The size of transmission shafts for ski and chair lift gear assemblies did not become a critical issue until the late fifties. Previously, the drive sheave had nearly always been engaged with an open gear transmission and thus rotated around a rigid shaft. With the introduction of closed drive units with cantilevered sheaves on stub shafts, the problem of transmission shaft safety gained significance. Initially, a minimum static safety factor of 5 was considered satisfactory. Soon, however, engineers realized that they were dealing with a fatigue strength problem which required an entirely different approach. The new types of transmission shafts were subject to a composite alternating bending and torsional stress. The literature available at that time was rather vague and did not supply any tangible figures. In his publication “Maschinenelemente“, G. Niemann had given experimental data but the results were not sufficient for reliable interpolation, let alone extrapolation.
When the 3rd edition of “Neue Festigkeitsberechnung fur den Maschinenbau” by Hanchen & Decker was published in 1967, it finally became possible to implement detailed calculations. The author described the respective calculation method in a 1973 issue of SAM.
A short procedural review for background purposes: The form strength of the component being investigated—in our case the transmission shaft—is compared with the stresses to which it is exposed (bending and torsional stresses).
Safety SD = Form strength / Stress
The magnitude of form strength depends on the following factors:
- Material: fatigue strength, diameter, surface roughness
- Shape of component: notches, shoulders, radii, slots, press-fitted parts etc.
To provide extra safety, the following factors must be considered for all ski and chair lift transmissions:
1.1 for computational uncertainty
1.5 for human life involvement — total 1.65
The required minimum safety SDmin depends on the frequency with which the maximum load occurs, such as
SDmin = 1.15 for 25% frequency of maximum load occurrence
SDmin = 2.0 for 100% frequency of maximum load occurrence
For better comprehension, it should be mentioned that fatigue failure would have to occur with a safety factor of 1.0 irrespective of the frequency of maximum load occurrence; an imaginary computational minimum safety factor of 0.61 is obtained by derating with the above mentioned additional factors of 1.1 and 1.5.
This calculation method was first discussed between gear manufacturers and licensing authorities in 1969. In view of the complex correlations involved, technical literature does not provide any convenient answers. Many questions and uncertainties remained and the discussions between industry and public authorities extended over several years until a satisfactory method of computation derived from the Hanchen & Decker proposal was found. The current consensus—always subject to amendments and modifications as new insights are gained—is as follows:
Output shafts with cantilevered sheaves are to be calculated for two load cases:
- Up full – down empty: Frequency of maximum load occurrence 25%
Minimum safety limit 1.15 - Up empty – down empty: Frequency of maximum load occurrence 100%
Minimum safety limit 2.0
In the first case, the torque will be about 3 to 5 times higher than in the second case. Contrarily, the sum S1 + S2 will be greater in the second case than in the first, primarily for ski lifts. This double calculation is based on the fact that the maximum load of a lift is known to occur at most during 25% of its total operating time (early and late rides for personnel, material handling, slope preparation, etc., included). During the remaining operating time, the torque decreases and the sum S increases accordingly. For extra safety, this remaining time has tentatively been taken as 100% although the complement to the above mentioned 25% is only 75%.
Interestingly enough, the strength reserve in the shafts for the second load case is usually smaller than for the first load case. This is primarily attributable to the massive increase of the safety limit from 1.15 to 2.0, strongly upgrading the influence of sum S in the calculation and degrading the influence to the heavily reduced torque.
The Regional Construction Bureau of the Tyrolean Government with Messrs. dipl. ing. Rhomverg and Judmaier was the first authority which became deeply involved in fatigue strength calculations for shafts. In collaboration with this bureau, the new method was employed to recalculate about 40 transmission output shafts with a long operating history with the intention of obtaining statistical information for the future. Simultaneously, 50 additional transmission shafts known to be subject to highest loads were calculated according to this method.
The recalculation of these approx. 90 transmission shafts (three of which failed in the second group) yielded the following results:
| Nominal value | 40 shafts varying usage | 50 shafts critical usage | |
|---|---|---|---|
| Safety factor first case | 1.15 | 1.06 to 3.45 | 0.75 to 1.55 |
| Safety factor second case | 2.0 | 1.70 to 4.84 | 1.02 to 4.80 |
| Year of manufacture | 1962 to 1976 | 1965 to 1976 | |
| failures | none | three |
All 3 failures in the second group occurred in load case 2; the safety factor at the critical cross sections of these three shafts ranged from 1.02 to 1.20. The safety factors of seven further shafts were about 1.20, the others exceeded 1.30 (for the sake of completeness, it should be mentioned that all shafts with the extremely low safety limits were not standard models. They had been built in an abnormal manner to meet specific customer requirements and the altered notch factors in the cursory calculations performed for their design were much lower than would be permissible with the information available today.)
Comments:
The safety factors of 1.15 for load case 1 and 2.0 for load case 2 seem to be adequate. The failures in load case 2 occurred at safety factors which were below 60% of the nominal value. For load case 1, however, it is only possible to note that no failures occurred down to 65% of the nominal safety limit.
Another realization pertains for the fact that the coefficients for computational uncertainty (1.1) and human life involvement (1.5) appeared to be necessary apart from the purely calculatory values in order to ascertain installations with the desired safety margin.
None of the shafts with a safety factor exceeding 65% of the nominal values (in both cases) failed. Since 106 is a satisfactory approximation of the number of load reversals per year in such transmissions, all transmission shafts installed prior to 1973 proved to be fatigue resistant and are thus relevant indicators with respect to durability.
In summary, it is reasonable to say that the current method of calculation used to determine safety limits will provide useful results in that among the 90 shafts analyzed, failures occurred only where this could have been expected according to the new method of calculation.
In this context, reference should be made to the reinspection of transmission shafts for cracks after a certain number of years in operation as demanded by various jurisdictions. Experience gained with some 5000 Kissling gear units with cantilevered sheaves indicates that the crack test after the installation of the transmission shaft is a disproportionate measure. None of the shafts even with only 65% of the currently demanded safety limit has failed so far even though ultrasonic testing of the shafts prior to machining was not performed until about 1967 (ultrasonic testing after machining even today is effected only in individual cases). Furthermore, ultrasonic tests on installed shafts are not very reliable; the reliable detection of cracks is not even possible with the shaft dismantled unless bearings, bushing, and gears are also removed. The complete disassembly of the shaft including the removal of all press-fitted parts represents a substantial effort because—primarily where large transmissions are involved—such operations can only be performed in the factory. The current calculating method provides such a margin of safety that subsequent tests are unnecessary. The objection related to scoring during operation due to oxidation has proven to be untenable while slippage (between frictional torque transmission elements) must be inspected or prevented by other means.
In cases of doubt, it is meaningful to reanalyze older transmission shafts according to the current calculating method and to replace such shafts if the safety margin thus obtained proves to be too low. For chair lift transmissions which are subject to the greatest risks, the limit must be applied at 100% of the nominal safety factor while 85 to 90% of the nominal factor can be deemed satisfactory in older surface lift transmissions particularly if they have already withstood more than 10 x 106 load reversals.
