Behaviour of concrete beams under torsion: NSC plain and hollow beams

Transcription

1 Materials and Structures (28) 41: DOI /s ORIGINAL ARTICLE Behaviour of concrete beams under torsion: NSC plain and hollow beams Luís F. A. Bernardo Æ Sérgio M. R. Lopes Received: 24 May 27 / Accepted: 2 September 27 / Published online: 19 October 27 Ó RILEM 27 Abstract A simple computation procedure is developed to predict the general behaviour of reinforced concrete beams under torsion. Both plain and hollow normal strength concrete beams are considered. Different theoretical models are used to reflect the actual behaviour of the beams in the various phases of loading. To pass from a phase to the following one, transition criteria need to be taken into consideration. Such criteria are explained. The theoretical predictions are compared with result from reported tests. Conclusions are presented. The main conclusion is that the calculation procedure described in this paper gives good predictions when compared with the actual behaviour of the plain and hollow beams. Keywords Reinforced concrete Beams Torsion behaviour L. F. A. Bernardo (&) Departamento de Engenharia Civil, University of Beira Interior, Edifício II das Engenharias, Calçada Fonte do Lameiro, Covilha, Portugal luis.bernardo@ubi.pt S. M. R. Lopes Departamento de Engenharia Civil, University of Coimbra, Polo II da Universidade, Pinhal de Marrocos, 33, Coimbra, Portugal sergio@dec.uc.pt 1 Introduction The first studies on torsion of reinforced concrete beams were published in the beginning of the past century. models were developed. These 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 code (since 1978). The newest theory is the Variable Angle Truss-Model which firstly aimed to unify the torsion design of small and large sections (for instance, small sections in buildings structures and large sections in bridges) and of reinforced and prestressed concrete. The first version of the model was presented by Rausch [29]. Other authors have contributed for new updated versions of the Model, such as: Andersen [3], Cowan [11], Lampert and Thurlimann [19], Elfgren [12], Collins and Mitchell [1]. The first versions of the model were complemented with the influence of the diagonal cracking on the compressive strength of the concrete struts (Softening Effect), as presented by Hsu and Mo [17, 18]. In 1973, Collins [9] has developed the Variable Angle Truss-Model using a different path, which was based on the compatibility of deformations, instead of the theory of plasticity. The next developments of the model by Mitchell and Collins [1, 23, 24] and Vecchio and Collins [32, 33] led to the so called Modified Compression Field Theory (MCFT). In 1995 and in 1996, Rahal and Collins [25, 27] have reported

2 1144 Materials and Structures (28) 41: results from an experimental project which indicated that the concrete cover did have influence on the ultimate torque due to the possibility of premature break off of such cover. As a consequence, the concrete cover should not be taken into account when computing the ultimate torque. The Variable Angle Truss-Model 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 Plasticity Compression Field Theory is the base of the exact method as presented by the European Code (MC9 [7] and EC 2 [8]), while MCFT is used in the Canadian Standard (CAN3-A [3]). When looking to the countries and their codes, the Plasticity Compression Field Theory is more widespread than other theories. The Space Truss Analogy has a historical value and gives good results for high levels of loading. However, for low levels of loading, the Method does not give good predictions. From the Space Truss Model by Rausch, and considering an angle of 45 for the concrete struts, Hsu [14] derived equations for the shear modulus and the torsion stiffness of a cracking beam. The Skew-Bending Theory, developed by Hsu [15], should also be referred, since it was the base of the American Code up to 1995 [1]. The provisions were changed afterwards, as proposed by MacGregor and Ghoneim [21], based on the Variable Angle Truss- Model, as the European Codes. Some late developments of MCFT were proposed by Rahal and Collins from 1995 onwards, both for the computing of the maximum torque [27] and for the prediction of the global behaviour of sections under torsion and other internal forces [26, 28]. The objective of this study is to develop a computational procedure, a simple algorithm, to predict the behaviour of plain and hollow beams under torsion. The behaviour of beams under torsion is modelled by means of T (Torque) h (Twist) curves. The theoretical prediction to the actual behaviour is developed for 3 phases that results from different theories: Linear elastic analysis in non cracked state (State I): Theory of Elasticity, Skew-Bending Theory and Bredt s Thin-Tube Theory; Linear elastic analysis in cracked state (State II): Space Truss Analogy with an angle of 45 for the concrete struts; Non linear analysis: Variable Angle Truss-Model, considering a non linear behaviour for the materials and the Softening Effect. This paper presents the theories used to model the different phases. It covers normal strength concrete (NSC) beams under torsion. Cross sections are rectangular, both plain and hollow. A computation procedure to predict the behaviour of the beams is also developed with the help of a spreadsheet (EXCEL) and taking advantage of VBA. Visual Basic for Applications (VBA) is a programming tool, derived from BASIC, and used for Windows applications. The aim is the automation of tasks involving objects (combinations of codes and data that can be manipulated as a unit). Graphs, spreadsheets and images are examples of EXCEL objects. VBA enables quick developments of simple computing procedures and the visual representation of data and results. The theoretical results are compared with results of tests of beams under pure torsion which are available in bibliography. It should be pointed out that this study deals exclusively with pure torsion. In actual structures, torsion normally occurs associated with other internal forces, such as, bending, shear and axial forces. However, in some cases, such as in curved bridges, torsion might constitute an important action. On the other hand, the behaviour of the members under pure torsion needs to be well known before other situations (interaction between torsion and other internal forces) can be studied. 2 Research significance The reasons for the study presented here are: The importance of a state-of-the art review on the current models for the theoretical prediction of the behaviour of concrete beams under torsion for all the loading stages; The non existence of a computing procedure to predict the overall behaviour (not only the ultimate behaviour) of a reinforced concrete beam under torsion; The difficulty in finding studies that analyse the current models, discuss their adequability and

3 Materials and Structures (28) 41: propose corrections to cover the whole loading and not exclusively the high levels of loading, near to the failure load, particularly for hollow beams. 3 idealisation of T h curve In general, the T h curves obtained from laboratorial tests on reinforced concrete beams (for normal reinforcement ratios) under pure torsion up to failure can lead to a typical T h curve, as presented in Fig. 1. This curve shows 3 different phases (zones 1 3 of Fig. 1). The key points to limit the 3 phases of a T h curve are fully defined by their (h;t) coordinates. These coordinates are (see Fig. 1): T cr = torque; I h cr = Twist corresponding to T cr for the top limit of Zone 1 of Fig. 1 (top limit for linear elastic analysis in non cracked stage); h II cr = Twist corresponding to T cr for the down limit of Zone 2b of Fig. 1 (sown limit for linear elastic analysis in cracked stage); T ly = Torque corresponding to yielding of longitudinal reinforcement; h ly = Twist corresponding to T ly ; T ty = Torque corresponding to yielding of transversal reinforcement; h ty = Twist corresponding to T ty ; T r = Ultimate (maximum) torque; h Tr = Twist corresponding to T r ; h max = Maximum twist at beam s failure. For the characterisation of the T h curve, the following properties were also considered (see Fig. 1): (GC) I = torsional stiffness of Zone 1 (for linear elastic analysis in non cracked stage); (GC) II = torsional stiffness of Zone 2.b (for linear elastic analysis in cracked stage). Zone 1 represents the beam s behaviour before cracking. The slope of the curve represents the elastic St. Venant stiffness ((GC) I ). In this phase the curve can be assumed as a straight line with origin in the point (;) and end in (h I cr ;T cr ). The theoretical model considered in this study for this Zone was based on: Theory of Elasticity, Skew-Bending Theory, and Bredt s Thin-Tube Theory. After cracking, the beams suffers a sudden increase of twist after what it resets the linear behaviour. This phase, identified as Zone 2.a in Fig. 1, starts at (h I cr ;T cr ) and ends at a certain level of twist (h II cr ). The slope of Zone 2.b represents the torsional stiffness in cracked stage ((GC) II ). The Fig. 1 Typical T h curve for a reinforced concrete beam under pure torsion T T max T r T ty ; T ly T cr 1 (GC) II T c 1 (GC) I I cr II cr 2.a 2.b ty; ly Tr max

4 1146 Materials and Structures (28) 41: model considered for Zone 2.b was based on the space truss analogy with 45 inclined concrete struts and linear behaviour for the materials. The point of the T h curve from which the linear behaviour starts not to be valid is defined by means of two different criteria. The first one corresponds to finding the point for which at least one of the torsion reinforcements (longitudinal or transversal) reaches the yielding point and the beam enters Zone 3 of Fig. 1. The second criterion corresponds to finding the point for which the concrete struts starts to behave non linearly, due to high levels of loading (this situation may occur before any reinforcement bar yields). Zone 3 of the curve was plotted by using the Variable Angle Truss-Model, with non linear behaviour of the materials and considering the Softening Effect. The three zones identified in the T h curve of Fig. 1 are characterised separately. The criteria used to make the transitions from one zone to the following one are also explained. 4 models 4.1 Linear elastic analysis in non cracked phase (State I) The Theory of Elasticity may be used to characterise the behaviour of a beam before cracking (State I). A slight correction needs to be introduced and the twist per length s unity is given by: T h ¼ KðGCÞ I ¼ T Kt I where: T = torque applied to the beam; (GC) I = torsional stiffness (State I); K = correcting factor (K&.7 1). ð1þ The correcting factor, K, takes into account the loss of stiffness observed in laboratorial tests. This loss, approximately of 2 4% of the (GC) I value was already reported by Leonhardt [5]. The reinforcement can be neglected when computing (GC) I. Since G (shear modulus) is equal to E c /[2(1 + m)], where E c is the Young Modulus of concrete and m is the Poisson Coefficient, for K&.7 and m =.2 in State I, an approximate value for K t I can be obtained from: K I t ¼ :3E c C I ð2þ For rectangular sections, the stiffness factor, C, is computed from St. Venant Theory (Fig. 2): C ¼ bx 3 y (plain section) C ¼ 4A 2 h=u (hollow section) where: ð3þ ð4þ x,y = minor and major dimension of the plain section, respectively; b = a St. Venant s coefficient; A = area limited by the centre line of the wall of the hollow section (A = x 1 y 1 ); u = perimeter of area defined by x 1 and y 1 : u = 2x 1 +2y 1 ; h = wall thickness of the hollow section. As a result of torsion forces, tension stresses occur in concrete. The maximum shear stress (at the middle of the longest side of the cross section) can be computed by: s max ¼ T W T ¼ s max ¼ T W T ¼ T ax 2 (plain section) y ð5þ T (hollow section) 2Ah ð6þ where W T is the Elastic Modulus of Torsion and a is a St. Venant s coefficient. The cracking torque, T cr, may be calculated by assuming a maximum concrete stress of f ctm (average concrete strength in tension): T cr ¼ W T f ctm y Fig. 2 Rectangular plain and hollow section y 1 y h x 1 ð7þ x

5 Materials and Structures (28) 41: f ctm and E c may be calculated from the concrete compressive strength through correlation equations. In this study, clauses and of MC 9 [7] are used. T cr may also be calculated through Skew-Bending Theory for non reinforced concrete beams (Hsu [16], with fc in psi, and x and y in inches): pffiffiffi T cr ¼ T np ¼ 6yðx 2 3 þ 1Þ ð8þ For hollow sections with constant wall thickness (h) Eq. 8 becomes (Hsu [16], with fc in psi, and x and y in inches): pffiffiffi T cr ¼ 6ðx 2 3 þ 1Þy ð4h=xþ for h x=4 ð9þ f c If h [ x/4, then h should be assumed as h = x/4. T cr may also be calculated by means of Bredt s Thin- Tube Theory for tubular sections (Hsu and Mo [17], with fc in psi, t in inches and A c in inches 2 ): pffiffiffi T cr ¼ 2A c t 2:5 ð1þ where: f c f c A c = area limited by the outer line of the wall of the hollow section area; t = wall thickness of the hollow section. Equation 1 may also be used for plain sections by taking: t = 1.2A c /p c, where p c (in inches) is the perimeter of the outer line of the section (Hsu [16]). Some research studies show that the torsion reinforcement has a negligible influence on the torsional stiffness. However the presence of torsion reinforcement does delay the cracking point. In 1968, Hsu [13] showed that the effective cracking moment, T cr,ef, may be computed by: T cr;ef ¼ð1 þ 4q tot ÞT cr ð11þ The total percentage of torsion reinforcement, q tot, is the sum of the longitudinal and the transversal percentages (q l and q t ): A l = total area of the longitudinal reinforcement; A t =areaofonelegofthetransversal reinforcement; A c = area of the section limited by the outer perimeter (includes the hollow part); s = stirrups spacing; u = perimeter of the centre line of the stirrups. The twist, in Rad/m, is calculated by: h I cr ¼ T cr;ef K I t 4.2 Linear elastic analysis in cracked phase (State II) ð14þ The torsional stiffness after cracking ((GC) II ) for a concrete beam was proposed by Hsu [14] by considering a tube of reinforced concrete with constant thickness. The validity of this theory was proved by Hsu by means of laboratory tests which showed that the torsion behaviour after cracking was not affected by the concrete core of the section. Accordingly to the Bredt s Thin-Tube Theory, the shear stress may be calculated by: s ¼ T 2Ah where: ð15þ A = area limited by the middle line of the transversal reinforcement; h = wall thickness of the hollow section. To predict the cracking behaviour, the concrete tube may be idealised through the special truss analogy proposed by Rausch. Using this analogy Hsu [14] derived the following equation for the torsional stiffness (GC) II of rectangular sections: ðgcþ II ¼ E s x h 2 1 y2 1 xy i ð16þ ðx 1 þ y 1 Þ 2 2nxy ðx 1 þy 1 Þh þ 1 q þ 1 l q t q l ¼ A l A c q t ¼ A tu A c s where: ð12þ ð13þ where: x 1, y 1 = are, respectively, the shorter and the longer dimensions of the rectangular area defined by the centre lines of the stirrups; n = E s /E c.

6 1148 Materials and Structures (28) 41: In Eq. 16, h represents the uniform thickness of the concrete tube s wall. The effective thickness, h e, to be considered for the cases of plain or hollow sections may be calculated by the following empirical equation, proposed by Hsu [14]: h e ¼ 1; 4ðq l þ q t Þx ð17þ From tests, Hsu found that the interception of the post cracking straight line of the T h curve with the vertical axis (by extrapolating the line) may be obtained from gt c, being g a coefficient and T c a torque level carried by the concrete (fc in psi, x and y in in): T c ¼ x2 y 3 2:4 pffiffiffi fc g ¼ :57 þ 2:86 h=x ð18þ ð19þ h is the wall thickness and x is the lesser dimension of the section. For plain sections, g = 2 since h/x = Transition from linear elastic uncracked to linear elastic cracked phases Figure 3 shows the theoretical T h curves (plain sections) for linear elastic uncracked and linear elastic cracked phases (Zones 1 and 2 of Fig. 1). The horizontal line of the transition from non cracked to cracked states is hardly observed in tests of hollow beams. For this type of beams, the value h II cr hi cr is normally very small or zero. 4.4 Non linear analysis When the steel yields or the concrete shows a non linear behaviour, the linear analysis must be replaced by a non linear analysis. In this study, for the non linear phase, the Variable Angle Truss-Model was used and the Softening Effect was taken into account. To characterise the behaviour of the compressed concrete in the struts, the authors considered the r d e d Curve proposed by Vecchio and Collins [31], which is presented in Fig. 4 and takes into account the Softening Effect by means of a reduction coefficient, k. This curve is based on experimental tests on panels under pure shear. For NSC the usual values of e o =,2 (strain correspondent to the peak stress, fc ; in the r e curve) and e cu =,35 (ultimate strain) were assumed (Fig. 4). The ascending (e d e p ) and the descending (e d [ e p ) parts of the r d e d curve are expressed respectively by Eqs. 2 and 21: " r d ¼ fc 2 e d k e # 2 d ð2þ e o e o r d ¼ f p 1 g 2 ð21þ rffiffiffiffiffiffiffiffiffiffiffiffiffi c k ¼ m m ð22þ e d g ¼ e d e p 2e o e p ð23þ where: Fig. 3 Transition from uncracked linear phase to cracked linear phase fc = Compressive strength of concrete from standard cylinder tests; m = Poisson Coefficient which may be assumed to be.3; c m = maximum distortion in a panel element (c m = e l + e t +2e d ); e l = medium strain in the longitudinal direction; e t = medium strain in the transversal direction; e d = medium strain in the concrete strut direction. The ordinary steel is characterised by the usual r e bilinear curve.

7 Materials and Structures (28) 41: Fig. 4 Concrete r e curve with softening effect d fp = f ć / Eq. 2 Eq. 21 d p = o cu In 1985, Hsu and Mo [17, 18] concluded that the r d e d Curve proposed by Vecchio and Collins was suitable for reinforced concrete members under torsion. To plot the theoretical T h curve from the Variable Angle Truss-Model (Fig. 5) it requires the 3 following equilibrium equations to determine the torque T, the effective thickness, t d, of the equivalent tubular section and the angle of the inclined concrete struts, a, from the horizontal axis of the beam [16]: T ¼ 2A o t d r d sin a cos a 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 ð24þ ð25þ ð26þ where: A o = area limited by the centre line of the flow of shear stresses, which coincides with the centre of the wall s thickness, t d ; r d = stress in the diagonal concrete strut; A l = total area of the longitudinal reinforcement; f l = stress in the longitudinal reinforcement; A t = area of one leg of transversal reinforcement; f t = stress in the transversal reinforcement; s = spacing of the transversal reinforcement. The three following equations also need to be used to compute the strains of the transversal reinforcement (e t ), the strains of the longitudinal reinforcement (e l ) and the twist (h) [16]: A 2 o e t ¼ r d p o T tga 1 e ds ð27þ 2 e l ¼ h ¼ A 2 o r d p o T cotga 1 2 e ds 2t d sin a cos a e ds ð28þ ð29þ The stress of the inclined concrete struts, r d, is defined as the medium stress of a non uniform diagram (Fig. 6): r d ¼ k 1 ð1=kþf c ð3þ q T h t A l f l q cracking b d s s s s s Fig. 5 Three dimensional truss analogy with variable angle

8 115 Materials and Structures (28) 41: Fig. 6 Strains and stresses distribution in concrete strut concrete strut ds B centerline of shear flow neutral axis t t d t d/2 k 2 t d C 1 ε σ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e l þ e t þ e ds k ¼ :3 e ds =2 ð31þ The coefficient k 1 is given by the ratio between the medium stress and the maximum stress by taking into account the Softening Effect (k 1 = B/A, see Fig. 6), and can be calculated by integration of Eqs. 2 and 21 which give the r d e d curve for the concrete strut [16]. When e ds e o /k: k 1 ¼ k e ds k2 e o 3 e 2 ds ð32þ e o When e ds [ e o /k:! 1 k 1 ¼ 1 ð2k 1Þ e p 3 e ds 1 e ds þ ð2k 1Þ ð33þ e ds e p 3 e p The above equations lead to the iterative procedure presented in Fig. 7. The theoretical T h curve is plotted using this procedure. In 1985 Hsu and Mo [17] checked the Variable Angle Truss-Model, with the Softening Effect, against experimental results of known tests and have found that this theoretical model gives good results. The predictions by the Variable Angle Truss-Model for the T h curve are especially good for high levels of loading, when the concrete is extensively cracked. For low levels of loading, the theoretical truss is not the best model and the predictions are not very good when compared to the experimental results [17]. Consequently, for the current work, the authors have used the described theoretical method for loads higher than that corresponding to the yielding of at least one torsion reinforcement bar or to a high level of stresses in the concrete struts. In fact, from this level of loading, the beam can easily be assumed to be extensively cracked. 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. 6), reaches its maximum value (e cu =.35) or the steel strain, e s, reaches the usual value of e su = Transition between linear and non linear cracked phases Figures 8 and 9 show the T h curve for the linear cracked phase and for the non linear phase (Zones 2.b and 3 of Fig. 1) for the two possible situations: ductile and fragile failure, respectively. For the ductile failure (Fig. 8), it is assumed that the linear cracked phase is valid until one of the reinforcement bars (transversal or longitudinal) yields. After the transition point is defined, Dh is calculated (see Fig. 8), Dh ¼ h y h II y : Then, the non linear zone of the T h curve is horizontally shifted to the left by Dh to reset the continuity of the whole line. Figure 9 shows the procedure to do the transition when the concrete in the struts reaches its ultimate strain before any of the torsion reinforcement bars yields. The transition point is defined from the slopes of both phases: linear cracked and non linear (there is a point when the slope of non linear phase line equals the slope of the linear phase). This point is identified in Fig. 9 by T A. The corresponding rotations are: h II A and h A. Dh is the value of the horizontal shift of non linear branch of the curve to the left to make the whole line continuous.

9 Materials and Structures (28) 41: Select ds Estimate t d,, Calculate k 1 (Eq. 32, 33), d (Eq. 3) Calculate T (Eq. 24), t (Eq. 27), l (Eq. 28) Calculate t d (Eq. 26) Is t d = t d? Yes Calculate (Eq. 25) Is =? Yes Calculate (Eq. 31) Is =? Yes Calculate (Eq. 29) Is ds >? Yes END No No No No Fig. 7 Flowchart for the calculation of T h curve 5 Comparation with experimental results Based on the criteria presented before, a calculation procedure was developed in a spreadsheet (Excel) environment by using VBA tool. The computational procedure is able to plot the T h curve for reinforced concrete rectangular sections (both plain and hollow) under pure torsion. Such curves are compared with experimental results. The amount of experimental studies found in literature is limited. Furthermore, some of the tests reported in literature cannot be used due to various reasons. For instance, some older studies simply do not meet the design recommendations found in current codes of practice (minimum reinforcement, maximum spacing, minimum wall s thickness of hollow sections, etc.). In some other experimental studies, the authors presented a medium rotation and not the rotation of the critical section. This aspect is particularly important in slender beams. Tables 1 3 summarise the theoretical calculations and the comparative analyses for the beams covered by the current study. The most important characteristics can be observed in Figs Table 1 summarises the comparative analysis for State I. The experimental cracking torsional moment (T cr,exp ) and the torsional stiffness in State I (ðgcþ I exp ) are presented there. The theoretical values of the torsional moment are also presented for comparison. Different theories were used, namely: Theory of Elasticity (Tcr;calc TE ), Skew-Bending Theory (TSBT cr;calc ) and Bredt s Thin-Tube Theory (Tcr;calc BTT ). The theoretical torsional stiffness are also presented (ðgcþ I calc ) for each limit of the reducing factors K as defined in Sect. 4.1 (K =.7 and K = 1.). to theoretical ratios are also presented to facilitate comparisons. The average values of such ratios are also presented separately for plain and hollow sections. Table 1 shows that, for State I, Theory of Elasticity and Skew-Bending Theory give good predictions for the cracking torsional moments of plain sections, whereas Bredt s Thin-Tube Theory is the best theory as far as the hollow sections are concerned. Despite the rareness of the available studies, they seem to indicate that, for State I plain sections, the calculation procedure of considering a theoretical hollow section is not correct. As far as the torsional stiffness is concerned, the dispersion of the values is higher (one of the reasons might be due to the difficulty in taking good rotation readings at this level of loading). The best values are obtained for K =.7. Therefore, the microcracking of concrete, which comes before the cracking itself,

10 1152 Materials and Structures (28) 41: T T T max T r T max T r initial position T y (GC) II 1 T y (GC) II 1 final position T cr T cr Tr,fin = Tr - max,fin = max - cr II y y Tr max cr II y Tr,fin max,fin = y - II y Fig. 8 Transition from linear cracked phase to non linear phase ductile failure T T T max T r T max T r final position T A (GC) II 1 (GC) II 1 initial position T cr T cr Tr,fin = Tr - cr II A A Tr cr Tr,fin = - A II A Fig. 9 Transition from linear cracked phase to non linear phase fragile failure must not be neglected. This phenomenon is more noticeable in hollow beams. Probably, the concentration of the stresses near the outer surface of the concrete section might be more important due to the existence of the interior void, which might prevent any stress redistribution towards the centre of the beam. Table 2 summarises the comparative analysis for State II. The experimental values of (gt c ) and of h II cr (see Fig.1), designated by (gt c ) exp and h II cr;exp ; respectively, are presented (values of h II cr;exp only for plain sections). The torsional stiffness in State II ( ðgcþ II expþ are also presented. values of the above parameters are also presented on Table 2. Some columns show the comparative ratios between experimental and theoretical values and the average values, X m. Table 2 shows that, for State II, the theoretical predictions of (gt c ) calc and ðgcþ II calc are very closed to the values obtained from tests. Therefore, the Space Truss Model with a constant angle of 45 for the concrete struts and with an empirical parameter g defined by Hsu [14] (see Sect. 4.2) seems to be adequate for the prediction of the torsional behaviour of reinforced concrete beams in State II. Table 2 shows that the values of h II cr;calc for plain sections are not as good as all the other theoretical values. Table 3 summarises the comparative analysis for the non linear phase. The experimental values of T ty,exp and T ly,exp (torsional moments of transversal

11 Materials and Structures (28) 41: Table 1 Comparative analysis for elastic non cracked phase, State I K =.7 K = 1. Beam Section type T cr,exp (knm) TE T cr,calc (knm) T cr;exp T TE cr;calc SBT T cr,calc (knm) T cr;exp T SBT cr;calc BTT T cr,calc (knm) T cr;exp T BTT cr;calc I (GC ) exp (knm 2 ) I (GC ) calc (knm 2 ) ðgcþ I exp ðgcþ I calc I (GC ) calc (knm 2 ) ðgcþ I exp ðgcþ I calc B2 [13] Plain B3 [13] Plain B4 [13] Plain B5 [13] Plain G6 [13] Plain G8 [13] Plain M2 [13] Plain T4 [5] Plain A3 [22] Plain B2 [22] Plain B3 [22] Plain B4[22] Plain D4 [13] Hollow T2 [19] Hollow T1 [5] Hollow VH1 [2] Hollow A1 [4] Hollow ,64 131, , A2 [4] Hollow ,25 135, , A3 [4] Hollow , , , A4 [4] Hollow ,487 14, , A5 [4] Hollow ,77 139, , Plain X m Hollow X m and longitudinal reinforcement yielding, respectively) and of maximum torque, T r,exp, are presented if they are available. The corresponding experimental twists (h ty,exp, h ly,exp e h Tr,exp ) are also presented. The theoretical values for the same parameters (T ty;calc ; T ly;calc ; T r;calc ; h ty;calc ; h ly;calc and h Tr,calc ) are also presented. The amount of the Dh shifts, needed as saw in Figs. 8 and 9, are also presented. Finally, the ratios of experimental to theoretical values are also presented, as well as the average values, X m. Table 3 shows that, for the non linear phase, the theoretical predictions of the torques (T ty,calc, T ly,calc and T r,calc ) are very close to the experimental ones (T ty,exp, T ly,exp and T r,exp ) both for plain and hollow sections. Therefore, the Variable Angle Truss-Model taking into account the Softening Effect (Sect. 4.4) gives a good prediction of the torsional moments of the beams in the non linear phase. As far as the rotations are concerned (h ty,calc, h ly,calc and h Tr,calc ), the values are not as good as those of the torques. The deviations are bigger for plain beams when compared to hollow beams. The plain beams have the possibility of stress redistribution towards the centre of the beams and this might imply different ultimate deformations when compared with hollow beams. Despite the small number of tests, the use of the theoretical models and criteria described in Sect. 4 seem adequate to cover all the phases of the behaviour of the materials when a rectangular normal concrete beam (plain or hollow) is loaded in pure torsion, from zero to its maximum strength. Figures 1 3 show the T h curves for each reference beam of Tables 1 3. Each figure presents the theoretical and the experimental curves. To calculate the cracking torsional moment of the theoretical curves the Theory of Elasticity was used for plain beams and the Bredt s Thin-Tube Theory was used for hollow beams. To evaluate the torsional

12 1154 Materials and Structures (28) 41: Table 2 Comparative analysis for elastic cracked phase, State II Beam (gt c ) exp (knm) (gt c ) calc (knm) ðgt cþ exp ðgt cþ calc II h cr,exp ( /m) II h cr,calc ( /m) h II cr;exp h II cr;calc II (GC) exp (knm 2 ) II (GC) calc (knm 2 ) ðgcþ II exp ðgcþ II calc B2 [13] B3 [13] B4 [13] B5 [13] G6 [13] G8 [13] M2 [13] T4 [5] a A3 [22] B2 [22] B3 [22] B4[22] D4 [13] T2 [19] T1 [5] a VH1 [2] A1 [4] A2 [4] A3 [4] A4 [4] A5 [4] Plain X m Hollow X m a Not used for computing X m stiffness in State I, the value of K =.7 was used. The points corresponding to concrete s cracking and to longitudinal ( Asl ) and transversal ( Ast ) steel yielding are identified in the graphs. For the State II phase Beam T1 [5] (Fig. 24) seems to show an excessive twist in the transition between the two first phases of behaviour. This provokes a great difference between experimental and theoretical curves. Because of this, the ratio (gt c ) exp /(gt c ) calc of this beam was not taken into account for the value of X m (Table 2). Beam T4 [5] (Fig. 17) is equal to Beam T1 [5] (Fig. 24), except for the type of section (Beam T4 is plain and Beam T1 is hollow). Despite the similarity, the experimental T h curve of the first beam show a great loss of torsional stiffness in State II, when compared to the second beam. Therefore, the experimental value of the torsional stiffness of Beam T4 [5] should be taken with great precaution. Because of II II this, the value (GC) exp /(GC) calc of this beam was not taken into account for the calculation of X m. Beam A1 [4] (Fig. 26) has a reinforcement ratio very close to the lower limit of ACI [2], and the post cracking behaviour is quite different from the theoretical predictions. The aggregate interlocking might have some influence on the high value of its strength [4]. The values of h r and T r for this beam were not considered in the calculation of X m. As far as the ultimate deformation is concerned, Figs. 1 3 seem to indicate that the theoretical predictions for the T h curves are not as good for plain sections as for hollow sections. Figures 22 3 show that for hollow beams with small wall s relative thicknesses, as the reinforcement ratio increases, the torsional stiffness suffers a great reduction before they reach its maximum strength. This big reduction might be associated with the

13 Materials and Structures (28) 41: Table 3 Comparative analysis for non linear phase Beam Dh ( /m) T ty,exp (knm) T ty,calc (knm) Tty;exp Tty;calc h ty,exp ( /m) h ty,calc ( /m) hty;exp hty;calc T ly,exp (knm) T ly,calc (knm) Tly;exp Tly;calc B2 [13] B3 [13] B4 [13].37 B5 [13].21 G6 [13] G8 [13].21 M2 [13] T4 [5] A3 [22] B2 [22] B3 [22] B4[22].1 D4 [13].9 T2 [19] T1 [5] VH1 [2] A1 [4] A2 [4] A3 [4] A4 [4] A5 [4] Plain Xm Hollow X m

14 1156 Materials and Structures (28) 41: Table 3 continued Beam h ly,exp ( /m) h ly,calc ( /m) hly;exp hly;calc T r,exp (knm) T r,calc (knm) Tr;exp Tr;calc h Tr,exp ( /m) h Tr,calc ( /m) htr;exp htr;calc B2[13] B3 [13] B4 [13] B5 [13] G6 [13] G8 [13] M2 [13] T4 [5] A3 [22] B2 [22] B3 [22] B4[22] D4 [13] T2 [19] T1 [5] VH1 [2] A1 [4] a a A2 [4] A3 [4] A4 [4] A5 [4] Plain X m Hollow X m a Not used for computing X m

15 Materials and Structures (28) 41: Fig. 1 T h curves for Beam B2 [13] ,1 l =,83 % t =,82 % = 28,6 MPa YieldingAst YieldingAsl 25,4, 1, 2, 3, 4, (º/m) Fig. 11 T h curves for Beam B3 [13] ,1 l = 1,17 % t = 1,17 % = 28,1 MPa 1 25,4 (º/m) Fig. 12 T h curves for Beam B4 [13] ,1 l = 1,6 % t = 1,62 % = 29,2 MPa 25,4, 1, 2, 3, 4, 5, (º/m) microcracking of the concrete struts due to the high level of stresses in the struts (the level of stresses becomes higher as the amount of steel increases and the wall s relative thickness decreases). The sudden end of some theoretical T h curves in comparison to experimental curves is due to the failure criterion described in Sect. 4.4.

16 1158 Materials and Structures (28) 41: Fig. 13 T h curves for Beam B5 [13] ,1, 1,5 3, 4,5 6, (º/m) 25,4 l = 2,11 % t = 2,4 % = 3,6 MPa Fig. 14 T h curves for Beam G6 [13] ,8 l =,6 % t =,59 % = 29,9 MPa Yieding Ast 1 25,4, 1,5 3, 4,5 6, (º/m) Fig. 15 T h curves for Beam G8 [13] ,8 l = 1,32 % t = 1,31 % = 28,4 MPa 25,4, 1, 2, 3, 4, 5, In general, Figs. 1 3 show that the theoretical curves are good approximation to the experimental graphs, which validates the calculation procedure explained above. 6 parametric analysis In this section, some behaviour curves are presented. These curves were obtained from the computational

17 Materials and Structures (28) 41: Fig. 16 T h curves for Beam M2 [13] ,1 l = 1,17 % t =,78 % = 3,6 MPa 25,4 (º/m) Fig. 17 T h curves for Beam T4 [5] , l =,72 % t =,75 % = 35,3 MPa 4 5,, 1, 2, 3, 4, (º/m) Fig. 18 T h curves for Beam A3 [22] ,4 l = 1,24 % t = 1,22 % = 39,4 MPa 25,4, 2, 4, 6, 8, (º/m)

18 116 Materials and Structures (28) 41: Fig. 19 T h curves for Beam B2 [22] ,56 17,78, 2, 4, 6, 8, (º/m) l =,82 % t =,98 % = 39,7 MPa Fig. 2 T h curves for Beam B3 [22] 35,56 l = 1,27 % t = 1,26 % = 38,6 MPa 17,78 Fig. 21 T h curves for Beam B4 [22] ,56 l = 1,8 % t = 1,73 % = 38,5 MPa 17,78, 2, 4, 6, 8, procedure described before and is based on the Variable Angle Truss-Model taking into account the Softening Effect. This procedure was used to model the non linear phase of Beams A1 A5, tested by Bernardo in 23 [4]. Figure 31 shows the evolution of the effective thickness of the wall of the equivalent hollow section, t d, and the maximum compressive strain at the struts surface, e ds (see Fig. 6). The curves start at the point corresponding to e ds =.5 (first

19 Materials and Structures (28) 41: Fig. 22 T h curves for Beam D4 [13] ,1 l = 1,6 % t = 1,62 % = 3,6 MPa 6,35 YieldingAst YieldingAsl 25,4, 1, 2, 3, 4, Fig. 23 T h curves for Beam T2 [19] , l t =,75 % = 25,6 MPa 8, 5,,,8 1,6 2,4 3,2 Fig. 24 T h curves for Beam T1 [5] , l t =,75 % = 35,4 MPa 8, 5,,,7 1,4 2,1 2,8 3,5 (º/m) entry of the computational procedure for the non linear phase). Figure 31 shows that the effective thickness, t d, increases with the reinforcement ratio. This is understandable, since the increasing of the reinforced ratio implies an increase of the internal forces, which need to be equilibrated by the concrete struts. The curves show a visible change when the steel yields, the variation of t d starts to be very small. After the yielding of the steel, the stress in the reinforcement

20 1162 Materials and Structures (28) 41: Fig. 25 T h curves for Beam VH1 [2] ,4 32,4 l =,33 % t =,3 % = 17,2 MPa 6,5, 1, 2, 3, 4, Fig. 26 T h curves for Beam A1 [4] , l =,18 % t =,19 % = 48,4 MPa 1, 4 6,, 1, 2, 3, 4, (º/m) Fig. 27 T h curves for Beam A2 [4] , l =,39 % t =,37 % f = 47,3 MPa cm 1, 6,, 1, 2, 3, 4, bars starts to be almost constant and this fact provokes little changes on the value of t d. Figure 32 shows the evolution of the concrete struts angle, a (see Fig. 5), with e ds. The curves also start at e ds =.5. As can be seen on this graph, the concrete struts angle, a, starts with values close to 45. This is set from the equilibrium between longitudinal and transversal torsional reinforcement

21 Materials and Structures (28) 41: Fig. 28 T h curves for Beam A3 [4] , l t =,49 % f = 46,2 MPa cm 1, 6,,,5 1, 1,5 2, 2,5 Fig. 29 T h curves for Beam A4 [4] , l t =,65 % f = 54,8 MPa cm 1, 9 6, Fig. 3 T h curves for Beam A5 [4] , l =,85 % t =,83 % f = 53,1 MPa cm 1, 1 6, [4]. The angle a tends to decrease slowly as the load increases (or as e ds increases). The curves have some sudden changes in its trajectories when one of the reinforcement bars yields. Figures show the evolutions of the maximum strain at the struts surface (e ds ), of the strains in the longitudinal bars (e l ) and of the strains in the transversal bars (e t ), respectively, with twist.

22 1164 Materials and Structures (28) 41: Fig. 31 t d e ds theoretical curves for Series A Beams [4] t d A1 A2 A3 A4 A5 4 2,,1,2,3,4 (-) ds Fig. 32 a e ds theoretical curves for Series A Beams [4] α (º) 45,7 45,5 45,3 45,1 44,9 A1 A2 A3 A4 A5 44,7 44,5,,1,2,3,4 (-) ds Fig. 33 e ds h theoretical curves for Series A Beams [4],4,3,2,1 A1 A2 A3 A4 A5, Figure 33 shows that the deformation of the struts reaches more quickly its ultimate value as the reinforcement ratio increases. Figure 34 and 35 shows that the strains of the steel reinforcement vary almost linearly with twist. 7 Conclusions From the comparative analysis carried out by the authors on NSC beams under pure torsion, the main conclusions are the following:

23 Materials and Structures (28) 41: Fig. 34 e l h theoretical curves for Series A Beams [4] l (-),12,1,8,6 A1 A2 A3 A4 A5 ε,4,2, Fig. 35 e t h theoretical curves for Series A Beams [4] ε t (-),15,12,9,6 A1 A2 A3 A4 A5,3, For the cracking torque, (T cr ), both Theory of Elasticity and Skew-Bending Theory give good predictions for plain sections beams. However, for hollow beams, Bredt s Thin-Tube Theory gives the best predictions. This shows that, in non cracked phase (State I), the concrete core of the section is also important. To consider that all the stresses takes place in an outer concrete ring, neglecting the inner concrete core, conduces to some non negligible errors; To predict the torsional stiffness in State I ((GC) I ) a reducing factor K =.7 must be used. This shows that the microcracking of the concrete that takes place before the effective cracking of the beams, should not be neglected; For State II (post cracking point), a three-dimensional truss model with concrete struts inclined by 45 gives good predictions for the T h curves. The intersection with the T axis (gt c ) and the rotational stiffness ((GC) II ) are very close to the experimental values, both for plain and hollow sections. This shows that, in this phase, the concrete core of the plain sections can safely be neglected; For the non linear phase, the comparative analysis showed that the use of Variable Angle Truss- Model taking into account the Softening Effect gives good predictions for the maximum torque (T r )and for the identification of the type of failure (ductile of fragile), independently from the type of section (plain or hollow). As far as the ultimate twist is concerned, the model gives acceptable predictions, despite some differences between plain and hollow sections (for hollow sections the predictions are closer to the experimental values). These observations lead to the conclusion that the concrete core does not affect the ultimate strength, but it does affect the ultimate deformation. The plain beams might have the capacity of redistributing the stresses towards the centre of

24 1166 Materials and Structures (28) 41: the section, which might not affect the ultimate strength, but could have some influence on the ultimate deformation, allowing a bigger deformation when compared with hollow beams; The general procedure used to predict the behaviour of the beams under pure torsion, from the early stages of loading up to the ultimate load, seems to be adequate for rectangular NSC beams, both plain and hollow. This procedure can be used both by researchers and structural designers. Researchers might use this procedure to optimise key parameters of physical models to be tested under torsion, such as, geometry, materials and reinforcement layout, in accordance with the intended structural behaviour. For instance, when there is a limit for the maximum twist of the equipment, the procedure can be a useful tool to design the test beams. For structural designers the procedure can be useful in the cases with very important torsion moments (for instance in curved box beams of bridges). In such cases, the checking for serviceability, such as deformation limits, might imply the computing of the twist for a certain load level in order to compare it with a maximum value fixed by a code of practice. The procedure might also be used when there is an incremental load and the global behaviour of the structure need to be known at different load levels up to failure. References 1. ACI Committee 318 (1995) Building Code Requirements for Reinforced Concrete, (ACI ), American Concrete Institute, Detroit, MI 2. ACI Committee 318 (25) Building Code Requirements for Reinforced Concrete, (ACI 318-5), American Concrete Institute, Detroit, MI 3. Andersen P (1935) Experiments with concrete in torsion. Trans ASCE 1: Bernardo LFA (23) Torção em Vigas em Caixão de Betão de Alta Resistência (Torsion in reinforced highstrength concrete hollow beams). Ph.D. Thesis, University of Coimbra, Portugal 5. CEB (1969) Torsion. Bulletin d Information No. 71, Mars 6. CEB-FIP (1978) Model code for concrete structures. CEB- FIP International Recommendations, 3rd edn. Paris, 348 pp 7. CEB-FIP MODEL CODE (199) Comité Euro-International du Béton, Suisse 8. CEN pren (22) Eurocode 2: design of concrete structures Part 1: general rules and rules for buildings, April Collins MP (1973) Torque-twist characteristics of reinforced concrete beams. Inelasticity and non-linearity in Structural Concrete Study No. 8. University of Waterloo Press, Waterloo, Ontario, Canada, pp Collins MP, Mitchell D (198) Shear and torsion design of prestressed and non-prestressed concrete beams. J Prestressed Concrete Inst Proc 25(5): Cowan HJ (195) Elastic Theory for torsional strength of rectangular reinforced concrete beams. Mag Concrete Res Lond 2(4): Elfgren L (1972) Reinforced concrete beams loaded in combined torsion, bending and shear publication 71:3 division of concrete structures. Chalmers University of Technology, Goteborg Sweden, 249 pp 13. Hsu TTC (1968) Torsion of structural concrete Behavior of reinforced concrete rectangular members torsion of structural concrete SP-18. American Concrete Institute, Detroit, pp Hsu TTC (1973) Post-cracking torsional stiffness of reinforced concrete sections. J Am Concrete Inst Proc 7(5): Hsu TTC (1979) Discussion of pure torsion in rectangular sections A re-examination. In: McMullen AE, Rangan BV, J Am Concrete Inst Proc 76(6): Hsu TTC (1984) Torsion of reinforced concrete. Van Nostrand Reinhold Company 17. Hsu TTC, Mo YL (1985) Softening of concrete in torsional members Theory and tests. J Am Concrete Inst Proc 82(3): Hsu TTC, Mo YL (1985) Softening of concrete in torsional members Design recommendations. J Am Concrete Inst Proc 82(4): Lampert P, Thurlimann B (1969) Torsionsversuche an Stahlbetonbalken (Torsion tests of reinforced concrete beams) Bericht Nr Institut fur Baustatik, ETH Zurich, 11 pp 2. Leonhardt F, Schelling G (1974) Torsionsversuche an Stahl Betonbalken Bulletin No Deutscher Ausschuss fur Stahlbeton, Berlin, 122 pp 21. MacGregor JG, Ghoneim MG (1995) Design for Torsion. J Am Concrete Inst Proc 92(2): McMullen AE, Rangan BV (1978) Pure torsion in rectangular sections A re-examination. J Am Concrete Inst 75(1): Mitchell D, Collins MP (1974) The behavior of structural concrete beams in pure torsion. Civil Engineering Publication No. 74-6, Department of civil Engineering, University of Toronto, 88 pp 24. Mitchell D, Collins MP (1974) Diagonal compression field theory A rational model for structural concrete in pure torsion. J Am Concrete Inst Proc 71(8): Rahal KN, Collins MP (1995) Effect of thickness of concrete cover on shear-torsion interaction An experimental investigation. J Am Concrete Inst Proc 92(3): Rahal KN, Collins MP (1995) Analysis of sections subjected to combined shear and torsion A theoretical model. ACI Struct J 92(4): Rahal KN, Collins MP (1996) Simple model for predicting torsional strength of reinforced and prestressed concrete sections. J Am Concrete Inst Proc 93(6):