Engineering Structures

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1 Engineering Structures 42 (202) Contents lists available at SciVerse ScienceDirect Engineering Structures journal homepage: 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, Portugal b University of Coimbra, Portugal article info abstract Article history: Received 7 October 20 Revised 28 February 202 Accepted 26 April 202 Available online 3 June 202 Keywords: Beams Torsion Truss-model Softening effect Stiffening effect Theoretical model 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. In order to model the non-linear behavior of the compressed concrete in the struts and of the tensioned steel reinforcement several proposals for the stress (r) strain (e) relationships were tested. The theoretical predictions of the maximum torque and corresponding twist were compared with results from reported tests and with the predictions obtained from Codes. One of the tested theoretical models was found to give excellent predictions for the maximum torque when compared with those obtained from some codes of practice and with experimental values of NSC and HSC beams (plain and hollow). Ó 202 Elsevier Ltd. All rights reserved.. Introduction The first studies on torsion of reinforced concrete beams were published in the beginning of the last century. The developed theoretical models can be divided into two main theories: the Skew- Bending Theory which was the base of the American code between 97 and 995, and the Space Truss Analogy which is currently the base of the American code (since 995) and of the European model codes (since 978) [ 3]. The Variable Angle Truss-Model (VATM), which firstly aimed to unify the torsion design of small and large sections and of reinforced and prestressed concrete, is probably the most used theoretical truss model to predict the theoretical ultimate behavior of beams under torsion. This theory also allows a good physical understanding of the torsion problem in Reinforced Concrete (RC) elements and has an important historical value. The first simple version of the model was presented by Rausch in 929 [4]. Other authors have contributed to updated versions of the model, such as: Andersen in 935 [5], Cowan in 950 [6], Lampert and Thurlimann in 969 [7], Elfgren in 972 [8], Collins and Mitchell in 980 [9]. In 985 Hsu and Mo [0] developed a model with the influence of Softening Effect. Some developments and Corresponding author. Address: Departamento de Engenharia Civil e Arquitetura, Edifício II das Engenharias, Universidade da Beira Interior, Calçada Fonte do Lameiro, Covilhã, Portugal. Tel.: ; fax: addresses: luis.bernardo@ubi.pt, lfb@ubi.pt (L.F.A. Bernardo). alternative methods were also introduced by Jeng and Hsu in 2009 [], Jeng in 200 [2], Mostofinejad and Behzad in 20 [3], Jeng et al. in 20 [4]. Some recent experimental works brought more information on the actual behavior of beams under torsion. This is the case of works by Bernardo and Lopes in 2008, 2009 and 20 [5 8], Algorafi et al. in 2009 and 200 [9,20], Al Nuaimi et al. in 2008 [2] and Lopes and Bernardo in 2008 [22]. The VATM can be divided into two categories: Plasticity Compression Field Theory (Lampert and Thurlimann, Elfgren) and Compatibility Compression Field Theory (Collins, Hsu and Mo). While in the first theory the stresses are based on the theory of plasticity, the second theory is based on the deformations compatibility of the truss analogy. The Space Truss Analogy provides good results for high levels of loading. However, for low levels, the method does not give good predictions since the model assumes a cracked state from the beginning of the loading. Even for high levels of loading, the accuracy of the results will highly depend on the constitutive law that would characterize the non-linear behavior of the materials. The objective of this study is to develop and test a computational procedure, based on the VATM, and to predict the ultimate behavior of both NSC and HSC beams under torsion (plain and hollow). The behavior of beams under torsion is studied by means of T (Torque) h (Twist) curves. The non-linear behavior for the materials (concrete and steel reinforcement) is incorporated by mean of stress (r) strain (e) relationships found in the literature /$ - see front matter Ó 202 Elsevier Ltd. All rights reserved.

2 L.F.A. Bernardo et al. / Engineering Structures 42 (202) Fig.. Normal and softened r e curves for concrete in compression in struts [27]. s kf sy f sy Ε s (a) εsy (b) ε su ε s Fig. 2. Stiffened and idealized non-stiffened r e curve for reinforcement in tension [27]. To help with calculations, a computing tool is developed with the help of the computing programming language Delphi. The theoretical results are compared with the experimental results of several beams under pure torsion, which are available in the literature and also with codes provisions. 2. Previous studies and research significance In previous theoretical studies, several authors predicted the behavior of NSC beams under torsion by using the VATM. For example, Hsu in 984 [23] used the VATM to calculate the theoretical T h curve. Hsu showed that the ultimate values of the T h curves, experimental and theoretical, were quite similar. These observations were also confirmed by Bernardo and Lopes [5] for a larger group of tested beams, including some HSC hollow beams tested by the authors [6]. For NSC beams under torsion, Bernardo and Lopes showed that, in general, VATM is quite appropriate for the prediction of the ultimate behavior. However, for HSC beams these authors showed that VATM could not be considered adequate since resistances were highly overestimated for beams with high torsional reinforcement ratio [5,8]. The calculus procedure was reviewed by the authors in order to incorporate specific r e relationships for HSC. In fact, for HSC the shape of r e curve for concrete in compression is quite different from that of NSC. Thus, theoretical models that depend on this kind of relationships cannot be extrapolated from NSC to HSC. Based on the theoretical predictions obtained with the new calculus procedure, Bernardo and Lopes [8] concluded that, in order to obtain good predictions for the whole range of beams, including those with high reinforcement ratios, the calculus procedure needed to be reviewed. This observation led the authors to propose empiric coefficients to correct the theoretical results. In their theoretical models the authors only tested some r e relationships to characterize the mechanical behavior of the concrete in compression (struts) and the steel in tension (reinforcement), but they did not carry out a comprehensive study with regards to the large quantity of r e relationships that could be found in bibliography. For NSC beams and concrete in compression, the cited authors used the r e relationship by Vecchio and Collins proposed in 982 [24] from the results of NSC plates tested under shear. For HSC beams, Bernardo and Lopes used r e relationships proposed by Belarbi and Hsu in 99 [25] (based on the results of HSC plates tested under shear). For the reinforcement in tension, many authors use a common bilinear relationship with horizontal or inclined plastic landing. Since the first study by Vecchio and Collins, published in 98 [26], several other proposals for r e relationships, which had taken 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 predictions based on the several proposals was very high. The author defended that there was a need of checking the different r e relationships proposals before one could reach definitive conclusions. This is the case in this study, because the theoretical results from VATM strongly depend on the r e relationships for concrete and steel. All these aspects justify the need for a state-of-the art work on r e relationships (for concrete in compression and reinforcement in tension) for the theoretical prediction of the ultimate behavior of NSC and HSC beams (plain and hollow) under torsion.

3 280 L.F.A. Bernardo et al. / Engineering Structures 42 (202) Table r e Relationships for concrete in compression in struts. Vecchio and Collins [24] " f ¼ bf 0 c 2 e e # 2 for e be o be P be o o " f ¼ bf 0 c e # be 2 o for e 2e o be < be o o Collins and Poraz [30] Model A n e ep f ;base ¼ f p n þ e nk ; n ¼ 0:80 þ f pðmpaþ 7 ( k ¼ :0 e p < e < 0 k ¼ 0:67 þ fpðmpaþ 62 e < e p ep (i) for e 6 be o : f p ¼ bfc 0 and e p = be o ; (ii) for be o < e 6 e o : f ¼ f p ¼ bfc 0; (iii) for e > e o : f ¼ bf ;base ; f p ¼ fc 0 and e p = e o. Belarbi and Hsu (99) [25] " f ¼ b r f 0 c 2 e e # 2 for e b e e o b e e P b e e o o " f ¼ b r f 0 c e # b e e 2 o for e 4e o b e e < b e e o o Hognestad [28] " f c ¼ f 0 c 2 e c e # 2 c e o e o Vecchio and Collins [29] " f ¼ b r f 0 c 2 e e # 2 e o e o Collins and Poraz [30] Model B n e ep f ;base ¼ f p n þ e nk ; n ¼ 0:80 þ f pðmpaþ 7 ( k ¼ :0 e p < e < 0 k ¼ 0:67 þ fpðmpaþ 62 e < e p ep (i) for e 6 e o : f p ¼ bf 0 c and e p = e o ; (ii) for e > e o : f ¼ bf ;base ; f p ¼ f 0 c and e p = e o. Zhu et al. [3] a " f ¼ b r f 0 c 2 e e # 2 for e b e e o b e e P b e e o o " f ¼ b r f 0 c e # b e e 2 o for e 2e o b e e < b e e o o a Calibrated for HSC. Table 2 Reduction factors b r and b e. Vecchio and Collins [24] Vecchio and Collins [29] b ¼ b r ¼ b e ¼ 0:85 0:27 e b c r ¼ e 0:8 0:34 e 6 :0 c Collins and Poraz v [30] Vecchio et al. [32] b ¼ b r ¼ b e ¼ :0þKc K f 6 b r ¼ :0þKc K f 6 ; b e ¼ :0 0:80 K c ¼ 0:35 ec e 0:28 K c ¼ 0:27 ec 0:37 eo p K f ¼ 0:825 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fc 0ðMPaÞ p P :0 K f ¼ 2:55 0:2629 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fc 0ðMPaÞ 6 : Collins and Poraz v2 [30] Hsu [33] b r ¼ 6 ; b 0:9 :0þKc e ¼ :0 b r ¼ b e ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ600ec K c ¼ 0:27 e c 0:37 eo Vecchio v [34] Vecchio v2 [34] b ¼ b r ¼ b e ¼ þcsc d b r ¼ þcsc d ; b e ¼ :0 0:80 C d ¼ K c ¼ 0:35 ec e 0:28 C d ¼ K c ¼ 0:27 ec 0:37 eo C s ¼ 0:55 C s ¼ 0:55 Vecchio v [35] Vecchio v2 [35] b ¼ b r ¼ b e ¼ 0:8 6 b r ¼ 0:8 6 þ0:35 e c e 0:28 eo þ0:35 e c e 0:28 Belarbi and Hsu [36] Belarbi and Hsu [37] 0:9 b r ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi b þkre e ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi b c þkee r ¼ b e ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:9 c þ400e c Zhang and Hsu [38] a Miyahara et al. [39] 8 b r ¼ b e ¼ R ð f0 c q ffiffiffiffiffiffiffiffiffiffiffiffiffi Þ ; g ¼ q l f sy;l þ 400e q ; r c t h ¼ r fsy;t v ¼ 0 < b r ¼ :0 e c 6 0:002 b g r ¼ :5 25e c 0:002 < e c 6 0: : b r ¼ 0:6 e c > 0:0044 g 6 ) g 0 ¼ g g > ) g 0 ; Rf ¼ =g c 0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5:8 6 0:9 fc 0ðMPaÞ Mikame et al. [40] Ueda et al. [4] a b ¼ b r ¼ b e ¼ b ¼ b 0:27þ0:96 e c ð Þ 0:67 r ¼ b e ¼ 0:8þ0:6ð000ecþ0:2Þ 0:39 eo a Calibrated for HSC.

4 L.F.A. Bernardo et al. / Engineering Structures 42 (202) Table 3 Tested models for concrete in compression in struts. Model r e Relationship Reduction factors b r and b e c0 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] c0 Vecchio and Collins [29] Vecchio [35] c Belarbi and Hsu [25] Belarbi and Hsu [36] Belarbi and Hsu [25] Belarbi e Hsu [37] c3 Belarbi and Hsu [25] Hsu [33] c4 Belarbi and Hsu [25] Zhang and Hsu [38] a c5 Zhu et al. [3] a Zhang and Hsu [38] a c6 Vecchio and Collins [24] Mikame et al. [40] c7 Vecchio and Collins [29] Mikame et al. [40] c8 Vecchio and Collins [24] Ueda et al. [4] a c9 Vecchio and Collins [29] Ueda et al. [4] a 0 Vecchio and Collins [24] Miyahara et al. [39] Vecchio and Collins [29] Miyahara et al. [39] a Calibrated for HSC. Table 4 r e Relationships for reinforcement in tension. Model r0 EC 2 [2] r e Relationship f s ¼ E s e s for e s 6 e sy ¼ f sy =E s f s ¼ f syðk Þ e su e sy e s ; k ¼ f st =f sy for e sy < e s 6 e su r02 EC 2 [2] or Vecchio and Collins [29] f s ¼ E s e s para e s 6 e sy ¼ f sy =E s f s ¼ f sy para e s > e sy r03 Belarbi and Hsu [42] 0:975E s e s f s ¼ h þ 0:025E m i s e s ; m ¼ m 9B 0:2 6 25; þ :Eses f sy B ¼ :5 qffiffiffiffiffiffiffiffiffiffiffiffiffi f cr ; f q cr ¼ 3:75 fc 0ðpsiÞ f sy 3. r e Relationships Theoretical models for the behavior of cracked RC plates under shear usually consider the behavior of the concrete and the reinforcement independently through their r e relationships. In addition, experimental stresses and strains are measured along a sufficient width to include several cracks. For this reason, average r e relationships are proposed. Transversal tension strains influence the behavior of concrete in struts, mainly in the cracked state. This phenomenon, called softening effect (Fig. ), is important in some structural concrete elements, like plates under shear or beams under shear or torsion. For concrete in compression in the struts, average r e relationships take this aspect into account. In Fig., the meaning of the parameters are: e o is the strain corresponding to the peak stress fc 0, ec is the principal tension strain and e c = e is the principal compression strain in the principal direction of the compression stress (f c = f ). For reinforcement in tension, average r e relationships take into account the interaction between reinforcement and concrete in tension between cracks. This interaction is called stiffening effect (Fig. 2a). Other models do not incorporate this interaction (Fig. 2b). In Fig. 2: f s = r s is the tensile stress, f sy is the yielding stress, f st = kf sy is the tensile strength, f n is the stress level corresponding to the intersection of the two asymptotic limits (straight lines), f o is the stress level corresponding to the intersection of the upper asymptotic limit (straight line) with vertical axis, fy is the apparent yielding stress, e s is the tension strain, e sy is the yielding strain at the end of the elastic behavior, e su is the ultimate strain, E s is Young s Modulus in elastic stage and E p and is Young s Modulus in plastic stage. In this study, only r e relationships of concrete in compression calibrated for plates under shear with proportional loading (transversal tension increase as principal compression stress increase), and with similar detailing of reinforcement, usually adopted for webs or walls of RC beams, are tested: longitudinal reinforcement orthogonal to transversal reinforcement and reinforcement to 45, or approximately, with principal directions. Table presents the mathematical equations for the r e relationships for concrete in compression in struts that are tested in this study, and proposed by several authors. Table 2 presents the mathematical equations for the reduction factors for stress (b r ) and strain (b e ) corresponding to the r e relationships of Table. Table 3 presents all the tested combinations between r e relationship and reduction factors, taking into account the original relations between them. In some cases, the authors propose a r e relationship for concrete and also the reduction factors b r and b e. In other cases, A t t cracking T concrete strut section ε ds A B ε ds/2 k2t d q A l f l t t d t d /2 C α σ d s t wall section neutral axis strain stress centerline of shear flow Fig. 3. Three dimensional truss analogy with variable angle.

5 282 L.F.A. Bernardo et al. / Engineering Structures 42 (202) d d εds σd Area Area 2 εds σd f ć β σ B f ć β σ B=Area/ε ds Area εds < β ε εo B=(Area + Area 2) /ε ds B εds > β ε εo ε ds ε p= β ε ε o 2ε o ε d ε p = β ε ε o ε ds 2ε o ε d (a) (b) Fig. 4. Integration of r e curve for concrete strut. Select ε ds Estimate t d, α, β σ, β ε Calculate k for model ci (Table ) and σ d (Eq. 9) Calculate T (Eq. ), ε l (Eq. 4), ε t (Eq. 5), σ t and σ l for model ri (Table 4) Calculate t d (Eq. 3) t d = t d? Yes Calculate α (Eq. 2) No some authors propose their own r e relationships for the concrete, but they use reduction factors b r and b e proposed by other researchers. Other authors (for instance, Ricardo et al. [27]) also combine the r e relationships for concrete proposed by some authors with the reduction factors b r and b e proposed by other authors. This latter is based on compatibility criteria (type of structural element in study, loading type, concrete strength, etc.). These last cases correspond to the combinations in Table 3 where the names of authors are not the same in the two columns. In Table 3, the combinations are called c0 (c concrete). Model c0 (based on Hognestad r e relationship for concrete in uniaxial compression) does not take into account the softening effect, and it constitutes a reference model. Table 4 presents the mathematical equations for the three r e relationships for reinforcement in tension tested in this study. Each r e relationship to be tested is called r0 r03 (r reinforcement). Model r03 is an average r e relationship taking into account the stiffening effect, while r0 and r02 are classics bilinear r e relationships for reinforcement in uniaxial tension. With respect to r e relationships for reinforcement, it can be said that a clear relationship between the equations presented in Table 4 (reinforcement) and the equations presented in Table 3 (concrete) does not exist. So all the r e relationships for the reinforcement will be combined with each r e relationships for the concrete (Section 5). Each model c0 was tested separately with models r0 r03. Thus, 2 3 = 63 theoretical models were tested. α = α? Yes Calculate β σ, β ε for model ci (Table 2) β σ = β σ? β ε = β ε? Yes Calculate θ (Eq. 6) No No 4. Theoretical model based on VATM The use of VATM to predict the behavior of RC beams under torsion, instead of Skew Bending Theory, allows computing the evolution of the state of the beam (load level, twist, strain and stress in the reinforcement 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 compute the torque, T, the effective thickness of the concrete struts, t d, of the equivalent tubular section and the angle of the concrete struts, a, from the longitudinal axis of the beam [23]: T ¼ 2A o t d r d sin a cos a ðþ ε ds > ε cu? ε l or ε t > ε su? No cos 2 a ¼ A lf l p o r d t d t d ¼ A lf l p o r d þ A tf t sr d ð2þ ð3þ Yes END Fig. 5. Flowchart for the calculation of T h curve. where A o is the area limited by the center line of the flow of shear stresses, which coincides with center of the walls thickness, t d ; r d is the stress in the diagonal concrete strut; A l is the total area of the longitudinal reinforcement; f l is the stress in the longitudinal

6 L.F.A. Bernardo et al. / Engineering Structures 42 (202) Table 5 Properties of test beams. Beam Section type x (cm) y (cm) t (cm) x (cm) y (cm) A sl (cm 2 ) A st /s (cm 2 /m) B2 Plain [43] B3 Plain [43] B4 Plain [43] B5 Plain [43] G6 Plain [43] G8 Plain [43] M2 Plain [43] T4 [44] Plain A3 Plain [45] B2 Plain [45] B3 Plain [45] B4 [45] Plain D4 Hollow [43] T2 [46] Hollow T [44] Hollow VH Hollow [47] A2 Hollow [6] A3 Hollow [6] A4 Hollow [6] A5 Hollow [6] B2 Hollow [6] B3 Hollow [6] B4 Hollow [6] B5 Hollow [6] C2 Hollow [6] C3 Hollow [6] C4 Hollow [6] C5 Hollow [6] C6 [6] Hollow a Calculated with EC 2 [2]. q l (%) q t (%) f (cm) (MPa) f ctm a (MPa) f lym (MPa) f tym (MPa) E c a (GPa) e o a (%) e cu a (%) reinforcement; A t is the area of one leg of transversal reinforcement; f t is the stress in the transversal reinforcement. The three following compatibility equations are also needed to compute the strain of the longitudinal reinforcement, e l, the strain of the transversal reinforcement, e t, and the twist, h [23]:! A e 2 l ¼ or d p o T cotg a e ds ð4þ 2 e t ¼ A 2 or d p o T tg a 2 e ds h ¼ 2t d sin a cos a! e ds ð5þ ð6þ where p o is the perimeter of the center line of the flow of shear stresses. The strain at the surface of the diagonal concrete strut, e ds, and at the center line of the flow of shear stresses, e d, can be computed from (Fig. 3): e ds ¼ 2p ot d ðe t þ e d Þ tg a sin a cos a A o ð7þ e d ¼ e ds =2 ð8þ The stress of the diagonal concrete struts, r d, is defined as the medium stress of a non-uniform diagram (Fig. 3): r d ¼ k b r f 0 c ð9þ

7 284 L.F.A. Bernardo et al. / Engineering Structures 42 (202) Table 6 Tn,exp/Tn,th ratios. ci ri Plain section NSC Hollow section NSC Hollow section HSC Average r0 r02 r03 r0 r02 r03 r0 r02 r03 r0 r02 r03 c0 x s cv (%) c02 x s cv (%) c03 x s cv (%) c04 x s cv (%) c05 x s cv (%) c06 x s cv (%) c07 x s cv (%) c08 x s cv (%) c09 x s cv (%) c0 x s cv (%) c x s cv (%) x s cv (%) c3 x s cv (%) c4 x s cv (%) c5 x s cv (%)

8 L.F.A. Bernardo et al. / Engineering Structures 42 (202) c6 x s cv (%) c7 x s cv (%) c8 x s cv (%) c9 x s cv (%) x s cv (%) x s cv (%) where b r is the reduction coefficient for the stress to take into account the softening effect; k is the ratio between the medium stress (B, see Fig. 3) and the maximum stress (A, see Fig. 3). Parameter k will be calculated by numerical integration of r e equations for the concrete strut (Table ) with the help of the computing programming language Delphi. Integration procedure to calculate k is illustrated in Fig. 4 for e d values below (Fig. 4a) and above (Fig. 4b) e o (strain for maximum stress). Based on a strain state analysis with Mohr circumference, the principal tension strain e c in the perpendicular direction to the concrete strut, to be introduced in some r e equations on Table 2, can be calculated by [23]: e c ¼ e l þ e t þ e d ð0þ The above equations and the equations from Tables 4 lead to the iterative procedure presented in Fig. 5 (for the general case b r b e ) in order to calculate the theoretical T h curve. In such calculus procedure, all the variables t d, a, b r and b e are interdependent. e o was calculated from EC 2 [2], by using Eq. () valid for NSC and HSC. e o e c ¼ 0:7f 0:3 cm < 2:8 ð Þ ðþ The theoretical failure of the sections was defined from the maximum strains of the materials (concrete and steel). Either the strain of the concrete struts, e ds (Fig. 3), reaches its maximum value (e cu ) or the steel strain, e s, reaches the usual maximum value of e s = 0. For NSC (f ck = f cm 8 (MPa) 6 50MPa as defined by EC 2 [2]), e cu was assumed to be the usual value e cu = 3.5. For HSC, e cu was calculated with Eq. (2) from EC 2 [2]. e cu ¼ 2:8 þ 27½ð98 f cm ðmpaþþ=00š 4 ð Þ 5. Comparison of experimental results ð2þ Based on the formulation of VATM and on the calculus procedure presented in Fig. 5, a computing tool (TORQUE_VATM) was developed with the help of the computing programming language Delphi to compute the T h curve for RC beams under pure torsion. The theoretical results obtained with TORQUE_VATM are compared with results of tests of beams under pure torsion which are available in the bibliography. The comparative analysis will focus on the ultimate behavior of the beams. In this section, the same beams used by Bernardo and Lopes in 2008 [5,8] were used for the comparative analysis. The experimental results of such beams can be considered trustworthy for comparative analysis with global theoretical results, as justified by the referred authors. In fact, not all the experimental results available in the bibliography can be used due to various reasons. For instance, some older studies do not have sufficient information nor meet basic design recommendations found in current codes of practice. In this earlier situation, such beams behave atypically under torsion. In other experimental studies, including recent ones, the authors present a medium twist for all the beam length, and not the twist of the critical section. Theoretical twists, based on a cross-section analysis, cannot be compared with these experimental twists. This aspect is particularly important in slender beams. Table 5 summarizes the geometrical and mechanical properties of 29 beams found in the bibliography, including the external width (x) and height (y) of the cross-section, the thickness of the walls of the cross-hollow sections (t), the distances between centerlines of legs of the closed stirrups (x and y ), the area of longitudinal reinforcement (A sl ), the distributed area of one branch of the transversal reinforcement (A st /s, where s is the

9 286 L.F.A. Bernardo et al. / Engineering Structures 42 (202) Table 7 h n,exp /h n,th ratios. ci ri Plain section NSC Hollow section NSC Hollow section HSC Average r0 r02 r03 r0 r02 r03 r0 r02 r03 r0 r02 r03 c0 x s cv (%) c02 x s cv (%) c03 x s cv (%) c04 x s cv (%) c05 x s cv (%) c06 x s cv (%) c07 x s cv (%) c08 x s cv (%) c09 x s cv (%) c0 x s cv (%) c x s cv (%) x s cv (%) c3 x s cv (%) c4 x s cv (%) c5 x s cv (%)

10 L.F.A. Bernardo et al. / Engineering Structures 42 (202) c6 x s cv (%) c7 x s cv (%) c8 x s cv (%) c9 x s cv (%) x s cv (%) x s cv (%) spacing of transversal reinforcement), the longitudinal reinforcement ratio (q l = A sl /A c, with A c = xy) and the transversal reinforcement ratio (q t = A st u/(a c s), with u = 2(x + y )), the average concrete compressive and tensile strength (f cm fc 0 and f ctm), the average yielding stress of longitudinal and transversal reinforcement (f lym and f tym ), the concrete Young Modulus s (E c ), the compressive strains for concrete (peak stress value, e o, and maximum value, e cu ). For the reinforcement, usual values were adopted for maximum tensile strain (e lu = e tu = 0 ) and Young Modulus s (E s = 200 GPa). Table 5 does not include HSC plain beams. Two experimental studies were found in the bibliography with HSC plain beams under torsion: Rasmussen and Baker in 995 [48] and Fang and Shiau in 2004 [49]. Those beams were not included in Table 5 for the same reasons previously presented. However, they will be included in Section 6 for the comparative analysis with codes (only resistant torque will be analyzed). From the theoretical curves, some key points to characterize the ultimate behavior of the test beams were obtained, namely the maximum torque (T n,th ) and the corresponding theoretical twist (h n,th ). From these key points, comparative analyses are made with the corresponding experimental values (T n,exp and h n,exp ). For this purpose, the ratios of experimental to theoretical values of the referred parameters were calculated (T n,exp /T n,th and h n,exp /h n,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 theoretical T h curves are based on conventional ultimate strains, comparative analysis with experimental results will be not consistent. Tables 6 and 7 summarize the results and the comparative analysis for the aforementioned parameters. For each ratio of experimental to theoretical value, three statistical parameters were calculated: the average value ðxþ, the sample standard deviation (s) and the variability coefficient (cv) in order to study the degree of dispersion of the results. Tables 6 and 7 relate to T n,exp /T n,th and h n,exp /h n,th ratios, respectively. Three groups of test beams were considered for presenting the results: NSC plain beams, NSC hollow beams and HSC hollow beams. Each line of the tables is relative to a model ci for concrete struts. For each group of beams, results are presented in columns for each model ri for tension steel. The last column presents the global results for all the tested beams. A global analysis of Table 6 shows that the variability between the results for each tested model is high and independent from the cross-section type (plain or hollow). For HSC beams, such variability was already expected because many of the r e relationships for the concrete struts were only calibrated for NSC. The global analysis of Table 6 shows that the range of values for the average ratio ðxþ is The distance between extremes values is not very high, showing that all the theoretical models give relatively goods previsions for the resistance of the test beams. Model c gives the lower values for x. This model do not incorporate reduction factors to take into account for the softening effect. The consequence is that the resistances are overestimated. For the other concrete models, and for those that incorporate reduction factors calibrated only for NSC, the same type of deviation observed for c0 is now observed in these models when comparative analysis is carried out with HSC. For such models and for HSC beams, strength is generally overestimated, unlike for NSC beams. From those observations, it is possible to conclude that the softening effect has more severe consequences in HSC beams under torsion when compared to NSC beams. Theoretical r e models for concrete in compression with reduction factors calibrated for NSC and HSC presents similar x values, regardless of the type of cross-section (plain or hollow).

11 288 L.F.A. Bernardo et al. / Engineering Structures 42 (202) Table 8 Results obtained with model c4 + r (plain beams). Table 9 Results obtained with model c4 + r (hollow beams). Hollow Beams T n,exp T n,th T n,exp / T n,th kn.m/m kn.m/m D4 NSC T2 NSC T NSC VH NSC A2 NSC A3 NSC A4 NSC A5 NSC B2 HSC B3 HSC B4 HSC B5 HSC C2 HSC C3 HSC C4 HSC C5 HSC C6 HSC = NSC s = 0.03 cv = 3% = 0.90 HSC s = cv = 8% = s = cv = 7% To help finding the most appropriate r e relationships for the concrete struts, in order to compute torsional strengths, and regardless of the concrete strength class, the results for the theoretical concrete models calibrated for NSC and HSC must be examined in detail. For this purpose, the last column of Table 6 helps to analyze the results regardless of the type of cross-section. In this column, the concrete models c03, c c5, c8 and c9 are those with x values closest to (over 0.90), particularly models c3, c4, c8 and c9 with x values greater than Those results do not depend on the tested theoretical reinforcement model being r or r2 (with or without hardening of steel after yielding). In contrast, the results obtained with the incorporation of model r3 show an increase of x values (getting closer to ), showing that the stiffening effect seems to have a non-negligible influence. In fact, r e curve for model r3 develops at a lower stress level compared to the r e curve for models r2 and r. Therefore, the steel reaches its yield point a bit faster for model r3 and the torsional strength for beams with moderate torsional reinforcement level (ductile failure) is lower compared to predictions obtained with models r or r2. In conclusion, from the average value ðxþ for T n,exp /T n,th, the best predictions of torsional strength are obtained with models c3, c4,

12 L.F.A. Bernardo et al. / Engineering Structures 42 (202) c8 and c9 for concrete struts, using model r3 for reinforcement in tension. From the analysis of the variation coefficient (cv) from Table 6, it can be observed that for the concrete models c03, c c3, c8 and c9, and for HSC hollow beams, the cv values are over 0%, showing a remarkable dispersion for T n,exp /T n,th ratios regarding x. From this point of view, models c4 and c5, with cv values generally below 0%, or slightly above this value, appear to be the best models, although HSC hollow beams show cv values slightly higher when compared with plain beams. In conclusion, concrete model c4 shows the lower value for cv (6.% for reinforcement models r or r2 and 8.02% for model r3). From the analysis of Table 6, an interesting aspect must be pointed out: models r3 + c4 provide the best results regarding x, while models r + c4 or r2 + c4 provide the best results regarding cv parameter. Thus, model r3 for reinforcement in tension leads to a slight higher dispersion of the T n,exp /T n,th values. According to the previous analysis of Table 6, the best theoretical model to compute the torsional strength of NSC or HSC beams with plain or hollow section, is the one that incorporates the model Table 0 Results obtained with model c4 + r3 (plain beams). Plain Beams T n.exp T n.th T n.exp / T n.th kn.m/m kn.m/m B2 NSC B3 NSC B4 NSC B5 NSC G6 NSC G8 NSC M2 NSC T4 NSC A3 NSC B2 NSC B3 NSC B4 NSC =.034 NSC s = 0.07 cv = 7% Table Results obtained with model c4 + r3 (hollow beams). Hollow Beams T n,exp T n,th T n,exp / T n,th kn.m/m kn.m/m D4 NSC T2 NSC T NSC VH NSC A2 NSC A3 NSC A4 NSC A5 NSC B2 HSC B3 HSC B4 HSC B5 HSC C2 HSC C3 HSC C4 HSC C5 HSC C6 HSC =.03 NSC s = cv = 6% = HSC s = 0.06 cv = % = s = cv = 9%

13 290 L.F.A. Bernardo et al. / Engineering Structures 42 (202) c4 for concrete struts, with the r e relationship proposed by Belarbi and Hsu in 99 [25] and the reduction factors proposed by Zhang and Hsu in 998 [38] (see Tables and 2). Also, with model c4 for concrete struts, the use of models r or r2 from EC 2 [2] (see Table 4) provides the best results, especially with regards to the dispersion of results. However, the use of model r3 (see Table 4) provides the best results regarding the average value for T n,exp /T n,th. Thus, the analysis of results should continue based on these two options (using models r/r2 or r3 with model c4), as well as on other criteria. This will be done at a later stage. The analysis of Table 7, regarding h n,exp /h n,th ratios, shows a wide range for the average value ðxþ, with some values far from unity, as well as high values for the coefficient of variation (cv), with some values over 0%. The dispersion of the results is much larger than that observed for T n,exp /T n,th ratio. Generally, Table 7 shows that all the theoretical models appear to have some difficulty in estimating the deformation of the model beams for high loading levels. This is probably due to the fact that VATM assumes a fully cracked state of the beam from the beginning of loading. This hypothesis does not match reality. For concrete model c4, shown in the previous analysis as the most suitable concrete model for the prediction of the torsional strength, the deformations of NSC beams are underestimated and the deformations of HSC beams are overestimated. By analyzing the results corresponding to model c4 in the last column of Table 7, the same tendency is observed, respectively, for the reinforcement models r/r2 and r3. The analysis of Table 7 also shows that for NSC beams the accuracy of the theoretical deformations are generally smaller for plain beams in comparison with to hollow beams. This observation seems to show that, while concrete core does not Fig. 8. T h curves: Beam T4 [44]. Fig. 6. T h curves: Beam B2 [43]. Fig. 9. T h curves: Beam A3 [45]. Fig. 7. T h curves: Beam G8 [43]. Fig. 0. T h curves: Beam D4 [43].

14 L.F.A. Bernardo et al. / Engineering Structures 42 (202) seem to influence the torsional strength (as already expected), the same cannot be stated for the ultimate deformations. This is probably due to the greater capacity of plain sections to redistribute torsional stress transversally for the ultimate levels of loading. In order to find out whether the theoretical model should incorporate concrete model c4 with reinforcement models r/r2 or r3, some previous results will be more rigorously analyzed. As previously observed, conclusions were identical, whether the model incorporates models c4 + r or c4 + r2. For this reason, only model c4 + r will be analyzed in comparison with model c4 + r3, since model r (instead of r2) is more realistic for characterizing the behavior of the reinforcement after the yielding point. Tables 8 resumes the results obtained for T n,exp /T n,th ratios with the theoretical model that incorporates model c4 for concrete struts. Tables 8 and 9 (plain and hollow beams, respectively) are related to the theoretical model that incorporates model r for reinforcement in tension, while Tables 0 and (plain and hollow beams, respectively) are related to the theoretical model that incorporates the model r3 for reinforcement in tension. Tables 8 include bar graphs for T n,exp /T n,th ratios in order to allow a visual analysis of the dispersion of the results in comparison with the optimal unit value, and also to check the degree of safety of the theoretical predictions compared to the experimental values. By comparing bar graphs for T n,exp /T n,th ratios from Tables 8 0 (plain beams) with models r and r3, respectively, it can be seen that the results are quite similar. One exception is that the values for model r3 are slightly shifted to the right side, compared to the results for model r. Thus, model r3 seems to give more safety results. This observation is also confirmed by the average value (x ¼ :034 for r3 and x ¼ 0:986 for r). Moreover, the coefficient of variation (cv) is slightly smaller for r3 (7% instead of 8%). By comparing the bar graphs for T n,exp /T n,th ratios from Tables 9 and (hollow beams) the same conclusions for plain beams can be stated. Note, however, that for HSC hollow beams, the theoretical models appear to overestimate a little the torsional strength, especially for beams with higher torsional reinforcement ratios. This observation is consistent with earlier conclusions, namely made by Bernardo and Lopes in 20 [8] by testing r e relationship from Belarbi and Hsu [25] for the concrete. However, in the study of these authors the degree of overestimation was found to be higher than that observed in the present study. This observation seems to show that the theoretical model used in this study is more accurate for HSC beams. The overestimation of the resistance can be explained by the behavior of those beams, at ultimate stage, which essentially depends on the compressed concrete struts, instead of the tensioned reinforcement. For those beams the influence of softening effect is higher. This observation led Bernardo and Lopes [8] to recommend the reduction of the softening parameters, multiplying them by 0.9 in order to take into account this overestimation and to achieve more accurate previsions. Based on the new results obtained in this article, with new theoretical models, the Fig. 2. T h curves: Beam VH [47]. Fig. 3. T h curves: Beam C3 [6]. Fig.. T h curves: Beam T [44]. Fig. 4. T h curves: Beam C6 [6].

15 292 L.F.A. Bernardo et al. / Engineering Structures 42 (202) authors consider that the aforementioned recommendation of Bernardo and Lopes is not longer justified. In fact, the results of this study demonstrate that the theoretical c4 + r03 gives more accurate results for HSC beams than the model tested by Bernardo and Lopes [8]. The previous analysis shows that the theoretical model, which incorporates models c4 and r03, seems to be the best. This conclusion is theoretically satisfying, since the reinforcement model r03 is a non-linear model and taken into account for the stiffening effect, while reinforcement models r0 or r02 characterize uniaxial tensile states. It should be noted that the results also appear to show that the influence of the stiffening effect is not very large. Figs. 6 4 present both experimental and theoretical T h curves for some selected beams. Theoretical T h curves for models c4 + r0 and c4 + r03 are highlighted to confirm the previous conclusions. From the whole set of the theoretical curves (except curves of model c4) only those that correspond to the upper and lower limits for the maximum theoretical torque were plot (in gray color). These limits define the range where the other hidden theoretical curves are located. Figs. 6 4 clearly confirm the high dispersion among T h curves, especially for high levels of loading since the distance between the limit curves is high. This observation is essentially associated to the several tested r e relationship for the compressed concrete in the struts. A global analysis of the theoretical curves of Figs. 6 4 for models c4 + r0 and c4 + r03 seems to show that the second one provides torsion strengths slightly below the first one (more safety values). Furthermore, some of the other tested models give unreliable figures for the torsional strength. Based on the analysis of the results in this section, a suitable theoretical model was found for calculating the ultimate torsional behavior (mainly torsional strength) of reinforced concrete NSC and HSC beams (plain and hollow). This model incorporates the concrete model c4 (r e relationship for compressed concrete in struts proposed by Belardi and Hsu in 99 [25] with softening factors proposed by Zhang and Hsu in 998 [38]) and reinforcement Table 2 Properties of new test beams. Beam Section type x (cm) y (cm) x (cm) y (cm) A sl (cm 2 ) A st /s (cm 2 /m) f (cm) (MPa) f lym (MPa) f tym (MPa) B50. [48] Plain B50.2 [48] Plain B50.3 [48] Plain B70. [48] Plain B70.2 [48] Plain B70.3 [48] Plain B0. [48] Plain B0.2 [48] Plain B0.3 [48] Plain H [49] Plain H-06-2 [49] Plain H-2-2 [49] Plain H-2-6 [49] Plain H [49] Plain H-07-0 [49] Plain H-4-0 [49] Plain H-07-6 [49] Plain N [49] Plain N-06-2 [49] Plain N-2-2 [49] Plain N-2-6 [49] Plain N [49] Plain N-07-0 [49] Plain N-4-0 [49] Plain N-07-6 [49] Plain Table 3 Equations to compute torsion strength. ACI 38R-05 [3] MC 90 [] EC 2 [2] A l ¼ A t s p f yv h cotg 2 h! h f yl T n ¼ 2A oa t f yv s Brittle failure: cotg h T n ¼ 0:67 p ffiffiffi fc 0 :7A 2 p h oh sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A swi f ywd z i =s tg h i ¼! h A si f yd T n ¼ T Rtwi ¼ 2F RtwiA ef z i F Rtwi ¼ A Swi f yd cotg h i z i s Brittle failure: T Rcwi ¼ 2F Rcwisinh i A ef z i F Rcwi ¼ f cd2 t i z i cosh i sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A st;i f ywd u k tg h ¼ s P! h A sl f yd T n ¼ T Rd2 ¼ 2A k ðf ywd A sw =sþ cotg h Brittle failure: T Rd ¼ 2mf cd ta k sinh cosh