Diagonal Tensile Failure Mechanism of Reinforced Concrete Beams

Transcription

1 Journal of Advaned Conrete Tehnology Vol., No. 3, 37-34, Otober 4 / Copyright 4 Japan Conrete Institute 37 Diagonal Tensile Failure Mehanism of Reinfored Conrete Beams Yasuhiko Sato, Toshiya Tadokoro and Tamon Ueda 3 Reeived 7 February 4, aepted June 4 Abstrat The mehanism of diagonal tensile failure of RC beams without shear reinforement, whih is diffiult to solve by means of experimental and analytial study, is investigated. The lose relationship between the frature modes and the transfer stress at shear raks is larified, and the experimental results are verified by the finite element method taking into onsideration the influene of a splitting tensile rak and dowel ation. In RC members without shear reinforement, the width of a shear rak inreases owing to the ourrene of a splitting tensile rak along the main bars. As a result, the transfer stress of a shear rak annot be generated and the shear rak grows and propagates rapidly. This paper also puts forth that FE analysis, in whih not only shear rak but also splitting tensile rak are modeled disretely, an predit the size effet on diagonal tensile failure strength of RC beams without shear reinforement.. Introdution Shear failure, whih is a typial failure mode of reinfored onrete (RC) members, must be prevented so as to ensure struture safety for users and third parties. It is desirable therefore to develop a rational shear design method. However, failure mode and apaity predition remains diffiult due to the existene of several failure modes and the high omplexity ohe failure proesses involved and their mehanisms. Shear failure an be lassified into two modes, diagonal tensile failure and shear ompression failure. Many researhers have already reported that shear ompression failure an be simulated by FE analysis with two-dimensional onstitutive models (Tadokoro et al. ; Yamaya et al. 995; Kaneko et al. ; Asin et al. 995). It is extremely diffiult, however, to simulate diagonal tensile failure, whih is a brittle failure, by FE analysis. In this ase, the harateristis of deformation and shear stresses generated at shear raks must be aurately identified. However reports with adequate information to larify suh harateristis are not available. Aording to previous studies, the diagonal tensile failure proess and its strength an be simulated by FE analysis, adopting the rotational rak model (Yamaya et al. 995) and shear stress softening model (An et al. 998). However, modeling of phenomena suh as the rotation of shear raks and the softening of shear stress is not assumed based on atual fats observed in experiments. Researh Assoiate, Division of Strutural and Geotehnial Engineering, Hokkaido University, Japan. ysato@eng.hokudai.a.jp Researher, Strutures Tehnology Division, Railway Researh Tehnial Institute, Japan. 3 Professor, Division of Strutural and Geotehnial Engineering, Hokkaido University, Japan. Firstly, this study larifies the shear transfer mehanism through experiment using RC beams in whih the presene/absene of shear reinforement, and ratio of shear span to effetive depth ratio are seleted as parameters. The observed shear deformation and shear transfer mehanism are ompared and examined with numerial results predited by FE analysis with a disrete rak model. Finally, the diagonal failure mehanism of RC beams without shear reinforement is desribed. Size effet simulation of diagonal tensile failure is also arried out.. Deformational harateristis at shear rak. Preliminary remarks Clarifiation of deformational and shear stress transfer harateristis at a shear rak is ruially important for the simulation of diagonal tensile failure by FE analysis. It is onsidered that splitting tensile raks have a onsiderable influene on the mehanial behavior of RC members (Noguhi et al. 98; Sanada et al. 995; Kim et al. 999). Splitting tensile raks an develop due to both the dowel fore ating vertially on main bars and the tensile fore. In this hapter, shear and opening deformations at shear and splitting tensile raks in beams are measured and shear transfer stresses are examined.. Outline of experiment Three RC beams of retangular ross setion were tested in order to losely observe deformation of shear and splitting tensile raks under stati loading. Details ohe speimens are listed in Table and the reinforement arrangements are shown in Fig.. The mix proportions ohe onrete are listed in Table. Speimen S is an RC beam with shear reinforement, while speimens T and T do not ontain any shear reinforement. The sole differene between speimens T and T is the shear span to effetive depth ratio (a/d).

2 38 Y. Sato, T. Tadokoro and T. Ueda / Journal of Advaned Conrete Tehnology Vol., No. 3, 37-34, 4 Table Details of speimens. Speimen S T T ad f f t.7.8. f y ρ ad:shear span to effetive depth ratio f :Conrete ompressive strength (MPa) f t :Conrete tensile strength(mpa) f y :Yield strength(mpa) ρ :Main reinforement ratio =5@ 35 (a) Speimen S 5 5 Table Mix proportions of onrete. Speimen S,T T D max w 6 5 air 5 5 W 6 6 C 6 3 S G 55 5 AE water reduting agent.5.4 D max :Maximum aggregate size(mm) w :Water to ement ratio(%) air:entrained air(%) :per volume(kgf / m 3 ) The ross-setion dimensions and tensile reinforement ratio are idential for all three speimens. A load was applied at the enter of a span on ontat with a steel plate of -mm width. Steel balls for measurement by ontat miron strain gauge were attahed on the tested shear span to measure deformation at the raks (see Fig. ). In the study, loading was periodially stopped for several minutes at a given load level to measure elongation between ontat points. As a result, relaxation behavior an be seen in the urves shown in Fig. 4 beause loading was stopped at a given load level. Crak opening and shearing displaements of shear raks were alulated in the following manner. Elongation between two steel balls an be alulated as follows (see Fig. 3): = + ω sinθ + δ osθ (a) = + ω sinθ + δ osθ (b) The rak opening and shear displaements an be (b) Speimen T 5 () Speimen T given by solving the above simultaneous equations. ( )os θ ( )osθ ϖ = (a) sinθ osθ sinθ osθ ( )sin θ ( )sinθ δ = (b) osθ sinθ osθ sinθ Shear transfer stresses are alulated using w and δ given by Eq. () as desribed later. Shear transfer stresses along a rak interseting with a triangular area demarated by three balls represents average stress in the rak length..3 Failure proess () Speimen S with shear reinforement The applied shear fore - deformation urve and raking pattern of speimen S are shown in Fig. 4(a) and Fig. (a), respetively. A shear rak ourred at a shear fore of approximately 8kN and propagated towards the loading point. Finally the beam failed in shear ompression after some raks around the loading point were observed. It an be learly seen in Fig. that the shear rak 5 5 Fig. Experimental speimens. 5 5

3 Y. Sato, T. Tadokoro and T. Ueda / Journal of Advaned Conrete Tehnology Vol., No. 3, 37-34, 4 39 K E I H G F J (a) Speimen S G F E C D (b) Speimen T E F () Speimen T A B C D A B C D A B C L C L C L TEST(S) (a) Speimen S TEST(T) Shear raking 3 (b) Speimen T Shear raking Fig. Craking pattern and arrangement of steel balls. A C C A θ θ θ δ ω C C TEST(T) Shear raking B B θ ω B B δ Positive shearing sliding diretion Positive rak opening diretion Fig. 3 Definition of deformations. intersets the shear reinforement and propagates toward supporting points along the main reinforements. The rak named splitting tensile rak was developed at the same load level as the shear raking load. The shear raking load was the same as that alulated by Niwa s equation (Niwa et al. 987) below. /3 ( ) ( ) / 4 ( ( )) V =. f pt d/ / a/ d bd (3) () Speimen T Fig. 4 Relationship between applied shear fore and displaement. where f is the onrete ompressive strength in MPa, ρ t is the main reinforement ratio, d, a, and b are the effetive depth, shear span, and beam width in mm, respetively. () Speimen T without shear reinforement The applied shear fore deformation urve and raking pattern of speimen T are shown in Fig. 4(b) and Fig. (b), respetively. The applied shear foredeformation urve before shear raking is idential to

4 33 Y. Sato, T. Tadokoro and T. Ueda / Journal of Advaned Conrete Tehnology Vol., No. 3, 37-34, 4 Zone A Crak Width (a) Zone A Zone C (b) Zone C Zone G Crak Width Displaement(mm) () Zone G Crak Width Fig. 5 Deformation at shear rak and splitting tensile rak in speimen S. that of speimen S, whih has the same dimensions, while after shear raking, the stiffness rapidly dereases beause speimen T does not have any shear reinforement. The applied fore ould be inreased after shear raking sine the failure mode was a kind of mixed mode with both shear ompression failure and diagonal tension failure harateristis, rather than a pure diagonal tension failure mode. Stresses on shear rak (MPa) Shear resisting fore (kn) Shear transfer stress Compressive stress Crak opening or shear displaement (mm) 5 5 (a) Shear transfer stress Shear fore arried by aggregate interloking 5 5 (b) Shear fore arried by aggregate interloking Fig. 6 Shear transfer stresses and shear resisting fores in speimen S. (3) Speimen T without shear reinforement The applied shear fore deformation urve and raking pattern of speimen T are shown in Fig. 4() and Fig. (), respetively. Shear raking ourred at a shear fore of approximately kn and propagated slowly up to the viinity ohe loading point while the applied shear fore was inreased up to 9 kn. The beam ould still sustain the load for a while and failed at 3kN just after rushing ohe onrete around the loading point was observed. Shear raks with an angle of 45 degrees ould be observed but splitting tensile raks were not deteted..4 Deformation and shear transfer mehanism at shear rak () Speimen S with shear reinforement The rak opening and shear displaements at the raks in zones A, C and G marked in Fig. (a), are shown in Fig. 5. Those displaements start to inrease at a shear raking load of approximately 8kN and linearly inrease as the applied shear fore inreases. It an be learly seen that rak opening displaement (Mode I) and shear displaement (Mode II) take plae simultaneously. Shear transfer behavior will now be disussed. In this

5 Y. Sato, T. Tadokoro and T. Ueda / Journal of Advaned Conrete Tehnology Vol., No. 3, 37-34, 4 33 study, stresses developed at shear raks are alulated by the ontat density model proposed by Li and Maekawa (988). The shear transfer model is as follows; δ τ = 3.83 f = σ ( w δ ) ( δ / w) ( δ w) f Zone A (a) Zone A w δ = 3.83 f.5π tan Crak Width Zone B (b) Zone C Crak Width () Zone G Zone G Crak Width Fig. 7 Deformation at shear rak and splitting tensile rak in speimen T. ( / ) wδ ( w + δ ) (4a) (4b) Stresses on shear rak (MPa) Shear resisting fore (kn).3.. Shear transfer stress Compressive stress.5 Crak opening or shear displaement (mm) 5 (a) Shear transfer stress Shear fore arried by aggregate interloking 5 (b) Shear fore arried by aggregate interloking Fig. 8 Shear transfer stresses and shear resisting fores in speimen T. where f is the onrete ompressive strength in MPa, and w and δ are the rak opening and shear displaements in mm, respetively. Aording to this model, stresses at a rak are governed by the ratio of shear displaement to rak opening displaement. In this sense, shear softening behavior an be readily observed when the value ohe ratio of opening to shear displaement with inreases in both shear and rak opening displaements rises. Figure 6(a) shows the alulated shear and ompressive stresses at the shear rak in zone C. On the other hand, Fig. 6(b) shows the shear fore arried by aggregate interloking at the shear rak. The shear fore is a vertial omponent ootal fore that is given by the summation of shear transfer fore along a rak interseting with eah triangular zone. One an establish from these figures that higher shear and ompressive stresses develop and that shear fore arried by aggregate interloking play a very important role for the shear resisting behavior of RC beams with shear reinforement. () Speimen T without shear reinforement The opening and shear displaements at zones A, C, and

6 33 Y. Sato, T. Tadokoro and T. Ueda / Journal of Advaned Conrete Tehnology Vol., No. 3, 37-34, Zone B Crak Width.5.5 (a) Zone B Zone C Crak Width.5.5 (b) Zone C Stresses on shear rak (MPa) Shear resisting fore (MPa) Crak opening or shear displaement (mm) 5 5 Shear transfer stress Compressive stress (a) Shear transfer stress Shear fore arried by aggregate interloking Zone D Crak Width.5.5 () Zone D Fig. 9 Deformation at shear rak and splitting tensile rak in speimen T. (b) Shear fore arried by aggregate interloking Fig. Shear transfer stresses and shear resisting fores in speimen T. G marked in Fig. (b) are shown in Fig. 7. Only the rak opening displaement starts to inrease at a shear raking load of approximately 8kN and linearly inrease as the applied shear fore inreases, but shear displaement does not (see Figs. 7(a) and 7(b)). On the other hand, the rak opening and shear displaements ohe splitting tensile rak rapidly inrease just after the shear rak ours (see Fig. 7()). Figure 8(a) shows the alulated shear and ompressive stresses at the shear rak. On the other hand, Fig. 8(b) shows the shear fore arried by aggregate interloking at the shear rak. The shear transfer stress is only one tenth or less ohat estimated in speimen S with shear reinforement. The same tendeny was observed at all measured zones with splitting tensile raks. Therefore, the shear fore arried by aggregate interloking is negligibly small. It an be onluded that a slender beam without shear reinforement would fail immediately after the ourrene of a shear rak beause shear resisting fore by aggregate interloking annot be developed. (3) Speimen T without shear reinforement The rak opening and shear displaements at the shear raks in zones B, C, and D marked in Fig. () are shown in Fig. 9. Opening and shear displaements start to inrease at a shear raking fore of approximately kn and linearly inrease as the applied shear fore inreases. The opening displaement is muh larger than the shear displaement. Figure shows the shear and ompressive stresses alulated by the Li and Maekawa model and the shear fore arried by aggregate interloking. The observed stress level and the shear fore arried by aggregate

7 Y. Sato, T. Tadokoro and T. Ueda / Journal of Advaned Conrete Tehnology Vol., No. 3, 37-34, interloking are less than those in speimen S with shear reinforement but signifiantly greater than those in speimen T, whose a/d ratio is larger than that of T. Those results indiate that, in the ase of a deep beam without shear reinforement, shear fore arried by aggregate interloking an be expeted and may ontribute to the formation of an arh ation mehanism. 3. Numerial simulation by smeared rak model 3. Outline of analysis In this study, a two-dimensional nonlinear finite element program, WCOMR (Okamura et al. 99), was used. In this analysis, 8 node iso-parametri elements with 9 gauss points were adopted for the representation of plain and reinfored onrete elements. The nonlinear iterative proedure was ontrolled by the modified Newton-Raphson method. In this proedure, onvergene is judged by the ratio of Σ(Residual fore) to Σ(Internal fore) and the iteration proedure is repeated until the ratio beomes less than -6. The elasto-plasti frature model developed by Maekawa (983) is used for the onrete model before raking. In this model, stresses and strains in a plain stress ondition are represented by an equivalent stress and equivalent strain, respetively. Aoyanagi and Yamada s model (Okamura et al. 99), Niwa s model (Okamura et al. 99), and a model based on previous experiment (Kupfer et al. 969) are used as raking riteria for the tension-tension domains (4 in Fig. ), for the tension-ompression domains ( and 3 in Fig. ), and for the ompression-ompression domains ( in Fig. ), respetively. The tension softening model proposed by Reinhardt (986) is adopted for onrete after raking (see Fig. ). 3 σ t δ δ δ 3 = + exp ( + ) exp( ) ft δ δ δ (3) where and are 3. and 6.93, respetively, is the onrete tensile strength in MPa, and δ is the ritial rak opening (rak width at zero stress). The ritial rak opening is determined by Eq. (6), whih represents the relationship between the frature energy and the rak width, proposed by Hordijk (99). G f δ = 5.4 ft (6) where is the onrete tensile strength in MPa and G f is the frature energy in N/mm. The frature energy for ompression is determined by JSCE ode equation (JSCE ). σ σ σ σ Crushing σ σt ft Crushing Crushing Crushing Gf δ δt Fig. Tension softening model. σ 3 3Craking σ σ σpeak.8. εp (a) Tension ompression σ f σ Fig. Craking riteria for bi-axial stress state..8 σ σpeak.. σ σ εp σ σpeak εu εp (b) Compression ompression εu Fig. 3 Compression softening model. f 4Craking 4 Craking σ Gf f leq Crushing Craking εu G f f leq Crushing G f f leq Crushing ε ε ε

8 334 Y. Sato, T. Tadokoro and T. Ueda / Journal of Advaned Conrete Tehnology Vol., No. 3, 37-34, 4 G D f 3 max f = (7) where f is the onrete ompressive strength in MPa and D max is the maximum aggregate size in mm. The tension stiffening model proposed by Tamai et al. (998) is applied for reinfored onrete elements under tension. σ ε tu = ft ε where is the onrete tensile strength in MPa, ε tu is the strain as softening starts, σ is the average stress in MPa, and ε is the average strain. In this analysis, the fixed rak model is adopted and shear transfer stresses are alulated using Li and Maekawa s model shown in Eq. (). After raking, the onrete linear softening model (Fig. 3(a)) for the diretion parallel to the rak is introdued to onsider the effet of raking on ompression-softening. In this model, ompressive stress is redued to zero at limited strain ε u. The gradient of strain softening is defined by ompressive frature energy onsumed in ompressive stress parallel to the rak in the tension-ompression area. In other words, the softening urve is defined in suh a way that the area surrounded by the envelope urve ohe stress-strain relation is equal to the frature energy as in tension softening. However the redued stress has a limit that is % ohe ompressive strength. When ompressive stress at raking is greater than 8% ohe ompressive strength, ompression-softening immediately starts. When it is less than 8% ohe ompressive strength, namely in stress states dominated by tensile stress, however, it is assumed that ompressive stress inreases up to 8% ohe ompressive strength and softening starts. This is beause onrete parallel to ompressive stress suffers less damage. Besides the ompression-softening property, it is assumed that elements that fail in ompression lose shear transferability to a onsiderable extent. Therefore, the shear transfer stiffness is redued to % ohe stiffness in the original onrete. Similarly, normal stress transferred by aggregate interloking is redued like shear transfer stress. Therefore, there is no effet of aggregate interloking in elements that fail in ompression. When deep beams failing in shear ompression were analyzed, it was found neessary to develop a model of ompression softening in the biaxial ompression area (Tadokoro et al. ). In RC members that fail in shear ompression, the diagonal raks propagated towards the loading point, and then the onrete near the loading point under biaxial ompressive stress states are gradually damaged. A model of ompression-softening in biaxial ompressive stress states is neessary to onsider this damage due to raking and improve the auray of predition ohe failure mode in the ultimate state, (8) the failure zone, ultimate defletion, and strength for RC members that fail in shear ompression. Aording to Kupfer s experimental investigation on onrete plates subjeted to biaxial ompression (Kupfer et al. 969), a rak ourred in the diretion parallel to the free surfae (see the left figure in Fig. 3(b)). In the biaxial ompression area, therefore, it is onsidered that the energy is onsumed by rushing of onrete (or ompression-softening) in both diretions of ompressive stress after raking. On the other hand, in the tension-ompression area, the energy was onsumed by rushing only in the diretion parallel to the rak (see the left figure in Fig. 3(a)). It is assumed that the sum ohe frature energy onsumed by onrete rushing in both diretions under a biaxial ompressive stress state is equal to the energy onsumed by rushing in one diretion under tensile and ompressive stress states. Thus the following equation is derived for the biaxial ompressive stress state (see Fig. 3(b)): G = G + G (9a) G G f f f σ = (9b) σ f, rak f, rak Table 3 Shear redution fator. Case Shear redution fator SMD. SMD.5 SMD3.5 where G f is the ompression frature energy in the tension-ompression stress state in N/mm, G f and G f are the ompression frature energy in the ompression-ompression stress state in N/mm, and σ,rak and σ,rak are stresses when softening starts in MPa. In this study, the ompression frature energy used is set to 5N/mm based on the study of Nakamura et al. (999). 3. Overview of numerial results FE analyses of speimens T and T were arried out. As is well known, the shear retention fator, β, for redution of raked onrete shear stiffness would be used to avoid stress loking and this fator greatly affets omputational results (Rots et al. 985). In the analysis of speimen T, whose failure mode was diagonal tensile failure, therefore, a parametri study on sensibility ohe shear stiffness was arried out by onduting three omputations with β =.5,.5, and. (see Table 3). The finite element meshes and raking pattern omputed using FE analysis are shown in Figs.4 and 5, respetively. The applied shear fore and deformation urves are shown in Fig. 6. As shown in Fig. 6(a), FE

9 Y. Sato, T. Tadokoro and T. Ueda / Journal of Advaned Conrete Tehnology Vol., No. 3, 37-34, =4@ =5@5 RC PLAIN CONCRETE RC PLAIN CONCRETE (a) Speimen T (b) Speimen T Fig. 4 Finite element meshes. (a) Speimen T (b) Speimen T Fig.5 Craking pattern predited by FE analysis TEST(T) SMD σ γ τ σtεt ( ) σtεt ( ) τ δ ω= γ ε σ L L δ γ εt εt γ ω Fig. 7 Deformations and strains. δ= L γ ω = L ε 5 (a) Speimen T 5 TEST(T) SMD SMD SMD (b) Speimen T Fig. 6 Applied shear fore displaement urves. analysis an predit deformational behavior and its apaity of shear ompression failure type beause the ompression softening model for ompression-ompression stress state was adopted in the program. However, FE analysis annot simulate the apaity of diagonal tension failure as shown in Fig. 6(b). Besides, numerial results are greatly influened by shear transferring stiffness. 3.3 Deformational harateristis at shear rak The total shear displaement or the total rak opening displaement in a onrete element an be represented by shear and tensile strains, respetively, as shown in Fig. 7 (Okamura et al. 99). That is, the ratio of shear strain to tensile strain an be represented by the ratio of shear displaement to rak opening displaement.

10 336 Y. Sato, T. Tadokoro and T. Ueda / Journal of Advaned Conrete Tehnology Vol., No. 3, 37-34, 4 Crak Width (mm) Tensile strain H=.5 (m) H=5. (m) Shear (a) Experiment Shear strain (b) Analysis H=.5 (m) H=7.5 (m) Shear sterss (MPa) Compressive stress (MPa) H=.5 (m) H=7.5 (m) Shear strain (a) Experiment H=.5 (m) H=7.5 (m) Tensile strain (b) Analysis Fig. 9 Shear transfer stresses at a rak. Fig. 8 Shear deformations at a rak. Figure 8 shows the relationships between rak opening and shear displaements observed in the experiment and the relationships between tensile and shear strains omputed by FE analysis. H in the figure represents the distane in height from the bottom fiber of a beam. No shear displaement is found in this experiment. However, larger tensile strain is found in the analysis and as a result shear transfer stresses muh greater than those observed in the experiment are generated, as shown in Fig Numerial simulation by disrete rak model 4. Influene of splitting tensile rak () Outline of analysis Rational raking riteria for splitting tensile raks have not yet been developed beause ohe large number of influening fators, suh as diameter and number of bars, thikness of onrete over, onrete strength, and so on. Dowel fore ating on main reinforements generates tensile stresses in onrete, whih in turn ause splitting tensile raks. When splitting tensile raks are represented by a disrete rak model using spring elements, tensile stress and deformation relationship must be established in the elements as for the onrete model. As a simple way for onsidering the influene oensile stress in main bars upon the ourrene of splitting tensile raks, redution of onrete strength in elements ontaining main bars may be allowed. In this ase, the influene of ross diretion non-uniform deformation aused by dowel ation must also be taken into aount. In this study, the influene of splitting tensile raks σ v Fig. Tensile stress - dowel displaement model. δ v

11 Y. Sato, T. Tadokoro and T. Ueda / Journal of Advaned Conrete Tehnology Vol., No. 3, 37-34, (a) Model A (w/o splitting tensile rak) (b) Model B (with splitting tensile rak) Fig. Finite element meshes. is investigated by using disrete rak elements in whih the simplified tensile stress displaement model (ut-off model) shown in Fig. is adopted. In the model, the asending part is modeled by a linear urve whose stiffness is high enough to avoid the ourrene of loalized deformation in onrete generated by dowel fore before raking. Eah disrete rak that represents a splitting tensile rak is modeled by a spring element with two springs, i.e. vertial and shearing springs, at eah node. As a rule, the simplified model shown in Fig. is adopted for the vertial spring, while an elasti model with high stiffness is used for the shearing spring. After a splitting tensile rak ours, the shear stresses on the splitting tensile rak are evaluated using Li and Maekawas model shown in Eq. (4). The objetive of investigation with the simplified model is to larify the influenes of horizontal splitting raks on the shear strength of a beam. The appliability ohe simplified model is disussed through parametri analysis in the next setion. Speimen T that failed in diagonal tension was used for investigation ohe influene of splitting tensile raks. Li and Maekawas model was used in disrete rak link elements that represent diagonal tensile raks. Isoparametri 8-node elements with 9 gauss points were used for onrete and main bars. Two analytial meshes were prepared: model A having only shear raks and model B having both shear and splitting tensile raks, as shown in Fig.. The thik lines in Fig. represent disrete rak link elements. Splitting tensile raks are loated along the main bar. Reinforement elements were attahed to the onrete elements by spring elements in whih the following bond model (Shima et al. 987) was installed exept 5D (D: diameter of bar) from the point interseted by the shear rak that represents a bond deterioration zone. { (.6 )} τ =.9 exp 4s () where τ is the bond stress in MPa, f is the onrete ompressive strength in MPa, s is normalized slip (=S/D), S is slip in mm, and D is the bar diameter in mm. 5 5 Propagation of shear rak Propagation of horizontal rak TEST(T) Model A Model B 3 Fig. Applied shear fore - displaement urves. () Analytial results Figure shows the relationship between applied shear fore and deformation observed in analysis. The blak irles represent the numerial result of model, in whih splitting tensile raks are not onsidered, while the white irles represent the result in whih splitting tensile raks are modeled. Flexural raks ourred at deformation of.6 mm. Then shear and splitting tensile raks ourred at deformation of.3 mm. Stiffness was signifiantly redued and shear fore ould not be inreased any more when splitting tensile raks propagated toward supporting points. In the ase of model, the same result an be observed until shear raking load, but after shear raking the response ompletely hanged, i.e., the applied shear fore ould be stably inreased beause an arh mehanism ould be reated in the ase of model without splitting tensile raks. This result indiates that splitting tensile raks must be onsidered so as to simulate diagonal tensile failure using a disrete rak model. Figure 3 shows the deformation and raking patterns of analyzed beams. The deformation is magnified 8 times. Displaement at shear raking load owo ases is the same. However, the rak width and rak tip loation in model B is muh wider and higher than in model A aused by the propagation of splitting tensile rak (see Fig. 3 at displaement of.6 mm). Figure 4 shows the relationships between rak

12 338 Y. Sato, T. Tadokoro and T. Ueda / Journal of Advaned Conrete Tehnology Vol., No. 3, 37-34, 4 Ourrene of flexural rak (δ=.6 mm) Ourrene of flexural rak (δ=.6 mm) Ourrene of shear rak (δ=.3 mm) Ourrene of shear rak (δ=.3 mm) Propagation of shear rak (δ=.6 mm) Propagation of shear rak (δ=.6 mm) No splitting tensile rak (δ=.6 mm) (a)model A Propagation of splitting tensile rak (δ=.6 mm) (b) Model B width and shear displaement observed in the experiment and analysis for the raks in zones A and B of speimen T shown in Fig. (b). The analytial results for H = 5. m and H =.5 m in Fig. 3 orrespond to the experimental results for zone A and zone B, respetively. The tendeny for the rak width to grow under small shear displaement an be found in both the experiment Fig. 3 Deformation predited by FE analysis. and analytial results. It is onluded that FE analysis, in whih not only shear raks but also splitting tensile raks are modeled disretely, an satisfatorily predit atual deformational harateristis for shear raks. As shown in Fig. 5, whih represents stresses at a shear rak observed in the FE analysis, the shear transfer stress is very small and shear stress softening behavior annot be seen at all.

13 Y. Sato, T. Tadokoro and T. Ueda / Journal of Advaned Conrete Tehnology Vol., No. 3, 37-34, Crak width (mm) Experiment H=.5 (m) H=5. (m) Analysis H=.5 (m) H=9. (m) Shear displaement (mm) Stresses on shear rak (MPa) Shear transfer stress Compressive stress H=.5 (m) H=9. (m) Crak opening or shear displaement (mm) Fig. 4 Deformation at shear rak. Fig. 5 Shear transfer stresses. σv σv σv モテ ル Model モテ ル Model δ σv δv σv δ モテ ル 3 モテ ル 4 Model 3 Model 4 V δ モデル 5 Model 5 δ δv δv δv 5 5 TEST(T) Model Model Model3 Model4 Model5 3 Fig. 6 Models for splitting tensile rak. 4. Influene of modeling of splitting tensile rak To evaluate the influene ohe type of model on splitting tensile raks, the five models shown in Fig. 6 were prepared for this study. In model, the splitting tensile rak strength and ritial displaement are assumed to be uniaxial tensile strength of onrete and 6µm, respetively, where the ritial displaement is defined based on Eq. (4). Splitting tensile rak strength might be smaller than uniaxial tensile strength of onrete sine not only dowel fore but also tension fore is applied. Therefore, in model, approximately half of the onrete tensile strength is assumed to be splitting tensile rak strength. On the other hand, models 3 and 4 have the same raking strength as models and, respetively, but those models do not ontain a stress-softening portion (ut-ofype). In the model 5, raking strength is set to half of onrete tensile strength and this model has a nonlinear softening portion whose urve is determined by Reinhardt s model. Stiffness in those models are given by Eq. (9) as well. The applied shear fore displaement urves predited by FE analysis with those models are shown in Fig. 7 Applied shear fore displaement urves fousing on influene of splitting tensile rak model. Shear strength (MPa) 3 Eq. FEM 5 5 Effetive depth (mm) Fig. 8 Size effet omparison. Fig. 7. Not only splitting tensile rak strength but also softening stress greatly affet stiffness and apaity after the ourrene of splitting tensile raks. That is, the greater raking strength and softening stress, the

14 34 Y. Sato, T. Tadokoro and T. Ueda / Journal of Advaned Conrete Tehnology Vol., No. 3, 37-34, 4 greater stiffness and apaity. FE analysis with model 5 gives the results that most losely math the experimental results. Development of a preise model remains an issue in this study. 4.3 Size effet analysis Size effet analysis is arried out here. Speimen T is used as the referene beam for parametri analysis. Virtual beams with heights of,, 4, 8, and 6 mm were prepared. The size ohe mesh and bar diameter were similarly transformed and the loation and angle of shear and splitting horizontal raks were assumed based on the results obtained from speimen T. Model 5 in Fig. 6 was used as the splitting tensile raking model in the analysis. The omputed shear strengths are shown in Fig. 8 along with the urve alulated by Eq. (4), in whih the size effet is proportional to d -/4. The analysis an satisfatorily simulate the size effet on shear strength although the predited strengths are slightly greater than those yielded by Eq. (4). One possible reason for this may be the influenes ohe shear rak loation and angle. To predit shear strength more preisely, the shear rak loation angle that produe minimum shear strength must be used. 5. Conlusions The following onlusions an be drawn based on the findings ohis study. () When an RC beam fails in shear ompression, greater shear transfer stresses at a rak an be generated beause rak opening displaement (Mode I) and shear displaement (Mode II) take plae simultaneously. () When an RC beam fails in diagonal tension, shear transfer stresses are negligibly small beause only rak opening displaement inreases with no inrease in shear displaement. (3) FE analysis using the smeared rak model annot simulate the atual deformational behavior at shear raks beause suh analysis annot onsider loal deformation and frature due to dowel ation. More onretely, FE analysis overestimates shear transfer stiffness and, as a result, annot aurately predit the shear apaity beause Mode I and II deformations simultaneously our in the analysis, unlike the atual phenomena. (4) The presene/absene of splitting tensile raks was found to greatly affet the mehanial behavior of RC beams without shear reinforement. (5) FE analysis in whih shear raks, splitting tensile raks, and main bars are modeled disretely an predit the diagonal tensile failure strength and the size effet on shear strength. However, modeling of dowel ation, whih an be onsidered to influene the number of main bars, bar diameter and spaing, and onrete over thikness, remains an important issue that will have to be addressed in the future. Referenes Kupfer, H. and Hilsdorf, K. H. (969). Behavior of onrete under biaxial stresses. ACI Journal, No. 65-5, Noguhi, H. and Maruta, M. (98). Nonlinear finite element analysis of shear resisting mehanism of reinfored onrete beams. Proeedings of JCI Colloquium on Shear Analysis of RC Strutures, Maekawa, K. and Okamura H. (983). Elasto-plasti frature model for onrete under plane stress states. Conrete Journal of JCI, (5), Okamura, H. and Maekawa, K. (985). Non-linear finite element analysis of reinfored onrete. JSCE Journal of materials, onrete strutures and pavements, 36/V(3), -. Rots, G. J., Nauta, P., Kusters, G. M. A. and Blaauwendraad, J. (985). Smeared rak approah and frature loalization in onrete. Helon, Vol. 3, No. Reinhardt, W. H., Cornelissen, W. A. H. and Hordijk, A. D. (986). Tensile tests and failure analysis of onrete. ASCE Journal of Strutural Engineering, (), Niwa, J., Yamada, K.,Yokozawa, K. and Okamura, H. (987). Reevaluation ohe equation for shear strength of reinfored onrete beams without web reinforement. Conrete Library International of JSCE, 9, Shima, H., Chou, L. L. and Okamura, H. (987). Miro and Maro Models for Bond in Reinfored Conrete. Journal ohe Faulty of Engineering, The University of Tokyo, (B) Vol. 39, No. Li, B. and Maekawa, K. (988). Contat density model for shear transfer of raked onrete. Conrete Journal of JCI, 6(), Tamai, S., Shima, H., Izumo, J. and Okamura, H. (988). Average stress-average strain relationship of steel in uniaxial tension member in post-yield range. Conrete Library International of JSCE,, 7-4 Okamura, H. and Maekawa. K. (99). Nonlinear Analysis and Constitutive Models of Reinfored Conrete. Giho-do. Hordijk, A. D. (99). Loal Approah to Fatigue of Conrete. Delft University of Tehnology. Asin, M., Walraven, J. C. (995). Numerial behavior of reinfored ontinuous deep beams. Heron, 4(), Yamaya, A., Nakamura, H. and Higai, I. (995). Shear behavior analysis of RC beams using rotating rak model. JSCE Journal of Materials, Conrete Strutures and Pavements, 6/V(43), (in Japanese) Sanada, O., Furuuhi, H., M., Ueda, T. and Kakuta, Y. (995). Analytial study on diagonal tension failure of reinfored onrete beams. Proeedings of JCI,

15 Y. Sato, T. Tadokoro and T. Ueda / Journal of Advaned Conrete Tehnology Vol., No. 3, 37-34, 4 34 Vol.6, Maekawa, K. and Quresh, J. (997). Stress transfer aross interfaes in reinfored onrete due to aggregate interlok and dowel ation. Conrete Library International of JSCE, 3, An, X., Maekawa, K. and Okamura, H. (998). Numerial simulation of size effet in shear strength of RC beams. Conrete Library International of JSCE, 3, Nakamura, H. and Higai, T. (999). Compressive frature energy and frature zone length of onrete. Seminar on Post-peak Behavior of RC Strutures Subjeted to Seismi Loads,, Kim, W. and White, N. R. (999). Hypothesis for loalized horizontal shearing failure mehanism of slender RC beams. Journal of Strutural Engineering, 5 (), Tadokoro, T., Sato, Y. and Ueda, T. (). Influene of ompression softening on ultimate deformation of reinfored onrete members. JSCE Journal of Strutural Engineering, 47A(3), 39-34, (in Japanese). Kaneko, Y. and Mihashi, H. (). Influene of onstitutive laws on shear failure of onrete beams without web reinforement in several shear-span ratios. Finite Element Analysis of Reinfored Conrete Strutures, ACI, SP5, pp Tadokoro, T., Sato, Y. and Ueda, T. (). Influene of ompression softening on shear apaity of reinfored onrete members. Proeedings ohe Japan Conrete Institute, Vol. 3,, pp JSCE(), Standard Speifiations for Conrete Strutures Strutural Performane Verifiation. Tokyo, Japan. (in Japanese)