Under gravity loading, the cracking of RC bridge piers is typically minor and can generally be neglected, assuming gross-section stiffness is both reasonable and sufficient. However, most RC piers either reach or approach yielding, leading to significant cracking and stiffness degradation under strong seismic excitations. Therefore, it is crucial to consider the realistic stiffness of bridge piers in seismic stability analyses. To reflect the cracked state of RC piers during seismic events, major design codes (i.e., China JTG (JTG/T2 231-01-2020, 2008), Caltrans (2019), Eurocode 8 (2005), and AASHTO (2015)) recommend the use of effective section stiffness in place of gross stiffness. Generally, it can be derived from moment–curvature (M-ϕ) analysis as the effective section stiffness of RC pier and would be determined from secant slope of the idealized elastic–plastic M-ϕ curve between the origin and the idealized yield point as Equation 1 (JTG/T2 231-01-2020, 2008; Caltrans, 2019): E I eff − sec = M y ϕ y (1) where M _y and ϕ _y are the idealized yield moment and yield curvature of the idealized yield point in Figure 1, respectively. This point shall be obtained by balancing the areas between the actual and the idealized M-ϕ curves beyond the first reinforcing bar yield point, as shown in Figure 1.

The rotational degrees of freedom of a pier bottom are assumed to be fixed, and the moment and curvature vary linearly over the height of the pier. Namely, the effective section stiffness is constant over the height of the pier. Hence, the effective section stiffness EI _eff-sec can be regarded as the effective flexural stiffness EI _eff-flex.

There is wide consensus that the estimation of yield displacement and ES for seismic analysis of bridge piers should consider the bending deformation, shear deformation, and bar slip deformation. Therefore, the effective flexural stiffness derived from moment–curvature analysis will overestimate the ES of the bridge pier due to ignoring the influence of shear effect and slip effect. In this study, the ES of bridge piers is defined following the method proposed by Park (1989), based on the measured lateral force–displacement (F-Δ) envelope of the member level. This approach considers the stiffness reduction caused by the shear and slip effect and is considered more representative in capturing the actual yield state of reinforced concrete piers under seismic loading.

As shown in Figure 2, when the force–displacement envelope curve of CHRCPs is known, the detailed procedure for determining its equivalent yield point using Park’s method is depicted below. On the ascending branch of the curve, first identify point A corresponding to the lateral force F = 0.75Fmax, where Fmax represents the specimen’s maximum lateral force capacity. Point B is then determined as the intersection between line OA (connecting origin O and point A) and the horizontal line passing through point D (the peak point of the backbone curve). The equivalent yield point C (Δ_y, F _y) is finally identified as the intersection between the backbone curve and the vertical line passing through point B. The effective stiffness of CHRCPs can subsequently be calculated using the Δ_y and F _y values from line OC.

In the elastic state, the yield displacement Δ_y and effective stiffness (ES) EI _eff of cantilever pier can be defined as (Zheng and Li, 2013). Δ y = F y × L 3 3 E I e f f (2) E I eff = F y × L 3 3 Δ y (3) where L is the equivalent cantilever length of the bridge pier. Here, the EI _eff of bridge pier under cyclic lateral loading is taken as the mean value of the positive (push) and negative (pull) directions.

It should be noted that the yield displacement obtained from the F-Δ envelope is the sum of the bending deformation, shear deformation, and slip deformation. Then, the assumption that the shear deformation and slip deformation are transformed into equivalent flexural deformation is implicit when calculating the yield displacement of the cantilever pier with Equation 2. Therefore, the ES determined by Equation 3 will be smaller than the effective section stiffness (i.e., effective flexural stiffness) that only considers the bending deformation. Obviously, the factors that affect flexural stiffness also affect ES, and the influencing trend is consistent.

In the existing literature (FEMA 356, 2000; ASCE 41-06, 2007; ACI 318-19, 2019; Paulay and Priestley, 1992; Kumar and Singh, 2010; Haselton et al., 2008; Elwood and Eberhard, 2009; Berry et al., 2008; Zheng and Li, 2013; Wei et al., 2019), the ES of concrete columns is generally expressed as a fraction (EI _eff/E _c I _g) of the gross-section stiffness E _c I _g. Here, E _c is the elastic modulus of the concrete, I _g is the moment of inertia of the gross section. Based on this, various design specifications and researchers have proposed different models for estimating the ES of RC piers, with 10 representative models summarized in Table 1. In Table 1, models M1–M4 are primarily developed for building structures, while M7–M10 are more relevant to bridge applications. Models M5 and M6 apply to both buildings and bridges. In terms of parameter usage frequency across these models, the most commonly considered variables include axial load ratio, shear-span ratio, longitudinal reinforcement ratio, the normalized reinforcement index f _y d _b/L f c ′ , longitudinal reinforcement diameter, and section height.

Notably, M1–M8 were developed for solid-section piers, while model M9 addresses rectangular hollow sections, but the influence of the section hollow ratio is not considered in the M9 model. Therefore, it is necessary to evaluate the applicability and accuracy of existing ES models when applied to CHRCPs.

Many studies (Zahn et al., 1990; Yeh et al., 2001; Lee et al., 2015; Li et al., 2020; Whittaker et al., 1987; Unjoh and ASAZU, 1999; Chung et al., 1999; Hoshikuma and Priestley, 2000; Ranzo and Priestley, 2001; Zhu et al., 2009; Kim and Kang, 2012; Kim et al., 2014; Kim et al., 2016; Liang et al., 2021a; Liang et al., 2021b) have provided the quasi-static test data needed to evaluate the accuracy of various ES models to CHRCPs. To limit in the analyses to bridge pier, the selection of test data meets the following requirements: (1) cantilever pier; (2) flexural failure mode; (3) 2.5 ≤ L/D; (4) 0.006 ≤ ρ _l ≤ 0.06; (5) 0 ≤ P/A _g f _c '≤ 0.35; (6) 20 Mpa ≤ f _c ' ≤50 Mpa, where P is the axial force, A _g is the net cross-sectional area of pier, D is the outer diameter of pier, and the meanings of other symbols are the same as before. Based on these criteria, a total of 52 test specimens were selected from the literature. Among them, 50 piers were utilized to evaluate the performance of existing ES models and for calibrating the regression model proposed in this study. The remaining two specimens, named PS1-C (Yeh et al., 2001) and HC-O-100 (Kim and Kang, 2012), were reserved for independent verification of the proposed model.

It should be noted that although all selected CHRCPs experienced flexural failure, the maximum shear-span-to-depth ratio in the tests was limited to 6.1 due to experimental constraints. The ES values of bridge piers with significantly larger shear-span-to-depth ratios (such as tall-pier bridges) may exhibit some differences. All specimens were tested under idealized fixed-base conditions at the pier bottom. If the soil–foundation interaction is pronounced in actual bridges, the ES of piers may exhibit substantial variations. Furthermore, the test specimens did not account for factors such as the effects of concrete deterioration, steel corrosion, or construction quality on ES. Therefore, when applying the ES model proposed in this study to the aforementioned conditions, further appropriate modifications to the ES are necessary. However, such adjustments fall beyond the scope of this study.

The envelope of the measured lateral load–displacement relationship was corrected for P-delta effects to obtain the effective lateral force–displacement envelope for each pier. Then, the measured effective stiffness ratio (EI _eff/E _c I _g) was calculated using the method proposed by Park (1989). The maximum, minimum, mean, median, and coefficient of variation properties of the main design parameters and measured effective stiffness ratio of 52 CHRCPs are reported in Table 2. Table 2 shows that the measured ES ranges from 10% to 50% of the gross-section stiffness E _c I _g, and the average value of measured effective stiffness ratio is 0.21.

Table 3 presents the statistical results for the ratio of calculated-to-measured effective stiffness of 50 CHRCP specimens. Most of the existing ES models tend to overestimate the measured ES of CHRCPs; the calculated stiffness is 1.41–2.94 times that of the measured stiffness, and only the stiffness calculated by Wei is less, at 0.9 times that of the measured stiffness. Of these existing procedures, Haselton and Zheng provide the minimum overestimation of the measured ES, but still approximately 40% higher; bridge seismic codes (JTG/T2 231-01-2020, 2008; Caltrans, 2019; Eurocode 8, 2005) significantly overestimate the ES, approximately 1.8 times that of the measured stiffness; the Paulay model is 2.94 times that of the measured stiffness, which is the most severely overestimated.

In addition, the scatter of the results of the existing models is large. According to the test data of 50 CHRCPs, the coefficient of variation for all of these models ranges from 0.25 to 0.41. The coefficient of variation provided by bridge seismic codes is 0.25, and the other models are 0.32–0.41. Considering the maximum, minimum, mean, and coefficient of variation, Zheng and Wei provide relatively good models compared to the existing ES models. Because the mechanical behavior of a rectangular hollow pier is similar to a CHRCP, the average value of ES calculated by the Wei model derived from rectangular hollow piers is the closest to the measured result, but the coefficient of variation is as high as 0.36. To sum up, the existing ES models appropriate for design applications generally tend to overestimate the measured ES of CHRCP and are unacceptably inaccurate. To obtain a more accurate and stable ES model, the main factors affecting the ES of CHRCP must be further studied in detail.

To investigate the factors that influence flexural deformation (or effective flexural stiffness), taking the full-scale CHRCPs in China as the prototype (Yeh et al., 2001), a parameter analysis was carried out to consider the axial load ratio and longitudinal reinforcement ratio for a typical bridge pier. The following basic data were assumed: (1) pier outer diameter:1.5 m; (2) pier inner diameter:0.9 m; (3) cover to longitudinal reinforcement:4 cm; (4) transverse reinforcement diameter and spacing: 13 mm, 100 mm; (5) concrete compressive strength: f _c ' = 32 MPa; (6) yield strength of longitudinal reinforcement: f _y = 420 MPa; (7) axial load ratio: P/A _g f _c ' = 0 to 0.35 (8 levels); (8) longitudinal reinforcement ratio: ρ _l = 0.5%–4.0% (5 levels).

Based on the computed moment–curvature relationship, the effective flexural stiffness of the pier EI _eff-flex can be determined using the idealized yield moment M _y and idealized yield curvature ϕ _y, as shown in Figure 1. Therefore, the factors influencing effective flexural stiffness can be identified through the parameter analysis of M _y and ϕ _y. The moment–curvature curve was determined based on plane-section analysis using the concrete constitutive model by Mander et al. (1988) and a linear constitutive model for steel. The section division of CHRCP is shown in Figure 3. For the parametric study, eight levels of axial load ratio were considered for each longitudinal reinforcement ratio, resulting in a total of 40 analysis cases. To facilitate comparison and interpretation, the dimensionless equivalent yield moment M _Dy and dimensionless equivalent yield curvature ϕ _Dy are respectively defined as follows (Equations 8, 9): M Dy = M y f c ′ D 3 (8) ϕ Dy = ϕ y D ε y (9) where ε _y is the yield strain of longitudinal reinforcement, and the meanings of other symbols are the same as before.

As shown in Figure 4a, the equivalent yield moment is significantly influenced by both the axial load ratio and the longitudinal reinforcement ratio, exhibiting a clear increasing trend with the rise of either parameter. In contrast, it is observed that the dimensionless equivalent yield curvature is comparatively insensitive to variations in the axial load ratio and longitudinal reinforcement ratio. Figure 4b shows the average value of dimensionless equivalent yield curvature (ϕ _Dy = 2.18), along with reference lines representing ±15% of this mean. It is seen that most data, except those for low reinforcement ratios coupled with very high axial load ratios, fall within the ±15% limits. This suggests that the equivalent yield curvature remains essentially stable across a wide range of design parameters and is not affected by the flexural capacity of the section.

It should be noted that although the data were generated from specific CHRCP sizes and material strengths, the dimensionless results are expected to be broadly applicable to other CHRCP configurations and material strengths within the typical range used in standard design practice (Priestley et al., 2007).

Combined with the data in Figure 4 and Equation 1, the axial load ratio and longitudinal reinforcement ratio can be identified as the governing parameters for the effective flexural stiffness of CHRCPs. To directly quantify their influence, the ratio of effective flexural stiffness to gross-section stiffness β _g is defined by Equation 10. β g = E I eff − flex E c I g (10) where E _c is the concrete modulus of elasticity (E _c can be taken as 5000 f c ′ MPa (Priestley et al., 2007)), and I _g is the gross-section moment of inertia. The results are shown in Figure 5 for the ranges of axial load ratio and longitudinal reinforcement ratio considered. It will be seen that the effective flexural stiffness ratio varies between 0.09 and 0.84. The data shown in Figure 5 can be used to determine the effective flexural stiffness of the piers as a function of axial load ratio and longitudinal reinforcement ratio. Therefore, the ES (EI _eff) of CHRCPs will increase with the increase in axial load ratio (P/A _g f _c ^’ ) and longitudinal reinforcement ratio (ρ _l).

Note that Figure 5 shows that as the longitudinal reinforcement ratio increases, its influence on the effective flexural stiffness tends to weaken with higher axial compression ratios, indicating a certain interactive effect between the longitudinal reinforcement ratio and the axial compression ratio on the effective flexural stiffness. This interaction largely depends on how the longitudinal reinforcement ratio and axial compression ratio affect the idealized yield curvature, which can be intuitively observed from the consistent trends between Figures 4b, 5.

Compared with solid piers, the shear capacity of hollow piers is significantly reduced due to the presence of a hollow section, making the contribution of shear deformation to the equivalent yield displacement more pronounced (Sun et al., 2013). As the degree of hollowness increases, the shear capacity decreases accordingly. Thus, the effect of hollowness on the ES of the CHRCP should be considered. To describe the hollowness of hollow section, a dimensionless parameter α _g (hollow ratio) is introduced, which is defined as the ratio of the area of hollow part to the cross-sectional area of the hollow pier as if it were solid. For the CHRCP, α _g is given by Equation 11. α g = d / D 2 (11) where d is the inner diameter, and D is the outer diameter.

For the cantilever CHRCP, after introducing the dimensionless parameters α _g and β _g, Equation 5, used to calculate the flexural deformation of the yield displacement, can be rewritten as Equation 12. Δ f l e x = 64 M y L 2 3 β g π D 4 1 − α g 2 E c (12)

Equation 6 used to calculate the shear deformation of the yield displacement can be rewritten as Equation 13. Δ s h e a r = 4 M y k v π D 2 1 − α g G e f f (13)

For a circular section, the shape coefficient (k _v) can be taken as 0.85 (Elwood and Eberhard). The concrete effective shear modulus (G _eff) depends on the modulus of elasticity of concrete and Poisson’s ratio of concrete (υ); that is, G _eff = E_c/[2 (1+υ)]. For normal-weight concrete, the value of Poisson’s ratio can be taken as 0.2. Based on these assumptions, the ratio of shear deformation to flexural deformation can be simplified as Δ shear Δ flex = 0.53 β g 1 / 1 + α g × L / D 2 (14)

The parameter (1/ 1 + α g ) before the shear-span ratio (L/D) in Equation 14 is defined as the equivalent shear-span ratio coefficient, which can be regarded as the reduction coefficient of the shear-span ratio when calculating the ratio of shear deformation to flexural deformation. The hollow ratio is between 0 and 1; thus, the equivalent shear-span ratio coefficient is between 0.71 and 1.00; that is, the effect of reducing the hollow ratio on the shear-span ratio is limited. Figure 6 shows the relationship between hollow ratio and equivalent shear-span ratio coefficient. The analyzed CHRCP specimens exhibited hollow ratios of 0.25–0.77 (mean = 0.45) and equivalent shear-span ratio coefficients of 0.75–0.89 (mean = 0.83).

From Equation 14 and Figure 6, it can be seen that the contribution of shear deformation to equivalent yield displacement decreases with the increase in shear-span ratio (L/D), and increases with the increase in hollow ratio (α _g). Therefore, the ES of the CHRCP increases with the increase in the shear-span ratio and decreases with the increase in the hollow ratio. The coupling effect of shear-span ratio and hollow ratio can be reflected through the equivalent shear-span ratio 1/ 1 + α g (L/D), and the ES (EI _eff) increases with the increase in the equivalent shear-span ratio.

For the purpose of this study, a uniform bond stress of u _b = 1.0 f c ′ MPa is assumed in the elastic range (Sezen and Setzler, 2008). Based on Equations 5,7, the ratio of slip deformation to flexural deformation can be expressed as Δ slip Δ flex = 3 8 × d b f y L f c ′ (15)

It can be seen from Equation 15 that the contribution of slip deformation to equivalent yield displacement increases with the increase of parameter f _y d _b/L f c ′ . Therefore, the ES (EI _eff) of the CHRCP decreases with the increase of f _y d _b/L f c ′ .

According to the aforementioned parametric study based on the three-component yield displacement model, the main factors influencing the ES of CHRCP are the axial load ratio (P/A _g f _c '), the longitudinal reinforcement ratio (ρ _l), the equivalent shear-span ratio 1/ 1 + α g (L/D), and f _y d _b/L f c ′ . Furthermore, the ES increases with the increase of all influencing factors except f _y d _b/L f c ′ . To verify the theoretical results, Figure 7 shows the influencing trend and correlation coefficient of main factors on the measured ES ratio (EI _eff /E _c I _g) of 50 CHRCPs. It can be seen from Figure 7 that the variation trend of measured ES with the axial load ratio, longitudinal reinforcement ratio, and equivalent shear-span ratio is consistent with the theoretical analysis.

The research on solid piers by Zheng and Li (2013) shows that the correlation coefficient between the parameter f _y d _b/L f c ′ and the measured ES ratio is −0.54, indicating that the experimental results are consistent with the theoretical analysis. However, for the CHRCPs in this work, the correlation coefficient between the parameter f _y d _b/L f c ′ and the measured ES ratio is only 0.02, as shown in Figure 7, indicating that their correlation is extremely weak and inconsistent with the theoretical results.

The possible reasons are as follows: first, the calculation model of slip deformation is derived from a solid pier with a single layer longitudinal reinforcement, which is not necessarily suitable for a hollow pier with inner and outer longitudinal reinforcement; second, the cross-section is discontinuous due to the hollowness; thus, the assumption that the bottom section of pier rotates around its neutral axis is not completely consistent with the actual situation when the longitudinal reinforcement slipping. Thus, the impact of hollowness on slip deformation remains unclear, requiring further theoretical and experimental investigation (Wang et al., 2019).

Based on the theoretical analysis and experimental validation of the main factors influencing the ES conducted in this study, as well as a comprehensive review of the governing parameters considered in existing models, a new four-parameter calibration model for the ES of CHRCPs is proposed. E I eff E c I g = λ 1 + λ 2 P A g f c ′ + λ 3 ρ l + λ 4 1 1 + α g L D ≤ 1.0 (16) where λ ₁∼λ ₄ are the parameters to be calibrated. Because the axial load ratio, longitudinal reinforcement ratio, hollow ratio, and shear-span ratio of hollow pier are relatively easy to determine, Equation 16 is suitable as the calculation model of ES of CHRCP.

Based on the measured ES ratio EI _eff/E _c I _g, axial load ratio P/A _g f _c ^’ , longitudinal reinforcement ratio ρ _l, and equivalent shear-span ratio 1/ 1 + α g (L/D) of 50 CHRCPs collected in this manuscript, the parameters λ ₁∼λ ₄ are calibrated using multiple linear regression in SPSS version 25 software package, and the calibration results are as follows: E I eff E c I g = − 0.192 + 1.014 P A g f c ′ + 6.680 ρ l + 0.058 1 1 + α g L D ≤ 1 . 0 (17)

Because the regression coefficients in Equation 17 were determined using ordinary least squares (OLS) linear regression, the independent variables were not normalized or standardized prior to regression. Figure 8 shows the comparison between the ES ratio of CHRCP calculated by Equation 17 and the measured values. It can be seen from Figure 8 that the calculated results are in good agreement with the experimental results, and the linear correlation coefficient is 0.94. Therefore, within the range of design parameters picked up in this work, Equation 17 gives an accurate estimate of ES of CHRCPs.

It should be noted that the linear model was adopted for the ES model in this study due to its advantages of concise form and convenience for engineering applications. Although Equation 17 incorporates the coupling effect of shear-span ratio and void ratio through the equivalent shear-span ratio, the linear model does not account for potential interactions or nonlinearities between parameters such as axial compression ratio and longitudinal reinforcement ratio. This may somewhat compromise the accuracy of the ES estimation. To address this limitation, future research should collect more experimental data and employ methods such as machine learning algorithms or nonlinear regression. This would enable a more profound insight into the potential interaction effects and nonlinear influences among parameters, facilitating the development of a more precise predictive model for ES.

To compare the proposed model with the Zheng model (Zheng and Li, 2013) and the Wei model (Wei et al., 2019), which give relatively accurate results in the existing ES models, Table 4 presents statistical parameters comparing the calculated-to-experimental ES values of 50 CHRCPs using Zheng’s formula, Wei’s formula, and the proposed formula. The evaluated metrics include maximum (Max), minimum (Min), median (Med), mean (Mean), coefficient of variation (CV), root mean square error (RMSE), mean absolute percentage error (MAPE), and coefficient of determination (R²). Figure 9 shows the ratio of calculated-to-measured ES for each pier. It can be observed from Table 4 and Figure 9 that: (1) The ES of CHRCPs calculated by the proposed formula Equation 17 demonstrates the closest agreement with the experimental results. The corresponding mean value and coefficient of variation are 1.04 and 0.21, respectively. Moreover, the proposed formula achieves the smallest RMSE (0.04) and MAPE (16.7%), along with the highest R ² (0.88), indicating superior error statistics and significantly enhanced prediction accuracy compared to the other two. For CHRCP, the calculated mean of the Wei model is close to the experimental value, which is 0.90, but the coefficient of variation is as high as 0.36; the results of the Zheng model are relatively poor in terms of mean and coefficient of variation, which are 1.40 and 0.32, respectively. (3) The fluctuation trends of the ratio of calculated to the measured ES given by the Zheng model, the Wei model, and Equation 17 are basically the same, and the fluctuation amplitudes (the difference between the maximum and minimum value) are 1.44, 1.14, and 1.09, respectively. Furthermore, most of the calculation results of Equation 17 fall between the results of the Zheng model and the Wei model. (4) It is noteworthy that, although discrepancies exist in prediction accuracy among the Zheng, Wei, and proposed models, all three models exhibit consistent increasing and decreasing trends across the 50 circular hollow pier specimens (as shown in Figure 9). This coherence demonstrates that these models collectively capture the sensitivity of ES to dominant parameters (axial compression ratio, longitudinal reinforcement ratio, shear-span ratio, etc.), reflecting the fundamental mechanical principles governing circular hollow piers. Moreover, this consistency implicitly validates the reliability of both the experimental dataset and data processing methodology.

Therefore, compared with the existing models, the model proposed in this work is more reasonable and stable for estimation of the ES of CHRCPs. In addition, the ES model proposed in this work is easy to apply and only requires determination of the axial load ratio, the longitudinal reinforcement ratio, the hollow ratio, and the shear-span ratio. It does not require determination of the longitudinal reinforcement diameter and yield strength, etc., and does not require complex moment–curvature analysis.

To further illustrate the validity of Equation 17, except for the 50 CHRCPs used for model calibration, another two piers (full-scaled pier PS1-C (Yeh et al., 2001) and scaled pier HCO-100 (Kim and Kang, 2012)) are used to verify the validity of Equation 17 on the simulation of lateral force–displacement response. The main design parameters of piers PS1-C and HCO-100 are shown in Table 2.

The overall force–displacement hysteretic curves for both piers were simulated based on a static reversed-cyclic analysis using OpenSees. In this manuscript, the piers are modeled as a Beam with Hinges element (OpenSees, 2024) proposed by Scott and Fenves (2006). The model assumes that the inelastic deformation is concentrated in the length L _p of the plastic hinge at the bottom of the pier, while the upper part of the plastic hinge remains elastic. When a plastic hinge forms at the pier base (i.e., when the base-section curvature exceeds the idealized yield curvature), the equivalent plastic hinge length L _p remains constant, as illustrated in Figure 10. To match the experimental loading conditions, the bottom node of the Beam with Hinges element was fully fixed, while the top node remained unconstrained to allow application of axial force and lateral displacement (or force). It should be noted that the Beam with Hinges element is particularly suitable for quasi-static pushover analysis of piers exhibiting typical flexural failure modes. The two most critical parameters of this model are the plastic hinge length L _p and the ES EI _eff, as shown in Figure 10. In the numerical simulation of fiber element based on the principle of curvature integration, it is more reasonable to use the plastic hinge length that matches the plastic curvature derived from moment–curvature analysis (Li et al., 2016). Therefore, in the static reversed-cyclic analysis, the plastic hinge length L _p is determined by Equation 18 suggested by Li et al. (2016), and the ES is calculated by Equation 17. The calculated values of the ES ratios (EI _eff /E _c I _g) of piers PS1-C and HCO-100 are 0.236 and 0.161, respectively, while the corresponding measured values are 0.216 and 0.177, and the relative errors are not larger than 10%. L p = 5.65 ρ l L + 0.325 D + 0.09 f y d b f c ′ (18)

For each pier, a fiber section composed of unconfined concrete, confined concrete, and steel material was initially established. In this study, the Concrete02 material model in the software of OpenSees was adopted for unconfined concrete and confined concrete, and the Reinforcing Steel uniaxial material model was used to simulate the longitudinal reinforcement. This model considers the mechanical effects of strain softening, low-cycle fatigue, and tensile fracture of the bars (Kunnath et al., 2009). The division of fiber section and the constitutive relationship of uniaxial material are shown in Figure 10.

Figure 11 shows the comparison between the simulated results and the experimental results of the force–displacement hysteresis curves of the full-scale pier PS1-C and scaled pier HCO-100. For full-scale and scaled bridge piers, the calculated results are in good agreement with the experimental results, and the initial stiffness, unloading stiffness, and reloading stiffness under low-cycle loads can be simulated accurately. These results indicate that the ES model proposed in this manuscript is reasonable and reliable to estimate the ES of CHRCPs.

Bridge seismic design codes, such as Caltrans, Eurocode 8, and China’s JTG, typically recommend using the effective flexural stiffness as the ES of the bridge pier. For the two representative CHRCP specimens, PS1-C and HC-O-100, the effective flexural stiffness ratio (EI _eff-flex /E _c I _g) are 0.383 and 0.278, respectively. These values indicate that the ES values prescribed by the seismic codes are approximately 1.62 and 1.73 times greater than the ES estimated by the model proposed in this work. This overestimation implies that using effective flexural stiffness as a proxy for overall ES can lead to a substantial underestimation of displacement demands under the same level of horizontal seismic loading. In practice, this means the predicted displacements will be smaller than those observed experimentally, which may compromise the reliability of deformation assessments and increase the risk of unseated spans. While a detailed investigation into span unseating risk is beyond the scope of this study, it highlights the critical importance of using a more accurate estimation of pier stiffness in seismic design. Moreover, the need for accurate ES estimation becomes even more critical in nonlinear time-history analyses, where the displacement response of bridge structures is highly sensitive to the peaks and troughs of the ground motion response spectrum.

Round-ended hollow piers are widely employed in railway bridges in China. To investigate the seismic performance of such piers, Shao et al. (2019) conducted low-cycle reversed loading tests on five 1/6-scale specimens with varying volumetric stirrup ratios and axial load levels, with specimen geometry and reinforcement layout illustrated in Figure 12. Jiang et al. (2024) performed similar tests on 1/10-scale specimens, considering variations in pier height, axial load ratio, longitudinal reinforcement ratio, and volumetric stirrup ratio. This section evaluates the applicability of the proposed ES model Equation 17 for round-ended hollow piers based on estimated ES values from 11 specimens in Shao et al. (2019) and Jiang et al. (2024). The key design parameters are summarized in Table 5, and the measured ES values were computed using the method proposed by Park (1989), as previously described.

Table 6 presents a comparison between the measured ES ratios (EI_eff/E_cI_g) and predictions from the Zheng model, the Wei model, and Equation 17. The results demonstrate that Equation 17 provides significantly better predictions, showing a mean ratio of calculated-to-measured ES of 0.976 with relative errors (RE) ranging from 4% to 21% (mean RE = 14%). In contrast, the Zheng and Wei models substantially overestimate the ES, yielding mean ratios of 1.510 (mean RE = 54%) and 1.225 (mean RE = 29%), respectively. This confirms the proposed model’s superior accuracy for round-ended hollow piers compared to existing approaches.

However, Equation 17’s predictions exhibit some variability, with a maximum RE of 21%, indicating notable sensitivity to certain parameters. Given the limited sample size, further experimental or numerical validation is recommended to better understand this sensitivity. Based on test conditions in Shao et al. (2019) and Jiang et al. (2024), the following parameter ranges are suggested for application of Equation 17 to round-ended hollow piers: a shear-span ratio of 2.5–6.5, a section hollowness ratio of 0.55–0.65, an axial load ratio of 0.05–0.15, and a straight segment length of section not exceeding the outer radius of the rounded ends.