# A New Buckling-Restrained Brace with a Variable Cross-Section Core

Authors: Liang Li; Tianhua Zhou; Junwu Chen; Jianfeng Chen
Journal: Advances in Civil Engineering
 (2019)

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## Abstract

In this paper, a new type of buckling-restrained brace characterized by a variable cross-section core (BRB-VCC) is proposed and investigated. The practical design equations of the BRB-VCC are derived based on mechanical and mathematical theories. Six specimens are designed and tested to clarify the mechanical behaviours of the BRB-VCC and to validate the reliability of the proposed equations. The test results show that (1) none of the specimens buckle under compression, as expected, and their ductilities and energy dissipation capacities are satisfactory; (2) the derived formulas are reliable and can be conveniently used in engineering practice; and (3) the yielding displacement of the BRB-VCC is approximately 70% that of the traditional TJ-1 buckling-restrained brace (BRB-TJ-1), which may yield earlier than the BRB-TJ-1 in concrete structures under the action of an earthquake.


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## Body

## 1. Introduction

A buckling-restrained brace (BRB) is typically composed of an exterior restrainer and an inner core, and the core does not buckle under compression due to the lateral support of the exterior restrainer [1–3]. The BRB has won the widespread favour of researchers in the past several decades, and many different types of BRBs have been developed by scholars worldwide [4, 5].In practical engineering, BRBs are widely used in concrete structures, and the failure mode of a concrete structure equipped with BRBs has attracted the attention of many researchers [6–10]. Wu et al. [11] investigated the seismic behaviour of a concrete frame with BRBs by performing a pseudo-static test; the results indicate that the RC frame is damaged earlier than BRBs. Bazaez and Dusicka [12] carried out large-scale experiments on RC frames equipped with diagonal BRBs, and it is found that the concrete components are severely damaged, although the BRBs resist a considerable amount of internal force. Liang and Li [13] presented a comprehensive investigation regarding the design approach and seismic behaviour of a steel-concrete hybrid structure, and the study results suggest that the BRBs effectively increase the stiffness and bearing capacity of the steel frame and decrease the seismic response of the structure in the form of energy dissipation devices.The destruction of the main concrete structure critically threatens the overall safety of a structure. Therefore, many scholars propose that the damage should be concentrated on the BRBs, and the main structure should remain in the elastic state [14–17]. Tabatabaei et al. [18] developed a new reduced length BRB (RLBRB) by dividing the core into a yielding core and nonyielding nonbuckling core; the results show that the relative length of the yielding core is related to the magnitude of the yielding displacement. Bruneau et al. [19] proposed a damage-controlled design approach, which assumes that all the damages are concentrated on the BRBs and the main vertical loading structure is in the elastic state. Guerrero et al. [20] proposed a performance-based design method of the building reinforced by BRBs, which can be used to promptly calculate the response of low-rise regular buildings. Barbagallo et al. [21] proposed a seismic design method for the concrete reinforcement structure with BRBs based on the EC8 code.From what has been discussed above, the yielding displacement of a BRB exerts an important influence on the seismic performance of the whole structure. However, the primary concerns of scholars are whether BRBs exhibit stable hysteric behaviour and high ductility [12, 22, 23], and methods to reduce the yielding displacement of BRBs are rarely considered [24–27].To summarize, the analysis of BRBs with a small yield displacement has important research significance and practical application value. Therefore, a novel buckling-restrained brace with a variable cross-section core (BRB-VCC) is proposed in this paper, whose yielding displacement is reduced because certain parts of the core yield earlier than the remaining parts of the core. The theoretical equations for predicting the yield strength, yield displacement, and axial stiffness of BRB-VCC are derived. Six specimens are tested to clarify the mechanical behaviours and energy dissipation capacities. The reliability of theoretical formulas is verified by experimental results. Finally, the behaviour of BRB-VCC and BRB-TJ-1 is compared by FEM analysis.


## 2. Composition of a Typical BRB-VCC

As shown in Figure1, a typical BRB-VCC is composed of an inner core and an outside restraining tube. The core is composed of several segments: connection segment, transition segment, flat-plate segment, and stiffened-plate segment. The axial sliding between the ribs and the tube can occur freely under reciprocating axial force. The ribs are welded on the core plate. It is known that welding process will generate welding stress, which may reduce the fatigue strength of steel. In order to guarantee the quality of the welding seams, the following three methods are suggested in this paper: (1) the welding seams distributed on both sides of the core plate should be welded symmetrically so that the welding deformations may offset each other; (2) block welding sequence should be taken to minimize the welding deformation of the core; and (3) preheating before welding or tempering after welding can also be used to relieve the welding stress.Figure 1
Schematics of a typical BRB-VCC. (a) BRB-VCC. (b) Core.
(a)
(b)


## 3. Theoretical Formulas

### 3.1. Mechanical Model

For simplicity, the bending stiffness of the external tube is assumed to be infinite, and thus, the external tube is replaced by a group of sliding supports in the mechanical model, as shown in Figure2. The origin of the reference coordinate system, x-y, is at the left side of the inner core. The distance x represents the distance from the cross section to the origin. The axial force F is positive during compression and negative during tension.Figure 2
Mechanical model of the core.


### 3.2. Buckling Capacity of the Flat-Plate Segment

As shown in Figure1, the flat-plate segment is adjacent to the stiffened-plate segment. Thus, the flat-plate can be simplified as a rectangular plate with free unloading edges and fixed loading edges, and the buckling of the flat-plate segment, as shown in Figure 3, should be prevented in the design process.Figure 3
Buckling of the flat-plate segment.According to the theory of elastic stability [2], the differential equation of a buckled plate under uniform compression can be expressed as(1)D∂4ω∂x4+2∂4ω∂x2∂y2+∂4ω∂y4+qx∂2ω∂x2=0,where D=Et3/121−ν2, which is the flexural rigidity of the plate; E is the elastic modulus; t is the thickness of the flat-plate segment; ν is Poisson’s ratio; ω is the deflection function; and qx is the magnitude of the compressive force per unit length of the edge.The deflection functionω can be represented by the following expression:(2)ω=A1−cos2πxl4,where A is constant and l4 is the length of the flat-plate segment along the x-axis.Considering the fixed boundary conditions of the flat-plate segment at the edges, specificallyx = 0 and x = l4, the buckling capacity of the flat-plate segment can be obtained as(3)Pcr,4=π2bD0.5l42,where b represents the width of the flat-plate.


### 3.3. Axial Strength of the Flat-Plate Segment

The yield strengthFy,4 and ultimate strength Fu,4 of the flat-plate segment can be obtained by the following equations [15]:(4)Fy,4=fyA4,Fu,4=fuA4,where fy is the yield strength of steel, fu is the ultimate strength of the core material, and A4 is the cross-section area of the flat-plate segment.


### 3.4. Maximum Length of the Flat-Plate Segment

From equation (3), we find that the buckling capacity of a flat-plate segment is a reciprocal function of the effective length 0.5l4. The buckling capacity gradually increases with decrease in the effective length. For a sufficiently short plate, the resistance capacity is determined by the yielding strength of the core. When Fy,4 = Pcr,4, the maximum length of the flat-plate segment can be obtained using the following expression:(5)l4,max=2πbDfyA4.


### 3.5. Maximum Cross-Section Area of the Stiffened-Plate Segment

The cross-sectional areas of the stiffened-plate segments within the length ofl3 and l5 are equal (see Figure 1(b)):(6)A3=A5,where A3 and A5 denote the cross-section areas of the inner core at segments l3 and l5, respectively.To ensure that the stiffened-plate segment yields before the split of the flat-plate segment, the axial yield strength of the stiffened-plate segment must be smaller than the ultimate strength of the flat-plate segment:(7)Fy,3=Fy,5=fyA3=fyA5<Fu,4.Therefore, the maximum cross-sectional area of the stiffened-plate segment can be determined:(8)A3,max=A5,max=fufyA4.


### 3.6. Axial Stiffness of BRB-VCC

#### 3.6.1. IdealizedF-δ Diagram

The idealizedF-δ diagram of the core is shown in Figure 4. The core initially exhibits a linear relationship between F and δ in the elastic state, in which the elastic stiffness is denoted by ke. When loading is continued, a point (Fy1, δy1) is reached at which the flat-plate segments yield; this point is called the first yield point. When the load F exceeds Fy1, the flat-plate segments reach the strain hardening stage, and the first plastic stiffness is symbolized by kp1. The axial force F is increased until the stiffened-plate segments yield, and this instant represents the second yield point (Fy2, δy2). If loading is continued, the stiffened-plate segments reach the strain hardening stage, and the second plastic stiffness kp2 can be identified.Figure 4
IdealizedF-δ diagram of the core.


#### 3.6.2. Elastic Stiffnesske

As shown in Figure2, the cross-sectional areas A2,x vary along the x-axis, and they can be obtained by(9)A2,x=A1−A1−A3x−l1l2,where A1 is the sectional area of the connection segment, l1 is the length of the connection segment, and l2 is the length of the transition segment.By integrating over the length ofli individually, the axial stretching or shortening of each segment under the action of axial F can be obtained using the following expressions:(10)δi=∫∑li−1∑liεidx=FliAiE,i=1,3,4,5,δ2=∫l1l1+l2ε2,xdx=FE∫l1l1+l21A2dx=Fl2ElnA3−lnA1A3−A1.For each segment, the elastic stiffness can be obtained by the formula(11)ki=Fδi=AiEli,i=1,3,4,5,k2=Fδ2=El2A3−A1lnA3−lnA1.Therefore, the elastic stiffness can be expressed as(12)ke=k1k2k3k4k5k1k2k3k4+2k5k2k3k4+k1k3k4+k1k2k4+k1k2k3.


#### 3.6.3. First Plastic Stiffnesskp1

By integrating over the length ofl4, the change in the plastic length of the flat-plate segment along the axial direction can be obtained:(13)δ4p=∫l1+l2+l3l1+l2+l3+l4ε4dx=∫l1+l2+l3l1+l2+l3+l4FEtA4dx=Fl4EtA4,where Et is the tangent modulus of the core material.The plastic stiffness of the flat-plate segment can be expressed as(14)k4p=EtA4l4.Substituting equation (14) in equation (12), the first plastic stiffness of the core equals(15)kp1=k1k2k3k4pk5k1k2k3k4p+2k5k2k3k4p+k1k3k4p+k1k2k44p+k1k2k3.


#### 3.6.4. Second Plastic Stiffnesskp2

When the axial forceF becomes equal to the second yield force Fy2, both the flat-plate segment and the stiffened-plate segment pass into the plastic state. The change in length of the stiffened-plate segment equals(16)δip=FliEtAi,i=3,5.The plastic stiffness of the stiffened-plate segment is(17)kip=EtAili,i=3,5.Therefore, the second plastic stiffness of the core can be obtained using(18)kp2=k1k2k3pk4pk5pk1k2k3pk4p+2k5pk2k3pk4p+k1k3pk4p+k1k2k4p+k1k2k3p.


## 3.1. Mechanical Model

For simplicity, the bending stiffness of the external tube is assumed to be infinite, and thus, the external tube is replaced by a group of sliding supports in the mechanical model, as shown in Figure2. The origin of the reference coordinate system, x-y, is at the left side of the inner core. The distance x represents the distance from the cross section to the origin. The axial force F is positive during compression and negative during tension.Figure 2
Mechanical model of the core.


## 3.2. Buckling Capacity of the Flat-Plate Segment

As shown in Figure1, the flat-plate segment is adjacent to the stiffened-plate segment. Thus, the flat-plate can be simplified as a rectangular plate with free unloading edges and fixed loading edges, and the buckling of the flat-plate segment, as shown in Figure 3, should be prevented in the design process.Figure 3
Buckling of the flat-plate segment.According to the theory of elastic stability [2], the differential equation of a buckled plate under uniform compression can be expressed as(1)D∂4ω∂x4+2∂4ω∂x2∂y2+∂4ω∂y4+qx∂2ω∂x2=0,where D=Et3/121−ν2, which is the flexural rigidity of the plate; E is the elastic modulus; t is the thickness of the flat-plate segment; ν is Poisson’s ratio; ω is the deflection function; and qx is the magnitude of the compressive force per unit length of the edge.The deflection functionω can be represented by the following expression:(2)ω=A1−cos2πxl4,where A is constant and l4 is the length of the flat-plate segment along the x-axis.Considering the fixed boundary conditions of the flat-plate segment at the edges, specificallyx = 0 and x = l4, the buckling capacity of the flat-plate segment can be obtained as(3)Pcr,4=π2bD0.5l42,where b represents the width of the flat-plate.


## 3.3. Axial Strength of the Flat-Plate Segment

The yield strengthFy,4 and ultimate strength Fu,4 of the flat-plate segment can be obtained by the following equations [15]:(4)Fy,4=fyA4,Fu,4=fuA4,where fy is the yield strength of steel, fu is the ultimate strength of the core material, and A4 is the cross-section area of the flat-plate segment.


## 3.4. Maximum Length of the Flat-Plate Segment

From equation (3), we find that the buckling capacity of a flat-plate segment is a reciprocal function of the effective length 0.5l4. The buckling capacity gradually increases with decrease in the effective length. For a sufficiently short plate, the resistance capacity is determined by the yielding strength of the core. When Fy,4 = Pcr,4, the maximum length of the flat-plate segment can be obtained using the following expression:(5)l4,max=2πbDfyA4.


## 3.5. Maximum Cross-Section Area of the Stiffened-Plate Segment

The cross-sectional areas of the stiffened-plate segments within the length ofl3 and l5 are equal (see Figure 1(b)):(6)A3=A5,where A3 and A5 denote the cross-section areas of the inner core at segments l3 and l5, respectively.To ensure that the stiffened-plate segment yields before the split of the flat-plate segment, the axial yield strength of the stiffened-plate segment must be smaller than the ultimate strength of the flat-plate segment:(7)Fy,3=Fy,5=fyA3=fyA5<Fu,4.Therefore, the maximum cross-sectional area of the stiffened-plate segment can be determined:(8)A3,max=A5,max=fufyA4.


## 3.6. Axial Stiffness of BRB-VCC

### 3.6.1. IdealizedF-δ Diagram

The idealizedF-δ diagram of the core is shown in Figure 4. The core initially exhibits a linear relationship between F and δ in the elastic state, in which the elastic stiffness is denoted by ke. When loading is continued, a point (Fy1, δy1) is reached at which the flat-plate segments yield; this point is called the first yield point. When the load F exceeds Fy1, the flat-plate segments reach the strain hardening stage, and the first plastic stiffness is symbolized by kp1. The axial force F is increased until the stiffened-plate segments yield, and this instant represents the second yield point (Fy2, δy2). If loading is continued, the stiffened-plate segments reach the strain hardening stage, and the second plastic stiffness kp2 can be identified.Figure 4
IdealizedF-δ diagram of the core.


### 3.6.2. Elastic Stiffnesske

As shown in Figure2, the cross-sectional areas A2,x vary along the x-axis, and they can be obtained by(9)A2,x=A1−A1−A3x−l1l2,where A1 is the sectional area of the connection segment, l1 is the length of the connection segment, and l2 is the length of the transition segment.By integrating over the length ofli individually, the axial stretching or shortening of each segment under the action of axial F can be obtained using the following expressions:(10)δi=∫∑li−1∑liεidx=FliAiE,i=1,3,4,5,δ2=∫l1l1+l2ε2,xdx=FE∫l1l1+l21A2dx=Fl2ElnA3−lnA1A3−A1.For each segment, the elastic stiffness can be obtained by the formula(11)ki=Fδi=AiEli,i=1,3,4,5,k2=Fδ2=El2A3−A1lnA3−lnA1.Therefore, the elastic stiffness can be expressed as(12)ke=k1k2k3k4k5k1k2k3k4+2k5k2k3k4+k1k3k4+k1k2k4+k1k2k3.


### 3.6.3. First Plastic Stiffnesskp1

By integrating over the length ofl4, the change in the plastic length of the flat-plate segment along the axial direction can be obtained:(13)δ4p=∫l1+l2+l3l1+l2+l3+l4ε4dx=∫l1+l2+l3l1+l2+l3+l4FEtA4dx=Fl4EtA4,where Et is the tangent modulus of the core material.The plastic stiffness of the flat-plate segment can be expressed as(14)k4p=EtA4l4.Substituting equation (14) in equation (12), the first plastic stiffness of the core equals(15)kp1=k1k2k3k4pk5k1k2k3k4p+2k5k2k3k4p+k1k3k4p+k1k2k44p+k1k2k3.


### 3.6.4. Second Plastic Stiffnesskp2

When the axial forceF becomes equal to the second yield force Fy2, both the flat-plate segment and the stiffened-plate segment pass into the plastic state. The change in length of the stiffened-plate segment equals(16)δip=FliEtAi,i=3,5.The plastic stiffness of the stiffened-plate segment is(17)kip=EtAili,i=3,5.Therefore, the second plastic stiffness of the core can be obtained using(18)kp2=k1k2k3pk4pk5pk1k2k3pk4p+2k5pk2k3pk4p+k1k3pk4p+k1k2k4p+k1k2k3p.


## 3.6.1. IdealizedF-δ Diagram

The idealizedF-δ diagram of the core is shown in Figure 4. The core initially exhibits a linear relationship between F and δ in the elastic state, in which the elastic stiffness is denoted by ke. When loading is continued, a point (Fy1, δy1) is reached at which the flat-plate segments yield; this point is called the first yield point. When the load F exceeds Fy1, the flat-plate segments reach the strain hardening stage, and the first plastic stiffness is symbolized by kp1. The axial force F is increased until the stiffened-plate segments yield, and this instant represents the second yield point (Fy2, δy2). If loading is continued, the stiffened-plate segments reach the strain hardening stage, and the second plastic stiffness kp2 can be identified.Figure 4
IdealizedF-δ diagram of the core.


## 3.6.2. Elastic Stiffnesske

As shown in Figure2, the cross-sectional areas A2,x vary along the x-axis, and they can be obtained by(9)A2,x=A1−A1−A3x−l1l2,where A1 is the sectional area of the connection segment, l1 is the length of the connection segment, and l2 is the length of the transition segment.By integrating over the length ofli individually, the axial stretching or shortening of each segment under the action of axial F can be obtained using the following expressions:(10)δi=∫∑li−1∑liεidx=FliAiE,i=1,3,4,5,δ2=∫l1l1+l2ε2,xdx=FE∫l1l1+l21A2dx=Fl2ElnA3−lnA1A3−A1.For each segment, the elastic stiffness can be obtained by the formula(11)ki=Fδi=AiEli,i=1,3,4,5,k2=Fδ2=El2A3−A1lnA3−lnA1.Therefore, the elastic stiffness can be expressed as(12)ke=k1k2k3k4k5k1k2k3k4+2k5k2k3k4+k1k3k4+k1k2k4+k1k2k3.


## 3.6.3. First Plastic Stiffnesskp1

By integrating over the length ofl4, the change in the plastic length of the flat-plate segment along the axial direction can be obtained:(13)δ4p=∫l1+l2+l3l1+l2+l3+l4ε4dx=∫l1+l2+l3l1+l2+l3+l4FEtA4dx=Fl4EtA4,where Et is the tangent modulus of the core material.The plastic stiffness of the flat-plate segment can be expressed as(14)k4p=EtA4l4.Substituting equation (14) in equation (12), the first plastic stiffness of the core equals(15)kp1=k1k2k3k4pk5k1k2k3k4p+2k5k2k3k4p+k1k3k4p+k1k2k44p+k1k2k3.


## 3.6.4. Second Plastic Stiffnesskp2

When the axial forceF becomes equal to the second yield force Fy2, both the flat-plate segment and the stiffened-plate segment pass into the plastic state. The change in length of the stiffened-plate segment equals(16)δip=FliEtAi,i=3,5.The plastic stiffness of the stiffened-plate segment is(17)kip=EtAili,i=3,5.Therefore, the second plastic stiffness of the core can be obtained using(18)kp2=k1k2k3pk4pk5pk1k2k3pk4p+2k5pk2k3pk4p+k1k3pk4p+k1k2k4p+k1k2k3p.


## 4. Design Process

To realize a successful design, the practical design procedure of the inner core, which includes the following steps, is described in a flowchart, as shown in Figure5.(1)
Select the initial geometric parameters of the BRB-VCC.(2)
Calculate the mechanical parameters by the theoretical formulas presented in Section3.(3)
Identify the failure mode of the flat-plate segment: ifFy4 ≤ Pcr,4, the BRB-VCC fails by the yielding of the flat-plate segments, as expected. Otherwise, the BRB-VCC fails by the buckling of the flat-plate segments. If so, return to step (1) to reselect the geometric parameters of the flat-plate segment, and iterate until Fy4 > Pcr,4.(4)
Check if the stiffened-plate segment yields before the flat-plate segment splits. In this case, if equation (7) is satisfied, the stiffened-plate segments pass into the yield state before the failure of the flat-plate segment. Otherwise, the BRB-VCC fails before the yielding of the stiffened-plate segments. In case this occurs, return to step (1) to reselect the cross section of the flat-plate segment and the stiffened-plate segment, and repeat until Fy3 = Fy4 ≤ Fu,4 is satisfied.Figure 5
Design process flowchart of BRB-VCC.


## 5. Test Programme

### 5.1. Specimen Details

The main objective of the specimen design is to make sure that the specimens may reveal the mechanical properties of BRB-VCC, and the dimensions of specimens should guarantee that well ductility and stable hysteretic behaviors can be achieved, and local buckling or overall buckling will not occur. The details of the specimens are shown in Figure6, and the geometric parameters of the six specimens are listed in Table 1. The properties of the material Q235B used for manufacturing the specimens are listed in Table 2.Figure 6
Details of specimens.Table 1
Geometric parameters of specimens (length unit: mm).
Series
Specimen ID
Length of segments
Rib
Core plate
Restraining tube
l
l
0
l
1
l
2
l
3
l
4
l
5
b
r1
b
r2
t
r
b
b
c
t
c
b
t
t
t
l
t
S-I
BRB-VCC-1 (2, 3)
360
39
16
10
30
10
150
12
9
6
30
24
6
32
4
230
S-II
BRB-VCC-4 (5, 6)
339
20
16
10
30
10
167
10
7
6
26
20
6
28
4
247Table 2
Material properties of Q235B.
Steel grade
Elastic modulusE (MPa)
Yield strengthfy (MPa)
Yield strainεy
Ultimate strengthfu (MPa)
Poisson’s ratioν
Q235B
1.91 × 105
251.3
0.0015
440
0.3


### 5.2. Test Setup and Loading Protocol

As shown in Figure7, the specimen is loaded at the push-pull fatigue tester (MTS 880). The quasi-static loading protocols for the specimens are shown in Figure 8. The axial loading is conducted at a loading speed of 0.1 mm/s. The quasi-static loading protocols for the specimens in the same series are different, as shown in Figure 8. Noticeably, BRB-VCC-1 experiences more loading circles than BRB-VCC-2 and BRB-VCC-3, and BRB-VCC-4 also experiences more loading circles than BRB-VCC-5 and BRB-VCC-6.Figure 7
Loading of the specimen.Figure 8
Cyclic loading protocol.


### 5.3. Test Results

#### 5.3.1. Damage Phenomenon

During testing, the inner core can shrink or stretch freely inside the restraining tube. No damage is found on the welding seams between the ribs and the flat-plate. The core does not buckle locally or globally under compression, and the restraining tube does not deform significantly, as shown in Figure9.Figure 9
BRB-VCC-1 after test. (a) Core. (b) Restraining tube.
(a)
(b)


#### 5.3.2. Hysteresis Response

The hysteresis curves of the six specimens are presented in Figures10 and 11. The energy dissipation capacities of the specimens are satisfactory. The hysteresis loop of BRB-VCC-1 is different from that of BRB-VCC-2 and BRB-VCC-3, and the hysteresis loop of BRB-VCC-4 is also different from that of BRB-VCC-5 and BRB-VCC-6. The primary reason is that BRB-VCC-1 and BRB-VCC-4 endured more loading circles, which led to more cumulative damage than other specimens in S-I and S-II, and the hysteresis loops of BRB-VCC-1 and BRB-VCC-4 pinch more severely.Figure 10
Hysteresis loops of S-I. (a) BRB-VCC-1. (b) BRB-VCC-2. (c) BRB-VCC-3.
(a)
(b)
(c)Figure 11
Hysteresis loops of S-II. (a) BRB-VCC-4. (b) BRB-VCC-5. (c) BRB-VCC-6.
(a)
(b)
(c)


#### 5.3.3. Skeleton Curves

The skeleton curves of the six specimens are shown in Figure12. It can be noted that the ductilities of the specimens are satisfactory. Table 3 lists the main mechanical parameters of the specimens. By comparing the specimens in two series, it is known that the axial capacity of BRB-VCC-1 is larger than that of BRB-VCC-4. It can be concluded that the capacity of entire BRB-VCC is bigger when the cross-sectional area of the flat-plate segment is bigger. The strength ratios Fc,u/Ft,u of the specimens do not exceed 1.3, which satisfies the criteria for BRBs specified in section K3 of ANSI/AISC 341-10 [9].Figure 12
Skeleton curves of specimens. (a) S-I (b) S-II.
(a)
(b)Table 3
Main experimental results of the specimens.
Series
Specimen
Tension
Compression
F
c,u/Ft,u
F
t,u (kN)
δ
t,u (mm)
Circles
F
c,u (kN)
δ
c,u (mm)
Circles
S-I
BRB-VCC-1
51.78
1.70
51
52.27
1.50
45
1.01
BRB-VCC-2
55.65
2.21
33
61.19
1.80
27
1.10
BRB-VCC-3
54.87
2.41
24
58.63
2.10
21
1.07
S-II
BRB-VCC-4
35.33
1.20
36
35.95
1.20
34
1.02
BRB-VCC-5
37.02
2.00
30
38.75
2.01
29
1.05
BRB-VCC-6
41.35
2.11
22
51.68
2.10
20
1.25


## 5.1. Specimen Details

The main objective of the specimen design is to make sure that the specimens may reveal the mechanical properties of BRB-VCC, and the dimensions of specimens should guarantee that well ductility and stable hysteretic behaviors can be achieved, and local buckling or overall buckling will not occur. The details of the specimens are shown in Figure6, and the geometric parameters of the six specimens are listed in Table 1. The properties of the material Q235B used for manufacturing the specimens are listed in Table 2.Figure 6
Details of specimens.Table 1
Geometric parameters of specimens (length unit: mm).
Series
Specimen ID
Length of segments
Rib
Core plate
Restraining tube
l
l
0
l
1
l
2
l
3
l
4
l
5
b
r1
b
r2
t
r
b
b
c
t
c
b
t
t
t
l
t
S-I
BRB-VCC-1 (2, 3)
360
39
16
10
30
10
150
12
9
6
30
24
6
32
4
230
S-II
BRB-VCC-4 (5, 6)
339
20
16
10
30
10
167
10
7
6
26
20
6
28
4
247Table 2
Material properties of Q235B.
Steel grade
Elastic modulusE (MPa)
Yield strengthfy (MPa)
Yield strainεy
Ultimate strengthfu (MPa)
Poisson’s ratioν
Q235B
1.91 × 105
251.3
0.0015
440
0.3


## 5.2. Test Setup and Loading Protocol

As shown in Figure7, the specimen is loaded at the push-pull fatigue tester (MTS 880). The quasi-static loading protocols for the specimens are shown in Figure 8. The axial loading is conducted at a loading speed of 0.1 mm/s. The quasi-static loading protocols for the specimens in the same series are different, as shown in Figure 8. Noticeably, BRB-VCC-1 experiences more loading circles than BRB-VCC-2 and BRB-VCC-3, and BRB-VCC-4 also experiences more loading circles than BRB-VCC-5 and BRB-VCC-6.Figure 7
Loading of the specimen.Figure 8
Cyclic loading protocol.


## 5.3. Test Results

### 5.3.1. Damage Phenomenon

During testing, the inner core can shrink or stretch freely inside the restraining tube. No damage is found on the welding seams between the ribs and the flat-plate. The core does not buckle locally or globally under compression, and the restraining tube does not deform significantly, as shown in Figure9.Figure 9
BRB-VCC-1 after test. (a) Core. (b) Restraining tube.
(a)
(b)


### 5.3.2. Hysteresis Response

The hysteresis curves of the six specimens are presented in Figures10 and 11. The energy dissipation capacities of the specimens are satisfactory. The hysteresis loop of BRB-VCC-1 is different from that of BRB-VCC-2 and BRB-VCC-3, and the hysteresis loop of BRB-VCC-4 is also different from that of BRB-VCC-5 and BRB-VCC-6. The primary reason is that BRB-VCC-1 and BRB-VCC-4 endured more loading circles, which led to more cumulative damage than other specimens in S-I and S-II, and the hysteresis loops of BRB-VCC-1 and BRB-VCC-4 pinch more severely.Figure 10
Hysteresis loops of S-I. (a) BRB-VCC-1. (b) BRB-VCC-2. (c) BRB-VCC-3.
(a)
(b)
(c)Figure 11
Hysteresis loops of S-II. (a) BRB-VCC-4. (b) BRB-VCC-5. (c) BRB-VCC-6.
(a)
(b)
(c)


### 5.3.3. Skeleton Curves

The skeleton curves of the six specimens are shown in Figure12. It can be noted that the ductilities of the specimens are satisfactory. Table 3 lists the main mechanical parameters of the specimens. By comparing the specimens in two series, it is known that the axial capacity of BRB-VCC-1 is larger than that of BRB-VCC-4. It can be concluded that the capacity of entire BRB-VCC is bigger when the cross-sectional area of the flat-plate segment is bigger. The strength ratios Fc,u/Ft,u of the specimens do not exceed 1.3, which satisfies the criteria for BRBs specified in section K3 of ANSI/AISC 341-10 [9].Figure 12
Skeleton curves of specimens. (a) S-I (b) S-II.
(a)
(b)Table 3
Main experimental results of the specimens.
Series
Specimen
Tension
Compression
F
c,u/Ft,u
F
t,u (kN)
δ
t,u (mm)
Circles
F
c,u (kN)
δ
c,u (mm)
Circles
S-I
BRB-VCC-1
51.78
1.70
51
52.27
1.50
45
1.01
BRB-VCC-2
55.65
2.21
33
61.19
1.80
27
1.10
BRB-VCC-3
54.87
2.41
24
58.63
2.10
21
1.07
S-II
BRB-VCC-4
35.33
1.20
36
35.95
1.20
34
1.02
BRB-VCC-5
37.02
2.00
30
38.75
2.01
29
1.05
BRB-VCC-6
41.35
2.11
22
51.68
2.10
20
1.25


## 5.3.1. Damage Phenomenon

During testing, the inner core can shrink or stretch freely inside the restraining tube. No damage is found on the welding seams between the ribs and the flat-plate. The core does not buckle locally or globally under compression, and the restraining tube does not deform significantly, as shown in Figure9.Figure 9
BRB-VCC-1 after test. (a) Core. (b) Restraining tube.
(a)
(b)


## 5.3.2. Hysteresis Response

The hysteresis curves of the six specimens are presented in Figures10 and 11. The energy dissipation capacities of the specimens are satisfactory. The hysteresis loop of BRB-VCC-1 is different from that of BRB-VCC-2 and BRB-VCC-3, and the hysteresis loop of BRB-VCC-4 is also different from that of BRB-VCC-5 and BRB-VCC-6. The primary reason is that BRB-VCC-1 and BRB-VCC-4 endured more loading circles, which led to more cumulative damage than other specimens in S-I and S-II, and the hysteresis loops of BRB-VCC-1 and BRB-VCC-4 pinch more severely.Figure 10
Hysteresis loops of S-I. (a) BRB-VCC-1. (b) BRB-VCC-2. (c) BRB-VCC-3.
(a)
(b)
(c)Figure 11
Hysteresis loops of S-II. (a) BRB-VCC-4. (b) BRB-VCC-5. (c) BRB-VCC-6.
(a)
(b)
(c)


## 5.3.3. Skeleton Curves

The skeleton curves of the six specimens are shown in Figure12. It can be noted that the ductilities of the specimens are satisfactory. Table 3 lists the main mechanical parameters of the specimens. By comparing the specimens in two series, it is known that the axial capacity of BRB-VCC-1 is larger than that of BRB-VCC-4. It can be concluded that the capacity of entire BRB-VCC is bigger when the cross-sectional area of the flat-plate segment is bigger. The strength ratios Fc,u/Ft,u of the specimens do not exceed 1.3, which satisfies the criteria for BRBs specified in section K3 of ANSI/AISC 341-10 [9].Figure 12
Skeleton curves of specimens. (a) S-I (b) S-II.
(a)
(b)Table 3
Main experimental results of the specimens.
Series
Specimen
Tension
Compression
F
c,u/Ft,u
F
t,u (kN)
δ
t,u (mm)
Circles
F
c,u (kN)
δ
c,u (mm)
Circles
S-I
BRB-VCC-1
51.78
1.70
51
52.27
1.50
45
1.01
BRB-VCC-2
55.65
2.21
33
61.19
1.80
27
1.10
BRB-VCC-3
54.87
2.41
24
58.63
2.10
21
1.07
S-II
BRB-VCC-4
35.33
1.20
36
35.95
1.20
34
1.02
BRB-VCC-5
37.02
2.00
30
38.75
2.01
29
1.05
BRB-VCC-6
41.35
2.11
22
51.68
2.10
20
1.25


## 6. Verification of Theory

Table4 lists the main mechanical properties of the specimens, as obtained from the test and formulas. The experimental results agree well with the calculation results. Therefore, the suggested formulas can be considered reliable, and they can be used to predict the main mechanical parameters of BRB-VCCs in practical engineering.Table 4
Comparison of theoretical and test results.
Series
Specimen
k
e (kN/mm)
F
y (kN)
δ
y (mm)
Test
Formula
Error (%)
Test
Formula
Error (%)
Test
Formula
Error (%)
S-I
BRB-VCC-1
130.2
131.5
0.10
35.2
36.1
2.74
0.29
0.28
1.82
BRB-VCC-2
133.5
1.49
36.0
0.33
0.30
2.18
BRB-VCC-3
139.4
5.63
36.6
1.33
0.28
4.36
S-II
BRB-VCC-4
110.4
115.2
4.19
30.7
30.1
1.83
0.23
0.24
4.16
BRB-VCC-5
115.6
0.29
31.5
4.48
0.24
0
BRB-VCC-6
109.4
5.02
31.7
5.02
0.23
4.16


## 7. Finite Element Analysis

### 7.1. Reliability of ABAQUS Analysis

Specimens are simulated using the ABAQUS software by employing the solid element C3D8R. The interaction between the core and the tube is simulated by defining the contact. The normal behavior of the contact is defined as “hard” contact, and the tangential behavior of the contact is smooth. The stress-strain relation of Q235B is bilinear in ABAQUS, and the material properties are listed in Table2.The hysteresis loops of BRB-VCC-3 by test and by ABAQUS are in good agreement, as shown in Figure13, which proves that the FEM result is reliable. The Mises stress distribution of the restraining tube and core is shown in Figure 14, which demonstrates that the flat-plate segments indeed yield earlier than the rest of core.Figure 13
Comparison of hysteresis loops of BRB-VCC-3.Figure 14
Mises stress contours of the specimen. (a) Restraining tube (σmax = 95.36 N/mm2 < fy). (b) Inner core (σmax = 403.5 N/mm2 > fy).
(a)
(b)


### 7.2. Comparison with BRB-TJ-1

#### 7.2.1. Design of Models

The BRB-VCCs are compared with a traditional TJ-I buckling-restrained brace (BRB-TJ-I) by FEM analysis. The BRB-TJ-I, invented by Professor G-Q Li at Tongji University, has been successfully applied in many important projects in China. The constitutions of BRB-TJ-I and BRB-VCCs are presented in Figure15, and their geometric parameters are listed in Table 5. The steel used is Q235B, and the material properties are listed in Table 2.Figure 15
Model details. (a) BRB-TJ-I. (b) BRB-VCC.
(a)
(b)Table 5
Design parameters of models (unit: mm).
Model
Restraining tube
Core
End rib
Middle rib
Flat-plate segment
L
t × B1 × B2 × t1
L × b1× b2 × t2
L
d1 × Ld2 × H1 × H2 × t3
L
s × H2 × t3
L
f
Number
BRB-TJ-I
1080 × 102 × 94 × 12
1600 × 100 × 80 × 10
180 × 60 × 45 × 30 × 8
—
—
—
BRB-VCC-7
520 × 30 × 8
80
1
BRB-VCC-8
346 × 30 × 8
40
2
BRB-VCC-9
208 × 30 × 8
20
4The exact method for establishing the finite element model in this Section is the same as the method in Section7.1. The whole model is shown in Figure 16.Figure 16
Finite element model in software ABAQUS.


#### 7.2.2. Yield Regions in Models

The yield points of models can be identified by observing the dynamic images of parameter AC YIELD in software ABAQUS. For BRB-TJ-I, all regions of the core yield simultaneously when the axial force arrives at the yield capacity, as shown in Figure17.Figure 17
AC yield of the BRB-TJ-I core (δy = 1.86 mm and Fy = 195.6 kN). Note: AC YIELD is 1.0 in the plastic region and is 0 in the elastic region.For BRB-VCCs, the flat-plate segment yields firstly while the rest of the segments of the core are still in elastic, as shown in Figures18(a), 19(a), and 20(a). The stiffened-plate segments yield when the axial force arrived at its yield capacity, as shown in Figures 18(b), 19(b), and 20(b).Figure 18
AC yield of the BRB-VCC-7 core. (a) Flat-plate segments yield firstly (δy1 = 1.35 mm and Fy1 = 201.6 kN). (b) Stiffened-plate segments yield soon later (δy2 = 1.52 mm and Fy2 = 333.4 kN). Note: AC YIELD is 1.0 in the plastic region and is 0 in the elastic region.
(a)
(b)Figure 19
AC YIELD of the BRB-VCC-8 core. (a) Flat-plate segments yield firstly (δy1 = 1.37 and Fy1 = 198.4 kN). (b) Stiffened-plate segments yield soon later (δy2 = 1.81and Fy2 = 325.9 kN). Note: AC yield is 1.0 in the plastic region and is 0 in the elastic region.
(a)
(b)Figure 20
AC yield of the BRB-VCC-9 core. (a) Flat-plate segments yield firstly (δy1 = 1.37 mm and Fy1 = 224.2 kN). (b) Stiffened-plate segments yield soon later (δy2 = 1.92 mm and Fy2 = 300.9 kN). Note: AC YIELD is 1.0 in the plastic region and is 0 in the elastic region.
(a)
(b)


#### 7.2.3. Comparing of Mechanical Performance

The hysteretic curves of models are shown in Figure21. There is only one yield point in hysteretic curves of BRB-TJ-I, as marked clearly in Figure 21(a). Noticeably, there are two yield points in the curves of BRB-VCCs, and their exact positions have been marked in Figures 21(b)–21(d).Figure 21
Hysteresis curves of models. (a) BRB-TJ-I. (b) BRB-VCC-7. (c) BRB-VCC-8. (d) BRB-VCC-9.
(a)
(b)
(c)
(d)The main mechanical properties of the models are listed in Table6. The following conclusions are found: (1) the yielding displacements of BRB-VCCs are approximately 73% compared to those of the BRB-TJ-I. (2) The interforces of BRB-VCCs increase faster than BRB-TJ-I after the yield of the flat-plate segment. (3) Different from the BRB-TJ-I, the hysteretic curves of BRB-VCCs show slight pinch after the second yield point, as shown in Figure 21.Table 6
Comparisons of FEM analysis results.
Model
k
e (kN/mm)
First yield point
Second yield point
Ratio of BRB-VCCs to BRB-TJ-І (%)
F
y1
δ
y1
F
y2
δ
y2
k
e
F
y1
δ
y1
BRB-TJ-I
117.8
195.6
1.86
—
—
—
—
—
BRB-VCC-7
154.9
201.6
1.35
333.4
1.52
1.31
1.03
72.5
BRB-VCC-8
148.5
198.4
1.37
325.9
1.81
1.26
1.01
73.6
BRB-VCC-9
141.2
200.2
1.37
300.9
1.92
1.20
1.02
73.6


## 7.1. Reliability of ABAQUS Analysis

Specimens are simulated using the ABAQUS software by employing the solid element C3D8R. The interaction between the core and the tube is simulated by defining the contact. The normal behavior of the contact is defined as “hard” contact, and the tangential behavior of the contact is smooth. The stress-strain relation of Q235B is bilinear in ABAQUS, and the material properties are listed in Table2.The hysteresis loops of BRB-VCC-3 by test and by ABAQUS are in good agreement, as shown in Figure13, which proves that the FEM result is reliable. The Mises stress distribution of the restraining tube and core is shown in Figure 14, which demonstrates that the flat-plate segments indeed yield earlier than the rest of core.Figure 13
Comparison of hysteresis loops of BRB-VCC-3.Figure 14
Mises stress contours of the specimen. (a) Restraining tube (σmax = 95.36 N/mm2 < fy). (b) Inner core (σmax = 403.5 N/mm2 > fy).
(a)
(b)


## 7.2. Comparison with BRB-TJ-1

### 7.2.1. Design of Models

The BRB-VCCs are compared with a traditional TJ-I buckling-restrained brace (BRB-TJ-I) by FEM analysis. The BRB-TJ-I, invented by Professor G-Q Li at Tongji University, has been successfully applied in many important projects in China. The constitutions of BRB-TJ-I and BRB-VCCs are presented in Figure15, and their geometric parameters are listed in Table 5. The steel used is Q235B, and the material properties are listed in Table 2.Figure 15
Model details. (a) BRB-TJ-I. (b) BRB-VCC.
(a)
(b)Table 5
Design parameters of models (unit: mm).
Model
Restraining tube
Core
End rib
Middle rib
Flat-plate segment
L
t × B1 × B2 × t1
L × b1× b2 × t2
L
d1 × Ld2 × H1 × H2 × t3
L
s × H2 × t3
L
f
Number
BRB-TJ-I
1080 × 102 × 94 × 12
1600 × 100 × 80 × 10
180 × 60 × 45 × 30 × 8
—
—
—
BRB-VCC-7
520 × 30 × 8
80
1
BRB-VCC-8
346 × 30 × 8
40
2
BRB-VCC-9
208 × 30 × 8
20
4The exact method for establishing the finite element model in this Section is the same as the method in Section7.1. The whole model is shown in Figure 16.Figure 16
Finite element model in software ABAQUS.


### 7.2.2. Yield Regions in Models

The yield points of models can be identified by observing the dynamic images of parameter AC YIELD in software ABAQUS. For BRB-TJ-I, all regions of the core yield simultaneously when the axial force arrives at the yield capacity, as shown in Figure17.Figure 17
AC yield of the BRB-TJ-I core (δy = 1.86 mm and Fy = 195.6 kN). Note: AC YIELD is 1.0 in the plastic region and is 0 in the elastic region.For BRB-VCCs, the flat-plate segment yields firstly while the rest of the segments of the core are still in elastic, as shown in Figures18(a), 19(a), and 20(a). The stiffened-plate segments yield when the axial force arrived at its yield capacity, as shown in Figures 18(b), 19(b), and 20(b).Figure 18
AC yield of the BRB-VCC-7 core. (a) Flat-plate segments yield firstly (δy1 = 1.35 mm and Fy1 = 201.6 kN). (b) Stiffened-plate segments yield soon later (δy2 = 1.52 mm and Fy2 = 333.4 kN). Note: AC YIELD is 1.0 in the plastic region and is 0 in the elastic region.
(a)
(b)Figure 19
AC YIELD of the BRB-VCC-8 core. (a) Flat-plate segments yield firstly (δy1 = 1.37 and Fy1 = 198.4 kN). (b) Stiffened-plate segments yield soon later (δy2 = 1.81and Fy2 = 325.9 kN). Note: AC yield is 1.0 in the plastic region and is 0 in the elastic region.
(a)
(b)Figure 20
AC yield of the BRB-VCC-9 core. (a) Flat-plate segments yield firstly (δy1 = 1.37 mm and Fy1 = 224.2 kN). (b) Stiffened-plate segments yield soon later (δy2 = 1.92 mm and Fy2 = 300.9 kN). Note: AC YIELD is 1.0 in the plastic region and is 0 in the elastic region.
(a)
(b)


### 7.2.3. Comparing of Mechanical Performance

The hysteretic curves of models are shown in Figure21. There is only one yield point in hysteretic curves of BRB-TJ-I, as marked clearly in Figure 21(a). Noticeably, there are two yield points in the curves of BRB-VCCs, and their exact positions have been marked in Figures 21(b)–21(d).Figure 21
Hysteresis curves of models. (a) BRB-TJ-I. (b) BRB-VCC-7. (c) BRB-VCC-8. (d) BRB-VCC-9.
(a)
(b)
(c)
(d)The main mechanical properties of the models are listed in Table6. The following conclusions are found: (1) the yielding displacements of BRB-VCCs are approximately 73% compared to those of the BRB-TJ-I. (2) The interforces of BRB-VCCs increase faster than BRB-TJ-I after the yield of the flat-plate segment. (3) Different from the BRB-TJ-I, the hysteretic curves of BRB-VCCs show slight pinch after the second yield point, as shown in Figure 21.Table 6
Comparisons of FEM analysis results.
Model
k
e (kN/mm)
First yield point
Second yield point
Ratio of BRB-VCCs to BRB-TJ-І (%)
F
y1
δ
y1
F
y2
δ
y2
k
e
F
y1
δ
y1
BRB-TJ-I
117.8
195.6
1.86
—
—
—
—
—
BRB-VCC-7
154.9
201.6
1.35
333.4
1.52
1.31
1.03
72.5
BRB-VCC-8
148.5
198.4
1.37
325.9
1.81
1.26
1.01
73.6
BRB-VCC-9
141.2
200.2
1.37
300.9
1.92
1.20
1.02
73.6


## 7.2.1. Design of Models

The BRB-VCCs are compared with a traditional TJ-I buckling-restrained brace (BRB-TJ-I) by FEM analysis. The BRB-TJ-I, invented by Professor G-Q Li at Tongji University, has been successfully applied in many important projects in China. The constitutions of BRB-TJ-I and BRB-VCCs are presented in Figure15, and their geometric parameters are listed in Table 5. The steel used is Q235B, and the material properties are listed in Table 2.Figure 15
Model details. (a) BRB-TJ-I. (b) BRB-VCC.
(a)
(b)Table 5
Design parameters of models (unit: mm).
Model
Restraining tube
Core
End rib
Middle rib
Flat-plate segment
L
t × B1 × B2 × t1
L × b1× b2 × t2
L
d1 × Ld2 × H1 × H2 × t3
L
s × H2 × t3
L
f
Number
BRB-TJ-I
1080 × 102 × 94 × 12
1600 × 100 × 80 × 10
180 × 60 × 45 × 30 × 8
—
—
—
BRB-VCC-7
520 × 30 × 8
80
1
BRB-VCC-8
346 × 30 × 8
40
2
BRB-VCC-9
208 × 30 × 8
20
4The exact method for establishing the finite element model in this Section is the same as the method in Section7.1. The whole model is shown in Figure 16.Figure 16
Finite element model in software ABAQUS.


## 7.2.2. Yield Regions in Models

The yield points of models can be identified by observing the dynamic images of parameter AC YIELD in software ABAQUS. For BRB-TJ-I, all regions of the core yield simultaneously when the axial force arrives at the yield capacity, as shown in Figure17.Figure 17
AC yield of the BRB-TJ-I core (δy = 1.86 mm and Fy = 195.6 kN). Note: AC YIELD is 1.0 in the plastic region and is 0 in the elastic region.For BRB-VCCs, the flat-plate segment yields firstly while the rest of the segments of the core are still in elastic, as shown in Figures18(a), 19(a), and 20(a). The stiffened-plate segments yield when the axial force arrived at its yield capacity, as shown in Figures 18(b), 19(b), and 20(b).Figure 18
AC yield of the BRB-VCC-7 core. (a) Flat-plate segments yield firstly (δy1 = 1.35 mm and Fy1 = 201.6 kN). (b) Stiffened-plate segments yield soon later (δy2 = 1.52 mm and Fy2 = 333.4 kN). Note: AC YIELD is 1.0 in the plastic region and is 0 in the elastic region.
(a)
(b)Figure 19
AC YIELD of the BRB-VCC-8 core. (a) Flat-plate segments yield firstly (δy1 = 1.37 and Fy1 = 198.4 kN). (b) Stiffened-plate segments yield soon later (δy2 = 1.81and Fy2 = 325.9 kN). Note: AC yield is 1.0 in the plastic region and is 0 in the elastic region.
(a)
(b)Figure 20
AC yield of the BRB-VCC-9 core. (a) Flat-plate segments yield firstly (δy1 = 1.37 mm and Fy1 = 224.2 kN). (b) Stiffened-plate segments yield soon later (δy2 = 1.92 mm and Fy2 = 300.9 kN). Note: AC YIELD is 1.0 in the plastic region and is 0 in the elastic region.
(a)
(b)


## 7.2.3. Comparing of Mechanical Performance

The hysteretic curves of models are shown in Figure21. There is only one yield point in hysteretic curves of BRB-TJ-I, as marked clearly in Figure 21(a). Noticeably, there are two yield points in the curves of BRB-VCCs, and their exact positions have been marked in Figures 21(b)–21(d).Figure 21
Hysteresis curves of models. (a) BRB-TJ-I. (b) BRB-VCC-7. (c) BRB-VCC-8. (d) BRB-VCC-9.
(a)
(b)
(c)
(d)The main mechanical properties of the models are listed in Table6. The following conclusions are found: (1) the yielding displacements of BRB-VCCs are approximately 73% compared to those of the BRB-TJ-I. (2) The interforces of BRB-VCCs increase faster than BRB-TJ-I after the yield of the flat-plate segment. (3) Different from the BRB-TJ-I, the hysteretic curves of BRB-VCCs show slight pinch after the second yield point, as shown in Figure 21.Table 6
Comparisons of FEM analysis results.
Model
k
e (kN/mm)
First yield point
Second yield point
Ratio of BRB-VCCs to BRB-TJ-І (%)
F
y1
δ
y1
F
y2
δ
y2
k
e
F
y1
δ
y1
BRB-TJ-I
117.8
195.6
1.86
—
—
—
—
—
BRB-VCC-7
154.9
201.6
1.35
333.4
1.52
1.31
1.03
72.5
BRB-VCC-8
148.5
198.4
1.37
325.9
1.81
1.26
1.01
73.6
BRB-VCC-9
141.2
200.2
1.37
300.9
1.92
1.20
1.02
73.6


## 8. Conclusions

According to the theoretical, experimental, and FEM study results obtained in this paper, the following conclusions can be drawn:(1)
The proposed design equations are reliable and can be used conveniently in practical engineering(2)
The hysteretic behaviours and ductilities of the BRB-VCC specimens are satisfactory, and the strength ratiosFc,u/Ft,u of the specimens do not exceed 1.3, which satisfies the criteria for BRBs defined in section K3 of ANSI/AISC 341-10(3)
The yielding displacement of BRB-VCC is approximately 70% that of BRB-TJ-I, which may yield earlier than BRB-TJ-I in concrete structures under earthquake action



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*Source: 4620430-2019-12-26.xml*