mechanical failure

MechanicalEngineeringDesign

MechanicalFailure

Dirk Pons

Mechanical FailureThird Edition, 2011

This paper describes themechanisms wherebymaterial fail, and themechanical engineeringprinciples to design againstfailure. Various theories offailure are presented.Another effect thatinfluences the failure of apart is the shape of thegeometry, particularly thesharpness of features,which concentrates thestresses to above nominalvalues. Thereafter the effectof fatigue is presented.

This material is provided under aCreative Commons license(AttributionNon-Commercial No Derivatives), seebelow for details. The Author[s] acceptno liability for the use or inability to usethe material in this book.

Published in New Zealand518 Hurunui Bluff RdHawardenNew Zealand

Copyright © Dirk Pons

About the AuthorDirk Pons PhD CPEngMIPENZ MPMI isprofessional EngineerTohunga Wetepangaa n d a C h a r te r e dProfessional Engineer inNew Zealand. Dirk is aSenior Lecturer at theUniversity of Canterbury,New Zealand. He holds aPhD in mechanicalengineering and amasters degree inbusiness leadership. TheA u t h o r w e l c o m e sc o m m e n t s a n ds u g g e s t i o n [email protected]

Mechanical failure1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 THEORIES OF FAILURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1 Why they are useful . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Theories using stress or strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Theories using strain energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Other Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 STATIC FAILURE OF DUCTILE AND BRITTLE MATERIALS . . . . . . . . . . . . . . . . . . . . . . . 104 GEOMETRIC STRESS CONCENTRATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.1 Mechanism for stress concentrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 Geometric Stress concentration factors for stepped shafts . . . . . . . . . . . . . . . . 124.3 Geometric Stress concentration factors for semicircular notch in a circular shaft

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4 Geometric Stress concentration factors for a U notch in a circular shaft . . . . 164.5 Other stress concentrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.6 Ways of avoiding stress concentrations in shaft shoulders . . . . . . . . . . . . . . . . 185 FATIGUE FAILURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.1 Mechanism of Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.2 Endurance limit of rotating beam specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.3 Fatigue Strength of Actual Machine Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.4 Low Cycle Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 CUMULATIVE FATIGUE DAMAGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.1 Manson’s approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.2 Miner’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.3 Cycle counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.3.1 Rainflow cycle counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.3.2 Reservoir cycle counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 FLUCTUATING STRESSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 FATIGUE IN BIAXIAL STRESS SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 SURFACE FATIGUE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.1 Hertz Contact Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.2 Buckingham's Contact Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4210 CORROSION FATIGUE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211 DESIGNING AROUND FATIGUE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.1 Changes to Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.2 Design Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.3 Surface Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.4 FATIGUE APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4411.4.1 REVERSED BENDING AND STATIC TWISTING OF SHAFTS . . . . . . . . . . . . . . . . 45

4

Mechanical failure

1 INTRODUCTION: WHAT ARE WE DESIGNING AGAINST?

Means of failureMachine parts fail by one (or more) of the following means:M ABUSE

Someone willfully uses the machine or part in a way which could not beexpected of a reasonable person. Typical of vandalism.

M OVERUSEDuty is more severe than the part can tolerate. However the application iscorrect. Eg using a small electric drill in a building construction industry. Manyconsumer products like this fit into this category. The cause is one or more of: * overuse by user, * under specification at concept stage, * design fault

M FAIR WEAR AND TEARMachines and parts have finite lives, after which they fracture or show grosswear.

M CORROSIONA machine can sometimes be designed to last forever against wear andfracture, but something else like corrosion will get it in the end. It is usually notgood marketing to produce machines which last too long.

Design strategiesThe mechanical designer has to take into account the number of factors whendesigning a machine or part. Perhaps the most critical factors are:* technical: expectation that the user has for the product life, and performance* manufacturing: cost of producing and selling the partThere is often some conflict between these, and the following strategy could befollowed, stopping where enough had been done.

(1) Static and Brittle failurePrevent gross fracture on first application of load. This is static failure, and willbe discussed below. Typically it is necessary to keep stresses below theultimate tensile strength of the material, or even below the yield. Classicalstructural mechanics is used to determine the stresses in the part, ornumerical calculation, or testing. A safety factor is taken into account, of atleast 2.

(2) Fatigue failurePrevent fatigue failure happening unreasonably early in the expected productlife. There are two ways for the designer to achieve this:(a) Use yield strength, with a safety factor of 4 to 12. This was the only

method available before the effects of fatigue were quantified, and it is

5

Design criteria:1: Avoid fracture2: Avoid yield3: Limit deflection

a method still used in some non-critical (cost/performance)applications.

(b) Calculate fatigue strength of material. This method will be shown in thisbook. Use a safety factor of 2 (standard), 1,5 (product use underclosely defined and controlled conditions) or 1,2 (cutting as close aspossible to the bone, only suitable for precision designs, where testingwill be used to verify performance). Certain types of wear are alsofatigue phenomena.

(3) DeflectionEnsure that deflection is acceptable. This only applies to structures that aresensitive to deflection. Typical examples are gears (where face contact isaffected), and gas turbines (where inter blade clearance, and blade-shroudclearance is affected). Classical structuralmechanics is used for these designs,sometimes with the assistance of finiteelement analysis. Creep can also be aproblem in these cases. In other types ofdesign problem the possibility of buckling(instability) needs to be considered, eg longcolumns, thin walled tubes, flat parts(especially plastic).

(4) CorrosionAvoid corrosion failures of the part. This is usually done by selection ofappropriate material.

(5) StickingOne of the causes of failure of mechanical machines is the sticking of amotion. This needs to be considered during the design stages. Parts may stickfor a number of reasons, including:

C excessively loose fit permits parts to change orientation and jamC tight fit causes frictionC debris in joint, from wear, corrosion or originating externallyC thermal expansion/contraction as parts change temperatureC loss of lubricant

The designer needs to complete sufficient of these design calculations to besatisfied. Thereafter will come the detailed drawing, and the considerations ofmanufacturing.

6

2 THEORIES OF FAILURE

Where a part is subject to uniaxial tensile stress only, then it will begin to fail whenthe imposed stress equals the yield strength of that material. If the imposed stress isincreased still further, up to the ultimate tensile strength of the material, then the partwill fracture completely. The yield strength Re and the ultimate tensile strength Rm arematerial properties, and are independent of the size of the sample.

Please note that the theories of failure apply to static loading. Static means that thestresses (or strains) don’t change with time.

2.1 Why they are useful

In engineering design the function of the part usually requires that fracture beavoided, and hence that imposed stresses be kept below the ultimate tensilestrength. In addition it is usual to design so that imposed stresses are below the yieldstrength of the material. This is because permanent deformation occurs at stressesgreater than the yield strength, and such deformation disrupts the function of thepart.

However not all machine parts are subject to simple tension. Instead, and moretypically, they may be subject to three dimensional stress patterns, that iscombinations of Fx, Fy, Fz, Jxy, Jxz, Jyz, (or the corresponding strains).

As appropriate tests cannot always be made so that the material is subjected to thereal conditions of stress, it is usual to convert the three dimensional loading into asingle effective tensile stress, which can then be compared to the results froma tensile test. In order to make this conversion, it is necessary to have an equationthat combines the various components of the three dimensional loading. Variousequations have been developed to account for various load cases, and the differentsensitivities of material to particular components of the loading. These equations arecalled theories of failure.

The selection of an appropriate theory of failure is based largely on the type ofmaterial: brittle or ductile, as described in following sections. Note that these theoriesof failure apply to static loading, that is loading that does not change with time. Thereare a number of theories which are described below. Not all of these are valid for allmaterial and load cases.

Note the following terminology:Rm ULTIMATE TENSILE STRENGTHRe YIELD STRENGTH in simple tension 8 or < POISSON'S RATIO e.g. 0,3 for steelE MODULUS OF ELASTICITY e.g. 206 x 109 Pa for steelG MODULUS OF RIGIDITY e.g. 82,6 x 109 Pa for steel

7

K BULK MODULUSFx, Fy, Fz direct stresses in x, y and z axesF1, F2, F3 principal stresses in x, y and z axes (no shear stresses present)Jxy, Jxz, Jyz shear stresses across z, y and x axes, direct stress in x, y or z direction( shear strain in xy, xz or yz plane

Note also the relationships between the fundamental elastic constants:

2.2 Theories using stress or strain

Maximum principal stressThis theory has that fracture occurs when the maximum principal stress

reaches the yield strength in simple tension.

Maximum Shear stressBy this theory fracture occurs when the maximum shear stress

reaches the shear yield strength. This is also called the Tresca theory.

Maximum Strain Here fracture is assumed to occur when the maximum strain

reaches the strain at yield in a simple tension test.

8

2.3 Theories using strain energy

Total strain energyThis theory provides for failure when the total strain energy of the part:

reaches that in a part under simple tension, namely:

Distortion energyThis criterion gives failure when the distortion energy (using principal stresses) of thepart

reaches that in simple tension, namely:

This is also called the von Mises theory.

Note that (distortion energy) = (total strain energy) - (dilation energy). The Distortionenergy is the energy required to change the shape without changing the volume. TheDilation energy changes the volume but not the shape.

The distortion energy theory is one of the better ones. It permits the use of thefatigue strength in place of the yield strength. Distortion energy is also sometimescalled shear strain energy, or the volumetric strain energy.

9

Octahedral shear stressThis theory assumes failure when the octahedral shear stress in the part:

reaches that in simple tension, namely

2.4 Other Theories

Mohr's theoryThis theory accounts for reversal of stresscomponents. It accommodates materials thathave different ultimate tensile andcompressive strengths. For a two dimensionalstress system, the permissible combinationsare as shown in the shaded area shown.

10

3 STATIC FAILURE OF DUCTILE AND BRITTLE MATERIALS

Ductile MaterialsFor ductile materials, failure is by yield rather than fracture. Planes of atoms aremoved by the distance of the lattice spacing. The mechanism is one wheredislocations (imperfections in the lattice) move atoms one by one under lower forcethan would be required to move the whole plane. Shear stress drives thedislocations.

Work hardening occurs as the dislocations tangle up, and stress relieving as theysmooth away. Interstitial atoms may diffuse to the dislocations and pin them, therebyhardening the material. Some dislocations may flow earlier than others, causingplastic stretch bands.

The shearing yield strength for a ductile material is typically about 0,57 of the tensileyield strength.

The Distortion energy theory (von Mises) and the Octahedral shear stress theoryare the most satisfactory. Maximum shear stress (Tresca) produces conservativeresults. Theories of maximum principal stress and maximum principal strain shouldnot be used.

Brittle MaterialsThese materials fail by fracture rather than by yielding. The mechanism isuncontrolled crack growth after cracks exceed a critical length. Thus the designershould try to keep these materials in compression.

Mohr's theory and theories of Maximum principal stress and Maximum principalstrain may be used here, but should not be used for yield failures.

Brittle materials typically have greater compressive that tensile strength. This isaccommodated in Mohr's theory.

11

4 GEOMETRIC STRESS CONCENTRATION

An important consideration in design is that parts are not uniform in shape, like thestructural mechanics equations generally assume. The shape of the geometry,particularly the sharpness of features, concentrates stresses to above nominalvalues, and we need to be able to take this into account.

4.1 Mechanism for stress concentrations

Stress concentrations arise where forces (or stresses) are concentrated at aboveaverage values in small regions. The nominal load on a part may usually be readilydetermined from the applied load and the minimum cross sectional area. Howeverthe geometry of the part (eg a hole) may disrupt the microscopic load bearing paths,causing them to crowd at some places. In particular the load carrying path cannotcross air gaps or voids (such as holes). The load carrying paths usually bunch upcloser in order to get round the obstacle. As a result the force distribution across thesection will become non-uniform. Stress is a measurement of the severity of theforce distribution, and thus local high stresses will occur.

The stress concentration effects of different geometry are determined by experiment,and presented in graphs. Generally the sharper the cut into the load carrying path,the greater the stress concentration. Stress concentration values are always greaterthan 1,00. The local stress at the most heavily loaded region is given by the productof the nominal stress (determined on the basis of the smallest cross section), and thestress concentration factor. The part is likely to break at the region of highest loading,which is usually at the stress concentration. These stress concentration factors aregeometric stress concentration factors (called Kt), as they depend only on thegeometry of the part: they are independent of the material.

The designer should attempt to reduce stress concentrations where ever possible.The means to do this are:* provide large fillet radii* specify smoother surface texture in critical regions* avoid scratches, surface and inclusion defects, especially those that cut

across the load carrying path* provide for gradual changes in section, or where this is not possible, consider

providing smaller stress raisers at the sides of the main stress raiser

Particular care should be taken with welding, where several disadvantageous stressmechanisms are combined: residual stresses, rough surfaces, possibility ofinclusions, modified metallurgy (eg heat affected zone), and sharp geometry.

Stress concentrations also occur where point forces are applied to a structure. Truepoint loading is impossible with materials of finite stiffness, and the force is insteadcarried over a small but finite area. If load(s) are applied within a region of size L,then stress at distances very much greater than L are unaffected by the precise loadplacement within L.

12

Figures follow for stress concentration factors for shaft shoulders, and various othertypes of geometry commonly encountered in design. For geometry not shown here,consult other handbooks, or use finite element analysis.

IMPORTANTSharp notched features have an infinitely high geometric stress concentration factor.A typical example is the groove that is cut into a shaft for a circlip. This groove issharp, and has no fillet radius in the corners. The sharp edge causes theoretically infinitively high stresses, since the force that passes through this region is taken byan infinitely small region of material. This means that it is impossible to define ageometric stress concentration factor for such parts. Finite element analysis is alsono help: although it will give a stress result for the region, if you were to refine themesh spacing around the sharp feature, you would find the stress rising. The finerthe FEA mesh, the greater the stress, and there is no limit.

Many people come unstuck in this matter because they fail to realise that infinitelysmall fillet radii produce infinitely high stress concentrations. This applies to thecirclip grooves already mentioned, as well as to sharp steps in shafts, cracks, Vgrooves (eg impact test specimens). However these comments do not apply to theexternal shoulders of shafts, since these regions are stress free.

If the stress concentration is infinite, then even a tiny force should generate infinitelyhigh stresses at sharp grooves. We should see all such parts fail immediately, but wedon’t. Why not? The answer is that the stresses do start to rise as soon as load isapplied, until the material starts to yield at the sharp places. Once the material yields,then there is plastic deformation, and the sharp feature is rounded out. If a higherforce is subsequently applied, then the feature will again go into yield, and rounditself out further. In this way the stresses are at most yield, and the part will not failimmediately, at least while there is still ductility in the material.

The geometric stress concentration factor only takes the geometry into account. Itdoes not account for the plasticity that materials have. The more ductile a material,the more tolerant it is of geometric stress concentration. Less ductile materials, likeglass, are still very sensitive to notches, and this can be seen in the way glass is cut:by cutting a shallow scratch and then applying a relatively light load to break it alongthe mark.

It will be shown later that there is a factor called the notch sensitivity, which takesinto account the ductility of a material.

4.2 Geometric Stress concentration factors for stepped shafts

The case of shoulders on a shaft occurs often in design, because of the need toprovide shoulders for bearings. The stress concentration factors may be determinedby referring to a diagram, or using an equation. Data are provided below. Note that itis important to distinguish between the different types of loading: axial, bending, andtorsion, since the results are not the same. To find the stress concentration factor,

13

AXIAL Stress concentration factor for round shaft with shoulder. Tensile stress is F =KtF/A, where A = Bd2/4

determine the ratio of the diameters, and also the ratio of the fillet radius to the minordiameter. Using this information, select the appropriate D/d line and find theintersection with r/d. The stress concentration factor is read off the left side.

Alternatively the stress concentration factors may be calculated. For axial tension:

where

and where K1, K2, K3, and K4 values are determined as follows.

For 0,25 # h/r # 2,0

use the following values

For 2,0 # h/r # 20,0


14

BENDING Stress concentration factor for round shaft with shoulder. Bending stressis F = KtMy/I, where y = d/2 and I = Bd4/64

Reference: YOUNG WC, 1989, Roark’s Formulas for stress and strain, McGraw-Hill.

Alternatively the stress concentration factors may be calculated. For bending:


For 0,25 # h/r # 2,0


For 2,0 # h/r # 20,0


15

TORSION Stress concentration factor for round shaft with shoulder. Torsional stressis J = KtTr/J, where r = d/2 and J = Bd4/32


For a round shaft with a shoulder fillet, the geometric stress concentration factor forbending is sometimes also given as

where the acos values must be in radians. However this equation is only anapproximation.

Alternatively the stress concentration factors may be calculated. For torsion:


For 0,25 # h/r # 4,0


16

Bending

Axial Tension

Torsion


4.3 Geometric Stress concentration factors for semicircular notch in acircular shaft

The geometric stress concentration factor is:

4.4 Geometric Stress concentration factors for a U notch in a circularshaft

This geometry is similar to that of a circlip groove, except that the circlip groove canhave very sharp corners. The geometric stress concentration factor is:

Where the values of K1, K2, K3, and K4 are determined as follows.

17

AXIALFor 0,25 # h/r # 2,0


K1 = 0,455 + 3,354 (h/r)0,5 - 0,769 h/rK2 = 3,129 - 15,955 (h/r)0,5 + 7,40 h/rK3 = -6,909+29,286 (h/r)0,5 -16,104h/rK4 = 4,325 - 16,685 (h/r)0,5 + 9,469 h/r

AXIALFor 2,0 # h/r # 50,0


K1= 0,935 + 1,922 (h/r)0,5 + 0,004 h/rK2 = 0,537 - 3,708 (h/r)0,5 + 0,040 h/rK3 = - 2,538 + 3,438 (h/r)0,5 - 0,012 h/rK4 = 2,066 - 1,652 (h/r)0,5 - 0,031 h/r

BENDING For 0,25 # h/r # 2,0


K1 = 0,455 + 3,354 (h/r)0,5 - 0,769 h/rK2 = 0,892 - 12,721 (h/r)0,5 + 4,593 h/rK3 = 0,286 + 15,481 (h/r)0,5 - 6,392 h/rK4 =-0,632 - 6,115 (h/r)0,5 + 2,568 h/r

BENDING For 2,0 # h/r # 50,0


K1 = 0,935 + 1,922 (h/r)0,5 + 0,004 h/rK2 = -0,552 - 5,327 (h/r)0,5 + 0,086 h/rK3 = 0,754 + 6,281 (h/r)0,5 - 0,121 h/rK4 = -0,138 - 2,876 (h/r)0,5 + 0,031 h/r

TORSION For 0,25 # h/r # 2,0


K1 = 1,245 + 0,264 (h/r)0,5 + 0,491h/rK2 = -3,030 + 3,269 (h/r)0,5 - 3,633 h/rK3 = 7,199 - 11,286 (h/r)0,5 + 8,318 h/rK4 = -4,414 + 7,753 (h/r)0,5 -5,176 h/r

TORSIONFor 2,0 # h/r # 50,0


K1 = 1,651 + 0,614 (h/r)0,5 + 0,040 h/rK2 = -4,794 - 0,314 (h/r)0,5 - 0,217 h/rK3 = 8,457 - 0,962 (h/r)0,5 + 0,389 h/rK4 = - 4,314 + 0,662 (h/r)0,5 - 0,212 h/r


4.5 Other stress concentrations

Geometric Stress concentration factors Kt for Threaded elements

THREAD FORMWitworthISO and UNIFIED

Geometric Stress concentration Kt

3,865,00

18

Geometric Stress concentration factors for keyways

KEYWAY TYPEEnd milled keyway Sled-runner keyway Combined bending and torsion

Geometric Stress concentration Kt

1,791,383,00

4.6 Ways of avoiding stress concentrations in shaft shoulders

Almost all shafts have shoulders, that is step changes in diameter. The shoulders atbearings are particularly severe stress raisers. Bearings have sharp corners (eg R =0,8 mm), and therefore the fillet radius at the shoulder has to be even sharper inorder to avoid interference. Therefore stress concentration factors of 2,5 arerelatively typical in such cases.

The diagram shows some design practices that are used to reduce the stressconcentration. Figure A represents the worst case: a sharp shoulder, with a roughsurface texture, and the texture marks at right angles to the line viewed (i.e. circularmarks). The first improvement (B) is to increase the fillet radius. Next (C), try to haveless abrupt change in section. Smoother texture is shown in (D), and axial marks.Note that this modification does not affect the geometric stress concentration, (whichis concerned with large scale effects), but it does improve the fatigue life of the partby reducing the number of microscopic places where cracks can start.While a larger fillet radius is the best and easiest way to decrease stressconcentration, it is not always practical because of the problem with small bearingcorner radii. The next few diagrams show some solutions in this particular case. (E)is an undercut shoulder: the radius of the undercut can be made relatively larger,thereby reducing the stress concentration factor. There is plenty of clearance for thecorner of the bearing, however sharp it might be. In practice shoulders are often too

19

small to accommodate an undercut, and undercutting the shaft (F) is the next option.This obviously removes material from the load carrying cross section, but theadvantage of a reduced stress concentration is more than worth it. This is arelatively common design. The next case (G) uses a spacer to provide a sharpcorner for the bearing, while still allowing a generous fillet radius. There are howevertwo difficulties with this option: firstly the shoulder must be high enough, andsecondly, if the spacer is assembled the wrong way round then it will bite into thefillet and may initiate failure there. The last design (H) shows the addition of anotherstress raiser. This might not seem a very good idea, but curiously it does reduce theoverall stress concentration. It does this by constraining the stress lines so that theydo not change direction abruptly.

Other shaft stress raisersAnother common source of stress concentration in shafts is a circlip groove. Thecirclip is used to provide axial location, typically for a bearing. The grooves cut intothe load bearing section, and they also have sharp corners, hence the stressconcentration. Figure A) below shows the standard design for a circlip groove.Improvements are shown in B) and C). In B) there are side grooves, which help alignthe stress paths so that they don’t have to suddenly make all their change at thecirclip groove. Turning down the shaft achieves a similar effect. The samemechanism works to reduce stress concentration in the machine screw in D), whichis turned down to the root diameter of the thread.

It is important to note that the stress concentration effect is one that occurs atchanges of shape, and the more abrupt the change the higher the factor. The effectis not so much caused by reduction in cross section as change in shape, andtherefore even increases in cross section can cause stress concentration. Thereforematerial that is not carrying load actually weakens the structure. It only provides atemptation for the load bearing lines to wander, thereby distorting the stressdistribution.

20

Fatigue only occurs where there isdynamic loading, that is forces thatchange with time. Dynamic loadingoccurs frequently, particularly in movingmachines. Failure by static loadingnormally only occurs in machines that aremisused, overloaded, or under designed.A design that is adequate for staticloading may still fail by fatigue.

5 FATIGUE FAILURE

Fatigue is the term that is used to describe the failure of a part at loads well belowthose predicted by the static theories of failure. Basically a low load appliedrepetitively for many cycles, can cause failure. Design against fatigue failure isimportant, since many parts, such as shafts and gears, are exposed to this type ofloading.

The way we go about designing against fatigue is to determine the stresses in thepart (using standard structural mechanics). Then we determine the “fatigue strength”of the material that we intend to use in the part. If the fatigue strength is substantiallygreater than the applied stress, then we are safe. Here is how we determine thefatigue strength: first determine the “endurance limit”, and then apply modifyingfactors.

5.1 Mechanism of Failure

Static failure and fatigue are very different failure mechanisms. In static loading (likea tensile test specimen) the load increases slowly, and a large amount of plasticdeformation occurs before final fracture. However fatigue occurs under changingloading, and it gives rise to cracks, even when the nominal stress is in the elasticregion (i.e. stresses are well below yield, no plastic flow).

Fatigue failure is the progressivefracture of a part. The fracture starts atone point, and progresses through thebulk of the material. Eventually somuch of the cross section has beenfractured, that the remainder breakssuddenly. The final failure may beafter a considerable time of otherwisesatisfactory service. Fatigue failuretypically occurs at stress levels wellbelow the yield strength of thematerial.

The mechanism of fatigue failure is that localised plastic deformation occurs at smallflaws in the material. Such flaws include microscopic features such as latticeimperfections, surface scratches, weld ripples, and machining marks. Larger scaleflaws include notches, geometrical changes in section, holes, keyways, threads,casting inclusions, and corroded areas. These flaws exist in all materials to someextent, either internally or on the surface.

The loading on the material creates a general strain (or stress) pattern in the wholepart. This distribution can be determined by classical structural analysis, or testing.The average strain (stress) may be well below the yield point of the material, but high

21

The larger the part, the more flaws it cancontain where fatigue may start.Conversely, small parts like glass fibres,have fewer flaws and therefore greaterresistance to fatigue failure.

strain (i.e. localised stress) can still exist around the stress concentrating flaw. Thiscauses the flaw to grow into a crack.

After being started, the crack grows with each load cycle. It progresses through thegrain in the direction of weakest resistance, until it gets to the grain boundary. Here itmeets resistance to growth, and is arrested. However if the loading is high enoughthe crack can break through the barrier and into the next grain. Here it will need tofollow the weakest path again, which may necessitate a change in direction.Afterwards will be other grain boundaries and grains, probably at differentorientations. The crack propagates through these, taking a winding three-dimensional path.

Eventually the extent of the crack is a significant part of the loaded cross sectionalarea. The deformation at the tip of the crack is increased, and therefore the splittingability of the crack is increased: it begins to cut right through grains, regardless oftheir orientations. Each load cycle now causes significant crack growth, which isvisible as microscopic striations on the surface. There are also larger scale "beachmarks", which are visible with the naked eye. These are a typical characteristic offatigue, the marks being similar to those left on a beach by the receding tide. Theyare caused by changes in the rate of crack growth.

Once enough of the cross section is lost, then one last load cycle causes the crack topropagate rapidly through to total fracture. This final mode of failure is brittle fractureunder static overload, and it produces a rough granular surface, with low distortion.This even occurs in materials which wouldotherwise be considered ductile. The granularappearance is not due to brittleness in thematerial, but to brittle mode of failure.

Early analysis of such fractures led to the falseconclusion that something had caused thematerial to go "brittle". The material waspresumed to have tired, or "fatigued", and hence the name developed. "Progressivefailure" would be a more appropriate name given the understanding that we nowhave of the mechanism.

The way we go about designing against fatigue is to determine the stresses in thepart (using standard structural mechanics). Then we determine the “fatigue strength”of the material that we intend to use in the part. If the fatigue strength is substantiallygreater than the applied stress, then we are safe. Here is how we determine thefatigue strength: first determine the “endurance limit”, and then apply modifyingfactors.

The fine details of fatigue are still actively debated, and from the perspective of thematerial scientist, the problem is far from solved. However from the engineer'sperspective, it does not matter if the material science theories are not yet reliable,since we have a job to do, and anyway there is already enough information for thepractical design of machines and structures. Engineers have available a large body

22

Moore rotating beam test

General S-N data and curve

of empirical knowledge of fatigue. This is based on experiment, and is independentof any underlying theory. Even if the fatigue theories eventually change, the designmethods won’t change much, since they are based on observation. It is to beexpected that consistent data should emerge from fatigue tests, since everyengineering part contains vast numbers of flaws, at least some of which will probablybe in the right location and orientation to initiate a fatigue crack. These data areexplained in the next section, and thereafter is shown how the information is adaptedfor design purposes.

5.2 Endurance limit of rotating beam specimens

The standard fatigue test is rotating bending, without transverse shear. This purebending loading is created in a Moore fatigue testing machine. The specimen iscarefully prepared to standard dimensions: N0,300", and with a large radius ofcurvature R 9 7/8") to prevent stress concentration. The surface is polished.The specimen is loaded with a givenweight, and rotated until failure. Thenumber of cycles to failure is recorded.Tests are made with different weights.A switch on the weights stops themotor when the specimen fails. Thetest is done for different weights.

Large number of specimens arerequired for each change in loading,due to the statistical nature of fatigue.Results are applied stress [S], plottedagainst number of stress cycles [N]. Usually log-log axes are used rather than linear.There is scatter in the results, more so than in static tensile tests, which is to beexpected given the nature of the fatigue mechanism.

For most materials, especially ferrous metals, there is a certain stress below whichfatigue failure will not occur however long the alternating stress is applied. Thisstress is called the endurance limit Rn, and it usually occurs at about 106 load cycles.

The standard deviation (a measure of datascatter) of the endurance limit is typicallyabout 8% of the value of endurance limit.

The essence of preventing fatigue is tokeep the stresses below the endurancelimit so low that no crack growth occurs atall. Alternatively the part can bedeliberately designed for a finite life, if thisis acceptable.

23

At 3000 rpm, a continuously running shaftwould clock up 108 cycles after a time of

108/3000 = 33 333min = 23 days

Ferrous (iron alloys) and titanium alloys exhibit an endurance limit. Unfortunately, fornon-ferrous metals there is no knee in the S-N curve, and thus no endurance limit.Instead the fatigue strength is usually based on 108 cycles for design purposes. If thepart is critical, then it is withdrawn from service after a predetermined period of use,whether or not it shows damage. Alternatively it is necessary to regularly inspect thepart using X-ray photography or other non-destructive testing.

Ideally the endurance limit for a materialshould be determined by tests. However inthe absence of test data, an acceptable approximation may still be made, since theendurance limit depends simply on the ultimate tensile strength Rm of the material.The relationships are as follow:

Material Endurance limit Rn for rotating beam specimen

STEELS, where Rm <1400 MPa Rn= 0,5 Rm

STEELS, where Rm >1400 MPa Rn = 700 MPa

CAST IRON Rn = 0,4 Rm

TITANIUM ALLOYS Rn = 0,45 Rm to 0,65 Rm

CAST ALUMINIUM ALLOYS Rn = 0,3 Rm [for 108 cycles]

WROUGHT ALUMINIUM ALLOYS Rn = 0,4 Rm [for 108 cycles]

WROUGHT & CAST MAGNESIUM ALLOYS Rn = 0,35 Rm [for 108 cycles]

COPPER ALLOYS Rn = 0,25 Rm to 0,50 Rm [for 108 cycles]

NICKEL ALLOYS Rn = 0,35 Rm to 0,50 Rm [for 108 cycles]

POLYMERS Rn = 0,4 Rm

It is important to remember that the endurance limit is the fatigue strength of apolished specimen of certain geometry, and loaded in only bending. Practicalengineering parts are obviously not identical in geometry or loading. The next sectionshows how to quantify these differences.

5.3 Fatigue Strength of Actual Machine Elements

The fatigue strength Rf of an actual machine element will be different to theendurance limit for a rotating beam specimen because of the differences in geometryand load. These differences are accommodated by applying modifying factors to Rn

as follows:

24

This equation is valid for 106 or 108 cycles as the case may be. Thus in an actualmachine part, the maximum permissible stress in order to avoid fatigue failure is thefatigue strength Rf. This value will always be less than the endurance limit Rn. Thefactors are determined as follow.

LOAD FACTOR Cl

The load factor accounts for types of load other than rotating bending. At 106 cyclesthe factor is:Rotating Bending Cl = 1Reversed Bending Cl = 1 (conservative)Axial Cl = 0,85 (no eccentricity)Torsion Cl = 0,58

To understand the reasons behind these factors, note that rotating bending producesapplies maximum stresses all around the perimeter at some time or another. This isthe standard test case. In reversed bending the maximum stresses are generatedonly at the top and bottom, at the worst flaw may not coincide with either of thesepositions. However the difference is small, and is conservatively neglected. In axialloading the entire cross section is subject to the maximum stress, and thus thechances of a flaw being in a position of stress is increased. If there is eccentricitythen there will be a bending stress as well. If the eccentricity is unknown then it iscommon to use a value of Cl = 0,80 to 0,70.

SURFACE FACTOR Cs

This factor accounts for the surface texture, which is not always the polishedcondition. The factor depends on the material.Cast iron: Cs = 1 (since even polished cast iron has defects due to the carbon

flakes) Non-ferrous materials Cs = 1Steel: Cs depends on machining process and tensile strength, and is shown

below.

25

Surface factor for steels

SIZE FACTOR Cd

The size factor depends on the diameter (or depth of section for non-round sections).Conservative values of Cd are

Diameter BENDING TORSION AXIAL

d < 7,6 mm7,6 mm # d # 50 mmd > 50 mm

Cd = 1Cd = 0,85Cd = 0,75

Cd = 1Cd = 0,85Cd = 0,75

Cd = 1Cd = 1Cd = 1

RELIABILITY FACTOR Cr

To determine endurance strength Rn from experimental data, it was necessary to fit aline between the points. This line is usually positioned in the middle of the group ofdata points, and termed 50% reliability. It means that a part has an equal chance offailing, or of lasting. A greater chance of survival is usually required, but to positionthe line below all possible data points would correspond to 100% reliability (which isstatistically unattainable). Reliability factors are given below.

Cr Reliability

1 50%

0,897 90%

0,868 95%

0,814 99%

26

Notch sensitivity for materials in reversed bending or reversed axial loading.

0,753 99,9%

TEMPERATURE FACTOR Ct

The operating temperature affects the fatigue strength.For Steels: Ct = 353/(273 + T[deg C]), with a maximum of Ct = 1,00

FATIGUE STRESS CONCENTRATION FACTOR Kf

The fatigue stress concentration factor is

Kf = 1 + q(Kt - 1)

where

Kt geometric stress concentration factor q notch sensitivity, which depends on the material, ultimate strength, and

loading. See the figures below.

This equation permits a reduction in the stress concentration. This is because somematerials are less sensitive to stress concentration, as they are able to yield inregions of high stress, and thereby reduce the sharpness of the cut. Toughermaterials have lower notch sensitivity. Notch sensitivity depends on the material, andalso on the type of loading and the notch radius. If in doubt, a value of q = 1 may beused, that is Kf = Kt.

27

Notch sensitivity for materials in reversed torsional loading

What is different about the fabricationthat rolled threads should be betterthan cut threads?

Some Fatigue Stress concentration factors Kf are given below. Note that these donot need any further correction for notch sensitivity q.

Kf for Threaded elements:The table below gives typical values for Geometric- and Fatigue -Stressconcentration factors. The fatigue factors depend on the hardness of the material,and on the manufacturing process.

THREADFORM

GeometricStressconcentration

Kt

Fatigue Stress concentration, Kf

ROLLED THREADS CUT THREADS

<200Bhn >200Bhn < 200 Bhn >200Bhn

Witworth 386 140 260 176 332

ISO andUNIFIED

500 220 300 284 385

where Bhn refers to Brinell hardness

Kf for keywaysThe Fatigue Stress concentration factor dependson the way in which the keyway is cut. The factor for keyways may be avoided byusing friction mount devices instead. Note that keys are not generally recommendedfor reversed shaft rotation.

28

Weld loading

Fatigue Stress concentration, Kf

Sled runner keyway End-milled keyway

Bending Torsion Bending Torsion

Annealed steel<200Bhn

1,3 1,3 1,6 1,3

Quenched &drawn steel>200Bhn

1,6 1,6 2,0 1,6

where Bhn refers to Brinell hardness

Kf for WeldsWelds are particularlyvulnerable to fatigue failure,because of the multitude offlaws internally (porosity, slaginclusions, and incompletepenetration) and externally(roughness), and the adverseheat treatment that the partreceives. The toe (edge of theweld bead) is a common fatigueinitiator. For fatigue resistance,welds should be ground flushwith the surface. Undercut andreinforced welds are bothundesirable. While it is possible to give fatigue stress concentration factors for welds,this is not usually done. Instead there are welding codes which provide thepermissible fatigue stress for a given type of weld.

29

Low cycle fatigue is stress loading between onethousand and one million cycles. This diagram isfor bending.

Low cycle fatigue: Axial loading

Some approximate fatigue stress concentration factors are given below, with thediagram showing how the loading is defined.

5.4 Low Cycle Fatigue

Low cycle fatigue refers tofatigue failure between 103

and 106 load cycles. (Forless than 103 cycles, treatas static failure.) If lowcycle fatigue failure ispermissible, then higherstresses may be acceptedthan for infinite life. Lowcycle fatigue also dependson the type of loading.Correction for surfacetexture does not have tobe made at low cyclefatigue.

BendingThe S-N line is drawn from 0,9 Rm at 103 cycles to Rf at 106 cycles, both axes logscaled. Therefore for bending

Which may be rearranged to determine the low cycle endurance limit at N cycles,RfN:

Axial The S-N line is drawnfrom 0,75 Rm at 103

cycles to 0,85Rf at 106

cycles, both axes logscaled.

30

Thus low cycle endurance limit at N cycles is:

TorsionThe S-N line is drawn from 0,9 Rms

at 103 cycles to 0,58Rf at 106 cycles,both axes log scaled. Note that iftest data is not available, then Rms =0,577 Rm for ductile materials(distortion energy theory).

Thus low cycle endurance limit at N cycles is:

NoteRm ultimate tensile strengthRms ultimate strength in shear (if this is unknown and cannot practically be

determined from tests, then use an appropriate theory of failure)Rn endurance limit (106 cycles)KfN low cycle fatigue stress concentration factor , given by:

KfN = Sr(Kf - 1) + 1where Sr is determined from the graph below.

Other modifying factors are the same as previously defined.

31

Sr factor for 103 cycles.

6 CUMULATIVE FATIGUE DAMAGE

Certain machines and structures (like automobile suspensions) are subject torandomly varying loads. The methods presented here can be used to analyse thesecases, providing a plot of actual or assumed stress vs time is available. Once thisdata is known, then add up the number of stress cycles at various intensity ranges:eg 1000 cycles at 0-20MPa, 3000 cycles at 20-40MPa, etc. There are severalmethods after this.

6.1 Manson’s approach

1 Select intersection of S = 0,9 Rm and N = 103 cycles (approximate).2 All lines, for virgin and damaged material, converge at 0,9Rm at 103 cycles.

Lines to be constructed in the same historical order as the stresses. 3 Determine endurance limit Rn for virgin material (at 106 cycles).4 Draw S-N line.5 For stress Fi (applied ni times) find predicted life Ni.6 Calculate remaining life Ni - ni (at stress Fi). 7 Connect this point to 0,9 Rm at 103 cycles.8 Extend line to meet 106 cycles and intersection determines new endurance

strength of (damaged) material Rni

9 Repeat as necessary for each other stress Fj (applied nj times) to find remaining life Nj

32

For stress Fi (greater than the endurance limit), the maximum number of cycles thatcan be taken is

If only ni cycles are applied, then the life remaining isNi+1 = Ni - ni

And the new (damaged) endurance limit due to the application of stress Fi is

6.2 Miner’s Rule

This is also sometimes called the Palmgren-Miner rule. The assumption is that if astress of (say) 30MPa would cause failure after 5x104 cycles, then 104 cycles of30MPa would use up 1/5 of the total cycles of life. If another stress of say 40MPaacted for 500 cycles, where it would normally have a life of 103 cycles, then it woulduse up ½ of the life. The lives used up so far would be the total, i.e. 1/5 + ½ =0,6.More stresses could be applied, and when the total got to 1,0 then failure would beassumed to occur. Mathematically this is stated as

33

where stress i (which acts for ni cycles) would have a life of Ni cycles if it acted alone.Stresses less than the fatigue strength are ignored in this method.

CommentManson's is by some considered to be superior to Miner’s Rule, since it takes intoaccount the order in which the stresses are applied. For example, one hard landingon an aircraft undercarriage early in service, will cause any crack to progress quicklypast the grain barriers, so that it subsequently even grows under light loading. If thesame landing occurred later in service it would also cause increased crack growth,but by that time the aircraft might have had many trouble free landings. The earlier inservice a major incident occurs, the shorter the total life. Manson's approach takesthis into account, but not Miner's rule. However Miner's rule is easier to use whererandom loading occurs, because it does not worry about the order in which the loadsoccur.

The accuracy of both these methods is limited by the scatter in data points makingup the fatigue test. For a given stress, the accuracy with which the life can bepredicted is particularly low. Neither of these methods appears to have very goodtheoretical justification, but they seem to be the best we have at the moment. Ofcourse testing is ultimately the best provider of design data, and many firms do thiswhen there are sufficiently large production volumes involved. The problem with testresults is that they are only valid for the applied conditions. Also, they are onlyavailable after fabrication, so they can’t be used for preliminary design. ThusManson’s and Miner’s methods are still in use.

6.3 Cycle counting

In many practical fatigue applications, such as automotive suspensions, wave andtraffic loading, the stress changes considerably with time. There is no one value ofstress that can be used for fatigue calculations, and the spectrum is too wild to beable to identify cycles. The solution is to use one of the cycle counting methods.They are procedures that converts a complex stress - time graph into identifiablecycles that can then be used in Miner’s Rule.

Cycle counting is applied to stress histories that are obtained from actual structuresin service, usually from strain gauges.

Two methods of counting cycles are given below.

6.3.1 Rainflow cycle counting

M Turn the stress history on its side, with the time axis running verticallydownwards.

M Decide on the stress range that you want to use, eg 10MPa intervals in thisexample. The finer the range the more work you will have to do, but the moreaccurate the results will be.

34

M Move the stress history so that you start where stress is zero.M Imagine that this is a roof, and that rain falls on the top.

M The water runs from 0 to A to F to G, and then off to the ground. The totalhorizontal distance covered (range) is 70MPa positive. Make a note of this in atable.

M Water also runs from A to B, down to D and then to E. It falls off at E, and isassumed to stop (the rainflow analogy is not quite watertight!), because abigger stream (G-H) chops it off. Range for this flow is A-E, which is 50MPanegative.

M B-C is 20 MPa+, since it is chopped off by the larger EG.M C-D is 20MPa-, and is stopped at D because A-B is a larger flow.M E-F is 50MPa+, and is stopped at F because 0-A is a larger flow.M G-H-L is 140MPa-, and it stops TU because it is largerM H-J is 20MPa+, and stops because it meets a larger flowM J-K is 20MPa-, and stops because it meets a larger flowM L-M-S-T is 120MPa+, and it stops all the smaller flows N-SM M-N is 40MPa-, and stops because it meets a larger flow TU

35

Rainflow cycle counting

M N-P-S is 40MPa+, and stops because it meets a larger flow at SM P-Q is 30 MPa-, and stops because it meets a larger flow at QM Q-R is 30MPa+, and stops because it meets a larger flow N-P-SM T-U is 120MPa-, and stops because it meets a larger flow G-H-LM U-V is 70MPa+

36

The ranges are then put into a table. A full cycle is made when there is one negativeand one positive cycle of the same size.

Stress range[MPa]

Negative - Positive + Total (-) + (+) = 11x(-) = ½1x(+) = ½

140 GH The cycles OAG and UV areactually just part of one largercycle, since the placing of thetime origin cut this cycle in half.

1

130 0

120 TU LT 1

110 0

100 0

90 0

80 0

70 OAGUV

see 140

60 0

50 AE EF 1

40 MN NS 1

30 PQ QR 1

20 CDJK

BCHJ

2

10 0

Therefore, for the purposes of Miner’s rule, this stress waveform is equivalent to onecycle at 140MPa, one at 120MPa, one at 50MPa, one at 40MPa, one at 30MPa, andtwo at 20MPa.

6.3.2 Reservoir cycle counting

M Orient the stress history with time on the horizontal axis, running towards theright.

M Shift the waveform along so that it starts and ends at the same high point.This limitation makes the method best suited for stress histories of short andcyclical duration.

M Imagine that you are looking at the cross section through a reservoir. M Fill up the reservoir with water.M Imagine that there are plugs at the bottom of all the troughs.M Remove the plug from the deepest reservoir. Make a note of the change in

water level. If there are two or more deepest troughs, then drain them in anyorder, but one at a time.

M Drain the next deepest trough, and note the change in level.M Continue until all the water is out.

37

Reservoir cycle counting

M Drain 1 gives 140MPaM Drain 2 gives 120MPaM Drain 3 gives 20MPaM Drain 4 gives 30MPaM Drain 5 gives 40MPaM Drain 6 gives 50MPaM Drain 7 gives 20MPa

Therefore, for the purposes of Miner’s rule, this stress waveform is equivalent to onecycle at 140MPa, one at 120MPa, one at 50MPa, one at 40MPa, one at 30MPa, andtwo at 20MPa.

Reference: Maddox SJ, 1991, Fatigue strength of Welded structures, Abington.

38

7 FLUCTUATING STRESSES

The previous sections on fatigue have all assumed an alternating stress, with zeromean stress. In other words the assumption so far has been complete stressreversal. However it often occurs in practice that a steady load is combined with analternating load. For example a rotating shaft with bending moment and axial force:rotation creates an alternating bending stress, and the axial force creates a steadystress.

In problems with fluctuating stress it is necessary to determine mean and alternatingcomponents of the stress:mean stress FFFFm = (FFFFmax + FFFFmin)/2alternating stress FFFFa = (FFFFmax - FFFFmin)/2

These two stresses are then plotted on a Goodman diagram. There are two methodswhich can be followed.

(a) Goodman diagram from test dataThis diagram provides a graphical conversion from Fmax and Fmin to Fm andFa. It also shows permissible combinations for various lives (eg 103 ... 106).The diagram is drawn from test results on the material. The example below isfor alloy steels generally. For a given mean and alternating stress, the life isthe nearest line above the data point.

39

Similar Goodman diagrams may be available for other materials.

(b) Modified Goodman diagramIf data from an actual test is unavailable, the next option is to approximate theGoodman. The method constructs an approximate Goodman diagram for thematerial, but only for 106 cycle loading. It shows graphically whichcombinations of Fm and Fa are permissible. A diagram is shown below, andimplicit in it are the boundary conditions.

40

In some design applications, the ratio of Fa/Fm may be known, but not the individualvalues. Solve this problem constructing a line from the origin, with slope equal to theknown stress ratio. Intersect this line with the envelope, and determine maximumsafe Fa and Fm.

For fluctuating torsion it is unnecessary to construct a Goodman Diagram. Insteaduse the following criteria of failure:

* fatigue occurs when: JJJJa = Rns (fatigue strength in shear)

* static failure occurs when: JJJJmax = JJJJa + JJJJm = Res (yield strength in shear) = 0,577Re

Apply safety factors.

8 FATIGUE IN BIAXIAL STRESS SYSTEMS

The previous sections dealt with uniaxial stress systems (with mean and alternatingcomponents). Biaxial systems have stresses in two directions, and each stress mayhave mean and alternating components. The method is to convert the system toequivalent uniaxial stresses, and then proceed with the Goodman method asdescribed above. The problem typically arises with a shaft that is subject to torsion (t)and bending (b), possibly also with axial load (n).

(1) Determine stresses for each type of loading, eg Fb = My/I, Fn = F/A and J =Tr/J.

(2) Consider a small element in the most highly loaded region. Determine meanand alternating stresses in each direction x, y, and z. Eg Fxm=Fn, Fxa= Fb, Fym=0, Fya= 0, Jxym=J, Jxya=0

(3) Then calculate principal mean alternating stresses F1m and F2m from Mohrcircle, or with the following (biaxial stress only):

Also, calculate principal alternating stresses F1a and F2a from Mohr circle, or with thefollowing (biaxial stress only):

41

(4) Then calculate equivalent stresses:FFFFm = ( FFFF1m

2 - FFFF1m. FFFF2m + FFFF2m2)0,5

FFFFa = ( FFFF1a2 - FFFF1a. FFFF2a + FFFF2a

2)0,5

where 1 and 2 refer to the principal stresses, and a and m refer to alternating andmean components.

If there is only one normal stress, Fx, and shear stress Jxy, (each with alternating andmean components), then the equations simplify to: FFFFm = ( FFFFxm

2 + 3JJJJxym2)0,5

FFFFa = ( FFFFxa2 + 3JJJJxya

2)0,5

(5) Create a Goodman diagram based on bending stresses (only). Plot the abovemean and alternating stress point, and ensure that it lies inside the envelopefor safe life.

9 SURFACE FATIGUE

Contacting surfaces under load, generate stresses in and under the surfaces.Surface fatigue failures result from repetition of such loads. Cracks propagate, untilsmall pieces of material are separated. Small pits or spalls thus form on thesurfaces. Under continued operation these areas grow in size, until function isimpaired. This type of failure is typical of gears, bearings, and cams. The greater theinterface pressure, the shorter the life of the parts.

9.1 Hertz Contact Stresses

The Hertz Contact stress analysis assumes frictionless contact (no sliding friction). Basically elastic deformation occurs when curved surfaces are pressed together, sothat a surface compressive stress is generated (perpendicular to the surface). Due tothe Poisson effect the material also tries to expand in the other directions, and thuscreates stresses in the plane of the surface. The maximum shear stress occursunder the surface, and it reverses sign as a rotating load approaches and thenpasses. This alternating stress induces fatigue failure.

Hertz contact pressure alone is not entirely adequate for situations where slidingfriction also occurs. The friction introduces a tangential normal force and a tangentialshear force. The normal force reverses sign, and the surface tensile stress is themore damaging.

The equations for Hertz contact stresses are given in the chapter on structuralmechanics.

42

9.2 Buckingham's Contact Stresses

Rolling or sliding motion of surfaces may cause fatigue failure of substrate material.Surface endurance limit Snf, is the contact pressure which will cause eventual failureof the surface. For steels the value is:Snf = 2,76 Hb - 70 [MPa]where Hb is the Brinell Hardness

For contacting cylinders, from Snf above, and the moduli of elasticity E1 and E2 of thetwo contacting materials, calculate Buckingham's load-stress factor K:K = 2,857 Snf

2(1/E1 + 1/E2)

Surface fatigue failure occurs at 108 cycles as follows:

wheren safety factor F contact forcer1 and r2 cylinder radii, w cylinder width

10 CORROSION FATIGUE

This is the combined action of corrosion and cyclic loading. Failure occurs quickerthat predicted by either mechanism acting alone. The explanation seems to be thatcorrosion pits act as stress raisers, and the dynamic loading breaks off the (brittle)protective films, causing the pits to develop rapidly into cracks. The final crackedsurfaces show corrosion stains, which are not evident in plain fatigue.

Corrosion fatigue depends on elapsed time (for more corrosion), and number ofcycles (for more fatigue). Specific test data is required for design, as corrosionfatigue strength does not depend on tensile strength. Heat treatment is not useful.Design approaches include:* use a more corrosion resistant material rather than one with greater fatigue

strength* apply coatings to reduce corrosion * use sacrificial anodes or coatings (eg Zn)* create residual compressive stresses

43

11 DESIGNING AROUND FATIGUE

Naturally, stronger materials provide greater resistance to crack growth. Howeverthere is a limit to the strength properties of engineering materials, and when the bestavailable material has been used, and is still inadequate, then the designer will haveto consider other ways of preventing fatigue failure. Some of these methods areprobably less troublesome and costly than going straight to the best materials.

11.1 Changes to Loading

A fundamental requirement for crack growth is that tensile loading should exist. If thedesigner can so arrange that the part, or at least the critical section, be loaded incompression, then crack growth can be arrested. Pretension is often used to achievethis, though it requires of course that some other part take even more tension that itwould otherwise have, eg bolted joints. The related topic of residual stress isdiscussed below.

11.2 Design Changes

Since fatigue failures start at small flaws or features, the elimination of such featurescan suppress fatigue. In this regard the designer can consider * smoother surface texture eg (polish rather than grind), * better material quality, eg materials cast under vacuum (less casting

inclusions)* wrought rather than cast material (inclusions in wrought materials are broken

up in the forming process)* less severe changes in geometry, eg larger fillet radii* avoiding keyways, threads, and holes, by using other features to perform the

function (eg friction mounts instead of keys)* moving stress raisers out of the highly stressed regions, eg sleeve between

bearing and shoulder

11.3 Surface Treatment

Bending and torsion stresses are larger on the outside of a part than internally, andthus surface flaws are more significant than internal ones. Therefore surfacecondition is an important parameter in fatigue design. The parts that benefit mostfrom surface treatment are those with steep stress gradients from surface to core.These include parts those in bending or torsion, especially under high loads. Axialloading produces a uniform stress distribution, and surface treatments are relativelyless effective. There are three parameters of interest:

(1) Surface texture (roughness), the effect of which is quantified previously in theCs factor and its graph

44

(2) Surface strength, which should be greatest at the surface. This is achievedby processes such as heat treatment, flame hardening, induction hardening,carburizing, and nitriding.

Note that surface weakening occurs in steels that are processed hot (egforging, hot rolling), as decarburisation of the surface occurs. hydrogenembrittlement can occur in chrome and nickel plated steels

(3) Residual surface stress, which is best if compressive, since this closes thecracks. Conservatively put Cs = 1,0 in such cases.

One of the methods of creating a compressive surface stress is shot peening.In this process ferrous shot is blasted at the part from an air nozzle.Consequently the surface is stretched, creating a reaction from the core whichputs the surface in compression. The depth is about a millimetre, and thecompressive stress is about half the yield strength. Greater effect is createdby peening the part while it is held in tension. Another method is cold rolling,which is done during the production of many linear products (wire, bar, tube).It can also be done on machine parts such as axles, after fabrication. Cold rollforming operations, eg to produce threads, and cold pressing are alsoeffective.

Certain of the heat treatment operations also cause residual compressivesurface stress. These include flame hardening, induction hardening,carburizing, and nitriding.

With any type residual stress it is important that the material have a relativelyhigh yield strength, otherwise loads in service will easily be able to erase theresidual stress.

Processes which cause harmful tensile surface stresses include heavygrinding, welding, and flame cutting.

Note that a compressive residual stress on the surface is maintained by atensile residual stress in the rest of the core. Thus internal flaws, such asporosity and inclusions, become more significant as fatigue initiators.

11.4 FATIGUE APPLICATIONS

The principles of fatigue design are applied to many machine parts. In some caseslike bearings, the manufacturer digests the data and provides it in convenient tablesfor the user. In other applications the designer will have to do the calculations, andmake the decisions along the way. Below are shown some applications thatcommonly arise.

45

11.4.1 REVERSED BENDING AND STATIC TWISTING OF SHAFTS

A shaft that rotates while subject to a bending moment will be loaded in reversedbending. This loading, combined with steady torsion, occurs frequently. Reversedbending typically occurs where the shaft rotates under a force of constant magnitudeand direction, such as that of a gear or a belt. An element of the shaft experiencesreversed bending as it rotates into and out of the neutral plane.

SINES EQUATIONFor this type of combined loading, Sines permits the bending stress to go as high asthe fatigue strength.

For a solid shaft this may be rewritten as

whereM amplitude (half total height) of bending momentn safety factorRf fully corrected fatigue strength in bendingd diameter

However there is one condition: that the shear stress J is less than 1,5 x torsionalyield strength. (For a ductile material torsional yield strength = 0,57 x yield strength).

Furthermore, when using the Sines equation it is necessary to also check forstrength against static failure, as the equation is blind to this mode of failure.

46

SODERBERGThe Soderberg equation is based on the maximum shear stress theory of failure, andproduces a more conservative result than Sines. The diameter d is:

where n reserve factorT steady torqueRe yield strengthM amplitude of bending momentRf fully corrected fatigue strength in bending

BENDING AND TWISTING WITH ALTERNATING AND MEAN COMPONENTSThe most general loading for a shaft is where both the bending and twistingmoments have alternating (a) and mean (m) components. An axial load and rotatingbending will produce this type of bending moment, in which case Fm = axial stressand Fa = bending stress. Fluctuating torque also occurs, typically with reciprocatingmachines.

GENERAL SODERBERG EQUATIONThe general Soderburg equation for the diameter d is:

where n reserve factorRe yield strengthRf fully corrected fatigue strength in bending

47

COMBINED GOODMAN - DISTORTION ENERGYFor the general case where moment M, torque T and axial load P act on a solidshaft, each with altenating and mean components, then the combined Goodman anddistortion energy requires that the following equation be solved:

where n reserve factorm subscript refers to mean stressa subscript refers to alternating stressRm maximum material stress (subscript s for shear)Rf fully corrected fatigue strength bending or subscript s for shear

This equation is solved by numerical means.

12 CONCLUSION

The following procedures may be appropriate in fatigue design

9 Project definition: Determine technical specifications: performance, duty,dynamic/static loading, operating environment, user expectations, reasonableuse, desired life

9 Determine manufacturing requirements: ease of manufacture, cost

9 Select material that is appropriate to (corrosive) environment.

9 For static loading then use appropriate theory of failure

9 If structure is sensitive to deflection, then do these calculations now, (usuallygives greater dimensions than even fatigue), then check fatigue

9 If significant dynamic loading exists then use fatigue design* determine fatigue strength of material* combine loading as necessary* use Goodman or other criterion

48

Design tip 1 Stronger materials provide greater resistance to crack growth. However there is alimit to the strength properties of engineering materials.

Design tip 2 Consider other ways of preventing fatigue failure, before using the ultimate material.Geometry changes may be beneficial.

Design tip 3 In many cases fatigue is inevitable, especially when product use cannot be predictedby the designer. Design for slight overuse by reasonable user. Do what youreasonably can to design against fatigue, and then ensure:T product fails in a safe place or safe wayT user is adequately warnedT liability anticipated: act in a reasonable engineering way, consider losses

Design tip 4 There is scatter behind the neat Fatigue data. There is a finite possibility that yourdesign will fail before its time. No amount of design or manufacturing or user care willreduce the risk to zero.

49

Creative Commons license: Attribution Non-Commercial No Derivatives

SummaryYou are free to copy, distribute and transmit the work, under the following conditions: Attribution — You must attribute the workto the Author but not in any way that suggests that they endorse you or your use of the work; Noncommercial — You may notuse this work for commercial purposes; No Derivative Works — You may not alter, transform, or build upon this work. Any ofthe above conditions can be waived if you get permission from the copyright holder.

THE WORK (AS DEFINED BELOW) IS PROVIDED UNDER THE TERMS OF THIS CREATIVE COMMONS PUBLICLICENSE ("CCPL" OR "LICENSE"). THE WORK IS PROTECTED BY COPYRIGHT AND/OR OTHER APPLICABLE LAW.ANY USE OF THE WORK OTHER THAN AS AUTHORIZED UNDER THIS LICENSE OR COPYRIGHT LAW IS PROHIBITED.

BY EXERCISING ANY RIGHTS TO THE WORK PROVIDED HERE, YOU ACCEPT AND AGREE TO BE BOUND BY THETERMS OF THIS LICENSE. TO THE EXTENT THIS LICENSE MAY BE CONSIDERED TO BE A CONTRACT, THELICENSOR GRANTS YOU THE RIGHTS CONTAINED HERE IN CONSIDERATION OF YOUR ACCEPTANCE OF SUCHTERMS AND CONDITIONS.

1. Definitions 1. "Adaptation" means a work based upon the Work, or upon the Work and other pre-existing works, such as atranslation, adaptation, derivative work, arrangement of music or other alterations of a literary or artistic work, orphonogram or performance and includes cinematographic adaptations or any other form in which the Work may berecast, transformed, or adapted including in any form recognizably derived from the original, except that a work thatconstitutes a Collection will not be considered an Adaptation for the purpose of this License. For the avoidance ofdoubt, where the Work is a musical work, performance or phonogram, the synchronization of the Work intimed-relation with a moving image ("synching") will be considered an Adaptation for the purpose of this License. 2. "Collection" means a collection of literary or artistic works, such as encyclopedias and anthologies, orperformances, phonograms or broadcasts, or other works or subject matter other than works listed in Section 1(f)below, which, by reason of the selection and arrangement of their contents, constitute intellectual creations, in whichthe Work is included in its entirety in unmodified form along with one or more other contributions, each constitutingseparate and independent works in themselves, which together are assembled into a collective whole. A work thatconstitutes a Collection will not be considered an Adaptation (as defined above) for the purposes of this License. 3. "Distribute" means to make available to the public the original and copies of the Work through sale or othertransfer of ownership. 4. "Licensor" means the individual, individuals, entity or entities that offer(s) the Work under the terms of thisLicense. 5. "Original Author" means, in the case of a literary or artistic work, the individual, individuals, entity or entities whocreated the Work or if no individual or entity can be identified, the publisher; and in addition (i) in the case of aperformance the actors, singers, musicians, dancers, and other persons who act, sing, deliver, declaim, play in,interpret or otherwise perform literary or artistic works or expressions of folklore; (ii) in the case of a phonogram theproducer being the person or legal entity who first fixes the sounds of a performance or other sounds; and, (iii) in thecase of broadcasts, the organization that transmits the broadcast. 6. "Work" means the literary and/or artistic work offered under the terms of this License including without limitationany production in the literary, scientific and artistic domain, whatever may be the mode or form of its expressionincluding digital form, such as a book, pamphlet and other writing; a lecture, address, sermon or other work of thesame nature; a dramatic or dramatico-musical work; a choreographic work or entertainment in dumb show; a musicalcomposition with or without words; a cinematographic work to which are assimilated works expressed by a processanalogous to cinematography; a work of drawing, painting, architecture, sculpture, engraving or lithography; aphotographic work to which are assimilated works expressed by a process analogous to photography; a work ofapplied art; an illustration, map, plan, sketch or three-dimensional work relative to geography, topography,architecture or science; a performance; a broadcast; a phonogram; a compilation of data to the extent it is protectedas a copyrightable work; or a work performed by a variety or circus performer to the extent it is not otherwiseconsidered a literary or artistic work. 7. "You" means an individual or entity exercising rights under this License who has not previously violated theterms of this License with respect to the Work, or who has received express permission from the Licensor to exerciserights under this License despite a previous violation. 8. "Publicly Perform" means to perform public recitations of the Work and to communicate to the public those publicrecitations, by any means or process, including by wire or wireless means or public digital performances; to makeavailable to the public Works in such a way that members of the public may access these Works from a place and at aplace individually chosen by them; to perform the Work to the public by any means or process and the communicationto the public of the performances of the Work, including by public digital performance; to broadcast and rebroadcastthe Work by any means including signs, sounds or images.

50

9. "Reproduce" means to make copies of the Work by any means including without limitation by sound or visualrecordings and the right of fixation and reproducing fixations of the Work, including storage of a protectedperformance or phonogram in digital form or other electronic medium.

2. Fair Dealing Rights. Nothing in this License is intended to reduce, limit, or restrict any uses free from copyright orrights arising from limitations or exceptions that are provided for in connection with the copyright protection undercopyright law or other applicable laws.

3. License Grant. Subject to the terms and conditions of this License, Licensor hereby grants You a worldwide,royalty-free, non-exclusive, perpetual (for the duration of the applicable copyright) license to exercise the rights in theWork as stated below: 1. to Reproduce the Work, to incorporate the Work into one or more Collections, and to Reproduce the Work asincorporated in the Collections; and, 2. to Distribute and Publicly Perform the Work including as incorporated in Collections.The above rights may be exercised in all media and formats whether now known or hereafter devised. The aboverights include the right to make such modifications as are technically necessary to exercise the rights in other mediaand formats, but otherwise you have no rights to make Adaptations. Subject to 8(f), all rights not expressly granted byLicensor are hereby reserved, including but not limited to the rights set forth in Section 4(d).

4. Restrictions. The license granted in Section 3 above is expressly made subject to and limited by the followingrestrictions: 1. You may Distribute or Publicly Perform the Work only under the terms of this License. You must include a copy of,or the Uniform Resource Identifier (URI) for, this License with every copy of the Work You Distribute or PubliclyPerform. You may not offer or impose any terms on the Work that restrict the terms of this License or the ability of therecipient of the Work to exercise the rights granted to that recipient under the terms of the License. You may notsublicense the Work. You must keep intact all notices that refer to this License and to the disclaimer of warrantieswith every copy of the Work You Distribute or Publicly Perform. When You Distribute or Publicly Perform the Work,You may not impose any effective technological measures on the Work that restrict the ability of a recipient of theWork from You to exercise the rights granted to that recipient under the terms of the License. This Section 4(a) appliesto the Work as incorporated in a Collection, but this does not require the Collection apart from the Work itself to bemade subject to the terms of this License. If You create a Collection, upon notice from any Licensor You must, to theextent practicable, remove from the Collection any credit as required by Section 4(c), as requested. 2. You may not exercise any of the rights granted to You in Section 3 above in any manner that is primarily intendedfor or directed toward commercial advantage or private monetary compensation. The exchange of the Work for othercopyrighted works by means of digital file-sharing or otherwise shall not be considered to be intended for or directedtoward commercial advantage or private monetary compensation, provided there is no payment of any monetarycompensation in connection with the exchange of copyrighted works. 3. If You Distribute, or Publicly Perform the Work or Collections, You must, unless a request has been madepursuant to Section 4(a), keep intact all copyright notices for the Work and provide, reasonable to the medium ormeans You are utilizing: (i) the name of the Original Author (or pseudonym, if applicable) if supplied, and/or if theOriginal Author and/or Licensor designate another party or parties (e.g., a sponsor institute, publishing entity, journal)for attribution ("Attribution Parties") in Licensor's copyright notice, terms of service or by other reasonable means, thename of such party or parties; (ii) the title of the Work if supplied; (iii) to the extent reasonably practicable, the URI, ifany, that Licensor specifies to be associated with the Work, unless such URI does not refer to the copyright notice orlicensing information for the Work. The credit required by this Section 4(c) may be implemented in any reasonablemanner; provided, however, that in the case of a Collection, at a minimum such credit will appear, if a credit for allcontributing authors of Collection appears, then as part of these credits and in a manner at least as prominent as thecredits for the other contributing authors. For the avoidance of doubt, You may only use the credit required by thisSection for the purpose of attribution in the manner set out above and, by exercising Your rights under this License,You may not implicitly or explicitly assert or imply any connection with, sponsorship or endorsement by the OriginalAuthor, Licensor and/or Attribution Parties, as appropriate, of You or Your use of the Work, without the separate,express prior written permission of the Original Author, Licensor and/or Attribution Parties. 4. For the avoidance of doubt: 1. Non-waivable Compulsory License Schemes. In those jurisdictions in which the right to collect royaltiesthrough any statutory or compulsory licensing scheme cannot be waived, the Licensor reserves the exclusive right tocollect such royalties for any exercise by You of the rights granted under this License; 2. Waivable Compulsory License Schemes. In those jurisdictions in which the right to collect royalties throughany statutory or compulsory licensing scheme can be waived, the Licensor reserves the exclusive right to collect suchroyalties for any exercise by You of the rights granted under this License if Your exercise of such rights is for apurpose or use which is otherwise than noncommercial as permitted under Section 4(b) and otherwise waives theright to collect royalties through any statutory or compulsory licensing scheme; and, 3. Voluntary License Schemes. The Licensor reserves the right to collect royalties, whether individually or, in theevent that the Licensor is a member of a collecting society that administers voluntary licensing schemes, via thatsociety, from any exercise by You of the rights granted under this License that is for a purpose or use which isotherwise than noncommercial as permitted under Section 4(b). 5. Except as otherwise agreed in writing by the Licensor or as may be otherwise permitted by applicable law, if YouReproduce, Distribute or Publicly Perform the Work either by itself or as part of any Collections, You must not distort,mutilate, modify or take other derogatory action in relation to the Work which would be prejudicial to the OriginalAuthor's honor or reputation.

51

5. Representations, Warranties and DisclaimerUNLESS OTHERWISE MUTUALLY AGREED BY THE PARTIES IN WRITING, LICENSOR OFFERS THE WORK AS-IS ANDMAKES NO REPRESENTATIONS OR WARRANTIES OF ANY KIND CONCERNING THE WORK, EXPRESS, IMPLIED,STATUTORY OR OTHERWISE, INCLUDING, WITHOUT LIMITATION, WARRANTIES OF TITLE, MERCHANTIBILITY,FITNESS FOR A PARTICULAR PURPOSE, NONINFRINGEMENT, OR THE ABSENCE OF LATENT OR OTHER DEFECTS,ACCURACY, OR THE PRESENCE OF ABSENCE OF ERRORS, WHETHER OR NOT DISCOVERABLE. SOMEJURISDICTIONS DO NOT ALLOW THE EXCLUSION OF IMPLIED WARRANTIES, SO SUCH EXCLUSION MAY NOTAPPLY TO YOU.

6. Limitation on Liability. EXCEPT TO THE EXTENT REQUIRED BY APPLICABLE LAW, IN NO EVENT WILL LICENSORBE LIABLE TO YOU ON ANY LEGAL THEORY FOR ANY SPECIAL, INCIDENTAL, CONSEQUENTIAL, PUNITIVE OREXEMPLARY DAMAGES ARISING OUT OF THIS LICENSE OR THE USE OF THE WORK, EVEN IF LICENSOR HAS BEENADVISED OF THE POSSIBILITY OF SUCH DAMAGES.

7. Termination 1. This License and the rights granted hereunder will terminate automatically upon any breach by You of the terms ofthis License. Individuals or entities who have received Collections from You under this License, however, will nothave their licenses terminated provided such individuals or entities remain in full compliance with those licenses.Sections 1, 2, 5, 6, 7, and 8 will survive any termination of this License. 2. Subject to the above terms and conditions, the license granted here is perpetual (for the duration of the applicablecopyright in the Work). Notwithstanding the above, Licensor reserves the right to release the Work under differentlicense terms or to stop distributing the Work at any time; provided, however that any such election will not serve towithdraw this License (or any other license that has been, or is required to be, granted under the terms of thisLicense), and this License will continue in full force and effect unless terminated as stated above.

8. Miscellaneous 1. Each time You Distribute or Publicly Perform the Work or a Collection, the Licensor offers to the recipient alicense to the Work on the same terms and conditions as the license granted to You under this License. 2. If any provision of this License is invalid or unenforceable under applicable law, it shall not affect the validity orenforceability of the remainder of the terms of this License, and without further action by the parties to this agreement,such provision shall be reformed to the minimum extent necessary to make such provision valid and enforceable. 3. No term or provision of this License shall be deemed waived and no breach consented to unless such waiver orconsent shall be in writing and signed by the party to be charged with such waiver or consent. 4. This License constitutes the entire agreement between the parties with respect to the Work licensed here. Thereare no understandings, agreements or representations with respect to the Work not specified here. Licensor shall notbe bound by any additional provisions that may appear in any communication from You. This License may not bemodified without the mutual written agreement of the Licensor and You. 5. The rights granted under, and the subject matter referenced, in this License were drafted utilizing the terminologyof the Berne Convention for the Protection of Literary and Artistic Works (as amended on September 28, 1979), theRome Convention of 1961, the WIPO Copyright Treaty of 1996, the WIPO Performances and Phonograms Treaty of1996 and the Universal Copyright Convention (as revised on July 24, 1971). These rights and subject matter take effectin the relevant jurisdiction in which the License terms are sought to be enforced according to the correspondingprovisions of the implementation of those treaty provisions in the applicable national law. If the standard suite ofrights granted under applicable copyright law includes additional rights not granted under this License, suchadditional rights are deemed to be included in the License; this License is not intended to restrict the license of anyrights under applicable law.

mechanical failure

Documents