applied
sciences
Article
Piecewise Function Hysteretic Model for
Cold-Formed Steel Shear Walls with Reinforced
End Studs
Jihong Ye * and Xingxing Wang
School of Civil Engineering, Southeast University, Nanjing 210096, China; [email protected]
* Correspondence: [email protected]; Tel.: +86-258-379-5023
Academic Editor: Zhong Tao
Received: 9 November 2016; Accepted: 11 January 2017; Published: 19 January 2017
Abstract: Cold-formed steel (CFS) shear walls with concrete-filled rectangular steel tube (CFRST)
columns as end studs can upgrade the performance of mid-rise CFS structures, such as the vertical
bearing capacity, anti-overturning ability, shear strength, and fire resistance properties, thereby
enhancing the safety of structures. A theoretical hysteretic model is established according to a
previous experimental study. This model is described in a simple mathematical form and takes
nonlinearity, pinching, strength, and stiffness deterioration into consideration. It was established
in two steps: (1) a discrete coordinate method was proposed to determine the load-displacement
skeleton curve of the wall, by which governing deformations and their corresponding loads of the
hysteretic loops under different loading cases can be obtained; afterwards; (2) a piecewise function
was adopted to capture the hysteretic loop relative to each governing deformation, the hysteretic
model of the wall was further established, and additional criteria for the dominant parameters of
the model were stated. Finally, the hysteretic model was validated by experimental results from
other studies. The results show that elastic lateral stiffness Ke and shear capacity Fp are key factors
determining the load-displacement skeleton curve of the wall; hysteretic characteristics of the wall
with reinforced end studs can be fully reflected by piecewise function hysteretic model, moreover,
the model has intuitional expressions with clear physical interpretations for each parameter, paving
the way for predicting the nonlinear dynamic responses of mid-rise CFS structures.
Keywords: CFS structures; CFS shear wall with reinforced end studs; concrete-filled rectangular steel
tube columns; hysteretic behavior; hysteretic model
1. Introduction
As the main load-bearing components of cold-formed steel (CFS) structures, the definition of a CFS
shear wall’s hysteretic model is essential for nonlinear dynamic analysis of the structures. At present,
full-scale cyclic loading tests are the main way to investigate the shear performance of CFS shear wall,
and the definition of the walls’ hysteretic characteristics depends largely on test results [1–8], because
those walls have complex configurations and their load-displacement curves exhibit high nonlinearity
and pinching together with strength and stiffness deterioration. However, cyclic loading tests fall short
with high costs and long experimental periods, amongst other shortcomings; furthermore, the test
curves are too complicated to be directly applied in nonlinear analyses of the structures. Therefore, it is
necessary to establish a hysteretic model, which can not only be described with simple mathematical
expressions, but also reflect the hysteretic characteristics of the wall in order to predict the nonlinear
dynamic responses of CFS structures. So far, numerous numerical studies have been conducted on the
seismic performance of traditional CFS shear walls with coupled C section end studs [9–14].
Appl. Sci. 2017, 7, 94; doi:10.3390/app7010094
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Appl. Sci. 2017, 7, 94
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Currently available hysteretic models for the shear behavior of CFS shear wall include
Bouc-Wen-Baber-Noori (BWBN) model, evolutionary parameter hysteretic model (EPHM), pivot
model, and three-segment nonlinear pinching hysteretic model [5–9]. The BWBN model was proposed
by Foliente [15] and Nithyadharan and Kalyanaraman [16] based on a single-degree-of-freedom (SDOF)
mechanical system of a CFS shear wall; the model can fully reflect the wall’s hysteretic characteristics
although its mathematical expression which adopts the use of a differential equation cannot obtain
an analytical solution, and parameter identification requires a specific algorithm. EPHM, which
was proposed by Pang et al. [17] can reveal the effect of loading history on a wall’s lateral stiffness
although it lacks a good reflection of pinching, also, loading stiffness increases monotonically with the
displacement of the wall growing, which differs from reality. Huang et al. [18] and Zhou et al. [19]
presented the degraded four-line pivot model and three-segment nonlinear pinching hysteretic model,
respectively, both of which can describe the hysteretic characteristics of the wall more completely,
although their parameter identification processes depend on experimental results.
Traditional CFS shear walls with coupled C section end studs are apparently not applicable
to mid-rise CFS structures, for both shear strength and fire resistance of the wall cannot satisfy the
requirements of mid-rise residential structures [20]. Hence, Wang and Ye [21] proposed a CFS shear
wall with continuous concrete-filled rectangular steel tube (CFRST) columns as end studs (see Figure 1),
in which double-layer wallboards were arranged on both sides with a staggered configuration in order
to enhance the vertical bearing capacity, anti-overturning ability, shear strength, and fire resistance
of the wall, thereby allowing it to be applied to mid-rise CFS structures. The subjects of the above
hysteretic models focus on the traditional CFS shear walls and are thus confined to low-rise CFS
structures, narrowing their applications. Consequently, it is necessary to establish a hysteretic model
with a good reflection of hysteretic characteristics of walls according to the configuration of CFS
shear walls with reinforced end studs in order to facilitate the nonlinear dynamic analysis of mid-rise
CFS structures.
A piecewise function hysteretic model of CFS shear walls with reinforced end studs was proposed
according to previous experimental hysteretic laws. The very model can fully reflect the relevant
hysteretic characteristics (i.e., stiffness deterioration, strength deterioration, and pinching). Approaches
to the load-displacement skeleton curve, hysteretic model, and decisive parameters of the wall are
presented in detail. Finally, the proposed hysteretic model is validated by the experimental results of
Xu [22].
2. Construction and Shear Behavior of CFS Shear Wall with Reinforced End Studs
As shown in Figure 1, a typical CFS shear wall with reinforced end studs consists of a steel
frame, sheathings, and screw connections. The steel frame is framed with continuous CFRST columns,
C-section interior studs, and U-section tracks; the CFRST column is welded face-to-face along the
vertical direction to create a rectangular tube using double CFS lipped channel section studs that are
conveniently filled with fine aggregate concrete as soon as the wallboards have been fixed on the frame;
a sheathing combination of gypsum wallboards (GWB) and bolivian magnesium boards (BMB), which
exhibits good fire performance, is adopted with a staggered configuration; sheathings are attached to
the frame using self-drilling screws with a 4.8 mm diameter, and all fastener spacings are 150 mm. In
addition, hold downs are placed at the bottom and top of the wall to further strengthen the connection
between the CFRST columns and tracks.
An experimental investigation of full-scale CFS shear walls with reinforced end studs, as shown
in Figure 1, subjected to cyclic loading was conducted by Wang and Ye [21]. The experimental results
indicate that, for screw connections in the perimeter of the walls, the failure modes were either screws
being sheared off or screws being pulled through wallboards, both of which were the main reasons
for causing the walls’ failure in shear; whereas, no obvious deformation was observed in the screw
connections of interior studs. In addition, when the wall became damaged, there were a large number
of cross inclined cracks in the base-layer wallboards. It is worth mentioning that the experimental
Appl. Sci. 2017, 7, 94
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results of CFS shear walls sheathed with ribbed steel plates in shear that were demonstrated by
Li et al. [23] also show that the ribbed steel plates had obvious oblique shear buckling. It can be
concluded that "tension fields" that were similar to those of the steel plate shear wall were formed in
Appl. Sci. 2017, 7, x
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the wallboard when the wall with reinforced end studs was in shear.
Figure
hysteretic curve
curve of
of the
the wall
Figure 22 describes
describes aa typical
typical load-displacement
load-displacement hysteretic
wall with
with reinforced
reinforced end
end
studs
subjected
to
cyclic
loading.
It
is
observed
that
the
curve
of
the
wall
with
reinforced
end
studs
studs subjected to cyclic loading. It is observed that the curve of the wall with reinforced end studs
has
with that
that of
of aa traditional
has similar
similar pinching
pinching characteristic
characteristic with
traditional CFS
CFS shear
shear wall
wall with
with coupled
coupled C-section
C-section
end
Screw connections
connections play
play aa governing
governing role
role in
the shear
shear behavior
behavior of
of the
the wall;
wall; once
once maximum
maximum
end studs.
studs. Screw
in the
historical
thethe
screws
willwill
squeeze
andand
frictionate
the
historical displacement
displacementof
ofscrew
screwconnections
connectionsare
areexceeded,
exceeded,
screws
squeeze
frictionate
wallboard
in
the
vicinity
of
the
screw
holes,
resulting
in
stiffness
deterioration
of
the
wall;
looseness
the wallboard in the vicinity of the screw holes, resulting in stiffness deterioration of the wall;
of
those screw
holes during
the slipping
and pinching
characteristics
of the
looseness
of those
screw cyclic
holes loading
during generates
cyclic loading
generates
the slipping
and pinching
hysteresis
curve.
characteristics of the hysteresis curve.
C-section interior
stud
Double-layer wallboards
with a staggered configuration
3
1
Bottom
track
Hold downs being
used as the connections
between CFRST column
and tracks
Continuous
CFRST column
2
4
Screw
connections
Joist
ne per row
Floor
slab
Top track
ns per column
Figure 1.
1. Construction
reinforced end
end studs.
studs. CFRST,
Figure
Construction of
of a cold-formed
cold-formed steel
steel (CFS) shear wall with reinforced
CFRST,
concrete-filled rectangular
rectangular steel
steel tube.
tube.
concrete-filled
Load/kN
140
120
100
80
60
40
20
0
-20 -20 0
-40
-60
-80
-100
-120
-140
-80
-60
-40
20
40
60
80
Displacement/mm
Figure 2.
Copyright 2015, with
2. Typical load-displacement
load-displacement hysteretic
hysteretic curve.
curve. Reprinted from [21].
[21]. Copyright
permission from Elsevier.
Elsevier.
3. Hysteretic Model of CFS Shear Walls with Reinforced End Studs
Based on the hysteretic characteristics of the load-displacement curves discussed in the study by
Wang and Ye [21], a hysteretic model of the wall with reinforced end studs will be established in two
steps: (1) the load-displacement skeleton curve, which can describe both strength deterioration and
stiffness deterioration of the wall, will first be determined as the backbone curve of the hysteretic
model; (2) hysteretic loops which can reflect both slipping and pinching characteristics of the wall
will be captured in succession, and the hysteretic model will be further established.
Appl. Sci. 2017, 7, 94
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3. Hysteretic Model of CFS Shear Walls with Reinforced End Studs
Based on the hysteretic characteristics of the load-displacement curves discussed in the study
by Wang and Ye [21], a hysteretic model of the wall with reinforced end studs will be established in
two steps: (1) the load-displacement skeleton curve, which can describe both strength deterioration
Appl. Sci. 2017, 7, x
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and stiffness deterioration of the wall, will first be determined as the backbone curve of the hysteretic
model; (2) hysteretic loops which can reflect both slipping and pinching characteristics of the wall will
3.1. Modeling of Backbone Curve
be captured in succession, and the hysteretic model will be further established.
3.1.1. Degradation Analysis of the Lateral Stiffness of the Wall
3.1. Modeling of Backbone Curve
During horizontal loading, the lateral stiffness of the wall degraded gradually because of the
3.1.1.
Degradation
Analysis
of connections
the Lateral Stiffness
ofaggravated
the Wall with the growing load. In this study,
degree
of damage to
the screw
that were
lateral
stiffness
deterioration
was
by tracing
the wall
secant
stiffnessgradually
of the wall;
that is,
the
During
horizontal
loading,
thereflected
lateral stiffness
of the
degraded
because
of the
average
ratio
of
the
loads
in
two
opposite
directions
of
the
skeleton
curve
to
their
corresponding
degree of damage to the screw connections that were aggravated with the growing load. In this study,
displacements,
respectively, was
under
each load
level. Figure
3 depicts
theofdegradation
trends
secant
lateral
stiffness deterioration
reflected
by tracing
the secant
stiffness
the wall; that
is, theofaverage
stiffness
Ksloads
relative
to opposite
elastic lateral
stiffness
, in which
coincident
degradation trends
of each
ratio
of the
in two
directions
of theKeskeleton
curve
to their corresponding
displacements,
specimen
can
be
observed.
Therefore,
degradation
coefficient
γ
was
introduced,
which
is
the
ratioKofs
respectively, under each load level. Figure 3 depicts the degradation trends of secant stiffness
Ks to Ke,to
toelastic
describe
the developing
of thecoincident
lateral stiffness
of the wall.
The
through
relative
lateral
stiffness Ketrend
, in which
degradation
trends
of coefficient,
each specimen
can
fitting
the testTherefore,
results, is denoted
as coefficient γ was introduced, which is the ratio of Ks to Ke , to
be
observed.
degradation
describe the developing trend of the lateral stiffness of
the wall. The coefficient, through fitting the test
γ  Ks / Ke  0.623e 0.098β  0.732e0.535β
(1)
results, is denoted as
γ = Ks /Ke = 0.623e−0.098β + 0.732e−0.535β
(1)
 K
β  ∆  ∆ · Kee
ee = 0.4
Fpp
∆
0.4F
(2)
(2)
β=
where
to its
its corresponding
corresponding displacement
where KKee is the ratio of conventional
conventional elastic
elastic strength
strengthlimit
limitFFee = 0.4Fpp to
∆
β, which
which can
can be
be calculated
calculated by Equation (2), which is the ratio of lateral displacement ∆
Δee; β,
Δ of the wall
to ∆
which is
is the
the shear
shear capacity
capacity of
of the
the wall.
wall.
Δe; Fpp which
1.0
WA1[21]
WA2[21]
WA3[21]
WB1[21]
WB2[21]
WC1[21]
WC2[21]
WC3[21]
Equation(1)
0.8

0.6
0.4
0.2
0.0
0
5
10
15
20

25
30
35
40
stiffness.
Figure 3. Degradation trends of secant stiffness.
3.1.2. Discrete
Discrete Coordinate
Coordinate Method
Method
3.1.2.
As shown
shown in
in Figure
Figure 4,
4, the
the load-displacement
load-displacement skeleton
skeleton curve
curve was
was discretized
discretized into
into aa series
series of
of
As
coordinates
(a,
b,
c
…).
Coordinate
a
is
defined
as
the
elastic
point,
where
it
is
assumed
that
the
coordinates (a, b, c . . . ). Coordinate a is defined as the elastic point, where it is assumed that the
skeleton curve
curve isislinear
linearuntil
untilitithas
hasreached
reachedthe
theelastic
elasticdisplacement
displacement
e corresponding
coordinate
skeleton
∆eΔcorresponding
toto
coordinate
a,
a,
with
the
load
relative
to
the
remaining
coordinates
being
a
function
of
the
lateral
displacement
and
with the load relative to the remaining coordinates being a function of the lateral displacement and
secant stiffness
stiffnessofof
wall.
be observed
from Equations
(2)lateral
that displacement
given lateral
secant
thethe
wall.
It canItbecan
observed
from Equations
(1) and (2)(1)
thatand
given
displacement Δ, the secant stiffness corresponding to Δ can be obtained according to the elastic lateral
stiffness Ke and shear capacity Fp of the wall, the discrete coordinates can be further determined, and
that the load-displacement skeleton curve of the wall will be formed by connecting each discrete
coordinate. Hence, Ke and Fp are the basis for determining the load-displacement skeleton curve of
the wall.
Appl. Sci. 2017, 7, 94
5 of 16
∆, the secant stiffness corresponding to ∆ can be obtained according to the elastic lateral stiffness Ke
and shear capacity Fp of the wall, the discrete coordinates can be further determined, and that the
load-displacement skeleton curve of the wall will be formed by connecting each discrete coordinate.
Appl. Sci. K
2017,
7, xFp are the basis for determining the load-displacement skeleton curve of the wall.5 of 16
Hence,
e and
KSb
F
b(b,KSbb)
KSc
c (c,KScc)
Ksa=Ke
a(e,Ksae

O
method.
Figure 4. Discrete coordinate method.
3.1.3. Determination
Determination of
of Elastic
Elastic Lateral
Lateral Stiffness
Stiffness K
Kee and
Fpp
3.1.3.
and Shear
Shear Capacity
Capacity F
1. Determination
Determination
of elastic
lateral
stiffness
1.
of elastic
lateral
stiffness
Ke Ke
A simplified
calculating
thethe
elastic
lateral
stiffness
of theof
wall
with doubleA
simplifiedmethod
methodfor
for
calculating
elastic
lateral
stiffness
thesheathed
wall sheathed
with
layer
wallboards
on
both
sides
was
proposed
by
the
Ye
et
al.
[20]
based
on
the
equivalent-bracing
double-layer wallboards on both sides was proposed by the Ye et al. [20] based on the
model. The method,
whichThe
avoids
the high
cost
and time-consuming
of full-scale
shear
equivalent-bracing
model.
method,
which
avoids
the high cost andshortcomings
time-consuming
shortcomings
wall
tests, can
predict
lateral
stiffness
accurately
according
to sheathing
material,
sheathing
of
full-scale
shear
wallelastic
tests, can
predict
elastic
lateral stiffness
accurately
according
to sheathing
thickness,
and
the
shear
performance
of
screw
connections
only.
Consequently,
this
method
was
material, sheathing thickness, and the shear performance of screw connections only. Consequently,
followed
to
calculate
the
elastic
lateral
stiffness
of
the
wall
with
reinforced
end
studs.
As
shown
in
this method was followed to calculate the elastic lateral stiffness of the wall with reinforced end
FigureAs
5, the
simplified
model
is simplified
based on the
following
assumptions:
(1) theassumptions:
end studs and
studs.
shown
in Figure
5, the
model
is based
on the following
(1)top
thetrack
end
are
simplified
as
rigid
bars,
and
the
end
studs
are
hinged
to
the
track
on
the
top;
additionally,
the
studs and top track are simplified as rigid bars, and the end studs are hinged to the track on the top;
bases of the end
assumed
to be
ends in
of consideration
hold-downs; (2)
the coupling
additionally,
the studs
bases are
of the
end studs
arefixed
assumed
to consideration
be fixed ends in
of hold-downs;
of
the
wallboards
and
interior
studs
is
equivalent
to
diagonal
braces
considering
axial
force
only,axial
and
(2) the coupling of the wallboards and interior studs is equivalent to diagonal braces considering
the
equivalent
braces
are
assumed
to
be
the
primary
components
for
resisting
lateral
load;
(3)
the
force only, and the equivalent braces are assumed to be the primary components for resisting lateral
axial stiffness
of the
bracesofare
load;
(3) the axial
stiffness
thecumulative.
braces are cumulative.
The
elastic
lateral
stiffness
of
the wall
wall with
with reinforced
reinforced end
end studs
studs can
can be
be deduced
deduced on
on the
the basis
basis of
of
The elastic lateral stiffness of the
the
above
assumptions,
elastic
theories,
structural
mechanics
theories,
and
force-balance
principles,
the above assumptions, elastic theories, structural mechanics theories, and force-balance principles, as
as expressed
in Equation
detailed
derivation
is described
in the
study
Yeal.
et [20].
al. [20].
expressed
in Equation
(3).(3).
TheThe
detailed
derivation
is described
in the
study
by by
Ye et
2
2 E A
2 L22 
bi bi
2L ∑ Ebi Abi
P 6i
Ke P  6icc2  2ii=11 2 1.5
Ke = = H2 + ( L  H ) 1.5
∆
H
( L2 + H 2 )
1.5
11
H22 +
 LL22 ))1.5
((H
E
A


i bi =
·
EbibA
bi
2deie i 22
22 HL
HL
((H
 f2d
H + HL
HL))
Gi ti +
Gi ti fii nnsese
H+L
Sa =
2ns H
+
neL− 5
Sa 
2nsH ne  5
nse =
+1
Sa
H
nse 
1
Sa
(3)
(3)
(4)
(4)
(5)
(5)
(6)
(6)
where, Ke is the elastic lateral stiffness of the wall; P is lateral load; Δ is lateral displacement of the
wall; ic is line rigidity of end studs; H is wall height; L is wall length; EbiAbi is axial stiffness of the
brace, which is simplified from the i-layer wallboards on one side of the wall and can be determined
by Equation (4); Gi and ti are shear modulus and thickness of the i-layer wallboard, respectively; fi
Appl. Sci. 2017, 7, 94
6 of 16
where, Ke is the elastic lateral stiffness of the wall; P is lateral load; ∆ is lateral displacement of the
wall; ic is line rigidity of end studs; H is wall height; L is wall length; Ebi Abi is axial stiffness of the
brace, which is simplified from the i-layer wallboards on one side of the wall and can be determined
by Equation (4); Gi and ti are shear modulus and thickness of the i-layer wallboard, respectively; fi
and Sci.
dei 2017,
are shear
Appl.
7, x strength and elastic deformation of the screw connection corresponding to the i-layer
6 of 16
wallboard, respectively; during the calculation of Ebi Abi , due to the arrangement of double-row screws
for the end
stud
and
single-line
screws for
the track,
as track,
shownasinshown
Figure in
1, Figure
in which
ne are
screw
screws
for the
end
stud
and single-line
screws
for the
1, nins and
which
ns and
ne
numbers
the vertical
andvertical
horizontal
on one
side of
average space
are
screwalong
numbers
along the
andedges
horizontal
edges
onthe
onewall,
siderespectively,
of the wall, an
respectively,
an
Sa of those
screw
in theconnections
perimeter ofinthe
was calculated
Equation
(5), and
average
space
Sa connections
of those screw
thewall
perimeter
of the first
wallby
was
calculated
firstthen
by
Equation
(5), and
then an
number
se of those
screws
on either
or was
rightdetermined
side of the
an equivalent
number
nseequivalent
of those screws
on neither
the left
or right
side ofthe
theleft
wall
wall
was determined
to Equation (6).
according
to Equationaccording
(6).
2.
Determination
shear
capacity
2. Determination
of of
shear
capacity
FpFp
Numerous
experimentalresults
resultsshow
showthat
that
shear
capacity
of CFS
shear
depends
onshear
the
Numerous experimental
shear
capacity
of CFS
shear
wallswalls
depends
on the
shear
strength
the screw
connections
in the perimeter
of theThe
wall.
The model,
elastic model,
is
strength
of the of
screw
connections
in the perimeter
of the wall.
elastic
which iswhich
usually
usually
to determine
shear capacity
of CFS
shear
is notfor
suitable
wall with
used to used
determine
the shearthe
capacity
of CFS shear
wall
[24],wall
is not[24],
suitable
a wall for
witha reinforced
reinforced
end
It can in
be that
explained
in model
that the
elastic
model
load thaton
screw
end studs. It
canstuds.
be explained
the elastic
takes
the load
thattakes
screwthe
connections
four
connections
onwall
fourexperience
corners of as
thethey
wallreach
experience
as they shear
reach strength
their ultimate
as the
corners of the
their ultimate
as the shear
wall’sstrength
shear capacity,
wall’s
shear
capacity,
the rest of the
connections
onreach
the end
studs
do not
reach their
ultimate
the rest
of the
screw connections
on screw
the end
studs do not
their
ultimate
strength
because
their
strength
because
their horizontal
force components
are defined
by lineartoconversion
according
to the
horizontal
force components
are defined
by linear conversion
according
the four screw
connections’
four
connections’
shearmode
strength,
the on
failure
mode
of being
all screws
on off
thethereby
end studs
being
shearscrew
strength,
and the failure
of alland
screws
the end
studs
sheared
cannot
be
sheared
thereby
cannot
be reflected
during
cycliccalculated
loading, resulting
lower
results.
It
reflectedoff
during
cyclic
loading,
resulting
in lower
results. Itinalso
cancalculated
be concluded
from
also
can
be concluded
from results
Table 1based
that the
on the elastic
modellower
are generally
Table
1 that
the calculated
oncalculated
the elasticresults
modelbased
are generally
20%–35%
than the
20%–35%
lowertest
than
the corresponding test results.
corresponding
results.


H ic
E bA b
P
B
B'
D
D'

E bA b
'
Lb
L
(a)
A
C
(b)
Figure 5. Equivalent-bracing model. (a) Simplification model; (b) calculation model. Reprinted from
Figure 5. Equivalent-bracing model. (a) Simplification model; (b) calculation model. Reprinted
[20]. Copyright 2015, with permission from Elsevier.
from [20]. Copyright 2015, with permission from Elsevier.
Table 1. Comparison of the shear capacity Fp of the wall with reinforced end studs.
Table 1. Comparison of the shear capacity Fp of the wall with reinforced end studs.
Specimen Number [21]
Specimen Number
WA1 [21]
WA2
WA1
WA3
WA2
WA3
WB1
WB1
WB2
WB2
WC1
WC1
WC2
WC2
WC3
WC3
Test Results/kN
Test Results/kN
100.0
101.5
100.0
109.0
101.5
109.0
126.5
126.5
60.3
60.3
109.8
109.8
95.4
95.4
67.0
67.0
Calculated Results Based on Elastic Model/kN
Calculated Results Based
79.7 on Elastic Model/kN
81.4
79.7
81.4
81.4
81.4
101.5
101.5
45.3
45.3
70.6
70.6
70.6
70.6
45.3
45.3
Relative Errors
Relative
−20.3%Errors
−19.8%
−20.3%
−25.3%
−19.8%
−25.3%
−19.8%
−19.8%
−24.9%
−24.9%
−35.7%
−35.7%
−26.0%
−26.0%
−32.4%
−32.4%
Compared to the strip model [25] for steel plate shear walls, it can be concluded that screw
Compared to the strip model [25] for steel plate shear walls, it can be concluded that screw
connections in the perimeter of CFS shear walls can be taken as the end fields of the equivalent strips.
connections in the perimeter of CFS shear walls can be taken as the end fields of the equivalent
Consequently, in considering the strip model, a simplified calculation model for the wall with
strips. Consequently, in considering the strip model, a simplified calculation model for the wall
reinforced end studs in shear, as shown in Figure 6, was established and based on following
assumptions: (1) a wall’s shear capacity depends on the shear strength of the screw connections in
the perimeter of the wall, while the screw connections in the interior studs only act as stiffeners of
wallboards; (2) wallboards are divided into a series of isoclinic strips along wall length with an
average space of Sa (see Equation(5)), and these strips fail only in the end fields; (3) when the screw
connections in the perimeter of the walls reach their ultimate shear strength, screw holes become
Appl. Sci. 2017, 7, 94
7 of 16
with reinforced end studs in shear, as shown in Figure 6, was established and based on following
assumptions: (1) a wall’s shear capacity depends on the shear strength of the screw connections in
the perimeter of the wall, while the screw connections in the interior studs only act as stiffeners of
wallboards; (2) wallboards are divided into a series of isoclinic strips along wall length with an average
space of Sa (see Equation(5)), and these strips fail only in the end fields; (3) when the screw connections
in the perimeter of the walls reach their ultimate shear strength, screw holes become loose, and the
Appl. Sci. 2017, 7, x
7 of 16
strips can thereby become pinned at both ends due to the free rotation of the screws; (4) the strips can
transform tension only, whose direction is the same as the wall’s diagonal direction.
The simplified calculation model, as shown in Figure 6, indicates that the shear capacity of the
The simplified calculation model, as shown in Figure 6, indicates that the shear capacity of
wall with reinforced end studs is equal to the cumulative horizontal force components of the screw
the wall with reinforced end studs is equal to the cumulative horizontal force components of the
connections in the perimeter of the wall. As a result of the offset horizontal forces that become
screw connections in the perimeter of the wall. As a result of the offset horizontal forces that become
separated from the screw connections on the left and right ends, only horizontal force components of
separated from the screw connections on the left and right ends, only horizontal force components
the screw connections at either the top or bottom edges of the wall will be considered in the shear
of the screw connections at either the top or bottom edges of the wall will be considered in the shear
capacity calculation, which is denoted as:
capacity calculation, which is denoted as:
nee
nee
nee f i  nee f i cosθ
Fp  
 ei
ex
i
(7)
(7)
Fp = i∑
1 f ex = i∑
1 f e cos θ
i =1
i =1
where nee is equivalent to the number of screws either on the top or at the bottom of the wall; f ei is
where
neestrength
is equivalent
number
of screws
on theoftop
at with
the bottom
of the
f ei
the
shear
of the to
iththe
screw
connection
in theeither
perimeter
the or
wall
reinforced
endwall;
studs,
is the shear
of the ith
screw
connection
the perimeter
of theoffwall
with pulled
reinforced
end
whose
failurestrength
modes mainly
exhibit
in the
form of ainscrew
being sheared
or being
through
studs, whose
modes
mainly
into
the
form
of athickness
screw being
sheared
off or
being pulled
wallboard;
thefailure
two failure
modes
areexhibit
relative
both
steel
ts and
sheathing
strength,
when
through
wallboard;
the
two
failure
modes
are
relative
to
both
steel
thickness
t
and
sheathing
strength,
s
ts is larger or the sheathing strength is higher, the steel frame can effectively restrain screw tilt,
when ts isinlarger
or being
the sheathing
the steel
frame
effectively
restrain
screw tilt,
resulting
screws
sheared strength
off, and is
thehigher,
shear-off
strength
of can
the screw
is thereby
assigned
to
resulting
in
screws
being
sheared
off,
and
the
shear-off
strength
of
the
screw
is
thereby
assigned
to
f ei ,
i
f e , whereas when ts is relatively small or the sheathing strength is lower, screws will be pulled
whereas when ts is relatively small or the sheathing strength is lower, screws will be pulled through
through the wallboard
due to their tilt and will thus further sink into the wallboard, in this case, the
the wallboard due to their tilt and will thus further sink into the wallboard, in this case, the shear
shear strength accumulation of the connections with screws being pulled through corresponding to
strength accumulation of the connections with screws being pulled through corresponding
to each
each layer of wallboard is assignedi to i f ei ; f exi is the horizontal force component
of f ei ; cosθ is angle
layer of wallboard is assigned to f e ; f ex is the horizontal force component of f ei ; cosθ is angle cosine of
cosine of the strip.
the strip.
nee 
L
1
Sa
Sa
FP
f
i
ex
f
f eyi f
i
e
f eyi
i
e
f exi
H
Sa
L

f exj
H
fe f
j
j
ey
j
f eyj f e
f exj
N
N
L
Figure 6. Simplified calculation model for the wall with reinforced end studs in shear.
Figure 6. Simplified calculation model for the wall with reinforced end studs in shear.
3.2. Hysteretic Model of Load-Displacement Curve
3.2.
Hysteretic Model of Load-Displacement Curve
3.2.1. Model Proposition
3.2.1. Model Proposition
According to previous test results, typical hysteretic loops of a wall with reinforced end studs
According to previous test results, typical hysteretic loops of a wall with reinforced end studs
were extracted, as shown in Figure 7, where δ is the ratio of lateral displacement ∆ to wall height H; f is
were extracted, as shown in Figure 7, where δ is the ratio of lateral displacement Δ to wall height H;
f is the ratio of lateral load F to shear capacity Fp. It can be observed that a typical hysteretic loop
consists of an unloading segment, a slipping segment, and a loading segment; the ascending and
descending branches of the curve are essentially symmetrical; particularly, the strength deterioration
in successive cycles of the same amplitude is negligible.
Appl. Sci. 2017, 7, 94
8 of 16
the ratio of lateral load F to shear capacity Fp . It can be observed that a typical hysteretic loop consists
of an unloading segment, a slipping segment, and a loading segment; the ascending and descending
Appl.
Sci. 2017,
x curve are essentially symmetrical; particularly, the strength deterioration in successive
8 of 16
branches
of 7,
the
Appl.
Sci.
2017,
7,
x
8 of 16
cycles of the same amplitude is negligible.
1.0
1.0
0.5
0.5
f f
0.0
0.0
-0.5
-0.5
Ascending branch
Ascending branch Loading segment
Slipping segment
LoadingUnloading
segment segment
Slipping segment
Unloading
segment
Unloading segment
Slipping
segment
Unloading segment
Slipping segment
Loading segment
Descending branch
Loading segment
Descending branch
-1.0
-1.0
-0.02
-0.02
-0.01
-0.01
0.00
0.00
0.01
0.01
δ

0.02
0.02
0.03
0.03
Figure 7. Typical hysteretic loops.
Figure
Figure 7.
7. Typical
Typical hysteretic
hysteretic loops.
loops.
Hysteretic loops for different governing deformations can be captured based on the above
Hystereticand
loops
different
governing
canfurther
be captured
basedasondepicted
the above
above
the
characteristics,
thefor
f-δdifferent
hystereticgoverning
model of deformations
the wall can be
established,
in
hysteretic model
of thedeterioration
be further
further
established,
depicted
characteristics,
theof
ff-δ
-δsimplification,
wall can be
established,
in
Figure
8. As a and
means
strength
in successive
cyclesasofdepicted
the same
Figure8.8.As
Asanot
a considered,
means
of simplification,
strength
in successive
cycles
of amplitude
theofsame
Figure
means
of simplification,
strength
deterioration
in successive
cycles
the same
amplitude
is
each hysteresis
loop
hasdeterioration
its unique
slipping
axis. Inofconsideration
the
amplitude
is not
considered,
each
hysteresis
hasslipping
its for
unique
slipping
axis.
In consideration
of the
is
not considered,
each hysteresis
has itsloop
unique
axis.
In consideration
of the
symmetry
of
hysteretic
loops,
a loop
mathematical
model
the
ascending
branch
thesymmetry
curve
is
symmetry
ofloops,
hysteretic
loops,
a mathematical
model
for the
ascending
thebycurve
is
of
hysteretic
a mathematical
model
the
ascending
of the
curve
isofproposed
first
proposed
first
(see
Figure
8b), and
then
the for
descending
part
ofbranch
the
curve
canbranch
be obtained
origin
proposed
first
(see
Figure
8b),
and
then
the
descending
part
of
the
curve
can
be
obtained
by
origin
(see
Figure
8b),
and
then
the
descending
part
of
the
curve
can
be
obtained
by
origin
symmetry.
symmetry.
symmetry.
B
滑 滑 滑
C
F
f
D
f
E
dnP1 kb
d
fmu
C(0,f0)
kc
dn+
滑 滑 滑
d
fml
ka
P2
滑 滑 滑
A
B
D
滑 滑 滑 I
滑 滑 滑 II
A(dn-,fn-)
(a)
(b)
(a)
(b)
Figure 8. Hysteretic model of a wall with reinforced end studs. (a) Hysteretic laws; (b) calculation
Figure
Figure 8.
8. Hysteretic
Hysteretic model
model of
of aa wall
wall with
with reinforced
reinforced end
end studs.
studs. (a)
(a) Hysteretic
Hysteretic laws;
laws; (b)
(b) calculation
calculation
model for the ascending branch of the curve. Note: The symbols’ meanings are identical to those in
model
for
the
ascending
branch
of
the
curve.
Note:
The
symbols’
meanings
are
identical
model for the ascending branch of the curve. Note: The symbols’ meanings are identical to
to those
those in
in
Equations (8)–(16).
Equations
(8)–(16).
Equations (8)–(16).
3.2.2. Modeling of Hysteretic Loop
3.2.2.
3.2.2. Modeling of Hysteretic
Hysteretic Loop
Loop
It is difficult to describe the f-δ curve with turning points as a single equation with clear physical
It
to
the ff-δ
-δ curve with turning points as a single equation with clear physical
It is difficult
difficult
to describe
describe
interpretations.
Therefore,
f = the
0 is taken
as the first demarcation point to divide the ascending branch
interpretations.
Therefore,
ff = 0 is taken as the first demarcation point to divide the ascending branch
interpretations.
Therefore,
of the curve into unloading segment AB and loading segment BCD. Furthermore, δ = 0 is takenbranch
as the
of the curve into
into unloading
unloadingsegment
segmentAB
ABand
andloading
loadingsegment
segment BCD.
Furthermore,
0taken
is taken
as
δ =δ0=Iisand
as the
second demarcation
point to divide the
loading
segment BCDBCD.
intoFurthermore,
loading segment
loading
the
second
demarcation point
divide
loadingsegment
segmentBCD
BCD intoloading
loadingsegment
segment II and loading
second
demarcation
to to
divide
thethe
loading
segment
II (see Figurepoint
8b). Consequently,
the ascending
branch ofinto
the typical hysteretic
loop consists
segment
II
(see
Figure
8b).
Consequently,
the
ascending
branch
of
the
typical
hysteretic
loop
consists
segment
Consequently,
the
ascending
branch
of
hysteretic
loop loading
consists
of three segments with monotonic stiffness development, namely, the unloading segment,
of three
segments
with
monotonic
stiffness
development,
namely,
the
unloading
segment,
loading
three
segments
with
monotonic
stiffness
development,
namely,
the
unloading
segment,
loading
segment I, and loading segment II, where loading segment I (i.e., the slipping segment) is described
segment I, and loading
segment II, where
loading
segment
I (i.e.,
the
slipping
segment)
is
described
loading
where
loading
segment
(i.e.,
the
slipping
segment)
is
described
linearly, because its stiffness can be seen as a constant, and the unloading segment and loading
linearly, IIbecause
its stiffness
be seen as aexpressions
constant, and
segment
and loading
segment
are described
usingcan
Richard-Abbott
[26].the
In unloading
summary, the
ascending
branch
segment
II
are
described
using
Richard-Abbott
expressions
[26].
In
summary,
the
ascending
branch
of a typical hysteretic loop can be captured using the following piecewise function:
of a typical hysteretic loop can be captured using the following piecewise function:
Appl. Sci. 2017, 7, 94
9 of 16
linearly, because its stiffness can be seen as a constant, and the unloading segment and loading segment
II are described using Richard-Abbott expressions [26]. In summary, the ascending branch of a typical
hysteretic loop can be captured using the following piecewise function:
Unloading segment : f = h
(ka − kb )(δ − δn )
i 1 + k b ( δ − δn ) , f ≤ 0
(ka −kb )(δ−δn ) nu nu
1+
f mu
Loading segment I : f = kb δ + f 0 , 0< f ≤ f 0
Loading segment II : f = f 0 + h
( k b − k c )δ
i 1 + kc δ, f > f 0
( k b − k c )δ nl nl
1+ f
ml
(8)
(9)
(10)
where ka is the initial unloading stiffness at point A; kb is the asymptote slope of the unloading segment
in the vicinity of point C; kc is the asymptote slope of the loading segment at point D; δn (including δn+
and δn− in opposite directions) is the governing deformation of the unloading segment at its initial
point; f mu is the reference load of the unloading segment, which is the difference between f n− and
the load of point P1 , f ml is the reference load of loading segment, which is the difference between the
pinching load f 0 at zero displacement point and the load of point P2 , points P1 and P2 locate on the
asymptotes of kb and kc , respectively; nu and nl are the shape parameters of the unloading segment
and loading segment II, respectively.
Equations (8)–(10) show that the dominant parameters of the ascending branch of a typical
hysteretic loop include ka , kb , kc , f mu , f ml , nu , nl , and f 0 .
3.2.3. Criteria for the Dominate Parameters
1 Stiffness parameters ka , kb and kc
ka and kc are the stiffness of the f -δ curve in two opposite directions at each governing deformation
(see Figure 8b), respectively, whose values under different load levels can be determined according to
the test hysteretic loops measurement. The formulas of ka and kc can be obtained through the statistical
analysis of the experimental results in the study by Wang and Ye [21], given by:
ka = Ra δn + ka0
(11)
kc = Rc1 e Rc2 δn + Rc3 e Rc4 δn
(12)
where Ra is the gradient of ka with governing deformation δn ; ka0 is the value of ka at δn = 0; Rc1 and
Rc3 are the controlling coefficients of kc ; Rc2 and Rc4 are the index gradients of kc with δn .
It can be observed form Figure 8b that kb approximates the slipping segment slope of the hysteresis
loop. Therefore, the development process of those hysteretic loops obtained from previous experiments
is listed in Figure 9 in order to investigate the developing rule of the slipping segment slope. It is
indicated that during the initiation of the loading process, the wall is in elasticity, the curve is linear, and
the slipping segment slope is equal to the elastic lateral stiffness of the wall; when the load increased,
screw connections fail progressively, pinching appears in the curve because of the opening and closing
of the screw holes, and the slipping segment slope decreases continuously; after an even larger amount
of wall displacement, an aggravated extrusion deformation of screw connections appears, the loose
screw holes cause the appearance of some no-load slipping of the screw connections, which leads
the slipping axis to have a tendency towards a horizontal configuration. It can be concluded that the
developing rule of the slipping segment slope is governed by a process whereby the elastic lateral
stiffness of the wall decreases to a minimum value.
Appl. Sci. 2017, 7, 94
10 of 16
10 of 16
The shape of hysteretic loops
Appl. Sci. 2017, 7, x
Ke
Slipping axis
Slipping axis
Load level
Figure
9. Developmentprocess
process of
of the hysteretic
ofof
thethe
wall.
Figure
9. Development
hystereticloops
loops
wall.
Based on the assumption of a linear relationship between the slipping segment slope and load
Based
the assumption of a linear relationship between the slipping segment slope and load
level, kon
b can be determined:
level, kb can be determined:
 χ)
(13) (13)
b = ((1
kb k=
1−
χ)kk11
k1 k== ((1
1
−λ)
λ)kk0
1
0
(14) (14)
wherewhere
k1 is kthe
difference of the slipping segment slopes between elastic and destruction phrases;
1 is the difference of the slipping segment slopes between elastic and destruction phrases; k0
k0 is the
slipping
slopeininelastic
elastic
phrase,
which
is assigned
with
thestiffness
elastic of
stiffness
is the slippingsegment
segment slope
phrase,
which
is assigned
with the
elastic
f-δ curveof f -δ
curve(i.e.,
(i.e.,the
theratio
ratioofoffe f to
to
δ
that
corresponds
to
the
elastic
strength
limit
F
,
see
Figure
λ is the
corresponds to the elastic strength limit Fe, seee Figure 8b); λ8b);
is the
e δe that
e
coefficient
relative
to the
slipping
segmentslope
slope in
in the destruction
which
approximates
to to
coefficient
relative
to the
slipping
segment
destructionphrase,
phrase,
which
approximates
0.1, according
to statistical
the statistical
analysis
experimentalresults;
results; χ
in
which
C
i =C(δ=
n/δ
min /δ
−
1),
C
0.1, according
to the
analysis
ofof
experimental
χ ==CCi/C,
/C,
in
which
(δ
n
min − 1),
i
i
=
(δ
nmax/δmin − 1), and δnmax and δmin are the maximum governing deformation and the minimum load
C = (δnmax /δmin − 1), and δnmax and δmin are the maximum governing deformation and the minimum
level difference, respectively.
load level
difference, respectively.
2
Pinching load f0
2 Pinching load f 0
As shown in Figure 2, for each hysteretic loop, the reverse loading paths approximately direct
As
shownpoints,
in Figure
2, for each
hysteretic
loop,
the8a.
reverse
approximately
direct
to certain
like pinching
points
C and F in
Figure
Screwsloading
are in thepaths
no-load
slipping phrase
to certain
points,
like
pinching
points
C
and
F
in
Figure
8a.
Screws
are
in
the
no-load
slipping
before those pinching points are reached, at that moment, the wallboard has the least restraint on the
phrase
before
those
pinching
points
are reached,
at thatcan
moment,
haspinching
the least load
restraint
steel
frame,
and
the shear
capacity
of the frame
therebythe
be wallboard
taken as the
(approximately).
Pinching
load capacity
f0 is a direct
the thereby
pinching be
characteristics
of pinching
the curve. load
on the
steel frame, and
the shear
of reflection
the frameofcan
taken as the
Based on a statistical
analysis,
constant
0.11 is assigned
to f0 due tocharacteristics
the pinching load
each
(approximately).
Pinching
load af 0reliable
is a direct
reflection
of the pinching
ofofthe
curve.
specimen
fluctuating
around
that
value,
which
is
consistent
with
the
shear
capacity
of
the
steel
frames
Based on a statistical analysis, a reliable constant 0.11 is assigned to f 0 due to the pinching load of each
that were tested by the authors.
specimen
fluctuating around that value, which is consistent with the shear capacity of the steel frames
3
Reference
fmu and fml
that were
tested byloads
the authors.
Asloads
shown
in and
Figure
3 Reference
f mu
f ml8b, given kb, kc, and f0, the coordinates of points P1 and P2 for different
governing deformations can be calculated, and the reference load fmu of unloading segment can be
As
shown in
8b, given
kb ,ofkcpoint
, andPf10from
, the the
coordinates
P1 and P2 for
different
determined
byFigure
subtracting
the load
initial loadoffn−points
that corresponding
to each
governing
deformations
can
be
calculated,
and
the
reference
load
f
of
unloading
segment
can be
governing deformation; while the reference load fml of loading segment
II can be determined by
mu
subtracting
the pinching the
loadload
f0 from
load
P2 that
corresponds
tofeach
governing
deformation.
determined
by subtracting
of the
point
P1offrom
the
initial load
corresponding
to each
n− that
governing
deformation;
while
the
reference
load
f
of
loading
segment
II
can
be
determined
by
ml
4
Shape parameters nu and nl
subtracting the pinching load f 0 from the load of P2 that corresponds to each governing deformation.
Through extracting nu and nl from the hysteretic curves under different load levels in the study
4 Shape
parameters
nl be concluded that nu and nl are approximately linear relative to governing
by Wang
and Ye n[21],
it can
u and
deformations. The regression formulas are thereby given by:
Through extracting nu and nl from the hysteretic curves under different load levels in the study
 nu 0
(15)
nu δnn
by Wang and Ye [21], it can be concluded thatnun=u Rand
l are approximately linear relative to governing
deformations. The regression formulas are thereby given by:
nu = Rnu δn + nu0
(15)
Appl. Sci. 2017, 7, 94
Appl. Sci. 2017, 7, x
11 of 16
11 of 16
nln== R
Rnlδδn+nnl0
l
nl
n
(16)
(16)
l0
where R
Rnu
and Rnlare
arethe
thegradients
gradientsof
ofnnuu(i.e.,
(i.e.,the
theshape
shapeparameter
parameterof
ofunloading
unloadingsegment)
segment) and
and nnll (i.e.,
(i.e.,
where
nu and Rnl
the
shape
parameter
of
loading
segment
II)
with
governing
deformation
δ
,
respectively;
n
and
n l0
the shape parameter of loading segment II) with governing deformation δnn, respectively; nu0
u0 and nl0
are the
the values
values of
of n
nuu and
and nnllat
atδδnn==0,0,respectively.
respectively.
are
3.3. Model Verification
3.3. Model Verification
In order to verify the proposed hysteretic model, comparative analysis on the experimental results
In order to verify the proposed hysteretic model, comparative analysis on the experimental
of the CFS shear walls described by Xu [22] was carried out, where the end stud sections of specimens
results of the CFS shear walls described by Xu [22] was carried out, where the end stud sections of
include square-89 × 100 × 0.9 and square-140 × 140 × 1.5 (hereinafter referred to as 89-type and
specimens include square-89 × 100 × 0.9 and square-140 × 140 × 1.5 (hereinafter referred to as 89-type
140-type walls, respectively); all specimens were sheathed with gypsum wallboards combined with
and 140-type walls, respectively); all specimens were sheathed with gypsum wallboards
combined
bolivian magnesium boards on both sides, whose shear modulus are 484 and 124 N/mm2 , respectively;
with bolivian magnesium boards on both sides, whose shear modulus are 484 and 124 N/mm2,
the shear performance of screw connections was determined according to the test results of Ye et al. [27],
respectively; the shear performance of screw connections was determined according to the test results
as detailed listed in Table 2.
of Ye et al. [27], as detailed listed in Table 2.
Table 2. The shear performance of screw connections [27] 1 .
Table 2. The shear performance of screw connections [27] 1.
12 mm Bolivian
12 mm Bolivian Magnesium
Magnesium Board
f e /kN
Board
fe/kN
de /mm
f
/kN
de/mm
f/kN
0.9
0.70 0.70
0.660.66
0.63 0.63
0.74
1.40 1.40
0.9
0.74
Screwbeing
beingpulled
pulledthrough
Screw
1.2
0.62 0.62
0.730.73
0.50 0.50
0.70
1.43 1.43
through
1.2
0.70
Screw being sheared off
1.78
Screw being sheared off
1.78
1 f and d are the shear strength and elastic deformation of a screw connection sheathed with a single-layer
1 f and de eare the shear strength and elastic deformation of a screw connection sheathed with a singlewallboard, respectively; f e is the shear strength of a screw connection in the perimeter of the wall with reinforced
layer
wallboard, respectively; fe is the shear strength of a screw connection in the perimeter of the wall
end studs.
Steel
12 mm Gypsum
12 mm Gypsum
Wallboard
Steel
Thickness
Thickness
Wallboard
(mm)
(mm)
de /mm
f /kNf/kN
de/mm
Failure
Modes
Failure
Modes
with reinforced end studs.
Firstly, the
and shear
of each
each specimen
specimen was
was determined
determined
Firstly,
the elastic
elastic lateral
lateral stiffness
stiffness K
Kee and
shear capacity
capacity F
Fpp of
according to
to the
the simplified
simplified method
method that
that was
was described
described in
in Section
Section 3.1.3;
3.1.3; on
on this
this basis,
basis, the
the discrete
discrete
according
coordinate method
method was
was used
used to
to establish
establish the
the load-displacement
load-displacement skeleton
skeleton curves
curvesof
of those
thosespecimens.
specimens.
coordinate
Limited to
to space,
space,only
onlythe
thecalculated
calculatedcurves
curvesofoftypical
typical89-type
89-type
and
140-type
walls
were
depicted
Limited
and
140-type
walls
were
depicted
in
in Figure
It can
be observed
the calculated
results,
which
agree
the results,
test results,
Figure
10. 10.
It can
be observed
thatthat
the calculated
results,
which
agree
well well
withwith
the test
can
can reflect
shear
behavior
of the
wall
with
reinforcedend
endstuds
studslike
likenonlinearity,
nonlinearity, stiffness,
stiffness, and
and
reflect
the the
shear
behavior
of the
wall
with
reinforced
strength
deterioration.
It
also
can
be
concluded
from
Table
3
that
the
calculated
and
test
results
are
of
strength deterioration. It also can be concluded from Table 3 that the calculated and test results are
high
consistency
in both
elastic
stiffness
Ke and
shear
capacity
Fp , with
errors
being
within
11%11%
and
of
high
consistency
in both
elastic
stiffness
Ke and
shear
capacity
Fp, with
errors
being
within
15%,
respectively.
and 15%, respectively.
F/kN
F/kN
50
40
30
20
-80
-60
-40
B
10
-20
/mm
0
-10
0
20
40
60
80
A
-20
Test result [22]
Calculated result
-30
-40
-50
(a)
-80
-60
-40
120
100
80
60
40
20
0
-20 -20 0
-40
-60
-80
-100
-120
w/mm
20
40
60
80
Test result [22]
Calculated result
(b)
Figure 10. Comparisons of the load-displacement skeleton curves of typical specimens. (a) 89-type
Figure 10. Comparisons of the load-displacement skeleton curves of typical specimens. (a) 89-type
wall
W140-1).
wall (specimen
(specimen W89-2);
W89-2); (b)
(b) 140-type
140-type wall
wall (specimen
(specimen W140-1).
Appl. Sci. 2017, 7, 94
12 of 16
Table 3. Comparisons between the calculated and test results 1 .
Specimen
Number
[22]
Wall Size/m
(L × H)
W89-1
W89-2
W89-3
W140-1
W140-2
3.6 × 3.0
1.2 × 3.0
3.6 × 3.0
3.6 × 3.0
3.6 × 3.0
Elastic Lateral Stiffness Ke
(N/mm)
Total Energy Dissipation
E/KJ
Shear Capacity F p /kN
Ket
Kec
κ1
F pt
F pc
κ2
Et
Ec
κ3
9670
3488
9192
12085
12227
10177
3400
10177
11573
11573
0.05
−0.02
0.11
−0.04
−0.05
110.2
44.4
85.4
122.6
117.2
98.0
40.6
98.0
120.0
120.0
−0.11
−0.08
0.15
−0.02
0.02
25.7
10.0
24.5
32.0
22.6
20.6
13.7
19.2
32.9
25.2
0.25
0.27
0.28
0.03
0.10
1
Ket and Kec are the test and calculated results of elastic lateral stiffness, respectively; Fpt and Fpc are the test
and calculated results of shear capacity, respectively; Et and Ec are the test and calculated results of total energy
dissipation, respectively; κ1 , κ2 , and κ3 are the relative errors between Ket and Kec , Fpt , and Fpc , and Et and
Ec , respectively.
Secondly, a piecewise function was adopted to establish the hysteretic loop relative to each
governing deformation, and the hysteretic model of a wall was further established. Taking specimen
W89-1, for example, details are as follows: (1) dimensionless treatment was performed on the
load-displacement skeleton curve that was determined by the discrete coordinate method, δn and
its corresponding load f n of the hysteretic loop were determined based on the load case relative
to each governing deformation, and the slipping segment slope k0 in elastic phrase was calculated
(see Figure 8b); (2) on the basis of k0 , δn , and f n , the dominant parameters of the hysteretic model were
obtained by Equations (11)–(16), as listed in Table 4, the controlling factors of parameters ka , kc , and
nu , nl are listed in Table 5; (3) after substituting those dominant parameters that are listed in Table 4
into Equations (8)–(10), the f -δ hysteretic loops corresponding to different governing deformations
were established, and the hysteretic model of the wall was further obtained through substituting wall
height H and the calculated result of shear capacity Fpc into those f -δ loops.
Table 4. The dominant parameters used to the hysteretic model of specimen W89-1 1 .
Load
Case
10 mm
15 mm
20 mm
30 mm
40 mm
50 mm
60 mm
δn
fn
k0
ka
kb
kc
f mu
f ml
nu
nl
f0
0.0033
0.005
0.0067
0.01
0.0133
0.0167
0.02
0.847
0.948
0.998
1.0
0.916
0.791
0.657
125.0
125.0
125.0
125.0
125.0
125.0
125.0
788.4
827.3
866.2
941.8
1017.3
1095.1
1170.7
102.3
92.0
81.8
61.4
40.9
20.4
12.5
502.4
285.6
206.2
148.8
118.7
95.4
77.2
1.074
1.298
1.436
1.504
1.350
1.022
0.797
1.258
1.050
1.041
1.212
1.317
1.252
1.247
1.45
1.40
1.35
1.25
1.15
1.04
0.94
1.35
1.40
1.46
1.57
1.68
1.78
1.89
0.11
0.11
0.11
0.11
0.11
0.11
0.11
1
k0 is the slipping segment slope of the wall in elasticity (see Figure 8b); the rest of the symbols’ meanings are
identical to those in Equations (8)–(10).
Table 5. The controlling factors of parameters ka , kc , and nu , nl 1 .
Ra
ka0
Rc1
Rc2
Rc3
Rc4
Rnu
Rnl
nu0
nl0
22892.0
712.8
2813.4
−701.4
277.0
−63.9
30.2
32.5
1.55
1.24
1
The meaning of each symbol is identical to the definitions provided in Equations (11)–(16).
Comparisons of the hysteretic curves between the calculated results and the test results are shown
in Figure 11. Good agreement is observed between the test results and the calculated results, which
fully reflect the hysteretic characteristics of the walls during cyclic loading.
Appl. Sci. 2017, 7, 94
13 of 16
Appl. Sci. 2017, 7, x
13 of 16
F/kN
50
F/kN
100
40
30
50
20
-40
-20
0
20
40
60
-80
-60
-40
/mm
0
B
-60
10
/mm
0
-20
-10
0
20
Test result [22]
Calculated result
60
80
A
-20
-50
40
Test result [22]
Calculated result
-30
-40
-100
-50
(b)
100
F/kN
F/kN
(a)
120
50
90
60
30
-60
-40
-20
0
20
40
60
/mm
80
-80
-60
-40
B
-80
B
/mm
0
0
-20
0
20
40
60
80
-30 A
A
Test result [22]
Calculated result
-50
-60
Test result [22]
Calculated result
-90
-120
-100
(d)
F/kN
(c)
100
50
/mm
0
-60
-40
-20
0
20
40
60
-50
Test result [22]
Calculated result
-100
(e)
Figure 11. Hysteretic curves of test results and the proposed hysteretic model. (a) Specimen W89-1;
Figure 11. Hysteretic curves of test results and the proposed hysteretic model. (a) Specimen W89-1;
(b) Specimen W89-2; (c) Specimen W89-3; (d) Specimen W140-1; (e) Specimen W140-2.
(b) Specimen W89-2; (c) Specimen W89-3; (d) Specimen W140-1; (e) Specimen W140-2.
Energy dissipation is an important indicator to measure the seismic performance of a structure
or
a
component.
Theisproposed
hysteretic
modelto
was
therebythe
validated
comparing the
Energy
dissipation
an important
indicator
measure
seismicbyperformance
of energy
a structure
dissipation
of
the
walls.
Only
the
energy
dissipation
relative
to
the
first
cycle
of
each
load
level
or a component. The proposed hysteretic model was thereby validated by comparing
thewas
energy
considered,
strength
in successive
cycles
of first
the same
waslevel
not was
dissipation
of thebecause
walls. Only
thedeterioration
energy dissipation
relative
to the
cycle amplitude
of each load
involved in the model.
considered, because strength deterioration in successive cycles of the same amplitude was not involved
A comparison of the cumulative energy dissipation between the calculated results and test
in the model.
results for each specimen is shown in Figure 12. It can be observed that during each level of horizontal
Aload,
comparison
of the
cumulative
energy dissipation
betweenfor
the
calculated
and test
results
the calculated
results
of the cumulative
energy dissipation
140-type
wallsresults
(specimens
W140for each
specimen
is
shown
in
Figure
12.
It
can
be
observed
that
during
each
level
of
horizontal
1 and W140-2) are very close to their test results; for 89-type walls, the relative errors ε of theload,
the calculated
results
ofdissipation
the cumulative
energy
dissipation
for 140-type
walls
(specimens
W140-1
cumulative
energy
between
the calculated
results
and the test
results
are small
in the and
initiation
of loading
increase
constantly
with growing
Figure
12a,c).
It the
alsocumulative
can be
W140-2)
are very
close toand
their
test results;
for 89-type
walls,load
the (see
relative
errors
ε of
concluded
frombetween
Table 3 that
relative error
κ3 ofand
the the
totaltest
energy
dissipation
between
energy
dissipation
thethe
calculated
results
results
are small
in thecalculated
initiation of
results
test results
for 140-type
is within
is obviously
thanbethat
of 89-typefrom
loading
andand
increase
constantly
withwalls
growing
load10%,
(seewhich
Figure
12a,c). It lower
also can
concluded
Table 3 that the relative error κ3 of the total energy dissipation between calculated results and test
results for 140-type walls is within 10%, which is obviously lower than that of 89-type walls with a
maximum value of 28%. The reason is that the energy dissipating mechanism of a CFS shear wall
Appl.
Appl. Sci.
Sci. 2017,
2017, 7,
7, 94
x
14
14of
of16
16
walls with a maximum value of 28%. The reason is that the energy dissipating mechanism of a CFS
depends on the deformation of screw connections; due to a larger stiffness of the reinforced end
shear wall depends on the deformation of screw connections; due to a larger stiffness of the reinforced
studs in a 140-type wall, the lateral of deformation of the wall was effectively mitigated, and the
end studs in a 140-type wall, the lateral of deformation of the wall was effectively mitigated, and the
deformation
deformation of
of the
the screw
screw connections
connections in
in the
the fields
fields of
of the
the wall
wall were
were allowed
allowed to
to get
get full
full development,
development,
reaping
the
benefit
of
a
steady
growth
of
the
cumulative
energy
dissipation,
therefore,
reaping the benefit of a steady growth of the cumulative energy dissipation, therefore, the
the energy
energy
dissipation
displayed
minor
differences
between
the
calculated
results
and
the
test
results
during
dissipation displayed minor differences between the calculated results and the test results during the
the
loading
forfor
a 89-type
wallwall
withwith
reinforced
end studs
lower
besides
loadingprocess;
process;whereas,
whereas,
a 89-type
reinforced
end that
studshad
that
hadstiffness,
lower stiffness,
steel
frame
buckling,
the addition
of a supplementary
torsional
deformation,
that—except
for shear
besides
steel
frame buckling,
the addition
of a supplementary
torsional
deformation,
that—except
for
deformations
in
screw
connections,
which
were
caused
by
the
in-plane
rotation
of
wallboards—was
shear deformations in screw connections, which were caused by the in-plane rotation of wallboards—
involved
in energy
dissipation,
improved
the energy
dissipating
capacity
of theofwall
the later
was involved
in energy
dissipation,
improved
the energy
dissipating
capacity
the during
wall during
the
stage
of
loading;
however,
the
proposed
hysteretic
model
cannot
consider
both
steel
frame
buckling
later stage of loading; however, the proposed hysteretic model cannot consider both steel frame
and
the supplementary
torsional deformation
into the energy
the wall, and
buckling
and the supplementary
torsional deformation
intodissipation
the energyofdissipation
of the
thecalculated
wall, and
results
are
thereby
obviously
lower
than
the
test
results.
the calculated results are thereby obviously lower than the test results.
ItIt should
should be
benoted
notedthat,
that,in
inthe
theend
endof
ofthe
theloading
loadingprocess,
process,due
duetotoaalarger
largeraspect
aspectratio
ratio(H/L
(H/L == 2.5),
2.5),
there
there was
was no
no obvious
obvious damage
damage in
in specimen
specimen W89-2
W89-2 except
except screws
screws in
in the
the end
end studs
studs being
being sheared
sheared off
off
on
on aa single
single side
side of
of the
the wall
wall [22].
[22]. The
The wall’s
wall’s energy
energy dissipating
dissipating capacity
capacity was
was significantly
significantly reduced
reduced by
by
the
insufficient
deformation
of
screw
connections
(see
Figure
12a).
Consequently,
for
such
walls
with
the insufficient deformation of screw connections (see Figure 12a). Consequently, for such walls with
larger
larger aspect
aspectratio,
ratio,the
theenergy
energydissipating
dissipatingcapacity
capacityshould
shouldnot
notbe
beconsidered.
considered.
Cumulative energy dissipation /KJ
W89-2:
20 W89-3:
35
Test results[22];
Calculated results
Test results[22];
Calculated results
Test results[22];
Calculated results
Cumulative energy dissipation /KJ
W89-1:
25
15
10
5
W140-1:
Test results[22];
Calculated results
Test results[22];
Calculated results
30
W140-2:
25
20
15
10
5
0
0
1
2
3
4
5
6
7
8
1
9
2
3
4
5
6
7
8
Load level
Load level
(a)
(b)
30
89-type walls
140-type walls
25
(%)
20
15
10
5
0
0
1
2
3
4
5
6
7
8
9
Load level
(c)
Figure 12. Cumulative energy dissipation of test and the proposed hysteretic model. (a) 89-type walls;
Figure 12. Cumulative energy dissipation of test and the proposed hysteretic model. (a) 89-type walls;
(b)140-type
140-typewalls;
walls;(c)
(c)Relative
Relativeerrors
errorsεεbetween
betweencalculated
calculatedand
andtest
testresults.
results.
(b)
4. Summary and Conclusions
4. Summary and Conclusions
A theoretical hysteretic model, which is concise enough to be mathematically described and can
A theoretical hysteretic model, which is concise enough to be mathematically described and
reflect the shear behavior of the wall with reinforced end studs, was proposed. A degradation
can reflect the shear behavior of the wall with reinforced end studs, was proposed. A degradation
coefficient of secant stiffness was first introduced, and a discrete coordinate method was proposed to
coefficient of secant stiffness was first introduced, and a discrete coordinate method was proposed
establish the load-displacement skeleton curve of the wall; on this basis, a piecewise function was
to establish the load-displacement skeleton curve of the wall; on this basis, a piecewise function was
adopted to capture the hysteretic loop relative to each governing deformation, the hysteretic model
of the wall was further established, and the criteria for the dominant parameters of the model was
Appl. Sci. 2017, 7, 94
15 of 16
adopted to capture the hysteretic loop relative to each governing deformation, the hysteretic model of
the wall was further established, and the criteria for the dominant parameters of the model was stated.
Finally, the hysteretic model was validated by the experimental results of Xu [22]. The conclusions
drawn are as follows:
(1)
(2)
(3)
(4)
(5)
Elastic lateral stiffness Ke and shear capacity Fp are key factors determining the load-displacement
skeleton curve of a CFS shear wall; where Ke and Fp can be predicted by the equivalent-bracing
model and the proposed simplified calculation model, respectively, and the relative errors
between their predicted and test results are within 11% and 15%, respectively.
The load-displacement skeleton curves determined by the discrete coordinate method agree
well with the test results, which fully reflect the shear behavior (e.g., nonlinearity, stiffness, and
strength deterioration) of the walls with reinforced end studs l.
During cyclic loading, test hysteretic curves and the calculated results that were determined by
the proposed hysteretic model are of high consistency in both the pinching characteristic and
energy dissipating level of the walls. Due to a larger stiffness of the reinforced end studs, the
relative error of the total energy dissipation between calculated and test results for 140-type
walls is within 10%; whereas, for 89-type walls with reinforced end studs having lower stiffness,
both steel frame buckling and a supplementary torsional deformation in screw connections were
involved in energy dissipation, resulting in the calculated results being lower than the test results
during the later stage of loading.
Several hysteretic models (e.g., Pinching4 material in OpenSees, BWBN model, EPHM, Pivot
model, as well as the three-segment nonlinear pinching hysteretic model), were able to reproduce
the behavior of the traditional shear wall with coupled C section end studs; in contrast, the
proposed piecewise function hysteretic model is suitable for the wall with reinforced end studs
(i.e., continuous concrete-filled rectangular steel tube columns), which is more in line with the
requirements of mid-rise CFS structures, and the model has intuitional expressions with clear
physical interpretations for each parameter.
Due to the significant reduction in cumulative energy dissipation that is caused by insufficient
deformation of screw connections, the energy dissipating capacity of a CFS shear wall with
reinforced end studs should not be considered when the wall’s aspect ratio is larger (in this study,
H/L = 2.5, where H and L are wall height and wall length, respectively).
Acknowledgments: The work described in this paper was fully supported by the National Key Program
Foundation of China (51538002).
Author Contributions: Jihong Ye and Xingxing Wang conceived and designed the experiments; Xingxing Wang
performed the experiments; Jihong Ye and Xingxing Wang analyzed the data; Jihong Ye and Xingxing Wang wrote
the paper.
Conflicts of Interest: The authors declare no conflict of interest.
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