Experimental Study on Fatigue Performance of Q355D Notched Steel Under High-Low Frequency Superimposed Loading

Abstract

During the service life of steel bridges, the structural stress histories display combined cyclic characteristics due to the superposition of low-frequency thermal loading and high-frequency vehicle loading. To investigate the fatigue performance under such loading patterns, a series of constant-amplitude and high-low frequency superimposed loading fatigue (HLSF) tests were conducted on notched specimens fabricated from Q355D bridge steel. The influence of HLSF waveform parameters on fatigue life was systematically investigated. Based on the fracture evolution mechanism, a concept of low-frequency periodic damage acceleration factor was proposed to effectively model the block nonlinear damage effects, and the applicability of existing fatigue life prediction models was discussed. The results show that the effect of average stress on the fatigue life under HLSF can be effectively considered by Walker’s formula. Low-amplitude ratios and low-frequency ratios indicate unfavorable loading conditions that may accelerate the Q355D fatigue damage accumulation, and these conditions are not adequately accounted for in current life prediction models. Compared to constant amplitude loading, HLSF can lead to a 66% and 46% reduction in high-frequency life when the amplitude ratio reaches 0.12 and the frequency ratio reaches 100. Compared to Miner’s rule, the proposed damage correction method reduces the life prediction error for HLSF by 11%. These findings provide valuable references for the fatigue assessment of bridge steel structures under the coupled effects of temperature and vehicle loading.

1. Introduction

In practical engineering, numerous structures or mechanical components are simultaneously subjected to cyclic loads with significantly different frequencies. For example, gas turbine blades are exposed to low-frequency loads caused by centrifugal forces and high-frequency vibrations [1], while ship structures are affected by waves of varying frequencies [2]. For bridge steel structures, they are subjected to the superimposed cyclic stresses of low-frequency temperature loads and high-frequency vehicle loads during their service life [3,4,5,6,7,8,9,10] with the frequency ratio of these two load types reaching up to thousands of times per day. However, existing fatigue design codes for steel bridges [11,12] primarily focus on fatigue damage induced by vehicular loading, with limited attention paid to the effects of temperature effects. With the intensification of global climate change, the influence of thermal fluctuations on fatigue damage in bridge structures has attracted growing attention [13,14,15,16,17,18]. It is of significant engineering importance to investigate the damage mechanisms of bridge steel under high-low frequency superimposed loading fatigue (HLSF) and to develop corresponding fatigue life prediction models.

Current research on HLSF fatigue primarily focuses on materials used in the aerospace and maritime industries [1,2,19,20,21]. Sonsino et al. [22] and Schijve et al. [23] conducted extensive fatigue tests under HLSF and found that the inapplicability of Miner’s rule (Palmgren-Miner linear damage accumulation rule) becomes especially pronounced when the amplitude ratio between the two waveforms is large. Gao et al. [24] performed HLSF tests on two types of steel with varying frequency ratios and observed that both the frequency ratio and amplitude ratio interact to influence the fatigue response. However, in bridge engineering, there is a notable lack of relevant experimental research, and Miner’s rule remains the primary basis for fatigue assessments [4,25].

To better predict fatigue life under the HLSF loading mode, many models have been developed based on experimental studies of different metallic materials. These models can be broadly classified into three major categories: parameterized empirical models [2,26,27], modified versions of Miner’s rule that incorporate coupled damage increments [28,29,30], and modified versions of two-step nonlinear damage curve models [31,32,33]. The first category comprises parameterized empirical models. These models extract HLSF loading characteristics and calibrate model parameters based on extensive experimental data across various materials. The earliest research in this field was conducted by Trufyakov and Kovalchuk [26]. They performed HLSF tests on materials such as medium-carbon steel S43 and aluminum alloys, introducing a correction factor determined by both the frequency ratio and the amplitude ratio. They proposed a fatigue life prediction model for superimposed loading under dual-frequency conditions. Zhao et al. [27] proposed a modified version of T-K model by incorporating material properties such as yield strength and ultimate tensile strength, and the model demonstrated improved accuracy in predicting the HLSF fatigue life of TC11 titanium alloy under high-frequency ratios. Gan et al. [2] conducted HLSF tests on T-shaped notched specimens of Q345B steel. They examined the effects of load amplitude ratio, frequency ratio, stress ratio, and mean stress level on fatigue behavior. Based on their experimental results, they developed an empirical life prediction model for frequency ratios lower than 10. The second category is established based on Miner’s rule, with adding a damage component that accounts for high-low frequency coupling effects. Researchers such as Zhu et al. [28] and Wang et al. [29,30], through the analysis of a large data set involving superimposed loading tests on materials like Ti-6Al-4V and TC11 titanium alloys, have formulated additional damage component that consider factors including frequency ratio and amplitude ratio. The third category is proposed based on the two-step nonlinear damage curve models. Under the framework of nonlinear damage theory, fatigue damage can be equivalently accumulated across different loading level damage curves [31]. Yue et al. [32] and Hou et al. [33] further refined the two-step M-H model by incorporating the characteristic parameters of the HLSF stress waveform. Their proposed model exhibits enhanced accuracy in predicting the fatigue life of Ti-6Al-4V titanium alloy.

In summary, although research on HLSF has been conducted across various materials, the applicability of existing findings and models to bridge steels remains unconfirmed. Moreover, previous studies have not explored the effect of mean stress, which constitutes a loading condition in bridge structures. This study examines the effects of frequency ratio, amplitude ratio, and mean stress through a series of HLSF tests performed on notched specimens fabricated from Q355D bridge steel. By analyzing the fatigue fracture mechanisms, an exponential block damage correction factor was proposed to quantify the nonlinear interactions caused by high-low frequency coupling damage, thereby improving the predictive accuracy of Miner’s rule. The results provide experimental evidence and methodological guidance for assessing the fatigue life of bridge steel structures subjected to temperature and vehicle superimposed loading.

2. Experimental Design and Results

2.1. Material and Specimen

Q355D is a low-alloy structural steel widely used in railway and highway bridges due to its excellent ductility and high tensile strength. The letter D indicates the quality level. Compared to the Q355B and Q355C steels, Q355D has lower phosphorus and sulfur content and better low-temperature impact resistance [34]. The chemical composition of the material is presented in

Table 1. Chemical composition (%) of Q355D steel used in this study.

Steel	C	Si	Mn	Cr	Ni	Nb	Cu	Als	Ti	P
Q355D	0.09	1.24	1.57	0.20	0.04	0.04	0.12	0.03	0.02	0.02

. This steel is a hot-rolled material without normalizing treatment, and its main production processes after continuous casting include controlled hot rolling at a finishing rolling temperature of 800 °C to 870 °C, controlled cooling with a final cooling temperature of 620 °C to 660 °C, and heat straightening at temperatures between 550 °C and 600 °C. Smooth round-bar specimens (Figure 1a) were prepared in accordance with GB/T 228.1-2021 [35] and uniaxial tensile tests were conducted under a constant strain rate of ε = 10⁻³/s. Each test was repeated three times, and the average value was recorded. The basic mechanical properties, summarized in

Table 2. Basic mechanical properties of Q355D steel used in this study.

Mechanical Properties	E (GPa)	ν -	σy (MPa)	σu (MPa)	A (%)	KRO (MPa)	nRO -
Q355D	216.71	0.30	440.33	587.39	32.33	428.89	0.15
GB/T 714-2015 [36]	-	-	≥345	≥490	≥20	-	-

, confirm that the material satisfies the mechanical performance requirements specified in GB/T 714-2015 [36].

As illustrated in Figure 1b, a type of 90° V-notched specimen was designed for fatigue testing. The specimens were cut and machined from a 10 mm thick Q355D steel plate, with the surfaces polished to a roughness of approximately R_a = 0.8 μm. The minimum radius at the tip of the V-notch was approximately 60 μm. The fatigue behavior of these notched specimens can partially simulate crack-like defects commonly found in welded structures [37].

2.2. Loading Conditions and Results

2.2.1. Generalization of HLSF Stress Waveform Patterns

In actual bridge structures, although the stress histories at the fatigue details are typically complex and random, key waveform features can still be extracted to establish load spectra for modular loading studies [38,39]. As an initial investigation into such materials, and to identify more generalizable patterns, the waveforms in this study are simplified to a dual-frequency sine wave superposition, as illustrated in Figure 2. Mathematically, this waveform is composed of three components: the first and second terms represent the low-frequency and high-frequency load components, respectively, while the third term, σ_m, denotes the overall mean stress of the composite waveform. In this formulation, σ_aL and σ_aH denote the stress amplitudes of the low-frequency and high-frequency components, respectively. The envelope of the low-frequency stress amplitude is defined as σ_L = σ_aL + σ_aH, while the high-frequency stress amplitude is given by σ_H = σ_aH, f_L and φ_L represent the frequency and phase of the low-frequency load component, respectively, whereas f_H and φ_H correspond to the frequency and phase of the high-frequency load component. σ_m indicates the overall mean stress of the superimposed waveform. Based on the established literature, several key waveform parameters are introduced: the amplitude ratio α = σ_H/σ_L, the frequency ratio n = f_H/f_L, the maximum stress σ_max = σ_m + σ_L, the minimum stress σ_min = σ_m − σ_L, and the master stress ratio R_m = σ_min/σ_max. A complete low-frequency load cycle is defined as a load block, which can be discretized into n high-frequency cycles with amplitude σ_H and one low-frequency cycle with amplitude σ_L. The number of cycles to specimen failure is denoted as N_b. The linear damage caused by the high-frequency and low-frequency cycles within each load block is represented by D_H and D_L, respectively. The total damage induced by each load block is D_b = 1/N_b. This idealized waveform approximates, to some extent, the daily stress spectrum blocks experienced by bridge components. The low-frequency wave σ_aL represents the equivalent stress amplitude induced by standard thermal loading, while the high-frequency wave σ_aH corresponds to the equivalent stress amplitude due to standard fatigue vehicle loading. The frequency ratio n reflects the average daily number of standard fatigue vehicles, and σ_m denotes the mean stress under constant loading conditions for the component. Using a reference standard fatigue vehicle load and daily thermal loading cycles, the amplitude ratio α is defined, enabling the equivalent conversion of the random spectrum block based on its high-frequency damage component.

2.2.2. Constant Amplitude Tests

The fatigue tests were carried out using an HD105B-WANCE electro-hydraulic servo fatigue testing machine (Shenzhen, China), which can generate superimposed waveforms by integrating a dynamic electro-hydraulic servo system with a composite load coordination control algorithm. Constant amplitude fatigue tests were conducted under uniaxial force-controlled conditions at a loading frequency of 30 Hz. Fatigue life was determined as the number of cycles until either complete specimen fracture occurred or the loading was interrupted by the system’s protective mechanism due to axial displacement exceeding 10 mm. The fatigue test instrument is illustrated in Figure 3.

Three groups of constant amplitude fatigue tests were conducted under stress ratios of R = −1, R = 0.1, and R = 0.5. As illustrated in Figure 4a, the relationship between the nominal stress amplitude (defined as the ratio of the load amplitude to the cross-sectional area at the narrowest section of the notched specimen), denoted as σ_a, and the fatigue life N until complete fracture of the specimen was plotted in a double-logarithmic coordinate system.

Three S-N curves fitted from the experimental data are given by Equations (1)–(3):

For R = −1:

$${\left({\sigma }_a\right)}^{3.30}N=5.43\times {10}^{11}$$

(1)

For R = 0.1:

$${\left({\sigma }_a\right)}^{3.497}N=6.71\times {10}^{11}$$

(2)

For R = 0.5:

$${\left({\sigma }_a\right)}^{3.665}N=8.27\times {10}^{11}$$

(3)

Based on the above constant amplitude test results, the Walker’s formula [40] was adopted to construct a family of S-N curves that can account for the effect of mean stress. The expression of Walker’s formula is shown in Equation (4). By applying this formula, the stress amplitude σ_a under any given stress ratio R can be converted into the equivalent stress amplitude ${\sigma }_a^{\prime }$ under a reference stress ratio R_ref [41]. This transformation enables the estimation of fatigue strength and life across different mean stress conditions, thereby establishing a basis for subsequent fatigue life analysis under superimposed loading scenarios.

$${\sigma }_a^{\prime }={\sigma }_a{\left({\displaystyle \frac{1-R_{ref}}{1-R}}\right)}^{1-\gamma }$$

(4)

where γ represents the stress sensitivity coefficient. Figure 4b presents the fitted stress sensitivity coefficient γ, which is derived from the results of the other two stress ratios relative to the reference stress ratio R_ref = 0.1. It decreases as the stress amplitude σ_a increases.

2.2.3. Superimposed Loading Tests

As shown in

Table 3. Summary of HLSF test conditions.

Test Conditions	σL (MPa)	σH (MPa)	n	α	σm (MPa)	Rm	Nb	Average
C-1	176.1	21.1	1000	0.12	215	0.1	1881, 2084, 1998	1988
C-2	176.1	31.2	1000	0.18	215	0.1	1048, 1058, 1154	1087
C-3	176.1	38	1000	0.22	215	0.1	953, 942, 457	784
C-4	176.1	48.1	1000	0.27	215	0.1	549, 409, 656	538
C-5	176.1	56.5	1000	0.32	215	0.1	341, 347, 302	330
C-6	127.2	48.1	1000	0.38	155	0.1	569, 604, 751	641
C-7	101.7	48.1	1000	0.47	124	0.1	962, 619, 1190	924
C-8	89.6	48.1	1000	0.54	110	0.1	879, 808, 833	840
C-9	176.1	48.1	100	0.27	215	0.1	3210, 2966, 2644	2940
C-10	176.1	48.1	500	0.27	215	0.1	1056, 1146, 818	1007
C-11	176.1	48.1	1500	0.27	215	0.1	394, 373, 380	382
C-12	176.1	48.1	2000	0.27	215	0.1	441, 402, 334	392
C-13	176.1	48.1	500	0.27	95	−0.3	1356, 1372, 1328	1352
C-14	176.1	48.1	500	0.27	31	−0.7	1906, 1958, 2024	1963
C-15	176.1	48.1	500	0.27	0	−1	2534, 2398, 1954	2295

, fifteen HLSF test conditions were designed and applied to the Q355D V-notched specimens with three repeated tests. The experimental setup was appropriately extended to encompass a range of typical loading scenarios encountered by bridge components. Specifically, the amplitude ratio α ranges from 0.12 to 0.54, the frequency ratio n ranges from 100 to 2000, and the master stress ratio R_m ranges from −1 to 0.1. For each factor (σ_H, σ_L, n, R_m), 4 to 5 parameter levels were established as control variables. During the tests, the high-frequency component frequency f_H was fixed at 30 Hz, while the low-frequency load frequency was adjusted to achieve different frequency ratios.

Based on the results presented in Table 3, the effects of different amplitude ratios, frequency ratios, and mean stresses on the fatigue life of specimens were further analyzed. The focus is on the ratio of total high-frequency cycles n·N_b to constant amplitude life N_H. When the ratio is less than 1, it indicates that HLSF accelerates fatigue damage and reduces the high-frequency fatigue life. Conversely, when the ratio exceeds 1, it suggests that HLSF inhibits fatigue damage and enhances the high-frequency fatigue life. As shown in Figure 5a, under constant n = 1000 and R_m = 0.1, the ratio of n·N_b to N_H increases with increasing amplitude ratio α. As shown in Figure 5b, under constant α = 0.27 and R_m = 0.1, the ratio of total cycles n·N_b to N_H increases with the increase in frequency ratio n. When n reaches 2000, HLSF mainly suppresses fatigue damage. According to the results of the analysis of variance (ANOVA), both the amplitude ratio and frequency ratio have a significant impact on HLSF life, with p-values for both being less than 0.05. The influence of mean stress was investigated by varying the stress ratio, as depicted in Figure 5c. The ANOVA results indicate minimal fluctuation in the ratio of total cycles to N_H with increasing stress ratio, suggesting that the coupling effect between loads is not significantly affected by changes in mean stress. The experimental results presented above further extend the conclusions derived from high-low frequency superimposed loading tests, wherein high-frequency loading becomes dominant under larger frequency ratios. These findings align with those reported by Gan et al. [2] for T-type welded joints and Zhao et al. [27,28] for the aerospace material TC11. When integrated with the low-frequency dominated results of high-low frequency superimposed tests conducted by these researchers, the following conclusions can be drawn: compared to high-frequency constant amplitude conditions, smaller frequency ratios n and amplitude ratios α tend to accelerate fatigue damage under HLSF, whereas larger values of n and α tend to delay fatigue damage.

2.3. Fracture Characteristics

The fracture surfaces of the specimens were examined using a Gemini SEM360 field emission scanning electron microscope (Oberkochen, Germany), to analyze the differences in damage mechanisms under different HLSF test conditions. Near the critical crack depth, distinct crack propagation striation bands with low-frequency periodicity were observed in most of the superimposed fatigue test fractures (Figure 6a–c). The width of these bands increased with both the frequency ratio and the amplitude ratio. However, under the low mean stress condition (σ_m = 0), such prominent bands were not observed (Figure 6d), which may be attributed to the reduced effective amplitude for fatigue crack propagation under this condition. Another reason could be the plastic extrusion of the crack propagation surface under tension-compression cycles, as the contact pressure may lead to wear [42]. Mean stress, to some extent, acts as a parameter characterizing the low-frequency envelope waveform and influences the overall crack damage process. Under high mean stress, more pronounced microstructural plastic bulging on the crack propagation surface was observed compared to that under low mean stress (Figure 6a,d).

3. Fatigue Life Analysis and Discussion

3.1. Existing Life Prediction Models

A review of existing studies indicates that HLSF life prediction models can be broadly categorized into three types: linear cumulative damage model (Miner’s rule), nonlinear cumulative damage models, and empirical models. This study selects several representative models for analysis and comparison in terms of their accuracy in predicting fatigue life.

• Miner’s rule

The Miner’s rule [22] is the most widely used for variable amplitude fatigue life prediction. Under the framework of this rule, the fatigue damage of HLSF blocks accumulates independently and linearly. In the case of a superimposed loading spectrum, both high-frequency and low-frequency stress amplitudes remain constant. Consequently, the fatigue damage induced by a single load block can be calculated using Miner’s rule, as presented in Equation (5). The number of load cycles N_b is defined as the reciprocal of the damage caused by a single load block, which is mathematically expressed in Equation (6).

$$D_{\mathrm{b}}={\displaystyle \frac{1}{N_{\mathrm{L}}}}+{\displaystyle \frac{n}{N_{\mathrm{H}}}}$$

(5)

$$N_{\mathrm{b}}={\displaystyle \frac{1}{D_{\mathrm{b}}}}$$

(6)

where N_b represents the number of block cycles, N_L represents the life cycles under constant low-frequency loading with a stress amplitude of σ_L, N_H represents the life cycles under high-frequency loading with a stress amplitude of σ_H, and D_b represents the fatigue damage induced by a single block.

2.

T-K model

The T-K model, which is proposed by Trufyakov and Kovalchuk [27] based on extensive experimental data, is an empirical model for HLSF that correlates fatigue life with load amplitude, frequency ratio, and other parameters. Its expression is as follows:

$$N_{\mathrm{b}}=N_{\mathrm{L}}\cdot {\left({\displaystyle \frac{1}{n}}\right)}^{\nu \cdot {\displaystyle \frac{{\sigma }_{\mathrm{H}}}{{\sigma }_{\mathrm{L}}}}}$$

(7)

where ν is a material constant that typically ranges from 1.3 to 1.7. For low-alloy high-strength steel, a commonly used value of ν is about 1.3.

3.

Yue model

The Yue model [33] is a modified nonlinear cumulative damage model that draws on the damage equivalence principle of two-step loading. The damage model is expressed as shown in Equation (8), wherein an equivalent stress amplitude ratio, defined as α_eq = 2σ_H/(σ_m + σ_H) is incorporated to account for the effects of load interactions.

$$D_{\mathrm{b}}=\left({\displaystyle \frac{1}{n}}\right)\cdot {\left({\displaystyle \frac{n}{N_{\mathrm{L}}}}\right)}^{{\left({\displaystyle \frac{N_{\mathrm{L}}}{N_{\mathrm{H}}}}\right)}^{0.4\cdot {\alpha }_{\mathrm{eq}}}}+{\displaystyle \frac{n}{N_{\mathrm{H}}}}$$

(8)

Based on the HLSF tests conducted in this study, the prediction accuracy of each model for fatigue life under various conditions is evaluated using the life prediction error, denoted as P_error. The formula for calculating P_error is as follows [43]:

$$P_{\mathrm{error}}={\log }_{10}\left(N_{\mathrm{fp}}\right)-{\log }_{10}\left(N_{\mathrm{f}\mathrm{t}}\right)$$

(9)

3.2. Modeling of HLSF Block Nonlinear Damage Effects

The results of fracture analysis indicate that under superimposed loading conditions, fatigue cracks predominantly undergo a damage evolution process governed by low-frequency cyclic periods. The amplitude ratio, frequency ratio, and mean stress each contribute to crack propagation through distinct mechanisms. Consequently, it is feasible to develop a nonlinear damage accumulation model based on spectral block analysis. Firstly, modify the formulation of Miner’s rule (Equation (5)) to integrate the waveform characteristic parameters of HLSF:

$$D_{\mathrm{b}}=D_{\mathrm{L}}\left(1+{\alpha }^{m}n\right)$$

(10)

where D_L = 1/N_L represents the damage caused by the low-frequency fatigue loading, which is a function of the mean stress σ_m, m is the slope of the S-N curve.

In terms of cyclic damage, the term (1 + α^mn) in the aforementioned equation can be interpreted as an acceleration factor that amplifies the superimposed fatigue cycle damage D_b relative to the per-cycle damage D_L induced by low-frequency fatigue loading. However, under Miner’s rule, this acceleration factor is assumed to be linear and thus unable to account for the potential coupling effects among the various parameters of the superimposed waveform. To address this limitation, a nonlinear exponent q is incorporated into the model, leading to the development of the Block Nonlinear Damage Model (BNDM), which is expressed as follows:

$$D_{\mathrm{b}}=D_{\mathrm{L}}{\left(1+{\alpha }^{m}n\right)}^q$$

(11)

Unlike the T-K model and Yue model, the BNDM provides a certain physical interpretability for the coupling effect between low-frequency and high-frequency loads, and its formulation does not favor a particular frequency component.

Differing from the T-K model and Yue model, the BNDM provides a certain physical interpretability for the coupling effect between low-frequency and high-frequency loads, and its formulation is not biased towards particular frequency. The introduced parameter q represents the nonlinear damage effect under superimposed fatigue loading conditions. A higher value of q corresponds to a more pronounced nonlinear behavior, whereas values approaching 1 indicate that the fatigue damage accumulation tends toward linear characteristics. The D_L configuration is selected as the reference case instead of D_H, mainly due to the fact that, under identical low-frequency envelope waveforms, the fatigue crack propagation length is governed by the maximum stress σ_max of the stress history, as confirmed by fracture surface analysis. For a given specimen, the nonlinear damage exponent q should be considered as a function of the amplitude ratio α, frequency ratio n, and master stress ratio R_m (or mean stress σ_m). As formulated in Equation (12), the nonlinear component of q is defined as the product of three individual functions—q₁(α), q₂(n), and q₃(R_m)—which were derived through regression analysis of experimental data (as illustrated in Figure 7), leading to the expressions provided in Equations (13)–(15). Experimental findings demonstrate that q exhibits high sensitivity to variations in both amplitude ratio α and frequency ratio n, particularly under lower values of these parameters. In the fitting process, the nonlinear components of the exponents q₁(α) and q₂(n) consider only the values greater than 1, which correspond to a potentially underestimated fatigue damage according to the Miner’s rule. When the value of exponent q is less than 1, such as when n > 1500, this indicates an improvement in fatigue performance. Due to the inherent variability in fatigue test results and the limited range of loading conditions, the lower values are conservatively modeled using a linear approach. Regarding the mean stress effect, when the Walker formula is employed to account for this effect separately in N_H, it does not lead to significant variations in the exponent (as illustrated in Figure 7c), and thus q₃(R_m) is set to 1. However, if the mean stress effect is not explicitly considered in N_H, such as when N_H and N_L represent fatigue lives under the same stress ratio, the exponent increases with increasing mean stress. Equations (13)–(15) are applicable to the first correction method of mean stress effect.

$$q=q_1\left(\alpha \right)q_2\left(n\right)q_3\left(R_{\mathrm{m}}\right)$$

(12)

$$q_1\left(\alpha \right)=\left\{\begin{array}{cc}0.4{\alpha }^{-0.7},\hfill & \alpha <0.27\hfill \\ 1,\hfill & \alpha \ge 0.27\hfill \end{array}\right.$$

(13)

$$q_2\left(n\right)=\left\{\begin{array}{cc}-0.0004566n+1.228,\hfill & n<500\hfill \\ 1,\hfill & n\ge 500\hfill \end{array}\right.$$

(14)

$$q_3\left(R_{\mathrm{m}}\right)=1$$

(15)

3.3. Life Analysis and Comparison

Fatigue life analysis and comparison are performed using the Miner’s rule, the T-K model, and the Yue model. The Miner’s rule includes two conditions: one where N_H does not account for the mean stress correction, assuming the life is extrapolated from the S-N curve corresponding to the stress ratio of N_L; and another where N_H incorporates the mean stress correction, i.e., the case where N_L and N_H share the same mean stress σ_m. Additionally, the effectiveness of using the proposed BNDM to improve Miner’s rule will also be demonstrated, and this integrated method will be referred to as the Miner-BNDM method. A comparison of predicted and experimental lifetimes is shown in Figure 8a. When the predicted lifetime N_b equals the experimental lifetime, the data points lie on the 45° reference line. If the predicted lifetime is above this line, it indicates that the predicted life exceeds the experimental life, and the prediction is considered optimistic. Otherwise, the prediction is deemed conservative. Based on this comparison of lifetimes, a more detailed error analysis is presented in Figure 8b. When Miner’s rule does not incorporate mean stress correction, most predicted lifetimes tend to be overly optimistic. However, with the application of mean stress correction, the majority of predicted lifetimes fall within a ±20% error margin. The proposed Miner-BNDM method, which integrates mean stress correction, further improves predictive accuracy for non-conservative cases—such as those characterized by low amplitude and frequency ratios—by ensuring that the predicted lifetimes predominantly lie below the zero-error line. Regarding prediction scatter, neither the T-K model nor the Yue model demonstrates superior performance compared to Miner’s rule with mean stress correction. Notably, the predictions of Yue model tend to be overly conservative, particularly in scenarios involving low-frequency ratios n and low principal stress ratios R_m.

Figure 9 presents the ratio of the experimental and predicted fatigue damage D_b of each model with the Miner’s rule considering mean stress effects. Deviations of this ratio from unity reflect the nonlinear fatigue damage effects observed in the HLSF tests and indicate the applicability and accuracy of each model. As shown in the figure, the condition where D_b/D_b,Miner > 1 primarily occurs under low-amplitude ratios (α < 0.27) and low-frequency ratios (n < 500), which corresponds to accelerated fatigue damage accumulation as illustrated in Figure 5a–c. In contrast, the condition where D_b/D_b,Miner < 1 mainly occurs at high-frequency ratios (n > 1500), indicating a deceleration in fatigue damage accumulation (see Figure 5c). Among the existing models, the T-K model and the Yue model demonstrate certain biases in estimating the nonlinear effects of amplitude ratio and frequency ratio. Specifically, they tend to underestimate the impact of amplitude ratio while overestimating the influence of frequency ratio. For the T-K model, although this may result in seemingly reasonable overall fatigue life predictions (Figure 8), the optimistic and pessimistic expectations it brings are contrary to the Miner’s rule. Overall, the BNDM method offers an effective solution for addressing the nonlinear effects of HLSF damage.

3.4. Discussion

It can be seen that the Miner’s rule tends to yield overly optimistic predictions in cases with low-amplitude ratios and low-frequency ratios. While the fatigue design goals for steel bridges under low-frequency ratios (which correspond to lower heavy vehicle traffic) are relatively easy to achieve, it is important not to be overly optimistic. Specifications should account for design redundancy resulting from variations in traffic volume. For low-amplitude ratios, this corresponds to higher temperature load effects and lower vehicle load effects, which in the context of this study, represent a more unfavorable scenario. This can even lead to the failure of the original infinite life design strategy. Because in most cases, the stress amplitudes induced by high-frequency vehicle loads are lower than the constant amplitude fatigue limit. For the C-1 loading condition, its high-frequency amplitude 27.87 MPa modified by σ_m is already below the constant amplitude fatigue limit of 29.23 MPa. The ratio of total fatigue life from the experiment to the predicted fatigue life using Miner’s rule is n·N_b:N_H,Miner = 1,988,000 cycle:5,922,147 cycle. This difference in fatigue assessment is significant and requires attention.

The existing models do not appear to be optimal for the material and waveform parameters used in this study. This limitation may be attributed to the validation scenarios employed during their development. For example, the T-K model is primarily derived from tests involving low-frequency-ratio superimposed waveforms, whereas the Yue model is based on experimental data characterized by high mean stress and high-frequency ratios. As an enhancement of Miner’s rule, the Miner-BNDM model is a semi-empirical approach that retains the fundamental structure of the linear damage criterion while integrating insights from both experimental observations and mechanistic analysis. However, further refinement of this model necessitates additional research, including the investigation of variations in the exponent q across different types of bridge steels, notch geometries, and welding conditions, as well as the establishment of generalized correlations.

However, from the perspective of this study, the mean stress effect in HLFSL can be independently adjusted, regardless of the stress amplitude, which simplifies the evaluation process. If possible, it is recommended to use S-N curves under consistent mean stress conditions for both high- and low-frequency components. Alternatively, using the S-N curve corresponding to the higher stress ratio between the two frequency components can provide a more conservative evaluation.

4. Conclusions

This study conducted a series of high- and low-frequency superimposed loading fatigue tests on Q355D notched specimens, exploring the effects of HLSF stress waveform parameters such as amplitude ratio, frequency ratio, and average stress on the life and damage mechanism of the specimens. The rationality and correction strategies of existing life prediction models were discussed. The following conclusions have been drawn:

• When other factors are held constant, the total number of high-frequency cycles is positively related to the amplitude ratio and the frequency ratio. Lower frequency ratios and lower amplitude ratios tend to accelerate fatigue damage, whereas higher frequency ratios and greater amplitude ratios contribute to the reduction of fatigue damage. When the amplitude ratio reaches 0.12 and the frequency ratio reaches 100, there is a reduction of at least 66% and 46% in HLSF high-frequency load cycles compared to constant amplitude loading.
• Under HLSF test conditions, the fracture surface exhibit striation band crack propagation characteristics in units of low-frequency block cycles. The higher the amplitude ratio and frequency ratio, the more pronounced the band width becomes. Additionally, the plastic protrusion observed on the crack propagation surface increases with rising mean stress.
• For fatigue life prediction of Q355D, the mean stress effect can be effectively accounted for by Walker’s formula without the other adverse effects on the number of high-frequency cycles. The T-K model underestimates the impact of low-amplitude ratios, while the Yue model overestimates the impact of low-frequency ratios and high mean stresses. The Miner’s rule tends to underestimate the impact of loading conditions with low-amplitude ratios and low-frequency ratios.
• A proposed model BNDM is developed to modify the block nonlinear damage by introducing an exponential acceleration damage factor referenced to low-frequency cyclic damage. The application of this model achieved a reduction in the average prediction error of Miner’s rule for fatigue life from 19.9% to 7.5%, thus enabling a more accurate prediction of HLSF fatigue life.