Hyperworks Fatigue

8/10/2019 Hyperworks Fatigue

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Knowledge BaseSearch Knowledge Base User Manuals [Document 82125]

Go to Search KB

OptiStruct User Guide 13.0User's GuideFatigue Analysis Fatigue Analysis

Fatigue Analysis

Fatigue analysis, usingS-N (stress-life),E-N (strain-life),andDang Van Criterion

(Factor of Safety)approachesfor predicting the life (number of loading cycles) of astructure under cyclical loading may be performed by using OptiStruct.

The stress-life method works well in predicting fatigue life when the stress

level in the structure falls mostly in the elastic range. Under such cyclicalloading conditions, the structure typically can withstand a large number of
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line segments. Normally, a one or two segment idealization is used.

Figure 2: One segment S-N curves in log-log scale

for segment 1 (1)

Where,

Sis the nominal stress range

Nfare the fatigue cycles to failure

b1is the first fatigue strength exponentS1is the fatigue strength coefficient

The S-N approach is based on elastic cyclic loading, inferring that the S-N curve should beconfined, on the life axis, to numbers greater than 1000 cycles. This ensures that no significant

plasticity is occurring. This is commonly referred to as high-cycle fatigue.

S-N curve data is provided for a given material on a MATFAT bulkDATA ENTRY . It is

referenced through a Material ID (MID) which is shared by a structural material definition.

Damage Model

Palmgren-Miner's linear damage summation rule is used. Failure is predicted when:

(2)

Where,
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Equivalent Nominal Stress

Since S-N theory deals with uniaxial stress, the stress components need to be resolved into onecombined value for each calculation point, at each time step, and then used as equivalent

nominal stress applied on the S-N curve.

Various stress combination types are available with the default being Absolute maximumprinciple stress. In general Absolute maximum principle stress is recommended for brittlematerials, while Signed von Mises stress is recommended for ductile material. The sign on the

signed parameters is taken from the sign of the Maximum Absolute Principal value.

Parameters affecting stress combination may be defined on a FATPARM bulk data entry. Theappropriate FATPARM bulk data entry may be referenced from a fatigue subcase definition

through the FATPARM Subcase Information entry.

Mean Stress InfluenceGenerally S-N curves are obtained from standard experiments with fully reversed cyclicloading. However, the real fatigue loading could not be fully reversed and the normal meanstresses have significant effect on fatigue performance of components. Tensile normal mean

stresses are detrimental and compressive normal mean stresses are beneficial, in terms of fatigue

strength. Mean stress correction is used to take into account the effect of non-zero mean stresses.

The Gerber parabola and the Goodman line in Haigh's coordinates are widely

used when considering mean stress influence, and can be expressed as:

Gerber: (3)

Goodman: (4)

Where,

Mean stress Sm= (Smax + Smin) / 2

Stress amplitude Sa = (Smax - Smin) / 2

Seis the stress range for fully reversed loading that is equivalent to the

load case with a stress range SRand a mean stress Sm

Suis ultimate strength

The Gerber method treats positive and negative mean stress correction inthe same way that mean stress always accelerates fatigue failure, while theGoodman method ignores the negative means stress. Both methods give


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conservative result for compressive means stress. The Goodman method isrecommended for brittle material while the Gerber method is recommendedfor ductile material. For the Goodman method, if the tensile means stress isgreater than UTS, the damage will be greater than 1.0. For Gerber method,

if the mean stress is greater than UTS, no matter tensile or compressive, the

damage will be greater than 1.0.

A Haigh diagram characterizes different combinations of stress amplitude and mean stress for agiven number of cycles to failure.

Figure 4: Haigh diagram and mean stress correction methods

Parameters affecting mean stress influence may be defined on a FATPARM bulkDATA

ENTRY . The appropriate FATPARM bulk data entry may be referenced from a fatiguesubcase definition through the FATPARM Subcase Information entry.

The Strain-Life (E-N) Approach

Monotonic Stress-Strain Behavior

Relative to the current configuration, the true stress and strain relationship can be defined as:

(5)

(6)

Where, Ais the current cross-section area, l is the current specimenlength, l0is the initial specimen length, and and are the true stress and

strain, respectively, Figure 5 shows the monotonic stress-strain curve in

true stress-strain space. In the whole process, the stress continues
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increasing to a large value until the specimen fails at C.

Figure 5: Monotonic stress-strain curve

The curve in Figure 5 is comprised of two typical segments, namely the

elastic segment OA and plastic segment AC. The segment OA keeps thelinear relationship between stress and elastic strain following Hooke Law:

(7)

Where, Eis elastic modulus and is elastic strain. The formula can also be

rewritten as:

(8)

by expressing elastic strain in terms of stress. For most of materials, therelationship between the plastic strain and the stress can be represented bya simple power law of the form:

(9)

Where, is plastic strain, Kis strength coefficient, and nis work hardening

coefficient. Similarly, the plastic strain can be expressed in terms of stress

as:

(10)The total strain induced by loading the specimen up to point B or D is the

sum of plastic strain and elastic strain:

(11)


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Cyclic Stress-Strain Curve

Material exhibits different behavior under cyclic load compared with that of

monotonic load. Generally, there are four kinds of response.

stable state

cyclically hardening

cyclically softening

softeningor hardening depending on strain range

Which response will occur depends on its nature and initial condition of heattreatment. Figure 6 illustrates the effect of cyclic hardening and cyclicsoftening where the first two hysteresis loops of two different materials are

plotted. In both cases, the strain is constrainedto change in fixed range,while the stress is allowed to change arbitrarily. If the stress range

increases relative to the former cycle under fixed strain range, as shown inthe upper part of Figure 6, it is called cyclic hardening; otherwise, it is

called cyclic softening, as shown in the lower part of Figure 6. Cyclic

response of material can also be described by specifying the stress rangeand leaving strain unconstrained. If the strain range increases relative to

the former cycle under fixed stress range, it is called cyclic softening;otherwise, it is called cyclic hardening. In fact, the cyclic behavior of

material will reach a steady state after a short time which generally occupiesless than 10 percent of the material total life. Through specifying different

strain ranges, a series of hysteresis loops at steady state can be

obtained. By placing these hysteresis loops in one coordinate system, asshown in Figure 7, the line connecting all the vertices of these hysteresisloops determine cyclic stress-strain curve which can be expressed in thesimilar form with monotonic stress-strain curve as:


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Figure 6: Material cyclic response (a) Cyclic hardening; (b) Cyclic softening


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Figure 7: Definition of stable stress-strain curve

(12)

Where, K'is cyclic strength coefficient, n'is strain cyclic hardening exponent.

Hysteresis Loop Shape

Bauschinger observed that after the initial load had caused plastic strain,

load reversal caused materials to exhibit anisotropic behavior. Based onexperiment evidence, Massing put forward the hypothesis that a stress-

strain hysteresis loop is geometrically similar to the cyclic stress straincurve, but with twice the magnitude. This implies that when the

quantity is two times of , the stress-strain cycle will lie on the

hysteresis loop. This can be expressed with formulas:

(13)(14)

Expressing in terms of , in terms of , and substituting it into Eq.12, the hysteresis loop formula can be deduced as:

(15)


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Strain-Life Approach

Almost a century ago, Basquin observed the linear relationship betweenstress and fatigue life in log scale when the stress is limited. He put forward

the following fatigue formula controlled by stress:

(16)

Where, a is stress amplitude, fatigue strength coefficient, bfatigue

strength exponent. Later in the 1950s, Coffin and Manson independently

proposed that plastic strain may also be related with fatigue life by a simplepower law:

(17)

Where, is plastic strain amplitude, fatigue ductility coefficient, cfatigue

ductility exponent. Morrow combined the work of Basquin, Coffin andManson to consider both elastic strain and plastic strain contribution to the

fatigue life. He found out that the total strain has more direct correlationwith fatigue life. By applying Hooke Law, Basquin rule can be rewritten as:

(18)

Where, is elastic strain amplitude. Total strain amplitude, which is the

sum of the elastic strain and plastic stain, therefore, can be described byapplying Basquin formula and Coffin-Manson formula:

(19)

Where, is the total strain amplitude, the other variable is the same withabove. Figure 8 illustrates three methods in log scale in stress-life

space. Two straight lines, which represent Basquin formula and Coffin-Manson rule respectively, intersect at a point where elastic strain is equal to

the plastic strain and the fatigue life predicted by the two methods is thesame. The fatigue life at the intersection point is called transition lifeandcan be calculated as:

(20)

by combining Eq.17 and Eq.18, at the same time, applying the conditions:


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(21)

(22)

Where, Ntis the transition life. When fatigue life is less than the transition

life, plastic strain plays the controlling role in life prediction; otherwise,

elastic strain plays the key role.

Figure 8: Strain-life curve in log scale

Damage Accumulation Model

In the E-N approach, use the same damage accumulation model as the S-Napproach, which is Palmgren-Miner's linear damage summation rule.

Mean Stress Influence

The fatigue experiments carried out in the laboratory are always fully

reversed, whereas in practice, the mean stress is inevitable, thus the fatiguelaw established by the fully reversed experiments must be corrected before

applied to engineering problems. Morrow is the first to consider the effect of

mean stress through introducing the mean stress 0in fatigue strengthcoefficient by:

(23)

Thus the entire fatigue life formula becomes:

(24)

Morrow's equation is consistent with the observation that mean stress


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effects are significant at low value of plastic strain and of little effect at highplastic strain.

Smith, Watson and Topper proposed a different method to account for the

effect of mean stress by considering the maximum stress during one cycle

(for convenience, this method is called SWT in the following). In this case,the damage parameter is modified as the product of the maximum stressand strain range in one cycle. For a fully reversed cycle, the maximum stressis given by:

(25)

By multiplying Eq.19 with Eq.25, it can be rewritten as:

(26)

The SWT method will predict that no damage will occur when the maximumstress is zero or negative, which is not consistent with the reality.

When comparing the two methods, the SWT method predicted conservative

life for loads predominantly tensile, whereas, the Morrow approach providesmore realistic results when the load is predominantly compressive.

Neuber Correction

Strain-life analysis is based on the fact that many critical locations such as

notch roots have stress concentration, which will have obvious plastic

deformation during the cyclic loading before fatigue failure. Thus, the elastic-plastic strain results are essential for performing strain-life analysis. Neuber

correction is the most popular practice to correct elastic analysis results intoelastic-plastic results.

In order to derive the local stress from the nominal stress that is easier toobtain, the concentration factors are introduced such as the local stress

concentration factor , and the local strain concentration factor .

(27)

(28

)

Where, is the local stress, is the local strain, Sis the nominal stress, and eis

the nominal strain. If nominal stress and local stress are both elastic, the

local stress concentration factor is equal to the local strain concentration

factor. However, if the plastic strain is present, the relationship between

and no long holds. Thereafter, focusing on this situation, Neuberintroduced a theoretically elastic stress concentration factor Ktdefined as:


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(29)

Substitute Eq.27 and Eq.28 into Eq.29, the theoretical stress concentrationfactor Ktcan be rewritten as:

(30)

Through linear static FEA, the local stress instead of nominal stress isprovided, which implies the effect of the geometry in Eq.30 is removed, thusyou can set Ktas 1 and rewrite Eq.30 as:

(31)

Where, e, is locally elastic stress and locally elastic strain obtained from

elastic analysis, , the stress and strain at the presence of plasticstrain. Both and can be calculated from Eq.31 together with the

equations for the cyclic stress-strain curve and hysteresis loop.

Dang Van Criterion (Factor of Safety)

The Dang Van criterion is used to predict if a component will fail in its entire

load history. In certain physical systems, components may be required tolast infinitely long. For example, automobile components which undergo

multiaxial cyclic loading at high rotational velocities (like propeller shafts)reach their high cycle fatigue limit within a short operating life. The

conventional fatigue result that specifies the minimum fatigue cycles tofailure is not applicable in such cases. It is not necessary to quantify the

amount of fatigue damage, but just to consider if any fatigue damage will

occur during the entire load history of the component. If damage does occur,the component cannot experience infinite life. Fatigue analysis based on the

Dang Van criterion is designed for this purpose.

Fatigue crack initiation usually occurs at zones of stress concentration such

as geometric discontinuities, fillets, notches and so on. This phenomenontakes place in the microscopic level and is localized to certain regions like

grains which have undergone local plastic deformation in characteristic intra-

crystalline bands. The Dang Van approach postulates a fatigue criterionusing microscopic variables in the apparent stabilization state; this is a stateof elastic shakedown if no damage occurs. The main principle of the criterion

is that the usual characterization of the fatigue cycle is replaced by the localloading path and so damaging loads can be distinguished.

The general procedure of Dang Van fatigue analysis is:


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1.Evaluate the macroscopic stresses , for each location at a different point in time.

2.Split the macroscopic stress into a hydrostatic part and a deviatoric part .

3.Calculate the stabilized microscopic residual stress based on the following equation:

The expression is minimized with respect to and maximized with respect to t.

4.Calculate the deviatoric part of microscopic stress.

5.Calculate factor of safety (FOS):

Where, band aare material constants.

If FOS is less than 1, the component cannot experience infinite life.

OptiStruct Factor of Safety setup

1.The torsion fatigue limit and hydrostatic stress sensitivity values required for an FOS analysiscan be set in the optional FOS continuation line on the MATFAT bulkDATA ENTRY .

2.The Dang Van criterion type can be selected on the FATPARM bulkDATA ENTRY .

3.Factor of Safety output can be requested using the FOS I/O options entry.

Other Factors Affecting Fatigue

Surface Condition (Finish and Treatment)

Surface condition is an extremely important factor influencing fatigue

strength, as fatigue failures nucleate at the surface. Surface finish and

treatment factors are considered to correct the fatigue analysis results.

Surface finish correction factor Cfinishis used to characterize the roughness ofthe surface. It's presented on diagrams that categorize finish by means ofqualitative terms such as polished, machined or forged.
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Figure 9*: Surface finish correction factor for steels(* Source: Yung-Li Lee, Jwo. Pan, Richard B. Hathaway and Mark E. Barekey. Fatigue testing and analysis:Theory and practice, Elsevier, 2005)

Surface treatment can improve the fatigue strength of components.

NITRIDED, SHOT-PEENED, COLD-ROLLED are considered for surface

treatment correction. It is also possible to input a value to specify thesurface treatment factor Ctreat.

In general cases, the total correction factor is Csur=Ctreat*Cfinish.

If treatment type is NITRIDED, then the total correction

is Csur=2.0 *Cfinish(Ctreat=2.0).If treatment type is SHOT-PEENED or COLD-ROLLED, then the total

correction is Csur= 1.0. It means you will ignore the effect of surface finish.

The fatigue endurance limit FL will be modified by Csuras: FL' =FL *Csur. For

two segment S-N curve, the stress at the transition point is also modified bymultiplying by Csur.

Surface conditions may be defined on a PFAT bulkDATA ENTRY . Surface

conditions are then associated with sections of the model through the FATDEF bulkDATA

ENTRY .

Fatigue Strength Reduction Factor

In addition to the factors mentioned above, there are various others factors

that could affect the fatigue strength of a structure, e.g., notch effect, size

effect, loading type. Fatigue strength reduction factor Kfis introduced toaccount for the combined effect of all such corrections. The fatigue

endurance limit FLwill be modified by Kf as: FL' =FL/Kf
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The fatigue strength reduction factor may be defined on a PFAT bulkDATAENTRY . It may then be associated with sections of the model through the FATDEF bulk data

entry.

If both Csurand Kf are specified, the fatigue endurance limit FLwill be

modified as: FL' =FL * Csur/Kf.Csurand Kf have similar influences on the E-N formula through its elastic partas on the S-N formula. In the elastic part of the E-N formula, a nominalfatigue endurance limit FLis calculated internally from the reversal limit of

endurance Nc. FLwill be corrected if Csurand Kfare presented. The elastic partwill be modified as well with the updated nominal fatigue limit.

Setting Up a Fatigue Analysis

Linear Superposition of Multiple FEA/Load Time History LoadCases

When there are several load cases at the same time, all of which vary

independently of one another, the principle of linear superposition will beused to combine all load cases together to determine the stress variation ateach calculation point due to the combination of all loads. The formula is:

(32)

Where,

nis the total number of load cases

Pk(t)and are, respectively, the time variation of the k-thload time

history and the total stress tensor

PFEA,kand are, respectively, the k-thload magnitude and stress tensor

from FE analysis

Load Time History Compression

This option is used to save calculation time. It will remove small cycles

(defined by a gate value) and intermediate points.
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Figure10: Sample showing removal of small cycles

When removing small cycles, adjacent turning points, where the difference isless than the maximum range multiplied by relative gate value, will be

removed from each channel. However, phase relationship will bemaintained, when peaks and valleys occur on different channels at different

times. This is shown by the sample above. In the first channel (top), thepoints at time 4 and 5 will be removed when the absolute gate equals one,

while in the second channel (bottom), the points at time 1 and 2 will not beremoved in order to keep the phase relationship between channels.


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Figure11: Sample showing removal of intermediate points

Removing intermediate points is another important mechanism to savecomputation time. If any point on the load-time history is neither a peak

nor valley point, it will not contribute in determining any stress cycle. Suchpoints could be screened out in the fatigue computation without losing the

accuracy, but the computation time could be saved significantly. Forexample, the left column in Fig 11 shows three load-time histories of three

super-positioned loadcases, respectively. After removing the intermediatepoints, the three load-time histories are obtained as in the right column,

which can produce the same fatigue results as the left column, but use muchless time. This mechanism is built in OptiStructand is effective automatically.

Fatigue Loads, Events and Sequences

Fatigue loading is defined by scaling a static subcase with a load-time

history.

A fatigue event consists of one or more static loadcases applied

simultaneously in the same time duration scaled by load-time histories. For

fatigue events with more than one static loadcase stress, linearsuperposition is used.

A fatigue sequence consists of a number of fatigue events and repeated

instances of these events. A fatigue sequence can be made up of other sub


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fatigue sequences and/or fatigue events. In this way, you can define verycomplex events and sequences for fatigue analysis.

In OptiStruct, fatigue sequences defined in fatigue subcases (referred by

FATSEQ) are the basic loading blocks. The fatigue life results of these fatiguesubcases are calculated as the number of repeats of the loading block.

Below is an example of a "tree-like" fatigue sequence, which can be definedin OptiStruct, with FSEQ# identifying fatigue sequences and FEVN# identifying

fatigue events:

Figure 12: Example of a "tree-like" fatigue sequence

Fatigue loading is defined by a FATLOADbulkDATA ENTRY ,where a static

subcase and a load-time history are associated.

A fatigue loading event is defined by a FATEVNTbulkDATA ENTRY ,whereone or more fatigue loads (FATLOAD) are selected.

A fatigue loading sequence is defined by a FATSEQbulkDATA ENTRY ,

where a sequence of one or more fatigue loading events or other fatigueloading sequences is given. The appropriate FATSEQ bulk data entry may bereferenced from a fatigue subcase definition through the FATSEQSubcase

Information entry.

Fatigue Analysis Index Multi-body Dynamics Simulation

Product:
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