ACI STRUCTURAL JOURNAL TECHNICAL PAPER. -

Torsion in High-Strength Concrete Hollow Beams:Strength and Ductility Analysisby Luis F. A. Bernardo and Sergio M. R. Lopes

The ultimate behavior of high-strength concrete hollow beams isstudied with respect to their strength and ductility. Sixteen beamswere tested and the results are presented herein. The hollow beamshad a constant square cross section and were symmetricallyreinforced. The variable parameters were the concrete's compressivestrength, from 46.2 to 96.7 MPa (from 6699 to 14,022 psi), and thetotal amount of torsional reinforcement, from 0.30 to 2.68%. Thestudy presented in this paper shows that the torsional ductility islow and that the range of reinforcement ratio where ductility stilloccurs is very narrow. Different codes of practice were comparedin the light of the experimental results. As a consequence, theauthors found that AC1 Code is the most appropriate for predictingtorsional strength and limiting torsion reinforcement, therebyleading to ductile behavior.

Keywords: ductility; high-strength concrete; hollow beams; torsion;ultimate strength.

INTRODUCTIONThe use of high-strength concrete (HSC) in many special

structures (such as long bridges) is nowadays a rationaloption to fulfil many requirements, such as strength,durability, and economy. Even for normal structures (suchas buildings), the use of HSC can lead to competitiveeconomical solutions because the structural members aresmaller than normal-strength concrete (NSC) members. Thisis mainly true for members with a high level of compression(such as columns). In consequence, HSC reduces self-weightand inertia. Those reductions constitute an important advantagewhen structures are located in seismic regions.

Because of their economic advantages, HSC structureswere initially used without being sufficiently studied. Someaspects of its mechanical behavior were assumed to be anextension of NSC. Some of the code rules developed forNSC must be fully studied to check their applicability toHSC. Some of this work has already been carried out. As aresult, some codes already incorporate design rules forconcrete strengths higher than 50 MPa (7250 psi). TheNorweoian,1 the Canadian? the New Zealand 3 the European 4,5

0" 6 "and ACI codes are examples. Nevertheless, some codes4

still suggest that for concrete strengths above 50 MPa (7250 psi),the rules have to be used with caution. In fact, some aspectsof the structural behavior of HSC members are either notfully known or even completely unknown. This is the case ofbeams under torsion. The new design rules for torsion mayneed a large number of tests to confirm some theories.Therefore, the ongoing research efforts need to be continuedto be able to correct future version of the codes.

In actual structures, torsion forces are usually combinedwith moments, shear, and axial forces, but in some structuressuch as bridges, the torsion can become very important fo;the design. Furthermore, the design procedures based onforce interactions need to know the behavior under pure

torsion. Because HSC and hollow beams are frequentlybeing used in bridges, a research program on the behavior ofHSC hollow beams under torsion is very important.

The validity of plastic analysis requires that the structureundergoes a ductile behavior to allow the internal forces tobe redistributed to meet the theoretical forces obtained bycomputation. This should take place with no risk of a brittlepremature failure. In the last couple of decades, someexperimental programs have shown that concrete structures,if correctly reinforced, exhibit high values of plasticdeformations after yielding of the reinforcement. Thesevalues are often sufficient to allow the theory of plasticity tobe used. Ductility behavior is widely accepted for membersunder flexure. For shear and torsion, the assumption ofductile behavior is often questioned.

The failure by shear in shear elements would not be asbrittle as expected if the structures have adequate transversaland longitudinal reinforcement ratios. Beyond the maximumload, the structure behavior gradually develops the so-calledsoftening effect (influence of diagonal cracking on theconcrete struts) that leads to a relatively high value ofinternal energy dissipation through a sufficient level ofdeformation. In these cases, plastic behavior may beassumed. Conversely, if the reinforcement ratio is too highor too low, there is a risk of brittle failure. In this case, aplastic behavior cannot be assumed. A top and a bottomlimiting value for the reinforcing ratio is a normal procedureto lead to ductile behavior. This is true for shear as well asfor flexure.

The considerations indicated in the previous paragraph forshear are also valid for torsion. The shear stresses are asimportant for members under torsion as for those undershear. It is important to have sufficient ductility in thesections with the maximum torque. The risk of failures ofbeams with high torsion forces is real, and reports of failureof bridges can be found in some publications such as that byPriestley et al.7 The use of HSC makes this issue even moreimportant because this concrete is more brittle than NSC.

Several codes of practice were analyzed and the only one thathas explicit clauses to ensure a minimum degree of ductility. h CI 614IS teA Code.' Basically, the clauses impose amaximum and a minimum value for the amount of the torquereinforcement (for both transverse and longitudinal bars).The equations for the minimum amount of reinforcementhow~ver, are ~ainlg' empirical and sometimes lead t~questIonable solutIons, namely, negative minimum longitudinal

ACI Structural Journal, V. 106, No. I, January-February 2009.MS No. S-2()()7-147.R2 received December 2, 2007, and reviewed under Institute

pubbcanon pohc,es. Copyri.ght 2009, American Concrete Institute. All rights reserved,mcluding the makingof co",es unless pennission is obtained from the copyright proprietors.Pertment diSCUSSIOnmcluding author's closure, if any, will be published in the November-December 2009 ACI Structural Journal if the discussion is received by July 1,2009.

Luis F. A. Bernardo is an Assistanl Professor al Ihe University of Beira Jnlerior,Covilha, PorllIgal. He received a degree in civil engineering from Ihe TechnicalUniversity of Lisbon, Lisbon, Porlugal, and his PhD from Ihe Universily of Coimbra,Coimbra, Portugal. His research interests include flexural Gild torsional behavior ofreinforced concrete structures.

Sergio M. R. Lopes is an Associate Professor at tlte University of Coimbra. Hereceived a degree ;11 civil engineering from the University of Coimbra; his Msc fromthe Technical University of Lisbon; and his PhD from the University of Leeds,Leeds, UK. His research interests include flexural and torsional behavior of reinforcedconcrele structures.

reinforcement requirement or disproportional longitudinal andstirrup reinforcement.

The lack of specific rules for torsion in some codes(MC 904 and EC 25 from Europe and CAN3-A23.3-042from Canada) is compensated for by requiring that theminimum transversal and longitudinal reinforcement fortorsion be considered to be the corresponding values of theminimum amount of transversal reinforcement for shear andthe minimum amount of longitudinal reinforcement forbending moments.

As far as the maximum amount of torsion reinforcement isconcerned, the codes of practice do not generally offer anyrule for its explicit quantification. They normally indicate amaximum value for the compressive stress in the concretestruts to prevent the concrete from crushing before thetorsion reinforcement yields. Obviously, this limitation canbe used indirectly to compute the maximum amount of thetransversal reinforcement. Before 1995, the ACI Codeproposed an explicit rule for the maximum transversalreinforcement. As far as the torsion is concerned, after the1995 edition, this Code became very different from previouseditions and this explicit rule was replaced by the indirectchecking of the maximum stress in concrete struts.

For HSC beams, the rules of codes need firm confirmationand there have not been enough studies on torsion ductilityto permit unquestionable conclusions.

This paper presents the study of the ultimate strength andductility of HSC hollow beams under pure torsion. For thispurpose, 16 hollow beams were tested to failure.

RESEARCH SIGNIFICANCESo far, only a very limited number of studies on HSC

beams under pure torsion have been carried out, amongwhich the publications by Rasmussen and Baker9,IO andWafa et al. I are examples. These initial studies compriselaboratory programs testing only small rectangular plainbeams. The authors did not find any study focused on HSChollow beams. As far as torsional ductility is concerned,HSC beams are more problematic than NSC beams, andhollow beams are more problematic than plain beams. Theauthors are presenting a study on HSC hollow beams thatprovides much needed information.

Test specimensSixteen rectangular hollow beams 5.90 m (232.28 in.) long

were tested on a ring that fixed one end and applied a torsionforce on the other end. The beams had a hollow square crosssection 0.60 m (23.62 in.) wide with walls 0.10 m (3.94 in.)thick (Fig. I). These dimensions were selected so that therewould be some similarity with the beams tested by Lampertand Thurlimann.13 The ends of the test beams were designedto be fIxed to the heads of the testing equipment. The variablesof the experimental program were the compressive strengthof the concrete and the amount of steel reinforcement. Theconcrete strength of cylinder specimens varied from 46.2 to96.7 MPa (6699 to 14,022 psi). Three beam series were castand tested-Series A, Band C-classified according toconcrete strength. Within each series, the limit percentagesof torsion steel reinforcement were the recommendedmaximum and minimum values given by the Code (amongthe investigated codes) that allowed the largest range for thepercentage of steel reinforcement. 14 The purpose was touse the largest range of reinforcement ratio allowed by allthe analyzed codes.

Table 1 presents a summary of the properties of the testbeams, including the average thickness of the walls of thecross section (t), the identifIcation of the torsion reinforcement,the distances between centerlines of legs of the stirrups (XIand YI)' the area of longitudinal reinforcement (As/) and of

1012 1012 1020 1020

508 0 508 0 301608@7.5cm01012 1012 10203016

tL,.,0.34 ;,

Beam B-69.8-0.80508

!l.I~, 0.40 !).I~.,.J 0.60 J'/

Beam C-96.7-2.073016

Fig. i-Geometry and examples of test beams' details. (Note: i in. = 2.54 em; i m =38.37 in.).

r, Longitudinal Transversal xI. Yl. Asl' Ast' f~, PI. PI. PlOt,Beam cm reinforcement reinforcement, cjl@s,cm cm cm cm2 cm2 MPa % % % mb

A-48.4-0.37 9.8 4cjl8+ 16cjl6 cjl6@9 53.7 54.7 6.53 0.28 48.4 0.18 0.19 0.37 0.96

A-47.3-0.76 10.7 4cjll2 + 12cjll0 cjl8@8 53.8 53.1 13.95 0.50 47.3 0.39 0.37 0.76 1.04

A-46.2-1.00 10.9 16cjll2 cjlI0@9.5 54.0 53.5 18.10 0.79 46.2 0.50 0.49 1.00 1.02

A-54.8-1.31 10.4 4cjll6 + 20cjll0 cjl10@7 52.0 52.5 23.75 0.79 54.8 0.66 0.65 1.31 1.01

A-53.1-1.68 10.4 4cjll6 + 20cjll2 cjl12@8 52.8 52.8 30.66 1.13 53.1 0.85 0.83 1.68 1.03

B-75.6-0.30 10.1 20cjl6 cjl6@11 53.9 54.4 5.65 0.28 75.6 0.16 0.14 0.30 1.1 I

8-69.8-0.80 10.8 4cjll2 + 20cjl8 cjl8@7.5 53.3 53.4 14.58 0.50 69.8 0.41 0.40 0.80 1.02

8-77.8-1.33 10.9 4cjll6 + 20cjll0 cjl10@7 53.5 53.7 23.75 0.79 77.8 0.66 0.67 1.33 0.99

8-79.8-1.78 11.2 16cjll6 cjlI2@7.5 52.3 53.6 32.17 1.13 79.8 0.89 0.89 1.78 1.01

B-76.4-2.20 11.6 20cjll6 cjl12@6 51.8 51.8 40.21 1.13 76.4 1.12 1.09 2.20 1.03

C-91.7 -0.37 9.7 4cjl8+ 16cjl6 cjl6@9 54.0 54.9 6.53 0.28 91.7 0.18 0.19 0.37 0.96

C-94.8-0.76 10.0 4cjl12+ 12cjllO cjl8@8 53.2 53.3 13.95 0.50 94.8 0.39 0.37 0.76 1.04

C-91.6-1.29 10.3 4cjll6 + 20cjll0 cjl10@7 54.5 54.0 23.75 0.79 91.6 0.66 0.63 1.29 1.05

C-9 1.4-1.7 I 10.3 4cjll6 + 20cjll2 cjl12@8 54.6 54.5 30.66 1.13 91.4 0.85 0.86 1.71 0.99

C-96.7-2.07 10.4 4cjl20 + 12cjll6 cjlI2@6.5 54.0 54.3 36.69 1.13 96.7 1.02 1.05 2.07 0.97

C-87.5-2.68 10.4 24cjll6 cjl12@5 53.3 52.9 48.25 1.13 87.5 1.34 1.34 2.68 1.00

one branch of the transversal reinforcement (Asr)' theaverage concrete compressive strength fe', the longitudinalreinforcement ratio (PI = AstiAc with Ac = x . y and x = Y =600 mm [24 in.]) and the transversal reinforcement ratio (PI =ASI u/Ac . s with u = 2(xl + YI)), the total reinforcement ratio(Prol)' and the longitudinal versus transversal reinforcement ratiomb = Asi' s/(Asr . u). All the beams were designed to havea balanced volume of longitudinal versus transversalreinforcement (PI = P,),

The beams are identified by the series to which they belong:the average concrete compressive strengthfc' (first number) andthe ratio of total reinforcement Plor (second number).

Material propertiesThe average value of the concrete's compressive strength

for each test beam was obtained from five specimens cast atthe same time as the corresponding beam, cured in the samehumid environment as the test beams, and tested on the sameday the corresponding beams were tested. Concrete mixturesare given in Table 2.

The ordinary steel bars were hot-laminated ribbed barscommercially identified as Class S500 (500 MPa [72,500 psi])with diameters varying from 6 to 20 mm (0.24 to 0.79 in.).To obtain the actual values of the yield stresses and strainslfy and Ey, respectively), six specimens of each diameter weretested. The average values were fy = 686 MPa (99,470 psi)and Ey = 3430 x 10-6.

Testing procedureThe testing rig consisted of three main parts: 1) the test

frame where the jack was fixed; 2) the torsion apparatus thattransformed the load applied by the jack to the torsionmoment transmitted to the end of the test beams; and 3) thereaction wall, placed at the other end of the beam, keepingthat end fixed, with no transversal rotation.

Figure 2 shows how the parts were placed during a test.The reaction wall and the torsion apparatus were fixed tothe solid floor. This testing rig allowed the longitudinaldeformation of the beams and warping in the ends of the beams.

Mixture design, content per m3

Components Series A Series B Series C

Thin aggregate, kg 205 164 83

Coarse aggregate, kg 914 908 766

Crushed grani te 5f] 1, kg 718 734 780

Normal portland cement- 360 375 530C Type 1/42.5R, kg

Admixture, L 4.1 4.8 15

Silica fume, kg - 41 60

Water, L 145 145 146

W/(C+ SF) 0.40 0.35 0.25

The load was applied in imposed deformation steps by thejack. This procedure was useful for the study of beams afterthe maximum peak load.

Loading was controlled by several load cells placed at keypoints on the testing rig (under the point of applied load andbetween torsion apparatus/reaction wall and solid floor).Transversal rotations were measured at 10 different crosssections located along the beam (Sections A-A to J-J) atregular intervals, as illustrated in Fig. 2.

The steel reinforcement bars were instrumented at threecross sections of the beams (midspan and quarter spans). At

Table 3-T-am curves: key points and ductility indexesTCT' ee" (GC)I, (GC)Il, Tty' ety, elY' eT" ey, eu,

Beam kNm degree/m kNm2 kNm2 kNm degree/m Tly,kNm degree/m T"kNm degree/m degree/m degree/m mqA-48.4-0.37 104.08 0.071 84,070 3807 128.37 0.44 131.84 0.57 150.78 1.18 0.50 1.50 2.99

A-47.3-0.76 109.50 0.064 97,950 7559 239.36 1.08 247.06 1.17 254.77 1.66 1.13 2.40 2.12

A-46.2-1.00 113.27 0.057 113,315 8337 272.65 1.16 259.17 1.07 299.91 1.54 1.11 1.57 1.41

A-54.8-1.31 120.87 0.063 109,485 11,100 360.87 1.56 368.22 1.66 368.22 1.66 1.61 1.76 1.09

A-53.1-1.68 120.93 0.044 159,702 14,398 - - - - 412.24 1.53 - - -B-75.6-0.30 111.50 0.060 107,198 612 - - 115.40 0.21 115.95 0.23 0.21 0.48 2.29

B-69.8-0.80 116.72 0.044 151,621 6715 265.83 1.33 273.28 1.42 273.28 1.42 1.37 2.30 1.67

B-77.8-1.33 130.45 0.043 172,940 9790 - - - - 355.85 1.45 - - -B-79.8-1.78 142.93 0.061 134,251 16,590 - - - - 437.85 1.24 - - -B-76.4-2.20 146.26 0.066 126,968 15,698 - - - - 456.19 1.24 - - -C-91.7 -0.37 117.31 0.038 177,073 4523 149.96 0.46 - - 151.76 0.72 0.46 0.83 1.80

C-94.8-0.76 124.46 0.049 146,640 6924 244.78 1.07 246.55 1.10 266.14 1.44 1.09 1.66 1.54

C-91.6-1.29 131.93 0.064 118,473 9209 347.65 1.46 - - 351.16 1.53 1.46 1.52 1.04

C-91.4-1.71 132.60 0.051 148,258 12,989 - - - - 450.31 1.50 - - -C-96.7-2.07 138.34 0.051 156,268 16,371 - - - - 467.26 1.34 - - -

C-87.5-2.68 139.09 0.054 146,963 19,294 - - - - 521.33 1.27 - - -Note: - = beams with brittle failure; I in.-kips = 0.113 kNm; I degree/in. = 39.37 degrees/m; 1kNm2 = 348.42 in2-kips.

, I

A

.~..="~-~-'- -~.

"'""~, .. .. - ',-- ~>~~1"- ~~ .I ;S~-"'53.1-1.68 - brittle failur

Fig. 4-Failure zone.

each section, eight strain gauges were attached to thereinforcement steel-four on each longitudinal bar locatedat the comers of the section and four on the transversal bars(one per branch of the stirrup).

A data logger was used to read and record the values givenby linear variable displacement transducers (LVDTs), load

nmane((GtorIra8tY'bypoiandyielmal(GClinE

FexpforiTablsireTabreinlead

rim)o 0

0.0 0.5 1.0 1.5 2.0 2..5 3.0 0.0 0.5 1.0 15 2.0 2.5 3.0

ISeries A I ISeries B I600 _

: ~: / '.:' .~~\~---Jf:g;~300 ~" -to', =-=-~:~~.~~

;r _8'l.- ~ .200 .'~// '". '.

-- I100

0. rim)o

M Q5 I~ 1.5 U ~ 3n

ISeries cl

Fig. 5-T-Bm curves. (Note: 1 in.-kips = 0.113 kNm; 1degree/in. = 39.37 degrees/m.)

cells, and strain gauges. Figure 3 shows a general view of thebeam with all the reading devices in place. A more detailedexplanation of the instrumentation of the beams is presentedin Bernardo's PhD thesis.12 Figure 4 shows some examplesof the typical failure zone of the tested beams.

EXPERIMENTAL RESULTSTorque-versus-twist curves

Figure 5 shows the curves with torsion moment (torque) Tversus average angular deformation per unit length (twist)am for all the series of tested beams. The average angulardeformation am was obtained by dividing the transversalangles measured at Sections A-A by the distance betweenSections A-A and J-J sections (5.35 m [210.6 in.], asshown in Fig. 2). Each individual T-am curve presentssome important points: cracking (D), yielding of transversalreinforcement (0), and yielding of longitudinal reinforce-ment (Ll). The yielding points were identified from the recordedexperimental data obtained through the strain gauges

attached to the bars. The failure zone of the beams generallyoccurred in the middle span, located in the same sectionwhere strain gauges were fIXedin the bars. Experimental strainvalues used to define yielding points are the average values of themeasured strains in the longitudinal or transversal reinforcementlocated in the section where strain gauges were fIXedin the bars.

From a global analysis of T-8m curves, some conclusionscan be drawn about the influence of the concrete strength onthe behavior of the tested beams under torsion.

Figure 5 indicates that the final part of the ascendingbranch of the T-8m curves are almost linear until maximumload for Series Band C beams, compared with Series Abeams. This is more noticeable for higher values of the torqueand torsion reinforcement. This may be due to the behavior ofthe compressed concrete struts. In fact, it is known that HSCexhibits a more linear stress-strain relationship when comparedwith NSC, and also a higher modulus of elasticity.

Table 3 summarizes relevant points from the T-8m curves,namely: cracking torque and corresponding twist angle (Tcrand 8f,r); torsional stiffness for noncracked stage, Stage I,((CC) ); torsional stiffness for cracked stage, Stage II, ((CC)I1);torque levels of yield of longitudinal steel and yield oftransversal steel; and corresponding twist angles (Tty' T1y'8ty, and 8Iy). The values of (CC)I and (CC)II were calculatedby linear regression with the range of experimental T-8mpoints of the corresponding stage: Stage I (between zero loadand cracking point) and Stage II (between cracking point andyield point for ductile beams or between cracking point andmaximum torque for brittle beams). The values of (CC)1 and(CC)1l were related to the inclination of the line calculated bylinear regression in each stage.

For beams with similar reinforcement ratio, it should beexpected that the ultimate torque would be generally higherfor higher values of the concrete compressive strength. FromTable 3, this conclusion it is not clear (the influence of concretestrength is probably too small to be noticed). Finally,Table 3 also shows that, for a series of beams with similarreinforcement ratios, the use of higher concrete strengthleads to less ductile beams.

Torsion strength and ductilityFrom the T-811l curves presented in the previous section, it

is possible to obtain the maximum experimental values of Trand 8Tr (resistant torque and corresponding transversalrotation). These values are also presented in Table 3 for thebeams tested in this research program.

For flexural ductility, usually the best parameter to use isdeformation. 15 Generally, a ductility index can be defined by~ = !1j!1 , where !1u represents the deformation correspondingto the ultimate load, and !1y represents the deformationcorresponding to the yielding of the steel bars.

A torsional ductility index (~e) was defined for the beamsin this research program. To characterize the globaldeformation of the test beams, the torsional rotation (per unitlength) was computed. The equation is ~e= 8J8y, where 8urepresents the ultimate torsional rotation and 8y representsthe torsional rotation corresponding to the yielding of thesteel bars. The values of ~e for the test beams were obtaineddirectly from the T-8m curves from the tests.

Because the transversal and the longitudinal bars did notyield simultaneously in the failure zone, the yielding pointwas considered as the average value of both yielding points(the two yielding points in each beam were generally veryclose together). There were some exceptions for the first

beams of each series, with low reinforcement ratio values(only the yielding of transversal reinforcement or longitudinalreinforcement was recorded). For those exceptions, theyielding point was considered as the only value recorded.For brittle beams (beams with high reinforcement ratiosvalues), the ductility index was not computed.

The ultimate load corresponds to the point at which thebeam no longer resists high loads. To fIXthis point, a criterionmust be assumed. The subjectivity of this parameter waspreviously discussed by the authors 16for the case of beamsunder flexure. For this case, many authors use the same criterionto define the ultimate point when the load (P)-displacement(0) curves have a descending branch. In these cases, it isusually considered that the ultimate point corresponds to aload of 80% of the maximum load. For the case of torsionand for this research, the authors have considered the sameprocedure. For the beams with no descending branch(sudden failure), the authors considered the ultimate loadequal to the maximum load.

Table 3 presents the experimental parameters used tocompute the torsional ductility index, namely, the averageyield rotation 8y, the ultimate rotation 8U' and the torsionalductility index ~e.A visual and qualitative value analysis ofthe torsional ductility index is explained in the following.

TORSION DESIGNThis section presents a comparative analysis of the predictions

for the maximum torque given in some codes of practice.The codes' predictions are compared with the results for thetest beams. The following codes of practice were considered:ACI 318-89,14 ACI 318-05,6 CAN3-A23.3-04,2 MC 90,4and EC 2.5 The superseded ACI 318-8914 was analyzedbecause torsion procedures were based on the skew-bendingtheory. The skew-bending theory was used by the ACI Codeup to 1995; in that year, the design rules were changed. Then,from 1995, they were based on the plasticity compressionfield theory (PCFT) as the European codes. PCFT is theresult of the development of the well-known variable angletruss-model. This theory is now the basis of the majority ofthe most important codes. The variable angle truss-modelwas also developed in a different way in 1973. It used thecompatibility of deformations instead of the theory of plasticity.The following developments of this theory led to the modifiedcompression field theory (MCFT). The MCFT is the base ofthe Canadian code CSA A23.3-04.2

For the comparative analysis presented herein, the beamswith brittle failures were also considered. The experimentalprogram showed that the range of beams with ductility (~e > 1)was very small. Therefore, all the beams had to be includedin this analysis, even those with a high reinforcement ratio,as Rasmussen and Baker did.9,IO

If a given code anticipates brittle failure due to crushing ofthe concrete struts of a beam under analysis, then thecorresponding theoretical strength is computed by adoptingthe top limit of the compression stress in concrete struts.

Table 4 presents the equations given in the studied codesand used in this study to compute torsion strength of reinforcedconcrete beams and the limits for the transversal reinforcement.Table 5 presents the theoretical values of the maximumtorque, Tr,calc' calculated through the codes' procedures. Thecorresponding experimental values, Tr,exp = Tr> are alsogiven. The experimental to theoretical torque ratios are alsopresented. A visual and qualitative analysis is given in thefollowing. The beams with predicted brittle failure are highlighted.

ACI318-8914 ACI318-056 MC904 EC 2s CAN3-A23.3-042

Tn = T,+ T, T = T . = 2FR

[] ~.bcams(hollow section) I

RasmussmlDaker(plainsc:etion) ,

60 80 100

f, (MP,)1 ACI318-89141

i} 1.0 +--''---00---0''c------1

I-[] Exp. Beams (hollow Sed~ RasmusscnJBaker(plainsectm)

f, (MPa)IEC251

[] Up. beams(hollow section)

Ra~mussenlDaker(plainsec:tion)

60 80 100

f, (MP,)IACI318-0S61

1.5

J +-.;0=---_-=- ---1..... 1,0 []~ OJ CD ~ .. "

o Exp.Beams (holl~ect~ RasmusscnlBaku (plain section)

f, (MP,)

JCAN3-A23.3-M042I

Some of these conclusions can be easily observed inFig. 6, which give the Tr,exp ITr,calc ratio as a function ofconcrete strength/c'. These figures also present the points fora group of nine rectangular over-reinforced HSC plainbeams tested by Rasmussen and Baker9,Io (all the beams hadbrittle failure by concrete crushing) with 3.0 m (118.11 in.)between supports. The cross-sectional dimensions (0.16 x0.275 m [6.30 x 10.83 in.]) and the reinforcement (PI =3.47% and Pt = 1.41% in the effective span) were constantfor all beams. Concrete strength was the only parameter thatvaried (from 57.1 to 109.9 MPa [8280 to 15,936 psi]).

Figure 6 confirms the general conclusions drawn fromTable 5, corresponding to the hollow beams tested by theauthors. Such conclusions are also confirmed by the over-reinforced plain beams tested by Rasmussen and Baker.9,IOAs noted by Hsu,17 the skew-bending theory 14seems to bebetter suited to predicting the ultimate torque, especially forsmall plain beams. For this theory, Fig. 6 also shows thatthe deviations from the predicted values decrease as theconcrete strength increases.

Figure 6 shows that the dispersion of the values for ACI318-056 is slightly higher than for ACI 318-8914 and doesnot depend on the concrete strength. Nevertheless, bothcodes give safe predictions. For ACI 318-05,6 the higherdispersion of values, in each beams series, occurs for thebeams that have the smallest and the highest reinforcementratios. Except for these beams, the predictions of this codeare comparable with those of ACI 318-89.14 The maximumtorque predicted for the over-reinforced beams tested byRasmussen and Baker9,Io is notably underestimated, withvalues close to 1.9.

For MC 90,4 EC 2,5 and CAN-A23.3-04,2 Fig. 6 confirmsthat, for beams with the highest reinforcement ratios and forthe highest concrete strengths, these codes are not safe andthe deviations from the actual maximum torque can beconsiderable. For the beams with average-to-low reinforcementratios, these codes underestimate the maximum torque. ForCAN-A23.3-04,2 Fig. 6 shows that the deviations are similarto those of EC 2.5 The deviations from the test values areparticularly apparent when applying MC 904 to beams withlow reinforcement ratios. The graphic representation of the

over-reinforced beams tested by Rasmussen and Baker9,IOconfirms the tendency observed for the beams with highreinforcement ratios tested by the authors.

In brief, the best code for the ultimate torque predictionis ACI 318-89,14 which is no longer in use. It was replaced byACI 318-05,6 which is the best code of all those that are currentlyin use. The other codes that were studied were not always safewith respect to the prediction of the maximum torque.

DUCTILITY DESIGNThe maximum and minimum values of the reinforcement

ratio that are proposed by different codes are analyzed hereinwith respect to the ductility behavior of the test beams. Thispoint will be studied herein. The over-reinforced beamstested by Rasmussen and Baker9 are not included in thisductility study because all of them had brittle failure.

Before examining the ductility study, the following shouldbe noted. Even though the ductility indexes of the first beamsof Series A, B, and C (Beams A-48.4-0.37, B-75.5-0.30, andC-91.7-0.37) were the highest (refer to Table 3), these beamshad a brittle failure due to insufficient reinforcement (thesteel bars reached failure shortly after cracking). This brittlebehavior indicates that these beams should not be acceptedby codes of practice, by imposing a minimum percentagelimit for the reinforcement. From the studied codes, only theACI Codes explicitly recommend a minimum amount ofreinforcement for torsion loads. In this study, the limit willbe applied only for transversal reinforcement because thetransversal and longitudinal reinforcement of the testedbeams are balanced.

Beams A-53. 1-1.68, B-79.8-1.78, B-76.4-2.20, C-96.7-2.07,and C-87.5-2.68 had brittle failure by crushing of the concretestruts. These beams had the highest reinforcement ratio valueswithin their series. Beams B-77.8-1.33 and C-91.4-1.71 had apremature and brittle failure by comers breaking off, followingby crushing of the concrete struts. All these beams, with brittlefailures, should be considered inadmissible by codes of practice.This is achieved by fLXinga top limit on the reinforcement ratio.

For each code used in this study, Table 6 presents the topand bottom limits of transverse reinforcement (At,min1s and

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ACI318-8914 T ACI318-056 MC904 I BC 25 I CAN3-A23.3-042~ As, mill As,mox As,min As,max As,min As,max As,min As,mox A.~,min As,max, , ,

S,

SS S S S S S S S sBeam cm2/m cm2/m cm2/m cm2/m cm2/m cm2/m cm2/m cm2/m cm2/m cm2/m cm2/m

A-48.4-0.37 3.11 1.62 14.02 2.04 9.53 1.37 15.04 0.86 17.52 0.75 16.80

A 47.3-0.76 6.25 1.48 14.31 1.84 8.72 1.49 14.99 0.84 17.53 0.80 16.34

A-46.2-1.00 8.32 1.44 13.87 1.77 8.68 1.53 15.54 0.83 17.52 0.81 15.84

A-54.8-1.31 11.29 1.44 15.21 1.93 8.88' 1.56 15.98 0.86 18.79 0.86 17.99

A-53.1-1.68 14.13 1.54 15.61 2.02 9.02' 1.56 16.25 0.90 18.56 0.84 18.50

B-75.6-0.30 2.33 1.62 18.14 2.55t 11.59 2.03 21.42 1.11 23.78 1.00 27.07B-69.8-0.80 6.67 1.48 17.57 2.24 10.66 1.94 20.33 1.04 23.59 1.02 24.38

B-77.8-1.33 11.29 1.44 18.04 2.30 11.00' 2.18 20.92 1.08 24.61 1.09 26.71

B-79.8-1.78 15.07 1.54 20.40 2.48 11.18' 2.25 23.05 1.19 26.17 1.14 30.00

B-76.4-2.20 18.83 1.54 21.81 2.42 10.60' 2.33 23.81 1.21 26.40 1.16 29.84

C-91.7-0.37 3.11 1.62 18.86 2.81 13.23 2.23 22.36 1.17 26.03 l.06 31.43

C-94.8-0.76 6.25 1.48 19.10 2.61 12.27 2.31 22.05 1.12 25.77 1.12 30.79C-91.6-1.29 10.53 1.44 18.04 2.50 11.74 2.36 20.68 1.10 24.42 1.12 29.56C-91.4-1.71 14.13 1.54 18.98 2.65 12.42' 2.37 22.52 1.17 25.71 1.13 31.48

C-96.7-2.07 17.38 1.54 19.93 2.73 13.30' 2.49 23.80 1.21 28.00 1.17 33.59

C-87.5-2.68 22.60 1.43 18.40' 2.41 11.34' 2.28 22.76 1.07 24.73 1.11 28.24

'Brittle failure due to insufficient concrete compressive strength expected in these beams.tBrittle failure due to insufficient reinforcement expected in this beam.Note: I in.-kips = 0.1 13 kNm; and I in2lin. = 2.54 cm2/m.

With ACI 318-05,6 Beam B-75.6-0.30 would not beallowed because it has a reinforcement ratio below thebottom limit. Therefore, this code has detected that this beamwould have brittle failure due to insufficient reinforcement.Although ACI 318-056 proved to be more restrictive thanACI 318-89,14 it is still not restrictive enough because it didnot detect two other test beams (Beam A-48.4-0.37 andBeam C-91.7-0.37), which failed closely due to insufficientreinforcement. As far as the top limit of the transversalreinforcement is concerned, Table 6 shows that the two lastbeams of Series A and the three last beams of the SeriesBand C do not pass the restrictions of ACI 318-05.6Therefore, on this joint, ACI 318-056 is more restrictivethan ACI 318-89.1 For the beams specified, ACI 318-056anticipates brittle failure. The limitation seems to be a littleexcessive because Beam A-54.8-1.31 is one that would beexcluded and, in fact, this beam did not have brittle failure.The ductility index for this beam, however, is very low (referto Table 3), and the exclusion of this beam can be consideredacceptable. When compared with its predecessor, ACI 318-056is better for fixing a maximum value for the transversalreinforcement.

As far as the bottom limit of the transversal reinforcementis concerned, the remaining codes indicate limit values thatdo not ban any of the test beams that are shown to haveinsufficient reinforcement (refer to Table 6). It should be notedthat these codes give limitation rules based on minimum shearreinforcement. This procedure does not seem to be adequate.As far as the top limit for the transversal reinforcement is'concerned, Table 6 shows that MC 90,4 EC 2,5 and CAN3-A23.3-042 consider that all the beams have acceptable values.Therefore, these three codes do not have good top limits ofthe maximum transversal reinforcement. Some of the beamsallowed by the codes did have brittle failure due to excessivecompressive stresses in concrete struts.

The conclusions derived from the tables are highlighted inFig. 7. These figure present, for all the analyzed codes andindependently of the concrete strength of the test beams, theevolution of the torsional ductility index /la, with thetransversal torsion reinforcement ratio Pt. Figure 7 alsoshows the maximum and minimum boundaries of thetrans versal reinforcement ratios (dotted lines) computedfrom the codes' rules. These values are computed from themaximum and minimum reinforcement areas (At,minls andAt,ma!s, respectively). The beams that, despite having highductility indexes, did have brittle failure due to insufficientreinforcement are identified by the symbol x. The beams thathad brittle failure due to crushing of concrete struts (with/la = 1) are also shown in Fig. 7, located on the Pt axis.Despite /la = 1 having no rational explanation, this value wasconventionally adopted for all brittle beams to allow for theirinclusion in Fig. 7.

Figure 7 shows that the interval defined by the maximumand the minimum values of transversal torsion reinforcementfound by ACI 318-8914 is too large. In fact, not only theacceptable beams, but most of the undesirable beams, arealso within this interval.

Figure 7 confirms that, comEared with ACI 318-8914 andthe other codes, ACI 318-05 indicates an interval muchcloser to the real behavior of the test beams. The authorsargue that only the lower limit needs a minor correction toensure that all the beams with brittle failure due to insufficientreinforcement are excluded.

Figure 7 also confirms that the other studied codes defineintervals of reinforcement that are too wide. The lower limitsare not specific for torsion (they are adopted from shearprovisions) and the experimental results clearly show that aspecific rule for torsion would be very desirable. The top

limits should be corrected to avoid cases of brittle failure byexcessive compression in concrete struts.

CONCLUSIONSThe study reported herein is based on the results of 16 test

beams. Although this kind of experimental work is ratherexpensive, it would be better to have more tests. Nevertheless,the number of tests is sufficient to draw some conclusions.The results clearly show that some codes are excessivelypermissive and could lead to the acceptance of brittle beamsor unsafe values of the predicted maximum torque. Of all thestudied codes in use, ACI 318-056 is the only one that hasacceptable rules with respect to minimum and maximumreinforcement of members under torsion. The other codesneed to be corrected, especially for the type of beams testedin this research (HSC hollow beams under pure torsion). Theauthors suggest that two main actions need to be taken, asexplained in the following.

First, all the undesirable beams that are at risk of brittlefailure should be banned. This is ensured by means of anadequate limit value for both the maximum and theminimum amount of torsion reinforcement, that is, for fc' :2:40 MPa (5800 psi).

Second, the codes need to be more reliable in predictingthe maximum torque, and the deviations should diminish.This is done by using a criterion better suited to predictingthe compressive failure of the concrete struts. This objectiveis related to the limitation of the maximum amount oftransversal reinforcement. It should be noted that the need toreduce the maximum value of the transversal reinforcementhas already been stated by Rasmussen and Baker10 for aprevious version of the European and Canadian codes.

Due to the limited number of available experimentalresults of HSC beams, the authors find it a little premature topropose new design rules. At this stage, they are onlyindicating the direction that those new rules should take.The experimental work on this subject must continuewith more tests to obtain the correct rules for both NSCand HSC.

AcAslAs,A"e!AtmaxA,:millfti1;,(GC)I(GC)'IsTTcrTlyTr,calcTr~Tr,expTryt

NOTATIONarea of cross section (including hollows)area of longitudinal reinforcementarea of one leg of transversal reinforcementeffective area of transversal reinforcementmaximum area of transversal reinforcementminimum area of transversal reinforcementuniaxial compressive strength of concreteyield strength of reinforcementtorsional stiffness (noncracking stage)torsional stiffness (cracking stage)longitudinal spacing of transversal reinforcementapplied torquecracking torqueyield torque of longitudinal reinforcementexpected torsion strengthexperimental torsion strengthyield torque of transversal reinforcementwall thickness of hollow sectionperimeter of stirrupsexternal dimensions of cross sectiondistance between axis of stirrups legsyield strain of reinforcement

~eScrSly8/11

STr8,v8,;8yPIP,Pl,max

PI,mill

PIO'

torsional ductility indexcracking angle of twist correspondent to Tcrangle of twist correspondent to Tlyaverage angle of twistangle of twist corresponding to Trangle of twist correspondent to Tryangle of twist corresponding to ultimate point of T-8 curveangle of twist corresponding to yielding point of T-8 curvelongitudinal reinforcement ratiotransversal reinforcement ratiomaximum ratio of transversal reinforcementminimum ratio of transversal reinforcemenltotal reinforcement ratio

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Stockholm, Sweden, 1989,78 pp.2. Standards Council of Canada, "Design of Concrete Structure for

Buildings (CAN3-A23.3-04)," Canadian Standards Association, Mississanga,Canada, Dec. 2004, 240 pp.

3. Standards Association of New Zealand, "Concrete Structures NZS3101-Part I: Design," 1995,256 pp.

4. CEB-FIP, "Model Code 1990," Comite Euro-International du Beton,Lausanne, Switzerland, 1990,461 pp.

5. CEN prEN 1992-1-1, "Eurocode 2: Design of Concrete Structures-Part I: General Rules and Rules for Buildings," Brussels, Belgium, Apr.2002, 225 pp.

6. Acr Committee 318, "Building Code Requirements for StructuralConcrete (AC! 318-05) and Commentary (3 I8R-05)," American ConcreteInstitute, Farmington Hills, MI, 2005, 443 pp.

7. Priestley, M. J. N.; Seible, E; and Wang, C. M., "The NorthridgeEarthquake of January 17, 1994-Damage Analysis of Selected FreewayBridges," Reporl No. SSRP-94/06, University of California, San Diego,CA, Feb. 1994, 266 pp.

8. Ali, M. A., and White, R. N., "Toward a Rational Approach forDesign of Minimum Torsion Reinforcemenl," ACt Structural Journal,V. 96, No. I, Jan.-Feb. 1999, pp. 40-45.

9. Rasmussen, L. J., and Baker, G., "Torsion in Reinforced Normal- andHigh-Strength Concrete Beams-Part I: Experimental Test Series," AClStructural Journal, Y. 92, No. I, Jan.-Feb. 1995, pp. 56-62.

10. Rasmussen, L. J., and Baker, G., "Torsion in Reinforced Normal- andHigh-Strength Concrete Beams-Part 2: Theory and Design," ACt StructuralJournal, Y. 92, No.2, Mar.-Apr. 1995, pp. 146-156.

II. Wafa, E E; Shihala, S. A.; Ashour, S. A.; and Akhtaruzzaman, A. A.,"Prestressed High-Strength Concrete Beams under Torsion," Journal of theStructural Engineering, ASCE, Y. 121, No.9, Sept. 1995, pp. 1280-1286.

12. Bernardo, L. EA., "Torao em Vigas em Caixao de Betao de AltaResistencia (Torsion in Reinforced High-Strength Concrete HollowBeams)," PhO thesis, University of Coimbra, Portugal, 692 pp. (in Portuguese).

13. CEB, "Torsion," Bulletin d'tnjornration, No. 71, Lausanne, Switzerland,Mar. 1969, 207 pp.

14. ACI Committee 318, "Building Code Requirements for ReinforcedConcrete (ACI 318-89) and Commentary (318R-89)," American ConcreteInstitute, Farmington Hills, MI, 1989,369 pp.

15. Shin, S.-W.; Kamara, M.; and Ghosh, S. K., "Flexural Ductility,Strength Prediction, and Hysteretic Behavior of Ultra-High-StrengthConcrete Members," High Strength Concrete, Second InternationalSymposium, SP-121, W. T. Hester, ed., American Concrete Institute,Farmington Hills, MI, 1990, pp. 239-264.

16. Bernardo, L. E A., and Lopes, S. M. R, "Flexural Ductility of High-Strength Concrete Beams," Struclural Concrete, V. 4, No.3, 2003,pp. 135-154.

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18. Mitchell, D., and Collins, M. P., "Diagonal Compression FieldTheory-A Rational Model for Structural Concrete in Pure Torsion," ACIJOURNAL,Proceedings Y. 71, No.8, Aug. 1974, pp. 396-408.