Introduction

Lateral displacement structures are affected by both \(P - \Delta\) and \(P - \delta\)effects. Because the deflection second-order effect of axially stressed reinforced concrete columns is geometric nonlinear, concrete structures often present material nonlinearity due to cracking, crushing, and steel reinforcement yield. Geometrical nonlinearity and material nonlinearity are the basic characteristics exhibited by reinforced concrete columns during loading. Currently, the elastic finite element method (FEM) for calculating the elastoplastic second-order effect is primarily used, and the accuracy of the FEM depends primarily on the accuracy of the reduced stiffness of a component. The American specification ACI 3181 and Canadian specification CSA-A23.3.042 propose a flexural rigidity reduction factor of 0.70 for concrete columns, whereas the New Zealand Specification NZS31013 specifies different rigidity reduction factors based on the axial compression ratio range. Meanwhile, the Chinese specification GB500104 proposes a rigidity reduction factor of 0.6 for concrete columns. These specifications provide the unified equivalent rigidity reduction factor of reinforced concrete columns, which is primarily used to calculate the flexural rigidity reduction when the second-order \(P - \Delta\) effect is calculated, and cannot reflect the change rule wherein the flexural rigidity of reinforced concrete columns gradually decreases with the development of plasticity. Currently, studies regarding the rigidity reduction of the second-order \(P - \delta\) effect of reinforced concrete compression columns are few. Haselton5 and Elwood6 conducted quasi-static tests on reinforced concrete columns and bridge piers, considering the axial compression ratio and shear span ratio as the primary influencing factors, and proposed a calculation formula for the effective stiffness of piers, the analysis is based on specific test conditions, and its general applicability as well as the level of detail in characterizing stiffness degradation throughout the entire loading process remain to be validated. Sapkota7 constructed an optimized machine learning algorithm model for predicting the effective stiffness of reinforced concrete columns based on a large number of nonlinear dynamic result samples, to evaluate the effect of reducing the stiffness of cracked sections under nonlinear dynamic loads. however, it fails to adequately address stiffness degradation throughout the full plasticity development process—encompassing yielding and crushing—under static or quasi-static conditions. Kumar et al.8, BERRY et al.9, Zheng Gang et al.10, Wei Wang et al.11, and Hou Zequn et al.12 analyzed the influence of factors such as axial compression ratio, shear span ratio, and reinforcement ratio on the bending stiffness of bridge piers using finite element numerical calculation methods, and proposed different forms of effective stiffness regression models. However, it lacks a clear theoretical framework characterizing the mechanism of stiffness degradation. Based on the principle of equal inelastic lateral displacement and using the variable-order frame columns of single-span industrial buildings, Liu13compared the calculation results of lateral displacement of the top of a frame based on nonlinear and elastic second-order analyses, consequently, a rigidity reduction factor of 0.6 was proposed for the frame, this value is specifically targeted at bent columns within particular structural configurations and cannot be directly extrapolated to general frame columns. Liu14 used a nonlinear FEM to analyze the change rule of the critical load of an inelastic single rod model corresponding to a frame column under the effects of various main factors and derived a corresponding formula for calculating the stability bearing capacity to consider the change rule of the elastoplastic rigidity of the frame column. Manigandan15,16,17 et al. analyzed the influence laws of local buckling, concrete stress and ductility of the steel pipe diameter thickness ratio in the high hollow ratio CFDST column through axial compression tests and finite element simulation, and corrected the AISC bearing capacity formula to consider the stiffness change implied by the strength reduction after buckling. The strength reduction coefficient model and the limit value of the clearance ratio were proposed to evaluate the constraint failure and performance degradation caused by the clearance. The influence of the diagonal bracing stiffening scheme and nanomaterials on the bearing capacity, stiffness and ductility was explored.

In summary, current research still lacks a practical calculation method with good theoretical basis and applicability that can accurately reflect the dynamic attenuation law of elastoplastic flexural stiffness of reinforced concrete eccentrically compressed columns under load (accompanied by the coupled development of geometric and material nonlinearities). Accordingly, addressing the deficiencies of current code simplification methods and existing research in describing the dynamic degradation of elastoplastic stiffness of reinforced concrete columns, this paper studies and analyzes the main factors influencing the elastoplastic flexural stiffness of reinforced concrete columns and conducts experiments on eccentrically compressed reinforced concrete columns. By analyzing the relationship between bending moment and curvature, the variation law of elastoplastic flexural stiffness of reinforced concrete compressed columns is investigated. A calculation formula for the reduction coefficient of elastoplastic flexural stiffness of eccentrically compressed reinforced concrete columns is proposed, which can consider the influence of geometric nonlinearity and material nonlinearity of reinforced concrete columns under load. This can deepen the understanding of the complex behavior of reinforced concrete columns under combined compression-bending loads, and also provides an important theoretical basis and practical computational tool for more accurately evaluating their load-bearing capacity and deformation capacity, particularly the influence of second-order bending effects.

Main factors affecting flexural rigidity of reinforced concrete columns

The loading process of reinforced concrete columns presents geometric and material nonlinearities. The essence of geometric nonlinearity is to consider the additional internal force caused by component deformation and that an increase in the internal force increases deformation18. During loading, the concrete will crack, the steel bar will yield, and other inelastic characteristics will cause reinforced concrete columns to present material nonlinearity, which further reduces the flexural rigidity of concrete components and increases deformation. Additionally, nonlinearity interactions between material and geometry significant reduce the rigidity of components, causing the latter to lose their bearing capacity eventually19.

Fig. 1
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Simply supported constrained model column.

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Figure 1 shows a simple supported model column with equal eccentricity under load at both ends, where e0 and \(\delta\) indicate the first- and second-order eccentricity, respectively, and the deflection in the middle of column is the second-order \(P - \delta\) effect. The effects of geometric and material nonlinearities must be considered when investigating the flexural rigidity of reinforced concrete columns under load. The different position of load \(P\) on a concrete column will result in different failure forms, i.e., large eccentric compression and small eccentric compression failure. Eccentricity \(e_{0} /h\)was introduced to account for the effect of the position of load \(P\) on the flexural rigidity of the column. Reinforced concrete columns were equipped with steel bars, and the number of steel bars affected the flexural rigidity of the column. Considering the effect of the reinforcement amount on the flexural rigidity of the column, a reinforcement ratio \(\rho {\text{ = }}A_{s} /bh\) was introduced. The strength of steel bars and concrete affected the cracking of the concrete column and the yielding of the steel bar, and hence the flexural rigidity of the reinforced concrete column.

Fig. 2
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Strain of column concrete and steel bars.

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Based on Fig. 2, the geometric relationship of strain in the rectangular section of the reinforced concrete column can be obtained as follows20:

$$\Phi = h\phi = \varepsilon _{{c1}} - \varepsilon _{{c2}}$$
(1)
$$EIy^{{''}} = - M = EI\phi = EI\frac{{\varepsilon _{{st}} - \varepsilon _{{sc}} }}{{h_{0} - a_{s}^{'} }}$$
(2)

Where \(\Phi\) is the dimensionless curvature; \(\phi\) is the curvature; \(\varepsilon _{{c1}}\) and \(\varepsilon _{{c2}}\)are the tensile and compressive strains of concrete, respectively; \(\varepsilon _{{st}}\) and \(\varepsilon _{{sc}}\) are the tensile and compressive strains of steel bars, respectively; and \(EI\)is elastoplastic flexural rigidity of reinforced concrete columns.

Based on Eq. (2), the change in the bending moment \(M\) and curvature \(\phi\) of the reinforced concrete eccentric compression column can reflect the change in the flexural rigidity of the column, and the slope of the bending moment-curvature change curve represents the section flexural rigidity. As the load increased, the \(M\) of the reinforced concrete column increased gradually. When the strain at the edge of the concrete exceeds the ultimate tensile strain of concrete, the concrete column will crack and present plasticity. Meanwhile, as the load increases, the strain of reinforced steel bars will gradually increase or the steel bar will yield. During column loading, the change rule of flexural rigidity of the reinforced concrete column can be investigated by analyzing the slope of the \(M\) vs. \(\phi\) change curve of the column. To determine the change rule of flexural rigidity of the reinforced concrete column, an eccentric compression column test of the reinforced concrete was designed.

Test for eccentric compression column of reinforced concrete

Owing to the effects of the strengths of the steel bar and concrete on reinforced concrete columns, the bearing capacity of the column increased, cracks appeared later, and the crack width was smaller; therefore, two basic factors, i.e., eccentricity and reinforcement ratio, were mainly considered in the test. Nine reinforced concrete specimens with a length of 2200 mm were fabricated. The specimens were symmetrically reinforced with rectangular cross-section, i.e., C25 for concrete, and C 8@75/150 for the stirrup. The parameters of the test components are shown in Table 1.

Table 1 List of test parameters of reinforced concrete eccentric compression components.
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To prevent the local compressive failure of concrete at the end of the specimen during loading, 20-mm-thick steel plates were attached at the upper and lower ends of the specimen21. A schematic diagram of the reinforced concrete specimens is shown in Fig. 3a, and the test loading device is illustrated in Fig. 3b.

Fig. 3
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Sample reinforcement and loading device.

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Load-deflection relationship curve

In this test, monotonic loading and equal displacement loading were performed, and the specimens showed typical characteristics of large eccentric compression and small eccentric compression in the reinforced concrete components. The failure mode of the specimen is shown in Fig. 4.

Fig. 4
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Damage mode of test samples A-1 to A-9.

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During the test, the reinforced concrete specimens were deflected laterally during the vertical loading. The mid-span deflection value was used to measure the second-order effect of components22. To better understand the elastoplastic flexural rigidity change rule of the reinforced concrete eccentric compression columns, a curve depicting the relationship between the load and deflection was constructed using the loading values at all levels and the corresponding mid-span deflection values obtained during the test, as shown in Fig. 5.

Fig. 5
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Load-deflection curve of tests on A-1 to A-9.

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Based on Fig. 5, A-1 to A-6 are components with large eccentric compression, whose load-deflection relationship curve can be classified into four stages: the first stage is the elastic segment before the cracking of the concrete column, i.e., when the curve is the steepest, the slope is the largest, and the column flexural rigidity is the largest; the second stage is the cracking of the concrete column, where the nonlinear characteristics of the material are presented and the deflection increases nonlinearly. As the load increases, the number of cracks and crack width increase gradually, and the plastic characteristics of the column become increases evident. At this time, the tangential slope of the curve decreases, the deflection increases faster than the increase rate of the load, and the flexural rigidity of the column decreases; the third stage is the approximate horizontal stage, during which the reinforcing bars yield. At this point, the load increases slightly, and the deflection will increase significantly; meanwhile, when the slope of the curve is approximately zero, the flexural rigidity of the column decreases significantly.

A-7 to A-9 are components with small eccentric compression. Their load-deflection relationship curve is not classified clearly in the ascending segment and the tangent slope of the curve does not change significantly. Meanwhile, the concrete column is in the elastic stage before cracking, and the flexural rigidity of the concrete column is elastic at this time; the concrete column indicates plasticity after cracking, and the plastic development accelerates as the load is further increased. At this time, the tangent slope of the curve as well as the flexural rigidity of the column decrease. The characteristics of the small eccentric compression column in the horizontal stage are not obvious, and the concrete column enters the descending section after yielding to the limit value and loses its rigidity.

Bending moment-curvature curves for different eccentricities

A change in the position of the load \(P\) exerting on the concrete column results in different damage modes of the concrete column. The effect of this change on the rigidity of the concrete column can be reflected by the eccentricity \(e_{0} /h\). To compare the effect of eccentricity on the flexural rigidity of the reinforced concrete eccentric compression columns, two sets of bending moment-curvature relationship curves with different eccentricities based on the same reinforcement ratio were constructed using the data measured from the aforementioned reinforced concrete eccentric compression column tests, and the relevant specimens selected were shown in Table 2.

Table 2 Selected test components with different eccentricities.
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Specimens for groups ① and ② were selected, and the bending moment-curvature relationship curve from zero to the limit load under vertical loading for the groups was plotted, as shown in Figs. 6 and 7, respectively.

Fig. 6
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Bending moment-curvature curve of Group ①.

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Fig. 7
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Bending moment-curvature curve of Group ②.

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Based on Figs. 6 and 7, each moment- curvature curve is in the elastic stage before the component cracked, and the slope of the curve is the largest at this stage. The slope of the curve represents the flexural rigidity of the component, therefore, the flexural rigidity of the component is the largest at the abovementioned stage; as the component cracks into the elastoplastic stage, the slope of the curve becomes smaller, which implies that the flexural rigidity of the component decreases. In addition, as the eccentricity of the component increases, the slope of the three curves decreases gradually, which implies that the flexural rigidity of the component decreases; meanwhile, as the eccentricity decreases, the slope of the three curves increases gradually, which implies that the flexural rigidity of the component increases.

Bending moment-curvature curves for different reinforcement ratios

To compare the effect of reinforcement ratio on the flexural rigidity of the reinforced concrete eccentric compression columns, two sets of bending moment-curvature relationship curves with different reinforcement ratios at the same eccentricity were constructed using the data measured in the aforementioned tests, and the relevant specimens selected were shown in Table 3.

Table 3 Selected test components with different reinforcement ratios.
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Specimens for group ① and ② were selected, and the bending moment-curvature relationship curve from zero to the limit load under vertical loading was plotted for both groups, as shown in Figs. 8 and 9, respectively.

Fig. 8
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Bending moment-curvature curve of Group ①.

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Fig. 9
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Bending moment-curvature curve of Group ②.

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Based on Figs. 8 and 9, the slope of each curve is larger before the component cracked. As mentioned earlier, the slope of the curve represents the rigidity of the component. As the crack progresses into the elastoplastic stage, the slope of the curve becomes smaller, which implies that the rigidity of the component decreases. In addition, as the reinforcement ratio increases, the slope of the three curves increases gradually, which implies that the rigidity of the components increases; meanwhile, as the reinforcement ratio decreases, the slope of the three curves decreases gradually, which implies that the rigidity of the components decreases.

Calculation of flexural rigidity reduction factor of reinforced concrete columns

Trilinear model of flexural rigidity change

The flexural stiffness of the reinforced concrete columns in specimens A1 to A9: During the loading process, before cracking occurs, both the concrete and the reinforcing bars are in an elastic state. At this time, the flexural rigidity of reinforced concrete column is elastic, and the slope of bending moment-curvature curve is the largest. As the load increases, cracking occurs in the concrete column, the stress of the reinforced steel bar increases gradually, and the elastoplastic stage is reached owing to concrete cracking. At this time, the flexural rigidity of the reinforced concrete column is elastoplastic, and the slope of the bending moment-curvature curve decreases gradually as plasticity develops further. When the cracks of the reinforced concrete column are fully developed via loading and the deformation amplitude of the component increases significantly after the column steel bar yields, the bending moment-curvature curve is approximately horizontal and the slope of the curve is approximately zero, similarly, the flexural rigidity of the reinforced concrete column is approximately zero. Because the slope of the bending moment-curvature curve represents the flexural rigidity of the reinforced concrete column, the change rule of flexural rigidity of the reinforced concrete column during loading can be expressed by the trilinear model of bending moment-curvature, as shown in Fig. 10.

Fig. 10
figure 10

Trilinear model of bending moment-curvature.

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The bending moment-curvature curve of the reinforced concrete column section can be regarded as a monotonic increasing curve from the origin, and its slope represents the flexural rigidity of a column section. Based on Fig. 10, before the concrete column cracks, the column represents the OA segment of the trilinear model, the abscissa of point A is the cracking curvature \(\phi _{c}\), and the ordinate is the cracking bending moment \(M_{c}\). When the concrete column is in the elastic range, the slope of the curve is the largest and the flexural rigidity of the column is the elastic. As the applied load increases, the bending moment of the concrete column increases, and the column presents the characteristics of elastoplastic through the cracking of point A, which belongs to the AB segment of the trilinear model; the abscissa of point B is the yield curvature \(\phi _{y}\), the ordinate is the yield bending moment \(M_{y}\), and the slope of the curve of segment AB decreases, i.e., the flexural rigidity of the column decreases as cracking progresses in the column. As the applied load increases further, the bending moment of the concrete column increases, and the steel bar strain reaches the yield strain and thus yield occurs. This segment is the BC segment of the trilinear line model, the abscissa of point C is the ultimate curvature \(\phi _{u}\), the ordinate is the yield bending moment \(M_{u}\), the BC segment is approximately horizontal, and a slight increase in the bending moment will significantly increase the column curvature. The slope of this segment approaches zero, which means that the rigidity is lost gradually. When determining the ultimate bearing capacity of the column, the increment in the segment bending moment is not a dominant factor, and the yield bending moment of point B represents the ultimate bearing capacity of the section. Therefore, when calculating the elastoplastic rigidity of the reinforced concrete column, points O and B in the trilinear line model can be connected, and the slope of this line represent the elastoplastic flexural rigidity of the column, which can reflect the crack of the reinforced concrete column as well as the effect of material nonlinearity.

Formula for calculating the flexural rigidity reduction factor of reinforced concrete columns

When calculating the elastoplastic flexural rigidity of reinforced concrete columns, a flexural rigidity reduction factor can be introduced to reduce the elastic flexural rigidity of columns23. Based on the experimental results, this rigidity reduction factor is related to the eccentricity, reinforcement ratio, and vertical load \(P\). By fitting the bending moment and curvature relation of specimens A-1 to A-9, the flexural rigidity reduction factor for the reinforced concrete column can be expressed as follows.

$$\xi {\text{ = }}\frac{{EI}}{{E_{c} I_{c} }} = (0.85 + 25\frac{{A_{s} }}{A})(1 - \frac{{e_{0} }}{h} - 0.5n)$$
(3)

Note

This formula is derived based on specific test conditions and simplified assumptions, without taking into account factors such as strain hardening of reinforcing bars, constrained path dependence, internal force redistribution, and material variability.

Where \(\xi\)is the rigidity reduction factor of the reinforced concrete column, i.e., \(0.35 \le \xi \le 0.85\) ; \(EI\)is the elastoplastic flexural rigidity of the reinforced concrete column; \(E_{c} I_{c}\) is the elastic flexural rigidity of the reinforced concrete column, where \(E_{c}\)refers to the elastic modulus of concrete, and \(I_{c}\)refers to moment of inertia of the concrete column; \(\frac{{A_{s} }}{A}\) is the ratio of the longitudinal reinforcement area of the column to the cross-sectional area of the column, i.e., the reinforcement ratio of the column; \(\frac{{e_{0} }}{h}\)is eccentricity of the column; \(n\) is the dimensionless load ratio calculated as \(n = P/P_{0}\), where \(P_{0}\) is the section bearing capacity of the reinforced concrete axial compression column, calculated as \(P_{0} {\text{ = }}0.9\varphi (f_{c} A + f_{y}^{\prime} A_{s}^{\prime} )\); \(\varphi\)is the stability coefficient of the reinforced concrete column, which is inversely proportional to the slenderness ratio of the column and can be calculated as \(\varphi {\text{ = [1 + 0}}{\text{.002(}}\frac{{l_{{\text{0}}} }}{b} - {\text{8)}}^{{\text{2}}} {\text{]}}^{{{\text{ - 1}}}}\), where \(b\) refers to the short-side dimension of the rectangular cross-section column, and \(P\) refers to the bearing load of the reinforced concrete column.

Based on Eq. (3), the elastoplastic rigidity of the reinforced concrete column is related to the reinforcement ratio, eccentricity, load ratio, and slenderness ratio. As the reinforcement ratio increases, the rigidity reduction factor and flexural rigidity of the reinforced concrete column increase; As the eccentricity and load ratio increase, the rigidity reduction factor and flexural rigidity of the reinforced concrete column decrease.

Calculating example of rigidity reduction factor

Figure 11 shows a reinforced concrete eccentric compression column with a column section measuring 300 mm x 400 mm and a column height of 5000 mm. The length coefficient of the column was calculated to be 1.25, the concrete strength grade was C30, \(f_{c}\)= 14.3 N/mm2, the steel bar strength grade was HRB400, and \(f_{y}\)= 360 N/mm2. ① The longitudinal reinforcement is 8C16, the designed value of axial force under applied load was N = 500 kN, and the bending moment was M = 110 kN·m. ② The longitudinal reinforcement is 8C12, the designed value of the axial force under applied load was N = 400 kN, the bending moment was M = 40 kN·m, and the flexural rigidity reduction factor of the reinforced concrete column in operation was determined based on two cases.

Fig. 11
figure 11

Reinforced concrete compression column.

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① Parameter calculation.

\(\varphi {\text{ = [1 + 0}}{\text{.002(}}\frac{{1.25 \times 5000}}{{300}} - {\text{8)}}^{{\text{2}}} {\text{]}}^{{{\text{ - 1}}}} {\text{ = 0}}{\text{.752}}\)

\(\begin{gathered} P_{0} {\text{ = }}0.9 \times 0.752 \times (14.3 \times 300 \times 400 + 360 \times 1608) \hfill \\ \quad {\text{ = }}1553kN \hfill \\ \end{gathered}\)

Substituting the calculated values above into Eq. (3), then:

\(\xi {\text{ = }}(0.85 + 35\frac{{A_{s} }}{A})(1 - \frac{{e_{0} }}{h} - 0.5n)\)

\(\begin{gathered} {\text{ = }}(0.85 + 35 \times \frac{{1608}}{{120000}})(1 - \frac{{220}}{{400}} - 0.5 \times \frac{{500}}{{1553}}) \hfill \\ {\text{ = }}0.381 \hfill \\ \end{gathered}\)

Met: \(0.35 < \xi {\text{ = }}0.381 < 0.85\)

Therefore, the rigidity reduction factor of the reinforced concrete column in operation \(\xi {\text{ = }}0.381\).

②Parameter calculation.

\(\varphi {\text{ = [1 + 0}}{\text{.002(}}\frac{{1.25 \times 5000}}{{300}} - {\text{8)}}^{{\text{2}}} {\text{]}}^{{{\text{ - 1}}}} {\text{ = 0}}{\text{.752}}\)

\(\begin{gathered} P_{0} {\text{ = }}0.9 \times 0.752 \times (14.3 \times 300 \times 400 + 360 \times 888) \hfill \\ \quad {\text{ = }}1378kN \hfill \\ \end{gathered}\)

Substituting the calculated values above into Eq. (3), then:

\(\xi {\text{ = }}(0.85 + 35\frac{{A_{s} }}{A})(1 - \frac{{e_{0} }}{h} - 0.5n)\)

\(\begin{gathered} {\text{ = }}(0.85 + 35 \times \frac{{888}}{{120000}})(1 - \frac{{100}}{{400}} - 0.5 \times \frac{{400}}{{1378}}) \hfill \\ {\text{ = }}0.671 \hfill \\ \end{gathered}\)

Met: \(0.35 < \xi {\text{ = }}0.381 < 0.85\)

Therefore, the rigidity reduction factor of the reinforced concrete column in operation \(\xi {\text{ = }}0.671\).

The flexural rigidity reduction factor of the reinforced concrete column in operation was fixed as 0.70 and 0.60 based on American code ACI 318 and Chinese code GB50010, respectively, regardless of the load action type on the reinforced concrete column and the change of reinforcement. In actual engineering, the rigidity reduction degree of reinforced concrete column differs depending on the operational stress states. Currently, this difference cannot be considered in the standard value selection method. In the calculation method presented herein, the column rigidity reduction factors of ① 0.381 and ② 0.671 are more consistent with the actual stress state and can account for the effects of the load action type (load value and load action position) and reinforcement ratio.

Conclusions

1) Reinforcement ratio, eccentricity, axial compression ratio, and slenderness ratio are important factors that affect the elastoplastic flexural rigidity of reinforced concrete columns. An analysis of the change in the bending moment-curvature of the reinforced concrete column during loading revealed the change rule of elastoplastic flexural rigidity of the reinforced concrete column, i.e., the elastoplastic flexural rigidity is directly proportional to the reinforcement ratio and inversely proportional to the eccentricity, axial compression ratio, and slenderness ratio.

2) Based on the load-deflection and moment-bending curves of 9 groups of reinforced concrete specimens with different reinforcement ratios (1.33%, 2.41% and 3.77%) and eccentricities (0.175, 0.40 and 0.70), a three-line model characterizing the flexural stiffness variation of reinforced concrete compression columns was proposed. An approximate calculation formula for calculating the elastic-plastic flexural stiffness reduction coefficient of reinforced concrete eccentrically compressed columns was established. This formula can quantitatively reflect the influence of the geometric nonlinearity of the component (second-order deflection effect) and the nonlinearity of the material (concrete cracking, steel bar yield) on the stiffness reduction, and can be used to predict the deflection deformation of reinforced concrete compressed columns and verify the reliability of the finite element calculation results. Provide references for engineering design and theoretical calculation. This paper mainly focuses on rectangular cross-section columns with hinged ends. However, the applicability to non-rectangular cross-section columns, other types of end constraints, cyclic loads or columns with a high aspect ratio still needs further research.