# Performance of the Cold-Bending Channel-Angle Buckling-Restrained Brace under Cyclic Loading

Authors: Kun Wang; Junwu Xia; Xiaomiao Chen; Bo Xu; Xiangzhou Liang; Jian Wang
Journal: Advances in Civil Engineering
 (2019)

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## Abstract

In this study, three restricted cold-bending channel-angle buckling-restrained brace (CCA-BRB) specimens were experimentally characterised by a low-reversed cyclic loading test. Three specimens had steel cores with cruciform cross section. Two restraining units were assembled to form an external constraint member, each of which was composed of an equilateral cold-bending channel and two equilateral cold-bending angles via welding. A gap or a thin silica gel plate was set between the internal core and the external constraint member to form an unbonded layer. Several evaluation parameters on the seismic performance, hysteretic behaviour, and energy dissipation capability of the CCA-BRB was investigated, including hysteresis curve, skeleton curve, compression strength adjustment factor, measured and computed stiffness, energy dissipation coefficient, equivalent viscous damping ratio, ductility coefficient, and cumulative plastic deformation. The test results and evaluation indices demonstrated that the hysteretic performance of braces with a rigid connection was stable. A Ramberg–Osgood model and two model parameters were calibrated to predict, with fidelity, the skeleton curve of CCA-BRB under cyclic load. The initial elastic stiffness of the brace used in practice should contain overall portions of the brace instead of the yielding portion of the brace. Finally, all the tested CCA-BRBs exhibited a stable energy absorption performance and verified the specimens’ construction was rational.


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## Body

## 1. Introduction

As a kind of structure system with excellent ductility, the braced frame is often used in seismic structures. However, the general brace itself will suffer from comprehensive problems such as strength stiffness degradation and low-cycle fatigue fracture after buckling when under the action of cyclic load. A buckling-restrained brace (BRB) is an energy dissipation damper of metal yielding which has advantages of stable energy dissipation capacity, easy construction and fabrication with low cost, etc. [1]. A BRB can provide stable lateral stiffness and load-carrying capacity for a frame structure. As a result, BRBs have been used extensively in existing and new structures to enhance the earthquake resistance [2–6].Diverse experiments show that BRBs have a substantial energy absorption capability under cyclic loading [7, 8]. In 1976, an early buckling-restrained attempt to propose a brace with dissipating energy yet does not buckle was reported in an experimental study [9]. The brace consisted of a single flat plate as the internal core and a square steel pipe filled with mortar as the external constraint member. Since then, various structural forms and experimental studies of traditional BRBs were proposed by investigators around the world [10–13]. However, there were a series of issues affecting the traditional BRBs, in which the core buckling restrained by steel tube filled with concrete or mortar, such as the need for higher precision control between the external concrete member and the core steel member, and the complex processes of wet concrete pouring, further increasing the complexity of component fabrication and prolonging the production cycle. Based on the above reasons, scholars have put forward various forms of all-steel BRBs [14–21], which only require clamping the inner core with section steel or composite steel members to impose constraints, and the all-steel BRBs have the advantages of simple construction, convenient assembly, and lighter weight. Kuwahara et al. proposed the first representative all-steel buckling-restrained brace in 1993 of which circular steel tubes were used to support the inner core (the inner circular tube) and the outer constraint members (the outer circular tube) [14]. The all-steel BRBs had steel cores with rectangular cross sections and were studied through a uniaxial test program [15] or subassemblage testing [16, 17]. The all-steel BRBs with cruciform cross-section core are generally used to provide large lateral restraint force to frame structures, and several axial cycle tests have been conducted to study the hysteresis behaviour of the all-steel BRBs with cruciform cross-section core [18–21]. In addition, several shaking table tests have been conducted on concrete and steel frames equipped with BRBs [22, 23]. A novel type of BRB, called cold-bending channel-angle buckling-restrained brace (CCA-BRB), has been proposed and tested by the authors. This paper describes the characteristics, hysteresis behaviour, and energy dissipation capability of the new BRB.


## 2. Layouts of the CCA-BRB

Each CCA-BRB consists of an internal steel core, external constraint members, and an unbonded layer between them (Figure1). The steel core is subjected to the axial load while constraint members provide lateral support to the steel core and prevent the core from buckling. The external constraint members of the CCA-BRB are composed of two restraining units, each of which consists of an equilateral cold-bending channel and two equilateral cold-bending angles via welding (Figure 2). The ratio of web height to flange plate width of the channel is 2 : 1. Each CCA-BRB has the steel core with cruciform cross section which consists of three portions: a yielding portion, a transition portion, and a connection portion (Figure 3). A gap must be present between the internal core and the external constraint members while the gap can be adjusted to form an unbonded layer. The thin layer of unbonded material along the internal steel core eliminates shear transfer during the elongation and contraction of the core and also accommodates its lateral expansion when in compression.Figure 1
Exploded view of the CCA-BRB.Figure 2
Diagram of the restraining units via welding.Figure 3
Schematic diagram of the CCA-BRB. (a) Midspan section; (b) end section; (c) dimensions of the CCA-BRB.
(a)
(b)
(c)Figure3 shows the cross-sectional schematic diagram of the CCA-BRB and the dimension parameters: hc denotes the width and tc the thickness of the steel core; hcc denotes the web height, bcc the flange width, and tcc the thickness of the cold-bending channel steel; hca denotes the vertical limb height, bca the horizontal width and tca the thickness of the cold-bending angle steel.


## 3. Component Testing

### 3.1. Specimen Design

The geometric dimension parameters of the specimens are listed in Table1, and the meaning of each parameter is shown in Figure 3, in which Ly represents the yielding portion length of the core, Ltr the transition portion length of the core, Lcon the connection portion length of the core, Ltotal the total length of the core, and Lem the length of the external constraint members.Table 1
Geometric dimension parameters and loading protocol of the specimens.
Specimen
h
c × tc (mm)
h
cc × bcc × tcc (mm)
h
ca × bca × tca (mm)
L
y (mm)
L
tr (mm)
L
con (mm)
L
total (mm)
L
em (mm)
Loading protocol
BRB-1
68 × 8
140 × 70 × 5
60 × 60 × 5
800
50
150
1200
1160
Loading protocol 1
BRB-2
68 × 8
140 × 70 × 5
60 × 60 × 5
800
50
150
1200
1160
Loading protocol 2
BRB-3
68 × 8
140 × 70 × 5
60 × 60 × 5
800
50
150
1200
1160
Loading protocol 3A construction drawing of the BRB-1 specimen is shown in Figure4. The internal steel core of CCA-BRB is composed of three flat plates via full-fillet welding by carbon dioxide arc welding. The core of CCA-BRB was a Q235 steel member with a variable cross section, in which the dimensions of the yielding portion section were 68 mm × 68 mm × 8 mm and connection portion section 128 mm × 128 mm × 8 mm. The yielding portion length Ly of the core was 800 mm, the transition portion length Ltr 100 mm (50 mm on each side), the connection portion length Lcon 300 mm (150 mm on each side), and the total length of the core Ltotal 1200 mm. The external restraining members of CCA-BRB are composed of two cold-bending channels and four cold-bending angles in which all components were made of Q235 steel. The cross-section dimensions of the channel were 140 mm × 70 mm × 5 mm and the angle 60 mm × 60 mm × 5 mm. The design length of all restraining components Lem was 1160 mm which is shorter than the core 40 mm; that is, each side is reserved the axial compression space of 20 mm to meet the 2% compressive strain (each end splits an average of 1% deformation) of the core and left a certain margin to consider the machining error. Each channel and two angles were bound via fillet welds.Figure 4
Construction drawing of the BRB-1. (a) Steel core; (b) top view; (c) front view; (d) A-A section; (e) B-B section.
(a)
(b)
(c)
(d)
(e)Note that 1 mm gap was set between the internal core and the external constraint members of the specimens on each side with pasting 1-millimeter-thick silica gel plate along the yielding portion of the core to reduce the friction when in compression. A 25-millimeter-thick square plate was welded at the end of the core via full penetration weld, and the dimensions of the plate were 200 mm × 200 mm. Based on the realistic consideration of the test fixture, a 18-millimeter-thick holding plate (the dimensions of the holding plate were 150 mm × 100 mm) and the square end plate were bound via fillet weld to form a fixed joint to satisfy test loading.


### 3.2. Experimental Setup

Figure5 indicates the experimental setup of the specimens. Both the ends of the specimens are connected to a PWS-500 electrohydraulic servo actuator at the Jiangsu Key Laboratory of Environmental Impact and Structural Safety in Engineering of China University of Mining and Technology by which an axial cyclic load was applied. The actuator had a cyclic load capacity of ±500 kN and a displacement capacity of ±200 mm. To observe the axial deformation, two displacement meters were set at the upper end of the specimens and two at the lower end.Figure 5
Experimental setup schematic diagram.


### 3.3. Loading Protocol

The loading protocols adopted in this low-reversed cyclic loading test are shown in Figure6. The parameter used to define the test program was the core axial strain, and three loading protocols were adopted corresponding to three specimens. Corresponding to the specimen BRB-1, two loading cycles were performed to an amplitude increment in the core axial strains of 0.5%, 1.0%, and 1.5% followed by five loading cycles of 2% strain to complete the experiment (see the loading protocol 1). Corresponding to the specimen BRB-2, two loading cycles were performed with 0.5%, 1.0%, and 1.5% core axial strain followed by five loading cycles of 2% strain to complete the experiment (see the loading protocol 2). The difference of loading protocols between BRB-1 and BRB-2 was the loading type of BRB-1 traction after compression and BRB-2 compression after traction. According to the requirements of GB50011-2010 [24], the parameter corresponding to the specimen BRB-3 was defined at levels of 1/300, 1/200, 1/150, and 1/100 bracing length under tension-compression three times (see the loading protocol 3). On the basis of the loading protocols, when the specimens BRB-1 and BRB-2 completed five 2% axial strain loadings, the value of CPD, which is a normalized expression of the cumulative plastic deformation, was measured at 212 which satisfies the plastic deformation capability requirements on the BRB specified in ANSI/AISC 341-10 [25]. The core axial displacement loading protocol of each specimen is shown in Table 2. The core length was 1200 mm, and the core yielding portion length was 800 mm.Figure 6
Loading protocols of the specimens.
(a)
(b)
(c)Table 2
Axial displacement loading protocols of the specimens.
Specimen
Loading sequence
Axial strain (%)
Axial displacement (mm)
Bracing length ratio
Cycle number
BRB-1, BRB-2
1
±0.5
±4
1/300
2
2
±1.0
±8
1/150
2
3
±1.5
±12
1/100
2
4
±2.0
±16
1/75
5
BRB-3
1
±0.5
±4
1/300
3
2
±0.75
±6
1/200
3
3
±1.0
±8
1/150
3
4
±1.5
±12
1/100
3


### 3.4. Material Properties

Based on the request of GB/T228.1-2010 [26], the material properties tests of the internal steel core and external constraint members of varying thicknesses were measured. The corresponding material property index is shown in Table 3, in which E denotes elastic Young’s modulus, fy the steel yield strength, fu the steel tensile strength, and fy/fu the steel yield-tensile strength ratio.Table 3
Material properties of steel.
Plate position
Steel type
Thickness (mm)
E (MPa)
f
y (MPa)
f
u (MPa)
f
y/fu
Core member
Q235
8
2.01 × 105
265
425.6
0.62
External channel
Q235
5
2.03 × 105
270.7
430.4
0.63
External angle
Q235
5
2.03 × 105
270.1
433.8
0.62
Square end plate
Q235
25
2.05 × 105
256.8
415.2
0.62
Clamping plate
Q345
18
2.06 × 105
368.2
512.4
0.72


## 3.1. Specimen Design

The geometric dimension parameters of the specimens are listed in Table1, and the meaning of each parameter is shown in Figure 3, in which Ly represents the yielding portion length of the core, Ltr the transition portion length of the core, Lcon the connection portion length of the core, Ltotal the total length of the core, and Lem the length of the external constraint members.Table 1
Geometric dimension parameters and loading protocol of the specimens.
Specimen
h
c × tc (mm)
h
cc × bcc × tcc (mm)
h
ca × bca × tca (mm)
L
y (mm)
L
tr (mm)
L
con (mm)
L
total (mm)
L
em (mm)
Loading protocol
BRB-1
68 × 8
140 × 70 × 5
60 × 60 × 5
800
50
150
1200
1160
Loading protocol 1
BRB-2
68 × 8
140 × 70 × 5
60 × 60 × 5
800
50
150
1200
1160
Loading protocol 2
BRB-3
68 × 8
140 × 70 × 5
60 × 60 × 5
800
50
150
1200
1160
Loading protocol 3A construction drawing of the BRB-1 specimen is shown in Figure4. The internal steel core of CCA-BRB is composed of three flat plates via full-fillet welding by carbon dioxide arc welding. The core of CCA-BRB was a Q235 steel member with a variable cross section, in which the dimensions of the yielding portion section were 68 mm × 68 mm × 8 mm and connection portion section 128 mm × 128 mm × 8 mm. The yielding portion length Ly of the core was 800 mm, the transition portion length Ltr 100 mm (50 mm on each side), the connection portion length Lcon 300 mm (150 mm on each side), and the total length of the core Ltotal 1200 mm. The external restraining members of CCA-BRB are composed of two cold-bending channels and four cold-bending angles in which all components were made of Q235 steel. The cross-section dimensions of the channel were 140 mm × 70 mm × 5 mm and the angle 60 mm × 60 mm × 5 mm. The design length of all restraining components Lem was 1160 mm which is shorter than the core 40 mm; that is, each side is reserved the axial compression space of 20 mm to meet the 2% compressive strain (each end splits an average of 1% deformation) of the core and left a certain margin to consider the machining error. Each channel and two angles were bound via fillet welds.Figure 4
Construction drawing of the BRB-1. (a) Steel core; (b) top view; (c) front view; (d) A-A section; (e) B-B section.
(a)
(b)
(c)
(d)
(e)Note that 1 mm gap was set between the internal core and the external constraint members of the specimens on each side with pasting 1-millimeter-thick silica gel plate along the yielding portion of the core to reduce the friction when in compression. A 25-millimeter-thick square plate was welded at the end of the core via full penetration weld, and the dimensions of the plate were 200 mm × 200 mm. Based on the realistic consideration of the test fixture, a 18-millimeter-thick holding plate (the dimensions of the holding plate were 150 mm × 100 mm) and the square end plate were bound via fillet weld to form a fixed joint to satisfy test loading.


## 3.2. Experimental Setup

Figure5 indicates the experimental setup of the specimens. Both the ends of the specimens are connected to a PWS-500 electrohydraulic servo actuator at the Jiangsu Key Laboratory of Environmental Impact and Structural Safety in Engineering of China University of Mining and Technology by which an axial cyclic load was applied. The actuator had a cyclic load capacity of ±500 kN and a displacement capacity of ±200 mm. To observe the axial deformation, two displacement meters were set at the upper end of the specimens and two at the lower end.Figure 5
Experimental setup schematic diagram.


## 3.3. Loading Protocol

The loading protocols adopted in this low-reversed cyclic loading test are shown in Figure6. The parameter used to define the test program was the core axial strain, and three loading protocols were adopted corresponding to three specimens. Corresponding to the specimen BRB-1, two loading cycles were performed to an amplitude increment in the core axial strains of 0.5%, 1.0%, and 1.5% followed by five loading cycles of 2% strain to complete the experiment (see the loading protocol 1). Corresponding to the specimen BRB-2, two loading cycles were performed with 0.5%, 1.0%, and 1.5% core axial strain followed by five loading cycles of 2% strain to complete the experiment (see the loading protocol 2). The difference of loading protocols between BRB-1 and BRB-2 was the loading type of BRB-1 traction after compression and BRB-2 compression after traction. According to the requirements of GB50011-2010 [24], the parameter corresponding to the specimen BRB-3 was defined at levels of 1/300, 1/200, 1/150, and 1/100 bracing length under tension-compression three times (see the loading protocol 3). On the basis of the loading protocols, when the specimens BRB-1 and BRB-2 completed five 2% axial strain loadings, the value of CPD, which is a normalized expression of the cumulative plastic deformation, was measured at 212 which satisfies the plastic deformation capability requirements on the BRB specified in ANSI/AISC 341-10 [25]. The core axial displacement loading protocol of each specimen is shown in Table 2. The core length was 1200 mm, and the core yielding portion length was 800 mm.Figure 6
Loading protocols of the specimens.
(a)
(b)
(c)Table 2
Axial displacement loading protocols of the specimens.
Specimen
Loading sequence
Axial strain (%)
Axial displacement (mm)
Bracing length ratio
Cycle number
BRB-1, BRB-2
1
±0.5
±4
1/300
2
2
±1.0
±8
1/150
2
3
±1.5
±12
1/100
2
4
±2.0
±16
1/75
5
BRB-3
1
±0.5
±4
1/300
3
2
±0.75
±6
1/200
3
3
±1.0
±8
1/150
3
4
±1.5
±12
1/100
3


## 3.4. Material Properties

Based on the request of GB/T228.1-2010 [26], the material properties tests of the internal steel core and external constraint members of varying thicknesses were measured. The corresponding material property index is shown in Table 3, in which E denotes elastic Young’s modulus, fy the steel yield strength, fu the steel tensile strength, and fy/fu the steel yield-tensile strength ratio.Table 3
Material properties of steel.
Plate position
Steel type
Thickness (mm)
E (MPa)
f
y (MPa)
f
u (MPa)
f
y/fu
Core member
Q235
8
2.01 × 105
265
425.6
0.62
External channel
Q235
5
2.03 × 105
270.7
430.4
0.63
External angle
Q235
5
2.03 × 105
270.1
433.8
0.62
Square end plate
Q235
25
2.05 × 105
256.8
415.2
0.62
Clamping plate
Q345
18
2.06 × 105
368.2
512.4
0.72


## 4. Test Results and Performance Analysis

### 4.1. Hysteresis Curve

The hysteresis curves of the specimens are plotted with the recorded axial force in the brace as ordinate against the axial displacement measured across the entire brace including connections as abscissa (Figure7). The vertical coordinate is the measured axial load of the specimens, and its sign convention is positive tension and negative compression. The horizontal coordinate is the measured axial deformation, and the sign convention is the same. It is noteworthy that the flexibility of the connections contributes only slightly to the overall displacement; further, the axial deformation measured by the displacement meter mainly reflects the axial deformation of the yielding segment of the brace.Figure 7
Hysteresis curves of the specimens. (a) BRB-1; (b) BRB-2; (c) BRB-3.
(a)
(b)
(c)The hysteresis loops resulting from the low-reversed cyclic loading test show that all the test braces exhibited stable hysteretic behaviour for all displacement amplitudes, with a plump hysteresis curve and substantial energy absorption capability. It can be seen that there was no significant stiffness or strength degradation when the test completed, and the loading and unloading stiffnesses were basically the same. In addition, the force-displacement loops showed a distinct cyclic hardening characteristic and an asymmetrical feature between tension and compression. It must be pointed out that the BRB with a rigid connection produce a moment on the loading setup which should be considered in the structural design of the frame.


### 4.2. Skeleton Curve

The skeleton curves of the specimens are shown in Figure8, in which the curves are almost identical. It is seen that the skeleton curves of three specimens present distinct bilinear characteristics, and there is an obvious inflexion between the elastic stage and the plastic stage. The core of each specimen has a noticeable strengthened segment after entering the plastic stage.Figure 8
Skeleton curves of the specimens.In order to concise the variation law of the skeleton curve of the CCA-BRB, the Ramberg–Osgood model [27] is used to fit the skeleton curves of the specimens. The formula of the Ramberg–Osgood model is given by(1)Δε2=Δεe2+Δεp2=Δσ2E+Δσ2K′1/n,where Δε represents total strain amplitude; Δεe and Δεp represent, respectively, elastic strain amplitude and plastic strain amplitude; Δσ represents stress amplitude; K′ and n represent, respectively, cyclic hardening coefficient and cyclic hardening exponent. The fitting parameters of the Ramberg–Osgood model are shown in Table 4.Table 4
Fitting parameters.
Specimen
Fitting parameter
K
′ (MPa)
n
BRB-1
439.6
0.165
BRB-2
464.5
0.17
BRB-3
373.2
0.13Figure9 plots the fitting skeleton curves. It is seen that the fitting curve was little different from that of test points curve, indicating that the Ramberg–Osgood model can be used to simulate the skeleton curve of CCA-BRB under cyclic loading.Figure 9
Fitting skeleton curves of the specimens. (a) BRB-1; (b) BRB-2; (c) BRB-3.
(a)
(b)
(c)


### 4.3. Compression Strength Adjustment Factor

Due to the Poisson effect, the lateral cross section of the internal core tended to increase under compression; thus, the internal core may contact the external constraint member. Further, a frictional force at the contact surface was generated to increase the axial force of the core under the compression stage. In addition, the bearing capacity under compression is greater than that under tension, affecting the asymmetry characteristic of tension and compression of a brace.The asymmetry between tension and compression of a brace is generally expressed by the compression strength adjustment factorβ [28], which is defined by(2)βi=NCi,maxNTi,max,where NCi,max and NTi,max represent the maximum axial compression and the maximum axial tension, respectively, during the ith hysteresis loop.The values of compression strength adjustment factor during the distinct loading displacement amplitudes are shown in Table5. Based on the requirements of ANSI/AISC 341-10, the compression strength adjustment factor of a BRB must not exceed 1.3. It can be seen that the increase factors of BRB-1 and BRB-2 were 1.092 and 1.118, respectively, which remained lower than the limit value 1.3 of the code under the loading amplitude of 2% core axial strain. The coefficient value of BRB-3 was 1.050 which is still below 1.3 when the core axial strain remained within 1.5%. The above-described indicated that the CCA-BRB still exhibited a stable and symmetrical hysteretic behaviour, with an adequate energy dissipation capacity even when the core axial strain reached 2%.Table 5
Compression strength adjustment factor of the specimens.
Specimen
Compression strength adjustment factor under each core axial strain
0.5%
1%
1.5%
2%
BRB-1
1.024
1.043
1.079
1.092
BRB-2
1.012
0.979
1.038
1.118
BRB-3
0.5%
0.75%
1.0%
1.5%
1.007
1.037
1.027
1.050


### 4.4. Measured and Computed Stiffness Values

Based on the experimental data, the initial elastic stiffness and the secondary stiffness which are of interest in the mechanical characterization of the brace are identified in this section. As shown in Figure3(c), the core of the brace with a cruciform cross section consists of a yielding portion, a transition portion, and a connection portion throughout, with the cross section of the yielding stage smaller than the cross sections of the transition portion and the connection portion. The total elastic stiffness of the brace is the summation in series of the individual stiffness of the segments described above. The formula for the total elastic stiffness of the core is given by(3)Ktotalc=11/Kyc+21/Ktrc+21/Kconc,where Ktotalc represents the total elastic stiffness, Kyc represents the elastic stiffness of the yielding portion, Ktrc represents the elastic stiffness of the transition portion, and Kconc represents the elastic stiffness of the connection portion.The values of computed elastic stiffness for the yielding portion, the transition portion, and the connection portion are, respectively,(4)Kyc=AyELy≈1,024mm2×201GPa800mm=275.3kN/mm,(5)Ktrc=Acon−AyElnAcon−lnAyLtr≈1,984−1,024mm2×201GPaln1,984−ln1,024×50mm=5,829.6kN/mm,(6)Kconc=AconELcon≈1,984mm2×201GPa150mm=2,658.6kN/mmwhere Ay and Acon represent, respectively, the cross-sectional area of the yielding portion and the connection portion. Based on the above values given by equations (4)–(6), the total elastic stiffness Ktotalc, given by equation (3), is 200.8 kN/mm.Table6 lists the computed stiffness, measured stiffness, and the difference between these values for each specimen. The computed elastic stiffness values given in columns 2, 3, 4, and 5 were obtained from equations (3)–(6), respectively. The measured stiffness values given in columns 6 and 7 were computed by fitting the tensile section of the skeleton curve (using a double broken line model) of each brace. Column 6, labelled Ktotalm, corresponds to the slope of the axial force-displacement relationship measured for the elastic portion of the fitting skeleton curve, whereas, column 7, labelled K2, corresponds to the slope measured for the plastic portion of the fitting skeleton curve.Table 6
Comparison of computed and measured stiffness values.
Specimen
Computed elastic stiffness (kN/mm)
Measured stiffness (kN/mm)
Difference
K
y
c equation (4)
K
tr
c equation (5)
K
con
c equation (6)
K
total
c equation (3)
K
total
m
K
2
α
K
total (%)
BRB-1
275.3
5,829.6
2,658.6
200.8
199.86
8.15
0.041
−0.47
BRB-2
275.3
5,829.6
2,658.6
200.8
210.85
9.01
0.043
5
BRB-3
275.3
5,829.6
2,658.6
200.8
192.22
7.01
0.036
−4.27As can be seen from Table6, the computed total elastic stiffness value given by equation (3) is in close accordance with the value measured from the test data. In theory, the stiffness values used in the design should contain the entire portions of the brace. If the stiffness value for yielding portion be used in the design instead of the overall stiffness, then the stiffness value is overestimated by 37%. In addition, if assuming that the cross area over the entire length is equal to Ay, then the stiffness value is calculated to be 171.5 kN/mm and the stiffness value is still figured to be undervalued by 15%.


### 4.5. Secondary Stiffness

The secondary stiffness which usually can also be written as postyielding stiffness is of interest for the mechanical properties of the BRB and depends in general on the loading course. The measured values of secondary stiffness and postyielding ratio,α=K2/Ktotalm, of each specimen are given in columns 7 and 8 of Table 6, respectively. Column 8, labelled α, corresponds to the ratio of elastic stiffness to secondary stiffness.


### 4.6. Energy Dissipation Index

Energy dissipation capacity is an important index for evaluating the performance of CCA-BRB. In this section, the energy dissipation capacity of each specimen is evaluated via parameters which contain energy dissipation coefficient and equivalent viscous damping ratio. The parameters of the specimens were obtained from the hysteresis curves and are listed in Table7. According to Section 4.5.6 of JGJ/T101-2015 [29], the values for the energy dissipation coefficient and the equivalent viscous damping ratio are calculated by, respectively,(7)ψ=SBEDFBS△OAB+S△OCD,ζ=SBEDFB2πS△OAB+S△OCD,where SBEDFB, as shown in Figure 10, represents the hysteresis loop area of a brace enveloped by one displacement amplitude, S△OAB and S△OCD represent the area of triangle OAB and triangle OCD, respectively. It can be seen from Table 7 that the energy dissipation coefficient and the equivalent viscous damping ratio of each specimen show a trend of gradual increase with the increase of loading displacement amplitude. Further, the gradual increasing trend of energy dissipation indices indicates that the brace has a substantial and repeatable capability to absorb earthquake energy.Table 7
Ductility indices and energy dissipation indices of the specimens.
Specimen
Energy dissipation coefficient
Equivalent viscous damping ratio
Ductility coefficient
Cumulative plastic deformation
0.5%
1.0%
1.5%
2.0%
0.5%
1.0%
1.5%
2.0%
BRB-1
2.646
2.894
2.902
2.854
0.421
0.461
0.462
0.454
11.4
311.2
BRB-2
2.609
2.851
2.900
2.835
0.415
0.454
0.461
0.451
12.4
326.6
0.5%
0.75%
1.0%
1.5%
0.5%
0.75%
1.0%
1.5%
BRB-3
2.612
2.770
2.896
3.050
0.416
0.441
0.461
0.485
7.4
167.6Figure 10
Calculation diagram of energy dissipation index.


### 4.7. Ductility Coefficient

The deformation capacity which determines the maximum displacement of a brace is an important index for evaluating the seismic performance of CCA-BRB and is expressed via the ductility coefficient. The ductility coefficient of a brace refers to the ratio of the maximum displacement to the yield displacement of the brace before the obvious strength degradation of the hysteresis curve. The coefficients of each specimen are listed in Table7. According to Section 4.5.4 of JGJ/T101-2015 [29], the formula is given by(8)μ=dmaxdy,where dmax and dy represent maximum displacement and yield displacement, respectively, of the brace. It can be seen from Table 7 that the ductility coefficient values of BRB-1 and BRB-2 appear to be 11.4 and 12.4, respectively, once the core axial strain reaches 2.0%, and the value of BRB-3 appears to be 7.4, once the core axial strain reaches 1.5%. In addition, the specimens have a large and substantial plastic deformation capability.


### 4.8. Cumulative Plastic Deformation Capability

An index used in practice to evaluate the plastic deformation capability of a brace is the cumulative plastic strain or, alternatively, the cumulative plastic deformation. The cumulative plastic deformation [30] of the brace can be defined by(9)η=∑i=1n2dmax+i+dmax−idy−4,where dmax+i and dmax−i represent the absolute values of the maximum tensile displacement and the maximum compressive displacement, respectively, during each visit i into the inelastic range. The last column of Table 7 lists the cumulative plastic deformation values of each specimen. It can be seen from Table 7 that the values of BRB-1 and BRB-2 appear to be 311 and 327, once the core loading displacement amplitude reaches the axial strain of 2.0%, and the value of BRB-3 appears to be 167.6, once the core loading displacement amplitude reaches the axial strain of 1.5%. Further, it is noted that the cumulative plastic deformation values of BRB-1 and BRB-2 during the testing protocol exceed the requirement of the minimum limit (200) set by the ANSI/AISC 341-10, indicating that the brace has an excellent plastic deformation capability.


## 4.1. Hysteresis Curve

The hysteresis curves of the specimens are plotted with the recorded axial force in the brace as ordinate against the axial displacement measured across the entire brace including connections as abscissa (Figure7). The vertical coordinate is the measured axial load of the specimens, and its sign convention is positive tension and negative compression. The horizontal coordinate is the measured axial deformation, and the sign convention is the same. It is noteworthy that the flexibility of the connections contributes only slightly to the overall displacement; further, the axial deformation measured by the displacement meter mainly reflects the axial deformation of the yielding segment of the brace.Figure 7
Hysteresis curves of the specimens. (a) BRB-1; (b) BRB-2; (c) BRB-3.
(a)
(b)
(c)The hysteresis loops resulting from the low-reversed cyclic loading test show that all the test braces exhibited stable hysteretic behaviour for all displacement amplitudes, with a plump hysteresis curve and substantial energy absorption capability. It can be seen that there was no significant stiffness or strength degradation when the test completed, and the loading and unloading stiffnesses were basically the same. In addition, the force-displacement loops showed a distinct cyclic hardening characteristic and an asymmetrical feature between tension and compression. It must be pointed out that the BRB with a rigid connection produce a moment on the loading setup which should be considered in the structural design of the frame.


## 4.2. Skeleton Curve

The skeleton curves of the specimens are shown in Figure8, in which the curves are almost identical. It is seen that the skeleton curves of three specimens present distinct bilinear characteristics, and there is an obvious inflexion between the elastic stage and the plastic stage. The core of each specimen has a noticeable strengthened segment after entering the plastic stage.Figure 8
Skeleton curves of the specimens.In order to concise the variation law of the skeleton curve of the CCA-BRB, the Ramberg–Osgood model [27] is used to fit the skeleton curves of the specimens. The formula of the Ramberg–Osgood model is given by(1)Δε2=Δεe2+Δεp2=Δσ2E+Δσ2K′1/n,where Δε represents total strain amplitude; Δεe and Δεp represent, respectively, elastic strain amplitude and plastic strain amplitude; Δσ represents stress amplitude; K′ and n represent, respectively, cyclic hardening coefficient and cyclic hardening exponent. The fitting parameters of the Ramberg–Osgood model are shown in Table 4.Table 4
Fitting parameters.
Specimen
Fitting parameter
K
′ (MPa)
n
BRB-1
439.6
0.165
BRB-2
464.5
0.17
BRB-3
373.2
0.13Figure9 plots the fitting skeleton curves. It is seen that the fitting curve was little different from that of test points curve, indicating that the Ramberg–Osgood model can be used to simulate the skeleton curve of CCA-BRB under cyclic loading.Figure 9
Fitting skeleton curves of the specimens. (a) BRB-1; (b) BRB-2; (c) BRB-3.
(a)
(b)
(c)


## 4.3. Compression Strength Adjustment Factor

Due to the Poisson effect, the lateral cross section of the internal core tended to increase under compression; thus, the internal core may contact the external constraint member. Further, a frictional force at the contact surface was generated to increase the axial force of the core under the compression stage. In addition, the bearing capacity under compression is greater than that under tension, affecting the asymmetry characteristic of tension and compression of a brace.The asymmetry between tension and compression of a brace is generally expressed by the compression strength adjustment factorβ [28], which is defined by(2)βi=NCi,maxNTi,max,where NCi,max and NTi,max represent the maximum axial compression and the maximum axial tension, respectively, during the ith hysteresis loop.The values of compression strength adjustment factor during the distinct loading displacement amplitudes are shown in Table5. Based on the requirements of ANSI/AISC 341-10, the compression strength adjustment factor of a BRB must not exceed 1.3. It can be seen that the increase factors of BRB-1 and BRB-2 were 1.092 and 1.118, respectively, which remained lower than the limit value 1.3 of the code under the loading amplitude of 2% core axial strain. The coefficient value of BRB-3 was 1.050 which is still below 1.3 when the core axial strain remained within 1.5%. The above-described indicated that the CCA-BRB still exhibited a stable and symmetrical hysteretic behaviour, with an adequate energy dissipation capacity even when the core axial strain reached 2%.Table 5
Compression strength adjustment factor of the specimens.
Specimen
Compression strength adjustment factor under each core axial strain
0.5%
1%
1.5%
2%
BRB-1
1.024
1.043
1.079
1.092
BRB-2
1.012
0.979
1.038
1.118
BRB-3
0.5%
0.75%
1.0%
1.5%
1.007
1.037
1.027
1.050


## 4.4. Measured and Computed Stiffness Values

Based on the experimental data, the initial elastic stiffness and the secondary stiffness which are of interest in the mechanical characterization of the brace are identified in this section. As shown in Figure3(c), the core of the brace with a cruciform cross section consists of a yielding portion, a transition portion, and a connection portion throughout, with the cross section of the yielding stage smaller than the cross sections of the transition portion and the connection portion. The total elastic stiffness of the brace is the summation in series of the individual stiffness of the segments described above. The formula for the total elastic stiffness of the core is given by(3)Ktotalc=11/Kyc+21/Ktrc+21/Kconc,where Ktotalc represents the total elastic stiffness, Kyc represents the elastic stiffness of the yielding portion, Ktrc represents the elastic stiffness of the transition portion, and Kconc represents the elastic stiffness of the connection portion.The values of computed elastic stiffness for the yielding portion, the transition portion, and the connection portion are, respectively,(4)Kyc=AyELy≈1,024mm2×201GPa800mm=275.3kN/mm,(5)Ktrc=Acon−AyElnAcon−lnAyLtr≈1,984−1,024mm2×201GPaln1,984−ln1,024×50mm=5,829.6kN/mm,(6)Kconc=AconELcon≈1,984mm2×201GPa150mm=2,658.6kN/mmwhere Ay and Acon represent, respectively, the cross-sectional area of the yielding portion and the connection portion. Based on the above values given by equations (4)–(6), the total elastic stiffness Ktotalc, given by equation (3), is 200.8 kN/mm.Table6 lists the computed stiffness, measured stiffness, and the difference between these values for each specimen. The computed elastic stiffness values given in columns 2, 3, 4, and 5 were obtained from equations (3)–(6), respectively. The measured stiffness values given in columns 6 and 7 were computed by fitting the tensile section of the skeleton curve (using a double broken line model) of each brace. Column 6, labelled Ktotalm, corresponds to the slope of the axial force-displacement relationship measured for the elastic portion of the fitting skeleton curve, whereas, column 7, labelled K2, corresponds to the slope measured for the plastic portion of the fitting skeleton curve.Table 6
Comparison of computed and measured stiffness values.
Specimen
Computed elastic stiffness (kN/mm)
Measured stiffness (kN/mm)
Difference
K
y
c equation (4)
K
tr
c equation (5)
K
con
c equation (6)
K
total
c equation (3)
K
total
m
K
2
α
K
total (%)
BRB-1
275.3
5,829.6
2,658.6
200.8
199.86
8.15
0.041
−0.47
BRB-2
275.3
5,829.6
2,658.6
200.8
210.85
9.01
0.043
5
BRB-3
275.3
5,829.6
2,658.6
200.8
192.22
7.01
0.036
−4.27As can be seen from Table6, the computed total elastic stiffness value given by equation (3) is in close accordance with the value measured from the test data. In theory, the stiffness values used in the design should contain the entire portions of the brace. If the stiffness value for yielding portion be used in the design instead of the overall stiffness, then the stiffness value is overestimated by 37%. In addition, if assuming that the cross area over the entire length is equal to Ay, then the stiffness value is calculated to be 171.5 kN/mm and the stiffness value is still figured to be undervalued by 15%.


## 4.5. Secondary Stiffness

The secondary stiffness which usually can also be written as postyielding stiffness is of interest for the mechanical properties of the BRB and depends in general on the loading course. The measured values of secondary stiffness and postyielding ratio,α=K2/Ktotalm, of each specimen are given in columns 7 and 8 of Table 6, respectively. Column 8, labelled α, corresponds to the ratio of elastic stiffness to secondary stiffness.


## 4.6. Energy Dissipation Index

Energy dissipation capacity is an important index for evaluating the performance of CCA-BRB. In this section, the energy dissipation capacity of each specimen is evaluated via parameters which contain energy dissipation coefficient and equivalent viscous damping ratio. The parameters of the specimens were obtained from the hysteresis curves and are listed in Table7. According to Section 4.5.6 of JGJ/T101-2015 [29], the values for the energy dissipation coefficient and the equivalent viscous damping ratio are calculated by, respectively,(7)ψ=SBEDFBS△OAB+S△OCD,ζ=SBEDFB2πS△OAB+S△OCD,where SBEDFB, as shown in Figure 10, represents the hysteresis loop area of a brace enveloped by one displacement amplitude, S△OAB and S△OCD represent the area of triangle OAB and triangle OCD, respectively. It can be seen from Table 7 that the energy dissipation coefficient and the equivalent viscous damping ratio of each specimen show a trend of gradual increase with the increase of loading displacement amplitude. Further, the gradual increasing trend of energy dissipation indices indicates that the brace has a substantial and repeatable capability to absorb earthquake energy.Table 7
Ductility indices and energy dissipation indices of the specimens.
Specimen
Energy dissipation coefficient
Equivalent viscous damping ratio
Ductility coefficient
Cumulative plastic deformation
0.5%
1.0%
1.5%
2.0%
0.5%
1.0%
1.5%
2.0%
BRB-1
2.646
2.894
2.902
2.854
0.421
0.461
0.462
0.454
11.4
311.2
BRB-2
2.609
2.851
2.900
2.835
0.415
0.454
0.461
0.451
12.4
326.6
0.5%
0.75%
1.0%
1.5%
0.5%
0.75%
1.0%
1.5%
BRB-3
2.612
2.770
2.896
3.050
0.416
0.441
0.461
0.485
7.4
167.6Figure 10
Calculation diagram of energy dissipation index.


## 4.7. Ductility Coefficient

The deformation capacity which determines the maximum displacement of a brace is an important index for evaluating the seismic performance of CCA-BRB and is expressed via the ductility coefficient. The ductility coefficient of a brace refers to the ratio of the maximum displacement to the yield displacement of the brace before the obvious strength degradation of the hysteresis curve. The coefficients of each specimen are listed in Table7. According to Section 4.5.4 of JGJ/T101-2015 [29], the formula is given by(8)μ=dmaxdy,where dmax and dy represent maximum displacement and yield displacement, respectively, of the brace. It can be seen from Table 7 that the ductility coefficient values of BRB-1 and BRB-2 appear to be 11.4 and 12.4, respectively, once the core axial strain reaches 2.0%, and the value of BRB-3 appears to be 7.4, once the core axial strain reaches 1.5%. In addition, the specimens have a large and substantial plastic deformation capability.


## 4.8. Cumulative Plastic Deformation Capability

An index used in practice to evaluate the plastic deformation capability of a brace is the cumulative plastic strain or, alternatively, the cumulative plastic deformation. The cumulative plastic deformation [30] of the brace can be defined by(9)η=∑i=1n2dmax+i+dmax−idy−4,where dmax+i and dmax−i represent the absolute values of the maximum tensile displacement and the maximum compressive displacement, respectively, during each visit i into the inelastic range. The last column of Table 7 lists the cumulative plastic deformation values of each specimen. It can be seen from Table 7 that the values of BRB-1 and BRB-2 appear to be 311 and 327, once the core loading displacement amplitude reaches the axial strain of 2.0%, and the value of BRB-3 appears to be 167.6, once the core loading displacement amplitude reaches the axial strain of 1.5%. Further, it is noted that the cumulative plastic deformation values of BRB-1 and BRB-2 during the testing protocol exceed the requirement of the minimum limit (200) set by the ANSI/AISC 341-10, indicating that the brace has an excellent plastic deformation capability.


## 5. Conclusions

In this study, low-reversed cyclic loading tests of three CCA-BRB specimens under three loading protocols were conducted to consider the seismic performance, hysteretic behaviour, and energy dissipation capability of CCA-BRBs. Based on the experimental results and several evaluation indices, the main conclusions drawn are the following:(1)
The observed hysteretic behaviour of the CCA-BRB with a rigid connection was excellent for all displacement amplitudes, indicating that the brace had a stable capability to absorb seismic energy.(2)
The skeleton curves of three CCA-BRB specimens present distinct bilinear characteristics. The Ramberg–Osgood model was adopted to approximate the skeleton curve of the brace, and two model parameters were found to fit the skeleton curves with little difference, indicating that the Ramberg–Osgood model can be used to simulate the skeleton curve of CCA-BRB under cyclic load.(3)
The computed initial elastic stiffness was in close agreement with the value measured from the test data, indicating that the stiffness used in the design should contain overall portions of the brace.(4)
All CCA-BRB specimens showed a stable energy absorption performance to enhance the resistance of existing and new structures during strong earthquake shaking.



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*Source: 9710529-2019-04-01.xml*