Chapter 2. State of the art

Transcription

1 Chapter 2 State of the art Thi chapter trie to contribute to our undertanding of the mechanim of hear trength in reinforced concrete beam with or without hear reinforcement. Conceptual model howing the internal force in a beam are preented, and a hitorical introduction of different approache for hear deign of reinforced concrete beam i made. Some current code propoal are reviewed both for member with and for thoe without web reinforcement. The main characteritic of High-Strength Concrete and how they affect the hear repone are alo dicued. Main attention i focued on the deign of B-region, a defined by Schlaich et al. (1987). Figure 2.1 how the ditribution of D and B region, where D tand for dicontinuity or diturbed, and B tand for beam or Bernoulli. In D region, the ditribution of train i ignificantly nonlinear along the depth and trut-and-tie model are particularly relevant. However, in B region the train ditribution i linear and the repone of the concrete member will be principally due to beam action which implie that the lever arm (z) i contant. The other extreme occur if the tenion in the longitudinal reinforcement remain contant and the lever arm varie (Figure 2.2). Thi occur if the hear flow cannot be tranmitted becaue the teel i unbounded, or if the tranfer of hear flow i prevented by an inclined crack extending from the load to the reaction. In uch a cae the hear i tranferred by arch action. 5

2 Chapter 2 D B B D B D B D B C D D B B z (varie) D D T Figure 2.1: B-region and D-region (Schlaich et al.,1987) Figure 2.2: Arch action in a beam 2.1 Introduction to High-Strength Concrete (HSC) In the pat decade there ha been a rapid growth in interet in high-trength concrete whoe compreive trength, f c, i higher than 50 MPa. Concrete of trength up to 100 MPa can be produced not only in precat plant but alo in the field, with carefully elected but commonly available cement, and, and tone, uing a very low watercement ratio and careful quality control during production. The neceary workability i achieved by high-range water-reducing admixture, the o-called uperplaticier. The main application for high-trength in itu concrete appear to be in offhore tructure, column for tall building, long-pan bridge and other highway tructure. In precat concrete, application are mainly in pretreed element. For intance, the pecified concrete trength for the in itu column of the 58-torey Two Union Square Centre in Seattle (1989) wa 120 MPa (Figure 2.3). Two pedetrian bridge contructed in Barcelona for the 1992 Olympic Game, hown in Figure 2.4, were the firt contruction to be deigned and built uing HSC in Spain. 6

3 State of the art Fig. 2.3: Two Union Square Centre, Seattle Figure 2.4: Montjuïc pedetrian bridge, Barcelona. Extenive experimentation ha greatly improved our undertanding of the fundamental behaviour and baic engineering propertie of the material. While mot of concrete propertie improve a it compreive trength increae, ome of it characteritic require pecial attention. To enure the afety and erviceability of tructural concrete, certain eentially empirical deign procedure and equation, baed on the characteritic of concrete of much lower trength, mut be re-examined. The hear capacity of reinforced high-trength concrete beam i an important iue. Proviion for hear deign are baed mainly on experimentally derived equation. Tet providing the baic data for thee equation were conducted on member whoe concrete trength were mainly below 40 MPa. Thee current approache will be analyed in thi thei. Section 2.4 introduce the main characteritic of the hear trength of high-trength concrete beam. The tre-train behaviour of HSC in uniaxial compreion ha been reported by many reearch centre. Figure 2.5 plot the main difference between the tre-train curve of normal and high-trength concrete, which are: - a more linear tre-train relationhip up to a higher % of the maximum tre - a lightly higher train at the maximum tre - a teeper hape in the decending part of the curve 7

4 Chapter 2 Figure 2.5: Typical tre-train relationhip for high-trength concrete (from FIP/CEB Working Group on HSC, 1990) A illutrated in Figure 2.6, both cement pate and natural rock aggregate are brittle material. The concrete made up of thee material ha an obviou ductile behaviour. Thi apparently paradoxical property can be explained a a reult of the difference in rigidity that normally exit between the cement pate and the aggregate. Thi difference will reult in tre concentration in the contact zone. Conequently, at a certain overall tre level, a ditributed microcrack pattern will begin to form. A the overall tre increae, an increaing part of the applied energy will be conumed a the crack pattern develop. At thi tage, the tre-train curve will tend to deviate from the linear-elatic coure, a hown in the figure. After the ultimate tre level ha been reached, the microcrack pattern will provide an efficient internal reditribution of the tre, and hence a tough failure. a) Normal trength concrete b) High-trength concrete Figure 2.6: Principal tre-train curve for cement pate, aggregate and concrete in compreion (from FIP/CEB Working Group on HSC, 1990) 8

5 State of the art The difference in rigidity between cement pate and aggregate i far le in HSC than in normal trength concrete, a illutrated in Figure 2.6. Conequently, the internal tre-ditribution i more homogeneou. A the tendency toward early microcracking i reduced, the tre-train curve become more linear. A le developed microcrack pattern alo reult in a more udden failure, becaue the ability to reditribute tre i reduced. Although the tenile trength of concrete i neglected in calculating the trength of reinforced and pretreed concrete tructure, it i generally an important apect during the development of cracking, and therefore, for the prediction of deformation and the durability of concrete. Other characteritic uch a bond and development length of reinforcement and the concrete contribution to the hear and torion capacitie are cloely related to the tenile trength of concrete. The tenile trength generally increae along with the compreive trength. However, thi increae i not directly proportional to the compreive trength. 2.2 Shear trength in reinforced concrete beam without web reinforcement Mechanim of hear tranfer The 1973 ASCE-ACI Committee 426 Report identified the following four mechanim of hear tranfer: hear tree in uncracked concrete; interface hear tranfer, often called aggregate interlock or crack friction ; the dowel action of the longitudinal reinforcing bar; and arch action. The 1998 ASCE-ACI Committee 445 Report highlight a new mechanim, reidual tenile tree, which are tranmitted directly acro crack. Opinion vary about the relative importance of each mechanim in the total hear reitance, reulting in different model for member without tranvere reinforcement. The force tranferring hear acro an inclined crack in a beam without tirrup are illutrated in Figure

6 Chapter 2 C 1 V a V ax V ay V cz V d T 2 V ax V ay V a V d V cz C 1 T 2 T 1 Figure 2.7: Internal force in a cracked beam without tirrup (adapted from MacGregor and Bartlett 2000) The hear tree in uncracked concrete are not a very important mechanim for lender member without axial compreion becaue the depth of the compreion zone i relatively mall. On the other hand, at location of maximum moment for le lender beam, much of the hear i reited in the compreion zone, particularly after ignificant yielding of the longitudinal reinforcement. a x a x a y µ σ p ay σ p Figure 2.8: Walraven model of crack friction Shear tranfer in the interface wa due primarily to aggregate interlock, and hence caued by thoe aggregate that protruded from the crack urface and provided reitance againt lip. However, a crack go through the aggregate in lightweight and high-trength concrete yet till have the ability to tranfer hear, the term friction i more appropriate. The four baic parameter involved are the crack interface hear tre, normal tre, crack width, and crack lip. Walraven (1981) made numerou tet and developed a model that conidered the probability that aggregate particle, idealied a phere, would project from the crack interface (Figure 2.8). A lip develop, the matrix phae deform platically, coming into contact with projecting aggregate. The tree in the contact zone are compried of a contant preure, σ p, and a contant hear, µσ p. The geometry of the crack urface i decribed tatitically in term of the 10

7 State of the art aggregate content of the mix and the probabilitie of particle projecting out at different degree. Dowel action i not very ignificant in member without tranvere reinforcement, a the maximum hear in a dowel i limited by the tenile trength of the concrete cover upporting the dowel. Neverthele, it may be ignificant in member with large amount of longitudinal reinforcement, particularly when the longitudinal reinforcement i ditributed in more than one layer. The relative importance of the arch action i direct related to the hear pan-to-depth ratio, a/d (i.e. the ditance from the upport to the load over the effective depth). Beam without tirrup, with an a/d ratio of le than 2.5 develop inclined crack and, after a reditribution of internal force, are able to carry an additional load due in part to arch action. Figure 2.9 how how the failure hear trength of a imply-upported reinforced concrete beam loaded with two-point load change a the hear pan change. For thee erie of beam, teted by Kani (1979), the ultimate hear trength wa reduced by a factor of about 6 a the a/d ratio increaed from 1 to 7. A the beam contained a large amount of longitudinal reinforcement, flexural failure at midpan did not become critical until a hear pan-to-depth ratio of about 7. Thi doctoral diertation focue on member whoe a/d ratio i over a V V a 610 mm V ' bdf c 0.15 V f c = 27 MPa a = 19 mm V 0.10 d = 538 mm b = 155 mm 0.05 A = 2277 mm trut and tie model ectional model a/d Figure2.9: Predicted and oberved trength of a erie of reinforced concrete beam teted by Kani (adapted from Collin and Mitchell 1997) 11

8 Chapter 2 The baic explanation of reidual tenile tree i that when concrete firt crack, mall piece of concrete bridge the crack and continue to tranmit tenile force a long a crack do not exceed mm in width. The application of Fracture Mechanic to hear deign i baed on the premie that reidual tenile tre i the primary mechanim of hear tranfer Hitorical development Prior to cracking, the maximum hear tre at the web can be calculated by uing the traditional theory for homogeneou, elatic and uncracked beam, developed by the 35- year-old Ruian railway engineer D.J. Jourawki in 1856 (Collin, 2001): V Q τ = (2.1) I b where I i the moment of inertia of the cro ection, Q the firt moment about the centroidal axi of the part of the cro-ectional area lying farther from the centroidal axi than the point where the hear tree are being calculated, and b the width of the member where the tree are being calculated. Figure 2.10:Principal compreive tre trajectorie in an uncracked beam and photograph of a cracked reinforced concrete beam. Figure 2.10 how the principal compreive tre trajectorie in an uncracked beam and a photograph of a cracked reinforced concrete beam. Although there i a imilarity between the plane of maximum principal tenile tre and the cracking pattern, they 12

9 State of the art are by no mean exactly alike. The flexural cracking, which precede the inclined cracking, dirupt the elatic tre field to uch an extent that inclined cracking occur at a principal tenile tre, baed on the uncracked ection, of roughly a third of the tenile trength of the concrete (MacGregor and Bartlett 2000). In 1902 Mörch derived the hear tre ditribution for a reinforced concrete beam containing flexural crack. Mörch predicted that hear tre would reach it maximum value at the neutral axi and would then remain contant from the neutral axi down to the flexural teel (Figure 2.11). The value of thi maximum hear tre would be where b w i the web width and z the flexural lever arm. V τ = (2.2) b z w b w d x C C+ C z τ M T = z τ = V b z w T T+ T T T+ T x x Figure 2.11: Shear tre ditribution in a reinforced concrete beam with flexural crack (adapted from Collin and Mitchell, 1997). Mörch recognied that thi wa a implification, a ome of the tranvere force could be reited by an inclination in the main compreion, which would caue the rib of the concrete between flexural crack to bend, producing dowel force in the main teel. T T + T Figure 2.12: Kani comb model for cracked beam ubjected to hear 13

10 Chapter 2 In 1964, Kani attempted a more realitic approach by addreing the problem of the bending of the teeth of the concrete between flexural crack. The concrete between two adjacent flexural crack wa conidered to be analogou to a tooth in a comb (Figure 2.12). The concrete teeth were aumed to be free cantilever fixed in the compreion zone of the beam and loaded by the horizontal hear from bonded reinforcement. Although thi theory did not cover mot of the hear tranfer mechanim, it wa probably the tart of more rational approache. Fenwick and Paulay (1968), working with tooth model, pointed out the ignificance of the force tranferred acro crack in normal beam by crack friction. Taylor (1974), alo evaluating Kani model, found that for normal tet beam the component of hear reitance were: compreion zone hear (20-40%), crack friction (35-50%) and dowel action (15-25%). Hamadi and Regan (1980), baed on extenive experimental work on interface hear, publihed an analyi of a tooth model. It wa aumed that the crack were vertical and that their pacing wa equal to half the effective depth of a particular beam. Reineck (1991) further developed the tooth model, taking all the hear tranfer mechanim into account, carrying out a full nonlinear calculation including compatibility. Reineck (1991), baed on hi mechanical model, derived an explicit formula for the ultimate hear force, which matched with the reult of the tet a well a with thoe of many empirical formula. bd V ' f c (MPa) mm max. aggregate PCA Tet of Model Air Force Warehoue 2.5 mm max. aggregate d d V calculated here 12 d d 25 mm max. aggregate ize ρ l = Air Force f c = 24 MPa Warehoue f y = 386 MPa d (mm) Figure 2.13: Influence of member depth and maximum aggregate ize on hear tre at failure (tet by Shioya et al.1989) d 14

11 State of the art Kani raied the ize effect ubject in 1967, when he demontrated that a the depth of the beam increae the hear tre at failure decreae. A the depth of the beam increae, the crack width at point above the main reinforcement tend to increae. Some author think that thi lead to a reduction in the aggregate interlock acro the crack, reulting in earlier inclined cracking. Collin and Kuchma (1999) demontrated that the ize effect diappear when beam without tirrup contain well-ditributed longitudinal reinforcement. Other author (Bazant and Kim, 1984) believe that the mot important conequence of wider crack i the reduction in reidual tenile tree. Figure 2.13 how the reult of the tet performed by Shioya et al. (1989). The influence of the concrete compreive trength on the ize effect will be dicued in The application of imple trut-and-tie model, which have their theoretical bai in the lower-bound theorem of platicity, require a minimum amount of ditributed reinforcement in all direction to enure ufficient ductility in order for internal tree to be reditributed after cracking. However, it i poible to extend thi imple trut-andtie model to member without web reinforcement by uing a clearly different approach. Marti (1980) extended the platicity approach by uing a Coulomb-Mohr yield criterion for concrete that include tenile tree. In 1987, Schlaich uggeted a refined trutand-tie approach that include concrete tenion tie. Reineck howed that uch tru model comply with the tooth model he had propoed. Empirically derived equation have been very important in the development of procedure ued for deigning member without tranvere reinforcement. The implet lower-bound average hear tre at diagonal cracking i given by the equation V f c c =τ = (2.3) bd 6 Thi well-known ACI equation, bai for the Spanih EH-91 hear proviion, i a reaonable lower bound for maller lender beam that are not ubjected to axial load and have at leat 1% longitudinal reinforcement (ACI-ASCE Committee 445, 1998). 15

12 Chapter 2 However, it may be unconervative for lowly-reinforced member and high-trength concrete member. The CEB-FIP Model Code (1990) ugget a more ophiticated empirical formula baed on Zutty (1968, 1971) equation and adding an extra term to account for the ize effect (equation 2.4). It hould be noted that the formula implicitly include the concrete afety factor. To diregard thi factor, we hould ue 0.15 a the contant rather than Vc b d 1 / d 3 = ( 100ρ ) 1 / fck 0.15σ cd d a (2.4) where σ cd equal N d /A c, N d being the factored axial force that include the pretre (tenile poitive) force and A c, the cro ectional area of the concrete Zutty equation took into account the influence of the compreion trength of the concrete and the longitudinal reinforcement ratio. When the teel ratio i mall, flexural crack extend higher into the beam and open wider than would be the cae with large value of ρ w. The MC-90 equation take the influence of compreion force a a factor. However, member without hear reinforcement ubjected to large axial compreion and hear may fail in a very brittle manner at the firt intance of diagonal cracking (Gupta and Collin, 1993). A a reult, a conervative approach hould be ued for thoe member. Gatebled and May (2001) recently developed a fracture mechanic model for the flexural-hear failure of reinforced concrete beam without tirrup. They aumed that that the ultimate hear load i reached when a plitting crack at the level of the longitudinal reinforcement tart to propagate. If we adopt the format of the CEB-FIP formula, their equation become 1 / 3 Vc d = 0.15 ( 100ρ ) 1 / ( ) 2 / 1 ρ fc b d d a (2.5) 16

13 State of the art It i worthy a mention that the analytical and the empirical formula compare very well (Gatebled et al. 2001). However, Gatebled equation give more importance to the ize effect than the CEB-FIP formula doe. Other different fracture mechanic model have been propoed to account for the fact that a peak tenile tre i near the tip of a crack and a reduced tenile tre (oftening) i located in the crack zone. Thi approach offer a poible explanation for the ize effect in hear. Two well known model are the fictitiou crack model (Hillerborg et al. 1976), and the crack band model (Bazant and Oh,1983). The Modified Compreion Field Theory (MCFT, Vecchio and Collin 1986) i a general model for the load-deformation behaviour of two-dimenional cracked reinforced concrete ubjected to hear. The MCFT, a it will be referred to later in thi chapter, i formulated in term of average tree and require an additional check to enure that the load reited by the average tree can be tranmitted acro the crack. For member without tranvere reinforcement, the local tree at a crack alway control the capacity of the member, and the average tre calculation i ued only for etimating the inclination of the critical diagonal crack. ASCE-ACI Committee 445 (1998) emphaied that, although the refined tooth model and the modified compreion field theory take different approache to the problem, the end reult of thee two method i very imilar for member without tranvere reinforcement. Both method conider that the ability of diagonal crack to tranfer interface hear tre play an important role in the determination of the hear trength of member without tranvere reinforcement Code review The Spanih EHE-99 Code The EHE code of practice adopted the CM-90 formula with a minor variation: 1 / 3 [ 0.12ξ ( 100ρ f ) 0.15σ ' ] b d Vc ck cd 0 = (2.6) where, f ck i in MPa and f ck 60 MPa, 17

14 Chapter 2 ξ = where d i in mm, d Al ρ l = 0. 02, b d A l b 0 w i the area of the anchored tenile reinforcement, i the width of the cro-ection (in mm), σ cd = N d /A c, N d being the factored axial force, including the pretre (tenile poitive) force and A c, the cro ectional area of concrete, V Rd i in Newton. The concrete afety factor i alo factored into equation 2.6. The contant in that equation (0.12) hould be changed to 0.15 to eliminate the afety factor from the equation. Eurocode 2: April 2002 Final Draft The final verion of the new draft of Eurocode 2 preent a different hear procedure than it predeceor. It i baed, with ome variation, on the MC-90 equation. The deign value for the hear reitance in non-pretreed member not requiring deign hear reinforcement i given by: V Rd,c 0.18 = k l ck σ γ c 3 ( 100ρ f ) 1 / b d cp w (2.7) with a minimum of 3 / 2 2 [ 0.035k f ] 1 / b d VRd,min = ck w (2.8) where, f ck i in MPa and f ck 100 MPa, 200 k = , where d i in mm, d Al ρ l = 0. 02, b d A l w i the area of the anchored tenile reinforcement, 18

15 State of the art b w σ cp V Rd i the mallet width of the cro-ection in the tenile area (in mm), = N Ed /A c < 0.2 f cd (MPa). N Ed i the axial force in the cro-ection due to loading or pretreing in Newton (N Ed > 0 for compreion). The influence of impoed deformation on N E can be ignored. A c i the area of the concrete cro ection (mm 2 ). i in Newton. AASHTO LRFD 2000 The AASHTO-LRFD hear deign procedure i baed on the modified compreion field theory. The nominal hear reitance for a non-pretreed member without hear reinforcement i given by: V c = β f ' b d (2.9) c v v the value of β and θ, depend on the equivalent crack pacing parameter, xe, where 35 = (2.10) a + 16 xe x where a i the maximum aggregate ize, and x i the crack pacing parameter a defined in Figure The longitudinal train in the web, ε x, can be derived from the longitudinal train in the flexural tenion flange, ε t, where M f + V f φ pv p + 0.5N f Ap f p0 dv ε t = (2.11) E A + E p Ap and f p0 can be taken to be 0.7f pu at typical level of pretre. For member without tirrup ε x can be taken to be ε t (ee Figure 2.14). 19

16 Chapter 2 z β θ β θ β θ β θ β θ β θ β θ β θ ε x Figure 2.14: Value of β and θ for ection without hear reinforcement. ACI Code The ACI code of practice preent two different procedure for calculating the failure hear trength for concrete beam without hear reinforcement. The implified method, equation 11-3, i a follow: 20

17 State of the art V c f ' c bwd 6 = (2.12) The econd procedure, equation 11-5, applie for thoe member whoe a/d 1.4: V c d bwd = f ' c + 120ρ w 0.3 f ' c bwd (2.13) a 7 where f c < 70 MPa, and all the other variable are a defined previouly. 2.3 Member with web reinforcement Force in member with web reinforcement Ideally, the purpoe of web reinforcement i to enure that hear failure doe not occur and that the full flexural capacity can be ued. Prior to inclined cracking, the train in the tirrup i equal to the correponding train in the concrete and, therefore, the tre in the tirrup prior to inclined cracking will be relatively mall. Stirrup do not prevent inclined crack from forming a they come into play only after crack have formed. A C 1 V cz V a V ay V ax T 2 V d V ax V ay V a V d V V V cz C 1 T 2 T 1 A Figure 2.15: Internal force in a cracked beam with tirrup (adapted from MacGregor et al. 2000). Figure 2.15 how the force in a beam with tirrup after the development of an inclined crack. The hear tranferred by tenion in the tirrup i V. Since V doe not 21

18 Chapter 2 diappear when the crack open, to verify equilibrium there will alway be a compreion force, C 1, and a hear force, V cz, acting on the part of the beam below the crack. Thu, T 2 will be le than T 1 and the difference will depend on the amount of web reinforcement. However, the force T 2 will be larger than T = M/z baed on the moment at A-A Hitorical development In the early 20th century, tru model were ued a conceptual tool in the analyi and deign of reinforced concrete beam. Ritter (1899) and Mörch (1902) potulated independently that after a reinforced concrete beam crack due to diagonal tenion tree, it can ideally be thought of a a parallel chord tru with compreion diagonal inclined at 45 with repect to the longitudinal axi of the beam. Several year later, Mörch (1920,1922) introduced the ue of tru model for torion. In thee tru model, in which the contribution of the concrete in tenion i neglected, the diagonal compreive concrete tree puh apart the top and bottom face of the beam, while the tenile tree in the tirrup pull them together (Figure 2.16 and 2.17). Equilibrium require thee two effect to be equal. According to the 45 tru model, the hear capacity i reached when the tirrup yield and will correpond to a hear tre of A f = f y (2.14) b v y τ = ρ v w where A v i the area of the tranvere reinforcement, the pacing of the tranvere reinforcement, f y the teel yielding tre and b w the web width. Figure 2.16: Ritter tru model. 22

19 State of the art Figure 2.17: Mörch tru analogy. In the United State an extra term wa added to improve correlation with tet reult, but it wa never explained in phyical term. It ha generally been taken to be the trength of a imilar beam without tirrup but, in light of the very different ultimate load behaviour in each cae, thi equation i phyically mileading (Regan, 1993). From 1921 to 1951 each new edition provided omewhat le conervative deign procedure (ASCE-ACI Committee 445), even though Talbot (1909) had pointed out that the value of the hear tre at failure varied with the amount of reinforcement, the relative length of the beam and the quality and the trength of the concrete in addition to other factor that affect the tiffne of the beam. However, ACI only pecified that web reinforcement mut be provided for the exce hear if the hear tre at ervice load exceeded 0.03 f c. Figure 2.18: Wilkin Air Force Depot in Shelby, Ohio. 23

20 Chapter 2 The Augut 1955 brittle hear failure of beam in a warehoue at Wilkin Air Force Depot in Shelby, Ohio, (Eltner et al. 1957, Anderon 1957) brought traditional hear deign procedure into quetion. The collape wa caued by the hear failure of 914 mm deep beam that did not contain tirrup at the location of failure and only had 0.45 percent of longitudinal reinforcement. The beam failed at a hear tre of le than 0.5 MPa, a low working tre compared with that obtained uing the ACI proviion of that day. Experiment (Eltner et al. 1957) conducted at the Portland Cement Aociation on 305 mm deep beam indicated that the beam could reit a hear tre of about 1.0 MPa prior to failure. However, application of an axial tenile tre of about 1.4 MPa reduced the hear capacity by about 50 percent. It wa concluded that tenile tree caued by the retraint of hrinkage and thermal movement were the reaon for the beam failure at uch low hear tree. However, a can be een in Figure 2.13, the ize effect, which wa the real reaon for the failure, wa not taken into conideration. Shear/compreion theorie tarted to be developed in the The idea behind them i that beam failure i caued by cruhing of the concrete compreion zone, the depth of which ha been reduced by a hear crack. The limiting compreive tree may alo be reduced by the effect of hear in the compreion zone. In 1958, Walther propoed what wa probably the bet known of thee theorie. However, the complexity of thi theory reulted in the impoibility of finding an explicit olution. The early work by Ritter and Mörch received new impetu in the period during the three decade from 1960 to In Stuttgart, Leonhardt and Walther (1961) carried out an extenive experimental campaign on beam failing in hear and developed a model that combined the beam and the arch effect. It wa hown that thee two reitant mechanim interact and that the relative importance of each one varie depending on the lenderne of the beam. Attention wa alo focued on tru model with diagonal having a variable angle of inclination ( 2.3.3) a a viable model for hear and torion in reinforced and pretreed concrete beam (Kupfer, 1964). Kupfer provided a olution for the inclination of the diagonal crack by conidering linearly elatic member and ignoring the concrete tenile trength. Further development of platicity theorie extended the applicability of the model to non-yielding domain (Nielen and Braetrup, 1975). Schlaich et al. (1987) 24

21 State of the art extended the tru model for beam with uniformly inclined diagonal. Thi approach i particularly relevant in D-region where the ditribution of train i ignificantly nonlinear along the depth. Modified tru model are ued in more recent deign code. For example, ACI Building Code till add a concrete contribution term to the hear reinforcement capacity obtained, auming a 45 tru. Another procedure involve the ue of a tru with a variable angle of inclination for the diagonal. The inclination of the tru diagonal i allowed to deviate from 45 within certain limit baed on the theory of platicity. The CEB-FIP model code for concrete tructure (1978), and many code of practice derived from it, adopted a combination of the variable-angle tru and concrete contribution. Mitchell and Collin (1974) developed the diagonal Compreion Field Theory for member ubjected to pure torion. The Compreion Field Theory (Collin 1978) and the Modified Compreion Field Theory (MCFT, Vecchio and Collin 1986) extended the firt theory, dating from 1974, to hear. The MCFT (Figure 2.19) i a further enhancement of the CFT that account for the influence of the tenile tree in the cracked concrete. They take into account the overall load/deformation repone of element in which the reinforcement act in uniaxial tenion and the concrete work in biaxial tenion/compreion. The principal tree and train in the concrete are aumed to be coincident. The equilibrium equation, the compatibility relationhip, the reinforcement tre-train relationhip, and the tre-train relationhip for the cracked concrete in compreion and tenion enable one to determine the average tree, the average train, and the angle θ for any load level up to failure. Failure of the reinforced concrete element may not be governed by average tree, but rather by local tree that occur at a crack. Thi o-called crack check i a critical part of the MCFT and the theorie derived from it. The crack check involve limiting the average principal tenile tre in the concrete to a maximum allowable value determined by conidering the teel tre at a crack and the ability of the crack urface to reit hear tree. 25

22 Chapter 2 a) Average tree b) Stree at a crack Figure 2.19: MCFT: average tree and tree at a crack Hu and hi colleague from the Univerity of Houton (Berlabi and Hu 1994, 1995) preented the Rotating-Angle Softened-Tru Model (RA-STM). Like the MFCT, thi method aume that the inclination of the principal tre direction, θ, coincide with the principal train direction. For typical element, θ will decreae a the hear i increaed, hence the name rotating angle. Pang and Hu (1995) limited the applicability of the rotating-angle model to ituation in which the rotating angle doe not deviate from the fixed angle by more than 12. Outide thi range they recommend the ue of a fixed angle model where it i aumed (Pang and Hu, 1996) that hear crack are parallel to the principal direction of compreive tre a defined by the applied load. The Diturbed Stre Field Model (DSFM), developed by Vecchio (Vecchio 2000, Vecchio 2001) a an extenion of the MCFT, explicitly incorporate rigid lipping along crack urface into the compatibility relation for the element. Thi allow for a divergence of the angle of inclination of average principal tre and apparent average principal train in the concrete. The model repreent crack a gradually rotating, but typically lagging behind the reorientation of the principal train. Vecchio et al. (2001) conclude that the corroboration tudie for the DSFM alo reaffirmed the trength of the MCFT a a imple model providing good accuracy over a wide range of condition. Although the MCFT aumption of coaxiality of tree and train i hown to have ome fault [..] it influence on predicted behaviour i minor in mot cae. 26

23 State of the art Tru model Tru model are provide an excellent conceptual model for howing the force that exit in a cracked concrete beam. The 45 Mörch model can be made more accurate by accounting for the fact that θ i typically le than 45. Figure 2.19 ummarie the equilibrium condition for the variable-angle tru-model. The required magnitude for the principal compreive tree, f 2, can be derived from the free-body diagram hown in Figure 2.20: f V = ( tanθ cotθ ) (2.15) b z 2 + w M = N v z co θ z f 2 A v f v in θ 0.5 N v Figure 2.20: Equilibrium condition for a variable-angle tru (adapted from Collin and Mitchell, 1991). The tenile force in the longitudinal reinforcement due to hear force i N v = V cotθ (2.16) The compreive tree, f 2, in the web tend to caue the top and bottom flange to eparate. To prevent thi from happening, the tenile force of the tirrup mut be equal to the vertical component of the compreion force in the web: A f V tanθ z v y = (2.17) The equilibrium equation hown above are inufficient, however, for gauging the tree caued by a given hear in a beam. There are four unknown (i.e., the principal 27

24 Chapter 2 compreive tre, the tenile force in the longitudinal reinforcement, the tre in the tirrup and the inclination, θ, of the principal compreive tree). In the traditional tru model the failure hear trength of a beam i determined uing an equilibrium equation by auming that at failure the tirrup yield and that θ = 45. Otherwie, it can be aumed a compreive tre, f 2, in the concrete at failure and then find V and θ. Alternatively, it could be aumed that, at failure, both the longitudinal reinforcement and the tirrup yield, and then determine V and θ from thi. Thee approache, which conider the mechanim of failure, are referred to a platicity method. Nielen (1984) ummaried thee method. The EHE code of practice aume that a concrete contribution, V c, can be added to the teel contribution. Thi contribution i taken to be approximately 85% of the cracking hear trength of a beam without tirrup. It hould be emphaied that taking V c to be equal to the hear at inclined cracking i approximately true if it i aumed that the horizontal projection of the inclined crack i d (MacGregor et. al, 2000). If a flatter crack i ued, o that z cotθ i greater than d, a maller value of V c mut be ued. For value of θ approaching 30, ued in the platic tru model, V c approache zero, a repreented in the EHE Code by the contant β, by which the concrete contribution, V c, i multiplied: V = (2.18) V + βv c For non-pretreed member without axial force β equal 1 if θ i taken to be 45. If cot θ i aumed to be equal to 2 (thu θ 26.6 ), then β = 0. From a tru model it i poible to identify the different hear failure mode that may caue the failure of a beam: Failure due to the tirrup yielding. Auming that all the tirrup croing a crack yield at failure, the hear reited by the tirrup i 28

25 State of the art A f d v y V = (2.19) However, tirrup are unable to reit hear unle they are croed by an inclined crack. It i poible for a 45 crack to cro the web without interecting a tirrup if the tirrup pacing exceed d. Therefore, the maximum tirrup pacing hould be d or le. In a wide beam with tirrup around the perimeter, the diagonal compreive tree in the web tend to be upported by the longitudinal bar in the corner, a hown in Figure The ituation i improved if there are more than two tirrup leg. The CEB-FIP 1990 ugget that the maximum tranvere pacing of the tirrup leg hould be limited to the maller of 2d/3 or 800 mm. Serna et al. (2000) concluded that for wide beam the ue of two leg tirrup hould be banned and that the maximum ditance in the tranvere direction between leg hould be limited to d. Figure 2.21: Flow of the diagonal compreive force in cro-ection of wide beam. Adapted from MacGregor et al. (2000). Equation 2.19 i baed on the aumption that the tirrup will yield before failure. Thi i true only if the tirrup are well anchored. Becaue the available development length between the inclined crack and the end of the tirrup can be very hort, the ue of mall diameter tirrup a well a hoop with the adequate geometry i recommended. Moreover, wide crack in beam are unightly and may allow water to penetrate the beam, poibly cauing the tirrup to corrode. Crack width i maller with very cloelypaced mall-diameter tirrup than with widely-paced large-diameter tirrup. The ue of horizontal teel ditributed near the face of beam web i alo effective in reducing crack width. Some code, uch a the Canada CSA-94, attempt to guard againt exceive crack width by limiting the maximum hear that can be tranmitted by tirrup to V,max = 0.8φ f ' b d (2.20) 29 c c w

26 Chapter 2 where φ c = 0.60 i the concrete afety factor. Shear failure due to cruhing of the web. A indicated earlier and hown in equation 2.15, compreion tree exit in the web of a beam. In very thin-walled beam, thee may lead to cruhing of the web. In predicting the hear trength of beam uing variable-angle tru model, it i neceary to ue an effective concrete compreive trength maller in value than the cylinder cruhing tre. A value of 0.6f c i frequently recommended. Shear failure initiated by failure of the tenion chord. The longitudinal component of the diagonal compreive force mut be counteracted by an equal tenile force in the longitudinal reinforcement. Thi tenion increae may caue the longitudinal reinforcement to yield, producing the failure of the beam. The tru analogy how that the force in the longitudinal tenile reinforcement at a given point in the hear pan i a function of the moment at a ection located approximately d v cotθ cloer to the nearet ection of maximum moment Modified Compreion Field Theory Mörch (1922) tated that it wa abolutely impoible to mathematically determine the lope of the econdary inclined crack to deign the tirrup. The German engineer Wagner (1929), however, olved an analogou problem when dealing with the potbuckling hear reitance of thin-webbed metal girder. Wagner aumed that after the thin metal kin buckled, it could continue to carry hear by a field of diagonal tenion, uppoing that it wa tiffened by tranvere frame and longitudinal tringer. He aumed that the angle of inclination of the diagonal tenile tree in the buckled thin metal kin would coincide with the angle of inclination of the principal tenile train a determined from the compatibility of the deformation of the kin, the tranvere frame and the longitudinal tringer. The compreion field approache alo determine the angle θ by conidering the compatibility of the deformation of the tranvere reinforcement, the longitudinal 30

27 State of the art reinforcement, and the diagonally treed concrete. Therefore, thee method atify equilibrium, train compatibility and tre-train relationhip. The firt method for determining θ that wa applicable over the full loading range and baed on Wagner procedure wa developed for member in torion by Mitchell and Collin (1974). Further development led to the Modified Compreion Field Theory (Vecchio and Collin, 1986). The MCFT i a general model for the load-deformation behaviour of two-dimenional cracked reinforced concrete ubjected to hear. A dicued earlier, it model concrete conidering concrete tree in the principal direction ummed with reinforcing tree aumed to be only axial. The concrete tre-train behaviour in compreion and tenion wa derived originally from tet performed by Vecchio (Vecchio and Collin, 1982). The key aumption the MFCT ue to implify i that the principal train direction coincide with the principal tre direction. Thi aumption i confirmed by experimental meaurement, which how that the principal direction of tre and train are parallel within ±10º. Concrete trut are alo at a hallower angle than crack, and the compreive tre field mut be tranferred acro the crack, which caue the concrete trength to be reduced from it uncracked tate and inducing hear tre acro the crack face. Thi produce tenile tree in the cracked concrete. Local tree in both the concrete and the reinforcement are recognied to vary from point to point in the cracked concrete, with high reinforcement tree but low concrete tenile tree taking place at the location of the crack. In the MCFT the compatibility condition relating the train in the cracked concrete with the train in the reinforcement are expreed in term of average train, where the train are meaured over bae length that are greater than the crack pacing. The equilibrium condition, which relate the concrete tree and the reinforcement tree to the applied load, are alo expreed in term of average tree. 31

28 Chapter 2 Similarly, the train ued for the tre-train relationhip are average train, that i, they conider together the combined effect of local train at crack, train between crack, bond-lip, and crack lip. The calculated tree are alo average tree in that they implicitly encompa the tree between crack, tree at crack, interface hear on crack and dowel action. In thi model, the cracked concrete in reinforced concrete i treated a a new material with empirically defined tre-train behaviour. Thi behaviour can differ from the traditional tre-train curve of a cylinder, for example. The equilibrium equation, the compatibility relationhip, the reinforcement tretrain relationhip, and the tre-train relationhip for the cracked concrete in compreion and tenion enable the average tree, the average train, and the angle θ to be determined for any load level up to failure. Failure of reinforced concrete element may be governed not by average tree, but rather by the local tree occurring at a crack. A o-called crack check i a critical part of the MCFT and the theorie derived from it. The crack check involve limiting the average principal tenile tre in the concrete to a maximum allowable value determined by conidering the teel tre at the crack and the ability of the crack urface to reit hear tree. f y ν 0.5 x f 2 ρ v f y Shear ε y ε 2 θ ν f x ν 2 2θ cr 1 ε 1 ε y ε x 2 ε 2 θ θ 2θ ε x Normal 0.5 γ m ρ x f x f 1 ε 1 Fig. 2.22: Equilibrium in term of average tree Fig 2.23: Compatibility in term of average train Figure 2.22 i ued to etablih the equation of equilibrium between crack. Shear in the ection i reited by the diagonal compreive tree, f 2, together with the diagonal tenile tree, f 1. The tenile tree vary from 0 at the crack to a maximum between crack. A ha been mentioned, the average value i ued in the equilibrium formula. ρ f = f + ν tanθ f (2.21) y y y 1 32

29 State of the art ρ f = f + ν cotθ f (2.22) f x x x ( tanθ + cot ) 1 2 = θ f 1 ν (2.23) The compatibility equation for the average concrete train are etablihed uing the geometrical tranformation repreented by Mohr Circle of Strain a hown in Figure ε ε x y 2 2 ( ε1 tan θ + ε 2 )/ ( 1 + tan θ ) 2 2 ( ε + ε tan θ )/ ( 1 + tan θ ) = (2.24) = (2.25) γ tan 2 1 xy 2 ( ε ε )/ tanθ = (2.26) 2 x 2 ( ε ε )/ ( ε ε ) θ = (2.27) x 2 y 2 The reinforcement tre-train relationhip i a typical bilinear diagram: f f x y = E ε f (2.28) x y x yield = E ε f (2.29) y yield The concrete web act not only in compreion in direction 2, but alo in tenion in direction 1. Therefore, the following average tre-train relationhip, baed on Vecchio experiment (Vecchio and Collin, 1982), are adopted: f 2 f ' c ε 2 = ε1 ε' c f f ε 2 ε' c 2 (2.30) = cr ε (2.31) 1 where f cr i the cracking trength of concrete. Figure 2.24 and 2.25 repreent the above equation. 33

30 Chapter f c f 1 f cr ε c f 2 ' c f Repone of Cylinder eq ε 1 /ε 2 = 5 ε 1 = 6.0x10-3 eq eq ε 2 (x10-3 ) Fig. 2.24: Compreive tre-train relationhip for cracked concrete ε cr ε 1 (at crack lip) Fig. 2.25: Average tre-train relationhip for concrete in tenion ε 1 In checking the condition at a crack, the actual complex crack pattern i idealied a a erie of parallel crack, all occurring at an angle θ to the longitudinal reinforcement and paced a ditance θ apart. The reinforcement tree at a crack, deduced from Figure 2.26, can be determined by the equation ρ f = f + ν cotθ ν cotθ (2.32) x xcr x + ρ f y ycr ci = f + ν tanθ ν tanθ (2.33) y ci The ability of the crack interface to tranmit the hear tre, ν ci, depend on the crack width, ω. The limiting value of ν ci propoed by Vecchio and Collin i ' 0.18 fc ν ci (2.34) 24ω a + 16 where a i the maximum aggregate ize in mm. Thi equation, baed on Walraven (1981) experiment, wa performed on variou concrete whoe cube trength were 13, 37, and 59 MPa. Neverthele, a the aggregate may fracture for high f c, and for low f c fracture goe around the aggregate, thi formula will require further invetigation (Duthinh et al., 1996),. The above formula require an etimation of the crack width, taken to be the crack pacing multiplied by the principal tenile train, ε 1 : 34

31 State of the art f ycr ρ v f ycr ν ci y ν θ f xcr ν 2θ cr x 2θ 1 ν ci c x x dbx c c v d by ν ρ x f xcr Fig. 2.26: Equilibrium in term of local tree at a crack Fig. 2.27: Parameter influencing crack pacing (Collin and Mitchell, 1997) mθ ω = ε (2.35) mx 1 mθ 1 = (2.36) inθ coθ + mv Finally, crack pacing, mx and mv are etimated uing the formula given by the CEB- FIP Model Code (1990) mx my 2 c 0.25k d x bx = x (2.37a) 10 ρ x dby = 2 c y k1 (2.37b) 10 ρ y where d b = bar diameter, c = ditance to reinforcement, = bar pacing, ρ y = A y /(b w ), ρ x = A x /A c, and k 1 = 0.40 for deformed bar or 0.8 for plain bar.. At high load, the average train of the tirrup, ε y, will typically exceed the yield train of the reinforcement. In thi ituation, both f y in equation 2.21 and f ycr in 2.33 will equal the yield tre in the tirrup. Equating the right-hand ide of thee two equation and ubtituting for ν ci from equation 2.34 give 35

32 Chapter 2 f ' fc tanθ 24ω a + 16 (2.38) Although in thi chapter the full analytical model ha been decribed, there are few implified method baed on the MCFT that have been adopted in a number of code (Canada, Norway, and in the AASHTO LRFD) Tru Model v. Modified Compreion Field Theory The Modified Compreion Field Theory can be explained a a tru model in which the hear trength i the um of the teel and concrete contribution. The main difference from a claic tru model with concrete contribution i that the concrete contribution in the MCFT i the vertical component of the hear tre tranferred acro the crack, υ ci (Figure 2.28), and not the diagonal cracking trength: A v f y A v f y f 1 υ ci θ θ Figure 2.28: Average tree and tree at a crack. V = V c + V (2.39) Av V = f ydv cot gθ (2.40) V c = υ b d (2.41) ci w v Figure 2.28 how a ection cut halfway between two crack. The average tenile tre tranvere to the trut i f 1, which give rie to a vertical force equal to f 1 b w d v cotgθ. A the two et of tree hown in Figure 1.b and 1.c mut equilibrate the ame vertical hear in both cae, we ee that 36

33 State of the art thu, V = υ b d f b d cot gθ (2.42) c ci w v = 1 w v f 1 = υ tanθ ci (2.43) Therefore, after the tirrup yield the beam will not collape if the hear friction increae and the angle θ decreae. The failure of the beam i governed by the cruhing of the compreion trut between the crack, or by the crack lip. The tre, f 2, in a compreion trut i given by f 1 ( tanθ cotθ ) 2 f υ + = (2.44) The concrete i ubjected to normal tenile tree, and a the principal tenile train, ε 1, increae, the maximum compreive trength decreae. Therefor, after the tirrup yield, the hear trength of a concrete beam can be increaed. Generally, for a normal trength concrete beam, the cruhing of the concrete govern the beam failure. In a high-trength concrete beam, the trut are able to carry more compreive tre, and failure i mot likely initiated by the crack lip. Finally, it i important to highlight the main difference between the tru model and the MCFT concrete contribution: - The tru model concrete contribution i conidered equal to the hear trength of a imilar beam without hear reinforcement. The MCFT take into account a concrete contribution baed on the actual collape mechanim of a reinforced concrete beam. - The tru model concrete contribution doe not vary with the amount of tranvere reinforcement. The MCFT concrete contribution depend on the crack width. The more hear reinforcement, the leer the crack width, and the greater the concrete contribution will be. 37

34 Chapter Code review Spanih Code EHE-99 The EHE code of practice aume that a concrete contribution, V c, can be added to the teel contribution. Hence V = V + βv c (2.45) and / 3 = ( 100 fck ) 0.15 ' cd b d d ρ σ (2.46) Vc 0 where all the parameter have the ame meaning a for member without web reinforcement (equation 2.6). The teel contribution i given by the following equation: Aw V = z f ywd cotθ (2.47) where cotθ i compreed between 0.5 and 2. For non-pretreed member without axial force β equal 1 if θ i taken to be 45. If cotθ i aumed to be equal to 2 (thu, θ 26.6 ), then β = 0. Eurocode 2: April 2002 Final Draft For member requiring deign hear reinforcement, their deign i baed on a tru model. For member with vertical hear reinforcement, the hear reitance, V Rd,, hould be taken to be the leer, either: Aw VRd, = z f ywd cotθ (2.48) or V = α b zν f / cotθ tanθ (2.49) ( ) Rd,max c w cd + The recommended limiting value for cotθ are given by the expreion 1 cotθ 2.5 (2.50) where A w i the cro-ectional area of the hear reinforcement, 38

35 State of the art f ywd ν α c i the pacing of the tirrup, i the yield trength of the hear reinforcement, may be taken to be 0.6 for f ck 60 MPa, and 0.9-f ck /200 for high-trength concrete beam and = 1, for non-pretreed tructure. AASHTO LFRD 2000 The hear trength of a reinforced concrete ection i expreed a follow V A f v y = β f c' bvd v dv cotθ (2.51) n + The value of β and θ lited in Figure 2.29 are baed on a calculation of the tre that can be tranmitted acro diagonally-cracked concrete containing at leat the minimum amount of tranvere reinforcement required for crack control. The hear tre in Figure 2.29 can be defined a V V n p ν = (2.52) b v d v For member with tirrup ε x can be taken to be 0.5 ε t, a i demontrated in Figure ε t i calculated uing equation ACI Code For member requiring deign hear reinforcement, their deign i baed on a 45º tru model plu a concrete contribution. Hence and V = V c + V (2.53) A w V Rd, = z f ywd (2.54) and the concrete contribution i equal to the failure hear trength of an identical beam without web reinforcement, given by equation

36 Chapter 2 ν f φ f' c c β θ β θ β θ β θ β θ β θ β θ β θ ε x Figure 2.29: Value of β and θ for ection containing at leat the minimum amount of hear reinforcement. 2.4 Shear trength in high-trength concrete beam Introduction A far a hear trength i concerned, Duthinh et al. (1996) ae that high-trength concrete preent u four main challenge: - Current code proviion for hear trength deign rely on empirical rule whoe databae i largely below 40 MPa. New deign rule would have to rely on either rational method or on tet that cover a higher range of trength. Much progre ha been made in the lat 25 year on rational method for hear deign and there i hope that the rule can be made more undertandable from firt principle of mechanic, uch a ha been achieved 40

37 State of the art for flexure. Moreover, it i likely that the rule can be made imple enough that they will gain adoption by the deign community in the not-too-ditant future. - Shear failure urface in high-trength concrete member are moother than in normal-trength concrete member, with crack propagating through coare aggregate particle rather than around them (Figure 2.30). Since one of the hear tranfer mechanim acro crack i by aggregate interlock, thi mechanim need to be re-examined for high-trength concrete. Tet reult to date indicate that hear friction in HSC can be a low a 35% of that in NSC (Walraven, 1995). Figure 2.30: Crack in high-trength concrete - In the cracked web of a beam under hear, the portion of concrete between crack act a compreion trut that are alo ubjected to tranvere tenion, which reduce their compreion capacity. Modelling of thi oftening behaviour i baed on tet. Softening how a dependence on concrete trength that need to be extended to HSC. However, tet reult to date indicate no marked difference in biaxial tenion-compreion behaviour between HSC and NSC. - Minimum hear reinforcement mut prevent udden hear failure on the formation of firt diagonal tenion crack and, in addition, mut adequately control the diagonal tenion crack at ervice load level. To prevent a brittle 41