LI ECC - 看图王 PDF
LI ECC - 看图王 PDF
Victor C. Li
Advanced Civil Engineering Materials Research Laboratory,
Department of Civil and Environmental Engineering,
University of Michigan, Ann Arbor, MI 48109-2125, USA
*
To appear in Fiber Reinforced Concrete: Present and the Future, Eds: N. Banthia,
A. Bentur, and A. Mufti, Canadian Society of Civil Engineers, 1997.
INTRODUCTION
The application of fiber reinforced concrete (FRC) can be grouped into two
general classes: Thin sheet products and bulk structures. When FRCs are used in
thin sheet products, such as cladding walls, roofing tiles, or pipes, they typically
have the following characteristics (1): A high fiber volume fraction is used, often in
the range of 3-10% with special processing methods (e.g. Hatschek, spray up, lay-
up, extrusion and pultrusion processes) to accommodate the high fiber volume
fraction. The fibers are often in continuous and aligned form (although one of the
most successful form -- asbestos cement, has short random fibers in high volume),
to take advantage of the simple geometric shape of the thin sheet product and to
optimize the reinforcement efficiency of the fibers. As a result of this high
efficiency and fiber volume fraction, this type of FRC often exhibits excellent
mechanical performance in tension and bending, e.g. (2,3), to the extent that
primary steel reinforcement is not needed. Alternatively, one can view the thin
sheet product as one in which primary steel reinforcement is difficult to place, and
therefore the FRC must be able to serve as primary load carrier. The excellent
performance of this type of FRC is often reflected by its strain-hardening behavior
beyond first cracking in tension. Despite this excellent performance, application of
this type of FRC is limited by the simple geometric shape requirement. The pre-
cast nature and special processing needs also restrict the extension of this class of
FRC to applications in cast-in-place and bulk structures.
FRC applications in bulk structures can be subdivided into approximately
three different classes, according to their fiber volume fraction and intended
functions of the reinforcing fibers. Fibers in low fiber volume fraction (<1%) FRCs
are often used for plastic shrinkage crack control. The fibers usually do not serve
any structural functions. Moderate fiber volume fraction (between 1 and 2%) FRCs
are versatile materials which can be found in both cast-in-place and pre-cast
structural members. Because the fibers used are typically chopped and because of
the low volume percentage, regular concrete mixing and casting processes can be
employed. This type of materials are characterized by their improved modulus of
rupture (MOR), fracture toughness, fatigue resistance, impact load resistance and
other desirable mechanical properties (1,4). The fibers in such FRCs are often
regarded as secondary reinforcement used in conjunction with main reinforcing
steel. Attempts at partial replacement of shear steel reinforcement (e.g. 58) and
reinforcement for crack width control (9,10) have been successful. Even so, the
application of this type of FRC often find obstacles in cost/performance
justifications. Recent progress at integrating structural performance with tensile
characteristics of this type of FRC should place the cost/performance at an
increasing advantage. For example, in an ongoing EU project, the tensile stress-
crack width relationship of the FRC is taken into account in the design of a
continuous FRC pavement overlay (11) for handling the expected thermal and
mechanical tensile load. Similarly, in the design of tunnel linings (12), the tension-
softening curve of the FRC is taken into account.
Table 1: Cement Based Fiber Composite Material Constituents and Their Properties
CONSTITUENTS PROPERTIES
Fiber Elastic modulus, tensile strength,
length, diameter, volume fraction
Matrix Fracture toughness, elastic modulus, initial flaw size
Interface Bond properties, snubbing coefficient
Stress
Strain-hardening
Quasi-brittle
B Brittle
A
Strain
the same order of magnitude as that for brittle materials. Strain-hardening materials
are characterized by their ability to sustain increasing levels of loading after first
cracking while undergoing large deformation (curve C). The ultimate strain value
(at peak tensile load) of a strain-hardening material can be orders of magnitude
higher than that of brittle or quasi-brittle material.
One of the most important conditions for the transition of quasi-brittle to
strain-hardening failure mode is the presence of 'steady state' cracking (2225). In
fiber composites, the extension of a matrix crack is accompanied by fiber bridging
across the crack flanks. As the matrix crack extends, the bridging zone increases in
length. During crack opening, the bridging stress increases as fiber/matrix
interfaces debond and the debonded segments of fibers stretch. When the bridging
stress increases to the magnitude of the applied load, the crack flanks flatten to
maintain the constant applied stress level (26). This load level is termed the steady
state cracking stress. The crack has now gone into the steady state cracking mode,
extending without the need of further increase in applied load. Thus during steady
state cracking, the tensile load is independent of crack length. This is in contrast to
the well known Griffith residual strength concept, which relates a decreasing tensile
load to increasing crack size.
Based on a J-integral analysis of a steady state crack, Marshall and Cox (26)
showed that
ss
Jc = ss ss ( )d (1)
0
ss o (2)
Eq. (1) and (2) together provide a general condition for transition from quasi-
brittle to strain-hardening failure mode. Apart from steady state cracking condition,
it is also necessary for the critical flaw size dependent first crack strength to be less
than the maximum bridging stress (22). Otherwise, the bridging fibers will not be
able to bear the tensile load shed by the matrix at first crack.
For Eq. (2) to be useful in fiber, matrix and interface tailoring, it will be
necessary to determine the bridging law specific for a given composite system. In
fiber reinforced cementitious composite in which the fibers are randomly oriented
and in which pull-out (rather than fiber rupture) are expected, Li (27) shows that the
bridging law can be derived as
o [ o o] for
2( / )1/ 2 ( / )
o
( ) = o (1 2 / L f )
2
for o L f / 2 (3)
0 for Lf / 2
1 L
o = g V f f (4)
2 df
Corresponding equations for cases where fibers can rupture and for cases
where fibers are of variable length can be found in (28, 29). In Eqs. (3) and (4), Vf,
Lf, df, and Ef are the fiber volume fraction, length, diameter and Young's Modulus,
respectively. is the fiber/matrix frictional bond strength, and the snubbing factor
2
g= 2 (1+ e )
f / 2
(5)
(4 + f )
12J c
crit
Vf (6)
g (L f / d f ) o
becomes exact only for low Vf case due to the -term in o) of Lf. (Note that o
scales with the square of Lf). While long fibers are preferred, difficulty in
processing because of poor workability puts a limit on the choice of fiber length.
Further, long fiber length can lead to fiber rupture and poor post-peak behavior. On
cu = o (7)
Hence the ultimate composite strength can also be controlled by tailoring the fiber
volume fraction, the interface frictional bond strength, and the fiber aspect ratio
Lf/df, according to (4) and (7). The ultimate strength is not affected by the matrix
properties.
To facilitate composite constituent selection, design charts (30) such as that
shown in Fig. 2 can be used in lieu of Eq. (6).
0.10
2
Jc = 0.02 kJ/m
0.08
2
0.015 kJ/m
0.06 0.010 kJ/m 2
Vfcrit
2
0.005 kJ/m
0.04
0.02
0.00
0.4 0.6 0.8 1.0
Bond Strength (MPa)
Fig. 2: Matrix Toughness & Interfacial Property Effect on Critical Vf. (Ef = 120
GPa, Lf = 12.7 mm, df = 0.038 mm, g = 2.0, Em = 25 GPa)
MATERIAL
ECCs can be formed with a variety of fibers, including polymeric (23), steel
(31) and carbon (29). The matrices used are mostly cement and mortar. So far,
most research has been conducted with a high modulus polyethylene fiber (Trade
name Spectra 900) in a cement matrix, and this material forms the basis of
discussions in this paper. The properties reviewed below involves Spectra ECCs
the composition of which has varied somewhat from test to test, carried out over the
last six years at the University of Michigan. Typical material composition and mix
proportions are given in Table 2. For the exact mix design, the reader is referred
back to the original publications. Fiber properties are given in Table 3. Most
mechanical properties described below are for a composite with 2% of fibers by
The interfacial bond property has been measured (32) both by single fiber
pull-out test as well as by back calculation based on ultimate strength measurement
(using Eq. (4)). The range of = 0.5 to 0.7 appears typical for the type of matrices
used. It is known to depend on age, matrix composition, and even fiber volume
fraction. This low bond property can be increased by a factor of two or more by
means of plasma treatment (32). The snubbing factor g has not been measured for
this material system, although polypropylene and nylon fibers in cementitious
systems show snubbing factors of 1.8 and 2.3, respectively (33). For the present
purpose, a value of g = 2 has been assumed.
The fracture toughness Km measured using LEFM techniques (34) yields a
value of 0.33 MPaP 7KH HODVWLF PRGXOXV Em has been estimated (30) using
Hirch's formula to be about 23 GPa, based on its age and w/c ratio.
Using Eq. (6) and the above parametric values, the critical fiber volume
fraction is estimated between 0.5 % and 1%. It should be understood that this is a
rough estimate, since the exact in-situ values of matrix and interface parameters has
not be measured directly. At any rate, the critical fiber volume fraction is well
below 2%. Hence a composite with 2% fiber should satisfy the condition of pseudo
strain-hardening, and exhibit high strain capacity after first cracking.
The polyethylene fibers are supplied by the manufacturer in bundle-like
form. Prior to mixing, the fibers were dispersed using air pressure for
approximately one minute. Then the amount of fibers needed for the mix was
weighted. After measuring the weight of all mix constituents, the cement was
poured into a three speed (Hobart) mixer with a planetary rotating blade. Silica
fume was slowly added to the cement when the mixer has been started. Then water
and superplasticizer were mixed together and slowly added. When all water and
superplasticizer were added and the cement paste mix became uniform, the
LVDT Output
LOAD LOAD
LVDT
Specimen
205 304.8
LVDT Holder
Epoxy
LOAD
Aluminum Plate
LOAD
12.7
76.2
FRC
5
ECC
4
Stress (MPa)
0
0 2 4 6 8
Strain (%)
5
(c)
4 (b)
(d)
(a)
Stress (MPa)
0
-1 0 1 2 3 4 5 6 7
Strain (%)
Fig. 5: Uniaxial Tensile Stress-Deformation Record for ECC
Compressive Strength
Compression cylinders (7.62mm x 15.24mm) were tested in an Instron
Model 8000 test system with a 2500 kN capacity loading frame (30). Each cylinder
was tested under displacement control at a loading rate of 0.0254 mm/s.
Compressive stress-strain curves for the ECC and FRC are shown in Fig. 7.
The compressive strength of this ECC, about 68.5 MPa, is not significantly higher
than that of the FRC (55 MPa). The compressive strain capacity has been observed
to increase by approximately 50%-100% over normal concrete and FRCs. Post-
peak ductility of ECCs are expected to be similar to that of normal FRCs.
The modulus of ECC, as in ordinary concrete, depends on the amount of
aggregates used. However, the presence of aggregates also changes other properties
80
70 FRC
60 ECC
Stress (MPa)
50
40
30
20
10
0
0 0.5 1 1.5 2 2.5 3
Strain (%)
Modulus of Rupture
The geometry and loading configuration of the flexural beam specimens are
shown in Fig. 8. This experimental set-up is recommended in ASTM C78-75,
standard test method of flexural strength of concrete (using simple beam with third-
point loading). The flexural tests were conducted in the same MTS testing system
as the uniaxial tensile tests. The specimens were loaded to complete failure with a
constant cross head speed (0.01 mm/s). The load, head displacement of the
machine, and deflection of the beams at the middle point were recorded in each test.
More details of the test set-up can be found in (35).
P (unit : mm)
76.2
101.6
measuring point
15
FRC
ECC
Flexural Stress (MPa)
10
0
0 2 4 6 8 10
Deflection (mm)
The crack pattern of the ECC is distinctly different from plain concrete or
normal FRC. The first crack started inside the mid-span at the tensile face, and
multiple cracks developed from the first cracking point and spread to the outside of
the mid-span. The multiple cracks in the outside of the mid-span were inclined
cracks similar to shear cracks in steel reinforced concrete (R/C) beams. As the
MOR is approached, one of the cracks inside the mid-span started to open up after a
large damage zone had been developed. The through-thickness damage zone can
2
reach an areal dimension of 200 cm . Fig. 10 shows a typical cracking pattern that
develops in the beam middle span around the peak load.
For ideally brittle material, the MOR to tensile strength ratio is unity. For
quasi-brittle material such as concrete or FRC, this ratio lies between 1 and 3. The
upper limit describes the case of a elastic-perfectly plastic material. For the case of
ECC, this ratio can be expected to be higher than 3 due to the strain-hardening
nature after first crack. This expectation is confirmed by the test results (35, 36)
which show that the ratio is equal to 5.0 for the ECC, compared to 2.5 for the FRC.
Fig. 10: Cracking Pattern in ECC Beam Mid-Span Around Peak Load
E399-78) was employed at both the top and bottom of the DCB specimens to allow
rotation as the specimens were loaded. The specimens were loaded to complete
failure with a constant cross head speed; the testing time was typically 40 minutes
for all tests. The load-line displacement L was measured using two LVDTs. The
total fracture energy was determined by means of the J-based technique described
by Li et al (38) and using a set of DCB specimens with different notch lengths.
Concurrently with the tests, damage evolution on the specimen surface was
recorded using a camera. The size of the specimen has been chosen to ensure
steady state crack growth.
P LVDT
L H
t
B
W Holder
(c)
(b) (d)
150
(a)
100
FRC (W = 127 mm, a = 75 mm)
ECC (W = 490 mm, a = 241 mm)
50
0
0 5 10 15 20 25 30
Load Line Displacement (mm)
POTENTIAL APPLICATIONS
ECCs are relatively new materials. Although further refinement and
optimization are expected, preliminary considerations suggest that the material can
Shear
panel 210
A
50 50 50
185 170 185
150
540
50
50 150
50
210
50 330 160
The Ohno shear beams were tested in an Instron Model 8000 test system with
a 500 kN capacity loading frame. Tests were run under displacement control at a
loading rate of 0.381 mm/min. Total test time was approximately 10 minutes. The
loads were applied through rollers resting on 25.4 mm wide thin aluminum spreader
plates glued to the specimen. The beams were placed on roller supports.
The shear load versus deflection curve is shown in Fig. 15. Beam deflection
was measured by a LVDT located under the interior load point. After first crack
strength, the ram load continued to increase. The pseudo strain-hardening behavior
of ECC revealed itself in the form of multiple diagonal cracks (Fig. 16) with small
crack widths of less than 0.1 mm even up to ultimate load. In contrast, the FRC
beam failed shortly after first crack load with a single crack opening as the crack
width increased at continuously softening load. It is clear from Fig. 15 that the
ductility of the ECC beam is extensive both pre-peak and post-peak. Indeed Li et al
200
FRC
ECC
150
Ram Load (kN)
100
50
0
0 1 2 3 4 5
Interior Load Point Deflection (mm)
Fig. 15: Ram Load versus Deflection Curves of ECC and FRC Ohno Shear Beams
The average shear strength in the Ohno shear beams was estimated as the
shear force at the beam center line (which is one-third of the ram load) divided by
the cross-sectional area resisting the shear force. The ECC system failed at a stress
of 5.09 MPa, compared to 3.03 MPa for the FRC. The unique ductility gain in the
ECC beam is reflected by the average shear strain of 2.6% at ultimate load
compared to 0.6% for the FRC.
The ductile shear response in ECC suggests that ECCs can be utilized in
structures where intense shear loading can be expected, such as experienced by
some concrete bridge decks (45), in concrete elements connected by steel anchors,
and where conventional shear reinforcement is desired but prevented from adoption
due to reinforcement congestion or otherwise.
Fig. 16: Crack Pattern of Shear Panel, After Peak Load Reached
In the proposed design of the R/C member, a layer of ECC is substituted for
the concrete that surrounds the main flexural reinforcement (Fig. 17). This ECC
layered beam has the same cover thickness as for a control specimen, which has a
regular concrete cover. Two performance requirements are imposed on the ECC
material to serve its intended purpose: (1) the ultimate tensile strain capacity of the
ECC material should be greater than the maximum strain that can be developed in
the outermost fiber at the tensile face of the R/C beam, and (2) the crack width at
the ultimate strain capacity of the ECC material (hereafter referred to as ultimate
crack width) should be less than the maximum crack width allowed in a particular
environment. The first condition ensures that no strain localization will take place
in the ECC layer, and the second condition ensures that the crack opening in the
ECC is maintained below the allowable value. Assuming that at ultimate load, the
strain in the extreme compression fiber of the concrete is equal to 0.003, and that
plane sections remain plane, the strain in the extreme tension fiber of the ECC is
found to be equal to 0.013. Therefore, the ECC that should be selected should have
an ultimate strain capacity at least equal to 0.013. In addition, suppose that the
member is to be exposed in an environment of seawater and seawater spray under
wetting and drying. In this case, according to ACI Committee 224, the crack width
should be limited to 0.15 mm. Therefore, the ultimate crack width of ECC should
be less than 0.15 mm. The 2% Spectra ECC discussed above was found to satisfy
114
16
=5
102
127
= 10
152
25
16
13
(unit = mm)
152
Fig. 17: Geometry of the R/C beam with ECC layer and reinforcement details
Figure 18 shows the test results, in the form of moment curvature and crack
width curvature diagrams, for both beams. As shown in this figure, there is no
significant difference between the moment curvature response of the two beams.
The beam with the ECC layer shows a 10 % higher load and curvature at failure.
The crack width-curvature response of the two beams is, however, significantly
different. Fig. 18 shows that the crack width in the control specimen increases
almost linearly as function of curvature. Before yielding of the reinforcement
(moment N1-m) the width of the crack at the bottom of the beam is
maintained below 0.20 mm. After yielding, the load starts to increase at a much
slower rate while the crack width continues to increase at the same rate. At peak
load the width of the crack is approximately equal to 1.52 mm. For the ECC
layered beam, the crack width maintains a small value at all times. The crack width
first starts to increase almost linearly as function of curvature, and then actually
decelerates at higher curvatures. Before yielding of the reinforcement (moment
15
1.2
10
0.8
0 0
0 0.05 0.1 0.15 0.2 0.25
Curvature (1/m)
Fig. 19 shows two series of pictures illustrating the increase of crack width as
function of load for both the control R/C beam (series a) and the R/C beam with the
ECC layer (series b). The load level is indicated on the bottom of each picture.
These series of pictures indicate a significant difference in the cracking behavior of
the two beams. They illustrate that for a given load level, the crack width in the
ECC layered R/C beam is always smaller than that in the control specimen, and that
the rate of crack width increase as function of load is much higher for the latter.
The overall crack patterns of the beams are shown in Fig. 20. For the control
R/C beam, the commonly observed crack pattern (Fig. 20a) with tensile cracks
emanating from the concrete cover was observed. For the ECC layered beam, the
cracks in the concrete material diffused into many fine cracks when they met the
ECC layer (Fig. 20b). This phenomenon appears similar to the crack pattern
observed in Double Cantilever Beam (DCB) fracture specimens (Fig. 13). When a
large crack develops in concrete it is accompanied by a strain concentration at the
location where this crack meets the ECC material. Because of the stress transfer
capability of the reinforcing fibers in the ECC material, stress redistribution occurs
so that localized fracture is delayed. In fact, localized fracture may never develop in
the ECC layer if the maximum strain in the layer is kept below the material ultimate
strain capacity. Consequently, an expanded zone of matrix cracking must develop
in the ECC layer prior to localized fracture.
(a) (b)
Fig. 19: Variation of Crack Width as Function of Load: (a) Control R/C Beam; (b)
R/C Beam with ECC Layer
(a)
(b)
Fig. 20: Crack Pattern (a) Control R/C Beam; (b) R/C Beam with ECC Layer
10 in.
3"
12.5 in. 3" 10 in.
1"
3"
11 in.
Results and discussions. The load vs. deflection hysteretic behavior is shown
in Fig. 24. For the PC hinge, the displacement ductility factor defined as the ratio
of ultimate deflection (corresponding to a failure load that is about 20% lower than
the maximum load carrying capacity) to yield deflection of about 4.8. For the ECC
hinge, the displacement ductility factor increases to 6.4, with less amount of
pinching and a much reduced rate of stiffness degradation (50). The cracking
pattern (Fig. 25) was distinctly different with more cracking taking place in the
plastic hinge zone with ECC rather than the zone outside as in the case of the PC
control specimen. The damage is mostly in the form of diagonal multiple cracking
in perpendicular direction. Unlike the control specimen which fail in a
predominantly shear diagonal fracture, the ECC specimen fails by a vertical flexural
crack at the interface between ECC plastic hinge zone and the plain concrete at the
column face. No spalling was observed in the ECC hinge, whereas the concrete
cover mostly disintegrated in the control. The cumulative energy over the load
cycles for the two specimens are compared in Fig. 26 which shows that the ECC
hinge absorbs about 2.8 times as much energy as the control. The control specimen
does behave in a manner similar to the ECC hinge specimen in its range of
deflection. However the ECC specimen far out-performs the control specimen in
the deflection regime beyond 1.2".
#6
#2 d = 8.63 in.
h = 10 in.
#6
2.5 in.
5/8 in.
5/8 in.
#3
#6 #5
#2 at 4 in. c/c
10 in.
16 in.
RESEARCH NEEDS
Further research in ECCs are needed at both the material and structural
levels. On the material level, additional property characterization and
improvements are needed. These include, for example, the characterization of high
and low cycle fatigue behavior, shrinkage behavior and freeze-thaw durability.
2.0
9 y
1.5
1.0
Head Displacement (in.)
0.5
y
0.0
y
-0.5
-1.0
-1.5
9 y
-2.0
0 100 200 300 400 500
Reading No.
Research have been conducted in first crack stress and elastic modulus
enhancement by addition of aggregates (30). This involves, however, careful
design of matrix properties since the addition of aggregates leads to an increase in
Jc which can rapidly increase the amount of fibers needed in satisfying the
condition for pseudo strain hardening given in Eq. 6. Hence it is necessary to
balance the various composite property needs by properly adjusting the material
constituents. The delicate balance is greatly aided by the availability of the
micromechanical models.
On a more micro-level, it is necessary to systematically investigate the
tailoring of fiber, interface and matrix, guided by micromechanical principles.
Fiber properties are continually improved, and new fiber types arise on an almost
monthly basis. However, very few commercial fibers are especially geared towards
applications in
(a)
(b)
Fig. 25: Photograph of final failure, in the plastic hinge zone of
600
Sp #1 (PC) 548.0
400
300
200 196.0
100
0
0.0 0.1 0.4 0.8 1.2 1.6
Displacement (in.)
CONCLUDING REMARKS
This article reviews the various types of FRCs in use today, and suggest the
need for developing a new class of FRCs which has the strain-hardening property
but which can be processed with conventional equipment. It is demonstrated that
such a material, termed engineered cementitious composites or ECCs, can be
designed based on micromechanical principles. The result is a moderately low fiber
volume fraction (<2%) composite which shows extensive strain-hardening, with
strain capacity extending to several percent (up to 6% has been demonstrated), with
compressive strength in the typical high strength concrete range (about 68.5 MPa),
with fracture energy several times above typical FRCs (about 27 kJ/m2). Under
bending and shear loads, the ECC beams show extensive ductile behavior both pre-
and post-peak. In addition, the ECC material has indicated notch insensitivity in
double edged uniaxial tensile specimens, suggesting that high reliability can be
achieved in this type of composite. ECC has isotropic mechanical properties.
Further improvement in ECCs, guided by micromechanical principles, are under
investigations in the area of matrix modifications, interface tailoring, and fiber
design. Because of the processing flexibility, the material can be used for pre-cast or
cast-in-place structures, and broad classes of potential applications, such as those
which require structural ductility, energy absorption, and crack width control under
high strain imposition, are identified to be suitable to take advantage of the unique
properties of ECCs. Some additional potential applications include
1. High energy absorption structures/devices:
EQ resistant structures Columns, short span beam, beam-column connections
Seismic retrofits shear walls, dampers
Steel structures joints
Hybrid structures steel/RC connections
2. Structures subject to impact loads:
Pavements durability, reflective cracking
Building core element
Lightweight durable bridge decks
3. Large deformation structures:
Underground structures; conformability to soil deformation, leakage prevention
Concrete pipes
4. Others:
REFERENCES
1. Balaguru, P.N. and Shah, S.P., Fiber Reinforced Cement Composites, McGraw
Hill, 1992.
2. Krenchel, H., Synthetic Fibres for Tough and Durable concrete, in
Developments in Fiber Reinforced Cement and Concrete, RILEM Symposium,
R. N. Swamy, R. L. Wagstaffe and D.R. Oakley eds., 1986, pp. 333-338.
3. Krenchel, H. and Stang, H., Stable Microcracking in Cementitious Materials, in
Proceedings of 2nd Int'l Symp. on Brittle Matrix Composites, Brandt, A.M.,
and I.H. Marshall, 1988, pp. 20-33.
4. Bentur, A. and Mindess, S., Fiber Reinforced Cementitious Composites,
Elsevier Applied Science, 1990.
5. Batson, G., Jenkins, E. and Spatney, R., Steel Fibers As Shear Reinforcement In
Beams, ACI Journal, 69(10) 1972, pp. 640-644.
6. Sharma, A. K., Shear Strength Of Steel Fiber Reinforced Concrete Beams, ACI
Journal Proceedings, 83(4) 1986, pp. 624-628.
7. Swamy, R. N. and Bahia, H.M., The Effectiveness Of Steel Fibers As Shear
Reinforcement, Concrete International, 1985 pp. 35-40.
8. Li, V.C., Ward, R. and Hamza, A.M., Steel And Synthetic Fibers As Shear
Reinforcement, J. Materials, American Concrete Institute, 89(4) 1992, pp. 499-
508.
9. Stang, H. and Aarre, T., Evaluation Of Crack Width In FRC With Conventional
Reinforcement, Cement & Concrete Comp. 14(2) 1992, pp. 143-154.
10. Stang, H., Li, V.C. and Krenchel, H., Design And Structural Applications Of
Stress-Crack Width Relations In Fiber Reinforced Concrete, RILEM J. of
Materials and Structures, 28 1995, pp. 210-219.
11. Stang, H., Personal communication, 1994.
12. Horii, H. and Nanakorn, P., Fracture Mechanics Based Design Of SFRC Tunnel
Lining, in Proceeding of JCI International Workshop on Size Effect in Concrete
Structures, Oct. 31 - Nov. 2, 1993, Sendai, JAPAN, pp. 347-358.