Hysteresis Model of RC Column Considering Cumulative Damage Effect under Variable Axial Load

Abstract

The axial force will be altered as a result of the overturning influence exerted by both horizontal and vertical seismic events, as well as the secondary effects induced by gravitational loads. The variation of the axial force will greatly affect the seismic performance of reinforced concrete (RC) columns, thus warranting close attention. This paper proposes a hysteresis model of RC columns considering the cumulative damage effect under the action of variable axial force. First, three groups of cyclic loading tests were performed across three distinct groups. Subsequently, numerical analysis models were constructed, employing fiber-based finite element methods. Furthermore, according to the test and finite element simulation results, the existing damage value was modified to describe the degradation of the stiffness and load-bearing capacity. Next, through a regression analysis, the skeleton curve was established. Finally, the hysteresis behavior under the influence of variable axial load was ascertained. The results, when compared with the experimental data, show that the proposed hysteresis model can accurately describe the seismic performance of RC columns under the influence of variable axial force.

1. Introduction

Currently, the seismic performance and analysis models of most RC columns are determined based on cyclic loading tests of piers under constant axial force, which is a reasonable approach for the seismic design of piers with little axial force variation. However, under the combined action of vertical and horizontal seismic motions, the piers are in a complex stress state; seismic analysis and design must also factor in the influence of variable axial loads [1,2]. During an actual earthquake, the axial load may change due to the overturning effect caused by horizontal and vertical earthquakes [3], as well as the second-order effect of gravity load, in addition to the horizontal shear force. Ignoring the impact of variable axial load could lead to a tendency toward unsafe conditions.

Most proposed hysteresis models are based on constant axial loads [4]. A relatively small amount of research has been dedicated to examining the performance of RC columns under conditions of variable axial loads [5,6,7]. However, there is a dearth of research concerning the influence of variable axial loads on the seismic performance of RC columns. Xu and Zhang [5] proposed a restoring force model that can simulate the hysteresis performance of RC members under variable axial loads and horizontal variable loads. The model considered the axial loading and lateral loading separately. Wu et al. [6] concluded that the role of horizontal and vertical ground motions in the fluctuation of axial forces was disclosed by assessing the axial forces acting on the RC frame columns and bridge piers during the influence of near-fault ground motions. Karalar and Dicleli [7] utilized the magnitude of the axial load as a parameter to investigate its effects on the cyclic behavior of jointless bridge steel H-piles. The experimental and numerical analysis outcomes indicated that, for the same levels of cyclic flexural strain amplitudes, steel H-piles designed for weak-axis bending typically exhibit enhanced low-cycle fatigue life when subjected to axial loads, as opposed to those configured for strong-axis bending.

Structures or components can suffer damage during an earthquake due to the cumulative dissipation of energy, resulting in a degradation of seismic performance, including hysteresis properties [8]. Previous studies have mainly relied on displacement-energy two-parameter damage values to build hysteresis models [9]. Deng et al. [10] established a hysteresis model of RC piers in a saline soil environment by introducing damage parameters based on seismic damage, considering the overall structural degradation caused by corrosion. Zheng et al. [11] established five scale models of steel-reinforced high-strength and high-performance concrete frame joints and performed low-cycle repeated loading, and they showed that energy dissipation and deformation of the pier column will be affected under variable axial load. A trilinear load–displacement hysteresis mode considering cumulative damage was proposed, which can reveal the load-bearing capacity and stiffness degradation.

Based on previous experiments [12] and combined with finite element simulation, this paper presents the hysteresis model for RC columns under the action of variable axial forces by using a modified damage value to quantitatively describe the exact impact of dynamic axial forces on seismic performance.

2. Quasi-Static Test of RC Columns under the Action of Variable Axial Load

2.1. Introduction of Test

An experimental study was undertaken to examine the impact of fluctuating axial loads on the seismic responses of RC columns. Thanks to the adequate maximum capacity of the loading apparatus, the size-related effects of the concrete [13] on the dependability of the outcomes were mitigated. Three specimens, namely SC1, SC2, and SC3, were crafted, each comprising an RC column, an upper plate, and a lower plate. The geometrical proportions and steel reinforcement layouts of these specimens are depicted in Figure 1. The three columns tested were exactly the same.

The mechanical properties of reinforcement bars were assessed using MTS testing machinery. After testing [12], the yield strength f_y, ultimate strength f_u, and elastic modulus E_s of longitudinal bars were 450 MPa, 620 MPa, and 2 × 10⁵ MPa. The yield strength f_y, ultimate strength f_u, and elastic modulus E_s of transverse bars were 435 MPa, 607 MPa, and 2 × 10⁵ MPa. Moreover, three concrete cubes with an edge length of 150 mm and three concrete prisms measuring 150 × 150 × 300 mm were cast and cured under the same conditions as the specimens, and then subjected to compressive testing at the end of the experimental phase. The mechanical characteristics of the concrete were ascertained using the YAW2000 device, and the cubic compressive strength f_cu was 45.6 MPa, prismatic compressive strength f_c was 30.6 MPa, and elastic modulus E_c was 3.8 × 10⁴ MPa. The design for each specimen conformed to the specifications outlined in the Chinese standard.

The load tests were carried out utilizing the Multiple Usage Structures Tester (MUST), an integrated system for simulating seismic loads in structural testing, as shown in Figure 2. This testing system facilitates independent three-dimensional loading, enabling the separation of vertical axial forces from horizontal loads without the need to account for tension stress in the concrete. The displacement readings of the test specimens were recorded by employing a linear variable differential transformer (LVDT) and digital image correlation methods. To monitor stress variations in the reinforcement and concrete, strain gauges for longitudinal reinforcement, stirrups, and concrete were strategically positioned at the top and bottom. Detailed specifications of the test loading apparatus and measurement instruments can be found in Figure 3. The testing procedure was divided into two phases: vertical force application and horizontal displacement actuation, with each independently managed and coordinated. SC-1 was subjected to lateral drift loading under a sustained constant axial load. An axial load of 750 kN was imposed prior to testing (axial load ratio 0.2). This aligns with the seismic design specifications for the axial load ratio of actual bridge piers. SC-2 and SC-3 were subjected to the same lateral drift loading as SC-1 under a varying axial load, with the fluctuation frequency of the axial load being in sync with the lateral drift. The axial load imposed was 750 ± 325 kN (axial load ratio of 0.2 ± 0.1). However, the fluctuation frequency of the axial load for SC-3 was twice that of SC-2.

Test	Specimen	Axial load	Horizontal displacement load
Constant axis force	SC-1		Using displacement control, cycling once at displacement amplitudes of 5, 10, and 15 mm, twice at 20~90 mm, and once at 100~160 mm.
Variable axial load test 1	SC-2
Variable axial load test 2	SC-3

Given that the primary focus of this study is on hysteretic models under variable axial loads, the results obtained with SC-2 and SC-3 shall predominantly be analyzed. Upon analysis and computation, the characteristic point information was obtained for SC-2 and SC-3, as shown in

Table 1. Characteristic point information.

	Direction	Δy/mm	Py/kN	Δm/mm	Pm/kN	Δu/mm	Pu/kN
SC-2	+	24.5	217.1	50	250.9	119.6	213.3
	-	33.6	191.2	60	215.4	190	183.1
SC-3	+	18.3	201.8	50	241.3	164	205.1
	-	42	216.4	60	235.2	154.8	199.9

.

2.2. Finite Element Simulation

According to the literature [12], the effect of variable axial load on the seismic performance of RC columns depends on the axial compression ratio at peak displacement. If the axial compression ratio at peak displacement remains the same, the hysteretic characteristic will be similar, resulting in a similar hysteretic curve. Therefore, the seismic performance under variable axial load can be converted into seismic performance under constant axial force for a simplified study. Apart from the axial compression ratio discussed in the numerical simulation, the axial compression ratio mentioned in the subsequent text refers to the equivalent axial compression ratio, aligning with the maximum displacement in both the positive and negative directions.

In order to expand the test samples and obtain the general rules, the finite element simulation of SeismoStruct was used [14,15]. Compared to traditional fixed-mode pushover analysis, the software’s built-in adaptive pushover analysis with displacement-based transformation capacity allows for updating lateral displacement modes based on the evolving system modes. This approach overcomes the limitations of fixed modes in traditional methods, providing more accurate and valid estimates of structural response [6]. The fiber-based finite element framework leverages the unidirectional stress–strain correlation of fibers to ascertain the cross-sectional force–deformation nonlinearity. In the case of RC columns, the numerical simulation incorporated an inelastic force-driven plastic hinge framework element. This hinge was localized within a predefined segment length, commonly designated as 16.67% of the member’s length. The cross-section was dissected into three distinct regions: the confined core concrete, the unconstrained concrete, and the reinforcing steel, each represented by 150 fiber elements. For the simulation of both confined and unconstrained concrete portions, the unilateral nonlinear concrete model (con_ma) [16] was selected, while the uniaxial strain-hardening material model (stl_mp) put forth by Menegotto and Pinto [17] was utilized to represent the steel reinforcement. The circular column model’s elements were distributed as depicted in Figure 4. The material parameters for these models can be found in the literature [12].

The finite element models are subjected to loading analogous to that applied in the experimental procedures. Figure 5 presents a comparison between the simulated and actual hysteretic curves. It is evident that there is a satisfactory correspondence between the simulated and experimental curves.

Previous research on constant axial load has shown that axial compression ratio, shear-span ratio, and reinforcement ratio are key factors affecting the seismic performance of RC columns [18,19,20]. Since concrete with a strength of C40 was used in the test, the same concrete was used in the subsequent numerical simulations. Using the same model as above, vary the magnitude of axial force, the length of the specimen, and the internal reinforcement. Allow the parameters to vary within a specific range in a regular pattern while meeting the requirements for specimen strength and other aspects. Based on the specimens mentioned above, a total of 125 RC columns (n-λ-ρ) with constant axial compression ratio n (=0.1, 0.2, 0.3, 0.4, 0.5), shear-span ratio λ (=3.0, 3.5, 4.0, 4.5, 5.0), and reinforcement ratio ρ (=1.43%, 1.64%, 1.84%, 2.05%, 2.25%) were designed for finite element simulation. The horizontal loading system should remain unchanged while carrying out finite element simulations under constant axial load and the same variable horizontal displacement as the test.

3. Study on Damage Value

3.1. Modification of Damage Value

The current damage values do not consider the effect of variable axial load and, therefore, require modification. The models in [21,22,23] were applied to the test components.

Park-Ang’s damage value is as follows:

$$D={\displaystyle \frac{{\delta }_m}{{\delta }_u}}+{\beta }_e{\displaystyle \frac{{\displaystyle \int dE}}{F_y{\delta }_u}}$$

(1)

where δ_m is the maximum deformation of the structural component under cyclic loading; dE is the energy dissipation increment; F_y is the yield load; δ_u is the ultimate deformation of the component under monotonic loading; and β_e is non-negative parameter.

Chen’s damage value is as follows:

$$D=\left(1-\beta \right){\displaystyle \frac{{\delta }_m}{{\delta }_u}}+\beta {\displaystyle \frac{{\displaystyle \int dE}}{E_{\mathrm{mon}}}}$$

(2)

where E_mon is the hysteresis energy dissipation when the structure or component is damaged under monotonic loading; β is non-negative parameter.

In Zeng’s damage model,

$$D=1-\left(1-{\displaystyle \frac{{\delta }_m-{\delta }_{\mathrm{y}}}{{\delta }_u-{\delta }_u}}\right)/\left(1+{\displaystyle \frac{E_h}{F_y{\delta }_u}}\right)$$

(3)

where E_h is the total area enclosed by the force-displacement loops in the seismic response cycle.

When the damage value (D) = 0, this means that the structure or component remains in its original undamaged state, fully functional; when D ≥ 1, this indicates that the structure or component has suffered complete failure; when 0 < D < 1, this means that damage has started but has not reached the level of complete failure, covering the entire range from slight damage to near-complete failure. In this study, the state in which the bearing capacity decreases to 85% of the maximum bearing capacity is defined as the unity state, and the D is 1 at this time, which means that the structure or component is in an unusable state, but does not imply that it cannot continue to bear loads. When the load exceeds the unity state, the damage will continue to increase, that is, D exceeds 1.

Table 2. Energy dissipation, at different displacement amplitudes, of SC-2 and SC-3.

	Direction	20	30	40	50	60	70	80	90	100	110	130	150	170
SC-2	+	332.9	1037.5	2129.6	3537.5	5134.9	6847.2	8808.3	10,917.2	13,212.7	15,573.2	20,821.4	25,982.9	30,347.8
	-	136.1	759.2	1787.2	3358.1	4831.7	6579.6	8905.6	10,835.7	13,006.2	15,295.6	20,568.5	25,082.7	28,657.9
SC-3	+	78.9	704.9	1701.5	3100.7	4664.6	6401.7	8188.2	10,086.3	12,307.1	14,422.5	19,293.8	23,764.2	27,945.9
	-	744.2	1681.9	3336.7	5563.5	7901.5	10,266.8	12,434.8	15,286.8	18,267.1	20,545.6	26,435.8	32,263.9	36,032.6

shows the energy dissipation of SC-2 and SC-3 at various displacement amplitudes during the test process. Three damage values were evaluated with the experimental data, and the results of the comparison are shown in Figure 6. The methods for modifying the damage values were further analyzed. In Park-Ang’s model, the energy term controls the damage value under large displacements, resulting in an excessively high energy dissipation factor [24]. During the testing, cracks appeared in the pier column when the horizontal displacement exceeded 10 mm, as shown in Figure 7, yet the damage value for this model remains at 0, which is inconsistent with the test results. Zeng’s model therefore deviates from reality. Chen’s model closely approximates the development trend and converges to 1 under large displacements but presents high damage values for the same reason as Park-Ang’s model. Previous studies [25] have shown that the energy dissipation capacity of RC pier columns under large displacements is reduced due to the influence of variable axial load. The energy term in the dual-parameter damage value needs to be reduced. In conclusion, modifying according to Chen’s model involves reducing the coefficient of the energy term to improve the description of energy consumption. The correction formula is as follows:

$$D=\left(1-\beta \right){\displaystyle \frac{{\delta }_{\mathrm{m}}}{{\delta }_{\mathrm{u}}}}+\beta {\displaystyle \frac{k{\displaystyle \int dE}}{Q_y\left({\delta }_u-{\delta }_y\right)}}$$

(4)

Through the same method as that in the literature [26], considering the effects of horizontal displacement, crack length, concrete spalling, and bending of the reinforcement on damage, the damage value of the specimen is judged under different displacement amplitudes, and the correction coefficient k is deduced in terms of a given damage value. The given damage value is shown in

Table 3. Given damage value.

Variable Axial Load Test 1		Variable Axial Load Test 2
Displacement	Damage Value	Displacement	Damage Value
0.0	0.00	0.0	0.00
12.0	0.02	16.0	0.03
24.0	0.10	31.9	0.09
36.0	0.22	47.9	0.17
48.0	0.32	63.8	0.26
60.0	0.43	79.8	0.40
72.0	0.56	95.7	0.53
84.0	0.70	111.7	0.66
96.0	0.83	127.6	0.78
108.0	0.92	143.6	0.89
120.0	1.00	159.5	1.00

.

The modification of the damage values is entirely based on the results of the test, while only two test columns can serve as a reference. Different displacement amplitudes would result in distinct damage values. By means of regression analysis, the value of k was fitted to be 0.82. According to reference [23], the value of β is 0.1.

The modified damage values were validated through experimental data and numerical simulation data. Figure 8 compares the damage values calculated by Chen’s model and the modified damage value with the test damage values. Due to the large number of samples in the numerical simulation, five sets of data were randomly selected for validation.

Table 4. The damage values calculated from the finite element simulation results.

Displacement (mm)	0.5-4.0-1.84		0.4-4.5-1.64		0.3-3.5-2.05		0.2-4.0-1.84		0.1-4.5-1.64
	D+	D−	D+	D−	D+	D−	D+	D−	D+	D−
20	0.22	0.21	0.19	0.19	0.18	0.18	0.12	0.12	0.07	0.07
30	0.33	0.33	0.29	0.29	0.28	0.27	0.18	0.18	0.11	0.11
40	0.45	0.44	0.39	0.38	0.37	0.37	0.24	0.24	0.15	0.15
50	0.57	0.56	0.49	0.49	0.47	0.46	0.31	0.31	0.19	0.19
60	0.69	0.67	0.59	0.59	0.57	0.56	0.37	0.37	0.22	0.22
70	0.80	0.79	0.69	0.69	0.67	0.66	0.44	0.44	0.27	0.26
80	0.92	0.90	0.79	0.79	0.76	0.75	0.50	0.50	0.31	0.30
90	1.04	1.02	0.90	0.89	0.86	0.84	0.57	0.57	0.35	0.34
100	1.16	1.17	0.99	0.99	0.96	0.94	0.63	0.63	0.39	0.38
110	1.28	1.25	1.10	1.09	1.06	1.04	0.70	0.70	0.43	0.42
130	1.52	1.49	1.30	1.29	1.26	1.23	0.83	0.83	0.51	0.50
150	1.76	1.73	1.51	1.50	1.46	1.43	0.96	0.96	0.59	0.58
170	2.00	1.96	1.72	1.70	1.66	1.62	1.09	1.09	0.67	0.66
190	2.24	2.20	1.92	1.91	1.86	1.81	1.23	1.22	0.75	0.74
Unity state	0.98	0.97	0.96	0.96	0.99	0.98	0.99	0.99	1	1

shows the results of these numerical simulations using the modified damage value calculation. It can be found that this damage value satisfies the general requirements for damage values, including monotonic increase and convergence of damage values to the state of complete failure. Additionally, the modified damage value was found to be closer to the experimental damage value than Chen’s.

3.2. Establishment of Relationship between Stiffness Degradation and Damage Value

Existing experimental studies have demonstrated [22] that the performance degradation of components relates not only to the maximum deformation, but also to the components’ responses under cyclic loads. The frequency of occurrences is of paramount importance. Thus, this article is dedicated to examining the impact of load cycling on. The cyclic stiffness degradation is due to the component coefficient’s performance impairment. Accordingly, the stiffness degradation coefficient γ is defined as follows:

$${\gamma }^{\pm }={\displaystyle \frac{k_{ui}^{\pm }}{k_0^{\pm }}}$$

(5)

where k_ui is the unloading stiffness when a certain displacement amplitude is unloaded for the first time, k₀ is the initial stiffness, and ± represents the positive and negative directions.

Figure 9 shows the relationship between the stiffness degradation coefficient and the damage value. Firstly, from the hysteresis curve obtained from numerical simulation, the skeleton curve and the coordinates of the yield state (Δ_y, P_y), the ultimate state (Δ_u, P_u), and the energy dissipation of each hysteresis loop can be observed. The method discussed above will be used to calculate the damage value (D) of the component after this cycle. Secondly, during the unloading process, the residual deformation after unloading (horizontal coordinate Δ_ri of the blue point) and the load value before unloading (vertical coordinate P_i of the red point) can be read from the graph. Finally, the unloading stiffness k_ui = P_i/(Δ_i − Δ_ri), Δ_i is the amplitude of the horizontal displacement for each cycle determined by the loading system. P_y/Δ_y represents the initial stiffness k₀, and the stiffness degradation coefficient γ = k_ui/k₀. Thus, a functional relationship between γ and D is established. Fifteen data sets of results were selected for further analysis. It can be observed that the degradation forms of positive and negative stiffness are the same, and the degree of degradation is similar. When D approaches 0, γ approaches 1. As D increases, γ gradually decreases and the rate of change slows down, converging to a certain value. This is consistent with the characteristics of exponential functions. Based on this, the functional relationship between the two is established, the fitting results are presented in Figure 9, and the fitting formula is as follows:

$${\gamma }^{\pm }=0.6\times {\exp }^{\left[-2.75\times \left(D^{\pm }-D_Y^{\pm }\right)\right]}+0.4$$

(6)

where D_Y is the damage value that reaches the yield state in a certain direction. When D^± < $D_Y^{\pm }$, D^± takes $D_Y^{\pm }$.

An analysis of Equation (6) reveals that when the damage value in a specific direction is less than the damage value corresponding to the yield state in that direction, the stiffness does not degrade before the component is loaded to the yield state, and the unloading stiffness is equivalent to the initial stiffness (γ = 1). From the transformation of Equation (5), it can be seen that the unloading stiffness of any hysteresis in a specific direction can be expressed as the product of the initial stiffness and the stiffness degradation coefficient, which is shown in Equation (7). Since γ can be represented by D, this allows for the quantitative description of stiffness degradation using the damage value.

$$k_{ui}^{\pm }=k_0^{\pm }\times {\gamma }^{\pm }$$

(7)

3.3. Establishment of the Relationship between Load-Bearing Capacity Degradation and Damage Value

Component load-bearing capacity degradation can be considered according to two methods [26], based on a hysteresis model in which the reloading curve points towards the maximum load state. The second method, shown in Figure 10, is more numerous. The adjustment of the skeleton curve reflects the change in the load-bearing capacity of the structure after experiencing cyclic loads. Wang and Shah [27] established the relationship between load-bearing capacity degradation and the damage value by changing the skeleton curve of the hysteresis model to account for load-bearing capacity degradation.

Figure 11 shows a partial loop in the hysteresis curve, where the red dots represent the points of the maximum bearing load capacity for each cycle (P_i, P_i+1), the black lines represent the strengthening section (k_s) of the bearing load capacity, and the green dots represent the displacement amplitude of the previous cycle corresponding to the load capacity of the current cycle (P′_i, P′_i+1). Δ_i and Δ_i+1 represent the horizontal coordinates corresponding to P_i and P_i+1. Δ_ri represents residual deformation of the specimen in the current cycle. The ratio of P′ to P is shown in Figure 12. As the damage progresses and the cycle advances, this ratio also changes. However, to simplify the analysis, 0.96 is taken as the coefficient of variation. The load-bearing capacity degradation is expressed as the degradation of the stiffness of the strengthening section, and the load-bearing capacity degradation coefficient η is defined as

$${\eta }^{\pm }={\displaystyle \frac{k_{si}^{\pm }}{k_p^{\pm }}}$$

(8)

where k_si is the slope of the strengthening section, and k_p is the line between the peak load state and the yield state.

The damage value was calculated and the corresponding data were extracted from the hysteresis curve, as described in the previous section. The results are shown in Figure 13. It is evident that the change in load-bearing capacity degradation coefficient gradually decreases with an increase in the damage value. Positive- and negative-intensity degradation forms exhibit similar degradation degrees, and a clear logarithmic function relationship exists between the load-bearing capacity degradation coefficient and the damage value. Based on this, a functional relationship is established, and the fitting results are presented in Figure 13. The fitting formula is as follows:

$${\eta }^{\pm }=-0.37\ln \left(D^{\pm }\right)+0.08$$

(9)

where D^± > $D_Y^{\pm }$, otherwise the component is not loaded to the yield state, and there is no strengthening section.

From the transformation of Equation (8), the derivation of the maximum load value from the current hysteresis to the next hysteresis can be achieved. As shown in Figure 11, P_i+1 can be calculated as follows:

$$P_{i+1}=k_p\times \eta \times \left({\Delta }_{i+1}-{\Delta }_i\right)+0.96P_{\mathrm{i}}$$

(10)

4. Establishment of Hysteresis Model

4.1. Determination of Skeleton Curve

The skeleton curve was obtained from the test’s hysteresis curve and then normalized to the peak point to obtain a dimensionless skeleton curve, as shown in Figure 14.

From the beginning of the loading until the formation of the cracks, the development of the yielding of the tensile reinforcement is the elastic stage. After the yielding of the tensile reinforcement, as the load increases, the structure enters the plastic development stage, and the stress distribution in the compression zone becomes fuller, marking the formation and rotation of plastic hinges. From the yielding of the longitudinal tensile reinforcement to the compression and bending failure, the concrete in the compression zone may crush or even spill, as shown in Figure 15. Consider the specimen failure when the ultimate bearing capacity reaches 85% of the maximum bearing capacity. Three key characteristic points may be defined along these stages’ corresponding load transformation curves. The skeleton curve is considered to be a three-line type, as shown in Figure 16, where characteristic point Y represents the yield state (Δ_Y, P_Y), characteristic point M represents the maximum load capacity state (Δ_M, P_M), and characteristic point U represents the ultimate state (Δ_U, P_U).

4.2. Calculation of Characteristic Points (Y, M, U)

Zhang et al. [28] fitted the expressions of each characteristic point on the skeleton curve into relevant functions of the axial compression ratio, shear-span ratio, and reinforcement ratio. Using the same method as described in that article, this paper fits the expressions of characteristic points into relevant functions of the axial compression ratio, shear-span ratio, and reinforcement ratio. The axial stress ratio used is the equivalent axial compression ratio [12], which is the magnitude of the axial stress ratio corresponding to the peak displacement. Through a regression analysis of the finite element simulation results, it was found that there are exponential or linear relationships. These relationships are summarized in Figure 17.

After multivariate nonlinear fitting analysis, the following results were obtained:

• The yield load and displacement to correspond to the yield load:

$$P_Y=n^{0.16}\times {\lambda }^{-1.1}\times \left(368\rho +545\right)$$

(11)

$${\Delta }_Y=\left(-0.3n+1.2\right)\times {\lambda }^2\times {\rho }^{0.6}$$

(12)

2.

The peak load and displacement to correspond to the peak load:

$$P_M=n^{1.8}\times {\lambda }^{-1.1}\times \left(397\rho +650\right)$$

(13)

In the process of the finite element simulation, it was found that the peak displacement Δ_M is only related to the shear-span ratio and does not show an apparent functional relationship, which is a stepwise increase with the increase in the shear-span ratio, as shown in Figure 18 (each interval is closed on the left and open on the right). For the specimen in the test, the peak displacement Δ_M is 40 mm.

3.

The ultimate bearing capacity and displacement correspond to the ultimate bearing capacity:

$$P_U=0.85P_M$$

(14)

$${\Delta }_U=n^{-0.84}\times \left(0.4\lambda +8\right)\times \left(1.9\rho +1.1\right)$$

(15)

4.3. Determination of Hysteresis Rules

By considering the stiffness and load-bearing capacity degradation of the RC column under the variable axial load and the analysis of its characteristic points, the proposed hysteresis model is obtained, as shown in Figure 19. The hysteresis rules are as follows:

• When the specimen is loaded without reaching the yield state of the column, the loading and unloading paths are along the elastic section of the skeleton curve (OA and OD);
• After the force exceeds the yield strength, the loading path travels along the skeleton curve (AB). The unloading stiffness (BC, EF, HI, KL) from the skeleton curve is determined by Equation (7). The skeleton curves are calculated from the axial compression ratio, shear span ratio, and reinforcement ratio. Furthermore, the initial stiffness k₀ and the parameters required to calculate the stiffness degradation coefficient γ are obtained.
• Reversed load cycles and reloading paths: The stiffness of the reloading and reversed load cycle sections at the initial stage is equivalent to the unloading stiffness of this stage. The inflexion point is located on the descending section of the skeleton curve that passes through the origin and is parallel to the current direction (the slope of ON and OL equals the calculated positive and negative skeleton curve strength descending segment slopes. The skeleton curves of the RC columns under variable axial loads are often asymmetrical, resulting indifferent slopes of strength descending segment). When the reverse load does not reach the yield state of the column in the other direction, the reversed load cycle path points from the inflexion point to the yield point in that direction (CD). When the displacement amplitude loaded exceeds the displacement amplitude of the yield state in the current loading direction, the loaded reference point needs to be distinguished and discussed. (i) If n ≥ 0.3 (take F-G-H as an example), first, it points to G (Δ_B, 0.96P_B) during loading, and then it points to point H (Δ_H, 0.96P_B + k_P × η × (Δ_H − Δ_B)). The x-coordinate of H is the displacement amplitude corresponding to the current cycle, and the y-coordinate is the bearing capacity value calculated by Formula (10) on the skeleton curve. (ii) If n < 0.3 (take N-J-K as an example), first, it points to J (Δ_E, 0.96P_E) during loading, and then it points to K, NJ and JK are on the same line, and point K is on the skeleton curve.
  The hysteresis rule is explained with the route A-I, where i represents A-F, i + 1 represents F-I, and +,- represent positive and negative directions.

  Loading path	Bearing capacity	Unloading path	Unloading stiffness/
  O→A	Py+	/	/
  A→B	Py+ + (Δi − Δy+) × ks+	/	/
  /	/	B→C until ON	γi × k0+
  C→D	Py-	/	/
  D→E	Py- + (Δi − Δy-)×ks-	/	/
  /	/	E→F until OF	γi × k0−
  F→G	0.96PB	/	/
  G→H	ks+ × η × (Δi+1 − Δi) + 0.96PB	/	/
  		H→I until ON	γi+1 × k0+

• When the displacement amplitude in a particular direction matches the displacement amplitude of the previous hysteresis loop, the unloading stiffness is equivalent to that of the previous loop (MN has the same unloading stiffness as the HI). The inflexion point of reversed load cycles and reloading points to the point where the maximum displacement amplitude of the previous cycle in this direction is 0.97 (through calculation, the peak bearing capacity at the second occurrence of a certain displacement amplitude was found to be approximately 0.97 times the peak bearing capacity at the first occurrence of that displacement amplitude) times the maximum load value of the previous cycle (P_M = 0.97P_H).

When studying the axial compression ratio as a variable, the range of its change was found to be 0.1–0.5. Under high axial compression ratios (≥0.9), the linearity of the skeleton curve and hysteresis characteristics will be different, and there will be significant errors in using the hysteresis model proposed in this paper.

4.4. Test-Based Validation

The above calculation model and test component parameters were utilized to calculate the characteristic points of the specimen’s skeleton curve. The resulting skeleton curve was compared to the test results, as shown in Figure 20. The parameters required for calculating the skeleton curve are shown in

Table 5. Parameters required for calculating the skeleton curve.

	Direction	n	l	r
SC-2	+	0.3	3.75	1.84
	-	0.1	3.75	1.84
SC-3	+	0.2	3.75	1.84
	-	0.2	3.75	1.84

. It was observed that the calculated values were in good agreement with the experimental values, and the skeleton curve fitted well.

The above-established hysteresis model was used to verify the hysteresis curve of the experimental RC column, as shown in Figure 21. As can be seen from the figure, the hysteresis model established in this paper is in good agreement with the experimental hysteresis curve in terms of stiffness, strength degradation, and other hysteresis characteristics. Therefore, the hysteresis model enables an effective prediction of the hysteresis characteristics of RC piers under the action of variable axial load.

5. Conclusions

• From the results of the numerical simulations, it can be inferred that the degradation of stiffness and strength follows an exponential or logarithmic function concerning damage development. Therefore, quantitatively describing the degradation of stiffness and strength is feasible and relatively accurate. Introducing stiffness and strength degradation functions into the hysteresis rule can effectively predict the seismic performance of RC pier columns under variable axial loads.
• According to the results of the numerical simulation, the characteristic points of the skeleton curve show power or linear relationships with the axial compression ratio, shear span ratio, and reinforcement ratio, and specific functional expressions can be obtained through a regression analysis. Specifically, the displacement at the peak load point only has a segmented increasing relationship with the shear span ratio. The equivalent axial compression ratio corresponding to the peak displacement is determined by analyzing the loading system of the horizontal displacement. By analyzing and calculating the basic parameters of the specimen, the shear-span ratio and reinforcement ratio can be determined. Subsequently, the skeleton curve of the specimen can be obtained.
• By integrating the characteristic points of the skeleton curve and introducing damage values to describe the hysteresis pattern of stiffness and strength degradation, this paper proposes a hysteresis model that can effectively reflect the seismic performance of RC piers under variable axial loads. However, due to the lack of comparable samples, its accuracy and scope of application still have certain limitations, and more reliable variable axial load tests need to be carried out for verification.