Aby SAAJEDI-DNV ABD OFFICE March 2009

UNDERSTANDING BASICS OF FATIGUE

Summary

This module calculates the fatigue life of a part under variable amplitude loading using constant amplitude fatigue life data.

The fatigue life of the structure can be represented by the following equation:

where:

- Total fatigue cycles to failure.

- Number of cycles to initiate a crack.

- Number of cycles to propagate a crack to failure. 

This module deals with fatigue damage during the crack initiation phase.  Damage during the initiation phase can be related to dislocations, slip bands, micro-cracks, etc.  Since these phenomena can only be measured in a highly controlled laboratory environment, most damage summation approaches for the initiation phase are empirical in nature. These methods relate damage to the expended life for a small laboratory specimen.  For this purpose, life is defined as the separation of a specimen, which is equivalent to the formation of a small crack in a large component or structure.

Background Information:

The service loading time history of an actual engineering part can be quite complex, as shown in Figure 1. Several methods have been developed to deal with variable amplitude loading using baseline data generated from constant amplitude tests. These damage summation methods can be used in conjunction with either the Stress-Life or Strain-Life methods of constant amplitude fatigue analysis to calculate the fatigue life of a component subject to variable amplitude loading.

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Figure 1:  Variable amplitude time history

 There are a number of methods such as rain flow cycle counting for extracting constant amplitude cycles from a non-uniform time history.  These algorithms decompose an irregular sequence of peaks and valleys into an equivalent set of loading blocks, as shown in Figure 2. These constant amplitude blocks are used by the damage summation techniques described below.

 

Figure 2:  Stress spectrum

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Linear Damage Rule (Miner's Rule)

The Linear Damage Rule was first proposed by Palmgren in 1924 and was further developed by Miner in 1945. Today the method is commonly known as Miner's Rule.  

The Linear Damage Rule is based on the concept of fatigue damage.  A damage fraction, D, is defined as the fraction of life used up by an event or a series of events.   Failure is predicted to occur when:

where:

- Index of each set of applied load cycles at constant stress level Si

- Damage fraction accumulated during the load cycles of interval i at constant stress level Si

- Damage criterion (a constant).  

The Linear Damage Rule states that the damage fraction at a given constant stress level Si is equal to the number of applied cycles n at stress level Si divided by the fatigue life N at stress level Si.  That is, 

where:

- Damage fraction accumulated during the load cycles of interval i at constant stress level Si

- Number of applied load cycles at constant stress level Si

- Fatigue life at constant stress level Si, obtained from the S-N curve. 

For Miner's Rule, the damage criterion X is assumed to be equal to 1.0, and failure is predicted to occur when:

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Considerable test data has been generated in an attempt to verify Miner's Rule. Most test cases use a two step load history. This involves testing at an initial stress level S 1

for a certain number of cycles, then the stress level is changed to a second level S 2

until failure occurs. If S1 > S2, it is called a high-low test, and if S1 < S2, a low-high test. The results of Miner's original tests showed that the damage criterion X corresponding to failure ranged from 0.61 to 1.45. Other researchers have shown variations as large as 0.18 to 23.0, with most results tending to fall between 0.5 and 2.0. In most cases, the average value is close to Miner's proposed value of 1.0.

One problem with two-level step tests is that they do not accurately represent many service load histories. Most load histories do not follow any step arrangement and instead are made up of a random distribution of loads of various magnitudes. However, tests using random histories with several stress levels show good correlation with Miner's rule. Even so, for conservative estimates of the life of a structure an X value of less than 1.0 is usually used.

The Linear Damage Rule has two main shortcomings when it comes to describing observed material behaviour:

Load sequence effects are ignored. The theory predicts that the damage caused by a stress cycle is independent of where it occurs in the load history.  An example of this discrepancy was discussed earlier regarding high-low and low-high tests.

The rate of damage accumulation is independent of the stress level. This trend does not correspond to observed behaviour. At high strain amplitudes cracks will initiate in a few cycles, whereas at low strain amplitudes almost all the life is spent initiating a crack (i.e., very little propagation fatigue).

Despite these limitations, the Linear Damage Rule is still widely used.  This is due both to its simplicity and the fact that more sophisticated methods do not always result in better predictions.  

 Miner's rule in brief

In 1945, M. A. Miner popularised a rule that had first been proposed by A. Palmgren in 1924. The rule, variously called Miner's rule or the Palmgren-Miner linear damage hypothesis, states that where there are k different stress magnitudes in a spectrum, Si

(1 ≤ i ≤ k), each contributing ni(Si) cycles, then if Ni(Si) is the number of cycles to failure of a constant stress reversal Si, failure occurs when:

C is experimentally found to be between 0.7 and 2.2. Usually for design purposes, C is assumed to be 1.

This can be thought of as assessing what proportion of life is consumed by stress reversal at each magnitude then forming a linear combination of their aggregate.

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Though Miner's rule is a useful approximation in many circumstances, it has two major limitations:

1. It fails to recognise the probabilistic nature of fatigue and there is no simple way to relate life predicted by the rule with the characteristics of a probability distribution.

2. There is sometimes an effect in the order in which the reversals occur. In some circumstances, cycles of low stress followed by high stress cause more damage than would be predicted by the rule. It does not consider the effect of overload or high stress which may result in a compressive residual stress. High stress followed by low stress may have less damage due to the presence of compressive residual stress.

Nonlinear Damage Rule - Marco and Starkey Method

Many nonlinear damage theories have been proposed which attempt to overcome the shortcomings of Miner's Rule. The following is a general description of a nonlinear damage approach which was proposed by Richard and Newmark and was developed further by Marco and Starkey. 

The theory uses a nonlinear damage exponent P to augment Miner's Rule. The damage fraction is thus given by:

The value of P is considered to be greater than 0.0 and less than or equal to 1.0, with the value increasing with stress level. Note that with P = 1.0, this method is equivalent to Miner's Rule.

The nonlinear method described has good correlation to observed material behaviour and can be used to sum damage in high temperature applications where there is interaction between creep and fatigue. However, like all nonlinear theories, it requires a material constant that requires a considerable amount of testing to determine and may not be available for a given material or application.

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STRESS LIFE

Summary

This module calculates the fatigue life of a part under constant amplitude oscillatory loading assuming the stress range controls fatigue life.

The Stress-Life method (also referred to as the S-N method) was the first approach used in an attempt to understand and quantify metal fatigue. The Stress-Life approach is generally categorized as a high-cycle fatigue methodology, and is still widely used in design applications where the applied stress is primarily within the elastic range of the material and the resulting fatigue lives (cycles to failure) are long.

The Stress-Life method does not work well in low-cycle applications, where the applied strains have a significant plastic component due to high load levels. For these applications, a Strain-Life Fatigue Analysis is more appropriate. 

Roughly speaking, low-cycle fatigue applications are those with less than 10,000 loading cycles during the component life.  High-cycle fatigue is associated with component lives greater than 100,000 cycles.  The transition life between low- and high-cycle fatigue depends on the material being considered, and is usually between 10,000 and 100,000 cycles.  In general, the Stress-Life approach should not be used to estimate fatigue lives below 10,000 cycles.

Background Information

Stress-Life Diagram (S-N Diagram)

The basis of the Stress-Life method is the Wohler S-N diagram, shown schematically for two materials in Figure 1.  The S-N diagram plots nominal stress amplitude S versus cycles to failure N.  There are numerous testing procedures to generate the required data for a proper S-N diagram.  S-N test data are usually displayed on a log-log plot, with the actual S-N line representing the mean of the data from several tests.

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Figure 1:  Typical S-N Curves

Endurance Limit

Certain materials have a fatigue limit or endurance limit which represents a stress level below which the material does not fail and can be cycled infinitely.  If the applied stress level is below the endurance limit of the material, the structure is said to have an infinite life.  This is characteristic of steel and titanium in benign environmental conditions. A typical S-N curve corresponding to this type of material is shown Curve A in Figure 1.

Many non-ferrous metals and alloys, such as aluminium, magnesium, and copper alloys, do not exhibit well-defined endurance limits.  These materials instead display a continuously decreasing S-N response, similar to Curve B in Figure 1.  In such cases a fatigue strength Sf for a given number of cycles must be specified.  An effective endurance limit for these materials is sometimes defined as the stress that causes failure at 1x108 or 5x108 loading cycles.

The concept of an endurance limit is used in infinite-life or safe stress designs. It is due to interstitial elements (such as carbon or nitrogen in iron) that pin dislocations, thus preventing the slip mechanism that leads to the formation of micro-cracks.  Care must be taken when using an endurance limit in design applications because it can disappear due to:

Periodic overloads (unpin dislocations) Corrosive environments (due to fatigue corrosion interaction) High temperatures (mobilize dislocations)

The endurance limit is not a true property of a material, since other significant influences such as surface finish cannot be entirely eliminated.  However, a test values

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(Se') obtained from polished specimens provide a baseline to which other factors can be applied. Influences that can affect the endurance limit include:

Surface Finish Temperature Stress Concentration Notch Sensitivity Size Environment Reliability

Such influences are represented by reduction factors, k, which is used to establish a working endurance strength Se for the material:

Power Relationship

When plotted on a log-log scale, an S-N curve can be approximated by a straight line as shown in Figure 3.  A power law equation can then be used to define the S-N relationship:

where b is the slope of the line, sometimes referred to as the Basquin slope, which is given by:

Given the Basquin slope and any coordinate pair (N, S) on the S-N curve, the power law equation calculates the cycles to failure for a known stress amplitude.  

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Figure 3:  Idealized S-N Curve

The power relationship is only valid for fatigue lives that are on the design line.  For ferrous metals this range is from 1x103 to 1x106 cycles.  For non-ferrous metals, this range is from 1x103 to 5x108 cycles.   Note the empirical relationships and equations described above are only estimates. Depending on the level of certainty required in the fatigue analysis, actual test data may be necessary.

Fatigue Ratio (Relating Fatigue to Tensile Properties)

Through many years of experience, empirical relations between fatigue and tensile properties have been developed.  Although these relationships are very general, they remain useful for engineers in assessing preliminary fatigue performance.

The ratio of the endurance limit Se to the ultimate strength Su of a material is called the fatigue ratio.  It has values that range from 0.25 to 0.60, depending on the material.  

For steel, the endurance strength can be approximated by:

and:

 

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In addition to this relationship, for wrought steels the stress level corresponding to 1000 cycles, S1000, can be approximated by:

Utilizing these approximations, a generalized S-N curve for wrought steels can be created by connecting the S1000 point with the endurance limit, as shown in Figure 4.

Figure 4: Generalized S-N Curve for Wrought Steels

Mean Stress Effects

Most basic S-N fatigue data collected in the laboratory is generated using a fully-reversed stress cycle.  However, actual loading applications usually involve a mean stress on which the oscillatory stress is superimposed, as shown in Figure 5.  The following definitions are used to define a stress cycle with both alternating and mean stress.

The stress range is the algebraic difference between the maximum and minimum stress in a cycle:

The stress amplitude is one-half the stress range:

The mean stress is the algebraic mean of the maximum and minimum stress in the cycle:

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Two ratios that are often defined for the representation of mean stress are the stress ratio R and the amplitude ratio A:

For fully-reversed loading conditions, R is equal to -1.  For static loading, R is equal to 1.  For a case where the mean stress is tensile and equal to the stress amplitude, R is equal to 0.  A stress cycle of R = 0.1 is often used in aircraft component testing, and corresponds to a tension-tension cycle in which the minimum stress is equal to 0.1 times the maximum stress.

Figure 5: Typical Cyclic Loading Parameters

The results of a fatigue test using a nonzero mean stress are often presented in a Haigh diagram, shown in Figure 6.  A Haigh diagram plots the mean stress, usually tensile, along the x-axis and the oscillatory stress amplitude along the y-axis.  Lines of constant life are drawn through the data points.  The infinite life region is the region under the curve and the finite life region is the region above the curve. For finite life calculations the endurance limit in any of the models can be replaced with a fully reversed (R = -1) alternating stress level corresponding to the finite life value.

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Figure 6:  Example of a Haigh Diagram

A very substantial amount of testing is required to generate a Haigh diagram, and it is usually impractical to develop curves for all combinations of mean and alternating stresses.  Several empirical relationships that relate alternating stress to mean stress have been developed to address this difficulty.  These methods define various curves to connect the endurance limit on the alternating stress axis to either the yield strength, Sy, ultimate strength Su, or true fracture stress Sf on the mean stress axis. 

The following relations are available in the Stress-Life module:

Goodman (England, 1899):

Gerber (Germany, 1874):

Soderberg (USA, 1930):

Morrow (USA, 1960s):

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A graphical comparison of these equations is shown in Figure 7.  The two most widely accepted methods are those of Goodman and Gerber.  Experience has shown that test data tends to fall between the Goodman and Gerber curves.  Goodman is often used due to mathematical simplicity and slightly conservative values.  Other observations related to the mean stress equations include:

All methods should only be used for tensile mean stress values.

For cases where the mean stress is small relative to the alternating stress (R <<

1), there is little difference in the methods.

The Soderberg method is very conservative.  It is used in applications where

neither fatigue failure nor yielding should occur.

For hard steels (brittle), where the ultimate strength approaches the true

fracture stress, the Morrow and Goodman curves are essentially equivalent.

For ductile steels (Sf > Su), the Morrow model predicts less sensitivity to mean

stress.

As the R approaches 1, the models show large differences. There is a lack of

experimental data available for this condition, and the yield criterion may set

design limits.

Figure 7:  Comparison of Mean Stress Equations

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NotchesThus far, the discussion of Stress-Life fatigue analysis assumed a smooth, unnotched specimen. However, in practice most fatigue failures occur at notches or stress concentrations. 

Almost all machine components and structural members contain some form of geometrical or micro-structural discontinuities. These discontinuities, or stress concentration factors, often result in maximum local stresses Smax that are many times greater than the nominal stress S of the member.  In ideally elastic members, the ratio of these stresses is designated the theoretical stress concentration factor KT:

The theoretical stress concentration factor is solely dependent on the geometry and the mode of loading (axial, in-plane bending, etc.)

In the Stress-Life approach, the effect of notches is accounted for by the fatigue notch factor Kf (also known as the fatigue stress concentration factor).  The fatigue notch factor relates the unnotched fatigue strength (the endurance limit for ferrous metals) of a member to its notched fatigue strength:

In almost all cases, the fatigue notch factor is less than the stress concentration factor, and is less than 1.  That is:

The stress concentration factor KT can be related to the fatigue notch factor Kf.

Unlike the stress concentration factor KT, the fatigue notch factor Kf is dependent on the type of material and notch size. To account for these additional effects, a notch sensitivity factor q was developed:

The values of q range from 0 (no notch effect, Kf = 1) to 1 (full theoretical effect, Kf = KT). 

A number of researchers have proposed analytical relationships for the determination of q, based on correlation to experimental data. The most common relationships are those proposed by Peterson and Neuber. Both the Peterson and Neuber relations are empirical curve fits to data.  When used for analysis there is little difference to the approaches. Both methods show that q is related to material, notch geometry, and notch size. Thus, two notches can have the same KT value, but different Kf values

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because of the different q values. Both approaches also appear to support a limiting value for Kf.  The limiting value is dependent on material, but is generally between 5 and 6. 

The Peterson Equation for notch factor sensitivity is:

where r is the notch root radius and a is a material constant. The constant a depends on the material strength and ductility and is obtained experimentally from long life fatigue tests on both notched and unnotched specimens. Sharp notches in hard metals tend to be notch sensitive.  For ferrous-based wrought metals, the constant a is given by:

The Neuber Equation for notch factor sensitivity is:

where r is the notch root radius and is a material constant that is related to the grain size of the material.

The notch factor Kf is usually used to correct the fatigue strength for the notched member. 

Although Kf can be used to correct the entire S-N curve, results tend to be conservative. Note that this module will calculate Kf, but will not incorporate it into any solutions. The user must divide the unnotched fatigue strength (and S1000 if desired) by the calculated Kf value to account for the effect of notches.

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Stress Life Technical Background

The stress life method is the classical method for fatigue analysis of metals and has its origins in the work of Wöhler in 1850. Stresses in the structure or component are compared to the fatigue limit of the material. The basis of the method is the materials S-N curve which is obtained by testing small laboratory specimens until failure.

Material Properties

Historically, these tests have been conducted in rotating bending. Today, it is often common to find test data for axial loading as well.

The fatigue limit, SFL, is the stress below which failures do not occur in the laboratory. Wöhler called this a safe stress level for design. Today we know that failures will occur below the safe stress level but it will take a very large number of cycles, longer than the 106 or 107 cycles used in normal fatigue testing. The finite life portion of the curve is fit to a power function relating the stress amplitude, S/2, and fatigue life in cycles, Nf. Be careful, some people use reversals, 2Nf, instead of cycles for plotting fatigue data. There are 2 reversals in one cycle. The intercept at 1 cycle, S f’, and slope, b, are taken as material constants.

Many times the fatigue behaviour of a material is unknown and must be estimated from the tensile properties. There is a strong correlation between fatigue strength and tensile strength. 

(From Forrest, Fatigue of Metals, Pergamon Press, London, 1962)

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The fatigue limit is approximated as one half of the tensile strength.

SFL = 0.5 Su

Many metal alloys are heat treatable and the hardness rather than the tensile strength are known. Another well known approximation, with just as much scatter as shown above, is that the tensile strength is approximately one half of the Brinell hardness, BHN. Combining these two approximations provides an estimate of the fatigue strength from the hardness.

SFL = 0.25( BHN )

It has been observed that the fatigue strength at 1000 cycles is approximately 0.9 S u. This gives two points on the SN curve, both in terms of hardness that can be used to estimate the entire SN curve.

Four material parameters are used to describe the materials stress life curve, Sf', b, SFL

and NFL , only three of which are independent. Leaving any one of the parameters

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blank will result in the fourth one being directly computed. If all four are entered, S FL

will be ignored.

Modifying Factors

Materials, as they are tested, are always in a different condition (surface finish, residual stress, etc.) from the materials as they are actually used. An important part of the analysis is to “correct” the basic materials data to obtain an estimate of the fatigue limit of the material in the component or structure of interest. Fatigue cracks usually nucleate on the surface so that the condition of the surface plays a major role in the fatigue resistance of a component. Test specimens are polished to eliminate the effects of surface finish. The degree of surface damage depends not only on the processing but also on the strength of the material. Higher strength materials are more susceptible to surface damage. To account for this in the analysis the material fatigue limit is reduced by a surface finish factor, KSF.

The original data for constructing this curve is shown below. The factors tend to provide conservative estimates for fatigue lives.

(From Noll and Lipson, “Allowable Working Stresses”, Society for Experimental Stress Analysis, Vol. III, no. 2, 1949) 

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These data are fit to a simple power function to obtain an estimate of the surface factor for any hardness steel.

 Ground 1.58 -0.085

 Machined 4.51 -0.265

 Hot Rolled 57.7 -0.718

 Forged 272 -0.995

Fatigue limits have historically been determined from small specimens, 6 mm in diameter, tested in rotating bending. High strength materials tested in tension tend to have lower fatigue limits. An empirical load factor, kL, is introduced for other types of loading.

kL

 Tension, Su ≤ 1500 MPa 0.92

 Tension, Su > 1500 MPa 1.0

 Bending 1.0

 Torsion 0.58

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Experiments show that smaller components tend to have higher fatigue limits than larger ones. This is accommodated in the analysis by introducing a size factor, ksize. Some people call this a gradient factor. One of the most widely used corrections is based on the diameter of a bar.

One of the problems associated with this simple approximation is what to do when the section is not round. This problem is overcome by defining an effective diameter. The volume of material subjected to 95% of the maximum stress in any shaped cross section is equated to a round bar of the same highly stressed volume. When these volumes are equated, the length cancelled and the ratio becomes one of the relative areas. If we define the cross sectional area of a non-circular section subjected to 95% of the maximum stress as A0.95, then the effective diameter is given by

Once these correction factors are determined, the fatigue limit of the machine component in the condition that it is being used in can be evaluated from the standard test specimen.

SFL (component) = SFL (material) · kSF · kL · ksize

These generalized empirical factors have the greatest confidence when applied to steel because they were all derived from on extensive database on steel accumulated from over 100 years of testing. The figures above have shown considerable scatter in the test data so that the factors must be regarded as approximate and are no substitute for actual test data for a critical application. That said, they do provide a good first approximation when no test data is available.

 Stress Concentrations

Stress concentrations are one of the most important factors affecting the fatigue life of any component or structure. These stress concentrations may be intentional in the design or unintentional such as deep machining marks or other processing related flaws. It seems reasonable to directly compare the maximum stress at a stress concentration to the estimated fatigue limit for that component.

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(From MacGregor and Grossman, “Effects of Cyclic Loading on Mechanical Behaviour of 24S-T4 and 75S-T6 Aluminium Alloys and SAE 4130 Steel”, NACA TN 2812, 1952) 

Test data for an unnotched bar, such as the one above pictured in blue, are plotted as blue circles.  This geometry has a stress concentration factor, Kt, equal to 1. The red dashed line represents estimated data for a geometry that has a stress concentration factor of 3.1, such as the notched bar pictured in red. The presence of the notch reduces the allowable nominal stress amplitude at any fatigue life by a factor equal to Kt. Notice that all of the actual test data, red triangles, lie above this estimate. This is because the effective stress concentration in fatigue is less than that predicted by the stress concentration factor, Kt. This effective stress concentration factor is called the fatigue notch factor, Kf. The variation between Kf and Kt is dependant on the size of the notch and strength of the material. A material that is very sensitive to notches will have Kf equal to Kt. If the material is very insensitive to notches, K f will be close to 1. A notch sensitivity factor, q, is introduced to quantify this sensitivity.

(From Peterson “Notch Sensitivity”, Metal Fatigue, Sines and Waisman, McGraw Hill, 1959) 

Smaller radii are less effective in fatigue than larger ones. Fatigue notch factors can be computed from Kt and q.

Kf = 1 + ( Kt – 1 )q

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Peterson fit the available test data for steel and aluminium to obtain an expression for Kf in terms of ultimate strength, Su, and notch radius, in mm.

for steel

for aluminium

Plastic deformation reduces the effect of a stress concentration at shorter lives. The fatigue notch factor becomes smaller as the lives become shorter. In ductile materials, stress concentrations play only a small role in determining the overall strength of the structure. One unfortunate consequence of this is that static strength testing often provides little information about the likely fatigue performance of a component. An estimate of the notched component SN curve may be obtained dividing the materials fatigue strength, including any modify factors, at 106 cycles by Kf and drawing a line to the unnotched fatigue strength at 1 cycle.

This has the effect of changing the slope of the material’s SN curve and leaving the intercept unchanged. If we include the modifying factors this new slope, bnotch, can be computed as

and the new notched SN curve will be given by

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Mean Stresses

Tensile mean stresses are known to reduce the fatigue strength of a component. Compressive mean stresses increase the performance and are frequently used to increase the fatigue strength of a manufactured part. The most common method for accounting for mean stresses is the Goodman Diagram. It was first proposed in 1890. Goodman writes ".. whether the assumptions of the theory are justifiable or not …. We adopt it simply because it is the easiest to use, and for all practical purposes, represents Wöhlers data." The fatigue limit for zero mean stress is plotted on one axis and the ultimate strength on the other.

The diagram can be described by

Stress concentration effects can be directly included in the Goodman diagram.

For finite lives, an equivalent completely reversed stress, Seq, is computed and compared with the component SN curve.

Here Kf is used to modify the mean stress but not the stress amplitude because stress concentration effects are already included in the component SN curve.

Direct use of these equations poses problems for situations involving high mean stresses or in situations that result in short fatigue lives. This problem is caused by the calculated elastic stress being higher than the actual stress and in some cases even higher than the ultimate strength. A preloaded bolt is a good example. A common way

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to overcome this problem is to apply the stress concentration factor only to the alternating stress and not to the mean stress. This has the potential to lead to non-conservative estimates at long lives with moderate mean stresses. Plastic deformation limits the stress around a stress concentration to some value between the yield and ultimate strength of the material. A more reasonable method is to apply the stress concentration factor to both mean and alternating stresses and then made a correction to the maximum stress Kf ( Sa + Sm ) when it exceeds the material’s yield strength. The maximum stress should be limited to about 80% of the ultimate strength.

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Definitions - Strain Life Analysis

Nominal Stress

The nominal stress is the stress away from any local stress concentrations.

 

Nominal Strain GlossaryClose

The nominal strain is the strain away from any local stress concentrations. 

Stress Range, ΔS GlossaryClose

The stress range, ΔS, is the peak to peak stress.

 

Mean Stress, SmGlossary

Close

The mean stress, Sm, is the average value of the stresses.

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Alternating Stress, SaGlossary

Close

The alternating stress, Sa, is the amount the stress deviates from the mean.  It is sometimes called the stress amplitude.

 

 

Stress Amplitude, SaGlossary

Close

The stress amplitude, Sa, is the amount the stress deviates from the mean.  It is sometimes called the alternating stress.

 

Strain Amplitude GlossaryClose

The strain amplitude is one half the strain range.

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Strain Range, GlossaryClose

The strain range, , is the maximum strain minus the minimum strain in a cycle.

 

Neuber's Rule GlossaryClose

Neuber’s rule is used to convert an elastically computed stress or stain into the real stress or strain when plastic deformation occurs. For example, we may compute a stress with elastic assumptions at a notch to be KtS and this stress exceeds the strength of the material. The real stress will be somewhere on the materials stress-strain curve at some point .

Neuber’s rule states, with some mathematical proof, that the product of the elastic solution is equal to the product of the real elastic plastic solution. Mathematically this is expressed as

KtS·Kte=·

 

Strain Life Curve GlossaryClose

The strain life curve is determined by testing materials in strain control. The strain range is controlled and the corresponding stress range and fatigue life are determined. It is convenient to consider the elastic and plastic strain amplitudes separately when curve fitting the test data. Two data points are plotted for each test, one for the elastic strain amplitude vs life and another one for the plastic strain amplitude vs life.

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Test data is then fit to a simple power function to obtain the material constants; fatigue ductility coefficient, ’f , fatigue ductility exponent, c , fatigue strength coefficient, ’f , and fatigue strength exponent, b ,

The total strain is then obtained by adding the elastic and plastic portions of the strain.

 

Cyclic Stress-Strain Curve GlossaryClose

The materials deformation during a fatigue test is measured in the form of a hysteresis loop. After some initial transient behavior the material stabilizes and the same hysteresis loop is obtained for every loading cycle. Each strain range tested will have a corresponding stress range that is measured. The cyclic stress strain curve is a plot of all of this data.

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The curve describes the behavior of the material after it has been plastically deformed. This behavior is usually different than the initial behavior that is measured in a traditional tensile test. A simple power function is fit to this curve to obtain three material properties; cyclic strength coefficient, K’ , cyclic strain hardening exponent, n’ , and elastic modulus, E.

 

Hysteresis Loop GlossaryClose

The stress-strain response of a material that is cyclically loaded is in the form of a hysteresis loop.

The hysteresis loop is often characterized by its stress range, , and strain range, .  The strain range is often broken up into the elastic part and the plastic part. 

Stress Concentration Factor, KtGlossary

Close

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Stress concentrations arise from any abrupt change in the geometry of a part under load. As a result, the stress distribution is not uniform throughout the cross-section.

For example, is often necessary to drill a hole in a plate. When a load, P, is applied, the presence of the hole disturbs the uniform nominal stress in the plate.

The profile of the stress in the cross-section through the center of the hole has the form shown in bottom of the figure. Notice that the maximum stress, max, is 3 and occurs at the edge of the hole. The factor of 3 is known as the stress concentration factor, Kt. It can be seen from the figure that any stress values larger than are localized in a region with a diameter approximately equal to D.

 

Fatigue Notch Factor, KfGlossary

Close

Experiments have shown that the effect of small notches is less than that estimated from the traditional stress concentration factor, Kt. The fatigue notch factor, Kf, can be thought of as the effective stress concentration in fatigue. It depends on the size of the stress concentration and the material. Small stress concentrations are more effective in high strength materials. This effect is dealt with using a notch sensitivity factor, q.

Kf = 1 + (Kt - 1)q

The notch sensitivity factor q is an empirically determined constant that depends on the notch radius and material strength.

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Ultimate Strength, SuGlossary

Close

The ultimate strength, Su, is also called the tensile strength. It is the maximum stress reached in an engineering stress strain diagram.

Glossary of Definitions

AAlternating Stress Sa

C Cyclic Stress-Strain Curve  

E Endurance Limit SFL

FFatigue LimitFatigue Notch Factor Fatigue Strength

SFL

Kf  

G Goodman Diagram  

H Hysteresis Loop  

L Loading Factor kL

M Mean Stress Sm

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NNeuber's RuleNominal StrainNominal Stress

 

S

Safety FactorSize FactorStrain Life CurveStrain RangeStress AmplitudeStress Concentration FactorStress Life CurveStress RangeSurface Finish Factor

 ksize

 Sa

Kf

 ΔSKSF

U Ultimate Strength Su

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