CIDECT Final Report 8G-10_06(3of4)

8/13/2019 CIDECT Final Report 8G-10_06(3of4)

1/64

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--------------------------------–=


2/64

7-2


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


3/64

7-3


(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-----------------------------------------–=


4/64


5/64

7-5


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


6/64

7-6


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


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'


7/64

7-7


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)


8/64

7-8


(7-8)


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)


equation (7-7)

FE results /

equation (7-8)



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 ⁄ ≤


9/64

7-9


a)

Data corresponding to FE connections with .


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)


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'


10/64

7-10


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'


11/64

7-11



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)


(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---------------------------------------–=


12/64

7-12


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)


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


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≥


13/64

7-13


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 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


14/64

7-14




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


equation (7-2)

FE results /

equation (7-12)

FE results /

equation (7-14)


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

+

--------------------------------------------------=


15/64

7-15


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'


16/64

7-16


(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)


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)


17/64

7-17


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.


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' ≤


18/64

7-18



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)


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' ≤


19/64

7-19


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)


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


20/64

7-20


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 ⁄


21/64

7-21


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


22/64

7-22


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)


23/64

7-23


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)


24/64

7-24


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


25/64

7-25


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


26/64

7-26


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.


27/64

7-27


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)


28/64

7-28


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


29/64

7-29


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


30/64

7-30


(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.


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)


(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≤=


31/64

7-31


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

≤


32/64

7-32


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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7-33


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'


34/64


35/64

7-35


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


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'


36/64

7-36


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


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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7-37


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


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


38/64

7-38


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.


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.


Table 7.20 Calculated resistance factors for slotted CHS connections


equation (7-2)

FE results /

equation (7-3)



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


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≥


39/64

7-39



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).


to CHS.

7.6.1.4 Reduction factors for suggested equations for slotted CHS connections (AISC


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).


Table 7.22 Calculated resistance factors for slotted gusset plate to CHS connections using anultimate deformation limit state of 0.03D


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)


equation (7-7)

FE results /

equation (7-8)



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≤


40/64

7-40



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.



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.


to CHS.

Table 7.24 Calculated resistance factors for slotted gusset plate to CHS connections (AISC)


equation (7-7)

FE results /

equation (7-9)


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)


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


41/64

7-41


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


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)


7.6.2.2 Reduction fac

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