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 Article history: Received 17 October 2011 Revised 28 February 2012 Accepted 26 April 2012 Available online 3 June 2012 Keywords: Beams Torsion Truss-model Softening effect Stiffening effect Theoretical model abstract 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 theoret- ical 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 2012 Elsevier Ltd. All rights reserved. 1. Introduction The first studies on torsion of reinforced concrete beams were published 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 between 1971 and 1995, and the Space Truss Analogy which is currently the base of the American code (since 1995) and of the European model codes (since 1978) [1–3]. The Variable Angle Truss-Model (VATM), which firstly aimed to unify 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 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 1929 [4]. Other authors have contributed to updated versions of the model, such as: Andersen in 1935 [5], Cowan in 1950 [6], Lampert and Thurlimann in 1969 [7], Elfgren in 1972 [8], Collins and Mitchell in 1980 [9]. In 1985 Hsu and Mo [10] developed a model with the influence of Softening Effect. Some developments and alternative methods were also introduced by Jeng and Hsu in 2009 [11], Jeng in 2010 [12], Mostofinejad and Behzad in 2011 [13], Jeng et al. in 2011 [14]. 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 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: 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 accu- racy 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 computa- tional procedure, based on the VATM, and to predict the ultimate behavior 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 of stress (r)–strain (e) relationships found in the literature. 0141-0296/$ - see front matter Ó 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, Edifı ´ cio II das Engenharias, Universidade da Beira Interior, Calçada Fonte do Lameiro, 6201-001 Covilhã, Portugal. Tel.: +351 275 329729; fax: +351 275 329969. E-mail addresses: luis.bernardo@ubi.pt, lfb@ubi.pt (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