2.1. Chassis Geometry

Three-axle truck chassis (Figure 3) is the vehicle’s bearing component. where the engine is installed the transmission system assemblies, the operative part, the control mechanism, the lodge, and the load. Three-axle chassis has a total length of 7280 mm, consisting of 9 horizontal bars and 2 major bars. Each side’s primary bar structure consists of double-channel section chassis, from behind the chassis, the inner main bar is 5540 mm long and the outside main bar is 7130 mm long [16].

Figure 3. Detailed dimensional drawing of the three-axle truck chassis [16].

Figure 4 depicts the main bar No. 1’s structure at the cross-sections.

Figure 4. Structure of main bar No. 1 at the cross-sections.

Crossbar 13 is connected to the main frame via components 2 and 3. The front-leaf-spring front shackle (4), (10), and rear-leaf-spring rear shackle (12) connect the vehicle frame to the suspension system. Figure 5 shows their designs.

Details 5 and 8 are mounting points for installing the engine onto the chassis frame, while detail 9 serves as a support for the vehicle cabin.

The cross members 14, 16, 18, and 19 in Figure 3 are used to increase the stiffness of the chassis frame and to provide mounting points for various auxiliary components. These members have a circular tubular cross-section and are connected to the longitudinal beams through flanges bolted to the side walls of the frame rails. Cross member 15 is used to mount the transfer gearbox support. It has a C-shaped cross-section and is connected to the longitudinal beams via C-shaped plates bolted to the frame’s side walls. Cross member 17 in Figure 3 supports the rear part of the cabin and is connected to the longitudinal beams using bolted joints. Cross member 20 in Figure 3 enhances the structural rigidity of the chassis at the mounting location of the rear suspension equalizer axle bracket. It consists of two C-shaped members connected to the main cross member through cover plates and I-shaped brackets. These cover plates and I-shaped brackets are connected to the rear leaf-spring shackle using bolts, while the cover plates are welded to the main member. Cross member 21 in Figure 3, which forms the rear end beam of the chassis. It has a C-shaped cross-section and is connected to the main frame members using rivets.

Figure 5. Structure of the suspension hangers.

2.2. Finite Element Model

A mesh convergence study was performed before the final numerical simulations to determine an appropriate finite element discretisation and to ensure that the numerical results were sufficiently independent of the mesh density. Four mesh configurations with progressively refined characteristic element sizes were investigated. The geometry, material properties, boundary conditions, and applied loads were kept identical for all mesh configurations so that the influence of mesh density could be evaluated independently.

SOLID186 and SOLID187 were selected according to the geometrical characteristics of the chassis components. Both are second-order (quadratic) three-dimensional structural solid elements. SOLID186 is a 20-node structural solid element and was primarily used in regular regions with relatively simple geometries, whereas SOLID187 is a 10-node tetrahedral structural solid element and was employed in geometrically complex regions where tetrahedral discretisation provides greater flexibility for representing irregular and curved features. The combination of the two element types provides a suitable balance between geometric representation, numerical accuracy, and computational efficiency. At the interfaces between the SOLID186 and SOLID187 regions, geometric connectivity was maintained through shared nodes between adjacent elements, ensuring continuous displacement and load transfer across the element-type transition.

The mesh convergence was evaluated using the maximum von Mises stress and the maximum total deformation obtained from the static structural analysis. The relative variation between two consecutive mesh configurations was calculated as

Table 1 presents the results obtained during the mesh verification process. As the mesh was progressively refined, the calculated stress and deformation approached stable values. The final mesh was selected when the variation of the principal response between two consecutive mesh configurations satisfied the adopted 5% criterion while maintaining a reasonable computational cost. The final model consisted of 174,223 elements and 376,982 nodes (Figure 6).

Figure 6. Finite element analysis.

All mesh refinement steps resulted in a variation of less than 5% in the maximum von Mises stress. For the final refinement from a 15 mm to a 10 mm element size, the variation was only 1.89%, indicating that the numerical results had reached satisfactory mesh convergence. Therefore, the 10 mm mesh was selected for the subsequent numerical analysis, with 174,223 elements and 376,982 nodes. The corresponding maximum von Mises stress and maximum deformation were 382.39 MPa and 5.325 mm, respectively.

The study used the combination of element models SOLID186 (Figure 7) and SOLID187 (Figure 8). In complex locations, angled corners use the SOLID187 element model for better geometric accuracy. The SOLID186 element model for straight bars is used to speed up computation. The mesh parameters of the three-axle truck chassis, including the number of elements, nodes, and mesh quality indicators, are summarized in Table 2. The mechanical properties of 40Cr steel used for the chassis, including density, elastic modulus, Poisson’s ratio, yield strength, and ultimate tensile strength, are presented in Table 3.

Figure 7. SOLID186 element model.

Figure 8. SOLID187 element model.

Research by Teo Han Fui [17] and Monika S. Agrawal [18] has proposed a method for static strength testing of truck frames and a corresponding approach for defining boundary conditions. The leaf spring mounts serve to constrain frame displacement in the three translational directions (x, y, z) and rotational directions; therefore, fixed supports are applied at these mounting locations. In the present finite element model, the suspension mounting locations were simplified as fixed constraints rather than explicitly modeling the suspension components and their stiffness characteristics. Thus, the translational and rotational degrees of freedom at these mounting locations were constrained.

We reduce the forces acting on the vehicle frame to specific locations based on the respective force groups. In reality, the frame is subjected to numerous forces acting in complex longitudinal and transverse directions; for the sake of simplicity, we categorize them into the following primary groups of forces:

Cabin assembly: Comprising the engine, transmission, and cabin, this unit exerts a vertical load on the chassis evenly distributed across both sides with a weight of 81,138 N (Figure 9).

Cargo unit: The total mass of the cargo box and its mounted components; when fully loaded, this assembly exerts force on the entire rear section of the vehicle body. The weight of the cargo box is evenly distributed across the surfaces of the rear frame rails on both sides, with a load value of 189,323 N (Figure 9).

Figure 9. Boundary conditions and loads.

3.1. Static Structural Analysis

Before the dynamic and fatigue assessments, the structural integrity and load-bearing capacity of the three-axle truck chassis were evaluated under maximum static loads using ANSYS static structural analysis. The present analysis aims to determine the critical stress distribution, stress concentration zones, and the maximum deformation of the chassis structure. The resulting static stress levels provide a fundamental baseline for assessing whether the structural material remains within its allowable yield limits and are a necessary input for subsequent modal and dynamic fatigue evaluations.

Figure 10 illustrates the deformation of the chassis under maximum static load. Analysis shows that the maximum deformation reaches 5.325 mm at the rear of the vehicle due to deflection under the maximum load. This deformation is relatively small compared to the overall length of the 3-axle chassis, ensuring static rigidity.

Figure 10. Deformation of the chassis.

Figure 11 and Figure 12 illustrate the strain and stress of the chassis under maximum static load. Analysis results show that the maximum von Mises stress is 382.39 MPa and the maximum elastic strain is 0.001949 mm/mm at the rear suspension mounting area, which is the weakest point in the vehicle’s chassis structure. The concentration of stress at the rear suspension mounting region can be physically attributed to the load-transfer characteristics of the chassis. The suspension mounting points provide the main load-transfer path between the road-induced axle forces and the longitudinal frame members. Consequently, these regions simultaneously experience vertical reaction forces, local bending, and torsional effects generated by the uneven distribution of loads between the axle and the two longitudinal members. In addition, the presence of suspension hangers, cross-member connections, bolted joints, and changes in local cross-sectional geometry introduces local stiffness discontinuities. These geometric and load-path discontinuities promote stress concentration and make the rear suspension mounting region more critical than the relatively uniform portions of the longitudinal members. This interpretation is consistent with previous numerical studies of truck chassis structures, which have reported that stress concentrations commonly occur near suspension mounting and cross-member connection regions [9,11,15].

Figure 11. Strain of the chassis.

Figure 12. Stress of the chassis.

The maximum deformation, equivalent elastic strain, and von Mises stress of the chassis under maximum static loading are summarized in Table 4.

3.2. Modal Analysis of the Chassis

Modal analysis was performed using the finite element model developed in ANSYS Mechanical to determine the natural frequencies and corresponding mode shapes of the three-axle truck chassis. The modal characteristics were obtained by solving the undamped eigenvalue problem expressed as [16,17,18]:

The same structural constraints as those used in the static structural analysis were retained for the modal analysis, with the leaf-spring mounting locations constrained to represent the chassis–suspension connections.

The structural mass of the chassis was determined from the finite element model using the density of the 40 Cr material, with ρ = 7850 kg/m³. An undamped modal analysis was performed; therefore, damping was not included in the eigenvalue extraction. The extracted eigenvectors were normalized according to the modal normalization procedure used in ANSYS. Therefore, the deformation values displayed for the individual modes represent normalized relative modal amplitudes for visualization of the corresponding mode shapes and should not be interpreted as actual physical displacement amplitudes under operational loading. The modal results are therefore used to identify the natural frequencies and compare the relative deformation patterns of the different vibration modes.

Table 5 provides the results of the analysis of the mode of vibration and mode of the chassis’s vibration by Ansys software.

The first natural frequency is 17.882 Hz, with a maximum modal displacement of 3.4786 mm. At mode 2 (31.759 Hz, Figure 13), the structure predominantly exhibits a combination of global bending and torsional distortion, yielding a localized relative deformation of 2.4030 mm. Between Modes 3 and 7 (54.209Hz to 64.404 Hz), the modes form complex coupled bending-torsion modes. This is evident from large modal displacements of 11.0610 mm and 12.7880 mm for Modes 4 (58.849 Hz) and 5 (62.387 Hz), respectively. This indicates higher structural vulnerability to dynamic excitation in this mid-frequency range. Mode 6 (63.545 Hz, Figure 14) is a localized bending mode with a relative deformation of 2.2708 mm. At higher frequencies (Modes 8–10, ranging from 78.633 Hz to 87.278 Hz), local structural resonances become predominant. The largest relative displacements are localized at the rear crossmembers and front chassis overhangs, reaching up to 9.0127 mm as shown in Figure 15 (Mode 8, 78.633 Hz) and Figure 16 (Mode 10, 87.278 Hz). The modal results further support the identification of these regions as structurally important. The coupled bending and torsional modes observed in the intermediate-frequency range indicate that the chassis does not deform purely in global bending; instead, local structural regions near the cross-members and suspension interfaces participate significantly in the vibration response. In particular, the rear cross-member and suspension-support regions are subjected to repeated relative deformation under dynamic excitation. Therefore, these regions are more susceptible to cyclic stress accumulation than regions undergoing mainly rigid-body-like motion. This provides a physical link between the modal response and the subsequent fatigue results under random road excitation.

Figure 13. Natural oscillation at frequency 31.759 Hz.

Figure 14. Natural oscillation at frequency 63.545 Hz.

Figure 15. Natural oscillation at frequency 78.633 Hz.

Figure 16. Natural oscillation at frequency 87.278 Hz.

3.3. Access the Fatigue Strength of Chassis

The dynamic load values placed on the chassis are taken based on the moving vehicle model [19]. In this study, the fatigue strength of the chassis is investigated when vehicles are moving on D and E roads based on ISO 8608:2016 [12].

3.3.1. Fatigue Criteria

In this study, the fatigue strength of the three-axle chassis is evaluated using the Goodman, Gerber, and Soderberg mean-stress correction criteria. These criteria are used to account for the effect of mean stress on the fatigue strength of the chassis under cyclic loading. The alternating stress and mean stress are calculated from the maximum and minimum stresses as follows [16]:

Figure 17 shows the isolife curves at the fatigue limit according to 3 criteria. The values of the 3 criteria are determined as follows [16,20]:

(1) $\mathrm{Soderberg’s\; criterion:}\frac{{\sigma }_a}{S_e}+\frac{{\sigma }_m}{S_y}=1$

(2) $\mathrm{Goodman’s\; criterion:}\frac{{\sigma }_a}{S_e}+\frac{{\sigma }_m}{S_u}=1$

(3) $\mathrm{Gerber’s\; criterion:}\frac{{\sigma }_a}{S_e}+{\left(\frac{{\sigma }_m}{S_u}\right)}^2=1$

Figure 17. Graph of fatigue limit lines.

The top three lines represent the fatigue strength limit, and they all share the same features: the fatigue failure zone lies to the right of the limit line after a specific number of impact cycles, and the safe zone lies to the left of the limit line (no fatigue failure). Any point on these lines represents the limiting values of stress, where S_a is the limit value of σ_a, and S_m is the limit value of σ_m.

Assuming that n is a safety factor, we have S_a = n.σ_a and S_m = n.σ_m. The conditions are expressed as follows [16]:

(4) $\mathrm{Soderberg’s\; criterion:}\frac{{\sigma }_a}{S_e}+\frac{{\sigma }_m}{S_y}=\frac{1}{n}$

(5) $\mathrm{Goodman’s\; criterion:}\frac{{\sigma }_a}{S_e}+\frac{{\sigma }_m}{S_u}=\frac{1}{n}$

(6) $\mathrm{Gerber’s\; criterion:}\frac{n{\sigma }_a}{S_e}+{\left(\frac{n{\sigma }_m}{S_u}\right)}^2=1$

3.3.2. Road Model according to ISO 8608:2016

ISO 8608:2016 [12] is the most commonly used pavement standards in automobile dynamics calculations. This standard distinguishes the linearly according to the density of the energy spectrum (PSD) and divides them into 8 categories with conventional symbols A through H. In this study, two types of roads according to ISO 8608:2016 have been tested, including D (bad) and E (very bad) curves, which were tested. These are also the actual types of roads that small trucks regularly operate. Road bumpers are placed evenly on 2 wheel tracks. Figure 18 and Figure 19 show the road surface profile over a length of 250 m. The vehicle is moving straight an evaluation speed of 20 km/h, additionally, this is the speed at which the car typically travels on the road.

Figure 18. Street surface imperfections D (v = 20 km/h).

Figure 19. Street surface imperfections E (v = 20 km/h).

3.3.3. Assess Fatigue Strength When Vehicles Are Moving on the D Road

Figure 20, Figure 21 and Figure 22 describe the total amount of fatigue life that appear within the chassis as the vehicle travels along the D road according to Goodman, Gerber, Soderberg standards, the findings indicate that the maximum fatigue life according to the three criteria above are all 106 cycles, minimum fatigue life under standard Goodman and Soderberg is 8338.5 standard cycle, the fewest cycle count of tiredness under standard Gerber is 8290.4, the damaged chassis due to fatigue is about 2074 km. Figure 23, Figure 24 and Figure 25 describe the fatigue damage appearing on the chassis when the car moves on the D road according to Goodman, Gerber, Soderberg standards, the results show that the smallest fatigue damages according to the above three standards is equal to 1000, the largest fatigue damages according to Goodman standard and Soderberg standard is 1.1993 × 10⁵, the largest fatigue damages according to Gerber standard is 1.2062 × 10⁵.

Figure 20. Predicted fatigue life of the chassis using the Goodman criterion [cycles] on class D.

Figure 21. Predicted fatigue life of the chassis using the Gerber criterion [cycles] on class D.

Figure 22. Predicted fatigue life of the chassis using the Soderberg criterion [cycles] on class D.

Figure 23. Predicted fatigue damage of the chassis using the Goodman criterion on class D.

Figure 24. Predicted fatigue damage of the chassis using the Gerber criterion on class D.

Figure 25. Predicted fatigue damage of the chassis using the Soderberg criterion on class D.

Table 6 shows results obtained from Fatigue analysis D road in three fatigue theories, the results of the minimum number of fatigue life, and the maximum fatigue damage to Goodman and Soderberg standards are the same. The smallest number of fatigue life of Gerber standard is smaller than that of Goodman standard, the largest fatigue damage to Gerber standard is larger than Goodman standard.

Table 6.

Results for Fatigue analysis D road.

Theory	Fatigue Life		Fatigue Damage
	Max	Min	Max	Min
Goodman	106	8338.5	1.1993 × 105	1000
Gerber	106	8290.4	1.2062 × 105	1000
Soderberg	106	8338.5	1.1993 × 105	1000

3.3.4. Assess Fatigue Strength When Vehicles are Moving on the E Road

As the automobile travels along the E road, the dynamic load acting the chassis has larger than as it travels along the D road, in this study also determines the number of cycles and fatigue damage as in the case of the automobile traveling through D road. The findings (see Figure 26, Figure 27, Figure 28, Figure 29, Figure 30 and Figure 31) indicate that the maximum fatigue life according to the above three standards is equal to 10⁶ cycles, the minimum number of fatigue life according to Goodman standard and Soderberg standard is 6349.9 cycles, the smallest amount of fatigue life as Gerber standard is 6306.2, the damaged chassis due to fatigue is about 1557 km. The smallest fatigue damages according to the above three standards is equal to 1000, the largest fatigue damages according to Goodman standard and Soderberg standard is 1.5748 × 10⁵, the largest fatigue damages according to Gerber standard is 1.5857 × 10⁵.

Figure 26. Predicted fatigue life of the chassis using the Goodman criterion [cycles] on class E.

Figure 27. Predicted fatigue damage of the chassis using the Goodman criterion on class E.

Figure 28. Predicted fatigue life of the chassis using the Gerber criterion [cycles] on class E.

Figure 29. Predicted fatigue damage of the chassis using the Gerber criterion on class E.

Figure 30. Predicted fatigue life of the chassis using the Soderberg criterion [cycles] on class E.

Figure 31. Predicted fatigue damage of the chassis using the Soderberg criterion on class E.

Table 7 shows results obtained fatigue analysis E road in three fatigue theories, if the automobile keeps moving on the E road, the amount of damage is greater, the smallest number of endurance cycles is smaller than if the vehicle is traveling while on the D road.

Table 7.

Results for Fatigue analysis E road.

Theory	Fatigue Life		Fatigue Damage
	Max	Min	Max	Min
Goodman	106	6349.9	1.5748 × 105	1000
Gerber	106	6306.2	1.5857 × 105	1000
Soderberg	106	6349.9	1.5748 × 105	1000

Overall, the results indicate that road roughness has a stronger influence on the predicted fatigue life than the selection among the three mean-stress correction criteria for the present chassis and loading conditions. Increasing the road class from D to E reduces the minimum fatigue life from 8338.5 to 6349.9 cycles for the Goodman and Soderberg criteria, corresponding to a reduction of approximately 23.8%. A similar reduction is obtained with the Gerber criterion. This behavior is physically reasonable because the Class E profile generates more severe dynamic excitation and consequently larger cyclic stresses in the chassis. In contrast, the differences among Goodman, Gerber, and Soderberg remain below 1% for the minimum predicted fatigue life. Therefore, for the present stress state, the road-induced dynamic loading is the dominant factor governing the fatigue-life prediction, whereas the choice of mean-stress correction criterion has a comparatively minor effect.