In the offshore oil and gas industry, jack-up platforms play a critical role in drilling and production operations, particularly in regions like the Bohai Sea, which has become a major crude oil production base. The rack and pinion gear lifting system is a fundamental component of these platforms, responsible for elevating and supporting the entire structure. Given its importance in ensuring operational safety and stability, assessing the integrity of the rack and pinion system is paramount. This study focuses on the safety evaluation and maximum wear analysis of a rack and pinion gear system used in a jack-up platform. Through field inspections, finite element simulations, and theoretical calculations, we aim to determine the system’s current safety status and predict the maximum allowable wear to prevent catastrophic failures. The rack and pinion gear system is subjected to heavy loads and harsh environmental conditions, leading to gradual wear and plastic deformation over time. Understanding the limits of this wear is essential for maintenance planning and risk mitigation.
The rack and pinion gear system in jack-up platforms operates under low-speed, high-load conditions, making it susceptible to bending stress failures at the tooth root. Unlike standard gear systems, the large module size of these components necessitates specialized analysis. In this work, we employ ANSYS software for finite element analysis (FEA) to simulate stress distributions and combine this with theoretical formulas to evaluate safety. We explore three distinct approaches for predicting maximum wear: the failure criterion for heavy-duty gears, the principle of maximum wear based on geometric changes, and the principle of maximum bending stress at the tooth root. By integrating these methods, we provide a comprehensive assessment that can guide inspection protocols and operational decisions for similar rack and pinion systems.
The rack and pinion gear system parameters are critical for accurate analysis. The module of the gear is 135 mm, with a pressure angle of 25°. The rack has a tooth width of 127 mm, while the pinion gear has a tooth width of 206 mm and 7 teeth. The material used is DILLIMAX690E, which has an elastic modulus of 2.06 × 10^5 N/mm², a Poisson’s ratio of 0.3, and a bending strength of at least 630 MPa. These parameters form the basis for our simulations and calculations. The operational loads on the rack and pinion system vary depending on the platform’s conditions, such as normal lifting, emergency operations, pre-loading, and survival scenarios. Each of these conditions imposes different stresses on the gear teeth, which must be accounted for in the safety assessment.
To quantify the working loads, we consider the maximum allowable loads per leg and the number of teeth engaged during each operation. The platform is supported by three legs, each with two rack and pinion gears on either side. The table below summarizes the loads under various operational conditions, which are derived from the platform’s operational manual. These loads are used as inputs for our finite element simulations to determine the bending stresses in the rack and pinion gear system.
| Operational Condition | Maximum Allowable Load per Leg (t) | Number of Engaged Teeth | Load per Tooth (t) |
|---|---|---|---|
| Normal Lifting | 1179.0 | 4 | 294.7 |
| Emergency Lifting 1 | 1300.0 | 4 | 325.0 |
| Emergency Lifting 2 | 1000.0 | 3 | 333.3 |
| Pre-loading | 1514.0 | 4 | 378.5 |
| Operational State 1 | 1489.1 | 4 | 372.3 |
| Operational State 2 | 1347.6 | 4 | 336.9 |
| Survival Condition | 1442.5 | 4 | 360.9 |
| Leg Extraction | 747.2 | 4 | 186.8 |
Field inspections of the rack and pinion gear system revealed varying degrees of wear and plastic deformation across all teeth, with the most severe wear occurring in the 20–40 meter zone of the leg. Detailed measurements were taken at distances of 58 mm, 127 mm, and 187 mm from the tooth tip to assess the actual tooth thickness. For instance, tooth number 13 exhibited the maximum plastic deformation, with measured thicknesses of 128.2 mm, 192.5 mm, and 251.0 mm at these points, indicating a wear of approximately 10 mm when compared to the基准齿形. This data was used to create a 3D model of the worn rack and pinion gear for further analysis, ensuring that our simulations accurately reflect the current state of the system.
The safety evaluation primarily focuses on the bending stress at the tooth root, as this is the critical failure mode for low-speed, heavy-duty rack and pinion gears. The engagement between the rack and pinion gear involves regions of single and double tooth contact, with the maximum bending stress typically occurring at the upper end of the single-tooth engagement zone near the tooth root. To simplify the analysis, we apply the load at the tooth tip in our finite element models and then multiply the resulting stress by the contact ratio coefficient to obtain a more accurate estimate of the maximum bending stress. The contact ratio coefficient, denoted as $$ Y_{\epsilon} $$, is calculated using the formula:
$$ Y_{\epsilon} = 0.25 + \frac{0.75}{\epsilon} $$
where $$ \epsilon $$ is the contact ratio, given by:
$$ \epsilon = \frac{1}{2\pi} \left[ z (\tan \alpha_1 – \tan \alpha) + \frac{4h_a^*}{\sin(2\alpha)} \right] $$
In this equation, $$ z $$ is the number of teeth on the pinion gear, $$ \alpha $$ is the pressure angle, $$ \alpha_1 $$ is the pressure angle at the tip of the pinion gear, and $$ h_a^* $$ is the addendum coefficient. For our rack and pinion gear system, with a pressure angle of 25° and 7 teeth, the contact ratio coefficient remains constant at 0.77 throughout the wear process, as the pressure angles do not change significantly with uniform wear.
Using ANSYS, we performed contact analysis on the 3D model of the rack and pinion gear with a wear amount of 10 mm. The contact pairs were set between the pinion gear tip and the corresponding rack surface, with a bonded contact type. The mesh was generated with a size of 10 mm, consisting of hexahedral elements, and the average mesh quality exceeded 0.96. Constraints were applied by fixing the back of the rack and restricting all degrees of freedom of the pinion gear except for vertical displacement, while applying downward loads corresponding to different operational conditions. The stress distribution from the simulation, such as in the leg extraction condition, showed a bending stress of 347.99 MPa at the tooth root. After multiplying by the contact ratio coefficient, the calculated bending stress was 267.95 MPa. The table below summarizes the bending stresses for all operational conditions, confirming that the current wear state does not exceed the material’s bending strength, indicating the system is safe for now.
| Operational Condition | Average Load per Tooth (t) | Calculated Bending Stress (MPa) |
|---|---|---|
| Normal Lifting | 294.7 | 467.4 |
| Emergency Lifting 1 | 325.0 | 515.9 |
| Emergency Lifting 2 | 333.3 | 529.0 |
| Pre-loading | 378.5 | 603.7 |
| Operational State 1 | 372.3 | 590.6 |
| Operational State 2 | 336.9 | 534.4 |
| Survival Condition | 360.9 | 572.9 |
| Leg Extraction | 186.8 | 268.0 |
For the maximum wear analysis, we first consider the principle based on geometric changes and safety factors. This approach is straightforward and suitable for initial assessments during platform operations without detailed inspections. The maximum wear amount $$ \Delta S $$ is related to the original tooth thickness $$ S_C $$ and the design safety factor $$ n $$ by the formula:
$$ n = \frac{S_C^2}{(S_C – \Delta S)^2} $$
According to ISO standards, the minimum safety factor for gear strength should not be less than 1.25. Substituting the tooth tip thickness of 127 mm into the equation, we solve for $$ \Delta S $$:
$$ 1.25 = \frac{127^2}{(127 – \Delta S)^2} $$
Solving this, we find that the total wear on both sides of the rack and pinion gear should not exceed 21 mm. This provides a quick reference for operational limits, but it must be complemented with more detailed analyses for accurate safety margins.
The second approach involves the principle of maximum bending stress at the tooth root. We developed 3D models of the rack and pinion gear with varying wear amounts from 0 mm to 20 mm in increments of 2 mm. Using ANSYS, we simulated the pre-loading condition, which imposes the highest operational load, to determine the bending stress at the tooth root. The contact settings, mesh parameters, and constraints were consistent with the safety evaluation. The results, as shown in the table below, indicate that as wear increases, the bending stress generally rises due to the reduction in tooth thickness. The material’s bending strength is 630 MPa, so the wear limit is reached when the stress approaches this value. From the data, the bending stress exceeds 630 MPa at a wear amount of 20 mm, but to ensure a safety margin, we consider the maximum allowable wear to be 18 mm, where the stress is 625.2 MPa, slightly below the strength limit.
| Wear Amount (mm) | Calculated Bending Stress at Tooth Root (MPa) |
|---|---|
| 0 | 558.2 |
| 2 | 566.7 |
| 4 | 565.2 |
| 6 | 575.2 |
| 8 | 583.7 |
| 10 | 603.7 |
| 12 | 596.8 |
| 14 | 620.6 |
| 16 | 618.3 |
| 18 | 625.2 |
| 20 | 639.9 |
The third method is based on the failure criterion for heavy-duty gears, as outlined in gear handbooks. The rack and pinion gear system in jack-up platforms is classified as a high-safety requirement device (Category III) because failures could lead to equipment damage or personal injury. For gears operating at speeds below 10 m/s, the failure criterion states that the total wear amount $$ S $$ as a percentage of the module $$ m $$ should not exceed 15%. With a module of 135 mm, this gives:
$$ M = \frac{S}{m} \times 100\% \leq 15\% $$
Solving for $$ S $$:
$$ S \leq 0.15 \times 135 = 20.25 \text{ mm} $$
Thus, the total wear on both sides of the rack and pinion gear should not exceed 20.25 mm according to this criterion. However, this approach provides a general guideline and may not account for specific load conditions or material properties, so it should be used in conjunction with other methods.
In conclusion, our safety evaluation confirms that the current wear state of the rack and pinion gear system, with a maximum wear of 10 mm, is within safe limits under all operational conditions. For maximum wear prediction, the three methods—maximum wear principle, maximum bending stress principle, and heavy-duty gear failure criterion—yield similar results, with allowable wear amounts of 21 mm, 18 mm, and 20.25 mm, respectively. The most conservative estimate comes from the maximum bending stress analysis, which suggests a limit of 18 mm. This is recommended for detailed safety assessments during platform maintenance. The rack and pinion gear system’s integrity is crucial for overall platform safety, and regular inspections combined with finite element simulations can help prevent failures. Future work could explore dynamic loading effects and environmental factors on wear progression in rack and pinion systems.