Influence of High Axial Tension on the Shear Strength of non-shear RC Beams

Transcription

1 Influence of High Axial Tension on he Shear Srengh of non-shear RC Beams Henrik B. JOERGENSEN PhD candidae Univ. of Souhern Denmark Joergen MAAGAARD Associae Professor Univ. of Souhern Denmark Linh C. HOANG Professor Univ. of Souhern Denmark Lars S. FABRIN M.Sc. Uni. of Souhern Denmark Summary This paper deals wih he influence of high axial ension on he shear srengh of beams wihou shear reinforcemen. An experimenal program wih shear-ension ess was carried ou. The experimenal resuls have been used o evaluae he applicabiliy of he Eurocode 2 (EC2) design formula in cases wih large normal forces. In addiion, he experimens have been used o evaluae an exension of he plasiciy based Crack Sliding Model (CSM) o cover cases wih large normal forces. The es resuls show, ha even in he presen of very high axial ensile sresses and srains, he member is sill able o carry significan shear sresses. The analysis reveals ha he EC2 formula is over conservaive in his regard. Keywords: Shear ess, Concree beams, Axial ension, Eurocode 2, Crack Sliding Model. 1. Inroducion I is well-known ha he shear capaciy of reinforced concree beams wihou sirrups can be enhanced by a compressive normal force. This enhancemen effec is incorporaed in mos design codes, including he EC2 [1]. Even hough he EC2-formula was developed o accoun for axial compression, i is also used for axial ension. This is apparen when reading he accompanying code ex and he background documen, see [2]. The formula predics a linear reducion of he shear srengh for members wih axial ension. ery lile experimenal work has been available for verificaion of his approach. Furher, in he few exising ess, he applied axial ension was raher low, [3], [4], [5], [6]. Some sudies have indicaed ha he shear srengh is no dramaically reduced as prediced by he codes provided ha he member is properly reinforced for he applied ension, see e.g. [5] and [6]. The quesion is wheher his also holds for high axial ension, measured boh in erm of sress and srain. The issue of members subjeced o shear in combinaion wih high axial ension is relevan in many siuaions in pracice. This could for insance be in a concree slab which is a par of a coninuous seel-concree composie bridge deck. A he inermediae suppor, he negaive momen from he global acions may resul in large ension in he slab, which a he same ime could be subjeced o shear from he local acion of raffic loads. In his paper he influence of high axial ension on he shear srengh is invesigaed. A es series has been conduced and he resuls are compared wih he EC2-design formula. Furher, he es resuls have also been used o evaluae, wheher he plasiciy based CSM could be applicable o model he effec of axial ension.

2 2. Experimens Fig. 1: (op) Specimen geomery, (boom) frame for applicaion of axial ension. The experimenal program consised of 23 beams wihou sirrups esed in combined shear-ension, [7]. The beams were esed for axial ension σ = N/bh ranging from 0 o 50% of he concree compressive srengh. Maximum N corresponded o 83% of he yield force of he longiudinal reinforcemen. This is equivalen o a maximum average ensile srain of ~ 0,40 %. The beam geomery is shown in Figure 1 (op). The widhs of he suppor and load plaes were 50mm. The beams, ogeher wih 27 cylinders wih a diameer x heigh measure of 100mm x 200mm, were cas from one single concree bach. The concree was made from a Rapid Porland Cemen and aggregae wih maximum size of 16 mm. According o [8] a conversion facor equal o 0,97 were used o conver he srengh o a sandard cylinder wih diameer x heigh = 150mm x 300mm. The specimens were longiudinally reinforced wih four Ø15mm DYWIDAG hread bars wih measured yield sress f y = 1027 MPa and ulimae srengh f u = 1227 MPa. The c-c disance of he longiudinal reinforcemen was in boh direcions 129mm. The high srengh made i possible o conduc ess wih high axial ensile srains. The beams were esed in symmeric hree poin bending wih shear span o deph raio, a/h, equalling eiher 2,25 or 2,75. The ensile normal force was firs applied and hen kep consan while he specimen was subjeced o ransverse loading unil failure. A U-shaped seel frame was used o induce ension ino he beam. The arrangemen is schemaically shown in figure 1 (boom). In one end, wo hydraulic jacks were placed beween a hick seel plae and he seel frame. The hick seel plae funcioned as anchorage plae for wo DYWIDAG Ø26,5mm hread bars, which were posiioned horizonally hrough oversized holes in he flanges of he frame leg. These bars were hen - via wo smaller seel plaes conneced o he four hread bars sicking ou of he concree beam end (he Ø15mm reinforcemen bars were exended 250 mm beyond he beam ends). A he oher end, a similar sysem for force exchange was used. Here, however, he DYWIDAG bars were anchored direcly o he seel frame. Pressure from he hydraulic jacks hen induced ension in he beam while subjecing he U-shaped frame o a closing momen. Since he specimen was placed on rollers i was ensured ha he same force was applied from boh ends. The srains in he DYWIDAG bars were moniored o ensure ha he axial load was applied cenrally o he beam. The frame ogeher wih he specimen fied inside he 2000 kn Amsler esing rig used o apply ransverse loading.

3 The main es resuls have been summarized in Table 1. Generally, i was observed ha he shear srengh was no significanly affeced by ensile normal forces less han abou 40% of he specimen ensile yield srengh. This corresponds o an average normal srain of ~ 0,20 %. Beyond his level of ension, a decrease in he shear srengh was generally observed for increasing axial ension. Figure 2 shows wo examples of he crack paerns observed a he sage of failure. Typically, a sysem of ensile crack appeared during he applicaion of he ensile normal force. Then, during ransverse loading, diagonal cracks crossing he ensile cracks sared o develop. This ook place already in he beginning of he ransverse loading process. Finally, shear failure would ake place in one of he diagonal cracks. For specimen ST-7 shown in Fig. 2 (boom), developmen of diagonal cracks iniiaed when he ransverse load reached ~16 kn (corresponding o a shear force of 8 kn), which was abou 20% of he ulimae load. ST-3 ST-7 Fig. 2: Typical crack paerns: (op) specimen ST-3 wih N/A s f y ~ 0,27, (boom) specimen ST-7 wih N/A s f y ~ 0,55. Indicaions of flexural failure, for insance in he form of wide open flexural cracks a mid span, were no observed even hough some of he ess wih very high axial ension heoreically had a bending srengh slighly smaller han he shear srengh (see e.g. Table 1 ST-23). 3. Calculaions Calculaions by use of he EC2-formula have been conduced and compared wih es resuls. This invesigaion is ineresing from a pracical poin of view since his formula a he presen momen forms he basis for shear srengh verificaion in many European counries. Moreover, he ess are also compared wih calculaions by use of he plasiciy based CSM, see [9] and [10]. The CSM is ineresing in his conex because i allows a direc modelling of he influence of normal forces. Correlaion of he model wih ess on members wih axial compression has been invesigaed by Zhang [9] and Jensen and Hoang [11]. In his paper, he model is exended o cover he case of large axial ension. 3.1 Calculaion using Eurocode 2 According o EC2, he shear srengh of a beam wihou shear reinforcemen should be aken as: u 1/3 ( 0,18 ( 100ρl c ) 0,15σ cp ) k f + bd = max 3/2 ( 0, 035k fc + 0,15σ cp ) bd Here k = 1 + [ 200/d ] (d in mm), ρ l = A s /bd 0,02 and N E /A c < 0,2f c (in MPa). As formulaed in he explanaory ex in EC2, N E is he axial force in he cross-secion due o loading or (1)

4 presressing (N E > 0 for compression). Hence, alhough inended o primarily cover he effec of compressive normal force, he formula is also be used when a ensile normal force is presen. Table 1: Summary of main es resuls Specimen ID f c [MPa] a/h N u,es u,csm u,ec2 u,flexural u, es u, CSM u, es u, EC 2 ST-1 24,8 2,25 426,9 39,5 43,9 0 55,6 0,90 - ST-2 26,1 2,25 96,8 43,5 45,1 29,7 96,4 0,97 1,46 ST-3 26,1 2,25 200,2 45,4 45,1 14,2 84,0 1,01 3,19 ST-4 26,1 2, ,1 49,0 44,3 108,6 0,90 1,00 ST-5 26,8 2, ,3 49,7 44,6 108,6 0,85 0,95 ST-6 26,8 2,25 300,0 43,0 45,6 0 70,8 0,94 - ST-7 27,4 2,25 401,1 40,8 46,2 0 58,4 0,88 - ST-8 27,4 2,25 498,6 39,8 46,2 0 46,2 0,86 - ST-9 27,4 2,25 299,2 45,4 46,2 0,1 70,8 0,98 360,34 ST-10 28,1 2,25 401,2 36,9 46,8 0 58,4 0,79 - ST-11 28,1 2, ,2 47,7 45,4 90,5 0,82 0,86 ST-12 28,1 2,25 200,3 44,0 46,8 15,3 83,8 0,94 2,87 ST-13 28,8 2,75 99,9 33,9 42,9 30,8 80,2 0,79 1,10 ST-14 28,8 2,75 200,5 40,7 39,3 15,7 69,8 1,03 2,59 ST-15 28,8 2,75 301,2 44,6 39,3 0,6 59,2 1,13 79,08 ST-16 29,5 2, ,3 49,0 46,1 90,3 0,76 0,81 ST-17 29,5 2,75 100,3 39,9 43,5 31,1 80,0 0,92 1,28 ST-18 29,5 2,75 200,3 32,8 39,8 16,1 69,8 0,82 2,04 ST-19 29,5 2,75 300,3 32,0 39,8 1,1 58,8 0,80 29,95 ST-20 29,5 2,75 500,3 30,2 39,8 0 38,8 0,76 - ST-22 30,2 2,25 500,7 37,5 48,5 0 46,0 0,77 - ST-23 30,2 2,25 601,6 34,5 38,8 0 33,6 0,89 - ST-24 30,2 2,75 600,4 35,2 32,3 0 28,0 1,09 - Noaions: f c : Tesed concree compressive srengh. a/h: shear span o deph raio. N: Applied axial ension in es. u,es : Tesed shear srengh. u,csm : Calculaed shear srengh, CSM. u,ec2 : Calculaed shear srengh, EC2. u,flexural = M u / a: Calculaed bending srengh.

5 3.2 Calculaion by he Crack Sliding Model The CSM developed by Zhang [9] is a refinemen of he classical upper bound plasiciy mehod for beams wihou sirrups, see [10]. Similar o he original plasiciy formulaion, CSM operaes wih concree as a rigid plasic maerial obeying he Modified Coulomb failure crierion and he associae flow rule of plasic heory. The menioned refinemen lies in he fac ha a cracking crierion (see below) is inroduced in order o deermine he posiion of he criical shear crack. This crierion supplemens he original concep of energy minimizaion when looking for he mos criical shear yield line. Fig. 3: Shear failure mechanisms in beam wih axial compression. In CSM, a shear failure is assumed o ake place as sliding in a linear diagonal crack characerised by is horizonal projecion x, see Figure 3. The considered shear mechanism involves a displacemen u of par I relaive o par II. Depending on he magniude of he axial ension N, he rae of displacemen u may be verically direced or i may have a componen in he direcion of N as indicaed. By following he procedure for calculaion of he raes of inernal energy dissipaion and exernal work, see [10], he upper bound soluion below is obained from he considered failure mechanism: 2 x σ σ x Φ Φ Φ σ ν fcbh h ν ν fc ν ν fc h ν ν fc , ( a) 2 2 u ( x) = (2) 2 1 x x Φ σ 1 ν fcbh 1 +, ( b) 2 h h ν ν fc 2 Here, b and h are he cross secional dimensions, Φ = A s f y /(bhf c ) is he mechanical degree of longiudinal reinforcemen including boh op and boom rebars, σ = N/bh is he average applied ensile sress. The parameer ν is he so-called effeciveness facor, which is required when applying rigid plasic heory o srucural concree. For crack sliding, Zhang [9] assumed: 0,44 1 ν = f h c ( ρ ) 1, in meers and h f c in MPa (3) Here he longiudinal reinforcemen raio ρ = A s /bh is calculaed on he basis of boh op and boom rebars. According o CSM, he horizonal projecion x of he criical shear crack mus be deermined by inroducing a cracking crierion. Basically, he crierion saes ha formula (2) provides a valid crack sliding soluion if he crack considered exiss. This means ha he load required o develop he crack (here ermed he cracking load cr (x)) mus be less or equal o he load ha is needed o cause sliding failure in he crack. Hence, o deermine he shear srengh from his upper bound model, u (x) mus be minimized wih respec o x and a he same ime be subjeced o wo condiions: x a and cr (x) u (x). The firs condiion is an obvious geomerical resricion and he second condiion ensures ha he crack exiss. For a more comprehensive explanaion as well as deailed discussions of he physical aspecs of he model, see [9] and [10]. Since CSM is based on plasic heory, Zhang [9] also adoped a plasiciy approach o esimae he cracking load. In his approach, he cracking load is simply deermined from a cracking mechanism involving roaion of par I abou he crack ip. By including he conribuion from he ensile normal force o he rae of exernal work in his mechanism, he original expression for he cracking load as a funcion of x may be exended as follow:

6 2 1 x fef bh 1+ N 0,30 2 h 2/3 h cr ( x) =, fef = 0,156 fc a 1 L (4) o h 2 h Here, he effecive ensile srengh f ef of concree is calculaed by insering h in meers and f c in MPa. The simpliciy of formulas (2) and (4) makes he process of finding he shear srengh raher easy. Since formula (2) decreases monoonically wih x while formula (4) increases monoonically, one jus have o solve he equaion cr (x) = u (x) wih respec o x. This can be done numerically. If he deermined x-value saisfied x a, hen his x-value is insered ino formula (2) o calculae he shear srengh. If he soluion of cr (x) = u (x) is larger han a, hen x = a mus be insered ino (2) for calculaion of he shear srengh. Graphical explanaions of he soluion procedure have been given in [9] and [10]. 4. Comparison of Tes Resuls wih Calculaions In Table 1 he calculaed resuls using EC2 and CSM are lised ogeher wih he observed shear srengh. As can be seen, he EC2-formula predics oal loss of shear capaciy for N > 300 kn corresponding o σ = 7,5 MPa. This is obviously no in agreemen wih experimenal observaions. I appears ha he resuls obained from CSM are in beer agreemen wih ess. The average value of es over calculaion is 0,90 wih a sandard deviaion of 0,10. The correlaion has been illusraed in Figure 4. Fig. 4: All ess resuls compared wih calculaions by CSM. Coloured dos mark he ess wih a heoreical lower bending srengh. EC2 CSM Fig. 5: Tes resuls for a/h = 2,25 compared wih CSM using average value of fc = 27,3 MPa. Coloured dos mark he ess wih a heoreical lower bending srengh. In Table 1, he calculaed flexural capaciy has also been shown. These values are based on cross secional ulimae bending momens deermined by he program Response-2000 developed by Benz [12]. I appears ha four of he specimens heoreically had smaller flexural srengh han shear srengh. The values are however close o each oher. The four ess have been emphasised in Figure 4 by coloured dos. Even hough a classical flexural failure mode was no observed in he ess, i should be menioned ha a combined shear-flexural mechanism, i.e. boh roaion and ranslaion in he diagonal crack, may possibly have aken place. Figure 5 depics he ess wih a/h = 2,25. In addiion, τ u /νf c = u /bhνf c versus N/A s f y as deermined by he CSM has also been ploed. The calculaions were performed by assuming f c = 27,3 MPa, which is he average value for he shown ess. Even hough he individual es had a slighly differen compressive srengh han he average srengh, he plo illusraes raher well he endency of boh ess and heory. The shear srengh calculaed from he EC2- formula has also been ploed in he figure. I is clearly seen ha his formula is no applicable for high axial ension. The ess wih a/h = 2,75 shows he same endency and a plo of hese resuls gives a very similar figure. This conclusion can also be drawn from Fig. 4; where all es resuls are ploed.

7 5. Conclusion The influence of high axial ension on he shear srengh of beams wihou sirrups has been sudied. Shear ess combined wih a high axial ensile load have been carried ou. The es resuls indicae ha he shear capaciy remains almos inac even when he axial ension corresponds o ~ 0,20 % srains. The EC2-formula was no developed for cases wih high axial ension even hough i is used for his purpose in pracice also. Comparison wih ess shows ha he EC2-formula is conservaive for high axial ension. Comparison wih ess indicaes ha he CSM seems o work well, even for members subjeced o axial ensile srains up o 0,30 0,40 %. 6. References [1] European Commiee for Sandardizaion. EN , Eurocode 2: Design of Concree Srucures - Par 1-1: General Rules and Rules for Buildings, [2] NARAYANAN, RS, Edior. Eurocode 2 Commenary. The European Concree Plaform ASBL, Brussels, [3] MATTOCK, AH. Diagonal Tension Cracking in Concree Beams wih Axial Force, Journal of he Srucural Division ASCE, 95(9), 1969, pp [4] HADDADIN, MJ, HONG, ST and MATTOCK, AH. Sirrup Effeciveness in Reinforced Concree Beams wih Axial Force, Journal of he Srucural Division ASCE, 97(9), 1971, pp [5] REGAN, PE. Shear in Reinforced Concree An Experimenal Sudy. Technical Noe 45, CIRIA, London, [6] BHIDE, SB and COLLINS, MP. Influence of Axial Tension on he Shear Capaciy of Reinforced Concree Members. ACI Srucural Journal,.86, No.5, Sepember-Ocober 1989, pp [7] JOERGENSEN, HB and FABRIN, LS. Shear Srengh of Non-Shear Reinforced Concree Beams wih Axial Tension. M.Sc. hesis, Deparmen of Indusrial and Civil Engineering, Universiy of Souhern Denmark, [8] NIELSEN, MP. Concree 1 par 1, 2nd Ediion, Technical Universiy of Denmark [9] ZHANG, JP. Diagonal Cracking and Shear Srengh of Reinforced Concree Beams. Magazine of Concree Research, 49(178), 1997, pp [10] NIELSEN, MP and HOANG, LC. Limi Analysis and Concree Plasiciy, CRC Press, 3rd Ediion, [11] JENSEN, UG and HOANG, LC. Shear srengh predicion of circular RC members by he crack sliding model. Magazine of Concree Research,.9, No.61, 2009, pp [12] BENTZ, EC. Secional analysis of reinforced concree members. PhD hesis, Graduae Deparmen of Civil Engineering, Universiy of Torono, Canada, 2000.