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8/13/2019 CIDECT Final Report 8G-10_06(3of4)
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7-1
SLOTTED END CONNECTIONS TO HOLLOW SECTIONS, CH 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
CHAPTER 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
The weakness shown by current design provisions to provide accurate connection
efficiency factors, close to the experimental data and the parametric analysis results, for slotted
CHS and EHS connections, has been illustrated in previous chapters. Therefore, new equations
that model the connection behaviour more accurately, based on either the Lw/w or /Lw ratio,
are suggested in this chapter. Moreover, the efficiency factors based on these new equations
are then compared to previous experimental programs and are shown to correlate favourably.
In addition, as seen throughout the experimental program and the parametric analysis of
the slotted gusset plate connections, the attainment of the maximum connection strength was
always associated with excessive distortion of the tube cross-section. For this reason, the use
of a distortion limit has been recommended to control the maximum strength for this connection
type. Hence, ultimate capacity equations providing efficiency factors based on this limit are also
suggested herein.
7.1 CHS connections in tension - CF failure
7.1.1 Shear lag equations suggested for CSA design provision format
7.1.1.1 Equation suggested for slotted CHS to gusset plate connections
In order to provide an equation representing the trend shown for CF failure by parametricanalysis results, where a gradual increase in the connection efficiency (U) occurred at the same
time as the weld length, several equation formats were examined. A very good fit to this data is
given by equation (7-1), which only utilizes the Lw/w ratio as suggested by the current CSA
(2001), which presumes that this has the primary influence. Moreover, this same behaviour was
seen during both the experimental program and the parametric analysis.
(7-1)
The coefficients a and b were determined by nonlinear regressions with the use of the
software Sigmaplot 9.0 (Systat Software 2004). Despite this software being able to provide
coefficients with four significant figures, it was found that these could be rounded to two or only
one significant figure (in order to simplify the use of these equations for design provisions)
without affecting the accuracy of the equations to predict the connection efficiency.
x'
U 11
1Lww------
a
+
b--------------------------------–=
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7-2
SLOTTED END CONNECTIONS TO HOLLOW SECTIONS, CH 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
The use of coefficients a=2.4 and b=5.7 can provide a very good prediction of the
connection efficiency for slotted CHS connections, with a mean actual-to-predicted ratio equal
to 1.00 and a coefficient of variation (COV) of 4%. Hence, the recommended expression for a
CSA format can be written as:
(7-2)
Despite the fact that the application of this equation seems appropriate for Lw/w ratios
below 0.70 (see Figure 7.1), its range of validity has been limited to ratios of Lw/w , in
accordance with the parametric analysis results, because the transition from a TO failure to a
CF failure took place in a region near this value. Moreover, the failure mechanism involved in a
TO failure diverges from the CF approach. Nevertheless, a convergence on the actual
connection strength with either prediction approach is expected in this low Lw/w region.
Besides the influence of the Lw/w ratio on connection efficiency, the parametric analysis
results have also illustrated some effect of the tube D/t ratio on this efficiency. During these
analyses, connections having similar Lw/w ratios but a lower D/t ratio attained a higher
efficiency than their counterparts with a high D/t ratio. Hence, equation (7-2) can be modified as
follows:
UCS A sl ot ted– tube– 11
1Lww------
2.4
+
5.7---------------------------------------–=
0.7≥
N
/ A
F
u F E
n
u
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5
U suggested
Slotted CHS (no weld return)
Slotted CHS (weld return)
TO Failure
CFShear
LagPresent
CF
L /w w
1 1
1L
ww
------( )2.4
+
5.7-------------------------------------- – =
CSA-slotted-tube
U
Figure 7.1 Suggested efficiency factor and parametric analysis results from slotted CHS
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7-3
SLOTTED END CONNECTIONS TO HOLLOW SECTIONS, CH 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
(7-3)
The equation format and coefficients offering the best-fit are given by equation (7-3). The
use of this modified equation does not have a significant impact on the connection efficiency as
the mean remained the same (1.0) and only represented an improvement to the COV (as it
decreased from 4% to 3.6%). Once again, the range of validity for this equation corresponds to
the ratios previously defined for equation (7-2), namely 15 D/t 45 and Lw/w 0.7. Based on
the results from equation (7-3), it is possible to infer that the connection Lw/w ratio has the main
influence on the connection efficiency and equation (7-2) favourably represents this general
behaviour. Since the inclusion of the D/t ratio in equation (7-3) does not have a significant
impact on the predicted connection efficiency, the use of the simplified equation (7-2) seems
appropriate for design provisions. The results of these comparisons are summarized in
Table 7.1.
a) Data corresponding to FE connections with .
7.1.1.2 Equation suggested for slotted gusset plate to CHS connections based on
ultimate strength
In an attempt to generalize the use of equation (7-2), it was also applied to the parametric
analysis results of slotted gusset plate connections resulting in a mean actual-to-predicted ratio
of 1.00 and a COV of 2.9%. Despite this result, a new regression was carried out exclusively
with the data corresponding to the parametric analysis results from the slotted gusset plate
connections as a means to provide an equation following the general trend of this data.
Equation (7-4) shows these new coefficients based on these data points.
(7-4)
The use of coefficients 1.4 and 4.65 provided a mean actual-to-predicted ratio of 1.00 and
a COV of 2.6%. These results are almost identical to the correlation provided by equation (7-2).
Table 7.1 Evaluation of potential equations for slotted CHS
FE results a)FE results /
equation (7-2)
FE results /
equation (7-3)
Slotted CHS (no weld return) and
Slotted CHS (weld return)146
1.00 1.00 Mean
4.0% 3.6% COV
UCSA s lot te d– tube– 11
1Lww------
2.4
+
5.7---------------------------------------
–
1.18D
t----
0.05–
1.0≤=
≤ ≤ ≥
Lw w ⁄ 0.7≥
UC SA s lot te d– gusset– 11
1Lww------
1.4
+
4.65-----------------------------------------–=
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7-5
SLOTTED END CONNECTIONS TO HOLLOW SECTIONS, CH 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
7.1.1.3 Equation suggested for slotted gusset plate connections based on deformation
limit (0.03D)
As seen throughout the parametric analysis, the attainment of the maximum strength for
this connection type was always accompanied by excessive distortion in the tube cross-section.
Therefore, the use of an ultimate deformation limit of 0.03D (which typically occurs before the
presence of any failure mechanism) has been suggested herein as a means to control this
distortion. Based on the connection strength calculated at this limit throughout the parametric
analysis, equation (7-5) was derived to give a connection efficiency, which is also dependent on
the connection Lw/w ratio.
(7-5)
The best-fit for the coefficients in equation (7-5) corresponded to 0.23 and 0.55 which
produced a mean actual-to-predicted ratio of 1.00 and a COV of 7.1%. Figure 7.3 shows the
suggested efficiency factor and the parametric analysis results.
The validity of equation (7-5) has been limited to ratios Lw/w 0.40. Below this range, it is
expected that TO failure will govern the behaviour of the connection. In contrast, at a ratio of Lw/
w=1.95, 100% of the connection efficiency is achieved, but the connection efficiency is clearly
limited to 1.0. This follows the behaviour that was seen throughout the parametric analysis
UCSA s lo tt ed– gusset– 0.
·03D–
0.23Lww------
0.55 1.0≤+=
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5
L / ww
N
/ A
F
u F E
n
u
UCSA-slotted-gusset-0.03D
0.23Lww------( ) 0.55+=U suggested for 0.03 D
Slotted gusset plate at D0.03
TO failure
No failure of connection
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7-6
SLOTTED END CONNECTIONS TO HOLLOW SECTIONS, CH 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
results, where the use of a long weld allowed the attainment of the connection maximum
strength at almost the same time as the distortion limit.
It was observed throughout the parametric analysis that the use of stockier tubes
postponed the deformation of the cross-section and permitted the attainment of higher
connection strength at the deformation limit. The influence that the D/t ratio may have on the
connection efficiency has thus been considered, resulting in equation (7-6):
(7-6)
Equation (7-6) provided a mean and COV of 1.01 and 3.5% respectively. Once again, a
lower bound (a ratio of Lw/w =0.40) has been suggested as a validity limit for this equation since
TO failure will govern for lower values. The attainment of the full efficiency for thinner tubes may
require longer welds than for thicker tubes, but the maximum possible efficiency will be always
limited to 100% of AnFu. Although the inclusion of the D/t ratio in equation (7-6) improved the
calculated COV (see Table 7.3), it is suggested to use the simplified equation (7-5).
a) Data corresponding to the connection strength at a deformation limit of 0.03D, for slotted gusset plate connections
to CHS.
7.1.2 Shear lag equations suggested for AISC design provision format
As seen during the parametric analyses, the transition from a TO failure to a CF failure
and the attainment of tube necking were clearly defined by the connection Lw/w ratios. In order
to transfer these limits (based on Lw/w) to the AISC design provision, the use of the ratio /Lw
(as used by AISC) was found to be inappropriate since a number of Lw/w ratios can be
calculated (from a single /Lw ratio) once several plate thicknesses are considered. Therefore,
the use of a reduced eccentricity ( ) is considered more appropriate because its calculation
provides a unique /Lw ratio since it accounts for the plate thickness (as the Lw/w ratio
Table 7.3 Evaluation of potential equations for slotted gusset plate connections using anultimate deformation limit state of 0.03D
FE results a)FE results /
equation (7-5)
FE results /
equation (7-6)
Slotted gusset plate to CHS 991.00 1.01 Mean
7.1% 3.5% COV
UCSA sl ot te d– gusset– 0.03D– 0.23Lww------
0.55+ 1.81 D
t----
0.18–
1.0≤=
x
x
x'
x'
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7-7
SLOTTED END CONNECTIONS TO HOLLOW SECTIONS, CH 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
approach also does). Hence, equations making use of the reduced eccentricity ( ) are
suggested herein.
7.1.2.1 Equation suggested for slotted CHS to gusset plate connections
Equation (7-7) shows the suggested equation to calculate the connection efficiency based
on the /Lw ratio. The use of coefficients of 3.2 and 9.9 provided a mean actual-to-predicted
ratio of 1.00 and a COV of 3.9%.
(7-7)
This equation is suitable for ratios /Lw< 0.245. This limit corresponds to the point of
change from a CF to a TO failure during the parametric analysis (see Figure 7.4). Beyond this
ratio, it is expected that a TO failure mechanism governs.
The influence of the D/t ratio was then included in equation (7-8) and a further regression
was carried out to determine the coefficients providing the best-fit. In contrast to previous
experience, the addition of this ratio only provided just a slight modification to the COV (see
Table 7.4). Hence, the sole use of equation (7-7) is suggested in order to simplify the calculation
of the connection efficiency.
x'
x'
U AI SC sl ot te d– tube–1
1x'
Lw------
3.2
+
9.9---------------------------------------=
x'
0.0
0.1
0.2
0.30.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
0.0 0.1 0.2 0.3 0.4 0.5
x'/Lw
N
/ A
F
u F E
n
u
U suggested
Slotted CHS (no weld return)Slotted CHS (weld return)
TO Failure
CF
Shear
Lag
Present
CF
1
1Lw------( )
3.2 9.9--------------------------------------=U
+ X’
0.245
AISC-slotted-tube
Figure 7.4 Suggested efficiency factor and parametric analysis results from slotted CHS (AISCdesign provision format)
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7-8
SLOTTED END CONNECTIONS TO HOLLOW SECTIONS, CH 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
(7-8)
a) Data corresponding to FE connections with .
7.1.2.2 Equation suggested for slotted gusset plate to CHS connections based on
ultimate strength
A regression was carried out with the slotted gusset plate connection data and equation
(7-9) shows the new coefficients based on these data.
(7-9)
The use of the coefficients 2.4 and 3.2 in equation (7-9) provided a mean actual-to-
predicted ratio of 1.01 and a COV of 2.4%. In addition, equation (7-7) was also applied to this
data resulting in a similar mean and COV (1.0 and 2.4% respectively). The range of validity for
these equations corresponds to 15 D/t 45 and . In a similar manner to section
7.1.1.2, where the equation suggested for slotted tubes was applied to slotted gusset plates for
the CSA format, these two equations showed closeness for small ratios (see Figure 7.5).
Since the use of either equation (7-7) or (7-9) provided almost the same mean (1.0 &
1.01) and a very small COV, it was decided not to develop an equation including the effect of the D/t ratio. Based on these results (see Table 7.5), the use of equation (7-7) is recommended
to calculate the connection efficiency of slotted gusset plate connections within the validity limits
of 15 D/t 45 and 0.245.
Table 7.4 Evaluation of potential equations for slotted CHS connections (AISC)
FE results a)FE results /
equation (7-7)
FE results /
equation (7-8)
Slotted CHS (no weld return) and
Slotted CHS (weld return)146
1.00 1.00 Mean
3.9% 3.8% COV
U AI SC sl ot te d– tube–1
1x'
Lw------
3.2+
9.9---------------------------------------
D
t----
1.95–
+ 1.0≤=
x' Lw ⁄ 0.245≤
U AISC sl ot te d– gusset–1
1x'
Lw------
2.4+
3.2---------------------------------------=
≤ ≤ x' Lw ⁄ 0.245≤
x' Lw ⁄
≤ ≤ x' Lw ⁄ ≤
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7-9
SLOTTED END CONNECTIONS TO HOLLOW SECTIONS, CH 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
a)
Data corresponding to FE connections with .
7.1.2.3 Equation suggested for slotted gusset plate connections based on deformation
limit (0.03D)
In contrast to section 7.1.1.3, the data herein described a nonlinear variation when it was
plotted against the /Lw ratio. Therefore, a new equation format was suggested and the best-fit
to the data corresponds to equation (7-10). The coefficients 0.26 and 0.40 produced a mean
actual-to-predicted ratio 1.00 and a COV of 7.1%.
(7-10)
This equation is suitable for connections with ratios /Lw< 0.44. Above this ratio, a TO
failure will govern the behaviour of these connections (see Figure 7.6). In order to attain the full
Table 7.5 Evaluation of potential equations for slotted gusset plate connections (AISC designprovision format)
FE results a)FE results / equation
(7-7)
FE results /
equation (7-9)
Slotted gusset plate to CHS 631.00 1.01 Mean
2.4% 2.4% COV
N
/ A
F
u F E
n
u
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
U suggested (7-7)
Slotted gusset plate
U suggested (7-9)
X' w/ L
TO Failure
CFShear
Lag
Present
Necking
0.245
Figure 7.5 Suggested efficiency factor and parametric analysis results from slotted gussetplate to CHS connections (AISC design provision format)
x'
Lw ⁄ 0.245≤
x'
U AI SC sl ot te d– gusset– 0.03D– 0.26 11
x'
Lw------
0.4------------------+
=
x'
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7-10
SLOTTED END CONNECTIONS TO HOLLOW SECTIONS, CH 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
connection efficiency, a ratio of /Lw=0.073 will be required. This equation will demand the use
of a longer weld length than that recommended by equation (7-5). This discrepancy can be
attributed to the difference in the formulation of these equations; while the first shows a linear
variation, this second one does not.
The influence of the D/t ratio is included in equation (7-11) where the coefficients 1.77 and
-0.17 produced a mean actual-to-predicted ratio equal to 1.00 and a COV of 3.7%.
(7-11)
In a similar manner as equation (7-10), the validity of this equation has been limited to
ratios /Lw< 0.44 and the maximum attainable connection efficiency is defined as 100% of
AnFu. In the same way as equation (7-6), the inclusion of the D/t ratio in equation (7-11) has
captured the scatter in the data and improved the prediction of the efficiency factor (see
Table 7.6).
x'
N
/ A
F
u F E
n
u
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
U suggested for 0.03D
Slotted gusset plate at 0.03D
X w'/ L
TO failure
U AISC-slotted-gusset-0.03D
0.26 1 1
x'
Lw------( ) 0.4
--------------------+
( )=
No failure of connection
0.44
Figure 7.6 Suggested efficiency factor and parametric analysis results from slotted gusset plateto CHS connections at 0.03D deformation limit (AISC design provision format)
U AISC sl ot te d– gusset– 0.03D– 0.26 11
x'
Lw------
0.4------------------+
1.77D
t----
0.17–
1.0≤=
x'
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7-11
SLOTTED END CONNECTIONS TO HOLLOW SECTIONS, CH 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
a) Data corresponding to the connection strength at a deformation limit of 0.03D, for slotted gusset plate connections
to CHS.
7.2 EHS connections in tension - CF failure
As seen in the previous section 7.1, the use of equations initially suggested to predict the
efficiency of slotted CHS connections provided acceptable results (within their validity limits)
when they were applied to slotted gusset plate connections. Therefore, the applicability of these
equations to results from EHS connections is studied herein. As a result, several new equations
are suggested to be used specifically for EHS connections.
7.2.1 Shear lag equations suggested for CSA design provision format
7.2.1.1 Equation suggested for slotted EHS to gusset plate connections
Even though equation (7-2) appears close to the data representing the response of
slotted EHS connections, especially for ratios of Lw/w ranging from 0.6 to 0.9 (see Figure 7.7), it
shows an early attainment of the full efficiency that contrasts with the trend shown by the EHS
data. Equation (7-2) can provide a mean actual-to-predicted ratio equal to 0.97 and a COV of
4.6% for data having ratios of Lw/w>0.6. A regression carried out exclusively with the data from
EHS connections produced the coefficients of 1.3 and 3.8 in equation (7-12) which resulted in a
mean actual-to-predicted ratio of 1.00 and a COV of 3.7%.
(7-12)
In Figure 7.7, equation (7-12) exhibits a better prediction of the connection efficiency than
equation (7-2), particularly showing accuracy for connections with large Lw/w ratios. Moreover,
in contrast to the trend shown by equation (7-2), this proximity continues occurring for Lw/w
ratios below 0.6 where a TO failure is expected to govern. Note also that Figure 7.7 includes
data for slotted EHS in both the major and minor axis directions.
Table 7.6 Evaluation of potential equations for slotted gusset plate connections using an
ultimate deformation limit state of 0.03D (AISC design provision format)
FE results a)FE results / equation
(7-10)
FE results / equation
(7-11)
Slotted gusset plate to
CHS 99
1.00 1.00 Mean
7.1% 3.7% COV
UC SA s lot te d– tube– EH S– 11
1Lww------
1.3
+
3.8---------------------------------------–=
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7-12
SLOTTED END CONNECTIONS TO HOLLOW SECTIONS, CH 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
A further regression was undertaken to include the influence of the Davg/t ratio in equation
(7-13). However, this new equation (with a coefficient of -2.9) resulted in a similar mean actual-
to-predicted ratio and COV as equation (7-12). Since this clearly does not represent an
improvement with respect to the initial equation, the use of the simplified equation (7-12) for this
connection type is recommended. Finally, the results from these comparisons are shown in
Table 7.7.
(7-13)
a) Data corresponding to FE connections with .
7.2.1.2 Equation suggested for slotted gusset plate to EHS connections based on
ultimate strength
The use of equation (7-2) with data corresponding to slotted gusset plate to EHS
connections provided a mean actual-to-predicted ratio of 0.99 and a COV of 4.1%. Even though
Table 7.7 Evaluation of potential equations for slotted EHS connections
FE results a)FE results /
equation (7-2)
FE results /
equation (7-12)
FE results /
equation (7-13)
Slotted EHS having small
and large143
0.97 1.00 1.00 Mean
4.6% 3.7% 3.7% COV
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6
N
/ A
u F E
n
u
F
L /ww
Slotted EHS with small x
Slotted EHS with large x
Suggested for CHS (7-2)
Suggested for EHS (7-12)
TO Failure
CF
Shear
Lag
Present
CF
Figure 7.7 Suggested efficiency factor and parametric analysis results from slotted EHS
UCSA sl ot te d– tube– EH S– 11
1Lww------
1.3
+
3.8---------------------------------------
–
Davg
t-----------
2.9–
+ 1.0≤=
x'
Lw w ⁄ 0.6≥
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7-13
SLOTTED END CONNECTIONS TO HOLLOW SECTIONS, CH 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
this appears to be a good result, it is mainly a consequence of the close correlation of data
located in the range of Lw/w from 0.6 to 1.0 (see Figure 7.8). Moreover, for ratios of Lw/w < 0.6
equation (7-2) does not follow the data trend, as has been the philosophy of the equations
provided previously. On the contrary, equation (7-12) can follow this trend, however it
underestimates the efficiency of these connections and provides a mean actual-to-predicted
ratio of 1.03 and a COV of 3.1%. Despite this, the use of this equation may still be accepted as
it offers a lower bound. A further regression undertaken with the data from slotted gusset plate
connections produced the coefficients of 1.2 and 4.3 in equation (7-14), with a mean actual-to-
predicted ratio of 1.00 and a COV of 2.5%.
(7-14)
Since the use of equation (7-14) provided an excellent mean (1.0) and a very small COV,
it was decided not to developed an equation including the effect of the Davg/t ratio. Although
equation (7-2) may be also used for this connection type, the use of an equation that is able to
follow the trend of the data (including beyond the CF failure region), such as (7-12) but
especially (7-14), is strongly recommended. The results from these comparisons are shown in
Table 7.8.
UCS A sl ot ted– gusset– EH S– 11
1Lww------
1.2
+
4.3---------------------------------------–=
0.0
0.10.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6
N
/ A
F
u F E
n
u
L /ww
TO Failure
CFShear
LagPresent
CF
Slotted gusset plate to EHS
Suggested for CHS (7-2)
Suggested for slotted gussetGplate to EHS (7-14)
Suggested for slotted EHS (7-12)
Figure 7.8 Suggested efficiency factor and parametric analysis results for slotted gusset plateto EHS connections
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7-14
SLOTTED END CONNECTIONS TO HOLLOW SECTIONS, CH 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
a) Data corresponding to FE connections with .
7.2.1.3 Equation suggested for slotted gusset plate connections based on deformation
limit (0.03D2)
In contrast to slotted gusset plate connections to CHS, where a clear linear increase in the
connection efficiency occurs as the weld length increases (as well as the possibility of attaining
100% of AnFu), the data from EHS connections did not show a clear tendency (see Figure 7.9).
Despite a gradual increase in the connection efficiency (as the weld length increases), hereinthe maximum efficiency attained never exceeded 90% of AnFu. Moreover, the presence of
shear lag produced a data scatter in the region where this phenomenon has a major influence.
Therefore, a new equation format was suggested in equation (7-15) to fit this data. This
equation has a range of validity between the Lw/w ratios of 0.2 and 1.5. A regression with this
data provided the coefficients 0.75, 1.5 and 1.2, which resulted in a mean actual-to-predicted
ratio of 1.01 and a COV of 6.8%.
(7-15)
Due to the complex nature of the data distribution, it was decided not to present an
additional equation including the effect of the Davg/t ratio. The results from this comparison are
shown in Table 7.9.
a) Data corresponding to the connection strength at a deformation limit of 0.03D2, for slotted gusset plate connections
to EHS.
Table 7.8 Evaluation of potential equations for slotted gusset plate to EHS connections
FE results a)FE results /
equation (7-2)
FE results /
equation (7-12)
FE results /
equation (7-14)
Slotted gusset plate
to EHS76
0.99 1.03 1.00 Mean
4.1% 3.1% 2.5% COV
Table 7.9 Evaluation of suggested equation for slotted gusset plate to EHS connections usingan ultimate deformation limit state of 0.03D
2
FE results a) FE results / equation (7-15)
Slotted gusset plate to CHS 1021.01 Mean
6.8% COV
Lw w ⁄ 0.6≥
UCS A sl ot te d– gusset– EH S– 0.03D2
–
0.75
1
Lww------
1.2–
1.75--------------------------
2
+
--------------------------------------------------=
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SLOTTED END CONNECTIONS TO HOLLOW SECTIONS, CH 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
7.2.2 Shear lag equations suggested for AISC design provision format
7.2.2.1 Equations suggested for slotted EHS to gusset plate connections
In contrast to the study of slotted EHS for the CSA design format, where all the data
showed close correlation as they were located in a region bounded approximately by L w/w
ratios of 0.3 and 1.5, a plot of these results against the /Lw ratio produces a greater data
scatter as they are grouped based on the eccentricity of each connection type (see Figure
7.10). Several attempts were made in order to provide a single equation format which could
predict the connection efficiency based on the ratio /Lw and the connection eccentricity.
However, this option was reconsidered because of the differences in applicability limits for each
connection and their data trends. Hence, two equations have been suggested here which can
be applied depending of the connection eccentricity.
For EHS connections with a small eccentricity ( ), a further regression carried out
exclusively with the data of this connection type produced the coefficients of 2.15 and 9.3 inequation (7-16), with a mean actual-to-predicted ratio of 1.00 and a COV of 2.7%. The range of
validity for this equation is 0 /Lw 0.14 (where 0.14 /Lw corresponds to 0.7 Lw/w, which
delineated the transition in connection behaviour during the parametric analysis, with values
below this producing a TO failure). Moreover, equation (7-16) can also provide a good
prediction of the efficiency of connections near this limit but in the TO failure region.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6
Slotted gusset plate to EHS at 0.03 D2
Suggested for slotted gusset plate to EHS at 0.03 D2
N
/ A
F
u F E
n
u
L /ww
Figure 7.9 Suggested efficiency factor and parametric analysis results for slotted gusset plate
to EHS connections at 0.03 D2
x'
x'
x'
≤ x' ≤ x'
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SLOTTED END CONNECTIONS TO HOLLOW SECTIONS, CH 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
(7-16)
For EHS connections with a large eccentricity, a further regression produced the
coefficients 2.1 and 2.8 in equation (7-17), with a mean actual-to-predicted ratio of 1.00 and a
COV of 2.4%. The range of validity for this equation is 0 /Lw 0.33 (where 0.33
corresponds to the transition region defined by 0.6 Lw/w, beyond which a TO failure is
expected). Nevertheless, equation (7-17) may also provide a close prediction for connections
showing a larger /Lw ratio but still near this limit (as did equation (7-16)).
(7-17)
Figure 7.10 shows the connection efficiency given by equation (7-7), in an attempt to
generalize this equation (initially developed for CHS). Its use is inappropriate for connections
having a small eccentricity as it would over estimate their real behaviour, but equation (7-7) has
close correlation to the data of EHS connections having a large eccentricity. Here, the
application of equation (7-7) resulted in a mean actual-to-predicted ratio of 0.98 and a COV of
2.8%.
U AISC sl ot te d– EHS– s ma ll x'( )–1
1x'
Lw------
2.15+
9.3-----------------------------------------=
≤ x' ≤
x'
U AI SC sl ot te d– EH S– l e x'( )arg–1
1x'
Lw------
2.1+
2.8---------------------------------------=
N
/ A
F
u F E
n
u
x'/Lw
CF
TO FailureSlotted EHS with large x'
Slotted EHS with small x'
Suggested for large x' (7-17)
Suggested for small x' (7-16)
Suggested for CHS (7-7)
TO Failure
Shear lag present
Shear lagpresent
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
0.14 0.33
Figure 7.10 Suggested efficiency factor and parametric analysis results from slotted EHSconnections (AISC design provision format)
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SLOTTED END CONNECTIONS TO HOLLOW SECTIONS, CH 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
Since both equations (7-16) and (7-17) provide a good mean (1.0 for both equations) and
a very small COV, it was decided not to develop additional equations including the effect of the
Davg/t ratio. The results from these comparisons are shown in Table 7.10.
a) Data corresponding to FE connections with .
b) Data corresponding to FE connections with .
Based on these results, it is noted that the parameter defining the connection efficiency is
principally the Lw/w ratio, and the eccentricity merely determines the capacity of each
connection to attain a higher efficiency if Lw/w is low. Furthermore, the use of the /Lw ratio to
explain the behaviour of these connections may be inappropriate as it requires two separate
equations.
7.2.2.2 Equations suggested for slotted gusset plate to EHS connections based on
ultimate strength
In an attempt to generalize the use of the equation (7-7), it was applied to this data. Figure
7.11 shows that this equation can provide an acceptable result as it produced a mean and COV
of 1.03 and 4.2% respectively. Equations (7-16) and (7-17) were also extended to this
connection type. While the prediction from equation (7-16) clearly disagrees with the data trend,
equation (7-17) was able to emulate the behaviour of this connection type. The use of this
equation produced a mean actual-to-predicted ratio of 1.05 and a COV of 3.3%. Since the use
of equation (7-17) can provide a lower bound, its use is recommended for this connection type.
The range of validity of this equation corresponds to 0 /Lw 0.33 (The upper limit
corresponds to a ratio of Lw/w=0.6 which defined the transition in behaviour during the
parametric analysis, where beyond this value a TO failure is expected to govern). The results
from this comparison are shown in Table 7.11.
Table 7.10 Evaluation of potential equations for slotted EHS connections (AISC designprovision format)
FE
results
FE results/
eq(7-16)
FE
results
FE results/
eq(7-7)
FE results/
eq(7-17)
Slotted EHS
with small
61 a)1.00 Slotted EHS
with large76 b)
0.98 1.00 Mean
2.7% 2.8% 2.4% COVx' x'
x' Lw ⁄ 0.14≤
x' Lw ⁄ 0.33≤
x'
≤ x' ≤
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SLOTTED END CONNECTIONS TO HOLLOW SECTIONS, CH 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
a) Data corresponding to FE connections with .
7.2.2.3 Equation suggested for slotted gusset plate to EHS connections based on
deformation limit (0.03D2)
As shown during the study of this data for the CSA format, this data lacks a defined trend
(see Figure 7.12). However, when this data was plotted against the /Lw ratio, it was possible
to apply a simplified equation format. The use of coefficients of -0.35 and 0.8 in Equation (7-18),
resulted in a mean actual-to-predicted ratio of 1.00 and a COV of 6.7%.
(7-18)
The range of validity of this equation is 0.13 /Lw 0.52, which corresponds to the
region where the data is available. Beyond these boundaries the use of equation (7-18) is not
recommended. Due to the complex nature of the data distribution, it was decided not to present
an additional equation including the effect of the Davg/t ratio. The result from this comparison is
shown in Table 7.12.
Table 7.11 Evaluation of potential equation for slotted gusset plate to EHS connections (AISCdesign provision format)
FE results a) FE results / equation (7-7) FE results / equation (7-17)
Slotted gusset plate
to EHS76
1.03 1.05 Mean
4.2% 3.3% COV
N
/ A
F
u F E
n
u
x /Lw'
TO Failure
CFShear
LagPresent
CF
0.0
0.1
0.2
0.3
0.4
0.5
0.60.7
0.8
0.9
1.0
1.1
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40
Slotted gusset plate to EHS
Suggested for CHS(7-7)
Suggested for small x' (7-16)
Suggested for large x' (7-17)
0.33
Figure 7.11 Suggested efficiency factor and parametric analysis results from slotted EHS(AISC design provision format)
x' Lw ⁄ 0.33≤
x'
U AISC sl ot te d– gusset– EH S– 0.03D2– 0.35x'
Lw------
– 0.8+=
≤ x' ≤
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SLOTTED END CONNECTIONS TO HOLLOW SECTIONS, CH 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
a) Data corresponding to FE connections with 0.13 /Lw 0.52.
7.3 CHS and EHS connections in tension - TO failure
To determine the factored tensile resistance (Tr ) of connections failing by “block shear”,
there are various models currently advocated in design provisions. However, only the accuracy
of the model suggested by CSA and AISC, which combines the tensile resistance of the area in
tension and the shear resistance of the area in shear, is evaluated here. This tensile resistance
is given by:
Tr = [Ant Fu + 0.6 Agv Fy Ant Fu + 0.6 Anv Fu] (7-19)
where, = resistance factor (this is defined by the design provision, but was equated to 1.0 in
this section), Agv= gross area subject to shear, Anv= net area subject to shear (where Anv= Agv
since these are welded connections), Ant= net area subject to tension, Fu= specified minimum
tensile strength and Fy= specified minimum yield stress.
Table 7.12 Evaluation of potential equation for slotted gusset plate to EHS connections using anultimate deformation limit state of 0.03D2 (AISC design provision format)
FE results a)FE results /
equation (7-18)
Slotted gusset plate to EHS 961.00 Mean
6.7% COV
Slotted gusset plate to EHS at 0.03 D2
Suggested for slotted gusset plate to EHS at 0.03 D2
N
/ A
F
u F E
n
u
x’/Lw0.0
0.1
0.2
0.3
0.4
0.50.6
0.7
0.8
0.9
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55
Figure 7.12 Suggested efficiency factor and parametric analysis results from slotted gussetplate to CHS connections at 0.03D2 (AISC design provision format)
≤ x' ≤
φ ≤
φ
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SLOTTED END CONNECTIONS TO HOLLOW SECTIONS, CH 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
From the moment when the first model was suggested by Birkemoe and Gilmor (1978),
subsequent research has recognized the importance that the connection type may have on the
stress distribution and consequently the connection strength. Therefore, that model
experienced several modifications (as described in Chapter Two) and nowadays various
coefficients affecting the tensile resistance are recommended in design provisions. In addition,
Cunningham et al. (1995) and Topkaya (2004) have also suggested alternative models for
specific connection types, however their use may be too complicated for practical engineers.
Equation (7-20) shows a modification to the original model recently suggested by Driver et al.
(2006), following the philosophy of adding resistance terms, that combines effective stresses
and mean stress correction factors affecting the tension area (R t) and the shear area (Rv), to
account for non-uniform stress distributions characteristic to each connection type.
Tr = [Rt Ant Fu + Rv Agv ] (7-20)
Even though all this research has significantly improved the accuracy of design equations
for bolted connections, the lack of data corresponding to welded connections has been found to
be a potential limitation to the use of these equations for these connection types. Furthermore,
the applicability of these models to slotted end connections to HSS may also need further
attention. Therefore, the FE analysis results failing by TO have been compared against the
predicted strength by equations (7-19) and (7-20) to assess their accuracy. In all cases, the
weld size was considered to calculate the tensile resistance of the net area in tension (Ant).
These two models presume that the attainment of the maximum load occurs by fracture
along Ant and by shear yielding along Agv. As a general rule, these connections show non-
uniform stress distributions along Ant as the load increases (see Figure 7.13), however this
distribution becomes fairly uniform at the maximum load as excessive deformation occurs in this
small region (see Figure 7.14). In order to attain fracture of Ant the material there must reach a
strain equivalent to the ultimate tensile strain whence it becomes unable to redistribute load to
Agv. This increases the magnitude of stresses at Agv which continues until fracture of Ant occurs.
At fracture the load level at Ant is below its expected value ( ) and stresses along Agv
exceed 0.6 Fy or , as suggested by current models. The stress level along Agv ranges
from 0.6 Fy to Fu and in some cases necking will start there (at the beginning of the weld).
φFy Fu+
2 3------------------
Ant Fu⋅
Fy 3 ⁄
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SLOTTED END CONNECTIONS TO HOLLOW SECTIONS, CH 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
Finally, once the material breaks, a rapid load decrease takes place as the crack continues
propagating to the weld toe and then towards the tube end (along the weld).
(MPa)
Figure 7.13 Typical stress distribution (von
Mises) at the slot region
(MPa)
Figure 7.14 Stress distribution (von Mises) at themaximum load
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SLOTTED END CONNECTIONS TO HOLLOW SECTIONS, CH 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
Figure 7.15 shows an ultimate strength comparison between the slotted CHS connections
and the loads predicted by equation (7-19). Even though this data shows some scatter, its trend
clearly exhibits a linear variation that stays close to a ratio of 1.0 (providing a mean actual-to
predicted ratio of 0.99 and a COV of 1.8%). Based on these results, it is possible to suggest that
this model predicts acceptably the strength of connections failing by TO in the range of Lw/w
ratios from 0.40 to 0.70. Beyond these limits, it is expected that this equation will produce
unsafe results as the governing failure mechanism changes.
As explained before, the attainment of fracture at Ant will always be associated with a non-
uniform stress distribution along Agv, which in most cases exceeds a value of 0.6 Fy. For
connections fabricated with materials having a considerable difference in their mechanical
properties (Fy and Fu), the model suggested by equation (7-20) can provide a better prediction
as it will raise the effective stress acting along Agv. In contrast, this equation will not provide a
significant improvement for connections fabricated with materials showing closeness in their
properties, such as those from the CHS herein (where Fu/Fy=1.08). The calculated mean and
COV for correlation with equation (7-20) is 0.99 and 1.8% respectively, which represents no
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80
L /ww
N
/ ( A
F
+ 0
. 6 A
F
)
u F E
n t
u
n v
y
D/t=15
D/t=20
D/t=25
D/t=30
D/t=35
D/t=40
D/t=45
TO Failure
CF
Shear
Lag
Present
Figure 7.15 Correlation of equation (7-19) for slotted CHS connections (no weld return)
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SLOTTED END CONNECTIONS TO HOLLOW SECTIONS, CH 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
improvement from the previous prediction. In order to increase the mean value to 1.0, a
correction factor Rv=0.99 could be applied to this equation. Nevertheless, stress correction
factors equal to 1.0 are suggested herein as a means of simplifying the use of equation (7-20)
for this connection type (see Figure 7.16).
For slotted CHS connections having a weld return, the model suggested by equation (7-
19) can also provide a good prediction of the connection strength (see Figure 7.17) in the range
of Lw/w ratios from 0.40 to 0.70. Once again, the trend shown for the data herein shows a linear
variation which almost becomes constant. Even though the mean and COV are 0.93 and 1.7%
respectively, the inability of these connections to attain a higher mean value (e.g. 1.0) is due to
the inclusion of a Heat Affected Zone (HAZ) during the FE modelling. (This HAZ emulated a
lack of ductility in this region which triggered the premature material failure there affecting the
overall connection strength (especially for small weld lengths)). On the other hand the use of
equation (7-20), with correction factors equal to 1.0, provided a similar outcome as it gave no
significant improvement. The use of a correction factor of Rv=0.91 would enhance this
prediction equation, changing the average ratio to 1.0 and the COV to 1.5%. Nevertheless, it is
suggested that a further study of this connection type, and the importance that the HAZ may
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80
N
/ [ R
A
F
+ R
A
( ( F
+ F
) / 2 3
) ]
u F E
t
n t
u
v
g v
y
u
D/t=15
D/t=20
D/t=25
D/t=30
D/t=35
D/t=40
D/t=45
TO Failure
L /ww
CF
Shear
Lag
Present
Rt = 1.0 Rv = 1.0
Figure 7.16 Correlation of equation (7-20) for slotted CHS connections (no weld return)
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SLOTTED END CONNECTIONS TO HOLLOW SECTIONS, CH 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
have on the connection strength. be conducted before a stress correction factor is
recommended.
Figure 7.18 shows how equation (7-19) can provide an unsafe strength prediction for
slotted gusset plate connections to CHS having thick tubes and Lw/w ratios approaching 0.70.
The reason for this behaviour is due to the philosophy behind this model, which expects a
gradual increase in the connection strength as the weld length increases. Nevertheless, it does
not consider the negative effect that the bowing outwards of the gusset plate has on the overall
connection response. As seen during the study of the CF mechanism, gusset plate bowing will
increase the level of strains at the Ant region precipitating fracture of the material there (which in
this case defines the attainment of the maximum load by TO failure). For connections with Lw/
w
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SLOTTED END CONNECTIONS TO HOLLOW SECTIONS, CH 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80
Lw /w
D/t=15
D/t=20
D/t=25
D/t=30
D/t=35
D/t=40
D/t=45
TO FailureTension
Failure
Shear LagPresent
N
/ ( A
F
+ 0
. 6 A
F
)
u F E
n t
u
n v
y
Figure 7.18 Correlation of equation (7-19) for slotted gusset plate to CHS connections
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
1.20
0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80
D/t=15
D/t=20
D/t=25
D/t=30
D/t=35
D/t=40
D/t=45
TO Failure
TensionFailureShear LagPresent
Rt = 0.7
Rv = 1.65 N
/ [ R
A
F
+ R
A
( ( F + F
) / 2 3 ) ]
u F E
t
n t
u
v
g v
y
u
L /ww
Figure 7.19 Correlation of equation (7-20) for slotted gusset plate to CHS connections
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SLOTTED END CONNECTIONS TO HOLLOW SECTIONS, CH 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
When these factors were applied to equation (7-20), the mean and COV changed to 1.04
and 3.6% respectively and it also adjusted the overall correlation. Despite this, the predicted
strength for thick tubes with D/t=15 still remains unsafe. Nonetheless, connections fabricated
with this ratio consistently had weld failure (for small Lw/w ratios) throughout the parametric
analysis. Thus, the application of a different welding procedure such as full penetration groove
welds may be required for thick tubes.
In contrast to the trend displayed during the strength correlation from slotted CHS
connections (where it almost became constant), the correlation for slotted EHS connections has
exhibited contrasting trends. For EHS connections having a large eccentricity, the comparison
showed that equation (7-19) generally provides conservative strengths values. The actual-to-
predicted ratio will decrease as the Lw/w ratio increases. This trend is approximately linear (see
Figure 7.20) except for thick tubes. The trend for connections with a small eccentricity follows a
more complex variation (see Figure 7.21). In both cases, the model suggested by equation (7-
19) provides an adequate mean actual-to-predicted ratio and COV. For connections with Lw/w <
0.60 and a large eccentricity, this equation provided values of 1.06 and 5.8% respectively;
alternatively, a mean of 1.12 and a COV of 5.9% for connections with a small eccentricity.
Despite these favourable results, the predicted strength for connections having a wall
slenderness of D/t=15 will be unsafe.
As could be expected, equation (7-20) predicted better strengths for slotted EHS
connections. The EHS material properties had a ratio of Fu/Fy=1.258 which increased the
expected stress acting at Agv from 0.6 Fy to 0.65 Fy. For connections with a large eccentricity,
the use of stress correction factors (Rt, Rv) equal to 1.0 produced a mean actual-to-predicted
ratio of 0.99 and a COV of 5.9%. Even though this mean is practically 1.0, this equation will also
predict unsafe strengths for connections fabricated with thick tubes. In the same way, the use of
factors (Rt, Rv) equal to 1.0 (providing a mean and COV of 1.05 and 5.6% respectively) in
connections with a small eccentricity predicts unsafe strengths for thick tubes.
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SLOTTED END CONNECTIONS TO HOLLOW SECTIONS, CH 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
1.20
1.25
1.30
0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70
D /t=15avg
D /t=20avg
D /t=25avg
D /t=30avg
D /t=35avg
D /t=40avg
D /t=45avg
TOFailure
CFShear Lag
Present
N
/ ( A
F
+
0 . 6
A
F
)
u F E
n t
u
n v
y
L /ww
Figure 7.20 Correlation of equation (7-19) for slotted EHS connections (large eccentricity)
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
1.20
1.25
1.30
0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70
D /t=15avg
D /t=20avg
D /t=25avg
D /t=30avg
D /t=35avg
D /t=40avg
D /t=45avg
TO failureCF
Shear Lag
Present
N
/ ( A
F
+ 0
. 6 A
F
)
u F E
n t
u
n v
y
L /ww
Figure 7.21 Correlation of equation (7-19) for slotted EHS connections (small eccentricity)
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SLOTTED END CONNECTIONS TO HOLLOW SECTIONS, CH 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
In a similar manner to slotted gusset plate to CHS connections, the bowing outwards of
the gusset plate also modified the strength of their EHS counterparts, as it induced premature
fracture at Ant. In general, the trend shown by the comparison with equation (7-19) is
approximately quadratic (see Figure 7.22). Tubes with Davg/t = 15 are not considered herein as
weld failure governed the response of these connections when they had a ratio of Lw/w < 0.60.
For this data, the mean and COV values were 0.89 and 5.4% respectively with the actual-to-
predicted ratio ranging form 0.75 to 0.95.
The use of equation (7-20), with stress correction factors equal to 1.0 produced a mean
and COV of 0.85 and 5% respectively. This equation lowers the mean (from 0.89 to 0.85) and
also decreases the data scatter. The parametric analysis showed that weld failure will govern for
low Davg/t ratios and short welds, thereby requiring the need for a different welding procedure.
As a final note, even though the predicted connection strength can be modified by
application of these Rt and Rv correction factors, a better TO failure model capable of accurately
capturing the response of slotted gusset plate connections is preferable. Also, as noted before,
equal Rt and Rv factors serve the same purpose as one resistance factor applied to the whole
prediction equation.
0.700.75
0.80
0.85
0.90
0.95
1.00
1.051.10
1.15
1.20
1.25
1.30
0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70
Lw /w
D /t=15avg
D /t=20avg
D /t=25avg
D /t=30avg
D /t=35avg
D /t=40avg
D /t=45avg
TO failure
CFShear Lag
Present
N
/ ( A
F
+ 0
. 6 A
F
)
u F E
n t
u
n v
y
Figure 7.22 Correlation of equation (7-19) for slotted gusset plate to EHS connections
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7.4 CHS connections in compression
7.4.1 Equation suggested for slotted CHS to gusset plate connections (under
compression loading)
As explained during the parametric analysis, the strain concentration at the slot may
encourage the development of a local buckle there. When small Lw/w ratios are used, this may
potentially prevent the attainment of the member design load. However, this phenomenon can
be minimized to the point where the connection strength only depends on the tube D/t ratio and
the slot length (if ratios Lw/w > 0.92 are used). Equation (7-21) shows the coefficients calculated
based on the data for slotted CHS connections under compression loading. The use of 1.7 and
3.7 herein resulted in a mean and COV of 1.0 and 5.3% respectively.
(7-21)
Figure 7.23 shows the equation following the trend of the data within the validity range
defined here as 15 D/t 45 and 0.4 Lw/w 1.26.
Since the data used herein corresponds to FE models having a slot length equivalent to
the gusset plate thickness, the use of this equation is only valid for connections emulating this
condition. The use of a longer slot will negatively impact the connection efficiency, as explained
in Section 6.6.1. Considering that the D/t ratio had a significant influence on the efficiency of this
connection detail, this ratio was included in equation (7-22).
UCSA s lo tt ed– tube– compression– 11
1
Lw
w------
1.7
+
3.7---------------------------------------–=
≤ ≤ ≤ ≤
N
/ A
F
u F E
g
y
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
Slotted CHS (C)
U suggested for slotted CHS (C)
Shear lagpresent
Tube local buckling of theentire cross-section
L /w w0.92
Figure 7.23 Suggested efficiency factor and parametric analysis results from slotted CHS
under compression loading
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(7-22)
The use of the coefficients 1.58 and -0.13 herein resulted in a mean and COV of 0.99 and
3.0% respectively. The ranges of validity for this equation are the same as for equation (7-21).
Moreover, the maximum efficiency is limited to 100% of AgFy. Even though the use of a
simplified equation such as (7-21) may be suggested (especially for small Lw/w ratios (where
the data tends to gather), the use of equation (7-22) is optional as the inclusion of the D/t ratio
tends to positively improve the calculated efficiency. The results from these comparisons are
shown in Table 7.13.
a) Data corresponding to FE connections with .
7.4.2 Equation suggested for slotted gusset plate connections (under compression
loading)
Since the efficiency of these connections increased at a constant rate as the weld length(see Figure 7.24), it was feasible to simplify the format of the equation used here. The use of the
coefficients 0.24 and 0.5 in equation (7-23) produced a mean and COV of 0.99 and 6.9%
respectively.
(7-23)
In contrast to slotted CHS connections, here the D/t ratio did not have significant influence
on the attainment of a higher efficiency. On the contrary, the data was mainly dependent on the
gusset plate width (B) at the slot region. To consider this, it was decided to relate this parameter
(B) to the tube thickness (t). As a result, the data was ordered in a range of B/t ratios from 11 to
20. Then, further regressions were undertaken with equation (7-23) but with this equation
modified by the B/t ratio according to several formats. However, the final outcome always
resulted in very minor modifications to equation (7-23), which minimized the influence of this
Table 7.13 Evaluation of potential equations for slotted CHS connections (under compressionloading)
FE results a)FE results / equation
(7-21)
FE results / equation
(7-22)
Slotted CHS 541.00 0.99 Mean
5.3% 3.0% COV
UCS A sl ot te d– tube– compression– 11
1Lww------
1.7
+
3.7---------------------------------------–
1.58D
t----
0.13–
1.0≤=
0.4 Lw w ⁄ 1.26< <
UCS A s lo tt ed– gusset– compression– 0.24Lww------ 0.5+
1.0≤=
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parameter. The range of validity for equation (7-23) has been limited to 15 D/t 45, 0.4 Lw/
w and 11 B/t 20. The result from this comparison is shown in Table 7.14.
a) Data corresponding to FE connections with 0.4 Lw/w.
7.5 Evaluation of recommended equations against experimental data
The results from previous sections have shown that equation (7-2), initially developed for
CF in slotted CHS connections using the CSA format, may also be applied to slotted gusset
plate connections with good results. Even though the use of this equation for EHS connections
may also provide acceptable results within the defined validity limits, the use of the CF equation
(7-12) is recommended as it can offer a continuous trend in the predicted efficiency for Lw/w
ratios near the transition region. Figure 7.25 shows the CF connection efficiency suggested by
equation (7-2) in comparison to the data from previous experimental programs. Herein, the ratio
Lw/w = 0.70 is defined as the transition point from a CF to a TO failure. Nevertheless, it should
be remembered that this value corresponds to a lower bound defined by the CHS connections
and a ratio Lw/w = 0.60 was exhibited by EHS connections.
Table 7.14 Evaluation of potential equation for slotted gusset plate to CHS connections (undercompression loading)
FE results a) FE results / equation (7-23)
Slotted CHS 660.99 Mean
6.9% COV
≤ ≤ ≤
≤ ≤
N
/ A
F
u F E
g
y
L /ww0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.80.9
1.0
1.1
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5
Slotted gusset plate to CHS (C)
U suggested for slotted gusset plate to CHS (C)
Figure 7.24 Suggested efficiency factor and parametric analysis results from slotted CHSunder compression loading
≤
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In contrast, CF equations recommended for AISC design provision format offered less
encouraging result. It was found that equation (7-7) can be extended to slotted EHS
connections with a large eccentricity and slotted gusset plate connections. However, its use in
slotted EHS connections with a small eccentricity is clearly inappropriate. The connection
efficiency proposed by CF failure equation (7-7) is plotted in combination with the data from
previous experimental programs in Figure 7.26. Here a ratio of /Lw = 0.245 specifies the
transition from a CF to a TO failure. Nonetheless, different ratios have been established for EHS
connections.
L /w w
U
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6
British
Korol
Cheng
UofT
Zhao-1995
Zhao-1999
CF
CF
Shear
Lag
PresentTO FailureU - CSA
Figure 7.25 Suggested efficiency factor by equation (7-2) and experimental results
x'
x'/Lw
U
TO Failure
CF
Shear
Lag
Present
CF
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60
British
Korol
Cheng
UofT
U-AISC
Zhao-1995
Zhao-1999 0.245
Figure 7.26 Suggested efficiency factor by equation (7-7) and experimental results
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7.5.1 Experimental program by British Steel (1992)
Table 7.15 shows the experimental data by British Steel (1992) and the best prediction
given by the modified AISC(2005) design provision when is used. In addition, the predicted
connection strengths from equations (7-2) and (7-7) are presented. Even though these new
equations seem to have a somewhat poor correlation herein (with a mean of 0.92), there are
several factors directly related to the data affecting these results that must be considered. The
experimental data showed considerable scatter in the region where a high efficiency may be
expected (see Figures 7.25 and 7.26). Based on the data from other experimental programs,
and especially from the parametric analyses, one could expect high efficiency values (i.e.
values above 90% of AnFu) when a connection has a ratio of Lw/w > 0.9 or /Lw < 0.2.
However, the data here exhibited lower values that even reached an efficiency as low as
0.72AnF
u. The inability of these connections to attain higher values has been related to the
likely presence of welding defects, as these connections were fabricated with a weld return,
which may have triggered their premature fracture during testing. Despite these facts, the new
equations provided a better prediction for CHS connections than AISC (modified). The main
reason for the higher AISC mean is that the AISC specification does not allow the full efficiency
attainment for RHS and SHS connections (even with long weld lengths), unlike for CHS
connections. Hence, the AISC predicted efficiency stayed close to the low experimental data for
RHS/SHS connections in the data group.
Table 7.15 Actual and predicted connection strength for British Steel (1992) data
Specimen Lw/w /Lw
Test
Capacity
Nux [kN]
Nux/AnFu
(An=Ag)
Circumferential Tensile Fracture
AISC
(2005)
using
Nu [kN]
Nux/ NuEq(7-2)
Nu [kN]
Nux/
Nu
Eq(7-7)
Nu [kN]
Nux/
Nu
C-Sep-1 0.94 0.18 256 0.90 285 0.90 277 0.92 274 0.93
C-Sep-2 0.97 0.16 326 0.90 362 0.90 353 0.92 351 0.93
C-Sep-3 1.01 0.14 371 0.89 416 0.89 408 0.91 408 0.91
C-Sep-4 0.91 0.19 522 0.93 561 0.93 541 0.96 533 0.98
C-Sep-5 0.94 0.18 652 0.85 763 0.85 741 0.88 734 0.89C-Sep-6 0.94 0.18 795 0.84 952 0.84 924 0.86 916 0.87
S-Sep-2 0.94 0.14 274 0.94 251 1.09 283 0.97 286 0.96
S-Sep-3 0.94 0.12 505 0.96 462 1.09 508 0.99 518 0.97
S-Sep-4 0.94 0.16 478 0.85 472 1.01 544 0.88 546 0.88
S-Sep-5 0.96 0.14 833 0.94 759 1.10 864 0.96 868 0.96
S-Sep-6 0.96 0.13 949 0.90 911 1.04 1023 0.93 1033 0.92
R-Sep-3 0.94 0.08 475 0.89 492 0.96 519 0.91 533 0.89
x'
x'
x'
x'
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equations (7-2) and (7-7). Even though these equations provided a mean actual-to-predicted
ratio similar to AISC, there are several advantages related to their use that should be
considered. As seen throughout the parametric analyses, the connections showed a gradual
increase in their efficiency and a clear tendency to converge as their Lw/w ratios approached
1.0. Moreover, the influence of the D/t ratio during this process was also noted. In general, this
contrasts with the approach by AISC where a ratio of 1.3 Lw/D guarantees the attainment of the
connection full efficiency but a slightly lower ratio is heavily penalized. Since the first eight
specimens had a ratio of Lw/D > 1.3, AISC provided them with an efficiency factor of 1.0. But,
AISC clearly failed to suggest an acceptable efficiency for the last specimen (spec2). In
contrast, equations (7-2) and (7-7) predict a close efficiency for the first specimens (as all they
have a ratio Lw/w > 1.0) and a much better prediction for spec2, which finally resulted in a better
COV.
Table 7.17 Actual and predicted connection strength for Cheng et al. (1996) data
7.5.4 Experimental program by the Authors
Table 7.18 shows the data from the experimental program undertaken at the University of
Toronto, the predicted connection strength by AISC(2005) when is used and the predicted
connection strength based on equations (7-2) and (7-7). In order to facilitate the comparisons,
the data is gathered here based on the structural shape (i.e. CHS versus EHS). In contrast to
the AISC design provision, the use of equations (7-2) and (7-7) for CHS connections predicted
Specimen Lw/w /Lw
Test
Capacity
Nux [kN]
Nux/
AnFu
Nux/
AgFu
Circumferential Tensile Fracture
AISC
(2005)
using
Nu [kN]
Nux/
Nu
Eq(7-2)
Nu [kN]
Nux/
Nu
Eq(7-7)
Nu [kN]
Nux/
Nu
pwc1 1.14 0.16 830 1.06 0.98 781 1.06 775 1.07 759 1.09
pwc2 1.14 0.16 869 1.02 849 1.02 843 1.03 825 1.05
pwc3 1.14 0.16 849 1.00 849 1.00 843 1.01 825 1.03
pwc4 1.14 0.16 875 1.03 849 1.03 843 1.04 825 1.06
pwc5 1.00 0.18 645 1.03 624 1.03 612 1.05 598 1.08
pwc6 1.00 0.18 634 1.02 624 1.02 612 1.04 598 1.06
pwc7 1.00 0.18 631 1.01 624 1.01 612 1.03 598 1.06
spec1 1.06 0.17 2160 1.01 2141 1.01 2114 1.02 2064 1.05
spec2 0.85 0.22 2157 1.01 1674 1.29 2025 1.06 1986 1.09
Mean 1.05 1.04 1.06
COV 8.6% 1.9% 1.9%
x'
x'
x'
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strengths close to the experimental data, including connections with ratios below the validity
limit (such as specimens A1 and C1). Moreover, the use of either equation (suggested for
different design provision format) can provide comparable results. Despite equations (7-2) and
(7-7) being developed for CHS connections, their application to EHS connections provides
acceptable results, including specimens located beyond the validity limit (E1 and E3). However,
it must be remembered that the use of equations (7-2) and (7-7) may result in a overestimation
of the connection capacity (the first equation for connections with Lw/w ratios approaching 0.6,
and the second equation for connections with small eccentricity). Because of this, the use of
equations (7-12), (7-16) and (7-17) is still recommended for these connection types to address
the CF limit state. The evaluation of equations (7-20), (7-22) and (7-23) is also included in this
table Table 7.18.
Table 7.18 Actual and predicted connection strength for data by the Authors
A further comparison is shown in Table 7.19 for equations specifically developed for EHS
connections, which show an improved mean relative to the AISC design provision format.
Specimen Lw/w /Lw
Test
Capacity
Nux [kN]
Nux/
AnFu
Nux/
AgFu
Circumferential Tensile Fracture
AISC (2005)
using
Nu [kN]
Nux/
Nu
Eq(7-2)
Nu [kN]
Nux/Nu
Eq(7-7)
Nu [kN]
Nux/
Nu
A1 0.66 0.26 1032 0.87 0.77 880 1.17 988 1.04 1042 0.99
A2 0.81 0.21 1154 0.97 0.86 939 1.23 1109 1.04 1112 1.04
B1 0.71 0.24 1087 0.91 0.81 1013 1167 1203
B2 0.87 0.20 1211 1.02 0.91 1073 1.13 1274 0.95 1265 0.96
C1 0.68 0.25 1107 0.83 999 1.11 1133 0.98 1185 0.93
C2 0.82 0.21 1196 0.90 1055 1.13 1247 0.96 1249 0.96
Mean 1.15 0.99 0.98
COV 4.1% 4.5% 4.1%
E1 0.62 0.32 1109 0.81 0.69 914 1.21 1067 1.04 1047 1.07
E2 0.78 0.26 1236 0.90 0.76 1019 1.21 1256 0.98 1298 1.02
E3 0.62 0.32 1336 0.83 1102 1.21 1275 1.05 1257 1.06
E4 0.74 0.27 1400 0.86 1188 1.18 1447 0.97 1404 1.00
E5 0.79 0.13 1282 0.94 0.79 1187 1.08 1253 1.02 1393 0.96
Mean 1.18 1.01 1.02
COV 4.9% 3.5% 4.6%
Nux/ AgFy Eq(7-20) Nux/Nu Eq(7-22) Nux/Nu Eq(7-23) Nux/NuB1 0.71 0.24 1087 0.81 1221 0.89
A3C 0.87 0.20 -1145 0.93 -1174 0.98
C3C 0.84 0.20 -869 0.71 -935 0.93
x'x'
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Table 7.19 Actual and predicted connection strength by the Author, continued
a) Equation (7-12) for slotted EHS connections.b) Equation (7-14) for slotted gusset plate to EHS connections.c) Equation (7-16) for EHS connections with small eccentricity.d) Equation (7-17) for EHS connections with large eccentricity.
7.6 Derivation of reduction (resistance) factors for the recommended equations
In order to provide an adequate level of safety to the equations developed in previous
sections, a simple reliability analysis was undertaken to derive reduction factors ( ) suitable for
these equations which will provide an adequate level of safety. The calculation of the reduction
factors is based on the classic equation derived by Ravindra and Galambos(1978):
(7-24)
Herein , known as the safety index or “reliability index”, represents the target probability
of failure during a structure’s service life. In the study by Ravindra and Galambos, a target value
of 4.5 was generally suggested for connections. However, several values were studied herein
as some design bodies now advise smaller values. A value of = 4.0 is now advocated by
AISC specification committee for connections. represents the mean actual-to-predicted ratio
of data against the prediction by equation(s) and represents the coefficient of variation
about that mean.
7.6.1 Reduction factors for CHS connections in tension - CF failure
7.6.1.1 Reduction factors for suggested equations for slotted CHS connections (CSA
design provision format)
Table 7.20 shows the suggested resistance factors for slotted CHS connections and
several safety indices (3.5, 4.0 and 4.5). In contrast to what may be expected, the use of these
Specimen Lw/w /Lw
Test
Capacity
Nux [kN]
Nux/
AnFu
Nux/
AgFu
Circumferential Tensile Fracture
AISC (2005)
using Nu
[kN]
Nux/
Nu
Eq(7-12)a
or (7-14)b
Nu [kN]
Nux/Nu
Eq(7-16)c
or (7-17)d
Nu [kN]
Nux/Nu
E1 0.62 0.32 1109 0.81 0.69 914 1.21 1084 1.02 1052b 1.05
E2 0.78 0.26 1236 0.90 0.76 1019 1.21 1196 1.03 1172b 1.05
E3 0.62 0.32 1336 0.83 1102 1.21 1373 0.97 1269b 1.05
E4 0.74 0.27 1400 0.86 1188 1.18 1451 0.94 1367b 1.02
E5 0.79 0.13 1282 0.94 0.79 1187 1.08 1190 1.08 1220a 1.05
Mean 1.18 1.01 1.05
COV 4.9% 4.7% 1.3%
x'x'
φ
φ 0.55 β VR⋅ ⋅–( )exp ϕm⋅=
β
β
ϕm
VR
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different safety indices resulted in extremely similar resistance factors. As a result, the use of a
reduction factor equal to 0.90 for both equations is recommended, as this value surpass all
target safety levels considered.
a) Data corresponding to FE connections with .
7.6.1.2 Reduction factors for suggested equations for slotted gusset plate to CHS
connections based on ultimate strength (CSA design provision format)
The use of the suggested equations ((7-2), (7-3) and (7-4)) for the data from slotted
gusset plate connections produced means equal to 1.0 and very small COVs. As a result, the
calculated reduction factors also are also very close (see Table 7.21). Hence, a reduction factor
of 0.90 is again suggested in an attempt to achieve uniformity with resistance factors previously
suggested for slotted CHS connections.
a) Data corresponding to FE connections with .
Table 7.20 Calculated resistance factors for slotted CHS connections
FE results a)FE results /
equation (7-2)
FE results /
equation (7-3)
Slotted CHS (no weld return) and
Slotted CHS (weld return)146
1.00 1.00 Mean
4.0% 3.6% COV
3.5 0.923 0.932
4.0 0.913 0.923
4.5 0.903 0.914
Table 7.21 Calculated resistance factors for slotted gusset plate to CHS connections
FE results a)FE results /
equation
(7-2)
FE results /
equation
(7-3)
FE results /
equation
(7-4)
Slotted gusset plate to CHS 631.00 1.00 1.00 Mean
2.9% 3.1% 2.6% COV
3.5 0.942 0.940 0.954
4.0 0.934 0.932 0.949
4.5 0.927 0.924 0.942
β φ φ
Lw w ⁄ 0.7≥
β φ φ φ
Lw w ⁄ 0.7≥
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7.6.1.3 Reduction factors for suggested equations for slotted gusset plate to CHS
connections based on deformation limit (CSA design provision format)
The larger COV associated with equation (7-5) ranged produced lower factors ranging
from 0.84 to 0.87 (see Table 7.22). As a result, the use of a reduction factor of 0.80 for this
equation is conservative. On the other hand, the inclusion of the D/t ratio, enhancing the
prediction accuracy of equation (7-5), also improved the reduction factors for equation (7-6), a
reduction factor of 0.90 is conservative (see Table 7.22). Since equation (7-6) can provide a
better prediction and a higher resistance factor, its use may be considered over equation (7-5).
a) Data corresponding to the connection strength at a deformation limit of 0.03D, for slotted gusset plate connections
to CHS.
7.6.1.4 Reduction factors for suggested equations for slotted CHS connections (AISC
design provision format)
The equations suggested for the AISC design provision format have shown their capacity
to predict the connection strength of CHS connections accurately. As a result, the mean and
COV generated reduction factors close to those given by the equations suggested for the CSA
format (always above 0.90). Hence, a resistance factor equal to 0.90 is also suggested for these
equations (see Table 7.23).
a) Data corresponding to FE connections with .
Table 7.22 Calculated resistance factors for slotted gusset plate to CHS connections using anultimate deformation limit state of 0.03D
FE results a)FE results /
equation (7-5)
FE results /
equation (7-6)
Slotted gusset plate toCHS
99 1.00 1.01 Mean7.1% 3.5% COV
3.5 0.869 0.944
4.0 0.853 0.935
4.5 0.836 0.926
Table 7.23 Calculated resistance factors for slotted CHS connections (AISC)
FE results a)FE results /
equation (7-7)
FE results /
equation (7-8)
Slotted CHS (no weld return) and
Slotted CHS (weld return)146
1.00 1.00 Mean
3.9% 3.8% COV
3.5 0.926 0.926
4.0 0.917 0.916
4.5 0.907 0.907
φ
β φ φ
β φ φ
x' Lw ⁄ 0.245≤
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7.6.1.5 Reduction factors for suggested equations for slotted gusset plate to CHS
connections based on ultimate strength (AISC design provision format)
The good results provided by the use of equations (7-7) and (7-9) for slotted gusset plate
connections (see Table 7.24) also suggest the use of a resistance factor at least of 0.9.
a) Data corresponding to FE connections with .
7.6.1.6 Reduction factors for suggested equations for slotted gusset plate to CHS
connections based on deformation limit (AISC design provision format)
In a similar manner to the CSA equations, equation (7-11) might be preferred over
equation (7-10) as it is more accurate (lower COV) and hence has a higher resistance factor,
with = 0.9 for equation (7-11) still being conservative recommendation.
a) Data corresponding to the connection strength at a deformation limit of 0.03D, for slotted gusset plate connections
to CHS.
Table 7.24 Calculated resistance factors for slotted gusset plate to CHS connections (AISC)
FE results a)FE results /
equation (7-7)
FE results /
equation (7-9)
Slotted gusset plate to CHS 631.00 1.01 Mean
2.4% 2.4% COV
3.5 0.953 0.960
4.0 0.947 0.954
4.5 0.941 0.947
Table 7.25 Calculated resistance factors for slotted gusset plate to CHS connections using an
ultimate deformation limit state of 0.03D (AISC)
FE results a)FE results /
equation (7-10)
FE results /
equation (7-11)
Slotted gusset plate to
CHS99
1.00 1.00 Mean
7.1% 3.7% COV
3.5 0.869 0.929
4.0 0.852 0.920
4.5 0.836 0.910
β φ φ
x' Lw ⁄ 0.245≤
φ
β φ φ
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SLOTTED END CONNECTIONS TO HOLLOW SECTIONS, CH 7: ANALYSIS OF FE AND EXPERIMENTAL RESULTS
7.6.2 Reduction factors for EHS connections in tension - CF failure
7.6.2.1 Reduction factors for suggested equations for slotted EHS connections (CSA
design provision format)
The calculated reduction factors are shown in Table 7.26, and again a resistance factor of
= 0.9 would be conservative for equations (7-12) and (7-13)
a) Data corresponding to FE connections with .
7.6.2.2 Reduction fac