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Displacement-based seismic performance of RC bridge pier

Abstract

To correctly manage the road infrastructure before and after an earthquake, it is necessary to estimate and even predict the seismic performance of the bridge. The quantification of the bridge's seismic performance response was present in terms of displacement and also based on previous research of reinforced concrete bridge pier models. The displacement did define from a force lateral-displacement response diagram corresponding to the capacity curve, calculated through a non-linear static pushover analysis of the reinforced concrete bridge pier model for each limit state, from intact state to collapse. Thus, six defined displacements correspond to the cracking displacement, the yielding displacement, the spalling displacement, the crushing displacement, the buckling displacement, and the fracturing displacement. The six defined limit states correspond to the cracking limit state, the yielding limit state, the spalling limit state, the crushing limit state, the buckling limit state, and the fracturing limit state. Also, parametric analysis did carry out to evaluate the influence, relative importance, and trend of the input parameters in response to the seismic performance of the reinforced concrete bridge pier model. Eleven input parameters did analyze as the concrete compressive strength, the yield stress of reinforcing steel, the concrete cover thickness, the pier aspect ratio, the configuration of the transverse reinforcement, the spacing of the transverse reinforcing steel, the transversal diameter of the transverse reinforcing steel, the longitudinal reinforcement ratio, the transversal diameter of the longitudinal reinforcing steel, the axial load ratio, and coefficient of subgrade reaction.

1 Introduction

The bridges collapse due to the ground and earthquakes; the bridges collapse due to the ground can be controlled by selecting the appropriate soil and site; however, the bridges collapse due to the earthquake cannot be predicted yet. Vehicular overload and earthquakes are the main factors of bridge damage (Tan et al. 2020; Liu et al. 2022; Li et al. 2020a, b).

The tendency of the collapse of bridges due to earthquakes increases year after year; therefore, it is challenging for engineering to adequately assess the response of the seismic performance of the bridge.

1.1 Response of the seismic performance of the bridge

The piers are the most critical components that define the bridges’ overall seismic performance; the pier’s vulnerability would lead to the bridge's collapse. The seismic performance of the bridge pier corresponds to each damage state expressed in terms of performance levels, generally qualitatively defined as fully operational (damage as cracking only), operational (repairable damage), life safety (damage as resistance degradation), and collapse prevention. Previous works in this regard have used damage indices to quantify the state of damage. Thus, an index value of 0 indicates an undamaged state, and 1 indicates a fully damaged state or failure. Damage states may or may not take into account the effect of cyclic load effect (non-cumulative or cumulative damage indices). The energy-based cumulative damage indices use energy spectra to estimate the influence of cumulative demand on element damage.

The evaluation of the deformation corresponding to the different states of damage requires the evaluation of the force–displacement response envelope (force–displacement model). Thus, the main objective of this paper corresponds to quantify the response of the seismic performance of the reinforced concrete (RC) bridge pier in terms of displacement (u). The RC bridge pier model (RCBPM) was proposed for the present work, as schematized in Fig. 1(a). The scaled RC bridge pier specimen consists of the following referential measurements: height of 2.05 m, circular section pier with a diameter of 0.35m, longitudinal reinforcement ratio of 3%, transversal reinforcement ratio of 2%, concrete compressive strength of 21 MPa, yield stress of reinforcing steel of 412 Mpa, square footing with the size of 0.9m × 0.90 × 0.50m, cap beam with the size of 0.35m × 1.30m.

Fig. 1
figure 1

RCBPM typical capacity curve

1.1.1 Capacity curve

Generally, due to its simplicity, the intact analytical model of the RCBPM corresponds to an equivalent system of a single degree of freedom (SDOF) (Su et al. 2019, 2015); as schematized in Fig. 1(b). The damaged analytical model of the RCBPM corresponds to the same equivalent system of a simple degree of freedom plus a rotational spring and angle of rotation; the rotational spring exhibits limit state level, rotational spring intact limit state → ∞, damaged limit state close to rotational spring collapse → 0; as schematized in Fig. 1(c). The nonlinear static pushover analysis of the RCBPM did consider, as schematized in Fig. 1(d). The collapse mechanism of the RCBPM corresponds to each progressive damage state from intact to collapse, as illustrated in Fig. 1(e) (Li et al. 2020a, b). The material laws consider the nonlinearity of the concrete material, as schematized in Fig. 1(f), and the nonlinearity of the steel material, as schematized in Fig. 1(g). The hysteresis curve or the force–displacement-response corresponds to the mathematical representation of the base lateral force versus the lateral displacement of the seismic resistant performance of the RCBPM in a random sequence of loading and unloading as is the case of an earthquake, as schematized in Fig. 1(h). Thus, the capacity curve or the envelope response did obtain from the peak value of the first cycle for each drift ratio in the force–displacement-response; the base shear or lateral force (F) did normalize to the weight on the pier, and u did normalize to height, i.e., drift ratio in percentage as schematized in Fig. 1(i).

1.1.2 Reference displacements

The previous similar research conducted by other authors in the literature in which the RC bridge pier's experimental displacement did define as a response parameter did find and ordered lists of some available research in Table 1. Some authors, such as Cassese et al. 2019 only defined two types of displacement (u-1: intact and u-2: damaged), and others, such as Pang and Li 2018 defined up to four types of displacement (u-1: yielding, u-2: crushing, u-3: buckling, and u-4: fracture).

Table 1 Reference displacements

In this regard, drawbacks of the reference displacements were considered, such as the heterogeneity in the type or range of displacements studied, the discontinuity of the research work, and above all, it did not correlate with all damage states (from intact state to collapse). Therefore, in the present manuscript, it did propose that the response of the seismic performance of the RC bridge pier expressed in terms of displacement must correlate with all damage states (from intact state to collapse).

1.2 Damage state of RC bridge pier

The performance levels of a bridge correspond to the different states of progressive damage during the seismic performance of the RC bridge pier. During the sequence of damage states, from intact to collapse, there are stages of progressive degradation of the strength and stiffness of the bridge's structural components. In the present work, the damage states did consider limit states (LS) of the response of the seismic performance of the RCBPM, as schematized in Fig. 2. It is controversial regarding the appearance sequences of concrete cover spalling and longitudinal reinforcement buckling. Some studies assumed that the buckled longitudinal bar would push the concrete cover and then exacerbate spalling of the concrete cover due to its outward deformation trend; hence concrete cover delays or inhibits bar buckling simultaneously. Others indicated that the concrete cover had already spalled off prior to the initiation of bar buckling. Thus, the concrete cover's effects on bar buckling did reasonably ignored. Traditionally, bridge piers' limit states did qualitatively define by damage states such as light, moderate, extensive, and complete. However, these now need to be made explicit; future work should focus on studying various levels of earthquakes and structures to establish a good correlation between local and global damage indices.

Fig. 2
figure 2

Limit states of RC bridge pier

1.2.1 Reference limit state

The previous similar research conducted by other authors searched the literature in which the RC bridge pier's limit state defined, found, and ordered lists some available research in Table 2. Some authors, such as Srivastava et al. 2022 only defined two types of LS (LS-1: delamination, and LS-2: loss of confinement), and others, such as Kashani et al. 2019 defined up to four types of limit states (LS-1: slight, LS-2: extensive, LS-3: complete, and LS-4: aftershock). In this regard, it did consider that the drawbacks refer to a diversity of qualitative classifications, there is no homogeneity or continuity, and it is not related to the damage in materials (concrete, reinforcement). Therefore, in the present manuscript, it did propose that the limit states of the RCBPM, based on the reference limit states of previous experimental research by other authors, correlated with the damage in the concrete and the reinforcing steel, and that can serve as expected limit states in the future works on RC bridge piers.

Table 2 Reference limit states

1.3 Input parameters of the RC bridge pier

In the experimental and analytical investigation, initial parameters or variables did study to characterize the response of the seismic performance of the RC bridge pier to lateral loads. The influence of individual parameters on the force–displacement response of RC bridge piers elements is analyzed. Thus, in the present work, the parameters were considered input parameters (IP).

1.3.1 Reference input parameters

The previous similar research conducted by other authors searched the literature in which input parameters of the RC bridge pier did define, found, and ordered lists some available research in Table 3. Some authors, such as Anand and Satish Kumar 2018 only defined two types of input parameters (IP6: spacing of the transverse reinforcing steel, and IP11: coefficient of the subgrade reaction), and others, such as Afsar Dizaj and Kashani 2020 defined up to five types of input parameters (IP4: pier aspect ratio, IP5: configuration of the transverse reinforcement, IP6: spacing of the transverse reinforcing steel, IP7: transversal diameter of the transverse reinforcing steel, IP8: longitudinal reinforcement ratio, and IP9: transversal diameter of the longitudinal reinforcing steel). In this regard, the drawbacks refer to the fact that these generally qualitative conclusions are not yet correlated with the displacements nor with the limit states, besides a limited number of parameters analyzed. Therefore, in the present manuscript, it did propose to analyze the most significant possible number of parameters and analyze their individual and group influence on the response of the seismic performance of the RCBPM and that they can serve as a categorization of the relevant and irrelevant parameters in future work of RC bridge piers.

Table 3 Reference input parameters

1.3.2 Regression analysis

A parametric study through the simple linear regression analysis procedure did propose as a statistical technique to model and investigate the relationship between u (a dependent variable) and input parameters (one independent variable) (Carrasquilla-Batista et al. 2016). Thus, a simple linear regression analysis is performed for only one independent variable, as indicated in Eq. (1).

$$u={\beta }_{0 }+{\beta }_{1}IP$$
(1)

where: u is the dependent variable is also called the explained variable, or response; IP variable independent variable, also called predictor variable, explanatory variable, regressor; β0 is the intersection of the trend line; β1 is the slope of the line, y ϵ represents the random error.

Therefore, in the present manuscript, the impact of each input parameter was determined based on the value of the coefficient of determination (R2). The coefficient of determination is a measure used to explain how much variability of a factor can be caused by its relationship with another related factor. This correlation or goodness of fit is a value between 0 and 1. A value of 1 indicates a perfect fit and, therefore, a very reliable model for future forecasts, while a value of 0 would indicate that the calculation fails to model the data accurately.

Also, a parametric study through the multiple regression analysis procedures did propose a statistical technique to model and investigate the relationship between u (a dependent variable) and IPk (two or more independent variables) (Carrasquilla-Batista et al. 2016). Thus, multiple linear regression analysis did perform for two or more independent variables) Eq. (2).

$$u={\beta }_{0 }+{\beta }_{1}{IP}_{1}+\dots +{\beta }_{k}{IP}_{k}+ \in$$
(2)

where: IP1…IPk independent variables, also called predictor variables, explanatory variables, regressors; β1 … β1 is the slope of the line.

Therefore, in the present manuscript, the displacement predictor equation was derived based on all the parameters analyzed. This derived multiple linear regression formula can be used in preliminary design when most of the design details are still unknown to determine the feasibility of using the proposed pier to reduce residual displacement (Ou et al. 2022).

Thus, in the present manuscript, the irrelevance of each input parameter was evaluated based on the adjusted coefficient of determination (Rr2). The variation of its value represents this correlation or goodness of fit. It is irrelevant if the Rr2 value increases by removing the input parameter from the multiple linear regression analysis. It is relevant if the Rr2 value decreases by removing the input parameter from the multiple linear regression analysis.

2 Selection of input parameters and analysis cases

2.1 Material

The f'c and fy did consider input parameters corresponding to the material. The increase in the value of the f'c reduced the lateral displacement, as well as improving the capacity to resist the lateral load, the dissipation of the hysteretic energy, and the viscous damping index, according to work developed in this regard by Abdallah and El-Salakawy 2022 and Ren et al. 2022. The nominal f'c analyzed 35 MPa, 18 MPa, 21 MPa, 27 MPa, and 35 MPa.

The increase in the value of fy increased the lateral displacement, it did not significantly influence the seismic performance, it reduced the amount of steel, it increased the load capacity and hysteretic energy dissipation; according to the works developed in this regard by Barbosa 2015, Su et al. 2021, and Su et al. 2017a, b. The nominal fy analyzed 300 MPa, 350MPa, 412MPa, and 549MPa.

2.2 Geometry

The r and L/D did consider input parameters that correspond to the geometry. The increase in the value of r did not significantly influence the hysteretic energy dissipation and the strain responses of longitudinal reinforcement. However, it significantly impacted the post-yield stiffness ratio, according to the works developed in this regard by Ding et al. 2021 and Ou et al. 2022. The nominal r analyzed 0.02 m, 0.03 m, 0.04 m, and 0.05 m.

The increase in the value of the L/D produced an increase in lateral displacement and a reduction in the basal shear force, as well as the piers with L/D between 3.8 and 5.4, failed due to flexion, the shorter piers with an L/D around 2.2 fell per shear; according to the works developed in this regard by (Tran et al. 2020, and Aamir Baig et al. 2022). The nominal L/D analyzed 4, 6, 8,10.

2.3 Transverse reinforcing steel

The Ø, the s, and the dt did consider input parameters corresponding to the transverse reinforcing steel. The Ø or types of stirrups in the pier improved ductility and shear, as well as providing greater confinement, prevented premature buckling of longitudinal reinforcing steel, and increased compressive strength, according to the works developed in this regard by Zignago and Barbato 2019, and Shelman and Sritharan 2014. The nominal Ø analyzed spiral stirrups (ss) and discrete hoops (sh).

The lower value of s improved ductility and compressive strength. It produced a significant change in the failure mode, according to the works developed by Xiamuxi et al. 2019 and Su et al. 2015. The nominal s analyzed 0.05 m, 0.10 m, 0.15 m, 0.20 m.

The increase in the dt value influenced the deformation, and the adhesion between the concrete and the reinforcing steel, according to work carried out by Ali et al. 2018, and Su et al. 2020. The nominal dt analyzed, such as #2, #3, #4, and #5.

2.4 Longitudinal reinforcing steel

The ρl and the dl did consider input parameters corresponding to the longitudinal reinforcing steel. The increase of the value of ρl increased the lateral displacement; the curvature did reduce, the ductility decreased, and the length of the plastic hinge increased; according to the works developed in this regard by Bouazza et al. 2022, Fu et al. 2022, and Li et al. 2020a, b. The nominal ρl analyzed 0.7%, 1.5%, 3%, and 4.5%.

The increase in value of dl reduced the lateral displacement, as well as adherence slightly decreased; there was an evident influence on the buckling as a function of the relationship between the diameter and the length of the reinforcing steel, according to the works developed in this regard by Wu et al. 2022, and Wang et al. 2022. The nominal dl analyzed #2, #3, #4, and #5.

2.5 Axial load

The P is the selected input parameter that corresponds to the weight of the superstructure or the axial load; in this regard, the increase in P reduced the displacement, as the plastic hinge length increased with axial load, the hysteresis curves for a lower axial load level were slightly tighter than those with a higher axial load level, increased effectiveness of energy dissipation due to higher axial load; according to the works developed in this regard by Jiao et al. 2019, and Kehila and Remki 2015. The nominal P analyzed 5%, 15%, 20%, and 25%.

2.6 Soil-structure interaction

The ks is the selected input parameter that corresponds to the soil-structure interaction; in this regard, lower stiffness of the foundation ground resulted in reduced displacement and resistance, as well as the natural frequency of the bridge pier decreased considerably when the pier was damaged, or the foundation stiffness did reduce; according to the works developed in this regard by Hung et al. 2011, Anand and Satish Kumar 2018, Shi et al. 2022, and d´Avila et al. 2022. The nominal ks analyzed flexible foundation soil (5 × 105 N/m3), rigid foundation soil (5 × 1010 N/m3), 1 × 1011 N/m3, and 2 × 1011 N/m3.

As mentioned earlier, the conclusions of the previous research generally correspond to qualitative conclusions. Only a few input parameters did in the study, and they only corresponded to the displacement of a specific limit state. Therefore, another of the proposed objectives of this work corresponds to studying the most significant number of input parameters to analyze their relevance and their trend, corresponding to the displacement of all limit states (from intact to collapse).

The selection of input parameters corresponds to the most studied characteristics of the model of the RC bridge pier on an analytical and experimental basis – most of them were covered. The cases of every parameter correspond to the usual values used in RC issues. The cases analyzed result from combining every parameter and the number of values allocated to each parameter. The eleven input parameters corresponding to the material, geometry, reinforcing steel, axial force, and soil-structure interaction did select, as summarized in Table 4 and Fig. 3.

Table 4 Selected input parameters and analysis cases of the RCBPM
Fig. 3
figure 3

Selected input parameters of the RCBPM

3 Materials and models

3.1 Material model

3.1.1 Unconfined concrete

Unconfined concrete (UC) refers to concrete (C) that did not confine by reinforcing steel (S); it did also call covered concrete thickness because it covers the longitudinal reinforcing steel (LS) and transverse reinforcing steel (TS) with concrete; it constitutes one of the components of the concrete as specified in Fig. 4(b), and also contributes to the strength of the RCBPM as specified in Eq. (3) and Eq. (4) (Zhong et al. 2022; Zhou and Kunnath 2022).

Fig. 4
figure 4

Material model

3.1.2 Confined concrete

Confined concrete (CC) refers to concrete that is confined by longitudinal reinforcing steel and transverse reinforcing steel, and it did also call concrete core because it did confine by reinforcing steel; it constitutes one of the components of the concrete as specified in Fig. 4(b), and also contributes to the strength of the RCBPM as specified in Eq. (3) and Eq. (4) (Zhong et al. 2022; Zhou and Kunnath 2022).

3.1.3 Longitudinal reinforcing steel

Longitudinal reinforcing steel (LS) refers to the primary steel along the bridge pier, and it did also measure by amount; it constitutes one of the components of reinforcing steel as specified in Fig. 4(c) and also contributes to the strength of the RCBPM as specified in Eq. (3) and Eq. (4) (Zhong et al. 2022; Zhou and Kunnath 2022).

3.1.4 Transverse reinforcing steel

Transverse reinforcing steel (TS) refers to the transverse, horizontal, or shear steel that, together with the longitudinal steel, confines the concrete core; it is another component of reinforcing steel as specified in Fig. 4(c) and also contributes to the strength of the RCBPM as specified in Eq. (3) and Eq. (4) (Zhong et al. 2022; Zhou and Kunnath 2022).

$$R= C+S$$
(3)
$$R=(UC+CC)+(LS+TS)$$
(4)

3.2 Limit states

3.2.1 The cracking limit state (LS1)

It corresponds to the first defined limit state, also called damage control, minimal, and slight (Jiao et al. 2019; Park et al. 2020; Todorov and Billah 2021; Hung et al. 2011). The damage status did define according to the material model as the UC with cracking initiation, the CC undamaged, the LS undamaged, and the TS undamaged. The resistance is defined as R = UC + CC + LS + TS, as illustrated in the second column of Table 5.

Table 5 Limit state definition

3.2.2 The yielding limit state (LS2)

It corresponds to the second defined limit state, called minor spalling, repairable, and moderate (Todorov and Billah 2021; Zhou and Kunnath 2022; Todorov and Muntasir Billah 2022a). The damage status did define according to the material model: the UC with extensive cracking, the CC with cracking initiation, the LS with yielding initiation, and the TS undamaged. The resistance is defined as R = CC + LS + TS, as illustrated in the third column of Table 5.

3.2.3 The spalling limit state (LS3)

It corresponds to the third defined limit state, life safety, and significant spalling, representing damage with loss of concrete cover (Srivastava et al. 2022). The damage status is defined according to the material model as the UC with full spalling, the CC with cracking initiation, the LS with yielding, and the TS with yielding initiation. The resistance is defined as R = CC + LS + TS, as illustrated in the fourth column of Table 5.

3.2.4 The crushing limit state (LS4)

It corresponds to the fourth defined limit state, life safety, crushing the concrete core (Hung et al. 2011; Todorov and Muntasir Billah 2022b; Srivastava et al. 2022). The damage status is defined according to the material model as the UC with full spalling, the CC with full cracking, the LS with extensive yielding, and the TS with yielding. The resistance is defined as R = LS + TS, as illustrated in the fifth column of Table 5.

3.2.5 The buckling limit state (LS5)

It corresponds to the fifth defined limit state, also called limited safety, near collapse, extensive, and severe (Kashani et al. 2019; Su et al. 2019; Todorov and Billah 2021). The damage status did define according to the material model: the UC with full spalling, the CC with full cracking, the LS with buckling, and the TS with complete yielding. The resistance is defined as R = LS + TS, as illustrated in the sixth column of Table 5.

3.2.6 The fracturing limit state (LS6)

It corresponds to the sixth and last defined limit state, also called collapse, fracture, irreparable, and replacement; it represents the state of collapse damage with the fracture of the reinforcement and collapse steel (Kashani et al. 2019; Su et al. 2019; Todorov and Billah 2021). The damage status did define according to the material model: the UC with full spalling, the CC with full cracking, the LS with fracturing, and the TS with buckling. The resistance is defined as R = 0, as illustrated in the seventh column of Table 5.

3.3 Displacement

3.3.1 Cracking displacement ([u 1])

The first defined displacement interval corresponds to LS1 (Cassese et al. 2019). The tensile strain does not exceed the maximum strain of concrete and does not exceed the yield strain in reinforcing steel. The compressive strain does not exceed the maximum strain of concrete and does not exceed the yield strain in reinforcing steel. The tensile stress distribution does not exceed the allowable stresses in the concrete and does not exceed the allowable stresses in the reinforcing steel, as schematized in Fig. 5(a) and (b). The displacement interval corresponds to the first line of the trilinear relationship that passes from zero and ends at the idealized yield point (initial stiffness) of the idealized capacity curve (proposed trilinear relationship similar to Federal Emergency Management Agency (FEMA) Standard 356 (2000) (Ou et al. 2022)); as schematized in Fig. 5(c) and (d).

Fig. 5
figure 5

Displacement definition

3.3.2 Yielding displacement ([u 2])

The second defined displacement interval corresponds to LS2 (Kashani et al. 2019; Afsar and Kashani 2020). The tensile strain reaches the maximum strain of concrete and the yield strain in reinforcing steel. The compressive strain does not exceed the maximum strain of concrete and does not exceed the yield strain in reinforcing steel. The tensile stress distribution exceeds the allowable stresses in the concrete and the allowable stresses in the reinforcing steel, as schematized in Fig. 5(a) and (b). The displacement interval corresponds to the first stretch of the second line, beginning from the idealized yield point and ending at the maximum lateral force on the envelope response (post-yield stiffness) of the idealized capacity curve (proposed trilinear relationship similar to the Federal Emergency Management Agency (FEMA) Standard 356 (2000) (Ou et al. 2022)); as schematized in Fig. 5(c) and (e).

3.3.3 Spalling displacement ([u 3])

The third defined displacement interval corresponds to LS3 (Cassese et al. 2019). The tensile strain does exceed the maximum strain of concrete and does exceed the yield strain in reinforcing steel. The compressive strain reaches the maximum strain of concrete and the yield strain in reinforcing steel. As schematized in Fig. 5(a) and (b), the tensile stress distribution reaches maximum stresses in the concrete and yields stresses in the reinforcing steel. The displacement interval corresponds to the second stretch of the second line, beginning from the idealized yield point and ending at the maximum lateral force on the envelope response (post-yield stiffness) of the idealized capacity curve (proposed trilinear relationship similar to the Federal Emergency Management Agency (FEMA) Standard 356 (2000) (Ou et al. 2022)); as schematized in Fig. 5(c) and (f).

3.3.4 Crushing displacement ([u4])

The fourth defined displacement interval corresponds to LS4 (Hung et al. 2011). The tensile strain does exceed the maximum strain of concrete and does exceed the yield strain in reinforcing steel. The compressive strain does exceed the maximum strain of concrete and does exceed the yield strain in reinforcing steel. The tensile stress distribution exceeds the maximum stresses in the concrete and the maximum stresses in the reinforcing steel, as schematized in Fig. 5(a) and (b). The displacement interval corresponds to the first stretch of the third line, beginning from the maximum lateral force point and ending at the ultimate drift or collapse on the envelope response (post-maximum lateral force stiffness) of the idealized capacity curve (proposed trilinear relationship similar to the Federal Emergency Management Agency (FEMA) Standard 356 (2000) (Ou et al. 2022)); as schematized in Fig. 5(c) and (g).

3.3.5 Buckling displacement ([u 5])

The fifth defined displacement interval corresponds to LS5 (Kashani et al. 2019). The tensile strain does exceed the maximum strain in reinforcing steel. The compressive strain does exceed the maximum strain of strain in reinforcing steel. As schematized in Fig. 5(a) and (b), the tensile stress distribution exceeds the maximum stresses in the reinforcing steel. The displacement interval corresponds to the second stretch of the third line, beginning from the maximum lateral force point and ending at the ultimate drift or collapse on the envelope response (post-maximum lateral force stiffness) of the idealized capacity curve (proposed trilinear relationship similar to the Federal Emergency Management Agency (FEMA) Standard 356 (2000) (Ou et al. 2022)); as schematized in Fig. 5(c) and (h).

3.3.6 Fracturing displacement ([u 6])

The sixth and last defined displacement interval corresponds to LS6 (Kashani et al. 2019). The tensile strain does exceed the maximum strain in reinforcing steel. The compressive strain does exceed the maximum strain of reinforcing steel. As schematized in Fig. 5(a) and (b), the tensile stress distribution exceeds the maximum stresses in the reinforcing steel. The displacement interval corresponds to the third and last stretch of the third line, beginning from the maximum lateral force point and ending at the ultimate drift or collapse on the envelope response (post-maximum lateral force stiffness) of the idealized capacity curve (proposed trilinear relationship similar to Federal Emergency Management Agency (FEMA) Standard 356 (2000) (Ou et al. 2022)); as schematized in Fig. 5(c) and (i).

4 Model results

4.1 Model displacement ([u cal])

The input parameters, as well as their respective nominal values, are taken into account in the calculation. They are shown in bold and italics in Table 4, with which 4608 model displacement results did achieve. Thus, to evaluate the amplitude of the [ucal] found, the minimum model displacement (umin) and the maximum model displacement (umax) were summarized in the second and third rows of Table 6, respectively. The nominal minimum model displacement results from umin-1 = 0.1, and the nominal maximum model displacement results from umax-1 = 1.7 corresponding to LS1, the nominal minimum displacement umin-2 = 0.4 and the nominal maximum displacement umax-2 = 2.3 corresponding to the LS2, up to the nominal minimum displacement umin-6 = 0.4 and the nominal maximum displacement umax-6 = 3.8 corresponding to the LS6. An increasing trend of their respective amplitude of intervals did observe in the last three limit states. These results did like attributed to the instability of the last damage states.

Table 6 Displacement results

To configure the idealized capacity curve, the mean displacement (ume) for each limit state, summarized in the fourth row of Table 6, as follows: the nominal mean displacement ume-1 = 0.6 corresponding to the LS1, the nominal mean displacement ume-2 = 1.2 corresponding to the LS2, up to the nominal mean displacement ume-6 = 2.5 corresponding to the LS6. The model's respective lateral force-mean displacement response diagram was recorded and schematized in Fig. 6(a). More excellent dispersion of the results did observe in the last three limit states. These results did attribute to the last damage states corresponding to the model collapse.

Fig. 6
figure 6

Model displacements

The residual displacement range ([ur]) corresponding to each limit state was defined proportionally to the inelastic displacement of the model (μ), as summarized in the fifth row of Table 6. Thus, the nominal residual displacement ur-2 = 0.30μ corresponding to the LS2, the nominal residual displacement ur-3 = 0.25μ corresponding to the LS3, up the nominal residual displacement ur-6 = 0.10μ corresponding to the LS6. The model's respective lateral force-residual displacement response diagram was recorded and schematized in Fig. 6(b). The residual displacement results exhibited did not remain constant for all limit states. In the first limit states, up to 55% of the μ did develop, and 45% of the μ did develop in the last three limit states. These results did likely attribute to the fact that in the first three limit states, the material model is concrete and reinforcing steel, which contributes to strength, unlike the last three limit states in which the material model did only reinforce steel; in addition to the large dispersion of results that were observed in the amplitude and mean displacement calculations indicated above.

The [ucal] was defined based on the mean displacement and standard deviation, [ume -σ, ume + σ], for each limit state and has been summarized in the sixth row of Table 6. Thus, the nominal model displacement [ucal-1] = [0, 0.9], corresponding to the LS1; the nominal model displacement [ucal-2] = [0.3, 1.5], corresponding to the LS2; up to the nominal model displacement [ucal-6] = [1.9, 2.9] corresponding to the LS6. The model's respective lateral force-model displacement result response diagram was recorded and schematized in Fig. 6(c). The [ucal] reflected the dispersion of results and lower inelastic displacement in the last three limit states indicated above; however, it exhibited homogeneity in the amplitude of the displacement intervals, on average between 1 and 1.1. These results did attribute to the fact that it would only reflect the theoretical, analytical model.

4.2 Reference displacement ([u ref])

The displacements of experimental models of reference RC bridge piers indicated in Table 1 did adapt to the proposed six displacement intervals and did summarize in the seventh row of Table 6. Thus, the nominal reference displacement [uref-1] = [0, 0.4], corresponding to the LS1; the nominal reference displacement [uref-2] = [1.2, 1.9], corresponding to the LS2; up to the nominal reference displacement [uref-6] = [9.3, 10] corresponding to the LS6. The model's respective lateral force-model displacement result response diagram was recorded and schematized in Fig. 7. A homogeneous amplitude of the displacement intervals was not observed as in the displacement intervals of the model, especially in the last three limit states. These results, too, were probably attributed to the last damage states corresponding to the model's collapse.

Fig. 7
figure 7

Reference displacements

4.3 RCBPM displacement ([u])

Note that the results revealed that the nominal model displacement [ucal-1] = [0, 0.9] was not that different from that of the nominal reference displacement [uref-1] = [0, 0.4]; the nominal model displacement [ucal-2] = [0.3, 1.5] was not that different from that of the nominal reference displacement [uref-2] = [1.2, 1.9]; and the nominal model displacement [ucal-3] = [0.9, 2], was not that different from that of the nominal reference displacement [uref-3] = [1.9, 2.9]. These results did probably attribute to the fact that only in the first three limit states did the developed theoretical, analytical model have a good approximation with the reference experimental models that did already note the uniform amplitude of the displacement intervals in the lesser dispersion of the average displacements and the more remarkable development of the inelastic displacement. Therefore, the seismic performance response based on the displacement of the RCBPM was quantified, thus: the nominal displacement [u1] = [0, 0.7] mean of [0, 0.9] and [0, 0.4], corresponding to the LS1; the nominal displacement [u2] = [0.8, 1.7] mean of [0.3, 1.5] and [1.2, 1.9], corresponding to LS2; and up to nominal displacement [u3] = [1.4, 2.5] mean of [0.9, 2] and [1.9, 2.9], corresponding to LS3; as specified in the last row of Table 6 and as illustrated in Fig. 8.

Fig. 8
figure 8

RCBPM displacements

While the nominal reference displacement [uref-4] = [3.9, 5.4] was higher than that of the nominal model displacement [ucal-4] = [1.3, 2.4]; also the nominal reference displacement [uref-5] = [5.8, 8.3], was higher than that of the nominal model displacement [ucal-5] = [1.7, 2.7]. The nominal reference displacement [uref-6] = [9.3, 10] was higher than that of the nominal model displacement [ucal-6] = [1.9, 2.9]. These results did probably attribute to the fact that the developed theoretical, analytical model could have better approximated the last three limit states with the reference experimental models; what did already note in the greater amplitude of the displacement intervals, the more excellent dispersion of the mean displacements, and the lesser development of the inelastic displacement. However, the seismic performance response based on the displacement of the RCBPM was also quantified, thus: the nominal displacement [u4] = [2.6, 3.9] mean of [1.3, 2.4] and [3.9, 5.4], corresponding to the LS4; the nominal displacement [u5] = [3.8, 5.5] mean of [1.7, 2.7] and [5.8, 8.3], corresponding to the LS5; and up to the nominal displacement [u6] = [5.6, 6.5] mean of [1.9, 2.9] and [9.3, 10], corresponding to the LS6; as specified in the last row of Table 6 and as illustrated in Fig. 8.

4.4 Displacement prediction

A multiple linear regression analysis derived a formula to relate the displacement with the eleven input parameters. The derived multiple linear regression formula is as Eq. (5), shown as the displacement prediction equation for RCBPM corresponding to the [u3].

With Eq. (5), 13,830 displacement results did obtain and compared with the 4608 displacement results of the developed model, finding discrepancies of around 20%, as illustrated in Fig. 9.

Fig. 9
figure 9

Discrepancy in lateral displacement results

$${u}_{3}=-(8*{10}^{-6})\frac{L}{D}-0.00002{d}_{l}-0.0003\Phi +0.0001{{f}{\prime}}_{c}+ 0.0004s+(3.2\times {10}^{-6}){f}_{y}+0.0022{\rho }_{l}-0.0376r- 0.00004P- 0.0001dt-(6.1\times {10}^{-14}){k}_{s}+0.0170$$
(5)

As an example of the application of the respective prediction equation, the displacement of the RC bridge pier was calculated (Fig. 6 (a), experimental specimen A, studied by Hung et al. (2011). An experimental study on the rocking response of bridge piers with spread footing foundations. Earthquake Engineering and Structural Dynamics. https://doi.org/10.1002/eqe.1057). The specified displacement of the specimen (u3-specimen) was 1.8%; however, the measurement results revealed that the displacement calculated with the respective prediction equation Eq. (6) was u3 = 2.1%. Therefore, the displacement of the experimental model of the RC bridge pier, u3-specimen = 1.8%, was close to that calculated with the respective prediction equation as u3 = 2.1%; in addition, it did found that the displacements u3-specimen and u3, within the displacement interval proposed as [u3] = [1.4%, 2.5%].

As another example of the application of the respective prediction equation, the displacement of the RC bridge pier was calculated (Fig. 7 (a), studied by Barbosa (2015). Seismic performance of high-strength steel RC bridge columns. American Society of Civil Engineers. https://doi.org/10.1061/(ASCE)BE.1943-5592.0000769). The specified displacement of the specimen (u3-specimen) was 2.1%; however, the measurement results revealed that the displacement calculated with the respective prediction equation Eq. (7) was u3 = 2%. Therefore, the displacement of the experimental model of the RC bridge pier, u3-specimen = 2.1%, was close to that calculated with the respective prediction equation as u3 = 2%; in addition, it did found that the displacements u3-specimen andu3, within the displacement interval proposed as [u3] = [1.4%, 2.5%].

$${u}_{3-specimen}=-\left(8*{10}^{-6}\right)*6-0.00002*6-0.0003*1+ 0.0001*29+0.0004*0.1+\left(3.2*{10}^{-6}\right)*420+ 0.0022*2.63-0.0376*0.0254-0.00004*15- 0.0001*3-\left(6.1*{10}^{-14}\right)*\left(5*{10}^{10}\right)+0.0170 = 0.021\approx 0.018$$
(6)
$${u}_{3-specimen}=-\left(8*{10}^{-6}\right)*6-0.00002*6-0.0003*1.1+ 0.0001*32.1+0.0004*0.05+\left(3.2*{10}^{-6}\right)* 420+0.0022*2.19-0.0376*0.0318-0.00004* 5-0.0001*3-\left(6.1*{10}^{-14}\right)*\left(5*{10}^{10}\right)+ 0.0170 =0.020\approx 0.021$$
(7)

4.5 Parametric study

4.5.1 Influence of IP of response of seismic performance of RCBPM

The values of R2 obtained from the respective simple linear regression analysis of the 13,830 results corresponding to the LS3, such as R2 = 0.6366 for the ρl input parameter, R2 = 0.464 for the ks input parameter, R2 = 0.0553 for the f'c input parameter, R2 = 0.0053 for the dl input parameter, R2 = 0.0021 for the input parameter; up to R2 = 7*10–5 for the fy input parameter. Therefore, the ρl input parameter and the ks input parameter were the ones with the most significant influence on the response of the seismic performance of the RCBPM. The other nine (f'c, P, L/D, Ф, r, dt, s, fy, dl) input parameters reached shallow values of R2 (less than 0.07), which have no statistical significance, practically meaningless, as specified in Fig. 10 and third row of Table 7.

Fig. 10
figure 10

Relative importance factors for input parameters

Table 7 Input parameters relevant and irrelevant

The values of Rr2 did obtain from the respective multiple linear regression analysis of the 4608 results corresponding to the LS3. The Rr2 = 0.5128 considering all eleven input parameters (ρlksf'cPL/DФrdtsfydl); the Rr2 = 0.5129 considering ten input parameters (ρlksf'cPL/DФrdtsfy); the Rr2 = 0.5095 considering nine input parameters (ρlksf'cPL/DФrdts); up to Rr2 = 0.3239 a single input parameter (ρl). The values of Rr2 obtained are less than 1, which indicates that it is not so easy to predict the displacement of the RCBPM.

By ignoring the input parameter Ф in the multiple linear regression analysis performed, the value Rr2 = 0.5058 increased to Rr2 = 0.5059; by ignoring the input parameter s in the multiple linear regression analysis performed, the value Rr2 = 0.5095 increased to Rr2 = 0.5096; by ignoring the input parameter dl in the multiple linear regression analysis performed, the value Rr2 = 0.5128 increased to Rr2 = 0.5129. Therefore, the irrelevance of the four parameters (Ф, s, and dl) cannot be determined since these increases reached shallow values of Rr2 (less than 0.01), which have no statistical significance, practically meaningless, as the specified fifth row of Table 7.

4.5.2 IP trend analysis

A clear increasing trend did found; as the longitudinal reinforcement ratio increased, the displacement tended to increase, as illustrated in Fig. 11(a). It is likely because excessive longitudinal steel bars did not use (0.7%, 1.5%. 3%, and 4.5%), and the failure mode did not change from ductile to fragile, which did not deteriorate displacement capacity.

Fig. 11
figure 11

Linear regression analysis

The coefficient of subgrade reaction refers to the settlement ratio of the ground caused by the load acting on the foundation. It represents how much the ground can withstand when subjected to loads from the structure, assuming the ground behaves like a spring. The coefficient of subgrade reaction is a constant that represents the stiffness of the ground. A larger coefficient of subgrade reaction indicates that the soil is more stable, dense, less compressible, and has a higher bearing capacity.

Also, a clear decreasing trend did found; as the reaction coefficient of the subgrade increased, the displacement tended to decrease, as illustrated in Fig. 11(b). It was likely because a low value of the reaction coefficient of the subgrade represented a flexible soil that makes the foundation of an RC bridge pier unstable.

Thus, the Fig. 11(c) up (k) could not define the trend line for the other nine input parameters (f'c, dlrP, dt, L/D, s, Ф, fy), since the data is sparse, as reflected in the shallow values of R2 (less than 0.07), which have no statistical significance, practically meaningless.

5 Conclusions

This article presents the response of the seismic performance of the reinforced concrete bridge pier model was evaluated in terms of displacement, taking into account eleven input parameters for six limit states, obtaining the following conclusions:

  1. (i)

    The range of the displacement in terms of drift was found: for the cracking displacement [u1] = [0, 0.7%], corresponding to the cracking limit state; for the yielding displacement [u2] = [0.8%, 1.7%], corresponding to the yielding limit state; for the spalling displacement [u3] = [1.4%, 2.5%], corresponding to the spalling limit state; for the crushing displacement [u4] = [2.6%, 3.9%], corresponding to the crushing limit state; for the buckling displacement [u5] = [3.8%, 5.5%], corresponding to the buckling limit state; and for the fracturing displacement [u6] = [5.6%, 6.5%], corresponding to the fracturing limit state. It can do use to evaluate expected displacements in reinforced concrete bridge piers.

  2. (ii)

    The derived prediction equation for spalling displacement showed an excellent approximation, around 20%, compared to the model displacement. It can do use for the initial calculation of referential spalling displacements.

  3. (iii)

    The pier aspect ratio and coefficient of subgrade reaction; were the two most influential in the response of the seismic performance of the reinforced concrete bridge piers, with coefficient of determination values of 64% and 46%, respectively; however, it did not approach 100%, indicating that displacement is not as easy to predict.

  4. (iv)

    The concrete compressive strength, the yield stress of reinforcing steel, the concrete cover thickness, the configuration of the transverse reinforcement, the spacing of the transverse reinforcing steel, the transversal diameter of the transverse reinforcing steel, the longitudinal reinforcement ratio, the transversal diameter of the longitudinal reinforcing steel, and the axial load ratio; were the nine input parameters less influential in the response of the seismic performance of the reinforced concrete bridge piers, with the coefficient of determination values up to a total of 7%, indicating their low influence on displacement and were statistically insignificant.

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The data and materials generated or used during the study are available from the corresponding author by request.

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Acknowledgements

The authors would like to acknowledge the National University of San Cristobal de Huamanga (Peru) for the financial support provided for carrying on his doctoral studies. Also, we would like to thank the Engineering Doctoral Program at the Pontifical Catholic University of Peru for the support received.

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Javier Taipe conceived this study, participated in its design and coordination, and reviewed and revised the manuscript. Victor Fernandez-Davila participated in the analysis of the test data, and revised of the manuscript. All the authors read and approved the final manuscript.

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Correspondence to Javier F. Taipe.

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Taipe, J.F., Fernandez-Davila, V.I. Displacement-based seismic performance of RC bridge pier. ABEN 4, 17 (2023). https://doi.org/10.1186/s43251-023-00095-0

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