In the field of internal combustion engines, the reliability of bolted connections is paramount, especially in critical components like the gear shaft assembly. As an engineer specializing in engine design and failure analysis, I recently investigated a case where a bolt used to secure an intermediate gear shaft fractured during endurance testing. This gear shaft bolt, with specifications including a length of 180 mm and a grade of M22×2.5 10.9, was tightened using a two-step process: pre-tightening to 200 Nm followed by a 180° rotation to yield. The fracture occurred at the first engaged thread, prompting a comprehensive analysis to determine the root cause. Through this investigation, I focused on evaluating the bolt’s relative plastic deformation—a key factor often overlooked in such failures. By applying standards like VDI 2230 and conducting material and fractographic examinations, I identified that excessive relative plastic deformation was the primary driver of the fatigue fracture. This article details my analytical approach, incorporating formulas and tables to summarize the findings, and emphasizes the importance of design considerations for gear shaft bolts in high-stress environments.
The gear shaft in question is part of a large engine system, where it transmits torque and maintains alignment within the gear train. The bolt connects the intermediate gear shaft to the gear housing and engine block, as illustrated in the simplified diagram. This connection relies on the bolt’s axial force to generate friction that counteracts radial loads from the gear operation. The bolt’s dimensions include a shank length of 124 mm, a shank diameter of approximately 22 mm, and a threaded section with an engaged length of 30 mm and an unengaged length of 26 mm. The materials involved are 42CrMo for the bolt and gear shaft, with elastic moduli of 208 GPa, while the gear housing and engine block are made of HT250 and RT450, with moduli of 117 GPa and 144 GPa, respectively. The tightening process, which involves yielding the bolt, is critical because it induces plastic deformation in the bolt’s weakest region, typically the unengaged threads. In this case, the short unengaged thread length raised concerns about the relative plastic deformation, which I calculated to be significantly high, leading to the fracture.
My analysis began with a thorough examination of the bolt material and fracture surface. The bolt, manufactured from 42CrMo steel and heat-treated to achieve a 10.9 grade, underwent metallographic inspection, which revealed a uniform microstructure of tempered sorbite with minimal ferrite content. No inclusions, original cracks, or decarburization were observed, indicating that the material itself met the required standards. The hardness was measured at approximately 36.0 HRC, falling within the specified range of 32–39 HRC for 10.9-grade bolts. The threading process involved rolling, which enhances fatigue strength by preserving material flow lines and avoiding stress concentrations from machining. The fracture surface exhibited clear beach marks, characteristic of fatigue failure, with the fatigue origin located at the root of the first engaged thread. This suggested that the failure was not due to material defects or manufacturing issues but rather to operational stresses. As I delved deeper, I considered factors like hydrogen embrittlement and raw material flaws, but these were ruled out based on the evidence. Thus, my focus shifted to the bolt’s loading conditions and plastic deformation behavior.
To quantify the plastic deformation, I applied principles from VDI 2230, a standard for high-strength bolted connections. The key parameter here is the relative plastic deformation, defined as the ratio of the plastic deformation amount to the length over which plastic deformation occurs. For this gear shaft bolt, the unengaged thread length was only 26 mm, which limited the region available for plastic deformation. I calculated the bolt’s stiffness and that of the connected components, as these influence the deformation under load. The bolt stiffness, \( K_{\text{bolt}} \), is the inverse of the bolt compliance, \( \delta \), which I derived by summing the compliances of different sections: the bolt head, shank, unengaged threads, and engaged threads. The formulas used are as follows:
Bolt head compliance: $$ \delta_{sk} = \frac{0.5 \cdot d}{E \cdot A_N} $$
Shank compliance: $$ \delta_1 = \frac{L_1}{E \cdot A_1} $$
Unengaged thread compliance: $$ \delta_{\text{Gew}} = \frac{L_{\text{Gew}}}{E \cdot A_{d3}} $$
Engaged thread compliance: $$ \delta_{\text{GM}} = \frac{1}{E} \cdot \frac{0.5 \cdot d}{A_{d3}} + \frac{1}{E_{\text{BI}}} \cdot \frac{0.33 \cdot d}{A_N} $$
Where \( E \) is the elastic modulus of the bolt (208 GPa), \( d \) is the nominal diameter (22 mm), \( A_N \) is the cross-sectional area based on the nominal diameter, \( L_1 \) is the shank length (124 mm), \( A_1 \) is the shank cross-sectional area, \( L_{\text{Gew}} \) is the unengaged thread length (26 mm), \( A_{d3} \) is the cross-sectional area based on the minor diameter (18.933 mm), and \( E_{\text{BI}} \) is the elastic modulus of the internal threads (assumed equal to the gear shaft material, 208 GPa). Using these, I computed the total bolt compliance and stiffness:
$$ \delta = \delta_{sk} + \delta_1 + \delta_{\text{Gew}} + \delta_{\text{GM}} = 2.47 \times 10^{-6} \, \text{mm/N} $$
$$ K_{\text{bolt}} = \frac{1}{\delta} = 4.05 \times 10^5 \, \text{N/mm} $$
For the connected components (gear shaft and housing), I used finite element analysis to determine the stiffness. Applying a 3000 N axial force to the area equivalent to the bolt head contact, the deformation was 9.61 × 10^{-4} mm, yielding a stiffness of:
$$ K_{\text{shaft}} = \frac{3000}{9.61 \times 10^{-4}} = 3.12 \times 10^6 \, \text{N/mm} $$
Next, I analyzed the tightening process to estimate the plastic deformation. The bolt yield strength, \( R_{p0.2} \), ranges from 940 MPa to 1070 MPa, and the friction coefficient, \( \mu \), varies between 0.08 and 0.14. I considered extreme cases: maximum plastic deformation occurs with the lowest yield strength (940 MPa) and lowest friction (0.08), and minimum with the highest yield strength (1070 MPa) and highest friction (0.14). The tightening involves a pre-torque of 200 Nm ± 20 Nm and a rotation of 180° ± 10°. The axial force during pre-tightening is calculated as:
$$ F_{M1} = \frac{M}{0.16P + \mu \cdot (0.58 \cdot d_2 + 0.5 \cdot D_{km})} $$
Where \( M \) is the torque, \( P \) is the pitch (2.5 mm), \( d_2 \) is the pitch diameter (20.376 mm), and \( D_{km} \) is the average friction diameter (32.4 mm). For the maximum deformation scenario (torque of 220 Nm, friction 0.08):
$$ F_{M1} = \frac{220}{0.16 \times 2.5 + 0.08 \times (0.58 \times 20.376 + 0.5 \times 32.4)} = 83287.7 \, \text{N} $$
The elastic deformation at this stage is:
$$ \Delta s_{\text{bolt1}} = \frac{F_{M1}}{K_{\text{bolt}}} = \frac{83287.7}{4.05 \times 10^5} = 0.206 \, \text{mm} $$
During the rotation phase, the bolt is tightened to yield, with an equivalent stress of approximately 1.05 times the yield strength. The reduced stress, considering shear from torsion, is derived using:
$$ \frac{\tau_M}{\sigma_M} \approx \frac{3 \cdot d_2}{2 \cdot d_0} \left( \frac{P}{\pi \cdot d_2} + 1.155 \mu \right) $$
Where \( d_0 \) is the stress diameter (19.654 mm for non-reduced shank bolts). Solving for \( \tau_M \) and \( \sigma_M \) using the von Mises criterion:
$$ \sigma_{\text{red}} = \sqrt{\sigma_M^2 + 3\tau_M^2} = 987 \, \text{MPa} $$
I found \( \sigma_M = 930.41 \, \text{MPa} \) and \( \tau_M = 190.18 \, \text{MPa} \), giving a final axial force of:
$$ F_{M2} = \frac{\sigma_M \cdot \pi \cdot d_0^2}{4} = 282281.76 \, \text{N} $$
The total deformation from rotation is:
$$ \Delta s_{\text{total2}} = 190^\circ \times \frac{2.5}{360} = 1.319 \, \text{mm} $$
The deformation of the connected components is:
$$ \Delta s_{\text{shaft2}} = \frac{F_{M2} – F_{M1}}{K_{\text{shaft}}} = \frac{282281.76 – 83287.7}{3.12 \times 10^6} = 0.064 \, \text{mm} $$
Thus, the bolt deformation during rotation is:
$$ \Delta s_{\text{bolt2}} = \Delta s_{\text{total2}} – \Delta s_{\text{shaft2}} = 1.256 \, \text{mm} $$
The total bolt deformation is:
$$ \Delta s_{\text{bolt total}} = \Delta s_{\text{bolt1}} + \Delta s_{\text{bolt2}} = 1.462 \, \text{mm} $$
The maximum elastic deformation before yield is:
$$ \Delta s_{\text{bolt elastic}} = \frac{R_{p0.2} \cdot \pi \cdot d_0^2}{4 \cdot K_{\text{bolt}}} = \frac{940 \times \pi \times (19.654)^2 / 4}{4.05 \times 10^5} = 0.705 \, \text{mm} $$
Therefore, the plastic deformation is:
$$ \Delta s_{\text{bolt plastic}} = \Delta s_{\text{bolt total}} – \Delta s_{\text{bolt elastic}} = 0.757 \, \text{mm} $$
With an unengaged thread length of 26 mm, the relative plastic deformation is:
$$ P_d = \frac{\Delta s_{\text{bolt plastic}}}{L_{\text{plastic}}} = \frac{0.757}{26} = 2.91\% $$
Similarly, for the minimum deformation scenario (yield strength 1070 MPa, friction 0.14, torque 180 Nm, rotation 170°), I calculated a relative plastic deformation of 1.53%. The average value of 2.22% far exceeds the recommended limit of 0.6% for safe operation, indicating a high risk of fracture. To address this, I proposed changing the bolt to a full-threaded design, increasing the unengaged length to 150 mm, which reduces the relative plastic deformation to approximately 0.5%, as estimated from similar calculations.
The following table summarizes the key parameters and results for the gear shaft bolt analysis:
| Parameter | Value | Unit |
|---|---|---|
| Nominal Diameter, \( d \) | 22 | mm |
| Pitch, \( P \) | 2.5 | mm |
| Pitch Diameter, \( d_2 \) | 20.376 | mm |
| Minor Diameter, \( d_3 \) | 18.933 | mm |
| Stress Diameter, \( d_0 \) | 19.654 | mm |
| Bolt Length | 180 | mm |
| Unengaged Thread Length | 26 | mm |
| Elastic Modulus, \( E \) | 208 | GPa |
| Yield Strength, \( R_{p0.2} \) | 940–1070 | MPa |
| Friction Coefficient, \( \mu \) | 0.08–0.14 | – |
| Bolt Stiffness, \( K_{\text{bolt}} \) | 4.05 × 10^5 | N/mm |
| Shaft Stiffness, \( K_{\text{shaft}} \) | 3.12 × 10^6 | N/mm |
| Max Relative Plastic Deformation | 2.91 | % |
| Min Relative Plastic Deformation | 1.53 | % |
| Average Relative Plastic Deformation | 2.22 | % |
In conclusion, my investigation into the gear shaft bolt fracture highlights the critical role of relative plastic deformation in bolted connections subjected to yield tightening. For bolts not used to yield, this factor may be negligible, but in high-stress applications like gear shaft assemblies, it must be carefully evaluated. By switching to a full-threaded bolt, the plastic deformation zone is extended, significantly reducing the relative deformation and enhancing reliability. This approach aligns with engineering best practices and standards such as VDI 2230, ensuring that gear shaft components operate safely under dynamic loads. Future designs should prioritize this aspect to prevent similar failures and improve overall engine durability.