Rotating Shaft Design Combined Loading Formulae and Calculator

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Rotating Shaft Design Combined Loading Formulae and Calculator

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ASME Design Code for the design of Transmission Shafting

Preview ASME Design Code for the design of Transmission Shafting Calculator

According to Fuchs and Stephens (1980), between 50% and 90% of all mechanical failures are fatigue failures. This challenges the engineer to consider the possibility of fatigue failure at the design stage. Figure 1 shows the characteristic variation of fatigue strength for steel with the number of stress cycles. For low strength steels, a “leveling off” occurs in the graph between 106 and 107 cycles under noncorrosive conditions and, regardless of the number of stress cycles beyond this, the component will not fail due to fatigue. The value of stress corresponding to this leveling off is called the endurance stress, or the fatigue limit.

Figure 1 A typical strength cycle diagram for various steels.

In order to design hollow or solid rotating shafts under combined cyclic bending and steady torsional loading for limited life, the ASME design code for the design of transmission shafting can be used. The ASME procedure ensures that the shaft is properly sized to provide adequate service life, but the designer must ensure that the shaft is stiff enough to limit deflections of power transfer elements, such as gears, and to minimize misalignment through seals and bearings. In addition, the shaft's stiffness must be such that it avoids unwanted vibrations through the running range

The equation for determining the diameter for a solid shaft is given by

Eq. 1

d = {[\frac{32 n_{s}}{π} \sqrt{{(\frac{M}{σ_{e}})}^{2} + \frac{3}{4} {(\frac{T}{σ_{y}})}^{2}}]}^{(1 / 3)}

where

d = diameter (m),
n_s = factor of safety,
M = bending moment (N-m),
σ_e = endurance limit of the item (N/m² ),
σ_y = yeild strength (N/m² ),
T = torque (N-m).

There is usually some uncertainty regarding what level a component will actually be loaded to, how strong a material is, and how accurate the modeling methods are. Factors of safety are frequently used to account for these uncertainties. The value of a factor of safety is usually based on experience of what has given acceptable performance in the past. The level is also a function of the consequences of component failure and the cost of providing an increased safety factor. As a guide, typical values for the factor of safety based on strength recommended by Vidosek (1957) are as follows:

1.25 to 1.5 for reliable materials under controlled conditions subjected to loads and stresses known with certainty,
1.5 to 2.0 for well-known materials under reasonably constant environmental conditions subjected to known loads and stresses,
2.0 to 2.5 for average materials subjected to known loads and stresses,
2.5 to 3.0 for less well-known materials under average conditions of load, stress, and environment,
3.0 to 4.0 for untried materials under average conditions of load, stress, and environment, and
3.0 to 4.0 for well-known materials under uncertain conditions of load, stress, and environment.

The endurance limit, σ_e , for a mechanical element can be estimated ( Marin, 1962 ) by Eq. 2. Here a series of modifying factors is applied to the endurance limit of a test specimen for various effects such as size, load, and temperature.

Eq. 2

σ_e = k_ak_bk_c k_d k_e k_gσ'_e

where

k_a = surface factor,
k_b = size factor,
k_c = reliability factor,
k_d = temperature factor,
k_e = duty cycle factor,
k_f = fatigue stress concentration factor,
k_g = miscellaneous effects factor, and
σ'_e = endurance limit of test specimen (N/m² ).

If the stress at the location under consideration is greater than σ'_e, then the component will fail eventually due to fatigue (i.e. the component has a limited life).

Mischke (1987) has determined the following approximate relationships between the endurance limit of test specimens and the ultimate tensile strength of the material (for steels only):

σ'_e = 0.504 σ_uts = for σ_uts ≤ 1400 MPA

σ'_e = 700 MPa = for σ_uts ≥ 1400 MPA

The surface finish factor

Eq. 3

k_a = aσ^b_uts

Surface finish factors (Noll and Lipson, 1946)

Surface finish	a (MPa)	b
Ground	1.58	-0.085
cold drawn	4.51	-0.265
Hot rolled	57.7	-0.718
Forged	272.0	-0.995

reliability factor k_c

Shaft nominal reliability	k_c
0.5	1.0
0.9 0.	897
0.99	0.814
0.999	0.753

For temperatures between -57 °C and 204 °C, the temperature factor, kd, can be taken as 1.0.
The ASME standard documents the values to use outside of this range.

Duty cycle factor k_e is used to account for cycle loading experienced by the shaft such as stops and starts, transient overloads, shock loading, etc. and requires prototype fatigue testing for its quantification.

k_e is taken 1.0 whne data is not valiable

k_f is the fatigue stress concentration factor kf is used to account for stress concentration regions such as notches, holes, keyways, and shoulders.

Eq. 4

k_f = 1 + q (k_t - 1)

where

q = notch sensitivity
k_t = geometric stress concentration factor.

k_g = miscellaneous effects factor - up to you to figure that one out, (use 1.0).

Peter R. N. Childs
2014