New model for Shear Failure of R/C Beam-Column Joints. Hitoshi Shiohara

Transcription

1 New model for Shear Failure of R/ Beam-olumn Joints Hitoshi Shiohara Dept. of Architectural Engineering, The University of Tokyo, Tokyo , Japan; PH +8(3) ; FAX+8(3) ; Abstract A new mathematical model for shear failure of R/ -column joint is introduced and the effects of the effects of anchorage strength of bars passing through joint are investigated with the model. The model assumes two independent deformation modes, i.e. mode, due to the opening of the crack at the end of, and joint mode, due to the opening of the diagonal crack in joint, which is equivalent to shear failure of -column joint. The model is demonstrated with example how anchorage capacidty affects the failure mode and strength by the comparison of strengths of the two modes. Introduction urrent building codes for an earthquake resistant design of reinforced concrete moment frame structure provide an empirical limit on joint shear force to avoid failure of -column joints exhibiting degradation of story shear and concentration of shear deformation in -column joint, (AI 995, AIJ 994, SANZ 995). However the limit value for joint shear is empirical and not based on mathematical models. The empirical joint shear strengths in the codes are usually function solely of concrete strength, as seen in the AI code typically, proportional to square root of the concrete compressive strength. In these codes, bond capacity is neglected as a factor which affects the joint shear strength nor failure mode. A new mathematical model for prediction of story shear and failure mode was proposed (Shiohara ) and it is introduced briefly and expanded in this paper. The model can taken into account many parameters including configuration, geometry, reinforcing arrangement, material properties and boundary conditions of various type of -column joints. In this paper, this model is demonstrated to investigate the effect of the bond capacity on the story shear and failure mode of interior -column joints. Modeling of Beam-olumn Joints Behavior at joint shear failure By the re examination of test data of past tests, fundamental causes of joint shear failure were investigated in the paper (Shiohara ). Most test data of -column joint specimen exhibiting joint shear failure showed following sequence of behaviors. At the stage the joint shear failure initiation; or the strength decay begins by cyclic loading, bars slipping into the joint at the in compressive side, while bars slipping out in at tensile side, as shown in the Fig. below. This cause compressive bars slip into the joint, lost the resistance to compressive force, the capacity of decreases and as a result, story shear decreased. Therefore the saturation of anchorage stress is an important factor which need to reflected in the mathematical model to explain the reason of reduction of story shear.

2 Tension ompression olumn Beam Beam end cracks Diagonal cracks Tension ompression olumn Beam slip in slip out B (Beam) Mode J (Joint) Mode Figure ause of story shear degradation Figure Two sets of flexural critical sections Before the initiation of story shear degradation, the longitudinal strain in bars usually distribute as shown in Fig. (a). If joint shear increase and bond slip begins, the bars slip into the joint at the end in compressive side, while bars slip out in tensile side with keeping constant bond resistance, because bond slip occurred in confined concrete. oncurrently, the elongation of bars cause opening of crack in concrete accompanied by bond-slip along bar to satisfy the deformation compatibility in bars and joint concrete. Figure shows two dominant sets of cracks. The elongation of bars may contribute to opening crack at column face as depicted in Fig. (a), which contribute to end rotation. Another possibility is that the elongation of bars may contribute to opening diagonal cracks in joint panel as shown in Fig. (b), which make the rotation of four triangular segments, which causes shear deformation of joint panes. Two deformation modes in -column joints Hereafter the deformation mode shown in Fig. (a) is named B (Beam)-mode. The other deformation mode shown in Fig. (b) is characterized with shear deformation in the joint cause by local crushing of concrete at the center of the joint due to large rotation of the segments. Hereafter the deformation mode is named J (joint)-mode. As shown in the reference, joint shear failure observed in laboratories are well portrayed by J- mode deformation. Flexural Strengths for B-mode and J-mode To predict the how the component of story drift is distributed to two modes of deformation, the following idea may be used. Two deformation modes have their own moment resisting mechanism and strength. Thus the lateral resistance of -column joint for each deformation mode are calculated assuming critical sections for B-mode and J- mode. If the calculated resistance for B-mode is smaller than that for J-mode, contribution of B-

3 mode deformation to total story drift is smaller. By this way, the deformation mode in -column joint may be predictable. The lateral resistance of -column joint subassembledge are discussed in the following sections. N c N b Story shear V b = L c V c L b V c olumn section j c L c / N b N c (Thrust force in columns) N b (Thrust force in s) V c (olumn shear; story shear) V b ( shear) L b (distance of inflection points) L c (distance of column inflection points) j b tanθ θ V b = L c L b V c L c / (Unit for length = column depth) Beam section V c L b / L b / N c Figure 3. Definitions and notations for applied forces to a -column joint substructure ritical Sections at oncrete Bondary Force in Rebars Beam Mode Joint Mode Quadra Flexural Actions Model σcj tanθ Left σcj Left Lower column tanθ σ cj σ cj σ cj σ cj Upper column σ cb Lower column σcj σcj tanθ σ cb Upper column Right tanθ θ tanθ tanθ θ tanθ D tanθ D Right Left A Lower column Left A T 4 T4 T3 T 3 T 3 T T 5 T 5 T T T T T T 4 D Upper column B Lower column Upper column Right Beam B D Right Figure 4. Two sets of critical secions and notations defining internal forces

4 Generic Beam-olumn Joint Substructure The geometry and dimensions of generic -column joint substructure including applied loads are defined in Fig. 3. The geometry and applied loads acting on this structure assumed to be symmetric for simplicity. The depth of columns is assumed unity, a unit of length, while the depth of is assumed tanθ, where θ is the angle of diagonal line of -column joint. The forces are represented by normalized value with coefficient (b c D c σ c ), where σ c is the concrete compressive stress at concrete compressive stress block. ritical Sections Figure 4 shows two sets of critical sections, internal forces and their notations, corresponding to Beam (B)-mode and Joint (J)-mode respectively. B-mode assumes critical sections at ends for flexural action of s, while J-mode assumes critical sections for coupled four flexural actions at diagonal lines in a -column joint. Strength for B-Mode Deformation mode Assumptions and Notations in Analysis By considering the equilibrium of horizontal force and moment at the critical section at the end of as shown in Fig. 4, the relation of the internal force, and moment M b at the critical section is derived as the Eq.. M b j b ( + + N b ) N b = The definitions of notations are shown in Figs. 3 and 4. The shear V b and the column shear V c are calculated from M b. Finally the pseudo joint shear stress B ; strength of B-mode is obtained using assumptions that the length of stress resultants at section j is equal to 7/8 of effective depth of d Effective area of -column joint A eff given by the AIJ Guidelines (AIJ 994) is used. Figure 5 is an example solution for B-mode deformation. () B /σ B ontour line of B /σ B ritical Section for B-mode Figure 5 An example of numerical solution for strength of B (Beam) -Mode

5 Strength for J-Mode Deformation Assumptions and Notations in Analysis Figure 4 includes the notations necessary to define the set of internal forces at the critical sections for the J-mode (Shiohara ). Equilibrium in Forces Acting on the Segments Five equations to define the equilibrium in the system. They are given as follows. + + N b T 5 = L c T 3 T 4 tanθ + tanθ V c = --L c V c L b + -- j c ( T 3 T 4 ) + -- j b ( ) tanθ tanθ θ + tan tan θ + tan θ = + V c = T 3 T 4 + tanθ + tanθ N c = () (3) (4) (5) (6) The simultaneous equations of second order from Eq. () to Eq. (6) yield two set of solution for five unknown variables, provided the value of the other variables are confirmed. By solving the equations from Eq.() to Eq.(6), the story shear V c is calculated. Hereafter, V c, T 3, T 4, and are chosen as an unknown variables, whereas, the, and T 5 is assumed to be given. Finally the pseudo joint shear stress J of J-mode is obtained as a function of and using same assumptions on length of stress resultants at section and effective area of joint used for B mode in the previous section.figure 6 is an example solution for J-mode deformation. contour lines of pseudo joint shear J / f' c ritical Section for J-mode J / f' c.4 T.4 T - - T.4 Figure 6. An example of numerical solution for strength of J (Joint) -Mode

6 : pseudo joint shear stress = : pseudo joint shear stress = B-region B < J = = B-region B < J = = B = J B = J - jj = jb B > J J-region T B > J J-region (a) T 4 =.4 (constant) with less joint hoops (b) in case joint hoop increased.5 times Figure 7. alculated joint shear stress by the new model Difference in Strengths for J-mode and B-mode Figure 7 shows the combination of contour lines by taking maximum value {J, B } for each set of values of (, ). The boundary curves are also plotted on (, ) plane, which connects the points at which J equals to B. Provided the point (, ) locates in the upper left of the boundary curve, the value of J is always larger than B. In this region, the strength of B-mode is smaller and B-mode deformation becomes dominant while the J mode deformation is minor. n the contrary, J-mode deformation becomes dominant if the point (, ) locates in the lower right region. Thus this boundary curves is useful to predict the deformation mode. The two regions divided by the boundary line are called B-region and J-region, where each joint shear deformation becomes dominant. Role of bond capacity in determining failure mode If straight bar has small bond capacity such as plain bar, the anchorage capacity is small and is almost zero. It means the points (, ) always stay on the 45 degree line passing through the origin of the plain. As shown in Fig. 8(a) and (b), the line always lies within the B-region. Therefore, the B-mode deformation is dominant for the -column joint with debonded bars passing through the joint. n the contrary, if anchorage strength is increased by some way including increased number of low strength steel or adding some special anchoring device in -column joint, the J- mode become more dominant. By this way, this model explains how the bond capacity affect the strength and the failure mode. Prediction of Failure Modes Loading path This model has a practical application, by considering a possible loading path in the plain of (, ). There are two physical constraint on the range for and. ne constraint is tensile strength of bars. The other constraint is anchorage capacity of straight bar through -column joint as shown in Fig. 8(a). Upper bound of the value of and is determined by the tensile strength of steel. The upper bound of anchorage force is determined by

7 the anchorage capacity. Figure 8 shows a typical loading pass in column joint. When a -column joint substructure is loaded monotonically, the and will increase proportionally, as far as the bond is capable to resist to the input bond force like a line - shown in Fig. 8. However once the bond capacity attains to its capacity, the loading path will turn the direction moving on the 45 degree line like a line -. If these constraint is taken into account, the behavior of -column joint is predictable when it is subjected monotonically increasing load. Provided bar yield while the point of (, ) remains within the B-region as shown in Fig. 8(a), the behavior is predicted as -yielding failure mode. If enough amount of bars are provided such that the bond strength B u is attained at point 3 and mode switches from B-mode to J-mode before bars yield, the loading pass will be change the direction and moves on the line 3-4 as shown in Fig. 8(b). This type of behavior is of J-mode dominant at ultimate stage, failure mode is described as joint shear failure after yielding. If more bars are provided, loading path reach the point 3 then heads to 4 at which maximum joint strength is reached. If it is loaded further, strength degradation in story shear follows. This type of behavior is of joint shear failure mode. The strength of -column joint of shear failure mode is calculated from the counter line of J-mode, if value of bond strength is given. As can be seen in Fig. 8(c), story shear of -column joint is enhanced by providing larger anchorage capacity to bars. If bars are deb- comp. tens. No load Beam bars yield 3 Anchorage capacity attained tens. No load comp. Mode switch from B to J 3 4 Beam bars yield Anchorage capacity attained (a) Beam yielding tens. (b) Joint shear failure after yielding tens. Story shear degradation starts Finally bars yield 4 5 tens. Story shear degradation and compressive flexural failure starts at ends 3 comp. No load Mode switch from B to J 3 Anchorage capacity attained No load comp. Anchorage capacity attained (c) Joint shear failure without yielding (d) Beam end failure without yielding Figure 8. Prediction of strength and failure mode

8 onded, loading path moves lines --3 as shown in Fig. 8(d) and always stays with in B-region. Hence the failure mode is of end failure without yielding. onclusions A new model is proposed to predicting the effect of anchorage strength affecting the strength and failure modes of rein-forced concrete -column joint. The prediction of the model is as follows,. The new model offer a unified view which enables us to understand the complex behavior of -column joint affected by possible factors, including joint shear force, type and strength of anchorage, material strength and amount of joint hoop, etc. The input joint shear and bond strength and their interaction.they are not independent. Each factor have independent influences on the strength and deformation mode of -column joint.. It is demonstrated that the model can predict that, the anchorage strength is more fundamental factor than joint shear and anchorage strength has influences on strength and failure mode of -column joint. It is also predicted that larger bond capacity increase the joint strength, while the joint shear deformation increase relative to deformation of or column, if sufficient amount of joint hoops is not provided. References American oncrete Institute (AI). (995). Building ode Requirements for Structural concrete and ommentary. AI 38-95, Farmington Hills, Michigan. Architectural Institute of Japan (AIJ), (994). AIJ Structural Design Guidelines for Reinforced oncrete Buildings. Tokyo, Japan. Standard Association of New Zealand (SANZ). (995). oncrete Structures Standard: Part - The Design of oncrete Structure. NZS 3, Wellington, New Zealand. Paulay, T., Park, R., and Priestley, M. J. N., (978). Reinforced oncrete Beam-olumn Joints Under Seismic Actions. J AI, 75(), Shiohara, H. () New Model for Shear Failure of R Interior Beam-olumn onnections, J Struct. Eng., ASE, 7, February, 5-6.