Proceedings of the 8 th U.S. National Conference on Earthquake Engineering pril 8-, 006, San Francisco, California, US Paper No. 73 NLYSIS F REINFRCED CNCRETE KNEE JINTS SED N QUDRUPLE FLEXURL RESITNCE Hitoshi Shiohara and Yong Woo Shin bstract The quadruple flexural resistance is a novel concept providing a unified view and explains the mechanics and failure modes, applicable to reinforced concrete interior, exterior as well as knee joint in a consistent way. The performance of knee joints is known to be affected by geometry, dimension, material strength as well as anchorage detailing of longitudinal bars. So supplemental criteria for eliminating knee joint with poor anchorage detailing are incorporated into the new theory by considering local failure of anchorage within knee joints. The validity of the new theory is examined by comparison of the predicted ultimate strengths and failure modes using test results of fifty six knee joint specimens reported by the researchers from US, Japan and New Zealand. Introduction eam-column joints can be a critical part in reinforced concrete frames designed for inelastic response to severe seismic attack. Recent design recommendations provide empirically derived upper limits for joint shear stress to preclude shear failure of reinforced concrete beam-column joints for seismic design. The reduction factor adopted in the US (CI35 00 for joint shear capacity of knee joints relative to that of interior joints is 0.6 while the value adopted in Japan (IJ 999 is 0.4. So design recommendations in both US and Japan recognize that the joint shear capacity is significantly affected by different configuration, while New Zealand code (SNZ 995 does not require such reduction of joint shear strength for knee joints. The recommended values for reduction of shear capacity are such varied in countries, because the adopted reduction factors are empirical and no theory or mathematical models are considered. This is a challenging issue to be solved in future. To solve this issue, a novel model was proposed (Shiohara 00, 00, 003, 004. The model is comprehensive and unified in which joint shear failure for all types of beam-column joints with different configurations is intrinsically incorporated and successfully applied ssociate professor, Dept. of rchitecture, The University of Tokyo, Tokyo 3-8656, Japan Graduate student, Dept. of rchitecture, The University of Tokyo, Tokyo 3-8656, Japan
(Shiohara 004. This theory requires no empirical assumptions accounting the difference in strength and type of failure, despite of various geometries of beam-column joints. However the modeling of knee joint is more challenging than that for interior joints, because various anchorage detailing for longitudinal bars exists. So supplemental criteria are incorporated into the theory to extend the model such that it could eliminate knee joints with poor performance due to the problems of anchorage detailing. This paper also reports a correlation study of the theoretical prediction with the test results of fifty six knee joint specimens. Quadruple Flexural Resistance in R/C eam-column Joints The Concept of Quadruple Flexural Resistance simple mathematical model for interior beam-column joint was first introduced by the first author (Shiohara 00. The basic b c concept of that model is called b c quadruple flexural resistances (Shiohara 004 depicted in Fig.. It considers the kinematics of the four segments divided by flexural critical sections in a joint and their equilibrium by the analogy of flexural resistance each other segment by a pair of force resultants in tension and compression c c b b arising in reinforcing bars and concrete. Figure. Quadruple flexural resistance in reinforced The failure criteria of material used for concrete beam-column joint the analysis of interior beam-column joint are ( yielding of steel reinforcement, ( compressive failure of concrete and (3 upper bound of bond strength of the longitudinal bars passing through the joint (Shiohara 004. The original model of quadruple flexural resistance for beam-column joint (Shiohara 004 is not necessarily capable to predict the outcome of inappropriate anchorage detailing which cause local premature failure or shear failure as a result. Two Kinematic Modes efore discussing how the inappropriate anchorage detailing affects the capacity of joints, let us presume that the anchorage of longitudinal reinforcement in the joint are anchored without deficiency. Two kinematic modes for a knee joint under both closing and opening loading were shown in Fig.. The former is hereafter called -mode which represents the flexural resistance mechanism at beam and/or column end, and the latter is hereafter called J-mode which represents quadruple flexural resistance mechanism on four diagonal flexural sections in joint. The objective of the introduction of the two kinematic modes is to obtain two joint shear strengths calculated for both J-mode and -mode. The two modes of J-mode and -mode usually give different strength reflecting a difference in critical sections. The smaller value of the strengths gives the real strength and its kinematic mode becomes dominant. Thus prediction of
dominant deformation mode is feasible by comparing the two strengths. Symmetric knee joint substructure shown in Fig. 3 is considered. To reduce the number of independent equations and simplify analysis, the geometry of joint is assumed to be symmetric in horizontal and vertical direction. The joint is square with size of D and with the thickness of t. The distance between the center of joint and inflectional points in beam and (eam mode J(Joint mode (a Loading and deformation in closing direction (eam mode J(Joint mode (b Loading and deformation in opening direction y x Figure. Two kinematic modes for reinforced concrete knee joint N L D jd Critical sections D jd T 4 c N L (a boundary conditions C D c b (b geometries b 45 T 4 T 5 T 3 T 4 T 3 C T 3 (c equibrium in free body (closing C C 0.85 T 5 T 3 T C 3 T Critical sections T D 0.85 N C T 4 T 4 T 5 Critical sections T 3 T C T 3 T C Critical sections N D (with idealization of stress blocks (d J-mode (closing directional loading (e -mode (closing directional loading T 5 C 6 0.85 C 5 T 0.85 T 3 T D T N T T 4 T 3 T 4 T 5 C 4 T T C 5 C T 3 T Critical section N D (with idealization of stress blocks (f J-mode (opening directional loading (g -mode (closing directional loading Figure 3. Model for knee joints loaded to closing and opening direction
column is L. The equal column shear ( N acts on the inflectional point of beam and column. The distance of tensile and compressive reinforcement is jd in both beam and column. Critical Sections and Stress Notations Figure 3 shows the assumed critical sections for -mode and J-mode of knee joints. The notations for resultant stress of reinforcing steel and concrete block at critical sections for J-mode of knee joint subjected to closing and opening loading are shown in Fig. 3 (d and Fig. 3 (f. The notations for T, T, T 3 and T 4 represent the resultant tensile forces of longitudinal bars, while C, C and C 3 represent the resultant compressive forces of concrete block subjected to closing loading and C 4, C 5 and C 6 represent those of concrete block subjected to opening loading. The values from C to C 6 are equal to the x component of compressive resultant in concrete and the directions of them are assumed to be normal to the critical sections. The distributions of them are assumed to be rectangular block with compressive stress of σ c. The forces of vertical and horizontal joint hoops are represented as T 5. They are assumed to act at the center of joint panel and same in vertical and horizontal direction. Equilibrium Conditions for J-mode and -mode To predict the capacities of the J-mode and -mode by applying the failure criteria of materials, equilibrium conditions are used to relate the applied load to internal stresses. Three equilibrium equations are necessary for each free body. Thus, six equations are needed to satisfy equilibrium of four free bodies if considering the symmetry. However, one of the equations depends on the other five equations. Finally, five independent equations remain for closing loading and opening loading respectively. y solving them, the resultant stresses for reinforcing bar, concrete stress block and joint shear force can be obtained. For the case under closing moment, the independent equilibrium equations are from Eqns. ( to (5, T T + C + C T N 0 ( 3 3 5 T4 + C + C T5 0 T C + C3 0 T ( T (3 C C jd(t 4 T T3 + C 0 D tσ c tσ c C C jd( T T 3 + T3 + C3 D L tσ c tσ c whereas, for the case under opening moment, the independent equilibrium equations are Eqns. (6 to (0. T T + C + C T + N 0 (6 3 5 6 5 T4 + C4 + C5 T5 0 T (7 0 (4 (5
T T C + C + 0 (8 5 6 C 4 C jd(t 5 4 T T3 + C5 0 D tσ c tσ c C 6 C jd( T 5 T + T3 + C5 + 0 D L tσ c tσ c The equilibrium condition for the free body for -mode are given by Eq. ( for horizontal direction and moment around the beam-column joint by Eq. (, where sign (+ stand for ( for closing and (+ for opening respectively. T T + C ± N 0 ( 3 C C jd ± jd( T T3 + ( D ( L 0 t σ c y solving the simultaneous equations above, the column shear C ( : beam shear can be obtained by considering the geometry. Failure Criteria of Materials Concrete It is assumed that diagonal cracks transfer no tensile force across the crack. Resultant forces in compression are transferred by compressive reinforcement and/or across concrete cracks. In J-mode, each direction of principle stress on the critical section is assumed parallel to the diagonal direction of the joint panel. n the critical sections, distribution of the concrete stress is assumed as a rectangular stress block, where the concrete stress is σ c and 85% of concrete compressive strength, typical value for flexural analysis. Reinforcing Steel Tensile force is transmitted by the longitudinal bars. Thus it is assumed that they do not exceed the yielding force. The typical restrictive conditions are given by Eq. (3. T 3 a t f (3 y where, Σ a t : total cross section area of tensile reinforcements, and f y : tensile yield point of longitudinal reinforcement. Joint shear reinforcing bars are assumed that they are concentrated at the mid-height of the joint and always equal to the yielding force and given by Eq. (4, T p t( jd 5 w f (4 sy (9 (0 ( where, p w : joint shear reinforcement ratio, f sy : tensile yield point of joint reinforcement.
Table. Prediction of failure mode of knee joint Failure type Condition Explanation J M uj < M ub Joint shear deformation dominant M uj > M ub eam end deformation dominant Prediction of Strength and Failure Modes When the equilibrium equations are solved, longitudinal bars in beam and columns are assumed perfectly anchored for the simplicity. Under closing moment, the value of T, T ; the resultant forces in longitudinal bars in compression are assumed to be zero. Then, five unknown variables, T 4, C, C, and C 3 are determined as a function of T 3 by solving the simultaneous equations from ( to (5. Under opening forces, the values of C 4, T ; the resultant forces in concrete and on longitudinal bar are assumed to be zero. s a result, five unknown variables, T 3, T 4, C 5, C 6 are obtained as functions of T by solving the simultaneous equations from (6 to (0. In both cases, the force T 5, confinement due to horizontal and vertical joint reinforcement are assumed to be equal to the yield stress. y substituting the yielding force of reinforcing bars into T 3, beam shear of J-mode is obtained as J-mode. The moment capacity of J-mode is hereafter defined as M j L x J-mode. y substituting the value T 3 and T into Eqns. ( and (, considering N, beam shear of -mode is obtained as -mode. The moment capacity of -mode is hereafter defined as M b L x -mode. The predicted strength M uj of beam-column joint is obtained as smaller one of J-mode and -mode. ased on the calculated hierarchy of the strengths of J-mode and -mode, type of failure mode of J, or is determined by applying rules given in Table. Consideration of Premature Failure due to Deficiency in nchorage Detailing Four possible types of premature failure shown in Fig. 4 due to deficiency in anchorage detailing of longitudinal reinforcement at the outer side of the joint are taken into account. They includes, ( anchorage failure of longitudinal bars at outer corner (see Fig. 4(a, ( crushing of bearing concrete at 90 degree bend anchorage of longitudinal reinforcements at outer corner (see Fig. 4(b, (3 bond failure of longitudinal reinforcement anchored in the outer cover concrete (see Fig. 4(c, and (4 flexural failure by yielding of longitudinal reinforcement at outer corner before yielding at column or beam face (see Fig. 4(d. To tell whether the premature failure should happen, force T 4 (force of longitudinal bar in outer bar is compared with each failure criteria shown in Table. The capacities of anchorage (T s (see Fig. 4 (a, bear (P bear (see Fig. 4 (b, bond capacity ( u (see Fig. 4 (c are adopted from Cui et. al. (00, IJ (994 and Paulay et. al. (99, respectively. The force T 4 is approximated from equilibrium of forces in x and y directions acting on the triangular free body as shown in Fig. 3(c with assumption that the T 5 is relatively small and could be neglected. It is concluded that the estimated tensile force T 4 is always nearly equal to the half of the value of T 3 by considering the equilibrium.
Discussion ehavior of Knee Joint with Ideal nchorage Detailing Diagram for the Predicted Strength for J-mode and -mode The calculated relationship between the most influential parameters T 3 (for closing moment, T (for opening moment and joint shear stress τ for a typical knee joint are illustrated in Fig. 5. It is assumed that there is no anchorage deficiency in this case. It is observed that the capacity under closing moment is always larger than that under opening moment. It is also observed that under closing moment (Fig. 5 (a the rate of increase in joint capacities gradually decrease with increasing of tensile stress T 3 at critical section. The strength of -mode shows similar relation to that of J-mode. In addition to that, strength of -mode is always smaller than that of J-mode with a little margin. Thus, it is predicted that there is no J-mode failure if there is no anchorage deficiencies. n the contrary, under opening moment (Fig. 5 (b, the joint capacities of -mode linearly increases, while the rate of increase for J-mode capacity gradually decreases with increasing of Tmod T4 tensile stress T at critical section. Thus, the order of the J-mode strength and -mode strength switches at certain point. It means that joint shear failure initiates with beam or column bar yielding if the amount of longitudinal bars is higher than the certain limit point. The maximum joint shear strength without yielding of longitudinal bars is estimated 4.7% of concrete compressive strength as shown in Fig. 5(b. The allowable upper limit by CI (reduction factor of 0.85 is applied is greater than that of IJ about.5 times. These facts are summarized that knee joint shear panel is generally vulnerable to opening moment if there are no deficiency in longitudinal bars anchorage detailing. Comparison of Strength and Failure Modes with Inventory Test Results Fifty six specimens were chosen from the literature published in the US, Japan and New Zealand. Failure modes are classified into four types, including ( flexural yielding at beam or/and column end ( failure, ( joint shear failure after flexural yielding at beam or/and column end (J failure, (3 joint shear failure without bar yielding (J failure and (4 anchorage ' ' (a nchorage failure of outer bar modt4 T s u T3 T3 ' ' (c ond failure of outer bar ' ' (b earing failure of outer bar Ty Pmod T4 T3 Pbear Tymod T4 Ty T3 ' ' (d Yielding of bend radius Figure 4. Four types of premature failures under closing moment
Table. Criteria for premature failure of anchorage of longitudinal bars in outer side Type of Failure 4 s s T s : anchorage force (kn s: average where anchorage stress (MPa L s : vertical anchorage length of beam bar (mm : s 7.45 + ( 3.0 + (.3 Ls / db perimeter of bar (mm n: number of bar 0.3 +.6 (9 d b d b : diameter of bar (mm : compressive concrete strength (MPa 5 + 78 (9 d b P bear : bearing force (kgf d b : diameter of P bar (cm r: radius of bend (cm, l : mod T4 (/ T3 Pbear w db fbear development length (cm s : area of lateral where joint reinforcement (cm, s: distance of 0.84 w r lateral joint reinforcement (cm,, ( r /3d, 6.C : b 0 / db compressive concrete strength (kgf/cm, C 0 : f, width of concrete cover from column side bear + 30 s /( l s face to center of main bar (cm Fig. 4 (c u : bond strength (N u.35 (MPa nchorage Fig. 4 (a Cui et.al (00 Concrete earing Fig. 4 (b IJ (994 ond Paulay et.al (99 Yield Fig. 4 (d T T (/ T3 mod mod T s / T l ' L ( 3 u u T T 4 y mod T y Failure Criteria n Notations : perimeter of bar (mm l ': development length (mm T y : yield strength of beam or column bar 0.35 τ/σ J-mode 0.35 τ/σ (joint shear stress σ (concretecompressive strength τ 0.3 0. 0. 5 0 J-mode(T3 -mode transverse joint hoop ratio ρ IJ 0.(minimum % -mode T3/Dtσ c 0. 0. 0.3 0.4 Steel tensile strength T3 at critical section (a Joint shear stress for closing loading τ (joint shear stress σ (concrete compressive strength 0.3 0. 0. 5 0 CI(.50 0.46(τ /σ (max J-mode(T IJ(0.98 -mode 56 transverse joint hoop ratio ρ IJ 0.(minimum % 96 T/Dtσ c 0. 0. 0.3 0.4 Steel tensile strength T at critical section (b Joint shear stress for opening loading Figure 5. Diagram for the predicted strength for J-mode and -mode failure or bond failure due to slippage of main reinforcement in beam or/and column (T failure in this paper. Figure 6 shows the comparison of test values with calculated ones using the theory in this report. ll the capacities are determined by the yielding of the longitudinal reinforcement. The horizontal axis represents the calculated ratio of J-mode to -mode (M j /M b, while the vertical axis represents the ratio of experimental value M jexp to the value from this theory (min(m uj, M ub : the smaller of M uj and M ub. The average ratio of the observed to the analysis is.09 and.0 for closing moment (Fig. 6(a and for opening moment (Fig. 6(b respectively. There seems to be good correlation between test and analysis if those specimen exhibited T failure are excluded. It should be noted that for opening moment, all specimens reported exhibited J failure are located left side of the line of.0 (i. e. (M j /M b <.0, while most of failure specimens are positioned right side of the line of.0 (i. e.(m j /M b.0 for opening moment. The specimens
M jexp / min (M j,m b.0.6. 0.8 0.4 J failure failure J failure T failure M jexp / min (M j,m b.0.6. 0.8 0.4 J failure failure J failure T failure 0. 0.4 0.6 0.8.0..4.6.8.0 0. 0.4 0.6 0.8.0..4.6.8.0 M j / M b M j / M b (a under closing moment (b under opening moment Figure 6. Comparison between test and theory using 56 knee joint test specimens M jexp / min (M j,m b M jexp / min (M j,m b.0.5.0.0.5.0.0.5.0.5 3.0 3.5 4.0 laue of anchorage force (T s / alue of anchorage force from model J failure failur J failu T failure J failure failure J failure T failure.0.5.0.5 3.0 laue of bond force ( u / alue of bond force from model 3.5 M jexp / min (M j,m b.0.5.0.0.5 J failure failure J failure T failure.0.5 laue of bearing force (P bear / alue of bearing force from model Figure 7. Correlation with anchorage, bearing and bond failure exhibited J failure scatter evenly around the line of.0. Thus, the observed failure mode is distinguishable by the ratio of J-mode strength to -mode strength from this model for opening moment. n the other hand, for the prediction of failure mode under closing moment, the ratio of strength of -mode to J-mode is useless. Consideration of Deficiency in nchorage Detailing s shown in Fig. 4, the knee joints under closing moment are vulnerable due to the deficiency in anchorage detailing of longitudinal bars in outer side of joint. To examine how the four premature failures criteria listed in Table correlate to the test results, the same inventory test data are used again. The ratios of the demand stress calculated by the model to the capacities calculated by the failure criteria listed in Table are plotted in Fig. 7. The ratios are plotted against the ratio of moment capacity in test to that by the analysis. The shaded area means the demands for anchorage is larger than their capacities. It is observed from Fig. 7 (a that T failure (i. e. pull-out failure specimens certainly fell
into anchorage failure zone, while, the precision of Eq. (5 needs to be examined more. It is also observed from Fig. 7 (b that all J, J or T failure specimens fell into area of bearing failure and show the strong correlation with failure mode observed. This fact suggests that concrete bearing failure could play an important role to determine joint shear failure under closing moment. It is presumed that the bearing failure of concrete could cause crushing concrete inside of the anchorage hook and accelerate opening of diagonal cracks. s a result the specimen is reported as joint shear failure happened. ll specimens fell into bond failure zone also judged that bearing failure of concrete also happen. It is observed that the bond failure does not much correlate with determination of failure mode. It could be because that the loss in bond resistance is not crucial because redistribution of force is possible and final failure is governed by bearing failure of hook anchorage. Conclusions mathematical model for reinforced concrete knee joint for predicting strength and failure mode is proposed based on the concept of quadruple flexural resistance, incorporated with failure criteria of anchorage detailing. It is concluded from the discussion as followings.. The new model is useful to understand quantitatively how those factors, such as material properties, joint reinforcement, amount of reinforcement direction of loading as well as anchorage detailing, affect the shear strength and failure mode of knee joints.. The new model is applied for the analysis of fifty six specimens of knee joints. It is demonstrated that the prediction of strength and failure mode by the proposed model shows good correlation with the test results. References merican Concrete Institute (CI. (00. uilding Code Requirements for Structural Concrete and Commentary, CI 38R-0. Farmington Hills, Michigan. rchitectural Institute of Japan (IJ. (994. IJ Structural Design Guidelines for Reinforced Concrete uildings. Tokyo, Japan. Standard ssociation of New Zealand (SNZ, (995. Concrete Structures Standard: Part - The Design of Concrete Structure, NZS 30. Wellington, New Zealand. Shiohara, H. (00. "New Model for Shear Failure of RC Interior eam-column Connections", Journal of Structural Engineering, SCE 7 (, February, 5-60. Shiohara, H. (00. "Effects of Interaction between Joint Shear and nchorage Force within R/C beam-column Joints on their Strength and Failure Modes", Proceedings 7NCEE, July -5, 00, oston, Massachusetts. Shiohara, H. (003. "New model for Shear Failure of R/C eam-column Joints", Proceedings of the 003 Structures Congress & Exposition, 9-3 May 003, Seattle, Washington U.S.. Shiohara, H. (004. "Quadruple Flexural Resistance in R/C eam-column Joints", Proc. of 3th World Conference on Earthquake Engineering, Paper No. 49, ancouver, Canada, 004. Cui, J., Fujii, S., Nishiyama, M., Watanabe, F. (003. "Modeling of Load-Resistance Mechanism on Corner Joint", Journal of Structure and Construction, IJ 567, May, 0-09. (in Japanese Paulay, T. and M. J. N. Priestly (99. Seismic Design of Reinforced Concrete and Masonry uildings, John Willy & Sons. Inc., Christchurch and San Diego.