Fatigue Life Prediction and Experimental Study of Landing Gear Components via FKM Local Stress Approach

Abstract

This study focuses on high-cycle fatigue (HCF) of aircraft landing gear (LG) components, covering material testing, full-scale component experiments, finite element (FE) modeling, life-prediction comparison, and probabilistic assessment. Fully reversed axial fatigue tests on forty 300M steel specimens were conducted to establish a reliable S-N curve. Full-scale fatigue experiment conducted on the upper torque link components showed that the one cracking at approximately 184,000 cycles (at the filet), while another remained undamaged after 166,000 cycles, providing a benchmark for model validation. FE simulations using ANSYS accurately captured the stress field within the component, with a maximum error of less than 10% compared to experimental strain measurements. Based on the FKM guideline, this work developed an improved FKM local-stress approach (LSA) for HCF life prediction, which integrates load-dependent stress gradients, FKM mean stress correction, and interpolated surface-condition factors for S-N curve adjustment specific to the component’s surface treatment. It predicts the fatigue life as 174,000 cycles (−5.4% error relative to test), outperforming standard FKM-LSA calculations and nCode software simulations. Furthermore, by augmenting the experimental data and constructing p-S-N curves, the improved LSA was extended to predict fatigue life under different survival probabilities and confidence levels, providing a practical tool for reliability-based design.

1. Introduction

Aircraft landing gear (LG) is a critical system that ensures safety and reliability during takeoff, landing, and ground operations like braking and taxiing. As the aircraft’s primary load-bearing assembly [1,2,3], the LG constantly bears cyclic loads [4,5] from the ground, so its fatigue performance directly determines the safety. For the large passenger aircraft’s LG, it is required that the fatigue life of its main components be the same as that of the aircraft body (30 calendar years), with a service life generally between 40,000 and 75,000 landings, which is a regime characterized by high-cycle fatigue (HCF) [3,6]. Notably, the European Aviation Safety Agency (EASA) explicitly states in its CS-25 amendment [7] “AMC 25.571 (a), (b), and (e)” that LG is a typical structure that is not conducive to damage-tolerance design and complies with the concept of safe-life fatigue evaluation [8,9]. Therefore, the design and certification of LG structures place high demands on the accurate and reliable HCF prediction for safe-life (the duration of fatigue cracking of the component) to preclude both excessive design (increasing weight) and inadequate design (causing catastrophic safety risks) [6,9].

The critical components for transmitting force in LG, such as the upper torque link and outer cylinder, are frequently fabricated from ultra-high-strength alloy steels like AISI 300M, prized for its excellent mechanical properties [2,5,10]. For metallic LG structures, the stress-life (S-N) approach is usually used to predict the HCF life. Yet, the HCF life analysis of 300M steel components in this paper faces unique challenges. Firstly, the high strength and rapid crack propagation of this material results in its service life being dominated by crack initiation rather than crack growth [1]. Secondly, LG components often operate within the elastic stress range, allowing plasticity effects to be reasonably neglected in safe-life estimations [2]. Moreover, detailed geometric features (e.g., notches, chamfers, and filets) can introduce stress concentration effects and are the preferred locations for fatigue crack initiation [5]. Finally, surface treatment of components (e.g., shot peening, electroplating) alters near-surface characteristics (such as roughness, residual stress), making fatigue life assessment more complex [11,12]. These factors collectively highlight the need for a tailored HCF life prediction framework for LG components.

The traditional safe-life analysis tool is the nominal stress method [13,14], but it is not suitable for handling the LG components with irregular geometries because defining key parameters such as notch depth, radius, and theoretical stress concentration factor is impractical [9]. With the widespread adoption of the finite element method (FEM) in LG fatigue design, it is more convenient to obtain local stress/strain data at critical nodes rather than nominal stress. For example, our previous research [3,4] analyzed the cyclic characteristics of ground load spectra through the FEM of the whole LG, established a stress spectrum calculation framework for dynamic changes in shock absorber stroke, and quantified the HCF life of the main LG based on the S-N method and Miner cumulative damage criterion [3]. Kang et al. [15] obtained the dynamic stress history of key components through transient FE structural analysis and calculated the fatigue life by the empirical formula of the S-N method and rain-flow counting method. Wang et al. [16] studied the HCF life of the LG outer cylinder using the S-N curve and critical plane method. Although these studies account for local stress via FE analysis, they not only lack comparison with full-scale component fatigue tests, but also typically treat key parameters such as stress gradient and fatigue notch coefficient as load independent constants, which cannot explain the load direction dependence of asymmetric LG structures. For instance, Chen et al. [17] and Mei et al. [18] noted that stress gradients and notch sensitivity in non-symmetric components can vary significantly under tension versus compression, which directly affects the fatigue crack initiation behavior. These also emphasize the necessity of improving traditional methods to meet the HCF life prediction of LG.

Experimental verification remains indispensable for the fatigue design and certification of LG [1], and the “building block” testing strategy—progressing from material specimens to structural components and ultimately to full-scale aircraft—has become an industry-standard for mitigating technical risks and controlling development costs [2,5,6]. This tiered verification logic disperses uncertainties across different design stages, enabling progressive validation of fatigue performance. In the study, this philosophy underpins an integrated experimental program: axial fatigue tests on forty 300M steel specimens establish a foundational S-N database, while full-scale tests on LG’s typical components furnish benchmark life data and identify critical crack-initiation sites—information unattainable from coupon-level tests alone. These experimental outcomes not only expose potential fatigue vulnerabilities but also supply essential data for calibrating and refining predictive models.

Despite the necessity and value of physical testing, it is difficult to fully consider all potential fatigue critical positions in complex components solely through experiments. For example, due to the limited number and location of strain gauges, stress characterization at geometric discontinuities may be missed [19,20,21]. Consequently, high-fidelity computational methods are indispensable for mapping the complete stress field. Among available analytical tools, the FKM local-stress approach (LSA)—rooted in the German FKM Guideline [22] for mechanical component strength assessment—has shown pronounced promise and has been widely adopted in European engineering practice [23]. It systematically incorporates corrections for geometry, surface condition, and load type, addressing several limitations of conventional nominal stress methods [24]. Recent applications by Rennert et al. [25] and Fazili et al. [26] demonstrated FKM-LSA’s efficacy in evaluating fatigue performance of surface-treated and geometrically complex metallic parts and the derived fatigue life calculation formula. Repplinger et al. [27] combined a nonlinear FEM and FKM-LSA to analyze the fatigue life and crack initiation of a complex aluminum component, ultimately achieving optimization of fatigue performance. Leonetti et al. [28] and Bagherifard et al. [23] focused on the application of FKM-LSA in the life assessment of regularly shaped specimens. Furthermore, Knabner et al. [29,30] extended the FKM guideline to the fretting fatigue assessment of components, obtained the corresponding fretting factors of materials through experiments or FE analysis, and integrated them into the guideline developed a safe-life design framework. These studies suggest that FKM-LSA is feasible and applicable for LG fatigue life assessment. However, critical gaps remain in its application to 300M steel LG components. That is, it relies on generic preset S-N curve parameters (e.g., slope k = 5 for steels) that do not reflect 300M steel’s unique fatigue characteristics, and it is likely to treat stress gradient and notch sensitivity as constants independent of the load [17,31], while ignoring the load direction dependence of asymmetric LG components.

To address these gaps, an integrated computational–experimental workflow tailored to 300M steel LG components is proposed herein: material S-N characterization via axial tests (to establish a 300M steel specific fatigue database), a full-scale test of the upper torque link using a dummy outer cylinder (to obtain benchmark life data and identify crack initiation sites), carefully validated FE stress field simulation, and integrating these data into the FKM guideline to improve its LSA. The improved LSA framework incorporates load-dependent stress gradients, FKM mean stress correction, and surface-condition interpolation for S-N curve adjustment. Meanwhile, this framework can also be used to combine deterministic FE stress mapping with probabilistic p-S-N curves, aiming to ensure the accuracy of HCF life prediction and provide a technical basis for the safe-life design and certification of aerospace LG structures.

The motivation of this work is to integrate experimental data and FE analysis to develop an improved fatigue life prediction framework for LG components. A key step in this is validating the FEM via experiment comparison, which ensures the stress field at critical locations is accurate, laying the foundation for reliable life prediction. The remainder of this paper is structured as follows: Section 2 details the material properties and experimental procedures, including specimen-level and full-scale typical structural fatigue tests. Section 3 presents the comparative work between FE simulations and experiments. Section 4 elaborates the improved FKM-LSA methodology. Section 5 discusses the life prediction of different methods and probabilistic life results, while concluding remarks are provided in Section 6.

2. Material and Experiment

2.1. Material

The material in this work is 300M ultra-high-strength alloy steel used for the LG of a large passenger aircraft [3,10]. The 40 specimens used in fatigue tests were cut from the 300M die forgings of the LG. After vacuum quenching and two tempering processes, the material exhibits a yield strength of 1650 MPa, a tensile strength of 1950 MPa, and an elastic modulus of 197 GPa [3,10,32]. These blank samples are subjected to rough machining, cleaning, CNC precision turning, grinding, etc., and low-temperature tempering to relieve residual stress introduced during machining. Then, surface shot peening (SP) was performed on these substrate samples, followed by cadmium–titanium (Cd/Ti) plating.

To characterize the fatigue life of the structural components as accurately as possible, the surface processing technology of the material specimens was consistent with that of the structural components. That is, the specimen and landing gear typical structure had the same processing techniques, such as heat treatment and surface processing, except for differences in size. The typical structures of the LG, such as the torque link, first underwent SP strengthening on the outer surface layer, followed by low hydrogen embrittlement Cd/Ti plating. Among them, SP was mainly used to improve fatigue performance [11,12], while Cd/Ti plating was used to enhance the corrosion resistance of the substrate [33,34]. Here, the Cd/Ti electroplating process first involves degreasing and sand-blasting the substrate surface, followed by cleaning, dust removal, acid pickling, and drying. Afterwards, the substrate was placed in the electrolyte and coated at a temperature of 25 °C and a current density of 2.5 A/dm² until the coating thickness meets the design specifications. Finally, a dehydrogenation treatment was carried out at 190 °C for more than 12.5 h.

2.2. Fatigue Test of Specimens

For specimens subjected to SP followed by Cd/Ti plating, conventional axial stress-controlled fatigue tests were performed in line with the Chinese national standard GB/T 3075-2021 [35] using a QBG-100 electromagnetic resonance high-frequency fatigue tester [36] (load range: ±50 kN), as illustrated in Figure 1a,b. The tests employed a fully reversed axial force with a sinusoidal waveform, operating at a frequency of 110 Hz. In compliance with the Chinese national standard GB/T 24176-2009 [37], the fatigue tests were executed using the staircase method and group method, respectively. The fatigue test results for 40 SP + Cd/Ti specimens are presented in Figure 1c, while the corresponding test data are listed in

Table 1. Fatigue test data of SP + Cd/Ti under stress ratio R = −1.

Smax (MPa)	Nf (Thousand Cycles)	lgN50	std	CV
1100	27, 70, 37, 41, 29, 64	4.6210	0.1723	0.0373
1050	67, 68, 131, 113	5.0224	0.2370	0.0472
1000	212, 217, 512, 802, 195, 195	5.4761	0.2641	0.0482
950	394, 650, 1087, 2185, 417	5.8809	0.3116	0.0530
900	5616, 1074, 2532, 6165, 2187	6.4627	0.3140	0.0486
870	5700, 1060, 2755, 3640, 10,000	specified life N = 1 × 107 fatigue limit Se = 838 MPa
840	6857, 6037, 10,000, 10,000, 10,000
810	2873, 10,000, 10,000
780	10,000

where, lgN₅₀ represents the average logarithmic life of a stress level, std denotes the standard deviation of logarithmic life; CV is the coefficient of variation.

. Among these, the maximum nominal stress applied to the specimens reached 1100 MPa, which corresponds to a load of 42.33 kN—confirming that the fatigue tester satisfied the load range requirements.

Specifically, the staircase method [38] was adopted to determine the fatigue limit, with an interval of 30 MPa between adjacent stress levels, and data processing was conducted using the stress pairing approach. Additionally, the experiment specified that the fatigue life corresponding to the fatigue limit was set to 1 × 10⁷ stress cycles. In other words, each specimen was tested continuously under the designated stress until either failure occurred or the specified cycle count (run-out) was achieved. Second, five stress levels were selected for fatigue testing via the group method [39,40], yielding medium-to-high cycle fatigue test data. Lastly, the S-N curve was derived by fitting the data acquired from both the staircase method and the group method [41,42].

From Table 1, it can be seen that the standard deviation (std) [40] of fatigue life of 300M steel material increases with the decrease in stress level, indicating that the higher the stress, the more dispersed the life. The coefficient of variation (CV) [41] under five stress levels is larger than that specified in GB/T 24176-2009, so the number of samples needs to be increased to obtain a high reliability S-N curve. Further data augmentation techniques are described in Section 5.2. In addition, near the fatigue limit of 838 MPa, there is a clear slope step of the life curve.

It is worth noting that according to the authoritative material manuals [43], the following three-parameter power function is recommended for fitting the S-N curve for 300M steel, which is also the equation recommended by the Chinese military standard GJB/Z 18A-2020 [44].

$$N_f·{\left(S_{max}-S_0\right)}^m=C$$

(1)

where m and C are material constants, and S₀ represents the theoretical fatigue limit.

2.3. Full-Scale Fatigue Test of Typical Components

In this work, the main LG’s typical structure such as torque link was fabricated according to the actual dimensions and production process. Moreover, the outer surface of this structure was strengthened by SP according to the internal process LPS-16003 of the author’s company, and then the low hydrogen embrittlement Cd/Ti plating was carried out according to the process LPS-23070. For the sake of experimental reliability, two upper torque links were made using the same process for fatigue testing. Meanwhile, a portion of the LG’s outer cylinder connected to the torque link was cut out as a test dummy to simulate the real connection relationship and support.

As shown in Figure 2a, the full-scale fatigue test frame was equipped with an upper torque link, an outer cylinder dummy, some connecting structures, etc. The experiment used a hydraulic servo actuator to apply cyclic loads to the torque link, and a position limiter was installed as depicted in Figure 2b to prevent the torque link from swinging. In addition, a hinge assembly device was mounted at the force transmission point between the hydro-cylinder and the torque link to release rotational freedom and mitigate the impact of force transmission deviations induced by structural deformation. Furthermore, to monitor the deformation during the fatigue test, a vertical displacement sensor was arranged, as shown in Figure 2c. Two rectangular strain gauge rosettes (No. 11 and No. 12) were affixed to the surface of the tested piece to monitor the stress–strain state, as illustrated in Figure 2d.

Set the loading cycle of the hydraulic servo actuator to T = 5 s, with the main and secondary peak values of 210 kN and 200 kN, and the main and secondary valley values of −210 kN and −100 kN, respectively. Through its force feedback signal, the actual loading curve can be monitored. As shown in Figure 3a, the actual values of the main and secondary peaks of a loading task segment were 204.7 kN and 193.8 kN, respectively, with relative errors of 2.52% and 3.10% compared to the design load curve. This means that the test ran stably and the error met the requirements of the dynamic loading system. During the fatigue test monitoring, the data acquisition frequency for the displacement sensors and strain gauge signal channels was set to 100 Hz. As shown in Figure 3b, the vertical displacement was consistent with the waveform of the load curve, which also indicates that the test system was reliable. The deformation corresponding to the main peak and the main valley values were 12.717 mm and −11.287 mm, respectively. That is to say, the structural deformation caused by tension and compression under the same load is different, indicating that the tested piece is geometrically asymmetric, and the structural stiffness is affected by the load direction.

The data from the rectangular strain rosettes were converted von Mises stress using Equation (2) [20,45], as follows:

$${\sigma }_{vm}=E·\sqrt{{\displaystyle \frac{{({\epsilon }_0+{\epsilon }_{90})}^2}{4(1-v{)}^2}}+{\displaystyle \frac{3{({\epsilon }_0-{\epsilon }_{45})}^2}{2(1+v{)}^2}}+{\displaystyle \frac{3{({\epsilon }_{45}-{\epsilon }_{90})}^2}{2(1+v{)}^2}}}$$

(2)

where E and $v$ are the elastic modulus and Poisson’s ratio of the material, respectively. ${\epsilon }_0$, ${\epsilon }_{45}$, and ${\epsilon }_{90}$ represent the strain values of the rectangular strain gauge rosette.

Then the stress spectra corresponding to the No. 11 and No. 12 were calculated, as shown in Figure 3c,d, respectively. It can be seen that the equivalent stress at the position of the No. 11 gauge is much lower than that of No. 12. Moreover, the changes in the peak and valley values of the stress of the No. 12 gauge are relatively small and consistent, but the consistency of the No. 11 gauge is poor. The reason may be that No. 12 strain gauge is affixed to a relatively flat surface of the upper torque link, while No. 11 is attached to the large filet.

In total, this work carried out a total of 350,000 loading cycles according to the requirements of the fatigue test outline, and non-destructive testing (magnetic particle inspection) was performed on the test specimens every 50,000 cycles. During the experiment, two fixing screws, one pin, and the first torque link suffered fatigue damage, and the same spare parts were promptly replaced. Note that for auxiliary components like screws and pins, their failure is an acceptable occurrence, as they are standard connecting components in the testing device. Once monitoring equipment and personnel detect abnormal data (e.g., force feedback of the hydraulic servo actuator), the experiment will be stopped for inspection or replaced with the same standard spare parts in order to minimize the impact on the testing device and loading conditions. Finally, in this fatigue test, the first torque link cracked at the chamfering position near the contact surface with the fixed bolt after 184,000 cycles, which provides the essential benchmark data for model validation. The second torque link remained uncracked after 166,000 cycles, and the outer cylinder dummy experienced a total of 350,000 cycles without cracks. At this time, the experiment reached the maximum set number of cycles and it was terminated. These results confirm the test’s repeatability and reliability.

3. Comparison Between FE Simulation and Experiment

To accurately identify the critical failure location and obtain the complete stress field required for subsequent life prediction, this section details the FE and fatigue life simulation, including the software, geometry, boundary conditions, and so on.

3.1. FE Model and Boundary Conditions

In engineering, the HCF assessment of the LG’s typical structures generally adopts stress-life fatigue analysis approach within safe-life concept [6,7]. The prerequisite for this is to determine the stress history at the critical points [3,16,46]. However, in the full-size experimental setup shown in Figure 2, it is obvious that the strain rosettes can only provide limited measurements at the pre-selected positions. These may not necessarily be the sites of the maximum fatigue damage [19,20], as cracks often originate from subtle geometric discontinuities (e.g., notches, filets) that are difficult to detect or predict a priori [21,47]. Therefore, relying solely on experimental strain data may lead to inaccurate life prediction and miss the actual critical point.

To identify the critical failure location and obtain the complete stress field required for subsequent accurate life prediction, a high-fidelity three-dimensional FEM of the whole fatigue testing device was established using the ANSYS 2022R1 software^®, with the software license provided by the High-Performance Computing Center of Central South University (this license covers all necessary modules for structural mechanics and fatigue analysis, ensuring the validity of the simulation environment). This modeling has two main purposes: (1) to verify the accuracy of the numerical model by correlating its results with experimental measurements (load, displacement, and strain); (2) to precisely quantify stress distribution, especially the stress gradient at critical locations.

Firstly, geometric model processing: directly imported the geometries of the upper torque link, the outer cylinder dummy, and connecting components (e.g., pins, bolts, screws) without simplification from their CATIA models into SpaceClaim 2022R1 (a module of ANSYS) for assembly, as shown in Figure 4a, preserving all critical geometric features, such as filets and chamfers, which are potential sites for stress concentration. Secondly, mesh generation: the geometric model was meshed into tetrahedral elements SOLID 187 with a target size of 3 mm and conducted the mesh refinement in the local details of the key investigation. In order to ensure stress convergence, the mesh convergence analysis was conducted on the key areas of concern, such as stress concentration areas like the chamfers, and the minimum element size in these areas was determined to be 0.5 mm. The final model for the main structures (excluding the test framework) comprised approximately 2.486 million nodes and 1.518 million elements, and the torque link alone was discretized into 1.038 million nodes and 0.619 million elements, with local refinement applied in critical regions like the chamfers, and filets. Next, contact settings: the real connection relationship of the structure was simulated by friction contact (i.e., surface-to-surface friction contact), where the friction coefficient of oil lubrication was set at 0.05 and that of the other contacts was set at 0.2. Finally, boundary conditions and loads applying: the outer cylinder dummy was fixed on the test framework with bolts. Here, to replicate real-world experimental conditions, all bolts were not omitted and the corresponding contact relationship was set. The hinge assembly was also fully modeled and considered with contact settings. Therefore, the force from the hydraulic servo actuator could be simulated using remote force and applied it to the cylinder end inside the actuator. That is, the force was transmitted to the hinge device through this cylinder and then applied to the torque link. Here, FEM was simulated with a compressive force Fz = +210 kN and a tensile force Fz = −210 kN to the model through the hydraulic servo actuator, respectively.

In summary, this approach of modeling the whole setup, as shown in Figure 4a, rather than applying loads directly to an isolated upper torque link, was crucial to accurately capture the load paths, boundary constraints, and potential secondary stresses arising from the interaction between components.

3.2. FEM Calibration

With the validated FE model described in Section 3.1, the stress and displacement fields under compressive and tensile loads were obtained. Specifically, the vertical displacement and equivalent stress results of FE simulation under the two working conditions are illustrated in Figure 4b–e, respectively. Among them, the absolute maximum displacement and stress under compression conditions are 13.298 mm and 1004.2 MPa, while under tension conditions, they become −11.706 mm and 1285.8 MPa. The maximum stresses of the two working conditions not only have different values but also are not in the same position.

Finally, the test values in Figure 3 were compared with the FE simulation results in Figure 4, and the maximum relative errors were calculated as shown in

Table 2. Comparison of simulation and experimental values.

Designation	Description	Simulation	Experimental	Relative Error
+Fz1	Load main peak	210 kN	204.7 kN	2.52%
+Fz2	Load secondary peak	200 kN	193.8 kN	3.10%
+dz1	Displacement main peak	13.298 mm	12.717 mm	4.37%
−dz1	Displacement main valley	−11.706 mm	−11.287 mm	3.58%
Seq,11	Maximum stress at No.11	631.4 MPa	582.1 MPa	8.47%
Seq,12	Maximum stress at No.12	473.3 MPa	440 MPa	7.62%

below.

From Table 2, it can be seen that the maximum difference in dynamic load error between the experiment and simulation is 3.10%, the maximum difference in dynamic displacement is 4.37%, and the maximum difference in dynamic stress error is 8.47%. The main reason for the stress discrepancy may be the strain-gauge placement deviation relative to FE nodes and the presence of certain dynamic load deviation during the testing. Nonetheless, the stress relative error of the experiment and FE simulation is within 10%, indicating that the FEM is accurate and reliable.

3.3. Fatigue Life Verification

Import the S-N curve obtained from Figure 1 (i.e., the simulation work uses the tested median S-N curve) and the FE stress results obtained from Figure 4 into ANSYS nCode DesignLife 2022R1 (nCode for short), set the load spectrum to 10 cycles, and use Goodman mean stress theory to simulate the fatigue life of the typical structure—upper torque link. The comparison between the fatigue simulation results and experimental is shown in Figure 5.

From Figure 5a, it can be seen that the torque link cracks from the chamfered position near the contact surface with the fixed bolt and propagates roughly along the direction of 45° from the axis. The fatigue simulation result after loading 10 cycles in nCode software, as shown in Figure 5b, shows that the maximum damage location is also at the chamfer of the bolt contact surface in the upper left corner (red area), which is highly consistent with the experimental cracking location. Among them, the most dangerous position is at the FE node 1,558,168 (geometric coordinates: X = 736.31 mm, Y = 148.06 mm, Z = −696.23 mm), with a maximum damage of 1.033 × 10⁻⁴ generated every 10 cycles. Therefore, according to the Miner criterion [3,6], the simulated fatigue life of this node in Figure 5b is 96,800 cycles. But its experimental fatigue life is 184,000 cycles, and the simulated value is smaller than the experimental value, with a relative error of 47.4% between the two. This suggests that a potential design modification to improve fatigue life includes increasing the chamfer radius at the critical location to reduce stress concentration. Alternatively, the surface treatment process can be specifically optimized for this location (e.g., by introducing higher CRS) to delay crack initiation and extending the safe life. In addition, although the simulation successfully identified the starting point, future work can also extend the simulation work to include crack propagation based on ANSYS fracture mechanics technology (e.g., SMART Crack Growth module), providing a complete simulation process from crack initiation, propagation to failure.

4. Prediction of HCF Based on FKM Local-Stress Approach

4.1. Determining the Fatigue Limit of Components

The FKM guideline [22,24,25] considers factors that affect the strength (static and dynamic) of components (made of steel, cast iron, and aluminum materials), such as design dimensions, geometric shapes, surface roughness, surface treatment, and load loading conditions, etc. It can be used to evaluate the static or fatigue strength of welded and non-welded components through the nominal stress method or the LSA based on the degree of utilization and have been widely applied in various fields in Europe. For non-welded structures, LSA uses design factors (see Equation (3)) to modify the fatigue limit of the material, thereby constructing a new fatigue life curve and estimating the fatigue life of the structure [22].

$$K_{WK}={\displaystyle \frac{1}{n_{\sigma }}}\cdot \left[1+{\displaystyle \frac{1}{\widetilde{K_f}}}\cdot \left({\displaystyle \frac{1}{K_R}}-1\right)\right]\cdot {\displaystyle \frac{1}{K_V\cdot K_S\cdot K_{NL,E}}}$$

(3)

where the notch sensitivity n_σ is the ratio of K_t to K_f, which is related to the structural stress gradient. $\widetilde{K_f}$ is the estimated fatigue notch factor for the component. Roughness factor K_R accounts for the influence of the surface roughness on the fatigue strength of the component. Surface treatment factor K_V represents the influence of surface strengthening process on the fatigue strength of components. Coating factor K_S considers the impact of surface coating on the component. $K_{NL,E}$ accounts for the influence of nonlinear effects of materials on components.

For the 300M steel of the LG in this work, based on the material characteristics and tensile strength limit, the FKM manual shows that the approximate value of the estimated fatigue notch factor is $\widetilde{K_f}$ = 2 [22,23,29], and the material nonlinear factor is $K_{NL,E}$ = 1. Since the surface of the upper torque link is treated with Cd/Ti plating after shot peening, K_V = 1.1 and K_S = 1 are taken according to the FKM guideline. Moreover, the roughness factor K_R can be calculated using the following equation [22]:

$$K_R=1-a_{R}lg\left(R_z\right)·lg\left({2R}_m/R_{m,N,min}\right)$$

(4)

where R_z denotes the mean roughness depth, R_m represents the ultimate tensile strength, a_R and R_m,N,min are material-dependent constants. Per the FKM guideline [22,25], the material constants for 300M steel are a_R = 0.22 and R_m,N,min = 400 MPa.

According to the upper torque link design drawing, the requirement for arithmetic roughness average is Ra = 3.2 um. Then check the Ra/Rz conversion table to obtain Rz = 12.5 um. Substituting it into Equation (4) gives K_R = 0.7613.

Through the approach outlined in the FKM guideline [22,25], the notch sensitivity n_σ can be determined as a function of the associated stress gradient G_σ, as presented in Equation (5).

$$n_{\sigma }=\left\{\begin{array}{c}1+G_{\sigma }·{10}^{-\left(a_G-0.5+{\displaystyle \frac{R_m}{b_G}}\right)} 0\le G_{\sigma }\le 0.1\\ 1+\sqrt{G_{\sigma }}·{10}^{-\left(a_G+{\displaystyle \frac{R_m}{b_G}}\right)} 0.1<G_{\sigma }\le 1\\ 1+\sqrt[4]{G_{\sigma }}·{10}^{-\left(a_G+{\displaystyle \frac{R_m}{b_G}}\right)} 1<G_{\sigma }\le 100\end{array}\right.$$

(5)

where G_σ represents the related stress gradient, a_G and b_G are material-group constants. For the 300M steel material, according to FKM guideline, the constants are a_G = 0.5 and b_G = 2700.

The relative stress gradient G_σ can be calculated by the stress of the critical node and nearby node [25,26], see the equation below.

$$G_{\sigma }={\displaystyle \frac{1}{\mathrm{\Delta }s}}\cdot (1-{\displaystyle \frac{{\sigma }_{2a}}{{\sigma }_{1a}}})$$

(6)

where ${\sigma }_{1a}$ is the stress amplitude of the surface critical node, ${\sigma }_{2a}$ is the stress amplitude of its adjacent subsurface nodes, and Δs represents the distance between adjacent node and the surface.

For example, for the hazardous location shown in Figure 5b of the torque link (corresponding to the node 1,558,168), the FE analysis in Figure 4 above shows that the relative stress gradients G_σ under compression and tension conditions calculated by Equation (6) are 0.6392 and 0.7441, respectively. Then, substituting into Equation (5), the notch sensitivity n_σ can be obtained as 1.0479 and 1.0517, respectively. Its value is close to 1, explaining that the support effect from the stress gradient at this location is relatively small. More importantly, it is not difficult to observe a slight deviation in n_σ between tension and compression load, indicating that it is load direction-dependent. This finding validates the necessity of calculating n_σ separately for different loading conditions in nonsymmetric components, rather than treating it as a constant like the standard FKM approach.

4.2. Considering the Mean Stress Effect

The LG components typically endure complex loads or asymmetric stress cycles [19], so it is necessary to consider the mean stress effect and correct the fatigue strength. According to the damage equivalence principle, the FKM guideline provides a correction method based on the Haigh diagram and material mean stress sensitivity factor, which converts the actual stress cycle into an equivalent stress amplitude σ_ar of the fully reversed stress state [22,25], as follows:

$${\sigma }_{ar}={\sigma }_a/K_{AK}$$

(7)

where K_AK is mean stress factor, calculated by the following Equation (8) [22]. It depends on the type of overload and is related to the stress ratio R.

$$K_{AK}=\left\{\begin{array}{c}{\displaystyle \frac{1}{1-M_{\sigma }}} 1<R\\ {\displaystyle \frac{1}{1+M_{\sigma }\cdot {\sigma }_m/{\sigma }_a}} R\le 0\\ {\displaystyle \frac{3+M_{\sigma }}{(1+M_{\sigma })\cdot (3+M_{\sigma }\cdot {\sigma }_m/{\sigma }_a)}} 0<R\le 0.5\\ {\displaystyle \frac{3+M_{\sigma }}{3\cdot {(1+M_{\sigma })}^2}} 0.5<R\le 1\end{array}\right.$$

(8)

where σ_m and σ_a denote the mean stress and stress amplitude of the structure, and mean stress sensitivity M_σ is calculated by the following equation [22,25].

$$M_{\sigma }=a_M·R_m·{10}^{-3}+b_M$$

(9)

For the 300M steel, using the material-specific constants a_M = 0.35 and b_M = −0.1 recommended in FKM and substituting the material’s ultimate tensile strength R_m = 1950 MPa into Equation (9) gives M_σ = 0.5825.

Furthermore, the stress amplitude at critical locations is essential for damage calculation and can be determined by the peak stress σ_max of the actual load cycle as follows [13,14]:

$${\sigma }_a={\sigma }_{max}·\left(1-R\right)/2$$

(10)

The peak stress of complex structures is generally derived through FE simulation, and then combined with FKM mean stress correction, which provides a reliable basis for fatigue damage assessment of critical points (non-fully reverse stress state).

4.3. Improving FKM-LSA Framework for Finite-Life Prediction

The FKM guideline provides a robust analytical framework for fatigue assessment, widely adopted in engineering practice due to its industrial applicability and particularly valuable for preliminary fatigue assessments in the absence of experimental data. Its core strength lies in the fatigue strength evaluation and primarily focuses on determining the fatigue limit (endurance strength) of notched parts. However, it does not directly provide a methodology for calculating life. Additionally, the FKM-LSA relies on preset Basquin or double-line S-N curves grouped by broad material categories (e.g., steel, aluminum alloy), utilizing the same slope exponent and intercept parameters across materials within a group. This overlooks the unique fatigue characteristics of advanced alloys like 300M steel, for which extensive experimental data were obtained in this study (Section 2.2, Figure 1). Crucially, key parameters within FKM, particularly the relative stress gradient and the notch sensitivity factor, are treated as static properties and load-independent constants. But the FE analysis (Section 3, Figure 4) reveals that G_σ and n_σ varies with loading conditions (e.g., tension vs. compression) for geometrically asymmetric LG components. Neglecting this variability undermines the accuracy of stress field quantification at critical locations (e.g., notches, chamfers, filets). Therefore, when extended to higher accuracy prediction of finite fatigue life, an improved FKM-LSA framework was developed, centered on two core pathways as elaborated below:

The first pathway focuses on accurate acquisition of the load-dependent stress spectra of critical locations via refined FE simulation. Instead of indirectly deriving notch stress via nominal stress and fatigue notch factor K_f (also known as fatigue strength reduction factor [14,23,31]), the improved LSA directly utilizes the peak stress σ_max from high-fidelity FE simulations, which inherently includes stress concentration effects. The load-dependent notch support effect is then incorporated through the calculated notch sensitivity n_σ to determine the effective notch stress for life prediction, as follows [13,14,48]:

$${\sigma }_{notch}={\sigma }_n{K}_f={\displaystyle \frac{{\sigma }_{max}}{K_t}}{K}_f={\displaystyle \frac{{\sigma }_{max}}{n_{\sigma }}}$$

(11)

where σ_n denotes nominal stress. K_t is the theoretical stress concentration factor. $n_{\sigma }=K_t/K_f$ ≥ 1 is a factor related to the stress gradient and represents the support effect of local stress gradient [13,18].

This is because the nominal stress and theoretical stress concentration factor are generally difficult to quantify for the geometric details of typical LG components such as notches and filets (usually in a multiaxial stress state [16]). By contrast, refined FE simulation can accurately obtain the peak stress at stress concentration locations in complex structures, and when combined with notch stress gradient correction, the effective notch stress can be readily derived [31].

The second pathway involves rational correction of S-N curves via logarithmic interpolation of surface factors. For the prediction of finite fatigue life of LG’s non-welded notch components, it seems insufficient to merely correct the fatigue limit based on the design factor obtained from Equation (3). Clearly, further correction of the S-N curve is needed, as the fatigue notch factor is related to the fatigue life. For example, the Heywood model [13,14,48] corrects the S-N curve with fatigue notch factors at fatigue life Nc = 10³ and Ne = 10⁶ cycles, as shown in Figure 6a, and the correction coefficients in the middle interval is obtained through double logarithmic linear interpolation. Inspired by this viewpoint, the actual tested S-N curve corrected by the fatigue influence factors involved in LSA of the FKM guideline is shown in Figure 6b. Specifically, the design factor K_WK of Equation (3) is divided into two parts: one is the notch sensitivity factor n_σ related to fatigue load conditions, and the other is the integrated surface state factor K_WS (accounting for surface roughness, coating, and strengthening processes), where K_WS = K_WK/n_σ.

Here Nc represents the boundary between high and low cycle fatigue life, and Ne is the endurance fatigue life.

According to the double logarithmic linear coefficient relationship, interpolate between Nc and Ne to calculate the surface influence factor K_WS as follows:

$${\displaystyle \frac{lg{K}_{ws}^N-lg{K}_{ws}^c}{lgN-lg{N}_c}}={\displaystyle \frac{lg{K}_{ws}^{\mathrm{e}}-lg{K}_{ws}^c}{lg{N}_e-lg{N}_c}}$$

(12)

Solving Equation (12) gives the corrected stress amplitude of the component at life N:

$${\sigma }_{aN}=S_{aN}/K_{ws}^N$$

(13)

where S_aN is the material stress amplitude at life N (from Section 2.2 material S-N curve).

This correction strategy avoids the conservatism of the original FKM-LSA’s default parameters by integrating 300M steel’s experimental data, ensuring the component S-N curve accurately reflects the upper torque link’s actual service behavior.

By integrating the two pathways above—accurate stress spectrum acquisition via refined FE simulation and rational S-N curve correction via surface state factor K_WS interpolation—the improved FKM-LSA framework is established, as illustrated in Figure 7 with the following key steps:

Step 1: Build a geometrically refined FE model of the upper torque link, perform mesh convergence verification.

Step 2: Calculate load-dependent stress parameters (σ_max, G_σ, n_σ, etc.) at critical locations.

Step 3: Apply the FKM mean stress criterion (Section 4.2) to convert the actual stress cycle to an equivalent fully reversed cycle (R = −1) and obtain the equivalent stress amplitude σ_ar.

Step 4: Calibrate factors such as K_R, K_S, K_V, etc.

Step 5: Obtain 300M steel’s S-N curve via axial fatigue tests (Section 2.2, stress ratio R = −1), and then correct the material S-N curve to the component’s curve using $K_{wk}^N$ (from Equations (12) and (13)).

Step 6: Substitute σ_ar into the component S-N curve to calculate its finite fatigue life.

Figure 7. Improved FKM-LSA framework for HCF life prediction: integration of test data, FE simulation, and the FKM guideline.

Figure 7. Improved FKM-LSA framework for HCF life prediction: integration of test data, FE simulation, and the FKM guideline.

This framework achieves high-precision prediction by integrating test data, refined FE simulation, and load-dependent correction—providing a targeted solution for fatigue life assessment of complex LG components like the upper torque link. This improvement not only inherits the standardized correction advantages of FKM for engineering factors such as structural geometry and surface technology but also solves the problem of high conservatism of FKM default parameters for finite life prediction of 300M steel through a data-driven approach, providing a more targeted quantitative tool for the LG life design.

5. Discussion

5.1. Comparison and Analysis of Fatigue Life Prediction Methods

Accurate fatigue life prediction is critical for the safety design of the upper torque link, key load-bearing parts of aircraft LG.

Table 3. Prediction life of the upper torque link of different methods.

Method	fatigue Life Nf (Thousand Cycles)	Relative Error δ
nCode (Goodman, no gradient)	98	−46.7%
nCode (FKM, no gradient)	67	−63.6%
nCode (Goodman, with gradient)	316	+71.7%
nCode (FKM, with gradient)	157	−14.7%
Standard FKM-LSA	506	+175%
Improved LSA framework	174	−5.4%

summarizes the comparison of fatigue life predictions obtained using different methods, and the relative error is calculated via $\delta =(N_{pred}-N_{test})/N_{test}$, with the full-scale fatigue test result (184,000 cycles for the first specimen) as the reference.

The nCode software is widely used in engineering for fatigue analysis, and its prediction results depend on two key settings: the mean stress correction model and whether stress gradient correction is enabled. For consistency, all nCode simulations adopted the 300M steel median S-N curve obtained from material fatigue tests (Section 2.2, Figure 1), with the load spectrum consistent with the full-scale test. When stress gradient correction was disabled, using the Goodman mean stress model, the predicted life was about 98,000 cycles, corresponding to a relative error of −46.7%, while adopting FKM mean stress correction decreased the life to approximately 67,000 cycles (−63.6%). When considering stress gradient correction in nCode simulation, the fatigue life combined with Goodman method becomes 316,000 cycles (a large overestimate, +71.7%), while with FKM mean stress treatment the prediction is about 157,000 cycles (−14.7%). These results indicate that the FKM mean stress model is more conservative compared to the Goodman model, and neglecting the notch stress gradient “support effect” (subsurface stress alleviating surface damage [49]) will also lead to an underestimation of the life.

Secondly, the standard FKM-LSA method adjusts the fatigue limit via a fixed design factors (Section 4.1, Equation (3)), uses a generic preset S-N curve parameters (i.e., for 300M steel material, slope exponent k = 5 and knee point N_D = 10⁶ recommended by the FKM guideline), and then calculates the life as 506,000 cycles (+175% error) using Formula (21) from Fazili et al. [26]. This method is useful for initial safety checks, but the results have significant deviations and unsuitable for precise life estimates of 300M steel components.

Finally, the improved LSA framework (Section 4.3, Figure 7) developed in this work integrates load-dependent notch sensitivity, FE-derived local stress gradients, interpolated surface state factors, and the FKM mean stress correction model. It predicted the upper torque link’s life as 174,000 cycles (−5.4% error). This result is very close to the nCode software prediction with stress-gradient correction and the FKM mean stress model, which further validates that the proposed approach correctly captures the combined effects of mean stress, surface treatment, and notch support. The close agreement between the two independent prediction methods indicates that the improved FKM-LSA is a reliable method for fatigue life assessment of the LG.

Overall, fixed-parameter methods failed to capture the upper torque link’s load-dependent stress state or 300M steel’s fatigue characteristics, while the improved LSA balances accuracy and practicality for safety-critical landing gear design.

5.2. Probabilistic Fatigue Life Prediction

As demonstrated by the fatigue experimental data of 300M steel mentioned earlier, fatigue life exhibits inherent scatter. For safety-critical components of LG, a deterministic assessment based solely on the median life is insufficient to support high-reliability design requirements. Therefore, this section introduces a probabilistic fatigue analysis method, built upon the improved deterministic FKM-LSA framework, to quantify component life under different reliability levels. As is well known, the probability stress-life (p-S-N) curve serves as the foundation for reliability assessment [39]. But strictly speaking, accurately constructing p-S-N curves typically requires large-sample experiments and statistical analysis [38]. To address the small-sample problem, the fatigue test data in Table 1 were augmented based on the following assumptions [40,41]: (1) at each stress level, the logarithm of fatigue life follows a normal distribution; and (2) the failure rate of the sample remains consistent across different stress levels. The specific procedure is as follows:

Firstly, the fatigue test data at the five stress levels were augmented using the method proposed by Bai et al. [41,42] to obtain a sufficient dataset for statistical analysis. Subsequently, referring to GJB/Z 18A-2020, the conditional probability fatigue life for a specified confidence level and survival probability was estimated using the median offset Formula (14) and one-sided tolerance factor Formula (15). For the long-life region (staircase method area), the same statistical method was applied to calculate the conditional fatigue endurance strength under different probabilities. Finally, the three-parameter S-N model (Equation (1)) was used to fit the augmented probabilistic data, constructing a family of p-S-N curves for 300M steel under different combinations of survival probability and confidence level (p-c), as shown in Figure 8.

$$lg{N}_{p,c,i}=\widehat{{\mu }_i}+k_{p,c}·\widehat{{\sigma }_i}$$

(14)

$$k_{p,c}={\displaystyle \frac{u_P-u_C\sqrt{\frac{1}{n}\left(1-\frac{{u_C}^2}{2(n-1)}\right)+\frac{{u_P}^2}{2(n-1)}}}{1-\frac{u_C^2}{2(n-1)}}}$$

(15)

where $\widehat{{\mu }_i}$ and $\widehat{{\sigma }_i}$ are the mean and standard deviation of the logarithmic life normal distribution at the i-th stress level. $k_{p,c}$ represents one-sided tolerance factor and n is the number of sub-samples. $u_P$ is a standard normal deviation related to survival probability p, and $u_C$ represents the standard normal deviation related to confidence level c.

By replacing the median S-N curve in the improved FKM-LSA framework (established in Section 4) with the p-S-N curves for different p-c levels from Figure 8, the probabilistic fatigue life of the torque link under different reliability requirements was calculated, as shown in Figure 9.

Figure 9 clearly shows that the fatigue life of the upper torque link’s critical node is closely related to the material’s reliability requirements. As the required survival probability P and confidence level C increase, the predicted life decreases significantly. This trend quantifies the trade-off between reliability and service life: to achieve higher confidence and component survival rates (e.g., from 50% to 95%), the design allowable life must become more conservative. This analysis provides a clear quantitative basis for engineers to conduct fatigue-resistant design while meeting specific reliability targets, highlighting the significant engineering value of integrating probabilistic analysis into the life assessment process for critical LG components.

6. Conclusions

In this study, the high-cycle fatigue performance of 300M steel was investigated by material axial fatigue test, and the full-scale fatigue test frame of the upper torque link of the landing gear was discussed in detail. Subsequently, an improved FKM local-stress approach was introduced to predict the fatigue life of the typical component, and validated against experimental data and commercial software simulations. The main conclusions are as follows:

(1)

A reliable S-N curve of 300M steel electroplated with cadmium–titanium after shot peening was established through axial fatigue tests of 40 specimens. The full-scale structure fatigue experiment revealed that after approximately 184,000 cycles, the bolt filets of the upper torque link cracked, providing a benchmark for model verification.

(2)

The finite element simulation reproduced the stress distribution at the critical location, with a maximum error of 8.47% compared to the measured data from strain gauges, which is less than 10%, confirming the accuracy of the numerical model for subsequent life prediction.

(3)

The improved FKM-LSA framework developed in this study incorporates load-dependent stress gradients and an S-N curve corrected by logarithmic interpolation of surface factors. The predicted life of 174,000 cycles is in close agreement with the full-scale test result and also consistent with nCode predictions that include gradient correction and the FKM mean stress model.

(4)

The probabilistic fatigue analysis underscores the importance of accounting for fatigue scatter in design. The p-S-N curve was successfully derived using small-sample data augmentation techniques based on sample aggregation theory. Combining the improved LSA framework, the calculation results indicate that higher survival rates and confidence levels lead to more conservative life predictions, quantifying the trade-off between reliability and service life.

The proposed framework, encompassing both deterministic and probabilistic analyses, provides a practical approach for accurate fatigue life assessment of critical landing gear components. Its ability to integrate experimental data, finite-element results, and reliability requirements makes it suitable for design evaluation in aerospace applications.

To further enhance the fatigue performance and engineering application value of the structure, subsequent research will combine fatigue life for topology optimization design, reduce redundant materials, and thus improve the overall performance of the landing gear.