RC DEEP BEAMS ANALYSIS CONSIDERING LOCALIZATION IN COMPRESSION

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1 RC DEEP BEAMS ANAYSIS CONSIDERING OCAIZATION IN COMPRESSION Manakan ERTSAMATTIYAKU* 1, Torsak ERTSRISAKURAT* 1, Tomohiro MIKI* 1 and Junihiro NIWA* ABSTRACT: It has been found that RC deep beams usually fail by the loalized ompressive failure of onrete. In this paper, the appliation of onept of loalized ompressive failure as the material model of onrete, based on the parameters suh as the loalized ompressive failure length, p, and the ompressive frature energy, G F, is performed. The analytial results using the lattie model and Mander s truss model onsidering the proposed material model show the satisfatory preditions in shear analysis of RC deep beams with and without transverse reinforement. KEYWORDS: RC deep beam, loalized ompressive failure, lattie model, Mander s truss model 1. INTRODUCTION The RC deep beam is a RC strutural member in whih the ratio of the shear span to effetive depth, a/d, is less than or equal to unity. At the failure stage of RC deep beams, along the diagonal raks onneting the loading point and supports, rushing of onrete at the upper portion of the beams in the viinity area of the loading point is usually observed, whih is alled loalized ompressive failure [1]. To obtain more aurate predition of the shear behavior of RC deep beams, not only the onept of tension softening, but the onept of loalized ompressive failure of onrete should also be inorporated. In this study, the loalized ompressive failure length, p, and the ompressive frature energy, G F, proposed by ertsrisakulrat, et al. [1] have been applied to formulate the stress-strain relationship of the onrete in ompression. The analytial methods applied in this study are the lattie model and Mander s truss model. The lattie model is onsidered as a simplified analytial model to larify the hange in shear resisting mehanism of the RC beams []. Alternatively, Mander s truss model is the modified truss model with a smaller number of degrees of freedom ompared with the lattie model. It an be used to assess the shear behavior of RC members by onsidering the interation between the shear and flexure mehanisms [3]. In this paper, the lattie model and Mander s truss model onsidering the onept of loalized ompressive failure have been utilized to evaluate the shear test of RC deep beams with and without the transverse reinforement arried out by ertsrisakulrat, et al. [1].. ATTICE MODE ANAYSIS Figure 1 shows the shemati diagram of a RC deep beam whih is modeled by the lattie model into an assembly of the truss omponents. The onrete and the reinforements are modeled into the members as shown in Fig. 1. The modeling of onrete onsists of flexural ompression members, flexural tension members, diagonal ompression members, diagonal tension members, vertial ompression members and an arh member. The reinforement is modeled into the horizontal and vertial members. The diagonal ompression and the diagonal tension members have an inlination of 45 and 135, respetively. By onsidering the onrete diagonal tension member, whih is one of the major outstanding points of the lattie model, the shear behaviors of onrete beams before and after the initiation of the diagonal raking an be aptured appropriately. *1 Department of Civil Engineering, Tokyo Institute of Tehnology, Member of JCI * Department of Civil Engineering, Tokyo Institute of Tehnology, Prof. Dr., Member of JCI

2 Flexural ompression member Diagonal ompression member a C V Width of truss member : b(1-t)/ Flexural ompression zone Diagonal tension member Vertial ompression member Vertial member Arh member Horizontal member d/8 d Conrete member Reinforement member Node Flexural tension member Fig. 1 Shemati diagram of RC deep beam in the lattie model (with transverse reinforement) b Flexural tension zone Width of arh member : bt Fig. Cross setion of a onrete beam in the lattie model The web onrete is divided into truss member and arh member as shown in Fig.. By assuming the parameter t as the ratio of the width of the arh member to beam width, b, the width of the arh member and the truss member is equal to bt and b(1-t), respetively, where <t<1. The value of t is determined in suh a way that it minimizes the total potential energy of the entire struture, Π. Π is alulated from the summation of the strain energy of eah element and the work done by externally applied load based on the elasti analysis. With the onsiderations of mesh disretization and the omplexity of flow of internal stresses in the member, a RC deep beam has been suitably modeled in half of the speimen in whih the horizontal spaing of eah adjaent nodes is equal to d/8 and the value of a/d is equal to 1. mathing to the atual speimen as shown in Fig. 1. The arh member is onneting between the loading point and the support with the thikness assumed to be (.3d+r) sin 45, where r is the bearing plate width. The tension softening model named one-forth model and the tension stiffening model has been applied to the diagonal tension members and the flexural tension members, respetively. For the reinforement members, the bilinear elasto-plasti model of steel is applied. 3. MANDER S TRUSS MODE ANAYSIS In the shear mehanism analysis, a RC strutural member may be onsidered as a strutural element of ombined mehanism between shear and flexure mehanisms [3]. Hene, the total deformation of the member an be expressed as: = u + f (1) where, u and f are total, shear and flexure deformations of a RC member, respetively. And the shear resisting apaity should be the lesser of: M y V u = Vs + V and V f = () where V u and V f represent shear fore resisted by shear and flexure mehanisms, orrespondingly. V s and V are the shear resistane due to the ontributions of transverse reinforement and onrete, respetively. M y is the yielding moment of the RC member and is the member s length. In the division of web onrete for V s and V model, it was found that, similar to the lattie model, the ratio of the model width whih minimizes the potential energy, gives a good predition.

3 3.1 SHEAR MECHANISM Figures 3(a) and (b) show the shemati diagram of one half of a RC deep beam, whih is modeled by applying Gauss -point quadrature with the normalized oordinate parameter, x 1, to Mander s truss model, for evaluating V s and V in shear mehanism, respetively [3]. In V s model, the transverse reinforements have been modeled perpendiularly to the beam axis. In V model, the inlined onrete ties have been modeled orresponding to the ineffetive zones in whih the effet of flexural raking in these regions should be eliminated. The diagonal struts represent the onrete ompression field stabilizing the truss model. In the traditional truss model, the shear resistane is assessed from the effets of transverse reinforement and onrete tensile strength in whih the shear resisted by the strut along the diagonal raks is negleted. However, for RC deep beams, the arh ation, whih is reated from the diagonal raks and longitudinal reinforement, beomes the signifiant resistane in governing the frature mehanism after the diagonal rak ourred. Thus, for simpliity, the shear resistanes by the struts in V s model (V sd ) and in V model (V d ) should be taken into the onsideration. By utilizing the virtual work method to the model, the relationships between the deformation and eah shear resistane omponent an be derived. The rak angle is simply proposed to be equal to α = tan -1 (jd/a) for RC deep beams without transverse reinforement. Whereas, for RC deep beams with transverse reinforement, the rak angle an be determined from Eq. 3 proposed by Mander, et al. [3]. 1 4 rw Av rwn + ζ p 1 t A g θ = tan 1 rwn (3) + where A g = bd, A v = b jd, p t = A sh /A g, n = E s /E, ζ = boundary ondition onstant = 1.574, A sh = ross-setional area of the longitudinal reinforement, and r w = transverse reinforement ratio. 3. FEXURE MECHANISM In order to evaluate the relationship between the shear fore due to the flexure mehanism and the deformation, for simpliity, the following way of thinking was applied. As shown in Fig. 4, before the onset of rak, the rigidity due to the flexural onrete should be onsidered. With the inrease in moment, the flexural rigidity of the setion is reduing by raking of onrete. The behavior of the setion after rak is dependent mainly on the reinforement ontent. Strut Transverse reinforement 1s 1 a a s (1-x 1 )a Inlined onrete tie Strut 1 1 Shear fore (Vf) jd tanα Steel behavior C Considered strut (a) V s model (b) V model C Considered strut V s +V sd jd=(7/8)d x 1 =.11349= V +V d Consider the flexural onrete Ineffetive zone jd Fig. 3 Modeling of RC deep beam in Mander s truss model.deformation Fig. 4 Shear fore deformation relationship for V f

4 4. CONCEPT OF OCAIZED COMPRESSIVE FAIURE OF CONCRETE 4.1 APPICATION OF THE CONCEPT In loalized ompressive failure of onrete, the loalized ompressive frature length, p, an be determined by Eq. 4 [1]. = 1.36 ; D * < 1 p = D * ; 1 D * * 18 (4) D =.57 ; D * > 18 * where D = A : the equivalent ross-setional width (mm) and A is the ross-setional area of the onrete member (mm ). The loalized ompressive failure volume, V p, an be derived by p A. The ompressive frature energy, G F, is defined as the energy required to ause ompressive failure per unit volume of failure onrete. Aording to ertsrisakulrat, et al. [1], G F obtained from the RC deep beam tests is equivalent to G F from the uniaxial ompressive tests and an be alulated from the empirial equation in terms of f as Eq. 5. F 1 4 ( N / mm ) G =.86 f (5) At this junture, by assuming V p the energy onsumed by the failure portion, E net, whih is equivalent to the area under the load-deformation urve (Ρ ), an be derived by multiplying G F with V p. To apply this onept to the material model of onrete in ompression, E net should be transformed to energy per unit volume, e net, whih is equal to the area under the ompressive stressstrain relationship (σ-ε) as expressed in Eq. 6. Here, it is noteworthy that the empirial fator K is introdued in order to take into aount the transverse reinforements and the effet of the energy onsumed by the frition and assumed to be 1.14 and 1.8 for beam without and with transverse reinforement, respetively. P d Enet G FVP p enet = K σdε = K = K = K = KGF (6) A V V 4. FORMUATION OF MATERIA MODE From the obtained parameters, the stress-strain relationship for ompression members an be proposed as shown in Fig. 5. The formula of the material model based on onept of loalized ompressive failure of onrete has been simply proposed as summarized in Eq. 7 [4]. For the asending branh (pre-peak), the relationship proposed by Vehio, et al. [5] has been applied as Eq. 7a and the onsumed energy of this part is set to be e 1 (Eq. 8a). For the desending branh (post-peak), the bilinear model has been proposed. By subtrating e net by e 1, the onsumed energy in the post-peak region, e, is obtained (Eq. 8b). By applying the empirial fator m (Eq. 8), the ε last an be derived (Eq. 8d). ine A an be obtained as Eq. 7b. Assuming the slope of ine B to be equal to E /1, ine B an be derived as Eq. 7. Finally the stress-strain relationship for ompression members an be proposed. Vehio s Equation, ε ε σ = f ; ε < ε ε ε f mf σ e 1 Vehio s Eq. ε e e net = e 1 + e ine A ε last ine B 1 E ε Fig. 5 Proposed stress-strain urve for onrete in ompression (7a)

5 ine A; ine B; = A1ε + A ; σ ε < ε < ε = B1ε + B ; last σ ε < ε The important fators in the alulations are reapitulated in Eq. 8 as e ε 1 = [ Eq.(7a )] dε = ε f 3 1 = 14 1 rw + 3 m d last 1 1 r d w = + 3 ( m 1) f ( m 1) f : A1 ; A f ε e e E : B1 = ; B = mf + ε last 1 (% ) (8a) (8) e ε F p = K ε f 3 last = E 1 G e = ε mf (7b) (7) (8b) (8d) 5. ANAYTICA RESUTS AND DISCUSSION The experimental data of 6 RC deep beams, tested by ertsrisakulrat, et al. [1] are adopted and ompared with the analytial results using the lattie model and Mander s truss model as tabulated in Table 1. It is noted that all ases of the speimens failed in the shear ompressive mode. Figures 6 and 7 show the omparisons between the experimental and the analytial results in ases where the effetive depths are, respetively, 4 and 6 mm with r w of %,.4% and.84%. The solid irles represent the experimental results (Exp.). For the lattie model analysis, the blak thin lines represent the analytial results when the original equation proposed by Vehio (Vehio) has been inorporated to the ompression member. While the blak bold lines represent the results inorporating the proposed material model (attie). The gray bold lines represent the results using Mander s truss model applying the proposed material model (Mander). In the lattie model analysis, it beomes apparent that analytial results in pre-peak region give the perfet preditions in most ases. For the post-peak region, the results applying the proposed material model show the better preditions of load-deformation relationship ompared with ones in whih Vehio s equation was used. Similarly, by inorporating the onept of loalized ompressive failure to Mander s truss model analysis, the analytial results show the aeptable tendeny to the experimental results. However, the analytial results show somewhat differene. Sine the flow of internal stresses in RC deep beams is omparatively ompliated, it is diffiult for the simplified model with a small number of members suh as Mander s truss model to predit the shear behavior aurately as the lattie model. Nevertheless, it is shown that Mander s truss model an be used to evaluate the shear resisting apaity and its deformation in an aeptable degree. Table 1 Outline of the experimental data arried out by ertsrisakulrat, et al. [1] Speimen d (mm) Beam height (mm) r w (%) f (MPa) D D D D D D r (mm) ongitudinal reinforement f y (MPa) 1 PC-φ PC-φ3 16 Transverse reinforement f wy (MPa) D6 331

6 8 6 4 oad (kn) D4 Exp. Vehio attie Mander oad (kn) D oad (kn) D Mid-span defletion (mm) Mid-span defletion (mm) Mid-span defletion (mm) Fig. 6 oad-midspan defletion (d = 4 mm) oad (kn) D6 oad (kn) D64 oad (kn) D Mid-span defletion (mm) Mid-span defletion (mm) Mid-span defletion (mm) Fig. 7 oad-midspan defletion (d = 6 mm) 6. CONCUSIONS (1) The stress-strain relationship of onrete in ompression onsidering the onept of loalized ompressive failure of onrete has been proposed based on the loalized ompressive failure length, p, and the ompressive frature energy, G F. () For RC deep beams with and without transverse reinforement failed by the loalized ompressive failure of onrete, the lattie model inorporating the proposed material model provides the high aurate predition of shear behavior until the ultimate stage. (3) By omparing with the lattie model, Mander s truss model is simpler but yields an aeptable predition on the load-deformation relationship, however, with a lower auray. REFERENCES 1. ertsrisakulrat, T., Niwa, J., Yanagawa, A., and Matsuo, M., Conepts of oalized Compressive Failure of Conrete in RC Deep Beams, J. of Materials, Conrete Strutures, Pavement, JSCE, No. 697, Vol.54, Feb., pp Niwa, J., Choi, I.K., and Tanabe, T., Analytial Study on Shear Resisting Mehanism of Reinfored Conrete Beams, J. of Materials, Conrete Strutures, Pavement, JSCE, No.58, V-6, Feb.1995, pp Mander, J.B., Kim, J.H., and Dutta, A., Shear-Flexure Interation Seismi Analysis and Design, Modeling of Inelasti Behavior of RC Strutures under Seismi oads, 1999, pp ertsrisakulrat, T., Conept of oalized Compressive Failure of Conrete and Its Appliations, Ph.D Thesis, Tokyo Institute of Tehnology, Tokyo,. 5. Vehio, F.J., and Collins, M.P., The Modified Compression Field Theory for Reinfored Conrete Elements Subjeted to Shear, ACI Journal, Vol. 83, No., 1986, pp