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Experimental and numerical studies of a controlled rocking steel frame with buckling-restrained columns
Structures 24 (2020) 690–704
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Experimental and numerical studies of a controlled rocking steel frame with buckling-restrained columns
T
⁎
Qing Jianga,b, Hanqin Wanga, Yulong Fenga,b, , Xun Chonga,b, Xiaowei Wanga, Yi Zhua a b
School of Civil Engineering, Hefei University of Technology, Anhui Province 230009, China Anhui Civil Engineering Structures and Materials Laboratory, Anhui Province 230009, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Controlled rocking steel frame Buckling-restrained column Seismic performance Pseudo-static test Out-of-plane deformation
A non-uplifting rocking steel frame with buckling-restrained columns (RSFB) is formed by replacing two side columns in the bottom story of a steel concentrically braced frame with two buckling-restrained columns (BRCs). During major earthquakes, the BRCs first yield to dissipate energy while the other part of the system remains elastic and rotates around the bottom of the middle column to control structural deformation. A pseudo-static test was conducted on the earthquake-resilient RSFB, observing larger out-of-plane deformation occurred in the RSFB first story and the embedment portions of BRCs due to the failure of the out-of-plane restraints. The results indicate that the RSFB needs a stronger out-of-plane restraint to ensure structural performance. A design suggestion for the out-of-plane restraint is subsequently provided. A multi-scale finite element model was established in ABAQUS and validated by the test results, and simulation analyses of the test specimen with full out-ofplane restraints were conducted. The simulation results of this model showed that the hysteresis curve is full under cyclic loads, the lateral drifts are uniformly distributed along the structural height, and the plastic deformation is mainly concentrated in the BRCs energy dissipating portion, implying that the proposed RSFB can yield excellent seismic performance, as expected. Finally, the influences of the BRC embedment lengths on the structural performance and out-of-plane deformation were studied to enhance the out-of-plane stability of the RSFB without full out-of-plane restraints.
1. Introduction Moment-resisting frames (MRFs) and concentrically braced frames (CBFs) have been widely used in steel buildings as lateral force resisting systems. However, some studies noted that damage concentration issues and soft story failure exist in the aforementioned systems, which has led to severe structural damage under horizontal seismic loads [1–3]. Accordingly, various rocking systems have been developed and added to steel buildings to control the damage distribution in steel MRFs and CBFs. To reduce structural damage, Midorikawa et al. [4] proposed an uplifting steel rocking system, which releases the constraint between the columns and the foundation and is connected with a yielding base plate. Shaking table test results showed that the columns can uplift to prompt yielding in the yielding base plate, thereby dissipating energy. Tremblay et al. proposed a similar system [5–6]. Based on the uplifting system, Eatherton et al. [7] and Ma et al. [8] and Sause et al. [9] proposed a series of controlled rocking systems with posttensioning strands to construct a self-centering system that not only
⁎
controls the structural damage distribution but also reduces the structural residual lateral drift, and they performed a series of analyses that verified the capacity of this self-centering system. López-Barraza et al. [10] and Reyes-Salazar et al. [11–12] proposed a self-centering system with posttensioned strands to reduce the residual deformation of MRF structure after earthquakes. To reduce structural seismic responses amplified by the rocking method of uplifting under high structural modes [13], Blebo and David [14] developed a pin-supported rocking core system. In contrast to the uplifting system, this structure can rotate around the stationary pinned support at the middle column base under horizontal seismic loads. To further enhance the energy dissipating capacity of the steel rocking system, Takeuchi et al. [15] and Blebo and David [16] employed two buckling-restrained braces (BRBs) on the side of the bottom story to use the displacement generated by the rocking system rotation. Feng et al. [17] installed two replaceable BRBs at the base of a rocking wall to form an earthquake-resilient shear wall and used this wall to enhance the seismic performance of the steel MRF. This study found that the rocking wall with BRBs in the base can effectively control the structural
Corresponding author at: School of Civil Engineering, Hefei University of Technology, Anhui Province 230009, China. E-mail address: [email protected] (Y. Feng).
https://doi.org/10.1016/j.istruc.2020.02.005 Received 19 November 2019; Received in revised form 4 February 2020; Accepted 8 February 2020 2352-0124/ © 2020 Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.
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Fig. 1. Description of an RSFB and a structure composed of a main frame and the RSFB.
connected with the foundation by the same method); the foundation and rigid support are steel members. The cross-section of the foundation and the design of the rigid support are shown in Fig. 2 and Fig. 4d, respectively. Fig. 3 schematically shows the overall composition of a BRC, which contains a core plate, restraining plates and filling plates (the material types and member sizes of all compositions can be seen in Table 2 and Fig. 4, respectively). The BRC is divided into the elastic portion, embedment portion and energy dissipating portion, and the lengths of these three portions are L1 = 140 mm, Lep = 60 mm and L2 = 483 mm, respectively, which means that the total length of the BRC is 883 mm. The preset compression gap is the distance between the stiffener and the restraining plate, and the length of the compression gap (Lcg) is 20 mm. The restraining plate and filling plate are connected by class 10.9 high-strength bolts with a diameter of 10 mm. The gap between the core plate and the restraining plate or filling plate is 1 mm and is filled with butyl rubber [20], as shown in Fig. 5a. Chen [21] noted that to prevent slipping of the restraining members (i.e., the restraining plates and filling plates), the stopper is set at the middle by enlarging the width of the core plate. The influence of the arc-shaped stopper on the distribution of the core plate stress is lower than other methods; thus, the arc-shaped stopper is used in this test specimen, as shown in Fig. 5b.
lateral deformation and dissipate the seismic energy. This paper proposes a rocking steel frame with buckling-restrained columns (RSFB) based on the works of Takeuchi et al. [15], Blebo and David [16], and Feng et al. [17], as shown in Fig. 1a. An RSFB is constructed by replacing the side columns at the bottom of a steel CBF with BRBs that are rigidly connected to the columns, and these BRBs are defined as buckling-restrained columns (BRCs). Fig. 1b schematically shows the composite structure formed by an RSFB and main structure that are connected with a rigid link. Under horizontal seismic loads, an RSFB can rotate around the middle column base to control the lateral drift of the main structure and concentrate seismic energy in the BRCs, and the BRC core plate is deformed to dissipate energy and reduce the main structural damage. However, few experimental studies of RSFBs have been reported. This paper performs a pseudo-static test and many simulation analyses of the proposed RSFB. The test results showed that the capacity of the test specimen cannot be fully used because of structural out-ofplane instability. Further, the performance of the test specimen with full out-of-plane restraints is studied using a numerical simulation method as a supplement to the experimental research. Finally, the influences of the BRC embedment portion length on the structural seismic performance and out-of-plane deformation are analyzed to enhance the outof-plane stability of the RSFB without full out-of-plane restraints, which might be helpful for retesting the RSFB.
2.2. Material properties 2. Test program Table 3 presents the steel material properties of the main compositions, in which the material test specimens were plate samples that were cut from the corresponding compositions. Various properties of the steel are calculated in accordance with Chinese code (GB/T228.12010) [22].
2.1. Test specimen Fig. 2 shows the pseudo-static test specimen, and the dimensional parameters of the RSFB mainly include two spans (B = 1000 mm) and three stories (h1 = 1400 mm, h2 = h3 = 1000 mm). Table 1 shows the sizes of the members and material types of the specimen, which are selected in accordance with the Chinese codes GB/T11263-2010 [18] and GB50017-2003 [19], respectively. All beam-to-column and brace joints are welded to achieve rigid connections, and the connection between the BRCs and the columns or BRC supports are also welded. To better acquire the deformation of the rigid support flange when the RSFB rotates, the rigid support that has the same member size as the column is manufactured individually and then connected with the column and foundation by high-strength bolts (the BRC supports are
2.3. Test setup and loading protocol 2.3.1. Test setup Fig. 6a shows the test specimen installation, which is fixed on the channel on the ground of the laboratory. The loading point is located at the center of the beam-to-column joint in the third story of the RSFB, and the other side of the actuator is connected to the reaction wall. To prevent slipping of the specimen, a support beam and a reaction beam are set on the left and right sides of the specimen foundation and 691
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Fig. 2. Sketch of the test specimen (unit: mm).
Moreover, three SGs are set on one side of the rigid support flange; these SGs are numbered from SG 37 to SG 39 (SG 37A to SG 39A are on the other side of the rigid support flange). The loading force (F) and top displacement of the RSFB (Δ) are recorded by the loading system.
Table 1 Member information. Member name
Number
Member type
Size
Material
Column
KZ
H-beam
Q345B
Beam
KL
H-beam
Steel brace Stiffener Foundation BRC Rigid/BRC support
– – – BRC –
Double angle Steel plate – – –
HN 200 × 100 × 5.5 × 8 (mm) HN 200 × 100 × 5.5 × 8 (mm) 2L90 × 90 × 8 (mm) Plate thickness 8 (mm) Plate thickness 25 (mm) Shown in Table 2 –
2.3.2. Loading protocol The loading protocol is divided into two phases in accordance with the Chinese code (specification for the seismic test of buildings JGJ/ T101-2015) [23], as shown in Fig. 7. Force control and displacement control are used before and after structural yielding, respectively. For force control, the loading increment is 10 kN, and each amplitude is cycled one time. For displacement control, the loading amplitude displacements of each stage are 10 mm, 20 mm, 30 mm, 40 mm, 50 mm, and 60 mm, and each stage is cycled three times.
Q345B Q235B Q345B Q345B – Q345B
Notes: HN 200 × 100 × 5.5 × 8 means that the sectional height, width, the thickness of web and flange of H-beam are 200 mm, 100 mm, 5.5 mm and 8 mm, respectively. L90 × 90 × 8 means that the width and thickness of angle steel are 90 mm and 8 mm, respectively.
3. Test results and analyses 3.1. Specimen failure modes
connected with jacks. Four out-of-plane restraints are arranged at the first and second-story beams to prevent structural out-of-plane deformation. The construction details of the out-of-plane restraints are shown in Fig. 6b and 6c. The main data acquisition points are shown in Fig. 6a. The interstory drifts of the RSFB are obtained from linear variable displacement transducers (LVDTs) 2–4, whereas LVDTs 1, 5 and 6 are used to judge whether slipping and uplifting occurred in the specimen. Six strain gauges (SGs) are arranged on the brace surface at each span and story, as shown in Fig. 6a, and these SGs are numbered from SG 1 to SG 36.
Fig. 8 shows the deformation and failure of the out-of-plane restraints. Bending deformations of the screws that are welded on the pulley and used to connect the steel sleeve and pulley in the out-ofplane restraints are seen in the first story. Moreover, friction traces appear on the steel plate that is welded onto the KL (i.e., the beams in RSFB) and in contact with the pulley. The abovementioned phenomena mean that the designed out-of-plane restraints cannot provide effective constraint to prevent the occurrence of structural out-of-plane deformation. Fig. 8b and Fig. 8c show the deformation and failure modes 692
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Cross-section is shown in Fig.5a
BRC stiffener
Embedment portion (Lep)
Elastic portion (L1)
Filling plate
Restraining plate
Core plate
Compression gap (Lcg)
Energy dissipating portion (L2)
Lep
Lep
Elastic portion (L1)
Fig. 3. BRC structure.
Table 2 BRC component information.
10
Member name
Size
Thickness
Material
Core plate Restraining plate Filling plate BRC stiffener
Shown in Fig. 4a Shown in Fig. 4b Shown in Fig. 4c 90 mm × 180 mm
8 mm 16 mm 10 mm 16 mm
Q235B Q345B Q345B Q345B
Butyl rubber
20 20 10
(a) Cross-section of the BRC
(b) Stopper detail
Fig. 5. Details of the BRC cross-section and stopper (unit: mm).
Stopper is shown in Fig.5b
BRC stiffener
52
80
204 84 10 16
Ø25
Core plate Filling plate
of the BRC core plate at the loading displacement of +60 mm and after loading, respectively. Out-of-plane deformation occurred at the end of the embedment portions of the BRCs, and high-mode buckling did not occur in the energy dissipating portion under the loading, which means that the energy dissipating capacity of the BRC may not have been fully
10 84
10 5
Restraining plate
30 24 60 66
56 66 114
523 883
(a) BRC core plate size
1 30 30 2530 30 253030 16 40×4=160 43 40×4=160 1-1 593
275
2010 20
(c) Filling plate size
25
700
Welded
HN200×100×5.5×8
25
250
275
593
(b) Restraining plate size
300
2 49
200 400
(d) Rigid support Fig. 4. BRC components and rigid support sizes (unit: mm). 693
2
49
Ø12
12 37
Ø12
510
60
180
Ø12
20
180 45 50 45
20
1
10 2-2
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Fb = φ ·A·f y
Table 3 Steel material properties of the main compositions. Category
Material grade
Yield strength (MPa)
Maximum strength (MPa)
Elongation (%)
BRC core plate Angle H-beam web H-beam flange
Q235B Q235B Q345B Q345B
330 370 390 445
465 505 530 620
19.1 17.3 18.2 24.5
where φ and A are the stability factor and area of the screw, respectively; fy is the yield strength of the screw material. In this paper, the values of φ, A and fy are 0.974, 314 mm2 and 235 MPa, respectively. To provide design suggestions for the out-of-plane restraint, a dimensionless parameter (β) is defined as the ratio of the total buckling force to the BRC yield force (Fy, BRC), as shown in Eq. (3). A larger β represents that the carrying capacity of the out-of-plane restraint might be higher than the RSFB out-of-plane demand.
used in the test.
β=
(3)
where ABRC is the area of the BRC core plate (640 mm ); fy, BRC is the yield strength of the material of the BRC core plate (330 MPa); n is the number of screws. Because the out-of-plane restraint in one side has two screws, the value of n is 2. Therefore, the value of r is 0.681 corresponding to the test. Because the out-of-plane restraints fail during loading, it is suggested that the value of β should be more than 0.681 in the design of the out-of-plane restraint. Note that this is an approximate suggestion, which needs to be further studied.
Fig. 9 shows the hysteresis curve of F-Δ obtained from the loading system, where θtop and Mc are the top drift ratio and overturning moment, respectively. θtop is equal to Δ divided by 3400 mm (i.e., the total structural height of the test specimen), and Mc is equal to F multiplied by 3400 mm. The hysteresis curve exhibits pinching effects when the loading displacement is greater than 40 mm because of the structural out-of-plane instability, and the shape of the hysteresis curve changes from a bow shape to an S shape, which means that the structural capacities are influenced by the out-of-plane deformation of the RSFB and the BRCs. The energy dissipation coefficient (E) is used to evaluate the structural energy dissipating capacity, and this coefficient can be calculated by Eq. (1) [23]:
4. Finite element model 4.1. Finite element model building Fig. 14 shows the RSFB finite element model established in ABAQUS. The beam element was used to build the steel braces in the second and third stories and all beams and columns of the rocking steel frame. Solid elements were used to model the BRCs, rigid support, BRC supports, connection joints (including beam-column and column-rigid support) and steel braces in first story of the rocking steel frame. This approach can accelerate the calculations and accurately capture the stress/strain responses of the BRCs and rigid support in RSFB, in which the solid elements C3D8I and C3D8R were adopted to model the core plate of the BRCs and the other solid parts of the finite element model, respectively. Thus, a multi-scale numerical model was established, and multi-point constraints (MPCs) were used to connect the beam and solid elements. Tie constraints were used to connect the restraining and filling plates of the BRCs because the high-strength bolts did not fail in the test, and the coefficient of friction between the core plate and the restraining/filling plates was set to 0.1 [24]. Out-of-plane deformation occurred in the first story of the RSFB because of the failure of the out-of-plane restraints in the first story. Thus, the out-of-plane restraints were added only to the second and third stories of the RSFB finite element model to approximately simulate the boundary conditions of the real test.
S(ABC + CDA) S(OBE + ODF )
n·φ ·A·fy nFb = Fy, BRC ABRC RC·f y, BRC 2
3.2. Hysteretic behavior and energy dissipation
E=
(2)
(1)
where S(ABC+CDA) is the area of the hysteresis curve shown in Fig. 10 and S(OBE+ODF) is the area of the triangles OBE and ODF shown in Fig. 10. Fig. 11 shows the values of E at each loading amplitude. The values of E increase with increasing loading displacement when the loading amplitude is less than 30 mm. As the loading amplitude reaches 40 mm, the BRC energy dissipating portion cannot normally deform to dissipate energy, which is caused by the larger out-of-plane deformations concentrated in the first story of the RSFB and at the end of the embedment portions of the BRCs. Thus, the value of E for the RSFB decreases when the loading amplitude reaches 50 mm and 60 mm. 3.3. Strain distribution Fig. 12 shows the maximum strain of the steel brace in each story of the RSFB. None of the braces experience a strain exceeding 900 με, which is less than the theoretical yield strain (the yield strength (370 MPa) divided by the elastic modulus (200000 N/mm2), which equals 1850 με); therefore, all braces remain elastic in the test. Fig. 13 shows the maximum strain for different measuring points of the rigid support. The theoretical yield strain of the rigid support flange equals 445 MPa divided by 200000 N/mm2, which is 2225 με. Plastic deformation occurs only in the bottom of the rigid support (SG 39), while the middle and top (SG 38 and SG 37) remain elastic. The strains gradually decrease from the bottom to the top of the rigid support.
4.2. Material parameters An elastic material with a Young’s modulus of 200 GPa was employed in members of the rocking steel frame built by the beam element, the restraining and filling plates of the BRCs and the connecting plates between the BRCs and the columns. The bilinear material model was used for the rigid support, BRC supports and the other parts of the rocking steel frame. The Young’s modulus of this material model was 200 GPa, and the yield and maximum strength were based on the true stress and strain. The true stress (σtrue) and strain (εtrue) can be calculated by using Eqs. (4) and (5), respectively:
3.4. Design suggestion for the out-of-plane restraint RSFB out-of-plane deformation causes extrusion between the steel beams and pulleys of out-of-plane restraint. The failure of the out-ofplane restraint is caused by the buckling of the screw that is welded to the pulley and used to connect the steel sleeve and pulley. The buckling force (Fb) of one screw can be regarded as the ultimate carrying capacity of the out-of-plane restraint, and it can be calculated using Eq. (2) according to Ref. [19].
σtrue = σnom (1 + εnom)
(4)
εtrue = ln(1 + εnom)
(5)
where σnom and εnom are the nominal stress and strain of the material, respectively, which can be obtained from Section 2.2. 694
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Fig. 6. Test setup.
The combined hardening model [25], including the isotropic and kinematic hardening rule, was employed for the BRC core plate to describe the strength development of the core plate under cyclic loading. First, the isotropic hardening model defines the relationship between the yield surface size (σ0) and the equivalent plastic strain (εpl), as shown in Eq. (6):
σ 0 = σ0 + Q∞ (1 − e−bε
pl
)
(6)
where σ|0 is the yield stress when the equivalent plastic strain reaches zero, Q∞ denotes the maximum change in the yield surface size, and b is the rate of change in the yield surface as the plastic strain increases. The parameters σ|0 = 330 MPa, Q∞ = 21 MPa and b = 1.2 are used in this paper for the core plate based on material properties and reference 695
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Displacement control
4.3. Validation of the finite element model
80
80
40
40
0
0
-40
-40
-80
-80
The loading protocol used in the simulation analyses was same as that in the displacement-controlled stage of the pseudo-static test, and each loading amplitude was cycled only one time to accelerate the calculations. Fig. 15a shows a comparison of the hysteresis curves of the test and simulation results, and the shapes of the hysteresis curves are basically consistent for the two results. The differences between the test results and the simulation results are found in the initial stiffness and lateral load-carrying capacity, which may have been caused by the initial bending during installation and dislocation between the specimen and the actuator. Fig. 15b-c shows the out-of-plane deformation and stress of the rocking steel frame (the units of U3 and S are mm and MPa, respectively), which are consistent with the test results and phenomena. Fig. 15d shows the BRC deformation of the simulation results when the top displacement load was +60 mm. The larger out-of-plane deformation occurred at the end of the embedment portion of the BRC, and high-mode buckling did not occur in the energy dissipating portion. This failure is also consistent with the test phenomena. In conclusion, the proposed multi-scale finite element model can reflect the hysteretic responses of the RSFB. However, this simulation does not fully capture all aspects of the test results, and it is difficult to simulate the imperfections and boundary conditions of the test model. Therefore, the initial stiffness and pinching effect of the finite element result are not fully consistent with those of the test results.
Displacement (mm)
Force (kN)
Force control
Fig. 7. Loading protocol.
[26]. Second, the kinematic hardening model defines the calculation method of backstress (αk), as shown in Eq. (7): N
αk =
∑ k=1
pl Ck (1 − e−γk ε ) γk
(7)
where Ck/γk expresses the maximum change in backstress and γk is the rate of change in backstress as the plastic strain increases. For the kinematic hardening model, the critical parameters of N = 3, C1 = 8 × 103 MPa, γ1 = 100, C2 = 1.0 × 105 MPa, γ2 = 3.0 × 103, C3 = 5.0 × 102, and γ3 = 0 were adopted for the core plate in this paper.
5. Numerical simulation results for the test under full out-of-plane restraints The aforementioned test phenomena and results were caused by the
(a) Deformation and failure of the out-of-plane restraints
(b) Deformation of the BRC core plate at the loading displacement of +60 mm
(c) Failure modes of the BRC core plate after loading
Fig. 8. Deformation and failure modes of the BRCs. 696
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Fig. 9. Hysteresis curve. Fig. 12. Maximum strain of the steel braces in each story.
Fig. 10. Calculation method of E.
Fig. 13. Maximum strain in different measuring points of the rigid support.
5.1. Deformation and damage forms of the BRCs Fig. 16a shows that high-mode buckling occurred in the BRC energy dissipating portion of the RSFB-FOR model when the top displacement reached +60 mm, and the out-of-plane deformations in the end of the BRC embedment portions of the RSFB-FOR model are apparently reduced and negligible in comparison to the results shown in Fig. 15d. Thus, the comparison of Fig. 16a and Fig. 15d indicates that if the outof-plane restraints in the first story do not fail, then the out-of-plane deformations of the BRCs will be constrained, thereby ensuring the outof-plane stability of the RSFB and enhancing the structural seismic performance. Fig. 16b-c shows the values of the equivalent plastic strain (the value is PEEQ in ABAQUS), wherein PEEQ was used to express the accumulation of plastic strain and represent the damage degree. The PEEQ values in the end of the embedment portions are apparently greater than those in the other part of the BRC core plates in the RSFB model. However, the PEEQ values in the energy dissipating
Fig. 11. The values of E at each amplitude.
failure of the out-of-plane restraints in the first story. Thus, for an RSFB with full out-of-plane restraints in each story (RSFB-FOR), a finite element model was built to investigate the structural seismic performance when the out-of-plane restraints are not failing. The loading point and protocol were same as those used in Section 4.3.
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Fig. 14. RSFB finite element model.
where θmax is the maximum inter-story drift ratio, utop is the loading displacement, and H is the structural height of the RSFB (RSFB-FOR) model. A comparison of the inter-story drift ratios and DCF values for RSFB and RSFB-FOR shows that the lateral drifts are uniformly distributed along the height of the RSFB-FOR model. Furthermore, a sudden change in inter-story drift occurs in the RSFB model, which is induced by the structural out-of-plane instability and leads to a larger DCF value compared to that of the RSFB-FOR model. The aforementioned results show that the RSFB with full out-ofplane restraints can achieve the expected seismic performance and energy dissipating capacity. The rocking steel frame remains elastic, the lateral drifts are uniform, the BRCs do not undergo buckling at the end of the embedment portions and the structural plastic deformation is mainly concentrated in the energy dissipating portion of BRCs.
portion for the RSFB-FOR model are obviously increased, which indicates that the addition of full out-of-plane restraints to the RSFB can prevent the BRCs buckling and prompt the plastic deformation to concentrate in the energy dissipating portion of the BRCs. Therefore, the BRCs in the test specimen under full out-of-plane restraints can be used to achieve the expected deformation and damage forms. 5.2. Hysteretic behavior and energy dissipation Fig. 17a shows a comparison of the hysteresis curves between the RSFB specimen test results and the RSFB-FOR model simulation results. Compared to the RSFB, the RSFB-FOR model has a fuller hysteresis curve that does not exhibit pinching effects and has a larger peak lateral load-carrying capacity. According to Fig. 17b, the value of E increases with increasing loading displacement and was greater than 1.5 at each loading amplitude except 10 mm. In contrast to the RSFB model, the RSFB-FOR model does not exhibit a reduction in energy dissipating capacity, which means that the capacity of the BRCs in the RSFB-FOR model can be fully used.
6. Influence of the BRC embedment length on the performance of RSFB without full out-of-plane restraints According to the analytical results shown in Section 5, full out-ofplane restraints can ensure the out-of-plane stability of the RSFB and the structural seismic performance. However, the full out-of-plane restraint is an ideal lateral boundary condition, which was hard to realize in the experiments. Moreover, it can be observed in the test results shown in Section 3 that out-of-plane deformation occurred in the first story of the RSFB. Section 4 verified that applying the out-of-plane restraints in the second and third stories can approximately simulate the out-of-plane boundary conditions of the real test, which is a condition deemed as “without full out-of-plane restraints” in the paper. Therefore, this section focuses on the factor affecting structural seismic performance and out-of-plane deformation of the RSFB without full outof-plane restraints. The BRC embedment length will be investigated because it might be related to the BRC and structural out-of-plane
5.3. Stress distribution and structural lateral drift Fig. 18 shows the stress of the rocking steel frame for the RSFB-FOR model at Δ = +60 mm. The rocking steel frame basically remains elastic, and plastic deformation occurs only in the rigid support. Fig. 19 shows the inter-story drift ratios at Δ = ± 60 mm and the drift concentration factor (DCF) values for the RSFB specimen test results and RSFB-FOR model simulation results. The DCF values were calculated by Eq. (8) [2], and the values for the RSFB and RSFB-FOR are 1.49 and 1.02 (at Δ = +60 mm), respectively.
DCF =
θmax utop H
(8) 698
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Fig. 15. Comparison of the experimental and simulated results.
enhanced by elongating the embedment length. Therefore, the embedment length might have an important influence on the RSFB out-ofplane stability and, further, influence the structural seismic performance. If the value of Lep is small, the weak position will be located at the end of the embedment portion after the core plate is elongated, and the weak position easily buckles under reverse compression because of the lack of out-of-plane restraints, as shown in Fig. 20a. Thus, the
stability. 6.1. Description of the RSFBs with different embedment lengths According to Fig. 8b, the apparent out-of-plane deformation was observed at the ends of the embedment portions of the BRCs. Takeuchi et al. [27] noted that the single BRC out-of-plane stability can be 699
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Fig. 16. Comparison of the BRC deformation and damage between the RSFB and RSFB-FOR models.
6.2. BRC out-of-plane deformation and failure modes
plastic deformation is concentrated in the end of the embedment portion of the BRCs, and the energy dissipating portion cannot be deformed to dissipate energy, which is similar to the pseudo-static test results. Fig. 20b shows the BRC with a Lep of 120 mm, which was formed by increasing the Lep value of 60 mm to 120 mm based on the test BRCs. Note that the total length of the BRCs and the compression gap of these BRCs were the same as those of the test BRCs, at 883 mm and 20 mm, respectively. Based on the validated RSFB model in Section 4.3, many RSFB models with different BRC embedment lengths were established by increasing the Lep value from 60 mm to 120 mm, and they are named in series from RSFB-Lep-60 (RSFB-Lep-60 is same as the RSFB finite element model in Section 4.3) to RSFB-Lep-120. These models were used to investigate the influence of the elongated value of Lep on the structural seismic performance and out-of-plane stability. The loading protocol, boundary conditions and out-of-plane restraints were same as those used in the RSFB finite element model.
Fig. 21 shows the BRC out-of-plane deformation of the RSFB-Lep-120 at Δ = +60 mm. Compared with the RSFB model shown in Fig. 15d, the RSFB-Lep-120 model exhibits less out-of-plane deformation in the embedment portion. Moreover, high-mode buckling occurs in the energy dissipating portion of the BRCs in the RSFB-Lep-120 model, which does not occur in the RSFB model. These results indicate that increasing Lep can also enhance the structural out-of-plane stability and ensure the seismic performance of the RSFB without full out-of-plane restraints. The BRC deformation mode is similar to that of the RSFB-FOR model. Fig. 22 shows the out-of-plane deformation in the end of the BRC embedment portion at Δ = +60 mm. The symbols in Fig. 22 indicate the calculations performed in this paper for RSFB models with different embedment lengths. The out-of-plane deformation decreases with increasing Lep values, and the decreasing rate apparently slows when the value of Lep exceeds 80 mm. Fig. 23 shows the values of PEEQ for the BRC core plates in the RSFB models with different Lep values (60, 65, 70, 80, 100, and 120 mm) at
Fig. 17. Hysteresis curves and energy dissipation of the RSFB and RSFB-FOR models. 700
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Fig. 18. Stress distribution of the rocking steel frame.
RSFB-FOR and RSFB-Led-120 simulation results. A comparison of Fig. 24 and Fig. 17a shows that the hysteresis curve of the RSFB-Lep-120 is fuller than that of the RSFB test specimen and similar to that of RSFBFOR. Fig. 25 shows the cumulative energy dissipation coefficients (i.e., the values of E accumulated from Δ = 10 mm to Δ = 60 mm) of RSFB models with different values of Lep. The cumulative energy dissipation coefficient increases with increasing embedment length and tends toward a constant value when the value of Lep exceeds 80 mm. Furthermore, when the value of Lep exceeds 80 mm, the cumulative energy dissipation coefficients of RSFB are basically consistent with that of RSFB-FOR, which indicates that the increase in the embedment lengths of BRCs in RSFB without full out-of-plane restraints can achieve a similar energy dissipating capacity to that of RSFB-FOR. Fig. 19. Structural inter-story drift ratios and DCF values.
7. Conclusions Δ = +60 mm. The values of PEEQ in the energy dissipating portion increase with the increase in Lep, which means that the plastic deformation is continuously concentrated in the energy dissipating portion. However, the values of PEEQ at the end of the embedment portions are always larger than those in the energy dissipating portion, which indicates that the anticipated failure position may be located in this portion for a BRC rigidly connected to the rocking steel frame. The influence of the method of connection between the BRC and the rocking steel frame on the structural and BRC capacities requires further research.
This paper performs a pseudo-static test for the proposed rocking steel frame with buckling-restrained columns (RSFB) without full outof-plane restraints. Then, the performance of the test specimen with full out-of-plane restraints is numerically studied. Finally, the influences of the buckling-restrained column (BRC) embedment portion length on the structural performance and out-of-plane stability of the RSFB without full out-of-plane restraints are discussed. The latter two contributions can be regarded as supplements to the test. The conclusions are listed as follows. (1) The pseudo-static test results show that the structural capacity cannot be fully used due to the structural out-of-plane instability caused by the failure of the out-of-plane restraints in the first story
6.3. Hysteretic behavior and energy dissipation Fig. 24 shows the hysteresis curves for the RSFB test results and 701
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Fig. 20. Description of the BRCs with different embedment lengths.
Out-of-plane deformation(mm)
200 160 120 80 40 0
60
70
80 90 100 110 120 Lep (mm)
Fig. 22. Out-of-plane deformation in the end of the BRC embedment portion for RSFB models with different Lep values at Δ = +60 mm.
rigid support. A design suggestion for the out-of-plane restraint is provided. (2) The simulation results for the RSFB test are consistent with the test results and phenomena, implying that the proposed multi-scale finite element model can reflect the hysteretic responses of the RSFB. (3) The simulation results for the RSFB with full out-of-plane restraints in each story (RSFB-FOR) show that the proposed RSFB with full out-of-plane restraints can achieve excellent seismic performance and energy dissipating capacity, as expected. The rocking steel frame remains elastic, lateral drifts are uniformly distributed along the structural height, and the structural plastic deformation is mainly concentrated in the BRCs energy dissipating portion. (4) The increase in the embedment lengths of the BRCs (Lep) (from 60 mm to 120 mm) can also enhance the out-of-plane stability and performance of the RSFB without full out-of-plane restraints. However, when the value of Lep reaches 80 mm, the decreasing rate of out-of-plane deformation is apparently slowed, and the cumulative energy dissipation coefficients of the RSFBs are basically
Fig. 21. BRC out-of-plane deformation of the RSFB-Lep-120 at Δ = +60 mm.
of the RSFB. When the loading amplitude reaches 40 mm (θtop approximately equals 1.2%), the visible out-of-plane deformation occurs at the end of the embedment portions of the BRCs; thus, the hysteresis curve exhibits pinching effects, and the values of the energy dissipation coefficient (E) decrease. The braces in the rocking steel frame always remain elastic, and plastic deformation occurs at the end of the embedment portions of the BRCs and the 702
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Fig. 23. Values of PEEQ for the BRC core plate in the RSFB models with different embedment lengths.
In a future study, the proposed RSFB with full out-of-plane restraints should be retested to further verify Conclusion (3), and the influence of the connection details on the structural and BRC performance should be studied. This paper is based on the pseudo-static test; thus, the main conclusions are obtained from the test results and relative simulation analyses. Further research is needed to determine whether these
consistent with that of the RSFB-FOR. (5) Although the plastic deformation in the energy dissipating portion of the BRC increases when full out-of-plane restraints are added or Lep is increased, the ends of the embedment portions always have the maximum values of the equivalent plastic strain (PEEQ). Therefore, for a BRC rigidly connected to the rocking steel frame, rupture may occur at the end of the embedment portions. 703
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References [1] Pollino M, Sabzehzar S, Qu B, et al. Research needs for seismic rehabilitation of substandard buildings using stiff rocking cores. Struct Congress Bridg Your Pass Your Profess 2013;2013:1683–93. [2] MacRae GA, Kimura Y, Roeder C. Effect of column stiffness on braced frame seismic behavior. J Struct Eng 2004;130(3):381–91. [3] Ji X, Kato M, Wang T, et al. Effect of gravity columns on mitigation of drift concentration for braced frames. J Constr Steel Res 2009;65(12):2148–56. [4] Midorikawa M, Azuhata T, Ishihara T, et al. Shaking table tests on seismic response of steel braced frames with column uplift. Earthquake Eng Struct Dyn 2006;35(14):1767–85. [5] Tremblay R, Ben Ftima M, Sabelli R. An innovative bracing configuration for improved seismic response. Proc. Recent Advances and New Trends in Structural Design International Colloquium. 2004. p. 419–30. [6] Tremblay R, Poirier LP, Bouaanani N, et al. Innovative viscously damped rocking braced steel frames. Proceedings of the 14th world conference on earthquake engineering, Beijing, China. 2008. p. 12–7. [7] Eatherton M, Hajjar J, Ma X, et al. Seismic design and behavior of steel frames with controlled rocking—part I: concepts and quasi-static subassembly testing. Struct Congress 2010;2010:1523–33. [8] Ma X, Eatherton M, Hajjar J, et al. Seismic design and behavior of steel frames with controlled rocking—Part II: Large scale shake table testing and system collapse analysis. Struct Congress 2010;2010:1534–43. [9] Sause R, Ricles JM, Roke DA, et al. Seismic performance of a self-centering rocking concentrically-braced frame. Proceeding of the 9th US National and 10th Canadian Conference on Earthquake Engineering. 2010. p. 25–9. [10] López-Barraza A, Bojórquez E, Ruiz SE, et al. Reduction of maximum and residual drifts on posttensioned steel frames with semirigid connections. Adv Mater Sci Eng 2013;2013. [11] Reyes-Salazar A, Llanes-Tizoc MD, Bojorquez J, et al. Force reduction factors for steel buildings with welded and post-tensioned connections. Bull Earthq Eng 2016;14(10):2827–58. [12] Reyes-Salazar A, Ruiz SE, Bojorquez E, et al. Seismic response of complex 3D steel buildings with welded and post-tensioned connections. Earthq Struct 2016;11(2):217–43. [13] Wiebe L, Christopoulos C, Tremblay R, et al. Mechanisms to limit higher mode effects in a controlled rocking steel frame. 1: Concept, modelling, and low-amplitude shake table testing. Earthq Eng Struct Dyn 2013;42(7):1053–68. [14] Blebo FC, Roke DA. Seismic-resistant self-centering rocking core system. Eng Struct 2015;101:193–204. [15] Takeuchi T, Chen X, Matsui R. Seismic performance of controlled spine frames with energy-dissipating members. J Constr Steel Res 2015;114:51–65. [16] Blebo FC, David AR. Seismic-resistant self-centering rocking core system with buckling restrained columns. Eng Struct 2018;173:372–82. [17] Feng Y, Wu J, Chong X, et al. Seismic lateral displacement analysis and design of an earthquake-resilient dual wall-frame system. Eng Struct 2018;177:85–102. [18] GB/T11263-2010 Hot rolled H and cut T section steel. Beijing: Standards press of China; 2010 (in Chinese). [19] GB50017-2003 Code for design of steel structures. Beijing: China Planning Press; 2003 (in Chinese). [20] Chen Q, Wang CL, Meng S, et al. Effect of the unbonding materials on the mechanic behavior of all-steel buckling-restrained braces. Eng Struct 2016;111:478–93. [21] Chen Q. Hysteretic performance of buckling-restrained braces and seismic performance of buckling-restrained braced frames. Southeast University; 2016. (in Chinese). [22] GB/T228.1-2010. Metallic materials-Tensile testing-Part 1: method of test at room temperature. Beijing: Standards Press of China; 2010. (in Chinese). [23] JGJ/T101-2015, Specification for seismic test of buildings. Beijing: China Architecture & Building Press, 2015. (in Chinese). [24] Wang CL, Usami T, Funayama J. Evaluating the influence of stoppers on the lowcycle fatigue properties of high-performance buckling-restrained braces. Eng Struct 2012;41:167–76. [25] Chaboche JL. Time-independent constitutive theories for cyclic plasticity. Int J Plast 1986;2(2):149–88. [26] Yongjiu Shi, Meng Wang, Yuanqing Wang. Experimental study of structural steel constitutive relationship under cyclic loading. J Build Mater 2012;15(03):293–300. (in Chinese). [27] Takeuchi T, Ozaki H, Matsui R, et al. Out-of-plane stability of buckling-restrained braces including moment transfer capacity. Earthq Eng Struct Dyn 2014;43(6):851–69.
Fig. 24. Comparison of the hysteresis curves.
Fig. 25. Cumulative energy dissipation coefficients for RSFB with different values of Lep.
conclusions can be applied to mid- and high-rise models. Meanwhile, the design of BRC embedment length needs to be further investigated. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The research described in this paper was financially supported by the National Natural Science Foundation of China (51708166; 51778201; 51878233), the Fundamental Research Funds for Central Universities of China (JZ2019HGTB0086) and the China Postdoctoral Science Foundation (2018M630706). Their support is gratefully acknowledged.
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Structures journal homepage: www.elsevier.com/locate/structures
Experimental and numerical studies of a controlled rocking steel frame with buckling-restrained columns
T
⁎
Qing Jianga,b, Hanqin Wanga, Yulong Fenga,b, , Xun Chonga,b, Xiaowei Wanga, Yi Zhua a b
School of Civil Engineering, Hefei University of Technology, Anhui Province 230009, China Anhui Civil Engineering Structures and Materials Laboratory, Anhui Province 230009, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Controlled rocking steel frame Buckling-restrained column Seismic performance Pseudo-static test Out-of-plane deformation
A non-uplifting rocking steel frame with buckling-restrained columns (RSFB) is formed by replacing two side columns in the bottom story of a steel concentrically braced frame with two buckling-restrained columns (BRCs). During major earthquakes, the BRCs first yield to dissipate energy while the other part of the system remains elastic and rotates around the bottom of the middle column to control structural deformation. A pseudo-static test was conducted on the earthquake-resilient RSFB, observing larger out-of-plane deformation occurred in the RSFB first story and the embedment portions of BRCs due to the failure of the out-of-plane restraints. The results indicate that the RSFB needs a stronger out-of-plane restraint to ensure structural performance. A design suggestion for the out-of-plane restraint is subsequently provided. A multi-scale finite element model was established in ABAQUS and validated by the test results, and simulation analyses of the test specimen with full out-ofplane restraints were conducted. The simulation results of this model showed that the hysteresis curve is full under cyclic loads, the lateral drifts are uniformly distributed along the structural height, and the plastic deformation is mainly concentrated in the BRCs energy dissipating portion, implying that the proposed RSFB can yield excellent seismic performance, as expected. Finally, the influences of the BRC embedment lengths on the structural performance and out-of-plane deformation were studied to enhance the out-of-plane stability of the RSFB without full out-of-plane restraints.
1. Introduction Moment-resisting frames (MRFs) and concentrically braced frames (CBFs) have been widely used in steel buildings as lateral force resisting systems. However, some studies noted that damage concentration issues and soft story failure exist in the aforementioned systems, which has led to severe structural damage under horizontal seismic loads [1–3]. Accordingly, various rocking systems have been developed and added to steel buildings to control the damage distribution in steel MRFs and CBFs. To reduce structural damage, Midorikawa et al. [4] proposed an uplifting steel rocking system, which releases the constraint between the columns and the foundation and is connected with a yielding base plate. Shaking table test results showed that the columns can uplift to prompt yielding in the yielding base plate, thereby dissipating energy. Tremblay et al. proposed a similar system [5–6]. Based on the uplifting system, Eatherton et al. [7] and Ma et al. [8] and Sause et al. [9] proposed a series of controlled rocking systems with posttensioning strands to construct a self-centering system that not only
⁎
controls the structural damage distribution but also reduces the structural residual lateral drift, and they performed a series of analyses that verified the capacity of this self-centering system. López-Barraza et al. [10] and Reyes-Salazar et al. [11–12] proposed a self-centering system with posttensioned strands to reduce the residual deformation of MRF structure after earthquakes. To reduce structural seismic responses amplified by the rocking method of uplifting under high structural modes [13], Blebo and David [14] developed a pin-supported rocking core system. In contrast to the uplifting system, this structure can rotate around the stationary pinned support at the middle column base under horizontal seismic loads. To further enhance the energy dissipating capacity of the steel rocking system, Takeuchi et al. [15] and Blebo and David [16] employed two buckling-restrained braces (BRBs) on the side of the bottom story to use the displacement generated by the rocking system rotation. Feng et al. [17] installed two replaceable BRBs at the base of a rocking wall to form an earthquake-resilient shear wall and used this wall to enhance the seismic performance of the steel MRF. This study found that the rocking wall with BRBs in the base can effectively control the structural
Corresponding author at: School of Civil Engineering, Hefei University of Technology, Anhui Province 230009, China. E-mail address: [email protected] (Y. Feng).
https://doi.org/10.1016/j.istruc.2020.02.005 Received 19 November 2019; Received in revised form 4 February 2020; Accepted 8 February 2020 2352-0124/ © 2020 Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.
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Fig. 1. Description of an RSFB and a structure composed of a main frame and the RSFB.
connected with the foundation by the same method); the foundation and rigid support are steel members. The cross-section of the foundation and the design of the rigid support are shown in Fig. 2 and Fig. 4d, respectively. Fig. 3 schematically shows the overall composition of a BRC, which contains a core plate, restraining plates and filling plates (the material types and member sizes of all compositions can be seen in Table 2 and Fig. 4, respectively). The BRC is divided into the elastic portion, embedment portion and energy dissipating portion, and the lengths of these three portions are L1 = 140 mm, Lep = 60 mm and L2 = 483 mm, respectively, which means that the total length of the BRC is 883 mm. The preset compression gap is the distance between the stiffener and the restraining plate, and the length of the compression gap (Lcg) is 20 mm. The restraining plate and filling plate are connected by class 10.9 high-strength bolts with a diameter of 10 mm. The gap between the core plate and the restraining plate or filling plate is 1 mm and is filled with butyl rubber [20], as shown in Fig. 5a. Chen [21] noted that to prevent slipping of the restraining members (i.e., the restraining plates and filling plates), the stopper is set at the middle by enlarging the width of the core plate. The influence of the arc-shaped stopper on the distribution of the core plate stress is lower than other methods; thus, the arc-shaped stopper is used in this test specimen, as shown in Fig. 5b.
lateral deformation and dissipate the seismic energy. This paper proposes a rocking steel frame with buckling-restrained columns (RSFB) based on the works of Takeuchi et al. [15], Blebo and David [16], and Feng et al. [17], as shown in Fig. 1a. An RSFB is constructed by replacing the side columns at the bottom of a steel CBF with BRBs that are rigidly connected to the columns, and these BRBs are defined as buckling-restrained columns (BRCs). Fig. 1b schematically shows the composite structure formed by an RSFB and main structure that are connected with a rigid link. Under horizontal seismic loads, an RSFB can rotate around the middle column base to control the lateral drift of the main structure and concentrate seismic energy in the BRCs, and the BRC core plate is deformed to dissipate energy and reduce the main structural damage. However, few experimental studies of RSFBs have been reported. This paper performs a pseudo-static test and many simulation analyses of the proposed RSFB. The test results showed that the capacity of the test specimen cannot be fully used because of structural out-ofplane instability. Further, the performance of the test specimen with full out-of-plane restraints is studied using a numerical simulation method as a supplement to the experimental research. Finally, the influences of the BRC embedment portion length on the structural seismic performance and out-of-plane deformation are analyzed to enhance the outof-plane stability of the RSFB without full out-of-plane restraints, which might be helpful for retesting the RSFB.
2.2. Material properties 2. Test program Table 3 presents the steel material properties of the main compositions, in which the material test specimens were plate samples that were cut from the corresponding compositions. Various properties of the steel are calculated in accordance with Chinese code (GB/T228.12010) [22].
2.1. Test specimen Fig. 2 shows the pseudo-static test specimen, and the dimensional parameters of the RSFB mainly include two spans (B = 1000 mm) and three stories (h1 = 1400 mm, h2 = h3 = 1000 mm). Table 1 shows the sizes of the members and material types of the specimen, which are selected in accordance with the Chinese codes GB/T11263-2010 [18] and GB50017-2003 [19], respectively. All beam-to-column and brace joints are welded to achieve rigid connections, and the connection between the BRCs and the columns or BRC supports are also welded. To better acquire the deformation of the rigid support flange when the RSFB rotates, the rigid support that has the same member size as the column is manufactured individually and then connected with the column and foundation by high-strength bolts (the BRC supports are
2.3. Test setup and loading protocol 2.3.1. Test setup Fig. 6a shows the test specimen installation, which is fixed on the channel on the ground of the laboratory. The loading point is located at the center of the beam-to-column joint in the third story of the RSFB, and the other side of the actuator is connected to the reaction wall. To prevent slipping of the specimen, a support beam and a reaction beam are set on the left and right sides of the specimen foundation and 691
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Fig. 2. Sketch of the test specimen (unit: mm).
Moreover, three SGs are set on one side of the rigid support flange; these SGs are numbered from SG 37 to SG 39 (SG 37A to SG 39A are on the other side of the rigid support flange). The loading force (F) and top displacement of the RSFB (Δ) are recorded by the loading system.
Table 1 Member information. Member name
Number
Member type
Size
Material
Column
KZ
H-beam
Q345B
Beam
KL
H-beam
Steel brace Stiffener Foundation BRC Rigid/BRC support
– – – BRC –
Double angle Steel plate – – –
HN 200 × 100 × 5.5 × 8 (mm) HN 200 × 100 × 5.5 × 8 (mm) 2L90 × 90 × 8 (mm) Plate thickness 8 (mm) Plate thickness 25 (mm) Shown in Table 2 –
2.3.2. Loading protocol The loading protocol is divided into two phases in accordance with the Chinese code (specification for the seismic test of buildings JGJ/ T101-2015) [23], as shown in Fig. 7. Force control and displacement control are used before and after structural yielding, respectively. For force control, the loading increment is 10 kN, and each amplitude is cycled one time. For displacement control, the loading amplitude displacements of each stage are 10 mm, 20 mm, 30 mm, 40 mm, 50 mm, and 60 mm, and each stage is cycled three times.
Q345B Q235B Q345B Q345B – Q345B
Notes: HN 200 × 100 × 5.5 × 8 means that the sectional height, width, the thickness of web and flange of H-beam are 200 mm, 100 mm, 5.5 mm and 8 mm, respectively. L90 × 90 × 8 means that the width and thickness of angle steel are 90 mm and 8 mm, respectively.
3. Test results and analyses 3.1. Specimen failure modes
connected with jacks. Four out-of-plane restraints are arranged at the first and second-story beams to prevent structural out-of-plane deformation. The construction details of the out-of-plane restraints are shown in Fig. 6b and 6c. The main data acquisition points are shown in Fig. 6a. The interstory drifts of the RSFB are obtained from linear variable displacement transducers (LVDTs) 2–4, whereas LVDTs 1, 5 and 6 are used to judge whether slipping and uplifting occurred in the specimen. Six strain gauges (SGs) are arranged on the brace surface at each span and story, as shown in Fig. 6a, and these SGs are numbered from SG 1 to SG 36.
Fig. 8 shows the deformation and failure of the out-of-plane restraints. Bending deformations of the screws that are welded on the pulley and used to connect the steel sleeve and pulley in the out-ofplane restraints are seen in the first story. Moreover, friction traces appear on the steel plate that is welded onto the KL (i.e., the beams in RSFB) and in contact with the pulley. The abovementioned phenomena mean that the designed out-of-plane restraints cannot provide effective constraint to prevent the occurrence of structural out-of-plane deformation. Fig. 8b and Fig. 8c show the deformation and failure modes 692
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Cross-section is shown in Fig.5a
BRC stiffener
Embedment portion (Lep)
Elastic portion (L1)
Filling plate
Restraining plate
Core plate
Compression gap (Lcg)
Energy dissipating portion (L2)
Lep
Lep
Elastic portion (L1)
Fig. 3. BRC structure.
Table 2 BRC component information.
10
Member name
Size
Thickness
Material
Core plate Restraining plate Filling plate BRC stiffener
Shown in Fig. 4a Shown in Fig. 4b Shown in Fig. 4c 90 mm × 180 mm
8 mm 16 mm 10 mm 16 mm
Q235B Q345B Q345B Q345B
Butyl rubber
20 20 10
(a) Cross-section of the BRC
(b) Stopper detail
Fig. 5. Details of the BRC cross-section and stopper (unit: mm).
Stopper is shown in Fig.5b
BRC stiffener
52
80
204 84 10 16
Ø25
Core plate Filling plate
of the BRC core plate at the loading displacement of +60 mm and after loading, respectively. Out-of-plane deformation occurred at the end of the embedment portions of the BRCs, and high-mode buckling did not occur in the energy dissipating portion under the loading, which means that the energy dissipating capacity of the BRC may not have been fully
10 84
10 5
Restraining plate
30 24 60 66
56 66 114
523 883
(a) BRC core plate size
1 30 30 2530 30 253030 16 40×4=160 43 40×4=160 1-1 593
275
2010 20
(c) Filling plate size
25
700
Welded
HN200×100×5.5×8
25
250
275
593
(b) Restraining plate size
300
2 49
200 400
(d) Rigid support Fig. 4. BRC components and rigid support sizes (unit: mm). 693
2
49
Ø12
12 37
Ø12
510
60
180
Ø12
20
180 45 50 45
20
1
10 2-2
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Fb = φ ·A·f y
Table 3 Steel material properties of the main compositions. Category
Material grade
Yield strength (MPa)
Maximum strength (MPa)
Elongation (%)
BRC core plate Angle H-beam web H-beam flange
Q235B Q235B Q345B Q345B
330 370 390 445
465 505 530 620
19.1 17.3 18.2 24.5
where φ and A are the stability factor and area of the screw, respectively; fy is the yield strength of the screw material. In this paper, the values of φ, A and fy are 0.974, 314 mm2 and 235 MPa, respectively. To provide design suggestions for the out-of-plane restraint, a dimensionless parameter (β) is defined as the ratio of the total buckling force to the BRC yield force (Fy, BRC), as shown in Eq. (3). A larger β represents that the carrying capacity of the out-of-plane restraint might be higher than the RSFB out-of-plane demand.
used in the test.
β=
(3)
where ABRC is the area of the BRC core plate (640 mm ); fy, BRC is the yield strength of the material of the BRC core plate (330 MPa); n is the number of screws. Because the out-of-plane restraint in one side has two screws, the value of n is 2. Therefore, the value of r is 0.681 corresponding to the test. Because the out-of-plane restraints fail during loading, it is suggested that the value of β should be more than 0.681 in the design of the out-of-plane restraint. Note that this is an approximate suggestion, which needs to be further studied.
Fig. 9 shows the hysteresis curve of F-Δ obtained from the loading system, where θtop and Mc are the top drift ratio and overturning moment, respectively. θtop is equal to Δ divided by 3400 mm (i.e., the total structural height of the test specimen), and Mc is equal to F multiplied by 3400 mm. The hysteresis curve exhibits pinching effects when the loading displacement is greater than 40 mm because of the structural out-of-plane instability, and the shape of the hysteresis curve changes from a bow shape to an S shape, which means that the structural capacities are influenced by the out-of-plane deformation of the RSFB and the BRCs. The energy dissipation coefficient (E) is used to evaluate the structural energy dissipating capacity, and this coefficient can be calculated by Eq. (1) [23]:
4. Finite element model 4.1. Finite element model building Fig. 14 shows the RSFB finite element model established in ABAQUS. The beam element was used to build the steel braces in the second and third stories and all beams and columns of the rocking steel frame. Solid elements were used to model the BRCs, rigid support, BRC supports, connection joints (including beam-column and column-rigid support) and steel braces in first story of the rocking steel frame. This approach can accelerate the calculations and accurately capture the stress/strain responses of the BRCs and rigid support in RSFB, in which the solid elements C3D8I and C3D8R were adopted to model the core plate of the BRCs and the other solid parts of the finite element model, respectively. Thus, a multi-scale numerical model was established, and multi-point constraints (MPCs) were used to connect the beam and solid elements. Tie constraints were used to connect the restraining and filling plates of the BRCs because the high-strength bolts did not fail in the test, and the coefficient of friction between the core plate and the restraining/filling plates was set to 0.1 [24]. Out-of-plane deformation occurred in the first story of the RSFB because of the failure of the out-of-plane restraints in the first story. Thus, the out-of-plane restraints were added only to the second and third stories of the RSFB finite element model to approximately simulate the boundary conditions of the real test.
S(ABC + CDA) S(OBE + ODF )
n·φ ·A·fy nFb = Fy, BRC ABRC RC·f y, BRC 2
3.2. Hysteretic behavior and energy dissipation
E=
(2)
(1)
where S(ABC+CDA) is the area of the hysteresis curve shown in Fig. 10 and S(OBE+ODF) is the area of the triangles OBE and ODF shown in Fig. 10. Fig. 11 shows the values of E at each loading amplitude. The values of E increase with increasing loading displacement when the loading amplitude is less than 30 mm. As the loading amplitude reaches 40 mm, the BRC energy dissipating portion cannot normally deform to dissipate energy, which is caused by the larger out-of-plane deformations concentrated in the first story of the RSFB and at the end of the embedment portions of the BRCs. Thus, the value of E for the RSFB decreases when the loading amplitude reaches 50 mm and 60 mm. 3.3. Strain distribution Fig. 12 shows the maximum strain of the steel brace in each story of the RSFB. None of the braces experience a strain exceeding 900 με, which is less than the theoretical yield strain (the yield strength (370 MPa) divided by the elastic modulus (200000 N/mm2), which equals 1850 με); therefore, all braces remain elastic in the test. Fig. 13 shows the maximum strain for different measuring points of the rigid support. The theoretical yield strain of the rigid support flange equals 445 MPa divided by 200000 N/mm2, which is 2225 με. Plastic deformation occurs only in the bottom of the rigid support (SG 39), while the middle and top (SG 38 and SG 37) remain elastic. The strains gradually decrease from the bottom to the top of the rigid support.
4.2. Material parameters An elastic material with a Young’s modulus of 200 GPa was employed in members of the rocking steel frame built by the beam element, the restraining and filling plates of the BRCs and the connecting plates between the BRCs and the columns. The bilinear material model was used for the rigid support, BRC supports and the other parts of the rocking steel frame. The Young’s modulus of this material model was 200 GPa, and the yield and maximum strength were based on the true stress and strain. The true stress (σtrue) and strain (εtrue) can be calculated by using Eqs. (4) and (5), respectively:
3.4. Design suggestion for the out-of-plane restraint RSFB out-of-plane deformation causes extrusion between the steel beams and pulleys of out-of-plane restraint. The failure of the out-ofplane restraint is caused by the buckling of the screw that is welded to the pulley and used to connect the steel sleeve and pulley. The buckling force (Fb) of one screw can be regarded as the ultimate carrying capacity of the out-of-plane restraint, and it can be calculated using Eq. (2) according to Ref. [19].
σtrue = σnom (1 + εnom)
(4)
εtrue = ln(1 + εnom)
(5)
where σnom and εnom are the nominal stress and strain of the material, respectively, which can be obtained from Section 2.2. 694
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Fig. 6. Test setup.
The combined hardening model [25], including the isotropic and kinematic hardening rule, was employed for the BRC core plate to describe the strength development of the core plate under cyclic loading. First, the isotropic hardening model defines the relationship between the yield surface size (σ0) and the equivalent plastic strain (εpl), as shown in Eq. (6):
σ 0 = σ0 + Q∞ (1 − e−bε
pl
)
(6)
where σ|0 is the yield stress when the equivalent plastic strain reaches zero, Q∞ denotes the maximum change in the yield surface size, and b is the rate of change in the yield surface as the plastic strain increases. The parameters σ|0 = 330 MPa, Q∞ = 21 MPa and b = 1.2 are used in this paper for the core plate based on material properties and reference 695
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Displacement control
4.3. Validation of the finite element model
80
80
40
40
0
0
-40
-40
-80
-80
The loading protocol used in the simulation analyses was same as that in the displacement-controlled stage of the pseudo-static test, and each loading amplitude was cycled only one time to accelerate the calculations. Fig. 15a shows a comparison of the hysteresis curves of the test and simulation results, and the shapes of the hysteresis curves are basically consistent for the two results. The differences between the test results and the simulation results are found in the initial stiffness and lateral load-carrying capacity, which may have been caused by the initial bending during installation and dislocation between the specimen and the actuator. Fig. 15b-c shows the out-of-plane deformation and stress of the rocking steel frame (the units of U3 and S are mm and MPa, respectively), which are consistent with the test results and phenomena. Fig. 15d shows the BRC deformation of the simulation results when the top displacement load was +60 mm. The larger out-of-plane deformation occurred at the end of the embedment portion of the BRC, and high-mode buckling did not occur in the energy dissipating portion. This failure is also consistent with the test phenomena. In conclusion, the proposed multi-scale finite element model can reflect the hysteretic responses of the RSFB. However, this simulation does not fully capture all aspects of the test results, and it is difficult to simulate the imperfections and boundary conditions of the test model. Therefore, the initial stiffness and pinching effect of the finite element result are not fully consistent with those of the test results.
Displacement (mm)
Force (kN)
Force control
Fig. 7. Loading protocol.
[26]. Second, the kinematic hardening model defines the calculation method of backstress (αk), as shown in Eq. (7): N
αk =
∑ k=1
pl Ck (1 − e−γk ε ) γk
(7)
where Ck/γk expresses the maximum change in backstress and γk is the rate of change in backstress as the plastic strain increases. For the kinematic hardening model, the critical parameters of N = 3, C1 = 8 × 103 MPa, γ1 = 100, C2 = 1.0 × 105 MPa, γ2 = 3.0 × 103, C3 = 5.0 × 102, and γ3 = 0 were adopted for the core plate in this paper.
5. Numerical simulation results for the test under full out-of-plane restraints The aforementioned test phenomena and results were caused by the
(a) Deformation and failure of the out-of-plane restraints
(b) Deformation of the BRC core plate at the loading displacement of +60 mm
(c) Failure modes of the BRC core plate after loading
Fig. 8. Deformation and failure modes of the BRCs. 696
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Fig. 9. Hysteresis curve. Fig. 12. Maximum strain of the steel braces in each story.
Fig. 10. Calculation method of E.
Fig. 13. Maximum strain in different measuring points of the rigid support.
5.1. Deformation and damage forms of the BRCs Fig. 16a shows that high-mode buckling occurred in the BRC energy dissipating portion of the RSFB-FOR model when the top displacement reached +60 mm, and the out-of-plane deformations in the end of the BRC embedment portions of the RSFB-FOR model are apparently reduced and negligible in comparison to the results shown in Fig. 15d. Thus, the comparison of Fig. 16a and Fig. 15d indicates that if the outof-plane restraints in the first story do not fail, then the out-of-plane deformations of the BRCs will be constrained, thereby ensuring the outof-plane stability of the RSFB and enhancing the structural seismic performance. Fig. 16b-c shows the values of the equivalent plastic strain (the value is PEEQ in ABAQUS), wherein PEEQ was used to express the accumulation of plastic strain and represent the damage degree. The PEEQ values in the end of the embedment portions are apparently greater than those in the other part of the BRC core plates in the RSFB model. However, the PEEQ values in the energy dissipating
Fig. 11. The values of E at each amplitude.
failure of the out-of-plane restraints in the first story. Thus, for an RSFB with full out-of-plane restraints in each story (RSFB-FOR), a finite element model was built to investigate the structural seismic performance when the out-of-plane restraints are not failing. The loading point and protocol were same as those used in Section 4.3.
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Fig. 14. RSFB finite element model.
where θmax is the maximum inter-story drift ratio, utop is the loading displacement, and H is the structural height of the RSFB (RSFB-FOR) model. A comparison of the inter-story drift ratios and DCF values for RSFB and RSFB-FOR shows that the lateral drifts are uniformly distributed along the height of the RSFB-FOR model. Furthermore, a sudden change in inter-story drift occurs in the RSFB model, which is induced by the structural out-of-plane instability and leads to a larger DCF value compared to that of the RSFB-FOR model. The aforementioned results show that the RSFB with full out-ofplane restraints can achieve the expected seismic performance and energy dissipating capacity. The rocking steel frame remains elastic, the lateral drifts are uniform, the BRCs do not undergo buckling at the end of the embedment portions and the structural plastic deformation is mainly concentrated in the energy dissipating portion of BRCs.
portion for the RSFB-FOR model are obviously increased, which indicates that the addition of full out-of-plane restraints to the RSFB can prevent the BRCs buckling and prompt the plastic deformation to concentrate in the energy dissipating portion of the BRCs. Therefore, the BRCs in the test specimen under full out-of-plane restraints can be used to achieve the expected deformation and damage forms. 5.2. Hysteretic behavior and energy dissipation Fig. 17a shows a comparison of the hysteresis curves between the RSFB specimen test results and the RSFB-FOR model simulation results. Compared to the RSFB, the RSFB-FOR model has a fuller hysteresis curve that does not exhibit pinching effects and has a larger peak lateral load-carrying capacity. According to Fig. 17b, the value of E increases with increasing loading displacement and was greater than 1.5 at each loading amplitude except 10 mm. In contrast to the RSFB model, the RSFB-FOR model does not exhibit a reduction in energy dissipating capacity, which means that the capacity of the BRCs in the RSFB-FOR model can be fully used.
6. Influence of the BRC embedment length on the performance of RSFB without full out-of-plane restraints According to the analytical results shown in Section 5, full out-ofplane restraints can ensure the out-of-plane stability of the RSFB and the structural seismic performance. However, the full out-of-plane restraint is an ideal lateral boundary condition, which was hard to realize in the experiments. Moreover, it can be observed in the test results shown in Section 3 that out-of-plane deformation occurred in the first story of the RSFB. Section 4 verified that applying the out-of-plane restraints in the second and third stories can approximately simulate the out-of-plane boundary conditions of the real test, which is a condition deemed as “without full out-of-plane restraints” in the paper. Therefore, this section focuses on the factor affecting structural seismic performance and out-of-plane deformation of the RSFB without full outof-plane restraints. The BRC embedment length will be investigated because it might be related to the BRC and structural out-of-plane
5.3. Stress distribution and structural lateral drift Fig. 18 shows the stress of the rocking steel frame for the RSFB-FOR model at Δ = +60 mm. The rocking steel frame basically remains elastic, and plastic deformation occurs only in the rigid support. Fig. 19 shows the inter-story drift ratios at Δ = ± 60 mm and the drift concentration factor (DCF) values for the RSFB specimen test results and RSFB-FOR model simulation results. The DCF values were calculated by Eq. (8) [2], and the values for the RSFB and RSFB-FOR are 1.49 and 1.02 (at Δ = +60 mm), respectively.
DCF =
θmax utop H
(8) 698
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Fig. 15. Comparison of the experimental and simulated results.
enhanced by elongating the embedment length. Therefore, the embedment length might have an important influence on the RSFB out-ofplane stability and, further, influence the structural seismic performance. If the value of Lep is small, the weak position will be located at the end of the embedment portion after the core plate is elongated, and the weak position easily buckles under reverse compression because of the lack of out-of-plane restraints, as shown in Fig. 20a. Thus, the
stability. 6.1. Description of the RSFBs with different embedment lengths According to Fig. 8b, the apparent out-of-plane deformation was observed at the ends of the embedment portions of the BRCs. Takeuchi et al. [27] noted that the single BRC out-of-plane stability can be 699
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Fig. 16. Comparison of the BRC deformation and damage between the RSFB and RSFB-FOR models.
6.2. BRC out-of-plane deformation and failure modes
plastic deformation is concentrated in the end of the embedment portion of the BRCs, and the energy dissipating portion cannot be deformed to dissipate energy, which is similar to the pseudo-static test results. Fig. 20b shows the BRC with a Lep of 120 mm, which was formed by increasing the Lep value of 60 mm to 120 mm based on the test BRCs. Note that the total length of the BRCs and the compression gap of these BRCs were the same as those of the test BRCs, at 883 mm and 20 mm, respectively. Based on the validated RSFB model in Section 4.3, many RSFB models with different BRC embedment lengths were established by increasing the Lep value from 60 mm to 120 mm, and they are named in series from RSFB-Lep-60 (RSFB-Lep-60 is same as the RSFB finite element model in Section 4.3) to RSFB-Lep-120. These models were used to investigate the influence of the elongated value of Lep on the structural seismic performance and out-of-plane stability. The loading protocol, boundary conditions and out-of-plane restraints were same as those used in the RSFB finite element model.
Fig. 21 shows the BRC out-of-plane deformation of the RSFB-Lep-120 at Δ = +60 mm. Compared with the RSFB model shown in Fig. 15d, the RSFB-Lep-120 model exhibits less out-of-plane deformation in the embedment portion. Moreover, high-mode buckling occurs in the energy dissipating portion of the BRCs in the RSFB-Lep-120 model, which does not occur in the RSFB model. These results indicate that increasing Lep can also enhance the structural out-of-plane stability and ensure the seismic performance of the RSFB without full out-of-plane restraints. The BRC deformation mode is similar to that of the RSFB-FOR model. Fig. 22 shows the out-of-plane deformation in the end of the BRC embedment portion at Δ = +60 mm. The symbols in Fig. 22 indicate the calculations performed in this paper for RSFB models with different embedment lengths. The out-of-plane deformation decreases with increasing Lep values, and the decreasing rate apparently slows when the value of Lep exceeds 80 mm. Fig. 23 shows the values of PEEQ for the BRC core plates in the RSFB models with different Lep values (60, 65, 70, 80, 100, and 120 mm) at
Fig. 17. Hysteresis curves and energy dissipation of the RSFB and RSFB-FOR models. 700
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Fig. 18. Stress distribution of the rocking steel frame.
RSFB-FOR and RSFB-Led-120 simulation results. A comparison of Fig. 24 and Fig. 17a shows that the hysteresis curve of the RSFB-Lep-120 is fuller than that of the RSFB test specimen and similar to that of RSFBFOR. Fig. 25 shows the cumulative energy dissipation coefficients (i.e., the values of E accumulated from Δ = 10 mm to Δ = 60 mm) of RSFB models with different values of Lep. The cumulative energy dissipation coefficient increases with increasing embedment length and tends toward a constant value when the value of Lep exceeds 80 mm. Furthermore, when the value of Lep exceeds 80 mm, the cumulative energy dissipation coefficients of RSFB are basically consistent with that of RSFB-FOR, which indicates that the increase in the embedment lengths of BRCs in RSFB without full out-of-plane restraints can achieve a similar energy dissipating capacity to that of RSFB-FOR. Fig. 19. Structural inter-story drift ratios and DCF values.
7. Conclusions Δ = +60 mm. The values of PEEQ in the energy dissipating portion increase with the increase in Lep, which means that the plastic deformation is continuously concentrated in the energy dissipating portion. However, the values of PEEQ at the end of the embedment portions are always larger than those in the energy dissipating portion, which indicates that the anticipated failure position may be located in this portion for a BRC rigidly connected to the rocking steel frame. The influence of the method of connection between the BRC and the rocking steel frame on the structural and BRC capacities requires further research.
This paper performs a pseudo-static test for the proposed rocking steel frame with buckling-restrained columns (RSFB) without full outof-plane restraints. Then, the performance of the test specimen with full out-of-plane restraints is numerically studied. Finally, the influences of the buckling-restrained column (BRC) embedment portion length on the structural performance and out-of-plane stability of the RSFB without full out-of-plane restraints are discussed. The latter two contributions can be regarded as supplements to the test. The conclusions are listed as follows. (1) The pseudo-static test results show that the structural capacity cannot be fully used due to the structural out-of-plane instability caused by the failure of the out-of-plane restraints in the first story
6.3. Hysteretic behavior and energy dissipation Fig. 24 shows the hysteresis curves for the RSFB test results and 701
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Fig. 20. Description of the BRCs with different embedment lengths.
Out-of-plane deformation(mm)
200 160 120 80 40 0
60
70
80 90 100 110 120 Lep (mm)
Fig. 22. Out-of-plane deformation in the end of the BRC embedment portion for RSFB models with different Lep values at Δ = +60 mm.
rigid support. A design suggestion for the out-of-plane restraint is provided. (2) The simulation results for the RSFB test are consistent with the test results and phenomena, implying that the proposed multi-scale finite element model can reflect the hysteretic responses of the RSFB. (3) The simulation results for the RSFB with full out-of-plane restraints in each story (RSFB-FOR) show that the proposed RSFB with full out-of-plane restraints can achieve excellent seismic performance and energy dissipating capacity, as expected. The rocking steel frame remains elastic, lateral drifts are uniformly distributed along the structural height, and the structural plastic deformation is mainly concentrated in the BRCs energy dissipating portion. (4) The increase in the embedment lengths of the BRCs (Lep) (from 60 mm to 120 mm) can also enhance the out-of-plane stability and performance of the RSFB without full out-of-plane restraints. However, when the value of Lep reaches 80 mm, the decreasing rate of out-of-plane deformation is apparently slowed, and the cumulative energy dissipation coefficients of the RSFBs are basically
Fig. 21. BRC out-of-plane deformation of the RSFB-Lep-120 at Δ = +60 mm.
of the RSFB. When the loading amplitude reaches 40 mm (θtop approximately equals 1.2%), the visible out-of-plane deformation occurs at the end of the embedment portions of the BRCs; thus, the hysteresis curve exhibits pinching effects, and the values of the energy dissipation coefficient (E) decrease. The braces in the rocking steel frame always remain elastic, and plastic deformation occurs at the end of the embedment portions of the BRCs and the 702
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Fig. 23. Values of PEEQ for the BRC core plate in the RSFB models with different embedment lengths.
In a future study, the proposed RSFB with full out-of-plane restraints should be retested to further verify Conclusion (3), and the influence of the connection details on the structural and BRC performance should be studied. This paper is based on the pseudo-static test; thus, the main conclusions are obtained from the test results and relative simulation analyses. Further research is needed to determine whether these
consistent with that of the RSFB-FOR. (5) Although the plastic deformation in the energy dissipating portion of the BRC increases when full out-of-plane restraints are added or Lep is increased, the ends of the embedment portions always have the maximum values of the equivalent plastic strain (PEEQ). Therefore, for a BRC rigidly connected to the rocking steel frame, rupture may occur at the end of the embedment portions. 703
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Fig. 24. Comparison of the hysteresis curves.
Fig. 25. Cumulative energy dissipation coefficients for RSFB with different values of Lep.
conclusions can be applied to mid- and high-rise models. Meanwhile, the design of BRC embedment length needs to be further investigated. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The research described in this paper was financially supported by the National Natural Science Foundation of China (51708166; 51778201; 51878233), the Fundamental Research Funds for Central Universities of China (JZ2019HGTB0086) and the China Postdoctoral Science Foundation (2018M630706). Their support is gratefully acknowledged.
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