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.

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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.

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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

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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,

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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

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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

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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.

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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