Seismic performance and damage properties of flexure–shear critical RC columns with various loading cycles

An experimental study was conducted to investigate the effects of loading cycles on seismic behaviors and damage properties of flexure–shear critical reinforced concrete columns. Two sets of circular columns (including eight specimens) with different shear span ratios were tested under quasi-static loads with various numbers of loading cycles, namely 0 (i.e., monotonic loading), 3, 10, and 20 cycles. The failure modes, load-deformation curves, ductility, and energy dissipation capacities of the flexure–shear critical columns with various loading cycles were explored. The test results revealed that the flexure–shear failures of columns were more likely to occur with an increase in loading cycles, as evidenced by intensified shear effects and cumulative damage effects induced by cyclic loading. The study examined that the ultimate deformation was the most crucial factor that affects the damage properties of the flexure–shear critical columns. The ultimate displacements diminished with the loading cycles increasing and reduced by 38%, 50%, and 62% when the loading cycles increased from 0 to 3, 10, and 20 cycles, respectively. Additionally, the damage properties of the flexure–shear critical columns were also pertinent to cumulative energy dissipation. The cumulative energy dissipation of the column specimens significantly increased after yielding and rose in proportion to the loading cycles. Two analytical models were proposed for the evaluation of equivalent viscous damping and seismic damage index in flexure–shear columns, which considered the effects of the number of loading cycles. These models could better provide an optimal analysis of the energy dissipation capacities and damage properties for flexure–shear critical columns subjected to seismic excitations, respectively.

The maximum displacement of RC members subjected to cyclic loading δ u

Introduction
The seismic hazard survey and experimental studies indicate that there are three main types of failure modes such as flexure, shear, and flexure-shear failure for RC columns subjected to seismic excitation (Sezen and Moehle 2006;Gong et al. 2010;Chen et al. 2015). For these failure modes, flexure failure is a kind of expected failure mode by virtue of its satisfactory ductility and energy dissipation capacity. Shear failure is a non-ductile failure with inferior energy dissipation capacity and will cause enormous damage to the structures. As for flexure-shear failure, its ductility and energy dissipation capacities are between those of the above two mentioned failure modes, which is a sort of failure mode with limited (or low) ductility and hard to precisely distinguish from other failure modes. In structural design, RC columns are recommended to be designed as flexure failure by checking the shear bearing capacity (MOHURD 2021;ICB 2021). However, the complex earthquake responses and excessive lateral deformations are prone to reduce the shear capacities of the RC columns, which will lead the columns to evolve into shear failure after experiencing the flexure responses, namely the typical flexure-shear failure of the columns occurs (Zhang et al. 2014a;Li et al. 2022). This type of seismic failure of structural RC columns is very common in active duty and marine RC structures, which is easy to cause a structural collapse in earthquakes (Vu and Bing 2018;Rinaldi et al. 2022). Therefore, it is noteworthy investigating the seismic performances and damage properties of flexure-shear critical columns. Certainly, many researchers have focused on the study of seismic performances and damage properties of flexure-shear failure columns subjected to the standard cyclic loading procedures (i.e., each loading displacement amplitude for 3-5 times cycles). Most of these studies have attached importance to the effects of their own design parameters such as shear span ratios, reinforcement ratios, and sectional dimension on the performance degradation of RC columns (Eric and Halil 2008;Zhang et al. 2013;Mohamed et al. 2022;Zhang et al. 2022a;Zhang et al. 2022b), whereas ignored the effects triggered by the complex cyclic loading procedures. Actually, the real seismic excitation is a kind of complex random perturbation (Ottavia et al. 2021), so the actual earthquake responses of the RC columns cannot be fully reflected by the traditional standard cyclic loading procedures. That is to say that the influences of damage accumulation triggered by the cyclic loading procedure should also be taken into account when analyzing seismic performances and damage properties of the RC columns.
More recently, some scholars have also carried out a series of research studies on the seismic effects of cyclic loading on RC columns. Raff et al. (2006) and Verderame et al. (2008a, b) compared the mechanical behaviors of RC columns under monotonic and cyclic loads, then discovered that the RC columns under cyclic loads had a tendency to fail in flexure-shear failure mode and that the deformation and load-bearing capacities under cyclic loading were significantly reduced compared with that under monotonic loading. In addition, many researchers (Gu et al. 2006;Goodnight et al. 2013;Xu et al. 2019) have further investigated the influences of loading histories (e.g., displacement amplitude, loading protocols, loading paths, etc.) under cyclic loading on the seismic performances of RC columns. Their study results indicated that the cumulative energy dissipation and ductility of the RC columns largely relied on loading displacement amplitude and also were influenced by loading cycles and the loading paths. Furthermore, the loading cycles have little influence on the seismic behaviors of RC columns in the response stage with small lateral displacement, but great effects on that with large lateral displacement (Ji et al. 2013;Qian and Feng 2014). In other words, the seismic performances of RC columns are strongly affected by the loading history after yielding (Acun and Sucuoğlu 2010). An experiment conducted by Zhu et al. (2019) also investigated that the post-peak deformation and energy dissipation capacities of the RC flexure-shear columns under cyclic loading decreased with an increase in loading cycles. Additionally, for figuring out the impacts of damage effects caused by cyclic loading on the structural seismic performance, the US standard (PEER/ ATC72-1 2010) recommended that the load-deformation curves of RC structures for monotonic and cyclic loading should be distinguished during seismic performance analysis. Also, according to the duration and pulse number of seismic waves, the bridge seismic design code of Japan (2002) divided earthquakes into type I (i.e., the seismic wave with long duration and many times of seismic pulses) and type II (i.e., the seismic wave with short duration and a few times of seismic pulses), so as to accurately determine the seismic effects and deformation capacity of bridge piers. Certainly, it should be noted that despite the tremendous impacts of loading history on the seismic performances and damage properties of RC columns have been discussed, these studies mainly focused on flexure-critical columns and rarely investigated the effects of loading history on the flexure-shear critical RC columns.

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Also, it is well known that the seismic performances of RC structures and members under earthquake are relevant to the damage properties. Hence, the quantitative assessment of the damage properties of RC structures and members is the key to seismic design (Bertero et al. 2002). The equivalent viscous damping could be employed to characterize the energy dissipation capacities of RC structures or members. Based on the quasi-static test results, various models proposed by scholars have demonstrated a response to damage properties of RC columns (Blandon and Priestley 2005;Zhang et al. 2017). Park and Ang (1985) put forward a damage model involving the combination of deformation and energy dissipation as early as 1985. Thereafter, many scholars have also proposed different damage models in terms of deformation, stiffness, low-cycle fatigue, and energy dissipation (Teran and Jirsa 2005;Zhao 2006;Jiang et al. 2012;Fu et al. 2013). These models could better reflect the damage laws for flexure-dominated RC structures, but whether they accurately reflected the damage laws for flexure-shear dominated structures and members was still an open question. Actually, the studies of references (Cao et al. 2019;Jiang et al. 2015) have indicated that most of the existing damage models considering energy and deformation factors for RC structures and members were established mainly on the basis of test results of flexure critical RC members, and these models could not be suitable for flexure-shear critical RC members. Thus, it is necessary to carry out some relevant research in the establishment of a damage model for flexure-shear critical columns.
In view of the above, this paper sought to study the seismic performances and damage properties of flexure-shear critical columns with different loading cycles. Eight RC columns were tested, subjected to monotonic and cyclic loading with various loading cycles. Subsequently, the effects of various loading cycles on the seismic performances of flexure-shear critical RC columns were analyzed in terms of failure modes, load-carrying capacities, ultimate deformation, and energy dissipation capacity. Besides, an optimized empirical equivalent viscous damping model was proposed to further investigate the energy dissipation capacities for flexure-shear columns. Finally, a new damage model for flexure-shear critical RC columns was developed that considered the ultimate deformation and energy dissipation involving the effects of various loading cycles.

Specimens design
This test gave priority to the effects of loading cycles on the seismic performances in flexure-shear critical RC columns. Eight RC circular columns were designed and produced. These column specimens were divided into two groups (i.e., groups I and II). Shear span ratios λ were the design variables of RC columns, which had dominant effects on the shear effects of specimens to lead the flexure-shear critical failures more susceptible to occur (Zhang et al. 2014a, b). The RC specimens were cast by commercial concrete with the strength grade of C35, whose average compressive strength of a concrete cube of 150 mm was 42.5 MPa. Further, the diameters of the columns section were designed as 300 mm with a 20-mm-thickness covering layer. The heights were designed as 1150 and 850 mm, and the corresponding λ was equal to 3.5 and 2.5, respectively. The column sections had configured six HRB400 bars (i.e., hot-rolled ribbed bars) of 16 mm diameter, and their actual yield and ultimate strength were 443.06 MPa and 628.32 MPa, respectively, and the corresponding reinforcement ratios ρ s of the columns were 1.71%. The HPB300 bars (i.e., hot-rolled plain bars) with a diameter of 6.5 mm were used for the transverse reinforcement, and the yield and ultimate strength were 338.95 MPa and 549.72 MPa, respectively. The space of 125 mm was set for transverse reinforcement in the columns and the corresponding transverse ratios ρ t of the columns were 0.43%. The section dimensions and reinforcement details of the column specimens are presented in Fig. 1.
One of the column specimens in each group was subjected to monotonic loading, while the other three specimens were subjected to variable-amplitude cyclic loads, and the number of load cycles N was set as 3, 10, and 20 cycles, respectively. 1 3 parameters of the column specimens. In this table, the critical variables are the number of load cycles and the shear span ratios of the RC columns. The alphabets 'M' and 'C' stand for the specimens under monotonic loading and cyclic loads, respectively. The following first number indicates the shear span ratio of 2.5 or 3.5, and the second number displays the number of load cycles. For instance, C2-10 means the specimen with a shear span ratio of 2.5 that cyclically loaded 10 times under each displacement amplitude.

Test setup and measurement method
In order to explore the effects of various loading cycles on the seismic performances of RC columns, the column specimens were subjected to a combination of constant axial compressive and cyclic/monotonic horizontal loading. The loading devices for quasi-static tests of RC columns are shown in Fig. 2. The axial loads were applied to the top of the column specimens by a hydraulic jack and remained constant, and the lateral cyclic/monotonic loading of the column specimens was applied using a hydraulic actuator with a capacity of 1000kN which was fixed on a reaction wall. Previous studies (Zhang et al. 2014a(Zhang et al. , 2022b suggested that the total lateral deformation of RC columns subjected to seismic loading is mainly comprised of flexure-deformation Δ b , shear-deformation Δ v , and slip-deformation Δ s . To measure the total deformation capacity and three deformation components, eleven LVDT sensors were installed on the plastic hinge zone of the column specimens as shown in Fig. 3. According to the formulas given in seismic codes (MOHURD 2021), the calculated heights of plastic hinge l p were assumed as 400 mm for specimens in group I and 550 mm for those in group II. The total lateral displacements of the column top were mainly measured by the LVDT δ 11 , and the LVDTs δ 1 to δ 10 were employed to measure the displacements of the deformation components for flexure, slip, and shear.
According to Fig. 3, the values of slip-deformation components Δ s could be calculated by the observed values of δ 1 and δ 2 , with the formula presented as follows: where b 2 was the distance of the LVDTs δ 1 and δ 2 .
where Δ ds was the displacement in the diagonal direction of the measured range; δ 5 -δ 8 were respectively the measured values of the corresponding displacement meters; θ was the angle between the diagonal and horizontal axis of the measured section. The method for calculating the shear-deformation Δ v (δ 9, δ 10 ) in the zone of LVDTs δ 9 -δ 10 was similar to the Eq. (2). Based on the above analysis, the shear-deformation at the column top Δ v was obtained using the following formula: where G was the shear elastic modulus of concrete, and A was the area of column section. The flexure-deformation Δ b could be determined by total deformation subtracting from the other two deformation components, as presented in Eq. (4).
where Δ was the total deformation of the columns. Note that, it could also obtain the flexure-deformation by integrating the mean curvature along the height of the column segments (Li et al. 2022), but this might lead to complex integrating calculations.

Loading protocol
The quasi-static tests were adopted to investigate the seismic performances of the RC columns. Firstly, the predetermined axial loads were applied at the top of the columns and kept constant throughout the loading process. Secondly, the lateral loading protocols of the column specimens were imposed using the displacement-control procedures, and the details of the loading protocols are reported in Fig. 5. For the monotonic loading protocol, the lateral displacements were applied slowly to the column specimens until the occurrence of failure. For the cyclic loading protocol, the lateral displacement amplitude of 1/2 times the yield displacement was applied to the column specimens before yielding firstly, then the lateral displacement amplitudes of integral multiples of the yield displacement (e.g., 1, 2, 3, and 4 times of the yield displacement) were applied to the column specimens after yielding. The columns were loaded for 3, 10, or 20 cycles at each displacement level, respectively. The yield displacements of the column specimens were determined according to the monotonic loading test results, and that of the column specimens with shear span ratios of 2.5 and 3.5 were approximately equal to 6.0 mm (θ = 0.80%) and 9.0 mm (θ = 0.85%) respectively in this study, where θ was the drift ratio and defined as the ratio of the lateral displacement to the effective height of RC columns. (3) Variable amplitude cyclic loading N=3/10/20

Damage observations
Test results indicated that the failure modes of all column specimens tended toward flexure-shear critical failure. The typical flexure-shear damage progression of test columns mainly included concrete cracking, longitudinal steel yielding, diagonal crack development, transverse steel yielding, concrete spalling, and final failure successively. The columns with a shear span ratio λ of 2.5 are taken as an example to illustrate the failure modes of flexure-shear critical columns subjected to different lateral loading sequences. All columns of λ = 2.5 exhibited an elastic behavior and did not show any visible damages when the drift ratios were relatively small (e.g., the θ was less than 0.3%). As the drift ratio increased, horizontal flexure cracks were first visible at the bottom of the columns, and then the cracks developed toward the loading points and generated diagonal shear cracks. Continuedly, the longitudinal steels began to yield and the plastic hinge formed in the stub zone of columns when the drift ratios were approximately up to 0.9%, and then the transverse steels which crossed diagonal shear cracks yielded as the lateral displacements increased. Subsequently, the diagonal shear cracks developed further and exploded next to the longitudinal axis of the columns in the post-peak loading stage, then the local concrete was crushed and fell off. After that, some of the longitudinal steels in the columns were exposed and buckling, and the columns lost their load-bearing capacities and seriously deformed until failure occurred.
It should be also noted that the damages and destruction of the columns were aggravated with the increase in loading cycles, especially for the columns with larger drift ratios (e.g., the columns entered into the post-yielding stage). The typical failure propagation of column specimens with λ = 2.5 at diverse drift ratios are presented in Fig. 6. It can be seen that the damages such as concrete spalling areas, diagonal crack width, and deformation amplitude of test columns under loading with more cycles were more serious than that under loading with fewer cycles (including monotonic loading). Furthermore, the ultimate deformation capacities of the columns reduced with the increase of loading cycles. For specimens M2-1, C2-3, C2-10, and C2-20, the failure drift ratios (i.e., the ultimate drift ratios) were equal to 5.6%, 4%, 3.2%, and 3.2%, separately. Although the test columns C2-10 and C2-20 were destroyed at the same lateral displacement amplitudes, they were destroyed at the 5th cycle and 3rd cycle at θ = 3.2%, respectively. The failure modes of the two columns are shown in Fig. 6 (c-III) and (c-IV). These suggest that the failure processes of the column specimens are closely related to the cumulative damage effects triggered by loading cycles at different displacement amplitudes.
Similarly, the flexure-shear failure progression of the columns with a shear span ratio of 3.5 was analogous to that of the columns with a shear span ratio of 2.5, and the distinction between them was that the shear effects of columns with λ = 3.5 were relatively weaker. Therefore, the failure drift ratios of the specimens M3-1, C3-3, C3-10, and C3-20 were equal to 7.4%, 4.2%, 4.2%, and 2.5%, respectively. What's more, the columns mentioned above were destroyed at different loading cycles, and the columns C3-3, C3-10, and C3-20 were destroyed at the 7th cycle, 3rd cycle, and 14th cycle, respectively. Overall, the failure degrees of the column specimens tended to be more severe with the increase in loading cycles and were also affected by the shear span ratios.  Figure 7 illustrates the load-deformation curves of each column specimen. In the figure, the dots are shear-failure points of the column specimens, and they are defined as the points where the peak load of a hysteresis loop begins to decline rapidly or the peak load of a hysteresis loop decreases by more than 10% compared with that of the previous loop under the same displacement level (Zhang et al. 2014a, b). It can be found that the load-deformation curves of the columns under lateral loading with more cycles are roughly below those of the same designed columns under lateral loading with fewer cycles or monotonic. The ultimate lateral displacements significantly decrease for the same designed columns with the increase of loading cycles. Also, the shapes of the hysteresis loops of the columns become thin as the number of load cycles increases, and the pinch effects of hysteresis loops tend to be remarkable. These conclude that the damage accumulation resulting from the increase in loading cycles aggravates the deterioration of seismic performances for flexure-shear critical columns. Furthermore, the skeleton curves of the RC columns subjected to cyclic loading and the load-deformation curves of the columns under monotonic loading are plotted together in Fig. 8. As presented in the figure, the monotonic load-deformation curves almost coincide with the skeleton curves of the same designed columns with different loading cycles before yielding. This suggests that the seismic damage effects triggered by loading cycles for the columns in the elastic response stage are so small that they even can be ignored. However, the damage effects of the loading cycles are significantly aggravated when the columns are loaded in the nonlinear response stage, especially in the post-peak softening stage. In the stage of load-bearing capacity degradation, the lengths of the load-deformation curves shorten obviously with the increase of loading cycles. Meanwhile, the slopes of curves also drop slightly with the loading cycles increase. This can be explained by that the shear failure of the columns appears in advance due to the cumulative damage effects or low-cycle fatigue effects triggered by cyclic loading, and the ultimate deformation capacities are weakening. Table 2 summarizes the recorded experimental results of lateral displacement and load corresponding to the yield points, peak points, and ultimate points (i.e., failure points) for the test columns. Note that the values of the displacement and lateral load listed in Table 2 are the mean values of the corresponding positive and negative loading directions, respectively. In the table, the yield points of the column specimens with the same design have slight differences, showing that the deformations and load-bearing capacities are almost not affected by the loading histories before yielding. However, the ultimate displacements and load-bearing capacities (i.e., peak load) are affected significantly by the loading cycles. Compared to the monotonic loading columns, the peak loads of the cyclic loading columns are reduced by 2.5% to 11.2%. For instance, the peak loads of the test columns with a shear span ratio of 2.5 are reduced by 6.1% at most, and those of the columns with a shear  Table 2 Summary of recorded experimental results of test columns δ y , F y are the yield displacement and the yield load, respectively; δ p is the peak displacement when the lateral load reaches the peak load F p ; δ u , F u are the ultimate displacement and lateral ultimate load corresponding to the shear failure point; μ is the displacement ductility factor, which is represented as the ratio of the ultimate displacement to the yield displacement (μ = δ u /δ y ) span ratio of 3.5 are reduced by 11.2% at most. Additionally, the ultimate displacements of each group's cyclic loading columns are much smaller than that of monotonic loading columns and gradually reduce with the increase of loading cycles. Taking the column specimens with a shear span ratio of 2.5 as an example, the ultimate displacements reduce by 24%, 39%, and 55% separately as the loading cycles increase from 0 cycle to 3, 10, and 20 cycles. Similarly, the displacement ductility factors μ of the columns also decrease with the loading cycles increase. The above-mentioned facts illustrate that the increase in the number of loading cycles exacerbates the cumulative damage effects and shear effects, so that the ultimate deformation capacities and the ductility capacities of flexure-shear critical columns are lessened. Figure 9 provides the variations in the proportion of each deformation component (flexure-deformation, slip-deformation, and shear-deformation) in the total lateral deformation of all column specimens. As evident from the figure, the proportions of the flexure-deformation component to the total deformation are larger than the proportions of the other two deformation components for flexure-shear critical columns. The proportions of the flexuredeformation component Δ b /Δ are around 50-70%, while the shear-deformation component Δ v /Δ and slip-deformation Δ s /Δ contribute around about 15-30% and 15-25%, respectively. Based on the aforementioned experimental results, it is evident that both total lateral deformation and each deformation component increase as lateral drift ratios increase. Additionally, the variation trends of the proportions for each deformation component indicate volatility to some extent with increases in total lateral displacements. For instance, the shear-deformation ratios (Δ v /Δ) of specimen C2-3 decrease from 48.5 to 18% with the increase of loading displacements after yield; whereas the Δ v /Δ of specimen C3-20 remain almost unchanged at about 23%. This can be explained by that the changes of the flexure and shear response components are interacted and fluctuated due to the abrupt changes of damage effects of the columns triggered by concrete cracking, steel bars yielding, and concrete spalling.

Load-bearing and deformation capacity
To further gain a more intuitive understanding of the impact of different loading cycles on shear effects and deformation capacities for flexure-shear critical columns, the average proportions of each deformation component to the total deformation ( Δ b ∕Δ , Δ v ∕Δ and Δ s ∕Δ ) for the column specimens after yielding are also provided in Fig. 9. It can be observed that the average shear-deformation ratios ( Δ v ∕Δ ) of the test columns with a shear span ratio of 2.5 are larger than that of 3.5. Furthermore, as the number of loading cycles increases, the average shear-deformation ratios of the specimens increase to a certain extent. For columns with a shear span ratio of 2.5, the average proportions of shear-deformation component in specimens subjected to cyclic loading can increase by up to 49.1% compared with that of specimens under monotonic loading; and for columns with a shear span ratio of 3.5, the values of Δ v ∕Δ in test specimens can increase by up to 41.7%. That is to say that the increase in loading cycles will exacerbate the cumulative damage effects, leading the shear effects of the RC columns to be more obvious.

Energy dissipation capacity
The hysteresis energy dissipation capacity of the RC columns under cyclic loading can be measured by calculating the areas enclosed by the hysteresis loop. The variation trends for the single-cycle energy dissipation E i and cumulative energy dissipation E cum (i.e., E cum = ΣE i ) of the column specimens are depicted in Fig. 10. As illustrated in the figure, the growth rates of energy dissipation of the column specimens are considerably accelerated after the drift ratios reach the yielding drift ratio (e.g., θ = 0.8% or θ = 0.85%), which reveals that the hysteresis energy dissipation capacities of the column specimens improve significantly with the increase of lateral deformation, especially at the post-yielding response stage. As shown in Fig. 10a, the E i with few numbers of cycles is larger than that with large numbers of cycles for the same designed columns under the same drift ratio. For instance, the E i of column C3-3 is 30.7% larger than that of C3-10, and 96.4% higher than that of C3-20 when the drift ratio is equal to 2.55%. Additionally, it can be also observed that it is characterized by the gradual weakening of the energy dissipation capacity of hysteresis loops under the same drift ratio, especially when the number of loading cycles is large. For instance, the E i in the first circle of specimen C3-20 is reduced by 30.8% compared with that in the 20th circle as the drift ratio is equal to 2.55%. This can be explained by the fact that the cumulative damage of column specimens increases as the increase of the loading cycles, which results in weakening of the hysteresis energy dissipation capacity. As presented in Fig. 10b, the total energy dissipation capacity of column specimens with the same designed parameters increases with the increase of the loading cycles. For instance, the total energy dissipation of specimen C2-20 is 54.2% greater than that of column C2-10 and 77.9% greater than that of C2-3 at the ultimate displacement. However, it should also be noted that the ultimate deformation of columns with the same designed parameters is reduced to varying degrees with the increase of the loading cycles. This indicates that the damage triggered by permanent deformation may be more dominant than damage triggered by energy dissipation.

Equivalent viscous damping
To further evaluate the energy dissipation capacities of flexure-shear RC columns effectively, it is imperative to establish a calculation model for equivalent viscous damping of flexure-shear RC columns. The equal-energy method is usually used to define equivalent viscous damping due to its intuitiveness and simplicity. The equivalent viscous damping ζ hys is defined as the ratio of hysteresis energy dissipation to the energy produced by the equivalent elastic body under the same displacement amplitude. The formula of equivalent viscous damping is given as follows: where S AD 2 A � + S AD � 2 A � presents the energy dissipated by the column specimen alternating for one cycle, which is expressed as the area of hysteresis loop A'D 2 AD 2 ' in Fig. 11. S OD 2 B + S OD � 2 B � indicates the energy absorbed by an equivalent ideal elasticity zone under the same deformation, and the area contains two triangles OD 2 B and OD 2 'B' in Fig. 11.
The equivalent viscous damping ζ hys versus the number of load cycles N is shown in Fig. 12. As plotted in the figure, the values of ζ hys increase with the increase of the loading drift ratios. Besides, the ζ hys declines with the increase of the loading cycles for the same designed column specimens with the same displacement amplitude. This also illustrates that the cumulative damages of flexure-shear critical columns increase with the increase of the loading cycles, resulting in a reduction in the energy dissipation capacities. In addition, the values of ζ hys grow rapidly in the last few cycles, that is because the columns are seriously damaged at the end of loading and the elastic absorbed energy decreases significantly, as seen from the triangles OEF and OE'F' in Fig. 11.
Obviously, it is impractical to directly calculate the equivalent viscous damping of flexure-shear critical columns due to the irregular shapes of the hysteresis loops and computational difficulties. Several previous studies have been conducted to assess the  Gulkan and Sozen 1974;Iwan 1980;Priestley 2003), whose calculate formulas are listed in Table 3.
As presented in Fig. 13, the single-parameter equations aforementioned underestimate the equivalent viscous damping of RC columns, while the Rosenbluth model overestimates its values. This indicates that existing models for estimating equivalent viscous damping may be overly inaccurate in analyzing energy dissipation in flexure-shear columns. Zhang et al (2017) and Shao and Wei (2021) have found that equivalent viscous damping ζ hys of flexure-shear critical columns mainly relate to the correlation between relative displacement ductility and stiffness degradation. Relative stiffness ratio γ' is defined as the ratio of the post-yield stiffness, which refers to secant stiffness between the maximum displacement point of a hysteresis loop and the yielding point, to the yield stiffness; and relative ductility factor μ' is defined as the ratio of the post-yielding displacement to the yield displacement. Figure 14 delineates the variations of ζ hys and γ', μ' of column specimens respectively in this study. It is noticed that the values of ζ hys of the column specimens decrease with an increase of γ', and the relation between them can be described roughly by a power exponential function; whereas the values of ζ hys increase with increases in the μ', and the relation between them can be described by a logarithmic function. Hence, the ζ hys herein can be represented as follows: where A and B are the coefficients relevant to the relative displacement ductility and relative stiffness ratio, respectively, which are equal to 0.1614 and − 0.0417 after optimized regression according to the experimental results of this study. Figure 15 presents a comparison between the calculated and experimental values of the equivalent viscous damping ζ hys , and the test data were derived from this study and Zhu's study (Zhu and Tang 2023). It can be visible that the calculated points are concentrated near the control datum line, and the vast majority of scattered points are within the ± 30% control lines. Furthermore, the mean error ME between the calculated and experimental results is equal to 0.014, with a maximum absolute error AE max of 0.052. The standard error SE is equal to 0.058, and the correlation coefficient R 2 between calculated values and test results is 0.90. This indicates that the proposed Eq. (6) demonstrates acceptable agreement with experimental values. It should be noted that the constant-energy method heavily relies on the selection of a hysteresis model when deriving equivalent viscous damping. In some cases, this estimation may be conservative compared to that from nonlinear dynamic analyses (Blandon and Priestley 2005;Aloisio et al. 2021), which requires further study.

Existing two-parameter seismic damage models
The hysteresis damage model can quantitatively describe the damage degrees of RC structures and members subjected to seismic excitation, which provides the theoretical basis for the damage assessment and improvement of the seismic design method. In the existing seismic codes for concrete structures (e.g., MOHURD 2021, IBC-2021, only single deformation indexes (e.g., displacement, ductility, and interlayer displacement angle) are utilized to characterize the damage degrees and limit states of structures. This can be explained that it is more direct and convenient to use the displacement index for structural design. However, this approach obviously underestimates the impacts of energy dissipation on such damages according to the above analysis. Considering the combined effects of deformation and energy dissipation, the two-parameters damage model proposed by Park and Ang (1985) has been recognized as the most classical so far. The Park-Ang model can be convenient to judge the damaged state of RC structures and members, but there are some limitations such as the model being proposed based on the incomplete available test data and some unproven hypotheses. For instance, the model assumed that elastic deformation will cause the RC structures or members to damage, which was somewhat contrary to the existing experimental results (Jiang et al. 2012;Fu et al. 2013). Also, the relation of damage triggered by deformation and that triggered by energy dissipation was assumed as a simple linear proportion in the Park-Ang model, which usually caused the values of total damage factors for RC members with failure state much higher than 1.0. In view of the above-mentioned deficiencies, some researchers have modified this model to improve its applicability and rationality (Jiang et al. 2012(Jiang et al. , 2015Fu et al. 2013;Rajabi 2013). For example, Rajabi et al. (2013) modified the model by removing the effect of elastic deformation and adopting more extensive test data of RC members. The existing damage models, as well as their characteristics are listed in Table 4.where D is the damage index of the RC structural members, and its value ranges from 0 to 1.0; D Δ and D E present the deformation damage term and energy dissipation damage term, respectively; D ' Δ is the modified deformation damage term after removing the effect of elastic deformation; δ m denotes the maximum deformation of RC members subjected to cyclic loading; δ u * means the ultimate displacement of RC members subjected to monotonic loading; ∫ dE indicates the cumulative energy dissipation of the structural members; β, β', β 1 , β 2 represent the combination coefficient of the energy damage term, which are obtained by fitting test results on seismic test data of RC members; e i represents the effective energy dissipation factor proposed by Fu et al. (2013).
To explore the damage properties of flexure-shear critical columns, the two models mentioned above were employed to quantitatively investigate the damage indexes of cyclic loading column specimens in this study. As shown in Fig. 16, it can be visible that the deformation damage terms and energy damage terms of the two models have an upward tendency with the increase of drift ratios. Both the values of the deformation terms D Δ and D' Δ are obviously below 1.0 until the loading begins to fail for the above two models. However, the values of the energy damage terms βD E and β ' D E much exceed 1.0 with the increase of the deformation and especially at the failure point, and both their values have excessive growth rates, illustrating a clear exponential relationship with the drift ratios. Furthermore, the variations of energy damage terms grow rapidly as the loading cycles increase under the same drift ratios, illustrating that the loading cycles have a perceptible Table 4 Existing two-parameter seismic damage models Scholars Damage models Characteristics Park and Ang (1985) Ignoring the effect of elastic deformation and improving by more extensive test data of RC members Elimination of non-normalization problem effect on the energy damage terms. From the above analyses, it concludes that the effects of the loading cycles on cumulative damage triggered by energy dissipation (i.e., energy damage term) for the columns have been overestimated by the traditional models, and the corresponding effects of loading cycles have also been ignored. Therefore, the damage models presented by Park and Ang (1985) and Rajabi et al. (2013) are unsuitable for the assessments of damage states for flexure-shear critical columns, and the damage model for flexure-shear critical column members should be further studied and presented.

Damage model for flexure-shear critical columns
To regulate the overestimation effects of energy dissipation on the damage degrees of flexure-shear critical columns, the energy damage terms in the proposed model are optimized by reducing with a nonlinear logarithmic relation according to the form of an inverse function. Therefore, the damage index D * for the flexure-shear critical columns can be indicated by Eq. (7) as follows: where β * is the adjustment coefficient of the energy damage term for the flexure-shear columns, and it can be determined by regression of test data. To effectively avoid data errors and to enhance the credibility of the formula, the experimental results in this study and similar test data for the flexure-shear critical columns derived from the articles (Zhang et al. 2014a;Zhu 2020) have been assembled to determine the coefficient β * , as listed in the Appendix Table 5. Both monotonic results and cyclic loading results of flexure-shear  Table 5. The range of the longitudinal reinforcement ratios ρ s is from 1.57 to 2.34%, that of the stirrup reinforcement ratios ρ t is from 0.33 to 1.09%, that of the shear span ratios λ is from 1.60 to 6.16, and that of the axial load ratios n 0 is from 0.2 to 0.25. Note that it is assumed that the adjustment coefficient β * is uniform in the proposed model Eq. (7), and the values of the β * can be derived from Eq. (7) based on the given damage index D * . The damage index D * is certainly assumed to be equal to 1.0 when the failure of flexure-shear critical columns occurs, so the values of the coefficient β * can be obtained, which is listed in the Appendix Table 5. The above analysis and results indicate that the values of energy damage terms are significantly influenced by control parameters such as stirrup reinforcement ratios (ρ t ), longitudinal steels ratios (ρ s ), shear span ratios (λ), axial load ratios (n 0 ) and the number of loading cycles (N). The relationships for the adjustment coefficient β * dependence of these parameters, as established from the Appendix Table 5, are depicted in Fig. 17. As indicated in Fig. 17, the adjustment coefficient β * has a liner relationship with ρ t , so the influence of the stirrup ratios ρ t is assumed to be a linear term in the proposed empirical formula. Similarly, the influences of ρ s , λ, n 0 , and N on the adjustment coefficient β * also should be included in the proposed empirical formula according to the relationships between β * and the above parameters in Fig. 17. The analytical approach employed can be informed by the study conducted by Zhang et al. (2022a) and Shen et al. (2023).
Thereby, the empirical formula of the adjustment coefficient β * considering the comprehensive influences of the key parameters mentioned above can express as follows: Fig. 17 The relationship between different variables and adjustment coefficient β * where K 1 to K 8 are impact coefficients, and they can be obtained according to optimal regression analysis based on the test data shown in the Appendix Table 5, and K 1 = 8.7, K 2 = -3.0, K 3 = − 1.0, K 4 = 2.1, K 5 = 1.2, K 6 = 2.8, K 7 = − 2.7, and K 8 = -0.033. It should be noted that Eq. (8) is mainly obtained based on the analysis of test data in the Appendix Table 5, hence the ranges of application for each design parameter are limited, and their values are presented as follows: 0.05 ≤ n 0 ≤ 0.25; 0.33% ≤ ρ t ≤ 1.09%; 1.60 ≤ λ ≤ 6.16; 1.57% ≤ ρ s ≤ 2.34%; 3 ≤ N ≤ 20.
The comparisons between the calculated values using the proposed Eq. (8) and the test results of the adjustment coefficient β * are given in Fig. 18. The scattered points between the calculated values and the test results of the adjustment coefficient β * are concentrated around the equal line (i.e., 45 degrees line), and their correlation coefficient R 2 reaches to 0.93, which indicates that the calculations have high correlations with the respective experimental data.
For the practical application aspect, the safety of the design values of the coefficient β * should be further considered. Hence, the safety factor γ s is adopted for judging the veracity of the proposed Eq. (8), and the design value of the adjustment coefficient β * could be calculated as follows: where β * des is the design value of the adjustment coefficient β * , and β * cal is the calculated value obtained by Eq. (8), γ s is the safety factor. Figure 19 presents the frequency histogram of the observations γ s , and it can be seen that the normal distribution can be used to represent the probability distribution of the γ s . Calculate the mean value μ γ and standard deviation σ γ of the factor γ s , and put them into the normal distribution function Eq. (10), so the values of γ s considering the given guarantee rate (or exceedance probability) can be obtained: where Φ −1 (⋅) is the inverse function of the cumulative distribution function of the standard normal distribution; p s is the probability that the design values are not greater than the (8) * = K 1 n 2 0 + K 2 n 0 + K 3 t + K 4 K 5 + K 6 s + K 7 e K 8 (N∕3−1) (9) * des = * cal s (10) s = + Φ −1 (p s ) Fig. 18 Comparisons between calculated values and test results of adjustment coefficient β * calculated values. In this study, γ s is taken as 1.2805 which can make it 90% probability that the design value is not greater than the actual-experiment value for the damage indexes. As shown in Fig. 18, the vast majority of data points are distributed within the γ s = 1.2805 Line and γ s = 0.7809 Line of the isoline, illustrating that the modified Eq. (8) predicts the adjustment coefficient β * well.
Therefore, the design value of the damage index D des for the flexure-shear critical columns considering different loading cycles can be expressed as follows To further verify the rationality of the proposed model (i.e., Eq. (11)), Fig. 20 compares the predicted damage indexes of flexure-shear critical columns using different damage models. As illustrated in the figure, the damage indexes calculated by various damage models show upward tendencies with the increase of drift ratios θ, but the rates of rise for the D * versus θ are different due to the discrepancy of the predicted results. The results of damage indexes predicted by the models proposed by Park and Ang (1985), Jiang et al. (2012), Fu et al. (2013), and Rajabi et al. (2013) increase dramatically with the increase of drift ratios, and the values of damage indexes outdistance the limit value of 1.0 before the occurrence of failure. Specifically, these models overestimate the effects of energy damage terms of the flexure-shear columns. For the model proposed in this study, the damage indexes are always kept between 0 and 1.0 during the process of loading to failure, proving that the prediction accuracy of the model proposed herein is excellent for the flexure-shear critical columns. In addition, the effect of loading cycles on energy damage terms of the RC columns is also integrated into the proposed model. Thereby, the proposed damage model herein is supposed to be feasible to evaluate seismic damage states for the flexure-shear critical columns subjected to different loading cycles.

Conclusions
This study investigated the seismic performances and damage properties of the RC flexure-shear critical columns subjected to monotonic and cyclic loading with various loading cycles. The following results were drawn from the study: (1) The failure modes of the RC columns were affected by the number of load cycles. As increasing the loading cycles, the shear effects of the column specimens were prone to be evident, and the diagonal cracks and crossing fractures developed at a faster rate. The increase in loading cycles aggravated the accumulative damage effects of the specimens, which led to the occurrence of flexure-shear failure of column specimens.
(2) With the number of loading cycles increased, the ultimate deformation capacities and ductility of test columns decreased apparently, as well as degraded the load-bearing capacities to a certain degree. Moreover, as the loading cycles increased, the pinch effects of the hysteresis hoops tended to be remarkable, and the slopes of the skeleton curves became steeper as well as the lengths of curves got shorter. (3) The hysteresis energy dissipation capacities of columns were significantly enhanced with increases in lateral deformations in the post-yielding response stage. Column specimens subjected to fewer loading cycles exhibited greater energy dissipation capacities than those subjected to more cycles at the same drift ratios due to the cumulative damage effects. An empirical expression for calculating equivalent viscous damping in flexure-shear critical columns was proposed for further investigation of energy dissipation capacities. (4) Several existing damage models, which were derived from the Park-Ang model, seemed to overestimate the influence of hysteresis energy dissipation capacities on damage states for flexure-shear critical columns. The damages triggered by permanent deformation might be more dominant than those triggered by energy dissipation. Therefore, an optimized two-parameter logarithmic damage model considering the effects of loading cycles and main parameters was proposed. The damage indexes of the proposed model were all along maintained between 0 and 1.0 for the flexure-shear critical columns subjected to seismic loading. The damage model updated herein was valuable for evaluating the seismic states of flexure-shear critical RC columns.