Methodology to evaluate the bending and contact allowable stress numbers of gears from rotating bending database

Gears must verify a fatigue load capacity concerning tooth bending failure mode at the tooth root and contact pressure or Hertzian fatigue failure mode on tooth flanks (called macropitting). Thus, current standards to evaluate load capacity of gears, such as ISO 6336, AGMA 2101, or DIN 3990, allow to compare the effective applied stress to the permissible stress provided by the tooth material and its associated heat and surface treatment.

On the one hand, ISO 6336-5, AGMA 2101, or DIN 3990 standards give a database of bending and pitting allowable stress for several metallic materials (steels and cast irons) and associated heat treatment for a given number of cycles, evaluated with 1% of failure probability. The given fatigue limits are usually evaluated using gear specimens on a back-to-back test bench (type FZG) for pitting and single tooth bending test (STBF) with a pulsator for bending. Alternatively, they also can be based on industrial experience from the field of application. Gear-testing evaluation represents significant cost and duration, mainly for pitting since the flank loading frequency corresponds to the frequency of rotation of the gear, which is lower than the frequency range that can be found on a pulsator (with hydraulic jack or resonance feature).

On the other hand, fatigue limit characteristics of materials can be evaluated on generic standardized test (for example plane or rotating bending) with standardized testing devices and process where simpler and cheaper specimens than gears are used on simpler and cheaper-to-use means. Outside of the field of gear applications, numerous (standardized) databases for most structural mechanical requirements already exist. On this basis, several fatigue data are already accessible with a larger range of material than those covered in ISO 6336-5, AGMA 2101, or DIN 3990. By this way, it would be easier and cheaper to evaluate fatigue requirement on a new material.

This article presents a methodology which consists of the transposition of standardized generic fatigue characteristics in order to define allowable stress fatigue limit (σ_Hlim and σ_Flim) of gears. After the development of the method, examples are given first in comparison by using it on ISO 6336-5 material fatigue limit database and then with fatigue test data developed at Cetim and NRIM.

Limits and the recommendation for the application of this methodology are also presented. For now, this method is only validated with steels and cast iron.

1 Introduction

1.1 Gears and fatigue failure

Gears must verify a fatigue load capacity concerning tooth bending failure mode at the tooth root and contact pressure or Hertzian fatigue failure mode on tooth flanks (called macropitting), see Figure 1. According to the ISO 10825-1:2022 [1], bending fatigue is a failure mode characterized by a fatigue crack at the tooth root fillet area. Macropitting is a failure mode characterized by a rolling/sliding fatigue crack initiated at the Hertzian depth under the active tooth flank surface. Thus, current standards to evaluate load capacity of gears, such as ISO 6336, AGMA 2101, or DIN 3990, allow to compare the effective applied stress to the permissible stress provided by the tooth material and its associated heat and surface treatment.

1.2 Gear fatigue test methods

ISO 6336-5, AGMA 2101, or DIN 3990-5 standards [2-4] give a database of bending and pitting allowable stress for several metallic material (steel and cast iron) and associated heat treatment for a given number of cycles, evaluated with 1% of failure probability and undefined confidence level. These allowable stresses primarily depend on the material composition, the cleanliness, the residual stresses, the microstructure quality, the heat treatment, and the manufacturing process. All these elements are integrated into fatigue curves provided by the aforementioned standards. For the remainder of this discussion, we will assume that these characteristics are similar for both “gear” and “generic fatigue” specimens. The given fatigue limits are usually evaluated using gear specimens on a rotating gear (RG) back-to-back test bench (type FZG) for pitting and single tooth bending test (STBF) with a pulsator for bending (see Figure 2). Alternatively, they can be also based on industrial experience from the field of application. Gear testing evaluation represents a significative cost and duration, mainly for pitting since the flank loading frequency correspond to frequency of rotation of the gear, which is lower than the frequency range that can be found on pulsator (with hydraulic jack or resonance feature).

The rotating gear test (RG) is a test performed on a rotating gear specimen, which is design such as its first failure mode is (generally) macropitting and not bending fatigue [5]. The advantage of this test is that it considers the gear geometry, manufacturing process, and all the physical effects involved during meshing. The drawback is those tests are long and expensive due to the need of one gear pair for one point of the fatigue SN curve and frequency limited by rotational speed. The load is equivalent to the real application stress ratio
R = stress_min/stress_max. Therefore, the RG test is considered as a method A according to ISO 6336-5:2016 [2] when it is performed directly on the industrial application or on a gear pair of similar size of the RG specimen. This method is expensive but necessary for POC (proof of concept).

The single tooth bending fatigue test (STBF) is a test performed on a single non-rotating gear with a pulsator applying a cyclic load on the fixed point of the tooth [5-7] generally closed to the highest point of single tooth contact. The advantage of this test is that it considers the gear geometry and manufacturing. It is quite fast and several tests can be performed with only one gear. The drawback is the loading point is not moving along the tooth profile, unlike the real gear meshing where the points of contact move from the SAP (point A) to the EAP (point E). Usually, a correction factor f_korr ≈ 0.9 is used to correlate this single-tooth bending test with the rotating gear test [7]. The load is generally applied alternatively with a stress ratio R = 0.1 (close to zero), but other values can be applied. A special mechanical assembly is required to apply for alternating stresses such as on planets of planetary gears or idle gears where R = –1 [8]. This method is generally used to evaluate accurately the Wöhler curve of the material for a gear rating according to ISO, AGMA, or DIN methods. The STBF test is considered as a method B according to ISO6336-5:2016 [2].

1.3 Generic fatigue test methods

Fatigue limit characteristics of materials can be evaluated according to generic fatigue test standards (see ASTM and ISO documents [9-14], (see Figure 3) where simpler and cheaper specimens than gears are used on more conventional test devices at higher frequency than gear rotation, so they are also faster. It results that, out of the field of gears, numerous (standardized) databases for most of structural mechanical solicitations already exists in a large panel of materials. On this basis, several fatigue data are already accessible with a larger range of material than those covered in ISO 6336-5, AGMA 2101, or DIN 3990. So, it can be an easiest and cheapest alternative way to derive a first assessment of fatigue data requirement for a gear on an existing or a new given material.

The plane bending test (PB): This is a test where the cyclic plane bending load is applied to a test specimen that may be ground and notched (see Figure 3a). The notch in the specimen can be standardized but also adjusted to have a notch effect near the reference gear tooth geometry. The load is generally applied by a 4-point configuration to have a constant bending moment in the center area of the bar. The stress ratio is usually R = 0 but other values can be applied, such as R = –1 for example. The specimens must be representative of machining and heat-treatment processes and eventually of a stress concentration factor of the gear, then the fatigue test results can be applied directly. Recently, ARGOUD et al. [6] showed the PB test could provide very similar results to STBF tests as shown in Figure 4.

The rotating bending test (RB): This is a test where a static bending moment is applied on a cylindrical specimen in rotation (see Figure 3b). The test specimen is generally ground and notched with a circular shape. The load is always applied in an alternating way with a stress ratio R = –1. This test method is widely used as it is a very simple, fast, and low-cost test compared to other tests.

The tension/compression test (TC): This is a test where a cyclic tension/compression is applied to a test specimen that is generally ground and notched (see Figure 3c). The load is generally applied alternatively with a stress ratio R = –1 but other values can be applied.

The Torsion test (TOR): This is a test where a cyclic torque is applied on a cylindric specimen that is generally ground and notched (see Figure 3d). The load may be applied alternatively with a stress ratio R = –1 but other values can be applied.

For steels, a relationship between the endurance limits of each a theses tests can be used to convert a type of loading to another [15] (see Equations 1, 2, and 3).

In the 1980s, Cetim, and then NRIM, gave databases to a rotating bending test for steels [15-17]. From both databases, a relationship has been established for 90% of tested steels between the ultimate tensile strength (R_m), and the rotating bending fatigue limit at 10⁷ cycles and 50% probability of failure (Equation 4) . In order to simplify the notation, the symbol σ^*_D,RB will be used. Figure 5 presents this relationship for steels with R_m between 500 MPa and 2,200 MPa. Similar graphics for aluminum alloys and magnesium alloys can be found in reference [15]. The fatigue limit to ultimate tensile strength relationships for other materials also are given in reference [15].

The ultimate strength can be evaluated from the Hardness Vickers, HV, using ISO18265:2013 [18] or by the Equation 5. This standard is valid for hardness below 650HV (see Figure 6).

1.4 Industrial need

Some industrial applications use gear materials not covered in the ISO 6336-5 standard, which are neither steels nor cast iron such as bronze, aluminum, or stainless steels. During a pre-project (pre-design) phase of a gearbox development, it cannot be considered to realize tests as complex as gear fatigue tests. This way, it is needed to identify methods to firstly evaluate macropitting and root bending endurance limit values from generic fatigue databases.

This article is organized as follows. First the methods to estimate the tooth bending and the macropitting fatigue limit from rotating bending tests are described. Second, an example of results is presented and compared to ISO 6336-5 standard experimental results. Lastly, a discussion and a conclusion are given.

2 From rotating bending fatigue test to tooth bending and macropitting strength

2.1 Root bending endurance limit according to ISO 6336-3:2019

2.1.1 Theory of root bending and definitions

Tooth root stress σ_F is the maximum tensile stress at the surface in the root fillet in the area of the critical section of the tooth s_Fn as defined in ISO 6336-3:2019 [19] (see Figure 7). This bending stress is a tensile stress type, tangential to the fillet surface. The stress concentration effect due to the curvature in the fillet is considered by the stress correction factor of the gear test specimen Y_ST according to ISO 6336-5:2016. Usually, Y_ST = 2 (–).

The nominal stress number (bending) s_F,lim is the bending stress limit value relevant to the influences of the material, the heat treatment and the surface roughness of the test gear root fillets. This limit strength is defined in ISO 6336-5:2016 from the material, the heat treatment and the surface hardness as an affine function (Equation 6).

Where A and B are constants given in the ISO 6336-5:2016 for each material and the heat treatment.

The allowable stress number (bending) σ_FE is the basic bending strength of the un-notched test piece, under the assumption that the material condition (including heat treatment) is fully elastic. It is defined for 3×10⁶ loading cycles (see Y_NT in ISO 6336-3), 1% probability of failure and R=0 stress ratio. This value is evaluated from the experimental bending nominal stress number by considering the stress correction factor due to the notch effect Y_ST (see Equation 7).

2.1.2 Hypothesis and development of the method

2.1.2.1 Hypothesis

The allowable bending stress number is a tensile plane bending type of stress. Thus, it is assumed the allowable stress number is approximatively equal to the plane bending stress number σ_D,PB of the material and heat treatment for a given probability of failure, number of cycle, and stress ratio (see Equation 8).This notably implies the metallurgical characteristics mentioned in section 1.2 are similar.

From this assumption, several correction factors must be applied to evaluate the bending nominal stress number from a rotating bending database to consider the differences in terms of number of cycles, stress ratio, notch effect, and probability of failure. All of them are defined from generic fatigue handbook [15].

2.1.2.2 Life correction factor (Bending) Y*_NT

Rotating bending databases are generally defined for an allowable strength at 10⁷ cycles. The bending life correction factor Y*_NT is defined as the ratio between the allowable bending stress number at 3×10⁶ cycles and at 10⁷ cycles (Equation 9)

Where Y_NT is the life factor (bending) as defined in the ISO 6336-3:2019.

It is noted the life factor is similar for all steels and cast irons within the scope of the ISO 6336-3 standard in these orders of magnitude of number of cycles (see Figure 8). Let’s assume that Y_NT (N = 1 ∙ 10⁷) = 0.978 and Y_NT(N = 3 ∙ 10⁶) = 1.0, then Y*_NT = (–).

Note : A lower limit (non-conservative) of the life correction factor is Y*_NT = 1.0(–).

2.1.2.3 Stress ratio effect and the mean stress influence factor, Y_M

The rotating bending applies an R = –1 stress ratio while the gear bending allowable stress number is evaluated for a R = 0 stress ratio. Thus, a correction must be applied that is called the mean stress influence factor Y_M according to ISO 6336-3:2019 and is the ratio between strength for a given stress ratio R and the R=0 (see Equation 10).

Where M is considers the mean stress influence on the endurance (or static) strength amplitudes (see ISO 6336-3:2019). For steels, M ≈ 0.4 ± 0.1 (–) and thus Y_M (R = –1) ≈ 0.7 ± 0.05 (–).

2.1.2.4 Notch effect

A rotating bending fatigue database is generally displayed to an un-notched allowable stress number. Thus, the rotating bending allowable stress can be compared to the gear-allowable stress. However, the ISO 6336-5:2016 gives a formula to calculate the nominal stress number (see Equation 6), which includes the gear-test specimen notch effect as explained in the section 2.1.1. For a direct comparison with the standard, the stress correction factor Y_ST = 2.0 (–) must be applied (Equation 7).

2.1.2.5 Probability of failure /reliability factor K_R

The rotating bending database is given for a probability of failure of 50% while the ISO 6336-5:2016 is given for a probability of failure of 1%. We define the reliability factor K_R as the ratio between the nominal stress number at a probability of failure of 1% and a given probability of failure 50% (see Equation 11)

According to ISO 12107:2012 [14] and assuming a gaussian distribution of endurance limits, Equation 12

is used to convert values from 50% to 1% probability of failure.

Where:

k(p, 1 1 – α, dof) is the one-sided tolerance limit for a normal distribution, as given in Table 2 in the annex.

p is the probability of failure.

1 – α is the confidence level with 0.5 ≤ α ≤ 0.1

dof (also noted v in the ISO 12107:2012) is the degree of freedom.

CV is the coefficient of deviation of the gaussian distribution.

Usually, 20 to 25 tests are used to evaluate the nominal stress number thus, assuming a confidence level of 90%, k(P = 1%, 1 – a, dof) ≈ 3.028 ± 0.144 (–). The coefficient of deviation of steels is generally within CV ≈ 7.5 ± 2.5%. Thus K_{R,ISO 12107:2012} (50%) ≈ 1.29 ± 0.16.

According to the NF E 23-015:1982 [20], the reliability factor K_R is given by Equation 14. Note that in this standard, the reliability coefficient is applied on the load and not on the stress number.

Table 1 presents the list of reliability factors according to AGMA 2001-D04 [3], which is based on first draft of ISO 6336-1980, and at that time it is also considered a reliability factor as the normal statistical distribution of failures found in materials testing is based on 10⁷ cycles tests. This factor is applied on the contact stress and bending stress number.

Figure 9 presents the comparison of these reliability factors. They are very close together, especially when considering the deviation coefficient uncertainty (see error bar at 50% probability of failure point).

2.1.2.6 Final application

Using Equations 4, 8, 9, 10, 7, and 11 altogether, the bending nominal stress number can be evaluated as follows (Equation 15):

Using the previously described evaluation of each factor, the numerical application gives (Equation 16):

2.2 Macropitting endurance limit

2.2.1 Theory of Hertzian contact applied to gears and definitions

The Hertzian contact theory is widely applied in contact mechanics [21] and especially on gears in ISO 6336-2:2019 [22]. The contact between tooth flanks in cylindrical gears can be assimilated as a cylinder-to-cylinder contact along their generatrixes resulting in a linear contact. The forces distributed along the contact line generate a contact pressure, which is calculated by Hertz’ theory. In a cross section to the contact line, the surface contact pressure distribution is parabolic along contact width 2b_H with a maximum pressure P_H,max (see Equations 17 and 18). It is for this value that the gear is rated according to ISO 6336-2:2019 [22] σ_H1,2 = P_H,max. This surface pressure distribution generates shear stresses in the subsurface. In the case of gears, the contact point is moving along the tooth flanks during gear mesh: This is equivalent for a given point under the surface to see changes in the local shear stress. The shear stresses are alternating before and after the contact point and repeated at the center of the contact point (see Figure 10). The semi-band width of contact b_H and the maximum contact pressure P_H,max are given by Equations 17 and 18.

The maximum alternating shear stress τ_{H,R= –1,max} in the subsurface is in relation with the maximum contact pressure P_H,max at the surface. It is at depth of maximum alternating shear stress z_{τH,R=–1,max} and its amount is τ_H,R–1,max (see Equations 19 and 20). This stress is alternating and parallel to the surface. It is the most soliciting stress from the point of view of contact fatigue.

The maximum repeated shear stress τ_H,R=0,max is in relation with the maximum contact pressure at the surface P_H,max. This stress is repeated and oriented at 45° with respect to the surface and at the depth of the maximum repeater shear stress z_τH,R=0,max (see Equations 21 and 22).

Alternating

Repeated

The allowable stress number (contact), σ_H,lim, is derived from the contact pressure that may be sustained for a specified number of cycles (standard number function of the material) without the occurrence of progressive pitting according to the ISO 6336-5:2016. For case carburized through hardened and induction-hardened material, the allowable stress number, 5×10⁷ cycles, is considered to be the beginning for the long life area according to ISO 6336-5:2017.

2.2.2 Hypothesis and development of the method

2.2.2.1 Hypothesis

The contact allowable stress number is a type of solicitation equivalent to shearing under the surface induced by contact pressure and can be derived from a torsion type of stress. Thus, the assumption that the allowable stress number is approximatively derived from the most critical value for the maximum alternated τ_{D,R= –1} and the maximum repeated allowable shear stress number τ_D,R=0 of the material and heat treatment, for a given probability of failure, number of cycle, and stress ratio is assumed (see Equation 8).

As mentioned in the introduction, the allowable alternated shear stress of a steel can be evaluated from the rotating bending test (Equation 3) at 50% probability of failure and 10⁷ cycles. From this assumption, several correction factors must be applied to evaluate the contact nominal stress number from a rotating bending database in order to take into account the differences in terms of the mode of solicitation, number of cycles, stress ratio, notch effect, and probability of failure. All of them are defined from the generic fatigue handbook [15].

2.2.2.2 Life correction factor (Macropitting) Z*_NT

Rotating bending test databases are generally defined for an allowable strength at 10⁷ cycles. The life cycle correction factor (macropitting) Z*_NT is defined as the ratio between the allowable bending stress number at 5×10⁷ cycles and at 10⁷ cycles (Equation 9)

Where Z_NT is the life factor (macropitting) as defined in the ISO 6336-2:2019.

It is noted that the life factor (macropitting) is similar for all steels and cast irons within the scope of the ISO 6336-2 standard in these orders of magnitude of number of cycles (see Figure 8). Unlike for bending life correction factor, the contact life correction factor differs a little bit between different type of steels and cast  irons (see Figure 12).  However,  it  is possible to evaluate this factor  within the range Z*_NT = 0.87 … 0.97 (–).

2.2.2.3 Stress ratio effect and the mean stress influence factor, Z_M

For a first approach, it is assumed the contact mean stress influence factor is equal to the bending mean stress influence factor. This assumption would deserve to be confirmed in future work.

Equation 25 is then applied to pass from alternative shear stress to repeated shear stress.

Note: It has to be noticed that the alternative shear stress is more affecting than the repeated shear stress.

2.2.2.4 Probability of failure/reliability factor K_R

The reliability factor can be applied for contact as for bending since it is a general statistical factor. In the case of the NF E 23-015:1982 [20], the K_R factor is applied to the load and not on the stress thus a square root is added for the pitting stress number. Figure 13 presents the comparison of the reliability factor for macropitting. It appears the NF E 23-015:1982 macropitting reliability factor curves move away from ISO 12107 and AGMA 2001-D04 in comparison with the bending reliability factor presented in Figure 9.

2.2.2.5 Final application

Using Equations 4, 3, 22, 23, 24, and 25 altogether, the contact nominal stress number can be evaluated (Equations 26 and 27):

Alternated

Repeated

Using the previously descried evaluation of each factor, and the minimum value of Z*_NT = 0.8655 (–) the numerical application gives (Equations 28, 29, and 30)

Alternated

Repeated

3 Results and discussions

The developed methodology is compared to the ISO 6336-5 nominal stress number for bending and macropitting. The considered material are:

• Through hardened wrought steels “V.”
• Through hardened cast steels “V cast.”
• Case hardened wrought steels “Eh.”
• Flame- or induction-hardened wrought and cast steels “IF.”
• Nitriding steels “NT and NV.”
• Wrought steels nitrocarburized steels “NV nitrocarburized.”

3.1 Root bending

Figure 14 presents the bending nominal stress number σ_F,lim = f(HV) as a function of the surface hardness for ISO 6336-5 materials (quality grades ML, MQ, ME quality grades) and for the proposed method. The proposal fits very well with the ISO 6336-5 data, especially in the scope 200 HV to 800 HV. It is noticed that the proposal is globally in the order of the ISO data; however, the proposed model slightly overestimates the allowable bending nominal stress number. This analysis confirms the results from [6] who observed a good correlation between generic plane bending fatigue test and single tooth bending test (see Figure 7).

3.2 Macropitting

Figure 15 presents the contact nominal stress number σ_H,lim = f(HV) as a function of the surface hardness. The proposal fits very well with the ISO 6336-5 data for all materials except for wrought steels that are nitrocarburized, which are slightly overestimated.

3.3 Conclusion

A methodology to convert rotating bending Wöhler curve to tooth bending and macropitting Wöhler curves was described. This method, combined with a relationship giving the rotational bending endurance limit as a function of Vickers hardness, gives a complete model of the bending and macropitting allowable stress number of steels and cast irons in the design phase. The proposed model has been preliminary compared to the experimental data given by the ISO 6336 standard.

Those results confirm that generic fatigue data can be used as a first approach to evaluate the bending and the contact allowable stress number.

It is also possible to look for a relation between the rotating bending allowable stress number and the ultimate tensile strength that could better fit the ISO values for steels (see Equation 31, which is a modification of Equation 4). Figure 16 shows nominal stress numbers (bending and contact), with a modified Equation 31.

3.4 Perspectives

This preliminary work is a proof of concept that provides an answer to the industrial need to have a method to evaluate fatigue limit of gear material during a pre-project. This method has been especially focused on steel because of the variety of experimental data presented in the ISO 6336-5. A first perspective would be to test and adapt this method to other non-standard materials (bronze, aluminum alloys, etc.).

This work also provides links between generic fatigue concepts and a specialized gear concept. It confirms that tooth bending failure is in the same nature as plane bending, and macropitting is in the same nature as alternating shear strength limit. A second perspective would be to evaluate and understand the gap between the proposed method, rotating bending test, STBT, and rolling contact fatigue test such as the twin- discs test and rotating gear tests.

The present method may be used to evaluate the local endurance limit as a function of the hardness. A further and last perspective is to evaluate the proposal in the risk for tooth flank fracture according to ISO/TS 6336-4 [23] where the permissible shear stress is established for 50% probability of failure, no life factor is developed from now, and a simpler linear correlation between the endurance limit and the hardness is used and may be inaccurate for high hardness.

4 Novelty statement

A methodology is presented to evaluate the limit stress of gear fatigue load capacity from simple fatigue test and avoid back-to-back pitting and single tooth bending fatigue tests. 

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^{Printed with permission of the copyright holder, the American Gear Manufacturers Association, 1001 N. Fairfax Street, Suite 500, Alexandria, Virginia 22314. Statements presented in this paper are those of the authors and may not represent the position or opinion of the American Gear Manufacturers Association. (AGMA) This paper was presented October 2023 at the AGMA Fall Technical Meeting. 23FTM06}