Mechanical Failure

Mechanical Failure

Mechanical Engineering Design

Dirk Pons

Mechanical Failure Third Edition, 2011

This paper describes the mechanisms whereby material fail, and the mechanical engineering principles to design against failure. Various theories of failure are presented. Another effect that influences the failure of a part is the shape of the geometry, particularly the sharpness of features, which concentrates the stresses to above nominal values. Thereafter the effect of fatigue is presented.

This material is provided under a Creative Commons license(Attribution Non-Commercial No Derivatives), see below for details. The Author[s] accept no liability for the use or inability to use the material in this book. Published in New Zealand 518 Hurunui Bluff Rd Hawarden New Zealand Copyright © Dirk Pons

About the Author Dirk Pons PhD CPEng MIPENZ MPMI is professional Engineer Tohunga Wetepanga and a Chartered Professional Engineer in New Zealand. Dirk is a Senior Lecturer at the University of Canterbury, New Zealand. He holds a PhD in mechanical engineering and a masters degree in business leadership. The Author welcomes comments and s u g g e s t i o n s [email protected]

Mechanical failure 1 2 2.1 2.2 2.3 2.4 3 4 4.1 4.2 4.3

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 THEORIES OF FAILURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Why they are useful . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Theories using stress or strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Theories using strain energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Other Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 STATIC FAILURE OF DUCTILE AND BRITTLE MATERIALS . . . . . . . . . . . . . . . . . . . . . . . 10 GEOMETRIC STRESS CONCENTRATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Mechanism for stress concentrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Geometric Stress concentration factors for stepped shafts . . . . . . . . . . . . . . . . 12 Geometric Stress concentration factors for semicircular notch in a circular shaft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.4 Geometric Stress concentration factors for a U notch in a circular shaft . . . . 16 4.5 Other stress concentrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.6 Ways of avoiding stress concentrations in shaft shoulders . . . . . . . . . . . . . . . . 18 5 FATIGUE FAILURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.1 Mechanism of Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.2 Endurance limit of rotating beam specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.3 Fatigue Strength of Actual Machine Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.4 Low Cycle Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 6 CUMULATIVE FATIGUE DAMAGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 6.1 Manson’s approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 6.2 Miner’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 6.3 Cycle counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6.3.1 Rainflow cycle counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6.3.2 Reservoir cycle counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 7 FLUCTUATING STRESSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 8 FATIGUE IN BIAXIAL STRESS SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 9 SURFACE FATIGUE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 9.1 Hertz Contact Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 9.2 Buckingham's Contact Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 10 CORROSION FATIGUE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 11 DESIGNING AROUND FATIGUE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 11.1 Changes to Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 11.2 Design Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 11.3 Surface Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 11.4 FATIGUE APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 11.4.1 REVERSED BENDING AND STATIC TWISTING OF SHAFTS . . . . . . . . . . . . . . . . 45

Mechanical failure 1

INTRODUCTION: WHAT ARE WE DESIGNING AGAINST?

Means of failure Machine 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 be expected of a reasonable person. Typical of vandalism. M OVERUSE Duty is more severe than the part can tolerate. However the application is correct. Eg using a small electric drill in a building construction industry. Many consumer 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 TEAR Machines and parts have finite lives, after which they fracture or show gross wear. M CORROSION A machine can sometimes be designed to last forever against wear and fracture, but something else like corrosion will get it in the end. It is usually not good marketing to produce machines which last too long. Design strategies The mechanical designer has to take into account the number of factors when designing 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 part There is often some conflict between these, and the following strategy could be followed, stopping where enough had been done. (1)

Static and Brittle failure Prevent gross fracture on first application of load. This is static failure, and will be discussed below. Typically it is necessary to keep stresses below the ultimate tensile strength of the material, or even below the yield. Classical structural mechanics is used to determine the stresses in the part, or numerical calculation, or testing. A safety factor is taken into account, of at least 2.

(2)

Fatigue failure Prevent fatigue failure happening unreasonably early in the expected product life. 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 4

(b)

a method still used in some non-critical (cost/performance) applications. Calculate fatigue strength of material. This method will be shown in this book. Use a safety factor of 2 (standard), 1,5 (product use under closely defined and controlled conditions) or 1,2 (cutting as close as possible to the bone, only suitable for precision designs, where testing will be used to verify performance). Certain types of wear are also fatigue phenomena.

(3)

Deflection Ensure that deflection is acceptable. This only applies to structures that are sensitive to deflection. Typical examples are gears (where face contact is affected), and gas turbines (where inter blade clearance, and blade-shroud clearance is affected). Classical structural mechanics is used for these designs, Design criteria: sometimes with the assistance of finite 1: Avoid fracture element analysis. Creep can also be a 2: Avoid yield problem in these cases. In other types of 3: Limit deflection design problem the possibility of buckling (instability) needs to be considered, eg long columns, thin walled tubes, flat parts (especially plastic).

(4)

Corrosion Avoid corrosion failures of the part. This is usually done by selection of appropriate material.

(5)

Sticking One of the causes of failure of mechanical machines is the sticking of a motion. This needs to be considered during the design stages. Parts may stick for a number of reasons, including: excessively loose fit permits parts to change orientation and jam tight fit causes friction debris in joint, from wear, corrosion or originating externally thermal expansion/contraction as parts change temperature loss of lubricant

C C C C C

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

5

2

THEORIES OF FAILURE

Where a part is subject to uniaxial tensile stress only, then it will begin to fail when the imposed stress equals the yield strength of that material. If the imposed stress is increased still further, up to the ultimate tensile strength of the material, then the part will fracture completely. The yield strength Re and the ultimate tensile strength Rm are material properties, and are independent of the size of the sample. Please note that the theories of failure apply to static loading. Static means that the stresses (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 be avoided, and hence that imposed stresses be kept below the ultimate tensile strength. In addition it is usual to design so that imposed stresses are below the yield strength of the material. This is because permanent deformation occurs at stresses greater than the yield strength, and such deformation disrupts the function of the part. However not all machine parts are subject to simple tension. Instead, and more typically, they may be subject to three dimensional stress patterns, that is combinations 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 the real conditions of stress, it is usual to convert the three dimensional loading into a single effective tensile stress, which can then be compared to the results from a tensile test. In order to make this conversion, it is necessary to have an equation that combines the various components of the three dimensional loading. Various equations have been developed to account for various load cases, and the different sensitivities of material to particular components of the loading. These equations are called theories of failure. The selection of an appropriate theory of failure is based largely on the type of material: brittle or ductile, as described in following sections. Note that these theories of failure apply to static loading, that is loading that does not change with time. There are a number of theories which are described below. Not all of these are valid for all material and load cases. Note the following terminology: ULTIMATE TENSILE STRENGTH Rm Re YIELD STRENGTH in simple tension 8 or < POISSON'S RATIO e.g. 0,3 for steel E MODULUS OF ELASTICITY e.g. 206 x 109 Pa for steel G MODULUS OF RIGIDITY e.g. 82,6 x 109 Pa for steel 6

K

Fx, Fy, Fz F1, F2, F3 Jxy, Jxz, Jyz , (

BULK MODULUS direct stresses in x, y and z axes principal stresses in x, y and z axes (no shear stresses present) 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 stress This theory has that fracture occurs when the maximum principal stress

reaches the yield strength in simple tension. Maximum Shear stress By 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.

7

2.3

Theories using strain energy

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

reaches that in a part under simple tension, namely:

Distortion energy This criterion gives failure when the distortion energy (using principal stresses) of the part

reaches that in simple tension, namely:

This is also called the von Mises theory. Note that (distortion energy) = (total strain energy) - (dilation energy). The Distortion energy is the energy required to change the shape without changing the volume. The Dilation energy changes the volume but not the shape. The distortion energy theory is one of the better ones. It permits the use of the fatigue strength in place of the yield strength. Distortion energy is also sometimes called shear strain energy, or the volumetric strain energy.

8

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

reaches that in simple tension, namely

2.4

Other Theories

Mohr's theory This theory accounts for reversal of stress components. It accommodates materials that have different ultimate tensile and compressive strengths. For a two dimensional stress system, the permissible combinations are as shown in the shaded area shown.

9

3

STATIC FAILURE OF DUCTILE AND BRITTLE MATERIALS

Ductile Materials For ductile materials, failure is by yield rather than fracture. Planes of atoms are moved by the distance of the lattice spacing. The mechanism is one where dislocations (imperfections in the lattice) move atoms one by one under lower force than would be required to move the whole plane. Shear stress drives the dislocations. Work hardening occurs as the dislocations tangle up, and stress relieving as they smooth away. Interstitial atoms may diffuse to the dislocations and pin them, thereby hardening the material. Some dislocations may flow earlier than others, causing plastic stretch bands. The shearing yield strength for a ductile material is typically about 0,57 of the tensile yield strength. The Distortion energy theory (von Mises) and the Octahedral shear stress theory are the most satisfactory. Maximum shear stress (Tresca) produces conservative results. Theories of maximum principal stress and maximum principal strain should not be used. Brittle Materials These materials fail by fracture rather than by yielding. The mechanism is uncontrolled crack growth after cracks exceed a critical length. Thus the designer should try to keep these materials in compression. Mohr's theory and theories of Maximum principal stress and Maximum principal strain may be used here, but should not be used for yield failures. Brittle materials typically have greater compressive that tensile strength. This is accommodated in Mohr's theory.

10

4

GEOMETRIC STRESS CONCENTRATION

An important consideration in design is that parts are not uniform in shape, like the structural mechanics equations generally assume. The shape of the geometry, particularly the sharpness of features, concentrates stresses to above nominal values, 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 above average values in small regions. The nominal load on a part may usually be readily determined from the applied load and the minimum cross sectional area. However the 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 cannot cross air gaps or voids (such as holes). The load carrying paths usually bunch up closer in order to get round the obstacle. As a result the force distribution across the section will become non-uniform. Stress is a measurement of the severity of the force 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 greater than 1,00. The local stress at the most heavily loaded region is given by the product of the nominal stress (determined on the basis of the smallest cross section), and the stress 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 are geometric stress concentration factors (called Kt), as they depend only on the geometry 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 stress mechanisms are combined: residual stresses, rough surfaces, possibility of inclusions, modified metallurgy (eg heat affected zone), and sharp geometry. Stress concentrations also occur where point forces are applied to a structure. True point loading is impossible with materials of finite stiffness, and the force is instead carried 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 load placement within L. 11

Figures follow for stress concentration factors for shaft shoulders, and various other types of geometry commonly encountered in design. For geometry not shown here, consult other handbooks, or use finite element analysis. IMPORTANT Sharp 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 is sharp, 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 by an infinitely small region of material. This means that it is impossible to define a geometric stress concentration factor for such parts. Finite element analysis is also no help: although it will give a stress result for the region, if you were to refine the mesh spacing around the sharp feature, you would find the stress rising. The finer the FEA mesh, the greater the stress, and there is no limit. Many people come unstuck in this matter because they fail to realise that infinitely small fillet radii produce infinitely high stress concentrations. This applies to the circlip grooves already mentioned, as well as to sharp steps in shafts, cracks, V grooves (eg impact test specimens). However these comments do not apply to the external shoulders of shafts, since these regions are stress free. If the stress concentration is infinite, then even a tiny force should generate infinitely high stresses at sharp grooves. We should see all such parts fail immediately, but we don’t. Why not? The answer is that the stresses do start to rise as soon as load is applied, 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 higher force is subsequently applied, then the feature will again go into yield, and round itself out further. In this way the stresses are at most yield, and the part will not fail immediately, at least while there is still ductility in the material. The geometric stress concentration factor only takes the geometry into account. It does 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, like glass, 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 along the mark. It will be shown later that there is a factor called the notch sensitivity, which takes into 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 to provide shoulders for bearings. The stress concentration factors may be determined by referring to a diagram, or using an equation. Data are provided below. Note that it is important to distinguish between the different types of loading: axial, bending, and torsion, since the results are not the same. To find the stress concentration factor, 12

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

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

13

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

BENDING Stress concentration factor for round shaft with shoulder. Bending stress is F = KtMy/I, where y = d/2 and I = Bd4/64 Alternatively the stress concentration factors may be calculated. For bending:

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 use the following values

14

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

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

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

TORSION Stress concentration factor for round shaft with shoulder. Torsional stress is J = KtTr/J, where r = d/2 and J = Bd4/32 Alternatively the stress concentration factors may be calculated. For torsion:

and where K1, K2, K3, and K4 values are determined as follows. For 0,25 # h/r # 4,0 use the following values

15

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

4.3

Geometric Stress concentration factors for semicircular notch in a circular shaft

The geometric stress concentration factor is: Bending Axial Tension

Torsion

4.4

Geometric Stress concentration factors for a U notch in a circular shaft

This geometry is similar to that of a circlip groove, except that the circlip groove can have very sharp corners. The geometric stress concentration factor is: Where the values of K1, K2, K3, and K4 are determined as follows. 16

AXIAL For 0,25 # h/r # 2,0 use the following values

AXIAL For 2,0 # h/r # 50,0 use the following values

K1 = 0,455 + 3,354 (h/r)0,5 - 0,769 h/r K2 = 3,129 - 15,955 (h/r)0,5 + 7,40 h/r K3 = -6,909+29,286 (h/r)0,5 -16,104h/r K4 = 4,325 - 16,685 (h/r)0,5 + 9,469 h/r

K1= 0,935 + 1,922 (h/r)0,5 + 0,004 h/r K2 = 0,537 - 3,708 (h/r)0,5 + 0,040 h/r K3 = - 2,538 + 3,438 (h/r)0,5 - 0,012 h/r K4 = 2,066 - 1,652 (h/r)0,5 - 0,031 h/r

BENDING For 0,25 # h/r # 2,0 use the following values

BENDING For 2,0 # h/r # 50,0 use the following values

K1 = 0,455 + 3,354 (h/r)0,5 - 0,769 h/r K2 = 0,892 - 12,721 (h/r)0,5 + 4,593 h/r K3 = 0,286 + 15,481 (h/r)0,5 - 6,392 h/r K4 =-0,632 - 6,115 (h/r)0,5 + 2,568 h/r

K1 = 0,935 + 1,922 (h/r)0,5 + 0,004 h/r K2 = -0,552 - 5,327 (h/r)0,5 + 0,086 h/r K3 = 0,754 + 6,281 (h/r)0,5 - 0,121 h/r K4 = -0,138 - 2,876 (h/r)0,5 + 0,031 h/r

TORSION For 0,25 # h/r # 2,0 use the following values

TORSION For 2,0 # h/r # 50,0 use the following values

K1 = 1,245 + 0,264 (h/r)0,5 + 0,491h/r K2 = -3,030 + 3,269 (h/r)0,5 - 3,633 h/r K3 = 7,199 - 11,286 (h/r)0,5 + 8,318 h/r K4 = -4,414 + 7,753 (h/r)0,5 -5,176 h/r

K1 = 1,651 + 0,614 (h/r)0,5 + 0,040 h/r K2 = -4,794 - 0,314 (h/r)0,5 - 0,217 h/r K3 = 8,457 - 0,962 (h/r)0,5 + 0,389 h/r K4 = - 4,314 + 0,662 (h/r)0,5 - 0,212 h/r

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

4.5

Other stress concentrations

Geometric Stress concentration factors Kt for Threaded elements THREAD FORM Witworth ISO and UNIFIED

Geometric Stress concentration Kt 3,86 5,00

17

Geometric Stress concentration factors for keyways KEYWAY TYPE End milled keyway Sled-runner keyway Combined bending and torsion

4.6

Geometric Stress concentration Kt 1,79 1,38 3,00

Ways of avoiding stress concentrations in shaft shoulders

Almost all shafts have shoulders, that is step changes in diameter. The shoulders at bearings 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 in order to avoid interference. Therefore stress concentration factors of 2,5 are relatively typical in such cases.

The diagram shows some design practices that are used to reduce the stress concentration. Figure A represents the worst case: a sharp shoulder, with a rough surface texture, and the texture marks at right angles to the line viewed (i.e. circular marks). The first improvement (B) is to increase the fillet radius. Next (C), try to have less abrupt change in section. Smoother texture is shown in (D), and axial marks. Note that this modification does not affect the geometric stress concentration, (which is concerned with large scale effects), but it does improve the fatigue life of the part by reducing the number of microscopic places where cracks can start. While a larger fillet radius is the best and easiest way to decrease stress concentration, it is not always practical because of the problem with small bearing corner 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 the corner of the bearing, however sharp it might be. In practice shoulders are often too 18

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 the advantage of a reduced stress concentration is more than worth it. This is a relatively common design. The next case (G) uses a spacer to provide a sharp corner for the bearing, while still allowing a generous fillet radius. There are however two difficulties with this option: firstly the shoulder must be high enough, and secondly, if the spacer is assembled the wrong way round then it will bite into the fillet and may initiate failure there. The last design (H) shows the addition of another stress raiser. This might not seem a very good idea, but curiously it does reduce the overall stress concentration. It does this by constraining the stress lines so that they do not change direction abruptly.

Other shaft stress raisers Another common source of stress concentration in shafts is a circlip groove. The circlip is used to provide axial location, typically for a bearing. The grooves cut into the load bearing section, and they also have sharp corners, hence the stress concentration. 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 align the stress paths so that they don’t have to suddenly make all their change at the circlip groove. Turning down the shaft achieves a similar effect. The same mechanism works to reduce stress concentration in the machine screw in D), which is turned down to the root diameter of the thread.

It is important to note that the stress concentration effect is one that occurs at changes of shape, and the more abrupt the change the higher the factor. The effect is not so much caused by reduction in cross section as change in shape, and therefore even increases in cross section can cause stress concentration. Therefore material that is not carrying load actually weakens the structure. It only provides a temptation for the load bearing lines to wander, thereby distorting the stress distribution.

19

5

FATIGUE FAILURE

Fatigue is the term that is used to describe the failure of a part at loads well below those predicted by the static theories of failure. Basically a low load applied repetitively for many cycles, can cause failure. Design against fatigue failure is important, since many parts, such as shafts and gears, are exposed to this type of loading. The way we go about designing against fatigue is to determine the stresses in the part (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 substantially greater than the applied stress, then we are safe. Here is how we determine the fatigue strength: first determine the “endurance limit”, and then apply modifying factors.

5.1

Mechanism of Failure

Static failure and fatigue are very different failure mechanisms. In static loading (like a tensile test specimen) the load increases slowly, and a large amount of plastic deformation occurs before final fracture. However fatigue occurs under changing loading, and it gives rise to cracks, even when the nominal stress is in the elastic region (i.e. stresses are well below yield, no plastic flow). Fatigue failure is the progressive fracture of a part. The fracture starts at one point, and progresses through the bulk of the material. Eventually so much of the cross section has been fractured, that the remainder breaks suddenly. The final failure may be after a considerable time of otherwise satisfactory service. Fatigue failure typically occurs at stress levels well below the yield strength of the material.

Fatigue only occurs where there is dynamic loading, that is forces that change with time. Dynamic loading occurs frequently, particularly in moving machines. Failure by static loading normally only occurs in machines that are misused, overloaded, or under designed. A design that is adequate for static loading may still fail by fatigue.

The mechanism of fatigue failure is that localised plastic deformation occurs at small flaws in the material. Such flaws include microscopic features such as lattice imperfections, surface scratches, weld ripples, and machining marks. Larger scale flaws include notches, geometrical changes in section, holes, keyways, threads, casting inclusions, and corroded areas. These flaws exist in all materials to some extent, either internally or on the surface. The loading on the material creates a general strain (or stress) pattern in the whole part. 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 20

strain (i.e. localised stress) can still exist around the stress concentrating flaw. This causes the flaw to grow into a crack. After being started, the crack grows with each load cycle. It progresses through the grain in the direction of weakest resistance, until it gets to the grain boundary. Here it meets resistance to growth, and is arrested. However if the loading is high enough the crack can break through the barrier and into the next grain. Here it will need to follow the weakest path again, which may necessitate a change in direction. Afterwards will be other grain boundaries and grains, probably at different orientations. The crack propagates through these, taking a winding threedimensional path. Eventually the extent of the crack is a significant part of the loaded cross sectional area. The deformation at the tip of the crack is increased, and therefore the splitting ability of the crack is increased: it begins to cut right through grains, regardless of their orientations. Each load cycle now causes significant crack growth, which is visible as microscopic striations on the surface. There are also larger scale "beach marks", which are visible with the naked eye. These are a typical characteristic of fatigue, the marks being similar to those left on a beach by the receding tide. They are 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 to propagate rapidly through to total fracture. This final mode of failure is brittle fracture under static overload, and it produces a rough granular surface, with low distortion. This even occurs in materials which would otherwise be considered ductile. The granular The larger the part, the more flaws it can appearance is not due to brittleness in the contain where fatigue may start. material, but to brittle mode of failure. Conversely, small parts like glass fibres, have fewer flaws and therefore greater

Early analysis of such fractures led to the false resistance to fatigue failure. conclusion that something had caused the material to go "brittle". The material was presumed to have tired, or "fatigued", and hence the name developed. "Progressive failure" would be a more appropriate name given the understanding that we now have of the mechanism. The way we go about designing against fatigue is to determine the stresses in the part (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 substantially greater than the applied stress, then we are safe. Here is how we determine the fatigue strength: first determine the “endurance limit”, and then apply modifying factors. The fine details of fatigue are still actively debated, and from the perspective of the material scientist, the problem is far from solved. However from the engineer's perspective, 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 the practical design of machines and structures. Engineers have available a large body 21

of empirical knowledge of fatigue. This is based on experiment, and is independent of any underlying theory. Even if the fatigue theories eventually change, the design methods won’t change much, since they are based on observation. It is to be expected that consistent data should emerge from fatigue tests, since every engineering part contains vast numbers of flaws, at least some of which will probably be in the right location and orientation to initiate a fatigue crack. These data are explained in the next section, and thereafter is shown how the information is adapted for design purposes.

5.2

Endurance limit of rotating beam specimens

The standard fatigue test is rotating bending, without transverse shear. This pure bending loading is created in a Moore fatigue testing machine. The specimen is carefully prepared to standard dimensions: N0,300", and with a large radius of curvature R 9 7/8") to prevent stress concentration. The surface is polished. The specimen is loaded with a given weight, and rotated until failure. The number of cycles to failure is recorded. Tests are made with different weights. A switch on the weights stops the motor when the specimen fails. The test is done for different weights. Large number of specimens are required for each change in loading, Moore rotating beam test due to the statistical nature of fatigue. Results are applied stress [S], plotted against 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 be expected given the nature of the fatigue mechanism. For most materials, especially ferrous metals, there is a certain stress below which fatigue failure will not occur however long the alternating stress is applied. This stress is called the endurance limit Rn, and it usually occurs at about 106 load cycles. The standard deviation (a measure of data scatter) of the endurance limit is typically about 8% of the value of endurance limit.

General S-N data and curve

The essence of preventing fatigue is to keep the stresses below the endurance limit so low that no crack growth occurs at all. Alternatively the part can be deliberately designed for a finite life, if this is acceptable.

22

Ferrous (iron alloys) and titanium alloys exhibit an endurance limit. Unfortunately, for non-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 the part 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 the part using X-ray photography or other nonAt 3000 rpm, a continuously running shaft destructive testing. 8 would clock up 10 cycles after a time of 108/3000 = 33 333min = 23 days

Ideally the endurance limit for a material should be determined by tests. However in the absence of test data, an acceptable approximation may still be made, since the endurance 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 Rn = 0,4 Rm

CAST IRON 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 a polished specimen of certain geometry, and loaded in only bending. Practical engineering parts are obviously not identical in geometry or loading. The next section shows 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 the endurance limit for a rotating beam specimen because of the differences in geometry and load. These differences are accommodated by applying modifying factors to Rn as follows: 23

This equation is valid for 106 or 108 cycles as the case may be. Thus in an actual machine part, the maximum permissible stress in order to avoid fatigue failure is the fatigue strength Rf. This value will always be less than the endurance limit Rn. The factors are determined as follow. LOAD FACTOR Cl The load factor accounts for types of load other than rotating bending. At 106 cycles the factor is: Rotating Bending Cl = 1 Reversed 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 produces applies maximum stresses all around the perimeter at some time or another. This is the standard test case. In reversed bending the maximum stresses are generated only at the top and bottom, at the worst flaw may not coincide with either of these positions. However the difference is small, and is conservatively neglected. In axial loading the entire cross section is subject to the maximum stress, and thus the chances of a flaw being in a position of stress is increased. If there is eccentricity then there will be a bending stress as well. If the eccentricity is unknown then it is common 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 polished condition. 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 = 1 Steel: Cs depends on machining process and tensile strength, and is shown below.

24

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 mm 7,6 mm # d # 50 mm d > 50 mm

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

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

Cd = 1 Cd = 1 Cd = 1

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

Reliability

1

50%

0,897

90%

0,868

95%

0,814

99%

25

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

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

geometric stress concentration factor 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 some materials are less sensitive to stress concentration, as they are able to yield in regions of high stress, and thereby reduce the sharpness of the cut. Tougher materials have lower notch sensitivity. Notch sensitivity depends on the material, and also on the type of loading and the notch radius. If in doubt, a value of q = 1 may be used, that is Kf = Kt.

26

Notch sensitivity for materials in reversed torsional loading

Some Fatigue Stress concentration factors Kf are given below. Note that these do not need any further correction for notch sensitivity q. Kf for Threaded elements: The table below gives typical values for Geometric- and Fatigue -Stress concentration factors. The fatigue factors depend on the hardness of the material, and on the manufacturing process. THREAD FORM

Geometric Stress concentration

Kt

Fatigue Stress concentration, Kf ROLLED THREADS <200Bhn

CUT THREADS

>200Bhn

< 200 Bhn

>200Bhn

Witworth

386

140

260

176

332

ISO and UNIFIED

500

220

300

284

385

where Bhn refers to Brinell hardness

What is different about the fabrication that rolled threads should be better than cut threads?

Kf for keyways The Fatigue Stress concentration factor depends on the way in which the keyway is cut. The factor for keyways may be avoided by using friction mount devices instead. Note that keys are not generally recommended for reversed shaft rotation.

27

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 Welds Welds are particularly vulnerable to fatigue failure, because of the multitude of flaws internally (porosity, slag inclusions, and incomplete penetration) and externally (roughness), and the adverse heat treatment that the part receives. The toe (edge of the weld bead) is a common fatigue initiator. For fatigue resistance, welds should be ground flush Weld loading with the surface. Undercut and reinforced welds are both undesirable. While it is possible to give fatigue stress concentration factors for welds, this is not usually done. Instead there are welding codes which provide the permissible fatigue stress for a given type of weld. 28

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

5.4

Low Cycle Fatigue

Low cycle fatigue refers to fatigue failure between 103 and 106 load cycles. (For less than 103 cycles, treat as static failure.) If low cycle fatigue failure is permissible, then higher stresses may be accepted than for infinite life. Low cycle fatigue also depends on the type of loading. Correction for surface texture does not have to be made at low cycle fatigue.

Low cycle fatigue is stress loading between one thousand and one million cycles. This diagram is for bending.

Bending The S-N line is drawn from 0,9 Rm at 103 cycles to Rf at 106 cycles, both axes log scaled. Therefore for bending

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

Axial The S-N line is drawn from 0,75 Rm at 103 cycles to 0,85Rf at 106 cycles, both axes log scaled. Low cycle fatigue: Axial loading 29

Thus low cycle endurance limit at N cycles is:

Torsion The S-N line is drawn from 0,9 Rms at 103 cycles to 0,58Rf at 106 cycles, both axes log scaled. Note that if test data is not available, then Rms = 0,577 Rm for ductile materials (distortion energy theory). Thus low cycle endurance limit at N cycles is:

Note Rm ultimate tensile strength Rms 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) + 1 where Sr is determined from the graph below. Other modifying factors are the same as previously defined.

30

Sr factor for 103 cycles.

6

CUMULATIVE FATIGUE DAMAGE

Certain machines and structures (like automobile suspensions) are subject to randomly varying loads. The methods presented here can be used to analyse these cases, providing a plot of actual or assumed stress vs time is available. Once this data 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 several methods after this.

6.1 1 2 3 4 5 6 7 8 9

Manson’s approach Select intersection of S = 0,9 Rm and N = 103 cycles (approximate). 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. Determine endurance limit Rn for virgin material (at 106 cycles). Draw S-N line. For stress Fi (applied ni times) find predicted life Ni. Calculate remaining life Ni - ni (at stress Fi). Connect this point to 0,9 Rm at 103 cycles. Extend line to meet 106 cycles and intersection determines new endurance strength of (damaged) material Rni Repeat as necessary for each other stress Fj (applied nj times) to find remaining life Nj

31

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

If only ni cycles are applied, then the life remaining is Ni+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 a stress of (say) 30MPa would cause failure after 5x104 cycles, then 104 cycles of 30MPa would use up 1/5 of the total cycles of life. If another stress of say 40MPa acted for 500 cycles, where it would normally have a life of 103 cycles, then it would use 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 be assumed to occur. Mathematically this is stated as

32

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. Comment Manson's is by some considered to be superior to Miner’s Rule, since it takes into account the order in which the stresses are applied. For example, one hard landing on an aircraft undercarriage early in service, will cause any crack to progress quickly past the grain barriers, so that it subsequently even grows under light loading. If the same 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 in service a major incident occurs, the shorter the total life. Manson's approach takes this into account, but not Miner's rule. However Miner's rule is easier to use where random loading occurs, because it does not worry about the order in which the loads occur. The accuracy of both these methods is limited by the scatter in data points making up the fatigue test. For a given stress, the accuracy with which the life can be predicted is particularly low. Neither of these methods appears to have very good theoretical justification, but they seem to be the best we have at the moment. Of course testing is ultimately the best provider of design data, and many firms do this when there are sufficiently large production volumes involved. The problem with test results is that they are only valid for the applied conditions. Also, they are only available after fabrication, so they can’t be used for preliminary design. Thus Manson’s and Miner’s methods are still in use.

6.3

Cycle counting

In many practical fatigue applications, such as automotive suspensions, wave and traffic loading, the stress changes considerably with time. There is no one value of stress that can be used for fatigue calculations, and the spectrum is too wild to be able 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 identifiable cycles that can then be used in Miner’s Rule. Cycle counting is applied to stress histories that are obtained from actual structures in service, usually from strain gauges. Two methods of counting cycles are given below.

6.3.1

M M

Rainflow cycle counting Turn the stress history on its side, with the time axis running vertically downwards. Decide on the stress range that you want to use, eg 10MPa intervals in this example. The finer the range the more work you will have to do, but the more accurate the results will be. 33

M M

Move the stress history so that you start where stress is zero. 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 total horizontal distance covered (range) is 70MPa positive. Make a note of this in a table. Water also runs from A to B, down to D and then to E. It falls off at E, and is assumed to stop (the rainflow analogy is not quite watertight!), because a bigger stream (G-H) chops it off. Range for this flow is A-E, which is 50MPa negative. B-C is 20 MPa+, since it is chopped off by the larger EG. C-D is 20MPa-, and is stopped at D because A-B is a larger flow. E-F is 50MPa+, and is stopped at F because 0-A is a larger flow. G-H-L is 140MPa-, and it stops TU because it is larger H-J is 20MPa+, and stops because it meets a larger flow J-K is 20MPa-, and stops because it meets a larger flow L-M-S-T is 120MPa+, and it stops all the smaller flows N-S M-N is 40MPa-, and stops because it meets a larger flow TU

M

M M M M M M M M

34

Rainflow cycle counting

M M M M M

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

35

The ranges are then put into a table. A full cycle is made when there is one negative and one positive cycle of the same size. Stress range [MPa]

Negative -

Positive +

Total (-) + (+) = 1 1x(-) = ½ 1x(+) = ½

140

GH

The cycles OAG and UV are actually just part of one larger cycle, since the placing of the time origin cut this cycle in half.

1

130 120

0 TU

LT

1

110

0

100

0

90

0

80

0

70

OAG UV

60

see 140

0

50

AE

EF

1

40

MN

NS

1

30

PQ

QR

1

20

CD JK

BC HJ

2

10

0

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

6.3.2

M M M M M M M M

Reservoir cycle counting Orient the stress history with time on the horizontal axis, running towards the right. 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 and cyclical duration. Imagine that you are looking at the cross section through a reservoir. Fill up the reservoir with water. Imagine that there are plugs at the bottom of all the troughs. 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 any order, but one at a time. Drain the next deepest trough, and note the change in level. Continue until all the water is out.

36

Reservoir cycle counting

M M M M M M M

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

Therefore, for the purposes of Miner’s rule, this stress waveform is equivalent to one cycle at 140MPa, one at 120MPa, one at 50MPa, one at 40MPa, one at 30MPa, and two at 20MPa. Reference: Maddox SJ, 1991, Fatigue strength of Welded structures, Abington.

37

7

FLUCTUATING STRESSES

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

In problems with fluctuating stress it is necessary to determine mean and alternating components of the stress: mean stress Fm = (Fmax + Fmin)/2 alternating stress Fa = (Fmax - Fmin)/2 These two stresses are then plotted on a Goodman diagram. There are two methods which can be followed.

(a)

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

38

Similar Goodman diagrams may be available for other materials.

(b)

Modified Goodman diagram If data from an actual test is unavailable, the next option is to approximate the Goodman. The method constructs an approximate Goodman diagram for the material, but only for 106 cycle loading. It shows graphically which combinations of Fm and Fa are permissible. A diagram is shown below, and implicit in it are the boundary conditions.

39

In some design applications, the ratio of Fa/Fm may be known, but not the individual values. Solve this problem constructing a line from the origin, with slope equal to the known stress ratio. Intersect this line with the envelope, and determine maximum safe Fa and Fm. For fluctuating torsion it is unnecessary to construct a Goodman Diagram. Instead use the following criteria of failure: *

fatigue occurs when: Ja = Rns (fatigue strength in shear)

*

static failure occurs when: Jmax = Ja + Jm = 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 alternating components). Biaxial systems have stresses in two directions, and each stress may have mean and alternating components. The method is to convert the system to equivalent uniaxial stresses, and then proceed with the Goodman method as described 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 mean and 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 Mohr circle, or with the following (biaxial stress only):

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

40

(4)

Then calculate equivalent stresses: Fm = ( F1m2 - F1m. F2m + F2m2)0,5

Fa = ( F1a2 - F1a. F2a + F2a2)0,5 where 1 and 2 refer to the principal stresses, and a and m refer to alternating and mean components. If there is only one normal stress, Fx, and shear stress Jxy, (each with alternating and mean components), then the equations simplify to: Fm = ( Fxm2 + 3Jxym2)0,5

Fa = ( Fxa2 + 3Jxya2)0,5 (5)

Create a Goodman diagram based on bending stresses (only). Plot the above mean and alternating stress point, and ensure that it lies inside the envelope for 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, until small pieces of material are separated. Small pits or spalls thus form on the surfaces. Under continued operation these areas grow in size, until function is impaired. This type of failure is typical of gears, bearings, and cams. The greater the interface 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, so that a surface compressive stress is generated (perpendicular to the surface). Due to the Poisson effect the material also tries to expand in the other directions, and thus creates stresses in the plane of the surface. The maximum shear stress occurs under the surface, and it reverses sign as a rotating load approaches and then passes. This alternating stress induces fatigue failure. Hertz contact pressure alone is not entirely adequate for situations where sliding friction also occurs. The friction introduces a tangential normal force and a tangential shear force. The normal force reverses sign, and the surface tensile stress is the more damaging. The equations for Hertz contact stresses are given in the chapter on structural mechanics.

41

Buckingham's Contact Stresses

9.2

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 failure of 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 the two contacting materials, calculate Buckingham's load-stress factor K: K = 2,857 Snf2(1/E1 + 1/E2) Surface fatigue failure occurs at 108 cycles as follows:

where n F r1 and r2 w

10

safety factor contact force cylinder radii, cylinder width

CORROSION FATIGUE

This is the combined action of corrosion and cyclic loading. Failure occurs quicker that predicted by either mechanism acting alone. The explanation seems to be that corrosion 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 cracked surfaces show corrosion stains, which are not evident in plain fatigue. Corrosion fatigue depends on elapsed time (for more corrosion), and number of cycles (for more fatigue). Specific test data is required for design, as corrosion fatigue 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

42

11

DESIGNING AROUND FATIGUE

Naturally, stronger materials provide greater resistance to crack growth. However there is a limit to the strength properties of engineering materials, and when the best available material has been used, and is still inadequate, then the designer will have to consider other ways of preventing fatigue failure. Some of these methods are probably 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 the designer can so arrange that the part, or at least the critical section, be loaded in compression, then crack growth can be arrested. Pretension is often used to achieve this, though it requires of course that some other part take even more tension that it would otherwise have, eg bolted joints. The related topic of residual stress is discussed below.

11.2

Design Changes

Since fatigue failures start at small flaws or features, the elimination of such features can 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, and thus surface flaws are more significant than internal ones. Therefore surface condition is an important parameter in fatigue design. The parts that benefit most from surface treatment are those with steep stress gradients from surface to core. These include parts those in bending or torsion, especially under high loads. Axial loading produces a uniform stress distribution, and surface treatments are relatively less effective. There are three parameters of interest: (1)

Surface texture (roughness), the effect of which is quantified previously in the Cs factor and its graph 43

(2)

Surface strength, which should be greatest at the surface. This is achieved by processes such as heat treatment, flame hardening, induction hardening, carburizing, and nitriding. Note that surface weakening occurs in steels that are processed hot (eg forging, hot rolling), as decarburisation of the surface occurs. hydrogen embrittlement can occur in chrome and nickel plated steels

(3)

Residual surface stress, which is best if compressive, since this closes the cracks. 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 which puts the surface in compression. The depth is about a millimetre, and the compressive stress is about half the yield strength. Greater effect is created by 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 roll forming operations, eg to produce threads, and cold pressing are also effective. Certain of the heat treatment operations also cause residual compressive surface stress. These include flame hardening, induction hardening, carburizing, and nitriding. With any type residual stress it is important that the material have a relatively high yield strength, otherwise loads in service will easily be able to erase the residual stress. Processes which cause harmful tensile surface stresses include heavy grinding, welding, and flame cutting. Note that a compressive residual stress on the surface is maintained by a tensile residual stress in the rest of the core. Thus internal flaws, such as porosity 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 cases like bearings, the manufacturer digests the data and provides it in convenient tables for the user. In other applications the designer will have to do the calculations, and make the decisions along the way. Below are shown some applications that commonly arise.

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11.4.1

REVERSED BENDING AND STATIC TWISTING OF SHAFTS

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

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

For a solid shaft this may be rewritten as

where M amplitude (half total height) of bending moment n safety factor Rf fully corrected fatigue strength in bending d diameter However there is one condition: that the shear stress J is less than 1,5 x torsional yield 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 for strength against static failure, as the equation is blind to this mode of failure.

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

where n reserve factor T steady torque yield strength Re M amplitude of bending moment Rf fully corrected fatigue strength in bending BENDING AND TWISTING WITH ALTERNATING AND MEAN COMPONENTS The most general loading for a shaft is where both the bending and twisting moments have alternating (a) and mean (m) components. An axial load and rotating bending will produce this type of bending moment, in which case Fm = axial stress and Fa = bending stress. Fluctuating torque also occurs, typically with reciprocating machines.

GENERAL SODERBERG EQUATION The general Soderburg equation for the diameter d is:

where n reserve factor yield strength Re Rf fully corrected fatigue strength in bending 46

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

where n reserve factor m subscript refers to mean stress a subscript refers to alternating stress Rm 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, reasonable use, 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, (usually gives 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

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Design tip 1 Stronger materials provide greater resistance to crack growth. However there is a limit 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 predicted by the designer. Design for slight overuse by reasonable user. Do what you reasonably can to design against fatigue, and then ensure: T product fails in a safe place or safe way T user is adequately warned T 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 your design will fail before its time. No amount of design or manufacturing or user care will reduce the risk to zero.

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