Shear strength model of large-scale reinforced concrete rectangular columns with light transverse reinforcement

Columns with low transverse reinforcement due to poor detail, small rebar size, and wide spacing were commonly used in old buildings. These types of columns often fail in shear with the brittle mode that could lead to a sudden collapse of the entire building. Thus, the accurate determination of the shear strength is significant for estimating the response of reinforced concrete (RC) columns under earthquake events. Numerous experiments were conducted to investigate the shear strength, however, there is a lack of tests on large-scale lightly RC columns under high axial compression load. Therefore, this study tested four large-scale RC columns with dimensions of 800 mm × 800 mm × 3200 mm. The paper highlights the main factors that influence the shear strength of the RC columns and based on the test results, an equation with high predictive power is established. In addition to the experimental results, the analytical/empirical models’ predictions from design codes or other researchers were evaluated. Furthermore, the interaction diagram between bending moment (M)—shear force (V)—axial compression force (N) was conducted to estimate the strength ability of RC columns. It was found that the simple processes to build the backbone curves and the hysteresis loops that were recommended had good agreement with the tested results and experimental database.


Introduction
Reinforced concrete (RC) columns that were designed to satisfy older versions of the construction codes have been identified as inadequate in seismic regions usually showing damages or even collapsing under extreme loads. These types of columns tend to fail in shear because of insufficient, poorly detailed, or widely spaced transverse reinforcement. The shear failure could not only cause the significant degradation of lateral strength, stiffness, and axial-load carrying capacity but also cause a drastic change in the elastic behavior of the structural member. The risk of failure increases when it is also considered the interacting effect that occurs in the strength envelope of the RC columns. The structural ability of RC columns is commonly expressed by the interaction relationship between moment (M), shear force (V), and axial (tension or compression) force (N). This relationship was studied by some researchers, for instance, Vecchio and Collins (1988) proposed sectional models to analyze prestressed and RC sections, and Rahal (2000) established a simplified approach to Vecchio's method showing that the simpler version of the M-V-N interaction diagrams had good agreement with its theoretical predecessor. In addition, Guner and Vecchio(2010) developed a computer-based analytical procedure for the non-linear analysis of framerelated structures in which it was possible to represent shear-related effects coupled with flexural and axial behaviors. Other research groups such as Recupero et al. (2003) developed an approximately physical model to evaluate the M-V-N interaction resistance domains for box and I-shaped concrete cross-sections while Mostafaei and Vecchio (2008) conducted a performance-based analysis of RC columns subjected to M-V-N. Although these two methods were established using different theories, both were able to predict the full load-deformation relationships of RC columns. In the unified theory developed by Hsu and Mo (2010), it was proposed two failure modes for beams or columns subjected to M-V or M-N, and simplified equations that expressed the relationship between M-V-N were developed.

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The response of the RC columns under cyclic loading is influenced by the correlation between lateral force and displacement (namely backbone curve) and the hysteresis behavior (namely hysteresis loop). So far, there have been some efforts to characterize the backbone curves, i.e., ASCE (2017), Halil Sezen (2002), Li and Hwang (2017), etc. ASCE (2017) recommended the backbone curve for shear-critical columns relying on obtaining only two points, the maximum strength capacity and its collapse. Other researchers have refined the calculation of the backbone curve to obtain more valuable information. Li and Hwang (2017) considered that the cracking point was also an important characteristic of the backbone curve and proposed a three-point backbone curve for short columns. In addition, Halil Sezen (2002) also included the cracking plus the yield point to establish a backbone curve by connecting three linear segments until it reaches its collapse.
As mentioned above, the hysteresis loop is also an important parameter to show the response of the RC columns under cyclic lateral loads. Numerous hysteresis models have been developed to simulate the response of RC columns (Clough, 1966;Ibarra et al., 2005;Lowes & Altoontash, 2003;Riddell & Newmark, 1979;Saatcioglu & Ozcebe, 1989;Saatcioglu et al., 1992;Takeda et al., 1970). Recently, Elwood (2004) developed a model that is capable of accurately predicting the shear failure of old RC columns. In this model, flexure behavior was simulated using fiber-section elements while shear behavior used an elastic shear spring element. However, the pinching and cyclic deteriorations in the strength and stiffness were not considered in this model. Besides, Halil Sezen and Chowdhury (2009) also pointed out that none of the separate spring models were verified for their accuracy by using separated response components extracted from column tests. The cyclic responses of flexure, bar slip, and shear components were predicted based on the hysteresis rules and then combined to obtain the total cyclic response of the column based on its identified failure mode. In addition, Halil LeBorgne and Ghannoum (2014) proposed a rotation-based shear spring element that can accurately simulate shear failure which has better performance than the lateral drift-based shear spring element model presented by Elwood (2004). Moreover, Lee and Han (2018) used the Pinching4 model in the OpenSees platform (Mazzoni et al., 2006) to simulate the cyclic behavior of shear dominant columns. The Pinching4 model could accurately and efficiently simulate the column behavior by considering pinching and cyclic deterioration. The calibration of the modeling parameters was based on a database of forty flexure-shear and shear critical columns.
Although many years of research have been devoted to the seismic study of lightly RC columns under the high axial load ratio (ALR), there is still a need to extend the applicability of the prediction models to large-scale columns. Some researchers believed that by increasing the cross-sectional size, the nominal compressive strength generally declines, while the brittleness of the columns increases Li et al., 2016). Jin et al. (2019;Shao Chia-Wei et al. 2021;Chung-Chan et al. 2016;Chung-Chan et al. 2021;Terry et al. 2022) investigated the size effect on RC columns and showed that increasing the structural size, the nominal moments, the ductility capacity, the energy dissipation capacity, and the rotation capacity would decline. They also suggested that the number of the tested columns was relatively small due to economic considerations, and the discreteness of test data were inevitable. They also suggested needing further studies on large-scale structures. Therefore, in this paper, an experimental program that included four large-scale RC columns was conducted and their results are thoroughly discussed. Moreover, based on the experimental results, a moment-shear-axial force interaction and revised equations were developed and checked for their accuracy by existing equations recommended by different researchers and codes with high accuracy. In addition, the simple processes to build the backbone curve and the hysteresis loop of the RC columns were proposed. An evaluation and comparison of these processes with tested results and experimental results of some researchers, showing efficiency and acceptable accuracy.

Some existing models for shear strength
In the past, numerous researchers and codes proposed shear equations to predict the shear strength of RC columns. Some early codes, i.e., ASCE-ACI Committee 426 (1977), SEAOC (1973), recommended the shear contribution equation of concrete that considers the effect of the shear stress (v c ) and axial compression load (P), as presented in Eqs. (1) and (2). The transverse reinforcement contribution is calculated by V s = A v f yv d∕s (where A v , f yv are the area and strength of transverse reinforcement, respectively; d is the effective depth; s is the spacing of transverse reinforcement) as follows: (1) Aschheim and Moehle (1992) used test data from the cantilever bridge column to establish the shear contribution of concrete as illustrated in Eq. (3) (where � = 0.006 v f yv ∕ ; v = A v ∕bs is the transverse reinforcement ratio). The results indicated that the shear strength could be a function of the displacement ductility, transverse reinforcement ratio, and axial compression load.
California Department of Transportation (1995) evaluated the shear strength of existing RC columns. The concrete contribution ( V c ) is expressed to be a function of the axial load, displacement ductility ( ) , and confinement level as expressed by Eqs. (4) and (5) (where is the transverse reinforcement ratio; F 2 is the axial load factor that varies between 1.0 to 1.5 for zero axial stress and compressive stress of 1000 psi, respectively).
Architectural Institute of Japan Structural Design Guidelines AIJ (1994) recommended the shear strength of RC columns that was calculated from the superposition of truss ( V t ) and arch (V a ) mechanisms ( V n = V t + V a ) . The contribution of truss and arch mechanism to shear strength is calculated by Eqs. (6) and (7), respectively (where j t is the distance between the top and bottom longitudinal bars; cot = min(2.0;j t ∕Dtan ; ∕D is the tangent of the strut angle in the arch mechanism; ν is the effectiveness factor for the compressive strength of the concrete, . Priestley et al. (1994) proposed that the shear strength was composed of following three components: i.e., concrete contribution ( V c ), shear contribution ( V s ), and arch mechanism ( V p ) as illustrated in Eqs. (8-10) and Fig. 1 (where A e = 0.8A g ; k is the parameter depending on the member displacement ductility; D′ is the distance measured parallel to the applied shear between centers of the peripheral hoop; is the inclination of diagonal compression strut; c is the neutral axis depth; D is the overall depth of the section). The contribution of transverse reinforcement to shear strength is based on a truss mechanism using a 30-degree angle between the diagonal compression struts and the column longitudinal axis.

Fig. 1
Concrete shear strength degradation with displacement ductility, and axial load contribution to shear strength (Priestley et al., 1994) Based on Priestley et al. (1994) and Kowalski et al. (1997) proposed the concrete contribution equation that considers the effects of column aspect ratio and longitudinal reinforcement, as expressed by Eq. (11) (where = (2 − a∕h) + 1;1 ≤ ≤ 1.5; = 0.5 + 20 l ≤ 1 accounts for the effect of longitudinal reinforcement; K is a factor considered the deterioration of shear strength at large displacement ductility; the truss mechanism component is modified slightly and is given by Federal Emergency Management Agency (1997) was based on a review of available experimental data for existing columns subjected to axial load and reversed cyclic lateral displacements. FEMA-273 considered the effect of the ductility in a simple manner as presented in Eq. (12) (where the ductility factor ( k ) is taken as 1.0 and 0 for low ductility demand, and moderate and high ductility demand, respectively); is the parameter that considered the effect of the type of concrete ( = 1.0 for normal weight concrete); P is the axial load, it was taken as zero in tension). The transverse reinforcement contribution is the same previous equation.
Based on the contribution of concrete and transverse reinforcement to shear strength, Halil Sezen (2002) and Sezen and Moehle (2002) proposed the shear strength of concrete ( V c ) calculated from the principal normal and shear stress equation by considering the effect of cross section and aspect ratio. The shear strength of transverse reinforcement ( V s ) was taken from the empirical equation for a 45-degree inclined crack angle. Both V c and V s are multiplied with a factor that considers the effect of displacement ductility. The proposed shear equation is presented in Eq. (13).
Abraham Christopher Lynn (2001) tested eight columns and found that the shear strength of transverse reinforcement should be gotten half. From Sezen's equation, Abraham Christopher Lynn (2001) proposed a revised equation that considered the effect of crack angle on transverse reinforcement contribution as illustrated in Eq. (14). In Eqs. (13) and (14), k is the parameter that considers the ductility factor ( ).
Recently, Tran and Li (2012) introduced an equation developed based on the strut-and-tie analogy to predict the shear strength of reinforced concrete columns with low transverse reinforcement ratios by the combination of arch action (V a ), concrete (V c ) , and transverse steel (V s ) contribution as presented in Eqs. (15) and (16) (where v c is the shear stress at the strain of concrete fiber reaching 0.002
Similarly, Pan and Li (2013) expressed the shear strength as the contribution of concrete, transverse reinforcement, and arch action mechanism, as illustrated in Fig. 3 and Eq. (17). This proposed model was based on the test of ninety shear-critical columns and was compared to some other models with acceptable accuracy. In Eq. (17), V ct and V ct are the contribution of concrete and transverse reinforcement to the shear in the truss model, respectively; k is the factor relating concrete contributions to shear strength in the truss model and concrete in the arch model to displacement ductility; V a is shear strength provided by arch action; K a is the shear stiffness of the arch model, K a = E c bc a sin 2 cos 2 ; K t is the shear stiffness of truss model, K t = (n v E c bd v cot 2 )∕(1 + n v csc 4 ) ; c a is the effective depth of strut in the arch model; c is the thickness of concrete cover; is the inclination of strut arch; is the angle of the inclined strut in cracked concrete concerning the longitudinal axis of the column in truss model; the effective shear depth (which need not be taken less than 0.9d ); n = E s ∕E c : Similarly, the ACI318-19 code (ACI Committee 318, 2019) based on the 45-degree truss model recommended that the shear strength of RC columns was determined by the sum of concrete and transverse reinforcement contribution(V n = V c + V s ). V c and V s were plotted in Eqs. (18) and 35bs∕f yv are the area, and the minimum area of transverse reinforcement within spacing s, respectively; V c calculated by Eq. (18) shall not be taken greater than 0.42 √ f ′ c bd ; P∕6A g in Eq. (18) shall not be taken greater than 0.05f ′ c ; s = √ 2∕(1 + 0.004d) ≤ 1 is the size effect modification factor; w = A s ∕A g , the value of A s to be used in the calculation of w may be taken as the sum of the areas of longitudinal bars located more than two-thirds of the overall member depth away from the extreme compression fiber; is the modification factor to reflect the type of the concrete, = 1 (17) Arch model (Pan & Li, 2013) 1 3 in this study for normal concrete (ACI Committee 318, 2019)). (18)

Specimen design
Four large-scale RC columns (named C1-0.58P, C2-0.58P, C3-0.1P, and C4-0.1P) were conducted and tested with the cross-section of (800 × 800)mm, and the clear height of 3.2 m. The actual strengths of concrete and reinforcement are presented in Table 1. During the test, the high axial ratio of 0.58 was applied for C1-0.58P and C2-0.58P columns, while the low axial ratio of 0.1 was applied for C3-0.1P and C4-0.1P specimens. The longitudinal bars were used 24#10 with the anchor heads SD420 that were attached at two ends to anchor the longitudinal bars into the top and  bottom bases. The transverse reinforcement was used #2 or # 3 with closed-hooks or cross-ties. To investigate the shear effect and avoid the buckling of longitudinal bars, a small spacing of 120 mm or 150 mm was applied for transverse reinforcement. In contrast, the transverse reinforcement of C4-0.1P was designed so that the amount of transverse steel is smaller than the requirement of the ACI318-19 code (ACI Committee 318, 2019). The details of all tested columns were presented in Fig. 4. The columns were designed as shear-controlled columns. To ensure the columns will fail in the shear failure mode, the ratio of nominal shear strength and the nominal flexure strength was controlled so that it would be approximately or greater than 1.0 (Anh Huy et al., 2022). The nominal shear strength was calculated according to ACI318-19 code (ACI Committee 318, 2019). And, the nominal flexure strength was determined by the fiber-based cross-sectional analysis via the OpenSees platform (University of Berkeley, 2020).

Test setup and instrumentation
The columns were tested by Multi Axial Test System (MATS) using double curvature, and lateral cyclic loading under constant axial load in NCREE in Taiwan. The layout of MATS is presented in Fig. 5. The MATS has a 6-DOF (Degree of Freedom) loading system for advanced seismic testing of structural components. And, it has two sets of vertical and horizontal actuators with capacities of 30MN and 4.5MN, respectively (Anh Huy et al., 2022;Chen et al., 2009). The columns were installed in the test system by fixing their top and bottom RC blocks to the steel plates of the MATS through multiple pre-stressed high-strength steel rods. A predetermined axial load was applied to the columns followed by the application of the prescribed horizontal displacement reversals on the foundation of the columns. During the test, the columns would deform in a double curvature configuration due to the boundary constraints applied by the MATS (Anh Huy et al., 2022). The applied axial load was controlled to be constant during the entire test. The displacement reversals were applied using a displacement-controlled loading protocol with a targeted displacement history conforming to FEMA461 (Federal Emergency Management Agency, 1997) shown in Fig. 6c. Each level was repeated twice to observe the stiffness and strength degradation. Specimens were applied the high and low axial compression load with the axial ratio of 0.58 and 0.1, as presented in Table 2. The top and bottom base was fixed by the strong platen by the high-strength rods. Sixty markers were used to record the displacement and rotation of column, top and bottom bases. The position of markers was arranged so that there were denser markers at near the top and bottom bases. The position of markers was presented in Fig. 6b. Besides, 24 strain gauges were attached on the longitudinal bars, and 21 strain gauges were attached on the transverse reinforcement to record the strains in reinforcement steel, as plotted in Fig. 6a. In addition, two LVDTs (linear variable differential transformer) and two incline meters were installed at the top and bottom bases to control the horizontal displacement and rotation.

Damage patterns
The damage patterns of all tested columns were presented in Fig. 6. For specimen C1-0.58P, the vertical cracks occurred at the drift of 0.25% and concentrated at the column ends. While the initial flexure and shear cracks appeared at the drift of 0.375%. The spalling of the concrete cover started at the drift of 0.5%, and significant spalling appeared at the drift of 0.75%. The peak load (V peak = 2543kN) occurred at the drift of 0.75%. From the drift of 1.0%, the number of cracks rapidly increased and lead to the failure at the drift of 1.5%. For specimen C2-0.58P, the vertical cracks were observed to occur at the drift of 0.25% and continued to develop until the drift of 0.75%. The shear and flexureshear cracks formed at the drift of 0.75%, and the spalling of the concrete cover was also observed in this cycle load. The column reached the peak load (V peak = 2286kN) at the drift of 0.88% and suddenly failed at the drift of 1.0%. For specimen C3-0.1P, the initial flexure-shear and shear cracks occurred at the drift of 0.25% with dominating by shear cracks. While the vertical cracks did not observe in this column. The spalling of the concrete cover started at the drift of 1.5% with large diagonal cracks. The peak load ( V peak = 1708kN) was obtained at the drift of 1.0%, and the test stopped at the drift of 2.0% due to failing. For specimen C4-0.1P, the shear cracks were mainly observed from the drift of 0.25%, while the initial flexure-shear cracks occurred at the drift of 0.375%. The spalling of concrete cover started at the drift of 0.5% and significantly spalled at the drift of 0.75% with some large and long diagonal cracks. Column C4-0.1P reached the peak load (V peak = 1468kN) at the drift of 0.718% and suddenly failed at 1.0% drift.
The tested results revealed that the presence of shear cracks was delayed in high ALR columns instead of the vertical cracks because of the effect of high compression load on closing the horizontal and inclined cracks. Furthermore, the effect of high ALR can be seen clearly by the failure patterns of columns C1-0.58P and C2-0.58P which mainly had vertical cracks until reaching failure, and the visible cracks were lesser than those of the counterparts. In contrast, excessively inclined cracks were observed in columns C3-0.1P and C4-0.1P. Especially, C2-0.58P was the only column in which the rupture of transverse reinforcement was observed after the tests, likely due to the combination of high shear and axial demands. The ultimate failure of columns C1-0.58P and C2-0.58P was caused by vertical cracks across the entire height of the columns accompanied by concrete crushing at the column ends. While the excessively inclined cracks were observed in low ALR columns.  Figure 6 presents the load-displacement hysteresis responses of all tested. It can be seen that all tested columns exhibited the poor hysteresis response presented by thin hysteresis loops and rapidly decrease after reaching the peak load. While high ALR columns (C1-0.58P, and C2-0.58P) exhibited rapid degrading the shear strength to compare with those of low ALR columns (C3-0.1P, and C4-0.1P), low ALR columns showed er drift capacity. This is likely due to the effect of high axial compression load. The significant pinching effect was found in the hysteresis loops after the strength declined below the nominal shear strength V n .

Load-displacement behavior
The drifts at which yielding of longitudinal and transverse reinforcement took place were also shown in Fig. 6. The tested results revealed that the transverse reinforcement yielded for all columns, and the longitudinal reinforcement yielded in tension for all columns except column C2-0.58P due to the large axial load demand and the shear-critical behavior (Fig. 7).    20) and (21), respectively, and illustrated in Fig. 8. (where N tly is the force in the top stringer at yielding; N bly is the force in the bottom stringer at yielding; M 0 , N 0 , V 0 are the pure bending, axial compression, and shear as presented in Fig. 8 respectively).
The interaction between bending-axial compression (M-N) was assumed as the parabolic function as presented in Eq. (22)   Parameters 1 and 2 are determined by tested, Abraham C Lynn et al. (1996), and H Sezen and Moehle (2002) data. To simplify, choosing 1 = 1 , 2 was solved by the conditions, i.e., V∕V 0 = 1 when M = 0 , satisfy Eq. (25), and be suitable with the experimental data. The interaction equation between M and V is illustrated in Eq. (27): Vh (shear span ratio for double curvatue) The bending-shear-axial compression interaction formula can be derived from Eqs. (23) and (27), and it can be rewritten using Eq. (28), where A, B, C are the constants that were solved from the satisfied conditions.  Fig. 10.

Equation verification
Using Abraham C Lynn et al. (1996), Sezen and Moehle (2002), and tested data to verify the accuracy of the proposed equation, as presented in Table 2. It can be seen that the results of the proposed equation had excellent agreement with experimental data (the mean of 1.01, and the COV of 0.07).

Concrete contribution
The RC columns fail in shear in two manners, i.e., diagonal tension and compression failure. While the diagonal tension failure was caused by the inclined crack due to the shear effect, the diagonal compression failure would occur crushing and spalling the concrete along the compressive strut. Based on the empirical stress equilibrium, Sezen and Moehle (2002) recommended that the contribution of concrete (V c ) would depend on the compressive strength of concrete (f � c ) , span-to-depth ratio (a∕d) , axial compression load (P) , and the effective area of cross-section (0.8A g ) , as illustrated in Eq. (31). Assuming that the confinement effect of poorly detail transverse reinforcement is very small so that it could be neglected.

Transverse reinforcement contribution
The experimental evidence indicated that the shear strength will increase with increasing the transverse reinforcement. MacGregor (1993) based on the truss model proposed the shear strength provided by transverse reinforcement as presented in Eq. (32). Federal Emergency Management Agency (1997), and ACI 318 (2019) (ACI Committee 318, 2019) take = 1 (corresponding to the crack angle is equal to 45-degree), while Priestley et al. (1994) take = cotg30 0 = 1.73 . However, the tested evidence showed that the crack angles were smaller than 45-degree. They also depended on the axial compression load, increasing the axial compression load leading to decreasing the crack angles. The crack angles of tested columns are presented in Fig. 12. (30)

Effect of displacement ductility
The displacement ductility ( ) was known as the ratio of the ultimate displacement (Δ u ) and yield displacement (Δ y ). Some researchers attempted to investigate the effect of displacement ductility on shear strength, e.g., Aschheim and Moehle (1992), Priestley et al. (1994), Abraham C Lynn et al. (1996), Halil Sezen (2002), and H Sezen and Moehle (2002). They concluded that the shear strength would be decreased when the displacement increased. Therefore, the shear strength contribution of concrete is multiplied by the k factor as presented in Fig. 11

Tested results and discussion
The measured shear strength of tested columns was presented in  (Sezen & Moehle, 2002;Halil Sezen, 2002) where s,test is the strain of transverse reinforcement that is measured by strain gauges, and is the average angle of inclined cracks. From the experimental evidence, the crack angle could be taken as ≈ 34 0 for low axial compression load, and ≈ 26 0 for high axial compression load, as illustrated in Fig. 12.
It can be found that the shear strength provided by concrete will increase when increasing the axial compression load. The shear strength comparison between tested results, ACI318 (2019), and Halil Sezen (2002)'s equation was implemented and presented in Table 3. The comparison results revealed that the ACI318 (2009)'s equation underestimated the shear strength for both high-and low-axial cases ( V test ∕V n,ACI = 1.09 and 1.38 for high and low ALR, respectively). While Sezen's equation underestimated ( V test ∕V n,Sezen = 0.85 ) and overestimated ( V test ∕V n,Sezen = 1.10 ) the shear strength for high and low ALR, respectively. The ratios V test ∕V ACI and V test ∕V Sezen in terms of concrete, transverse reinforcement, and total contribution were presented in Table 3. Figure 13 presents V c,test , V s,test , and V test versus drift ratios. It is worth noting that V c,test did not reach the maximum value at the peak lateral load. Furthermore, the strain in transverse reinforcement did not yield at the peak load. The average ratios of maximum stress and yield stress in transverse reinforcement ( test ∕f yv ) are 0.7 and 1.2 for high and low ALR, respectively.

Proposed shear equation
Based on the shear equation of Abraham C Lynn et al. (1996), Halil Sezen (2002, H Sezen and Moehle (2002), and experimental evidence, the author proposed the modified shear strength equation as illustrated in Eq. (34). From the comparison of V c and V s between ACI318-19 (ACI Committee 318, 2019), H Sezen and Moehle (2002), and tested results, and the structural effect, the shear strength provided by concrete and transverse reinforcement is modified by multiplying by the modified factor and , respectively, considering the size effect and inclined crack angles.
where k is the parameter considering the displacement ductility as presented in Halil Sezen (2002)

Verification of proposed equation
The accuracy of the proposed equation (Eq. (34)) was verified using the tested result and the 63-column test database

Fig. 13 V c,test , V s,test
, and V test of tested columns    (Anh Huy et al., 2022;H Sezen & Moehle, 2002). Moreover, its accuracy was compared to those of ACI318-19 (ACI Committee 318, 2019), H Sezen and Moehle (2002), and Abraham Christopher Lynn (2001)models, as expressed in Table 4. It can be seen that the proposed equation gave a better prediction for shear strength with the mean of 1.008 and the COV of 0.174.

Backbone curves for shear and flexure-shear critical columns
The force-displacement curve (backbone curve) of shearcritical columns was controlled by force with the peak force taken as the nominal shear strength ( V n ). In contrast, it would be the nominal flexure strength ( V mn ) for flexureshear critical columns. Therefore, they could be called the force-controlled curve and deformation-controlled curve for shear and flexure-shear critical columns, respectively. Based on the results of previous researchers that were presented in the literature part, the author proposed the backbone curve for shear and flexure-shear critical columns as presented in Fig. 14. The proposed curves were modeled by the simplified linear curve with three points or four points for shear critical and flexure-shear critical columns, respectively. The total displacements at each point were obtained from flexure (Δ f ) , shear (Δ shear ) , and slip (Δ slip ) displacements, as illustrated in Eq. (35).

Cracking point
The cracking point is known as the onset that the cracks begin occurring. At this point, the behavior of materials (concrete and reinforcement steel) is assumed as elastic behavior. Therefore, the flexure displacement ( Δ f ,cr ) could be obtained from the empirical structure equation, as presented in Eq. (36). The effective stiffness ( E c I e ) proposed from ACI318 (2019) and Li and Hwang (2017) as presented in Table 5. Similarly, the shear displacement at the cracking point ( Δ shear,cr ) derived from the empirical equation proposed by Park and Paulay (1975)  where V cr , M cr are the shear force and moment at crack point; V y is the yield load; L is the shear span ( L = H∕2 for double curvature); G is the shear modulus; d is the effective depth; c is the compression depth; d b is the longitudinal steel diameter; I, I e are the moment inertial of gross section and the effective moment inertial, respectively; u is the bond stress;

Peak point
The peak point is the onset when columns reached the peak shear force. The peak shear force of both shear and flexureshear critical columns was determined by the proposed equation, as presented in Eq. (34). The flexure displacement at peak point ( Δ f ,peak ) was obtained from Eq. (40) with E c I e taken from Table 5. The shear displacement at peak point ( Δ s,peak ) was calculated using Park andPaulay (1975)'s Eq. (1975), as presented in Eq. (41). The slip deformation ( Δ slip,peak ) was also determined by the proposed formula of (Sezen & Moehle, 2006;Halil Sezen & Setzler, 2008a, 2008b as plotted in Eq. (42).

Maximum strength point
The maximum strength point was only used for flexure-shear controlled columns with V peak gotten by the flexure strength (V mn = 2M n ∕H for double curvature, M n is the nominal flexure strength, and H is the clear height of the column). This point is the onset that columns reached the yield plateau. The flexure, shear, and slip displacement were also obtained by Eqs. (40-42).

Beginning of shear degradation point
The beginning of shear degradation is the onset that the shear strength of columns begins declining. While the flexure and slip displacement at beginning of shear degradation remain at their values, the shear displacement was calculated by the proposed equation of Elwood and Moehle (2005b) considering the effect of ALR (P∕A g f � c ) , shear stress ratio (v∕f � c ) , and transverse reinforcement ratio ( v ).

Collapse point
At the collapse point, the columns would be failed by the axial load. The displacement at this point was calculated by Elwood and Moehle (2005a). And, it is assumed that the shear force at the collapse point is equal to zero. The collapse displacement (Δ collapse ) was obtained from Elwood and Moehle (2005a) as presented in Eq. (44). Δ collapse considered the effect of the parameters, i.e., axial load (P) , area (A v ) and yield strength (f yv ) of transverse reinforcement, spacing of transverse steel (s) , compressive depth (d c ) , and angle of cracks ( ) ( was assumed 65-degree during the derivation of the model).

Verification of proposed backbone curve
The backbone curves of four tested columns and Lynn et al.'s experimental columns (Abraham C Lynn et al., 1996) were established by using the proposed process, as presented in Fig. 15. It is worth noting that the proposed curves were consistent with the tested and Lynn et al. (1996)'s curves.

Hysteresis loops for shear and flexure-shear critical columns
The tested columnswere simulated by the Opensees' model (University of Berkeley, 2020) with displacement-based beam-column element, as presented in Fig. 16. A series of three springs, i.e., shear, bar slip, and axial springs were put at the column ends. While the shear spring would be able to capture the stiffness and shear degradation, the slip spring will record the slip rotation, and the axial spring would determine the axial failure point. The shear, bar slip, and axial springs were used by 'Pinching4', 'Bond SP01', and ' limitCurve Axial ' materials, respectively. The Bond SP01 material is used to construct a uniaxial material object for capturing strain penetration effects at thecolumn-to-footing, while the ' limitCurve Axial ' material could conduct an axial limit curve that is used to define the point of axial failure for a ' LimitStateMaterial ' object (University of Berkeley, 2020).
According to Lee and Han (2018), the pinching behavior of monotonic and cyclic loading of the RC column are presented in Fig. 17. The pinching behavior is simulated by three parameters in the OpenSees platform, i.e., rDisp, rForce, and uForce . While rDisp and rForce control the pinching behavior of reloading and unloading branches, respectively, uForce will control the degree of pinching. Based on the previous experimental data, Lee and Han (2018) proposed algebraic equations to estimate these parameters, as illustrated in Eqs. (45)(46)(47). The shear and  where d max,i is the maximum deformation demand of the ith loading cycle; d f is the deformation at failure; E i is the dissipated energy at loading cycle i ; E t is an energy dissipation capacity up to the yield point in the monotonic backbone curve. Four parameters ( 1 , 2 , 3 , and 4 ) should be determined for estimating the degree of cyclic deterioration in strength, unloading, and reloading stiffness prior to simulating the cyclic curves. Furthermore, Lignos and Krawinkler (2011) and Haselton et al. (2016) implemented the simplified equation to estimate the damage index ( i ) , as illustrated in Eq. (49). Besides, Lee and Han (2018) also recommended the algebraic equation to calculate parameter 2 as expressed in Eq. (50): The pinching4 parameters of tested columns were calibrated as presented in Table 6, and the simulated hysteresis loops are illustrated in Fig. 18. Due to the consistency between the proposed shear equation (Eq. (34)) and test results, V max in Table 6 could be obtained from Eq. (34) or tested results. The residual strength (V r ) should be taken as 20%V max based on the recommendation of ASCE41 (2017). From Fig. 17, it can be seen that the hysteresis loops obtained from the proposed model matched well with the tested results and Lynn et al. (1996)'s experimental data.

Conclusions
The seismic response of four large-scale RC columns with low transverse reinforcement was investigated in this study. Based on the experimental results the following conclusions were drawn: 1. The high ALR columns failed under vertical cracks that occurred in entire columns, and the low ALR columns failed with large diagonal cracks. Increasing the ALR led to more brittle failure in all tested columns. 2. Based on the unified theory of Hsu and Mo (2010), it was found that the modified interaction equation describes the correlation between bending moment, shear, and axial load (M-V-N). The proposed equation gave accurate results that were verified by the tested results and experimental database (the mean of 1.01, and the COV of 0.07). 3. The axial compression load had a significant effect on shear strength and inclined crack angles. The shear strength increases when increasing the axial compression load. For the tested columns, the average value of inclined crack angles changed from 34-degree to 26-degree for axial load ratio varying from 0.1 to 0.58. 4. ACI 318 (2019) underestimated the shear strength of the tested columns for both low-and high-axial-compression load. On the other hand, Sezen (2002)'s equation underestimated the shear capacity under low axial compression load but overestimated the shear capacity under high axial compression load. 5. A revised shear strength equation was proposed considering the size effect and inclined crack angles. It could accurately estimate the shear strength of RC columns even when these columns had low transverse reinforcement. Its predictive capacity was verified using the tested results and an experimental database and validated with some other equation results (the mean of 1.008, and the COV of 0.174). 6. A simple process was proposed to build the backbone curves of shear and flexure-shear critical columns. The process required design engineers relying only on some simplified equations to linear connecting there or four relevant segments of the curve. 7. The proposed simulation model with three springs could capture the shear and stiffness degradation, bar slip effect, and transmission of axial compression load through the OpenSees platform. This model could simulate the hysteresis loops with good agreement with tested results and the experimental database.