A Review of Prediction Methods for Ultra-Low Cycle Fatigue Damage of Steel Piers Under Earthquakes

Abstract

Steel bridge piers are vulnerable to seismic damage, with ultra-low cycle fatigue commonly occurring at the base of the piers or at welded joints under strong earthquake conditions. The current methods for predicting ultra-low cycle fatigue in steel bridge piers include the empirical formula method, the cyclic void growth model (CVGM), and the continuous damage mechanics (CDM) model. This paper reviews the principles, development, advantages, and limitations of these three prediction methods. Compared to the empirical formula method, the CDM model offers improved accuracy for predicting ultra-low cycle fatigue life in welded joints under complex stress conditions. Unlike the CVGM, the CDM model requires fewer calibration tests, directly incorporates the effects of damage on material properties, and integrates well with finite element software. These benefits highlight the potential of CDM-based prediction methods for practical application.

1. Introduction

Steel structure bridges are known for their lightweight, high strength, compact area, and low energy consumption [1,2]. As steel production and labor costs rise, the advantages of steel bridges become more apparent. Recently, the number of steel bridges in China has increased significantly [3,4]. When compared to traditional reinforced concrete piers, steel bridge piers offer high strength, excellent seismic performance, and rapid construction, making them widely utilized in the United States, Japan, and Taiwan [5,6]. With China’s rapid economic growth and rising seismic performance standards for steel bridges, the application of steel bridge piers is expected to expand further.

Despite their benefits, steel piers remain vulnerable during earthquakes [7]. Past seismic events have shown that steel bridge piers primarily suffer from the local buckling failure of the steel plates near the bottom [8] and ultra-low cycle fatigue failure at welded joints [9]. There are three main reasons for the local buckling of the pier. First, the earthquake causes the horizontal movement of the pier, which causes the axial force at the top of the pier to become the eccentric axial force. Second, ground motion causes the resonance of the bridge pier, which changes the shape of the pier. Finally, the material becomes less strong in earthquakes. Notably, thick-walled steel bridge piers may experience ultra-low cycle fatigue failure prior to local buckling, representing a typical mode of seismic damage. Under the action of earthquakes, steel piers are not only subjected to axial load, but also to horizontal seismic load. Therefore, the welded joints of steel piers are subjected to axial force, horizontal shear force, and bending moment simultaneously, placing them in a complex stress state. For instance, during the 1995 Kobe earthquake in Japan, complex stress states at the joints of many steel piers led to ultra-low cycle fatigue failures, as illustrated in Figure 1 [9,10]. Due to the characteristics of temperature transfer, the welded joints have obvious gradient distribution of mechanical properties [11], which has an adverse effect on the fracture properties of welded joints. So far, researchers both domestically and internationally have extensively studied local buckling in steel plates and ultra-low cycle fatigue failure in welded joints. They have effectively addressed the local buckling issues by implementing longitudinal stiffeners, increasing plate thickness, and infilling concrete at the bases of steel bridge piers [12,13,14,15,16].

Recently, scholars identified ultra-low cycle fatigue failure as a failure mode characterized by high strain levels and fatigue lives ranging from just a few cycles to even less than ten. This mode is different from low cycle fatigue failure [17,18,19]. Through various studies on the failure mechanisms of ultra-low cycle fatigue and low cycle fatigue in steel, researchers have clarified that ultra-low cycle fatigue failure is fundamentally different. It is primarily a ductile failure, while low cycle fatigue failure tends to be brittle.

In 2006, Kermajani M [20] and Zhang [21] observed, using scanning electron microscopy of the fracture surface (Figure 2), that cracks resulting from ultra-low cycle fatigue failure exhibit bluntness, noticeable width, and significant opening. The fracture surface displays distinct ductile dimples, indicating a ductile failure morphology. In contrast, cracks due to low cycle fatigue failure are sharp, narrow, and deep, typically resulting in a cleavage fracture indicative of brittle failure. Based on the characteristics of ductile failure in steel, Anderson et al. [22] in 2005 and Kanvinde et al. [17] in 2007 proposed the microvoid coalescence fracture mechanism to explain the microscopic processes involved in ultra-low cycle fatigue failure.

The research on the mechanism of ultra-low cycle fatigue failure in structural steel reveals that this failure is characterized by ductile cracking [23]. Its fracture mechanism differs from that of low cycle fatigue failure [24,25,26,27]. Currently, research on predicting ultra-low cycle fatigue damage remains exploratory. The primary methods for predicting ultra-low cycle fatigue damage in steel include the empirical formula method [28,29], the cyclic void growth model (CVGM) [30], and the continuous damage mechanics (CDM) model [31].

The traditional fracture mechanics method is mainly suited for studying brittle or pseudo-brittle fractures with minimal local plastic damage [32]. It is not applicable to ductile fracture under cyclic loading [33]. Although the empirical formula method offers simplicity and ease of integration with finite element software, it requires extensive parameter calibration and cannot effectively predict the fatigue life of welded joints under complex stress states [34]. The CVGM can forecast the ultra-low cycle fatigue life of steel or steel structures under complex stress conditions; however, it demands numerous calibration tests, features significant parameter dispersion, and entails a complicated prediction process [35,36]. Although the CDM model has fewer calibration requirements, offers multiple prediction functions, and integrates easily with finite element software, it does not account for the influence of equivalent plastic strain on damage increment, indicating a need for improved prediction accuracy [37].

These prediction methods primarily focus on estimating the crack initiation life of ultra-low cycle fatigue in steel piers [38]. However, it has been observed that crack initiation typically does not impact the horizontal bearing capacity or displacement ductility of steel piers. Only once a crack reaches a certain length does the bearing capacity and ductility significantly decline. Thus, predicting ultra-low cycle fatigue in steel bridge piers should not rely solely on crack initiation limits [39].

Due to the lack of consensus among scholars on predicting ultra-low cycle fatigue life in steel, relevant standards domestically and internationally, such as China’s “General Specifications for Design of Highway Bridges and Culverts” [40] and Japan’s “Specifications for highway bridges-PartⅡ Steel Bridges” [41], do not provide methods for predicting ultra-low cycle fatigue damage in steel piers during strong earthquakes. Consequently, these standards are insufficient for evaluating the ability of steel piers to withstand ultra-low cycle fatigue damage.

2. Research Status and Development Trend at Home and Abroad

To establish a prediction method for ultra-low cycle fatigue damage of steel bridge piers during earthquakes, scholars have explored the life, damage locations, and damage processes of ultra-low cycle fatigue in steel and steel structures. Currently, there are three primary methods for predicting ultra-low cycle fatigue damage in steel structures: the empirical formula method, the CVGM, and the CDM model. This section reviews the current research on these three prediction methods, focusing on their principles, applicability, and prediction accuracy.

2.1. Empirical Formula Method

In 1954, Coffin [28] and Manson [42] proposed the Coffin–Manson Formula (1) to predict low cycle fatigue life based on the relationship between fatigue life and plastic strain amplitude.

$$\Delta {\epsilon }_p/2={\epsilon }_f^{\prime }{(2N_f)}^c$$

(1)

In the formula, Δε_p is the plastic strain amplitude; N_f is the fatigue life of the material; ${\epsilon }_f^{\prime }$ is the fatigue ductility coefficient; and c is the fatigue ductility exponent.

The Coffin–Manson formula remains the primary method for predicting low cycle fatigue life in steel. Its accuracy has been confirmed by numerous experimental results [43,44,45]; however, it tends to overestimate ultra-low cycle fatigue life in steel [34,46,47]. To enhance the prediction accuracy, researchers have developed empirical formulas suitable for both low and ultra-low cycle fatigue predictions, including the Kuroda model [48], Tateishi model [33], and Xue model [25]. Nonetheless, these models are limited to uniaxial stress states and do not account for stress triaxiality, making them inadequate for predicting ultra-low cycle fatigue life in welded joints under complex stress conditions. Additionally, the parameters in these empirical formulas vary with material properties and welding methods, necessitating extensive parameter calibration experiments [49].

Beyond predicting fatigue life in steel and welded joints, researchers have also investigated methods for predicting ultra-low cycle fatigue damage in steel piers, typically based on the Coffin–Manson formula [38,50,51]. For instance, Ge et al. [38] proposed a prediction method for ultra-low cycle fatigue damage in steel bridge piers, which combines the Coffin–Manson formula with the Miner cumulative damage criterion, as shown in Formula (2):

$$D=C^{\prime }{\displaystyle \sum _{i=1}^n{(\Delta {\epsilon }_{pi})}^m}$$

(2)

In the formula, D is the damage index; Δε_pi is the plastic strain amplitude; and the material parameters $m=1/k$ and $C\prime ={(1/C)}^m$ are determined according to the test results at fracture (D = 1). However, in the modeling process of steel bridge piers, the length of 2B (B is the width of the flange plate) or 3l_d (l_d is the spacing of the diaphragm) is shown according to Figure 3 and Figure 4, and the element division criteria are not established. It is relatively difficult to promote the application. To enhance the prediction accuracy and stability, Kang et al. [50] proposed a non-local damage variable in 2015 using an exponential weight function, which led to the development of a non-local damage method.

The aforementioned research indicates that the empirical formula method is only applicable to uniaxial stress states. It is unable to consider the effects of stress triaxiality and cannot predict the ultra-low cycle fatigue life of welded joints under complex stress states. This method requires numerous parameter calibration experiments.

2.2. Cyclic Void Growth Model

Recent micro-damage mechanism fracture models offer advantages over empirical formula methods. These models not only predict the fatigue life of steel structures but also describe how stress–strain fields affect the inherent microstructural characteristics of materials. As a result, they provide clear physical criteria for more accurate predictions of ductile cracks in steel structure nodes [52,53,54]. Key fracture models for predicting ultra-low cycle fatigue damage based on the micro-damage mechanism include the CVGM) [30] and degraded significant plastic strain (DSPS) [55]. The following discussion describes the development process of the cyclic void growth model (CVGM).

In 1969, Rice and Tracey [56] believed that when microvoids exceeded the critical void size, unstable necking would occur between voids and lead to void aggregation and crack initiation. Therefore, a void growth model (VGM) that was suitable for ductile cracking prediction under monotonic loading was proposed. In 1976, Hancock et al. [57] proposed a stress-corrected critical strain (SMCS) model based on VGM for predicting ductile cracking under monotonic loading, assuming that stress triaxiality remains basically unchanged during loading. Based on the VGM and the SMCS model, scholars have conducted research on the prediction of steel fracture under monotonic tensile load. In 2006, Chi et al. [58] corrected the SMCS model parameters of ASTM A572 Grade 50 steel through the tensile test and finite element analysis of notched steel bar specimens. They then verified the prediction accuracy of the SMCS model for the ductile failure of large-scale joints through beam–column joint tests. Researchers believe that SMCS can predict the ductile failure of metal materials without initial cracks. In 2007, Kanvinde [17] verified the effectiveness of VGM and the SMCS model in predicting the ductile cracking of steel through the tensile tests of 12 specimens. Based on the preceding analysis, VGM and the SMCS model are suitable for the ductile cracking of steel specimens under monotonic loading. Moreover, VGM can consider the change in stress triaxiality, and its prediction accuracy is higher than that of the SMCS model.

Based on VGM, Kanvinde et al. [17] proposed Formula (3) for calculating the cavity expansion index VGI_cyclic under tension and compression cyclic loading based on the cavity radius growth formula of Rice–Tracey in 2007.

$$VG{I}_{cyclic}={\displaystyle \sum _{tensile}{\displaystyle {\int }_{{\epsilon }_1}^{{\epsilon }_2}\exp \left(\left|1.5T\right|\right)d{\epsilon }_p}}-{\displaystyle \sum _{compressive}{\displaystyle {\int }_{{\epsilon }_1}^{{\epsilon }_2}\exp \left(\left|1.5T\right|\right)d{\epsilon }_p,}}$$

(3)

where ε₁ and ε₂ are the equivalent plastic strain at the beginning and end of the tension or compression cycle, respectively; ε_p is the equivalent plastic strain; and T is the stress triaxiality. The calculation formula of T is

$$T={\displaystyle \frac{1}{3}}\left({\sigma }_1+{\sigma }_2+{\sigma }_3\right)/\sqrt{{\displaystyle \frac{1}{2}}\left[{\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_2-{\sigma }_3\right)}^2+{\left({\sigma }_3-{\sigma }_1\right)}^2\right]},$$

(4)

where σ₁, σ₂, and σ₃ represent principal stress in the fracture zone.

The calculation formula of the critical hole expansion index $VG{I}_{cyclic}^{critical}$, which is used to compare with the hole expansion index VGI_cyclic, is given as

$$VG{I}_{cyclic}^{critical}=\eta \exp \left(-{\lambda }_{CVGM}{\epsilon }_p^{accumulated}\right),$$

(5)

where ${\lambda }_{CVGM}$ is the damage degradation parameter of the material under cyclic loading, ${\epsilon }_p^{accumulated}$ is the cumulative equivalent plastic strain, and $\eta$ is the material toughness parameter under monotonic tension. The calculation formula of $\eta$ is given as

$$\eta ={\displaystyle \underset{0}{\overset{{\epsilon }_p}{\int }}\exp \left(1.5T\right)}\times d{\epsilon }_p.$$

(6)

Finally, the calculation Formula (7) of the fracture index $F{I}_{CVGM}$ in the cyclic cavity expansion model (i.e., CVGM) can be obtained using Formulas (3) and (5).

$$F{I}_{CVGM}=VG{I}_{cyclic}/VG{I}_{cyclic}^{critical}\ge 1$$

(7)

When $F{I}_{CVGM}\ge 1$, the material point is determined to exhibit ultra-low cycle fatigue ductile cracking, and the fracture criterion must be satisfied on characteristic length l* to predict the fracture [35]. From the preceding formulas, the CVGM accounts for changes in stress triaxiality during loading, enhancing prediction accuracy for fatigue life in steel structure joints.

In the ultra-low cycle fatigue prediction process of the CVGM, when the $F{I}_{CVGM}$ value of the unit satisfies Formula (7), the finite element is deleted, and then the post-cracking path is predicted.

The CVGM accounts for changes in stress triaxiality during loading, enhancing prediction accuracy for fatigue life in steel structure joints. When stress triaxiality remains relatively constant during loading, the CVGM can be simplified to the DSPS, although this may reduce calculation accuracy.

Building on the CVGM, researchers have studied methods for predicting ultra-low cycle fatigue damage in steel structures [35,59,60,61,62,63]. In 2012, Liao Fangfang et al. [35] calibrated the parameter $\eta$ of the CVGM for Q345 steel and the damage degradation parameter ${\lambda }_{CVGM}$ under cyclic loading. Their findings indicated a need for numerous calibration tests, which often produced widely varying results. In 2022, Luo [64] calibrated the fracture toughness parameters of S220503 duplex stainless steel under the CVGM. At the component level, Xie et al. [62] established a mixed-element model in 2018 to predict ultra-low cycle fatigue damage in the high-strain areas of steel arch bridges, based on the CVGM fracture criterion. However, the CVGM assumes that the expansion and contraction rates of microvoids are equal, which does not accurately reflect the actual stresses in steel. The simulation agreed well with the test results, which verified the applicability of the CVGM to the numerical simulation of the ULCF fracture of structural steels [65]. Qiu [66] carried out experimental research on specimens made of high-strength steel at different positions (base metal, welding material, heat-affected zone). After calibrating the parameters of the kinematic–isotropic hybrid strengthening model, the finite element software was used for calculation. The results were substituted into the CVGM for fracture prediction. By comparing the experimental data, it was found that the CVGM can accurately predict the ultra-low cycle fatigue life. Based on the CVGM, Christopher Smith [67] proposed a new stress-weighted ductile fracture model (SWDFM) to simulate ductile fracture induced by ultra-low cyclic fatigue (ULCF). The SWDFM can accurately predict the cycle life of ULCF under a complex stress state and loading system.

The research indicates that although the CVGM effectively accounts for the influences of stress triaxiality and cumulative equivalent plastic strain on cavity expansion ratio, leading to a more accurate prediction of ultra-low cycle fatigue life in steel [59,60,61,62,63], it also has several shortcomings. The CVGM is both semi-empirical and semi-theoretical, relying on numerous parameter calibration tests. Additionally, variations in the materials of welds and heat-affected zones result in significant discrepancies in damage degradation parameters for welded specimens. Moreover, the model assumes that the expansion and contraction rates of micro-pores are identical, which does not reflect actual conditions.

2.3. Cyclic Multiaxial Fracture Strain Energy (CMFSE)

Based on the Multiaxial Fatigue Strain Energy (MFSE) model with the stress correction, Nam proposed the Cyclic Multiaxial Fracture Strain Energy (CMFSE) model. The formula of accumulated effective equivalent plastic strain energy can consider the influence of the expansion and contraction of microvoids during cyclic loading [68].

2.4. Continuum Damage Mechanics (CDM) Model

The CDM model is the third method for predicting ultra-low cycle fatigue damage in steel structures. This model describes the macroscopic mechanical behavior of materials and the evolution of damage by utilizing appropriate damage variables. Unlike empirical formula methods, the CDM model can take stress triaxiality into account and predict the ultra-low cycle fatigue life of welded joints under complex stress states. Additionally, compared to the CVGM, which is grounded in microscopic damage mechanisms, the CDM model requires fewer calibration tests. It also results in less variability in calibrated parameters. CDM can directly factor in the effects of damage on material constitutive behavior and integrates easily with finite element software, enhancing its practical applications [69].

2.4.1. The Principle and Development of Traditional CDM Model

In 1963, Rabotnov [70] proposed the concepts of damage factor and effective stress while formulating the creep constitutive equation for metals, laying the groundwork for the CDM model. In 1997, building on Lemaitre’s research [71], Bonora et al. [31,72] assumed that the damage strain threshold under uniaxial stress corresponds to that under multiaxial stress. They derived a CDM model suitable for predicting ductile fracture under monotonic loading, represented in Formula (8):

$$dD=c{\displaystyle \frac{{(D_{cr}-D_0)}^{1/c}}{\ln ({\epsilon }_f/{\epsilon }_{th})}}f(T){(D_{cr}-D)}^{(1-1/c)}{\displaystyle \frac{d{\overline{\epsilon }}^p}{{\overline{\epsilon }}^p}}$$

(8)

where $D$, $D_0$, and $D_{cr}$ are the damage, initial damage, and critical damage variables, respectively. c is the damage exponent characteristic of the material. ${\epsilon }_{th}$ represents the plastic strain threshold. ${\epsilon }_f$ represents the fracture-accumulated plastic strain under uniaxial stress. T and f(T) represent the stress triaxiality and stress triaxial function, respectively. ${\overline{\epsilon }}^p$ is the accumulated plastic strain.

The uniaxial damage strain threshold ε_th in Formula (8) is calibrated by Formula (9):

$${\overline{\epsilon }}_f^p={\epsilon }_{th}{({\displaystyle \frac{{\epsilon }_f}{{\epsilon }_{th}}})}^{1/f(T)}$$

(9)

${\overline{\epsilon }}_f^p$ represents the fracture-accumulated plastic strain under multiaxial stress.

It can be seen from the formulas above that the CDM model, suitable for the monotonic tensile loading of metals, requires only the calibration of the uniaxial damage strain threshold value ε_th and the uniaxial fracture strain to effectively predict the material’s damage process. The damage variable accounts for the effects of cumulative plastic strain and stress triaxiality, resulting in high prediction accuracy. In 2016, Tong et al. [73] suggested that the damage to steel under ultra-low cycle fatigue load results solely from tensile strain. They determined the tensile and compressive strains during loading based on the positive and negative signs of stress triaxiality. They extended Formula (8) to apply to ultra-low cycle fatigue fracture prediction, as shown in Formulas (10)–(12):

$$dD=c{\displaystyle \frac{{(D_{cr}-D_0)}^{1/c}}{\ln ({\epsilon }_f/{\epsilon }_{th})}}f(T){(D_{cr}-D)}^{(1-1/c)}{\displaystyle \frac{d{\overline{\epsilon }}^{p+}}{{\overline{\epsilon }}^p}}$$

(10)

$$d{\overline{\epsilon }}^{p+}=d{\overline{\epsilon }}^p\times H(T)$$

(11)

$$H(T)=\left\{\begin{array}{cc}0\hfill & T<0\hfill \\ 1\hfill & T\ge 0\hfill \end{array}\right\}$$

(12)

$d{\overline{\epsilon }}^{p+}$ is the cumulative plastic tensile strain increment, and $H(T)$ is the function describing the damage state.

By modifying the elastic modulus of steel, Formula (13) realizes the influence of the damage variable D on the macroscopic mechanics of steel.

$$E=E_0\left[1-D\times H(T)\right]$$

(13)

In the formula, E₀, E are the initial elastic modulus and the elastic modulus after damage, respectively.

Building on the traditional CDM model, Tong et al. [73] calibrated the parameters solely through monotonic tensile tests. They predicted the crack initiation life and fracture life of six beam–column joints, validating the formula through experiments. However, this prediction approach does not account for the influence of equivalent plastic strain on the damage increment, contributing to discrepancies between the predicted and test lives. The CDM model treats the entire system as solid elements, which affects calculation efficiency. There is also a lack of refined modeling methods that balance efficiency and accuracy.

In 2020, Tian et al. [37] investigated ultra-low cycle fatigue damage prediction in Q345qC steel-welded specimens using the solid element approach of the traditional CDM model. They found that while the CDM model could predict the crack initiation position, initiation life, and crack propagation path, the predicted life was lower than the test life, indicating a need for improved prediction accuracy.

2.4.2. The Method of Testing the Critical Value of Damage Variable

All of the aforementioned studies on the prediction of the ultra-low cycle fatigue damage of steel using the traditional continuum damage mechanics (CDM) model have assumed that the critical value of the damage and fracture of materials is constant at 1. That is, the complete fracture state of a specimen is set as the criterion for the critical value of damage and fracture, which is inaccurate. The reason is as follows: regardless of whether the monotonic tensile test of steel or the load–displacement curve measured by the ultra-low cycle fatigue test is used, the load drops sharply when the bearing capacity passes through a certain point, and the steel sample can be regarded as a failure, but the steel sample is not completely broken at this moment. The method of determining the critical value of damage and fracture in the traditional CDM model must be further studied [74]. At present, two major methods are used to measure damage variables. The first one directly measures various microscopic defects in the process of material damage from a microscopic point of view [75]. However, this method cannot reflect the relationship between microscopic damage and macroscopic mechanics. It also cannot predict the macroscopic mechanical damage characteristics of steel structures, and thus, it is difficult to apply. The other one is an indirect measurement method that can consider the influence of micro-damage on the constitutive relationship of materials. It can directly establish a constitutive relationship with macro-mechanical quantities, and thus, it is widely used. In Lemaitre et al. [75], an elastic modulus method for testing metal damage based on the coupling law of elasticity and damage of thermodynamic potential was proposed, as shown in Figure 5. This method measures the elastic modulus under different strains through repeated loading and unloading, and it calculates the development process of the damage variable D. Although the elastic modulus method is widely used, its measurement accuracy of strain is affected by the early microplasticity of the metal and the reversible movement and texture development of the dislocation, resulting in a considerable difference in the test results. Simultaneously, repeated loading and unloading lead to the complexity of the test process.

To simplify the test process and improve the test accuracy of the damage variable, Policella et al. [76] defined the effective cross-sectional area as the damage variable. Based on the relationship between the effective cross-sectional area of a material and the potential difference, the change in the effective cross-sectional area of a material is indirectly reflected by measuring the change in the potential difference between the two sides of the break point of a notched round bar sample. The basic principle and formula of test are as follows. In accordance with Ohm’s law, the two-point potential difference V without damage and the potential difference $\overline{V}$ in a damaged state can be computed using Formula (14).

$$\left\{\begin{array}{l}V=r{\displaystyle \frac{L}{A}}I\\ \overline{V}=r{\displaystyle \frac{L}{A_0}}I\end{array}\right.$$

(14)

where r is the resistivity of the specimen, L is the initial distance between the measuring points, and I is the applied current.

The damage variable D can be expressed by the potential difference through Formula (15).

$$D=1-{\displaystyle \frac{V}{\overline{V}}}$$

(15)

The alternating current potential damage measurement method is not required to consider the insulation between the specimen and the testing machine and the extensometer and the specimen during the test. The test results are not affected, and test accuracy is high. This method is only required to perform a monotonic tensile test to measure the potential difference between the measuring points when the steel sample is undamaged or damaged. Then, the evolution law of the damage variable can be determined. The test process is simple.

2.4.3. Simulation Method for Ultra-Low Cycle Fatigue Crack Propagation

The propagation of ultra-low cycle fatigue cracks involves the evolution of the plastic zone at the crack tip. Currently, fatigue crack propagation methods primarily rely on empirical formulas, such as the Paris formula. In 1961, Paris et al. [77] established a relationship between the stress intensity factor amplitude and the crack growth rate through experiments. They proposed the Paris formula, which can predict fatigue crack life and propagation length. However, this formula mainly focuses on the stress intensity factor amplitude as the control parameter. It is primarily effective for fatigue crack propagation in linear elastic stress conditions and not suitable for ultra-low cycle fatigue scenarios involving plastic deformation. To broaden the application of the traditional Paris formula, researchers worldwide have explored alternative elastic-plastic fracture mechanics parameters that resemble the stress intensity factor for describing low-cycle fatigue crack propagation. Commonly used parameters include crack opening displacement (CTOD) [78], plastic strain range (Δε) [79], and cyclic J integral range (ΔJ) [80]. These parameters have significantly advanced the study of low-cycle fatigue crack propagation, yet they possess certain limitations. Additionally, the existing methods for predicting low cycle fatigue crack propagation can be complex and inefficient. These methods often require fine element division at the crack tip to calculate stress intensity factors or plastic strain ranges, as well as adjustments to the finite element mesh according to the new crack’s propagation direction and size.

Before the crack initiation, the CDM model effectively predicts the ultra-low cycle fatigue damage evolution and crack initiation life of steel bridge piers [37,69]. Research on crack propagation primarily relies on fracture mechanics theory. When predicting the ultra-low cycle fatigue damage of steel bridge piers, it is essential to consider the relationship between the mechanical parameters in the two prediction stages. This overall process can be relatively complex. Continuing to simulate the crack propagation process and predict the crack propagation life using results from the CDM model prior to crack initiation is quite convenient. In 2019, Tian et al. [37] simulated crack propagation in Q345 steel under ultra-low cycle fatigue load on the basis of the CDM model and birth–death element method. However, research on crack propagation paths and life based on the CDM model for steel structures remains limited.

2.4.4. Advantages and Disadvantages of Prediction Methods

Advantages and disadvantages of prediction methods is shown as

Table 1. Advantages and disadvantages of prediction methods.

No	Name	Advantages	Disadvantages
1	Empirical formula method	Few parameter calibration experiments; easy to integrate with finite element software; high computational efficiency	Prediction of fatigue life cannot consider the influence of the spatial stress state
2	CVGM	Prediction of fatigue life can consider the influence of the spatial stress state; medium computational efficiency	Additional parameter calibration experiments; difficult to integrate with finite element software
3	CDM	Prediction of fatigue life can consider the influence of the spatial stress state; few parameter calibration experiments; easy to integrate with finite element software	Low computational efficiency

. The research indicates that the traditional CDM model is an effective method for studying ultra-low cycle fatigue damage in steel and steel structures. Compared to empirical formulas and the CVGM, the CDM model offers several advantages:

(1)

The CDM model, designed for ductile fracture prediction, requires parameter calibration from monotonic tensile tests, minimizing the number of tests needed.

(2)

This model features multiple prediction functions. It not only forecasts the ultra-low cycle fatigue life of steel and steel structures but also estimates damage cycles.

(3)

The CDM model accounts for damage effects on the material constitutive behavior, making it compatible with finite element software and practical for engineering applications.

Despite the promising development potential of traditional CDM models, they also have the following shortcomings. (1) The existing models do not consider the effect of equivalent plastic strain on damage increment. (2) There is a lack of numerical simulation methods grounded in the CDM model to predict the ultra-low cycle fatigue crack propagation in welded joints. (3) Predicting fatigue crack propagation requires a combination of solid elements for accurate results, resulting in increased computational demands. Furthermore, there is an absence of mixed-element division criteria for steel bridge piers that take into account ultra-low cycle fatigue damage.

2.5. Seismic Performance and Seismic Response Calculation Model of Steel Pier

Zhuge et al. [81] conducted an ultra-low cycle fatigue test on steel bridge piers. The material parameters of Q345qC are provided in

Table 2. Basic material parameters of Q345qC steel [81].

σy(MPa)	E(MPa)	μ	${\mathit{\epsilon }}_{\mathit{f}}^{\prime }$	c
391.2	2.045 × 105	0.3	0.4441	−0.6003

. ${\epsilon }_f^{\prime }$ is the fatigue ductility coefficient; c is the fatigue ductility exponent. Figure 6 shows the specimens and test devices. A schematic of the weld of specimens is presented in Figure 7. Figure 8 presents the loading form of forced horizontal displacement. δ₀ is the loading strain, and δ_y is the yield strain. In the test, the location of the crack initiation and cycle times of steel piers are analyzed. When a crack initiates, the bearing capacity does not decrease immediately but can withstand subsequent cyclic loads.

Current methods for predicting ultra-low cycle fatigue damage in steel bridge piers primarily focus on estimating crack initiation life [82], with limited consideration for crack propagation life. Based on quasi-static tests, researchers have found that cracks may initiate before reaching the maximum horizontal bearing capacity, without immediately affecting the horizontal thrust of the piers. A significant reduction in horizontal thrust occurs only when cracks reach a critical length. Thus, predictions of ultra-low cycle fatigue for steel bridge piers should extend beyond the crack initiation phase [39]. Existing seismic response models do not account for crack propagation effects, resulting in discrepancies between the predicted and observed horizontal load-displacement hysteresis and skeleton curves in the crack propagation stage [83]. The fatigue damage prediction limit (crack propagation length) under varying damage levels is closely tied to the seismic performance evaluation index and an accurate seismic response model for steel bridge piers. The following sections outline the current research and development trends in these areas.

2.5.1. Seismic Performance Evaluation Indicators for Steel Bridge Piers

In 2000, Ge and Usami et al. proposed an empirical formula for assessing the bearing capacity and ductility of steel piers subjected to horizontal loads influenced by local buckling of steel plates. This was based on findings from pseudo-static tests, pseudo-dynamic tests, and finite element analyses of steel piers [84,85]. Japan’s “Specifications for highway bridges-PartⅡ Steel Bridges” adopted the research results of Ge and Usami et al. and established an evaluation index system for the seismic performance of steel piers, as shown in Figure 9 [5]. Figure 9 shows the skeleton curve of a steel bridge pier under horizontal reciprocating load. δ_u is the ultimate displacement, and δ_y is the yield displacement. The size of the horizontal displacement determines the damage state of the steel pier. The state wherein the ultimate bearing capacity reduces by 5% is defined as the limit of damage state for the steel pier. After developing the seismic performance evaluation index for steel piers, it is crucial to create an accurate hysteresis model. This model will help determine the boundary for predicting ultra-low cycle fatigue damage across varying degrees of damage.

2.5.2. Seismic Response Calculation Model of Steel Bridge Pier

The primary seismic failure modes of steel piers include local buckling of steel plates and ultra-low cycle fatigue failure of welded joints. To address local buckling while ensuring calculation efficiency and accuracy, most researchers use shell elements to model the local buckling zones of steel piers and beam elements for other undamaged areas. This method has demonstrated effectiveness in calculating the seismic response and horizontal bearing capacity of steel piers [86,87]. However, simulating the ultra-low cycle fatigue failure of welded joints presents challenges, particularly in modeling fatigue crack propagation. As a result, the current seismic response calculation models for steel bridge piers often overlook the influence of crack propagation. This leads to discrepancies between the displacement hysteresis curve and skeleton curve of the steel bridge pier during the crack propagation stage and the corresponding experimental results [83].

The coexistence of ultra-low cycle fatigue failure in welded joints and local buckling failure in steel plates is common. The sequence of these failures is unpredictable, and their interaction affects the seismic response of steel piers. For instance, in 2012, Ge Hanbin et al. conducted various reciprocating displacement load tests on nine unstiffened box steel piers. They discovered that most thick-walled pier specimens developed fatigue cracks before experiencing local buckling. Notably, horizontal loads had not reached their maximum when these cracks appeared [39]. In 2019, Zhuge Hanqing et al. performed quasi-static tests on two box-type steel bridge piers. Their findings indicated that welded joints underwent ultra-low cycle fatigue cracking following the local deformation of the steel plate. Once the cracks grew to a certain length, the structural bearing capacity decreased [81].

Using mixed-element models is necessary to improve computational efficiency during the seismic analysis of steel bridge piers. This model requires the use of beam or plate shell elements in regions wherein ultra-low fatigue damage has not occurred and that of solid elements in regions wherein ultra-low cycle fatigue damage has occurred. The CDM programs representing ultra-low cycle fatigue damage are written in programming software, such as FORTRAN. Subsequently, the damage attributes are assigned to solid elements in finite element software, such as ABAQUS 6.14. Finally, the ultra-low cycle fatigue damage of steel bridge piers is calculated on the basis of multiscale models. The CDM model is successfully applied to predict the ULCF damage of steel beam-to-column connections [88].

In summary, current predictions for ultra-low cycle fatigue damage in steel bridge piers, based on crack initiation, lack accuracy. The predictions vary depending on the seismic performance evaluation indicators and seismic response calculation models for steel bridge piers. Existing models do not account for the effects of ultra-low cycle fatigue crack propagation and their interaction with local buckling, limiting their theoretical applicability in predicting damage.

3. Conclusions

(1)

The traditional CDM model is not highly accurate in predicting ultra-low cycle fatigue damage in steel and its welded specimens. Although this model can estimate the fatigue process and life of welded joints under seismic conditions, it fails to incorporate the effects of equivalent plastic strain on damage increments and lack a reliable method for determining the critical value of damage and fracture, leading to discrepancies between predictions and experimental results.

(2)

Current prediction methods lack a mixed-element partitioning criterion for steel bridge piers that accounts for the coupling effects of ultra-low cycle fatigue and local buckling. The modeling approach is not universally applicable and poses challenges for practical engineering applications.

(3)

Prediction limits for ultra-low cycle fatigue damage should not rely solely on crack initiation. The hysteresis model and skeleton curve utilized for evaluating the seismic performance of steel bridge piers do not incorporate the effects of crack propagation and its interaction with local buckling, which deviates from experimental findings.