Journal of Constructional Steel Research 183 (2021) 106718

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Journal of Constructional Steel Research

Cold-formed steel portal frame moment-resisting joints: Behaviour,

capacity and design
Xin Chen a,⁎, H.B. Blum b, Krishanu Roy a, Pouya Pouladi a, Asraf Uzzaman c, James B.P. Lim a,⁎
a
Department of Civil and Environmental Engineering, The University of Auckland, New Zealand
b
Department of Civil and Environmental Engineering, University of Wisconsin-Madison, USA
c
School of Computing, Engineering and Physical Sciences, University of the West of Scotland, UK

a r t i c l e i n f o a b s t r a c t

Article history: The work presented in this paper investigates the strength of moment-resisiting apex brackets of cold-formed
Received 7 December 2020 steel portal frames. The results of two previously reported full-scale portal frame tests, where the frames failed
Received in revised form 13 April 2021 through buckling of the apex bracket, were used to validate a finite element model that included material non-
Accepted 16 April 2021
linearity, geometric imperfections, and bolt bearing behaviour. From portal frame test results as well as the finite
Available online 28 April 2021
element results, the importance of appropriate detailing for the apex brackets for strength is demonstrated. A
Keywords:
simplified finite element model of the apex bracket is then presented, again validated against previously reported
Cold-formed steel experimental tests. The effects of geometrical parameters of the brackets, bolt-group configuration, and yield
Portal frames stress of brackets were investigated through a parametric study comprising 648 models. From the results of
Finite element analysis this parametric study, design equations are proposed that can be used by practicing engineers to predict the
Apex brackets strength of apex brackets used for cold-formed steel portal frames.
Design equations © 2021 Elsevier Ltd. All rights reserved.

1. Introduction Zhang [4] also conducted full-scale tests on a portal frame with back-
to-back channels, however, the frame was intentionally designed with
Portal frames composed entirely of cold-formed steel (CFS) (see slender columns to produce local and distortional buckling in the col-
Fig. 1), are a popular form of construction in New Zealand and umn members in order to analyse the interaction between local and dis-
Australia, and are commonly used for shelters, warehouses, garages, tortional buckling modes [8]. Additionally, the joints were designed so
and sheds [1–6]. In such portal frames, CFS lipped channel-sections that they would not fail before the columns and are therefore not repre-
are used for column and rafter members, with the eaves and apex joints sentative of typical connection designs.
formed through brackets, bolted to the webs of the channel-sections. In Blum [9] focused on back-to-back lipped channel members for the
New Zealand, spans typically range from 10 to 60 m. main framing members, which included columns, rafters, and knee-
Previous work by Lim and Nethercot [2] has been concerned with braces. The eaves and knee connections were formed by back-to-back
the full-scale testing of two portal frames. Lateral-restraints were pro- lipped brackets and were connected to the main framing members
vided at discrete positions along the back-to-back channel frame col- with bolted connections through the webs of all connected members.
umns and rafters (see Fig. 2), with failure of the frame as a result of Most of the tested frames had unbraced columns, and frame failure re-
the apex brackets buckling, as opposed to distortional or local buckling sulted from lateral-torsional buckling of a column member initiated by
of the channel members. From these tests, it was concluded that the out-of-plane movement of the knee-brace-to-column connection
semi-rigidity of the portal frame joints means that the bending moment bracket. One test was conducted with braced columns which consisted
at the apex is higher than that calculated assuming rigid joints, and that of girts on the columns between the frames. Failure was initiated by
joint flexibility should therefore be taken into account by practicing en- buckling of the apex bracket which caused large vertical displacements
gineers. Fig. 3 shows a photograph of a buckled apex bracket. It should at the apex, and subsequent frame failure occurred at both the apex con-
be noted that this photograph is taken from tests reported by Öztürk nection and the rafter at the knee-brace connection locations.
and Pul [7], but is similar to that observed by Lim and Nethercot [2]. Rinchen [6] focused on frames composed of single lipped channels
for the columns and rafters, with lipped brackets for the eaves and
apex connections. The connection brackets were connected to the
⁎ Corresponding authors.
main framing members with bolts through the webs, and screws
E-mail addresses: xche889@aucklanduni.ac.nz (X. Chen), james.lim@auckland.ac.nz connecting the bracket lips to the channel flanges. In subsequent tests,
(J.B.P. Lim). the screws were replaced by bolts to improve the strength of the

https://doi.org/10.1016/j.jcsr.2021.106718
0143-974X/© 2021 Elsevier Ltd. All rights reserved.
X. Chen, H.B. Blum, K. Roy et al. Journal of Constructional Steel Research 183 (2021) 106718

Notation

aa Width of triangular area of apex bracket

ab Length of bolt-group
ba Edge width of apex bracket
bb Breadth of bolt-group
CFS Cold-formed steel
ds Depth of stiffener
E Young's modulus of elasticity
Eq. (x) Equation (x)
eo Member imperfections
FE Finite element
FEA Finite element analysis
FEXP
u Ultimate experimental load
FFEA
u Ultimate load from FEA
FFEA
u _EB Ultimate load from FEA with elastic brackets
fy Yield stress
fu Ultimate stress
fod Elastic distortional buckling stress
fol Elastic local buckling stress
h Height to eaves of portal frame
kb_x Spring stiffness in x direction
kb_y Spring stiffness in y direction
kb_z Spring stiffness in z direction
L Member length
MP Plastic moment-capacity of bracket
MEXPu Ultimate moment-capacity of bracket from experiment
MEq.u
(x)
Ultimate moment-capacity of bracket from Equation (x)
FEA
Mu Predicted moment capacity of bracket from FEA
Ms,u FEA Predicted moment capacity of bracket from simplified
FE model Fig. 2. Full-scale portal frame test after Lim and Nethercot [2].
sol Imperfection multipliers for local buckling
sod Imperfection multipliers for distortional buckling
t Thickness Local buckling of the column occurred near the eaves connection. One
θa Pitch of portal frame test was performed with girts on the columns between the frames to
α Coefficient brace the columns, which resulted in higher frame strength and reduced
β Coefficient column twist, yet still ultimately failed by local buckling of the channels.
γ Coefficient Full-scale tests showed the importance of the connections on the
performance of the frame. Additionally, parametric studies conducted
using validated FE models [5] showed the significant influence of con-
connection. As the load increased, the columns twisted and experienced nection design and stiffness on the overall frame performance. Different
a gradual lateral-torsional buckling behaviour, with final failure due to stiffness of the column base, knee-brace, and apex connections affect
local buckling of the columns and rafters near the flange-web junction. the ultimate capacity of the frames and may change the failure mode.

Fig. 1. Cold-formed steel portal frame [7].

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X. Chen, H.B. Blum, K. Roy et al. Journal of Constructional Steel Research 183 (2021) 106718

Fig. 3. Buckling failure of apex brackets after Öztürk and Pul [7].

Fig. 4. Photograph of the laboratory test set-up of apex joint after Lim [3].

For frames without knee-braces, the eaves connection is also important Blum [11]. However, no design recommendations were proposed in
[6] to the frame performance. As a result, many researchers have inves- the form of equations.
tigated the performance of the connection through isolated component Other researchers have previously conducted component tests on
tests. joint connections. Öztürk and Pul [7] conducted two experimental
Following the design of the full-scale frame tests, component tests of tests. The apex brackets used were formed through welding steel plates
two apex brackets were also conducted by Lim [3] (see Fig. 4). The re- (4 mm plate for the bracket and 6 mm for the lips). The yield strength of
sults of the apex bracket tests were then used to validate a non-linear the plates was approximately 275 N/mm2. The buckling failure of the
elasto-plastic FE model, from which an equation to predict the strength apex brackets is shown in Fig. 3. Additionally, Stratan [12] conducted
of the apex bracket was proposed: apex connection tests, but used the Lindab system and included built-
up gussets in the apex connection. However, design recommendations
2 were not provided from either study. Recently, numerical studies on
f y ba t 2
MP ¼ ½2:88−2:88ðba =aa Þ þ 1:26ðba =aa Þ  ð1Þ the eave connections in single-channel portal frames containing both
4
screws and bolts were conducted by Pouladi et al. [13]. It is shown
that the screws failed in shear before the bolts had fully slipped. Only
Where aa and ba are defined in Fig. 5. However, this equation is lim- upper and lower bounds to the experimental failure load were
ited, only valid for 0.4 ≤ ba/aa ≤ 1.0, an apex bracket thickness of 3 mm, proposed.
an apex bracket yield stress of 280 N/mm2 and a frame pitch of 10°.
More recently, twelve component tests on apex connections were
also conducted at the University of Sydney by Peng et al. [10]. The
brackets tested were 2.4 mm thick and had a lip stiffener on the bottom
of the web. The apex tests considered three different thicknesses of
lipped channels and two channel member depths. Two bracket sizes
were considered: one design each for the two channels depths. The
tests were conducted to quantify the moment-rotation behaviour of
the connection and to determine the strength and initial stiffness of
the connection. All connections, regardless of the channel depth or
thickness, failed by buckling of the centre of the apex bracket web. No
bolt-slip or bolt-hole elongation was observed in any of the tests. The
experimental test results were used to validate a non-linear elasto-
plastic FE model, from which parametric studies were conducted by Fig. 5. Diagram showing parameters of apex bracket.

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X. Chen, H.B. Blum, K. Roy et al. Journal of Constructional Steel Research 183 (2021) 106718

Fig. 6. Detail of general arrangement of the test frame [2] (dimensions in millimeters).

Fig. 5 shows typical apex bracket arrangements used in practice in

New Zealand. The apex brackets are fastened to the webs of the
channel-sections using bolted connections. As can be seen, the apex
brackets are cut from flat plates, with top lip stiffeners folded. As
meationed previously, the Eq. (1) is only valid within a certain range
of design parameters. There is therefore an obvious need for a more uni-
versally applicable design methodology given the wide range of apex
brackets available in practice. This paper aims to address the issue and
propose design equations to predict the strength of apex brackets
used for portal frames composed of back-to-back channel.

2. Summary of experimental tests from the literature

2.1. Details of portal frame tests

Lim and Nethercot [2] tested two portal frames (see Fig. 2), to be re-
ferred to as Frames A and B. Fig. 6 shows details of the general arrange-
ment for the frame test and the provision of lateral restraint to prevent
premature failure of the back-to-back channel sections due to lateral
buckling, as described in Lim and Nethercot [14,15]. As can be seen,
the frame was loaded through a loading rig consisting of eight jacks
and the pinned supports were at the column bases. Fig. 7 shows the de-
tails of point loads applied to the portal frames. Fig. 8 shows the details
of the pinned column base plate which was bolted between the back-to-
back channel-sections at the ends of the column.
Both frames had a span of 12 m, column height of 3 m and pitch of
10°. The difference between the two frames was the size of the joints,
which was based around two different bolt-group sizes which were
315 mm × 230 mm and 615 mm × 230 mm for Frames A and B, respec- Fig. 8. Details of column base plate [3] (dimensions in millimeters).
tively (see Fig. 9). As can be seen from Fig. 6, double apex brackets were
bolted to the back-to-back channel sections. The bolts had a nominal di-
ameter of 16 mm and were fully threaded. The average dimensions of

Fig. 7. Detail of point loads applied to standard frame [2] (dimensions in millimeters).

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X. Chen, H.B. Blum, K. Roy et al. Journal of Constructional Steel Research 183 (2021) 106718

Fig. 9. Details of bolt-groups and corresponding sizes of eaves and apex brackets after Lim [3] (dimensions in millimeters).

the channel-sections used in the tests are shown in Fig. 10. Table 1
shows the average measured material properties of the channel sections
and brackets. All the portal frame tests failed as a result of the apex
brackets buckling [3].

2.2. Details of apex joint tests

Two apex joints were tested by Lim [3] under pure bending (see Fig. 4).
As a summary, these tests are briefly described in this section. Details of
pin-ended boundary conditions, lateral restraints on the web of the
channel-sections and loads applied to the apex joint tested are shown in
Fig. 11. To ensure the apex bracket failed prior to the channel sections,
only a single apex bracket was bolted to the back-to-back channel sections.
The average dimensions of the channel-sections used in the apex joint
tests are the same as those used in the portal frame tests (see Fig. 10). The
dimensions and the material properties of the apex brackets tested are
shown in Table 2. The difference between the tests was the size of the
apex bracket. The pitch of apex connections tested was same as that in
portal frame, and the value is 10°. All the joint tests failed as a result of pre-
mature buckling of the stiffened-edge of the apex bracket [3].

Table 1
Material properties of channel-sections and brackets used for frame tests [2].

Components fy fu t1
(N/mm2) (N/mm2) (mm)

Channel-sections 358 425 2.95

Brackets 200 305 2.98
Fig. 10. Average dimensions of back-to-back channel sections [3] (dimensions in 1
core thickness excluding galvanising.
millimeters).
X. Chen, H.B. Blum, K. Roy et al. Journal of Constructional Steel Research 183 (2021) 106718

Fig. 11. Details of lateral restraint and loading applied to apex joint tests after Lim [3] (dimensions in millimeters).

Table 2
Comparison of experimental and FE results of apex joint tests after Lim [3].

Test aa ba ds t1 fy MEXP
u MFEA
u Ms,u FEA MEq.
u
(3)
MFEA EXP
u /Mu Ms,u FEA/MEXP
u MEq.
u
(3)
/MEXP
u
(mm) (mm) (mm) (mm) (N/mm2) (kNm) (kNm) (kNm) (kNm)

7–12 406 340 80 2.99 272 32.5 30.75 30.63 30.23 0.95 0.94 0.93
7–22 700 340 80 2.95 262 35.0 33.57 33.30 33.07 0.96 0.95 0.94
1
core thickness excluding galvanising.
2
the test name in [3].

3. Finite element model of portal frames channel sections and the bracket sections. Inter-penetration of these
shell elements was prevented. The general contact algorithm uses a
3.1. Geometrical modelling ‘hard contact’ formulation, and the penalty method is used to approxi-
mate the hard pressure-over-closure behaviour. Friction was also
In the general purpose, finite element (FE) program ABAQUS [16] modelled, and the coefficient of friction was taken as 0.2 for all contact
was used to model CFS portal frames, which were composed with the interfaces [17–19].
columns, rafters, eaves brackets, apex brackets, and base plates. The In the portal frame tests, columns and rafters were connected to re-
overall dimensions of the frame layout plan and the average dimensions spective brackets using M16 bolts. For modelling the restraint provided
of the channel sections and brackets are given in Section 2.1. The full by bolted connections, linear translational “SPRING2” elements [16]
portal frame FE model is shown in Fig. 12. have been employed, placed at the position of the physical link between
the webs of channel sections and the bracket sections, with their
3.2. Contact and connection modelling assigned stiffness kb_x = kb_y = 6.21 kN/mm and kb_z = 6210 kN/mm
[20]. Typical shell element models of apex connections are shown in
Surface-to-surface contact using the finite-sliding tracking method Fig. 13.
was used to define the interaction relationship between the webs of

Fig. 12. FE model frame layout.

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X. Chen, H.B. Blum, K. Roy et al. Journal of Constructional Steel Research 183 (2021) 106718

Fig. 13. SPRING2 element on connections.

ultimate stress, as measured from the coupon tests reported in Lim

Table 3 and Nethercot [2], are summarized in Table 1.
Amplitudes of imperfections according to AS/NZS 4600 [24].

Cross-section Imperfection Member Frame 3.4. Initial geometric imperfections and residual stresses
Imperfection Imperfection
Local Distortional
buckling buckling As no geometric imperfections were measured by Lim and Nethercot
qffiffiffiffiffi qffiffiffiffiffiffi [20], the geometric imperfections recommended by AS/NZS 4600 [24]
fy f eo = L/1000 (mm) h/500 (mm)
sol ¼ 0:3t (mm) sod ¼ 0:3t f y (mm)
f ol od
are applied, namely, cross-sectional imperfection, member imperfec-
tion, and frame imperfection. The amplitude of imperfections is summa-
rized in Table 3. To model the cross-sectional imperfections, local and
3.3. Element type and material properties distortional buckling imperfections should be considered in advanced
structural engineering analysis. Local or distortional imperfections
All the frame members were modelled with the standard 4-noded were included in the columns and rafters, depending on which mode
doubly curved thin shell elements with linear interpolation, reduced in- had the lower critical buckling stress [25]. The value of member imper-
tegration, hourglass control, and finite membrane strains (S4R ele- fection shall be taken as 1/1000 of the member length, where the mem-
ments) [16,21,22]. The default setting of Drilling hourglass scaling ber length is the distance between connection points to adjoining
factor, Viscosity, Hourglass control, Element detetion and Max Degrada- members or supports. The frame imperfection refers to the top of the
tion were used. This particular element was previously shown by other column at the intersection between the centrelines of the column and
researchers when modelling CFS portal frames [5]. The columns and raf- rafter [26]. Considering the initial imperfections, the symmetric model
ters were modelled with a mesh size of 15 mm × 15 mm. For the base used in [20] will not be applicable, so the full-scale portal frame was
and eaves brackets, the mesh size of 10 mm × 10 mm was used. modelled in this paper.
As the tests failed by the buckling of apex brackets, a smaller mesh Residual stresses were not modelled as the through-thickness resid-
size 5 mm × 5 mm was used in the middle of apex brackets (see ual stress is accounted for in the FE models through the measured ma-
Fig. 12) and in the other part of apex brackets, the mesh size was terial properties determined from the coupon tests, and membrane
10 mm × 10 mm. The number of elements of Frame-A is 611470 and residual stresses are small relative to the yield stress of CFS sections [27].
that of Frame-B is 904588.
The material behaviour was modelled using a bilinear stress-strain 3.5. Boundary conditions, lateral restraint and loading
diagram with an initial elastic modulus (E) of 205 GPA, followed by a
linear hardening range with a slope of E/100. This model was previously As described in Section 2.1, the pinned column base boundary condi-
proposed by Haidarali and Nethercot [23]. The yield stress and the tion was applied to the bottom of the base plate by constraining two in-

Fig. 14. Boundary conditions and lateral restraint on the portal frame FE model (refer to Fig. 7 for dimensions).

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X. Chen, H.B. Blum, K. Roy et al. Journal of Constructional Steel Research 183 (2021) 106718

Table 4
Comparison of experimental [2] and FE results of tests conducted on the portal frame.

Frame FEXP
u FFEA
u FFEA
u _EB FFEA EXP
u /Fu FFEA FEA
u _EB/Fu MFEA
u MEq.
u
(3)
(kNm) MEq.
u
(3)
/MFEA
u
(kN) (kN) (kN) (kNm)

A 77.22 76.42 111.13 0.99 1.45 22.06 22.93 1.04

B 109.38 102.83 131.80 0.94 1.28 25.09 26.04 1.04

Note: moment capacity in Table 4 used for a single apex bracket.

corresponding nodes of the webs, and the values of loads are

shown in Fig. 7.

3.6. Analysis

A static geometric and material nonlinear analysis was performed

for the test frame model. A single load step was defined for frames sub-
jected to only vertical loads. A “static general” option featuring a full
Newton-Raphson solution technique was used for the static nonlinear
analysis of the frames.

3.7. Comparison of results from the frame tests and FE models

Finite element analysis (FEA) results were compared with the exper-
imental results to validate the suitability of the modelling method. The
ultimate strengths of frames predicted by the FEA are presented in
Fig. 15. Load vs. apex deflection curve comparison of experimental and FEA results. Table 4 along with the results from the frame tests. As observed, the ul-
timate loads predicted by FEA (FFEAu ) are close to those obtained from
the tests (FEXP
u ), and the differences are within 6% for both Frame A
plane translations (see Fig. 14). As can be seen, lateral restraints were and B. There are several contributing factors to the differences, which
applied on the same locations as shown in Fig. 6. include the likelihood of the introduction of imperfections in the
In the experiments as described in Section 2.1, vertical loads were frame tests during installation, varying degree of bolt pretension, and
applied to the frame through the plates connecting the web of the eccentric location of loading points. These factors are difficult to account
rafters. Therefore, in the model, loads were also applied to the for, in the numerical model.

Fig. 16. Stress contours of apex bracket.

Fig. 17. Stress contours of the right eave joint (brackets idealised as elastic).

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X. Chen, H.B. Blum, K. Roy et al. Journal of Constructional Steel Research 183 (2021) 106718

Fig. 18. FE model of apex connection showing loading and boundary conditions.

Fig. 19. FE model of simplified moment-resisting apex connection including loading and boundary conditions.

Fig. 15 shows the load-apex deflection curve comparison of the ex- the gradients of the experimental and FEA results are similar. Fig. 16
perimental and FEA results. Overall, there is a close agreement between shows the von Mises stress contours of the apex bracket. As can be
the results of experiments and the FE model. As can be seen, when the seen, the red area is the yielded area, so the portal frame FE model failed
FEA results are offset to account for the initial bolt-hole misalignment, at the apex bracket, which matches well with the experimental failure

Table 5
Selected variable for parametric studies.

d s1 Bolts Group1 ba/aa t fy

(mm) (mm) (N/mm2)

ba1=150 mm 50 ab
¼ bbba ¼ 0:8 [0.4– 1.2] [2, 2.5, 3] [250– 550]
aa
ba=200 mm 55
ba=250 mm 60
ba=300 mm 70

ba=350 mm 70
ba=400 mm 70

1
refer to Fig. 5

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X. Chen, H.B. Blum, K. Roy et al. Journal of Constructional Steel Research 183 (2021) 106718

mode. The FE model was validated against the two full-scale portal were applied to the nodes in the middle of the bracket. As the lateral dis-
frame tests. placements were prevented at those locations where bracing was put in
place, the stiffness contribution of the rafter channel to the apex bracket
3.8. Effect of brackets

Both experimental and FEA results show that the portal frames failed
due to the buckling of apex brackets. To study the influence of connec-
tion brackets on the capacity of portal frames, apex brackets and eaves
brackets were both set as linear elastic material in the FE model, and
the analyses were rerun.
Fig. 17 shows the von Mises stress contours of the column, when
brackets idenlised as elastic. The yielded area is shown as the red colour
and the failure mode is now buckling of the column near to the eaves
joint. Failure of the frame in this case occurs due to buckling of the
cold-formed steel channel sections, influenced by a bi-moment as de-
scribed by Lim et al. [28].
Fig. 15 also shows the load with the apex deflection curve when the
brackets were prevented from failure. The value of ultimate load from
FEA with elastic brackets (FFEA
u _EB) are summarized in Table 4. As can
be seen, the ultimate load increased by 45% for Frame A and 28% for
Frame B. Furthermore, as the connection brackets control the ultimate
strength of the frame, it is necessary to accurately predict the strength
of connections including apex brackets.

4. Finite element model of apex joints

In this study, the experimental work summarized in Section 2.2 was

used to validate a FE model with the purpose of further investigating the
capacity of apex connections. Advanced shell FE models were created in
ABAQUS [16] for the apex connections described in Section 2.2 of this
paper.
The geometry and boundary conditions of the experimental setup
were replicated in the FE model, as shown in Fig. 18. To simulate pin-
ended boundary conditions at the specimen ends, the bolt nodes on
the web of back-to-back channels were first coupled to the pin support
node where a reference point was defined. Simply supported boundary
conditions were then applied to the reference points at both ends. The
lateral displacements of the webs were prevented at those locations
where bracing was put in place during the test. Finally, point loads
were applied in a force-controlled manner to reference points that
were coupled to all the bolts nodes on the web of back-to-back channels
at the load application position. As described in section 2.2, the loading
and boundary conditions in the experiment were normal to rafters, so
two local coordinate systems were applied in the FE model, as shown
in Fig. 18.
Element type, mesh size, contact and connection modelling
were same with the FE model of portal frames as described in
Section 3. The material properties of the channel sections and apex
bracket were described in Section 2.2.
The two connections, representing apex connections with different
bolt group lengths, were analysed using a material and geometric non-
linear analysis. Table 2 demonstrates that the proposed FE model pre-
dicts the moment capacity (MFEA u ) of all apex connections very well,
when compared to the experimental results (MEXP u ), with an average
difference of 4.5%.

5. Simplification of the apex joint model

To simplify the FE model for further investigations, the apex connec-

tion in four-point bending was alternatively modelled as an apex
bracket subjected to pure bending. Fig. 19 illustrates the FE model and
indicates the loading and boundary conditions. All the bolt nodes, out-
side of the centroid, were coupled to the centroid point of bolt group,
where a reference point was defined. Bending moments were applied
to the two reference points. Simply supported boundary conditions Fig. 20. Variation of plastic moment-capacity Mp of apex bracket against ba/aa.

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X. Chen, H.B. Blum, K. Roy et al. Journal of Constructional Steel Research 183 (2021) 106718

can be ignored. Element type, mesh size, and material property were
2
same with the model described in Section 4. f y ba t
MP ¼ ð2Þ
The predicted moment capacities (Ms,FEA
u ) of the simplified apex 4
connection also agree well with the experimental results of the full
apex connection, as shown in Table 2, with an average difference Curve-fitting of the results for the apex bracket having a value for θa
of 5.5%. as 10o gives the relationship for Mp as shown in the following equation:
2 2
MP ¼ f y ba t ½αðba =aa Þ  βðba =aa Þ þ γ ð3Þ
6. Parametric studies
Table 6 summarizes the values of coefficients α, β, and γ of Eq. (3)
The simplified FE model described in the previous Section was fur- adapted to the apex bracket. Eq. (3) is used to predict the moment
ther used to conduct parametric studies. The purpose of parametric capacity of the practical apex brackets with the nominal dimensions
study is to investigate the effect of variables on the capacity of the given in Table 5. Due to the limited value ranges for the key parameters
apex bracket and then propose the equation to predict the moment ca- in parametric studies, the application scope of the proposed Eq. (3) valid
pacity of apex brackets. Table 5 summarized the selected variables for 0.4 ≤ ba/aa ≤ 1.2, an apex bracket thickness ranges from 2 mm to
based on the nominal dimension used in practice. The parameters of 3 mm, an apex bracket yield stress valid from 250 N/mm2 to 550
apex bracket were shown in Fig. 5. Two distinct bolt group arrange- N/mm2 and a frame pitch of 10°. This range is common for typical portal
ments were considered, containing bolts in 2 × 2 and 3 × 2 arrays. frame apex connections, based on the investigation of practical apex
Within each configuration, the ratio of the bolt group length to the brackets.
width of triangular area of apex bracket (ab/aa) is equal to the ratio of
the bolt group breadth to the edge width of apex bracket (bb/ba), and 7. Discussion of the proposed equation
the value of the ratio is 0.8. Six different cross-sectional geometries of
lipped channels were considered, based on the value of the edge A total of 72 practical apex brackets, where the ratio of ba/aa ranged
width of apex bracket (ba). The ratio of the edge width of the from 0.75 to 1.11, were used to compare the moment capacity results
apex bracket to the width of triangular area of the apex bracket predicted by Eq. (1) (MEq.(1)
u ), Eq. (3) (MEq.(3)
u ) and FEA (MFEA
u ), and
(ba/aa) varied from 0.4 to 1.2 at an interval of 0.1. In addition, three dif- the results are summarized in Table 7. The average ratio of MEq.(1) u /
ferent section thickness (t), namely 2, 2.5, and 3 mm were considered. MFEA
u was 1.15 with a standard deviation of 0.24, while the average
The values of yield stress of the bracket were varied from 250 N/mm2 ratio of MEq.(3)
u /MFEA
u was 1.00 with a standard deviation of 0.02. The
to 550 N/mm2 at an increment of 100 N/mm2. The elastic modulus E, proposed Eq. (3) provided rather accurate predictions for the moment
and the frame pitch θa were taken as 205 GPa, and 10o, respectively. capacity of apex brackets.
A total of 648 FE models were created to systematically investigate The value of coefficients (α = 0.13, β = 0.31, and γ = 0.49)
the effects of the aforementioned variables on the capacity of the apex from Table 6 based on the specimen “ba=350 mm, fy=250 N/mm2,
bracket. Fig. 20 shows the variation of Mp against ba/aa. As can be t = 3 mm” was used to predict the moment capacities of apex brackets
seen, the moment capacity (Mp) decreases as the value of ba/aa in- from the full-scale portal frame tests which were described in
creases, and yield stress (fy), thickness (t) and edge width of apex Section 2.1 of this paper. The moment capacities of apex brackets
bracket (ba) all have a positive influence on the moment capacity of predicted by Eq. (3) (MEq.(3)u ) were also summarized in Table 4,
apex bracket (Mp). compared with the values from full scale portal frame FEA, as there
From geometric considerations of the apex bracket, Mp is clearly a was no data about the moment capacity of apex brackets from the
function of ba, ba/aa, and t as well as fy. If the value of frame pitch θa is full-scale portal frame tests. The predicted moment capacities of apex
zero, then from simple plastic analysis considerations of a beam subject brackets using Eq. (3) have a good agreement with the FEA results,
to pure bending [3]: with an average difference of 4%.

Table 6
Coefficients of the proposed moment capacity of apex brackets.

Coefficient Thickness of apex bracket

t = 2 (mm) t = 2.5 (mm) t = 3 (mm)

fy (N/mm2)

250 350 450 550 250 350 450 550 250 350 450 550

ba1 = 150 mm α 0.21 0.23 0.18 0.12 0.27 0.23 0.23 0.23 0.32 0.28 0.25 0.23
β 0.52 0.55 0.44 0.31 0.63 0.55 0.54 0.53 0.71 0.64 0.58 0.54
γ 0.66 0.65 0.59 0.51 0.73 0.68 0.66 0.64 0.78 0.73 0.70 0.67
ba=200 mm α 0.21 0.13 0.09 0.07 0.24 0.19 0.16 0.14 0.26 0.24 0.25 0.19
β 0.47 0.32 0.24 0.19 0.55 0.44 0.38 0.34 0.60 0.55 0.54 0.44
γ 0.60 0.50 0.45 0.40 0.66 0.59 0.55 0.50 0.70 0.65 0.63 0.58
ba=250 mm α 0.13 0.09 0.07 0.06 0.20 0.13 0.08 0.08 0.22 0.21 0.15 0.11
β 0.31 0.21 0.18 0.16 0.43 0.31 0.20 0.20 0.49 0.46 0.36 0.27
γ 0.49 0.41 0.36 0.33 0.57 0.49 0.42 0.40 0.62 0.58 0.52 0.46
ba=300 mm α 0.08 0.08 0.05 0.05 0.14 0.10 0.08 0.07 0.19 0.14 0.10 0.07
β 0.19 0.18 0.14 0.12 0.33 0.23 0.20 0.17 0.42 0.32 0.25 0.18
γ 0.39 0.35 0.31 0.28 0.50 0.42 0.38 0.34 0.57 0.50 0.44 0.38
ba=350 mm α 0.07 0.05 0.06 0.05 0.10 0.09 0.07 0.05 0.13 0.08 0.09 0.08
β 0.16 0.13 0.13 0.11 0.24 0.20 0.17 0.12 0.31 0.19 0.20 0.18
γ 0.34 0.30 0.27 0.25 0.43 0.37 0.33 0.29 0.49 0.41 0.38 0.35
ba=400 mm α 0.05 0.05 0.05 0.05 0.07 0.07 0.05 0.05 0.08 0.09 0.05 0.04
β 0.12 0.12 0.12 0.10 0.16 0.17 0.12 0.12 0.21 0.21 0.13 0.11
γ 0.30 0.27 0.24 0.22 0.36 0.33 0.28 0.26 0.42 0.38 0.32 0.29
1
refer to Fig. 5

11
X. Chen, H.B. Blum, K. Roy et al. Journal of Constructional Steel Research 183 (2021) 106718

Table 7
Equation and FEA results of moment capacity on practical apex brackets.

ba/aa fy Thickness of apex bracket

(N/mm2)
t = 2 mm t = 2.5 mm t = 3 mm

MFEA
u
Eq:ð1Þ
Mu
Eq:ð3Þ
Mu MFEA
u
Eq:ð1Þ
Mu Mu
Eq:ð3Þ
MFEA
u
Eq:ð1Þ
Mu
Eq:ð3Þ
Mu
(kNm) MFEA
u MFEA
u (kNm) MFEA
u MFEA
u (kNm) MFEA
u MFEA
u

ba1=150 mm 0.75 250 4.43 0.91 0.99 5.82 0.86 0.99 7.16 0.84 1.01
350 5.86 0.96 0.99 7.89 0.89 0.99 9.76 0.86 0.99
450 7.32 0.99 1.00 9.85 0.92 0.99 12.30 0.88 1.00
550 8.56 1.03 1.00 11.50 0.96 1.00 14.60 0.91 1.00
ba=200 mm 0.82 250 6.90 0.99 1.03 9.27 0.92 1.00 11.70 0.88 0.98
350 9.12 1.05 1.00 12.30 0.97 1.02 15.50 0.93 0.98
450 11.10 1.11 1.02 15.20 1.01 1.02 19.30 0.96 0.99
550 12.90 1.16 0.99 17.70 1.06 0.98 22.70 0.99 1.01
ba=250 mm 1.02 250 9.66 1.01 1.00 13.00 0.94 1.02 16.60 0.88 0.99
350 12.40 1.11 1.02 16.90 1.01 1.00 21.70 0.95 1.00
450 14.60 1.21 0.96 20.80 1.06 1.01 26.70 0.99 0.98
550 16.50 1.31 0.96 24.00 1.12 1.00 31.30 1.03 0.99
ba=300 mm 1.00 250 12.70 1.12 0.99 17.60 1.01 0.99 22.50 0.95 1.02
350 15.70 1.26 1.00 22.70 1.09 1.28 29.60 1.01 1.02
450 18.20 1.40 0.98 26.90 1.19 0.98 35.70 1.07 0.99
550 20.40 1.53 1.02 30.30 1.29 0.98 41.30 1.13 0.97
ba=350 mm 0.97 250 15.50 1.26 0.99 22.60 1.08 0.99 29.20 1.00 0.98
350 19.10 1.43 0.99 28.10 1.21 0.99 37.70 1.08 1.03
450 22.30 1.57 0.99 32.60 1.34 0.98 44.90 1.17 1.00
550 24.80 1.73 1.03 37.00 1.45 1.00 50.90 1.26 1.00
ba=400 mm 1.11 250 17.80 1.35 1.03 26.50 1.15 1.01 35.10 1.08 0.98
350 21.70 1.56 1.02 32.30 1.36 0.99 44.50 1.20 0.97
450 25.40 1.83 0.95 37.50 1.48 1.00 51.90 1.30 0.99
550 28.40 1.81 1.06 42.50 1.64 0.98 58.50 1.42 0.98

Average (MEq.
u
(1)
/MFEA Eq. (1)
u ) =1.15; Standard deviation (Mu /MFEA
u ) =0.24;
Average (MEq.
u
(3)
/MFEA Eq. (3)
u ) =1.00; Standard deviation (Mu /MFEA
u ) =0.02
1
refer to Fig. 5

The value of coefficients (α = 0.13, β = 0.31, and γ = 0.49) Acknowledgement

from Table 6 based on the specimen “ba=350 mm, fy=250 N/mm2,
t = 3 mm” was used to predict the moment capacities of apex brackets The author wishes to acknowledge the Centre for eResearch at the
from the tests which were described in Section 2.2 of this paper. Table 2 University of Auckland for their help in facilitating this research.
summarized the moment capacities predicted by Eq. (3) (MEq. u
(3)
). http://www.eresearch.auckland.ac.nz
The predicted moment capacities of apex brackets using Eq. (3) have
a good agreement with the experimental results, with an average References
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