Experimental and numerical investigation on seismic performance of corroded RC columns of low-strength concrete

A huge number of reinforced concrete structures designed without aseismic consideration exist at the risk of both reinforcement corrosion and earthquake. They were generally made of low-strength concrete and loaded with a relatively large compression ratio. A total of 11 RC columns of low-strength concrete with different corrosion degrees were obtained by the impressed current method, and low-cycle repeated loading tests were carried out on the columns with axial compression ratios ranging from 0.25 to 0.75. The seismic performance of corroded RC columns of low-strength concrete was analyzed through a discussion of the impact of reinforcement corrosion on failure modes, hysteretic behavior, skeleton curve, and energy dissipation capacity. For a more comprehensive understanding of the seismic performance of corroded RC columns, a numerical model based on the multi-spring model was developed by introducing a restoring model of corroded rebars. The numerical simulation results showed a good agreement with experimental results. With the verified numerical model, the influence of concrete strength, reinforcement ratio, and axial compression ratio on the impact of reinforcement corrosion on the seismic performance of RC columns was further investigated. It is found that the seismic performance of RC columns decreases rapidly with the increase of axial compression ratio and corrosion degree. As the axial compression ratio increases, the impact of reinforcement corrosion on the seismic performance is gradually weakened owing to the decreased contribution of longitudinal rebars to the load-carrying capacity. For similar reasons, the impact of reinforcement corrosion is more obvious in RC columns of low-strength concrete with a larger reinforcement ratio.


Introduction
In practical engineering, a considerable number of existing reinforced concrete (RC) structures suffer from performance deterioration caused by reinforcement corrosion during their lifetime. Reinforcement corrosion leads to the reduction of cross-sectional areas (Zhang et al. 2014;Jiang et al. 2022), degradation of mechanical properties (Cairns et al. 2005), cracking or spalling of concrete cover (Chen et al. 2020;Zhang et al. 2019) and degradation of bond behavior between rebars and surrounding concrete (Zhang et al. 2016Fang et al. 2006;Almusallam et al. 1996;Tahershamsi et al. 2017). Meanwhile, many coastal regions are located in tectonically active areas, and deterioration of the seismic performance of marine structures will inevitably increase the risk of economic loss and casualties caused by earthquake events (Alipour et al. 2011;Sharma et al. 2016;Kaushik and Jain 2007). Therefore, it is of paramount importance to investigate the effects of reinforcement corrosion on the seismic performance of RC structures.
RC frames are one of the main structural types of multi-story and high-rise buildings (Zheng et al. 2022), in which columns mainly resist horizontal actions such as wind or earthquake, and reinforcement corrosion of columns will adversely affect the seismic performance of RC frames and may even lead to an overall collapse (Dai et al. 2021). A gradual reduction in nominal strength, a dramatic reduction in elongation, and brittle fracture with the shortened or even disappeared yielding plateau were observed in the tensile tests of corroded rebars (Zhang et al. 2012). It can be derived that reinforcement corrosion should have a detrimental impact on the seismic performance of RC structures. Therefore, experimental studies of corroded RC columns were abundantly conducted during the past few years since they provide easier and more direct investigations on the structural behavior of corroded RC buildings (Dai et al. 2021;Meda et al. 2014;Ma et al. 2012Ma et al. , 2018Yalciner and Kumbasaroglu 2020;Celik et al. 2022;Dai et al. 2020a, b;Rajput and Sharma 2018;Guo et al. 2015;Yuan et al. 2017a, b;Yuan et al. 2018;Deng et al. 2022;Zhou et al. 2020;Zhao et al. 2021;Koçer et al. 2021;Yang et al. 2016;Chang et al. 2020;Rinaldi et al. 2022;Li et al. 2018;Goksu and Ilki 2016). For instance, Meda et al. (2014) performed cyclic loading tests on two full-scale column specimens to investigate the influence of corrosion on the structural behavior and revealed that corrosion of rebars causes a significant reduction in structural ductility. Ma (2012), Yalciner (2020), Dai (2020a), and Rajput (2018) investigated the degradation of seismic performance for RC columns subjected to different corrosion degrees and axial compression ratios through a series of pseudo-static tests. The above experimental results highlighted that the seismic performance of corroded RC columns, especially the strength, ductility, and energy dissipation capacity, decreased significantly with the increase of corrosion degree and axial compression ratio. An obviously brittle failure had been observed in severely corroded RC columns. In addition, numerous numerical methods and modeling guidelines have been proposed for assessing the seismic performance of corroded RC columns (Deng et al. 2022;Chang et al. 2020;Carlo et al. 2017;Xu et al. 2020Xu et al. , 2022aXu et al. , b, 2021Dai et al. 2020a, b;Vu et al. 2016;Biswas et al. 2021;Cui et al. 2019;Yuan et al. 2017a, b), manifesting the consistent conclusions with the experimental results.
The axial compression ratio is a crucial parameter for columns, which determines the seismic capacity and failure mode of corroded RC columns (Vu and Li 2018a, b). As revealed in the abovementioned test results, with the increase of axial load, the failure mode of corroded columns changed with the decreased ductility and energy dissipation. The axial compression ratio of columns in many existing buildings constructed before the 1 3 issue and adoption of seismic design code often exceeds 0.6 or even much larger due to the use of low-strength concrete (Celik et al. 2022;Huan et al. 2020;Bru et al. 2018). However, in previous experimental research, most of the corroded RC columns were designed with low axial compression ratio, e.g., 0.1 in Guo et al. (2015), 0.13 in Yuan et al. (2017a, b) and Rinaldi et al. (2022), 0.17 in Zhou et al. (2020) and 0.2 in Goksu et al. (2016). To the authors' best knowledge, few experiments have been conducted previously on the hysteresis behavior of corroded RC columns with large axial compression ratios. Besides that, the concrete strength is higher than 25 MPa in most studies. It remains to be discussed whether the experimental results are applicable to existing RC columns of low-strength concrete with a large axial compression ratio. Therefore, there is a vital need for investigating the seismic performance of corroded RC columns with a large axial compression ratio.
An experimental investigation was presented in this paper on the seismic behavior of corroded RC columns of low-strength concrete with a relatively large axial compression ratio. The impressed current method was conducted to accelerate the corrosion of longitudinal rebars, and cyclic loading tests were carried out on 11 RC column specimens with different corrosion degrees, axial compression ratios, and stirrup spacings. A restoring force model for corroded RC columns was thereafter developed based on the multi-spring model considering damage accumulation, and parameter analysis was performed with the verified model.

Specimen design and fabrication
In order to study the seismic performance of corroded RC columns with different axial compression ratios and stirrup spacing, 11 specimens were designed and fabricated. As shown in Table 1, four target corrosion degrees, three axial compression ratios, and three spacings of stirrups were considered. In this study, the stirrups were expected to be uncorroded and the target corrosion degree in Table 1 refers to that of longitudinal reinforcements. The corrosion degree refers to the ratio between the mass loss caused by corrosion and the original mass of the uncorroded one. All specimens were inverted T-shaped and consisted of a column of 200 × 200 × 700 mm and a base beam of 300 × 300 × 800 mm, as shown in Fig. 1. Each column was reinforced with six longitudinal rebars and stirrups with different spacing, and each base beam was reinforced with eight longitudinal rebars and stirrups with a spacing of 80 mm. Deformed rebars with a diameter of 12 mm were used as longitudinal rebars, and plain rebars with a diameter of 6 mm were used as stirrups. The longitudinal rebars extended 100 mm beyond the top of the columns to connect with the electric wires during accelerated corrosion and with the horizontal loading device during the test. The tested yield strength and ultimate strength of deformed rebars were 298 MPa and 461 MPa, respectively.
Considering there are a large number of existing RC frame buildings made of concrete with relatively low strength, concrete with a cubic strength of 15 MPa was used in the test. The cement, water, sand, and, coarse aggregate proportions by weight were 1: 0.7: 2.96: 4.07. To depassivate the steel bars, 2% of sodium chloride by weight of cement was added to the mixtures.

Accelerated corrosion of RC columns
The impressed current method was utilized to accelerate the corrosion of longitudinal rebars. As shown in Fig. 2, the columns were partially immersed in a solution of 5% sodium chloride with longitudinal rebars approximately 15 mm above the solution surface. Insulating tape and epoxy resin were placed at the intersections between stirrups and longitudinal rebars to prevent stirrups from being corroded. Similar insulation measures were taken to prevent longitudinal rebars in the anchorage region from being corroded. The longitudinal rebars were connected to the positive electrode of a DC power supply as the anode, while the copper strip was connected to the negative electrode as the cathode. The implied current density was kept at 100 μA/cm 2 . According to Faraday's law (El Maaddawy and Soudki 2003;Almusallam 2001;Du et al. 2005), durations of impressed current were determined according to the target corrosion degrees of longitudinal rebars in RC columns listed in Table 1.
After the cyclic loading test, the columns were crashed and the corroded rebars were retrieved and collected. All of the corroded rebars underwent acid cleaning, drying, and weighting according to ASTM G1-03 (ASTM 2003). The mass loss ratio, i.e., the average corrosion degrees were calculated and given in Table 1.

Test setup and loading protocol
The cyclic loading tests of all columns were carried out on a large multifunctional structural experiment system (LMSES). In order to ensure that the top joints of half-column specimens can simulate the features of the inflection point in full-column specimens, all specimens were connected to the LMSES with a well-designed pin-hinge device at the top of the columns (Fig. 3a). The pin-hinge device consists of an L-shaped loading adapter and a steel base connected to the column specimen. As illustrated in Fig. 3a, the bottom of the pin-hinge device is connected to the steel plate on the top of the column by bolts. The vertical and the horizontal ends of the L-shaped loading adapter were connected with the vertical and horizontal loading device of the LMSES, respectively. Through the pin-hinge device connection, it can be considered that the pin-hinge device and the half-column specimen jointly formed an equivalent column. The measured distance from the top of the column to the center of the pin hinge was 340 mm, i.e., the total height of the equivalent column was 1040 mm.
It is noteworthy that the equivalent cross-sectional flexural stiffness of the pin-hinge base at the top of the column is approximately 8 times that of the column, and the corresponding lateral stiffness is about 1.03 of that of an idealized column with the equal crosssection along its length. The simplified analytical diagram is shown in Fig. 4. Even if the cross-sectional flexural stiffness of the pin-hinge base is assumed to be infinity, the corresponding lateral stiffness is only about 1.036 of the idealized column. Thus, the idealized column is used in the analysis of experimental results.
The vertical load was first applied to the specimen and kept unchanged. Then, the lateral cyclic loads were imposed on the specimens via the LMSES, according to the displacement-control loading protocol shown in Fig. 5. As illustrated in the figure, a loading cycle with a specific displacement was applied before the strain of the rebar reached the yielding strain, while three loading cycles with a specific displacement were applied after yielding until the column failed. The failure was declared if the load-carrying capacity dropped to 85% peak load or the concrete in plastic hinge region was crushed. In the loading process, the strains of longitudinal rebars were measured by the strain gauges, and the lateral displacement at the loading point was measured by the displacement transducer as shown in Fig. 3a. Figure 6 shows the damaged column specimens after loading. Characteristic displacements and loads of all specimens are summarized in Table 2.   Columns C1-1-C1-3 were uncorroded specimens with axial compression ratios of 0.25, 0.5, and 0.75, respectively. Concrete in the plastic hinge region of C1-1-C1-3 columns were crushed and longitudinal rebars were yielded and even buckled, and the failure modes changed from large-eccentricity compression failure to small-eccentricity compression failure with the increase of axial compression ratio (Fig. 6a-c). Before longitudinal rebars yielding and concrete crushing, horizontal cracks appeared and propagated on column C1-1 from the bottom upwards as the load increased, while fewer horizontal cracks were found in columns C1-2 and C1-3. The ultimate displacement when the specimen is destroyed of the C1-1 specimen was 56 mm with a ductility coefficient of 7.79, showing good ductility. Due to the larger axial compression ratio, C1-2 and C1-3 exhibited a brittle failure trend, with ultimate displacements of only 24 mm and 9 mm, and ductility coefficients of 6.8 and 1.86, respectively.

Failure modes of columns
The axial compression ratio of columns C2-1-C2-4 was 0.5, and the measured average corrosion degrees were 0.062, 0.063, 0.181, and 0.192, respectively. As the corrosion degree increased, the bond behavior between longitudinal rebars and concrete deteriorated. Compared with uncorroded column C1-2, the first horizontal crack occurred later, and a larger space was observed. The yielding and ultimate load/displacement and ductility coefficient decreased with the increasing corrosion degree, since the effective cross-sectional area, nominal yield strength, ultimate strength, and ultimate strain decreased significantly. All the columns failed with a mode of large-eccentricity compression.
Columns C3-1, C2-3, and C3-2 had a similar corrosion degree and different axial compression ratios of 0.25, 0.5, and 0.75. Because of the severe corrosion, the occurrence and development of horizontal cracks were relatively slow. The larger the axial compression ratio is, the fewer the horizontal cracks are. There is a big difference in failure mode between columns C3-1 and C3-2. Although column C3-1 had an average corrosion degree above 0.16, a typical large-eccentricity compression was observed with good ductility due to its low axial compression ratio (Fig. 6h). Due to the combined effects of severe corrosion and large axial compression ratio, brittle compression-shear failure occurred in column C3-2 (Fig. 6i). It was found that some stirrups in column C3-2 corroded severely, despite the insulation measures taken at the intersection between longitudinal reinforcement and stirrup. Before the failure, the longitudinal cracks of concrete in the middle of the column developed rapidly. Because of the severe corrosion of the stirrups, the stirrups lost their confinement on concrete and longitudinal rebars. The core concrete was immediately crushed and the longitudinal rebars were buckled. On one hand, the contribution of concrete to the resistance of vertical load increased due to the severe corrosion of longitudinal rebars; on the other hand, the compression performance of core concrete degraded due to the weakened confinement effects of corroded stirrups. Therefore, although the shear span  Table 2 List of test results η is corrosion degree; Δ cr , Δ y , and Δ u are cracking, yield, and ultimate displacement, respectively; P cr , P y , and P u are cracking, yield, and ultimate strength, respectively; λ is displacement ductility ratio; E is energy dissipation; l p is plastic hinge length; " + " and "−in the brackets represent the reinforcement yielded and the edge of the concrete in the compression zone reached its ultimate strain, respectively Specimen η Δ cr (mm) 0.78 -ratio of the specimen reaches 5.2, compression shear failure, which usually occurs in columns with a small shear span ratio, was observed in column C3-2. Columns C2-3, C4-1, and C4-2 were designed to have the same target corrosion degree of 0.15 and different stirrup spaces ranging from 100 to 200 mm. However, their measured average corrosion degrees were 0.181, 0.117, and 0.205, respectively, having obvious difference for randomness and some other reasons. Column C4-1 failed in small-eccentricity compression failure with few horizontal cracks, and the plastic hinge region was located between the first stirrup and the second stirrup above the column base (Fig. 6j). Column C4-2 failed in shear with an oblique crack and a fractured stirrup in the cracking zone due to the large stirrup spacing (Fig. 6k).

Load-displacement hysteresis curves
Load-displacement hysteretic curves of the columns are shown in Fig. 7. In general, the shape of the hysteretic curves of corroded columns are similar to those of uncorroded columns.
For the columns under the same axial compression ratio, as the corrosion degree increases, the shape of the hysteretic loop changes slightly but the number of hysteretic loops decreased gradually. The decrease for columns with a small axial compression ratio is more significant than those with a large axial compression ratio. As the axial compression ratio increases, for either uncorroded columns (C1-1, C1-2, and C1-3) or corroded columns (C3-1, C2-3, and C3-2), the number of hysteresis loops decreases rapidly and the seismic performance deteriorates promptly. Both uncorroded column C1-3 and corroded C3-2 have an axial compression ratio of 0.75, and the almost plastic stage was not observed in their hysteresis curves. With the increase of stirrup spacing, the number of hysteresis loops decreases prior to failure, and the failure mode may change from compression-flexure failure to shear failure.

Load-displacement skeleton curves
Based on the load-displacement hysteresis curves in Fig. 7, the maximum load point in each loading cycle was used to construct the load-displacement skeleton curves, as shown in Fig. 8.
The load-displacement skeleton curves for uncorroded columns and corroded columns with different axial compression ratios are compared in Fig. 8a and b, respectively. With the increase of the axial compression ratio, the horizontal load-carrying capacity and deformation capacity decrease rapidly. For uncorroded columns, the load-carrying capacity and ultimate displacement of column C1-3 with an axial compression ratio of 0.75 were 81% and 16% of column C1-1 with an axial compression ratio of 0.25, respectively. For columns with an average corrosion degree of around 0.16, the load-carrying capacity and ultimate displacement of column C3-2 with an axial compression ratio of 0.75 were only 52% and 12% of that of column C3-1 with an axial compression ratio of 0.25. Figure 8c, d and e shows the load-displacement skeleton curves for three sets of columns with different corrosion degrees under the same axial compression ratio. It is shown that the horizontal load-carrying capacity and deformation capacity of the columns decreased as expected with the increase of corrosion degree. For columns with an axial compression ratio of 0.25, the load-carrying capacity and ultimate displacement of column C3-1 with the average corrosion degree of 0.166 were 73.1% and 66.0% of 1 3 uncorroded column C1-1, respectively. For columns with an axial compression ratio of 0.50, the load-carrying capacity and ultimate displacement of column C4-1 with the average corrosion degree of 0.117 were 68.8% and 51.5% of uncorroded column C1-2, respectively. For columns with an axial compression ratio of 0.75, the load-carrying capacity and ultimate displacement of column C4-2 with the average corrosion degree of 0.205 were 78.0% and 51.2% of uncorroded column C1-3, respectively.  Figure 8f shows the load-displacement skeleton curves for columns with different stirrups spacing. It can be seen that the confinement effect of stirrups on core concrete degrades with the increase of stirrup spacing, leading to a decrease in load-carrying capacity, ultimate displacement, and potential failure mode transformation. Compared with column C2-3 with a stirrup spacing of 100 mm, the load-carrying capacity and ultimate displacement of column C4-1 with a stirrup spacing of 150 mm decreased by 15.4% and 28.0%, respectively; and those of column C4-2 with a stirrup spacing of 200 mm decreased 1 3 by 15.9% and 49.8%. The reduction in deformation capacity is much more obvious than that in lateral loading capacity.

Energy dissipation capacity
The area enclosed by a single hysteresis loop is always used to evaluate the energy dissipation capacity of a column, as shown in the shaded part of Fig. 9. The accumulative The calculation results of the accumulative energy dissipation capacity of columns are presented in Table 2 and Fig. 10. It can be found that the energy dissipation capacity decreased with the increase of corrosion degree, and the impact of reinforcement corrosion decreased with the increase of axial compression ratio. (1)

Multi-spring model for simulation of the load-displacement relationship of RC columns
A multi-spring model consists of a set of axial springs representing the stiffness of rebars and concrete material (Li and Otani 1993). Lai et al. (1984) defined the elastic element and plastic element of components. In elastic-plastic analysis, the column element model was composed of an elastic rod element and a plastic element with no length at the end, as shown in Fig. 11a. In this model, the plastic deformation of a column is concentrated in the plastic element at the bottom, simulating the plastic hinge region. The assumption of the plane section in the plastic element is still valid. The plastic element at the bottom of the column is composed of several rebar springs and concrete springs with no length as shown in Fig. 11b. The number and position of rebar springs and concrete springs can be determined according to the shape of the crosssection and the requirements of the model. Assuming that no shear failure occurs to the RC column, based on the research of Li and Otani (1993), the spring in plastic element concentrates the deformation of reinforcement and concrete within the assumed plastic hinge length ηh (h is the height of the column section, η equals to 1.5). The elastic element of the middle section is considered as a completely elastic body. A typical division of a cross-section is shown in Fig. 12, and there are three kinds of springs, including confined core concrete spring, unconfined concrete spring, and rebar spring. The initiation and propagation of rebar corrosion may induce cracking and even spalling of concrete cover. The damage to the concrete induced by the corrosion expansion cracks was mainly concentrated in the concrete cover, thus only the reduction in the area of unconfined cover concrete is considered. An efficient thickness of concrete cover can be obtained to consider the effects of corrosion-induced cover cracking. When a partial concrete cover spalls, the corresponding springs are withdrawn from work according to the actual situation. The analytical diagram of the load-displacement relationship for columns is schematically shown in Fig. 13. In the Multi-spring model, the displacement of the column top can be divided into two parts, which can be calculated as follows: (2) where P is the horizontal force; N is the vertical force; M is the bending moment at the bottom of the column subject to both the horizontal force and vertical force; E c is the elastic modulus of concrete; I e is the equivalent inertia moment of the cross-section considering the contributions of rebars; Δ 1 is the plastic displacement; Δ 2 is the flexure elastic displacement. In the process for calculating the load-displacement relationship, the rotation of the column bottom is given firstly, and then calculate the horizontal displacement (Δ 1 + Δ 2 ) (by Eq. (2) and 3) and the bending moment (by the constitutive relationship of rebar and concrete springs). Finally, the horizontal force at the top of the column was obtained by Eq. (4).

Damage accumulation model on the material scale
In order to quantify the influence of damage accumulation on structural performance, and to characterize the damage state of a component, the damage index D was introduced. The value of D range from 0 to 1, where 0 means intact state and 1 means failure state. A damage index model considering loading history was proposed by Mehanny and Deierlein (2001): where θ + p is the inelastic component deformation in the positive loading direction; θ + pu is the associated capacity under monotonic loading; i is the number of loading cycles; α and β are parameters; Primary half cycle (PHC) refers to any half cycle which amplitude exceeds that in all previous cycles, and following half cycle (FHC) refers to all subsequent cycles of smaller amplitude. Figure 14 shows a schematic loading history with designations of primary and following half cycles.
(5) For concrete springs, considering the inconsistency between the two directions when the concrete material was deformed in tension and compression, the cumulative damage indexes in the two directions of tension and compression were considered respectively under repeated loads, and the total damage index was calculated according to Eq. (7).
where γ is a parameter, which Mehanny suggested to take 6 for the reinforced concrete columns through experiments.

Restoring force model for corroded steel spring
A previously developed stress-strain relationship for corroded rebars by the authors (Zhang et al. 2006) is transformed into the load-deformation relationship as the skeleton curve for corroded rebar springs in the multi-spring model. In the process of transformation, on account of the perspective of improving computing efficiency, the skeleton curves for rebars with different corrosion ratios were simplified into the bilinear model, as shown in Fig. 15 and Eqs. (8-13). where P sc is the load on the corroded rebar spring; d sc is the deformation of corroded rebar spring; K 0 and K 1 are the elastic and hardening stiffness of the corroded rebar spring, respectively; P syc and d syc are the yielding load and deformation of the corroded rebar spring, respectively; P suc and d suc are the ultimate load and deformation of the corroded rebar spring, respectively; l p is the length of the plastic element at the bottom of the column, which was calculated as 1.5 times of cross-sectional height; f yc and f uc are the yielding and ultimate strengths of corroded rebar spring, respectively; f y0 and f u0 are the yielding and ultimate strengths of uncorroded rebar spring, respectively; ε cyc and ε cuc are the yielding and ultimate strain of corroded rebar spring; ε cu0 is the ultimate strain of uncorroded rebar spring; η s is the average corrosion degree; E s is the elastic modulus of corroded rebars; α yc and α uc are the relative yield and ultimate loads of corroded rebars, respectively; α δc is the ratio of the ultimate strain of corroded rebars to that of uncorroded rebars; A sc is the average cross-sectional area of the corroded rebar. The restoring force model for a corroded rebar spring is shown in Fig. 16. Segments OA and OD are both elastic stages, in which the rebar spring is considered to be intact. When the applied load exceeds point A or D and the rebar spring enters the yield stage, damage begins to occur. Assuming it is unloaded at point B with the deformation of d smax,1 and a load of P smax,1 , the corresponding damage index, D + s,1 , can be calculated by Eq. (6), representing the damage caused by the first positive loading. The corresponding unloading stiffness is defined in Eq. (14): Since there is no reverse loading and unloading process at this time, it is considered that D − s,1 equals 0. ξ Ks is a parameter used to control the amount of rebar spring stiffness degradation caused by the damage index. When unloaded to point C, the load is reversed and increased along segment CD. When the deformation of the rebar spring exceeds the corresponding deformation value of point D, the compression yield occurs. If unloaded at , representing the damage caused by the first negative loading. The rebar spring is unloaded along segment EF, and the corresponding unloading stiffness is defined in Eq. (15): The second cycle begins when positive reloading occurs after passing through point F. As a result of the effect of damage accumulation, the strength, stiffness, and ductility of the spring deteriorate. Therefore, the positive loading of the second cycle proceeds along segment FG. Point G has the same deformation as point B but a strength less than point B, with a degradation in strength, denoted as P D s max,1 . If the load function on the unit crosssectional area of the rebar spring under monotonic loading is f(d s ), P D s max,1 can be obtained by Eq. (16).
where D s,1 is the damage index after the first cycle, which is calculated by Eq. (6). ξ Ps is a parameter used to control the amount of rebar spring stiffness degradation caused by the damage index. When the rebar deformation exceeds the maximum deformation of the previous cycle, the rebar spring continues to be loaded along segment GH. The slope of segment GH is the second-stage stiffness K 1 , which is the same as that of segment AB. Subsequent loops follow the same way. Extending the above formula to the ith cycle, Eqs.

Restoring force model for concrete spring
In RC columns, the transverse deformation of concrete under compression is constrained by stirrups, which delays and restricts the occurrence and development of internal cracks along the axis. Therefore, the ultimate compressive strength and strain of confined core concrete are significantly increased. Roufaiel and Meyer (1987) idealized the compressive stress-strain relationship of confined core concrete into a tri-linear model shown in Fig. 17. The corresponding ultimate strength and strain can be obtained according to the uniaxial cylinder strength and the stirrup ratio by Eqs. (23-27).
(23) c0,core = 1 + 10 sv c0 (24) c0,core = 1 + 10 sv c0 where ρ sv is the volumetric stirrup ratio; σ c0 and ε c0 are the ultimate compressive strength and strain of unconfined concrete, respectively; σ c0,core and ε c0,core are the compressive stress and strain of confined core concrete, respectively; ε cu,core is the ultimate strain of confined core concrete. Based on the results of Roufaiel and Meyer (1987), Wu modified the load-deformation curves of unconfined concrete and confined core concrete springs, in which the improvement of compressive capacity and the reduction of tensile capacity of the confined core concrete spring were considered, as shown in Fig. 18 and Eqs. (28)(29)(30)(31)(32)(33). The modified model is adopted in this paper while the tensile strength of concrete is ignored. where b cor and h cor are the width and height of the confined core concrete zone, respectively; A sv1 is the cross-sectional area of a single stirrup; s is stirrup spacing; d t can be taken as 0.00012 times of the plastic hinge length.
Only the hysteretic rules of confined core concrete are depicted here as shown in Fig. 19, assuming that they also apply to unconfined cover concrete despite of differences in skeleton curves. Segment OA is an elastic stage, in which loading and unloading do not cause any damage. The concrete spring enters the plastic stage after passing point A and the damage begins to accumulate. If unloaded at point D, the corresponding unloading stiffness can be calculated by Eq. (34). The load and deformation at point D are defined as P core c max, 1 and d core c max, 1 respectively. Since the tensile strength of concrete is not considered, after being unloaded to point E, the deformation is unloaded along the horizontal coordinate axis to point O. Whenever the deformation is unloaded in the negative direction of the d core c axis, the reloading always starts at point O. In the second cycle, it is loaded along segment OD' where strength degradation is considered. The load P core c max, 1D at point D' can be calculated by Eq. (35). After the deformation exceeds d core c max, 1 , the curve moves along segment D'B' and B'C' which is parallel to the original skeleton curve. In the model, it is assumed that the deformation d core cv corresponding to the ultimate load of the concrete spring does not change with the accumulation of damage. The subsequent loops follow the same loading and unloading rules, and the strength and unloading stiffness considering the effects of damage accumulation in the ith cycle can be calculated by Eqs. (36) and (37).  Fig. 19 Schematic hysteretic rules for confined core concrete spring 1 3

Restoring force model for corroded RC columns
On account that the Multi-spring model cannot simulate the shear deformation, it is assumed that only flexural failure occurs in the uncorroded and corroded RC columns. Based on the restoring force model for corroded rebar spring and concrete spring, a brief description of the steps involved to simulate the load-displacement relationship of RC columns is provided herein: Step 1: Determine an increment of cross-sectional rotation as Δθ, and increase the rotation of the cross-section step by step from 0 (θ = θ + Δθ).
Step 2: The deformation at the center of the cross-section is assumed to be d middle , as shown in Fig. 20. The deformation at the center and its corresponding load of each concrete spring and the rebar spring can be calculated based on the assumption of the plane section and the deformation compatibility conditions.
Step 3: Compare the absolute value of the difference between the resultant force of each spring and the axial force N against the value k using Eq. (38) and readjust the deformation at the center of the cross-section (d middle ) until the absolute value does not exceed k.
Step 4: The bending moment of the cross-section can be calculated as: Fig. 20 Cross-sectional calculation diagram of columns Step 5: The horizontal load and displacement of the top of the column can be calculated using Eqs. (2-4). When the displacement of the top of the column reaches the amplitude of each cycle, the rotation increment is taken as −Δ and the deformation of the concrete and the steel spring is recorded. Calculate the damage index of each spring by Eq. (6). Based on the restoring force model for rebar and concrete spring, the degraded strength extreme value, unloading stiffness and other indicators are calculated with ξ Ks , ξ Kc , ξ Ps and ξ Pc equal 0.27, 0.27, 0.15 and 0.15, respectively.
Step 6: When all core concrete springs reach the ultimate strain or any one of the rebar springs reaches the ultimate strain, the column is declared invalid. Based on the above steps, the program flowchart is summarized in Fig. 21.

Model verification
Considering the effects of damage accumulation, deterioration of mechanical properties of corroded rebars, and corrosion-induced cover cracking, a numerical model is proposed and a program is developed to obtain load-displacement curves for corroded RC columns. Before the parametric analysis is carried out, the model is verified with the test results. According to the above analysis, the flexural stiffness of the pin-hinge base on the top of the columns can be regarded as infinite for simplicity, as shown in Fig. 22. And then the total displacement at the top of the column can be obtained by: where M = PL 0 ; L 0 = 1.04 m; L 1 = 0.70 m. Therefore, the total displacement can be calculated as follows: By comparing with Fig. 22a, it can be found that the plastic displacement caused by the plastic hinge is independent of the stiffness of the column, so only the elastic displacement needs to be modified. The comparison of calculation results with the test results is shown in Fig. 23. It can be found that the calculation results are basically consistent with the test results. As shown in Fig. 23k, since the shear failure was not considered in the model, the energy dissipation capacity of C4-2 is overestimated.
For further comparison, the experimental and calculated characteristic load and displacement are given in Table 3. Except that the calculated ultimate displacement of columns C1-2 and C2-1 is one cycle level lower than the experimental value, the calculation results for the other columns agree well with the experimental results. The calculated load-carrying capacity was generally slightly lower than the test results, and the yielding (1) The constitutive relationship for rebars was simplified from a tri-linear model into the bilinear model, which generally neglected yield point elongation.
(2) The increase in the randomness of mechanical properties after rebars were corroded.
(3) The bond-slip performance between rebar and concrete was neglected.
(4) The effect of shear failure was not taken into account in the proposed model.

Parametric analysis
Using the restoring force model proposed in this paper, the mechanical parameters such as axial compression ratio, concrete strength, reinforcement ratio, longitudinal reinforcement corrosion degree, and concrete cover damage were further numerically analyzed in this section. Each numerical RC column had a cross-section of 200 × 200 mm except T11-1 ~ T11-4 of 250 × 250 mm and T12-1-T12-4 of 300 × 300 mm. The numerical columns had a height of 2190 mm and were reinforced with six deformed longitudinal bars with a diameter of 12 or 14 mm and stirrups with a diameter of 6 mm and a spacing of 100 mm. The concrete cover depths used for the numerical columns were kept at 20 mm. The axial compression ratio of each column ranged from 0.2 to 0.8 The compressive strength of concrete was 15 or 25 MPa. The strength of the deformed rebars was the same as the tested specimens and the corrosion degree of deformed rebars was 0.1, 0.2, and 0.3. The total number of numerical RC columns was 51. The detailed parameters of numerical columns and the results of the numerical analysis were listed in Table 4. Especially, the effect of damage of concrete cover on the seismic performance of RC columns was studied by T10-1, T10-2, and T10-3. The location of spalling in the concrete cover was shown in Fig. 24. The damage of concrete cover was not considered in other specimens.  Table 4 that the number of hysteresis loops decreases gradually as the corrosion degree increases, and the load-carrying capacity, deformation capacity, and energy dissipation capacity decrease rapidly. As the axial compression ratio increases, the load-carrying capacity decreased gradually, and the deformation capacity and energy dissipation capacity decrease rapidly for both uncorroded and corroded columns. It can be found by comparing columns T1-T3 that, when subjected to a larger axial compression load, the impact of reinforcement corrosion is less significant due to the less contribution of longitudinal rebars to the load-carrying capacity of the column. For instance, compared with uncorroded columns, the energy dissipation of columns with a corrosion degree of 0.1 decreased by 34.0%, 33.5%, 30.3%, and 32.4% when subjected to the axial load with the compression ratio of 0.2, 0.4, 0.6, and 0.8.

Table 3
Comparison between the experimental and calculated results Δ y and Δ u are yield and ultimate displacement, respectively; P y and P u are yield and ultimate strength, respectively; " + " and "-" in the brackets represent the reinforcement yielded and the edge of the concrete in the compression zone reached its ultimate strain, respectively Specimen      0.26 Δ y and Δ u are yield and ultimate displacement, respectively; P y and P u are yield and ultimate strength, respectively; E is energy dissipation; " + " and " − " in the brackets represent the reinforcement yielded and the edge of the concrete in the compression zone reached its ultimate strain, respectively. It can be derived by comparing the columns with the same corrosion degree, axial compression ratio, and different concrete strength, that the impact of reinforcement corrosion is much more significant for columns of low-strength concrete due to the greater contribution of longitudinal rebars to the load-carrying capacity of the column. For instance, when subjected to an axial load of the same compression ratio of 0.2, as the corrosion degree increases from 0.1 to 0.3, the energy dissipation decreased by 36.7%, 58.8%, and 67.8%, respectively, for columns with the cubic concrete strength of 15 Mpa. They are only 10.9%, 33.5%, and 49.6% for columns with a cubic concrete strength of 25 Mpa.
By comparing columns T1-T3 and T4-T6, it can be derived that the seismic performance can be effectively improved with the increase of reinforcement ratio. For instance, when the reinforcement ratio of columns is increased by 26.5% with a corrosion degree of 0.1, and the axial compression ratio is 0.2, 0.6, and 0.8, the energy dissipation of columns is increased by 41.7%, 58%, and 80% respectively. For columns with a relatively larger reinforcement ratio, the impact of reinforcement corrosion is much more significant due to the greater contribution of longitudinal rebars to the load-carrying capacity.
Comparing columns T8, T11, and T12 with the same reinforcement, axial compression ratio, and corrosion degree, it can be found that the horizontal load-carrying capacity of the columns increases with the increase of the cross-section, and the yield displacement and ultimate displacement decrease. Compared column T10 with T1-2, as the cracking and spalling of the concrete cover continues to increase, although the deformation capacity of the columns decreases inconspicuously, the load-carrying capacity and energy dissipation capacity decrease gradually.

Conclusions
A total of 11 RC column specimens with different axial compression ratios and stirrup spacings were designed and tested under cyclic loading, and 8 of them were corroded by the impressed current method before being loaded. A restoring force model considering damage accumulation was numerically developed and applied to predict the seismic performance of corroded RC columns. The following conclusions can be drawn based on experimental and numerical studies: • With the increase of axial compression ratio and corrosion degree, the seismic performance of RC columns decreases rapidly. Under the combined effects of severe corrosion and large axial compression ratio, brittle compression-shear failure occurred in columns. However, as the axial compression ratio increases, the impact of reinforcement corrosion on the seismic performance is gradually weakened owing to the decreased contribution of longitudinal rebars to the load-carrying capacity. • The failure mode of both uncorroded and corroded RC columns tends to change from ductile flexure failure to brittle shear failure as expected with the increase of stirrup spacing. Correspondingly, the load-carrying capacity of RC columns decreased slightly, but the deformation capacity and energy dissipation capacity decreased obviously due to the less constraint from stirrups. In practical engineering, since stirrups have more severe corrosion than longitudinal rebars owing to thinner concrete cover, it can be derived that RC columns subjected to corrosion may fail in shear even if it is designed to fail in flexure. • A restoring force model for corroded RC columns was developed based on the multispring model in good agreement with the test results. Through the parameter analysis with the proposed model, it is found that the impact of reinforcement corrosion is more obvious in RC columns of low-strength concrete, or with a larger reinforcement ratio, owing to the greater contribution of longitudinal rebars to the seismic performance.

Data availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.