Parameter study on composite frames consisting of steel beams and reinforced concrete columns

Journal of Constructional Steel Research 77 (2012) 145–162

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Journal of Constructional Steel Research

Parameter study on composite frames consisting of steel beams and reinforced concrete columns Wei Li a,⁎, Qing-ning Li b, Wei-shan Jiang b a b

College of Architecture and Civil Engineering, Wenzhou University, Wenzhou 325035, PR China School of Civil Engineering, Xi'an University of Architecture and Technology, Xi'an 710055, PR China

a r t i c l e

i n f o

Article history: Received 20 August 2011 Accepted 23 April 2012 Available online 18 June 2012 Keywords: Composite frame Hysteretic model ABAQUS Finite element analysis Confined concrete High-strength ties

a b s t r a c t A host of tests and numerical analyses for frames consisting of reinforced concrete columns and steel beams (RCS) have been conducted in the US and Japan over the past decades. Most results have revealed the superior performance of these structures relative to that of traditional concrete and steel frames. However, few studies could be found about composite frame structures consisting of high-strength concrete columns confined by continuous compound spiral ties and steel beams (CCSTRCS), which requires the development of an accurate finite element model of composite CCSTRCS frames. The validity of the proposed model is examined by comparing with the test data presented in reference studies. With the proposed model, an extensive parametric study is carried out to investigate the behavior of composite CCSTRCS frames. Simplified hysteretic lateral load versus lateral displacement models are proposed for such composite frames. Crown Copyright © 2012 Published by Elsevier Ltd. All rights reserved.

1. Introduction It has been widely recognized that composite moment frames consisting of RC columns and steel (S) beams, or the so-called RCS system, can provide cost-effective alternative to traditional steel or RC construction in seismic regions. As opposed to conventional steel or RC moment frames, the problems associated with connections are greatly reduced, and the RCS frames are generally more economical than the purely steel or RC moment frames. The research program included extensive testing and finite element analyses of RCS beamto-column connections and subassemblies, testing of reduced-scale and full-scale RCS moment frames and finite element analyses, seismic design studies and analyses of RCS moment frames and development of guidelines and recommendations for detailed design work [1]. Despite its potential benefits in construction speed and structural excellent ductility due to the use of CCSTRC column, research was rarely conducted on composite moment frames consisting of continuous compound spiral tie reinforced concrete (CCSTRC) columns and composite steel beams. The experimental research on the CCSTRC column and CCSTRC-steel (S) composite connection has been conducted and the CCSTRC and steel (CCSTRCS) composite frames have great advantage due to the use of high‐strength concrete column confined with high‐ strength continuous compound spiral ties, which improves the strength

⁎ Corresponding author at: College of Architecture and Civil Engineering, Wenzhou University, Chashan University Town, Wenzhou City, Zhejiang Province, 325035, PR China. Tel.: + 86 57786689609. E-mail address: [email protected] (Q. Li).

and ductility of the column and reduces the section size of column, thereby increasing effective building space. As an “undefined structural system”, the composite CCSTRCS system cannot be easily adopted in design and construction practice. However, they have become recognized by more and more researchers and practicing professionals in recent years that though structural systems do not fully satisfy the prescriptive requirements of current building codes, they can provide satisfactory seismic performance. The desirable seismic characteristics should be validated by analyses and laboratory tests. However, since it is difficult to conduct a lot of experiments from an economical viewpoint, and due to unique features of the tested specimens and material heterogeneity, it is also difficult to understand the complex seismic behavior of beam–column connections and framed structures. Furthermore, the effect of several influencing parameters such as plate thickness, axial load and the effect of confining cannot be varied in a limited number of tests. In order to quantify and make clear the influence of critical design parameters, it is necessary to develop a robust numerical model. Following this understanding, a series of finite element analysis for composite structures was conducted by many researchers. Liu and Foster [2] developed a finite element model to investigate the response of concentrically loaded columns with concrete strength up to 100 Mpa. Yu et al. 2010 [3,4] presented a modified Drucker-Prager (D-P)-type model and a Plastic-damage model and then implemented it in ABAQUS. Hajjar et al. [5] proposed a 3D modeling of interior beam-to-column composite connections with angles by means of the ABAQUS code [6]. Salvatore et al. [7] studied seismic performance of exterior and interior partial-strength composite beam-tocolumn joints by using ABAQUS software. Hu et al. [8] proposed proper material constitutive models for concrete-filled tube (CFT) columns

0143-974X/$ – see front matter. Crown Copyright © 2012 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2012.04.007

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and they were verified by the nonlinear finite element program ABAQUS against experimental data. Zhao and Li [9] studied the nonlinear mechanical behavior and failure process of a bonded steel–concrete composite beam by using finite element program ABAQUS, The eight-node brick elements (C3D8) were employed to model the concrete slab and steel beam. An adhesive layer was modeled by the eight-node threedimensional cohesive elements (COH3D8). Bursi et al. [10] studied the seismic performance of moment-resisting frames consisting of steel– concrete composite beams with full and partial shear connection by using the ABAQUS program. Han et al. [11] presented a finite element modeling of composite frame with concrete-filled square hollow section (SHS) columns to steel beam, the finite element program ABAQUS was adopted. Wu et al. [12] studied the effect of wing plates numerically by simulating H-beams in bolted beam–column connections as cantilever beams using ABAQUS. Set against this background, Li et al. [13] applied the finite element program ABAQUS to simulate the behavior of composite CCSTRCS frames. The results show that continuous compound spiral highstrength ties can effectively improve the lateral deformation capacity of concrete, with a good constraint to the concrete in the core area, which increases the ultimate lateral bearing and deformation capacity of the composite CCSHRCS frame. However, the detailed influence factors for composite CCSTRCS frames are not clear. Therefore, this paper focuses on conducting an extensive parametric study on the behaviors of composite CCSTRCS frames. Through parametric analysis, the simplified hysteretic lateral load versus lateral displacement models for such composite frames is proposed.

2.2. Material modeling of concrete In conventional concrete models, the behavior under compressive stresses is usually represented by the plasticity model, while the behavior under tensile stresses is expressed by the smeared cracking model. The smeared cracking model, however, always encounters numerical difficulties on analysis under cyclic load. To circumvent this situation, the concrete damaged plasticity model (Lee and Fenves 1998) implemented in ABAQUS 2006 [6] is used herein. By experimental observations on most of quasi-brittle materials, including concrete, when the load changes from the tension to compression, compression stiffness recovers with the closure of crack. In addition, when the load changes from compression to tension, once the crushed micro-cracks occur, and the stiffness in tension will not be restored. This performance corresponds to the default value wt = 0 and wc = 1 in ABAQUS. Fig. 1 describes the default properties under uniaxial cyclic loading. 2.3. Material modeling of reinforcement and structural steel In this paper, in order to simplify the problem in the analysis of finite element method, assuming that the ties and longitudinal reinforcement in concrete columns are ideal elasto-plastic materials, regardless of the reinforcement service stage and Bauschinger effect in their stress–strain relations. The stress–strain curve is sloped before the steel yields, and it should be simplified to horizontal line after that, as shown in Fig. 2. The Von Mises yield criterion with isotropic hardening model is adopted for structural steel.

2. Finite element model 2.1. General descriptions

2.4. Interactive modeling between concrete and reinforcement, concrete and structural steel

In order to accurately simulate the actual behavior of RCS frame specimen, the main six components of the frames need to be modeled. They are the confined concrete columns, the interface and contact between the concrete in joint regions and the structural steel (e.g. face bearing plates, cover plates), the interface and contact between the shear connections of steel beam and concrete slab, the interaction of reinforcement and concrete, the connection details between RC columns and steel beam, and the steel beam. In addition to these parameters, the choice of the element types, mesh sizes, boundary conditions and load applications that provide accurate and reasonable results are also important in simulating the behavior of structural frames.

Since joints of steel beam and concrete column connect together by welding face bearing plate at the steel beam flange in the frame structure, beam–column and face bearing plate have a good confined with joint regions to make joint regions little slip. The details of the connection of steel beam and reinforced concrete column are shown in Fig. 3a. It is shown that concrete and steel in the joint regions can still work together until the destruction of the beam–column joint. Salvatore et al. [7] studied the seismic performance of exterior and interior partial-strength composite beam-to-column joints by using ABAQUS software, a hardening elasto-plastic material is modeled using discrete two-nodded beam elements, dimensionless bond-link

Fig. 1. Uniaxial load cycle (tension–compression–tension) assuming default values for the stiffness recovery factors: wt = 0and wc = 1.

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Steel beam and concrete, face bearing plate and concrete in joint regions are directly constrained by the module “Tie” command in the “Interaction” to make binding constraint, so there is no relative slip between them. 2.5. Bounding conditions and loading

Fig. 2. Stress–strain relationship of reinforcement.

elements are adopted to connect concrete and steel, friction between the steel bars and the concrete slab is not modeled because it has little influence on substructure responses. On the other hand, the joint adding bonding element, the analysis would become too complicated, so the bond-slip of the joint regions is not considered. In this paper, taking “Interaction” module “Constraint” command in ABAQUS as Embedded region/embedded in the reinforced concrete columns. In this condition, there is no relative slip between the steel and concrete.

(a) The connection details

FBP

(b) Load modeling of a composite RCS frame

The boundary conditions and loading manners of RCS frame structures are specific in this paper: concrete column foot is fixed constraint, axial load is imposed by the loading plate on top of the column, and lateral load is imposed on the beam. In the ABAQUS software, the boundary conditions are set as follows: three concrete columns with fixed boundary constraints, the steel beam and face bearing plates used the “Merge” command of the “Assembly” module to merge. In this case, steel beams and face bearing plate can be regarded as fixed constraints. The loading plate and the interface of column cap are constrained by the “Interaction” module “Tie” command. As shown in Fig. 3b. The loading of the frame divides into two categories: the axial load at the top of the framed column and the lateral load at the end of framed beam. The two load steps are required in the ABAQUS code. The specific methods are as follows: at the top of the three framed columns respectively applied to the axial load. Firstly applied to interior column, and then to two other exterior columns, set it as a load step. When axial loading is completed, the lateral load should be loaded at both ends of the framed beam, and displacement loading adopted in order to obtain the load–displacement curves of frame, that is, displacement applied at the end of the beam (applied displacement boundary conditions that known). In order to avoid stress concentration, the “Load” module “Pressure” of ABAQUS is adopted for the axial load and the analysis will not stop until the selected displacement is reached. Fig. 3b is the loading model diagram of composite RCS frame. 2.6. The selection of element type and meshing In order to simulate the detailed characteristics of steel beamsconcrete columns joint, steel beams and concrete adopted a threedimensional solid element with reduced integration eight-node formulation (C3D8R). Compared with the high-order isoparametric element, although the accuracy of this element is slightly lower, it can reduce to a lot of freedom degree, which can greatly reduce the computational cost. In order to understand the force characteristics of reinforcement, the ties and longitudinal reinforcement in concrete columns used a two‐node linear three-dimensional truss element (T3D2). Fig. 4 shows the cross-section mesh diagram of the finite element model for the concrete columns, steel, steel beams—the whole face bearing plate and beam–column joint region in this paper. Due to the complexity of the beam-column joint regions, these regions are subdivided to ensure the accuracy of the computational results. 3. Validity of FEM modeling

Fig. 3. The connection details and load modeling of a composite RCS frame.

In order to validate the finite element model that developed in this paper, the numerical results were compared with the test results in the literature [14]. The detail sizing and reinforcement are shown in Fig. 5. Details of the specimens are seen in Table 1, the corresponding material properties are shown in Table 2. In accordance with the specific parameters of specimens mentioned, the finite element model of the specimen is developed by ABAQUS, as shown in Fig. 6. The frame specimen model consists of 18,511 elements, 17,566 C3D8R solid elements and 945 T3D2 truss element. Since the beam–column joint region is under a larger force, in order to avoid distortions in the joint area, firstly, complex geometric models of joint regions are cut into simple geometric models, further element joints are subdivided by geographical mesh.

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(a) Concrete column

(b) Longitudinal reinforcementand tie

(c) Steelbeam and FBP

(d) beam-column joint

Fig. 4. Meshing sketch of section.

3.1. The load–displacement relation for composite RCS frame The relationship of the load–displacement for the RCS framed specimen is obtained from calculation, and it is shown in Fig. 7, at the initial of load for frame, when lateral displacement is within 20 mm, the structure has not yet reached the ultimate bearing capacity, load–displacement curve has a nearly linear relationship. When the lateral displacement is more than 20 mm, with the increase of the displacement for the load–displacement curve, the load slowly increases until yielding to the destruction of the structure. In this condition, the horizontal ultimate load and displacement are respectively 797 kN and 77.8 mm. In addition, when adjusted to the test procedures, the ANSYS software is used for analysis on load–displacement relationship for framed specimens, and the ANSYS model is shown in Fig. 8a; and the comparison with that calculated by ABAQUS is shown in Fig. 8b. The load–displacement curve calculated by ANSYS in Fig. 8b does not descend obviously and the load is more than 700 kN while the displacement is 15 mm. However, the ultimate load and displacement are 1021 kN and 47.6 mm, they are visibly higher than the calculated values by ABAQUS. The load–displacement curve calculated by ANSYS and ABAQUS is compared with the experimental results, and they are shown in Fig. 9. The calculated results by ABAQUS agree well with the experimental value, while the calculated values by ANSYS are obviously higher than the experimental value, because the crushing and the stiffness degradation of concrete under compression of concrete constitutive model in the ANSYS program haven't been considered, resulting in the calculated values being higher than the actual value. Table 3 shows the comparison between the relation of load–displacement and the value calculated by ABAQUS. From that, the calculated values are in good agreement with the experimental values, it is indicated that the ABAQUS software could simulate the load–displacement relations of this composite frame. However, it should note the load–displacement curve in the model that is influenced not only by the constitutive relation

of concrete materials, but also by that of different materials. If the choice of constitutive model of materials is inappropriate, it will result in larger deviation for the calculated results, or it will lead to the serious distortion of the calculated results. Therefore, when using the numerical model, the setting of each parameter should be grasped to make the model reflect the actual situation, it is also necessary to master the computational efficiency of the comprehensive computer. 4. Parametric analysis of composite CCSHRCS frame 4.1. Overview The ABAQUS software has been used in the previous section to analyze the performance of composite RCS frame, and the model is validated by the test results in the literature. A new finite element model is developed using the same method in the previous section, and it is used to analyze the composite CCSTRCS frame structure, in order to understand the influence of various parameters on the performance of composite CCSTRCS frame. The following parameters are analyzed: the ratio of longitudinal reinforcement of continuous compound spiral ties concrete columns (ρs) and the strength of longitudinal reinforcement (fys), the volumetric ratio of ties (ρv) and the strength of tie (fyv), compressive strength of concrete (fcu), the characteristic values of tie (λv), the yield strength of steel (fak), axial-load ratio (n), the linear stiffness ratio of the beam–column (K) and the slenderness of the column (λ). The calculation only changes one parameter while keeping other parameters unchanged. 4.2. Development of model for composite CCSTRCS frame In order to investigate the behavior of composite CCSTRCS frame structure, the various parameters in composite CCSTRCS frame model are changed. To take continuous compound spiral stirrups, a uniform

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Beam

Column

Through beam type (TB) joint Fig. 5. Specimens tested by Iizuka et al. 1997 [14].

spacing with 50 mm was used to simplify the model. The model is shown in Fig. 10.

Lateral stiffness at post-yield stage slightly enhances. None of the curves is descending, and lateral displacement is relatively large, it indicates that the frame has good ductility.

4.3. The ratio of longitudinal reinforcement (ρs) 4.4. The strength of longitudinal reinforcement (fys) The geometrical parameters are the same as that of the composite RCS frame model in the previous section. The axial load for the exterior column is 242.1 kN, and the interior column is 484.2 kN. For the specific parameters, see Table 4. While keeping other parameters unchanged, just the ratio of longitudinal reinforcement is different, Fig. 11 shows the lateral load– lateral displacement of composite CCSTRCS frame with different ratios of longitudinal reinforcement. With the increase of the ratio of longitudinal reinforcement, the lateral stiffness of the frame at the elastic stage slightly increases. It can be seen that the maximum ratio of longitudinal reinforcement increases by 56% to that of the minimum longitudinal reinforcement, but lateral stiffness at the yield stage doesn't improve.

To study the effect of the longitudinal reinforcement strength of the column on the behavior of the frame structures, the following longitudinal reinforcement strengths were used: HRB335, HRB400, and HRB500. The diameter and uniform ratio of longitudinal reinforcement are taken as 12-D18 (3.39%). While keeping other parameters unchanged, just strength of longitudinal reinforcement is different; Fig. 12 shows the relationship of lateral load–lateral displacement of composite CCSHRCS frame at different strength of longitudinal reinforcement. With the increasing of the strength of longitudinal reinforcement, the lateral stiffness of the frame at the elastic stage keeps the same, lateral stiffness at the yield stage is almost the

Table 1 Details and size of specimen. Detailed parameters of the specimen Column

Section Longitudinal reinforcement Tie Axial load

Beam Beam–column joints

Section Tie

b × h = 300 mm × 300 mm 12-D19 (ρt= 1.28%) 4D-10@50 (ρw= 1.9%) (Exterior column) 0.1BDσB= 242100 N (interior column) 0.2BDσB= 484200 N σB= 26.9 N/mm2 BH-200 × 100 × 12 × 16 (mm) 4D-6@50 (ρw=0.85%)

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Table 2 The properties of materials. Specimen

Compression strength (N/mm2)

Tensile strength (N/mm2)

Modulus of elasticity (104 N/mm2)

Column Foundation

26.9 29.4

1.91 2.62

2.21 2.19

Specimen Steel reinforcement

D6 D10 D19 12 mm 16 mm

Steel plate

Yield strength (N/mm2)

Tensile strength (N/mm2)

Modulus of elasticity (105 N/mm2)

386 375.3 385.4 312.4 299.8

543.5 516.3 556 463.9 425.4

1.79 1.69 1.77 1.97 1.95

same, and the yield load remains unchanged. Lateral stiffness at postyield stage increases slightly, it is indicated that ultimate load increases with the increasing of the strength of longitudinal reinforcement. None of the curves is descending, and lateral displacement is able to continue to increase, since the axial compression ratio of the columns is relatively small, and high-strength ties provide good constraints on the core concrete, thus greatly increasing the large deformation of the frame structure.

4.5. The volume ratio of tie (ρv) In order to study the influence of the volume ratio of tie of column on the behaviors of the frame structures, high-strength ties with diameters of 5 mm, 7 mm, 9 mm, respectively, were used. The yield and tensile strength remain the same; longitudinal reinforcement with HRB400 is adopted. While keeping other parameters unchanged, only the volume ratio of the tie is different. Fig. 13 shows the relationship of lateral load–lateral displacement of composite CCSHRCS frame at different volume ratios of tie. With the increase of the volume ratio of tie, the lateral stiffness of the frame at the elastic stage slightly increases, both the yield displacement and the yield load are increased, lateral stiffness at the yield stage increases, and lateral post-yield stiffness increases slightly. This indicates that the ultimate load increases with the increase of the volume ratio of tie. None of the curves is descending, and lateral displacement is able to continue to increase. Since the axial compression ratio of the columns is relatively small, and high-strength ties provide good constraints on the core concrete, the large deformation of the frame structure thus greatly increases.

T3D2

4.6. The strength of tie (fyv) In order to study the influence of the volume ratio of tie of column on the strength of tie of column on the behaviors of the frame structures, tie as HRB335, HRB400, HRB500 respectively and prestressed concrete steel barΦ PC are used [16]. The diameter and uniform ratio of the longitudinal reinforcement are taken as 12-D18 (3.39%). While keeping other parameters unchanged, only the strength of the tie is different. Fig. 14 shows the relationship of lateral load–lateral displacement of composite CCSTRCS frame at different strengths of tie. With the increase of the strength of tie, the lateral stiffness of the frame at the elastic stage remains the same, the lateral stiffness at the yield stage remains the same, and the lateral post-yield stiffness also almost remains the same. It indicates that ultimate load increases with the increase of the strength of tie. None of the curves is descending, and lateral displacement is able to continue to increase. Since the axial compression ratio of the columns is relatively small and the space of tie is small, the result in the strength of tie has not been fully applied. In addition, high-strength ties provide good constraints on the core concrete, thus greatly increasing large deformation of the frame structure. 4.7. The compressive strength of concrete (fcu) In order to study the influence of the compressive strength of concrete of column on the behaviors of the frame structures, concrete, C40, C60, C80 and C100 are used. The diameter and uniform ratio of longitudinal reinforcement are taken as 12-D18 (3.39%), and prestressed concrete steel bar Φ PC [15] is adopted. While keeping other parameters unchanged, just not the same compressive strength of concrete, Fig. 15 shows the relationship of the lateral load–lateral displacement of composite CCSTRCS frame at different compressive strengths of concrete. With the increase of the compressive strength of concrete, the lateral stiffness of the frame at the elastic, yield, and post-yield stages increases slightly, it is indicated that yield and ultimate load increase with the increase of the compressive strength of concrete. In a concrete strength from C40 to C100, the elastic modulus also increases, but only little, so it only has little effect on the lateral stiffness. None of the curves is descending, and lateral displacement is able to continue to increase, it shows that a further increase of concrete strength has little effect on the bearing capacity of the frame structure, in the case that highstrength ties provide good constraints on the core concrete. 4.8. The characteristic values of tie (λv)

C3D8R

Fig. 6. FEM model of framed specimen.

Reinforced concrete columns are the compression and bending component, the characteristic values of tie are vital to improve the deformation properties of concrete. Previous sections have analyzed behaviors of the composite CCSHRCS frame structures to be influenced on the volume ratio of tie, the strength of tie and the strength of concrete, the results show that the volume ratio of tie has an important

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151

800

600

Load/kN

Displacement 400

Load

200

0

0

20

40

60

80

Displacement/mm Fig. 7. Relationship of load–displacement of framed specimen based on FEM.

influence on lateral stiffness at yield and strengthening stages for the composite CCSTRCS frame structures, while the strength of tie and the strength of concrete strength are less affected. To understand the performance of these three combined effects of the frame, different characteristic values of tie have been used to reflect the influence on behaviors of composite CCSTRCS frame. While keeping other parameters unchanged, Fig. 16 shows the relationship of lateral load–lateral displacement of composite CCSTRCS frame with different characteristic values of tie. With the increasing of the characteristic values of tie, the lateral stiffness of the frame at the elastic stage slightly increases. This is mainly because all the concrete used C60 in the model; ties have few effects on the lateral stiffness at the elastic stage for composite CCSHRCS frame. The characteristic values of tie λv, which can be seen increasing from 0.308447 to 0.642598, the relationship of lateral load–lateral displacement of composite CCSTRCS frame descending from changing in circumstances, there is no dropoff situation, increasing the yield load, ultimate load has a larger increase. Because when the characteristic values of tie λv is 0.308447, the column crushing due to no enough confined to the core concrete. In addition to characteristic values of tie λv 0.308447 under the descending curve, none of the other curves descends, and the lateral displacement is relatively large. For rare earthquakes in more than elastic–plastic story-drift ratio, the frame is also able to continue to bear the load and deformation. It shows that the frame has a good ductility. With the increasing of characteristic values of tie, lateral stiffness at the yield stage increases slightly and lateral post-yield stiffness increases significantly, but when the characteristic values of the tie reach a certain level, there is no significant effect on the curve. And λv 0.308447 compared to 0.642598, 1.645052 and 2.082019, respectively, under the ultimate load increases by 3.65% and 21.69%, and 24.52%. It is indicated that the characteristic values of tie from small to large, the shape of the curve and the trends of lateral load–lateral displacement of frame are greatly influenced. When rising to a certain level, the effect on the curve decreases; it doesn't stabilize until the characteristic values of tie increase to a value. 4.9. The strength of steel beam (fak) In order to study the influence of the strength of steel beam on the behaviors of the frame structures, steels Q235, Q345, Q390, and Q420 were used. The diameter and uniform ratio of longitudinal reinforcement

are taken as 12-D18 (3.39%), and prestressed concrete steel bar ΦPC [15] is adopted. While keeping other parameters unchanged, just not the same strength of steel beam, Fig. 17 shows the relationship of lateral load–lateral displacement of composite CCSTRCS frame at different strengths of steel. With the increase of the strength of steel, the lateral stiffness of the frame at the elastic stage remains the same. Since the elastic modulus of the steel beam and its strength has little to do, it is indicated that the maximum yield strength of the steel beam than minimum yield strength 78.72% increases, but the lateral stiffness of the frame at the yield stage almost remains the same. Because the horizontal load of the yield load at the previous stage for RC column is play early to the steel beam. The lateral post-yield stiffness increases significantly from Q235 to Q345; it is after the yielding of the beams that the horizontal load is greatly affected. None of the curves is descending, and lateral displacement is relatively large, at the elastic–plastic story-drift ratio for the rare earthquake, the frame also can continue to carry load and deformation. It is shown that the frame has a good ductility. Compared with the Q235, Q345, Q390 and Q420, respectively, the ultimate load increased by 12.42%, 13.80%, and 15.91%. However, the shape of the curve and trends of lateral load–lateral displacement of frame almost remain the same. 4.10. The axial-load ratio (n) In order to study the influence of the axial-load ratio on the behaviors of the frame structures, axial-load ratio with 0, 0.1, 0.3, 0.5, 0.7, 0.9 and 1.05 was changed. The diameter and uniform ratio of longitudinal reinforcement are taken as 12-D18 (3.39%), and prestressed concrete steel bar Φ PC [15] is adopted. HRB400 is used in the longitudinal reinforcement. While keeping other parameters unchanged, just not the same axial-load ratio, Fig. 18 shows the relationship of lateral load–lateral displacement of composite CCSTRCS frame at different axial-load ratios. With the increase of the axial-load ratio, the lateral stiffness of the frame at the elastic, yield and strengthening stages improves. None of the curves is descending, and lateral displacement is relatively large, mainly because the model calculation of slenderness of column is relatively small, that at this time of the displacement ductility factor of is infinite. In the axial-load ratio increases from 0 to 0.5, the lateral stiffness of the frame at the elastic and yield stages increases obviously.

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(a)

(b) 1100 1000 900 800

Load/kN

700 600 500

ANSYS

400

ABAQUS

300 200 100 0 0

10

20

30

40

50

60

70

80

90

Displacement/mm Fig. 8. Relationship of load–displacement of framed specimen based on ANSYS and ABAQUS.

From 0.7 to 1.05, the lateral stiffness of the frame at the elastic and yield stage remains almost the same. The lateral post-yield stiffness of the frame increases slightly. The yield load in axial-load ratio increases from 0 to 0.5 than increases from 0.5 to 1.05 is more significant. Ultimate load also increases, but the ductility decreases.

ANSYS ABAQUS Expeirment

1200 1000

P (kN)

800 600

Table 3 Load-drift based on calculated and experiment.

400

Experimental results

Calculated results

Interior column

Interior column

639 1/113 787 1/25 885 1/33

658.015 1/114.17 776.45 1/28.50 797 1/30.64

200

Load(kN)

0 0

20

40

60

80

displacement (mm) Fig. 9. ABAQUS and ANSYS versus experimental relationship of load–displacement of framed specimen.

Location of yield

Column foot at the first story Column cap the second story Steel beam at the second story

Calculated/ experimental

1.030 0.990 0.986 0.877 0.901 1.078

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Fig. 10. The FEM model of CCSH.

4.11. The linear stiffness ratio of beam–column (K) In order to study the influence of linear stiffness ratio of beam– column on the behaviors of the frame structures, taking into account the convenience for model, the calculation input parameters remain the same, just changing the model in the modulus of elasticity of the steel beam to achieve the purposes of changing the linear stiffness ratio of the beam–column. The modulus of elasticity of steel beams is taken, 2.06 × 10 5Mpa, 4.12 × 10 5Mpa, 8.25 × 10 5Mpa, and 1.37 × 10 6Mpa, respectively. While keeping other parameters unchanged, just not the same linear stiffness ratio of beam–column, Fig. 19 shows the relationship of the lateral load–lateral displacement of composite CCSTRCS frame at different linear stiffness ratios of beam–column. With the increasing of the linear stiffness ratio of beam–column, the lateral stiffness of the frame at the elastic stage increases. With the increase of the linear stiffness ratio of beam–column, beam‐to‐column constraints are also increasing. However, increasing the ratio of its lateral stiffness increases not only blindly, but also by a significant difference. The degree of increasing for lateral stiffness at the elastic stage is a decreasing trend with the increase of linear stiffness ratio of beam–column. Especially when linear stiffness ratio of beam–column from K = 1.49 to 2.64, the lateral stiffness increases to a lower extent. It is observed that none of curves is descending, and lateral displacement is relatively large. At the elastic–plastic story-drift ratio for the rare earthquake, the frame also can continue to carry load and deformation. It is shown that the frame has a good ductility. With the increase of the linear stiffness ratio of beam–column, the yield load and ultimate load also increase. Compared with K = 0.396, K = 0.793, K = 1.59 and K = 2.64, respectively, the ultimate load increases by 8.42%, 13.52%, and 16.27%. The reason for the linear stiffness ratios of beam–column has an influence on the bearing capacity of the frame is that the effective length of framed column changes. The linear stiffness ratio of the beam–column reflects the level of constraint that beam to column, and the linear beam-column stiffness ratios at the end of a beam-column affect the capacity for lateral deformation of a

rotational framed column. Thus, the effective length of the framed columns is directly affected, leading to changes in the bearing capacity of a given frame. 4.12. The slenderness of column (λ) In the member of flexure and compression, in general according to the different slenderness of column, are divided into short columns and long columns. Columns in the loading process, its second-order effects can be ignored; long columns in the course of its secondorder effect cannot be ignored. Whether the lateral deformation at the end of columns, the second-order effects are very different. The ACI 318R-08 code [16] divides the column into without lateral sway and with lateral sway. For non-sway frames, because there is no lateral sway at both ends of the columns, the second-order effects only contain P–δ effect, the column buckles on its own, when both ends of the additional moment is zero. For sway frames, which include the P–δ effect and the P–Δ effect, the P–δ effect is the relation of the load and bending deformation, and the P–Δ effect is load, bending and lateral deformation, not only the additional bending moment at both ends of column is not zero, but also relatively large. This model can consider the constraint at both ends of the sway frames. Therefore, in order to more accurately study the influence of the slenderness of column on the behaviors of the frame structures, the American Concrete Specification ACI 318R-08 [16] equations was used to determine the effective length of framed columns, thus determining the slenderness of the column. To facilitate the modeling calculations, other parameters in the model are the same, only the length of the framed columns is changed to achieve the purpose of changing the slenderness of the column. To this end, the clear height of the column, respectively, is designed for 900 mm, 1800 mm, 3600 mm and 4500 mm. While keeping other parameters unchanged, just not the same slenderness of column, Fig. 20 shows the relationship of lateral load–lateral displacement of composite CCSHRCS frame at different slenderness of column. With the increase of the slenderness of column, the lateral stiffness of the frame at the elastic stage reduces.

Table 4 The properties of materials. Specimen

Compression strength (N/mm2)

Tensile strength (N/mm2)

Modulus of elasticity (104 N/mm2)

Column

26.9

1.91

2.21

Steel Reinforcement Steel plate

Tie ΦPC Longitudinal reinforcement HRB400 12 mm 16 mm

Yield strength (N/mm2)

Tensile strength (N/mm2)

Modulus of elasticity (105 N/mm2)

1000 400 312.4 299.8

1200 540 463.9 425.4

2.0 2.0 1.97 1.95

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W. Li et al. / Journal of Constructional Steel Research 77 (2012) 145–162

1200

1000

Load/kN

800

600

400

200

0 0

20

40

60

80

100

120

140

160

180

Displacement/mm Fig. 11. The relationship of load–displacement of composite CCSHRCS frame during the different ratios of longitudinal reinforcement.

overall frame instability due to the formation of institutions. Therefore, it is a good engineering design to control the slenderness of column.

The larger slenderness of column is more prone to tensile failure, leading to the premature yield of longitudinal reinforcement and concrete crushing. The lateral stiffness and bearing capacity at the yield stage significantly reduce, and that at post-yield stage also significantly reduces; this is due to the local instability of concrete columns. Although none of the curves is descending, and lateral displacement is relatively large, at this time the yield and ultimate load remain almost unchanged, it is indicated that the overall structure imminently fails. With the increase of the slenderness of column, the yield load and ultimate load decrease significantly. And λ = 23.0 compared to λ = 35.1, λ = 57.3 and λ = 67.7, respectively, under the ultimate load decreases 1.38, 2.15, and 2.82 times. It is shown that the slenderness of the column is more than 57.3, the

5. The load–displacement backbone curve of composite CCSTRCS frame structure 5.1. The development for load–displacement backbone curve of composite CCSTRCS frame structure The relationship of the lateral load–lateral displacement of composite CCSTRCS frame is based on the above parameters. The bilinear model and trilinear model are adopted to simulate non-descending

1200

HRB500 HRB400

1000

HRB335

Load/kN

800

600

400

200

0

0

20

40

60

80

100

120

140

160

180

Displacement/mm Fig. 12. The relationship of load–displacement of composite CCSHRCS frame during the different strengths of longitudinal reinforcement.

W. Li et al. / Journal of Constructional Steel Research 77 (2012) 145–162

v =6.36% v =3.85% v =1.96%

1000

800

Load/kN

155

600

400

200

0

0

20

40

60

80

100

120

140

160

180

Displacement/mm Fig. 13. The relationship of load–displacement of composite CCSHRCS frame during the different volume ratios of ties.

an important influence on the elastic stiffness of composite CCSTRCS frame, and it can be obtained by regression analysis, the empirical formula of the elastic stiffness Ke is,

and descending curves of the lateral load–lateral displacement of the composite CCSTRCS frame. The bilinear model represents skeleton curves of lateral load– lateral displacement with non-descending, as shown in Fig. 21. Where, Ke, Ks, represent the elastic stiffness, stiffness at the postyield stage of frame structures, respectively; Py is the yield load, Δy is the displacement corresponding to Py, that is the yield displacement. It can be seen that once the elastic stiffness Ke, the yield load Py and stiffness at the strengthening stage Ks are determined, skeleton curves of lateral load–lateral displacement with non-descending for composite CCSTRCS frame are developed. Finite element analysis results show that the axial-load ratio n, linear stiffness ratio of beam–column K and the slenderness of column λ have


 1 3E I 2 c 0 K e ¼ 6:93 þ 39:66n−40:35n β 3 l3

Where, n is the design axial-load ratio; β coefficient of effective length of column, their calculation formula see ACI 318R-08 [16]; l is the length of the framed column. The lateral stiffness at the post-yield stage of composite CCSTRCS frame is influenced by the following key factors: the characteristic values of tie (λv), yield strength of steel (fak), axial-load ratio (n),

1000

pc

800

HRB500 Load/kN

HRB400 600

HRB335

400

200

0

0

20

40

60

ð1Þ

80

100

120

140

160

180

Displacement/mm Fig. 14. The relationship of load–displacement of composite CCSHRCS frame during the different strengths of ties.

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W. Li et al. / Journal of Constructional Steel Research 77 (2012) 145–162

1000

800

Load/kN

C100 600

C80 C60 C40

400

200

0

0

20

40

60

80

100

120

140

160

180

Displacement/mm Fig. 15. The relationship of load–displacement of composite CCSHRCS frame during the different concrete strengths.

and the slenderness of column (λ). The empirical formula for stiffness at the post-yield stage Ks can be obtained by regression analysis, 1 −0:05407 þ 0:003829λv þ 0:001255λv 2 .
.  B 2 C 3Ec I 0 f ak f C Ks ¼ B −0:03136 ak @ þ0:098253 A l3 235 235 2 −0:05936n þ 0:098943n

hinge distribution is shown in Fig. 22. Therefore, the lateral ultimate load is obtained,

0

ð2Þ

Where, l is the length of framed column; λv is the characteristic values of tie; fak is standard value for the strength of steel; λ is the slenderness of the column; n is the axial-load ratio; Ec is the modulus of elasticity of concrete; and I0 is the moment of inertia of gross concrete column section about centroidal axis. Though analysis on the frame of this model shows 12 indeterminate numbers, under lateral load, there are 12 plastic hinges; the plastic

Pu ¼

12M s l

ð3Þ

Where, Ms is the plastic limit moment of the beam section; l is the length of framed column. According to the method in literature [17], if λ > 35 or n > np, then,  P y ¼ P u 1−

2N Kel þ N

 ð4Þ

1200

=2.082019 =1.645025

1100 1000

=0.642598

900

Load/kN

800

=0.308447

700 600 500 400 300 200 100 0

0

20

40

60

80

100

120

140

160

180

Displacement/mm Fig. 16. The relationship of load–displacement of composite CCSHRCS frame during the different characteristic values of tie.

W. Li et al. / Journal of Constructional Steel Research 77 (2012) 145–162

157

1000

Load/kN

800

600

Q420 Q390 Q345 Q235

400

200

0

0

20

40

60

80

100

120

140

160

Displacement/mm Fig. 17. The relationship of load–displacement of composite CCSHRCS frame during the different strengths of steel beam.

If λ ≤ 35 or n ≤ np, then, Py ¼ Pu

It can be seen that once the elastic stiffness Ke, the yield load Py, the ultimate load Pu, post-yield stiffness Ks, degradation stiffness Kc and the peak displacement ΔB are determined, the skeleton curves of lateral load–lateral displacement with descending for composite CCSTRCS frame will be developed. The empirical formula of the elastic stiffness can be obtained by regression analysis,

ð5Þ

Where, N is the axial compressive load; l is the length of framed column; Ke is the elastic stiffness; np = 1.05 − 0.024279λ. The trilinear model represents the skeleton curves of the lateral load–lateral displacement with descending, as shown in Fig. 23. Where, OA is at the elastic stage, AB is at the post-yield stage, BC is at the descending stage, Ke, Ks, and Kc, are elastic stiffness, post-yield stiffness and degradation stiffness, respectively. Py is the yield load, Δy is the displacement corresponding to Py, that is the yield displacement, ΔB is the peak displacement corresponding to the ultimate load Pu, the lateral load of A is based on the method in the literature [18], that is Py = 0.6 Pu, Pu obtained by Eq. (3).


 2 3E c I 0 1 K e ¼ −5:75107−1:58967λ−0:06849λ l3 β 3

Where, l is the length of the framed column; λ is the slenderness of column; Ec is the modulus of elasticity of concrete; I0 is the moment of inertia of gross concrete column section about centroidal axis, β is the coefficient of effective length of column, and its equation is seen in ACI 318R-08 [16].

1400

n=1.05 n=0.9 n=0.7 n=0.5 n=0.3

1200

Load/kN

1000

n=0.1 n=0

800

600

400

200

0

0

20

40

60

ð6Þ

80

100

120

140

160

180

Displacement/mm Fig. 18. The relationship of load–displacement of composite CCSHRCS frame during the different axial-load ratios.

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700

K=2.64 K=1.59 K=0.793 K=0.396

600

Load/kN

500

400

300

200

100

0

0

20

40

60

80

100

120

140

160

180

Displacement/mm Fig. 19. The relationship of load–displacement of composite CCSHRCS frame during the different linear stiffness ratios of beam–column.

1200

=23.0 1000

Load/kN

800

600

=35.1 =57.3

400

=67.7 200

0

0

20

40

60

80

100

120

140

160

180

Displacement/mm Fig. 20. The relationship of load–displacement of composite CCSHRCS frame during the different slenderness of column.

The post-yield stiffness:
 2 2 3Ec I 0 K s ¼ −1:2923 þ 0:099319λ−0:001619λ þ 0:8313n−1:9011n l3

ð7Þ Where, l is the length of framed column; λ is the slenderness of column; n is the axial-load ratio; Ec is the modulus of elasticity of concrete; and I0 is the moment of inertia of gross concrete column section about centroidal axis. The degradation stiffness of frame structures:

Fig. 21. Bilinear model.


 2 3Ec I 0 2 K c ¼ −0:7292−4:4809λv þ 11:9856λv þ 3:8491n−3:2449n l3 ð8Þ

W. Li et al. / Journal of Constructional Steel Research 77 (2012) 145–162

159

Fig. 22. The distribution of plastic hinge of frame under lateral load.

Where, l is the length of framed column; λv is the characteristic values of stirrup; n is the axial-load ratio; Ec is the modulus of elasticity of concrete; and I0 is the moment of inertia of gross concrete column section about centroidal axis. Fig. 23 shows the geometric relationship: Ks ¼

P u −P y ΔB −Δy

ð9Þ

Hence: ΔB ¼

P u −P y P −0:6P u 0:4P u þ Δy ¼ u þ Δy ¼ þ Δy Ks Ks Ks

ð10Þ

5.2. The validation for load–displacement backbone curve of composite CCSTRCS frame structure 5.2.1. The validation for non-descending skeleton curve model Fig. 24 shows the comparison of the skeleton curves of the lateral load–lateral displacement by the finite element method and simplified model conducted by the Technical Research and Development Institute, Nishimatsu Construction in 1997, for steel beam–concrete column frame specimens [14]. It is shown that the elastic stiffness and postyield stiffness agree well with them, but the yield load is larger than the FEA results, since the yield load of the simplified model is based on the determination of the ideal rigid-plastic limit analysis theory, and in the actual process frame stiffness of the structure will have degraded. In order to study the simplified model in a wider scope of applicability, a two-story and two-bay composite CCSHRCS frame is selected

to compared different parameters, the comparison for calculated results are shown in Fig. 25, it shows good agreement with elastic stiffness, slightly higher than the yield load of finite element analysis, strengthening stage is lower than the stiffness of the finite element calculation. As a simplified model, however, the overall good agreement between the two can meet general engineering requirements. 5.2.2. The validation for descending skeleton curve model Fig. 26 is the comparison of the finite element analysis and the simplified model, good agreement with elastic stiffness, yield load is slightly less than the finite element method, and stiffness at the post-yield stage of the simplified model is lower than that of the finite element method, degradation stiffness is in good agreement. As a simplified model, however, the overall good agreement between the two can also meet the general engineering requirements. 6. Summary and concluding remarks In this paper, the ABAQUS software is applied to analyze the influence of different parameters on behaviors of composite CCSTRCS frame structures. These parameters are as follow: the ratio of longitudinal reinforcement (ρs) and the volume ratio of tie (ρv), the strength of longitudinal reinforcement (fys) and the strength of tie (fyv), the compressive strength of cubic concrete (fcu), characteristic values of tie (λv), the yield strength of steel (fak), axial-load ratio (n), the linear stiffness ratio of beam–column (K) and the slenderness of column (λ). The results show that the above parameters have a corresponding effect on lateral load–lateral displacement of the composite CCSTRCS frame. In which, the axial-load ratio(n), the linear stiffness ratio of

1000

P/kN

800

600 N

N

N

400

FEM simplified model

P

200

0

0

20

40

60

80

/mm Fig. 23. Trilinear model with deterioration.

Fig. 24. Comparisons of load–displacement backbone curve of CCSHRCS tested frame between FEM and simplified model.

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(a)

(b)

1200

f ak

345 N / mm 2 ,

1000

500

2.082

v

23, n 0.5, K

345 N / mm 2 ,

f ak

0.128

35.1, n

0.5484

v

0.5, K

0.396

400

P/kN

P/kN

800 600

200

FEM

400

300

N

N

FEM

N

Simplified model

N

P

100

200

N

P

0

0 0

20

40

60

80

100

0

20

40

60

/mm

(c)

N

simplified model

80

100

/mm

(d)

600

600 500

f ak

345 N / mm 2 , 35.1, n 0.5, K

v

2.1285

345 N / mm 2 ,

f ak

0.793

500

35.1, n 0.5, K

2.1285

v

1.59

P/kN

P/kN

400 300 N

200

N

N

300 FEM

200

FEM

100

400

N

N

N

Simplified model

Simplified model

P

P

100 0

0 0

20

40

60

80

100

0

20

40

60

80

100

/mm

/mm

(e) 1000

f ak

345 N / mm 2 , 23, n 0.1, K

v

2.1285

0.128

P/kN

800

600

400

N

FEM

N

N

Simplified model

P

200

0 0

20

40

60

80

100

/mm Fig. 25. Comparisons of backbone curve of composite CCSHRCS frame between FEM and the simplified model.

beam–column (K) and the slenderness of column (λ) have a significant influence on elastic stiffness; and the characteristic values of tie (λv), the yield strength of steel (fak), axial-load ratio (n) and the slenderness of column (λ) have a significant influence on the post-yield stiffness. When the characteristic value of tie is more than 0.362, P–Δ curves are not descending; if less than 0.362, P–Δ curves are descending. Other conditions are the same, with the decrease of the characteristic value of tie, elastic stiffness is invariable, post-yield stiffness and ultimate load reduce, so is the degradation stiffness; when the axial-load ratio increases, the elastic stiffness changes little, post-yield stiffness and ultimate load significantly reduce, degradation stiffness also significantly reduces.

The relationship of the lateral load–lateral displacement of composite CCSTRCS frame is based on the above parameters, the bilinear model and trilinear model are adopted to simulate non-descending and descending curves of lateral load–lateral displacement of composite CCSTRCS frame. The parameters of the simplified model of P–Δ curve with descending and non-descending for composite CCSTRCS frame are derived by multiple linear regressions. Comparing the simplified model with finite element results, it shows that for the case of non-descending, elastic stiffness is of good agreement, yield load is slightly higher and postyield stiffness is lower than that of the finite element method. For the descending case, elastic stiffness is also of good agreement, yield load

W. Li et al. / Journal of Constructional Steel Research 77 (2012) 145–162

(a)

161

(b)

400

700

350

600

300

P/kN

35.1, n 0.9, K

500

0.362

v

0.396

P/kN

345 N / mm 2 ,

f ak

250 200 150

N

FEM

100

N

345 N / mm 2 ,

f ak

400

23, n 0.9, K

0.362

300

FEM

200

simplified model

N

N

N

N

P

simplified model

P

100

50

0

0 0

20

40

60

80

0

100

20

40

/mm

60

80

100

/mm

(c)

(d)

800

400

700

350 300

600 345 N / mm 2 , 23, n 0.7, K

0.362

v

250

0.128

P/kN

f ak

500

P/kN

v

0.128

400 300

FEM

200

simplified model

N

N

345 N / mm 2 ,

f ak

v

35.1, n 0.8, K

0.2142 0.396

200 150

N

N

FEM

100 P

simplified model

N

P

50

100

N

0

0

0

20

40

60

80

100

0

20

40

60

80

100

/mm

/mm

(e) 400 350 300 f ak

P/kN

250

345 N / mm 2 ,

v

0.2142

35.1, n 0.5, K

0.396

N

N

200 150

FEM

100

simplified model

N

P

50 0

0

20

40

60

80

100

/mm Fig. 26. Comparisons of load–displacement backbone curve of composite CCSHRCS frame between FEM and the simplified model.

is slightly less and post-yield stiffness is lower than that of the finite element method. The degradation stiffness also agrees well with each other. As a simplified model, the overall agree well between the two, and they are able to meet the general engineering requirements. In addition, if ties with high-strength and small spacing are adopted, the characteristic value of tie is generally greater than 0.362. Therefore, the P–Δ curves for high-strength concrete column confined with continuous compound spiral ties-steel beam frame structure, even when the axial-load ratio is more than 1.0, the curves are still not descending. Ductility coefficient is infinite in theory.

However, in practical engineering, it is necessary to consider the slenderness of column and axial-load ratio to ensure the ductility of the columns. Acknowledgments The authors would like to express their appreciation to Xi'An University of Architecture & Technology and their classmates. Simultaneously we thank the beneficial material provided by the Architectural Institute of Japan.

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The writers would like to acknowledge the help of Mei Li and Xiao-lei Li; they translated and revised the part of this paper. In addition, the writers appreciate the reviewers for their valuable comments and suggestions. References [1] Goel SC. United States–Japan Cooperative Earthquake Engineering Research Program on Composite and Hybrid Structures [J]. J Struct Eng ASCE 2004;130(2): 157–8. [2] Liu J, Foster SJ. Finite element model for confined concrete columns[J]. J Struct Eng ASCE 1998;124(9):1011–7. [3] Yu T, Teng JG, Wong YL, Dong SL. Finite element modeling of confined concrete-I: Drucker–Prager type plasticity model [J]. Eng Struct 2010;32:665–79. [4] Yu T, Teng JG, Wong YL, Dong SL. Finite element modeling of confined concrete-II: Plastic-damage model [J]. Eng Struct 2010;32:680–91. [5] Hajjar J, Leon R, Gustafson M, Shield C. Seismic response of composite moment-resisting connections. II: Behavior[J]. J Struct Eng ASCE 1998;124(8):877–85. [6] ABAQUS user's manual—version 6.8.1. Pawtucket, RI: Hibbit, Karlsson & Sorenson; 2006. [7] Salvatore W, Bursi OS, Lucchesi D. Design, testing and analysis of high ductile partial-strength steel-concrete composite beam-to-column joints[J]. Comput Struct 2005;83:2334–52. [8] Hu HT, Huang CS, Wu MH, Wu YM. Nonlinear Analysis of Axially Loaded Concrete-Filled Tube Columns with Confinement Effect[J]. J Struct Eng ASCE 2003;129(10):1322–9. [9] Zhao GZ, Li A. Numerical study of a bonded steel and concrete composite beam[J]. Comput Struct 2008;86:1830–8. [10] Bursi OS, Sun FF, Postal S. Non-linear analysis of steel–concrete composite frames with full and partial shear connection subjected to seismic loads[J]. J Constr Steel Res 2005;61:67–92. [11] Han LH, Wang WD, Zhao XL. Behavior of steel beam to concrete-filled SHS column frames: Finite element model and verifications [J]. Eng Struct 2008;30:1647–58. [12] Wu LY, Chung LL, Wang MT, Huang GL. Numerical study on seismic behavior of H-beams with wing plates for bolted beam–column connections [J]. J Constr Steel Res 2009;65:97–115. [13] Li W, Li QN, Jiang WS. Nonlinear finite element analysis of behaviors of steel beam-continuous compound spiral stirrups reinforced concrete column frame structures [J]. The Structural Design of Tall and Special Buildings. http://dx.doi. org/10.1002/tal.758,2012. [14] Li W, Li QN, Jiang WS, Jiang L. Seismic performance of composite reinforced concrete and steel moment frame structures-state-of-the-art[J]. Compos B 2011;42(2):190–206. [15] Technical specification for confined concrete structures and hybrid structures of concrete and steel (DB13(J)/T83-2009[S]. Shijiazhuang: Engineering Construction Standard in Hebei Province; 2009 [in Chinese].

[16] ACI Committee 318. Building code requirements for structural concrete, ACI318-08, and commentary, ACI R318-08[S]. Farmington Hills, Michigan: American Concrete Institute; 2008. [17] Zhong ST, Jiang WS, Liu WY. The theory and practice of steel and concrete composite structures [M]. Beijing: China Architecture and Building Press; 2008 [in Chinese]. [18] Wang WD, Han LH. Research on practical resilience model of load versus displacement for concrete filled steel tubular frame[J]. Eng Mech 2008;25(11):62–9 [in Chinese].

Nomenclature C: grade of compressive strength of concrete HPB: hot rolled steel reinforcing bars fcu: specified compressive strength of concrete cubes As ρs: ratio of longitudinal reinforcement ρs ¼ bh fys: specified yield strength of longitudinal reinforcement ρv: volumetric ratio of tie fyv: specified yield strength of tie fc: design compressive strength of concrete f λv: characteristic values of tie λv ¼ ρv fyv c fak: specified yield strength of steel beam K: linear stiffness ratio of beam–column λ: slenderness of column n: axial-load ratio n ¼ f NA c N np: peak axial-load ratio np ¼ f pA c Ke: elastic stiffness of frame structures Ks: post-yield stiffness of frame structures Kc: degradation stiffness of frame structures Py: yield load Pu: ultimate load △y: yield displacement △B: peak displacement corresponding to ultimate load Ec: modulus of elasticity of concrete l: unsupported length of framed column β: effective length coefficient of framed column I0: moment of inertia of gross concrete column section about centroidal axis Ms: plastic limit moment of beam section N: axial compressive load Wei Li (1981―), male, ethnic Han, native of Wannian, Jiangxi Province, born in November 1981. PhD, interested in the research of high-rise building structures and composite steel and concrete structures.