 Open Access
Development of a Shear Strength Equation for Beam–Column Connections in Reinforced Concrete and Steel Composite Systems
 YunChul Choi^{1},
 JiHo Moon^{2},
 EunJin Lee^{3},
 KeumSung Park^{4} and
 Kang Seok Lee^{5}Email author
https://doi.org/10.1007/s4006901701992
© The Author(s) 2017
 Received: 29 November 2016
 Accepted: 23 March 2017
 Published: 19 May 2017
Abstract
In this study, we propose a new equation that evaluates the shear strength of beam–column connections in reinforced concrete and steel beam (RCS) composite materials. This equation encompasses the effect of shear keys, extended face bearing plates (EFBP), and transverse beams on connection shear strength, as well as the contribution of cover plates. Mobilization coefficients for beam–column connections in the RCS composite system are suggested. The proposed model, validated by statistical analysis, provided the strongest correlation with test results for connections containing both EFBP and transverse beams. Additionally, our results indicated that Architectural Institute of Japan (AIJ) and Modified AIJ (MAIJ) equations should be used carefully to evaluate the shear strength for connections that do not have EFBP or transverse beams.
Keywords
 connection design
 RCS composite system
 shear strength equations
 statistical analysis
1 Introduction
Buildings are generally designed to have strong column–weak beam systems, where such connections are assumed to have sufficient strength and stiffness to support external loads. However, such connections can be destroyed when unexpectedly strong loads are applied, leading to catastrophic failure of the entire building. For example, forces induced by earthquakes can concentrate on a connection, causing damage that can lead to their failure and the potential collapse of the entire structure.
Design equations to estimate the shear connection strength have been proposed by several previous studies (Dierlein 1988; Sheikh and Deierlein 1989; Kanno 1993). The American Society of Civil Engineers (ASCE) Task Committee (1993) suggested a simple modified design equation based on an equation proposed by Deierlein (1988) and Sheikh and Deierlein (1989). Kanno (1993) suggested a shear strength equation for connections based on the ASCE equation that considers the effect of transverse beams and band plates. Several Japanese researchers (Nishimura 1986; Kei et al. 1990, 1991; Mikame 1990, 1992) proposed shear design equations for connections in RCS composite systems; these were subsequently adopted by Japanese Standards (1975 and 1987).
The above studies have resulted in various shear strength equations for connections in RCS composite systems that encompass different variables and thus give inconsistent results. Furthermore, the load transfer mechanism is not fully understood for such connections. Thus, accurate design equations that estimate the shear strength of the connections are still necessary.
In this study, we propose a model that estimates the shear strength of beam–column connections in RCS composite systems. After first analyzing previously proposed shear strength equations including existing researches regarding connections of framing system (Kim and Choi 2006, 2015; LaFave and Kim 2011; Yang et al. 2007; Lim et al. 2016), we developed a shear strength equation for general connections in RCS composite systems, encompassing the effect of extended face bearing plates (EFBP), transverse beams, and cover plates. Statistical analysis was conducted to verify the proposed equation. This analysis showed that our proposed equation accurately represented the shear strength in RCS composite system connections.
2 Background Theory: Evaluating Connection Shear Strength in RCS Composite Systems
The shear strength of reinforced concrete can be obtained by calculating the effective width of internal and external elements; these internal and external elements are divided on the basis of the face bearing plate (FBP). Conventional methods for evaluating the shear strength of concrete can also be used for internal elements, as there are no reinforcements within this material. On the other hand, shear strengths for external elements can be obtained by summing the strength of the concrete and stirrup.
Shear strength equations for connections in RCS composite systems.
ASCE  \( V_{b} = \frac{{\left[ {V_{s} \cdot d_{f} + 0.75V_{n} \cdot d_{w} + V_{n}^{'} \cdot \left( {d + d_{o} } \right)} \right]}}{{\left[ {\frac{{L_{b} }}{{L_{c} }}\left( {L_{c}  d} \right)  jh} \right]}} \) 
Kanno  \( V_{b} = \frac{0.9d}{{L_{b}  h}}\left( {V_{s} + V_{sf} + V_{n} + V_{n}^{'} } \right) \) 
MKanno  \( V_{b} = \frac{{\left( {V_{s} \cdot d_{f} + 0.75V_{n} \cdot d_{w} + V_{n}^{'} \cdot d} \right)}}{{\left[ {\frac{{L_{c} }}{{L_{b} }}\left( {L_{c}  d_{f} } \right)} \right]}} \) 
Parameters  Symbol  ASCE equation  Kanno equation  MKanno equation 

b _{ o }  C  C _{ wt }  max(C _{ wt }, C _{ t })  max(C _{ wt }, C _{ t }) 
C _{ wt }  \( \left( {x/h} \right) \cdot \left( {y/b_{f} } \right) \le 1 \)  \( \left( {\frac{x}{h}} \right) \cdot \left( {0.3 + 0.7\frac{y}{{\bar{b}_{f} }}} \right) \le 1 \)  \( \left( {\frac{x}{h}} \right) \cdot \left( {0.3 + 0.7\frac{y}{{\bar{b}_{f} }}} \right) \le 1 \)  
C _{ t }  –  \( \frac{1}{2.5}\left( {\frac{d}{{b  b_{i} }}} \right) \)  \( 0.7\left( {\frac{x}{h}} \right) \cdot \left( {0.3 + 0.7\frac{y}{{\bar{b}_{f} }}} \right) \)  
b _{ m }  \( \frac{{\left( {b_{f} + b} \right)}}{2} \)  \( \frac{{\left( {\bar{b}_{f} + b} \right)}}{1.5} \)  \( \frac{{\left( {\bar{b}_{f} + b} \right)}}{1.5} \)  
\( \bar{b}_{f} \)  b _{ f }  \( max(b_{i} ,\bar{y}) \le 2b_{f} \) \( \bar{y} = min\left( {y,b_{f} + n \cdot t_{s} } \right) \)  max(b _{ i }, y) ≤ 2b _{ f }  
V _{ s }  jh  Iteration procedure  \( 0.8\;h\;({\text{No shear key}}) \) \( h\;(E  FBP, \;{\text{Band Plate}}) \)  0.8 h 
V _{ sf }  V _{ sf }  –  4M _{ pf }/d _{ f }  – 
V _{ sf }, V _{ b }  d _{ j }  –  1.25d  1.1d 
V _{ n } ^{’}  V _{ c } ^{’}  \( 0.4\sqrt {f_{ck} } \cdot b_{o} \cdot h \)  \( 1.05\sqrt {f_{ck} } \cdot b_{o} \cdot h \)  \( 1.05\sqrt {f_{ck} } \cdot b_{o} \cdot h \) 
The ASCE design code (1993) adopted the ACIASCE 352 equation to evaluate the effective width of external elements (b _{ o }), as shown in Table 1. This ASCE design code limits the effective width by using one mobilization coefficient (C _{ wt }), regardless of the existence of transverse beams. However, two different mobilization coefficients, with coefficient factors that consider shear key (C _{ wt }) and transverse beams (C _{ t }), are used in the Kanno equations (Kanno 1993, 2002). Larger values for C _{ wt } and C _{ t } are also used, which means that only larger effects between the shear key and transverse beam can be captured by this equation. Additionally, the Kanno equations generally result in larger values of b _{ o } than the ASCE equation.
As shown in Table 1, the ASCE equation does not consider transverse beam effects when calculating b _{ o }. However, the transverse beam effect is included in Kanno equations for the calculation of C _{ t }. The mobilization coefficient for transverse beams (C _{ t }) is obtained from the original Kanno equation by considering the transverse beam twist resistance. Values for C _{ t } are simply calculated as 0.7C _{ wt } in the MKanno equation (2002).
The maximum effective connection width (b _{ m }) is assumed to be 1/2 of the total connection and flange width length within the ASCE equation. Both the Kanno and MKanno equations use 1/1.5 of the total length, based on a comparison with test results.
In Table 1, values for jh [which is needed to calculate the shear strength of the steel web (V _{ s })] represent the effective steel beam web panel width and are obtained by iteration to find the conversed value within the ASCE equation. In the Kanno equation, jh is defined depending on connection details. For example, when a small column or no shear key is used, jh is equal to 0.8 h; if EFBP or band plates are used, jh is assumed to be equal to h.
The shear strength of flanges within steel beams (V _{ sf }) is included within the Kanno equation (Table 1). However, this factor is small compared with other variables, and is thus not considered in other equations. To evaluate the concrete strut strength (V’ _{ n }), the ASCE equation (ACIASCE 352) uses a value of 0.4, while 1.05 is used in Kanno and MKanno equations (as shown in Table 1).
3 A New Equation for Connection Shear Strength in RCS Composite Systems
As previously discussed, shear strength equations proposed by previous studies differ both in terms of variables included and resulting estimates for strength. Mobilization coefficients, which represent the effects of a shear key and a transverse beam on shear strength, differ considerably among the proposed equations. In this study, we focused on the development of improved mobilization coefficients, while also considering the effects of cover plates on the shear strength equation. The following sections discuss details of how the equation was developed, as well as the proposed equation itself.
3.1 Effect of Shear Key (EFBP etc.)
The shear key (including a EFBP and band plate) transfers external forces to the inside of the connection, forming a compressive strut inside the connection (Fig. 3) that increases the connection’s shear strength within the RCS composite system. The effect of the shear key is included within mobilization coefficients C _{ wt } and C _{ wt } as a function of the effective width x and y (Fig. 3).
When EFBP is installed in the connection, the effective width x and y can be determined by considering the compressive strut formed by the shear key (as shown in Fig. 3b). Most of the EFBP is welded to the flange, where its width is not larger than the flange and y/b _{ f } is less than 1. However, when the band plate is used, the width of the shear key (y) can be larger than that of the flange (b _{ f }), as shown in Fig. 3c. In this case, the maximum y/b _{ f } value is limited to 1, as shown in Table 1.
Values of C _{ wt } for shear strength equations shown in Table 1 are a function of y/b _{ f }. However, the compression field varies depending on values of y and the total width of the connection (b), as shown in Fig. 3b and c (Lee et al. 2005). Therefore, the effective width should be determined from y/b, rather than y/b _{ f }. However, when y/b _{ f } is used, the shear strength can be overestimated. Test results show that an increase in shear strength upon EFBP installation is about 15–20% (Lee 2005). However, strength increases due to EFBP installation, as calculated by the Kanno equation (which uses y/b _{ f }), are as much as ~50%.
Increases in shear strength due to the EFBP.
Specimen  Test  Kanno equation  Proposed equation  

Strength increment (%)  C _{ wt }  Strength increment (%)  C _{ wt }  Strength increment (%)  
JL01  21.8  0.21  47  0.21  23 
JL02  1  0.56  
STI1  15.2  0.21  53  0.21  22 
STI2  1  0.56 
3.2 Transverse Beam Effect
Increases in shear strength due to transverse beam.
Specimen  Test  Kanno equation  Proposed equation  

Strength increment (%)  C _{ t }  Strength increment (%)  C _{ t }  Strength increment (%)  
SNI1  14.8  0.21  10.7  0.21  16 
STI1  0.51  0.64 
3.3 Effect of Cover Plates
Increases in shear strength due to the cover plate.
Specimen  Strength increment (test)  Strength increment (Kanno equation)  Strength increment (proposed equation) 

LCS1  26.2%  4.98%  27.53% 
LCC1 
3.4 Proposed Shear Strength Equation
 (1)Determination of effective width:

Effective internal element width:$$ b_{i} = \hbox{max} (b_{p} ,b_{f} ) $$(5)

Effective external element width:$$ b_{0} = C(b_{m}  b_{i} ) < 2d_{0} $$(6)$$ b_{m} = \frac{{(b_{f} + b)}}{2} < b_{f} + h < 1.75b_{f} $$(7)$$ C = \hbox{max} (C_{wt} ,C_{t} ) $$(8)In the above equations, x = 0.7 h and y = 0 without a shear key, x = h and y = the width of the shear key when it is present. b is the connection width.$$ C_{wt} = \left( {\frac{x}{h}} \right) \cdot \left( {0.3 + 0.7\frac{y}{b}} \right)$$(9)$$ C{}_{t} = \left( {\frac{1}{2.0}} \right) \cdot \left( {\frac{d}{{b  b_{i} }}} \right) $$(10)

 (2)Steel beam web shear strength:$$ V_{s} = \frac{1}{\sqrt 3 } \cdot F_{yw} \cdot t_{w} \cdot jh $$(11)
 (3)Internal concrete shear strength:$$ V_{n} = 1.65 \cdot \sqrt {f_{ck} } \cdot b_{p} \cdot h $$(12)
 (4)External concrete shear strength:$$ V_{n}^{'} = V_{c} + V_{s} \le 1.65\sqrt {f_{ck} } b_{o} h $$(13)$$ V_{c} = 1.05 \cdot \sqrt {f_{ck} } \cdot b_{o} \cdot h $$(14)$$ V_{s} = \frac{{A_{sh} \cdot F_{ysh} }}{{S_{h} }} \cdot 0.9h $$(15)
 (5)The cover plate shear strength can be calculated as$$ V_{cp} = 2 \cdot \alpha \cdot \frac{{F_{yc} }}{\sqrt 3 } \cdot t_{c} $$(16)
Statistical evaluation was conducted to verify the proposed equations, as described below.
4 Statistically Evaluating Shear Strength Equations
4.1 Comparing Results from Shear Strength Equations with Test Results
4.2 Sample size
The sample size within the statistical analysis must be defined to ensure reliability and range. In conventional statistical analyses, the sample size is defined to guarantee target reliability levels and range. However, sampling size was fixed in this study; thus, reliability level and range were calculated to validate sampling size.
We found that between 35 and 59 samples are needed to guarantee reliability levels of 95 and 99%, respectively. Thus, 49 samples were used in this study to satisfy a reliability level of 95%.
4.3 Statistical Analysis Results
Statistical analysis results.
Statistical parameter  ASCE  Kanno  MKanno  AIJ  MAIJ  Proposed 

Min.  0.47  0.68  0.64  0.40  0.37  0.61 
Max.  1.16  1.29  1.20  2.51  1.98  1.11 
Mean  0.73  0.94  0.90  0.86  0.81  0.88 
Std. error of mean  0.02  0.02  0.02  0.05  0.04  0.02 
Standard deviation  0.14  0.12  0.12  0.37  0.27  0.11 
Variance  0.02  0.01  0.01  0.14  0.07  0.01 
Skewness  0.20  0.24  0.39  2.61  2.42  0.32 
Kurtosis  0.68  0.90  0.35  9.34  9.04  0.26 
Frequency histograms and normal distributions are plotted in Fig. 6; in this figure, the xaxis represents the estimation/test ratio, and the yaxis the frequency. The histograms were in good agreement with normal distributions for the Kanno, MKanno, and our proposed equation, while the distribution of data from AIJ and MAIJ equations was far from normal. A 0.63–0.75 shear strength range was the most popular for the ASCE equation. Thus, our proposed equation provided the most accurate prediction of shear strength, given that it had the smallest standard deviation, averaged standard error, and a normal data distribution.
4.4 Factor Analysis
Factor analysis results.
Test or equations  Group  Factor  

1  2  3  4  5  
Test  A  0.17  0.00  0.20  0.96  −0.05 
B  0.28  0.93  −0.04  0.21  0.03  
C  −0.24  −0.11  0.92  −0.21  0.14  
D  0.95  0.16  −0.21  −0.14  0.11  
ASCE  A  −0.26  0.26  0.63  0.61  −0.21 
B  0.25  0.87  −0.17  0.38  0.04  
C  0.00  −0.17  0.95  0.26  0.01  
D  0.86  0.17  −0.02  −0.02  0.48  
Kanno  A  −0.09  −0.03  0.59  0.78  0.17 
B  −0.06  0.76  −0.20  0.55  −0.22  
C  0.16  −0.37  0.85  0.33  0.04  
D  0.99  −0.04  0.12  −0.07  −0.01  
MKanno  A  −0.20  0.37  0.21  0.88  0.11 
B  0.15  0.90  −0.38  0.02  −0.13  
C  0.12  −0.39  0.88  0.23  0.07  
D  0.99  0.04  0.07  −0.08  0.03  
AIJ  A  0.29  0.36  −0.20  −0.75  0.41 
B  0.06  0.92  −0.28  −0.09  0.22  
C  −0.26  −0.30  0.73  0.41  0.38  
D  0.86  0.31  −0.16  −0.15  −0.32  
MAIJ  A  0.74  0.23  −0.08  0.48  −0.40 
B  0.13  0.96  −0.12  −0.13  −0.07  
C  −0.12  −0.16  0.88  0.42  −0.01  
D  0.83  0.35  −0.18  −0.17  −0.35  
Proposed  A  −0.20  0.37  0.21  0.88  0.11 
B  0.24  0.95  −0.19  0.09  0.00  
C  0.07  −0.18  0.92  0.03  −0.34  
D  0.99  0.14  0.01  −0.04  0.08 
 (1)
Factor 1 was related to EFBP and transverse beams (Group D),
 (2)
Factor 2 was related to EFBP and no transverse beam (Group B),
 (3)
Factor 3 was related to transverse beam and no EFBP (Group C),
 (4)
Factor 4 was related to no EFBP or transverse beams (Group A).
Factors 1, 2, 3, and 4 are highly related to Groups D, B, C, and A, respectively. However, Group A is most related to Factor 3 for the ASCE equation. Thus, the ASCE equation may not be appropriate to predict the shear strength of Group Atype materials. Additionally, Group A showed ambiguous correlation with factors for AIJ and MAIJ equations; thus, AIJ and MAIJ equations should be applied carefully to Group Atype RCS connections.
4.5 Correlation Analysis
Correlation coefficients.
Equations  Group  Test  

A  B  C  D  
ASCE  A  0.65  0.27  0.41  −0.45 
B  0.35  0.97  −0.41  0.36  
C  0.44  −0.14  0.84  −0.26  
D  0.11  3416.00  −0.16  0.91  
Kanno  A  0.84  0.10  0.41  −0.30 
B  0.50  0.81  −0.37  0.02  
C  0.52  −0.26  0.73  −0.12  
D  0.12  0.22  −0.11  0.91  
MKanno  A  0.84  0.47  0.03  −0.28 
B  −0.02  0.90  −0.50  0.35  
C  0.42  −0.31  0.79  −0.16  
D  0.10  0.30  −0.16  0.94  
AIJ  A  0.74  0.28  −0.09  0.52 
B  −0.16  0.87  −0.35  0.29  
C  0.48  −0.28  0.73  −0.46  
D  −0.03  0.50  −0.41  0.88  
MAIJ  A  0.59  0.52  −0.43  0.65 
B  −0.11  0.91  −0.21  0.32  
C  0.54  −1318.00  0.74  −0.39  
D  −0.06  0.52  −0.42  0.86  
Proposed  A  0.84  0.47  0.03  −0.28 
B  0.09  0.98  −0.35  0.40  
C  0.25  −0.18  0.80  −0.20  
D  0.12  0.41  −0.22  0.97 
5 Conclusions
 (1)
The effect of EFBP and transverse beams on the shear strength of connections in RCS composite systems can be accurately evaluated by mobilization coefficients representing the shear key and transverse beams.
 (2)
The proposed equation considers the confining effects of adding a cover plate and, in doing so, provides a good estimation of the shear strength due to this cover plate.
 (3)
Statistical analysis showed that the proposed equation is the most accurate, compared with previously investigated equations. This analysis also categorized RCS composite system connections into four groups, based on factor analysis results.
 (4)
Correlation analysis showed that our proposed equation correlates well with connection details from the RCS composite system.
Declarations
Acknowledgements
This work was supported by a 2016 Chungwoon University Foundation Grant and a Grant from the Technology Advancement Research Program (15CTAPC09749001), funded by the Korean Ministry of Land, Infrastructure, and Transport Affairs.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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