2006 Shi - Analysis of the moment–rotation behaviour of stiffened end plate

Journal of Constructional Steel Research 63 (2007) 1279–1293
www.elsevier.com/locate/jcsr
Experimental and theoretical analysis of the moment–rotation behaviour of
stiffened extended end-plate connections
Yongjiu Shi ∗ , Gang Shi, Yuanqing Wang
Department of Civil Engineering, Tsinghua University, Beijing 100084, PR China
Received 19 January 2006; accepted 1 November 2006
Abstract
A new theoretical model to evaluate the moment–rotation (M–φ) relationship for stiffened and extended steel beam–column end-plate
connections has been derived in this paper. Based on a specific definition of the end-plate connection rotation, the end-plate connection is
decomposed into several components, including the panel zone, bolt, end-plate and column flange. The complete loading–deformation process of
each component is then analysed. Finally the loading–deformation process for the whole connection is obtained by superimposing the behaviour of
each component. In addition, 5 joint tests have been conducted to verify the proposed analytical model. By comparing it with the test results, it has
been concluded that this analytical model can evaluate the rotational behaviour of end-plate connections, as well as the moment–rotation (M–φ)
curve and the initial rotational stiffness accurately. Furthermore, it can analyse every contribution to the joint’s rotational deformation, such as the
shear deformation of the panel zone, the bolt extension, the bending deformation of the end-plate and column flange, etc. This analytical model
also provides moment–shear rotation (M–φs ) and moment–gap rotation (M–φep ) curves, which establish a reliable foundation for analysing the
detailed rotational behaviour of end-plate connections.
c 2006 Elsevier Ltd. All rights reserved.
Keywords: End-plate connection; Semi-rigid; Rotational stiffness; Moment–rotation
1. Introduction
Conventional analysis and design of steel frames are usually
carried out under the assumption that the connections joining
the beams to the columns are either fully rigid or ideally
pinned. In fact, as is evident from experimental observations,
all connections used in current engineering practice possess
rigidities which fall between the extreme cases of fully rigid
and ideally pinned [1], i.e., the connection is actually semirigid. The behaviour of semi-rigid connections significantly
influences not only the internal force distribution, but also
the deformation of steel structures [2]. For most connections,
the axial and shearing deformations are usually low compared
to the rotational deformation. Consequently, for practical
design, it is essential to determine the connection’s rotational
deformation. Therefore, almost all the steel design codes of
different countries from all over the world [3–6] require that the
effect of connection deformations should be taken into account
∗ Corresponding author. Tel.: +86 10 6278 2012; fax: +86 10 6278 8623.
E-mail address: [email protected] (Y. Shi).
0143-974X/$ - see front matter c 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jcsr.2006.11.008
in the global analysis and design of steel frames with semi-rigid
connections. Hence, much effort has been focused in recent
years toward determining connection moment–rotation (M–φ)
relationships [7].
Certainly, full-scale and carefully conducted joint experiments are the most reliable sources and direct method of obtaining M–φ relationships. While more than 800 tests of beamto-column connections have been performed around the world
today, only about 300 of them have provided currently useful
moment–rotation data [2]. Since the connection details consist
of a number of components, any changes in these connection
details may lead to significant variations in the connection characteristics [8]. In addition, many other variable parameters, for
example, details of the fabrication and assembly of the connections, also vary enormously and can affect their behaviour [9].
It is impossible to test all of the connections that might be used
in steel construction. Some researchers such as Goverdhan,
Nethercot, Kishi and Chen, have collected the available experimental results and constructed steel connection data banks that
provide the user with not only the test data, but also some predictive equations [1]. Where the connections detailing, beam,
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Y. Shi et al. / Journal of Constructional Steel Research 63 (2007) 1279–1293
Notation
Ae
E
Eb
Eh
E bh
G
Ic f,l
Ic f,r
M
M Rd,s
N1
Ni
1Ni
Vp
bep
ef
ew
fv
fy
f by
f bu
h bw
h cw
ht
1l
1l1
1l2
lb
tc f,l
tc f,r
tcw
tep
yi
y1
y2
α
εi
effective area of bolt shank threaded section;
elastic modulus of steel;
elastic modulus of bolt;
hardening modulus of steel;
hardening modulus of bolt;
shear modulus;
moment of inertia of the column panel zone left
flange;
moment of inertia of the column panel zone right
flange;
the joint moment;
the design moment resistance of the panel zone;
the transferred tension force of a single bolt in the
first bolt-row;
the transferred tension force of a single bolt in the
ith bolt-row;
the actual increase of the bolt tension force in the
ith bolt-row;
volume of the panel zone;
end-plate width;
distance from the bolt centre to the beam flange
surface;
distance from the bolt centre to the beam web
surface;
the steel shear strength;
the steel yield strength;
the bolt yield strength;
the bolt ultimate tension strength;
beam web height;
column web height;
the distance between the centre lines of the beam
top and bottom flanges;
deformation at the beam tension flange centreline
caused by the bolt extension;
the extension of a single bolt in the first bolt-row;
the extension of a single bolt in the second boltrow;
bolt shank length, equal to the thickness of
plates which the bolt clamped plus the washers
thickness;
thickness of the column panel zone left flange;
thickness of the column panel zone right flange;
column web thickness;
end-plate thickness;
distance from the ith bolt-row centreline to the
neutral axis;
distance from the first bolt-row centreline to the
neutral axis;
distance from the second bolt-row centreline to
the neutral axis;
the ratio of the compression distribution area
between the contacted surfaces around a bolt on
the bolt shank area;
the bolt tension strain of the ith bolt-row.
and column sizes used in frame analysis are significantly different from the available experiments, however, the connection
behaviour retrieved from a database may not correctly represent the actual connections. Furthermore, not every structural
engineer has access to the database of experimental results.
The most widely used semi-rigid connections are the angle
cleat and end-plate connections. For end-plate connections,
the common approaches for predicting the M–φ relationship,
besides experiments, are the T-stub analogy, the yield line
theory and the finite element analysis [10].
Early attempts to develop a design methodology for endplate connections were based on the T-stub analogy developed
by Douty and McGuire [11], Nair et al. [12], Kato and
McGuire [13], Agerskov [14] and others. More recently,
methods based on refined yield line analysis have been
suggested, in which the widely accepted design procedures
of end-plate connections are derived from [15–19]. Eurocode
3 [3] has formally recommended an M–φ curve for end-plate
connections based on the T-stub yield line theory.
The finite element analysis of end-plate connections was
first developed by Krishnamurthy [20–22]. An exhaustive
analytical study of four-bolt, unstiffened, extended end-plates,
along with a series of experimental investigations, leads to
the development of the design procedure found in Ref. [23].
Tarpy and Cardinal carried out an elastic finite element
study and experimental verification for unstiffened end-plate
connections and also proposed a design methodology [24].
Maxwell et al. [25] developed a prediction equation for the
ultimate moment of the connection and the M–φ relationships
based on the finite element method and experimental as well.
Sherbourne and Bahaari used 3-D finite elements to analyze
end-plate connections. In addition to the overall behaviour,
the contribution of the bolt, end-plate and column flange
flexibility to the connection rotation was singled out [26].
With the finite element method, they also studied the structural
properties of an extended end-plate connected to an unstiffened
column flange [27]. Based on 34 stiffened, extended end-plate
connections and 19 end-plate connections without stiffeners in
the tension region, they produced a single standardized M–φ
function for each of these two connection types by curve
fitting [28,29]. Shi et al. used many new functions of the finite
element method and simulated the mechanical behaviour of
end-plate connections and each component more accurately [9,
30,31].
Currently the widely accepted nonlinear M–φ relationship
formulae are the polynomial model, the power model and the
exponential model. From a survey of these existing models,
it was found that the coefficient and parameters involved in
these modeling formulae, by and large, were calibrated from
the relevant test results or finite element analysis results.
But curve fitting will not give any indications as to how
connection components deform or fail, and provides less help
to designers for improving connection design. In the design
M–φ curve recommended by Eurocode 3, the nonlinear part
is also determined by curve fitting, and the corresponding
research is mostly based on the end-plate connections with
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unstiffened columns, which may not be suitable for all the endplate connection types. Furthermore, the last part of this curve
is the horizontal line and this cannot consider the hardening
effect, which leads to great discrepancies with the actual
behaviour of the connection [32]. Finite element analysis is
precise and reliable, and can analyze complicated connection
profiles that may be difficult to investigate by experiment.
However, such finite element analysis generally requires a large
commercial package and is not therefore feasible for many
practical applications. Most importantly, the effects of torque,
lack of fit, construction imperfections and defects, etc., may be
difficult to include in a finite element model. A well-defined
theoretical method that can be easily carried out and is only
based on the connection’s details is needed indeed for the
analysis of the end-plate connection M–φ relationship.
Fig. 1. Test specimen and loading arrangement.
2. End-plate connection standard details
Many experiments have been performed to investigate
the influences of the connection details on the connection’s
behaviour. Ghobarah et al. [33–35] concluded that connections
with unstiffened columns showed very poor behaviour as
compared with those that were stiffened. Tsai and Popov [36]
pointed out that an end-plate rib stiffener and stronger bolts can
significantly improve the behaviour of end-plate connections
under large cyclic loading, and the extended end-plate moment
connections can be designed to develop the full plastic moment
capacity of the beam under cyclic loading, and the effect of
the prying force was reduced by the use of the end-plate rib
stiffener. Adey et al. [37] proposed that the application of
extension stiffeners increases the connection flexural strength,
yield rotation as well as energy dissipation capacity.
Summarising the available test results and other relevant
research results, standard details of end-plate moment
connections for multistorey steel frames, especially in seismic
regions, can be proposed as follows: The end-plate extends
on both sides; the column flange and end-plate are stiffened;
the thickness of the column flange stiffener and the end-plate
extension stiffener should be no less than the thickness of the
beam flange and web respectively; the thickness of the column
flange is equal to the end-plate within the range of 100 mm
above and below the extension edge of the end-plate.
In this paper, an analytical model for the M–φ relationship
of this type of end-plate connection has been proposed and
the corresponding joint tests have been conducted to verify the
analytical results.
3. Test specimens
Five specimens of stiffened and extended beam-to-column
end-plate connections with various details are tested under
monotonic loads. A sketch of a typical connection specimen
is shown in Figs. 1 and 2. The out-of-plane deformation of
specimens was restrained during tests. The details of these
5 specimens are shown in Table 1 and Fig. 3. The beam
and column sizes with welded I-shaped cross-sections used
for all these 5 specimens were identical. The section depths,
Fig. 2. Testing set.
Table 1
Types and details of specimens
Specimen number
End-plate thickness (mm)
Bolt diameter (mm)
EPC-1
EPC-2
EPC-3
EPC-4
EPC-5
20
25
20
25
16
20
20
24
24
20
web thicknesses and flange thicknesses of the columns and
beams are 300 mm, 8 mm and 12 mm, and the flange widths
are 250 mm and 200 mm, respectively. The thickness of the
column flange is equal to that of the end-plate within the range
of 100 mm above and below the extension edge of the endplate. The thicknesses of the column stiffener and end-plate rib
stiffeners are 12 mm and 10 mm respectively.
Full penetration welds are applied between the end-plate and
beam flanges as well as the column flange splices, and the other
welds, including the welds between the flanges and webs of
beams and columns, end-plates and beam webs, are fillet welds
with 8 mm leg size.
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Y. Shi et al. / Journal of Constructional Steel Research 63 (2007) 1279–1293
Table 2
Material properties
Material
Measured yield
strength (MPa)
Measured tensile
strength (MPa)
Measured elastic
modulus (MPa)
Design value of bolt
pre-tension force (kN)
Measured bolt average
pre-tension force (kN)
Steel (thickness ≤ 16 mm)
Steel (thickness > 16 mm)
Bolts (M20)
Bolts (M24)
391
363
995
975
559
537
1160
1188
190 707
204 228
–
–
–
–
155
225
–
–
185
251
Table 3
Test results
Specimen
number
Loading
capacity (kN)
Moment
resistance (kN m)
Moment resistance change
compared with EPC-1 (%)
Initial rotational
stiffness
S j,ini (kN m/rad)
Initial rotational
stiffness change
compared with
EPC-1 (%)
Failure mode
EPC-1
EPC-2
EPC-3
286.4
268.4
325.3
343.7
322.1
390.3
0.0
−6.3
13.6
52 276
46 094
46 066
–
−11.8
−11.9
EPC-4
342.3
410.8
19.5
47 469
−9.2
EPC-5
296.1
355.4
3.4
41 634
−20.4
Bolt fracture
Bolt fracture
Buckling of beam flange and
web in compression
Buckling of beam flange and
web in compression
Bolt fracture and buckling of
end-plate rib stiffener in
compression
Fig. 4. A gauged bolt.
Fig. 3. Details of connections.
The steel is grade Q345 (nominal yielding stress f y =
345 MPa), and the bolts are high strength friction-grip bolts
(Grade 10.9). The material properties of the steel and bolts
are obtained from tensile tests on coupons and from the bolts’
certificate of quality, as shown in Table 2. The proof elastic
modulus of the bolts is taken as 206 000 N/mm2 .
One line of bolts on each connection was instrumented using
strain gauges, and these bolts are numbered in Fig. 3. Two
shallow slots were grooved symmetrically on the unthreaded
portion of the bolt shank, and in each slot a strain gauge
was fixed and covered with resin for protection (Fig. 4).
The alignment of the two strain gauges was secured to be
perpendicular to the beam flange during the tightening of the
bolts, so that the maximum and minimum strain of the bolt
in the connection moment plane could be measured. The bolt
axial force can be calculated by taking the average of the
two strain gauges’ measured values. All bolts were tightened
by the calibrated wrench method. The design values of bolt
pre-tension forces and the actually applied bolt pre-tension
forces are listed in Table 2. The contact surface between the
end-plate and column flange was prepared by blasting, with a
slip coefficient 0.44.
Fig. 1 displays all the displacement transducers installed
to measure the joint deformation. No. 1 was used to monitor
the displacement at the loading point. Nos. 2–10 measure the
relative deformation between the end-plate and column flange.
Nos. 11 and 12 measure the inner shearing deformation of the
panel zone. Nos. 13 and 14 were arranged next to the column
stiffeners to measure the shearing deformation of the panel
zone. No. 15 measures the slippage between the end-plate and
column flange.
4. Test results and discussion
4.1. Rotational stiffness and M–φ, M–φep , M–φs curves
The test results of the specimens are summarised in Table 3.
The failure modes of specimens are shown in Fig. 5. In these
tests, the fracture and necking positions of all the failed bolts
appear on the threaded portion and not at the strain gauge slots,
as shown in Fig. 5, which indicates that these grooved slots do
not damage the bolt capacity.
With regard to the joint rotation φ, the conventional
definition is suggested as the angle change between the beam
Y. Shi et al. / Journal of Constructional Steel Research 63 (2007) 1279–1293
1283
Fig. 5. Failure modes of the specimens.
Fig. 6. Definition of joint rotation.
and the column from its original configuration [1]. In this paper,
the joint rotation φ of the beam-to-column end-plate connection
is defined as the relative rotation between the centre lines of the
beam top and bottom flanges at the beam end, and it usually
includes two parts: the shearing rotation φs , contributed by the
panel zone of the column, and the gap rotation φep , caused by
the relative deformation between the end-plate and the column
flange, including the bending deformation of the end-plate and
column flange as well as the extension of the bolts (Fig. 6).
The shearing rotation φs is calculated by ∆/ h t , and the gap
rotation φep is calculated by δ/ h t , giving φ = φs + φep , where
∆ is the displacement difference of the panel zone at the centre
lines of the top and bottom beam flanges at the beam end which
can be measured by displacement transducer Nos. 13 and 14;
δ is the gap between the end-plate and the column flange at
the beam tension flange centre line, which can be measured by
displacement transducer No. 4; and h t is the distance between
the centre lines of the top and bottom beam flanges, and is
288 mm. M–φ curves of all the specimens are shown in Fig. 7.
M–φs and M–φep curves of each specimen are also measured
to verify the analytical model, and the comparison results are
shown in Figs. 12 and 13.
The loading capacity presented in Table 3 is the maximum
pushing load applied. The moment resistance is calculated by
multiplying the load with the arm of the loading(1.2 m). It can
be seen from Fig. 7 that the M–φ curves of all the end-plate
connection specimens are almost linear when the moment is
less than 60 kN m; therefore the initial rotational stiffnesses,
Fig. 7. M–φ curves of the specimens.
S j,ini , of the connections are defined as the secant rotational
stiffnesses up to this bending moment. According to Eurocode
3 [3], S j,ini is compared to the flexural stiffness of the connected
beam EIb /L b . The connection is rigid when S j,ini is larger than
25EIb /L b for unbraced frames, nominally pinned when S j,ini
is less than 0.5EIb /L b , and semi-rigid when S j,ini is between
these two values. If we assume that the natural beam length of
specimens is 1.2 m × 2 = 2.4 m, then EIb /L b is 9751 kN m,
and S j,ini for all the specimens are about 4.3–5.4 times EIb /L b ,
so all the tested connections are semi-rigid. If the tested beam
section sizes are applied in 4–5 m span frames and EIb /L b is
4681–5851 kN m, the Sj,ini of all the connections are about 7.1–
11.2 times EIb /L b , and therefore all the connections are still
classified as semi-rigid.
The influence of connection details on the behaviour of endplate connections can be analyzed from Table 3 and Fig. 7.
EPC-1 is a reference specimen, and the other connection
specimens alter only one or two parameters from EPC-1. EPC2 has increased the end-plate thickness compared with EPC1, but its moment resistance decreases remarkably and its
initial rotational stiffness changes less; its rotational stiffness
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Y. Shi et al. / Journal of Constructional Steel Research 63 (2007) 1279–1293
Fig. 8. Bolt tension force generated by applied moment.
is much larger than EPC-1 when nonlinearity of the M–φ
curve occurs. EPC-3 has increased the bolt diameter, which also
means a larger bolt pre-tension force; its moment resistance
is much higher than EPC-1, which fails by bolt fracture.
EPC-4 has increased both the end-plate thickness and bolt
diameter; its moment resistance is the largest among all the
specimens; higher rotational stiffness is observed during its
loading process. EPC-5 has reduced the end-plate thickness,
but its moment resistance increases on the contrary, because the
end-plate is thin and its bending stiffness is smaller, leading
to the bolt tension force distribution among the four bolts in
tension being more uniform. Meanwhile, its initial rotational
stiffness decreases significantly, with excellent ductility and
rotation capacity.
4.2. Bolt force and distribution
The bolt tension forces generated by the applied moments
are shown in Fig. 8. In this paper, the bolt tension force
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Y. Shi et al. / Journal of Constructional Steel Research 63 (2007) 1279–1293
Table 4
Moment values of curve of Specimens 1, 2 and 3 (kN m)
Specimen number
Curve 1
Curve 2
Curve 3
EPC-1
EPC-2
EPC-3
EPC-4
EPC-5
80
80
100
100
70
199
210
224
226
209
248
248
300
289
230
indicates the tension force increment caused by the joint
moment, excluding the pre-tension force. The loading stages
corresponding to curve 1, 2 and 3 in Fig. 8 are listed in
Table 4. The applied moment at curve 1 is around the design
moment resistance of the connection from the Chinese codes [6,
38,39], where the connection is in the elastic stage. The
applied moment at curve 2 is around the knee point of the
M–φ curves of each specimen, where the nonlinearity and
partial plasticity occur. Curve 3 corresponds to the moment
where the nonlinearity is obvious and plasticity has developed
significantly.
From Fig. 8, it can be noted that, for all these connections,
generally the neutral axis of the connection is around the centre
line of the bolts group, which is on the centre line of the
connection when the bolts are arranged symmetrically about
the beam section centre line, and the bolt force generated by
the bending moment is approximately distributed linearly. The
negative values in the compression zone means that the bending
moment will reduce the resultant bolt force to levels below the
pre-tension level. For EPC-1, EPC-2, EPC-4 and EPC-5, the
maximal tension force appears on the first bolt-row, and the
tension force of the second bolt-row develops gradually. For
EPC-3, the tension force of the second bolt-row is maximal
at the initial loading stage. With the moment increasing, the
tension force of the first bolt-row exceeds the second bolt-row
and remains maximal up to failure.
5. Analytical model
As specified above, the end-plate joint rotation φ consists
of the shearing rotation φs and the gap rotation φep . The gap
rotation φep is contributed by the flexural deformation of the
end-plate and column flange as well as the extension of the
bolts. This analytical model will evaluate the M–φs and M–φep
curve respectively, and then superpose these two curves to get
the final M–φ curve.
5.1. Analytical model for M–φs curve
From the relevant research findings [32,40], the M–φs curve
adopting a trilinear model is recommended as shown in Fig. 9.
Some research [40] has indicated that the restraints around the
column panel zone, especially the column flange, will increase
the panel zone loading capacity and its stiffness. The Chinese
steel structural design code [6] simplifies this contribution
and presents the calculation equation for the design moment
resistance of the panel zone as:
M Rd,s =
4
4
4 fy
f v V p = f v h bw h cw tcw = · √ · h bw h cw tcw .(1)
3
3
3
3
Fig. 9. M–φs model.
When the joint applied moment M is equal to M Rd,s , the
panel zone yields, and the panel zone shear strain reaches
γ = γ y = f v /G. The shear rotation actually is the shear strain,
and is also the shearing rotation φsy = γ y . When M ≤ M Rd ,
the joint shearing rotation initial stiffness K s,ini is given by:
K s,ini =
=
M Rd,s
M Rd,s
=
φsy
γy
4
3 f v h bw h cw tcw
f v /G
=
4
· G · h bw h cw tcw .
3
(2)
From Eqs. (1) and (2), it is noted that the restraints around
the panel zone increase its resistance and stiffness by 1/3.
According to Krawinkler et al. [40], after yielding, the panel
zone rotational stiffness can be attributed to the bending of the
column flanges, and it can be computed as:
Kp =
12 EIc f,r
12 EIc f,l
+
5 tc f,l β
5 tc f,r β
(3)
where β is a factor intended to account for the beneficial effect
of column shear above and below the joint, whose calculation
method can be obtained from Ref. [40].
This post-yielding stiffness is developed up to the yielding
of the column flanges occurring for a shear deformation of the
panel zone, which can be approximately assumed to be equal to
4 γ y . Therefore, the corresponding moment is given by:
M p = K s, ini · γ y + K p · 3γ y .
(4)
After the development of this moment, i.e., the column
flange yielding, the panel zone rotational stiffness attributed to
the strain hardening can be computed as:
Ks =
Eh
K s,ini .
E
(5)
5.2. Analytical model for M–φep curve
(i) Analysis assumptions
From the test results obtained by this paper, and the typical
end-plate connections meeting the abovementioned standard
details and requirements, we can simplify and assume that
the bolt tension strain generated by the applied moment is
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Y. Shi et al. / Journal of Constructional Steel Research 63 (2007) 1279–1293
Substituting Eq. (9) into Eqs. (6)–(8), it can be obtained that
at this moment, the bolt tension strain is given by
ε0 =
1Ni
P
=
.
E b Ae
α E b Ae
(10)
The total bolt force is
Fig. 10. Bolt tension strain and force generated by the applied moment.
approximately a linear distribution, and the neutral axis of the
connection is located approximately at the beam mid-height,
and the bolt tension strains in each of the two rows of bolts
are considered as equal. Here the specified bolt tension strain
is the increment caused by the applied joint moment, excluding
the pretension strain. Until the end-plate and the column flange
separate, the bolt force contributed by the bending moment
can be assumed to be linearly distributed as shown in Fig. 10.
The tension force indicates an increment in the bolt force and
compression force indicates a decrement in the bolt force.
(ii) Joint rotation contributed by bolt extension
Before the moment is applied to the connection, the bolts
are tightened to the bolt pre-tension force P. From the static
equilibrium, the contact force C between the end-plate and the
column flange around this bolt is also equal to P.
With the joint moment developing, the bolts in tension can
experience the following three stages:
Stage 1: The first stage lasts until the end-plate and the
column flange separate around bolt location, i.e., εi ≤ ε0 .
During this stage, with the joint moment developing, the contact
force between the end-plate and the column flange decreases
while the bolt tension force increases. Assuming the applied
moment is M, and the contact force becomes C 0 = C − 1C,
and the total bolt tension force is increased to Ni = P + 1Ni ,
where C is the initial contact force between the end-plate and
column flange under pre-tension P; 1C, 1N are the contact
force decrement and bolt tension increment caused by applied
moment. From the static equilibrium, it can be obtained that
1C = α1Ni , and α is the ratio of the contact area around a bolt
to the bolt shank area, according to the analysis results [31], the
value of α can be taken as 10 conservatively.
The actual increment in bolt tension force can be expressed
as
1Ni = E b Ae εi
(6)
and the tension force on this bolt is given by
Ni = (α + 1)1Ni
(7)
(8)
Whenever the contact force between the end-plate and the
column flange around this bolt is reduced to zero, i.e., the endplate and the column flange separate around bolt, the decrement
of this contact force is given by 1C = α1Ni = P, so that
1Ni =
P
.
α
(11)
1li = ε0 · lb .
(12)
Stage 2: The second stage starts from the separation of the
end-plate and the column flange around the bolt and lasts until
bolt yielding in tension, i.e., ε0 < εi ≤ εby − ε p . During this
stage, the actual increase of the bolt tension force can still be
calculated by Eq. (6), and the total tension force on this bolt is
Ni = P + E b Ae εi
(13)
and when yielding, the total bolt strain is equal to the yielding
f
strain εby = Ebyb , and the actual bolt tension strain increment is
given by
εi = εby − ε p =
f by Ae − P
f by
P
−
=
.
Eb
E b Ae
E b Ae
(14)
The total tension force of this bolt is given by
Ni = f by Ae
(15)
where ε p is the bolt pre-tension strain taken as ε p = E bPAe ; f by
is the bolt yield strength.
Stage 3: The third stage starts from the bolt yielding in
tension until the bolt reaches its ultimate tension strength,
i.e., εby − ε p < εi ≤ εbu − ε p . During this stage, the actual
increment of the bolt tension force is given by
1Ni = f by Ae + E bh Ae (εi − εby ) − P.
(16)
The total tension force of this bolt is evaluated as
Ni = f by Ae + E bh Ae (εi − εby ).
(17)
When the bolt reaches its ultimate tension strength, its total
f −f
f
strain is equal to the ultimate tension strain εbu = Ebyb + buE bh by ,
and this bolt tension strain increment is
εi = εbu − ε p .
(18)
The total tension force of this bolt is given by
and the extension deformation of this bolt is given by
1li = εi · lb .
α+1
P
α
and the extension deformation of this bolt is
Ni =
(9)
Ni = f bu Ae .
(19)
The joint rotation contributed by bolt extension deformation,
which is also called bolt extension rotation in the following
text, can be obtained by the following pivotal points of
procedure (a)–(e). Linking these points with straight lines, the
moment–bolt extension rotation curve will be obtained:
(a) The contact force between the end-plate and the column
flange at the first bolt-row is reduced to zero, i.e., at the
stage where the end-plate separates from the column flange by
applying moment at the first bolt-row.
Y. Shi et al. / Journal of Constructional Steel Research 63 (2007) 1279–1293
1287
Fig. 11. End-plate stiffness.
Here the actual increments on the bolt tension force, the bolt
tension strain, the total tension force and the extension of the
bolt in the first bolt-row can be calculated by Eqs. (9)–(12)
respectively.
From the linear distribution assumption, the bolt tension
strain in tension zone εi can be calculated from
εi =
ε1
· yi .
y1
(20)
The tension force of this bolt can be calculated from Eqs. (6)
and (7), and can also be calculated as
Ni =
N1
· yi .
y1
(21)
(b) The contact force between the end-plate and the column
flange at the second bolt-row is reduced to zero, i.e., at the
moment the end-plate is separated from the column flange by
applying moment at the second bolt-row.
Here the actual increment on the bolt tension force, the bolt
tension strain, the total tension force and the extension of the
bolt in the second bolt-row can be calculated by Eqs. (9)–(12)
respectively.
Similarly, bolt tension strain in tension zone εi can be
calculated as
ε2
(22)
εi =
· yi .
y2
(c) The first bolt-row yields.
Here the bolt tension strain and the total tension force of the
bolt in the first bolt-row can be calculated by Eqs. (14) and (15)
respectively. The bolt tension strain in tension zone εi can be
calculated by Eq. (20).
(d) The second bolt-row yields.
Here the bolt tension strain and the total tension force of the
bolt in the second bolt-row can be calculated by Eqs. (14) and
(15) respectively. The bolt tension strain in tension zone εi can
be calculated by Eq. (22).
(e) The bolt of the first bolt-row reaching ultimate tension
strength.
Here the bolt tension strain and the total tension force of the
bolt in the first bolt-row can be calculated by Eqs. (18) and (19)
respectively. The bolt tension strain in tension zone εi can be
calculated by Eq. (20).
After the tension strain of the bolts(Fig. 10) in the tension
zone has been obtained, the actual increment on the bolt tension
force, the total tension force and the extension deformation of
this bolt can be calculated. The corresponding joint moment and
the bolt extension rotation can be calculated as:
X
M =2
(Ni · yi )
(23)
1l1
1l2
1li
1l
=
=
=
(24)
ht
2y1
2y2
2yi
P
where,
is summation for all the bolts in the tension zone.
From the above calculation procedure (a)–(e), we can get
some pivotal points for the joint moment and the corresponding
bolt extension rotation. Linking these pivotal points by the
sequence of the magnitudes of the corresponding bolt extension
rotation, the moment–bolt extension rotation curve can be
obtained, extending the straight line segment by its own slope
beyond the point (e).
(iii) Joint rotation contributed by the bending deformation of
the end-plate and column flange
The joint rotations contributed by the bending deformation
of the end-plate and column flange are called the end-plate
rotation and the column flange rotation respectively. The
bending deformation of the end-plate and column flange can be
calculated according to the bolt tension force obtained by the
above calculation procedure (a)–(e), taking into account only
the deformation of the end-plate and column flange around one
bolt-row on both sides of the beam tension flange.
Since the calculation procedure of the column flange
deformation is the same for the end-plate, the end-plate rotation
is given to introduce the calculation method.
For the end-plate connection meeting the above standard
details requirements, the beam flange, the beam web and the
end-plate extension rib stiffener can be considered as the fixed
restraints of the end-plate segment. The end-plate segment
around the bolt in tension can be separated and simplified into
a two-edge fixed plate [41] as shown in Fig. 11. The arrow is
the bolt clamp force. As the pre-tension bolt clamps the plates
effectively, it can be assumed that the end-plate segment is fixed
at the bolt centre line. Furthermore, this end-plate segment can
be decomposed into two plates with opposite ends fixed. The
stiffness of this end-plate segment k I or kII is equal to the
φb =
1288
Y. Shi et al. / Journal of Constructional Steel Research 63 (2007) 1279–1293
Fig. 12. Comparison of the M–φs curve from the analytical models and tests.
summation of stiffness of these two plates:
kI
or
kII = k1 + k2
(25)
where k I or kII is the stiffness of the end-plate segment around
one bolt of the first and second bolt-rows respectively. If b1 or
b2 is larger than ew + e f , b1 or b2 is set equal to ew + e f , as
shown in Fig. 11.
In particular, when the plate bending spans are of the
same order of magnitude as the plate thickness itself [42], the
plate’s shear deformation cannot be neglected. From material
Y. Shi et al. / Journal of Constructional Steel Research 63 (2007) 1279–1293
Fig. 13. Comparison of the M–φep curve from analytical model and tests.
1289
1290
Y. Shi et al. / Journal of Constructional Steel Research 63 (2007) 1279–1293
mechanics, the k1 and k2 can be expressed as
k1 = β1
k 2 = β2
1
3
ew
12EI
+
α s ew
GA
1
e3f
12EI
+
αs e f
GA
= β1
1
3
ew
3
Eb1 tep
+
α s ew
Gb1 tep
1
= β2
e3f
3
Eb2 tep
+
αs e f
Gb2 tep
(26)
(27)
where αs = 1.5, is a parameter taking into account the shear
deformation; β1 = 1 − A2 /(b1 ew ) and β2 = 1 − A1 /(b2 e f ) are
the reduction coefficients, because the stiffness of the areas A1
and A2 in Fig. 11 have been calculated repeatedly.
After yielding occurs at the section edge of the endplate segment and it reaches its bending resistance, according
to references [3,32], the end-plate segment stiffness is
approximate to 1/7 of its initial stiffness, i.e., k I /7 or kII /7.
After the total cross-section of the end-plate segment yields,
it reaches its ultimate bending resistance. Taking into account
the strain hardening, the end-plate segment stiffness is taken as
Eh
Eh
E k I or E kII .
From Ref. [41], the bolt tension force corresponding to the
bending resistance of a two-edge fixed end-plate segment is
given by
Ny =
2 f
bep tep
y
6e f
+
2 f
(e f + ew )tep
y
3ew
.
(28)
When this segment reaches its ultimate bending resistance, the
bolt tension force is taken as
Nu = 1.5N y .
(29)
Based on the bolt tension force calculated from the above
calculation procedure (a)–(e), the end-plate deformation at the
first bolt-row ∆ep1 and second bolt-row ∆ep2 can be calculated.
The deformation at the centre line of the beam tension flange
∆ep , is taken as the average of ∆ep1 and ∆ep2 , i.e., ∆ep =
(∆ep1 + ∆ep2 )/2. The corresponding end-plate rotation is given
by
φep =
∆ep
.
ht
(30)
From the above calculation procedure (a)–(e), the joint
moment and the corresponding end-plate rotation can be
calculated. The moment–end-plate rotation curve can be
obtained. After point (e), the line slope is taken as EEh K i , where
K i is the initial stiffness of this curve, i.e. the slope of the first
straight line segment of this curve.
For the actual behaviour of all the end-plate connection
moment–rotation (M–φ) curves, its tangent stiffness is
always decreasing. Accordingly, it is required to revise the
moment–end plate rotation curve obtained above. Whenever the
slope of any straight line segment is larger than its preceding
straight line segment, its slope is taken equal to its preceding
straight line segment, and the moments of the pivotal points do
not change.
For the calculation procedure (e), if the edge of the end-plate
segment does not yield, i.e. the bolt tension force corresponding
to the bending resistance of the end-plate segment is larger
than the bolt’s ultimate tension resistance, this indicates that
the end-plate is rather thick and it does not yield. Here, the
slope of the last straight line segment, i.e. the straight line
segment after point (e), is taken to be equal to the slope of its
preceding straight line segment. This circumstance should be
avoided during the design. According to the relevant tests and
investigation results [3,32,43], the failure of the end-plate in
bending can provide the joint with higher deformation capacity,
while the failure of bolts in tension is much less dissipative and
less deformable. The seismic design rule of “strong connection,
weak plate” should be adopted. It also suggested applying the
end-plate with moderate thickness and adequately stiffening.
The bolt diameter should be adequate so that the bolt ultimate
tension resistance is higher than the end-plate segment ultimate
bending resistance, in order to assure the joint ductility and
energy dissipation capacity.
For the end-plate connection with a strong bolt and weak
end-plate, it may happen that the bending resistance of the endplate segment at the first bolt-row is below the bolt tension
force of the first bolt-row when the end-plate and column
flange separate at this position, i.e. the end-plate segment yields
prior to the separation of the end-plate and column flange.
Here a pivotal point should be added to the moment–end-plate
rotation curve, which is the bolt tension force of the first boltrow corresponding to the yielding resistance of the end-plate
segment at this position. The relevant bolt transferred tension
force, the bolt extension, the bending deformation of the endplate and the corresponding end-plate rotation can then be
calculated.
(iii) Moment–gap rotation (M–φep ) curve
The moment–gap rotation (M–φep ) curve can be obtained
by superposing the moment–bolt extension rotation curve, the
moment–end plate rotation curve and the moment–column
flange rotation curve.
The calculation method for the moment–column flange
rotation curve is the same as that for the moment–end plate
rotation curve. If the column flange is wider than the end-plate,
the values of b1 and b2 for the column flange should be taken
as the values of the end-plate.
5.3. Moment–rotation (M–φ) curve
The moment–rotation (M–φ) curve can be obtained by
superposing the moment-shearing rotation (M–φs ) curve and
the moment–gap rotation (M–φep ) curve. During the process
of superposing, for the moment value of each pivotal point of
the M–φs curve and M–φep curve, adding the corresponding
shearing rotation φs and gap rotation φep , the joint rotation φ
is equal to φs + φep . Linking these pairs of M and φ, the M–φ
curve of the end-plate connection can be obtained.
6. Comparison of analytical and test results and discussion
The M–φ curves for test specimens EPC-1–EPC-5 have
been calculated using the above analytical model, and the
comparison between the M–φs curves, the M–φep curves and
Y. Shi et al. / Journal of Constructional Steel Research 63 (2007) 1279–1293
1291
Fig. 14. Comparison of the M–φ curves from the analytical model and tests.
the M–φ curves of the analytical models and test results are
shown in Figs. 12–14. The comparison between the joint initial
rotational stiffness, S j,ini , of the analytical models and test
results are listed in Table 5. The value of S j,ini from the
analytical model is taken as the slope of the first straight line
segment of its M–φ curve. In Figs. 12–14 and Table 5, Model
1 denotes that the analytical results are calculated according to
the material property values specified in the current Chinese
1292
Y. Shi et al. / Journal of Constructional Steel Research 63 (2007) 1279–1293
Table 5
Comparison of S j,ini of specimens from analytical and test results
Specimen number
Model 1
(kN m/rad)
Model 2
(kN m/rad)
Test
(kN m/rad)
Model 1/
Test
Model 2/
Test
EPC-1
EPC-2
EPC-3
EPC-4
EPC-5
49 690
53 911
49 845
54 136
43 441
46 811
50 577
39 712
50 775
40 331
52 276
46 093
46 066
47 469
41 634
0.95
1.17
1.08
1.14
1.04
0.90
1.10
0.86
1.07
0.97
1.08
0.98
Average
steel design code; the yield strength of Q345 steel is f y =
345 MPa; the ultimate strength is f u = 470 MPa; the yield
strength of grade 10.9 bolt is f by = 940 MPa and its ultimate
tension strength is f bu = 1040 MPa; the elastic modulus of
steel is E = 2.06 × 105 N/mm2 and its shear modulus G =
79 × 103 MPa. Model 2 denotes that the analytical results are
calculated according to the material actual properties obtained
from coupon tests which are listed in Table 2. The steel’s
hardening modulus is assumed to be E h = 0.04E; the bolt’s
hardening modulus is assumed to be E bh = 0.1E b ; Poisson’s
ratio is taken as µ = 0.3; and the elastic modulus of bolt is
taken as E b = 2.06 × 105 N/mm2 .
The other connection parameters are taken as h bw =
276 mm, h cw = 276 mm, tcw = 8 mm, Ae = 244.8 mm2
(M20) or 352.5 mm2 (M24); lb = 48 mm (EPC-1), or 58 mm
(EPC-2), or 50 mm (EPC-3), or 60 mm (EPC-4), or 40 mm
(EPC-5); α = 10, d f = 288 mm, y1 = 200 mm, y2 = 88 mm,
ew = 49 mm, e f = 50 mm, b1 = 100 mm, b2 = 95 mm,
β = 0.856.
From the comparison of the results in Figs. 12–14 and
Table 5, it can be concluded that:
(1) The M–φ curves, the M–φs curves and the joint
initial rotational stiffnesses obtained from the analytical model
coincide well with the test results if the actual material
properties are applied. The comparison verifies the accuracy
of this analytical model. In practical structural design, where
the nominal material property specified in current code is
applied, the analytical models also give satisfactory solutions
with adequate accuracy.
(2) The M–φep curves obtained from the analytical results
coincide well with the tests results at the initial loading stage.
After nonlinearity occurs in the loading curves, there are some
discrepancies between the analytical results and the test results.
The possible reasons behind these discrepancies may
be explained as follows: The end-plate and column flange
deformation is evaluated under the assumption that the endplate segment is fixed at the bolt centre line. The endplate and column flange contact closely due to the bolt pretension force at the initial loading stage, and this assumption
is reasonable. With the joint moment increasing, the contact
force between the end-plate and column flange is significantly
reduced and even if they separate, this assumption will lead to
some discrepancies, but the discrepancies do not considerably
influence the evaluation of joint total rotation (Fig. 14). The
final results of the M–φ curve are in accordance with the tests
generally.
7. Conclusions
(1) In this paper, the stiffened and extended beam–column
end-plate connection has been recommended for end-plate
moment connections in multistory steel frames, and its
standards details have been proposed. Five full-scale joint tests
of this type of end-plate connection have been conducted to
investigate the influences of bolt size and end-plate thickness on
the joint behaviour. The rotational stiffness, moment resistance
and moment–rotation (M–φ) curves are obtained. A clear
definition for the end-plate connection rotation has been
proposed. With a special method to measure the bolt strain, the
distribution and the development of bolt tension force have been
obtained during the tests.
(2) A new analytical model to evaluate the moment–rotation
(M–φ) relationship of this type of end-plate connection has
been proposed. The end-plate connection is decomposed into
several components, including the panel zone, bolt, endplate and column flange. The complete loading process of
each component is analysed. The moment–rotation curve of
the whole connection is obtained by superimposing each
component. Comparing with the test results, it has been verified
that this analytical model can sufficiently predict the rotational
behaviour of end-plate connections, such as the initial rotational
stiffness and the moment–rotation (M–φ) curve. Furthermore,
the contributions to the joint rotational deformation of each
component, such as shear deformation of the panel zone,
the bolt extension, bending deformation of the end-plate and
column flange etc. are provided. This analytical model can also
provide the moment–shear rotation (M–φs ) and moment–gap
rotation (M–φep ) curves, thus provide a reliable foundation
for analysing the detailed rotational behaviour of end-plate
connections.
Acknowledgements
The writers gratefully acknowledge the support for this
work, which was funded by the Tsinghua Basic Research
Foundation (Grant No. JCqn2005006) and the Natural Science
Foundation of China (No. 50578083).
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