behavior of reinforced concrete beams with minimum torsional reinforcement
TRANSCRIPT
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Engineering Structures 29 (2007) 21932205
www.elsevier.com/locate/engstruct
Behavior of reinforced concrete beams with minimumtorsional reinforcement
Hao-Jan Chiu, I-Kuang Fang, Wen-Tang Young, Jyh-Kun Shiau
Department of Civil Engineering, National Cheng Kung University, Tainan, 701, Taiwan, ROC
Received 17 February 2006; received in revised form 9 October 2006; accepted 8 November 2006
Available online 22 December 2006
Abstract
An experimental investigation was conducted on the behavior of thirteen high-(HSC) and normal-strength concrete (NSC) full-size beams
with relatively low amounts of torsional reinforcement. The crack patterns, the maximum crack widths at service load level, torsional strength,
torsional ductility, and post-cracking reserve strength results of the experiments are discussed. The main parameters include the volumetric ratio
of torsional reinforcements, the compressive strength of the concrete, and the aspect ratio of the cross section. It was found that the adequacy
of the post-cracking reserve strength for specimens with relatively low amounts of torsional reinforcement is primarily related to the ratio of the
transverse to the longitudinal reinforcement factors in addition to the total amounts of torsional reinforcement. The minimum requirements of
torsional reinforcement for NSC beams proposed by other researchers are also discussed on the basis of our test results of both HSC and NSC
beams.c 2006 Elsevier Ltd. All rights reserved.
Keywords:High strength concrete; Reinforced concrete beam; Torsion
1. Introduction
Structural elements such as spandrel beams in buildings,
curved beams, and eccentrically loaded box girders in bridges
are subjected to significant torsional moments that affect their
strength and deformation. The torsion design provisions in
the ACI Building Code before 1995 were based on the skew-
bending theory [1]. Since 1995, the design for torsion is based
on the thin-walled tube [2], and space truss analogy[3], which
covers both prestressed and nonprestressed concrete members.
The torsional cracking strength Tcr includes the effects of
concrete compressive strength, solid or hollow cross section,and level of axial or prestressing force.
Unlike the 1989 version of the ACI 318 Code [4], the
contribution of concrete to the ultimate torsional strength in a
structural concrete member was neglected, whereas the nominal
torsional moment strength specified in the ACI 318-05 Code [5]
is proportional to the amounts of transverse and longitudinal
Corresponding author. Tel.: +886 6 2757575x63163; fax: +886 6 2080565.E-mail address:[email protected](I.-K. Fang).
reinforcements, and the angle of the compression diagonals.
The code provisions also assume that both longitudinal and
transverse reinforcements yield prior to the ultimate strength
stage. Furthermore, the maximum shear stress is specified to
control the crack width. To prevent brittle and sudden failures
upon the formation of the first inclined cracking, the minimum
amount of transverse reinforcement specified in ACI 318-
05 Code [5] includes the effect of compressive strength of
concrete. Nevertheless, the test data used to validate the above
specification were primarily based on the beams subjected
to pure shear [68]. More details about the torsion design
provision in ACI 318-05 will be introduced in the followingparagraph.
Recently, Ali and White [9] proposed that the minimum tor-
sional reinforcement specified in the ACI 318-95 Code [10]
could result in a negative calculated minimum longitudinal rein-
forcement and cause unnecessary confusion to designers. Thus,
they suggested that the minimum required torsional reinforce-
ment should be a function of the torsional cracking strength.
Koutchoukai and Belarbi [11]investigated the effect of high-
strength concrete on the torsional cracking strength Tcr. They
also proposed the minimum required torsional reinforcement
0141-0296/$ - see front matter c 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.engstruct.2006.11.004
http://www.elsevier.com/locate/engstructmailto:[email protected]://dx.doi.org/10.1016/j.engstruct.2006.11.004http://dx.doi.org/10.1016/j.engstruct.2006.11.004mailto:[email protected]://www.elsevier.com/locate/engstruct -
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Notations
Acp area enclosed by outside perimeter of concrete
cross section, mm2
Ag gross area of concrete cross section, mm2. For a
hollow section,A gis the area of the concrete only
and does not include the area of void(s).Al total area of longitudinal reinforcement to resist
torsion, mm2
Al,min (ACI) minimum area of total longitudinal reinforce-
ment required for torsion, mm2
Ao gross area enclosed by shear flow path, mm2
Aoh area enclosed by centerline of the outermost
closed transverse torsional reinforcement, mm2
At area of one leg of a closed stirrup resisting torsion
within a distances , mm2
At,min (ACI) minimum cross-sectional area of one leg of
closed stirrups, mm2
bw web width, or diameter of circular section, mmfc specified compressive strength of concrete, MPa
fyl yield strength of longitudinal torsional reinforce-
ment, MPa
fyv yield strength of closed transverse torsional
reinforcement, MPa
pcp outside perimeter of the concrete cross sec-
tion, mm
ph perimeter of centerline of outermost closed
transverse torsional reinforcement, mm
s spacing of torsional reinforcement measured in
a direction parallel to longitudinal reinforce-
ment, mm
Tcr torsional cracking moment under pure torsion,kN m
Tn nominal torsional moment strength, kN m
x1 shorter overall dimension of rectangular part of
cross section, mm
y1 longer overall dimension of rectangular part of
cross section, mm
angle of compression diagonals in truss analogy
for torsion
associated with the minimum required torsional strength to the
torsional cracking strength.Experimental investigations on the torsional behavior of
reinforced concrete beams with relatively lower amounts of
transverse and longitudinal reinforcement are limited. The
effects of the ratio of transverse to longitudinal reinforcement
on the post-cracking reserve strength and crack control under
service conditions for members with the minimum amount
of torsional reinforcement still need to be discussed in the
literature. Therefore, this paper presents the test results of our
investigation of the behavior of reinforced concrete beams with
relatively low levels of torsional reinforcement and evaluates
the minimum torsional reinforcement provision in the ACI 318
Code.
2. Research significance
The crack patterns, crack width, post-cracking reserve
strength, and torsional ductility for NSC and HSC beams
with lower amounts of torsional reinforcement under pure
torsion were investigated. The main parameters included the
volumetric ratio of transverse to longitudinal reinforcement,compressive strength of concrete, aspect ratio of the cross
section, and hollow and solid sections. The minimum
requirements of torsional reinforcement for NSC beams
proposed by other researchers are also discussed according to
the test results.
3. Brief introduction of torsion design in the ACI 318-05
code
The design provisions for torsional cracking strength for
the nonprestressed concrete beam in ACI 318-05 Code [5]are
specified as follows:
Tcr =
fc
3
A2cp
pcp
for solid section (1)
Tcr =
fc
3
A2cp
pcp
Ag
Acp
for hollow section. (2)
Upon torsional cracking, the ACI 318-05 Code assumes that
the torsional resistance of a structural concrete member is pro-
vided mainly by closed stirrups, longitudinal reinforcements,
and compression diagonals, which construct a space truss. In
accordance with the space truss analogy and current torsion de-
sign provisions, the torsional strength and the required longitu-dinal reinforcement are specified as follows. The angle of the
compression diagonal is specified as varying from 30 to 60
deg.
Tn =2AtAofyv
scot (3)
Ao = 0.85Aoh (4)
Al =At
sph
fyv
fyl
cot2 . (5)
The ACI 318-05 Code requires a minimum amount of
torsional reinforcement to provide the torsional resistance when
the factored torsional moment exceeds the threshold torque
specified in Section 11.6.1 of the code. For pure torsion,
the minimum amount of closed stirrups is specified by the
following two equations, depending on whichever is greater:
2At,min (ACI) = 0.062
fcbws
fyv(6)
2At,min (ACI) 0.35bws
fyv. (7)
According to the Eq. (6), we find that the effect of the
compressive strength of concrete has been included in the
design of the minimum amount of transverse reinforcement.
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Fig. 1(a). Comparison of minimum transverse reinforcement requirements for
pure torsion.
The current design code also specifies the following minimum
longitudinal torsional reinforcement.
Al,min (ACI) =5
fcAcp
12fyl
At
s
ph
fyv
fyl. (8)
In order to ensure the development of the ultimate torsional
strength, to control crack width, and to prevent excessive
loss of torsional stiffness after the cracking of the reinforced
concrete member, the ACI 318-05 Code specifies the maximum
spacing of the torsional reinforcement in Section 11.6.6. The
spacing of transverse torsional reinforcement shall not exceed
the smaller of ph /8 or 305 mm. In addition, the provision of
the longitudinal reinforcement required for torsion is specified
in Section 11.6.6.2 of the ACI 318-05.
The effects of the concrete compressive strength on theminimum transverse, longitudinal, and total amount of torsional
reinforcement requirements specified in the current and older
versions of the ACI 318 Code are compared inFigs. 1(a)1(c).
4. Experimental program
4.1. Specimen details
Thirteen beam specimens, having rectangular cross sections
of 420 420 mm (y/x = 1.0), 350 500 mm (y/x = 1.43),
and 250 700 mm (y/x = 2.8), were constructed in the
laboratory and tested under pure torsion. The details, includingthe identification and design parameters of the specimens are
shown in Figs. 2(a) and 2(b) and Table 1. A clear concrete
cover to the outer surface of stirrups was 20 mm. Additional
transverse reinforcement was placed at both ends of the beam,
so that failure would occur in the central test region of the beam.
The test zone was 1.6 m wide to allow at least one complete
helical crack to form along each beam specimen.
The primary parameters consisted of the: (1) ratios of
transverse and longitudinal reinforcement (t= 0.13%0.61%,
l = 0.43%0.91%); (2) compressive strength of concrete
(fc = 3578 MPa); (3) aspect ratio of the cross section (A-
series (y/x = 1.0), B-series (y/x = 1.43), and C-series
Fig. 1(b). Comparison of minimum longitudinal reinforcement requirements
for pure torsion.
Fig. 1(c). Comparison of minimum torsional reinforcement requirements.
(y/x = 2.8)); and (4) hollow (H) and solid (S) sections.
In addition, we use the ratio of transverse to longitudinal
reinforcement factors t fyv /l fyl , the volumetric ratio of
the torsional reinforcements including the effect of the yield
strength of the reinforcement, to investigate the behavior of
the reinforced concrete beams with lower amounts of torsional
reinforcement subjected to pure torsion.
The HSC specimen HBS-82-13 inTable 1,designed with the
minimum amount of transverse reinforcement and maximum
spacing of transverse reinforcement (ph /8 = 190 mm) of
the ACI 318-05 Code [5], i.e., At/s = (At/s)min,(ACI)(t =
0.13%) and Al = 1.52 Al,min,(ACI)(l = 0.82%), hadits sum of torsional reinforcement ratios total = 0.95%.
Similarly, the NSC specimen NBS-82-13 was designed with the
maximum spacing of the transverse torsional reinforcements
(ph /8 = 190 mm), having At/s = 1.39(At/s)min,(ACI),
t = 0.13%, l = 0.82%, and total = 0.95%. Another
HSC specimen HBS-74-17 was designed with At/s =
1.35(At/s)min,(ACI), l = 0.74%, and total = 0.91%. The
ratios oft/l for the above three specimens ranged from 0.16
to 0.23.
The values of total for the other ten specimens, as shown
inTable 1, varied from 0.87% to 1.41%. The ratios oft/l for
these specimens varied from 0.43 to 1.0. Among them, the HSC
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Fig. 2(a). Elevation of the steel cage.
Fig. 2(b). Specimen details.
specimens HAS-51-50 and HCS-52-50 were designed with
Tn = 1.0Tcr and = 45 deg, which is equivalent to At/s =
1.99(At/s)min,(ACI). Similarly, the HSC specimen HBS-60-61hadTn = 1.2Tcr, = 45 deg, and At/s = 3.05(At/s)min,(ACI).
The NSC specimen NBS-43-44 was designed with Tn =
1.29Tcr and = 45 deg, and At/s = 3.02(At/s)min,(ACI). In
addition, the specimens HAH-81-35, NCH-62-33, and HCH-
91-42 with hollow sections were designed to compare with
those having solid sections.
4.2. Material properties
The concrete was supplied from a local ready mix plant. Two
types of concrete mixture, for the normal- and high-strength
concretes, were used and are shown in Table 2.For both types
of concrete, Type I Portland cement, Type F fly ash, slag, localcrushed aggregate with a maximum size of 10 mm, and local
river sand with a fineness modulus of 2.7 were used. Silica
fume (11% by weight of cement) with a specific gravity of
2.2 was used for the high-strength concrete. Superplasticizer
(ASTM C494 Type G) was used to improve the workability of
the mixtures for achieving the desired flow of 600 mm.
For each test beam specimen, six 150 300 mm concrete
cylinders and three 150 150 530 mm prisms were cast
as control specimens for basic material strength. The concrete
cylinders, prisms, and the test beams were stored together and
sprayed with curing compound several times during the curing
period until testing. The uniaxial compressive strength was
determined according to the average test results of three control
cylinders.
Mild steel bars were used as transverse and longitudinalreinforcements. The test yield strengths of the various sizes of
reinforcement used in the test beams are shown inTable 1.
4.3. Test setup and instrumentation
Details of the schematic test setup are shown in Figs. 3(a)
and3(b). Near the ends of the test region, the specimen was
clamped with steel torsional arms, which were loaded through
a steel transfer beam by the Shimatzu universal testing machine
to generate pure torsional loads. The support devices were
installed to ensure that the beam would be free to elongate in
the longitudinal direction and rotate in the transverse direction
during the test. At both ends of the central test region, aluminumrigs were tied to the surfaces of each specimen to measure the
rotation of its cross section. Four electronic dial gauges were
used to measure the relative deflections of the aluminum rigs,
which were transformed into the rotation of the cross section.
The twist of the test region was determined from the relative
rotations of the two aluminum rigs at the sides of the test
region.
Electrical resistance strain gauges were mounted on the
stirrups and longitudinal reinforcements in the test region to
monitor the strain variations of the reinforcements, as shown
in Fig. 2(a). As shown in Fig. 4, copper target points were
attached to the front, back, and top side of the test region of
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Table 1
Details of test specimens
Specimen number y/x fc fyv fyl Longitudinal bars Stirrupss (mm) totaltfyvl fyl
Comments
(MPa) (MPa) (MPa) (%)
HAS-51-50 76.0 396 6-No. 4 and 2-No. 3 No. 3@120 1.01 0.95 Tn = 1.0Tcr; = 45
(l = 0.51%) (t= 0.50%) t/l = 0.98
NAS-61-35 48.0 394 4-No. 5 and 4-No. 3 No. 3@170 0.96 0.56 At/s = 1.77(At/s)min,(ACI)1.0 385 (l = 0.61%) (t= 0.35%) t/l = 0.57
HAH-81-35 78.0 493 4-No. 6 and 4-No. 3 No. 3@170 1.16 0.34 At/s = 1.39(At/s)min,(ACI)(l = 0.81%) (t= 0.35%) t/l = 0.43
HAS-90-50 78.0 400 8-No. 5 No. 3@120 1.40 0.53 At/s = 1.97(At/s)min,(ACI)(l = 0.90%) (t= 0.50%) t/l = 0.56
NBS-43-44 35.0 385 400 6-No. 4 No. 3@140 0.87 Tn = 1.29Tcr; = 45
(l = 0.43%) (t= 0.44%) 0.98 t/l = 1.02
HBS-74-17 67.0 600 505 4-No. 6 and 2-No. 3 No. 2@140 0.91 0.27 At/s = 1.35(At/s)min,(ACI)(l = 0.74%) (t= 0.17%) t/l = 0.23
HBS-82-13 67.0 600 493 4-No. 6 and 4-No. 3 No. 2@190 0.95 0.19 At/s = (At/s)min,(ACI)1.43 (l = 0.82%) (t= 0.13%) t/l = 0.16
NBS-82-13 35.0 600 493 4-No. 6 and 4-No. 3 No. 2@190 0.95 0.19 At/s = 1.39(At/s)min,(ACI)(l = 0.82%) (t= 0.13%) t/l = 0.16
HBS-60-61 67.0 385 402 4-No. 5 and 2-No. 4 No. 3@100 1.21 0.97 Tn = 1.2Tcr; = 45
(l = 0.60%) (t= 0.61%) t/l = 1.02
HCS-52-50 76.0 396 6-No. 4 and 2-No. 3 No. 3@140 1.02 0.93 Tn = 1.0Tcr; = 45
(l = 0.52%) t= 0.50% t/l = 0.96
NCH-62-33 48.0 394 4-No. 5 and 4-No. 3 No. 3@210 0.95 0.52 At/s = 2.41(At/s)min,(ACI)2.8 385 (l = 0.62%) t= 0.33% t/l = 0.53
HCH-91-42 78.0 8-No. 5 No. 3@165 1.33 0.44 At/s = 2.40(At/s)min,(ACI)400 (l = 0.91%) (t= 0.42%) t/l = 0.46
HCS-91-50 78.0 8-No. 5 No. 3@140 1.41 0.53 At/s = 2.83(At/s)min,(ACI)(l = 0.91%) (t= 0.50%) t/l = 0.55
Note:t= AtPhAcp s 100%;l = AlAcp
100%;total = t+ l
#2:A s = 28.3 mm2; #3: A s = 71.3 mm
2; #4: A s = 126.7 mm2
#5:A s = 198.6 mm2; #6: A s = 286.5 mm
2.
beam specimens to provide full information about the average
surface deformations in the horizontal, vertical, 45 deg, and 135
deg directions. The relative displacements of the adjacent target
points were measured by an electronic digital caliper gauge
at each load stage during the test. The angles of the principal
compressive strain at mid-span during the test procedure were
obtained using the Mohrs strain circle. The electronic load cells
placed at the top of the steel torsional arms were used to monitor
the applied load. The data of load, twist, and reinforcement
strains of the beam were collected by a personal computer for
automatic data acquisitions.
4.4. Test procedure
During the tests, the torsional load was applied in a
controlled manner until several visible cracks occurred on the
surface of the specimen. The cracking torque Tcr and the
associated twist were recorded, and the specimen was then
loaded monotonically to failure. At every load stage after initial
cracking, the load was held constant for several minutes to
measure the crack widths. In addition, the crack propagations
were traced and marked on the surfaces of the specimens and
the maximum crack width was measured by using a magnifying
glass.
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Table 2
Concrete mixture proportions
Constituents(kg/m3) Target strength Target strength
70 MPa (HSC) 40 MPa (NSC)
Cement, 413 264
Silica fume, 44
Slag, 65 61Fly ash, 28 81
Sand, 622 725
Coarse aggregate, 988 1033
Water, 164 183
Superplasticizer, 12.1 4.9
(ASTM C 494 Type G)
Fig. 3(a). Schematic test setup.
Fig. 3(b). Schematic test setup at the end of specimen.
5. Test results and discussion
5.1. Crack patterns
The observed crack patterns of the test specimens are showninFig. 5.One major inclined crack initiated on the top and front
sides of the HSC specimen HBS-74-17 having relatively lower
ratio oft fyv /l fyl (total= 0.91%, tfyv /l fyl = 0.27), and
soon after that, the concrete on the back side of it was crushed as
shown inFigs. 5(a)and5(b).The crack pattern of this specimen
is similar to that assumed in the skewing bending theory[1].
According toFigs. 5(c)5(g),for the specimens with relatively
higher ratios of t fyv /l fyl , 0.440.97, we observe that the
smeared helical cracks were evenly distributed on the surface
in which the inclined concrete struts of the space truss analogy
Fig. 4. Location of targets on concrete surface.
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Fig. 5(a). Crack pattern of specimen HBS-74-17 after failure (front side).
Fig. 5(b). Crack pattern of specimen HBS-74-17 after failure (back side).
Fig. 5(c). Crack pattern of specimen NAS-61-35 after failure.
Fig. 5(d). Crack pattern of specimen HBS-60-61 after failure.
were developed to resist the external torque. Corner spallings
were observed on some of the test specimens.
The selections of the angle of the compression diagonal
for torsion design of reinforced concrete beams vary from
30 deg to 60 deg based on the current provisions of the
ACI 318-05 Code. If an angle of 45 deg is chosen for the
Fig. 5(e). Crack pattern of specimen NAS-61-35 after failure.
Fig. 5(f). Crack pattern of specimen HCH-91-42 after failure.
Fig. 5(g). Crack pattern of specimen NCH-62-33 after failure.
compression diagonal, it will end up with equal percentages
of reinforcement in the longitudinal and transverse directions,
i.e., t fyv = l fyl . However, if the selected angle deviates from
45 deg, the designed percentage of torsional reinforcement in
the longitudinal direction will differ from that in the transversedirection. The initial cracking angles of the specimens as shown
inFig. 5are about 4347 deg, except for the specimen HBS-74-
17, which failed shortly after its initial diagonal crack occurred.
The angles of the principal strain at the ultimate strength stage
of the thirteen specimens are about 3544 deg, which coincide
with the tendencies of the angles for the compression diagonals
calculated from the ACI 318-05 Code [5]. From Figs. 5(c)
and5(d), the angles of the principal strain at ultimate strength
stage for the specimens HAS-51-50 and HBS-60-61, having
tfyv /l fyl = 0.95 and 0.97, are very close to 45 deg. Also,
the deviations of the inclined angles at the ultimate strength
stage from those at the initial cracking stage are insignificant.
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However, as shown inFigs. 5(e)5(g),the angles of principal
strain at the ultimate strength stage for the specimens NAS-
61-35, HCH-91-42, and NCH-62-33, having t fyv /l fyl =
0.440.56, are approximately 3537 deg, which deviate about
79 deg from those at the initial cracking stages. The test results
validate the theory that the tendency of deviation of the angles
of the compression diagonal is mainly dependant on the ratio oft fyv /l fyl [12].
5.2. Crack width
For the crack control, there must be sufficient reinforcement
in the cross section to ensure that the distribution of cracks
can occur and the reinforcement does not yield at the first
cracking. According to the theory of elasticity, when the
specimens are subjected to pure torsion, the first inclined
crack normally initiates in the middle of the wider face of the
cross section. Therefore, during the test, the crack widths were
measured at that location. As mentioned above, for specimens
having similar amounts of torsional reinforcement, the torsionalcracking strength is lower for those with hollow sections or
greater aspect ratios. As a result, the reinforcement started to
resist external loads at an earlier load stage for such specimens.
From the test observations, the specimen HBS-82-13 (At/s =
(At/s)min,(ACI) and tfy v/l fyl = 0.19) approached its
ultimate strength stage shortly after the formation of diagonal
cracking. Furthermore, the deformations on the surface of the
specimens HBS-74-17 and NBS-82-13 were concentrated on
only a few cracks. Therefore, the crack control is inadequate for
the specimens containing relatively lower amounts of transverse
reinforcements.
In this investigation, we select the A (y/x = 1.0) andC-series (y/x = 2.8) specimens to discuss the development
of crack widths for specimens with lower amounts of
torsional reinforcement. Fig. 6 shows the relationships of
the T(test)/Tu(test) and the crack widths of A- and C-series
specimens. Each curve starts at the cracking torque and
terminates at the point when the reinforcement reaches its
yielding strain. In this paper, we adopted the 60% of the
nominal torsional strength calculated by the ACI 318-05
Code[5] as the service load level, which was also proposed by
Yoon et al. [7] and Ozcebe et al. [8] for reinforced concrete
beams subjected to shear. The horizontal and vertical dotted
lines in the figures represent the calculated service load level
and crack width criteria in a flexure of 0.30 mm in the ACI318-95 Code [10] and in Eurocode 2 [13] at the service
load level, respectively. Figs. 6(a) and 6(b) show that the
calculated service loads are less than the experimental cracking
loads; therefore, the specimens designed with relatively higher
ratios of t fyv /l fyl , 0.34 to 0.95, remain un-cracked at the
calculated service load level.
As shown in Fig. 6(a), the crack width of the specimen
HAH-81-35 with hollow section is greater than the HSC
specimen HAS-90-50 with solid section at the same load level.
A similar phenomenon is observed in Fig. 6(b) for the C-
series specimens HCH-91-42 and HCS-91-50. Therefore, the
developments of crack widths for the specimens with hollow
Fig. 6(a). External torque level versus crack width for A-series specimens.
Fig. 6(b). External torque level versus crack width for C-series specimens.
sections are more significant than those of the specimens
with solid sections. From Fig. 6(b), it can also been seen
that the crack width of HSC specimen HCH-91-42 is greater
than that of the NSC specimen NCH-62-33 at the same load
level. Similarly, the tendency can be observed in Fig. 6(a)for
HSC specimen HAS-51-50 and NSC specimen NAS-61-35 to
go beyond 80% of the experimental ultimate torque. This is
because the HSC beams have higher tensile strength and exhibit
fewer inclined cracks and larger torsional crack width than
the NSC beams. A comparison ofFigs. 6(a) and6(b) shows
a significant difference in the development of crack widths
between the A- and C-series specimens. The crack widths of the
C-series specimens HCS-52-50 and HCS-91-50 (y/x = 2.8)are larger than the corresponding specimens HAS-51-50 and
HAS-90-50 (y/x = 1.0) in the A-series, which indicates that
the crack widths increase with increases in the aspect ratio of
the cross section.
According to the numerical analysis and experimental
investigations conducted by Park et al. [14] the maximum
crack width was affected by the relative amounts of torsional
reinforcement in the transverse and longitudinal directions. The
crack widths of specimen HCS-91-50 are smaller than those of
specimen HCS-52-50 at the same external load level. A similar
result is also shown in Fig. 6(a) for specimens HAS-90-50
and HAS-51-50 after going beyond 80% of the experimental
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Table 3
Summary of test results of specimens
Specimen number Tcr(test) (kN m) Tu(test) (kN m) Tcr(test)
Tcr(ACI)
Tu(test)Tn (ACI)
Tn(ACI)Tcr(ACI)
Tu(test)Tcr(test)
0.85AuAy
HAS-51-50 62.10 84.86 1.15 1.56 1.01 1.37 4.12
NAS-61-35 50.03 74.71 1.17 1.49 1.18 1.49 4.06
HAH-81-35 44.42 94.31 1.39 1.46 2.02 2.12 3.88HAS-90-50 68.43 104.23 1.25 1.43 1.34 1.52 5.71
NBS-43-44 44.50 60.60 1.25 1.32 1.29 1.36 3.79
HBS-74-17 57.48 62.20 1.17 1.18 1.12 1.08 2.51
HBS-82-13 56.31 56.31 1.15 1.20 1.06 1.00 2.72
NBS-82-13 46.18 52.90 1.30 1.12 1.32 1.15 2.46
HBS-60-61 59.01 93.70 1.20 1.47 1.30 1.59 3.81
HCS-52-50 47.22 73.54 1.01 1.64 1.00 1.56 3.46
NCH-62-33 36.61 64.14 1.43 1.60 1.57 1.75 1.95
HCH-91-42 40.74 87.51 1.25 1.59 1.69 2.15 2.13
HCS-91-50 53.22 95.86 1.12 1.60 1.26 1.80 4.73
Average 1.22 1.44
ultimate torque. This indicates that an increase in the amount
of longitudinal reinforcement decreases the crack width forreinforced concrete beams subjected to pure torsion. The crack
widths at 60% ofTu(test) for specimens HAS-51-50 and HCS-
52-50 (total = 1.02%) are smaller than 0.3 mm. Thus, the
specimens designed with Tn = 1.0Tcr provide adequate crack
control.
5.3. Torsional strength
The experimental results of the torsional strength tests are
listed in columns 2 and 3 ofTable 3and compared with the
calculated values of the ACI 318-05 Code in columns 4 and 5.
The crack initiates as the maximum applied tensile stress arrivesat the tensile strength of concrete; therefore, the torsional
cracking strengths of the HSC specimens are higher than those
of the NSC specimens. The test results indicate that the average
value ofTcr(test)/Tcr(ACI)for HSC and NSC specimens are 1.19
and 1.29, respectively, and the average value ofTcr(test)/Tcr(ACI)for all specimens shown inTable 3is approximately 1.22.
As shown in Table 3, the experimental cracking strengths
of the hollow section specimens HAH-81-35 (y/x = 1.0) and
HCH-91-42 (y/x = 2.8) are 44.42 kN m and 40.74 kN m,
respectively, which are less than the 68.43 kN m and
53.22 kN m, respectively, of the corresponding solid section
specimens HAS-90-50 (y/x = 1.0) and HCS-91-50 (y/x =
2.8). In addition, the test results of the above four specimensalso reveal that the aspect ratio would affect the torsional
cracking strength. We further normalize the torisonal cracking
strength of the specimens with solid and hollow sections byfc as shown in Fig. 7. The normalized torsional cracking
strength decreased as the aspect ratios of specimens increased.
Furthermore, the experimental ultimate torsional strengths of
the specimens HAS-51-50 (y/x = 1.0, total = 1.01%) and
HAS-90-50 (y/x = 1.0, total = 1.40%) are 84.86 kN m
and 104.23 kN m, respectively, which are greater than the
73.54 kN m and 95.86 kN m, respectively, of the corresponding
solid section specimens HCS-52-50 (y/x = 2.8, total =
1.02%) and HCS-91-50 (y/x = 2.8, total = 1.41%). The test
Fig. 7. Normalized cracking torsional strengthaspect ratio relationships for
the test specimens.
results also reveal that the ultimate torsional strength decreases
with the increase of the aspect ratio of the specimens.
5.4. Torsional ductility
Fig. 8(a)(d) show the experimental torquetwist relation-
ships of the test specimens. The torsional ductility of the
specimen is defined as the ratio of the area enclosed by the
torquetwist curve between the origin and 85% of the peak
strength (A0.85Tu ) in the descending branch to that between the
origin and the first yielding of torsional reinforcement (Ay ).
The variations of torsional ductility among the specimens are
listed in column 8 of Table 3. The reinforcements of the all
specimens yielded prior to the ultimate strength stage, except
for the specimens HBS-74-17, HBS-82-13, and NBS-82-13
shown inFig. 8(a), which were designed with relatively lower
ratios of t fyv /l fyl . Only the transverse reinforcement of
the above three specimens yielded. The torquetwist curves of
the HBS-82-13 and NBS-82-13 (tfyv /l fyl = 0.19), shown
in Fig. 8(a), designed with the minimum amount of stirrups
and maximum spacing of the stirrups specified in ACI 318-
05 Code, respectively, had steeper strength decay than the
other specimens shown inFig. 8. FromTable 3, the ratios of
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(a) Beams HBS-74-17, HBS-82-13, and NBS-82-13. (b) Beams HAS-90-50 and HAH-81-35.
(c) Beams HCS-91-50 and HCH-91-42. (d) Beams HAS-51-50, HCS-52-50, and NBS-43-44.
Fig. 8. Experimental torquetwist relationships of the test specimens.
A0.85Tu /Ay for specimens HBS-82-13 and HBS-74-17, hav-
ing t fyv /l fyl = 0.19 and 0.27, are 2.72 and 2.51, respec-
tively, which are less than the 3.81 of the specimen HBS-60-
61 of the same cross section designed with a relatively higher
t fyv /l fyl ratio of 0.97.From Fig. 8(b) and (c), the test results reveal that the
ascending branches in the experimental torquetwist curves of
the specimens with solid sections are slightly steeper than those
with hollow sections. The ratios of A0.85Tu /Ay for specimens
HAH-81-35 and HCH-91-42 with hollow sections, shown in
Table 3,are 3.88 and 2.08, respectively, which are less than the
5.71 and 4.73 of the corresponding specimens HAS-90-50 and
HCS-91-50 with solid sections.According to the test results of Fang and Shiau [15], the
torsional ductility of HSC specimens is better than that of NSC
specimens. In this investigation, the ratios ofA0.85Tu /Ayfor the
HSC specimens HBS-82-13 and HCH-91-42 are 2.72 and 2.13,
which are greater than the 2.46 and 1.95 of the corresponding
NSC specimens NBS-82-13 and NCH-62-33.The experimental torquetwist curves of the specimens
HAS-51-50, HCS-52-50, and NBS-43-44 (t fyv /l fyl =0.930.98) in Fig. 8(d) show fairly ductile behavior in the
descending branches. The ratios of A0.85Tu /Ay for the above
three specimens are 4.12, 3.46, and 3.79, respectively. The test
results reveal that the specimens designed with t fyv = l fylcan provide better torsional ductility than those having lower
ratios oftfyv /l fyl .
5.5. Effect of t fyv /l fyl ratio on the post-cracking reserve
strength
According to the equilibrium equations of the space truss
analogy theory [3,16,17] for reinforced concrete members
subjected to pure torsion, the ratio of the amount of transverse
to longitudinal reinforcement (t/l ) significantly affects the
torsional strength and the angle of the compression diagonal.
Furthermore, Leu and Lee [18]and Rahal [19]found that the
ratio oftfyv /l fyl has a significant influence on the ultimate
torsional strength and failure mode of beams subjected to pure
torsion. The test results of this investigation indicated that all
of the torsional reinforcements of specimens yielded before
reaching their ultimate strength stages. Therefore, the result
ofTu(test)/Tcr(test) should be greater than 1.0, because the code
provisions assume that all of the torsional reinforcements yield
at the ultimate strength stage.
The effect of the tfyv /l fyl ratio on the post-cracking
reserve strength (Tu(test)/Tcr(test)) for specimens with lower
amounts of torsional reinforcement is investigated as follows.
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As shown in Table 3, the post-cracking reserve strength
Tu(test)/Tcr(test) for HSC specimen HBS-82-13 (with At/s =(At/s)min,(ACI) and l = 0.82%) and HBS-74-17 (with
At/s = 1.35(At/s)min,(ACI) and l = 0.74%), having
tfyv /l fyl = 0.19 and 0.27, are 1.00 and 1.08, respectively,
which are less than the corresponding code prediction values,
Tn(ACI)/Tcr(ACI), of 1.06 and 1.12, respectively. Similarly, theresult of Tu(test)/Tcr(test) for NSC specimen NBS-82-13, with
reinforcement ratio t fyv /l fyl = 0.19 and total = 0.95%
is 1.15, which is also less than the code prediction value of
1.32. Therefore, the specimens designed with lower ratios of
tfyv /l fyl , 0.19 and 0.27, did not provide adequate post-
cracking reserve strength even though they were designed with
torsional reinforcements oftotal >0.90%.
The following HSC specimens were designed with relatively
more transverse reinforcements, i.e., At/s = 1.39 to 2.83
(At/s)min,(ACI), l = 0.81%0.91%,t fyv /l fyl = 0.340.53
andtotal = 1.16%1.41%. The experimental reserve strengths
for the HSC specimens HAH-81-35, HAS-90-43, HAS-90-50,
HCH-91-42, and HCS-91-50 are 2.12, 1.48, 1.52, 2.15, and1.80, respectively, which are all greater than the corresponding
prediction values of Tn(ACI)/Tcr(ACI), 2.02, 1.24, 1.34, 1.69,
and 1.26, respectively. Similarly, for the NSC specimens
NAS-61-35 and NCH-62-33, with At/s = 1.77 and 2.41
(At/s)min,(ACI), t fyv /l fyl = 0.56 and 0.52, and total
0.96%, the test values of the reserve strengtsh are 1.49 and 1.75,
respectively, which are also greater than the associated values
ofTn(ACI)/Tcr(ACI), which are 1.18 and 1.57, respectively.
According to the code provisions of ACI 318-05 [5],
i.e., Eqs. (3) and (5) in this paper, the angle of the
compression diagonal is 45 deg for beams designed with equal
percentages of torsional reinforcement in the transverse andlongitudinal directions. FromTable 3,we find that the values
ofTu(test)/Tcr(test)for the HSC specimens HAS-51-50, HCS-52-
50, and HBS-60-61, with At/s =1.99 to 3.22(At/s)min,(ACI),
tfyv /l fyl = 0.930.97, and total = 1.01%1.21%, are
1.37, 1.56, and 1.59, respectively, which are all greater than
the prediction values ofTn(ACI)/Tcr(ACI), which are 1.01, 1.00,
and 1.20, respectively. Similarly, for the NSC specimen NBS-
43-44, having Tn = 1.29Tcr,At/s = 3.02(At/s)min,(ACI),
tfyv /l fyl = 0.98, and total = 0.87%, the value of
Tu(test)/Tcr(test)is 1.36, which is greater than the code prediction
value of 1.29.
To summarize the above comparisons of HSC and NSC
specimens designed with total = 0.87%1.21%, which
are close to the minimum amounts required by the current
design provisions, the experimental post-cracking strengths
are approximately 1.371.59 if tfyv /l fyl 1.0 is used.
Therefore, the lower post-cracking reserve strengths of the
specimens are primarily due to the design with t fyv
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that of the torsional reinforcement in the longitudinal direction
as specified in ACI 318 Code.
As mentioned previously, the inadequacy of the post-
cracking reserve strength for HSC specimens with a lower
ratio of total was primarily due to the greater difference
in the amounts of transverse and longitudinal reinforcements
(t fyv
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