Performance evaluation of RC frame with RC wall piers equipped with unbonded steel rod dampers subjected to in-plane loading

Engineering Structures 197 (2019) 109403

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Performance evaluation of RC frame with RC wall piers equipped with unbonded steel rod dampers subjected to in-plane loading ⁎

T

⁎

Qihao Hana, , Hiroyasu Sakatab, Yusuke Maidac, , Takayoshi Morid, Toshio Maegawae, Yongshan Zhanga a

School of Civil Engineering, Guangzhou University, Guangzhou 510006, China Department of Architecture and Building Engineering, Tokyo Institute of Technology, Tokyo 152-8550, Japan c Building Department, National Institute for Land and Infrastructure Management, MLIT, Ibaraki 305-0802, Japan d Structural Engineer Division, Kume Sekkei Co., Ltd., Tokyo 135-0052, Japan e Technical Division, Kumagai Gumi Co., Ltd., Tokyo 162-8557, Japan b

A R T I C LE I N FO

A B S T R A C T

Keywords: RC wall pier Shear failure Unbonded damper Numerical analysis Performance evaluation

Brittle shear failure of partial Reinforced Concrete (RC) wall pier system is usually observed in post-earthquake investigations, which affects the building functions seriously. A new RC wall pier system with unbonded steel rod dampers is introduced in this study. Numerical simulation using solid element method, validated by existing experimental results, was adopted firstly to investigate the behaviors of dampers, lateral force capacity, stiffness degradation, and ductility of the frame with the new RC wall pier system considering the influence of the dampers and the frame axial compression ratios. To decrease the huge computational costs of the solid element model, macro element model was further developed and verified. Performance evaluation of a twelve-story RC frame structure with three models of RC wall piers, namely damper model, slit model and rigid model, was conducted with the macro element model. Results show that RC frame with RC wall piers equipped with dampers possesses a stable lateral, and the designed dampers provide more than 20% of the lateral force to the frame. It is advised that the axial compression ratio of frame column shall be within 0.1 from the perspectives of plastic regions of dampers and lateral force capacity of frame. The nonlinear responses of the new RC wall pier system can be well captured by the macro element model with less computational costs. By comprehensively considering the structural drift ratios, the energy dissipation and the shear force in the frame beam, the damper model is suggested to be the best among the three models. The new RC wall pier system highlights an alternative for the conventional RC wall pier systems in the engineering practice.

1. Introduction RC frame structures are widely used in modern buildings with its relatively high cost-effectiveness performance compared with RC shearwall structures and steel structures. The structural components of RC frame, consisting of beams, columns, and slabs, are designed to resist horizontal and vertical loads. In Japan, RC frame structures are often constructed with partial RC wall (lightly RC wall) with openings (hanging/spandrel, wing and wall piers) for architectural design. The partial RC wall is not expected to resist the external force, which is typically 120–200 mm thickness, and horizontal reinforcement has no hook anchorage and boundary region has no confinement [1]. According to past post-earthquake investigations, severe damage on partial RC wall was observed in many RC buildings in and out of Japan,

⁎

such as the 2011 earthquake off the Pacific coast of Tohoku and 2016 Kumamoto earthquake, as shown in Fig. 1 [2,3]. Although the forceresisting beams and columns did not suffer damage, shear failure of partial RC wall affected the building functions seriously. In/before the 1990s in Japan, RC frame structures and partial RC wall piers were constructed simultaneously. The wall-frame interactions influenced the seismic behavior of RC frame structures. For example, shear failure of short columns with hanging/spandrel walls was reported in previous earthquake investigations [4,5]. However, influences of partial RC wall piers were generally neglected in structural design and evaluated under engineering judgments. Several researchers conducted experimental study, numerical and theoretical analysis on the partial RC walls that rigidly connected with surrounding structural components (such as Kono et al. [6,7]; Sengupta et al. [8]; Albidah et al.

Corresponding authors. E-mail addresses: [email protected] (Q. Han), [email protected] (Y. Maida).

https://doi.org/10.1016/j.engstruct.2019.109403 Received 7 April 2019; Received in revised form 12 June 2019; Accepted 13 July 2019 0141-0296/ © 2019 Elsevier Ltd. All rights reserved.

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Fig. 1. Photographs of damage on partial RC walls [2,3].

experimental specimens were performed with cyclic loading, and the tests showed that steel dampers provided stable energy dissipation capacity [15]. Damage of RC wall piers of specimen LD that dampers were installed in the lower wall pier was more severe, compared with that of specimen CD that dampers were installed in the center of wall pier. As shown in Fig. 3, serious damage on RC wall piers at the joints of damper and wall piers occurred with axial forces between the damper and wall piers arising under large drift ratio. The issue could be properly addressed by using an energy dissipation device that the axial force between the damper and wall piers is released. Partial RC wall piers equipped with unbonded steel rod dampers in the middle seismic slits were proposed by Mori et al. [16] and a series of components and RC frames with RC wall piers specimens were tested under cyclic loading. Although the experiment has been carried out, mechanical behaviors of the subassembly of a frame with RC wall piers are not fully understood and effects on the performance of RC frame structure with different kinds of RC wall piers are not yet known. In order to address these concerns, numerical analysis was performed. Behaviors of the equipped dampers, lateral force capacity, stiffness degradation and ductility of the frame with the new RC wall pier system considering the influence of the dampers and the frame axial compression ratios were investigated with solid element models. Macro element models were further developed and verified to decrease the huge computational costs of the solid element models. Performance evaluation of a twelvestory RC frame structure with three models of RC wall piers including damper model, slit model and rigid model was conducted.

[9]; Pecce et al. [10]), and some valuable conclusions were obtained. After the 1995 Kobe earthquake, the isolated walls from frame structural components were widely applied to RC buildings in Japan to avoid the unexpected wall-frame interactions. According to the Japanese construction guidelines for RC buildings [11], tie-bars (dowel bars) are often applied to the seismic slits between RC walls and primary frames, to prevent its falling or out-of-plane deformation excited by the external force. Yoon et al. [12] tested three 1/2.5-scale specimens under cyclic loading and proposed analytical models, which could evaluate the seismic behavior of RC frames with non-structural walls. Okubo et al. [13] conducted the tests of isolated non-structural walls using dowel bars, and it showed that the non-structural walls could be considered incorporated into actual buildings as a new passive control device. On one hand, negative wall-frame interactions effects on primary structural components can be avoided with the slit partial RC walls. On the other hand, if partial RC walls are considered to resist part of the external force, the force-resisting requirements of primary frames are decreased expectedly. The seismic performance of the whole structure can be improved accordingly. The seismic response control strategy that dampers applied in the seismic slits to provide stiffness and energy dissipation capacity is practical and effective. Lateral force-displacement characteristics of RC frame with different partial RC walls are illustrated in Fig. 2. It should be noted that rigid RC wall pier would suffer a brittle crack in the relatively small displacement. The available solution to control the damage of partial RC wall by introducing steel slit dampers in the seismic slit was proposed by Maida et al. [14]. Four

Fig. 2. Lateral force-displacement characteristics of RC frame with different partial RC walls. 2

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Fig. 3. Tests conducted by Suzuki et al. [15].

Ec Ec − fc / εc0

2. Introduction of the new RC wall pier system

n=

RC wall piers are often constructed in the mid-span of RC frame for architectural functions. Based on the previous researches on the issue of RC wall piers, Mori et al. [16] proposed a new RC wall pier system, which was equipped with steel rod dampers in the middle seismic slit of wall pier. RC frame with new RC wall pier system is demonstrated in Fig. 4, which is the existing experimental specimens. Configuration of the new RC wall pier system with unbonded dampers is illustrated in Fig. 5, steel rod dampers are inserted in the upper and lower RC wall piers. Half of the dampers are anchored in the lower RC wall pier with headed steel plates. The other half of the dampers attached with adhesive tapes, as unbonded material, are inserted in the upper RC wall pier. In addition, polystyrene foam provided in the slit of wall piers are to increase the flexible length of dampers and that provided on the end of dampers is to allow dampers to move up and down freely. It is expected that the steel rod dampers provide lateral stiffness and dissipate energy by plastic deformation of the dampers in the flexible region around the middle slit, as the relative displacement between the upper and lower RC wall piers occurs with the sway of frame. The axial force along the length direction of dampers is released because of the use of the unbonded material, which decreases the tension damage of RC wall piers prospectively. RC wall pier with bonded dampers system is used for comparison, in which the dampers are anchored at both ends and vertical sliding is restrained. The simple configurations and low cost of the proposed RC wall pier system make it practical and feasible in the engineering practice.

where fc and εc0 are the uniaxial compressive strength and the corresponding strain respectively; Ec is the concrete Young’s modulus. The concrete uniaxial tension response follows a linear elastic relationship until the tensile strength ft is reached. The failure stress corresponds to the onset of micro-cracks in the concrete. A softening stress-strain response is represented beyond the tensile strength. In order to decrease the mesh sensitivity, the stress-crack opening response rather than a stress-strain response was defined as the softening region. The Cornelissen model [21] was adopted for the tensile stress and crack opening behavior, as plotted in Fig. 6(a).

(2)

3

σt (u) =

u u u ⎫ ⎧⎡ 1 + ⎛c1 ⎞ ⎤ e−c2 utc − (1 + c13) e−c2 ft ⎥ ⎬ ⎨⎢ u utc tc ⎠ ⎦ ⎝ ⎭ ⎩⎣ ⎜

⎟

(3)

3.1.1. Material constitutive behavior

where c1 and c2 are constants with values of 3 and 6.93 respectively; ft is the uniaxial tensile strength; utc is the ultimate crack opening equaling to 5.14Gf/ft [22], in which Gf is the fracture energy, Gf = αf (0.1fc )0.7 , indicating the energy required to open a unit area of crack, in which αf is a coefficient related to the maximum aggregate size Dmax. The fracture energy coefficients are given in Table 1 according to the CEP-FIB [23]. The damage variables dc and dt define the degradation of the unloading stiffness, which can be calculated using the Eqs. (4) and (5). The damage variables have values from 0 (representing the undamaged stage) to 1 (representing the total material failure). The stiffness recovery factors wt and wc define the elastic modulus change extent from tension to compression and from compression to tension, as shown in Fig. 6(b). In this study, wt = 0 and wc = 1 are defined, indicating that the compressive stiffness is recovered upon crack closure as the load changes from tension to compression. The tensile stiffness is not recovered as the load changes from compression to tension once crushing micro-cracks have developed.

(1) Concrete

dc =

3. Numerical simulation with solid element model 3.1. Establishment of solid element model

The general-purpose nonlinear finite element software Abaqus/ Standard [17] was adopted to conduct the numerical analysis of this study. The Concrete Damaged Plasticity (CDP) model was used to predicate the behavior of concrete [18,19] which assumes two main failure mechanisms, namely compressive crushing and tensile cracking for the concrete. Stiffness recovery effect is considered by the CDP model during cyclic loading. The uniaxial stress-strain curve is illustrated in Fig. 6(a). The concrete uniaxial compressive stress-strain behavior consists of the elastic region, stress hardening region and strain softening region and can be described as [20]:

ε n ⎞ σc (ε ) = fc ⎛ ⎞ ⎛ ε n − 1 + (ε / εc 0)n ⎠ c 0 ⎝ ⎠⎝ ⎜

⎟⎜

dt =

εc,in (1 − βc ) Ec σc + εc, in (1 − βc ) Ec

(4)

ut (1 − βt ) Ec σt + ut (1 − βt ) Ec

(5)

where εc, in is the uniaxial compressive inelastic strain; βc is the ratio between compressive plastic strain and inelastic strain; ut is the crack opening; βt is the ratio between crack opening and ultimate crack opening. The material properties of concrete, such as uniaxial compressive strength fc, tensile strength ft, Young’s modulus Ec, and strain at compressive strength εc0, were obtained from the material experimental results, as given in Table 2, in which two specimens with different materials were designed and details can be found in the reference [16]. The initial yield point σc,e0 = fc/3, the maximum compressive strain of 0.003 were defined according to the ACI Code [24].

⎟

(1)

3

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Fig. 4. Details of RC frame with new RC wall pier system (units in mm).

because it only consists of an elastic region and hardening region based on the material test of steel. As illustrated in Fig. 7(a), where εy and σy denote the yield strain and the yield stress respectively, εu and σu denote the ultimate strain and the ultimate stress respectively. The trilinear

(2) Steel The isotropic bilinear model was adopted by rebar (short for reinforcing bar) D6 (representing the rebar with a diameter of 6 mm) 4

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Table 1 Aggregate size-based fracture energy coefficients [23].

model was adopted for dampers PL16 (representing the steel plate with a thickness of 16 mm) and rebar D10&D13 because the obvious yield plateaus were observed in the tests, as shown in Fig. 7(b). The properties of steel are given in Table 3. Engineering stress and engineering strain were obtained with the original dimensions of a specimen. However, length and cross-sectional area change in the plastic region. As shown in Fig. 8, the true stress-true strain relationship is given from the tensile coupon test by considering the actual (instantaneous) dimensions. The true stress σT and true strain εT are obtained by: (6)

εT = ln(1 + ε )

(7)

Coefficient (αf)

8 16 32

0.025 0.03 0.058

concrete and dampers. The maximum bond stress was assumed to be 0.3 fc [23]. For the part of unbonded material attached on dampers, hard contact relationship of normal behavior and penalty friction with zero friction coefficient (frictionless) of tangential behavior were applied on the interfaces of unbonded dampers and the corresponding concrete, as illustrated in Fig. 9. Boundary conditions were defined, as demonstrated in Fig. 10. The Z translational degrees of freedom (UZ) of the surface cut was restrained. To model the pin supports at the top and bottom of columns, the reference points were set at the corresponding pin connections firstly. Then, all degrees of freedoms at the surface of the columns sections were coupled with the corresponding reference points. Finally, Z rotational degrees of freedom (RZ) of the reference points were released. Connector element of CONN3D2 with “link” behavior, ensuring a constant distance between the two reference points and having no effect on the rotational degrees of freedom, was applied at the top of the two columns to simulate the rigid loading beam. Because of the stress concentration at the interface between concrete and dampers, elements of concrete are distorted seriously. In addition, the involving friction behavior raises severely discontinuous problems, which causes convergence difficulties. A monotonic loading with target drift ratio of 4% was applied at the loading point. The computer with eight Intel Core i7 CPUs and 8 GB memory was used to perform the analyses. The computational costs with solid element model are given in Table 4.

Fig. 5. The new RC wall pier system [16].

σT = σ (1 + ε )

Maximum aggregate size (Dmax, mm)

where σ is the engineering stress and ε is the engineering strain.

3.2. Validation of the solid element model

3.1.2. Numerical program Only half of the model was established with its symmetry to decrease the computational costs. Concrete and steel rod dampers were modeled by the eight-node reduced integration solid element C3D8R, and rebar was modeled by the two-node truss element T3D2. Rebar was embedded in the concrete, in which a perfect bond between rebar and concrete was assumed. Slip between the steel rod damper and the surrounding casting concrete was considered due to the relatively smooth surface of the dampers. A cohesive and damage behavior interaction was adopted for the slip behavior between cast-in-place

Comparisons of lateral force versus drift ratio between the experiment and simulation are shown in Fig. 11. Comparisons of the corresponding skeleton curves are shown in Fig. 12. It can be found that the lateral force obtained from the simulation agrees well with that of the experiment. The maximum error of the lateral force of the specimen MD-SS-B is 7.6%, as shown in Fig. 12(a). For the specimen MD-SS-U, the lateral force from simulation appears higher than that from the experiment at the previous two loading stages, and the maximum error is 10.7%, as shown in Fig. 12(b). As shown in Figs. 13 and 14, it demonstrated that concrete damage is mainly distributed at the ends of

Fig. 6. Constitutive models of concrete. 5

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Table 2 Properties of concrete [16]. Specimen

Compressive strength (fc, MPa)

Tensile strength (ft, MPa)

Young’s modulus (Ec, MPa)

Strain at compression strength (εc0)

MD-SS-B MD-SS-U

41.4 44.3

2.5 2.6

29,550 30,230

0.00247 0.00259

Fig. 7. Constitutive models of D6 and D10. Contact pressure Any pressure possible when in contact

Table 3 Steel properties. Steel

Yield strength (σs, MPa)

Ultimate strength (σu, MPa)

Young’s modulus (Es, MPa)

D6 (SD295A) D10 (SD295A) D13 (SD345) PL16 (SN400B)

351* 360 383 298

538 492 591 458

192,000 192,000 191,000 205,000

Unbonded

No pressure when no contact Clearance Contact stress bonded

tnmax (tsmax , ttmax )

Note: 351* is the 0.2% proof stress.

K n ( Ks , K t ) 1

beams first, followed by beam-column joints, and finally wall piers, indicating that the numerical result is consistent with experimental results.

δ nmax(δ smax, δ tmax)

Separation

Fig. 9. Interactions of elements (specimen MD-SS-U).

regions appears as the relative displacement between the upper and lower RC wall piers occurs. Fig. 16 shows a close-up of the yield of dampers, the yield area of the dampers of specimen MD-SS-B is border than that of specimen MD-SS-U because the dampers of the latter can move up and down freely. From the views of the concrete damage of RC wall piers and yield of rebar, the MD-SS-U is a relatively superior system.

3.3. Comparison of specimens As pointed out in Figs. 13 and 14, concrete damage of RC wall pier of the specimen MD-SS-U is significantly decreased than that of the specimen MD-SS-B because the axial force is released with the unbonded dampers. The yield state of rebar frame is shown in Fig. 15, where the AC YIELD is a flag indicating if the material is yielding or not. The longitudinal rebars of the wall pier at the interface between the wall pier and the lower beam of specimen MD-SS-B are yield. It can be inferred that the proposed RC wall pier system can avoid the rebar yield prematurely. Plastic deformation of the steel rod dampers at the flexible

3.4. Contribution of dampers to structural lateral stiffness The lateral force contribution to the frame structure from dampers is

Fig. 8. True stress-strain relationship. 6

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hinge is generated at the ends of frame beam, as shown in Fig. 13. 3.5. Effect of axial compression ratio Frame columns are typically subjected to vertical and lateral loads simultaneously. The vertical loads are generally respected by axial compression ratio (ACR) of column with equation of ACR = N/(fcA), in which N is the vertical load, fc is the compressive strength of concrete and A is the overall sectional area of column. ACR is normally in the range of [0, 0.2] and is one of the key factors of evaluating structural ductility [25]. Effects of ACR on behaviors of the equipped dampers, lateral force capacity, stiffness degradation, and ductility of RC frame were investigated and there are four kinds of ACRs, namely 0, 0.05, 0.1, 0.2. The simulation could be conducted with two analysis steps. Vertical loads were applied at the loading points firstly, and the horizontal displacement-controlled loading method was further performed under the constant vertical loads. The target drift ratio of all of the four frame models was defined as 2%.

Fig. 10. Boundary conditions.

Table 4 Computational costs with solid element model. Model

MD-SS-B (with damper) MD-SS-B (without damper) MD-SS-U (with damper) MD-SS-U (without damper)

Elements

41,153 38,274 40,659 37,980

Nodes

52,346 46,721 51,315 46,140

Output file size (GB)

Time (h)

7.2 3.5 15.5 3.3

12.1 5.4 23.6 5.2

3.5.1. Yielding of dampers Plastic hinges occur at the ends of beams with the sway of frame. As shown in Fig. 19, deformation of the ends of beams with ACR = 0.2 is relatively larger than that with ACR = 0.05. It indicates that the rotation of plastic hinge increases as ACR increases. Concrete damage of RC frame of MD-SS-U at the end of loading is shown in Fig. 20. It can be found that concrete damage is mainly distributed at the ends of beams and beam-column joints with an ACR of 0.05 while concrete damage mostly occurs at the ends of beams with an ACR of 0.2. It means that damage of frame tends to concentrate on the ends of beams as ACR increases. An equivalent result was demonstrated with the experimental study of Ref. [26], in which it reported that the weakening of the beam end and the failure process of the entire structure were accelerated as the ACR increased. Yielding of the unbonded steel rod dampers of specimen MD-SS-U with different ACRs are demonstrated in Fig. 21. It can be observed that yield zones of dampers broaden as ACR increases. The reason is that rotation of ends of beams are increased with the increase of ACR, causing a larger relative displacement between the upper and lower RC wall piers. The upper yield zones of dampers are beyond the flexible region (with a gap of 60 mm) before an ACR of 0.05. As ACR beyond to 0.1, the upper yield zones of dampers increase significantly and it extends to the flexible region. Yield zones of dampers with an ACR of 0.2 are almost the same as that with an ACR of 0.1. From the perspective of energy dissipation of dampers, it is advised that ACR shall be within 0.1 due to the broadest plastic regions of dampers, thus supplying an optimal energy dissipation. In addition, yielding of the bonded dampers of specimen MD-SS-B remains almost unchanged with the different ACRs due to the anchorage effects at its both ends.

shown in Fig. 17. It can be obtained that the designed dampers provide more than 20% of the lateral force to the frame. As for the specimen MD-SS-B, dampers resist more than 40% of structural lateral force before drift ratio of 0.5%. After that, contribution of dampers to the structural lateral force maintains about 30% as damper yielding. As for the specimen MD-SS-U, although the contribution of the dampers appears a little lower than that of the specimen MD-SS-B, there is also more than 20% for the contribution, indicating that the designed dampers can contribute stable lateral force to the frame. Lateral secant stiffness at each loading level is used to evaluate the stiffness degradation behavior of the two specimens, which is defined as Ki = Fi /Δi , in which Ki and Fi are the secant stiffness and the lateral force at the ith loading level, Δi is the lateral displacements corresponding to the lateral force. As shown in Fig. 18, it can be obtained that the lateral secant stiffness of the frame with dampers is higher than that without dampers and appears slower downward trend, meaning that the designed dampers contribute stable lateral stiffness to the frame and improve the structural lateral resistance in a certain extent. In addition, the lateral secant stiffness of the frame decreases with increasing drift ratio and appear significant decreasing before drift ratio of 1%, indicating that the concrete material begins to damage and the dampers are tending to yielding. Reduction of the lateral secant stiffness shows slow gradually when drift ratio beyonds to 1% because plastic

200

300

Experiment Simulation

Lateral force (kN)

Lateral force (kN)

300

3.5.2. Lateral force capacity and stiffness degradation Relationships between lateral force and drift ratio of the frame are

100 0 -100 -200 -0.04

-0.02 0.00 0.02 Drift ratio (rad)

200 100 0 -100 -200 -0.04

0.04

Experiment Simulation

-0.02 0.00 0.02 Drift ratio (rad)

Fig. 11. Lateral force versus drift ratio relationships of specimens. 7

0.04

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Fig. 12. Comparison of skeleton curves.

constitutive behavior of the materials. Although it can track the stress distribution and damage of structural members, it needs a significant calculating memory and is time-consuming. The results are sensitive to many parameters and modeling details, which makes it impractical for engineering analysis and design. Macro element model was further developed with the purpose of capturing structural response and saving computational costs at the same time to ensure engineering applications.

shown in Fig. 22. It can be obtained that the lateral force shows increasing to a certain extent as ACR varies from 0 to 0.1. When ACR reaches to 0.2, the lateral force appears obvious fluctuations with the loading drift ratio. The reason of this fluctuation is that concrete damage and plastic deformation mostly concentrate in the ends of beams as an ACR of 0.2 and the failure process of the entire frame are accelerated, as demonstrated in Figs. 19 and 20. Lateral stiffness versus drift ratio is shown in Fig. 23. It can be found that the lateral stiffness of the frame structure increases slightly as ACR increases.

4.1. Establishment of macro element model 3.5.3. Displacement ductility The displacement ductility coefficient μ is usually evaluated by the ratio of the ultimate displacement Δu to the yield displacement Δy. The maximum displacement Δm, with a value of 30 mm, is selected as the ultimate displacement because it has almost no descending branch of the lateral force-drift ratio curves in simulations. The equivalent yield point is determined by the geometric graph method [27], as shown in Fig. 24. Firstly, make a straight OA that tangent to the initial segment of the curve, and it intersects at point A with the horizontal line through the peak point M. Secondly, make a vertical line through point A, and it intersects at point B with the curve. Thirdly, connect straight OB, and it intersects at point C with the horizontal line through the peak point M. Finally, make a vertical line through point C, and it intersects at point Y with the curve. The obtained point Y is the equivalent yield point, thus the yield displacement Δy and the corresponding yield strength Py can be determined. The displacement ductility is summarized in Table 5, where Pm is the maximum lateral force. It shows that the displacement ductility decreases as ACR increases. However, as ACR reaches to 0.2, the displacement ductility increases. For instance, the displacement ductility of specimen MD-SS-B is decreased by 9% as an ACR of 0.05, compared with that as ACR of 0. However, the displacement ductility is increased by 5.9% as an ACR of 0.2, compared with that with an ACR of 0.1.

4.1.1. Determination of plastic zones of frame beams and columns Based on the experimental and numerical investigation, it can be obtained that there are obvious elastic and plastic zones during loading. The plastic zone is normally generated at the ends of frame beam, which can be evaluated by the stress state of main rebar. Length of the plastic zone of frame beams was assumed as the depth of beam section D2 based on the yield state of longitudinal rebars and cracks of concrete. Cracks of concrete of the specimen MD-SS-B are shown in Fig. 25, and the strain distribution of longitudinal rebars in beams of the specimen MD-SS-B is summed in Fig. 26 based on the experimental data. Although the longitudinal rebars in columns did not yield during the loading process, certain cracks for concrete could also be found at the ends of column. Length of the plastic zone for columns was assumed as 0.5D1. Moreover, large internal force was generated at the interaction between beam and wall pier. Cracks occur on the upper and lower beams, and the length of the plastic zone for the beams near the wall pier were assumed as 0.5D2. Beam-column connections were assumed as rigid joints. Rigid zones were defined in the range from each node to 1/4 of the depth of the corresponding structural member according to the Japanese standard [28]. The macro element model was established by the open-source software OpenSees [29], as shown in Fig. 27. 4.1.2. Modeling of beams and columns Beams and columns were modeled with the displacement-based beam-column fiber elements, which was thought to be with good accuracy and convergence efficiency [30]. As shown in Figs. 28 and 29(a),

4. Numerical simulation with macro element model The solid element model is investigated based on plastic damage

Fig. 13. Concrete damage of specimen MD-SS-B at the end of the loading. 8

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Fig. 14. Concrete damage of specimen MD-SS-U at the end of the loading.

Modeling of RC wall pier and damper are shown in Fig. 30. The three-vertical-line-element-model (TVLEM) was adopted to simulate the wall pier [32]. The rigid body was taken as the depth of beam at beam-wall joints. The wall pier was assumed as shear wall. The central wall segment was assumed to the 0.9Dw (630 mm) for a rectangular shear wall, in which Dw is the depth of wall section, and it was modeled with axial spring, shear spring, and bending spring. The section depth of boundary regions with supplementary reinforcement was taken as 0.05Dw (35 mm), and truss elements were used. In addition, the wall pier was divided into two segments along its height, wall I and II, based on the different reinforced ratios, shown in Fig. 30. Ideal elastic-plastic model was used by axial spring, as shown in Fig. 31(a). The compressive strength WNcy was combined by the compressive strength of both concrete and rebar, while the tensile strength WNty was only from the tensile strength of rebar. Following equations can then be obtained:

the confined concrete was defined in considering of the confining effect of the core concrete within the hoop/stirrup rebars and was modeled with the Mander model [31]. The Popovics model was adopted for the strain-stress relationship of the unconfined concrete. The longitudinal rebars were modeled with Menegotto-Pinto model, which was represented by the uniaxial Steel02 material model in OpenSees. The parameters R0, cR1, and cR2, which controls the nonlinear curve shape, were taken as 18.5, 0.925, and 0.15, respectively, as shown in Fig. 29(b).

ε n ⎞ σc (ε ) = fcc ⎛ ⎞ ⎛ n ⎝ εcc ⎠ ⎝ n − 1 + (ε / εcc ) ⎠

(8)

f ⎤ ⎡ εcc = εc 0 ⎢1 + 5 ⎜⎛ cc − 1⎟⎞ ⎥ f c 0 ⎝ ⎠⎦ ⎣

(9)

⎜

n=

⎟⎜

⎟

Ec Ec − fcc / εcc

7.94fl′ f′⎞ ⎛ fcc = fc 0 ⎜−1.254 + 2.2654 1 + −2 l⎟ fc 0 fc 0 ⎠ ⎝

fl′ =

1 ke ρ f 2 s yh

(10)

(11)

W Nty

= σy As

(13)

W Ncy

= σB Ac + σy As

(14)

where σy is the yield strength of rebar, As is the sectional area of longitudinal rebar, σB is the compressive strength of concrete, Ac is the sectional area of compressive concrete. The trilinear model with Takeda hysteretic principle was adopted by the shear spring with shear force-drift behavior and bending spring with moment-rotation behavior, as shown in Fig. 31(b) and (c). The shear cracking strength WQcr, shear ultimate strength WQu, bending cracking strength WMcr, and bending ultimate strength WMu were evaluated based on the Japanese standards [28]. The drift ratio at shear ultimate strength was assumed to be 0.004 rad. The hardening stiffness after the ultimate strength was taken as 0.001 of the elastic stiffness. The decreasing rate of bending stiffness αu can be calculated by Eq. (20)

(12)

where fcc and εcc are the compressive strength of confined concrete and the corresponding strain; fc0 and εc0 are the compressive strength of unconfined concrete and the corresponding strain; Ec is the concrete Young’s modulus; fl′ is the effective lateral confining stresses; ke is the confinement effectiveness coefficient; ρs is the ratio of volume of the transverse confining steel to the volume of the confined concrete core; f yh is the yield strength of the transverse rebar. 4.1.3. Modeling of RC wall pier and damper

W Qcr

(1) RC wall pier

= φ σT2 + σT σ0 bD / κ

Fig. 15. Yield of rebar at the end of the loading. 9

(15)

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Fig. 16. Yield of dampers (close-up).

Fig. 17. Contribution of the lateral force from the designed dampers.

Fig. 18. Stiffness degradation.

Fig. 19. Deformation of RC frame with RC wall piers at the end of loading.

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Fig. 20. Concrete damage of RC frame of MD-SS-U at the end of loading.

Fig. 21. Yielding of dampers with different ACRs of specimen MD-SS-U.

W Qu

0.053pt0.23 (σB + 18) + 0.85 pw σwy ⎫ bj =⎧ ⎬ ⎨ M /(Qd ) + 0.12 ⎭ ⎩

of pw = a w /(bs ) ; a w is the sectional areas of one group shear reinforcement; s is the spacing of shear reinforcement; σwy is the yield strength of shear reinforcement; j is the spacing of section stress center with j = 7d/8.

(16)

where φ is the capacity factor with φ = 1; σT is the concrete tensile strength, which can be calculated by σT = 0.33 σB ; σ0 is the axial stress with the external forces; b and D are the sectional width and depth respectively; κ is the section shape factor with κ = 1.5; pt is the ratio of tensile rebar (unit in %) with equation of pt = at /(bd ) ; at is the sectional areas of tensile rebar; d is the effective depth of cross-section; M /(Qd ) is the shear-span ratio; pw is the shear reinforcement ratio with equation

W Mcr

Ze =

2n·(at + ac )·ys2 bD 2 + 6 D

W Mu

11

= 0.56 σB Ze

= at σy l w + 0.5a w σwy l w

(17)

(18) (19)

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Fig. 22. Lateral force versus drift ratio with different ACRs.

Fig. 23. Stiffness degradation with different ACRs.

αu =

w Mu cn Ec Iw εy

(20)

where n is the ratio of Young’s modulus of rebar and concrete; ys is the distance between tensile rebar and neutral axis; σy is the yield strength of rebar at tensile boundary region; lw is the central distance of boundary region, which is taken as 0.9D for rectangular section; cn is the distance between tensile boundary region and neutral axis; Iw is the second moment of area of wall section; εy is the yield strain of longitudinal rebar at boundary region. (2) Damper Steel rod damper was modeled by bilinear model with kinematic hardening principle at axial, shear, and bending directions. The hardening stiffness was taken as 0.01 of the elastic stiffness, which agreed with the experimental result of steel material. The bonded part of the damper was modeled as axial, shear and bending springs, while the unbonded part was modeled as shear and bending springs as the axial force was released.

Fig. 24. Determination of the equivalent yield point.

Table 5 Displacement ductility. Specimen

ACR

Δy (mm)

Py (kN)

Δm (mm)

Pm (kN)

μ (Δm/ Δ y)

Increasing rate of ACR

MD-SS-B

0 0.05 0.1 0.2

7.32 8.04 8.39 7.92

113.3 121.8 125.2 122.0

30 30 30 30

179.9 190.3 195.2 189.8

4.10 3.73 3.58 3.79

– −9.0% −4.1% 5.9%

MD-SS-U

0 0.05 0.1 0.2

7.32 7.95 8.58 8.15

108.1 115.1 121.8 117.6

30 30 30 30

171.8 180.9 191.2 183.8

4.10 3.77 3.50 3.68

– −7.9% −7.3% 5.3%

4.2. Validation of the macro element model Fig. 32 shows the comparisons of the lateral force- drift ratio curves between the numerical results and the tests. It can be found that the numerical results agree well with that of the tests under both cyclic and monotonic loadings. The maximum errors between them for specimen MD-SS-B and MD-SS-U are 13.4% and 16.1% respectively, indicating that the macro element model is feasible to capture the general hysteretic behaviors of the frame, including lateral force, loading stiffness, and unloading stiffness. Lateral force obtained from numerical simulation is higher than that of the test after drift ratio of 0.01 because the

Note: Increasing rate of ACR0.2 = (ACR0.2-ACR0.1)/ACR0.1 × 100%, ACR0.1 represents an axial compression ratio of 0.1.

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Fig. 25. Plastic zone based on cracks of concrete of specimen MD-SS-B.

Fig. 26. Strain distribution of longitudinal rebars in beams of specimen MD-SS-B.

Fig. 27. Macro element models of the specimen.

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Fig. 28. Member cross-sections.

Fig. 29. Uniaxial stress-strain relationship.

5. Performance evaluation of RC frame based on the macro element model

concrete at the interface between damper and concrete is crushed in the test. However, the damage of concrete is ignored by the macro element model. Besides, results of the macro element model are higher than that of the solid element model relatively. The main reason is that the macro element model consists of mechanical components such as elastic and plastic zones of frame and TVLEM of RC wall piers based on some ideal assumptions. Moreover, the confined concrete was defined in the beams and columns for macro element model, while it was neglected by the solid element model. As pointed out in Fig. 26, strain of longitudinal rebar in beam is shown in Fig. 33. It can be obtained that the simulations agree with the experiment, meaning that the mechanical behaviors of rebar can also be captured by the macro element model. In a word, the developed macro element model can be thought with adequate reliability. The same computer used for the simulation with solid element model was adopted to perform the analysis and the computational costs with macro element model are given in Table 6. It shows that computational costs with macro element model are greatly decreased compared to that with solid element model, as listed in Table 4. The former indicating a huge advantage for engineering applications.

5.1. Prototype of RC frame Based on the verified macro element model, performance evaluation of a twelve-story frame structure with three kinds of RC wall pier models, namely damper model, slit model and rigid model, were further investigated through nonlinear time history analysis. The twelve-story RC frame structure is derived from the prototype of the Japanese design code [33]. Details of the prototype are shown in Fig. 34, RC wall piers, as the outer wall piers, are arranged at the middle span of frames Y1 × X2–X3, Y1 × X5–X6, Y4 × X2–X3, and Y4 × X5–X6. The span and thickness of the RC wall pier are 1500 mm and 150 mm respectively. Two vertical D13 rebars are adopted at end regions of wall pier and distributed horizontal/vertical D10 rebars are arranged with a spacing of 450 mm in a staggered pattern according to AIJ recommendations of nonstructural components [1]. 5.2. Analysis models and input ground motions To investigate the effect of RC wall pier on the performance of frame beam and column, three RC wall pier models were considered according to different connection methods between wall pier and frame beam, as shown in Fig. 35. (i) Slit model: full-length slit with a width of 14

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Fig. 30. Macro element models of RC wall piers and dampers.

Fig. 31. Behaviors of spring models for RC wall pier.

deformation behavior was the same with Section 4.1.3. Plastic hinges were inserted in the two ends of beams and columns. The M3 and interacting P-M2-M3 hinge types were adopted for beam and column respectively. Mechanical behaviors of the plastic hinges were defined as typical flexural hinge property based on FEMA 356 [34], as shown in Fig. 36(b). It can be found that linear response from A to effective yield point B is defined, followed by a slope of BC reflecting the strain hardening phenomena, CD reflecting significant strength degradation, DE reflecting reduced resistance, and finally, EF reflecting totally loss of the resistance. Building performance level is defined as Immediate Occupancy (IO), Life Safety (LS) and Collapse Prevention (CP). The finite element model of the Damper model is shown in Fig. 37. Nonlinear time history analysis was performed with seven ground motions, including five recorded ground motions and two artificial earthquakes. Two levels of ground motion intensity [35], corresponding to damage

40 mm was set between the wall pier and the lower frame beam. (ii) Rigid model: rigid connection between the wall pier and frame beam was adopted. (iii) Damper model: the proposed RC wall pier system with unbonded steel rod damper, as a superior RC wall pier that had been proved in both the experiment [16] and the simulation of this study, was considered as the Damper model. The Slit and Rigid models are the conventional RC wall pier systems adopted in RC frames, which have been constructed in many engineering projects nowadays. The same steel material parameters with full-scale dampers in the previous test were used. Modeling and analysis were conducted using software SAP2000. As shown in Fig. 36(a), beams and columns were modeled with frame element. The macro element with TVLEM model was adopted to model the wall piers, in which the link elements with the nonlinear Takeda hysteresis properties were selected and the definition of force15

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Fig. 32. Measured and computational lateral force versus drift ratio relationships.

Fig. 33. Comparison of strain of longitudinal rebar in beams.

drift ratio of the Rigid model and Damper model decrease by 25.3% and 19.9% compared with that of the Slit model under the ground motions in Level 2.

Table 6 Computational costs with macro element model. Model

Elements

Nodes

Output file size (KB)

Time (s)

MD-SS-B (with damper) MD-SS-U (with damper)

85 85

68 68

147 109

75 43

5.3.2. Hinge state of frames Plastic hinge state of RC frame in Level 2 is shown in Fig. 41. Number of plastic hinges in beams and column are summarized in Fig. 42, in which Flag B represents yielding; Flag IO represents the immediate occupancy status; Flag LS represents life safety status. At Level 1, most of the plastic hinges are mainly in IO status and few plastic hinges remain B status indicating the initial yield of columns. Sixteen plastic hinges with LS status occur in beams of the Slit model while it is changed to IO status in the Rigid model and Damper model. The reason is that the wall piers assure the integrity of the structure and the dampers dissipate parts of input energy with dampers, thus deformation of the whole structure is redistributed. At Level 2, as demonstrated in Fig. 41, distribution of plastic hinges in beams and columns is the desired damage state, in which damage mainly locates at ends of beams while minor damage on columns. Most of the plastic hinges occur at ends of beams and mainly in LS status while there is no plastic hinge in CP status, indicating that the structure has enough strength and stiffness, and the structure will not collapse. Damage on columns of the Rigid model and Damper model is more serious compared with that of the Slit model. The main reason is that lateral stiffness of the Rigid model and Damper model is larger than that of the Slit model, thus attracting greater seismic force. However, plastic hinges in columns are in B and IO statuses, maintaining an adequate gravity load carrying capacity.

Note: Output file size is counted with the data file of lateral force and drift ratio.

limit state (Level 1) and safety limit state (Level 2), were determined. Fig. 38 shows the spectrum of the selected input ground motions, indicating the spectrum of the selected ground motions agrees well with the target spectrum. It should be noted that the macro element models of RC wall piers were developed with constitutive behaviors based on the in-plane characteristics, meaning that out-of-plane behavior of RC wall pier was ignored. 5.3. Performance evaluation of RC frames 5.3.1. Distribution of maximum drift ratios Distributions of the maximum drift ratio under seven ground motions are shown in Figs. 39 and 40. It shows that drift ratio of the whole structure is distributed uniformly, indicating uniform lateral stiffness of the structure. At Level 1, average of the maximum drift ratio occurs at the fifth story with value of 1/242, 1/322, and 1/288 for the Slit model, Rigid model, and Damper model, respectively. Average of the maximum drift ratio of three models is less than the limit value of 1/200. At Level 2, average of the maximum drift ratio also occurs at the fifth story with value of 1/121, 1/162, and 1/151 for the Slit model, Rigid model, and Damper model, respectively. It can be found that the maximum drift ratio decrease for the Rigid model and Damper model because the wall piers provide lateral stiffness for the whole structure and dampers dissipate input energy of ground motions. Average of the maximum

5.3.3. Internal force of subassembly of the frame with RC wall piers Bending moment diagrams of the subassembly of frame with RC wall piers at the eighth story in Level 2, which is tagged in Fig. 41 with 16

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Fig. 34. Details of the prototype (units in mm).

Fig. 35. Configurations of analysis models.

Fig. 36. Details of the model.

the bending moment in the beams of the Rigid model is increased compared with that in the Slit model. However, the bending moment in some sections of beam for the Damper model is lower than that for the Slit model while the bending moment in other sections is higher than the corresponding one. Note that plastic hinges with LS status occur in

the dotted line, are plotted in Fig. 43. It can be found that the distribution of the bending moment along the beams for the Rigid model and Damper model is different from that for the Slit model, in which the points of contraflexure locate in the mid-span of beams for the latter while it occurs out of mid-span of beams for the former. It shows that 17

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Fig. 37. Finite element model (Damper model).

Fig. 38. Input ground motions.

Fig. 39. Maximum drift ratio (Level 1).

18

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Fig. 40. Maximum drift ratio (Level 2).

B IO LS

(a) Slit model

(b) Rigid model

(c) Damper model

Fig. 41. Hinge state of RC frame (El Centro, Level 2).

B

IO

500 400 300 200 100 0

600 Hinges in beams

LS

Hinge number

Hinge number

600 Hinges in beams

Hinges in columns Slit model

(a) Level 1

LS

400 300 200

0

Damper model

IO

500

100 Rigid model

B

Hinges in columns Slit model

Rigid model

Damper model

(b) Level 2

Fig. 42. Hinge number in beams and columns (El Centro).

model and Damper model increased by 63.7% and 39.3% respectively compared with that of the Slit model. It indicates that shear force in beams of the subassembly of frame of the Rigid model and Damper model is increased greatly compared that of the Slit model, which causes potentially shear failure of beams. However, the Damper model could decrease this negative effect with the lower increasing of shear

the lower beam of the Slit model while it is changed to IO status in the Damper model. It highlights that shear force in beams of the Rigid model and Damper model is increased greatly compared with that of the Slit model. For instance, shear force in the left ends of the upper beams of the Slit model, Rigid model, and Damper model is 582 kN, 953 kN, and 811 kN, respectively. Shear force in beams of the Rigid 19

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M=2363 Q=680 M=1825

Q=582 M=1833

Q=953 M=1842

Q=682 M=1851

Q=591 M=1853 IO

Q=1112 M=1858

Q=993 M=1858

LS

IO

(a) Slit model

M=897

Q=1068 M=1840

LS

Q=704 M=1828

Q=811 M=1820

Q=787 M=1834

Q=760 M=1845 IO

(b) Rigid model

LS

(c) Damper model

Fig. 43. Bending moment diagrams at the eighth story (El Centro, Level 2, units in kN for Q (shear force) and kN·m for M (bending moment)).

Energy (kN• m)

4000

ground motion intensity, indicating that dampers dissipate part of input energy. As demonstrated in Fig. 44, the added dampers dissipate about 11% of input energy. The equivalent viscous damping ratio β0 from the equipped dampers can be calculated with Eqs. (21) and (22) according to [36]. Results of the equivalent viscous damping ratio in Level 1 are given in Table 7. It shows that the dampers provide an equivalent viscous damping ratio of 1.05% for the whole structure. It should be noted that the arrangement of RC wall piers and the parameters of dampers consistent with the conducted experiments were designed for computational convenience, in which the new RC wall pier systems were used in only two spans of two frames of the structure. If more new RC wall pier system and optimum parameters of dampers are adopted, the dampers could provide a higher equivalent viscous damping ratio for the whole structure.

Input energy Damper energy

3500 3000 2500 2000 1500 1000 500 0

0

5

10

15

20

25 30

35

40

45

Time (s) Fig. 44. Time history of energy dissipation (Imperial, Level 1).

Shear force (kN)

15

β0 =

10 Wk =

5 0

4πWk

(21)

1 2

(22)

∑ Fi δi i

where Wj is the energy dissipated by damper j, namely the area enclosed by a single hysteresis loop; j is the number of dampers; Wk is the maximum strain energy in the frame; Fi is the inertia force at floor level i; δi is the corresponding floor displacement; i is the number of floors.

-5 -10 -15 -6

∑j Wj

-4

-2

0

2

4

6. Conclusions

6

Displacement (mm)

Numerical analyses were performed in this research, based on the existing experiment. Mechanical behaviors of RC frame with RC wall piers and effects on the performances of RC frame with different kinds of RC wall piers were investigated. The following conclusions can be drawn from the analyses.

Fig. 45. Shear force versus displacement relationship of a damper. Table 7 Equivalent viscous damping ratio (Level 1). Ground motion

∑j Wj (kN·m)

Wk (kN·m)

β0 (%)

El Centro Hachinohe Imperial Kobe Northridge Artificial 1 Artificial 2 Average

554.216 537.483 501.508 422.480 481.847 549.495 564.454 –

4874.217 4236.469 3587.333 2978.218 3755.406 4031.171 4114.165 –

0.90 1.01 1.11 1.13 1.02 1.08 1.09 1.05

(1) RC frame with RC wall piers equipped with dampers possesses a stable lateral stiffness subjected to lateral load, and the designed dampers provide more than 20% of the lateral force to the frame. It is advised that the axial compression ratio of frame column shall be within 0.1 because the broadest plastic regions of dampers and stable lateral force capacity without obvious fluctuations of RC frame can be ensured. (2) Solid element model needs huge calculating memory and is timeconsuming, while the macro element model can capture the general lateral force and strain of rebar versus drift ratio responses of the new RC wall pier system with less computational costs, which is practical for engineering applications. (3) Compared with the Slit model, the maximum drift ratio of the Rigid and Damper models decreased due to the lateral stiffness and energy dissipation provided by the RC wall piers and dampers. Shear force in beams of the subassembly of frame of the latter is significantly higher than the former, which causes potentially shear failure of beams. The Damper model is suggested to be the best by comprehensively considering the structural drift ratios, energy dissipation and the shear force in beams of the subassembly of

force. Although shear force in beams of the Rigid model and Damper model is increased compared to the Slit model, the Damper model is suggested to be the best by a comprehensive review on the drift ratio and shear force in beams. 5.3.4. Energy dissipation of the damper model Time history of energy dissipation under the ground motion Imperial (Level 1) is shown in Fig. 44. Shear force versus displacement relationship of a damper in the fifth story is shown in Fig. 45. It shows that the damper yield and have a stable hysteretic curve at Level 1 20

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frame with RC wall pier. (4) Based on the analyses results, the new RC wall pier system could be an alternative for the conventional RC wall pier systems in the engineering practice with its effectiveness and practicability.

[14]

[15]

Acknowledgments [16]

This work was jointly supported by a special fund to study at Sakata Laboratory of Tokyo Institute of Technology, National Natural Science Foundation of China (Grant No. 51778162, 51878191) and Graduate Student Fundamental Innovation Program of Guangzhou University (Grant No. 2017GDJC-D12). The financial supports are greatly appreciated.

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