Flexural to shear and crushing failure transitions in RC beams by the bridged crack model

Transcription

1 exura to shear and crushing faiure transitions in RC beams by the bridged crack mode A. Carpinteri & G. Ventura Poitecnico di Torino, Torino, Itay J. R. Carmona Universidad de Castia-La Mancha, Ciudad Rea, Spain ABSTRACT: The bridged crack mode has been deveoped for modeing the fexura behaviour of reinforced concrete beams and reated size effects expaining britte-ductie-britte faiure mode transitions. In the present paper the mode is extended to anayze shear cracks and concrete crushing, introducing a given shape for the hypothetica crack trajectory and determining the initia crack position and the oad versus crack ength curve for three point bending probems. The proposed formuation reproduces the pure Mode I fexura behaviour as a particuar case, so that the fexura and the diagona tension (shear) faiures modes can be immediatey compared to detect which one dominates and determine the reevant faiure oad. A concrete crushing criterion competes the mode. A the mutua transitions between the different coapse mechanisms can be predicted. In the paper these transitions are shown by varying the governing nondimensiona parameters. INTRODUCTION or a ong time, the transitions between fexura and diagona tension faiures in reinforced concrete eements inside a consistent theoretica framework have represented an unsoved probem. The main issue for the present anaysis is to get a consistent modeing of shear cracks behavior and diagona tension faiure as we as concrete crushing mechanisms. These probems, despite numerous extensive studies over the past 5 years, sti remain unsoved for a competey satisfying framework, unifying a the faiure modes, so that a direct reation between faiure mode transition coud be drawn. Shear crack propagation and diagona tension faiure have been addressed in the iterature by severa authors with different approaches. In the fied of racture Mechanics and using a cohesive mode to describe concrete behaviour, some anayses have been performed by Gustafson and Hierborg (Gustafsson and Hierborg 983) and Niwa (Niwa 997) among others. In the framework of Linear Eastic racture Mechanics and in order to avoid finite eement computations, some modes are especiay remarkabe in the ong ist of iterature contributions. In particuar, Jenq and Shah (Jenq and Shah 989) anaysed the diagona shear fracture superposing the contribution of concrete and stee bars, with a technique that is somehow conceptuay cose to the bridged crack mode (Carpinteri 984). Some further deveopment of this work with other origina contributions were made by So and Karihaoo (So and Karihaoo 993). The bridged crack mode has been originay proposed by Carpinteri (Carpinteri 98; 984) for the study of reinforced concrete beams by racture Mechanics. The probem of the size effect and the britteductie transition were anayzed with reference to the probem of minimum reinforcement (Carpinteri et a. 999; Bosco and Carpinteri 992). Subsequenty, the action of cohesive stresses has been introduced in addition to that of the reinforcing bars (Carpinteri et a. 23). More recenty, the mode has been further extended anaysing concrete crushing by racture Mechanics concepts (Carpinteri et a. 24) and eading to anayse in a consistent way the interaction between fexura (yieding) and crushing faiures. Moreover, whie imit state anaysis yieds ony the utimate oad, the bridged crack mode reveas in addition scae effects, instabiity phenomena and britte-ductie faiure transition of the structura member. In the present work, the behaviour of reinforced concrete beams without stirrups is anayzed, using the bridged crack mode. To extend the mode to account for the shear cracks behavior and to evauate diagona tension faiure oad, some additiona hypotheses

2 about the crack trajectory and for the evauation of the stress-intensity factors are assumed. In this way the different coapse modes are joined together into a unified genera mode, so that the simuation of the transitiona phenomena is naturay accompished. The mode is anaysed showing the infuence of the variation in the nondimensiona parameters on the mechanica response of the reinforced concrete eement and the reated faiure mode transitions. x P x Γ P V γ M M=Vx -V M =V cv igure : Cracked eement. 2 MODELLING O LEXURAL AND SHEAR CRACKS The bridged crack mode can be appied for studying the propagation of a crack in a reinforced concrete beam assuming as monotonicay increasing contro parameter the ength of the crack path. Linear Eastic racture Mechanics is assumed for concrete with a crack propagation condition rued by the comparison of the stress intensity factors K I to the concrete toughness K IC. Neither cosed form soution nor noninear regressions of numerica data are avaiabe for evauating the stress-intensity factors of the geometry given in igure. Therefore, some assumptions have been made to derive suitabe approximations. The adopted mode scheme is reported in igure aong with the used symbos. The geometric dimensions are converted into nondimensiona quantities, after dividing by the height h in the case of vertica distances and by the shear span in the case of horizonta distances. Thus, the foowing nondimensiona parameters are defined: et = x be the nondimensiona horizonta distance from the support to the crack tip, = a the crack depth, h = x the initia crack mouth position and ζ = c the reinforcement cover. A these nondimensiona parame- h ters range from to. Additionay, et λ be the shear span senderness ratio (λ = ). h The crack trajectory Γ is considered as formed by a first vertica segment Γ from the bottom to the reinforcement ayer. A second part Γ 2 is assumed being a power aw with some given exponent, going from the end of the first part to the oad point, ig. 2. The crack trajectory Γ=Γ Γ 2 is defined by the nondimensiona function: c A S b a h ( ) ζ µ (ζ, ) = ζ + ( ) ζ ζ µ=2 λ =2.5 b) µ=4 λ = µ=2 λ =4 () µ=4 λ = igure 2: Crack trajectories, ( λ =2.5; (b) λ =4. The constitutive reation for the reinforcement bars is assumed rigid-perfecty pastic with no upper imit to the maximum deformation. The maximum for the bridging reinforcement reaction is defined by P P = A s σ y, where A s is the reinforcement area and σ y the minimum between yieding and siding stress for the bars. With reference to igure, et K I be the stress intensity factor at the crack tip. By superposition, it is given by the sum of the stress intensity factor K IV, due to the bending moment associated to the shear force V, and K IPγ, due to the cosing force at the reinforcement position: K I = K IV K IPγ (2) To evauate the stress intensity factor due to the externa oad K IV, Jenq and Shah (Jenq and Shah 989) assumed that it can be approximated by the stressintensity factor of a bent beam with a symmetric edge notch of depth a subjected to the bending moment corresponding to the cross section at the mouth of the crack. Here a simiar approach is foowed, athough the variation of the bending moment at each section due to the crack path wi be accounted for. Therefore: V (ζ, ) K IV = Y h 3 M () = V 2 b h 2 b Y V (ζ, )λ (3) The stress-intensity factor produced at the crack tip by the appied forces P acting at the eve of the reinforcement is obtained from the case of vertica crack, K IP. Severa numerica anayses by boundary eements (Portea and Aiabadi 992) have been made to get an approximation to the stress-intensity factor for different positions of the crack tip. It is observed

3 that the stress intensity factor is a function of the ange γ, see igure. Consequenty, a function β(γ) to approximate the variation Y P with γ has been defined. inay, the stress intensity factor due to the reinforcement reaction P is given by K IPγ = P h 2 b Y P (ζ, )β(γ) = P h 2 b Y P γ (ζ, ) (4) with β(γ) = ( γ 9 ).2 and γ expressed in degrees. The functions Y M () and Y P (ζ, ) are given in the stress intensity factor handbook (Okamura et a. 975). Let ρ be the bar reinforcement percentage defined as ρ = A s and w the crack opening at reinforcement bh eve. The foowing nondimensiona parameters can be defined = σ yh 2 K IC ρ; w = we K IC h 2 (5) where is the britteness number defined by Carpinteri (Carpinteri 98; 984) and E is the Young s moduus of the materia. Substituting Eqs. (3) and (4) into Eq. (2), the foowing nondimensiona equiibrium equation is obtained Ṽ = + P YPγ (ζ, ) λ Y V () (6) where Ṽ = V /(K IC h 2 b) and P = P/P P. The crack opening, w at the nondimensiona coordinate ζ can be determined by adding the two contributions of shear Ṽ and bar reaction P. The nondimensiona opening evauated at the crack propagation shear V =V, presents the foowing expression: w =2λ Ṽ Y V (z)y Pγ (ζ, z)g(ζ, z) dz 2 P ζ ζ Y 2 P γ (ζ, z)g(ζ, z) dz (7) where g(ζ, ) is the Jacobian mapping the curviinear integra aong the crack trajectory onto the interva, (Carpinteri et a. 26). If the reative dispacement in the cracked crosssection at the eve of reinforcement is assumed to be equa to zero, up to the yieding or sippage of the reinforcement ( w=), we obtain the dispacement compatibiity condition that aows us to obtain the unknown force P as a function of the appied shear Ṽ. In fact, from Eq. (7), we may define: r (ζ, ) = λ Ṽ P = ζ Y 2 P γ (ζ, z)g(ζ, z) dz ζ Y V (z)y Pγ (ζ, z)g(ζ, z) dz (8) If the force transmitted by the reinforcement is equa to P P =σ y A s, in other words, if the reinforcement traction imit has been reached (Ṽ =ṼP ), from (6) we obtain: Ṽ P = + Y Pγ (ζ, ) (9) λ Y V () On the other hand, if Ṽ < ṼP, the foowing reation hods from Eqs. (6) and (8): Ṽ = λ Y V () Y P γ (ζ, ) () r (ζ, ) Therefore, according to the mode, when Ṽ < ṼP the shear of crack propagation Ṽ depends ony on the reative crack depth, and is not affected by the britteness number. 3 MODELLING CONCRETE CRUSHING The probem of concrete crushing in the upper part of the beam is anayzed evauating the compressive stress in the cracked eement. Concrete crushing wi be detected by comparing the stress σ c to the crushing strength σ cu. The compressive stress at the upper edge of the cracked section is the sum of the contributions due to shear and reinforcement reaction: σ c = σ V c + σ P c () Introducing two suitabe shape functions Yσ M () and Yσ P (ζ, ) (Carpinteri et a. 23) and etting Yσ V () = (ζ, )Yσ M (), the foowing expression is derived V σ c = λ bh Y σ V () P bh Y σ P (ζ, ) (2) Let V = V C be the concrete crushing oad, attained when σ c = σ cu. In nondimensiona form we may write: σ cu h 2 K IC = Ṽ λ Y V C σ () P Y P σ (ζ, ) (3) Consequenty, in the same way as for the stee yieding mechanism, a britteness number for the crushing faiure can be naturay defined: N C = σ cuh 2 K IC (4)

4 so that the nondimensiona shear for compression faiure is given by Ṽ C = N λ Yσ V C + P Y P () σ (ζ, ) (5) To eiminate the dependence on P in (5), we may observe that, at stee yieding, it is P = and therefore, from Eq. (5): Ṽ C = N λ Yσ V C + Yσ P (ζ, ) (6) () In the same way, when P <, from (8) it is: Yσ P (ζ, ) Ṽ C = N λ Yσ V C + (7) () Y V ()r (ζ, ) Y Pγ (ζ, ) The nondimensiona shear of Eqs. (6) and (7) produces, for a given crack depth, the crushing stress σ c = σ cu in the uppermost part of the beam. On the other hand, for equiibrium and compatibiity being satisfied, ony the non-dimensiona shear of crack propagation, Eqs. (9) and (), is compatibe with a given crack depth, so that crushing faiure occurs when the crack propagation non-dimensiona shear, Eqs. (9) and (), is equa to the non-dimensiona crushing shear, Eqs. (6) and (7) respectivey. or ṼC = Ṽ ṼP, we have: N C + Yσ P (ζ, ) Yσ V () = + Y P (ζ, ) Y V () as we as, for ṼC = Ṽ < ṼP, it is: N C + Y P σ (ζ,) Y V ()r (ζ,) Y P γ (ζ,) λ Y V σ () = Y V () Y P γ (ζ, ) r (ζ, ) (8) (9) Equations (8) and (9) determine the points of crushing faiure in in the crack depth vs. nondimensiona shear diagram. 4 LEXURAL AND SHEAR CRACK PROPAGA- TION In this section it wi be shown how the vaue of the initia crack position affects the mechanica response of the beam and impies the stabiity/instabiity of the cracking process. or the sake of carity, reference is made to a rea exampe, based on experimenta resuts. A more detaied expanation can be found in (Carpinteri et a. 26). The experimenta test has been performed by Bosco and Carpinteri (Bosco and Carpinteri 992), and it was abeed as B-6. The materia properties and beam geometry of this test are shown in igure 3a. c).5 Span=mm ( =5mm) h=2mm b=mm c=2mm A s =mm 2 K IC =63.4N/mm 3/2 σ y =58 Mpa Three point bending test. V =.7 Stabe Unstabe b).5 V Nondimesiona shear d).5 V =. Unstabe Stabe Nondimensiona crack depth, = Unstabe igure 3: Nondimensiona shear force vs. crack depth: ( materia properties and beam geometry; (b) initia crack position =.; (c) initia crack position =.7; (d) initia crack position =.3. The foowing nondimensiona parameters characterize the simuation case: λ = 2.5, ζ =., =.4 and a 4 th order crack trajectory curve (µ = 4) is assumed. igures 3b-d show the nondimensiona shear force vs. crack depth curves for the initia crack position in the interva.3,. and a sketch of the crack trajectories. It is we-known that a crack may present stabe or unstabe behaviour. When the crack is stabe, an increase in the crack depth requires a oad increase to fufi the mode equations. On the contrary, unstabe crack impies a oad decrease. Both kinds of behavior can occur at different oad eves during crack propagation, see igures 3b, c and d. When the initia crack position is at midspan ( =.), igure 3b, the mode converges to the origina bridged crack for beams in fexure (no shear). Immediatey after the crack crosses the reinforcement, an unstabe branch begins. This turns stabe for a crack depth.3. Then the nondimensiona shear force grows unti the yieding of stee takes pace (.7). Physicay the reinforcement reaction stabi-

5 izes the initia unstabe crack propagation and finay produces the stee yieding. The second pot, igure 3c, computed for an initia crack position =.7, shows the same characteristic behaviour for ow vaues of the crack depth, an unstabe branch foows the stabe branch for a crack depth vaue of.65. rom this point on the crack is unstabe eading the beam to faiure. In this case, as for the fexura crack =., the reinforcement reaction stabiizes the crack propagation for ow crack depth but there is a point where the propagation becomes unstabe. The change in the nature of the propagation provokes the reative maximum that is observed in igure 3c. This change for shear crack in reinforced concrete beams without stirrups has been reported experimentay by Carmona (Carmona et a. 26). V Nondimesiona shear 2.5 =. =.9 =.8 =.7 =.6 = =.4 = ,8,6 b).6.4 V igure 4: ( Nondimensiona shear force, Ṽ vs. crack depth, ; (b) detai, V,4 the thick ine is the curve of the minimum critica shear oad.,2 igure 4a shows a superposition of the pots for different initia crack positions. Observe that, negecting the singuarity at the reinforcement,2 position,,4,6 each,8 curve presents a reative maximum with the exception of cracks near the support which are unstabe during a the propagation process. The thick ine in igure 4b represents the crack having the property that its reative maximum is minimum among the maxima. This reative maximum is assumed as the shear faiure oad and the curve where it is ocated aows to determine the initia crack position as we. The minimum in the shear force when the crack initiation position is changed aong the shear span has been reported aso by Niwa (Niwa 997) in a inite Eement numerica study. Niwa fixed the shear span to depth ratio λ to 2.4 and assumed a inear crack path from the initiation point to the oad point. The position for the crack initiation reported for the minimum shear resistance in his study was.62 and the nondimensiona shear was.33. These resuts compare fairy we with the resuts of the present mode. 5 CONCRETE CRUSHING AILURE The equations (6) and (7) reported in Section 3, give the shear producing crushing faiure. But, to satisfy equiibrium and compatibiity, the crushing shear must be equa to the shear of crack propagation, as expressed by (8) and (9). Therefore, the crushing points expressed by (8) and (9) can be found by intersecting the crushing curve (6) and (7) with the crack propagation curve given by (9) and (). or a better carity, this is done in the hypothesis that faiure by crushing occurs at the centra crack ( =.), but the mode is not restricted to this situation. V. N = c N = 2 c N = 5 c = = 5 = 2 N p =,8,6 V,4 b),2,8,6 V,4,2 V. N = c N = 2 c N = 5 N p =,2,4,6,8 N p =2.5 c = = 5 = 2 V = = 2 = 5 = = 5 = 2,2,4,6,8 b),8,6 V,4 igure 5: Infuence of N C in nondimensiona shear as function of the crack depth for λ =2 and ζ=.; ( =; (b) =2.5. In ig. 5a the curve showing a discontinuity at the reinforcement position ( =.) represents the nondimensiona shear of crack propagation, Ṽ, for a vertica crack at the midspan ( =.). The other famiy of curves represent the shear of crushing faiure, Ṽ C, for different vaues of N C, Eqs. (6) and (7). As appears from ig. 5a, if N C is ess than 2 (see the two owest curves), the RC beam exhibits an unstabe behaviour, and the fracture process cannot occur because the shear necessary for crushing is ess than the shear necessary to the cracking process. or higher vaues of N C the beam exhibits first yieding ( ), and then crushing when the curves with varying N C intersect the thick curve of the oad vs. crack depth diagram. In this exampe it is therefore useess to increase the concrete strength above say,2

6 N C 5 as yieding wi aways precede crushing and the faiure mode is fexura. In ig. 5b the beam britteness number is varied from to 2.5 with respect to ig. 5a. or vaues of N C smaer that 2 the beam presents the same unstabe behaviour of the previous exampe. In contrast, for N C higher than 4, the crushing faiure occurs before yieding. As N C is increased, the crushing coapse progressivey approaches the yieding point. Ony for N C = 2 the yieding precedes crushing faiure, as the curve for N C = 2 intersects the thick curve of the cracking process ony after yieding. Therefore, a variation in the britteness number N C can change the coapse mechanism from yieding to concrete crushing and viceversa. 6 TRANSITION BETWEEN AILURES MODES The proposed mode covers the three fundamenta faiure mechanisms of RC beams: stee yieding (fexur, diagona tension (shear) and concrete crushing. As shown in the foowing, the transition between the aforementioned mechanisms is rued by the nondimensiona mode parameters, N C and λ. or the sake of carity, first the transition from fexura to shear faiure is anaysed, then the transition from shear to crushing faiure..6 Nondimesiona shear V V.2 =. =.9 =.8 =.7 =.6 = =.4 V c).6 Np=.4 V Np=.2 V =.8 V =.33 b).6 V.2 d) V.4 Np=.3 V Np=. V =.25.2 V V =.33 igure 6: Transition from fexura to diagona tension faiure in RC beams as function of the governing nondimensiona parameter ; ( =.2; (b) =.3; (c) =.4; (d) =.. igure 6 shows four Ṽ curves obtained by increasing the britteness number from.2 to. and keeping constant a the remaining parameters (λ = 2.5, ζ =., µ = 6). A sketch iustrating the crack trajectories at faiure is reported for each beam mode. In igure 6a the mode response for a britteness number =.2 is shown. When the nondimensiona shear force reaches a vaue of.4 the fexura crack ( =.) begins its stabe. As the oad is increased, some other neighboring cracks deveop with a stabe. The more marked ines in the pot represent the growing cracks. When the nondimensiona shear force is equa to.8 the stee yieds at the fexura crack. We assume that this vaue of the nondimensiona shear force represents the fexura faiure oad. Thus the beam modeed in igure 6a shows a fexura faiure due to the yieding of the stee at the midspan crack. In the same way, when the britteness number is increased to.3, the beam coapses by fexura faiure at the midspan crack, athough an increment in nondimensiona shear force from.8 to.25 is observed. Comparing the crack pattern sketches at faiure, we observe that the increment in the nondimensiona shear aso aows for new neighboring cracks to deveop through the reinforced concrete eement. If the britteness number is increased to.4, see igure 6c, initiay the cracking process is simiar to the previous cases. Nevertheess, when the nondimensiona shear force reaches the vaue.33, fexura and diagona tension faiure occur at the same time. In fact, as pointed out in Section 4, diagona tension (shear) faiure occurs when a shear crack deveops an instabiity process after a stabe crack. or higher vaues of the britteness number fexura faiure needs a higher nondimensiona shear than diagona tension faiure, as iustrated in ig. 6d, where the britteness number is set to.: the oad required to provoke fexura coapse for the crack situated at midspan is.8 whie the oad to provoke diagona tension faiure is.33 for the crack in =.6. Therefore, for ow vaues of, cracks at midspan (fexura cracks) need ower nondimensiona shear force to provoke fexura faiure than shear cracks situated aong the span to deveop diagona shear faiure. As is increased, the opposite case occurs: cracks aong the span need ower nondimensiona shear force to provoke beam coapse than the crack at midspan. Thus there is a point where the transition between these types of faiure takes pace. igure 7 shows a conceptua sketch of a the faiure mode transitions in reinforced concrete eements without stirrups predicted by the mode by varying the nondimensiona parameters. Based on the definition of the britteness number

7 ( (b) (c) REINORCEMENT AREA (d) (e) (f) (g) (h) (i) SCALE ( ρ = const.) SPAN SCALE (A = const.) s exura faiure Shear faiure Crushing faiure igure 7: The goba conceptua scheme iustrating faiure mode transitions., an increase in can be read as: an increase in the reinforcement area, transition from (d) to (e); a decrease in the scae with constant reinforcement area, transition from ( to (e); an increase in the scae with a constant reinforcement percentage, transition from (g) to (e). When a crushing faiure is considered, the behavior of the RC eement is controed by the three nondimensiona parameters, N C and λ. As defined in Section 3, crushing faiure occurs when the crushing vs. shear pot intersects the crack propagation vs. shear pot. To simpify the expanation of the transition and for reasons of space in the present paper, a direct transition from fexura to crushing faiure is iustrated, athough intermediate shear faiure transition can be demonstrated to exist as reported in the genera scheme of ig. 7. In ig. 8a the transition process is shown when is varied and the rest of parameters remains constant. The nondimensiona shear Ṽ at yieding increases as increases. At the same time, the shear for crushing faiure increases, athough in a smoother way. Thus the transition from fexura to crushing faiure appears ceary as shown in ig. 8b, where we can read the britteness number in the abscissas against the nondimensiona shear at faiure. Two different areas are deimitated. or ow vaues of the britteness number faiure is due to stee yieding (fexure). or.3, the transition takes pace and then shear for crushing faiure needs a ower vaue compared to shear for V Nondimesiona Shear V N =5 C λ =3 V (N =.2) C P V (N =.8) C P =.6 = b) V V (yieding) V =V C exura to crushing faiure igure 8: Transition from fexura to crushing faiure in RC beams as function of governing nondimensiona parameters, increment of ; ( Ṽ - curves; (b) Ṽ - curve. fexura faiure, i.e. crushing precedes yieding. Physicay this transition appears when the reinforcement ratio ρ is increased and the rest of parameters remains constant: we have the transition from (d) to (f) in the scheme of ig. 7. According to the scheme, another transition can be demonstrated when the beam is scaed keeping constant the reinforcement ratio ρ. Looking at the definitions of and N C, the condition of an increment in the scae can be expressed keeping constant the ratio N C. inay, the transition by size effect can be demonstrated in the hypothesis that the reinforcement area A s is constant. This condition can be expressed by considering the ratio N C as a inear function of the scaed beam depth h h, where h is a reference depth. The conceptua scheme in ig. 7 summarizes the faiure transitions predicted by the mode, from fexura crushing faiure.

8 The transitions to crushing can take pace as: an increase in the reinforcement area, transition from (d) to (e) and finay (f); an increase in the scae with constant reinforcement percentage, transition from (g) to (e) and finay (c); a decrease in the scae with constant reinforcement area, transition from ( to (e) and finay (i). In some cases, depending on materia and geometrica properties, the intermediate transition through (e) may be skipped and a direct transition from yieding to crushing can be observed. 7 CONCLUSIONS This paper presents an extension of the bridged crack mode to anayse fexura-shear-crushing faiure modes in R.C. beams. The faiure mode transitions have been iustrated by varying the controing nondimensiona parameters: the britteness numbers and N C and the senderness λ. The study demonstrates that the diagona tension faiure is a consequence of unstabe crack propagation. The shear faiure initiation point and coapse oad are determined anayticay by the present mode without using empirica parameters. The mode gives rationa expanation to the transitions between a the faiure modes and size effects in faiure transitions are shown by varying the britteness numbers, and N C. ACKNOWLEDGEMENTS Jacinto R. Carmona gratefuy acknowedges the financia support for this research provided by the Ministerio de Educación y Ciencia, Spain, under grant MAT23-843, and by the Ministerio de omento, Spain, under grant BOE35/23. REERENCES Bosco, B. and A. Carpinteri (992). racture mechanics evauation of minimum reinforcement in concrete structures. In Appications of racture Mechanics to reinforced concrete, London, pp A. Carpinteri, ed., Esevier Appied Science. Carmona, J. R., G. Ruiz, and de Viso. J. R. (submitted 26). Mixed-mode crack propagation through reinforced concrete. Engineering racture Mechanics. Carpinteri, A. (984). Stabiity of fracturing process in RC beams. Journa of Structura Engineering-ASCE, Carpinteri, A., J. R. Carmona, and G. Ventura (submitted 26). Propagation of fexura and shear cracks through reinforced concrete beams by the bridged crack mode. Magazine of Concrete Research. Carpinteri, A., G. erro, C. Bosco, and M. Ekatieb (999). Scae effects and transitiona faiure phenomena of reinforced concrete beams in fexure. In A. Carpinteri (Ed.), Minimum Reinforcement in Concrete Members, Voume 24 of ESIS Pubications, pp. 3. Esevier Science Ltd. Carpinteri, A., G. erro, and G. Ventura (23). Size effects on fexura response of reinforced concrete eements with a noninear matrix. Engineering racture Mechanics 7, Carpinteri, A., G. erro, and G. Ventura (24). A fracture mechanics approach to over-reinforced concrete beams. In V. Li, C. Leung, K. Wiam, and S. Biington (Eds.), Proceedings of the 5 th racture Mechanics of Concrete and Concrete Structures (ramcos-5), pp Gustafsson, P. and A. Hierborg (983). Sensitivity in shear strength of ongitudinay reinforced concree beams to fracture energy of concrete. ACI Structura Journa 85(3), Jenq, Y. S. and S. P. Shah (989). Shear resistance of reinforced concrete beams - a fracture mechanics approach. In racture Mechanics: Appications to Concrete, Detroit, pp V. Li and Bažant, Z.P., eds., American Concrete Institute. Niwa, J. (997). Size effect in shear of concrete beams predicted by fracture mechanics. In CEB Buetin d Information n Concrete Tension and Size Effects, Lausanne, Switzerand, pp Comite Euro-Internationa du Béton (CEB). Okamura, H., K. Watanabe, and T. Takano (975). Deformation and strength of cracked member under bending moment and axia force. Engineering racture Mechanics 7, Portea, A. and M. H. Aiabadi (992). Crack Growth Anaysis Using Boundary Eements. Southampton: Computationa Mechanics Pubications. So, K. O. and B. Karihaoo (993). Shear capacity of ongitudinay reinforced beams - A fracture mechanics approach. ACI Structura Journa 9, 59 6.