# Restoring Force Model Research on Joint of Circular Tubed Steel-Reinforced Concrete Column

Authors: Yan Dai; Shaofeng Nie; Tianhua Zhou; Yichun Xu; Jingrong Peng
Journal: Shock and Vibration
 (2021)

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## Abstract

In this paper, two joints of circular tubed steel-reinforced concrete (CTSRC) column were designed. The load-displacement hysteretic curve and skeleton curve of this new type of joint are obtained by the pseudostatic test under low cycle cyclic load on the top of the column. The results show that this new type of joint has good seismic energy dissipation performance. On the basis of the test, a three-fold skeleton curve model considering three characteristic points of yield, limit, and failure is proposed, and the expression of skeleton curve model is given. The load and unload stiffness degradation law of specimens under reciprocating load is studied, and the expression of stiffness degradation law is given. The hysteresis law of the new type joint specimens is described in detail. The validity of the model is verified by comparing the experimental curve with the model curve. The model can be used in the elastic-plastic seismic time-history analysis on the joint of circular tubed steel-reinforced concrete (CTSRC) column.


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## Body

## 1. Introduction

The restoring force model is used to explain the relationship between the load and displacement of the structure under repeated load, which can reflect the ability of the structure to recover the original deformation after unloading. In the study of the seismic performance of the structure, the restoring force model is the basis of the seismic stress analysis, which mainly reflects the performance of the structure, such as energy consumption, ductility, strength, and stiffness. Since 1940s, seismic engineering has carried out extensive research on structural restoring force curve modeling. Researchers have proposed the following models: the Ramberg–Osgood model [1, 2] proposed by Jennings is shown in Figure 1(a). The model is mainly used to describe the restoring force characteristics of metal materials and also used in soil and reinforced concrete flexural structures; as shown in Figure 1(b), Clough [3] proposed a relatively early model considering the degradation of loading and unloading stiffness, which is mainly proposed as the restoring force characteristic model of reinforced concrete flexural members. Takeda et al. [4] proposed a three-fold line stiffness degradation model, as shown in Figure 1(c). The model is obtained through the low cycle reciprocating loading test of reinforced concrete columns and based on the test results. Because it has one more crack point than the two-fold line model, it can more truly reflect the mechanical characteristics of steel reinforced concrete structures.Figure 1
Common restoring force models. (a) Ramberg–Osgood model. (b) Clough degenerate bilinear model. (c) Takeda three-fold line stiffness degradation model.
(a)(b)(c)In recent years, a series of research studies have been carried out on the load displacement relationship models of various new structures under repeated loads. Yan et al. established the hysteretic model of the joints between steel reinforced ultrahigh strength concrete columns and steel reinforced concrete beams and put forward the quantitative method of seismic damage and attenuation coefficient [5]. Zhang and Han proposed a prediction model for cyclic lateral load deformation response of RC columns subjected to axial compression and cyclic shear; comparison between the predicted cyclic response and experimental results indicates that the proposed model can predict the observed hysteretic response of flexure-shear critical RC columns well [6]. Yuka and Hideki proposed a hysteretic model of H-beam considering slenderness ratio and also proposed a hysteretic model suitable for uniform moment and antisymmetric moment [7]. Zhang et al. studied the hysteretic behavior of six glass bead insulation concrete frame columns (GIC columns) and one ordinary concrete frame column; the results can provide theoretical basis for elastic-plastic analysis of GIC column under earthquake action [8]. Hao et al. proposed an effective method to calculate the monotone curve and hysteretic curve of H-beam under bending moment. The accuracy of predicting the hysteretic behavior of H-beam is verified by comparing with the numerical results [9]. Wang et al. carried out nonlinear time-history analysis on a new type of energy dissipation joint of bamboo steel hybrid frame and proposed the joint restoring force model with the best performance [10]. Li et al. proposed an innovative method to consider the influence of dynamic effects and degradation on seismic performance analysis of reinforced concrete frame structures [11]. Xie et al. carried out quasistatic tests on three 1/10 scale pier specimens and established the hysteretic model of prefabricated pier, which was in good agreement with the test results [12]. Wu et al. studied the restoring force model of h-steel-reinforced concrete column composite joints (hereinafter referred to as MPSC joints), which has high accuracy and can be used for the design of MPSC joints [13]. Song et al. designed ten PVC-CFRP confined concrete column-ring beam interior joints (PCCC-RBIJs) and one PVC confined concrete column-ring beam interior joint (control interior joint) and studied their restoring force models [14]. Ni et al. conducted in-depth experimental research on the seismic performance of high-strength reinforced concrete shear walls and proposed a restoring force model of high-strength reinforced concrete shear walls composed of skeleton curves and a group of empirical hysteretic laws [15].The CTSRC column is a new type of composite structure. The relative research [16–18] shows that it has high compressive and shear capacity and superior seismic performance and can make full use of the performance of high-strength materials. On this basis, the authors put forward a new type joint of CTSRC column [19–21]. By applying low-cycle reciprocating load on the top of the column, the load displacement relationship under reciprocating load is studied.There are mainly two methods to analyze and study the restoring force model: one is based on the basic research of the steel bar, steel tube, and concrete and the bond relationship between steel bar and concrete, steel tube and concrete, to obtain the idealized material constitutive model and structural strain model and to obtain the hysteretic curve of the restoring force relationship by the iterative method; the second is to simulate the test results directly [22, 23]. In this paper, the second method is used to study the restoring force characteristics of the new type joint based on the quasistatic test and the measured hysteretic curves. The restoring force model suitable for the new type joint is established, which provides the basis for the future elastic-plastic time-history analysis of structures.


## 2. Test Preparation and Results

### 2.1. Specimen Design and Loading Scheme

In this paper, two full-scale joints in frame structure were designed. The specimen numbers are SH-1 and SH-2. The main differences between the two specimens are the reinforcement ratio of ring beam and the form of ring beam. The specimen numbers and parameters are shown in Table1.Table 1
Test piece number and parameter table.
Specimen numberSteel ratio (%)Thickness of circular tube steel pipe (mm)Form of ring beamReinforcement ratio of ring beam (%)SH-14.195.0Octagon3.55SH-2Circular2.05The height and width of the two specimens are 4375.0 mm and 3370.0 mm, respectively. The outer diameter of circular steel tube is 400.0 mm. In order to ensure that the circular steel tube does not bear the longitudinal load, the circular steel tube was disconnected in the joint area and a 15.0 mm ring opening is reserved from the upper and lower edges of the concrete ring beam. The other structures and dimensions are shown in Figure2 (taking SH-1 specimen as an example).Figure 2
Structural diagram of the specimen (mm). (a) Elevation. (b) Plan.
(a)(b)The quasistatic test is generally divided into column end loading and beam end loading. Because the second-order effect of gravity can be considered in the column end loading [24], the column-end-loading mode was adopted in this paper: First, the vertical load is gradually applied to 1500 kN by the jack at the top of the column in three stages and the load lasts for several minutes; second, the MTS servoactuator is used to apply the low-cycle reciprocating load on the top of the column, and the load control is used before yielding. After yielding, the load is carried out according to the multiple of the yield displacement of the top of the column, and each cycle is carried out three times until the specimen is damaged or the horizontal load of the top of the column drops below 85% of its peak load. The test device diagram and loading diagram are shown in Figure 3.Figure 3
(a) Test device diagram (① specimen, ② lifting jack, ③ Reaction beam, ④ Loading head, ⑤ Connector, ⑥ 5MTS(50t), ⑦ Reaction wall, ⑧ Rigid bearing, ⑨ Chain link, ⑩ Hinged support, ⑪ Reaction frame, ⑫ Lateral support of steel beam, ⑬ Sliding rail). (b) Loading diagram.
(a)(b)


### 2.2. Test Result

The load-displacement hysteretic curves of the two specimens are shown in Figure4. It can be seen from the figure that the hysteretic curves of the two specimens are full, showing a very good seismic energy dissipation performance.Figure 4
Load-displacement hysteretic curve of specimen column top. (a) SH-1. (b) SH-2.
(a)(b)


## 2.1. Specimen Design and Loading Scheme

In this paper, two full-scale joints in frame structure were designed. The specimen numbers are SH-1 and SH-2. The main differences between the two specimens are the reinforcement ratio of ring beam and the form of ring beam. The specimen numbers and parameters are shown in Table1.Table 1
Test piece number and parameter table.
Specimen numberSteel ratio (%)Thickness of circular tube steel pipe (mm)Form of ring beamReinforcement ratio of ring beam (%)SH-14.195.0Octagon3.55SH-2Circular2.05The height and width of the two specimens are 4375.0 mm and 3370.0 mm, respectively. The outer diameter of circular steel tube is 400.0 mm. In order to ensure that the circular steel tube does not bear the longitudinal load, the circular steel tube was disconnected in the joint area and a 15.0 mm ring opening is reserved from the upper and lower edges of the concrete ring beam. The other structures and dimensions are shown in Figure2 (taking SH-1 specimen as an example).Figure 2
Structural diagram of the specimen (mm). (a) Elevation. (b) Plan.
(a)(b)The quasistatic test is generally divided into column end loading and beam end loading. Because the second-order effect of gravity can be considered in the column end loading [24], the column-end-loading mode was adopted in this paper: First, the vertical load is gradually applied to 1500 kN by the jack at the top of the column in three stages and the load lasts for several minutes; second, the MTS servoactuator is used to apply the low-cycle reciprocating load on the top of the column, and the load control is used before yielding. After yielding, the load is carried out according to the multiple of the yield displacement of the top of the column, and each cycle is carried out three times until the specimen is damaged or the horizontal load of the top of the column drops below 85% of its peak load. The test device diagram and loading diagram are shown in Figure 3.Figure 3
(a) Test device diagram (① specimen, ② lifting jack, ③ Reaction beam, ④ Loading head, ⑤ Connector, ⑥ 5MTS(50t), ⑦ Reaction wall, ⑧ Rigid bearing, ⑨ Chain link, ⑩ Hinged support, ⑪ Reaction frame, ⑫ Lateral support of steel beam, ⑬ Sliding rail). (b) Loading diagram.
(a)(b)


## 2.2. Test Result

The load-displacement hysteretic curves of the two specimens are shown in Figure4. It can be seen from the figure that the hysteretic curves of the two specimens are full, showing a very good seismic energy dissipation performance.Figure 4
Load-displacement hysteretic curve of specimen column top. (a) SH-1. (b) SH-2.
(a)(b)


## 3. Establishment of the Restoring Force Model

The restoring force model for nonlinear analysis consists of three parts: skeleton curve, stiffness degradation law, and hysteretic characteristics. The results show that a reasonable restoring force model can reflect the hysteretic stress characteristics of actual structure or component and be simple and practical within the acceptable limit, so as to facilitate the effective elastic-plastic response analysis.


### 3.1. Skeleton Curve Model

The skeleton curve determines the characteristic points of the restoring force model, which is the connecting line of the characteristic points in the 1/4 hysteretic curve. In this test, the characteristic point loads and displacements of the two joint specimens are different. In order to facilitate the comparison, the skeleton curves of the two specimens are first treated as dimensionless. The formula is as follows:(1)±P±Pmax,±Δ±Δmax,where ±ΔPmax is the maximum positive and reverse load on the top of the column and ±Δmax is the displacement corresponding to the maximum positive and reverse load. The results are shown in Table 2, and the dimensionless skeleton curves of specimens are obtained, as shown in Figure 5.Table 2
Dimensionless treatment of characteristic points of skeleton curves.
SpecimenYield stateUltimate load stateFailure statePyΔyPmaxΔmaxPuΔuSH-1Positive155.8 kN41.7 mm168.4 kN60.0 mm143.4 kN178.5 mmDimensionless0.930.691.001.000.852.98Reverse127.7 kN32.4 mm144.3 kN60.0 mm126.9 kN154.2 mmDimensionless0.890.541.001.000.882.57SH-2Positive131.3 kN31.3 mm151.5 kN80.0 mm130.4 kN180.0 mmDimensionless0.870.391.001.000.862.25Reverse120.7 kN20.3 mm155.3 kN60.0 mm132.8 kN157.5 mmDimensionless0.780.341.001.000.862.92Figure 5
Dimensionless skeleton curves of specimens.It can be seen from the above figure that the skeleton curves of the two specimens after dimensionless treatment still show their own regular characteristics. By analyzing the hysteretic curves and dimensionless skeleton curves of specimens, it can be seen that the skeleton curves of the two specimens can be approximately fitted with a broken line. Therefore, the skeleton curve in this paper adopts the form of three broken lines considering yield load, ultimate load, and stiffness degradation and is simulated by the connecting line of three characteristic points. The three characteristic points are yield point, ultimate load point, and failure point (when the bearing capacity decreases to 0.85 times of the ultimate load point).According to the dimensionless values of the characteristic points of the specimens in Table2, the coordinates of the characteristic points corresponding to the specimens SH-1 and SH-2 are given as follows: (−Δu/|−Δmax|, −Pu/|−Pmax|), (1, 1), (−Δy/|−Δmax|, −Py/|−Pmax|), (0, 0), (+Δy/+Δmax,+Py/+Pmax), (1, 1), and (+Δu/+Δmax,+Pu/+Pmax).Taking these coordinate points as the control points of regression analysis and connecting the characteristic points, the three-fold line skeleton curve model of two specimens can be drawn, as shown in Figure6.Figure 6
Curve model of three-fold line skeleton of test piece. (a) SH-1. (b) SH-2.
(a)(b)In Figure6, the control points A and A' are the positive and reverse yield points, B and B' are the positive and reverse ultimate load points, and C and C' are the positive and reverse failure points. According to the control points on the skeleton curve, the three-fold line skeleton curve expression of the two specimens can be obtained, as shown in Table 3.Table 3
Three-fold line skeleton curve expression.
Line segmentSH-1SH-2OAP/+Pmax=1.3329Δ/+ΔmaxP/+Pmax=2.2174Δ/+ΔmaxABP/+Pmax=0.2451Δ/+Δmax+0.7549P/+Pmax=0.2184Δ/+Δmax+0.7816BCP/+Pmax=−0.0754Δ/+Δmax+1.0754P/+Pmax=−0.1112Δ/+Δmax+1.1112OA’P/−Pmax=1.6419Δ/−ΔmaxP/−Pmax=2.2988Δ/−ΔmaxA’B’P/−Pmax=0.2495Δ/−Δmax−0.7505P/−Pmax=0.3369Δ/−Δmax−0.6631B’C’P/−Pmax=−0.0772Δ/−Δmax−1.0772P/−Pmax=−0.0893Δ/−Δmax−1.0893


### 3.2. Stiffness Degradation Law

According to the hysteretic curve and skeleton curve, the loading and unloading stiffness of the specimen has a certain degree of degradation. The following is the degradation law of loading and unloading stiffness of the specimen under repeated loading [25]. As shown in Figure 7, K1 is the positive unloading stiffness, K2 is the reverse loading stiffness, K3 is the reverse unloading stiffness, K4 is the positive loading stiffness, a is the positive unloading point, b is the reverse loading point, c is the reverse unloading point, d is the positive loading point, Δ1 is the displacement corresponding to the positive unloading point a, Δ2 is the residual deformation when positive unloading to zero, Δ3 is the displacement corresponding to the reverse unloading point c, and Δ4 is the residual deformation when reverse unloading to zero, and the meanings of other symbols are the same as above.Figure 7
Stiffness degradation law.


#### 3.2.1. Degradation Law of Specimen Stiffness under Positive Unloading (K1)

As shown in Figure7, the slope of line ab is the positive unloading stiffness K1. By sorting out the hysteretic curve data of two specimens, the slope value of line ab under different displacement amplitudes of each specimen can be obtained, and it can be dimensionless. The abscissa is Δ1/Δmax+, and the ordinate is K1/K0+. The scatter diagram of the degradation law of positive unloading stiffness (K1) of each specimen can be obtained, where Δmax+ represents the displacement corresponding to the peak point when the joint specimen is under positive loading. K0+ represents the initial elastic stiffness of the joint under positive loading. After removing the obvious singular points, using the method of nonlinear data fitting, the degradation law curve and its regression formulas of the positive unloading stiffness K1 of each specimen can be obtained, as shown in Figure 8 and equations (2) and (3).Figure 8
Regression curve fitting (K1). (a) SH-1. (b) SH-2.
(a)(b)The regression formulas of K1 are as follows:(2)K1K0+=1.221e0.20164/Δ1/Δmax++0.16819,(3)K1K0+=1.11917e−Δ1/Δmax+/0.31645+1.23644.


#### 3.2.2. Degradation Law of Specimen Stiffness under Reverse Loading (K2)

As shown in Figure7, the slope of line bc is the reverse loading stiffness K2. Using the same method as in Section 3.2.1, the degradation law curve and its regression formulas of the reverse loading stiffness K2 of each specimen can be obtained, as shown in Figure 9 and equations (4) and (5).Figure 9
Regression curve fitting (K2). (a) SH-1. (b) SH-2.
(a)(b)The regression formulas of K2 are as follows:(4)K2K0−=e−1.4838Δ2/Δmax+,(5)K2K0−=e0.10061−2.83541Δ2/Δmax++0.8073Δ2/Δmax+2.


#### 3.2.3. Degradation Law of Specimen Stiffness under Reverse Unloading (K3)

As shown in Figure7, the slope of line cd is the reverse unloading stiffness K3. Using the same method as in Section 3.2.1, the degradation law curve and its regression formulas of the reverse unloading stiffness K3 of each specimen can be obtained, as shown in Figure 10 and equations (6) and (7).Figure 10
Regression curve fitting (K3). (a) SH-1. (b) SH-2.
(a)(b)The regression formulas of K3 are as follows:(6)K3K0−=e0.57368−0.32551Δ3/Δmax−+0.07056Δ3/Δmax−2,(7)K3K0−=e0.61056−0.49113Δ3/Δmax−+0.14038Δ3/Δmax−2.


#### 3.2.4. Degradation Law of Specimen Stiffness under Positive Loading (K4)

As shown in Figure7, the slope of line da is the positive loading stiffness K4. Using the same method as in Section 3.2.1, the degradation law curve and its regression formulas of the positive loading stiffness K4 of each specimen can be obtained, as shown in Figure 11 and equations (8) and (9).Figure 11
Regression curve fitting (K4). (a) SH-1. (b) SH-2.
(a)(b)The regression formulas of K4 are as follows:(8)K4K0+=e0.30267−1.95449Δ4/Δmax−+0.40772Δ4/Δmax−2,(9)K4K0+=e0.28584−1.57987Δ4/Δmax−+0.03819Δ4/Δmax−2.


### 3.3. Description of Hysteresis Rule

The actual restoring force curve on the joint of CTSRC column is hard to describe, so it is necessary to establish a simplified model which can reflect the actual restoring force characteristics. According to the skeleton curve model and stiffness degradation law, combined with the hysteretic curve, the restoring force model of the specimens was established, as shown in Figure12.Figure 12
Restoring force model.The hysteresis rule is described as follows:(1)
When the specimen is under positive loading, the relationship between load and displacement develops along the skeleton curveOABC. If the specimen is unloading in the OA section (the positive loading does not yield), the positive loading stiffness does not degenerate and the relationship between load and displacement returns along the AO section (the unloading line is AO). If the specimen is unloading in the AB section (the positive loading yields), the positive loading stiffness begins to degenerate and the relationship between load and displacement returns along the ab line (the unloading line is ab). If the specimen is unloading in the BC section (the load has passed the peak point), the relationship between load and displacement returns along eb section (the unloading line is eb).(2)
When the specimen is unloading tob point and then loading reversely, the relationship between load and displacement develops along blA'B'C'. If the specimen is unloading at 1A' section (the reverse loading does not yield), the reverse loading stiffness does not degenerate and the relationship between load and displacement returns along A'O line section (the unloading line is A'O). If the specimen is unloading at A'B' section (the reverse loading yields), the reverse loading stiffness will degenerate and the relationship between load and displacement returns along section cd (the unloading line is cd). If the specimen is unloading at B'C' section (the reverse loading has passed the peak point), the relationship between load and displacement returns along fd section (the unloading line is fd).(3)
When the specimen is unloading tod point and then loading again, the relationship between load and displacement develops along d2ABC, and the subsequent loading and unloading is the same as the first loading.


## 3.1. Skeleton Curve Model

The skeleton curve determines the characteristic points of the restoring force model, which is the connecting line of the characteristic points in the 1/4 hysteretic curve. In this test, the characteristic point loads and displacements of the two joint specimens are different. In order to facilitate the comparison, the skeleton curves of the two specimens are first treated as dimensionless. The formula is as follows:(1)±P±Pmax,±Δ±Δmax,where ±ΔPmax is the maximum positive and reverse load on the top of the column and ±Δmax is the displacement corresponding to the maximum positive and reverse load. The results are shown in Table 2, and the dimensionless skeleton curves of specimens are obtained, as shown in Figure 5.Table 2
Dimensionless treatment of characteristic points of skeleton curves.
SpecimenYield stateUltimate load stateFailure statePyΔyPmaxΔmaxPuΔuSH-1Positive155.8 kN41.7 mm168.4 kN60.0 mm143.4 kN178.5 mmDimensionless0.930.691.001.000.852.98Reverse127.7 kN32.4 mm144.3 kN60.0 mm126.9 kN154.2 mmDimensionless0.890.541.001.000.882.57SH-2Positive131.3 kN31.3 mm151.5 kN80.0 mm130.4 kN180.0 mmDimensionless0.870.391.001.000.862.25Reverse120.7 kN20.3 mm155.3 kN60.0 mm132.8 kN157.5 mmDimensionless0.780.341.001.000.862.92Figure 5
Dimensionless skeleton curves of specimens.It can be seen from the above figure that the skeleton curves of the two specimens after dimensionless treatment still show their own regular characteristics. By analyzing the hysteretic curves and dimensionless skeleton curves of specimens, it can be seen that the skeleton curves of the two specimens can be approximately fitted with a broken line. Therefore, the skeleton curve in this paper adopts the form of three broken lines considering yield load, ultimate load, and stiffness degradation and is simulated by the connecting line of three characteristic points. The three characteristic points are yield point, ultimate load point, and failure point (when the bearing capacity decreases to 0.85 times of the ultimate load point).According to the dimensionless values of the characteristic points of the specimens in Table2, the coordinates of the characteristic points corresponding to the specimens SH-1 and SH-2 are given as follows: (−Δu/|−Δmax|, −Pu/|−Pmax|), (1, 1), (−Δy/|−Δmax|, −Py/|−Pmax|), (0, 0), (+Δy/+Δmax,+Py/+Pmax), (1, 1), and (+Δu/+Δmax,+Pu/+Pmax).Taking these coordinate points as the control points of regression analysis and connecting the characteristic points, the three-fold line skeleton curve model of two specimens can be drawn, as shown in Figure6.Figure 6
Curve model of three-fold line skeleton of test piece. (a) SH-1. (b) SH-2.
(a)(b)In Figure6, the control points A and A' are the positive and reverse yield points, B and B' are the positive and reverse ultimate load points, and C and C' are the positive and reverse failure points. According to the control points on the skeleton curve, the three-fold line skeleton curve expression of the two specimens can be obtained, as shown in Table 3.Table 3
Three-fold line skeleton curve expression.
Line segmentSH-1SH-2OAP/+Pmax=1.3329Δ/+ΔmaxP/+Pmax=2.2174Δ/+ΔmaxABP/+Pmax=0.2451Δ/+Δmax+0.7549P/+Pmax=0.2184Δ/+Δmax+0.7816BCP/+Pmax=−0.0754Δ/+Δmax+1.0754P/+Pmax=−0.1112Δ/+Δmax+1.1112OA’P/−Pmax=1.6419Δ/−ΔmaxP/−Pmax=2.2988Δ/−ΔmaxA’B’P/−Pmax=0.2495Δ/−Δmax−0.7505P/−Pmax=0.3369Δ/−Δmax−0.6631B’C’P/−Pmax=−0.0772Δ/−Δmax−1.0772P/−Pmax=−0.0893Δ/−Δmax−1.0893


## 3.2. Stiffness Degradation Law

According to the hysteretic curve and skeleton curve, the loading and unloading stiffness of the specimen has a certain degree of degradation. The following is the degradation law of loading and unloading stiffness of the specimen under repeated loading [25]. As shown in Figure 7, K1 is the positive unloading stiffness, K2 is the reverse loading stiffness, K3 is the reverse unloading stiffness, K4 is the positive loading stiffness, a is the positive unloading point, b is the reverse loading point, c is the reverse unloading point, d is the positive loading point, Δ1 is the displacement corresponding to the positive unloading point a, Δ2 is the residual deformation when positive unloading to zero, Δ3 is the displacement corresponding to the reverse unloading point c, and Δ4 is the residual deformation when reverse unloading to zero, and the meanings of other symbols are the same as above.Figure 7
Stiffness degradation law.


### 3.2.1. Degradation Law of Specimen Stiffness under Positive Unloading (K1)

As shown in Figure7, the slope of line ab is the positive unloading stiffness K1. By sorting out the hysteretic curve data of two specimens, the slope value of line ab under different displacement amplitudes of each specimen can be obtained, and it can be dimensionless. The abscissa is Δ1/Δmax+, and the ordinate is K1/K0+. The scatter diagram of the degradation law of positive unloading stiffness (K1) of each specimen can be obtained, where Δmax+ represents the displacement corresponding to the peak point when the joint specimen is under positive loading. K0+ represents the initial elastic stiffness of the joint under positive loading. After removing the obvious singular points, using the method of nonlinear data fitting, the degradation law curve and its regression formulas of the positive unloading stiffness K1 of each specimen can be obtained, as shown in Figure 8 and equations (2) and (3).Figure 8
Regression curve fitting (K1). (a) SH-1. (b) SH-2.
(a)(b)The regression formulas of K1 are as follows:(2)K1K0+=1.221e0.20164/Δ1/Δmax++0.16819,(3)K1K0+=1.11917e−Δ1/Δmax+/0.31645+1.23644.


### 3.2.2. Degradation Law of Specimen Stiffness under Reverse Loading (K2)

As shown in Figure7, the slope of line bc is the reverse loading stiffness K2. Using the same method as in Section 3.2.1, the degradation law curve and its regression formulas of the reverse loading stiffness K2 of each specimen can be obtained, as shown in Figure 9 and equations (4) and (5).Figure 9
Regression curve fitting (K2). (a) SH-1. (b) SH-2.
(a)(b)The regression formulas of K2 are as follows:(4)K2K0−=e−1.4838Δ2/Δmax+,(5)K2K0−=e0.10061−2.83541Δ2/Δmax++0.8073Δ2/Δmax+2.


### 3.2.3. Degradation Law of Specimen Stiffness under Reverse Unloading (K3)

As shown in Figure7, the slope of line cd is the reverse unloading stiffness K3. Using the same method as in Section 3.2.1, the degradation law curve and its regression formulas of the reverse unloading stiffness K3 of each specimen can be obtained, as shown in Figure 10 and equations (6) and (7).Figure 10
Regression curve fitting (K3). (a) SH-1. (b) SH-2.
(a)(b)The regression formulas of K3 are as follows:(6)K3K0−=e0.57368−0.32551Δ3/Δmax−+0.07056Δ3/Δmax−2,(7)K3K0−=e0.61056−0.49113Δ3/Δmax−+0.14038Δ3/Δmax−2.


### 3.2.4. Degradation Law of Specimen Stiffness under Positive Loading (K4)

As shown in Figure7, the slope of line da is the positive loading stiffness K4. Using the same method as in Section 3.2.1, the degradation law curve and its regression formulas of the positive loading stiffness K4 of each specimen can be obtained, as shown in Figure 11 and equations (8) and (9).Figure 11
Regression curve fitting (K4). (a) SH-1. (b) SH-2.
(a)(b)The regression formulas of K4 are as follows:(8)K4K0+=e0.30267−1.95449Δ4/Δmax−+0.40772Δ4/Δmax−2,(9)K4K0+=e0.28584−1.57987Δ4/Δmax−+0.03819Δ4/Δmax−2.


## 3.2.1. Degradation Law of Specimen Stiffness under Positive Unloading (K1)

As shown in Figure7, the slope of line ab is the positive unloading stiffness K1. By sorting out the hysteretic curve data of two specimens, the slope value of line ab under different displacement amplitudes of each specimen can be obtained, and it can be dimensionless. The abscissa is Δ1/Δmax+, and the ordinate is K1/K0+. The scatter diagram of the degradation law of positive unloading stiffness (K1) of each specimen can be obtained, where Δmax+ represents the displacement corresponding to the peak point when the joint specimen is under positive loading. K0+ represents the initial elastic stiffness of the joint under positive loading. After removing the obvious singular points, using the method of nonlinear data fitting, the degradation law curve and its regression formulas of the positive unloading stiffness K1 of each specimen can be obtained, as shown in Figure 8 and equations (2) and (3).Figure 8
Regression curve fitting (K1). (a) SH-1. (b) SH-2.
(a)(b)The regression formulas of K1 are as follows:(2)K1K0+=1.221e0.20164/Δ1/Δmax++0.16819,(3)K1K0+=1.11917e−Δ1/Δmax+/0.31645+1.23644.


## 3.2.2. Degradation Law of Specimen Stiffness under Reverse Loading (K2)

As shown in Figure7, the slope of line bc is the reverse loading stiffness K2. Using the same method as in Section 3.2.1, the degradation law curve and its regression formulas of the reverse loading stiffness K2 of each specimen can be obtained, as shown in Figure 9 and equations (4) and (5).Figure 9
Regression curve fitting (K2). (a) SH-1. (b) SH-2.
(a)(b)The regression formulas of K2 are as follows:(4)K2K0−=e−1.4838Δ2/Δmax+,(5)K2K0−=e0.10061−2.83541Δ2/Δmax++0.8073Δ2/Δmax+2.


## 3.2.3. Degradation Law of Specimen Stiffness under Reverse Unloading (K3)

As shown in Figure7, the slope of line cd is the reverse unloading stiffness K3. Using the same method as in Section 3.2.1, the degradation law curve and its regression formulas of the reverse unloading stiffness K3 of each specimen can be obtained, as shown in Figure 10 and equations (6) and (7).Figure 10
Regression curve fitting (K3). (a) SH-1. (b) SH-2.
(a)(b)The regression formulas of K3 are as follows:(6)K3K0−=e0.57368−0.32551Δ3/Δmax−+0.07056Δ3/Δmax−2,(7)K3K0−=e0.61056−0.49113Δ3/Δmax−+0.14038Δ3/Δmax−2.


## 3.2.4. Degradation Law of Specimen Stiffness under Positive Loading (K4)

As shown in Figure7, the slope of line da is the positive loading stiffness K4. Using the same method as in Section 3.2.1, the degradation law curve and its regression formulas of the positive loading stiffness K4 of each specimen can be obtained, as shown in Figure 11 and equations (8) and (9).Figure 11
Regression curve fitting (K4). (a) SH-1. (b) SH-2.
(a)(b)The regression formulas of K4 are as follows:(8)K4K0+=e0.30267−1.95449Δ4/Δmax−+0.40772Δ4/Δmax−2,(9)K4K0+=e0.28584−1.57987Δ4/Δmax−+0.03819Δ4/Δmax−2.


## 3.3. Description of Hysteresis Rule

The actual restoring force curve on the joint of CTSRC column is hard to describe, so it is necessary to establish a simplified model which can reflect the actual restoring force characteristics. According to the skeleton curve model and stiffness degradation law, combined with the hysteretic curve, the restoring force model of the specimens was established, as shown in Figure12.Figure 12
Restoring force model.The hysteresis rule is described as follows:(1)
When the specimen is under positive loading, the relationship between load and displacement develops along the skeleton curveOABC. If the specimen is unloading in the OA section (the positive loading does not yield), the positive loading stiffness does not degenerate and the relationship between load and displacement returns along the AO section (the unloading line is AO). If the specimen is unloading in the AB section (the positive loading yields), the positive loading stiffness begins to degenerate and the relationship between load and displacement returns along the ab line (the unloading line is ab). If the specimen is unloading in the BC section (the load has passed the peak point), the relationship between load and displacement returns along eb section (the unloading line is eb).(2)
When the specimen is unloading tob point and then loading reversely, the relationship between load and displacement develops along blA'B'C'. If the specimen is unloading at 1A' section (the reverse loading does not yield), the reverse loading stiffness does not degenerate and the relationship between load and displacement returns along A'O line section (the unloading line is A'O). If the specimen is unloading at A'B' section (the reverse loading yields), the reverse loading stiffness will degenerate and the relationship between load and displacement returns along section cd (the unloading line is cd). If the specimen is unloading at B'C' section (the reverse loading has passed the peak point), the relationship between load and displacement returns along fd section (the unloading line is fd).(3)
When the specimen is unloading tod point and then loading again, the relationship between load and displacement develops along d2ABC, and the subsequent loading and unloading is the same as the first loading.


## 4. Verification of Resilience Model

### 4.1. Verification of Skeleton Curve

The skeleton curve model can be obtained by connecting the feature points in front, and the skeleton curve model is compared with the measured skeleton curve, as shown in Figure13. It can be seen that the skeleton curve model can better reflect the load displacement variation law of two joint specimens.Figure 13
Comparison between the model curve and test curve. (a) SH-1. (b) SH-2.
(a)(b)


### 4.2. Verification of Hysteresis Curve

The hysteresis curve drawn by the above hysteresis rule is compared with the experimental hysteresis curve, as shown in Figure14, and the results show that they are consistent.Figure 14
Comparison between the model curve and test curve. (a) SH-1. (b) SH-2.
(a)(b)


## 4.1. Verification of Skeleton Curve

The skeleton curve model can be obtained by connecting the feature points in front, and the skeleton curve model is compared with the measured skeleton curve, as shown in Figure13. It can be seen that the skeleton curve model can better reflect the load displacement variation law of two joint specimens.Figure 13
Comparison between the model curve and test curve. (a) SH-1. (b) SH-2.
(a)(b)


## 4.2. Verification of Hysteresis Curve

The hysteresis curve drawn by the above hysteresis rule is compared with the experimental hysteresis curve, as shown in Figure14, and the results show that they are consistent.Figure 14
Comparison between the model curve and test curve. (a) SH-1. (b) SH-2.
(a)(b)


## 5. Conclusions

Based on the load-displacement hysteretic curve and skeleton curve obtained from the test, we establish a restoring force model suitable for the new type joints. This model mainly consists of three parts: the three-fold line skeleton model obtained from the test data, the specific calculation formula of stiffness degradation, and the mathematical description of hysteretic rules.From the above results, it can be seen that the restoring force model on the joint of CTSRC column has the following characteristics:(1)
Under low cycle cyclic loading, the positive and reverse loading and unloading stiffness of the new type joint is equal to the initial stiffness of the joint in the elastic stage(2)
When the specimen enters the yield stage, the initial stiffness of positive and reverse loading and unloading gradually degenerates, and the degradation in the early stage is more obvious than that in the later stage(3)
Compared with the positive unloading stiffness and reverse unloading stiffness of the specimen, the stiffness degradation of positive loading stiffness and reverse loading stiffness is more obviousThe restoring force model can reflect the hysteretic behavior of the new type joint and can be used for elastoplastic seismic time-history analysis on the joint of CTSRC column.



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*Source: 9021167-2021-09-17.xml*