torsion

462 ACI Structural Journal/July-August 2001

ACI Structural Journal, V. 98, No. 4, July-August 2001.MS No. 00-132 received June 2, 2000, and reviewed under Institute publication

policies. Copyright © 2001, American Concrete Institute. All rights reserved, includ-ing the making of copies unless permission is obtained from the copyright propri-etors. Pertinent discussion will be published in the May-June 2002 ACI StructuralJournal if received by January 1, 2002.

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

To study the effect of high-strength concrete on the torsionalbehavior of reinforced concrete (RC) beams, nine full-size beamswere tested under pure torsion. The main parameters in this studywere concrete strength and amount of reinforcement. Concretestrength ranged from normal strength through all grades of high-strength concrete (defined as 50, 60, 80, and 95 MPa). The amountof reinforcement varied from less than the minimum to the so-called balanced condition (when expected crushing of concreteoccurs at the same time as yield of steel). With the intent of keepingthe inclination of the concrete struts approximately equal to45 degrees, equal percentages of reinforcement were provided inthe transverse and longitudinal directions. Results indicate that theminimum amount of reinforcement defined in ACI 318-99 is inade-quate for equilibrium torsion of high-strength RC beams, and anew expression is proposed. It was found that the torsional capac-ity of under-reinforced beams is independent of concrete strength,and the amount of longitudinal reinforcement was more effective incontrolling crack width than the amount of transverse reinforce-ment (stirrups).

Keywords: beam; concrete, high-strength; reinforcement; shear; strain;torsional stress.

INTRODUCTIONTorsion can become a predominant action in structures

such as eccentrically loaded box beams, curved girders,spandrel beams, and spiral staircases (Fig. 1). Prior to 1995,the design and analysis of such members were based onsemi-empirical provisions and were lacking rationality. Fur-thermore, prestress was not addressed in torsion design. In1995, ACI 318-951 adopted new torsion provisions thatseem to be more rational. This new method is based on thethin-wall tube/space truss analogy and is capable of address-ing both reinforced and prestressed concrete. In this method,the torsional member is idealized as a tube that, after crack-ing, becomes a space truss where transverse (stirrups) andlongitudinal reinforcement are in tension and concrete diag-onals are in compression (Fig. 2). Tests and theoreticalinterpretation2,3 have shown that once cracking has oc-curred, the concrete in the center of the member contributeslittle to the torsional strength of the cross section and canthus be ignored.

In previous versions of ACI building code (from 1971 to1989), torsional strength of beams was considered to becomposed of two parts: the concrete contribution Tc, and thereinforcement contribution Ts. While the torsional momentstrength provided by reinforcement Ts was obtained from theequilibrium of a space truss, assuming the diagonal compres-sion struts to have an inclination of 45 degrees, the procedureincluded a correction factor αt (empirical) that incorporatesthe effect of the cross-section aspect ratio. In ACI 318-95,torsional moment strength provided by concrete was elimi-nated. Furthermore, the simplification in the estimate of the

lever arm area Ao (Fig. 2) did not address the cross-sectionaspect ratio. Torsional design provisions are based on yieldof steel, meaning that design is based on the under-reinforcedbeam concept. All test data used to validate this method werebased on beams having a maximum concrete strength ofapproximately 50 MPa.4

Very few tests of high-strength reinforced concrete (RC)beams under torsion have been reported in the literature.5

For this reason, ACI 318-95 limits the use of the theory up toa concrete strength fc′ = 69 MPa. This paper reports the re-sults of an investigation on torsional behavior of RCbeams6,7 with emphasis on high-strength concrete as a con-tribution to the understanding of the behavior of such mem-bers and their failure modes, and it addresses some designissues specific to pure torsion of high-strength RC members,such as minimum reinforcement and effects of concretestrength.

Title no. 98-S44

Torsion of High-Strength Reinforced Concrete Beams and Minimum Reinforcement Requirementby Nasr-Eddine Koutchoukali and Abdeldjelil Belarbi

Fig. 1—Examples of structures where torsion is predomi-nant action.

Fig. 2—Space truss analogy.

463ACI Structural Journal/July-August 2001

RESEARCH SIGNIFICANCEStudies with experimental evidence are very limited for

torsion of high-strength concrete beams. The present exper-imental investigation addresses the existing research voidpertaining to the torsional behavior of high-strength, under-reinforced RC beams. With the increased use of high-strength concrete in structures such as bridges, where torsioncan be an important design factor, it seems necessary for anydesign code to include commercially produced concretestrengths up to 100 MPa. Design equations for minimum re-inforcement for torsion in the present code are discussed inthis paper, and changes are proposed for equilibrium torsion.

EXPERIMENTAL PROGRAMSpecimen details

In torsion, beams can be: 1) under-reinforced, when boththe stirrups and the longitudinal reinforcement yield beforecrushing of concrete diagonals; 2) partially over-reinforced,when either stirrups or longitudinal bars, but not both, yieldbefore crushing of concrete diagonals; and 3) over-reinforced,when crushing of concrete takes place before steel yields.Furthermore, a minimum amount of reinforcement should beprovided in both directions to ensure the post-cracking capacityand desirable postyielding ductility. Because the presenttorsional provisions are based on yield of steel, the RCbeams included in this test series were designed to be under-reinforced or near the balanced condition (when crushing ofconcrete occurs at the same time as yield of steel).

All test beams were 3.96 m long, having a cross section of305 by 203 mm (Fig. 3). A concrete cover of 19 mm wasused. To allow for failure to occur in the central test regionof the beam, additional stirrups were placed at both ends ofthe beam. The test region was selected to be 2.38 m long toallow at least two complete spirals of diagonal cracks to formalong the beam and also to accommodate the test-rig lengthin the laboratory. This made each of the two heavily reinforcedends 0.8 m long.

Two series of rectangular, full-size beams were tested. Theinvestigated parameters were concrete strength for Series 1,and amount of reinforcement for Series 2. Series 1 includedfive beams with design strengths selected to be 35, 48, 62,83, and 96 MPa. This range was chosen to cover most gradesof normal- and high-strength concrete.8 Identification of theindividual test beams is given by the form BXURN, where Xstands for the design concrete strength of a given beam inksi, UR refers to under-reinforced, and N is the beam numberof the same concrete strength, ordered from lowest to highestreinforcement ratio. Series 2 included four beams having aconstant concrete strength (approximately 76 MPa) and a totalreinforcement ratio that varied from 1.76 to 2.64%. Table 1summarizes the characteristics of reinforcement and con-crete for all beams included in this investigation.

Material propertiesTable 2 gives the mixture proportions for the various con-

crete strengths. Local river sand had a fineness modulus of2.3. Two different local crushed aggregates were used: a12.7 mm maximum aggregate size limestone for the threemixtures of fc′ equal to 35, 48, and 62 MPa; and a 12.7 mmmaximum aggregate size granite for the two mixtures of fc′equal to 83 and 96 MPa. A melamine-based high-range water-reducing admixture (HRWR) was used for the 62 MPa con-crete, and a modified naphthalene-sulfonate-based HRWRwas used for the 83 and 96 MPa mixtures. Type I portlandcement was used for all five mixtures, Type C fly ash wasused in the 48 MPa concrete, and silica fume in powder formwas used for the 83 and 96 MPa mixtures. An air-entrainingagent (AEA) was used in the 35 MPa mixture to reducebleeding water.

Typically, six to 12 100 x 200 mm cylinders and six to 12152 x 305 mm cylinders were cast with each test beam andused for compression and splitting tests. Beams B5UR1,B7UR1, and B9UR1 were made of two similar batches. Therest of the beams were made from three batches to allow forsmaller mixture quantities and better handling. A few hoursafter casting, the beams and cylinders were covered with wetburlap and plastic sheets. Form stripping was done about 24h after casting, and both beams and cylinders were kept un-der the same curing conditions. Compression and splittingtests on cylinders were carried out around the time of beamtesting. Table 1 gives the test results of the cylinders as wellas the age at testing (some specimens were tested beyond theaimed 28 days because of scheduling issues). Strengths ofthe 100 x 200 mm cylinders were taken as the referencestrength for all beams except B5UR1, which was based on152 x 305 mm cylinder results.

Test setup and instrumentationDetails of the test setup are shown in Fig. 4. One 267 kN

hydraulic actuator was used to apply the load near the eastsupport. The load had a 457 mm lever arm from the centroi-dal axis of the beam, giving the test rig a 122 kN-m torquecapacity. A 445 kN tension load cell was used to measure theapplied load. The actuator had a stroke length of 178 mmproviding a 19 degree twist capacity of the beam. A reactionarm was used near the west support to balance the appliedload by attaching the arm to the laboratory strong floor. Thereaction rod had a 457 mm eccentricity from the centroidalaxis of the beam as well. After cracking, the beam elongates

ACI member Nasr-Eddine Koutchoukali is a project engineer with HBE Corp., St.Louis, Mo., and a former graduate research assistant in civil engineering at the Univer-sity of Missouri-Rolla. He received his BSE from the University of Constantine, Algeria,his MS from the University of Washington, Seattle, and his PhD from the University ofMissouri-Rolla. His research interests include nonlinear behavior of reinforced concretemembers, seismic retrofit of structures, and structural renovation of buildings.

Abdeldjelil Belarbi, FACI, is an associate professor of civil engineering, Universityof Missouri-Rolla, Mo.

Fig. 3—Cross-section and reinforcement arrangement oftest beam of first series (dimensions in mm).


longitudinally. To avoid any longitudinal restraint and sub-sequent compression, the beam was allowed to slide andelongate freely. This was achieved by supporting the westend of the beam on rollers.

In the beams of Series 1, 15 electrical resistance straingages were used to measure strains on the reinforcing bars.Six strain gages were mounted on three stirrups within thetest region, with one stirrup located at midspan and two sym-metrically located at 648 mm from midspan. Each stirrupwas instrumented with two strain gages, one mounted at themiddle of the short leg (bottom face) and one at the middleof the long leg (side face; Fig. 5). Nine strain gages weremounted on longitudinal bars at three different sections of

the test region. One set of three gages was located in the mid-dle, and the other two sets symmetrically located at 648 mmfrom the middle on each side of the test beam. At each sec-tion, two gages were mounted on the bottom corner bars andone gage on an upper corner bar.

In beams of Series 2, all details of the instrumentation ofthe test beams were the same as those of the beams fromSeries 1, with one exception: the number of strain gages usedto measure strains in the longitudinal bars was reduced byone (eight strain gages instead of nine). Additionally, tomeasure the thickness of the shear flow zone, multilayeredstrain gage units developed at University of Missouri-Rolla9

were embedded in the concrete and used to measure com-pressive strains through the thickness of the concrete strut atthe outer skin of the beam and toward the inside of the beam.Two units of three multilayered strain gages were placed atmidspan of the beam: one mounted at the middle of the longside, and the other at the middle of the short side of thebeam’s cross section. When additional longitudinal barswere used at midheight of the long side (for Beams B12UR3and B12UR4), these units—at midheight of the long side ofthe beam—were shifted slightly upward. Figure 6 shows oneof these units and describes the location on test beams.

The twist of the beam was measured by a rotational varia-tional differential transducer (RVDT) having a gage lengthof 1.93 m and located in the middle of the test region. Linear

Table 1—Material properties

Series no. Test beam

Longitudinal reinforcement Transverse reinforcement Concrete

No. of bars fyl , MPa dv , mm fyv, MPa s, mm

Testing age, days fc′, MPa fct , MPa

1

B5UR1 4 No. 4 386 9.5 373 108 30 39.6 3.6

B7UR1 4 No. 4 386 9.5 399 108 34 64.6 4.5

B9UR1 4 No. 4 386 9.5 373 108 35 75.0 4.6

B12UR1 4 No. 4 386 9.5 399 108 54 80.6 5.3

B14UR1 4 No. 4 386 9.5 386 108 56 93.9 6.3

2

B12UR2 4 No. 4 386 9.5 386 102 51 76.2 5.5

B12UR34 No. 4 373 9.5

386 95 90 72.9 5.32 No. 3 386 9.5

B12UR4 6 No. 4 373 9.5 386 90 48 75.9 5.3

B12UR5 4 No. 5 380 9.5 386 70 81 76.7 5.5

Table 2—Concrete mixture proportions

Constituents

Concrete mixture proportions

35 MPa 48 MPa 62 MPa 83 MPa 96 MPa

Cement, kg/m3 345 508 613 568 503

Water, kg/m3 193 191 193 138 138

w/c 0.56 0.38 0.31 0.24 0.27

Fine aggregate, kg/m3 847 634 586 693 649

Coarse aggregate, kg/m3 831 1036 976 1112 1112

Fly ash, kg/m3 — 61 — — —

Silica fume, kg/m3 — — — 41.5 47.5

Water reducer, mL — 745 — — —

High-range water reduc-ing admixture, L — — 3.752 24.3 21.0

Air-entraining admixture, mL 130 — — — —

Slump, cm 11.5 11.5 7.5 Flowing Flowing

Fig. 4—Test setup for pure torsion.

Fig. 5—Location of electrical strain gages (dimensions in mm).

ACI Structural Journal/July-August 2001 465

variable differential transformers (LVDTs) were used tomeasure concrete surface strains as well as beam end elonga-tion. Three LVDTs were placed in a rosette format to mea-sure average concrete strains in three directions as shown inFig. 7. One LVDT was placed along the longitudinal axis ofthe beam and the other two, at 45 and 135 degrees, measuredcounter-clockwise from the longitudinal axis. A fourth

LVDT was placed near the support to measure the longitudi-nal elongation of the beam (Fig. 7).

Test procedureMeasuring devices for load, deformation, and strains were

read through a computer-driven data acquisition system. Priorto failure of the beam, data were recorded at a prescribedload increment. Smaller increments were used around crackingto accurately measure the value closest to actual crackingtorque. At every load stage after cracking, the load was heldconstant for several minutes before collecting the data, afterwhich the crack pattern was marked, crack width and spacingwere measured, and concrete spalling-off was checked. Thischeck was performed by knocking on the concrete with asteel hammer and listening to the sound. A hollow soundwould indicate separation of the concrete cover from theconcrete inside the stirrups. When the beam reached itstorsional capacity, data was continuously recorded until thehydraulic jack reached its maximum stroke, which corre-sponds to the maximum twist capacity of the setup.

TEST RESULTS AND DISCUSSIONSCracking characteristics

An accurate estimate of cracking torque is important in thecase of compatibility torsion. In indeterminate structures,torsional moment can be redistributed to the adjoining mem-bers after cracking.10, 11 For this reason, the ACI Code allows

Table 3—Experimental results

Test beam

At cracking At ultimate

Torque, kN-m

Twist, 10–2

degrees/mTorque, kN-m

Twist, degrees/m εds, 10–6 m/m εr, 10–6 m/m εl , 10–6 m/m

B5UR1 11.6 8.5 19.4 2.18 −1819 5010 1505

B7UR1 14.1 8.0 18.9 1.93 −2441 5827 1488

B9UR1 13.0 13.4 21.1 2.05 −3000 Not measured 1450

B12UR1 16.2 9.1 19.4 1.14 −1940 3218 1248

B14UR1 19.3 11.7 21.0 0.21 −292 260 233

B12UR2 17.8 11.1 18.4 0.75 −1488 5355 377

B12UR3 16.0 10.2 22.5 2.02 −2578 12,281 1658

B12UR4 16.9 14.0 23.7 1.93 −1971 7065 1425

B12UR5 13.6 3.56 24.0 2.50 −2650 9772 1733

Fig. 6—Embedded strain gages and their location on testbeams (dimensions in mm).

Fig. 7—Different deformation measurements of test beam.


the torsional moment to be reduced to the cracking torsionalmoment in members where redistribution of internal forcesis possible. Under combined stresses, ACI 318-99 providesan estimate of cracking torsional moment as

(1)

in which the units are MPa and mm.Up to cracking, the behavior of the beam is essentially

elastic. Torsion is mainly resisted by concrete. Prior tocracking, the measured surface concrete strains along the di-agonal at 45 and 135 degrees (Fig. 7) are essentially of thesame magnitude, but of the opposite sign, with tension beingalong the 135 degree direction. Measured values of torqueand twist at cracking are shown in Table 3. Table 4 providesthe calculated cracking torque of the nine tested beams basedon Eq. (1). The mean value of the ratio of the measuredtorque at cracking to the calculated torque is 1.45 with acoefficient of variation of 0.096. This leads to the conclusionthat the ACI Code underestimates the cracking torque forpure torsion by about 31%. Similar conclusions were report-ed by Ghoneim and MacGregor,4 based on 94 torsionalmembers. They found that the Code underestimates thecracking torque by approximately 32%.

Postcracking behaviorAfter cracking, concrete behaves as a nonlinear and dis-

continuous medium that leads to redistribution of internalstresses, forming a truss action in which reinforcement actsas tensile links, and concrete acts as compression diagonals.As applied torque increases, spiral cracks develop at about45 degrees and spread over the test region. Because thebeams were reinforced with equal amounts of reinforcementin the longitudinal and transverse directions, all cracks wereinclined at 45 degrees throughout the loading history. Thesecracks were evenly distributed in the test region, except forBeam B14UR1, which experienced one major crack (fol-lowed by failure), and Beam B12UR2, which experiencedseveral cracks in the central region of the beam but failed be-fore cracks became evenly distributed along the beam. Suchbehavior in the latter beams is attributed to the fact that thesebeams were reinforced with an amount of reinforcementclose to the minimum, and limited strength after cracking

Tcr 0.33 fc′Ac

2

pc

--------=

was available. This reduced the amount of cracking along thebeam as shown in Table 3 by the smaller amount of strain re-corded in the longitudinal direction εl.

Ultimate torque and twist—Torque, twist, and concretesurface strains are given in Table 3 for all test beams. For thefirst series, results show that as concrete strength increases,behavior changes from barely under-reinforced (close to bal-anced) to grossly under-reinforced (close to minimum rein-forcement).

For Beam B5UR1, yield of steel occurred just a few loadstages before failure of the beam (crushing of concrete),whereas for B14UR1, yield of steel occurred just after crackingof the beam. Although these beams were similarly reinforced,the torque at cracking for Beam B14UR1 was high, and theamount of reinforcement provided did not supply sufficientstrength after cracking to prevent a brittle failure. The datarecorded indicate that the ratios of ultimate to cracking

Fig. 8—Torque-twist relationships for tested beams: (a)Series 1; and (b) Series 2.

(a)

(b)

Table 4—Comparison of experimental results with ACI predictions

Beam

At cracking At ultimate

Tcr,ACI, kN-m Tcr,test/Tcr,ACI Tn,ACI, kN-m Tn,test/Tn,ACI

B5UR1 7.9 1.47 16.4 1.17

B7UR1 10.1 1.41 17.1 1.10

B9UR1 10.9 1.20 16.5 1.28

B12UR1 11.3 1.44 17.1 1.14

B14UR1 12.1 1.59 17.1 1.23

B12UR2 10.9 1.63 17.1 1.06

B12UR3 10.7 1.50 20.0 1.11

B12UR4 10.9 1.55 23.8 1.00*

B12UR5 11.0 1.24 24.0 1.00*

Mean 1.45 — 1.12*Based on crushing of concrete.

ACI Structural Journal/July-August 2001 467

torque for Beams B12UR1 and B14UR1 were only 1.2 and1.09, respectively.

The torque-twist relationships for beams of both series areshown in Fig. 8. For the second series, the behavior of beamsranges from minimum reinforcement (Beam B12UR2 withlimited ductility) to slightly over-reinforced (BeamsB12UR4 and B12UR5 with failure by crushing of concrete).For both Beams B12UR4 and B12UR5, yield of steel wasimpending at failure, meaning that the amount of reinforce-ment provided was close to the balanced condition.

Equation (2) provides the ACI 318-95 estimate of torsionalstrength of under-reinforced RC beams. Table 4 gives acomparison between the experimental ultimate torquesand those predicted by Eq. (2). The mean Tmax,test /Tn,ACIfor all beams is 1.12, which shows that the ACI Code un-derestimates the ultimate strength by approximately 11%.These results indicate that the ACI 318-95 estimate fortorsional strength is reasonably conservative for thesetested beams.

(2)

where cotθ is defined as

(3)

Crack width and failure modes of beams—Test results fromthe first series showed that when the amount of reinforcementis kept constant, the initial crack width increases with the in-crease of concrete strength (Fig. 9), which induces largerstrains in the reinforcing steel at beam cracking. As torqueapproaches the ultimate capacity of the beam, and shortly afteryield of the longitudinal reinforcement, the width of the maincrack (usually the first one to form) increases rapidly andcauses the longitudinal bar to kink, resulting in a sudden dropin the capacity of the beam. At this stage of the test, all de-

Tn ACI,2AoAt fyv

s--------------------- θcot=

θcotAl fyls

At fyvph

-----------------=

formations take place at the location of the main crack,where the rest of the beam is in an unloaded state. Thisphenomenon is probably more severe for high-strength RCbeams than for normal-strength RC beams because thesmooth-faced cracks are less effective in transmitting shearstresses for the former.

Concrete internal strains—As expected, the measuredconcrete strains through the cross section showed two dis-tinct stages, before and after cracking. Before cracking, allmeasured strains in the concrete strut at 45 degrees counter-clockwise from the longitudinal direction were compressivestrains. After cracking, the innermost strain gage measuredtension, suggesting warping of the concrete strut. As as-sumed by the theory, measured strains varied linearlythrough the thickness of the shear flow zone (Fig. 10).

Minimum torsional reinforcement for equilibrium torsion

A minimum amount of reinforcement is needed to ensurethat the beam does not fail at cracking. Observed behaviorsof Beams B12UR1 and B14UR1 provide valuable informa-tion regarding minimum reinforcement for high-strengthconcrete beams. Although Beam B14UR1 was reinforcedwith an amount much higher than that prescribed by the ACICode, the beam failed at a very small twist. This shows abrittle failure after cracking, meaning that there was notenough reinforcement for postcracking strength. Observa-tion of Beam B12UR1 also shows the following: to allow forthe truss action to form and to develop postcracking behavioras opposed to local brittle failure, the beam needed about20% reserve strength after cracking. This is probably due toa larger crack width at cracking for high-strength RC beamsthan for normal-strength RC beams (Fig. 9) and to the previ-ously mentioned smooth-faced-cracks that are less effectivein transmitting shear stresses.

Based on these observations, the following procedure isadopted to derive the minimum amount of reinforcement fortorsion. From the data obtained in this study, corroboratingthe average obtained by Ghoneim and MacGregor,4 an esti-mate of cracking torque is given by

(4)

Assuming a 45-degree angle for θ, 20% reserve in capaci-ty after cracking as in Beam B12UR1 (that is, Tn/Tcr = 1.2),and equating the right-hand sides of Eq. (2) and (4) times λ= Tn/Tcr = 1.2, gives

Tcr 0.46 fc′Ac

2

pc

--------=

Fig. 9—Torque-maximum crack width relationships of testbeams.

Fig. 10—Concrete internal strain showing linearly varyingstrain distribution and thickness of shear flow zone.


(5)

For θ equal to 45 degrees

(6)

and replacing At/s by the right-hand side of Eq. (5) results in

(7)

Use of Eq. (5) and (7) removes the unnecessary confusionthat can face a designer when using Eq. (11-24) of ACI 318-95,as shown by Ali and White.12 The amount of minimum rein-forcement proposed in this study is similar to that proposed byAli and White. Though the ratio λ was taken as 1.2 in thisstudy (as compared with 1.7 in Ali and White), this was com-pensated by the assumed magnitude of torsional torque. Theauthors assumed a magnitude of Tcr proposed by Ghoneimand MacGregor,4 while Ali and White used the magnitudeprovided by the ACI Code. The former is about 40% largerthan the latter. It should be noted that the ACI crackingtorque estimate is based on beams subjected to combinedstresses, whereas the present results are for beams subjectedto pure torsion.

For lightly reinforced beams, comparison of the behaviorsof Beams B12UR2 and B12UR3 shows that longitudinalreinforcement is more effective than transverse reinforcementin increasing the torsional capacity of the beam. This may beattributed to the contribution of the longitudinal bars, which,in keeping crack width smaller, is probably more significantfor high-strength concrete than for normal-strength concretebecause of the reduced contribution of aggregate interlockto shear. Furthermore, the theory is based on the smearedreinforcement concept as follows: for small beams likethose tested in this study, it is inappropriate to assume thatlongitudinal reinforcement can be visualized as smearedwhen only corner bars are used for reinforcement. There-fore, transverse reinforcement should be designed to yieldprior to longitudinal reinforcement to avoid large crackwidths. Achieving a larger amount of longitudinal thantransverse reinforcement in design can be accomplished byrequiring an angle θ smaller than 45 degrees.

Effect of concrete strengthPrior to 1995, in the ACI Code, a contribution by the con-

crete compression zone acting in torsional shear Tc was in-cluded in the torsional capacity of RC beams. Test resultsfrom PCA and the University of Stuttgart investigations2,13

support this inclusion. The strength of the concrete used inthose tests ranged approximately from 15 to 45 MPa and27 to 53 MPa, respectively. Based on the results of PCA in-vestigation, early versions (from 1971 until 1989) of theCode included a concrete contribution Tc that was taken to beproportional to √fc′. The results from the first series of thisinvestigation, however, in which concrete strength rangedapproximately from 40 to 94 MPa, do not substantiate suchfindings and tend to support the notion that torsional strength

is independent of concrete strength,14 as assumed in the trussmodel adopted in ACI 318-95.

CONCLUSIONS AND RECOMMENDATIONSResults of this study lead to the following conclusions:1. The minimum amount of reinforcement provided in

ACI 318-95 is inadequate for high-strength RC beams. Toavoid brittle failure (at the onset of cracking), a 20% reserveof strength after cracking must be available for the beam toexperience uniform cracks along its length. Based on thisassumption, a new expression for minimum torsionalreinforcement was developed;

2. Torsional strength of RC beams is independent of con-crete strength as long as the beam is under-reinforced (steelyielding); and

3. The observed failure of high-strength RC beams wasmainly controlled by the amount of strain of longitudinalreinforcement. When larger crack widths develop at yield ofthe longitudinal reinforcement, the beam forms a torsionalhinge at the yield/crack location and ceases on behaving as aunit. Localized twist increases at the hinge region, whereasother regions of the beam undergo an unloading process.This is probably due to the smooth-faced cracks, which areless effective in transmitting shear stresses and contribute toearly failure of beams.

Further research effort remains to be implemented to accu-rately predict the behavior of high-strength RC beams. Theeffects of parameters such as size, reinforcement ratios, andcombined actions (torsion, shear, and bending moments) stillneed to be clarified.

CONVERSION FACTORS

NOTATIONAc = area enclosed by outside perimeter of concrete cross-sectionAl = total area of longitudinal reinforcement to resist torsionAo = gross area enclosed by shear flow pathAoh = area enclosed by centerline of outermost closed transverse

torsional reinforcementAt = area of one leg of closed stirrup resisting torsion within spacing sfc′ = concrete compressive strengthfyl = yield strength of longitudinal reinforcementfyv = yield strength of closed transverse torsional reinforcementpc = perimeter of concrete cross sectionph = perimeter of centerline of outermost closed stirrups = spacing of transverse reinforcement in direction parallel to

longitudinal reinforcementTc = nominal torsional moment strength provided by concrete Tcr = cracking torsional moment of beamTn = nominal torsional moment strengthTs = nominal torsional moment strength provided by torsion

reinforcementtd = wall thickness of equivalent thin-walled tubeV1 to V4 = shear forces in Sides 1 to 4 of space truss due to torsionxo = center-to-center length of shorter side of closed rectangular

stirrupyo = center-to-center length of longer side of closed rectangular

stirrupεds = surface strain in diagonal concrete struts (45 degrees from

longitudinal axis of beam)εl = surface concrete strain along the longitudinal axis of beamεr = surface concrete strain perpendicular to diagonal concrete

strutsθ = angle of concrete diagonal strut in space truss analogyλ = ratio of nominal-to-cracking torques taken larger than 1

At

s-----

min

0.28fc′

fyv

---------Ac

2

Aopc

-----------=

Al

At

s-----ph

fyv

fyl

------=

Al( )min 0.28fc′

fyl

---------ph

Pc

-----Ac

2

Ao

--------=

1 kg/m3 = 1.686 lb/yd3

1 kN = 0.225 kip1 kN-m = 8.849 kip-in.

1 mm = 0.0394 in.1 MPa = 145 psi

469ACI Structural Journal/July-August 2001

REFERENCES1. ACI Committee 318, “Building Code Requirements for Structural

Concrete (ACI 318-95) and Commentary (318R-95),” American ConcreteInstitute, Farmington Hills, Mich., 1995, 369 pp.

2. Hsu, T. T. C., “Torsion of Structural Concrete Behavior of ReinforcedConcrete Rectangular Members,” Torsion of Structural Concrete, SP-18,American Concrete Institute, Farmington Hills, Mich., 1968, pp. 261-306.

3. Lampert, P., and Thurlimann, B., “Torsion Tests of Reinforced Con-crete Beams (Torsionsversuche an Stahlbetonbalken),” Report No. 6506-2,June 1968, 101 pp.

4. Ghoneim, M. G., and MacGregor, J. G., “Evaluation of Design Proce-dures for Torsion in Reinforced and Prestressed Concrete,” Report No.184, Department of Civil Engineering, University of Alberta, Edmonton,Feb. 1993, 301 pp.

5. Rasmussen, L. J., and Baker, G., “Torsion in Reinforced Normal andHigh-Strength Concrete Beams—Part 1: Experimental Results,” ACIStructural Journal, V. 92, No. 1, Jan.-Feb. 1995, pp. 55-63.

6. Koutchoukali, N., “Non-Linear Behavior of Reinforced ConcreteBeams Subjected to Pure Torsion,” PhD dissertation, Department of CivilEngineering, University of Missouri-Rolla, 1998, 127 pp.

7. Koutchoukali, N., and Belarbi, A., “Effect of Concrete Strength onthe Behavior of Reinforced Concrete Beams Subjected to Pure Torsion,”Proceedings of High Strength Concrete First International Conference,ASCE, 1997, pp. 38-51.

8. Goodspeed, C. H.; Vanikar, S.; and Cook, R., “High-PerformanceConcrete Defined for Highway Structures,” Concrete International, V. 18,No. 2, Feb. 1996, pp. 62-67.

9. Alkhardaji, T., “Experimental Investigation of the Shear Flow Zonein Torsional Members,” Master’s thesis, University of Missouri-Rolla,1998, 180 pp.

10. Collins, M. P., and Lampert, P., “Redistribution of Moments atCracking—The Key to Simpler Design?” Analysis of Structural Systemsfor Torsion, SP-35, American Concrete Institute, Farmington Hills, Mich.,1973, pp. 343-383.

11. Hsu, T. T. C., and Huang, C. S., “Torsional Limit Design of SpandrelBeams,” ACI JOURNAL, Proceedings V. 74, No. 2, Feb. 1977, pp. 71-79.

12. Ali, M. A., and White, R. N., “Toward a Rational Approach forDesign of Minimum Torsion Reinforcement,” ACI Structural Journal,V. 96, No. 1, Jan.-Feb. 1999, pp. 40-45.

13. Leonhardt, F.; Walther, R.; and Schelling, A., “Torsionsversuche anStahlbetonbalken,” Bulletin No. 239, Deutscher Ausschuss fur Stahlbeton,Berlin, 1974, 122 pp.

14. Lampert, P., and Collins, M. P., “Torsion, Bending, and Confusion—An Attempt to Establish the Facts,” ACI JOURNAL, Proceedings V. 69,No. 8, Aug. 1972, pp. 500-504.

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