Probabilistic Fatigue Life Assessment of a Heavy Truck Front Axle Beam under Various Operating Conditions Using a Combined FE-MBD Approach

The front axle assembly of trucks performs several important functions. It assists steering control, distributes loads evenly across the wheels, and receives and transmits interaction forces between the wheels and the chassis frame. Typically, it supports about 30–40% of the vehicle's total weight. There are two main types of front axles for trucks: the drive axle and the dead axle. The dead axle is more commonly used for trucks, with main components including a front axle beam (FAB) and steering knuckles, which are connected to each other via kingpins, as shown in Figure 1.

During vehicle operation, FAB is subjected to various types of loads, encompassing both static and dynamic loads. These are vertical forces from sprung masses, longitudinal inertial forces when braking and accelerating, lateral forces when turning, dynamic loads from road roughness, and so on. These loads create complex stress states in the structure, especially cyclic stresses, which can lead to fatigue failure. Fatigue failure often leads to serious consequences and seems to occur without any prior warning. When a structure is subjected to a single load, the static strength is evaluated based on the material's yield strength or ultimate strength. But when it is subjected to cyclic load, fatigue failure can occur at stress levels significantly lower than the material's static strength [1]. Therefore, ensuring the fatigue strength of the FAB throughout the vehicle's service life is always a top priority when designing the product.

Figure 1. Typical configuration of the front axle

As computer technology has advanced, the finite element (FE) method [2] is now employed in many fields, like structural mechanics, fluid mechanics, electromagnetic fields, etc., to design and analyze complex structures and environments [3, 4]. The FE method is an effective tool for stress analysis and identifying critical positions where failures may occur [5]. Surveys of the structural design and analysis of a truck's FAB show that all involve the application of the FE method.

Dey et al. [6] studied the load capacity for the FAB of a light commercial vehicle. The stress, strain, and deformation of the structure were determined under vertical, braking, and cornering load conditions. Datti et al. [7] calculated the vertical load on the front axle of the truck under various driving conditions, including uphill, downhill, and level roads. The stress and strain of the FAB were obtained from static analysis. Sivaraman et al. [8] studied how different shapes of alternative components affected stress and strain distribution in the FAB of a heavy commercial vehicle. The best shape of the FAB was selected from static analysis under specific load situations and driving torque needs. Avikal et al. [9] conducted fatigue life analysis for the FAB with 5 different materials: 50 steel, AISI 1045, AISI 4130, AISI 4140, and AISI 4150. The effects of states without and with cracks on fatigue life were taken into account. Verma and Pant [10] implemented a stress analysis for FAB with materials from AISI 1045 and AISI 1053. The fatigue life was predicted with different crack angles and directions. To make the FAB of a commercial vehicle lighter while ensuring mechanical strength, Ruban and Sivaganesan [11] conducted optimal analysis for three concept designs of the FAB with static load cases, including vertical, braking, and cornering loads. Fatigue analysis was carried out with sinusoidal vertical loads and combined vertical and braking loads. With 50 steel as the fabrication material, Zhang et al. [12] conducted stress and fatigue analysis for the FAB of a heavy-duty truck under sinusoidal vertical loading. The structure was improved to reduce stress concentration and increase fatigue life. Fatigue simulations with multi-working conditions and the effect of a crack's length and depth on its fatigue life were carried out. To determine the cause of premature failure at the kingpin hole and propose a structural improvement plan, Guo et al. [13] performed static strength and fatigue failure analysis for the FAB of a mining truck. Symmetrical dynamic loads were applied to the FE model for fatigue analysis. The failure fracture at the kingpin hole was studied by both simulation and experiment methods.

The aforementioned studies have focused on static strength and fatigue life analysis with typical load cases acting on the FAB under various operating conditions. Fatigue life analysis with different materials has also been conducted, enabling more effective material selection. However, the fatigue loads applied to the FAB are confined to static loads [9, 10] or sinusoidal dynamic loads [11-13]; they do not account for dynamic loads induced by random road roughness, which act continuously throughout the vehicle's operation. On the other hand, these studies use standard parameters for material fatigue curves with a default survival probability of 50%. For primary load-bearing structures such as FAB, it needs to be considered at higher survival probability levels to enhance safety.

As is known, while experimental methods are highly accurate for collecting fatigue load data, their high cost and long implementation time are disadvantages. In limited experimental conditions, combining software packages for structural simulation and analysis partially overcomes these disadvantages. This method offers several advantages, including the ability to generate multiple simulation scenarios with many influencing factors considered, reduced experimental costs, and so on. Among them, the combined FE analysis and multibody dynamics (MBD) simulation method has become the preferred approach for stress and fatigue life analysis of the structures on transport vehicles [14-17].

The research objective of this paper is to assess the fatigue life for the FAB of a heavy truck with a payload of 8.5 tons, taking into account different material types, operating conditions, and survival probability levels. First, the FE model of the front axle assembly is constructed using ANSYS software. Next, the MBD model of the entire vehicle is built using ADAMS/View software, with the object of the front axle assembly taken from the modal neutral file of the FE model. In the MBD model, road roughness as the excitation source causing fatigue loads is simulated based on the ISO 8608:2016 standard [18], considering operating conditions including various road classes and vehicle speeds. The dynamic loads obtained from the MBD simulations are used to perform a quasi-static analysis that finds out the structural stress-time histories. In the next step, the fatigue analysis models of AISI 1045 and AISI 4135 materials in the crack initiation period are developed based on the S-N fatigue curve approach using ANSYS nCode DesignLife software, which considers the survival probability, also known as the P_(%)-S-N curve [19]. The P_(50%)-S-N curves for these materials are established from fatigue experimental data of equivalent materials [20]. The fatigue life of critical positions at higher survival probability levels, such as P_(90%), P_(95%), and P_(99%), is estimated by extrapolating from the mean life of P_(50%). Finally, the fatigue life of the FAB made from the two typical materials is calculated, taking into account the effects of different road classes, vehicle speeds, and survival probability levels. The fatigue life in cycles is converted to operating time in years to assess the durability of the FAB against the vehicle's service life.

3.1 Vehicle multibody dynamics modeling

The MBD model of the entire vehicle is constructed using ADAMS/View software. Import the modal neutral file to create the front axle assembly's object, which is considered a rigid body. The other bodies are connected to each other and to the front axle assembly using appropriate joints and elements through the interface nodes. Figure 4 shows the vehicle's MBD model that has 7 degrees of freedom (DOFs). The front and rear axles each have two DOFs: the roll angle and the vertical displacement. The sprung mass (including the chassis, cabin, cargo body, payload, etc.) has three DOFs: the roll angle, the pitch angle, and the vertical displacement. Table 3 provides the basic parameters of this model [21, 22].

Table 3. Multibody dynamic (MBD) model parameters

Figure 4. Multibody dynamic (MBD) model of the entire vehicle

3.2 Road roughness simulation

In the MBD model, the uneven road surface is considered the only source of excitation causing vehicle vibrations. The road roughness continuously impacts the wheels, causing the vehicle to oscillate in the vertical, roll, and pitch angle directions, which are the main causes of fatigue loads on the FAB. To simulate road roughness, the studies have assumed that the road surface profile is a stable random process following a Gaussian distribution and utilized the vertical displacement power spectral density (PSD) as its representative [23-29]. The PSD of the road surface profile can be described in Eq. (1) according to ISO 8608:2016 [18]. On that basis, this standard has given eight road classes, from class A to class H, corresponding to the lower limit, upper limit, and geometric mean value of the PSD.

$G_q(n)=G_q\left(n_0\right)\left(\frac{n}{n_0}\right)^{-w}$         (1)

where, n and n₀ refer to the spatial frequency and the referenced spatial frequency, respectively, with n₀ set at 0.1 (m^-1); G_q(n₀) indicates the PSD of road surface roughness (m³) when n equals n₀; w represents the frequency index, which is typically w = 2. The value of n falls within the effective spatial frequency range (n₁, n₂), which must include the main natural frequency of the vehicle at an average speed.

Based on ISO 8608:2016 [18], there are many methods for simulating excitation signals from road surface profiles in the frequency and time domains. Among them, the Inverse Fast Fourier Transform (IFFT) method is often utilized when studying vehicle dynamics or road roughness simulation [23, 24, 26, 28]. This method not only generates a PSD that is more consistent with the specified one but also takes into account vehicle speed. The basic contents of the IFFT method are first converted from spatial frequency to time frequency. Next, road roughness data, according to the PSD, is discretized by the Fourier transform and added with certain rules. Finally, the IFFT method is employed to convert the existing data into road roughness in the time domain.

For truck vehicles, the operating road conditions are relatively diverse, including excellent-quality highways or good-quality interprovincial roads (which can be rated to road class A or B), normal urban or interprovincial roads (road class C), and poor-quality suburban roads (road class D). Following the recommendations for maximum speeds applied when analyzing vehicle vibration according to road classes [29], this paper selects operating conditions to simulate road surface roughness: v = 100 km/h (27.78 m/s) for road class B (denoted as the operating condition of B-V100); v = 60 km/h (16.67 m/s) for road class C (operating condition of C-V60); and v = 20 km/h (5.56 m/s) for road class D (operating condition of D-V20). Figure 5 presents a partial excerpt of the simulation results for road roughness over a 20-second period under three operating conditions.

Figure 5. Simulation results of road roughness under various operating conditions

3.3 Dynamic load extraction from simulation results

Using road roughness corresponding to operating conditions B-V100, C-V60, and D-V20 as the vibration excitation source, dynamic simulations are performed for the entire vehicle's MBD model. The simulation process is executed in 10 seconds with a sampling time step of 0.01 seconds. After the simulation, the “Export FEA Loads” feature in ADAMS/View is implemented to export the load file (*.lod). There are 1,000 load steps in each load file that act on the six interface nodes of the front axle assembly. Each load step contains data on force and torque, as well as inertial loads like angular velocity (omega), angular acceleration (domega), and acceleration along the X, Y, and Z axes.

5.1 The P_(50%)-S-N curve establishment

The stress analysis findings demonstrate that all nodal stress values are lower than the material's yield strength. Consequently, the fatigue analysis model is applied in the crack initiation period and is developed based on the S-N fatigue curve approach, commonly referred to as the stress-life method [19]. Figure 8 illustrates a typical S-N curve of the material, representing the relationship between the amplitude or range of cyclic stress (S) and the number of cycles (N) until a fatigue crack occurs, which is often plotted on a logarithmic coordinate system. S_RI1 and S_e denote the stress amplitudes associated with N = 1.0 and N_C1 = 1.0E+5 to 1.0E+7 cycles, respectively. N_C1 is considered the transitional point of life, while S_e represents the fatigue limit, and b₁ and b₂ represent the gradients of the line segments.

The standard S-N curve is often built by experimental data with different constant stress amplitude levels, each with a mean stress of S_m = 0 and a default survival probability of P_(50%). The collection of fatigue curves, taking into account various survival probability levels, is referred to as the family of P_(%)-S-N curves, which can be represented by Eq. (2) [19], where a_P and b_P denote the experimental regression coefficients.

Figure 8. Typical S-N curve

$\lg \left(N_{P(\%)}\right)=a_P+b_P \lg (S)$         (2)

The P_(50%)-S-N curves of AISI 1045 and AISI 4135 materials are established in ANSYS-nCode DesignLife software using experimental data for notched specimens of two equivalent materials, GB 45 and GB 35CrMo [20], as given in Table 4. The linear regression method is applied to estimate the coefficients a_P and b_P. With the choice of N_C1 = 1.0E + 7 cycles, S_RI1 and S_e values are calculated by Eq. (2), and b₁ and b₂ are derived by Eq. (3) [30]. The P_(50%)-S-N curves of AISI 1045 and AISI 4135 materials are shown in Figure 9.

$b_1=\frac{\left(\lg S_e-\lg S_{R I 1}\right)}{\left(\lg N_{C 1}-1\right)} ; b_2=\frac{b_1}{\left(2+b_1\right)}$         (3)

$\lg \left(N_{P(\%)}\right)=\lg \left(N_{P(50 \%)}\right)+S D_{P(\%)} S E$        (4)

Table 4. Fatigue experimental data and P_(50%)-S-N curve parameters of the equivalent materials

a) AISI 1045

b) AISI 4135

Figure 9. P_(50%)-S-N curves of the materials

In fatigue analysis, structures subjected to primary load-bearing, such as FAB, must account for higher survival probability values. Under limited experimental conditions, fatigue life with different survival probabilities is extrapolated via the mean life of P_(50%) using Eq. (4) [30]. The standard error (SE) is usually set at 0.1, whereas the standard deviation (SD) is derived from its correlation with P_(%), as illustrated in Figure 10 [30].

Figure 10. Correlation between standard deviation (SD) and P_(%)

5.2 Mean stress correction method

Most structures subjected to loads in practice have cyclic fatigue stresses that are often asymmetrical with S_m ≠ 0. This phenomenon has a significant impact on fatigue strength in the high-cycle fatigue region (N > 1.0E+4 cycles). When S_m > 0 (tension), it is detrimental, and when S_m < 0 (compression), it is beneficial for fatigue strength. Several classical correction methods have been proposed to consider the effect of mean stress on fatigue strength, such as those of Gerber, Goodman, and Soderberg [19]. The stress analysis results indicate that the Goodman correction method, described in Eq. (5) [19], is proposed as appropriate because it is linear and effective in cases of tensile mean stress at the fatigue limit.

$\frac{S}{S_e}+\frac{S_m}{\sigma_u}=1$         (5)

5.3 Fatigue damage evaluation

Random road roughness additionally affects the nodal stress-time histories in the FAB, which are characterized by variable amplitude and an asymmetric profile. In ANSYS-nCode DesignLife software, the rainflow counting method [19, 31] is employed to statistically compute the parameters of the stress cycle. The counting results will ascertain the number of stress cycles (n_i) corresponding to each stress amplitude level (S_i), from which the fatigue life (N_i) will be calculated based on the established fatigue curve. It enables the application of the Miner damage rule expressed in Eq. (6) [32] to determine fatigue damage (D_i). During the crack initiation period, a fatigue crack will appear when accumulated fatigue damage (D_Ʃ) attains the threshold value of 1.0.

$D_{\Sigma}=\sum D_i=\sum \frac{n_i}{N_i}$         (6)

This study was conducted to assess the fatigue life of a heavy truck's FAB. The structural fatigue stress was determined utilizing a combined method of FE analysis and MBD simulation. The fatigue analysis model in the crack initiation period was developed based on the P_(%)-S-N fatigue curves of typical materials, AISI 1045 and AISI 4135. The P_(50%)-S-N curves of these materials were generated from experimental fatigue data of equivalent materials. The fatigue life of the FAB was calculated, taking into account the effects of various operating conditions, such as road classes and vehicle speeds, and specified survival probability levels. The analysis results lead to the following main conclusions:

(1) AISI 1045 material appears appropriate for fabricating FAB under light and medium load conditions. Under heavier operating conditions, such as D-V20, and with a survival probability level of P_(99%), AISI 1045 material becomes unsuitable when considering the vehicle's service life of 20 years.

(2) AISI 4135 material has superior fatigue strength. Under various operating conditions and at all specified survival probability levels, the fatigue life of FAB is consistently longer than 20 years.

(3) Both material types satisfy the FAB fatigue life requirement of exceeding 20 years at all specified survival probability levels, assuming the vehicle operates in a combined mode.

Based on the methods and results obtained in this paper, optimizing the structure according to fatigue strength to reduce the weight of FAB is considered a direction for future investigation.