Marin Endurance Limit Equations and Calculator

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Marin Endurance Limit Equations and Calculator

Machine Design Applications
Strength and Mechanics of Materials

Marin Endurance Limit Equations and Calculator

Marin Factors for Corrected Endurance Limit Fatigue

The endurance limit (S'_e) determined using Eq. 2 that is established from fatigue tests on a standard test specimen must be modified for factors that will usually be different for an actual machine element. These factors account for differences in surface finish, size, load type, temperature, and other miscellaneous effects that may differ from those for the test specimen.

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The mathematical model commonly used to apply these factors is credited to Joseph Marin (1962) and is given as:

Eq. 1
S_e = k_a k_b k_c k_d k_e S'_e

Endurance Limit. A sufficient number of ferrous materials (carbon steels, alloy steels, and wrought irons) have been tested using the R. R. Moore rotating-beam machine so that the following relationship between the ultimate tensile strength (Sut ) and the endurance limit (Se ) that would have been obtained from a fatigue test can be assumed to give an accurate value even if the material has not been tested. This relationship is given in Eq. (2) for both the U.S. Customary and SI/metric system of units.

Eq. 2

U.S. Customary: S'_e =	0.504 S_ut	S_ut ≤ 200kpsi
U.S. Customary: S'_e =	100 kpsi	S_ut > 200 kpsi

SI/metric: S'_e =	0.504 S_ut	S_ut ≤ 1400 MPa
SI/metric: S'_e =	700 MPa	S_ut > 1400 MPa

Where:

S_e = endurance limit corrected so that it can predict fatigue resistance in a real component
S'_e = endurance limit obtained from guidelines from R.R. Moore test
S_ut = Endurance Limit
k_a = surface finish factor
k_b = size factor
k_c = load type factor
k_d = temperature factor
k = miscellaneous effects factor

Each of these five factors are used to provide an estimate of the endurance limit (Se) for a particular machine element design.

The first factor to discuss is the surface finish factor (k_a), probably the most important of the five factors.

Surface Finish Factor k_a

The surface finish of the R. R. Moore rotating-beam machine test specimen is highly polished, particularly to remove any circumferential scratches or marks that would cause premature failure and thereby corrupt the data. The actual machine element under investigation may have a relatively rough surface finish, thereby providing a place for a crack to develop, eventually leading to a fatigue failure.

Eq. 3
k_a = a S^b_ut

Where:

Coefficient a has units of stress abd exponent (b), which is negative and dimensionless is found below:

Surface finish	Factor a		Exponent b
Surface finish	kpsi	Mpa	Exponent b
Ground	1.34	1.58	-0.085
Machined	2.70	4.51	-0.265
Cold-drawn	2.70	4.51	-0.265
Hot-rolled	14.4	57.7	-0.718
As Forged	39.9	272	-0.995

Ultimate Tensile Strength S_ut

Surface finish	kpsi	Mpa	Surface factor k_a
Machined	65	455	0.89
As forged	65	455	0.63
Machined	125	875	0.75
As forged	125	875	0.33

Size Factor k_b

The size factor k_b accounts for the difference between the machine element and the test specimen. For axial loading, the size factor (kb) is not an issue, so use the following value:

k_b = 1

For bending or torsion, use the following relationships for the range of sizes

Eq. 4

k_b =	(d / 0.3)^-0.1133	0.11 in ≤ d ≤ 2 in
k_b =	(d / 7.62)^-0.1133	2.79mm ≤ d ≤ 51 mm

For bending and torsion of larger sizes, the size factor (k_b) varies between 0.60 and 0.75. For machine elements that are round but not rotating, or shapes that are not round, an effective diameter, denoted (d_e), must be used in Eq. (4).

For a nonrotating round or hollow cross section, the effective diameter (d_e) is given in Eq.

Eq 5
d_e = 0.370 D

where the diameter (D) is the outside diameter of either the solid or hollow cross section.

For a rectangular cross section (b × h), the effective diameter (d_e) is given in Eq. (5) as:

Eq. 6
d_e = 0.808 (bh)^1/2

Temperature Factor k_d.

For temperatures very much lower than room temperature materials like ductile steel become brittle. Materials like aluminum seem to be unaffected by similar low temperatures.

Eq. 7
k_d = S_T / S_RT

where ( S_T ) is the ultimate tensile strength at some specific temperature ( T ) and ( S_RT ) is the ultimate tensile strength at room temperature (RT).

Temperature Factors

TABLE 6 Temperature Factors

◦F	kd	◦C	kd
70	1.000	20	1.000
100	1.008	50	1.010
200	1.020	100	1.020
300	1.024	150	1.025
400	1.018	200	1.020
500	0.995	250	1.000
600	0.963	300	0.975
700	0.927	350	0.927
800	0.872	400	0.922
900	0.797	450	0.840
1000	0.698	500	0.766
1100	0.567	550	0.670

Miscellaneous Effects Factor. All the following effects are important in the dynamic loading of machine elements, however, only one can be quantified. These effects are residual stresses, corrosion, electrolytic plating, metal spraying, cyclic frequency, frettage corrosion, and stress concentration.

Residual stresses can improve the endurance limit if they increase the compressive stresses, especially at the surface through such processes as shot peening and most cold working. However, residual stresses that increase the tensile stresses, again especially at the surface, tend to reduce the endurance limit.

Corrosion tends to reduce the endurance limit as it produces imperfections at the surface of the machine element where the small cracks associated with fatigue failure can develop.

Electrolytic plating such as chromium or cadmium plating can reduce the endurance limit by as much as 50 percent.

Like corrosion, metal spraying produces imperfections at the surface so it tends to reduce the endurance limit.

Cyclic frequency is usually not important, unless the temperature is relatively high and there is the presence of corrosion. The lower the frequency of the repeated reversed loading and the higher the temperature, the faster the propagation of cracks once they develop, and therefore, the shorter the fatigue life of the machine element.

Frettage is a type of corrosion where very tightly fitted parts (bolted and riveted joints, press or fits between gears, pulleys, and shafts, and bearing races in close tolerance seats) move ever so slightly producing pitting and discoloration similar to normal corrosion. The result is a reduced fatigue life because small cracks can develop in these microscopic areas. Depending on the material, frettage corrosion can reduce the fatigue life from 10 to 80 percent, so it is an important issue to consider.

Stress concentration is the only miscellaneous effect that can be accurately quantified.

Reduced stress concentration factor ( K_f ) needed to be applied to the design of brittle materials. As fatigue failure is similar to brittle failure, stress concentrations need to be considered for both ductile and brittle materials under repeated loadings, whether they are completely reversed or fluctuating.

Reduced stress concentration factor ( K_f ) was determined from equation:

K_f = 1 + q (Kt - 1 )

where the geometric stress concentration factor ( K_t ) is modified or reduced due to any notch sensitivity (q) of the material. Values for the stress concentration factor ( K_t ) for various types of geometric discontinuities are given in any number of references.

The miscellaneous effects factor for stress concentration ( k_e ) is therefore the reciprocal of the reduced stress concentration factor ( K_f ) and given

k_e = 1 / K_f

Where as ( K_f ) is usually greater than one, the miscellaneous effects factor ( k_e ) will be less than 1 and thereby reduce the test specimen endurance limit ( S'_e ) accordingly.

Note that the miscellaneous effects factor ( k_e ) for stress concentration applies to the endurance limit ( S'_e ) at (N =10⁶) and greater. However, below (N =10³) cycles it has no effect, meaning (K_f = 1) or (k_e = 1). Similar to the process for finite life, between (N =10³) and (N =10⁶) cycles define a modified stress concentration factor ( K'_f ) where:

K'_f = a N^b

and the coefficients (a) and (b), both dimensionless, are given:

a = 1 / K_f

Source:

Marks' Calculations for Machine Design,
Thomas H. Brown, Jr. Ph.D., PE
Faculty Associate
Institute for Transportation Research and Education
NC State University
Raleigh, North Carolina