Keywords

1 Introduction

Steel tube confined concrete column is a new type of composite member in which the steel tube does not directly bear the longitudinal load1 [1]. In practical engineering, frame columns are usually subjected to the combined action of axial force, shear force and bending moment. Therefore, it is necessary to configure reinforcement in steel tube confined concrete columns to form steel tube confined reinforced concrete (STRC) [2]. In STRC column frame structure, because the outer steel pipe of the column is disconnected in the joints area, without the effective restraint of the steel pipe, the joints will become a relatively weak part. Therefore, strengthening measures should be taken for the joints area of STRC column frame to meet the seismic design requirements of “strong joints and weak members”.

In recent years, domestic and foreign scholars [3,4,5,6,7,8,9,10,11] have conducted a lot of research on the structure of steel tube confined concrete column and various joints strengthening measures. In terms of joints, Jianguo Nie proposed the form of steel tube non penetrating ring beam joints, and multiple reinforcement rings are arranged horizontally in the joints area, so that the concrete in the joints area is subject to multiple constraints [7]. The results show that under the action of transverse multiple constraints, the axial compressive strength of the core concrete in the joints area can make up for the strength weakening caused by the disconnection of the steel tube in the joints area. Liangcai Xiao conducted axial compression test research and nonlinear finite element analysis on 18 steel tube confined concrete joints with high longitudinal reinforcement ratio. The test results show that reducing the height of the joints area and increasing the number of beams can improve the axial compression bearing capacity [8]. Xuhong Zhou strengthened the circular steel tube confined concrete beam column joints through the structural measures of horizontal haunching and stirrup densification in the joints area. The test results show that this structural measure improves the axial compression performance of the steel tube confined concrete beam column joints, and the damage degree in the core area of the joints is relatively light [9]. Qingjun Chen and others have conducted a series of studies on the column steel pipe incomplete through joints in the joints area. These joints form mainly increases the area of the joints area by adding the ring beam or horizontal reinforcement cage in the joints area, so as to increase the bearing capacity of the joints. The results show that, under the constraints of ring beam and circumferential reinforcement, the steel tube incomplete through joint has high bearing capacity and good ductility, but the volume of outer ring beam is large, so it has some limitations when applied to corner joints [10, 11].

To sum up, the main problems of the joints strengthening measures proposed for STRC column joints are as follows: (1) the structure is complex; (2) The increase of node section size affects the appearance of the building and the construction is complex; (3) The strengthening measures have poor reinforcement effect on edge joints and corner joints. Therefore, the author of this paper proposes to build a section of build-in steel tube reinforced STRC column joints. Through finite element analysis, the restoring force characteristics of build-in steel tube reinforced STRC column joints with different lengths are analyzed and studied. In this paper, the author proposes a section of build-in steel tube reinforced STRC column joints. Through finite element analysis, the restoring force characteristics of build-in steel tube reinforced STRC column joints with different lengths are analyzed and studied. According to the characteristics of skeleton curve, a three fold restoring force model is proposed by using regression analysis method, and the stiffness degradation law and hysteretic rule of hysteretic curve are analyzed and compared with the simulation results, it provides a reference for the seismic performance prediction and structural design of the proposed new joints.

2 Establishment of Finite Element Model

2.1 Material Constitutive Relation

The steel used in the finite element analysis is mainly divided into reinforcement and steel tube. Assuming that they are ideal elastic-plastic materials, they are divided into elastic section and yield section according to the binary constitutive relationship. The slope of the elastic section is the elastic modulus of the corresponding reinforcement or steel, the slope of the yield section is 0, the poisson’s ratio is 0.3, the elastic modulus is 2.06 × 10 MPa, and the yield strength of the material is 235 MPa.

There are mainly two kinds of concrete constitutive models. The concrete outside the steel tube restraint area adopts the uniaxial tension and compression constitutive relationship of ordinary concrete recommended in GB50010-2010 code for design of concrete structures. The tensile constitutive of concrete in the confined area of steel tube is consistent with that of ordinary concrete. The compressive constitutive model adopts the equivalent stress-strain relationship of concrete considering different restraint levels proposed in document 12, and the specific expression is as follows:

$$y = \left\{ {\begin{array}{*{20}l} {2x - x^{2} } \hfill & {\left( {x \le 1} \right)} \hfill \\ {\frac{x}{{\beta_{{\text{o}}} \cdot \left( {x - 1} \right)^{2} + x}}} \hfill & {\left( {x > 1} \right)} \hfill \\ \end{array} } \right.$$
(1)

where: \(x = \frac{\varepsilon }{{\varepsilon_{co} }}\); \(\varepsilon_{co} = \varepsilon_{c} + 800 \cdot \xi^{0.2} \cdot 10^{ - 6} ;\xi = \frac{{f_{y} A_{s} }}{{f_{ck} A_{c} }};y = \frac{\sigma }{{\sigma_{co} }}\); \(\varepsilon_{c} = (1300 + 12.5f_{co} ) \cdot 10^{ - 6} ;\sigma_{co} = f_{co}\); fco is the axial compressive strength of concrete; σco and εco are the corresponding peak stress and peak strain in the equivalent stress-strain relationship, respectively; fy and fck are the standard values of yield strength of steel and compressive strength of concrete respectively; As and Ac are the cross-sectional areas of steel tube and concrete in the core area respectively [12].

2.2 Model Establishment

C3D8R eight node linear reduced integral hexahedron element is selected to simulate concrete; the S4R four node curved thin shell reduced integral element is used to simulate the steel tube; the reinforcement is simulated by two node linear three-dimensional truss element (T3D2). The elements size is taken as 1/50 of the height of the node model.

The interaction between beam and column reinforcement and concrete and the interaction between steel core embedded in concrete and concrete are simulated by embedded region, which has simple operation and high calculation efficiency. For the interaction between the steel tube wrapped around the column and the concrete surface, it is transmitted through the contact surface, which can be simulated by considering the normal and tangential interface properties of the contact surface. Normal upward, it is generally considered that the pressure at the interface can be completely transmitted, and the “hard contact” simulation is adopted; In the tangential direction, it can be considered that the interaction between concrete and steel tube conforms to the Coulomb friction model, and the shear stress transfer between interfaces can be described by the following formula:

$$\tau { = }\mu \cdot p \ge \tau_{bond}$$
(2)

where μ is the friction coefficient between interfaces; p is the normal pressure of the interface; τbond is the average bond force between steel tube and concrete interface. Cai Shaohuai et al. made a detailed analysis on the bonding mechanism and pointed out that the surface condition of steel tube, concrete strength and concrete curing conditions have a significant impact on the bonding strength of the interface. When the steel tube is mechanically derusted before pouring concrete, the average bonding force of the interface can be calculated according to formula (3) [13, 14]. Considering the test and engineering practice, the steel tube will be derusted before pouring concrete, which meets the application conditions of Eq. (3), so the parameter value will be taken as the basis.

$$\tau_{bond} = 0.1 \cdot (f_{cu} )^{0.4}$$
(3)

Referring to the boundary conditions and loading methods of the previous quasi-static test [15], the column end loading is adopted, and the boundary condition setting is divided into two steps: (1) apply axial force: the column bottom is set as hinged, the beam end is free end, and the axial force is perpendicular to the column top plane. (2) Apply horizontal displacement: the beam end is set as sliding hinge support, the column bottom remains hinged, and displacement loading is adopted.

2.3 Verified by Experiments

In order to verify the reliability of the finite element analysis method, the test piece in literature 15 is selected as the research object for finite element analysis of the test piece. The comparison between the test results and the results obtained by the finite element analysis method in this paper is shown in Fig. 1. It can be seen that the finite element analysis results are in good agreement with the test results.

Fig. 1
A line graph plots force-displacement versus delta for F E A results and test results. The lines are plotted between negative 120 and 90 on the y-axis.

Force-displacement curves comparison between test and FEA results referred to specimen

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On the basis of verifying the reliability of the finite element analysis method, six finite element analysis models of build-in steel tube reinforced STRC column joints with different lengths are established. All the test pieces have the same parameters except the length of the build-in steel tube. The height of the test piece is 2360 mm. The outer steel pipes of the upper and lower columns are made of Q235 steel, with a diameter of 273 mm and a thickness of 3 mm. In order to make the column steel pipe not bear the longitudinal load, the outer steel pipes of the column are disconnected from the column top and the node area respectively, and the joint width is 10 mm, 8C10 is configured in the column. Stirrup selection A8@100, 4 stirrups are set in the core area of the joint, with a spacing of 60 mm; The distance between the loading points of the left and right beams of the test piece is 3000 mm, and the section size of the beam is 200 mm × 350 mm, the longitudinal reinforcement of the beam is equipped with 16C16, and the stirrup is made of A8@100. The concrete strength is C60. The relevant parameters of the test piece are shown in Table 1.

Table 1 Summary of test specimens
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2.4 Finite Element Analysis Results

The skeleton curve is the envelope of peak points at all levels in the hysteretic curve obtained by repeated cyclic loading. The node hysteretic curve is obtained through finite element simulation, and the skeleton curves of 6 specimens are extracted, as shown in Fig. 2.

Fig. 2
A cartesian plane plots P versus delta for J D 1 to 6. The lines are plotted in the first and the third quadrant.

FEA results of skeleton curve

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Stiffness degradation refers to the phenomenon that the peak displacement increases with the increase of the number of cycles when the same peak load is maintained under the action of cyclic repeated load. In order to quantitatively describe the stiffness degradation, the ratio of the sum of the maximum horizontal load of N cycles to the sum of displacement amplitude is defined as the equivalent stiffness K, and the calculation formula is as follows:

$$K_{i} = \frac{{\sum\nolimits_{i = 1}^{n} {P_{i} } }}{{\sum\nolimits_{i = 1}^{n} {\Delta_{i} } }}$$
(4)

where: K is the equivalent stiffness; Pi is the horizontal load corresponding to the i-th displacement angle; Δi is the amplitude of stage I.

The stiffness degradation curve can be obtained after data processing, as shown in Fig. 3. By analyzing the skeleton curve and stiffness degradation curve of 6 specimens, it can be seen that: (1) The initial stiffness of each specimen before yield is roughly the same. (2) In the elastic–plastic stage, the stiffness of each specimen increases with the increase of the length of the build-in steel tube, and the ultimate bearing capacity increases slightly. The build-in steel tube of specimen JD-6 is the longest, so its ultimate bearing capacity is the highest. (3) The stiffness degradation rate decreases with the increase of loading displacement. In the failure stage, the stiffness degradation rate decreases with the increase of the length of the build-in steel tube.

Fig. 3
A line graph plots stiffness degradation versus delta for J D 1 to 6. The graph has a decreasing trend.

Stiffness degradation curve

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3 Establishment of Resilience Model

Structural members will produce a series of nonlinear performance strains under seismic conditions. In order to analyze the mechanical performance of members in the whole process of earthquake, the elastic–plastic time history analysis of the structure must be carried out. The relationship between the elastic and plastic restoring force of the structure is obtained from the simplified mathematical model, and the appropriate restoring force of the structure is obtained from the mathematical model of the seismic response, it can reflect the mechanical properties of joints in strength, stiffness, energy consumption and ductility.

3.1 Four Basic Assumptions

In order to establish the restoring force model of build-in steel tube reinforced STRC column joints with different lengths, the following four assumptions are adopted: (1) the maximum elastic load point is taken as the yield load point; (2) In the elastic stage, the unloading stiffness is the initial stiffness, and after the elastic stage, the stiffness begins to deteriorate; (3) The skeleton curve adopts the three line model considering stiffness degradation; (4) Repeated cyclic loading conforms to the fixed-point pointing law [16].

3.2 Establishment of Skeleton Curve

With the increase of the length of the build-in steel tube, the ultimate bearing capacity of the joint specimen increases slightly. At the same time, the bearing capacity of the specimen decreases slowly in the failure stage, the deformation capacity increases gradually in the later stage, and the ductility is improved. The restoring force model is simplified according to the dimensionless skeleton curve of build-in steel tube reinforced STRC column joints with different lengths and the data results obtained from numerical simulation analysis. The specific description is divided into the following three stages:

  1. (1)

    Elastic stage: Through the analysis of the skeleton curves of six specimens with different lengths of build-in steel tube reinforced STRC column joints, it is found that there are no obvious cracking points. The main reason is that the build-in steel tube in the joint area can well restrict the transverse deformation of the concrete in the joint core area, and resist shear together with the concrete, so the stiffness attenuation of the joint area is small in the elastic stage. Connecting the coordinate origin O and yield point a, the OA section is regarded as the elastic section of the build-in steel tube reinforced STRC column joint specimen. This stage refers to the process from the loading of the specimen to the yield of the specimen. The load displacement curve is basically linear. The slope of this stage is the initial stiffness of the member, and the calculation formula of its equivalent stiffness is:

    $$K_{e} = P_{y} /\Delta_{y}$$
    (5)
  2. (2)

    Elastoplastic stage: This stage refers to the process from the yield of the specimen to the peak point of the specimen. With the increase of displacement, the growth of load becomes slower than that in the previous stage, and the stiffness degradation is significant. The elastic–plastic stage of the skeleton curve is regarded as AB, and its equivalent stiffness is described as follows:

    $$K_{s} = \frac{{P_{\max } - P_{y} }}{{\Delta_{\max } - \Delta_{y} }}$$
    (6)
  3. (3)

    Plastic stage: This stage refers to the process from the maximum bearing capacity of the specimen to the failure of the specimen. At this stage, the load of the specimen begins to decrease with the increase of displacement, which is simplified as section BC in Fig. 4. The equivalent stiffness Kp of the specimen in the plastic stage is:

    $$K_{p} = \gamma_{d} K_{e}$$
    (7)
    Fig. 4
    A cartesian plane plots P versus delta for a three-line resilience model. Two definite points are highlighted in the first and the third quadrant.

    Degenerative three-line resilience model

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The regression analysis of the data points of the finite element analysis results shows that the relationship between the length of the build-in steel tube and the stiffness degradation rate in the plastic stage of the specimen is:

$$\gamma_{d} = - 0.05153 - 1.45512 \cdot e^{{ - 0.00548h_{e} }}$$
(8)

where γd is stiffness degradation rate; he is the length of build-in steel tube, unit: mm.

3.3 Determination of Unloading Stiffness and Reloading Stiffness

With the increase of loading displacement, the strength and stiffness of the specimen degenerate. Therefore, the slope changes of loading section and unloading section in hysteresis loop can be used for quantitative calculation. Based on the analysis of the simulation data of six specimens with different lengths of build-in steel tube reinforced STRC column joints, the positive and negative loading and unloading stiffness are dimensionless, and the following formula can be obtained by fitting:

$$K_{u} /K_{e} = \left( {\Delta_{y} /\Delta_{l} } \right)^{U}$$
(9)
$$K_{r} /K_{e} = \left( {\Delta_{y} /\Delta_{l} } \right)^{R}$$
(10)

where \(U = - 0.58541 + 5.51436 \cdot 10^{ - 4} \cdot h_{e} - 3.04018 \cdot 10^{ - 7} \cdot h_{e}^{2} ;R = - 2.22731 + 0.00485 \cdot h_{e} - 3.7909 \cdot 10^{ - 6} \cdot h_{e}^{2}\).

3.4 Hysteresis Loop Rule

By analyzing the characteristics of build-in steel tube reinforced STRC column joints with different lengths, the three fold linear restoring force model is shown in Fig. 4.

The hysteresis loop rules can be summarized as follows:

  1. (1)

    From the initial loading to the yield value of the specimen, the loading and unloading are carried out according to the elastic section of the skeleton curve (OA/OD section in Fig. 4), and the loading and unloading stiffness is the initial elastic stiffness Ke.

  2. (2)

    After the specimen reaches the peak load, it enters the strengthening stage along the AG load path. When unloading in the forward direction, the unloading direction changes from G to H, and the unloading stiffness begins to deteriorate, which can be determined by formula (9); When reverse loading, take the intersection of the curve with zero unloading and the coordinate axis as the starting point and carry out along the skeleton curve. The reverse loading path is HI, and the reverse loading stiffness is calculated by formula (10); The corresponding path during reverse unloading is the IJ segment in Fig. 4.

  3. (3)

    When the extreme load point is reached, if the load is unloaded at the peak point B, the corresponding unloading path is section BK in Fig. 4. When the load is unloaded to 0, the load is loaded reversely, and the loading path points to the reverse peak load point E. If it is unloaded at point M, the unloading path is MN section. During reverse loading, it is carried out along the path NQ, and during unloading, it is from point Q to point R.

4 Conclusion

  1. (1)

    The build-in steel tube has an obvious enhancement effect on the seismic performance of STRC column joints. When the length of the core steel tube increases, the stiffness of the joint specimen increases, the ultimate bearing capacity increases, the stiffness degradation rate slows down with the increase of loading displacement, and the stiffness degradation rate of the specimen in the plastic stage slows down with the increase of the length of the build-in steel tube.

  2. (2)

    On the basis of verifying the rationality and reliability of the finite element analysis method, through the analysis and fitting regression of the simulation results, the calculation formula of the skeleton curve characteristic parameters of steel tube STRC column joints with different lengths is obtained, and the three broken line skeleton curve model is proposed. The comparison between the calculation results and the simulation results shows that there is little difference between them.

  3. (3)

    Based on the skeleton curve model obtained by regression analysis, the calculation method of loading and unloading stiffness in hysteretic curve is determined. At the same time, the length of build-in steel tube is introduced as a parameter, and the calculation formula of loading and unloading stiffness is obtained. According to the finite element simulation results and the three fold skeleton curve, the stiffness degradation law and hysteretic rule of the model are analyzed. The established degraded three fold restoring force model can effectively reflect the hysteretic characteristics of steel tube STRC column joints with different lengths, which can provide a reference for the elastic–plastic analysis and engineering application of the structure.

  4. (4)

    This paper only studies the seismic performance of in-plane joints of steel tube reinforced STRC columns under low cyclic loading. The seismic performance of edge joints, corner joints, spatial joints and frame structures and the similarities and differences between them need to be further studied.

  5. (5)

    The restoring force model proposed in this paper is only based on the length of build-in steel tube. For the case of multi parameters in practical engineering, it needs more comprehensive and in-depth research.