for
By-Products
Utilization
SHEAR STRENGTH OF FULL SIZE HIGH
STRENGTH CONCRETE BEAMS
By Tarun R. Naik
Report No. CBU-2003-47
REP-540
December 2003
A CBU Report
Department of Civil Engineering and Mechanics
College of Engineering and Applied Science
THE UNIVERSITY OF WISCONSIN-MILWAUKEE
1
Shear Strength of Full Size High Strength Concrete Beams
By
Tarun R. Naik
Professor of Civil Engineering and Director, UWM Center for By-Products Utilization,
University of Wisconsin-Milwaukee, Milwaukee, P.O. Box 784, Milwaukee, WI 53211,
USA
ABSTRACT
Eighteen high strength full size concrete beams were tested under shear. The concretes
used had nominal compressive strengths of 10,000 and 12,500 psi. The other design
variables were shear span to depth ratio (a/d) and nominal shear reinforcement (vs). The
shear span to depth ratio (a/d) for nine beams each was 3, 4, and 5. The nominal shear
steel provided (vs) was varied from 50 to 120 psi. The analysis of the data from test
suggests that the minimum amount of shear steel required increases with an increase in
the concrete compressive strength. But it was also observed that a sudden increase in the
amount of shear steel as specified by the ACI-318-89 from 50 psi to 100 psi for concrete
with f‟c of 10,000 psi is not necessary. Instead a smooth transition shall be adopted.
2
INTRODUCTION
High-strength concrete usually refers to normal weight concrete, which has a uniaxial
compressive strength greater than 6000 psi at 28 days. Although 6000 psi is selected as
the lower limit by the ACI, it is not intended to imply that there is a drastic change in
material properties. There are distinct advantages in the use of concrete with higher
compressive strengths in both cast-in-place reinforced and precast/ prestressed concrete
construction. Many investigations on the theoretical and experimental aspects of the
behavior of reinforced flexural members using high-strength concrete have been carried
out. The shear strength of reinforced flexural members is an equally important design
area. Due to the abrupt nature of shear failures and complicated interdependent failure
mechanisms, it is extremely difficult to formulate reliable mathematical models for
designs for reinforced concrete. Therefore, many researchers have concentrated on using
empirical equations for the prediction of failure loads for such members. The ACI-318-
892 has revised the Section 11.1.2, on shear for high-strength concrete, and placed an
upper limit on all shear strength equations based on the concrete compressive strength
(f‟c) greater than 10,000 psi. The Code change is based on a very small sample of tests
conducted with high-strength concrete beams. Chapter-11 of the Code contains equations
to compute shear and torsional strengths provided by concrete. These equations are a
function of (f‟c)1/2
and have been verified experimentally for members with compressive
strengths of 3000 to 8000 psi. In the absence of test data for members with f‟c greater
than 10,000 psi, the values of (f‟c)1/2
are limited in the 1989 Code to 100 psi, except as
specified in Section 11.1.2.1. Section 11.1.2 does not prohibit the use of concrete with f‟c
3
greater than 10,000 psi. Based on few tests5-8
, the 1989 Code permits (f‟c)1/2
greater than
100 psi if a certain higher amount of minimum web reinforcement is provided, as
compared to that specified by Sections 11.5.5.3, 11.5.5.4 or 11.5.5.5 multiplied by
f‟c/5000. The multiplier f‟c/5000 is applied to concrete strength greater than 10,000 psi;
and is not to exceed a value of 3. Therefore, for 10,000 psi concrete, minimum shear
reinforcement (Av) computed by the ACI Equations 11-14, 11-16, is doubled. For f‟c =
15,000 psi or larger, minimum Av is tripled as compared to 1983 Code. Therefore, it was
desirable to determine the adequacy of the proposed revisions in the shear design for new
higher strengths concrete in the ACI 318-892 Code.
Studies on shear strength of reinforced concrete have been reported by many authors9-16
.
But in the recent past, few researchers17-23
have reported results of their investigations on
the shear strength of high strength concrete beams. In theses investigation17-23
, authors
have gone up-to concrete of compressive strength of 90 MPa, but their beams sizes were
mostly of proto-type only (not full scale beams).
This investigation was carried our to study the shear strengths of rectangular beams with
web reinforcement, using high-strength concrete. The concretes used were designed to
achieve nominal compressive strengths of 10,000 and 12,500 psi. The size of the beams
was chosen so that they were like real-life structures.
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RESEARCH SIGNIFICANCE
The test results reported in this paper are important and useful to understand the shear
behavior of high strength concrete full size beams.
EXPERIMENTAL PROGRAM
Materials and Specimens Details
A total of eighteen reinforced high-strength concrete beams were tested to determine the
effects of concrete compressive strength (f‟c), shear span to depth ratio (a/d), and the
strength contributed by the shear steel (vs) on the actual nominal shear strength (Vn) of
concrete beams. Two different concrete mixes were used; the concrete compressive
strengths (f‟c) selected were 10,000 psi and 12,500 psi. All the beams had different shear
span to depth ratio‟s (a/d), 3, 4 and 5. The levels of shear stress carried by the stirrups
(vs), ranged from 50 to 120 psi. For each concrete mix, the compressive strength (f‟c),
splitting tensile strength and modulus of elasticity were determined.
Beams had a dimension of 10” wide by 19.5” deep. The first nine beams had a f‟c of
10,000 psi, a/d ratio of 3, 4, and 5, and vs amounts of 50, 75, and 100 psi. The second set
of nine beams had f‟c of 12,500 psi, a/d ratio‟s of 3, 4 and 5, and vs amounts of 60, 90,
and 120 psi.
The design data are presented in Table 1, and typical reinforcement details of beam no. 1
is shown in Fig. 1. The concrete mix design details are given in Table 2.
5
Steel Reinforcement
The steel used in the beams served two main purposes: (a) to resist flexural stresses; and,
(b) to resist shear stresses. All the eighteen beams had flexural and web reinforcement.
The flexural reinforcement consisted of Grade 60 steel deformed bars. The size ranged
from #5 to #9 bars. All the beams had 12” of overhang beyond the knife-edge support
points on each end and the reinforcement extended 10” beyond the support points. This
provided adequate anchorage to avoid bond failure leading to loss in the dowel force,
which resists the shearing displacements along crack. All reinforcement was tied using 8”
and 12” long wire loop ties. The web reinforcement was in the form of the two legged
stirrups with 135o
hooks at the ends, and typical reinforcement detail is shown Fig 1.
Plain smooth bars of ¼ inch nominal diameter were used for the shear stirrups. The yield
strength of the steel used, was 52,500 ksi. The shear reinforcement was required to be ¼
inch diameter and have yield strength of 40 ksi or lower. The 52.5 ksi yield strength shear
reinforcement, therefore, was annealed to lower the yield strength of the shear steel.
Beam Casting
The beams were cast in wooden forms. Reinforcement cages were placed, and the
concrete was placed. Immediately after placing the concrete, it was vibrated using a
hand-held internal vibrator. After placing and finishing the concrete beams, all the beams
were sprayed with a curing compound for high-strength concrete. The curing compound
consists of 9 parts of water to 1 part “Confilm”. “Confilm” is a commercially marketed
6
chemical by Master Builders of Cleveland, Ohio. At seven days, the wood forms were
removed and the beams were completely covered with a thick layer of hay and then with
a plastic sheet from all the sides.
Instrumentation
Strain gauges were used to determine strains in steel and concrete during loading. The
strain readings were recorded by a data acquisition system. The following types of gauges
were used.
(i) Stirrup steel: EA-06-1258BT-120
(ii) Longitudinal steel: EA-06-20CBW-120
(iii) Concrete: EA-06-20CBW-120
Reinforcement Strains
In order to estimate the loads carried by the web and flexural reinforcement steel, strain
gauges were mounted on the respective reinforcement bars. All the eighteen beams had
gauges mounted at mid-depth of the stirrups. A minimum of five, and a maximum of six
stirrups were strain gauged in each shear span.
7
Concrete Strains
The strains in concrete were measured at mid-span of each beam. To make the surface
smooth for a strain gauge application, a thin layer of epoxy was applied. After the epoxy
had hardened, the gauge was mounted.
Load Measurement
The beams were loaded at two points using 100kip capacity hydraulic jacks. The amount
of load applied by each jack was measured using two compression load cells requiring a
6 volt DC excitation. Configuration “A” cables were used to convert the load cell signals
to load in “kips”. The load cells were accurately calibrated using the Tinus Olsen
compression-testing machine in UWM laboratory.
Deflection Measurement
The deflection of each beam due to the applied two-point loading was measured at the
mid-span. This was done using a 6-volt excitation power Linear Variable Differential
Transducer (LVDT). The LVDT had a +3” range. Configuration “A” cables were used to
convert LVDT signals to deflection in “inches”. Prior to the use of LVDT in the test, it
was accurately calibrated and tested using the DANA voltmeter and accurately milled
pieces in UWM Structures laboratory.
Properties of Fresh Concrete
A large variety of tests were conducted on fresh and hardened concrete. The temperature
of fresh concrete and air was measured at the time of casting of specimens. Slump,
8
density and air content of all the three concretes were measured in accordance with
ASTM standards. The results are listed in Table 2.
Mechanical Properties of Hardened Concrete
Mechanical properties of hardened concrete were determined. There were two diameters
of cylinders cast due to the capacity limitation (4,00,000 lbs maximum) of the
compression testing machine. Twenty-seven 6 x 12 inch cylinders were cast in cast iron
molds for measuring the compressive strength and modulus of elasticity of concrete.
Another twenty-eight 6 x 12 inch cylinders were cast in plastic molds for measuring the
splitting tensile strength of concrete. Another forty-eight 4 x 8 inch cylinders were cast
in cast iron molds for compressive strengths at various ages. All the specimens were
prepared in accordance with ASTM and then sprayed on the exposed surface with
“confilm” curing compound, which prevents evaporations of the mix water from the
concrete. The cylinders were then covered with plastic bags and placed in lime-saturated
water at temperature of 73o F ± 3
oF. All the specimens were stripped after 24 hours and
kept in the lime-saturated water tank until the time of test.
Compressive strength of concrete
Two sizes of cylindrical specimens were tested in accordance with ASTM C-39 to
determine the compressive strength of concrete. Three 4 x 8 inch cylinders were tested at
each of the following test ages: 1, 3, 7, 14, 28, 56, and 91 days, to determine the
compressive strength of the three concrete mixes. Three 6 x 12 inch cylinders were tested
at each test age for compressive strength until the concrete had approximately 10,000-psi
9
compressive strength. This was due to the limiting load capacity of the testing machine.
The test results are presented in Tables 3 & 4.
Tensile strength
The 6 x 12 inch cylinders were tested to determine the tensile strength of concrete.
Cylinders were tested in accordance with the ASTM C496 at 1,3,7,14, 28, and 56 days.
Results are given in Tables 3 & 4.
Modulus of elasticity
The 6 x 12 inch cylinders were tested to determine the static modulus of elasticity. All
the tests for the determination of the modulus of elasticity were carried out in accordance
with ASTM C-469. The tests were conducted at 1, 3, 7, 14, 28, and 56 days. Three
cylinders were tested at each age test. The strains in the concrete were measured up-to
approximately 70% of f‟c at that age. The secant modulus of elasticity was then
calculated by measuring the slope of the line joining the points with stress corresponding
to 40% of f‟c and stress at 50 millionths strain, as per ASTM C-469.This value was then
rounded off to the nearest 50,000 psi. The test results are shown in Table 5.
RESULTS AND DISCUSSION
General Failure Modes and Crack Development
All eighteen beams failed in shear. Very few flexural cracks were observed near mid-
span in the early stages of loading. For all beams the cracks near mid-span, in the zone
10
where there are only flexural stresses, were vertical and extended above mid-depth
starting from the bottom of the beam. With increasing load, additional flexural cracks in
each shear span were observed. These cracks then began to turn towards the loading
points due to combined shear and flexural stresses. On further loading, a primary shear
crack formed in one of the shear span. This is the load at which shear strength of concrete
(Vc) is defined. 15 out of 18 beams failed in shear compression. It is known fact that
beams with out web reinforcement having a/d ratio of 2.5 and greater fail in diagonal
tension of flexure. Therefore, it can be concluded that the inclusion of web reinforcement
generally changes the failure mode for the beams with a larger a/d ratio to a pre-warned
shear compression failure.
Beams with f’c = 10,000 psi
Three out of nine beams failed in diagonal tension. The other six failed in shear
compression. Results are shown in Table 6
Beams with a/d ratio of 3
For beams with an actual shear span to depth ratio (a/d) of 3.3, Beam 1 and 4
failed in diagonal tension and beam 7 failed in shear compression. Beam 1 and 4 failed at
a load 22% and 4% respectively lower than that predicted by ACI. All three beams (1, 4,
and 7) had a common significant defect, honeycombing at the bottom of the beam in the
failed shear span due to improper vibration and low slump at the time of concrete
placement. Beam 1 and 4 showed lower deflections immediately prior to failure as
11
compared to Beam 7. Beam 1,4, and 7 had final deflections of 0.42”, 0.64”, and 0.71”
respectively. The reason for this is that Beam 1 and 4 had lower Vs and pw than Beam 7.
Beams with a/d ratio of 4
For beams with an actual shear span to depth ratio (a/d) of 4.3, Beam 2 failed in
diagonal tension and Beam 5 and 8 failed in shear compression. Beam 2 failed at a load
16% lower than that predicted by ACI. Beam 2 and 5 had a significant honeycombing
defect at the bottom of the beam in the failed shear span. Beam 2 showed a significantly
lower deflection prior to failure as compared with Beam 5 and 8. Beam 2,5, and 8 had
final deflections of 0.82”, 1.56”, and 1.51”, respectively. Beams 5 and 8 had
approximately the same final deflection but beam 8 failed at a higher load. The amount
of stirrup strain beyond yield was lower for Beam 2 than for Beam 5 and 8.
Beams with a/d ratio of 5
For beams with an actual shear span to depth ratio of 5.3, all three beams failed in
shear compression. All of the beams carried more load than that predicted by ACI. None
of the beams had honeycombing defects. Beam 6 had more deflection at failure as
compared to Beam 9. Beam 6 and 9 had final deflections of 1.56” and 1.45” respectively
Beams with f’c = 12,500 psi
All beams in this category failed in shear compression.
Beams with a/d ratio of 3
All three beams had a higher load carrying capacity than predicted by ACI. The
number of inclined cracks increased in Beam 10, 13, and 16 respectively. In addition, the
12
length of the flexural cracks decreased in these beams. None of these had honeycombing
defects. Beam 13 showed a little lower deflection immediately prior to failure as
compared to Beam 10, and 16. Beams 10, 13, and 16 had a final deflection of 0.79”,
0.64”, and 0.78” respectively. The strains in the stirrups for beams 10, and 16 were
beyond the range of the strain gauge in the failure span.
Beams with a/d ratio of 4
Beam 11 failed at a load 9% lower than predicted by the ACI. The length of the
flexural cracks increased in Beam 11, 14, and 17 respectively. There were a large number
of small horizontal cracks just above the level of the flexural steel in beam 17. Beam 11
showed lower deflection prior to failure as compared to beams 14, and 17. Beams 11, 14,
and 17 had a final deflection of 0.83”, 1.18” and 1.04” respectively. It was observed that
the stirrup strains went farther beyond the yield point for beams 14 and 17 as compared to
beam 11.
Beams with a/d ratio of 5
All beams failed at a load higher than that predicted by ACI. Beam 18 carried a
40% higher load than that predicted by the ACI, which is the most under-predicted of aa
the 27 beams. The number of cracks increased in Beam 12, 15, and 18. Short horizontal
cracks were observed in all beams above the flexural steel. However Beam 18 had a
significantly larger number of the horizontal cracks as compared to the other beams. The
deflection immediately prior to failure successively increased in Beam 12, 15, and 18. It
13
was observed that more stirrups strained beyond the yield point for Beam 15 and 18 as
compared to Beam 12.
Diagonal Cracking Loads
All the 18 beams failed in shear as designed, although the failure mode varied
with the amount of shear steel. In this investigation, 15 beams failed in shear compression
and 3 beams failed in diagonal tension. Two of the three beams that failed in diagonal
tension had a vs of 50 psi, which was lowest of all the beams. The diagonal cracking
load, equal to shear strength of concrete (Vc) was determined by two methods. The first
method used to determine Vc was by carefully inspecting the load versus stirrup strain
graphs for each beam, and the second method used to determine Vc, is by observing
crack patterns. However, in this investigation, Vc calculated using the stirrup strains has
been used.
Fig. 2 shows the relationship between variation of ultimate shear capacity (Vn)
and diagonal cracking loads (Vc) versus a change in the concrete compressive strength
(f‟c). It is clear that the ultimate shear capacity (Vn) and the diagonal cracking load (Vc)
increased with an increase in the concrete compressive strength (f‟c). Furthermore, it is
observed that the ratio of Vn actual over Vn predicted by ACI, generally increases with
an increase in the concrete compressive strength 9f‟c) for beams with Vs equal to 50 psi
as shown in Fig. 3 The diagonal cracking shear (Vc) increased with an increase in f‟c
14
Effect of Shear Span to Depth Ratio
In this investigation, since beams had a/d ratio greater than 2.5, Vc increases with
an increase in a/d. In this investigation, pw did not vary by a significant amount from
one value of a/d to another value. The increase in the shear carrying capacity of the
beams with increasing a/d ratio cannot be attributed to the increase in pw because of the
negligible increase in pw from one value of a/d to another. With an increase in a/d, the
ratio of Vn actual to Vn predicted also increases. This can be attributed to the fact that
Vc increases with increase in a/d ratio.
Effect of Nominal Shear Stress (vs) Carried by Stirrups
The 10,000 psi beams had a nominal Vs of 50, 75, and 100 psi. The 12500 psi
beams had a nominal Vs of 60, 90, and 120 psi. Figs 4 and 5 show the relationship of the
ratio of Vn actual to Vn predicted versus Vs. For the beams of 10,000 psi, the ratio of Vn
actual to Vn predicted increased with increasing vs. For beams of 12500 psi, the ratio Vn
actual to Vn predicted decreased with increasing vs for the beams with a/d of 3, for other
a/d ratio, it increased. The probable cause of the increasing ration of Vn actual to Vn
predicted for 10,000 psi mix, and the decreasing ratio of Vn actual to Vn predicted for
12,500 psi mix , for a/d of 3, can be the fact that this value of a/d is very near to the point
of transition. At the point of transition, the load carrying mechanisms change from shear
compression to inclined cracking capacity as a/d decreases. However, for a/d of 4 and 5,
for all mixes, a generally increasing trend of the ratio of Vn actual to Vn predicted was
observed. Figs. 6 and 7 display failure load versus Vs. It is clear from these figures that
15
Vn increases with an increase in Vs. It is also clear from these figures that rate of increase
of Vn increases as a/d increases.
Deflections
Mid span deflections were recorded for all the 18 beams at every load increment
using a LVDT. Figs 8 to 13 show the plots of load increment versus mid-span deflection
for a given f‟c and a/d. Deflections to failure were recorded. It is apparent that beams with
the same a/d show very similar deflection behavior. For all the beams, with the same f‟c,
the deflection at failure increased with increasing a/d. Deflections of beams with
approximately the same moment of inertia and modulus of elasticity, are affected
significantly by a change in the span. From these figures, it is clear that for a given f‟c
and a/d, an increase in Vs results in a decrease in the deflection at the same load. Thus, it
can be concluded that the beam stiffness increases as Vs increases. This can be attributed
to the fact that as Vs increases, the propagation and widening of the cracks is reduced.
This results in higher stiffness of the beams
Stirrups Effectiveness
Stirrup effectiveness of the shear reinforcement is defined as the increase in the
ultimate shear stress (Vn) above the diagonal cracking stress (Vc). Figs 14 and 15 show
a relation between an increment of stress (Vn-Vc) versus the nominal shear stress
provided in the form of stirrups. It can be concluded from theses figures that for the
beams with f‟c of 10,000 psi, the stirrups effectiveness increases with an increase in the
amount of web reinforcement provided. Since Beam 1 and 2 failed in diagonal tension,
16
their stirrup contribution is extremely low. Hence the stirrup effectiveness increases at a
higher rate for beams with a/d of 3 and 4. However, the rate of increase of stirrup
effectiveness is lower for a/d of 5. This can be attribute to the fact that all the three beams
failed in shear compression.
For beams with f‟c of 12,500 psi, the stirrup effectiveness decreased slightly and then
increased for increasing Vs for beams with a/d of 3 and 5. The beams with a/d of 4
showed an increase of stirrup effectiveness with increase in Vs.
CONCLUSIONS
The following conclusions are made from this study:
1. The inclusion of web reinforcement generally changes the failure mode for beams
with a larger a/d ratio to a pre-warned shear compression failure.
2. Determination of Vc from stirrup strains or crack patterns results in
approximately the same value.
3. Ultimate shear capacity (Vn) and the diagonal cracking load (Vc) increases with
an increase in f‟c.
4. The ratio of Vn actual to Vn predicted generally decreases with an increase in f‟c;
the ratio of Vc actual to Vc predicted generally decreases with an increase in f‟c;
17
and as the compressive strength of concrete increases, the splitting tensile strength
of concrete decreases.
5. Vc and Vn, both increase with an increase in a/d ratio.
6. As a/d ratio increases, the number of cracks and the penetration of flexural cracks
at mid-span increased.
7. As Vs increases with f‟c and a/d constant, the rate of increase in the stirrup strains
decreases.
8. The minimum shear steel criteria according to ACI 318-83 is not conservative for
beams with concrete strength of 10,000 psi.
9. ACI 318-89 code is conservative for high strength concrete beams in shear.
RECOMMENDATIONS
1. It is recommended that a smooth transition of vs minimum, from low-strength to
high-strength concrete be used rather than the abrupt 50 psi increase at f‟c of
10,000 psi. More testing should be done to define this transition.
2. It is recommended that the value of 50 in ACI (318-89) equation (11-14), for f‟c
in the range from 10,000 to 15,000 psi be changed to (0.004f‟c + 40) and be
exempt from the f‟c/5000 correction (Section 11.1.2.1) for f‟c. 10,000 psi.
3. It is recommended that the minimum vs be based on the a/d factor as well as f‟c.
Further testing should be done to evelop the proper recommended changes.
4. The current ACI Code equation for Vc was developed on the basis of more than
400 tests of multiple beams, therefore, base upon this precedent more beams with
18
high-strength concrete and shear reinforcement should be tested. Since research
for shear high-strength concrete beams to date has been primarily on beams with
out shear reinforcement and most practical applications include beams with shear
reinforcement, it is recommended that future test include beams with shear
reinforcement.
19
NOTATIONS
a = Shear span, distance between the concentrated load and the
face of the support
a/d = Shear span to depth ratio
As = Area of non-prestressed tension reinforcement, in2
Av = Area of shear reinforcement within distance „s‟ in2
b = Width of compression face of member, in
bw = Web width, in
d = Effective depth of the beam, in
Ec = Modulus of Elasticity of concrete, psi
Es = Modulus of elasticity of reinforcement, psi
f‟c = Specified compressive strength, psi
fy = Specified yield strength of non-prestressed reinforcement,
psi
Mcr = Moment causing flexural cracking at section due to
externally applied loads
s = Spacing of shear or torsion reinforcement in direction
parallel to longitudinal reinforcement, in
Va = Shear force carried by aggregate interlock, psi
Vd = Shear force carried by dowel action
Vc = Nominal shear strength provided by concrete
vc = Permissible shear stress carried by concrete, psi
Vn = Nominal shear strength
Vs = Nominal shear strength provided by shear reinforcement
p = Ratio of non-prestressed tension reinforcement = As/bd
pw = As/bwd
vs = Vs/bd, the strength provided by the shear reinforcement in
nominal shear stress
20
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22
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20. Shin, Sung-Woo, Lee, Kwang-Soo, Moon, Jung-Ill, and Ghosh, S.K., “Shear
Strength of Reinforced High-Strength Concrete Beams with Shear Span-to-Depth
Ratios between 1.5 and 2.5,” ACI Structural Journal, V.96, No.4, July 1999, pp.
549-556.
21. Tan, Kang-Hai, Kong Fung-Kew, Teng, S., and Guan, Lingwei, “High-Strength
Concrete Deep Beams with Effective Span and Shear Span Variations,” ACI
Structural Journal, V.92, No.4, July-August, 1995, pp. 395-405.
22. Ozcebe, G., Ersoy, U., and Tankut, T., “Evaluation of Minimum Shear
Reinforcement Requirements for Higher Strength Concrete,” ACI Structural
Journal, V.96, No. 3. May 1999, pp. 361-368.
23. Oh, Jung-keun, and Shin, Sung-Woo, “Shear Strength of Reinforced High-
Strength Concrete Beams,” ACI Structural Journal, V.98, No.2, March 2001,
pp.164-173.
23
Table 1. Test program for high-strength concrete beams
Beam
No.
Width of
beams (b),
in.
Effective
depth (d),
in.
Comp.
Strength
(f‟c), ksi
Shear
span (a),
in.
Shear span-
depth ratio
(a/d)
vs, psi Tension steel
(As), square in.
No. of ¼
in Stirrups
1 10 16 10.18 54 3.375 50 2.64 (6#6) 30
2 10 16 10.18 70 4.375 50 3.60 (6#7) 44
3 10 16 10.18 86 5.375 50 4.74 (6#8) 42
4 10 16 10.18 54 3.375 75 3.02 (4#7+2#5) 28
5 10 16 10.18 70 4.375 75 4.04 (4#8+2#6) 34
6 10 16 10.18 86 5.375 75 5.16 (4#8+2#9) 38
7 10 16 10.18 54 3.375 100 3.28 (4#7+2#6) 32
8 10 16 10.18 70 4.375 100 4.36 (4#8+2#7) 38
9 10 16 10.18 86 5.375 100 5.58 (4#9+2#8) 44
10 10 16 12.25 54 3.375 60 3.02 (4#7+2#5) 34
11 10 16 12.25 70 4.375 60 4.04 (4#8+2#6) 40
12 10 16 12.25 86 5.375 60 5.16 (4#8+2#9) 46
13 10 16 12.25 54 3.375 90 3.28 (4#7+2#6) 34
14 10 16 12.25 70 4.375 90 4.74 (6#8) 40
15 10 16 12.25 86 5.375 90 5.58 (4#9+2#8) 46
16 10 16 12.25 54 3.375 120 3.78 (4#8+2#5) 40
17 10 16 12.25 70 4.375 120 5.16 (4#8+2#9) 46
18 10 16 12.25 86 5.375 120 6.54 (6#8+3#7) 54
24
Table 2 Concrete mix and test data
Mix No. 1 2 3
Nominal Strength 10,000 11,000 12,000
Cement, Type I, lb/yd3 600 700 700
Fly Ash, Type C, lb/yd3 350 100 100
Silica Fume, lb/yd3 (gallons) -- 70 (12.7) 100 (18.2)
Water, lb/yd3 303 240 274
Water to cementitious ratio 0.3 0.29 0.30
Sand, SSD, lb/yd3 1,200 1,280 1,250
½ inch Max. crushed limestone, SSD, lb/yd3 1,650 1,700 1,700
Slump, in. 6 7.5 10.5
Air Temperature, Deg. F 68 68 69
Concrete Temperature, Deg. F 72 69 68
Concrete Density, pcf 152 152 154
ASTM Type A Retarding Admixture, oz/yd3 28.5 20.8 21
ASTM Type F Superplasticizing Admixture,
oz/yd3
198 210 240
25
Table 3 Concrete strength test data for 10,000 psi specified strength
Test
Age,
Days
Compressive strength, psi Splitting tensile strength
4” x 8” cylinders 6” x 12” cylinders
Actual Average
Actual Average Actual Average
1 3519
3527
3343
3460
3731
3855
3183
3590
384
406
371
390
3 5095
5573
5175
5280
6667
--
6596
6630
424
539
565
510
7 8280
7643
8917
8280
7463
6438
7746
7220
508
548
486
510
14 8638
7245
8280
8050
8277
7728
9249
8420
592
570
574
560
28 10191
8280
10151
950
10350
10085
10209
10210
752
730
699
730
56 8837
10788
10800
10140
--
--
699
690
743
710
91 12900
13850
9160
11970
--
--
606
920
774
770
1 psi = 0.006895 MPa
26
Table 4 Concrete strength test data for 12,500 psi specified strength
Test
Age,
Days
Compressive strength, psi Splitting tensile strength
4” x 8” cylinders
6” x 12” cylinders Actual Average
Actual Average
Actual Average
1 3384
3503
3702
3530
4494
4565
4547
4590
354
358
385
370
3 6369
6449
5892
6240
5697
6016
7502
6900
367
429
376
390
7 7484
7803
8121
7800
8563
8581
8139
8430
557
584
601
580
14 9713
9475
11057
10090
10227
11058
10952
10750
690
659
760
700
28 10350
10948
9953
10420
10580
10828
10757
10720
836
849
915
870
56 10828
11544
11146
11170
--
--
924
902
937
920
91 11540
10788
11186
11180
--
--
841
1040
707
870
27
Table 5 Modulus of elasticity test data
Age, Days Modulus of Elasticity*, psi
f‟c = 10,000 psi f‟c = 12,500 psi
1 3,750,000 3,700,000
3 4,050,000 4,100,000
7 4,850,000 5,150,000
14 5,400,000 5,750,000
28 5,450,000 6,000,000
56 5,750,000 6,000,000
* Average of three specimens
1 psi = 0.006895 MPa
28
Table 6
Beam
No.
Actual
f‟c
pw pw/pb a/d Vc
Predict,
kips
Vc
Strains,
kips
Vc,
Cracks,
kips
Vc,
psi
Vs,
psi
Vn,
Calcu,
kips
Vu
(Actual),
kips
Ratio
of
Vu(act
and Vn
(predic
Final
Deflection,
in.
Type
of
failure
1 10180 0.017 0.333 3.375 32.6 30 30 204 52 40.9 32 0.78 0.42 DT
2 10180 0.023 0.444 4.375 32.7 30 35 205 55 41.5 35 0.84 0.82 DT
3 10180 0.030 0.598 5.375 32.9 35 35 205 52 41.2 44 1.07 0 SC
4 10180 0.019 0.381 3.375 32.9 32 35 206 75 45.0 43 0.96 0.64 DT
5 10180 0.025 0.581 4.375 33.0 33 35 206 75 45.0 51 1.13 1.56 SC
6 10180 0.032 0.551 5.375 33.1 36 35 207 75 45.1 60 1.33 1.67 SC
7 10180 0.021 0.414 3.375 33.1 36 35 207 98 48.8 53 1.09 0.71 SC
8 10180 0.027 0.550 4.375 33.2 34 30 207 98 48.9 62 1.26 1.51 SC
9 10180 0.035 0.704 5.375 33.3 40 40 208 98 49.0 57 1.16 1.45 SC
10 12250 0.019 0.317 3.375 35.9 40 38 224 57 45.0 55 1.22 0.79 SC
11 12250 0.025 0.424 4.375 36.0 32 30 225 57 45.1 41 0.91 0.83 SC
12 12250 0.032 0.541 5.375 36.0 33 40 225 57 45.2 50 1.11 1.49 SC
13 12250 0.021 0.344 3.375 36.1 38 30 225 90 50.5 50 0.99 0.64 SC
14 12250 0.030 0.497 4.375 36.4 35 30 227 90 50.8 54 1.06 1.18 SC
15 12250 0.035 0.585 5.375 36.2 41 35 227 90 50.7 56 1.10 1.48 SC
16 12250 0.024 0.396 3.375 36.4 35 30 228 121 55.9 59 1.06 0.78 SC
17 12250 0.032 0.541 4.375 36.6 38 40 229 121 56.0 61 1.09 1.04 SC
18 12250 0.041 0.686 5.375 36.7 44 45 229 90 56.1 78 1.39 1.85 SC
29