Fatigue strength assessment of preloaded cross-toothed flange connections based on the FKM guidelines

Cross-toothed flange connections are used in drivetrains for form-fit coupling of shafts. They belong to the group of non-shiftable couplings. Compared to frictional connections, tooth flange connections can transmit significantly higher torques in the same space. In contrast to other tooth couplings, the crossed tooth flange coupling investigated here is characterized by a particularly complex load distribution. The preload force, which is necessary to transmit operational loads, is generated by tensioning bolts, which can be located centrally or decentralized. Thus, the stress condition in the notch of the coupling is always determined by a superposition of stresses from preload force and operating loads. Extensive experimental and simulative investigations were carried out as a part of a research project. In addition to the development of a pragmatic, specially adapted calculation concept for cross-toothed couplings for practice, the experimental tests were accompanied by calculations with the FKM guideline “Analytical Strength Assessment” and the FKM guideline “non-linear”, The determination of local stresses was carried out with the help of elasticity theoretical and elastic-plastic FE full-contact calculations based on experimentally determined material data. This article deals with the possibilities of calculations based on the FE simulation results using the above mentioned FKM guidelines. The experimentally determined load-carrying capacity is critically compared to the calculation results. The focus is on the implementation of the fatigue strength analysis in the context of the corresponding boundary conditions. Topics such as the assumed knee point of the Wöhler curve to the fatigue strength range are discussed as well as the evaluation options for elastic-plastic FE results for the guideline „non-linear”.


Introduction
Fatigue assessment of preloaded cross-toothed flange connections plays a critical role in ensuring the reliability and performance of non-switchable couplings.These couplings are used to transmit high torques and they are essential in various applications where power transmission is required, such as connecting gearboxes to drive train or cardan shafts.The coupling is named after the layout of the teeth: crossed tooth flange connection.Unlike traditional flange connections, cross-toothed flange connections rely on a preload force to effectively transfer operating loads.This preload is achieved through the use of a pretensioning bolt, which applies a controlled level of compression to the connection.Without this preload, the coupling would be susceptible to separation during operation, resulting in critical failures.A mechanical clamping nut is used to avoid the effects of torque-controlled tightening.The fatigue strength assessment of cross-toothed flange connections has been the subject of extensive research, including an FVA (Research Association for Drive Technology) project [1].The main objective of this project was to develop an analytical calculation method to evaluate the fatigue strength of these connections.The research involved a combination of experimental tests and theoretical studies, supported by simulations.The aim was to fully understand the behavior and determine the fatigue strength of the connections.
However, the strength assessment concept was developed only for specified geometry parameters.Therefore, the calculation of fatigue strength using the local stress approach (FKM guidelines [2,3]) is now compared with these experimental results.

Experimental investigations
For the experimental investigations, parts for the component tests and specimens for material tests were manufactured from raw parts (one material, one batch, one equal heat treatment).

Determination of the fatigue strength
A torsion test rig has been specifically designed and built at the Institute to perform component testing, as shown in Figure 3.The component to be tested is centered and clamped using two clamping hubs, one top and one bottom.To generate the torsional moment, two forces are applied with their respective lever arrangements.
The coupling is preloaded using a central bolt.For monitoring purposes, the preload force is measured using strain gauges to ensure control and accuracy during the test.Furthermore, it is worth noting that the cylinders are connected with flexible elements, which apply an equal amount of force but in opposite directions.The flexible elements serve to accommodate and mitigate any potential misalignments or deformations during the torsion test.Additionally, beams are employed as supporting elements to provide stability and structural reinforcement to the setup.
The stair-case test method and the evaluation (Figure 4) according to Hück [4] is used, which consists of a series of individual tests.After each test, the components are thoroughly inspected to determine the presence or absence of cracks.The torsional moment is evaluated based on a survival probability of 50 % at 1 million cycles.To check for potential damage, the tested parts were inspected using a dye penetrant method [5].This technique proved effective in detecting cracks.Cracks were found in two locations, as shown in Figure 5 .The standard test gearing according to [6] and [7] is shown.

Determination of material properties
In addition to component tests, the same raw parts were used for material tests.The results are later used as material parameters for FE simulations.Standard values, which are often conservative, are therefore not required.The results from the tensile tests (according to standards [8,9]) are summarized in Table 1 In addition to tensile tests, strain-controlled constant amplitude tests were also performed.Through a series of individual tests conducted at various strain amplitudes, a stabilized stress-strain curve was determined (Figure 6).The elastic-plastic material behavior is described by the Ramberg-Osgood equation [10].And the resulting stress-strain curve can be formulated with the parameters from Table 2.As a result, elasticplastic material behavior can be represented in FE simulations.The pre-tensioning force is a constant load and is applied in the first load step.After that, the torsional moment is added as a swelling load.All load are applied are via pilot nodes and the toothing range is modeled as full contact.The friction contact has a coefficient of friction of µ = 0.1.Detailed investigations [11] have shown this value to be a reasonable basis for FE simulations.As shown in the experimental results, the damaged area is located symmetrically.Maximum stress values are also found in these two areas in the simulation.Because of the symmetry, it is sufficient to consider only one of the two critical areas (Figure 10), whereas, of course, the highly stressed area is calculated for both.Based on the material investigations, two model variants can now be created with the geometry of the test specimens (in particular the toothing).One with linear elastic material behavior and one with elastic-plastic material behavior.The evaluated stresses and strains form the basis for the strength assessment in the next section.On the one hand with the FKM-guideline [2] (linear elastic stresses) and on the other hand with the FKM-guideline non-linear [3] (elastic plastic stresses and strains).

Strength assessment
The strength assessment according to the FKM guidelines [2,3] was used to answer the question of how well the experimental values of the stair-case test can be recalculated.When fatigue loads occur, as in the example, a static strength assessment and a fatigue strength assessment must always be performed.
For the static strength assessment, the maximum local stresses of the FE model with linear elastic material behavior were evaluated.Also material properties from experiments were part of the calculation.In addition, the plastic notch factor was determined with an independent simulation according to FKM-Guideline [2].As a result, the degree of utilization is less than 1, which means that the cross-tooth flange connection will not break under maximum load.
For the fatigue assessment, stress amplitudes and mean values were determined from several load steps of the FE simulation.In addition, a mean stress factor KAK (according to FKM-Guideline [2]) is required.Two overload cases are considered.The overload case F2 is used if the stress ratio remains constant at overload.In practice, this means that if a higher torsional moment is to be transmitted, the preload force must also be increased.This case can be used in the design process.The reason for this is that the preload force can be adjusted.The overload case F3 (it is permitted, but only described in earlier edition of the guideline [2]) says that the minimum stress remains constant in case of overload.This case is suitable if only the torsional moment amplitude changes and the preload force remains constant.This is valid for the recalculation, where the preload force was the same for all specimens and the swelling torsional moment was changed in the stair-case test.
Also, the KAK can be calculated with equivalent mean stress or with single mean stress (single mean stress component).The rule given in the guideline [2] leaves some scope for interpretation.Therefore, the load factors for overload cases F2 and F3 were calculated once with equivalent mean stress and once with single mean stress.But the results are similar (Table 3.).Values greater than 1 mean that the load-carrying capacity is underestimated by this concept.Table 3. Cyclic degree of utilisation calculated with FKM guideline [2] (experimentally determined load value used for calculation, P = 50% failure probability) overload case F2 F3 KAK with equivalent mean stress aBK = 1.52 aBK = 1.48 KAK with single mean stress aBK = 1.41 aBK = 1.44 The stress and strain results of FE simulation with elastic plastic material behavior cannot be used in the FKM guideline [2].Therefore, the assessment is performed with the damage parameter PRAM of the FKM guideline nonlinear [3].If the maximum damage parameter PRAM,max is greater than the allowed value, the verification is not passed.The equation ( 2) is valid only for the evaluation of the fatigue limit.The fatigue strength is verified when the largest damage parameter that occurs is less than the material fatigue strength divided by the component factor.
In order to compare these results with other values, a degree of utilization must be calculated.Due to non-linearities, the degree of utilization cannot be determined in a simple way.Additional simulations are necessary.The simulation is now performed with a reduced load.This is repeated until the estimated fatigue strength of the component (PRAM,D,WS/fRAM) is reached.The ratio of the loads is now a degree of utilization.
There are two methods that can be used for crossed tooth flange connections.In the first, the preload force and the torsional moment are reduced proportionally (this would be a design calculation).And in the second method, only the torsional moment is reduced and the preload force remains constant (e.g., when recalculating experiments).
All results are summarized in Table 4 and assigned to the individual guideline.These calculations were all performed with a survival probability of 50 % and a safety factor of 1.

Conclusion
Torque amplitude is plotted against the number of load cycles in Figure 11.The plot of the FKM guideline is based on linear elastic simulation (dark blue line).It is marked by the knee point at 1 million cycles.To compare, the knee point to the fatigue limit in the nonlinear guideline is already at 82578 cycles.This knee point depends on tensile strength.And the elastic-plastic simulation results are used for this plot (light blue line).Additionally, the test values (staircase test) for 50 % survival probability are entered into the figure.These values are above the two graphs of the FKM guidelines.

Figure 1 .
Figure 1.Part of a crossed tooth flange connection for the component test (In practice, the body of the coupling is usually designed to be much thinner.)

Figure 4 .
Figure 4. Summary of the results from the stair-case test (load ratio = 0)

Figure 5 .
Figure 5. Crack inspection after 1 million load cycles

Figure 3 .
Figure 3. Test rig for components tests with high torque

3 .
Simulative studies FE simulations are used to evaluate stresses and strains in the damage-relevant area.The structure of the model is shown in Figure 8.

Figure 8 .
Figure 8. Simplified representation of FE model

Figure 10 .
Figure 10.FE model with stress results (very fine meshing) of the critical area

7th
International Conference of Engineering Against Failure Journal of Physics: Conference Series 2692 (2024) 012035

Table 1 .
. Results of the tensile tests

Table 2 .
Experimentally determined parameters of the stabilized stress-strain curve