For the fatigue strength, the stress amplitudes (Equation 3) of all bins are relevant, contributing to the damage accumulation. The equivalent amplitude Sequ used in FKM is a constant stress amplitude with an assigned cycle number equal to the knee point of the S-N curve, which is damage-equivalent to the related stress spectrum under consideration of the shape of the S-N curve, the required total number of cycles and the maximum amplitude in the spectrum. The description of the verification proceeding with all formulas would exceed the purpose of this paper. So only some of the important steps will be presented.

For every stress component (axial, bending, torsional, and shear) the same procedure is applied. In the first step, the mean stress value with amplitude (stress Sm.i and Sa,i) of every bin is converted in an equivalent alternating stress with amplitude SaW.i (with Sm.i = 0 and SaW.i). Then the range of bins must be resorted, such that the first bin has the biggest amplitude (SaW.1). In the third step, the variable amplitude factor must be found, using Equation 4 and Equation 5.

This permits obtaining the equivalent stress amplitude at the knee point of the SN-curve, Sequ, FKM Equation 2.4.27 (Ref. 5).



(6)

For the fatigue strength assessment, depending on the position of the considered shaft section, at locations with shaft shoulders, keyways, etc., the stresses are increased due to local stress concentration. In the FKM (Ref. 5), as in AGMA 6001 (Ref. 13) or DIN 743 (Ref. 8), for the verification the nominal stresses are used (not increased), but the fatigue strength is modified (reduced) by stress concentration factors. As mentioned in AGMA, “since the fatigue strength is largely influenced by physical conditions, environmental conditions, and application conditions as well as material conditions, the basic fatigue strength must be modified (Ref. 13)”. In FKM (Ref. 5) all these effects are documented and combined in the so-called “design factor” KWK. So, the component fatigue limit for axial, bending, torsional and shear are reduced by the design factor per component to obtain the effective nominal values of the component fatigue limit SWK, TWK.

In ISO or AGMA ratings, the final result of a verification is the safety factor obtained by division of the permissible stress by the effective stress. The resulting safety S must then be equal to or higher than the minimum requested safety Smin. The result obtained by FKM is not a safety factor, but the “degree of utilization.” The degree of utilization is just the inverted value of the safety factor. The advantage of the utilization concept is that formulas for the verification with combined stresses are simpler. For this paper, the safety concept is used, and the FKM formulas are adapted accordingly. As a symbol for safety, we use here SY, as the symbol S in the FKM guideline is used for nominal stress.

The basic equation for component safety SY, equal to the inverted value of the degree of utilization aBK according to FKM Equation 2.6.3 (Ref. 5), for axial, bending, torsional, and shear are:



(7)

So, based on the applied stress and the permissible stress per component, the safety factors obtained are SYBK,zd, SYBK,b, SYBK,t, and SYBK,s. For the consideration of the combined stresses in a shaft, these factors are then combined according to the von-Mises criterion (Equation 2.6.7, FKM [Ref. 5]) to obtain the final result.



(8)

In a shaft section, there are two locations where the combined stress may be highest, position W1 and W2 in Figure 5. In point W1, bending and torsion stresses are high, but shear is zero. In W2, shear and torsion stresses are high, but bending is zero. Thus, both positions must be checked to find the lowest safety factor.

The verification of whether the shaft section fulfills the request is then given by:



(9)

with the requested minimum safety factor jD for fatigue strength assessment. For cases where the consequences of shaft failure are high, this factor can be assumed to be 1.5.