A yield curvature model considering axial compression ratio

This study proposed a yield curvature prediction model considering axial compression ratio with exponential function which is improved on the basis of the specification effective yield curvature prediction model. Parametric moment–curvature curves approach was used to verify that the yield curvature is greatly affected by the section size, the yield strength of longitudinal steel bar and the axial compression ratio. On the basis, the yield curvature under different levels was obtained by Xtract and parametric moment–curvature curves approach, the result shows that with the increase of axial compression ratio, the yield curvature also increases, which is roughly a linear relationship. Subsequently, combined with the specification and considering the influence of axial compression ratio, a new yield curvature prediction model is proposed based on a massive sample space obtained by parametric moment–curvature curves approach. Moreover, by comparing with the experimental database of PEER, the accuracy of new prediction model is verified, and the result shows that the new prediction model is suitable for estimating yield curvature of square section columns.

Page 2 of 20 Zhu and Zhong Advances in Bridge Engineering (2023) 4:19 point corresponding to the yield curvature, but it is different from the definition of yield curvature, moreover, some scholars have considered the influence of axial compression ratio in the yield curvature prediction model (Hernández-Montes and Aschleim 2003;Zhong et al. 2022a).In the prediction model proposed by Hernández (2003), the yield curvature increases with the increase of axial compression ratio.The yield curvature prediction model proposed by Zhong (Zhong et al. 2022a) also considers the influence of axial compression ratio, the yield and effective yield curvature prediction model proposed by scholars and specifications is shown in Table 1.Although the yield curvature prediction models proposed by Hernández (2003) and Zhong (2022a) have been verified, there are still the following problems: (1) it is inconvenient to remember; (2) prediction model is not applicable when the axial compression ratio is 0.
Finite element analysis has become a popular choice among researchers for studying bridge columns due to its advantages in terms of resource efficiency and convenience, as compared to experimental methods.With the aid of finite element analysis software, scholars extensively investigate and optimize the design of bridge columns.Su (2019) obtained the yield curvature of bridge columns using Xtract and implemented it into a formula to calculate the displacement of steel reinforcement slip.Zhang (2020) studied the factors affecting the top displacement of high-strength column through ABAQUS.Aldabagh (2022) deduced the prediction model of drift ratio of different damage states by establishing samples with OpenSees.Therefore, Xtract was employed as a tool to validate the yield curvature in this study.
In this study, the yield curvature is first defined, and the section information of the columns is replaced by the expression by using the parametric moment-curvature (Mφ) curves approach (PMCA), and the M-φ curves can be obtained based on the plane hypothesis model and the stress-strain models of longitudinal reinforcement and concrete.On this basis, PMCA was employed to discuss the influence of section size (L), axial compression ratio (R ac ), longitudinal reinforcement ratio (ρ l ), stirrup reinforcement ratio (ρ s ), concrete compressive strength (f co ) and yield strength of longitudinal steel bar (f y ) on yield curvature, and then remove the parameters that have little effect on yield curvature.On this basis, some levels under different L, f y and R ac were set, and the yield curvature was obtained by Xtract and PMCA and compared.Moreover, some columns whose failure form is flexural failure are selected and the experimental yield curvature is obtained.The yield curvature prediction model considering the influence of R ac was Table 1 Summary of some prediction models ϕ * y = effective yield curvature; ϕ y = yield curvature;ε y = f y /E s ; f y = the yield strength of longitudinal steel bar;E s = the elastic modulus; L = section size; R ac = axial compression ratio; ρ s = stirrup reinforcement ratio; ρ l = longitudinal reinforcement ratio

Definition of yield curvature
The order of failure of the bridge column is as follows: the cracks first occur, then the first yielding of longitudinal steel bars is observed, followed by the spalling of the cover concrete, the crushing of the core concrete, and the fracture of the stirrup in the final stage.The section curvature corresponding to the first yielding of longitudinal steel bars is taken as the yield curvature (φ y ) of column in this study, as shown in Fig. 1.Strain can be used as an index to judge the damage state (Calvi and Kingsley 1995;Priestley et al. 1996).When the strain of the longitudinal steel bars reaches, it can be considered that the longitudinal steel bars has yielded for the first time, and the corresponding curvature is φ y .

Parametric M-φ curves approach
Zhong (2022a) proposed a parametric moment-curvature (M-φ) curves approach (PMCA), which transformed discrete longitudinal steel bars into 'steel loop' , the thickness of the 'steel loop' can be approximately represented to d 0 = (ρ l LW) / (2L + 2W-4C 0 ), where L and W are the size of the bridge column section, C 0 is cover concrete thickness, and so that three important parts (cover concrete, core concrete and longitudinal steel bars) are continuous sections.Each part can be represented by thickness information: T uc , T cc , and T s , as shown in Fig. 2.Then, based on the plane hypothesis model and the stress-strain model of concrete (proposed by Mander et al. (1988)) and steel (steel01), the balance between the axial force and the reaction force of the column is established and moment is obtained.The stress-strain model is shown in Fig. 3, where ε co , ε cc and ε cu are peak strain of cover concrete, peak strain of core concrete and ultimate strain of core The φ y prediction model proposed by Zhong (2020) does not consider the influence of f y , and the φ y cannot be calculated when the R ac is 0. Therefore, this study attempts to establish a φ y prediction model of six parameters, and then remove the parameters that have little influence on the yield curvature.9 6 (531,441) levels are established to fit the prediction model through PMCA, and the parameters information is shown in the Table 2.
The exponential function is used for fitting to directly obtain the influence of each parameter on φ y .The form of the expression is as follows: where a, a 1 , a 2 , a 3 , a 4 , a 5 , a 6 are coefficients of exponential function.The greater the absolute value of a 1 to a 6 , the greater the influence of parameters on φ y .The fitted expression is shown in Eq. 2.
(1) By comparing the fitted expression and PMCA in Fig. 4, the effect is not bad.From the fitted expression, φ y is significantly influenced by L and f y , and as f y increases or L decreases, the φ y increase accordingly, as reported in previous studies.Priestley ( 1998) reported that L and f y have a great influence on the curvature of the slight limit state, which is inversely proportional to L and proportional to f y .The coefficient of (1-R ac ) is also high, therefore, based on the effective yield curvature prediction model combined with the specification, this study proposes a yield curvature prediction model considering R ac .Hernández ( 2003) studied that as R ac increases, the curvature of the slight limit state also increases, which is consistent with Eq. 2. (2)

Finite element model
Xtract was used in this study to verify the accuracy of PMCA and provide a control group for subsequent studies.According to the main influence parameters obtained by PMCA, some levels of different L, f y and R ac are set, as shown in Table 3.
Xtract is an effective software for obtaining φ y , which can calculate the moment-curvature curve of column section and output φ y directly.Therefore, a square section column was established by Xtract, and the parameters of cover concrete, core concrete and longitudinal steel bars are defined.The cover and core concrete material model proposed by Mander et al. (Mander et al. 1988) are adopted in Xtract.In this study, the concrete grade is C50, the cover concrete strength is 32.4 MPa.The core concrete strength is 42.1 MPa in Test 1 and 42.8 MPa in Test 3. The yield strain of core concrete is 0.0035 in Test 1 and 0.00365 in Test 3. The elastic modulus of cover and core concrete is 3.45 × 10 4 MPa.The longitudinal reinforcement ratio is 1.13%, the yield strength of stirrup is 360 MPa, the stirrup ratio is 1%, and the thickness of cover concrete is 0.04 m, the other parameters are shown in Table 3, the units are divided according to the default length.

Comparison
By applying Xtract and PMCA, the yield curvature of all conditions in Table 3 is calculated and compared.On the one hand, the trend of yield curvature with the change of R ac can be seen.On the other hand, Xtract can be set as the control group to verify the accuracy of PMCA.The comparison results are shown in Fig. 5.The error rate = 100% × (PMCA-Xtract) / Xtract.
From the comparison of the two, the above two problems can be easily solved: (1) With the increase of R ac , the yield curvature also increases, which is roughly a linear relationship; (2) The calculation results of PMCA and Xtract are very close, the error between the two is about 3%, and the maximum is only 5%.Therefore, PMCA can replace Xtract as the main tool to obtain yield curvature in the follow study.In addition, considering that the influence of longitudinal steel bars into 'steel loop' , the yield curvature of PMCA is always smaller than Xtract.However, this is beneficial to engineering design and can leave a certain safety space for the project.
The research object of this paper is mainly the bridge column with flexural failure.In order to ensure that the bridge column is flexural failure, only the shear span ratio not less than 2.5 (H/L ≥ 2.5) and the square section bridge column of ordinary reinforced concrete are retained.Therefore, the stacked bar charts for the column physical parameters are shown in Fig. 7, f yh is the yield strength of stirrups; H/L is the column aspect ratio.

Fig. 6
The experimental columns parameter distribution of PEER database: a L; b R ac ; c f y According to the yield displacement method proposed by Priestley and Park (Eq. 3), the yield displacement of these experiments is marked in the hysteresis curve, and the results are shown in Fig. 8.

New prediction model
Considering that the effective yield curvature is close to the yield curvature, so the prediction model of yield curvature is derived based on the effective yield curvature prediction model of Chinese specification.Based on the accuracy of PMCA, and this study removes some less influential parameters to use easily, a new prediction model is proposed which combines the exponential function considering the influence of R ac with the Chinese specification.A large number of levels are obtained by PMCA, the parameters information is shown in Table 2.The new prediction model is not complex and is more suitable for researchers to use directly.The prediction model is as follows: where the first part is the Chinese specification effective yield curvature prediction model, b 1 and b 2 are the coefficients.The values of b 1 and b 2 are fitted by the least square method.Finally, the prediction model of φ y combined with Chinese specification is Eq. 5.The experimental axial compression ratios of the retained PEER database are all brought into the new prediction model, and the average value of the coefficients is calculated, the coefficient is 1.82, it shows that the yield curvature is about 0.1 times smaller than the coefficient of effective yield curvature (1.957). (3)

Application of the prediction model
In order to verify the accuracy of the new prediction model, the yield curvature obtained from the experiment is compared with the predicted, and the results are shown in the Table 4 and Fig. 9.
From the comparison in Fig. 9 and Table 4, it can be seen that the results of the experiment and the prediction are very close, and the error rate is kept within -20% ~ 20%, and most of them are less than 0%.This shows that the predicted yield curvature is often smaller than the experiment, which also provides a certain safety space for practical projects.In addition, the reason why some errors are relatively large may be that the data source of the new prediction model is determined according to the set condition range.There will be some errors outside the range, but the error is still guaranteed to be within a relatively small range.Therefore, the prediction model can be used to predict the yield curvature of practical engineering.

Conclusions
The main purpose of this paper is to study the influence of axial compression ratio on yield curvature, and propose a prediction model considering the axial compression ratio based on the effective yield curvature prediction model of Chinese specification.To achieve this goal, parametric moment-curvature curves approach (PMCA) is adopted to verify that section size, the yield strength of longitudinal steel bar and the axial compression ratio have great influence on yield curvature.Then, some levels under different section size, the yield strength of longitudinal steel bar and the axial compression ratio were set, and the yield curvature was obtained by Xtract and PMCA and compared, the result shows that the yield curvature increases with the increase of the axial compression ratio.Besides, a new prediction model is proposed by combining the exponential function with effective yield curvature prediction model of Chinese specification, the yield curvature of the database of PEER is compared with the new prediction model.Through the above research, the following conclusions can be obtained: (1) The numerical simulation result shows that yield curvature increases with the increase of the axial compression ratio, and the growth trend is obvious, which is different from the specifications effective yield curvature prediction model.
(2) The yield curvature prediction model proposed in this study can well predict the yield curvature of the experiment, and the error is between -20% and 20%, and most of them are concentrated near -10%.The negative error indicates that the prediction is smaller than the experiment, indicating that retaining a certain safety space and it is beneficial to the actual project.
In this paper, a simplified prediction model of yield curvature is proposed, which provides a convenient tool for the design of bridge columns in practical engineering.However, this paper only studies the square cross-section column and cannot predict the curvature of the limit state after the column yield.In the future research, we will make up for these deficiencies.

Fig. 3 a
Fig. 3 a Concrete04; and b Steel01 model

Fig. 7
Fig. 7 Stacked bar chart of column physical parameters

Fig. 8
Fig. 8 Yield displacement and yield curvature of experiment

Fig. 9
Fig. 9 Comparison between experiment and new model

Table 2
Sample space parameter information

Table 4
The comparison between experiment and new model