 Research
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Simplified Design Equation of Minimum Interior Joint Depth for Special Moment Frames with HighStrength Reinforcement
International Journal of Concrete Structures and Materials volume 12, Article number: 70 (2018)
Abstract
To avoid excessive slip of beam longitudinal bars at the joints of an earthquakeresisting moment frame, ACI 318 Building Code set a minimum joint depth of 20 times the diameter of the largest longitudinal beam bars passing through the joint, which is based on prior experimental verification of beamcolumn joints with Grade 420 MPa reinforcement. In view of that the 20bardiameter criterion cannot be simply extended for concrete frame joints with higher grade reinforcement, this paper summarizes international existing design criteria and proposes a simplified equation for the minimum joint depth. The equation applicability is assessed by evaluating the cyclic testing results of beamcolumn joints conducted in East Asian and Pacific Countries, where Grade 490, 590, and 690 MPa reinforcement have been used for earthquakeresistant concrete structures. Beamcolumn joints that satisfy the proposed equation can demonstrate satisfactory hysteresis behavior at an interstory drift of 4%.
Introduction
A welldesigned beamcolumn joint in a special momentresisting frame should be able to transfer beam and column moments and anchor the beam and column longitudinal reinforcement. Under the design basis earthquake (DBE) or the maximum considered earthquake (MCE), beam hinging adjacent to the faces of beamcolumn joints is anticipated, which results in a severe bond stress demand along the straight beam bars extending through the joint. As shown in Fig. 1, the straight beam bars in the joint are subjected to tension at one face of the joint and compression at the opposite face. If the column dimension or joint depth is relatively short, the beam bar tension cannot be fully developed in the joint because of excessive bond demands, but instead the beam bar must be anchored within the beam on the opposite side of the joint. This results in an increase in flexural compression at the opposite beam end, leading to crushing of the concrete. Figure 2 shows consequences of excessive bond demands at an interior joint with adequate transverse reinforcement (Lee et al. 2007). Moehle (2015) indicated that such bond deterioration along the beam bars in the joint and severe damage at the beam ends enable the bars to slip almost freely, resulting in excessive deformation of the beamcolumn joint for small lateral force changes (very pinched hysteresis behavior), and thereby reducing the energy dissipation capacity. From analytical and experimental observations, Hakuto et al. (1999) and Shiohara (2001) also indicated that the bond deterioration could cause degradation in beam flexural strength. The occurrence of bond deterioration or joint core anchorage failure in a DBE event may render the frame flexible and prone to large lateral drift in the later moderate earthquakes. Because severe bond deterioration is difficult to repair, and it should be avoided in the design of a special moment frame at the drift demand from the DBE event.
ACI 318 Building Code (ACI Committee 318 2014) has set a minimum joint depth of 20 \(d_{b}\), where \(d_{b}\) is the diameter of the largest longitudinal beam bars extending through a joint of a special moment frame. This 20 \(d_{b}\) criterion is based on an evaluation (Zhu and Jirsa 1983) of cyclic loading response for 18 beamcolumn joints made with normalstrength concrete and Grade 420 MPa reinforcement. Zhu and Jirsa (1983) concluded that a minimum column depth of 20–22 \(d_{b}\) is appropriate to avoid excessive bar slip at an interstory drift of 3% for Grade 420 MPa reinforcement.
Currently, several reinforcement producers in the United States are capable of producing Grade 550 and 690 MPa reinforcement with similar manufacturing costs per unit weight (Kelly et al. 2014). Therefore, the use of higher grade reinforcement has advantages of cost and labor savings in steel fabrication. However, ACI 31814 (2014) only permits Grade 420 MPa for primary reinforcement of special seismic systems because of insufficient data to confirm the applicability of existing code provisions for structures using higher grades. On the other hand, Grade 490 or 500 MPa reinforcement has been widely used as primary reinforcement of earthquakeresistant concrete structures in Japan (AIJ 2010), Taiwan (Ministry of the Interior 2017), and New Zealand (NZS 3101 2006). Even higher Grade 590 and 690 MPa reinforcement can be used in highrise buildings with peer review and special approval in Japan (Aoyama 2001; Nishiyama 2009). Whenever a higher grade reinforcement is used, the anchorage at a beamcolumn joint become a critical issue.
To address concerns about the excessive slip of higher grade bars at beamcolumn joints, the ACIASCE Joint committee 352 recommended a multiplier to the 20 \(d_{b}\) criterion as follows (ACIASCE Committee 352 2002).
More recently, Hwang et al. (2014) conducted cyclic loading tests of three cruciform beamcolumn joints made with Grade 600 MPa am bars and 32 MPa concrete. The column depths ranged from 20 \(d_{b}\) to 25 \(d_{b}\), which is less than 29 \(d_{b}\) required by Eq. (1). The test results exhibited very pinched hysteresis behavior due to excessive slip of beam bars in the joint core. The multiplier of \(f_{y} /\left( {420 {\text{MPa}}} \right)\) in Eq. (1) seems necessary for the joints made of normalstrength concrete and higher grade reinforcement; however, a greater \(h_{c} /d_{b}\) ratio needs either a large column depth or a small permissible diameter of beam bars, which would make the design or construction difficult. In practice, higher grade reinforcement is used together with highstrength concrete, particularly for columns with limited architectural dimensions and high axial load at the lower levels of highrise buildings (Aoyama 2001). Basically, Eq. (1) does not account for the beneficial effects of highstrength concrete and high axial load. Somewhat more complicated equations of minimum \(h_{c} /d_{b}\) ratios can be found in other international concrete design codes (AIJ 2010; CEN 2004; NZS 3101 2006).
This paper summarizes international existing design criteria and proposes a simplified design equation for the minimum joint depth of special moment frames. The applicability of the proposed equation is assessed using a test database of beamcolumn joints made with Grade 490, 590, and 690 MPa reinforcement. Hysteresis performance in terms of strength degradation, residual stiffness, and energy dissipation capacity of each joint test is evaluated according to the ACI standards for special moment frames. Referable design equations and experimental verification are provided. Areas needing further research experiments are also indicated in this paper.
Existing Design Codes and Recommendations
Generic Formula
During the formation of beam hinging adjacent to the joint faces, the beam bar may be subjected to a tensile stress of \(\alpha_{o} f_{y}\) at one face of the joint and a compressive stress of \(\kappa \alpha_{o} f_{y}\) at the opposite face of the joint, as shown in Fig. 1c. By assuming an average bond stress on the beam bar along the column depth, the bond demand should not exceed the available bond resistance in the joint to avoid excessive bar slip, as expressed below.
Rearranging the equation, one can obtain:
The design criteria of minimum \(h_{c} /d_{b}\) ratios or maximum \(d_{b} /h_{c}\) limits on the basis of Eq. (3) can be found in international concrete design codes and standards such as the NZS 3101 (2006), AIJ (2010), and Eurocode 8 (CEN 2004). Table 1 compares these design equations together with additional two sets of equations from Brooke and Ingham (2013) and Li and Leong (2015), both of which were amendments to the NZS 3101 (2006). These existing design criteria have been widely used for earthquakeresisting concrete buildings made with Grade 300–500 MPa reinforcement and normal strength concrete. Unfortunately, the expressions of \(\alpha_{s}\), \(\alpha_{p}\), and \(u_{b}\) listed in Table 1 are inconsistent. Notably, the design equations of Eurocode 8 (CEN 2004) and AIJ (2010) are similar, the difference being the application of reduction factors of 0.75 for the reinforcement compressive stress, 0.80 for the bond strength, and 0.80 for the column axial stress in the case of Eurocode 8. On the other hand, NZS 3101 (2006) uses a basic bond strength of 1.5 \(\sqrt {f_{c}^{'} }\) MPa, which is 60% of the peak local bond strength of 2.5 \(\sqrt {f_{c}^{'} }\) MPa observed by Eligehausen et al. (1983), and two additional modification factors, \(\alpha_{f}\) and \(\alpha_{t}\), to consider the bidirectional loading and the top bar effects, respectively.
Recently, Brooke and Ingham (2013) assembled a database of 93 interior beamcolumn joint tests to assess the suitability of existing design criteria for the minimum joint depth. The research concluded that the existing criteria cannot reflect the bond deterioration observed in experiments. Brooke and Ingham (2013) recommended to modify the basic bond strength to 1.25 \(\sqrt {f_{c}^{'} }\) MPa and the corresponding equations of \(\alpha_{s}\) and \(\alpha_{p}\) for NZS 3101 (2006), as shown in Table 1. In addition, Li and Leong (2015) performed experimental and numerical investigations on eight interior beamcolumn joint specimens and proposed somewhat different modifications. More detailed comparisons of the design equations listed in Table 1 are summarized elsewhere (Brooke 2011; Chen 2017).
Comparison of Existing Equations for Minimum Column Depth
In common design practice, the beam reinforcement ratio \(A_{s,bot} /A_{s,top}\) usually ranges between 0.5 and 1.0. If the top reinforcement area (\(A_{s,top}\)) exceeds the bottom reinforcement area (\(A_{s,bot}\)) at a beamcolumn joint, relatively larger flexural compression would be developed at the bottom beam ends in order to balance the greater flexural tension developed from the top beam bars at the same section. Thus, the compressive stress of the bottom beam bars at the joint face would be greater than that of the top beam bars at the opposite face. In other words, the stress gradient along the bottom bars is larger than that along the top bars in the joint, indicating a relatively severe bond demand for the bottom bars.
For a common beam reinforcement ratio of \(A_{s,bot} /A_{s,top} \le\) 0.75 and overstrength factor \(\alpha_{o} = 1. 2 5\), the outcomes of the stress gradient factor \(\alpha_{s} = \left( {1 + \kappa } \right)\) for bottom beam bars are equal to 2.0 per AIJ (2010), 1.75 per Eurocode 8 (CEN 2004), 1.80 per NZS 3101 (2006), 1.75 per Brooke and Ingham (2013), and 1.64 per Li and Leong (2015). Notably, the bar compressive stress \(\kappa \alpha_{o} f_{y}\) is unlikely to reach \(\alpha_{o} f_{y}\) as for the tensile stress, because of the limited compressive strain of the beam bar in the flexural compression zone at the beam ends. To conclude, taking \(\alpha_{s} = 2.0\) (or \(\kappa = 1.0\)) is the most conservative but unrealistic condition, and thereby \(\alpha_{s} = 1.8\) (or \(\kappa = 0.8\)) may be taken as a physically reasonable assumption.
To compare the minimum column dimensions required by various design codes and recommendations, a reference cruciform beamcolumn joint is assumed to have beam hinging adjacent to the joint faces, a typical overstrength factor \(\alpha_{o} = 1.25\), a minimum axial compression P = 0.2 \(A_{g} f_{c}^{'}\), a common beam reinforcement ratio \(A_{s,bot} /A_{s,top} = 0.75\), and an equal bar diameter \(d_{b}\) for the top and bottom beam bars. Because of the larger stress gradient (\(\alpha_{s}\)) along the bottom bars due to the unequal reinforcement ratio \(A_{s,bot} /A_{s,top} = 0.75\), the minimum column dimension given by Eq. (3) and Table 1 would be determined by the bottom beam bars passing through the joint.
For the reference beamcolumn joint, Fig. 3 shows the ratio of Eq. (3) to Eq. (1) for various values of \(f_{c}^{'}\) using the five sets of design equations listed in Table 1. Notably, Eq. (1) requires minimum \(h_{c} /d_{b}\) ratios of 20, 26, and 33 for Grade 420, 550, and 690 MPa reinforcement, respectively. The minimum \(h_{c} /d_{b}\) ratios for bottom beam bars per Eurocode 8 (CEN 2004) are the most conservative (Fig. 3) because of lower bond strength (Table 1). The requirements of the AIJ (2010) are also conservative at taking \(\alpha_{s} = 2.0\). As shown in Fig. 3, the recommendation of Li and Leong (2015) is more conservative than that of NZS 3101 (2006) because of reduced bond strength (Table 1). Brooke and Ingham (2013) also suggested a lower bond strength but allowed taking more advantage of the column axial load (Table 1), thus resulting in \(h_{c} /d_{b}\) ratios about 2% larger than those of NZS 3101 (2006) for P = 0.2 \(A_{g} f_{c}^{'}\). For very low or high axial load, such as P = 0.05 \(A_{g} f_{c}^{'}\) or 0.50 \(A_{g} f_{c}^{'}\), the \(h_{c} /d_{b}\) ratios recommended by Brooke and Ingham (2013) would be 17% larger than those of NZS 3101 (2006). For the normalstrength concrete of \(f_{c}^{'} \le\) 55 MPa, the minimum column dimensions required by the five sets of design equations are well above that obtained from Eq. (1) per ACI 352R02 (ACIASCE Committee 352 2002). By contrast, the requirement of Eq. (1) may be too conservative for a concrete compressive strength exceeding 70 MPa.
Simplification of Design Equation for the Minimum Joint Depth
Due to the longstanding use of \(\sqrt {f_{c}^{'} }\) in NZS3101 (2006) and ACI 318 (2014) for bond and shear strengths, this paper also recommends simplifying the design equation from the criterion of NZS 3101 (2006), which sets the basic bond strength u_{b} = 1.5 \(\sqrt {f_{c}^{'} }\) MPa for Eq. (3). As listed in Table 1 and discussed in the prior section, the stress gradient factor \(\alpha_{s}\) is affected by beam reinforcement ratios and bar overstrength factor \(\alpha_{o}\). Except for the AIJ (2010), all existing criteria give a \(\alpha_{s}\) factor not exceeding 1.80 for bottom beam bars with \(\alpha_{o} = 1.25\). For simplicity and conservativeness, \(\alpha_{s}\) = 1.80 is taken to simplify Eq. (3) as follows.
where a typical value of \(\alpha_{o} = 1.25\) is usually used in practice, and this paper recommends to use \(\alpha_{p} = 0.9 + 2.0\frac{P}{{A_{g} f_{c}^{'} }} \le 1.20\), which is proposed by Brooke and Ingham (2013).
All design equations listed in Table 1 allow designers to increase the reliable bond strength in the joint with different rates and limits as the column axial load increases. The effect of column axial load on the reliable bond strength in the joint was extensively investigated by Brooke and Ingham (2013). The data support the common view that increasing axial load increases the bond strength. However, tests on isolated bars embedded in concrete suggest that there is an upper limit to the bond strength enhancement that can be achieved by increasing the transverse compression stress acting on an anchorage. Therefore, Brooke and Ingham (2013) proposed to set an upper limit of 1.20 on the \(\alpha_{p}\) factor.
The seismic forces acting on a moment frame generally do not make large contributions to the axial load at interior columns (Moehle 2015). For a typical interior column in a multistory building, a minimum column axial load of 0.15 \(A_{g} f_{c}^{'}\) is a rational assumption. Substituting P = 0.15 \(A_{g} f_{c}^{'}\) into the equation of \(\alpha_{p}\) recommended in Eq. (4) results in a constant \(\alpha_{p} = 1. 20\). Accordingly, Eq. (4) can be further simplified as below.
The proposed Eq. (5) gives a relatively short depth for the reference beamcolumn joint shown in Fig. 3 and verified with the database presented later.
Database Investigation
Assessment of Anchorage Performance of Beam Bars in the Joint
The quasistatic reversed cyclic loading test is the most commonly used method for testing components of earthquakeresistant structures in laboratory. Numerous cyclic loading tests of reinforced concrete beamcolumn joints have been extensively reviewed and assembled in several databases (Brooke and Ingham 2013; Kim and LaFave 2007; Lee and Hwang 2013) for various evaluations of design equations or analytical models. Brooke and Ingham (2013) assembled a database of 93 interior beamcolumn joint tests to evaluate the minimum joint depth; however, only 32 specimens had reinforcement grades of 490 or higher. The anchorage performance of higher grade bars has not been fully clarified. For highstrength reinforcement and concrete, Lee and Hwang (2013) also assembled a more comprehensive database consisting of 202 cruciform beamcolumn joints to assess current ACI 318 design provisions for the joint strength and confinement.
For cyclic testing of a cruciform beamcolumn joint, the consequences of bond deterioration along the beam bars in the joint core includes excessive bar slip, severe concrete crushing at beam ends, strength degradation at peak displacement, very low stiffness at small displacement, and reduced energy dissipation capacity (hysteretic pinching). In the database of Lee and Hwang (2013), some test results with aforementioned characteristics were reported as “BJa” failure [joint core anchorage failure after beam yielding, or “bond” failure as called by Brooke and Ingham (2013)] by the original researchers. However, some test results may also have bond deterioration but not subjectively indicated as “BJa” failure because of the difficulty in detecting such damage in the joint core.
Therefore, this paper proposes to assess how well the beam bars are anchored in the joint by evaluating the hysteresis behavior of a beamcolumn joint. For acceptance, as illustrated in Fig. 4, the test results of the third complete cycle to a limiting drift ratio of 3.5% at least should satisfy the following acceptance criteria for testing components of special moment frames given by ACI 374.105 (2005).

1.
Strength degradation at the peak displacement of the limiting drift cycle shall not exceed 25% of the maximum load resistance in the same loading direction;

2.
Residual secant stiffness between ± 1/10 of the limiting drift ratio shall not be less than 5% of the initial stiffness obtained from the first cycle; and

3.
Energy dissipated in the limiting drift cycle shall not be less than 12.5% of the idealized elastoplastic energy of that drift ratio.
Unfortunately, only a few tests had three cycles at the limiting drift ratio. More recently, ACI 374.2R13 (ACI Committee 374 2013) reported that a minimum of two cycles at each drift ratio is sufficient to consider the damage associated with the number of cycles at a given drift ratio. To evaluate the hysteresis performance for each test specimen, the second (or third, if available) cycle at a drift ratio of 3.5% or 4% was used. Therefore, this study omitted the test data that did not have a minimum of two cycles at a drift ratio of 3.5% or 4%. In addition, the beamcolumn joints that failed in joint shear (socalled “J” failure) without yielding of beam bars were excluded, because the “J” failure is primarily dominated by the joint shear stress and indirectly related to the permissible bond stress of the beam bars in the joint. Finally, to evaluate the minimum joint depth, this study assembled a database of available test data with the following conditions.

1.
Reinforced concrete cruciform beamcolumn joints without transverse beams and slabs. The confinement of continuous transverse beams and slabs may enhance the bond strength of the joint core concrete.

2.
For each joint, all the beam longitudinal bars extended through the joint core are confined by transverse reinforcement.

3.
The hysteresis loop of the second (or third, if available) cycle at a drift ratio of 3.5% or 4% can be extracted data values from published graphs by a plot digitizer.

4.
Yielding of beam bars occurred at the joint faces, followed by beam flexure, joint shear, or anchorage failure (“B”, “BJ”, or “BJa” failures).
Finally, this paper presents the evaluation results of 61 cruciform beamcolumn joints reinforced with beam bars of Grade 490 MPa or higher, as listed in Tables 2, 3, and 4. Only nine specimens had three cycles at a drift ratio of 3.5% or 4%. The details and cyclic loading response for each beamcolumn joint are summarized elsewhere (Chen 2017).
Joints with Grade 490 or 500 MPa Beam Bars
Table 2 shows the evaluation of the data subset of joints with Grades 490 or 500 MPa beam bars. The hysteresis loops of these tests were obtained from references (Teraoka and Kanoh 1994; Nakachi and Tabata 1995; Hosoya et al. 2003; Maruta and Sanada 2004; Hori et al. 2004; Brooke et al. 2006; Umemura et al. 2006; Kimoto et al. 2006; Yagenji et al. 2009; Li and Leong 2015; Alaee and Li 2017) and evaluated according to the ACI standards. The joints having hysteresis performance that fully satisfy the three criteria of strength, stiffness, and energy dissipation (\(Q_{r} \ge 0.75Q_{m}\), \(K_{o} \ge 0.05K_{i}\), and \(E_{D} \ge 0.125E_{pp}\)) given by ACI 374.105 (ACI Committee 374 2005) are evaluated to be “acceptable”, while the others are evaluated as “unacceptable”. Figure 4 illustrates an unacceptable 4% drift cycle for a cruciform beamcolumn joint with bond deterioration initiating in the prior 3% drift cycles.
For various failure modes, the relations of the \(h_{c} /d_{b}\) ratio, measured concrete compressive strength \(f_{c}^{'}\), and evaluated hysteresis performance are shown in Fig. 5, wherein the symbols “o” and “x” denote the “acceptable” and “unacceptable” test results, respectively. The proposed minimum \(h_{c} /d_{b}\) ratios of Eqs. (1) and (5) for Grade 490 MPa are plotted for comparison. Ideally, the test data having \(h_{c} /d_{b}\) ratios above the minimum joint depth should demonstrate acceptable performance unless other design parameters do not conform to the code. In contrast, the test data having \(h_{c} /d_{b}\) ratios below the minimum joint depth would exhibit unacceptable performance.
From the comparison shown in Fig. 5, the minimum \(h_{c} /d_{b}\) ratio from Eq. (1) is obviously too conservative for test data with \(f_{c}^{'} \ge 70 {\text{MPa}}\). Comparatively, the proposed Eq. (5) is less conservative. Most of the test data having \(h_{c} /d_{b}\) ratios above the curve of Eq. (5) are evaluated as acceptable. Three “B” failure data [Table 2, Specimen 2B (Brooke et al. 2006); Specimens AS2 and AS4 (Li and Leong 2015)] performed well up to 3% drift but the strengths rapidly degraded in the later drift cycles due to buckling of the beam bars in the plastic hinge regions. Therefore, the unacceptable evaluation of three “B” failure specimens can be primarily attributed to beam flexural failure rather than bond deterioration.
Excluding “B” failure specimens, the remaining four unacceptable data had \(h_{c} /d_{b}\) ratios slightly above the curve of Eq. (5). Figure 6 shows the reproduced cyclic responses of Specimens NO. 5 and NO. 6 (Nakachi and Tabata 1995), Specimen NO. 1 (Hosoya et al. 2003), and Specimen JUS (Yagenji et al. 2009). The strength and stiffness of Specimen NO. 6 gradually degraded in 2%, 3%, and 4% drift cycles. This degradation can be attributed to the excessive joint shear stress (\(V_{jh, m} /V_{n,ACI} = 1.18\)) of Specimen NO. 6, as listed in Table 2. Note that either joint shear failure or bond deterioration would result in hysteretic strength and stiffness degradation; however, the joints with bond deterioration would typically display very low residual stiffness at small displacement (very low \(K_{o} /K_{i}\)). As shown in Fig. 6 and Table 2, Specimens NO. 5, NO. 1, and JUS exhibited very low \(K_{o} /K_{i}\) values in the evaluation of the 4% drift cycle because of the bond deterioration initiated in the previous 3% drift cycles. Since the limiting drift cycle evaluated in this study is 4% drift, which is slightly beyond the 3.5% drift given by ACI 374.105 (ACI Committee 374 2005), the performance of Specimens NO. 5, NO. 1, and JUS could be considered as marginally acceptable.
Notably, four “BJa” failure specimens [Specimens 3B and 4B (Brooke et al. 2006); Specimens I6C and I6P (Kimoto et al. 2006)] are still evaluated as acceptable (Table 2) at the 4% drift cycle, in which the excessive bar slip or joint core anchorage failure has not occurred. In other words, the “BJa” failure could be acceptable if the failure drift ratio is large enough.
Joints with Grade 590 or 600 MPa Beam Bars
To date, no experiment has been conducted for beamcolumn joints with ASTM A706 Grade 550 MPa reinforcement, for which the specified yield strength ranged from 550 to 675 MPa. Therefore, the available test data of beamcolumn joints with Grade 590 or 600 MPa beam bars were collected from literature (Oka and Shiohara 1992; Teraoka et al. 2004; Abe et al. 2006; Hori et al. 2006; Takamori et al. 2007; Hwang et al. 2014) and assembled in Table 3, where 11 out of 13 data had actual yield strength \(f_{ya}\) meeting the specification of ASTM A706 Grade 550 MPa. Thus, this subset of the database could be referred for the application of ASTM A706 Grade 550 MPa bars.
Table 3 and Fig. 7 show the relations of the evaluated hysteresis performance and the key test parameters. Obviously, the minimum \(h_{c} /d_{b}\) ratio of 28 obtained from Eq. (1) is too conservative for the eight acceptable data with \(f_{c}^{'} \ge 5 5 {\text{MPa}}\) and \(h_{c} /d_{b} \le 25\). Notably, although the three “BJa” failure specimens tested by Hwang et al. (2014) also used \(h_{c} /d_{b}\) ratios ranging from 20 to 25, they did not perform well within 3.5% drift cycles because of the low \(f_{c}^{'}\) value of 32 MPa. The last two data [Specimen J1 (Oka and Shiohara 1992); Specimen HJ12 (Teraoka et al. 2004)] had significant strength losses at the 4% drift cycle due to excessive shear stress and insufficient transverse reinforcement in the joint (Table 3). Excluding Specimens J1 and HJ12, only three “BJa” failure specimens had \(h_{c} /d_{b}\) ratios much smaller than that from Eq. (5) and exhibited unacceptable hysteresis performance. However, there are no test data with lower \(f_{c}^{'}\) values and larger \(h_{c} /d_{b}\) ratios in Fig. 7, neither for higher \(f_{c}^{'}\) values with smaller \(h_{c} /d_{b}\) ratios. Therefore, it is difficult to conclude whether the proposed Eq. (5) is conservative for various concrete strengths. More beamcolumn joint tests are needed for these areas.
Joints with Grade 690 MPa Beam Bars
The available test data of beamcolumn joints with Grade 690 MPa (100 ksi) beam bars are collected from literature (Noguchi and Kashiwazaki 1992; Watanabe et al. 2005; Hori et al. 2006; Kuo 2011; Lee et al. 2014, 2016; Alaee and Li 2017) and listed in Table 4. All the specimens were made with higher strength concrete. Figure 8 shows the relations of the evaluation results to the \(h_{c} /d_{b}\) ratio and measured concrete compressive strength \(f_{c}^{'}\). Definitely, the minimum joint depth from Eq. (1) is far above the acceptable test results with smaller \(h_{c} /d_{b}\) ratios. As listed in Table 4, six specimens did not perform well at the limiting drift ratio. Specimens OKJ1 and OKJ4 (Noguchi and Kashiwazaki 1992) were subjected to excessive joint shear stress (\(V_{jh,m} /V_{n,aci} \ge 1.2\)). Thus, the lateral strength degraded rapidly due to the occurrence of joint shear failure. Specimens B151 and B153 (Hori et al. 2006) used ultrahigh strength concrete for the joint but the amounts of transverse reinforcement (\(A_{sh,ratio}\), Table 4) were far below the code requirement. Thus, both joints failed in shear with a stress level below the nominal joint shear strength. Specimens IH60 and IH60A (Alaee and Li 2017) demonstrated severely pinched hysteresis loops due to the unmatched use of 60 MPa concrete with a joint depth of 24 \(d_{b}\), which is less than the minimum joint depth obtained from Eq. (5) for Grade 690 MPa beam bars.
On the other hand, the first author and his colleagues (Kuo 2011; Lee et al. 2014, 2016) also tested several cruciform beamcolumn joints at the laboratory of the National Center for Research on Earthquake Engineering (NCREE) in Taiwan. As shown in Fig. 9, each joint specimen had a unit column length of 3.2 m and a unit beam length of 6 m to simulate an interstory beamcolumn joint under reversed cyclic loading (Fig. 1b). The test specimens satisfied most of the ACI 318 (2014) and ACI 352R02 (ACIASCE Committee 352 2002) seismic provisions for special moment frames, except the limitations on the joint shear stress, bar \(f_{y}\), and \(h_{c} /d_{b}\) ratio. As listed in Table 4, six “BJ” failure specimens (X100, CG1, CG3, CG4, A24, and B24) were reinforced with 25mm Grade 690 MPa beam bars passing through a joint depth of 600 mm, resulting in a \(h_{c} /d_{b}\) ratio of 24. The design concrete strength was 100 MPa but the measured concrete compressive strength ranged from 80 to 120 MPa. Figure 10 shows the normalized cyclic loading response of Specimens X100, CG1, A24, and B24 with various ratios of \(A_{s,bot} /A_{s,top}\) and \(V_{jh,m} /V_{n,aci}\). The cyclic responses of Specimens CG3 and CG4 are not shown here, because they were almost identical to that of Specimen CG1 with the same reinforcing details, except the use of mechanical couplers for bottom beam bars. As demonstrated in Fig. 10, all tested specimens performed well up to the 4% drift ratio, although eventually they failed in joint shear in the later 6% drift cycles. Notably, the residual stiffness around zero drift of the 4% drift cycles were not too low. Excessive bar slip was not observed in these experiments within 4% drift cycles.
Based on Figs. 8 and 10, the proposed Eq. (5) seems adequate for concrete strength \(f_{c}^{'}\) ranged from 80 to 120 MPa, but it may be inadequate for higher strength concrete, where the bond strength of ultrahigh strength concrete is still questionable. Due to the limited test data with \(h_{c} /d_{b} \le 20\), this study also suggests to set a lower limit of 20 for Eq. (5). In other words, this study recommends the minimum joint depth to be the larger of 20 \(d_{b}\) and \({{\alpha_{o} f_{y} d_{b} } \mathord{\left/ {\vphantom {{\alpha_{o} f_{y} d_{b} } {\left( {4\sqrt {f_{c}^{'} } } \right)}}} \right. \kern0pt} {\left( {4\sqrt {f_{c}^{'} } } \right)}}\). This recommendation is conservative for cruciform beamcolumn joints reinforced with reinforcement grades not exceeding 690 MPa.
Overall Observation
For all the test data listed in Tables 2, 3, and 4, Fig. 11 displays the data distributions of unacceptable and acceptable data with respect to the ratio of experimentaltonominal shear strength and the providedtorequired column dimension ratio (i.e., the provided column depth divided by the minimum joint depth). Attention shall be drawn to the test data that fall in Quadrants 3 and 4, where \({{V_{jh,m} } \mathord{\left/ {\vphantom {{V_{jh,m} } {V_{n,aci} }}} \right. \kern0pt} {V_{n,aci} }} \le 1.0\) means the experimental shear stresses are below the nominal value of 1.25 \(\sqrt {f_{c}^{'} }\) MPa specified in ACI 31814 (2014), and thus, the premature joint shear failure data were precluded. The horizontal axis of Fig. 11 represents the ratio of the column depth provided in experiments to the proposed minimum joint depth. Therefore, the test data in Quadrant 4 of Fig. 11 are requirementconforming joints and are expected to perform well up to 4% drift ratio. Ideally, the joints with excessive bar slip or anchorage failure should not appear in Quadrant 4 of Fig. 11a, where only three BJafailure specimens (Specimens NO. 5, NO. 1, and JUS, as shown in Fig. 6) had marginallyacceptable hysteresis performance. The other three “x” data in Quadrant 4 of Fig. 11a are Bfailure specimens (Specimens 2B, AS2, and AS4 listed in Table 2), which had significant strength loss at the limiting drift cycle due to bar buckling in the beam hinging zone. In other words, these three Bfailure data can be precluded for bond assessment.
Notably, the recommended minimum \(h_{c} /d_{b}\) ratio is smaller and less conservative than those required by other design codes listed in Table 1. However, most of the test data that fall in Quadrant 4 of Fig. 11 had acceptable hysteresis performance, indicating that the recommended minimum \(h_{c} /d_{b}\) ratio could be acceptable, on the basis of the present database investigation. Notably, setting the basic bond strength \(u_{b} = 1.25\sqrt {f_{c}^{'} } \;{\text{MPa}}\) as suggested by Brooke and Ingham (2013) for Eq. (4) could give a relatively conservative \(h_{c} /d_{b}\) ratio for the presented data.
A larger \(h_{c} /d_{b}\) ratio and a lower joint shear stress (Quadrant 4 of Fig. 11b) can ensure better performance of joints under cyclic loading. Higher joint shear stress may result in premature joint shear failure, which may cause building collapse, and therefore should be avoided. A smaller \(h_{c} /d_{b}\) ratio may result in excessive bar slip in the joint core and pinching hysteretic performance, but it would not lead to any local collapse. Therefore, the minimum \(h_{c} /d_{b}\) ratios specified for special moment frames by codes and standards are based on the judgment of the expected hysteresis behavior at a design interstory drift.
Summary and Recommendations
This paper reviews existing design criteria in different codes for the minimum joint depth of special moment frames and proposes a simplified equation by omitting some minor variables. The applicability of the proposed equation is verified against a database assembling available experiments on the use of higher grade reinforcement in reinforced concrete beamcolumn joints subjected to quasistatic reversed cyclic loading. The hysteresis behavior of each beamcolumn joint was evaluated according to the ACI standards for special moment frames. Based on this review and database assessment, the following conclusions and recommendations can be drawn:

Relatively pinched hysteresis behavior can be observed in the beamcolumn joints with bond deterioration along the beam bars passing through the joint. Such damage in the joint core is unlikely to be easily repairable and therefore should be avoided in a design basis earthquake event.

The ACI 318 (2014) requirement of a minimum joint depth of 20 \(d_{b}\) is based on test data with Grade 420 reinforcement and may be too short for bar \(f_{y}\) exceeding 420 MPa. ACI 352R02 (ACIASCE Committee 352 2002) recommends a simple multiplier of \(f_{y} /420\) on the 20 \(d_{b}\) criterion without accounting for the various concrete strength, which is very conservative for highstrength concrete and unconservative for lowstrength concrete, according to the presented database investigation.

Based on the evaluation of the assembled test data, this study recommends that the joint depth, or the column dimension in parallel to the beam bars extending through the joint should not be less than the larger of 20 \(d_{b}\) and \(\alpha_{o} f_{y} d_{b} /\left( {4\sqrt {f_{c}^{'} } } \right)\). Cruciform beamcolumn joints with column dimension meeting this criterion can demonstrate acceptable hysteresis performance up to a limiting drift ratio of 3.5% at least.

The proposed design equation is empirical and should be used with limitations of \(f_{y}\) not exceeding 690 MPa and \(f_{c}^{'}\) not exceeding 100 MPa.

The simplified design equation is proposed for typical interior beamcolumn joints with column axial load exceeding 0.15 \(A_{g} f_{c}^{'}\). If the seismic forces lead to significant variation of the column axial load, the detailed design equations including the effects of column axial load and other parameters should be used.
Abbreviations
 A _{ g } :

Gross sectional area of column
 A _{ s } :

Area of the bar group \(A_{s,top}\) or \(A_{s,bot}\) containing the bar for which the development length is being calculated
 \(A_{s,bot}\) :

Area of bottom beam bars
 \(A_{s,top}\) :

Area of top beam bars
 \(A_{sh,ratio}\) :

Provided amount of joint transverse reinforcement divided by the amount required in ACI 318
 \(d_{b}\) :

Diameter of largest beam longitudinal bars extending through the joint
 \(E_{D}\) :

Area of hysteresis loop or energy dissipated in the limiting drift cycle
 \(E_{PP}\) :

Area of the elastoplastic loop for the limiting drift cycle
 \(f_{c}^{'}\) :

Compressive strength of concrete
 \(f_{s}\) :

Tensile stress in reinforcement
 \(f_{s}^{'}\) :

Compressive stress in reinforcement
 \(f_{y}\) :

Minimum specified yield strength of reinforcement, or reinforcement grade
 \(f_{ya}\) :

Actual yield strength of longitudinal reinforcement
 \(f_{yt}\) :

Yield strength of transverse reinforcement
 \(h_{c}\) :

Column depth or joint depth
 \(K_{o}\) :

Secant stiffness around zero drift, obtained for positive and negative loading directions between ± 1/10 of the limiting drift ratio in the hysteresis loop
 \(K_{i}\) :

Initial secant stiffness of the first drift cycle
 \(L_{b}\) :

Unit beam length of test module, or distance between the assumed inflection points
 \(L_{c}\) :

Unit column height of test module, or assumed story height
 \(P\) :

Axial compression on the column
 \(Q\) :

Lateral load or column shear force of test module
 \(Q_{m}\) :

Maximum lateral load of test module
 \(Q_{r}\) :

Peak lateral load of test module in the repeated cycle of the limiting drift ratio
 \(Q_{y}\) :

Yield strength of test module with beam yielding at both joint faces, determined using actual bar yield strength, measured concrete compressive strength, and nominal beam moment strength at the joint face
 \(u_{b}\) :

Basic bond strength on the beam bar over the column depth
 \(V_{b}\) :

Beam shear force
 \(V_{jh,m}\) :

Experimental shear force acting on the joint, backcalculated using the force couples in the beam resisting \(Q_{m}\)
 \(V_{n,aci}\) :

Nominal joint shear strength per ACI 318
 \(\alpha_{f}\) :

0.85 for bidirectional loading; 1.0 for unidirectional loading
 \(\alpha_{o}\) :

Overstrength factor of the beam bars
 \(\alpha_{p}\) :

Factor accounting for the effect of column axial compression on the bond strength
 \(\alpha_{s}\) :

1 + \(\kappa\), stress gradient factor accounting for the bar stress developed at the joint faces
 \(\alpha_{t}\) :

0.85 for a top beam bar where more than 300 mm of fresh concrete is cast below the bar; 1.0 for all other cases
 \({\updelta}\) :

Beam deflection
 \({\Delta }\) :

Interstory drift
 \(\uptheta\) :

Drift ratio, or angular rotation between the beam and column centerlines at the joint
 \(\kappa\) :

Ratio of the bar compressive stress to bar tensile stress developed at the joint faces
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Authors’ contributions
HJ reviewed all the data analyzed by HC and then wrote the manuscript. TC helped to review Japanese articles. All authors read and approved the final manuscript.
Acknowledgements
The authors are grateful for the financial support from the Ministry of Science and Technology in Taiwan.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
All data analyzed during this study are available in the Ph.D. thesis of Dr. H.C. Chen at National Yunlin University of Science and Technology in 2017.
Funding
Ministry of Science and Technology of Taiwan, Grant No. MOST 1052625M224001.
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Lee, HJ., Chen, HC. & Tsai, TC. Simplified Design Equation of Minimum Interior Joint Depth for Special Moment Frames with HighStrength Reinforcement. Int J Concr Struct Mater 12, 70 (2018). https://doi.org/10.1186/s4006901803032
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DOI: https://doi.org/10.1186/s4006901803032
Keywords
 beamcolumn joint
 bond
 cyclic testing
 highstrength reinforcement
 slip