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Beams prestressed with unbonded tendons at ultimate

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Beams prestressed with unbonded tendons at ultimate
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  Beams prestressed with unbonded tendons at ultimate Marco Andrea Pisani and Emilio Nicoli Abstract: This paper presents a numerical investigation on beams and slabs prestressed with either unbonded internal or external tendons. Twenty-three experimental tests (beams and slabs prestressed with unbonded internal tendons) were numerically simulated to verify the reliability of the numerical algorithm adopted. The good agreement established enables us to study the behaviour of these beams in depth and to compare it with the behaviour of similar beams prestressed with external tendons. The numerical analyses were then repeated after including the safety factors related to the materials. The outputs were then compared with the results adopted by making use of the simplified method suggested by Eurocode E.C.2 Part 1-5, to check the size of the error involved in the adoption of the latter. Key words: umerical analysis, unbonded internal tendons, external tendons, European Prestandard, prestressed concrete, beams, post-tensioned. Rksumk : Cet article prCsente une ttude nurnerique sur les poutres et dalles prkcontraintes avec tendons internes non adhCrants ou tendons externes. Vingt-trois essais experimentaux (poutres et dalles precontraintes avec tendons internes non adhCrants) ont CtC sirnults numtriquement pour verifier la fiabilitC de l'algorithme nurnkrique adopt$. La bonne corrClation Ctablie nous a permis dlCtudier en profondeur le comportement de ces poutres et de le cornparer avec le comporternent de poutres similaires precontraintes avec tendons externes. Les analyses numirique ont ensuite CtCreprises en incluant les facteurs de sCcuritC reliCs aux matkriaux. Les rCsultats ont ensuite CtC comparCs avec ceux obtenus en adoptant la mCthode simplifike suggCr6e par 1'Eurocode E.C.2 Part 1-5, afin de juger de l'importance de l'erreur impliquee par l'adoption de cette dernikre mCthode. Mots clis : analyse numCrique, tendons internes non adherants, tendons externes, PrCnorme EuropCenne, bCton precontraint, poutres, post-tensionne. [Traduit par la rCdaction] Introduction significantly different from that of bonded beams (Rozvany External prestressing is increasingly adopted both in the con- struction of new precast beams or bridge decks and in the rehabilitation or the strengthening of existing structures (see, for example, Falkner et al. (1995) and Godart (1995)), while unbonded internal prestressing is frequently used in slabs. A careful reading of the scientific literature on this topic is nevertheless somewhat confusing: because of the absence of bond, the load carrying capacity of the tendons is left partially unused, so that the breaking load of these structures could be considerably lower than the one of similar beams, prestressed with bonded tendons (Walter and Miehlbradt 1990). Experimental results indicate that the load-deflection behaviour of some unbonded prestressed concrete beams is Received November 8, 1995. Revised manuscript accepted June 25, 1996. M.A. Pisani. Department of Structural Engineering, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milan, Italy. E. Nicoli. Percassi Consulting Engineers, Via Bergamo 19, Clusone, Italy. Written discussion of this paper is welcomed and will be received by the Editor until April 30, 1997 (address inside front cover). and Woods 1969). It was found systematically that structures prestressed with either internal or external tendons behave essentially in the same way at all loading stages up to ultimate. In terms of structural behaviour and design methods, the experience gained during a period of now more than ten years has proven the method of external prestressing to be essentially equivalent to the method of internal prestressing (Muller and Gauthier 1990). Simple span and fully loaded continuous, unbonded, post- tensioned beams containing unprestressed bonded reinforce- ment and designed according to the provisions of ACI 318-63, will have Serviceability characteristics, ductility and strength, equal or better than those of comparable bonded post-tensioned beams (Mattock et al. 197 1). From a flexural analysis and design viewpoint, external tendons can be treated as unbonded tendons provided secondary effects and frictional forces at deviators are neglected (Naaman and Alkhairi 1991~). But both American and European standards make a distinc- tion between the two technologies. These differences in the iriterpretation of the experimental tests justify further research on the problem. This paper aims to give a contribution to the investigation by performing a numerical simulation that will both validate the algorithm adopted (when compared with the experimental Can. J. Civ. Eng. 23: 1220- 1230 (1996). Printed In Canada Impr~mC u Canada    C  a  n .   J .   C   i  v .   E  n  g .   D  o  w  n   l  o  a   d  e   d   f  r  o  m   w  w  w .  n  r  c  r  e  s  e  a  r  c   h  p  r  e  s  s .  c  o  m    b  y   D  e  p  o  s   i   t  o  r  y   S  e  r  v   i  c  e  s   P  r  o  g  r  a  m   o  n   0   6   /   0   5   /   1   3   F  o  r  p  e  r  s  o  n  a   l  u  s  e  o  n   l  y .  Pisani and Nicoli Fig 1 Discretization in the numerical model. 1 cracked region tests) and allow us to evaluate the size of the error involved in the use of the simplified method suggested by the Euro- pean Prestandard E.C 2 Part 1-5 (when performed including the safety factors) (CornitC EuropCen de Normalisation 1992b). The numerical algorithm From a general point of view, a beam prestressed with exter- nal tendons may be considered as the coupling of two distinct substructures: the concrete beam and the external tendons. The interaction between the two substructures is restricted to the points where anchorages and deviators are placed. To describe the behaviour of the beam, the numerical algorithm (Pisani 1996) evaluates the displacement of the lower edge of the concrete substructure, which has to be orthogonal to the concrete cross sections and rectilinear before bending. Through the hypothesis that plane sections remain plane after bending (shear deformation is neglected), it is possible to determine the displacements of every point of the beam and then the response of the tendons too. The changes in the shape of the concrete substructure involve changes in the relative position of the tendons, which can be significant in the equilibrium condition of the deformed beam. To avoid approximations related to this effect, the algo- rithm includes second-order effects, large deflections, and the change in length of the beam due to compression. Discretiza- tion is performed by means of the well-known finite difference method. Stating that we aim to evaluate the beam behaviour under increasing loads, a step-by-step method is adopted. The analysis of the concrete substructure requires the knowledge of the external forces acting on it, including the forces transferred by the deviators and the anchorages, which depend on the bent shape of the beam itself. This observa- tion, together with the assumption of geometrically nonlinear behaviour of the structure and mechanically nonlinear response of the materials, implies an iterative process, at each step of loading. The method brings with it essentially three limitations: (i) Cracking is spread over a segment of finite length in the concrete substructure (consequently precast segmental beams, especially if cast with dry joints, are excluded from this analysis). (ii) Shear deformation is neglected both before and after cracking (that is, the beam has the amount of vertical stirrups necessary to resist shear at all loading stages). (iii) Tensile strength of concrete is neglected (this assump- tion is expected to have no significant effect on the load carrying capacity of the beam, since the contribution of con- crete in the tensile zone becomes negligible at high loading levels, at least if a minimum amount of unprestressed rein- forcement is adopted). Creep effects are not included in the numerical algorithm. Beams prestressed with unbonded internal tendons do not exactly meet the specifications of this method, but can be suita- bly modelled by adopting a dense subdivision of the concrete substructure and by placing deviators in each cross section. The main difference between this method and that of Alkhairi and Naaman (1993) is essentially in the way the two methods compute the effects related to span-to-depth ratio. Alkhairi and Naaman neglect the horizontal displacements of deviators and anchorages but increase the bending moments of the beam to take into account the increase of tensile stress in the unprestressed longitudinal reinforcement due to shear and compute the cracking moment of the cross sections. The numerical algorithm we adopted neglects these effects, but includes large deflections and computes the horizontal dis- placements of the points connecting the two substructures. Comparisons with the experimental tests Twenty-three experimental tests, carried out by three distinct research teams, were reproduced. All the tests refer to simply supported post-tensioned beams and slabs with unbonded internal tendons. Since the strands that were adopted in these tests exhibit very low friction coefficients, in the numerical analysis friction was neglected. Performing this nonlinear structural analysis, all the parameters we adopted to describe the materials behaviour were those experimentally measured (or those given by Fintel (1974) and ACI (1970) when these values were not explicitly stated), and the shape of the con-    C  a  n .   J .   C   i  v .   E  n  g .   D  o  w  n   l  o  a   d  e   d   f  r  o  m   w  w  w .  n  r  c  r  e  s  e  a  r  c   h  p  r  e  s  s .  c  o  m    b  y   D  e  p  o  s   i   t  o  r  y   S  e  r  v   i  c  e  s   P  r  o  g  r  a  m   o  n   0   6   /   0   5   /   1   3   F  o  r  p  e  r  s  o  n  a   l  u  s  e  o  n   l  y .  Can. J. Civ. Eng. Vol. 23, 1996 222 Fig 2 Shape of the beams and loading arrangement. A) I A beams Concrete 152rn Prestressing steel @I slabs named N.l -N.9) m _---- line load line load fy;:m 120rnrn SU A~ L Reinforcing steel stitutive laws themselves were those suggested by Eurocode E.C.2, briefly described in Fig. 2D (continuous thick lines). Note that this analysis does not bring with it any safety factor, i.e., it simulates the experimental test. The first group of nine tests was performed by Du and Tao (1985). The shape of the beams and the loading arrange- ment are shown in Fig. 2A while the mechanical details are listed in Table 1. Figure 3, which compares the experimental and numerical rnidspan deflections, shows the good level of precision attained by the mathematical model. Similar approximations are shown in Fig. 4 referring to stress increase in unbonded reinforce- ment. In these tests collapse is always due to concrete crush- ing, whereas the tensile stress in the unbonded tendons at this stage (see Table 2) is smaller than or at least similar to the yielding stress, and considerably lower than the ultimate stress. This outcome strengthens the statement made by Walter and Miehlbradt (1990). Referring to the amplitude of the midspan deflection near collapse, it can be noticed that the deflection increases in direct proportion to the decrease in the amount of reinforcing or prestressing steel, but this means a reduction in the load carrying capacity of the beam. This outcome is a logical con- sequence of the observation that the deflection mainly depends on the curvature, llr, in the critical zone of the beams, which all have the same section, collapse because of concrete crush- ing, and were cast with the same concrete. Given that c is the depth of the concrete under compression and E the ultimate compressive strain in concrete, the relationship llr, = ~,,lc holds, so that increasing values of c means decreasing deflec- tions, but also an increase in the compressive stress resultant that has to be balanced, in the concrete substructure, by the stress resultant in the unprestressed reinforcement plus the axial force passed by the tendons. A reduction in one of these last two terms lowers the value of c and therefore increases the deflection of the beam near collapse. The second group of experimental tests, described in Fig. 2B and Table 1, was performed by Mattock et al. (1971). These tests confirm the ~revious utcomes: colla~se s due to concrete crushing, and high values of c mean small bend- ing and limited increase of the tensile stress in the tendons. The peculiar shape of the T sections makes it possible to markedly reduce c, so that midspan deflection increases con- siderably, as it is shown in Fig. 5 where T beams exhibit a final branch almost horizontal at ultimate. Du and Tao (1985) pointed out that the applied load versus midspan deflection diagrams of unbonded prestressed beams with bonded unprestressed steel can be approximated by three straight segments, namely the uncracked elastic stage, a cracked elastic stage, and a third stage representing the beam behaviour after yielding of unprestressed steel. The experi- mental data and-the numerical analysis prompted us to- add a fourth segment with a very small slope related to yielding of the tendons (see Fig. 6A). At this stage the increase in the    C  a  n .   J .   C   i  v .   E  n  g .   D  o  w  n   l  o  a   d  e   d   f  r  o  m   w  w  w .  n  r  c  r  e  s  e  a  r  c   h  p  r  e  s  s .  c  o  m    b  y   D  e  p  o  s   i   t  o  r  y   S  e  r  v   i  c  e  s   P  r  o  g  r  a  m   o  n   0   6   /   0   5   /   1   3   F  o  r  p  e  r  s  o  n  a   l  u  s  e  o  n   l  y .  Pisani and Nicoli Table 1 Details of test beams and slabs. Concrete Prestressing steel Reinforcing steel Beam1 f, A, APIAc f f As ASIAc f, Source slab (MPa) Llh (rnrn2) ( ) Lld (MPa) (MPa) (mrn2) ( ) (MPa) Du and Tao (1985) A1 30.6 15 58.8 0.131 19.1 960 1790 157 0.350 A2 30.6 15 98.0 0.219 19.1 904 1790 157 0.350 A3 30.6 15 156.8 0.350 19.1 820 1790 236 0.527 A4 30.6 15 58.8 0.131 19.1 869 1790 157 0.350 A5 30.6 15 78.4 0.175 19.1 810 1790 308 0.688 A6 30.6 15 156.8 0.350 19.1 854 1790 462 1.031 A7 30.6 15 39.2 0.088 19.1 885 1790 308 0.688 A8 33.1 15 58.8 0.131 19.1 894 1790 462 1.031 A9 33.1 15 156.8 0.350 19.1 920 1790 804 1.795 Mattock et al. (1971) TU1 27.6 28 197.4 0.225 33.6 1261 1930 353 0.402 348 TU2 27.6 28 197.4 0.225 33.6 1252 1930 127 0.145 348 TU3 27.6 28 197.4 0.225 33.6 1298 1930 55 0.063 1930 RU1 27.6 28 197.4 0.425 33.6 1262 1930 205 0.441 348 RU2 27.6 28 197.4 0.425 33.6 1287 1930 192 0.413 348 Cooke et al. (1981) N.l 30.1 25.6 279 0.439 38.3 1163 1765 N.2 30.1 25.6 279 0.220 38.3 1145 1765 N.3 30.1 25.6 116 0.055 38.3 1197 1839 N.4 34.4 18.9 279 0.439 28.3 1163 1765 - N.5 34.4 18.9 279 0.220 28.3 1154 1765 - N.6 34.4 18.9 116 0.055 28.3 1220 1839 - N.7 30.8 12.2 279 0.439 18.3 1164 1765 N.8 30.8 12.2 279 0.220 18.3 1168 1765 N.9 30.8 12.2 116 0.055 18.3 1204 1839 resisting moment derives from steel hardening and from small increases in the internal lever arm. Often these diagrams are incomplete, that is, the shape of the cross section, the reinforcement ratio, and the prestressing ratio lead to a pre- mature collapse because of concrete crushing (see, for instance, RU beams). The numerical model does not compute the delayed behaviour of concrete. The experimental midspan deflections were therefore drawn in Fig. 5 by ignoring the increment measured under a constant load. This deficiency should imply a significant error in the evaluation of both the load at ulti- mate and the stress in the unbonded prestressed reinforcement, but it does not (see Fig. 5 and Table 2). This result may be justified by observing that in TU beams c is very small (from 20 to 25 mm) and prestressed steel has reached its yielding stress, so that the load carrying capacity of these beams is almost independent of their deflections (stress resultants in the cross section cannot significantly vary both in value and in position as midspan deflection increases). In RU beams the stress in prestressing steel at ultimate is lower than yield stress and c is much greater than before (about 101 mm), but collapse occurs because of concrete crushing when midspan deflection is still small, as well as its increase due to the delayed behaviour of concrete (about 50 mm) and then the error related to an inaccuracy in the evaluation of the deformed shape of the beam is still small. The last group of nine experimental tests, performed by Cooke et al. (1981), refers to slabs that were cast without unprestressed reinforcement (see Fig. 2C and Table 1). All the slabs collapsed because of concrete crushing, but some of them (namely slabs N.3, N.6, and N.9) developed one or two cracks only. This condition does not fulfil the hypothesis, adopted by the numerical model, that cracking is spread over a segment of finite length of the beam axis, as on the contrary it happens in the other tests considered. The precision of the numerical results is nevertheless surprisingly good (see Figs. 7-9). This unexpected outcome may be accounted for by observing that midspan deflection and tensile strains in the tendons (the experimental data) depend on the behaviour of the entire beam, whereas the stress distribution around the crack should be considerably different from the one numer- ically evaluated. Moreover, these three beams exhibit a sudden decrease in the carrying capacity once cracking occurs (the peak of the diagrams). This behaviour, which should be avoided in usual practice, cannot be modelled because of lack of tensile strength of concrete in the numerical model. Furthermore, our final aim is to check the design criteria suggested by E.C.2, and in doing so we have to take into account the occurrence that a real slab bears some cracks, inherited from its previous life. These nine tests confirmed that even if unprestressed rein- forcement is absent, beams prestressed with unbonded tendons often bear good ductility, although it is not a consequence of yielding of the reinforcement, so that, for instance, slabs N. 1, N.4, and N.7 on unloading after collapse almost reverted back to their srcinal undeflected position, providing that    C  a  n .   J .   C   i  v .   E  n  g .   D  o  w  n   l  o  a   d  e   d   f  r  o  m   w  w  w .  n  r  c  r  e  s  e  a  r  c   h  p  r  e  s  s .  c  o  m    b  y   D  e  p  o  s   i   t  o  r  y   S  e  r  v   i  c  e  s   P  r  o  g  r  a  m   o  n   0   6   /   0   5   /   1   3   F  o  r  p  e  r  s  o  n  a   l  u  s  e  o  n   l  y .  1224 Fig 3 Applied load versus midspan deflection diagram for A beams. Can. J. Civ. Eng. Vol. 23, 1996 ..----.-- xperimental umerical urocode E.C.2 .-.- ---------------- E.C.2 approximate solution) w Midspan deflection (mm) Fig 4 Applied load versus stress increase in unbonded steel diagrams for A beams. -..-..--- xperimental Numerical w Stress increase in unbonded steel (MPa) spalled concrete did not prevent closure of the cracks by is quite clear and convincing. It is natural to wonder how becoming jammed in them. similar beams would behave if prestressed with external tendons. The previous analyses were then repeated after sub- Numerical analysis of similar beams stituting the internal unbonded tendons with kxternal tendons prestressed with external tendons whose shape is described in Fig. 10. The results of these analyses are plotted in Figs. 11 and The experimental tests provide a description of the behaviour 12, which show that at stages and 5 the beam behaviour is of beams prestressed with unbonded internal tendons, which independent of the prestressing technology adopted, owing to    C  a  n .   J .   C   i  v .   E  n  g .   D  o  w  n   l  o  a   d  e   d   f  r  o  m   w  w  w .  n  r  c  r  e  s  e  a  r  c   h  p  r  e  s  s .  c  o  m    b  y   D  e  p  o  s   i   t  o  r  y   S  e  r  v   i  c  e  s   P  r  o  g  r  a  m   o  n   0   6   /   0   5   /   1   3   F  o  r  p  e  r  s  o  n  a   l  u  s  e  o  n   l  y .
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