Behavior of Reinforced Concrete Beams With Minumum Torsional Reinforcement

8/22/2019 Behavior of Reinforced Concrete Beams With Minumum Torsional Reinforcement

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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
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2194 H.-J. Chiu et al. / Engineering Structures 29 (2007) 21932205

Notations

Acp area enclosed by outside perimeter of concrete

cross section, mm2

Ag gross area of concrete cross section, mm2. For a

hollow section, Ag is 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 distance s, 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 =2AtAo fyv

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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H.-J. Chiu et al. / Engineering Structures 29 (2007) 21932205 2195

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

12 fyl

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 in Figs. 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 in Table 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

in Table 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-60-61had Tn = 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 in Table 1.

4.3. Test setup and instrumentation

Details of the schematic test setup are shown in Figs. 3(a)

and 3(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 Stirrups s (mm) totalt fyvl 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: As = 28.3 mm2; #3: As = 71.3 mm

2; #4: As = 126.7 mm2

#5: As = 198.6 mm2; #6: As = 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 shownin Fig. 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%, t fyv /l fyl = 0.27), and

soon after that, the concrete on the back side of it was crushed as

shown in Figs. 5(a) and 5(b). The crack pattern of this specimen

is similar to that assumed in the skewing bending theory [1].

According to Figs. 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

in Fig. 5 are 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)

and 5(d), the angles of the principal strain at ultimate strength

stage for the specimens HAS-51-50 and HBS-60-61, having

t fyv /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 in Figs. 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 t fy 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 of Figs. 6(a) and 6(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% of Tu(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 of Table 3 and 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 in Table 3 is 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 in Fig. 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 (t fyv /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 in Fig. 8. From Table 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 of A0.85Tu /Ay for 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 oft fyv /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 oft fyv /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

of Tu(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 t fyv /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

t fyv /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

t fyv /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

and total = 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. From Table 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),

t fyv /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 of Tn(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),

t fyv /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 t fyv /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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[13] Eurocode 2. Design of concrete structural. Part 1. General rules and rules

for building. European committee for standardisation. ENV 1992-1-1.

[14] Park SK, Ko WJ, Kim HY. Estimation of torsional crack width for

concrete structural members. Magazine of Concrete Research 2001;53(5):

33745.

[15] Fang IK, Shiau JK. Torsional behavior of normal- and high-strength

concrete beams. ACI Structural Journal 2004;101(3):30413.

[16] Mitchell D, Collins MP. Diagonal compression field theoryA rational

model for structural concrete in pure torsion. ACI Journal Proceedings

1974;71(8):396408.

[17] Hsu TTC, Mo YL. Softening of concrete in torsional memberTheory

and test. ACI Structural Journal 1985;82(3):290303.

[18] Leu LJ, Lee YS. Torsion design charts for reinforced concrete rectangular

members. Journal of Structural Engineering ASCE 2000;126(2):2108.

[19] Rahal KN. Torsional strength of reinforced concrete beams. Canadian

Journal of Civil Engineering 2000;27(3):44553.

[20] Hsu TTC. ACI shear and torsion provisions for prestressed hollow girders.

ACI Structural Journal 1997;94(6):78799.

Behavior of Reinforced Concrete Beams With Minumum Torsional Reinforcement

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