Fillet Radius Variation and Stress in Thin Gear Rims
In an aircraft reduction gearbox, the tooth root fillet is a small geometric feature with a large structural influence. Its radius governs how smoothly load flows from the tooth into the rim, while the rim itself may be thin enough to deform, ovalise, or amplify local bending. A radius selected from a standard gear chart can therefore behave differently when the surrounding gear body has limited thickness.
For geared aircraft engines, this interaction matters at high rotational speed, where low mass, compact packaging, lubrication performance, and fatigue life must be balanced. The OPTIMIZE project’s combination of design of experiments, simulation, tolerance analysis, and physical testing provides a useful framework for studying how manufacturing variation in the root blend changes the stress concentration factor.
| Design condition | Typical stress behaviour | Main engineering concern | Useful evaluation method |
|---|---|---|---|
| Small root fillet radius | Sharp stress gradient and high peak stress | Bending fatigue crack initiation | Refined finite-element model |
| Larger radius with adequate clearance | Lower local concentration and smoother load transfer | Possible tooth interference or reduced manufacturing access | Parametric geometry study |
| Thin rim with small radius | Tooth and rim stresses interact strongly | Rim flexibility and local distortion | Coupled tooth-rim FEA |
| Variable radius along the face width | Non-uniform stress distribution | Localised fatigue damage | 3D analysis and inspection data |
| Radius near the manufacturing limit | Performance may depend on tool wear and process scatter | Production consistency | Tolerance analysis and sample testing |
Why Fillet Geometry Matters in a Thin Rim
The tooth root is already a natural stress raiser because the involute flank transitions into the dedendum and then into the gear body. Under transmitted torque, the tooth behaves broadly like a cantilever. Bending stress reaches a maximum near the root, and the fillet radius controls the sharpness of the transition. A smaller radius generally increases the local stress concentration factor, commonly written as (K_t), because the load path changes over a shorter distance.
A thin rim changes the problem from a relatively rigid tooth-body system into a coupled structure. Instead of transferring most of the bending load directly into a thick gear blank, the tooth may cause the rim to flex between adjacent teeth. The rim can develop circumferential bending and radial deformation, which alter the position and size of the peak stress. In some geometries, increasing the fillet radius reduces tooth-root stress while moving a portion of the critical stress into the rim.
This is particularly important for aircraft gearboxes, where weight reduction may lead to lower rim thickness and higher power density. A gear that performs well in a conventional solid-body calculation may show a different stress field after the rim is lightened. The design must therefore consider tooth-root bending, rim bending, contact loading, and local deformation together rather than treating the fillet as an isolated detail.
Australian aerospace organisations face a similar need to validate compact drivetrain designs under demanding operating conditions. A gearbox developed in Adelaide, Melbourne, or Sydney may combine locally produced components with specialised overseas materials, coatings, and inspection equipment. That supply chain makes robust geometric tolerances valuable because a small change in root blending can influence fatigue performance long before the gear reaches a flight-test programme.
How Radius Changes the Stress Concentration Factor
The relationship between fillet radius and stress concentration is not linear across every design range. When the radius is very small, a modest increase can produce a substantial reduction in peak stress because the notch becomes less severe. Once the transition is already smooth, further increases may deliver diminishing returns. At that point, the limiting factors may become rim thickness, tooth thickness, material strength, or contact fatigue rather than the root blend itself.
A simplified interpretation is that the local bending stress can be estimated by multiplying a nominal tooth-root stress by a geometry-dependent factor. In practice, the factor includes more than radius. Pressure angle, tooth size, profile shift, fillet form, face width, rim thickness, and load distribution all contribute. The stress concentration factor from a two-dimensional section may also differ from the effective fatigue stress factor used in design because surface finish, residual stress, notch sensitivity, and material condition affect crack initiation.
For a thin rim, radius variation can change the location of the maximum principal stress. With a small root blend, the peak may sit close to the dedendum transition. With a larger blend, the peak can spread along the fillet or shift towards the rim-side region. This shift matters when comparing designs because a lower maximum value does not automatically mean lower fatigue risk if the highly stressed zone becomes broader or more difficult to inspect.
The available material behind the tooth is another major variable. Research on gear backup ratio shows why rim support and tooth geometry should be assessed together. A generous fillet on a poorly supported thin rim may provide less benefit than expected, while an optimised backup ratio can improve load sharing and reduce sensitivity to local radius changes.
Using Simulation and Design Of Experiments
A practical investigation begins with a parameterised gear model. The root fillet radius can be varied alongside rim thickness, tooth thickness, profile shift, face width, and applied torque. Rather than running only a best-case and worst-case model, a design-of-experiments approach identifies interactions between variables. This can reveal, for example, that radius has a strong effect when the rim is thin but a minor effect when the gear body is sufficiently rigid.
Finite-element modelling should use adequate mesh refinement in the root region. Coarse elements can smooth the curvature artificially and underpredict the local peak. A convergence study is needed to distinguish a genuine geometric effect from a numerical artefact. Three-dimensional models are especially useful when the radius varies across the face width, when the gear has web openings, or when misalignment creates uneven tooth loading.
The model should report more than a single maximum stress. Useful outputs include the stress concentration factor, maximum principal stress, bending stress at defined reference points, rim deformation, tooth deflection, and the size of the high-stress region. Plotting these outputs against fillet radius helps designers identify a practical region where additional radius produces limited structural benefit but may create manufacturing or packaging problems.
The OPTIMIZE project’s members area offers a relevant route to broader project information and research context. For an engineering team, the value of such a resource is greatest when simulation findings are connected with test planning, design constraints, and the wider objective of reducing gearbox power loss without sacrificing durability.
Managing Manufacturing Variation And Inspection
The nominal radius in a CAD model is rarely the exact radius produced on every tooth. Cutter geometry, tool wear, machine accuracy, heat treatment distortion, finishing operations, and measurement uncertainty can all change the final root form. A nominal 0.80 millimetre radius may become a distribution of slightly different profiles across a production batch. For a thin-rim gear operating close to its fatigue limit, that distribution should be included in the structural assessment.
Tolerance analysis can treat the root radius as a random or bounded variable, depending on the available production data. A sensitivity study may show that radius variation has a stronger effect than pitch error or a smaller effect than rim thickness scatter. The result should guide inspection effort. If the root radius is a dominant contributor to stress variation, measurement systems must capture the actual blend form rather than relying only on general gear inspection results.
Manufacturing decisions also involve practical access. A larger fillet may require a different cutter, reduce tool clearance, or interfere with adjacent geometric features. A small radius may be easy to generate but difficult to finish consistently, particularly in high-strength aerospace gear steels. Australian operators supplying mining, defence, or aviation markets may also need to coordinate production across regional suppliers, where equipment capability and inspection practice are not identical.
Digital equipment used for metrology, data processing, and production monitoring is part of the same engineering chain. Procurement teams evaluating laboratory or workshop hardware can use guidance on refurbished electronics when selecting lower-cost computing equipment, provided that performance, cybersecurity, calibration, and support requirements are checked carefully. The central principle is traceability: the geometry used in the stress model must correspond to the geometry actually manufactured.
Translating Results Into Design Decisions
A useful design decision does not simply select the largest possible root radius. It balances stress reduction against tooth spacing, contact geometry, rim thickness, manufacturability, inspection capability, and mass. In a geared aircraft engine, every increase in radius can affect the dedendum shape and may reduce clearance for mating teeth. The preferred value is usually a robust range rather than a single theoretically optimal point.
The following checks help organise a radius-variation study:
- Compare nominal, minimum, and maximum manufactured fillet profiles.
- Track both tooth-root stress and thin-rim stress in every model.
- Confirm mesh convergence at the smallest and largest radii.
- Include torque variation, load sharing, and shaft misalignment.
- Assess whether the high-stress region becomes wider after radius growth.
- Relate predicted stress to material fatigue data and surface condition.
Testing should then be designed to discriminate between competing predictions. Strain gauges, digital image correlation, tooth-root replicas, or carefully located inspection methods can help confirm whether the critical region lies in the fillet or the adjacent rim. Spin testing is particularly valuable because centrifugal loading, thermal growth, lubrication behaviour, and dynamic effects may alter the static finite-element picture.
A focused validation programme can use gears with several controlled radius conditions while holding the remaining geometry as constant as possible. This is more informative than testing a single production component because it isolates the influence of the root blend. If destructive fatigue testing is impractical, subcomponent tests and correlated strain measurements can still establish whether the model captures the relative ranking of designs.
For Australian programmes, environmental conditions may also deserve attention. A gearbox intended for remote operations in Western Australia can encounter dust-control and maintenance constraints, while an engine supporting coastal operations may face different storage and corrosion conditions. These factors do not directly define (K_t), but they can influence surface damage, lubrication quality, inspection intervals, and the fatigue margin available to a thin-rim gear.
The engineering workflow can be summarised as follows:
- Define radius, rim thickness, backup ratio, and load parameters.
- Generate a design-of-experiments matrix rather than isolated cases.
- Run refined 2D and 3D finite-element analyses.
- Apply production tolerances to the most sensitive variables.
- Validate stress trends with physical measurements.
- Select a radius range that supports reliable manufacture and inspection.
For gearbox designers, the key outcome is a defensible link between geometry and service performance. A root fillet radius should be specified with awareness of its actual production distribution, its interaction with a flexible rim, and its effect on both local stress concentration and fatigue reliability. Access the OPTIMIZE project resources to connect this analysis with broader methods for improving aircraft gearbox efficiency, durability, and power density.