Development of a Shear Strength Equation for Beam–Column Connections in Reinforced Concrete and Steel Composite Systems
International Journal of Concrete Structures and Materials
Development of a Shear Strength Equation for Beam-Column Connections in Reinforced Concrete and Steel Composite Systems
Kang Seok Lee
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 (E-FBP), 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 E-FBP and transverse beams. Additionally, our results indicated that Architectural Institute of Japan (AIJ) and Modified AIJ (M-AIJ) equations should be used carefully to evaluate the shear strength for connections that do not have E-FBP or transverse beams.
connection design; RCS composite system; shear strength equations; statistical analysis
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.
The load transfer mechanism that occurs at reinforced
concrete and steel (RCS) beams is complex. Failure modes
at these connections can be categorized as: (1) shear and
bearing failure of the internal connection element (Fig. 1a)
and (2) bond and shear failure of the external connection
element (Fig. 1b). Shear failure is most significant among
these failure modes.
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 (E-FBP), 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.
Panel shear failure
Vertical bending failure
2. Background Theory: Evaluating
Connection Shear Strength in RCS
The typical load transfer mechanisms occurring at
connections in RCS composite systems are shown in Fig. 2. The
connection shear strength can be calculated by summing the
contribution of each connection element (including the steel
web, as well as inner and external concrete elements), in
which the reinforced concrete and steel beam web are the
main parts of the connection that resist shearing forces.
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.
Several studies have previously suggested shear strength
equations for connections in RCS composite systems, as
shown in Table 1. As observed within this table, each
equation takes a different approach to evaluate the
connection’s shear strength (Vb). The various attributes of equations
shown in Table 1 are compared below.
The ASCE design code (1993) adopted the ACI-ASCE
352 equation to evaluate the effective width of external
elements (bo), as shown in Table 1. This ASCE design code
limits the effective width by using one mobilization
Vb ¼ ½Vs df þ0½L:L7bc5ðLVcn ddwÞþVjhn0 ðdþdoÞ
Vb ¼ L0b:9dh Vs þ Vsf þ Vn þ Vn
Table 1 Shear strength equations for connections in RCS composite systems.
coefficient (Cwt), regardless of the existence of transverse
beams. However, two different mobilization coefficients,
with coefficient factors that consider shear key (Cwt) and
transverse beams (Ct), are used in the Kanno equations
(Kanno 1993, 2002). Larger values for Cwt and Ct 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 bo than the ASCE equation.
Values of Cwt represent the effective width of the external
element (bo), including the effect of shear keys. Larger Cwt
values denote increased strength of the external concrete, due
to an increase in the effective width of the concrete. To
calculate Cwt, the length of the stress block (x and y values in
Fig. 3) are first determined. A minimum value of Cwt = 0.25
is used within the ASCE equation when a shear key is not
present in the connection. However, in the Kanno equation
(1993), values for x are assumed to be 0.7, and values for y are
assumed to be 0 (Fig. 3a); this results in a Cwt of 0.21. When
E-FBP is used, x and y are equal to h and bf, respectively, as
shown in Fig. 3b. In this case, the mobilization coefficient
(Cwt) is equal to one, indicating a maximum effective width.
0:8 h ðNo shear keyÞ
FBP; Band PlateÞ
max(bi, y) B 2bf
As shown in Table 1, the ASCE equation does not
consider transverse beam effects when calculating bo. However,
the transverse beam effect is included in Kanno equations for
the calculation of Ct. The mobilization coefficient for
transverse beams (Ct) is obtained from the original Kanno
equation by considering the transverse beam twist resistance.
Values for Ct are simply calculated as 0.7Cwt in the
M-Kanno equation (2002).
The maximum effective connection width (bm) is assumed
to be 1/2 of the total connection and flange width length
within the ASCE equation. Both the Kanno and M-Kanno
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 (Vs)] 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 E-FBP or band plates are used,
jh is assumed to be equal to h.
The shear strength of flanges within steel beams (Vsf) 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 (ACI-ASCE 352) uses a
value of 0.4, while 1.05 is used in Kanno and M-Kanno
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 (E-FBP etc.)
The shear key (including a E-FBP 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 Cwt and Cwt as a function of
the effective width x and y (Fig. 3).
When E-FBP 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 E-FBP is welded to the flange, where its width is
not larger than the flange and y/bf 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 (bf), as shown in
Fig. 3c. In this case, the maximum y/bf value is limited to 1,
as shown in Table 1.
Values of Cwt for shear strength equations shown in
Table 1 are a function of y/bf. 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/bf. However, when y/bf is used, the
shear strength can be overestimated. Test results show that
an increase in shear strength upon E-FBP installation is
about 15–20% (Lee 2005). However, strength increases due
to E-FBP installation, as calculated by the Kanno equation
(which uses y/bf), are as much as *50%.
Table 2 shows test specimens used to compare calculated
strengths; all test specimens shown in Table 2 have the same
dimensions, except for E-FBP (Lee 2005). In Table 2, test
specimens JL0-1 and STI-1 do not contain E-FBP; JL0-2
and STI-2 are the same as JL0-1 and STI-1, except for their
inclusion of E-FBP. When y/b was used instead of y/bf in
strength calculations, the mobilization coefficient decreased
from 1 to 0.56, which matched well with test results.
Therefore, it is reasonable to use y/b rather than y/bf to
evaluate Cwt, where the proposed mobilization coefficient is
given by changing bf to b:
In this equation, Cwt is the mobilization coefficient
(including the shear key effect), h is the connection length, y is the
effective width in the y direction (including the shear key
effect), and b is the connection width.
3.2 Transverse Beam Effect
The addition of a transverse beam increases the
connection’s shear strength; for example, Kanno (1993) and Lee
et al. (2004) reported an *15% increase in shear strength
due to the transverse beam. However, according to the
Kanno equation, the increase in connection shear strength is
*10.7%, which is 29% lower than the test results shown in
Table 3 (Lee 2005). Thus, the Kanno equation
underestimates the transverse beam mobilization coefficient (Ct). We
modified Ct using the Kanno equation and test results, as
where Ct is the mobilization coefficient (including the
transverse beam effect), b is the connection width, bi is
effective internal connection width, and d is the beam height.
In this equation, a constant value of 1/2.5 used within the
Kanno equation (as shown in Table 1) was replaced by
1/2.0. This results in a 25% larger Ct value than calculated
by the Kanno equation. However, we do note that Eq. (2) is
based on limited test results, which are later verified using
the test results discussed in Sect. 4.
3.3 Effect of Cover Plates
The effect of cover plates on connection shear strength is
ignored in both ASCE and Kanno equations. However, Lee
et al. (2004) showed that the confining force exerted by the
cover plate is significant, and that the cover plate should be
considered when evaluating shear strength. The shear force
exerted by the cover plate can be calculated as
where a is the strength coefficient, tc the cover plate
thickness, and Fyc is the cover plate yield stress. Comparisons
between shear strength calculations (including the cover
plate effect) and test results are shown in Table 4 (Lee
2005); our proposed equation accurately predicts the
E-FBP extended face bearing plates.
Table 3 Increases in shear strength due to transverse beam.
Strength increment (%)
Table 2 Increases in shear strength due to the E-FBP.
Strength increment (%)
Strength increment (%)
Strength increment (%)
Table 4 Increases in shear strength due to the cover plate.
increase in shear strength upon addition of a cover plate. In
Table 4, LCS-1 and LCC-1 specimens have the same
dimensions (except for the cover plate).
3.4 Proposed Shear Strength Equation
We previously discussed the effects of a shear key,
transverse beams, and cover plates on shear strength. Based
on these results, we suggest the following equation for
calculating the shear strength of connections in RCS composite
Vs df þ 0:75Vn dw þ Vn0 d þ Vcp d
Determination of effective width:
Effective internal element width:
bi ¼ maxðbp; bf Þ
Effective external element width:
bm ¼ ðbf 2þ bÞ \bf þ h\1:75bf
C ¼ maxðCwt; CtÞ
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.
(2) Steel beam web shear strength:
Vs ¼ pffiffi Fyw tw jh
(3) Internal concrete shear strength:
(4) External concrete shear strength:
Vs ¼ Ash ShFysh 0:9h
(5) The cover plate shear strength can be calculated as
Statistical evaluation was conducted to verify the proposed
equations, as described below.
4. Statistically Evaluating Shear Strength
4.1 Comparing Results from Shear Strength
Equations with Test Results
To verify the proposed equations, results from our proposed
equation, as well as other shear strength equations, were
compared with that of test results. A total of 49 shear failure
RCS composite system connection test specimens were
selected from previous studies (Dierlein 1988; Sheikh and Deierlein
1989; Kanno 1993; Kei et al. 1990, 1991; Lee et al. 2004; Choi
et al. 2003). Test specimens were categorized into four types,
depending on connection details: Group A, no shear keys or
transverse beams; Group B, shear keys but no transverse
beams; Group C, transverse beams but no shear key; and
Group D, has both a shear key and a transverse beam. All
connection details (according to group) are shown in Fig. 4.
We next used these test specimens to compare shear
strength values derived from all equations with test results,
as shown in Fig. 5. In Fig. 5, the x-axis represents calculated
values (CalQc) and the y-axis the test results (ExpQc). These
comparisons show that the ASCE equation underestimates
the connection’s shear strength, with a larger discrepancy
between test results for specimens with transverse beams.
This is because the ASCE equation does not contain a
parameter related to the presence of a transverse beam in the
connection. Estimates of shear strength from the Kanno
equation more closely agree with the test results compared
with those of the ASCE. However, test results with a welded
cover plate show a large discrepancy for the Kanno and
M-Kanno equation, even if the simplified calculation method
is used. Results from the Architectural Institute of
JapanStandard for Structural Calculation of Steel Reinforced
Concrete Structures (AIJ-SRC) produced average accuracy
results of *86%, with a standard deviation of 0.37 (the
largest value among all equations). M-AIJ-SRC resulted in a
standard deviation of 0.27, which is greater than that from
ASCE and Kanno equations; this is because AIJ-SRC and
M-AIJ-SRC do not consider shear keys, such as an E-FBP or
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.
Sampling size (as a ratio) can be expressed as
where a = 0.3 (population standard deviation) and
e = ± 0.1 (permitted limit of error).
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 was conducted for the 49 test results to
investigate all shear strength equations (as shown in
Table 5). The ratios of estimated shear strength from the
ASCE equation to test results ranged from 0.47 to 1.16; this
equation underestimated shear strength by 27% compared
with test results. Alternatively, Kanno equations showed
only a 6% discrepancy compared with test results. Our
proposed equation showed the lowest value for standard
deviation (0.11) and best fit to test results with regard to
shear strength. The averaged standard error represents the
averaged error obtained using the standard deviation and
reliability coefficients. Our proposed equation showed the
smallest averaged standard error, as well as the smallest
standard deviation. Thus, our proposed equation accurately
Table 5 Statistical analysis results.
Kanno M-Kanno AIJ
0.7 0.8 0.9 1.0 1.1 1.2
Ratio of estimation and test results
0.7 0.8 0.9 1.0 1.1
Ratio of estimation and test results
0.9 1.2 1.5 1.8 2.1
Ratio of estimation and test results
predicts the shear strength of connections in RCS composite
Skewness represents the skewed angle and direction of
the data distribution, and kurtosis represents the
sharpness of the data distribution; the skewness and kurtosis
for each equation is shown in Table 5. The skewness was
greater than zero for all equations, such that most
estimated and test result values were larger than 1, indicating
that the equations provide a safe estimation of shear
strength. Skewness for the ASCE and M-AIJ equations
was greater than that for the other equations, indicating
that these equations underestimate shear strength to a
greater extent than the others. The kurtosis of all
equations was also larger than 0. This means that the data
distribution is sharper than standard normal distributions,
where data are concentrated around a specific value. The
kurtosis value for the Kanno equation was somewhat
large; however, this was not a significant problem as data
were concentrated around the average value, as shown in
the histogram in Fig. 6. On the other hand, data were
0.6 0.8 1.0 1.2 1.4 1.6
Ratio of estimation and test results
concentrated in several specific sections with an irregular
distribution for the AIJ and M-AIJ equations, making it
difficult for these equations to satisfy a normal
Frequency histograms and normal distributions are plotted
in Fig. 6; in this figure, the x-axis represents the estimation/
test ratio, and the y-axis the frequency. The histograms were
in good agreement with normal distributions for the Kanno,
M-Kanno, and our proposed equation, while the distribution
of data from AIJ and M-AIJ 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
We next conducted factor analysis to reduce parameters to
those with the greatest effect on connection shear strength. Five
major factors were selected from the factor analysis, and each
factor (Factor 1–5) was matched with test and shear strength
equations, as shown in Table 6. Table 6 also shows the degree
of correlation between each factor and connection group
(Fig. 4). For example, Group A showed a high degree of
correlation between the test results and Factor 4 (0.96). Groups
B, C, and D also showed a high degree of correlation with
Factors 2, 3, and 1 (respectively), as shown in Table 6. Factor 5
had a low degree of correlation with Groups A–D (the largest
degree of correlation was 0.41), such that only Factors 1–4
were considered in this study. For Factors 1–4, each showed a
strong relationship with a specific connection group:
Factor 1 was related to E-FBP and transverse beams
Factor 2 was related to E-FBP and no transverse beam
Factor 3 was related to transverse beam and no E-FBP
Factor 4 was related to no E-FBP or transverse beams
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
A-type materials. Additionally, Group A showed ambiguous
correlation with factors for AIJ and M-AIJ equations; thus,
AIJ and M-AIJ equations should be applied carefully to
Group A-type RCS connections.
4.5 Correlation Analysis
Correlation analysis between test results and shear strength
equations was conducted; the results are shown in Table 7.
ASCE, AIJ, and M-AIJ equations all exhibited low correlation
values (\0.7) for Group A, in which no shear key or
transverse beams were installed. Correlation with Group B was
weaker than the others for the Kanno equation. Additionally,
Group C (which does not contain E-FBP, but does contain
transverse beams) showed weak correlation with all
equations, even if the ASCE equation showed the strongest
correlation with Group C. This implies that parameters related to
E-FBP are not well represented in shear strength equations.
Our proposed equation showed the strongest correlation for
Table 7 Correlation coefficients.
Bold values in the table show the highest degree of correlation between test results and shear strength equations with respect to connection
group depicted in Fig. 4.
Group D; thus, our equation accurately encompassed the
effects of E-FBP and transverse beams in shear strength
calculations. Taken together, these results demonstrate that our
proposed equation best predicted the shear strength of RCS
connections, as compared across all equations.
In this paper, we introduced an equation for calculating
shear strength for connections in RCS composite systems,
with verification by statistical analysis. The major findings
of this paper are given below:
(1) The effect of E-FBP 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
This work was supported by a 2016 Chungwoon University
Foundation Grant and a Grant from the Technology
Advancement Research Program (15CTAP-C097490-01),
funded by the Korean Ministry of Land, Infrastructure, and
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