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Softened truss model for reinforced NSC and HSC beams under torsion:A comparative study

L.F.A. Bernardo a,!, J.M.A. Andrade a, S.M.R. Lopes b

a University of Beira Interior, Covilhã, Portugalb University of Coimbra, Portugal

a r t i c l e i n f o

Article history:Received 17 October 2011Revised 28 February 2012Accepted 26 April 2012Available online 3 June 2012

Keywords:BeamsTorsionTruss-modelSoftening effectStiffening effectTheoretical model

a b s t r a c t

A computing procedure is presented to predict the ultimate behavior of Normal-Strength Concrete (NSC)and High-Strength Concrete (HSC) beams under torsion. Both plain and hollow beams are considered. Inorder to model the non-linear behavior of the compressed concrete in the struts and of the tensionedsteel reinforcement several proposals for the stress (r)–strain (e) relationships were tested. The theoret-ical predictions of the maximum torque and corresponding twist were compared with results fromreported tests and with the predictions obtained from Codes. One of the tested theoretical models wasfound to give excellent predictions for the maximum torque when compared with those obtained fromsome codes of practice and with experimental values of NSC and HSC beams (plain and hollow).

! 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The first studies on torsion of reinforced concrete beams werepublished in the beginning of the last century. The developed the-oretical models can be divided into two main theories: the Skew-Bending Theory which was the base of the American code between1971 and 1995, and the Space Truss Analogy which is currently thebase of the American code (since 1995) and of the European modelcodes (since 1978) [1–3].

The Variable Angle Truss-Model (VATM), which firstly aimed tounify the torsion design of small and large sections and of rein-forced and prestressed concrete, is probably the most used theo-retical truss model to predict the theoretical ultimate behavior ofbeams under torsion. This theory also allows a good physicalunderstanding of the torsion problem in Reinforced Concrete(RC) elements and has an important historical value. The firstsimple version of the model was presented by Rausch in 1929[4]. Other authors have contributed to updated versions of themodel, such as: Andersen in 1935 [5], Cowan in 1950 [6], Lampertand Thurlimann in 1969 [7], Elfgren in 1972 [8], Collins andMitchell in 1980 [9]. In 1985 Hsu and Mo [10] developed a modelwith the influence of Softening Effect. Some developments and

alternative methods were also introduced by Jeng and Hsu in2009 [11], Jeng in 2010 [12], Mostofinejad and Behzad in 2011[13], Jeng et al. in 2011 [14]. Some recent experimental worksbrought more information on the actual behavior of beams undertorsion. This is the case of works by Bernardo and Lopes in 2008,2009 and 2011 [15–18], Algorafi et al. in 2009 and 2010 [19,20],Al Nuaimi et al. in 2008 [21] and Lopes and Bernardo in 2008[22].

The VATM can be divided into two categories: PlasticityCompression Field Theory (Lampert and Thurlimann, Elfgren) andCompatibility Compression Field Theory (Collins, Hsu and Mo). Whilein the first theory the stresses are based on the theory of plasticity,the second theory is based on the deformations’ compatibility ofthe truss analogy.

The Space Truss Analogy provides good results for high levels ofloading. However, for low levels, the method does not give goodpredictions since the model assumes a cracked state from thebeginning of the loading. Even for high levels of loading, the accu-racy of the results will highly depend on the constitutive law thatwould characterize the non-linear behavior of the materials.

The objective of this study is to develop and test a computa-tional procedure, based on the VATM, and to predict the ultimatebehavior of both NSC and HSC beams under torsion (plain and hol-low). The behavior of beams under torsion is studied by means of T(Torque)–h (Twist) curves. The non-linear behavior for the materi-als (concrete and steel reinforcement) is incorporated by mean ofstress (r)–strain (e) relationships found in the literature.

0141-0296/$ - see front matter ! 2012 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.engstruct.2012.04.036

! Corresponding author. Address: Departamento de Engenharia Civil e Arquite-tura, Edifıcio II das Engenharias, Universidade da Beira Interior, Calçada Fonte doLameiro, 6201-001 Covilhã, Portugal. Tel.: +351 275 329729; fax: +351 275 329969.

E-mail addresses: [email protected], [email protected] (L.F.A. Bernardo).

Engineering Structures 42 (2012) 278–296

Contents lists available at SciVerse ScienceDirect

Engineering Structures

journal homepage: www.elsevier .com/locate /engstruct

To help with calculations, a computing tool is developed withthe help of the computing programming language Delphi.

The theoretical results are compared with the experimentalresults of several beams under pure torsion, which are availablein the literature and also with codes provisions.

2. Previous studies and research significance

In previous theoretical studies, several authors predicted thebehavior of NSC beams under torsion by using the VATM. Forexample, Hsu in 1984 [23] used the VATM to calculate the theoret-ical T–h curve. Hsu showed that the ultimate values of the T–hcurves, experimental and theoretical, were quite similar. Theseobservations were also confirmed by Bernardo and Lopes [15] fora larger group of tested beams, including some HSC hollow beamstested by the authors [16]. For NSC beams under torsion, Bernardoand Lopes showed that, in general, VATM is quite appropriate forthe prediction of the ultimate behavior. However, for HSC beamsthese authors showed that VATM could not be considered ade-quate since resistances were highly overestimated for beams withhigh torsional reinforcement ratio [15,18].

The calculus procedure was reviewed by the authors in order toincorporate specific r–e relationships for HSC. In fact, for HSC theshape of r–e curve for concrete in compression is quite differentfrom that of NSC. Thus, theoretical models that depend on this kindof relationships cannot be extrapolated from NSC to HSC. Based onthe theoretical predictions obtained with the new calculus proce-dure, Bernardo and Lopes [18] concluded that, in order to obtaingood predictions for the whole range of beams, including thosewith high reinforcement ratios, the calculus procedure needed to

be reviewed. This observation led the authors to propose empiriccoefficients to correct the theoretical results.

In their theoretical models the authors only tested some r–erelationships to characterize the mechanical behavior of the con-crete in compression (struts) and the steel in tension (reinforce-ment), but they did not carry out a comprehensive study withregards to the large quantity of r–e relationships that could befound in bibliography. For NSC beams and concrete in compres-sion, the cited authors used the r–e relationship by Vecchio andCollins proposed in 1982 [24] from the results of NSC plates testedunder shear. For HSC beams, Bernardo and Lopes used r–e rela-tionships proposed by Belarbi and Hsu in 1991 [25] (based onthe results of HSC plates tested under shear).

For the reinforcement in tension, many authors use a commonbilinear relationship with horizontal or inclined plastic landing.

Since the first study by Vecchio and Collins, published in 1981[26], several other proposals for r–e relationships, which had ta-ken into account the softening effect and the stiffening effect,were published by various authors. In 2005 [27], Costa et al.showed that, for panels under shear, the variability of predictionsbased on the several proposals was very high. The authordefended that there was a need of checking the different r–e rela-tionships proposals before one could reach definitive conclusions.This is the case in this study, because the theoretical results fromVATM strongly depend on the r–e relationships for concrete andsteel.

All these aspects justify the need for a state-of-the art workon r–e relationships (for concrete in compression and reinforce-ment in tension) for the theoretical prediction of the ultimatebehavior of NSC and HSC beams (plain and hollow) undertorsion.

Fig. 1. Normal and softened r–e curves for concrete in compression in struts [27].

! s

s

!sy

fsy

1"s

kfsy

!su

(b)(a)Fig. 2. Stiffened and idealized non-stiffened r–e curve for reinforcement in tension [27].

L.F.A. Bernardo et al. / Engineering Structures 42 (2012) 278–296 279

Table 1r–e Relationships for concrete in compression in struts.

Vecchio and Collins [24]

fc2 ! bf 0c 2ec2

beo

! "" ec2

beo

! "2" #

for ec2 P beo

fc2 ! bf 0c 1" ec2 " beo

2eo " beo

! "2" #

for ec2 < beo

Hognestad [28]

fc ! f 0c 2ec

eo

! "" ec

eo

! "2" #

Vecchio and Collins [29]

fc2 ! brf 0c 2ec2

eo

! "" ec2

eo

! "2" #

Collins and Poraz [30] – Model A

fc2;base ! "fp

n "ec2ep

# $

n" 1# "ec2ep

# $nk ; n ! 0:80# fp$MPa%17

Collins and Poraz [30] – Model B

fc2;base ! "fp

n "ec2ep

# $

n" 1# "ec2ep

# $nk ; n ! 0:80# fp$MPa%17

k ! 1:0 "ep < ec2 < 0

k ! 0:67# fp$MPa%62 ec2 < "ep

(k ! 1:0 "ep < ec2 < 0

k ! 0:67# fp$MPa%62 ec2 < "ep

(

(i) for "ec2 6 beo: fp ! bf 0c and ep = beo;(ii) for beo < "ec2 6 eo: fc2 ! fp ! bf 0c ;

(iii) for "ec2 > eo: fc2 ! bfc2;base; f p ! f 0c and ep = eo.

(i) for "ec2 6 eo: fp ! bf 0c and ep = eo;(ii) for "ec2 > eo: fc2 ! bfc2;base; f p ! f 0c and ep = eo.

Belarbi and Hsu (1991) [25]

fc2 ! brf 0c 2ec2

beeo

! "" ec2

beeo

! "2" #

for ec2 P beeo

fc2 ! brf 0c 1" ec2 " beeo

4eo " beeo

! "2" #

for ec2 < beeo

Zhu et al. [31]a

fc2 ! brf 0c 2ec2

beeo

! "" ec2

beeo

! "2" #

for ec2 P beeo

fc2 ! brf 0c 1" ec2 " beeo

2eo " beeo

! "2" #

for ec2 < beeo

a Calibrated for HSC.

Table 2Reduction factors br and be.

Vecchio and Collins [24] Vecchio and Collins [29]b ! br ! be ! 1

0:85"0:27ec1ec2

br ! 10:8"0:34ec1

eo

6 1:0

Collins and Poraz – v1 [30] Vecchio et al. [32]b ! br ! be ! 1

1:0#Kc Kf6 1 br ! 1

1:0#Kc Kf6 1; be ! 1:0

Kc ! 0:35 "ec1ec2" 0:28

# $0:80 Kc ! 0:27 ec1eo" 0:37

# $

Kf ! 0:1825%%%%%%%%%%%%%%%%%f 0c$MPa%

pP 1:0 Kf ! 2:55" 0:2629

%%%%%%%%%%%%%%%%%f 0c$MPa%

p6 1:11

Collins and Poraz – v2 [30] Hsu [33]br ! 1

1:0#Kc6 1; be ! 1:0 br ! be ! 0:9%%%%%%%%%%%%%%%

1#600ec1

p

Kc ! 0:27 ec1eo" 0:37

# $

Vecchio – v1 [34] Vecchio – v2 [34]b ! br ! be ! 1

1#CsCdbr ! 1

1#CsCd; be ! 1:0

Cd ! Kc ! 0:35 "ec1ec2" 0:28

# $0:80 Cd ! Kc ! 0:27 ec1eo" 0:37

# $

Cs ! 0:55 Cs ! 0:55

Vecchio – v1 [35] Vecchio – v2 [35]b ! br ! be ! 1

1#0:35 "ec1ec2"0:28

# $0:8 6 1 br ! 1

1#0:35 "ec1ec2"0:28

# $0:8 6 1

Belarbi and Hsu [36] Belarbi and Hsu [37]br ! 0:9%%%%%%%%%%%%%%

1#Krec1

p be ! 1%%%%%%%%%%%%%1#Keec1

p br ! be ! 0:9%%%%%%%%%%%%%%%1#400ec1

p

Zhang and Hsu [38]a Miyahara et al. [39]

br ! be !R f 0c$ %%%%%%%%%%%%%%1#400ec1

g0

q ; g ! ql fsy;lqt fsy;t

; rh ! rv ! 0 br ! 1:0 ec1 6 0:0012br ! 1:15" 125ec1 0:0012 < ec1 6 0:0044br ! 0:6 ec1 > 0:0044

8<

:

g 6 1) g0 ! gg > 1) g0 ! 1=g

&; R f 0c' (! 5:8%%%%%%%%%%%%%%%

"f 0c $MPa%p 6 0:9

Mikame et al. [40] Ueda et al. [41]a

b ! br ! be ! 10:27#0:96 ec1

"eo$ %0:167 b ! br ! be ! 1

0:8#0:6$1000ec1#0:2%0:39


280 L.F.A. Bernardo et al. / Engineering Structures 42 (2012) 278–296

3. r–e Relationships

Theoretical models for the behavior of cracked RC plates undershear usually consider the behavior of the concrete and thereinforcement independently through their r–e relationships. Inaddition, experimental stresses and strains are measured along asufficient width to include several cracks. For this reason, averager–e relationships are proposed.

Transversal tension strains influence the behavior of concrete instruts, mainly in the cracked state. This phenomenon, called soft-ening effect (Fig. 1), is important in some structural concrete ele-ments, like plates under shear or beams under shear or torsion.For concrete in compression in the struts, average r–e relation-ships take this aspect into account.

In Fig. 1, the meaning of the parameters are: eo is the strain cor-responding to the peak stress f 0c

' (, ec1 is the principal tension strain

and ec = ec2 is the principal compression strain in the principaldirection of the compression stress (fc = fc2).

For reinforcement in tension, average r–e relationships takeinto account the interaction between reinforcement and concretein tension between cracks. This interaction is called stiffening ef-fect (Fig. 2a). Other models do not incorporate this interaction(Fig. 2b).

In Fig. 2: fs = rs is the tensile stress, fsy is the yielding stress,fst = kfsy is the tensile strength, fn is the stress level correspondingto the intersection of the two asymptotic limits (straight lines), fo

is the stress level corresponding to the intersection of the upperasymptotic limit (straight line) with vertical axis, f &y is the apparentyielding stress, es is the tension strain, esy is the yielding strain atthe end of the elastic behavior, esu is the ultimate strain, Es isYoung’s Modulus in elastic stage and Ep and is Young’s Modulusin plastic stage.

In this study, only r–e relationships of concrete in compressioncalibrated for plates under shear with proportional loading (trans-versal tension increase as principal compression stress increase),and with similar detailing of reinforcement, usually adopted forwebs or walls of RC beams, are tested: longitudinal reinforcementorthogonal to transversal reinforcement and reinforcement to 45",or approximately, with principal directions.

Table 1 presents the mathematical equations for the r–e rela-tionships for concrete in compression in struts that are tested inthis study, and proposed by several authors. Table 2 presents themathematical equations for the reduction factors for stress (br)and strain (be) corresponding to the r–e relationships of Table 1.

Table 3 presents all the tested combinations between r–e rela-tionship and reduction factors, taking into account the originalrelations between them.

In some cases, the authors propose a r–e relationship for con-crete and also the reduction factors br and be. In other cases,

Table 3Tested models for concrete in compression in struts.

Model r–e Relationship Reduction factors br and be

c01 Hognestad [28] –c02 Vecchio and Collins [24] Vecchio and Collins [24]c03 Vecchio and Collins [29] Vecchio and Collins [29]c04 Collins and Poraz [30] – Model A Collins and Poraz [30]c05 Collins and Poraz [30] – Model B Collins and Poraz [30]c06 Collins and Poraz [30] – Model A Vecchio et al. [32]c07 Collins and Poraz [30] – Model A Vecchio [34]c08 Collins and Poraz [30] – Model B Vecchio [34]c09 Vecchio and Collins [24] Vecchio [35]c10 Vecchio and Collins [29] Vecchio [35]c11 Belarbi and Hsu [25] Belarbi and Hsu [36]c12 Belarbi and Hsu [25] Belarbi e Hsu [37]c13 Belarbi and Hsu [25] Hsu [33]c14 Belarbi and Hsu [25] Zhang and Hsu [38]a

c15 Zhu et al. [31]a Zhang and Hsu [38]a

c16 Vecchio and Collins [24] Mikame et al. [40]c17 Vecchio and Collins [29] Mikame et al. [40]c18 Vecchio and Collins [24] Ueda et al. [41]a

c19 Vecchio and Collins [29] Ueda et al. [41]a

c20 Vecchio and Collins [24] Miyahara et al. [39]c21 Vecchio and Collins [29] Miyahara et al. [39]


Table 4r–e Relationships for reinforcement in tension.

Model r–e Relationship

r01 EC 2 [2]

fs ! Eses for es 6 esy ! fsy=Es

fs !fsy$k" 1%esu " esy

es; k ! fst=fsy for esy < es 6 esu

r02 EC 2 [2] or Vecchio and Collins [29]

fs ! Eses para es 6 esy ! fsy=Es

fs ! fsy para es > esy

r03 Belarbi and Hsu [42]

fs !0:975Eses

1# 1:1Esesfsy

# $mh i1m# 0:025Eses; m ! 1

9B" 0:26 25;

B ! 1q

fcr

fsy

! "1:5

; f cr ! 3:75%%%%%%%%%%%%%f 0c$psi%

q

Al f lq

T

#

s

$d

At tcracking

t

tt /2d

neutral axis

td

k2td

!ds

AB

C

concrete strut section

wall section

strain stress

centerline of shear flow

!ds /2

Fig. 3. Three dimensional truss analogy with variable angle.


some authors propose their own r–e relationships for the con-crete, but they use reduction factors br and be proposed by otherresearchers. Other authors (for instance, Ricardo et al. [27]) alsocombine the r–e relationships for concrete proposed by someauthors with the reduction factors br and be proposed by otherauthors. This latter is based on compatibility criteria (type ofstructural element in study, loading type, concrete strength,etc.). These last cases correspond to the combinations in Table 3where the names of authors are not the same in the two columns.

In Table 3, the combinations are called c01–c21 (c – concrete).Model c01 (based on Hognestad r–e relationship for concrete inuniaxial compression) does not take into account the softening ef-fect, and it constitutes a reference model.

Table 4 presents the mathematical equations for the three r–erelationships for reinforcement in tension tested in this study. Eachr–e relationship to be tested is called r01–r03 (r – reinforcement).Model r03 is an average r–e relationship taking into account thestiffening effect, while r01 and r02 are classics bilinear r–erelationships for reinforcement in uniaxial tension.

With respect to r–e relationships for reinforcement, it can besaid that a clear relationship between the equations presented inTable 4 (reinforcement) and the equations presented in Table 3(concrete) does not exist. So all the r–e relationships for the rein-forcement will be combined with each r–e relationships for theconcrete (Section 5).

Each model c01–c21 was tested separately with models r01–r03.Thus, 21 ' 3 = 63 theoretical models were tested.

4. Theoretical model based on VATM

The use of VATM to predict the behavior of RC beams under tor-sion, instead of Skew Bending Theory, allows computing the evolu-tion of the state of the beam (load level, twist, strain and stress in thereinforcement and in the concrete, etc.) with growing load levels.

Computing the theoretical T–h curve from the VATM (Fig. 3)requires the three following equilibrium equations to computethe torque, T, the effective thickness of the concrete struts, td, ofthe equivalent tubular section and the angle of the concrete struts,a, from the longitudinal axis of the beam [23]:

T ! 2Aotdrd sin a cos a $1%

cos2a ! Alfl

pordtd$2%

td !Alfl

pord# Atft

srd$3%

where Ao is the area limited by the center line of the flow of shearstresses, which coincides with center of the walls thickness, td; rd

is the stress in the diagonal concrete strut; Al is the total area ofthe longitudinal reinforcement; fl is the stress in the longitudinal

fc$%

d

!o!% 2!o !d!p =!ds

Area!dsB=Area/

!ds

B

$d

!ds < !o!%

d

2!o !d!ds

Area 1

!dsB=(Area 1 + Area 2) /

!ds

B

$dArea 2

!ds > !o!%

fc$%

!o!%!p =

(b)(a)Fig. 4. Integration of r–e curve for concrete strut.

Select !ds

Estimate td, #, %$, %!

Calculate k1 for model ci(Table 1) and $d (Eq. 9)

Calculate T (Eq. 1), !l (Eq. 4), !t (Eq. 5), $t and $l for model ri

(Table 4)

Calculate td´ (Eq. 3)

td = td´ ? No

Yes

Calculate #´ (Eq. 2)

# = #´ ? No

Yes

Calculate %$´, %!´ for model ci (Table 2)

%$ = %$´ ? %! = %!´ ?

No

Yes

Calculate & (Eq. 6)

!ds > !cu ? !l or !t > !su ?

No

Yes

END

Fig. 5. Flowchart for the calculation of T–h curve.


reinforcement; At is the area of one leg of transversal reinforce-ment; ft is the stress in the transversal reinforcement.

The three following compatibility equations are also needed tocompute the strain of the longitudinal reinforcement, el, the strainof the transversal reinforcement, et, and the twist, h [23]:

el !A2

ord

poT cotg a"12

!eds $4%

et !A2

ord

poT tg a"12

!eds $5%

h ! eds

2td sin a cos a $6%

where po is the perimeter of the center line of the flow of shearstresses.

The strain at the surface of the diagonal concrete strut, eds, andat the center line of the flow of shear stresses, ed, can be computedfrom (Fig. 3):

eds !2potd

Ao$et # ed% tg a sin a cos a $7%

ed ! eds=2 $8%

The stress of the diagonal concrete struts, rd, is defined as the med-ium stress of a non-uniform diagram (Fig. 3):

rd ! k1brf 0c $9%

Table 5Properties of test beams.

Beam Sectiontype

x(cm)

y(cm)

t(cm)

x1

(cm)y1

(cm)Asl

(cm2)Ast/s(cm2/m)

ql

(%)qt

(%)f(cm)

(MPa)fctm

a

(MPa)flym

(MPa)ftym

(MPa)Ec

a

(GPa)eo

a

(%)ecu

a

(%)

B2[43]

Plain 25.4 38.1 – 21.6 34.3 8.0 7.1 0.8 0.8 28.6 3.0 317 320 25.3 0.20 0.35

B3[43]

Plain 25.4 38.1 – 21.6 34.3 11.4 10.2 1.2 1.2 28.1 3.0 328 320 25.1 0.20 0.35

B4[43]

Plain 25.4 38.1 – 21.6 34.3 15.5 14.0 1.6 1.6 29.2 3.0 320 324 25.6 0.20 0.35

B5[43]

Plain 25.4 38.1 – 20.3 33.0 20.4 18.5 2.1 2.0 30.6 3.1 332 321 26.2 0.20 0.35

G6[43]

Plain 25.4 50.8 – 21.6 47.0 7.7 5.6 0.6 0.6 29.9 3.0 335 350 26.0 0.20 0.35

G8[43]

Plain 25.4 50.8 – 21.6 47.0 17.0 12.3 1.3 1.3 28.4 3.0 322 329 25.2 0.20 0.35

M2[43]

Plain 25.4 38.1 – 21.6 34.3 11.4 6.8 1.2 0.8 30.6 3.1 329 357 26.2 0.20 0.35

T4 [44] Plain 50.0 50.0 – 45.4 45.4 18.1 10.3 0.7 0.8 35.3 2.7 357 357 32.7 0.20 0.35A3

[45]Plain 25.4 25.4 – 21.9 21.9 8.0 8.9 1.2 1.2 39.4 3.5 352 360 29.7 0.20 0.35

B2[45]

Plain 17.8 35.6 – 14.6 32.4 5.2 6.6 0.8 1.0 39.7 3.5 380 286 29.8 0.20 0.35

B3[45]

Plain 17.8 35.6 – 14.3 32.1 8.0 8.6 1.3 1.3 38.6 3.5 352 360 29.4 0.20 0.35

B4[45]

Plain 17.8 35.6 – 14.3 32.1 11.4 11.8 1.8 1.7 38.5 3.5 351 360 29.4 0.20 0.35

D4[43]

Hollow 25.4 38.1 6.4 21.6 34.3 15.5 14.0 1.6 1.6 30.6 3.1 330 333 26.2 0.20 0.35

T2 [46] Hollow 50.0 50.0 8.0 44.2 44.2 18.1 10.5 0.7 0.8 25.6 2.0 357 357 29.4 0.20 0.35T1 [44] Hollow 50.0 50.0 8.0 45.4 45.4 18.1 10.3 0.7 0.8 35.4 2.7 357 357 32.7 0.20 0.35VH1

[47]Hollow 32.4 32.4 6.5 28.5 28.5 3.5 2.8 0.3 0.3 17.2 1.3 447 447 25.7 0.20 0.35

A2[16]

Hollow 60.0 60.0 10.7 53.8 53.1 14.0 6.3 0.4 0.4 47.3 3.5 672 696 36.1 0.20 0.35

A3[16]

Hollow 60.0 60.0 10.9 54.0 53.5 18.1 8.3 0.5 0.5 46.2 3.4 672 715 35.8 0.20 0.35

A4[16]

Hollow 60.0 60.0 10.4 52.0 52.5 23.8 11.2 0.7 0.7 54.8 3.9 724 715 37.9 0.20 0.35

A5[16]

Hollow 60.0 60.0 10.4 52.8 52.8 30.7 14.1 0.9 0.8 53.1 3.8 724 672 37.5 0.20 0.35

B2[16]

Hollow 60.0 60.0 10.8 53.3 53.4 14.6 6.7 0.4 0.4 69.8 4.1 672 696 39.4 0.21 0.33

B3[16]

Hollow 60.0 60.0 10.9 53.5 53.7 23.8 11.2 0.7 0.7 77.8 4.3 724 715 40.7 0.21 0.31

B4[16]

Hollow 60.0 60.0 11.2 52.3 53.6 32.2 15.1 0.9 0.9 79.8 4.4 724 672 41.0 0.21 0.31

B5[16]

Hollow 60.0 60.0 11.7 51.8 51.8 40.2 18.9 1.1 1.1 76.4 4.3 724 672 40.5 0.21 0.31

C2[16]

Hollow 60.0 60.0 10.0 53.2 53.3 14.0 6.3 0.4 0.4 94.8 4.9 672 696 43.2 0.22 0.28

C3[16]

Hollow 60.0 60.0 10.3 54.5 54.0 23.8 10.5 0.7 0.6 91.6 4.8 724 715 42.8 0.22 0.28

C4[16]

Hollow 60.0 60.0 10.3 54.6 54.5 30.7 14.1 0.9 0.9 91.4 4.8 724 672 42.7 0.22 0.28

C5[16]

Hollow 60.0 60.0 10.4 54.0 54.3 36.7 17.4 1.0 1.1 96.7 4.9 672 672 43.5 0.22 0.27

C6[16]

Hollow 60.0 60.0 10.4 53.3 52.9 48.3 22.6 1.3 1.3 87.5 4.7 724 724 42.2 0.22 0.29

a Calculated with EC 2 [2].


Tabl

e6

T n,e

xp/T

n,t

hra

tios

.

ciri Pl

ain

sect

ion

NSC

Hol

low

sect

ion

NSC

Hol

low

sect

ion

HSC

Ave

rage

r01

r02

r03

r01

r02

r03

r01

r02

r03

r01

r02

r03

c01

! x0.

818

0.81

90.

823

0.70

10.

711

0.82

40.

709

0.72

80.

727

0.74

30.

753

0.79

1s

0.07

80.

078

0.06

10.

171

0.18

00.

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0.12

30.

140

0.10

30.

124

0.13

30.

079

cv(%

)9.

539.

517.

4124

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25.3

38.

9617

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19.2

414

.17

17.0

718

.03

10.1

8

c02

! x0.

894

0.89

50.

940

0.88

10.

883

0.93

90.

765

0.76

90.

819

0.84

70.

849

0.90

0s

0.08

20.

082

0.08

30.

096

0.09

70.

099

0.13

30.

137

0.14

10.

104

0.10

50.

108

cv(%

)9.

209.

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8410

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10.9

710

.49

17.4

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.77

17.2

712

.49

12.6

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.20

c03

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0.97

01.

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0.94

90.

949

1.02

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0.81

50.

872

0.91

10.

911

0.97

3s

0.06

60.

066

0.07

80.

067

0.06

70.

105

0.11

60.

117

0.14

70.

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0.08

30.

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cv(%

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847.

647.

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0510

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14.2

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16.8

09.

369.

4111

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c04

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943

0.89

00.

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0.94

10.

800

0.80

30.

854

0.86

40.

866

0.91

3s

0.07

80.

077

0.07

90.

081

0.08

30.

082

0.11

50.

118

0.13

60.

091

0.09

30.

099

cv(%

)8.

618.

598.

339.

119.

318.

6814

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14.6

415

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10.6

810

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10.9

8

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0.93

70.

990

0.92

00.

921

0.98

30.

805

0.80

80.

860

0.88

80.

888

0.94

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0.06

80.

067

0.07

80.

067

0.06

80.

092

0.11

40.

116

0.13

80.

083

0.08

40.

103

cv(%

)7.

217.

207.

927.

297.

349.

3414

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14.4

216

.09

9.55

9.65

11.1

2

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0.94

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001

0.93

10.

932

0.99

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814

0.81

60.

869

0.89

70.

898

0.95

4s

0.06

60.

066

0.07

80.

060

0.06

10.

091

0.11

10.

113

0.13

80.

079

0.08

00.

102

cv(%

)6.

976.

977.

756.

496.

539.

1313

.58

13.8

315

.91

9.01

9.11

10.9

3

c07

! x0.

869

0.86

90.

899

0.85

60.

860

0.89

60.

748

0.75

40.

790

0.82

40.

828

0.86

2s

0.07

90.

079

0.07

40.

087

0.09

00.

079

0.12

80.

135

0.12

50.

098

0.10

10.

093

cv(%

)9.

159.

138.

1910

.21

10.5

08.

7817

.15

17.8

715

.88

12.1

712

.50

10.9

5

c08

! x0.

895

0.89

50.

942

0.88

10.

882

0.93

80.

773

0.77

60.

823

0.85

00.

851

0.90

1s

0.07

70.

077

0.08

00.

085

0.08

60.

091

0.12

50.

129

0.13

50.

096

0.09

70.

102

cv(%

)8.

578.

578.

469.

679.

789.

6516

.21

16.6

316

.44

11.4

811

.66

11.5

2

c09

! x0.

911

0.91

10.

958

0.89

00.

891

0.94

80.

768

0.77

20.

819

0.85

60.

858

0.90

8s

0.07

30.

073

0.07

60.

090

0.09

10.

096

0.12

80.

132

0.13

60.

097

0.09

80.

102

cv(%

)7.

997.

987.

9010

.13

10.2

210

.09

16.6

217

.05

16.5

911

.58

11.7

511

.53

c10

! x0.

922

0.92

20.

966

0.90

00.

902

0.95

60.

793

0.79

60.

846

0.87

20.

873

0.92

3s

0.07

30.

073

0.07

50.

089

0.09

00.

094

0.11

70.

120

0.13

60.

093

0.09

40.

102

cv(%

)7.

917.

907.

769.

8710

.00

9.86

14.7

715

.12

16.0

310

.85

11.0

111

.22

c11

! x1.

000

1.00

01.

046

0.95

90.

960

1.02

00.

818

0.82

00.

869

0.92

60.

926

0.97

8s

0.07

50.

075

0.06

60.

046

0.04

60.

077

0.10

10.

103

0.12

60.

074

0.07

40.

090

cv(%

)7.

467.

456.

344.

794.

817.

5412

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12.5

414

.44

8.20

8.27

9.44

c12

! x0.

979

0.97

91.

028

0.94

50.

945

1.00

80.

802

0.80

30.

857

0.90

90.

909

0.96

4s

0.07

10.

071

0.06

60.

045

0.04

50.

080

0.10

70.

108

0.13

10.

074

0.07

50.

092

cv(%

)7.

267.

256.

454.

794.

807.

8913

.32

13.4

615

.26

8.46

8.50

9.87

c13

! x1.

037

1.03

71.

085

0.99

90.

999

1.06

20.

835

0.83

50.

892

0.95

70.

957

1.01

3s

0.08

90.

089

0.07

20.

032

0.03

20.

067

0.09

20.

093

0.12

50.

071

0.07

10.

088

cv(%

)8.

548.

546.

673.

213.

226.

3111

.04

11.1

013

.97

7.60

7.62

8.98

c14

! x0.

986

0.98

61.

034

0.97

00.

970

1.03

10.

896

0.89

70.

957

0.95

10.

951

1.00

7s

0.07

60.

076

0.07

10.

027

0.02

70.

064

0.07

10.

071

0.10

60.

058

0.05

80.

080

cv(%

)7.

677.

676.

832.

752.

766.

177.

907.

9111

.06

6.11

6.11

8.02

c15

! x0.

963

0.96

31.

013

0.94

60.

946

1.00

50.

882

0.88

20.

942

0.93

00.

931

0.98

6s

0.06

60.

066

0.07

20.

042

0.04

20.

074

0.07

60.

076

0.10

80.

061

0.06

10.

085

cv(%

)6.

856.

857.

144.

454.

477.

328.

578.

5911

.50

6.63

6.64

8.65


where br is the reduction coefficient for the stress to take intoaccount the softening effect; k1 is the ratio between the mediumstress (B, see Fig. 3) and the maximum stress (A, see Fig. 3).

Parameter k1 will be calculated by numerical integration of r–eequations for the concrete strut (Table 1) with the help of thecomputing programming language Delphi. Integration procedureto calculate k1 is illustrated in Fig. 4 for ed values below (Fig. 4a)and above (Fig. 4b) eo (strain for maximum stress).

Based on a strain state analysis with Mohr circumference, theprincipal tension strain ec1 in the perpendicular direction to theconcrete strut, to be introduced in some r–e equations on Table2, can be calculated by [23]:

ec1 ! el # et # ed $10%

The above equations and the equations from Tables 1–4 lead tothe iterative procedure presented in Fig. 5 (for the general casebr – be) in order to calculate the theoretical T–h curve. In such cal-culus procedure, all the variables td, a, br and be areinterdependent.

eo was calculated from EC 2 [2], by using Eq. (11) valid for NSCand HSC.

eo ( ec1 ! 0:7f 0:31cm < 2:8 $‰% $11%

The theoretical failure of the sections was defined from the maxi-mum strains of the materials (concrete and steel). Either the strainof the concrete struts, eds (Fig. 3), reaches its maximum value (ecu) orthe steel strain, es, reaches the usual maximum value of es = 10‰.For NSC (fck = fcm " 8 (MPa) 6 50MPa as defined by EC 2 [2]), ecu

was assumed to be the usual value ecu = 3.5‰. For HSC, ecu was cal-culated with Eq. (12) from EC 2 [2].

ecu ! 2:8# 27)$98" fcm$MPa%%=100*4$‰% $12%

5. Comparison of experimental results

Based on the formulation of VATM and on the calculus proce-dure presented in Fig. 5, a computing tool (TORQUE_VATM) wasdeveloped with the help of the computing programming languageDelphi to compute the T–h curve for RC beams under pure torsion.

The theoretical results obtained with TORQUE_VATM arecompared with results of tests of beams under pure torsion whichare available in the bibliography. The comparative analysis willfocus on the ultimate behavior of the beams.

In this section, the same beams used by Bernardo and Lopes in2008 [15,18] were used for the comparative analysis. The exper-imental results of such beams can be considered trustworthy forcomparative analysis with global theoretical results, as justifiedby the referred authors. In fact, not all the experimental resultsavailable in the bibliography can be used due to various reasons.For instance, some older studies do not have sufficient informa-tion nor meet basic design recommendations found in currentcodes of practice. In this earlier situation, such beams behaveatypically under torsion. In other experimental studies, includingrecent ones, the authors present a medium twist for all the beamlength, and not the twist of the critical section. Theoretical twists,based on a cross-section analysis, cannot be compared with theseexperimental twists. This aspect is particularly important in slen-der beams.

Table 5 summarizes the geometrical and mechanical proper-ties of 29 beams found in the bibliography, including the externalwidth (x) and height (y) of the cross-section, the thickness of thewalls of the cross-hollow sections (t), the distances betweencenterlines of legs of the closed stirrups (x1 and y1), the area oflongitudinal reinforcement (Asl), the distributed area of onebranch of the transversal reinforcement (Ast/s, where s is the

c16

! x0.

899

0.89

90.

923

0.87

00.

874

0.90

80.

752

0.75

90.

777

0.84

00.

844

0.87

0s

0.06

60.

066

0.06

40.

086

0.08

80.

083

0.12

20.

131

0.11

10.

092

0.09

50.

086

cv(%

)7.

377.

356.

949.

8610

.04

9.16

16.3

017

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14.3

211

.18

11.5

410

.14

c17

! x0.

899

0.89

90.

934

0.87

00.

874

0.91

60.

752

0.75

90.

792

0.84

00.

844

0.88

1s

0.06

60.

066

0.06

50.

086

0.08

80.

084

0.12

20.

131

0.11

90.

092

0.09

50.

089

cv(%

)7.

377.

356.

919.

8610

.04

9.15

16.3

017

.22

15.0

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.18

11.5

410

.35

c18

! x1.

028

1.02

81.

068

0.97

10.

972

1.02

40.

839

0.84

10.

886

0.94

60.

947

0.99

3s

0.08

40.

083

0.06

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056

0.05

60.

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0.09

00.

093

0.11

20.

076

0.07

80.

085

cv(%

)8.

138.

116.

475.

775.

807.

2310

.72

11.0

212

.68

8.20

8.31

8.79

c19

! x1.

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81.

068

0.97

10.

972

1.02

40.

839

0.84

10.

886

0.94

60.

947

0.99

3s

0.08

40.

083

0.06

90.

056

0.05

60.

074

0.09

00.

093

0.11

20.

076

0.07

80.

085

cv(%

)8.

138.

116.

475.

775.

807.

2310

.72

11.0

212

.68

8.20

8.31

8.79

c20

! x0.

945

0.94

50.

973

0.89

70.

900

0.93

60.

769

0.77

60.

805

0.87

00.

874

0.90

4s

0.06

00.

060

0.06

00.

074

0.07

60.

075

0.10

60.

115

0.10

70.

080

0.08

40.

081

cv(%

)6.

396.

356.

168.

288.

408.

0113

.72

14.8

313

.25

9.46

9.86

9.14

c21

! x0.

945

0.94

50.

973

0.89

70.

900

0.93

60.

769

0.77

60.

805

0.87

00.

874

0.90

4s

0.06

00.

060

0.06

00.

074

0.07

60.

075

0.10

60.

115

0.10

70.

080

0.08

40.

081

cv(%

)6.

396.

356.

168.

288.

408.

0113

.72

14.8

313

.25

9.46

9.86

9.14


Tabl

e7

h n,e

xp/h

n,t

hra

tios

.

ciri Pl

ain

sect

ion

NSC

Hol

low

sect

ion

NSC

Hol

low

sect

ion

HSC

Ave

rage

r01

r02

r03

r01

r02

r03

r01

r02

r03

r01

r02

r03

c01

! x0.

529

0.53

00.

490

0.48

30.

544

0.40

20.

319

0.41

00.

305

0.44

40.

495

0.39

9s

0.25

80.

257

0.21

80.

167

0.09

80.

115

0.10

90.

041

0.09

50.

178

0.13

20.

143

cv(%

)48

.75

48.5

144

.51

34.6

317

.98

28.5

434

.01

9.98

30.9

939

.13

25.4

934

.68

c02

! x0.

892

0.89

40.

760

0.82

70.

866

0.68

50.

542

0.55

60.

517

0.75

40.

772

0.65

4s

0.19

80.

195

0.23

20.

131

0.10

30.

133

0.10

40.

086

0.09

70.

144

0.12

80.

154

cv(%

)22

.15

21.8

030

.54

15.8

611

.91

19.4

319

.20

15.4

818

.77

19.0

716

.40

22.9

1

c03

! x1.

160

1.16

21.

027

1.07

51.

083

0.92

20.

703

0.74

30.

635

0.97

90.

996

0.86

1s

0.22

90.

227

0.22

30.

154

0.16

50.

126

0.08

10.

066

0.10

40.

155

0.15

30.

151

cv(%

)19

.73

19.5

621

.69

14.3

715

.20

13.7

011

.54

8.86

16.3

015

.21

14.5

417

.23

c04

! x0.

849

0.85

10.

725

0.79

40.

828

0.68

80.

627

0.62

90.

600

0.75

70.

769

0.67

1s

0.21

70.

214

0.22

80.

192

0.16

40.

181

0.11

60.

113

0.11

70.

175

0.16

40.

175

cv(%

)25

.56

25.1

031

.52

24.1

619

.86

26.2

918

.48

17.9

719

.53

22.7

420

.98

25.7

8

c05

! x1.

082

1.08

20.

974

1.02

91.

041

0.88

90.

643

0.64

70.

611

0.91

80.

923

0.82

5s

0.18

70.

187

0.22

70.

123

0.12

40.

151

0.10

50.

102

0.11

40.

139

0.13

80.

164

cv(%

)17

.30

17.3

123

.30

12.0

011

.93

16.9

616

.33

15.8

418

.66

15.2

115

.03

19.6

4

c06

! x1.

106

1.10

80.

998

1.05

11.

066

0.90

90.

658

0.66

40.

627

0.93

90.

946

0.84

5s

0.18

70.

185

0.22

60.

131

0.13

20.

145

0.10

30.

097

0.11

40.

140

0.13

80.

161

cv(%

)16

.92

16.7

022

.60

12.4

912

.40

15.9

415

.62

14.6

318

.14

15.0

114

.58

18.8

9

c07

! x0.

719

0.72

10.

627

0.65

30.

680

0.56

80.

474

0.49

80.

448

0.61

50.

633

0.54

8s

0.22

20.

219

0.22

60.

186

0.17

20.

156

0.12

30.

090

0.11

00.

177

0.16

00.

164

cv(%

)30

.93

30.4

036

.10

28.4

825

.26

27.5

025

.94

18.0

524

.47

28.4

524

.57

29.3

6

c08

! x0.

939

0.94

00.

823

0.89

00.

908

0.75

60.

562

0.56

70.

531

0.79

70.

805

0.70

3s

0.20

10.

200

0.23

40.

129

0.11

90.

165

0.11

40.

104

0.11

30.

148

0.14

10.

171

cv(%

)21

.43

21.2

428

.45

14.4

613

.11

21.8

320

.31

18.3

421

.23

18.7

317

.56

23.8

4

c09

! x0.

935

0.93

60.

812

0.86

00.

890

0.72

40.

547

0.55

70.

520

0.78

10.

794

0.68

5s

0.20

60.

205

0.24

30.

129

0.10

30.

158

0.11

70.

100

0.11

00.

151

0.13

60.

170

cv(%

)22

.07

21.9

029

.89

14.9

711

.58

21.8

121

.36

17.9

721

.14

19.4

617

.15

24.2

8

c10

! x0.

828

0.82

90.

764

0.76

10.

767

0.70

70.

612

0.61

10.

584

0.73

40.

736

0.68

5s

0.21

50.

214

0.22

70.

137

0.13

20.

136

0.11

80.

119

0.11

80.

157

0.15

50.

160

cv(%

)25

.98

25.8

529

.72

17.9

517

.15

19.2

519

.30

19.5

120

.16

21.0

820

.83

23.0

4

c11

! x1.

165

1.16

51.

055

1.07

81.

092

0.93

70.

706

0.73

10.

626

0.98

30.

996

0.87

3s

0.22

50.

225

0.24

80.

152

0.14

50.

147

0.08

80.

090

0.13

00.

155

0.15

30.

175

cv(%

)19

.31

19.3

123

.51

14.1

413

.31

15.6

612

.50

12.2

620

.77

15.3

214

.96

19.9

8

c12

! x1.

272

1.27

51.

120

1.15

51.

161

0.98

40.

753

0.78

10.

623

1.06

01.

072

0.90

9s

0.26

40.

262

0.25

90.

179

0.18

60.

156

0.08

60.

075

0.13

40.

176

0.17

40.

183

cv(%

)20

.76

20.5

623

.14

15.4

816

.05

15.8

711

.38

9.54

21.4

715

.87

15.3

820

.16

c13

! x1.

376

1.37

61.

204

1.18

31.

190

1.05

90.

808

0.82

80.

663

1.12

21.

131

0.97

5s

0.32

80.

328

0.26

40.

217

0.23

40.

165

0.06

00.

075

0.12

70.

202

0.21

20.

185

cv(%

)23

.84

23.8

421

.94

18.3

419

.70

15.5

87.

449.

0219

.14

16.5

417

.52

18.8

8

c14

! x1.

293

1.29

31.

127

1.17

11.

178

1.00

40.

846

0.85

60.

728

1.10

41.

109

0.95

3s

0.29

10.

291

0.26

00.

165

0.17

00.

150

0.08

10.

097

0.13

00.

179

0.18

60.

180

cv(%

)22

.48

22.4

823

.07

14.0

914

.43

14.9

19.

5411

.35

17.8

515

.37

16.0

918

.61

c15

! x1.

176

1.17

60.

965

1.10

21.

116

0.85

10.

836

0.84

80.

713

1.03

81.

047

0.84

3s

0.28

20.

282

0.21

80.

168

0.19

20.

158

0.07

40.

099

0.13

30.

174

0.19

10.

169

cv(%

)23

.94

23.9

422

.60

15.2

317

.17

18.5

48.

8511

.67

18.6

116

.01

17.5

919

.92


spacing of transversal reinforcement), the longitudinal reinforce-ment ratio (ql = Asl/Ac, with Ac = xy) and the transversal reinforce-ment ratio (qt = Ast u/(Acs), with u = 2(x1 + y1)), the averageconcrete compressive and tensile strength (fcm ( f 0c and fctm), theaverage yielding stress of longitudinal and transversal reinforce-ment (flym and ftym), the concrete Young Modulus’s (Ec), the com-pressive strains for concrete (peak stress value, eo, and maximumvalue, ecu). For the reinforcement, usual values were adopted formaximum tensile strain (elu = etu = 10‰) and Young Modulus’s(Es = 200 GPa).

Table 5 does not include HSC plain beams. Two experimentalstudies were found in the bibliography with HSC plain beamsunder torsion: Rasmussen and Baker in 1995 [48] and Fang andShiau in 2004 [49]. Those beams were not included in Table 5for the same reasons previously presented. However, they willbe included in Section 6 for the comparative analysis with codes(only resistant torque will be analyzed).

From the theoretical curves, some key points to characterizethe ultimate behavior of the test beams were obtained, namelythe maximum torque (Tn,th) and the corresponding theoreticaltwist (hn,th). From these key points, comparative analyses aremade with the corresponding experimental values (Tn,exp andhn,exp). For this purpose, the ratios of experimental to theoreticalvalues of the referred parameters were calculated (Tn,exp/Tn,th

and hn,exp/hn,th).Comparative analysis with the last point of T–h curves or with

ductility (that generally depends on the last point of the curve)will be not included in this article. Since the last points of theoret-ical T–h curves are based on conventional ultimate strains, com-parative analysis with experimental results will be not consistent.

Tables 6 and 7 summarize the results and the comparativeanalysis for the aforementioned parameters. For each ratio ofexperimental to theoretical value, three statistical parameterswere calculated: the average value $!x%, the sample standard devi-ation (s) and the variability coefficient (cv) in order to study thedegree of dispersion of the results.

Tables 6 and 7 relate to Tn,exp/Tn,th and hn,exp/hn,th ratios, respec-tively. Three groups of test beams were considered for presentingthe results: NSC plain beams, NSC hollow beams and HSC hollowbeams. Each line of the tables is relative to a model ci for concretestruts. For each group of beams, results are presented in columnsfor each model ri for tension steel. The last column presents theglobal results for all the tested beams.

A global analysis of Table 6 shows that the variability betweenthe results for each tested model is high and independent from thecross-section type (plain or hollow). For HSC beams, such variabil-ity was already expected because many of the r–e relationships forthe concrete struts were only calibrated for NSC.

The global analysis of Table 6 shows that the range of values forthe average ratio $!x% is 0.7–1.0. The distance between extremesvalues is not very high, showing that all the theoretical modelsgive relatively goods previsions for the resistance of the testbeams. Model c1 gives the lower values for !x. This model do notincorporate reduction factors to take into account for the softeningeffect. The consequence is that the resistances are overestimated.

For the other concrete models, and for those that incorporatereduction factors calibrated only for NSC, the same type of devia-tion observed for c01 is now observed in these models when com-parative analysis is carried out with HSC. For such models and forHSC beams, strength is generally overestimated, unlike for NSCbeams. From those observations, it is possible to conclude thatthe softening effect has more severe consequences in HSC beamsunder torsion when compared to NSC beams. Theoretical r–emodels for concrete in compression with reduction factors cali-brated for NSC and HSC presents similar !x values, regardless ofthe type of cross-section (plain or hollow).

c16

! x0.

807

0.80

70.

774

0.69

30.

707

0.64

80.

488

0.51

30.

423

0.66

30.

676

0.61

5s

0.25

60.

255

0.27

30.

175

0.16

80.

192

0.14

10.

105

0.13

80.

190

0.17

60.

201

cv(%

)31

.68

31.5

735

.33

25.2

023

.80

29.6

928

.95

20.4

732

.58

28.6

125

.28

32.5

3

c17

! x0.

807

0.80

70.

733

0.69

30.

707

0.62

70.

488

0.51

30.

459

0.66

30.

676

0.60

7s

0.25

60.

255

0.26

50.

175

0.16

80.

168

0.14

10.

105

0.12

70.

190

0.17

60.

187

cv(%

)31

.68

31.5

736

.15

25.2

023

.80

26.8

528

.95

20.4

727

.76

28.6

125

.28

30.2

5

c18

! x1.

025

1.02

50.

953

0.95

30.

966

0.85

40.

674

0.67

30.

645

0.88

40.

888

0.81

7s

0.22

10.

221

0.24

70.

136

0.12

90.

165

0.13

70.

138

0.14

30.

165

0.16

30.

185

cv(%

)21

.57

21.5

825

.91

14.2

313

.40

19.2

820

.33

20.5

422

.21

18.7

118

.51

22.4

7

c19

! x1.

025

1.02

50.

953

0.95

30.

966

0.85

40.

674

0.67

30.

645

0.88

40.

888

0.81

7s

0.22

10.

221

0.24

70.

136

0.12

90.

165

0.13

70.

138

0.14

30.

165

0.16

30.

185

cv(%

)21

.57

21.5

825

.91

14.2

313

.40

19.2

820

.33

20.5

422

.21

18.7

118

.51

22.4

7

c20

! x0.

848

0.84

90.

800

0.70

90.

715

0.66

00.

523

0.55

00.

494

0.69

40.

704

0.65

1s

0.30

50.

305

0.32

30.

192

0.18

70.

195

0.16

80.

129

0.16

00.

222

0.20

70.

226

cv(%

)36

.01

35.9

240

.35

27.1

126

.15

29.6

132

.09

23.4

232

.32

31.7

428

.50

34.0

9

c21

! x0.

848

0.84

90.

800

0.70

90.

715

0.66

00.

523

0.55

00.

494

0.69

40.

704

0.65

1s

0.30

50.

305

0.32

30.

192

0.18

70.

195

0.16

80.

129

0.16

00.

222

0.20

70.

226

cv(%

)36

.01

35.9

240

.35

27.1

126

.15

29.6

132

.09

23.4

232

.32

31.7

428

.50

34.0

9


To help finding the most appropriate r–e relationships for theconcrete struts, in order to compute torsional strengths, andregardless of the concrete strength class, the results for the theo-retical concrete models calibrated for NSC and HSC must be exam-ined in detail. For this purpose, the last column of Table 6 helps toanalyze the results regardless of the type of cross-section. In thiscolumn, the concrete models c03, c11–c15, c18 and c19 are thosewith !x values closest to 1 (over 0.90), particularly models c13,c14, c18 and c19 with !x values greater than 0.95. Those results donot depend on the tested theoretical reinforcement model beingr1 or r2 (with or without hardening of steel after yielding). In

contrast, the results obtained with the incorporation of model r3show an increase of !x values (getting closer to 1), showing thatthe stiffening effect seems to have a non-negligible influence. Infact, r–e curve for model r3 develops at a lower stress level com-pared to the r–e curve for models r2 and r1. Therefore, the steelreaches its yield point a bit faster for model r3 and the torsionalstrength for beams with moderate torsional reinforcement level(ductile failure) is lower compared to predictions obtained withmodels r1 or r2.

In conclusion, from the average value $!x% for Tn,exp/Tn,th, the bestpredictions of torsional strength are obtained with models c13, c14,

Table 8Results obtained with model c14 + r1 (plain beams).

Table 9Results obtained with model c14 + r1 (hollow beams).

Hollow Beams Tn,exp Tn,th Tn,exp / Tn,thkN.m/m kN.m/m

D4 – NSC 47.925 50.268 0.953T2 – NSC 132.906 139.509 0.953T1 – NSC 140.006 145.110 0.965VH1 – NSC 21.793 21.404 1.018A2 – NSC 254.079 257.404 0.987A3 – NSC 299.915 319.223 0.940A4 – NSC 368.218 387.699 0.950A5 – NSC 412.236 416.100 0.991B2 – HSC 273.275 276.929 0.987B3 – HSC 355.845 434.801 0.818B4 – HSC 437.853 490.529 0.893B5 – HSC 456.192 523.375 0.872C2 – HSC 266.138 268.216 0.992C3 – HSC 351.165 437.613 0.802C4 – HSC 450.305 466.526 0.965C5 – HSC 467.26 551.915 0.847C6 – HSC 521.331 584.229 0.892

= 0.973NSC s = 0.031

cv = 3%= 0.910

HSC s = 0.076cv = 8%

= 0.932s = 0.070cv = 7%


c18 and c19 for concrete struts, using model r3 for reinforcement intension.

From the analysis of the variation coefficient (cv) from Table 6,it can be observed that for the concrete models c03, c11–c13, c18and c19, and for HSC hollow beams, the cv values are over 10%,showing a remarkable dispersion for Tn,exp/Tn,th ratios regarding !x.From this point of view, models c14 and c15, with cv values gener-ally below 10%, or slightly above this value, appear to be the bestmodels, although HSC hollow beams show cv values slightly higherwhen compared with plain beams. In conclusion, concrete model

c14 shows the lower value for cv (6.11% for reinforcement modelsr1 or r2 and 8.02% for model r3).

From the analysis of Table 6, an interesting aspect must bepointed out: models r3 + c14 provide the best results regarding !x,while models r1 + c14 or r2 + c14 provide the best results regardingcv parameter. Thus, model r3 for reinforcement in tension leads toa slight higher dispersion of the Tn,exp/Tn,th values.

According to the previous analysis of Table 6, the best theoret-ical model to compute the torsional strength of NSC or HSC beamswith plain or hollow section, is the one that incorporates the model

Table 10Results obtained with model c14 + r3 (plain beams).

Plain Beams Tn.exp Tn.th Tn.exp / Tn.thkN.m/m kN.m/m

B2 – NSC 30.186 31.111 0.970B3 – NSC 37.526 41.021 0.915B4 – NSC 47.36 48.610 0.974B5 – NSC 56.176 53.444 1.051G6 – NSC 39.18 36.891 1.062G8 – NSC 73.47 64.934 1.131M2 – NSC 40.323 36.143 1.116T4 – NSC 138.606 134.556 1.030A3 – NSC 28.216 27.893 1.012B2 – NSC 20.798 18.179 1.144B3 – NSC 25.101 25.630 0.979B4 – NSC 31.63 30.890 1.024

= 1.034NSC s = 0.071

cv = 7%

Table 11Results obtained with model c14 + r3 (hollow beams).

Hollow Beams Tn,exp Tn,th Tn,exp / Tn,thkN.m/m kN.m/m

D4 – NSC 47.925 50.444 0.950T2 – NSC 132.906 128.049 1.038T1 – NSC 140.006 134.556 1.041VH1 – NSC 21.793 19.042 1.144A2 – NSC 254.079 233.499 1.088A3 – NSC 299.915 295.424 1.015A4 – NSC 368.218 382.268 0.963A5 – NSC 412.236 410.130 1.005B2 – HSC 273.275 248.260 1.101B3 – HSC 355.845 403.017 0.883B4 – HSC 437.853 479.360 0.913B5 – HSC 456.192 514.918 0.886C2 – HSC 266.138 235.055 1.132C3 – HSC 351.165 394.330 0.891C4 – HSC 450.305 428.516 1.051C5 – HSC 467.26 543.874 0.859C6 – HSC 521.331 578.313 0.901

= 1.031NSC s = 0.064

cv = 6%= 0.957

HSC s = 0.106cv = 11%

= 0.992s = 0.094cv = 9%


c14 for concrete struts, with the r–e relationship proposed by Bel-arbi and Hsu in 1991 [25] and the reduction factors proposed byZhang and Hsu in 1998 [38] (see Tables 1 and 2). Also, with modelc14 for concrete struts, the use of models r1 or r2 from EC 2 [2] (seeTable 4) provides the best results, especially with regards to thedispersion of results. However, the use of model r3 (see Table 4)provides the best results regarding the average value for Tn,exp/Tn,th.Thus, the analysis of results should continue based on these twooptions (using models r1/r2 or r3 with model c14), as well as onother criteria. This will be done at a later stage.

The analysis of Table 7, regarding hn,exp/hn,th ratios, shows a widerange for the average value $!x%, with some values far from unity, aswell as high values for the coefficient of variation (cv), with somevalues over 10%. The dispersion of the results is much larger thanthat observed for Tn,exp/Tn,th ratio. Generally, Table 7 shows thatall the theoretical models appear to have some difficulty in esti-mating the deformation of the model beams for high loading levels.This is probably due to the fact that VATM assumes a fully crackedstate of the beam from the beginning of loading. This hypothesisdoes not match reality.

For concrete model c14, shown in the previous analysis as themost suitable concrete model for the prediction of the torsionalstrength, the deformations of NSC beams are underestimated andthe deformations of HSC beams are overestimated. By analyzingthe results corresponding to model c14 in the last column of Table

7, the same tendency is observed, respectively, for the reinforce-ment models r1/r2 and r3. The analysis of Table 7 also shows thatfor NSC beams the accuracy of the theoretical deformations are gen-erally smaller for plain beams in comparison with to hollow beams.This observation seems to show that, while concrete core does not

Fig. 6. T–h curves: Beam B2 [43].

Fig. 7. T–h curves: Beam G8 [43].

Fig. 8. T–h curves: Beam T4 [44].

Fig. 9. T–h curves: Beam A3 [45].

Fig. 10. T–h curves: Beam D4 [43].


seem to influence the torsional strength (as already expected), thesame cannot be stated for the ultimate deformations. This is prob-ably due to the greater capacity of plain sections to redistribute tor-sional stress transversally for the ultimate levels of loading.

In order to find out whether the theoretical model should incor-porate concrete model c14 with reinforcement models r1/r2 or r3,some previous results will be more rigorously analyzed. As previ-ously observed, conclusions were identical, whether the modelincorporates models c14 + r1 or c14 + r2. For this reason, only mod-el c14 + r1 will be analyzed in comparison with model c14 + r3,since model r1 (instead of r2) is more realistic for characterizingthe behavior of the reinforcement after the yielding point.

Tables 8–11 resumes the results obtained for Tn,exp/Tn,th ratioswith the theoretical model that incorporates model c14 for con-crete struts. Tables 8 and 9 (plain and hollow beams, respectively)are related to the theoretical model that incorporates model r1 forreinforcement in tension, while Tables 10 and 11 (plain and hollowbeams, respectively) are related to the theoretical model thatincorporates the model r3 for reinforcement in tension. Tables 8–11 include bar graphs for Tn,exp/Tn,th ratios in order to allow a visualanalysis of the dispersion of the results in comparison with theoptimal unit value, and also to check the degree of safety of thetheoretical predictions compared to the experimental values.

By comparing bar graphs for Tn,exp/Tn,th ratios from Tables 8–10(plain beams) with models r1 and r3, respectively, it can be seenthat the results are quite similar. One exception is that the valuesfor model r3 are slightly shifted to the right side, compared to theresults for model r1. Thus, model r3 seems to give more safetyresults. This observation is also confirmed by the average value(!x ! 1:034 for r3 and !x ! 0:986 for r1). Moreover, the coefficientof variation (cv) is slightly smaller for r3 (7% instead of 8%).

By comparing the bar graphs for Tn,exp/Tn,th ratios from Tables 9and 11 (hollow beams) the same conclusions for plain beams canbe stated. Note, however, that for HSC hollow beams, the theoret-ical models appear to overestimate a little the torsional strength,especially for beams with higher torsional reinforcement ratios.This observation is consistent with earlier conclusions, namelymade by Bernardo and Lopes in 2011 [18] by testing r–e relation-ship from Belarbi and Hsu [25] for the concrete. However, in thestudy of these authors the degree of overestimation was found tobe higher than that observed in the present study. This observationseems to show that the theoretical model used in this study ismore accurate for HSC beams.

The overestimation of the resistance can be explained by thebehavior of those beams, at ultimate stage, which essentiallydepends on the compressed concrete struts, instead of the

tensioned reinforcement. For those beams the influence of soften-ing effect is higher. This observation led Bernardo and Lopes [18]to recommend the reduction of the softening parameters, multi-plying them by 0.9 in order to take into account this overestima-tion and to achieve more accurate previsions. Based on the newresults obtained in this article, with new theoretical models, the

Fig. 13. T–h curves: Beam C3 [16].

Fig. 12. T–h curves: Beam VH1 [47].

Fig. 11. T–h curves: Beam T1 [44]. Fig. 14. T–h curves: Beam C6 [16].


authors consider that the aforementioned recommendation ofBernardo and Lopes is not longer justified. In fact, the results ofthis study demonstrate that the theoretical c14 + r03 gives moreaccurate results for HSC beams than the model tested byBernardo and Lopes [18].

The previous analysis shows that the theoretical model, whichincorporates models c14 and r03, seems to be the best. This conclu-sion is theoretically satisfying, since the reinforcement model r03is a non-linear model and taken into account for the stiffening ef-fect, while reinforcement models r01 or r02 characterize uniaxialtensile states. It should be noted that the results also appear toshow that the influence of the stiffening effect is not very large.

Figs. 6–14 present both experimental and theoretical T–h curvesfor some selected beams. Theoretical T–h curves for modelsc14 + r01 and c14 + r03 are highlighted to confirm the previousconclusions. From the whole set of the theoretical curves (exceptcurves of model c14) only those that correspond to the upperand lower limits for the maximum theoretical torque were plot

(in gray color). These limits define the range where the other hid-den theoretical curves are located. Figs. 6–14 clearly confirm thehigh dispersion among T–h curves, especially for high levels ofloading since the ‘‘distance’’ between the limit curves is high. Thisobservation is essentially associated to the several tested r–e rela-tionship for the compressed concrete in the struts.

A global analysis of the theoretical curves of Figs. 6–14 for mod-els c14 + r01 and c14 + r03 seems to show that the second one pro-vides torsion strengths slightly below the first one (more safetyvalues). Furthermore, some of the other tested models give unreli-able figures for the torsional strength.

Based on the analysis of the results in this section, a suitabletheoretical model was found for calculating the ultimate torsionalbehavior (mainly torsional strength) of reinforced concrete NSCand HSC beams (plain and hollow). This model incorporates theconcrete model c14 (r–e relationship for compressed concrete instruts proposed by Belardi and Hsu in 1991 [25] with softening fac-tors proposed by Zhang and Hsu in 1998 [38]) and reinforcement

Table 12Properties of new test beams.

Beam Section type x (cm) y (cm) x1 (cm) y1 (cm) Asl (cm2) Ast/s (cm2/m) f(cm) (MPa) flym (MPa) ftym (MPa)

B50.1 [48] Plain 16.0 27.5 12.0 23.5 15.27 8.73 61.8 612.0 665.0B50.2 [48] Plain 16.0 27.5 12.0 23.5 15.27 8.73 57.1 614.0 665.0B50.3 [48] Plain 16.0 27.5 12.0 23.5 15.27 8.73 61.7 612.0 665.0B70.1 [48] Plain 16.0 27.5 12.0 23.5 15.27 8.73 77.3 617.0 658.0B70.2 [48] Plain 16.0 27.5 12.0 23.5 15.27 8.73 76.9 614.0 656.0B70.3 [48] Plain 16.0 27.5 12.0 23.5 15.27 8.73 76.2 617.0 663.0B110.1 [48] Plain 16.0 27.5 12.0 23.5 15.27 8.73 109.8 618.0 655.0B110.2 [48] Plain 16.0 27.5 12.0 23.5 15.27 8.73 105.0 634.0 660.0B110.3 [48] Plain 16.0 27.5 12.0 23.5 15.27 8.73 105.1 629.0 655.0H-06-06 [49] Plain 35.0 50.0 30.0 45.0 11.92 7.13 78.5 440.0 440.0H-06-12 [49] Plain 35.0 50.0 30.0 45.0 20.27 7.13 78.5 440.0 410.0H-12-12 [49] Plain 35.0 50.0 30.0 45.0 20.27 14.26 78.5 440.0 410.0H-12-16 [49] Plain 35.0 50.0 30.0 45.0 28.65 14.26 78.5 440.0 520.0H-20-20 [49] Plain 35.0 50.0 30.0 45.0 34.38 23.04 78.5 440.0 560.0H-07-10 [49] Plain 35.0 50.0 30.0 45.0 17.19 7.92 68.4 420.0 500.0H-14-10 [49] Plain 35.0 50.0 30.0 45.0 17.19 15.84 68.4 360.0 500.0H-07-16 [49] Plain 35.0 50.0 30.0 45.0 28.65 7.92 68.4 420.0 500.0N-06-06 [49] Plain 35.0 50.0 30.0 45.0 11.92 7.13 35.5 440.0 440.0N-06-12 [49] Plain 35.0 50.0 30.0 45.0 20.27 7.13 35.5 440.0 410.0N-12-12 [49] Plain 35.0 50.0 30.0 45.0 20.27 14.26 35.5 440.0 410.0N-12-16 [49] Plain 35.0 50.0 30.0 45.0 28.65 14.26 35.5 440.0 520.0N-20-20 [49] Plain 35.0 50.0 30.0 45.0 34.38 23.04 35.5 440.0 560.0N-07-10 [49] Plain 35.0 50.0 30.0 45.0 17.19 7.92 33.5 420.0 500.0N-14-10 [49] Plain 35.0 50.0 30.0 45.0 17.19 15.84 33.5 360.0 500.0N-07-16 [49] Plain 35.0 50.0 30.0 45.0 28.65 7.92 33.5 420.0 500.0

Table 13Equations to compute torsion strength.

ACI 318R-05 [3] MC 90 [1] EC 2 [2]

Al !At

sph

fyv

fylcotg2h! h

Tn !2AoAtfyv

scotg h

tg hi !

%%%%%%%%%%%%%%%%%%%%%%%%Aswifywdzi=s

Asifyd

s

! h

Tn ! TRtwi !2FRtwiAef

zi

FRtwi ! ASwifyd cotg hizi

s

tg h !

%%%%%%%%%%%%%%%%%%%%Ast;ifywduk

sP

Aslfyd

s

! h

Tn ! TRd2 ! 2Ak$fywdAsw=s% cotg h

Brittle failure:

Tn !0:67

%%%%f 0c

p1:7A2

oh

ph

Brittle failure:

TRcwi !2FRcwisinhiAef

zi

FRcwi ! fcd2tizi coshi

Brittle failure:

TRd1 ! 2mfcdtAksinh cosh


model r03 (r–e relationship for tension reinforcement proposed byBelarbi and Hsu in 1994 [42]).

6. Comparative analysis with codes provisions

Based on TORQUE_MTEAV (models c14 and r03) and experi-mental results of test beams, this section presents a comparativeanalysis with the predictions for the torsional resistance calculatedwith codes of practice. The following were considered: ACI 318R-05 [3], MC 90 [1] and EC 2 [2]. The objective is to compare the

degree of safety of the theoretical model predictions in comparisonwith normative predictions expected to be much more conserva-tive. To compute the normative predictions of the torsionalstrength of the test beams no safety factors were used and thestrength of materials (steel and concrete) were characterized bytheir corresponding average values.

If a given code predicts brittle failure due to the crushing of con-crete struts of a beam under analysis, then the correspondingstrength is computed by adopting the top limit of the compressionstress in concrete struts of the code.

Table 14Torsion strengths of test beams.

Viga Tn,exp (kNm) Tn,th (kNm) ACI 318R-05 [3] MC 90 [1] EC 2 [2] TORQUE_VATM

Tn,calc (kNm) Tn;expTn;calc

(–) Tn,calc (kNm) Tn;expTn;calc

(–) Tn,calc (kNm) Tn;expTn;calc

(–) Tn;expTn;th

(–)

B2 [43] 30.19 31.11 28.65 1.054 27.80 1.086 26.51 1.14 0.970B3 [43] 37.53 41.02 29.35a 1.279 40.23 0.933 38.37 0.98 0.915B4 [43] 47.36 48.61 29.92a 1.583 52.86a 0.896 52.27 0.91 0.974B5 [43] 56.18 53.44 26.35a 2.132 56.40a 0.996 63.88 0.88 1.051G6 [43] 39.11 36.89 33.16 1.179 30.67 1.275 29.62 1.32 1.060G8 [43] 73.47 64.93 45.13a 1.628 64.26 1.143 62.07 1.18 1.131M2 [43] 40.32 36.14 30.62a 1.317 34.79 1.159 33.18 1.22 1.116T4 [44] 138.61 134.56 126.67 1.094 93.71 1.479 111.75 1.24 1.030A3 [45] 28.22 27.89 18.66a 1.512 26.57 1.062 25.05 1.13 1.012B2 [45] 20.80 18.18 15.93 1.306 15.31 1.358 14.79 1.41 1.144B3 [45] 25.10 25.63 15.93a 1.576 23.34 1.075 22.57 1.11 0.979B4 [45] 31.63 30.89 15.90a 1.989 31.49 1.005 30.50 1.04 1.024B50.1 [48] 19.95 20.06 9.95a 2.005 26.02a 0.767 34.08a 0.59 0.994B50.2 [48] 18.46 19.43 9.56a 1.931 24.60a 0.751 32.22a 0.57 0.950B50.3 [48] 19.13 20.05 9.94a 1.925 25.99a 0.736 34.04a 0.56 0.954B70.1 [48] 20.06 21.81 11.12a 1.804 29.81a 0.673 39.07 0.55 0.920B70.2 [48] 20.74 21.77 11.09a 1.870 29.74a 0.697 38.97 0.57 0.953B70.3 [48] 20.96 21.73 11.04a 1.899 29.63a 0.707 38.83 0.57 0.965B110.1 [48] 24.72 24.88 13.26a 1.864 34.67a 0.713 45.44 0.68 0.994B110.2 [48] 23.62 24.35 12.96a 1.823 34.04a 0.694 44.66 0.64 0.970B110.3 [48] 24.77 24.36 12.97a 1.910 34.05a 0.727 44.67 0.68 1.017H-06-06 [49] 92.00 90.14 76.16 1.21 84.65 1.087 79.96 1.15 1.021H-06-12 [49] 115.10 104.23 95.89 1.20 93.73 1.228 102.15 1.13 1.104H-12-12 [49] 155.30 144.32 121.60a 1.28 132.56 1.172 144.47 1.07 1.076H-12-16 [49] 196.00 160.39 121.60a 1.61 177.72 1.103 187.84 1.04 1.222H-20-20 [49] 239.00 201.42 121.60a 1.97 243.50a 0.982 267.44 0.89 1.187H-07-10 [49] 126.70 105.68 100.42 1.26 109.91 1.153 103.90 1.22 1.199H-14-10 [49] 135.20 140.21 113.51a 1.19 141.65 0.954 134.02 1.01 0.964H-07-16 [49] 144.50 106.86 113.51a 1.27 126.91 1.139 134.13 1.08 1.352N-06-06 [49] 79.70 82.41 76.16 1.05 84.65 0.941 79.96 1.00 0.967N-06-12 [49] 95.20 89.42 81.77a 1.16 93.73 1.016 102.15 0.93 1.065N-12-12 [49] 116.80 123.20 81.77a 1.43 132.56a 0.881 144.47 0.81 0.948N-12-16 [49] 138.00 120.14 81.77a 1.69 129.76a 1.064 169.97a 0.81 1.149N-20-20 [49] 158.00 138.85 81.77a 1.93 136.49a 1.158 179.68a 0.88 1.138N-07-10 [49] 111.70 88.04 79.44a 1.41 109.91 1.016 103.90 1.08 1.269N-14-10 [49] 125.00 119.22 79.44a 1.57 130.91a 0.955 134.02 0.93 1.049N-07-16 [49] 117.30 85.48 79.44a 1.48 106.42a 1.102 134.13 0.87 1.372

D4 [43] 47.93 50.44 30.65a 1.564 51.55a 0.930 58.63 0.82 0.950T2 [46] 132.91 128.05 123.50 1.076 112.55 1.181 134.22 0.99 1.038T1 [44] 140.01 134.56 126.67 1.105 111.07 1.261 132.46 1.06 1.041VH1 [27] 21.79 19.04 18.05 1.207 19.49 1.118 18.38 1.19 1.144A2 [16] 254.08 233.50 212.18 1.197 191.00 1.330 221.30 1.15 1.088A3 [16] 299.92 295.42 284.94 1.053 245.36 1.222 292.64 1.02 1.015A4 [16] 368.22 382.27 298.13a 1.235 356.66 1.032 411.02 0.90 0.963A5 [16] 412.24 410.13 302.90a 1.361 427.25 0.965 507.02 0.81 1.005B2 [16] 273.28 248.26 223.26 1.224 199.79 1.368 232.74 1.17 1.101B3 [16] 355.85 403.02 383.61a 0.928 351.60 1.012 405.19 0.88 0.883B4 [16] 437.85 479.36 373.33a 1.173 438.67 0.998 523.14 0.84 0.913B5 [16] 456.19 514.92 343.05a 1.330 530.00 0.861 645.08 0.71 0.886C2 [16] 266.14 235.06 211.00 1.261 194.67 1.367 225.56 1.18 1.132C3 [16] 351.17 394.33 386.48 0.909 346.45 1.014 399.25 0.88 0.891C4 [16] 450.31 428.52 438.19a 1.028 428.80 1.050 508.86 0.88 1.051C5 [16] 467.26 543.87 440.87a 1.060 515.84 0.906 593.58 0.79 0.859C6 [16] 521.33 578.31 395.44a 1.318 676.89 0.770 835.53 0.62 0.901

!x 1.434 !x 1.023 !x 0.940 1.038s 0.332 s 0.196 s 0.217 0.113

cv 23.18% cv 19.18% cv 23.11% 10.87%

a Brittle failure.


In addition to the reference beams analyzed in the previous sec-tion, other reference beams found in the bibliography are nowadded for this section. As previously referred in Section 5, thosebeams only allow the study of their torsional strength (maximumtorque), which is the purpose of this section. Nine HSC plain beamstested by Rasmussen and Baker in 1995 [48] and sixteen plainbeams (eight HSC beams) tested by Fang and Shiau in 2004 [49]are added to this section. The relevant characteristics of suchbeams to calculate the normative torsional strength are summa-rized in Table 12.

Table 13 summarizes the equations (in metric units) incorpo-rated in the codes of practice and used in this study to computethe torsional strength of the test beams. The meaning of eachparameter can be found in the codes. For beams with brittle failure(failure of the concrete struts), the equation to compute Tn, basedon ACI 318-05 [3], can be derived from Eqs. (11)–(18) or (11)–(19) of the code (Clause 11.6.3.1) considering Vu = Vc = 0, Tu = Tn

and / = 1.The results are presented in Table 14.According to Table 14, ACI 318R-05 [3] does not appear to

provide precisely the torsional strength of beams with a small

cross-section and with high torsional reinforcement ratios. Forsuch beams, ACI 318R-05 [3] appears to be quite conservative.One possible explanation for this observation is that ACI 318R-05[3] assumes a 45" angle for concrete struts to calculate torsionalstrength provided by the concrete strength in the struts.

Table 14 also shows that, based on the !x parameter analysis (al-most 1), MC 90 [1] seems to estimate the torsional strength of thereference beams very well. However, the cv parameter analysisshows that the dispersion of the results is quite remarkable sincetheir value is very high (+19%). For beams with lower torsionalreinforcement ratios, Tn,exp/Tn,calc are over that 1, while for beamswith intermediate and higher torsional reinforcement ratiosTn,exp/Tn,calc are now lower than 1. This observation explains the ob-served dispersion.

Table 14 shows that predictions of EC 2 [2] followed similartrends to those observed for MC 90 [1].

Figs. 15–17 graphically present the ratios Tn,exp/Tn,calc and Tn,exp/Tn,th as a function of the compressive strength of concrete fcm ( f 0c

' (

for each code and for the results presented in Table 14.Generally, Figs. 15–17 confirm the previously stated conclu-

sions based on the analysis of Table 14.Fig. 15 shows that ACI 318R-05 [3] is somewhat conservative,

especially for beams with high torsional reinforcement ratio.Figs. 16 and 17 show that both MC 90 [1] and EC 2 [2] appear to

be conservative for normal strength beams, but not for the major-ity of the HSC beams. Figs. 16 and 17 also show that the deviationsof results are high between the analyzed test beams.

Figs. 15–17 show that the results of TORQUE_VATM (modelsc14 and r03) are more rigorous and the dispersion is lower. So, itcan be concluded that, when compared to the codes studied, TOR-QUE_VATM (models c14 and r3) provides a more optimized tor-sional design.

7. Conclusions

In the first part of this article, several theoretical models weretested based on numerical simulations in order to calculate theultimate behavior of test beams subjected to pure torsion whoseexperimental results were found in the literature. The theoreticalmodels were based on VATM formulation. Several proposals ofthe r–e relationships for the materials (concrete and reinforce-ment) were tested. From the comparative analysis, the followingconclusions can be drawn:Fig. 16. Tn,exp/Tn,calc and Tn,expv/Tn,th ratios (MC 90 [1]).

Fig. 15. Tn,exp/Tn,calc and Tn,expv/Tn,th ratios (ACI 318R-05 [3]).

Fig. 17. Tn,exp/Tn,calc and Tn,expv/Tn,th ratios (EC 2 [2]).


– A global analysis of the results shown that for the testedmodel the dispersion of the results is high and indepen-dent from the cross-section type (plain or hollow).

– For HSC beams, the concrete models that incorporatereduction factors calibrated for NSC are not adequate. Ifthese models are used in HSC beams, the torsionalstrength is generally overestimated.

– From the obtained results, it was possible to conclude thatthe softening effect has more severe consequences in thestrength of HSC beams under torsion when compared toNSC beams.

– One given concrete model presents similar results for thetorsional strength of the beams, regardless of the type ofcross-section (plain or hollow).

– For HSC hollow beams, the theoretical models appear tooverestimate more the torsional strength than for NSChollow beams, especially for beams with relatively hightorsional reinforcement ratios.

– For concrete models with best predictions for torsionalstrength, the results do not depend on the geometry ofthe tested bilinear theoretical reinforcement model (withor without hardening of steel after yielding). In contrast,the results obtained with the incorporation of non-lineartheoretical reinforcement model shows that the stiffeningeffect seems to have a non-negligible influence.

– For the ultimate twists, the dispersion of the results ismuch larger than that observed for the torsionalstrengths. This shows that the theoretical models appearto have greater difficulty in estimating the deformationof the beams when the level of loading becomes higher.

– For concrete models with the best results for torsionalstrength, it was observed that the deformations of NSCbeams are underestimated and the deformations of HSCbeams are overestimated, regardless of the reinforcementmodel. It was also observed that the accuracy of the the-oretical deformations is generally smaller for plain beamsin comparison with hollow beams. This seems to indicatethat the concrete core has a non-negligible influence onthe ultimate deformations.

– It was found that one theoretical model in particular givesvery good predictions of the torsional strength andacceptable predictions of the corresponding twist forreinforced concrete NSC and HSC beams under pure tor-sion (plain or hollow). This theoretical model incorporatesthe concrete model c14 (r–e relationship for compressedconcrete in struts proposed by Belarbi and Hsu in 1991[25] with softening factors proposed by Zhang and Hsuin 1998 [38]) and steel reinforcement model r03 (r–erelationship for tension reinforcement proposed by Belar-bi and Hsu in 1994 [42]).

In the second part of this article, the theoretical and experimen-tal predictions from TORQUE_VATM (models c14 and r03) for tor-sional strength of the test beams were compared with the onescomputed from some important codes (ACI 318R-05 [3], MC 90[1] and EC 2 [2]). This comparative study led to some findings:

– ACI 318R-05 does not seem to provide an accurate valuefor the torsional strength of beams with small cross-sections and with high torsional reinforcement ratios.For such beams, ACI 318R-05 appears to be quiteconservative.

– MC 90 and EC 2 show a large dispersion of the results. Forbeams with lower torsional reinforcement ratios, the tor-sional strengths are underestimated, while for beamswith intermediate and high torsional reinforcement ratios

the torsional strengths are overestimated.– MC 90 and EC 2 seem to be conservative for NSC beams,

but not for the majority of the HSC beams.– When compared with the previsions from codes, TOR-

QUE_VATM (models c14 and r03) gives very good predic-tions for the torsional strength, conducting to a moreoptimized torsional design.

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