Design Procedure 7.2: Staircase Approach
7.8 The Modified Endurance Limit
As discussed in Section 7.5.1, fatigue experiments use the best possible circumstances for estimating fatigue performance.
However, this situation cannot be guaranteed for design ap- plications, so the component’s endurance limit must be mod- ified or reduced from the best-case scenario. This is done in practice by using endurance limit modification factors that take important factors into account. The endurance limit modification factors covered in this text are for completely reversed loading (σm = 0). Themodified endurance limit can be expressed as
Se=kfkskrktkmSe0, (7.18) whereS
0
e = endurance limit from experimental apparatus under idealized conditions, Pa
kf = surface finish factor ks = size factor
kr = reliability factor kt = temperature factor km= miscellaneous factor
Note that the type of loading has already been incorpo- rated intoSe0 as presented in Eq. (7.6). As discussed in Sec- tion 7.7, the effects of stress concentrations are not included, since these factors are used to increase stress but not to reduce allowable strength.
Equation (7.18) should not be taken as an accurate pre- diction of endurance limit for complicated situations, but merely a reasonable approximation of what should be ex- pected in practice. As will be seen, universally applicable correction factors do not exist, and those that are presented are experimentally based for controlled materials, loadings, and other parameters. This further confirms the observa- tion stated above that experimental confirmation or the use of large safety factors are unavoidable in fatigue design.
7.8.1 Surface Finish Factor
The specimen shown in Fig. 7.3 has a highly polished sur- face finish with final polishing in the axial direction to smooth any circumferential scratches. Most machine elements do not have such a high-quality finish. The modification factor to incorporate the surface finish effect depends on the process used to generate the surface and on the material. Given a manufacturing process, Fig. 7.11a estimates the surface finish factor when the ultimate strength in tension is known, or else the coefficients from Table 7.3 can be used with the equation
kf =eSutf , (7.19)
where
kf = surface finish factor
Sut= ultimate tensile strength of material, MPa eandf= coefficients defined in Table 7.3
(b) (a)
1.0 0.9 0.8 0.7 0.6 0.5
0.4200 1600
80 40 20 10
5 2.5
1.25
0.64 0.2
0.10.040.08
Surface finish Ra, µm
Ultimate strength in tension, Sut, (MPa) Surface finish factor, kf
400 600 800 1000 1200 1400 Tensile strength, Sut (MPa)
400 800 1200 1600 2000 Surface finish factor, kf
1.0
0 0.2 0.4 0.6
0.8 Machined, cold
forged, cold rolled Hot
rolled Hot forged Tap wa
ter corroded
Salt water corroded Fine polishing
Figure 7.11: Surface finish factors for steel. (a) As a function of ultimate strength in tension for different manufacturing processes; (b) as a function of ultimate strength and surface roughness as measured with a stylus profilometer.Source:(a) Adapted from Juvinall and Marshek [1991] and data from the American Iron and Steel Institute; (b) adapted from Johnson [1967].
Table 7.3: Surface finish factor. Source:Shigley and Mitchell [1983].
1.58 4.51 272.057.7 Manufacturing
process Factor e Exponent f Grinding
Machining or cold drawing Hot rolling As forged
-0.085 -0.265 -0.718 -0.995
Note that Eq. (7.19) and Fig. 7.11a can give different results, especially for hot-working processes. This can be under- stood, recognizing the different sources for the expressions and data, and also because of the wide range of properties that can occur in these processes.
If the process used to obtain the surface finish is not known but the quality of the surface is known from the mea- sured or prescribed arithmetic average surface roughnessRa, the surface finish factor can be obtained from Fig. 7.11b. Note also from the discussion of surface roughness that these val- ues ofkf are approximate; surfaces are not fully character- ized by their roughness, and deep and sharp circumferential scratches are the most detrimental to fatigue life, which may not be captured in theRaroughness value.
These approaches are all approximate and are used only for well-controlled manufacturing processes. It is mislead- ing to apply Table 7.3 for other circumstances or operations.
For example, plasma spray operations tend to provide an ex- tremely rough surface, but the fatigue properties are mainly determined by the surface layer beneath the plasma-sprayed coating. Further, the data in Table 7.3 are undoubtedly too stringent. With modern numerically controlled machine tools and improvements in tooling materials, superior finishes are routinely produced that will give slightly better performance from a fatigue standpoint.
7.8.2 Size Factor
The high-cycle fatigue apparatus used to obtain the en- durance limit Se0 was for a specific diameter, namely, 7.62 mm, and often uses extruded or drawn steel bar stock. For metals, such extrusions have pronounced grain elongation in the direction transverse to fatigue crack growth. Also, the de- gree of cold work is high and the likelihood of large flaws is low. Similar effects are seen for ceramics and castings but for different reasons (smaller shrinkage pores, etc.). However, it must be noted that the size, shape, and number of flaws in a given cross section are strongly dependent on the manufac- turing process.
Many researchers have suggested size factor expres- sions, but a simple approach suggested by Shigley and Mitchell [1983] is as follows for round bars. For bending or torsion the size factor is
ks=
1 d≤8mm
1.248d−0.112 8mm< d≤250mm (7.20) For axial loadingks= 1.
For components that are not circular in cross-section the size factor is difficult to determine. However, an ap- proach suggested by Kuguel [1969] is often referenced. This approach requires obtaining the cross-sectional area that is loaded above 95% of the maximum stress, denoted asA95. (This is admittedly difficult to obtain for complicated cross sections.) Equating this to the portion of a circular cross sec-
Table 7.4: Reliability factors for six probabilities of survival.
Probability Reliability of survival, factor,
percent kr
50 1.00
90 0.90
95 0.87
99 0.82
99.9 0.75
99.99 0.70
tion that is loaded above 95% of its maximum stress yields d=
r A95
0.0766. (7.21)
Equation (7.21) does not differentiate according to processing history, and this is a major shortcoming of the approach. For example, consider a steel bar that has been extruded to a di- ameter of 50 mm, and is then machined to produce a square cross section with a side length of 25 mm. Machining is a material removal process; it does nothing to improve the sub- strate microstructure. Any size effect present should be based on the 50 mm initial diameter, not the machined dimension or shape, since this reflects the material state with respect to initial flaws, grain sizes and shapes, etc.
7.8.3 Reliability Factor
Table 7.4 shows the reliability factor for various percentages of survival probability. This table is based on the endurance limit having a standard deviation of 8%, generally the upper limit for steels. The reliability factor for such a case can be expressed as
kr= 0.512 ln
1 R
0.11
+ 0.508, (7.22) whereRis the probability of survival. The reliability factor as obtained from Table 7.4 or Eq. (7.22) can be considered only as a guide because the distribution is generally not well char- acterized for high values of reliability. It can also be assumed that Table 7.4 can be applied to materials other than steel, but care must be taken if the standard deviation is greater than 8%.
7.8.4 Temperature Factor
Many high-cycle fatigue applications take place under ex- tremely high temperatures, such as in aircraft engines, where the material is much weaker than at room temperature. Con- versely, in some applications, such as automobile axles in cold climates, the metal is generally less ductile than at room temperature.
In either case, the Manson-Coffin relationship given by Eq. (7.5) would suggest that a major factor affecting fatigue life is the strength in one loading cycle,σf0. It is therefore rea- sonable to follow one of two approaches. The designer can either (a) modify the ultimate strength of the material based on its properties at the temperature of interest before deter- mining a material endurance limit in Eq. (7.18), or (b) use a temperature factor:
kt= Sut
Sut,ref (7.23)
where Sut is the ultimate tensile strength of the material at the desired temperature and Sut,ref is the ultimate ten- sile strength at a reference (usually room) temperature. Of
The Modified Endurance Limit 173 course, when fatigue experiments have been conducted at
operating temperatures, the resulting endurance limit inher- ently accounts for the temperature, andktcan be ignored.
7.8.5 Miscellaneous Effects
Several other phenomena can affect a component’s fatigue properties. Whereas the preceding sections have outlined methods for numerically approximating some effects, other considerations defy quantification. Among these are the fol- lowing:
1. Manufacturing history. Manufacturing processes play a major role in determining the fatigue characteristics of engineering materials. This role is partially manifested in the size factor discussed in Section 7.8.2, but there are other effects as well. Because fatigue crack growth is often more rapid along grain boundaries than through grains, any manufacturing process affecting grain size and orientation can affect fatigue. Because some form- ing operations, such as rolling, extrusion, and drawing, lead to elongated grains, the material’s fatigue strength will vary in different directions (anisotropy). With ex- trusion and drawing this effect is usually beneficial, since the preferred direction of crack propagation be- comes the axial direction and crack propagation through the thickness is made more difficult by grain orientation and elongation in metals. Annealing a metal compo- nent relieves residual stresses, causes grains to become equiaxed, and may cause increased grain growth rates.
Relieving tensile residual stresses at a surface is gen- erally beneficial, but equiaxed or larger grains can be detrimental from a fatigue standpoint.
2. Residual stresses can result from manufacturing pro- cesses. A residual stressis caused by elastic recovery after nonuniform plastic deformation through a compo- nent’s thickness. Compressive residual stress on a sur- face retards crack growth; tensile residual stress can en- courage crack growth.
Compressive residual stresses can be imparted through shot or laser peening and roller burnishing and may be obtained in forging, extrusion, or rolling.
Shot peening is a cold working process in which the surface of a part is impacted with small spherical me- dia called shot. Each impact leads to plastic deforma- tion at the workpiece surface, leading to compressive residual stress after elastic recovery. The layer under compressive residual stress is usually less than 1 mm thick, and the material bulk properties are unaffected.
Crack development and propagation are severely re- tarded by compressive residual stresses; for this reason, shot peening is a common surface treatment for fatigue- susceptible parts such as gears, springs, shafts (espe- cially at stress concentrations), connecting rods, etc.
The beneficial effect of shot peening on fatigue life can be seen in Fig. 7.12. Similar behavior can be found for other materials. This is an important tool for fatigue design because it represents one of the only strategies thatincreasesthe fatigue strength of materials, and this increase can be very large. For example, consider an air- craft landing gear, produced from steel with a 2068 MPa strength. Figure 7.12a shows that shot peening can in- crease the fatigue strength by a factor of 3 over a pol- ished surface. Similar benefits are possible with other materials, but as seen in the figure, the more typical fa- tigue strength improvement is 15 to 30%.
Fatigue strength at two million cycles (MPa)
1380
1035
690
345
0
Ultimate tensile strength, Sut, (MPa) 690 1380 2170 Peened - smooth
or notched
Not peened - notched (typical machined surface)
Not peened - smooth
(a) 483
414 345 276 207 Alternating stress, σ, MPaa 138
104 105 106 107 108 Number of cycles to failure, N'
Al 7050-T7651 Ti-6Al-4V Shot peened
Machined Polished
(b)
Figure 7.12: The use of shot peening to improve fatigue prop- erties. (a) Fatigue strength at2×106cycles for high-strength steel as a function of ultimate strength; (b) typical S-N curves for non-ferrous metals. Source: Courtesy of J. Champaigne, Electronics, Inc.
3. Coatings can significantly affect fatigue. Some opera- tions, such as carburizing, lead to a high carbon content in steel surface layers (and thus a high fracture strength) and impart a compressive residual stress on the surface.
Electroplated surfaces can be porous and promote crack growth, reducing fatigue strengths by as much as 50%.
Zinc plating is the main exception where the fatigue strength is not seriously affected. Anodized oxide coat- ings are also usually porous, reducing fatigue strength.
Coatings applied at high temperatures, such as in chem- ical vapor deposition processes or hot dipping, may in- duce tensile residual stresses at the surface.
4. Corrosion. It is not surprising that materials operating in corrosive environments have lower fatigue strengths.
The main adverse reactants in corroding metals are hy- drogen and oxygen. Hydrogen diffuses into a material near a crack tip, aided by large tensile stresses at the tip, embrittling the material and aiding crack propaga- tion. Oxygen causes coatings to form that are brittle or
porous, aiding crack initiation and growth. High tem- peratures in corrosive environments speed diffusion- based processes.