# Design of Machine Elements

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Design of Machine Elements
Fatigue Failure Theories Design of Machine Elements © Dr Moudar 2011 [extracted from different sources]

Failure Theories

Introduction Fluctuating stresses in machinery often take the form of sinusoidal pattern because of the nature of the nature of some rotating machinery. Other patterns some quite irregular do occur. The smooth parts of the fracture surface usually exhibit beach marks which occurs as a result of changes in the magnitude of the fluctuating fatigue load. Fatigue behavior of materials is usually described by means of the S-N diagram which gives the number of cycles to failure, N as a function of the max applied alternating stress.

Introduction Fracture surface which usually exhibits smooth areas which correspond to the gradual crack growth stage, and rough areas, which correspond to the catastrophic fracture stage.

Characterizing Fluctuating Stresses
In periodic patterns exhibiting a single maximum and single minimum of force, the shape of the wave is not important. The peaks on both sides (maximum, minimum) are important.

Characterizing Fluctuating Stresses
Fmax and Fmin in a cycle can be used to characterize the force pattern. A steady component and an alternating component can be constructed as follows:

Characterizing Fluctuating Stresses
Stress Range Mean (Midrange Stress) Stress Amplitude (Alternating Stress) Stress Ratio Stress Amplitude

Prepare a Presentation on Catastrophic Fatigue Failure in Aviation, Massive Structures and other Engineering Applications Groups will be assigned in the class on Sunday

Characterizing Fluctuating Stresses

Characterizing Fluctuating Stresses

Characterizing Fluctuating Stresses

Characterizing Fluctuating Stresses

Characterizing Fluctuating Stresses
The steady, or static, stress is not the same as the midrange stress. The steady stress may have any value between σmin and σmin . The steady stress exists because of a fixed load or preload applied to a part. The steady load is independent of the varying portion of the load.

Characterizing Fluctuating Stresses
A helical compression spring is always loaded into a space shorter than the free length of the spring. The stress created by this initial compression is called the steady, or static, component of the stress.

Characterizing Fluctuating Stresses
The equations use symbols σa and σm as the stress components at the location of scrutiny. In the absence of a notch, σa and σm are equal to the nominal stresses σao and σmo induced by loads Fa and Fm, respectively. With a notch they are σa = Kf σao and σm = Kf σmo , respectively.

Stress Concentration Kf = 1+q(Kt + 1)
Kt - geometric stress concentration factor q - notch sensitivity factor

Stress Concentration Kf = 1+q(Kt + 1)
Kt - geometric stress concentration factor q - notch sensitivity factor

Stress Concentration

Stress Concentration

Stress Concentration Neuber equation,
𝑎 = Neuber constant and is a material constant.

Design Against Fatigue
S-N curve is a graphical representation of the maximum applied stress versus the number of stress cycles N before the fatigue failure on a semi-log graph. For ferrous metals like steel the curve becomes asymptotic at 10^6 cycles. The completely reversed stress which a material can withstand 10^6 cycles without failure is called ENDURANCE LIMIT of the material. For non ferrous materials, the curve slopes gradually even after 10^6 cycles. These materials do not have a limiting value of endurance in true sense.

Design Against Fatigue

Design Against Fatigue
In the majority cases, the reported fatigue strength or endurance limits of the materials are based on the test of carefully prepared small samples under laboratory condition. Such values cannot be directly used for design purposes because the behavior of a component or structure under fatigue loading does depend not only on the fatigue or endurance limit of the material used in making it, but also an several other factors.

Design Against Fatigue
The Endurance Limit: The strength corresponding to the knee in the S-N diagram is called the endurance limit Se, or the fatigue limit. The Estimating Se’ or Sf’ The determination of endurance limits by fatigue testing is now routine, though a lengthy procedure. Generally, stress testing is preferred to strain testing for endurance limits. For preliminary and prototype design and for some failure analysis as well, a quick method of estimating endurance limits is needed.

Design Against Fatigue

Design Against Fatigue
For steels where Sut is the minimum tensile strength. The prime mark on S′e in this equation refers to the rotating-beam specimen itself.

Endurance Limit Modifying Factors
Size and shape of the component or structure Type of loading and state of stress Stress concentration Surface finish Operating temperature Service environment Method of fabrication

Endurance Limit Modifying Factors
Se = kakbkckdkekgkhSe’ Where Se = endurance limit of component Se’ = endurance limit experimental ka = surface finish factor (machined parts have different finish) kb = size factor (larger parts greater probability of finding defects) kc = reliability / statistical scatter factor (accounts for random variation)

Endurance Limit Modifying Factors
Se = kakbkckdkekgkhSe’ Where kd = operating T factor (accounts for diff. in working T & room T) ke = loading factor (differences in loading types) kg = service environment factor (action of hostile environment) kh = manufacturing processes factor (influence of fabrication parameters)

Endurance Limit Modifying Factors
ka = surface finish factor (machined parts have different finish)

Endurance Limit Modifying Factors
kb = size factor (larger parts greater probability of finding defects) Large engineering parts have lower fatigue strength than smaller test specimen Greater is the probability of finding metallurgical flaws that can cause crack initiation Following values can be taken as rough guidelines : kb = 1.0 for component diameters less than 10 mm kb = 0.9 for diameters in the range 10 to 50 mm kb = 1 – [( D – 0.03)/15], where D is diameter expressed in inches, for sizes 50 to 225 mm.

Endurance Limit Modifying Factors
kb = size factor (larger parts greater probability of finding defects) Large engineering parts have lower fatigue strength than smaller test specimen Greater is the probability of finding metallurgical flaws that can cause crack initiation Following values can be taken as rough guidelines : kb = 1.0 for component diameters less than 10 mm kb = 0.9 for diameters in the range 10 to 50 mm kb = 1 – [( D – 0.03)/15], where D is diameter expressed in inches, for sizes 50 to 225 mm.

Endurance Limit Modifying Factors
kc = reliability / statistical scatter factor (accounts for random variation) Accounts for random variation in fatigue strength. The following value can be taken as guidelines kc = for 90% reliability kc = for 99 % reliability kc = for 99.9 % reliability

Endurance Limit Modifying Factors
kd = operating T factor (accounts for diff. in working T & room T) Accounts for the difference between the test temperature and operating temperature of the component For carbon and alloy steels, fatigue strength not affected by operating temperature – 45 to 4500C kd = 1 At higher operating temperature kd = 1 – 5800( T – 450 ) for T between 450 and 550oC, or kd = 1 – 3200( T – 840 ) for T between 840 and 1020oF

Endurance Limit Modifying Factors

Endurance Limit Modifying Factors
kg = service environment factor (action of hostile environment) Accounts for the reduced fatigue strength due to the action of a hostile environment.

Endurance Limit Modifying Factors
kh = manufacturing processes factor (influence of fabrication parameters) Accounts for the influence of fabrication parameter Heat treatment, cold working, residual stresses and protective coating on the fatigue material. It is difficult to quantify, but important to included

Endurance Limit Modifying Factors
Endurance limit/Fatigue strength The endurance limit, or fatigue strength, of a given material can usually be related to its tensile strength, as shown in table (next slide) The endurance ratio, defined as (endurance limit/ tensile strength), can be used to predict fatigue behavior in the absence of endurance limits results. From the table shows, endurance ratio of most ferrous alloys varies between 0.4 and 0.6

Endurance Limit Modifying Factors

Endurance Limit Modifying Factors

Fatigue Failure Theories