Materials Selection and Design (2010) Part 11 doc

To account for uncertainties in material properties, the factor of safety is used to divide into the nominal strength S of the material to obtain the allowable stress Sa as follows: wher

Properties Needed for the Design of Static Structures

Mahmoud M Farag, The American University in Cairo

Introduction

ENGINEERING DESIGN can be defined as the creation of a product that satisfies a certain need A good design should result in a product that performs its function efficiently and economically within the prevailing legal, social, safety, and reliability requirements In order to satisfy such requirements, the design engineer has to take into consideration a large number of diverse factors:

• Function and consumer requirements, such as capacity, size, weight, safety, design codes, expected

service life, reliability, maintenance, ease of operation, ease of repair, frequency of failure, initial cost, operating cost, styling, human factors, noise level, pollution, intended service environment, and possibility of use after retirement

• Material-related factors, such as strength, ductility, toughness, stiffness, density, corrosion resistance,

wear resistance, friction coefficient, melting point, thermal and electrical conductivity, processibility, possibility of recycling, cost, available stock size, and delivery time

• Manufacturing-related factors, such as available fabrication processes, accuracy, surface finish, shape,

size, required quantity, delivery time, cost, and required quality

Figure 1 illustrates the relationship among the above three groups The figure also shows that there are other secondary relationships between material properties and manufacturing processes, between function and manufacturing processes, and between function and material properties

Fig 1 Factors that should be considered in component design Source: Ref 1

The relationship between design and material properties is complex because the behavior of the material in the finished product can be quite different from that of the stock material used in making it This point is illustrated in Fig 2, which

shows that in addition to stock material properties, production method and component geometry have direct influence on the behavior of materials in the finished component The figure also shows that secondary relationships exist between geometry and production method, between stock material and production method, and stock material and component geometry The effect of component geometry on the behavior of materials is discussed in the following section

Fig 2 Factors that should be considered in anticipating the behavior of material in the component Source: Ref

1

Reference

1 M.M Farag, Selection of Materials and Manufacturing Processes for Engineering Design, Prentice Hall,

London, 1989

Properties Needed for the Design of Static Structures

Mahmoud M Farag, The American University in Cairo

Effect of Component Geometry

In almost all cases, engineering components and machine elements have to incorporate design features that introduce changes in their cross section For example, shafts must have shoulders to take thrust loads at the bearings and must have keyways or splines to transmit torques to or from pulleys and gears mounted on them Under load, such changes cause localized stresses that are higher than those based on the nominal cross section of the part The severity of the stress concentration depends on the geometry of the discontinuity and the nature of the material A geometric, or theoretical,

stress concentration factor, Kt, is usually used to relate the maximum stress, Smax, at the discontinuity to the nominal

stress, Sav, according to the relationship:

The value of Kt depends on the geometry of the part and can be determined from stress concentration charts, such as those

given in Ref 2 and 3 Other methods of estimating Kt for a certain geometry include photoelasticity, brittle coatings, and

finite element techniques Table 1 gives some typical values of Kt

Table 1 Values of the stress concentration factor Kt

Round shaft with transverse hole

r/d = 0.20 1.25

Source: Ref 1

Experience shows that, under static loading, Kt gives an upper limit to the stress concentration value and applies it to high-strength low-ductility materials With more ductile materials, local yielding in the very small area of maximum stress causes some relief in the stress concentration Generally, the following design guidelines should be observed if the deleterious effects of stress concentration are to be kept to a minimum:

• Abrupt changes in cross section should be avoided If they are necessary, generous fillet radii or relieving grooves should be provided (Fig 3a)

stress-• Slots and grooves should be provided with generous run-out radii and with fillet radii in all corners (Fig 3b)

• Stress-relieving grooves or undercuts should be provided at the end of threads and splines (Fig 3c)

• Sharp internal corners and external edges should be avoided

• Oil holes and similar features should be chamfered and the bore should be smooth

• Weakening features like bolt and oil holes, identification marks, and part numbers should not be located

in highly stressed areas

• Weakening features should be staggered to avoid the addition of their stress concentration effects (Fig 3d)

Fig 3 Design guidelines for reducing the deleterious effects of stress concentration See text for discussion

Source: Ref 1

References cited in this section

1 M.M Farag, Selection of Materials and Manufacturing Processes for Engineering Design, Prentice Hall,

London, 1989

2 R.E Peterson, Stress-Concentration Design Factors, John Wiley and Sons, 1974

3 J.E Shigley and L.D Mitchell, Mechanical Engineering Design, 4th ed., McGraw-Hill, 1983

Properties Needed for the Design of Static Structures

Mahmoud M Farag, The American University in Cairo

Factor of Safety

The term factor of safety is applied to the factor used in designing a component to ensure that it will satisfactorily perform its intended function The main parameters that affect the value of the factor of safety, which is always greater than unity, can be grouped into:

• Uncertainties associated with material properties due to variations in composition, heat treatment, and processing conditions as well as environmental variables such as temperature, time, humidity, and ambient chemicals Manufacturing processes also contribute to these uncertainties as a result of variations in surface roughness, internal stresses, sharp corners, and other stress raisers

• Uncertainties in loading and service conditions

Generally, ductile materials that are produced in large quantities show fewer property variations than less ductile and advanced materials that are produced by small batch processes Parts manufactured by casting, forging, and cold forming are known to have variations in properties from point to point

To account for uncertainties in material properties, the factor of safety is used to divide into the nominal strength (S) of the material to obtain the allowable stress (Sa) as follows:

where ns is the material factor of safety

In simple components, Sa in the above equation can be viewed as the minimum allowable strength of the material However there is some danger involved in this use, especially in the cases where the load-carrying capacity of a component is not directly related to the strength of the material used in making it Examples include long compression members, which could fail as a result of buckling, and components of complex shapes, which could fail as a result of

stress concentration Under such conditions it is better to consider Sa as the load-carrying capacity that is a function of both material properties and geometry of the component

In assessing the uncertainties in loading, two types of service conditions have to be considered:

• Normal working conditions, which the component has to endure during its intended service life

• Limited working conditions, such as overloading, which the component is only intended to endure on

exceptional occasions, and which if repeated frequently could cause premature failure of the component

In a mechanically loaded component, the stress levels corresponding to both normal and limited working conditions can

be determined from a duty cycle The normal duty cycle for an airframe, for example, includes towing and ground handling, engine run, take-off, climb, normal gust loadings at different altitudes, kinetic and solar heating, descent, and normal landing Limited conditions can be encountered in abnormally high gust loadings or emergency landings Analyses of the different loading conditions in the duty cycle lead to determination of the maximum load that will act on the component This maximum load can be used to determine the maximum stress, or damaging stress, which if exceeded

would render the component unfit for service before the end of its normal expected life The load factor of safety (nl) in this case can be taken as:

where P is the maximum load and Pa is normal load

The total or overall factor of safety (n) that combines the uncertainties in material properties and external loading

conditions can be calculated as:

Factors of safety ranging from 1.1 to 20 are known, but common values range from 1.5 to 10

In some applications a designer is required to follow established codes when designing certain components, for example, pressure vessels, piping systems, and so forth Under these conditions, the factors of safety used by the writers of the codes may not be specifically stated, but an allowable working stress is given instead

Properties Needed for the Design of Static Structures

Mahmoud M Farag, The American University in Cairo

in determining the factor of safety and reliability of a given design Figure 4 shows that failure takes place in all the components that fall in the area of overlap of the two curves, that is, when the load-carrying capacity is less than the

external load This is described by the negative part of the (S - P) curve of Fig 4 Transforming the distribution ( - )

to the standard normal deviate z, the following equation is obtained:

z = [(S - P) - ( - )]/[( S)2 + ( P)2]1/2 (Eq 5)

From Fig 4, the value of z at which failure occurs is:

z = - (S - P)/[( S)2 + ( P)2]1/2 (Eq 6)

Fig 4 Effect of variations in load and strength on the failure of components Source: Ref 1

For a given reliability, or allowable probability of failure, the value of z can be determined from cumulative distribution function for the standard normal distribution Table 2 gives some selected values of z that will result in different values of

Knowing S, P, and the expected , the value of can be determined for a given reliability level As defined earlier,

the factor of safety in the present case is simply S/P The following example illustrates the use of the above concepts in

design; additional discussion of statistical methods is provided in the article "Statistical Aspects of Design" in this Volume

Example 1: Estimating Probability of Failure

A structural element is made of a material with an average tensile strength of 2100 MPa The element is subjected to a static tensile stress of an average value of 1600 MPa If the variations in material quality and load cause the strength and stress to vary according to normal distributions with standard deviations of S = 400 and P = 300, respectively, what is the probability of failure of the structural element? The solution can be derived as follows: From Fig 4, ( - ) = 2100 -

1600 = 500 MPa, standard deviation of the curve ( - ) = [(400)2 + (300)2]1/2 = 500; from Eq 6, z = -500/500 = -1

Thus, from Table 2, the probability of failure of the structural element is 0.1587 (15.87%), which is too high for many practical applications

One solution to reduce the probability of failure is to impose better quality measures on the production of the material and thus reduce the standard deviation of the strength Another solution is to increase the cross-sectional area of the element

in order to reduce the stress For example, if the standard deviation of the strength is reduced to S = 200, the standard deviation of the curve ( - ) will be [(200)2 + (300)2]1/2 = 360, z = -500/360 = -1.4, which, according to Table 2, gives

a more acceptable probability of failure value of 0.08 (8%)

Alternatively, if the average stress is reduced to 1400 MPa, ( - ) = 700 MPa, z = -700/500 = -1.4, with a similar

probability of failure as the first solution

Experimental Methods. As the above discussion shows, statistical analysis allows the generation of data on the probability of failure and reliability, which is not possible when a deterministic safety factor is used One of the difficulties with this statistical approach, however, is that material properties are not usually available as statistical quantities In such cases, the following approximate method can be used

In the case where the experimental data are obtained from a reasonably large number of samples, more than 100, it is possible to estimate statistical data from nonstatistical sources that only give ranges or tolerance limits In this case, the standard deviation S is approximately given by:

S = (maximum value of property

This procedure is based on the assumption that the given limits are bounded between plus and minus three standard deviations

If the results are obtained from a sample of about 25 tests, it may be better to divide by 4 in Eq 7 instead of 6 With a sample of about 5, it is better to divide by 2

In the cases where only the average value of strength is given, the following values of coefficient of variation, which is defined as ' = S/S can be taken as typical for metallic materials: ' = 0.05 for ultimate tensile strength, and ' = 0.07

for yield strength

Reference cited in this section

1 M.M Farag, Selection of Materials and Manufacturing Processes for Engineering Design, Prentice Hall,

London, 1989

Properties Needed for the Design of Static Structures

Mahmoud M Farag, The American University in Cairo

Designing for Static Strength

The design and materials selection of a component or structure under static loading can be based on static strength and/or stiffness depending on the service conditions and the intended function

In the case of ductile materials, designs based on the static strength usually aim at avoiding yielding of the component The manner in which the yield strength of the material affects the design depends on the loading conditions and type of component, as illustrated in the following cases

Designing for Axial Loading. When the component is subjected to uniaxial stress, yielding takes place when the local

stress reaches the yield strength of the material The critical cross-sectional area, A, of such a component can be estimated

as:

where Kt is the stress concentration factor (described in the section "Effect of Component Geometry" in this article), P is the applied load, n is the factor of safety, (described in the section "Factor of Safety" in this article), and YS is the yield

strength of the material

Designing for Torsional Loading. The critical cross-sectional area of a circular shaft subjected to torsional loading can be determined from the relationship:

where d is the shaft diameter at the critical cross section, max is the shear strength of the material, T is the transmitted torque, and Ip is the polar moment of inertia of the cross section (Ip = d4/32 for a solid circular shaft; Ip = ( -

)/32 for a hollow circular shaft of inner diameter di and outer diameter do.)

While Eq 9 gives a single value for the diameter of a solid shaft, a large combination of inner and outer diameters can satisfy the relationship in the case of a hollow shaft Under such conditions, either one of the diameters or the required thickness has to be specified in order to calculate the other dimension

The ASTM code of recommended practice for transmission shafting gives an allowable value of shear stress of 0.3 of the yield or 0.18 of the ultimate tensile strength, whichever is smaller With shafts containing keyways, ASTM recommends a reduction of 25% of the allowable shear strength to compensate for stress concentration and reduction in cross-sectional area

Designing for Bending. When a relatively long beam is subjected to bending, the bending moment, the maximum allowable stress, and dimensions of the cross section are related by:

Reference cited in this section

4 W.C Young, Roark's Formulas for Stress and Strain, 6th ed., McGraw-Hill, 1989

Properties Needed for the Design of Static Structures

Mahmoud M Farag, The American University in Cairo

Designing for Stiffness

In addition to being strong enough to resist the expected service loads, there may also be the added requirement of stiffness to ensure that deflections do not exceed certain limits Stiffness is important in applications such as machine elements to avoid misalignment and to maintain dimensional accuracy of machine parts

Design of Beams. The deflection of a beam under load can be taken as a measure of its stiffness, and this depends on the position of the load, the type of beam, and the type of supports For example, a beam that is simply supported at both

ends suffers maximum deflection (y) in its middle when subjected to a concentrated central load (P) In this case the maximum deflection, y, is a function of both E and I, as follows:

y = (P · L3) / (48 · E · I) (Eq 11)

where L is the length of the beam, E is Young's modulus of the beam material, and I is the second moment of area of the

beam cross section with respect to the neutral axis

Design of Columns. Columns and struts, which are long slender parts, are subject to failure by elastic instability, or

buckling, if the applied axial compressive load exceeds a certain critical value, Pcr The Euler column formula is usually

used to calculate the value of Pcr, which is a function of the material, geometry of the column, and restraint at the ends

For the fundamental case of a pin-ended column, that is, ends are free to rotate around frictionless pins, Pcr is given as:

where I is the least moment of inertia of the cross-sectional area of the column, and L is the length of the column

The above equation can be modified to allow for end conditions other than the pinned ends The value of Pcr for a column with both ends fixed built-in as part of the structure is four times the value given by Eq 12 On the other hand, the

critical load for a free-standing column one end is fixed and the other free as in a cantilever Pcr is only one-quarter of the value given by Eq 12

The Euler column formula given above shows that the critical load for a given column is only a function of E and I and is

independent of the compressive strength of the material This means that resistance to buckling of a column of a given material and a given cross-sectional area can be increased by distributing the material as far as possible from the principal

axes of the cross section to increase I Hence, tubular sections are preferable to solid sections Reducing the wall thickness

of such sections and increasing the transverse dimensions increases the stability of the column However, there is a lower limit for the wall thickness below which the wall itself becomes unstable and causes local buckling

Experience shows that the values of Pcr calculated according to Eq 12 are higher than the buckling loads observed in practice The discrepancy is usually attributed to manufacturing imperfections, such as lack of straightness of the column and lack of alignment between the direction of the compressive load and the axis of the column This discrepancy can be accounted for by using an appropriate imperfection parameter or a factor of safety For normal structural work, a factor of safety of 2.5 is usually used As the extent of the above imperfections is expected to increase with increasing slenderness

of the column, it is suggested that the factor of safety be increased accordingly A factor of safety of 3.5 is recommended

for columns with [L(A/I)1/2] > 100, where A is cross-sectional area

Equation 12 shows that the value of Pcr increases rapidly as the length of the column, L, decreases For a short enough column, Pcr becomes equal to the load required for yielding or crushing of the material in simple compression Such a case represents the limit of applicability of the Euler formula as failure takes place by yielding or fracture rather than elastic instability Such short columns are designed according to the procedure described for simple axial loading

Design of Shafts. The torsional rigidity of a component is usually measured by the angle of twist, , per unit length For a circular shaft, is given in radians by:

where G is the modulus of elasticity in shear, and:

where is Poisson's ratio

The usual practice is to limit the angular deflection in shafts to about 1° ( /180 radians) in a length of 20 times the diameter

Properties Needed for the Design of Static Structures

Mahmoud M Farag, The American University in Cairo

Selection of Materials for Static Strength

Static Strength and Isotropy. The resistance to static loading is usually measured in terms of yield strength, ultimate tensile strength, and compressive strength When the material does not exhibit a well-defined yield point, the stress required to cause 0.1 or 0.2% plastic strain, the proof stress, is used instead (This is usually called the 0.2% offset yield strength.) For most ductile wrought metallic materials, the tensile and compressive strengths are very close, and in most cases only the tensile strength is given However, brittle materials like gray cast iron and ceramics are generally stronger in compression than in tension In such cases, both properties are usually given For polymeric materials, which usually do not have a linear stress-strain curve, and whose static properties are very temperature dependent, other design methods must be used; additional information is provided in the article "Design with Plastics" in this Volume

Although many engineering materials are almost isotropic, there are important cases where significant anisotropy exists

In the latter case, the strength depends on the direction in which it is measured The degree of anisotropy depends on the nature of the material and its manufacturing history Anisotropy in wrought metallic materials is more pronounced when they contain elongated inclusions and when processing consists of repeated deformation in the same direction Composites reinforced with unidirectional fibers also exhibit pronounced anisotropy Anisotropy can be useful if the principal external stress acts along the direction of highest strength

The level of strength in engineering materials may be viewed either in absolute terms or relative to similar materials For example, it is generally understood that high-strength steels have tensile strength values in excess of 1400 MPa (200 ksi), which is also high strength in absolute terms Relative to light alloys, however, an aluminum alloy with a strength of

500 MPa (72 ksi) would also be designated a high-strength alloy even though this level of strength is low for steels

From the design point of view, it is more convenient to consider the strength of materials in absolute terms From the materials and manufacturing point of view, however, it is important to consider the strength as an indication of the degree

of development of the material concerned, that is, relative to similar materials This is because highly developed materials are often complex, more difficult to process, and relatively more expensive Figure 5 gives the strength of some materials both in absolute terms and relative to similar materials In a given group of materials, the medium-strength members are usually more widely used because they generally combine optimal strength, ease of manufacture, and economy The most-developed members in a given group of materials are usually highly specialized, and as a result they are produced in much lower quantities The low-strength members of a given group are usually used to meet requirements other than strength For example, electrical and thermal conductivities, formability, or corrosion resistance may be more important than strength in some applications

Fig 5 Comparison of various engineering materials on the basis of tensile strength Source: Ref 1

Frequently, higher-strength members of a given group of materials are more expensive However, using a stronger but more expensive material could result in a reduction of the total cost of the finished component This is because less material would be used, and consequently processing cost could also be less

Weight and Space Limitations. The load-carrying capacity of a given component is a function of both the strength

of the material used in making it and its geometry and dimensions This means that a lower-strength material can be used

in making a component to bear a certain load, provided that its cross-sectional area is increased proportionally However, the designer is not usually completely free in choosing the strength level of the material selected for a given part Other factors such as space and weight limitations could limit the choice

Weight limitations are encountered with many applications including aerospace, transport, construction, and portable appliances In such cases, the strength/density, or specific strength, becomes an important basis for comparing the different materials Figure 6 compares the materials of Fig 5 on the basis of specific strength, which is the tensile strength

of a material divided by its density The figure shows a clear advantage of the fiber-reinforced composites over other materials

Fig 6 Comparison of various engineering materials on the basis of specific tensile strength Source: Ref 1

Using stronger material will allow smaller cross-sectional area and smaller total volume of the component It should be noted, however, that reducing the cross-sectional area below a certain limit could cause failure by buckling due to increased slenderness of the part

Example 2: Materials Selection for a Cylindrical Compression Element

A load of 50 kN is to be supported on a cylindrical compression element of 200 mm length As the compression element has to fit with other parts of the structure, its diameter should not exceed 20 mm Weight limitations are such that the mass of the element should not exceed 0.25 kg Table 3 shows the calculated diameter of the compression element when made of different materials The diameter is calculated on the basis of strength and on the basis of buckling The larger value for a given material is used to calculate the mass of the element The results in Table 3 show that only epoxy-62% Kevlar satisfies both the diameter and weight limits for the compression element

Table 3 Comparison of materials considered for a cylindrical compression element

See Example 2 in text

MPa

Elastic modulus,

GPa

Specific

gravity

Diameter based on strength,

mm

Diameter based on buckling,

mm

Mass based on larger diam,

kg

Remarks

Steels

ASTM A 675, grade 45 155 211 7.8 20.3 15.75 Reject(a)

ASTM A 675, grade 80 275 211 7.8 15.2 15.75 0.3 Reject(b)

ASTM A 715, grade 80 550 211 7.8 10.8 15.75 0.3 Reject(b)

Aluminum

Plastics and composites

Epoxy-70% glass 2100 62.3 2.11 5.5 21.4 Reject(a)

Source: Ref 1

(a) Material is rejected because it violates the limits on diameter

(b) Material is rejected because it violates the limits on weight

Reference cited in this section

1 M.M Farag, Selection of Materials and Manufacturing Processes for Engineering Design, Prentice Hall,

London, 1989

Properties Needed for the Design of Static Structures

Mahmoud M Farag, The American University in Cairo

Selection of Materials for Stiffness

Deflection under Load. As discussed in the section "Design of Beams" in this article, the stiffness of a component may be increased by increasing its second moment of area, which is computed from the cross-sectional dimensions, and/or by selecting a high-modulus material for its manufacture

An important characteristic of metallic materials is that their elastic moduli are very difficult to change by changing the composition or heat treatment Using high-strength materials in attempts to reduce weight usually comes at the expense of reduced cross-sectional area and reduced second moment of area This could adversely affect stiffness of the component

if the elastic constant of the new strong material does not compensate for the reduced second moment of area

Selecting materials with higher elastic constant and efficient disposition of material in the cross section are essential in designing beams for stiffness Placing material as far as possible from the neutral axis of bending is generally an effective

means of increasing I for a given area of cross section See the discussion of shape factor in the article "Performance

Indices" in this Volume

When designing with plastics, whose elastic modulus is 10 to 100 times less than that of metals, stiffness must be given special consideration This drawback can usually be overcome by making some design adjustments These usually include increasing the second moment of area of the critical cross section, as shown in the following example

Example 3: Design Changes Required for Materials Substitution

This example considers the design changes required when substituting high-density polyethylene (HDPE) for stainless steel in making a fork for a picnic set while maintaining similar stiffness The narrowest cross section of the original stainless steel fork is rectangular with an area of 0.6 by 5 mm

Analysis:

• E for stainless steel = 210 GPa

• E for HDPE = 1.1 GPa

• I for the stainless steel section = 5 × (0.6)3/12 = 0.09 mm4

• From Eq 11, EI should be kept constant for equal deflection under load

• EI for stainless steel = 210 × 0.09 = 18.9

• EI for HDPE design = 1.1 × I

• I for HDPE design = 17.2 mm4 Taking a channel section of thickness 0.5 mm, web height 4 mm, and

width 8 mm, I = [8 × (4)3 - 7 × (3.5)3]/12 = 17.7 mm4, which meets the required value

• Area of the stainless steel section = 3 mm2

• Area of the HDPE section = 7.5 mm2

• The specific gravity of stainless steel is 7.8 and that of HDPE is 0.96

• Relative weight of HDPE/stainless steel = (7.5 × 0.96)/(3 × 7.8) = 0.3

Weight Limitations. In applications where both the stiffness and weight of a structure are important, it becomes necessary to consider the stiffness/weight (specific stiffness), of the structure In the simple case of a structural member

under tensile or compressive load, the specific stiffness is given by E/ , where is the density of the material In such cases, the weight of a member of a given stiffness can be easily shown to be proportional to /E and can be reduced by

selecting a material with lower density or higher elastic modulus When the component is subjected to bending, the

dependence of the weight on and E is not as simple From Eq 11 it can be shown that the weight w of a simply

supported beam of square cross-sectional area is given by:

(Eq 15)

Related information is provided in the article "Performance Indices" in this Volume

Equation 15 shows that for a given deflection y under load P, the weight of the beam is proportional to ( /E ) As E in

this case is present as the square root, it is not as effective as in controlling the weight of the beam It can be similarly

shown that the weight of the beam in the case of a rectangular cross section is proportional to ( /E1/3), which is even less

sensitive to variations in E This change in the effectiveness of E in affecting the specific stiffness of structures as the

mode of loading and shape change, is shown in Table 4

Table 4 Comparison of the stiffness of selected engineering materials

elasticity

(E), GPa

Density ( ), mg/m3

Steel (carbon and low alloy) 207 7.825 26.5 5.8 35.1

Aluminum alloys (average) 71 2.7 26.3 9.9 71.2

Magnesium alloys (average) 40 1.8 22.2 11.1 88.2

Titanium alloys (average) 120 4.5 26.7 7.7 50.9

Epoxy-73% E-glass fibers 55.9 2.17 25.8 10.9 81.8

Epoxy-70% S-glass fibers 62.3 2.11 29.5 11.8 87.2

Epoxy-63% carbon fibers 158.7 1.61 98.6 24.7 156.1

Epoxy-62% aramid fibers 82.8 1.38 60 20.6 146.6

Source: Ref 1

Buckling Strength. Another selection criterion that is also related to the elastic modulus of the material and

cross-sectional dimensions is buckling under compressive loading The compressive load, Pb, that can cause buckling of a strut

is given by Euler formula (Eq 12)

Equation 12 shows that increasing E and I will increase the load-carrying capacity of the strut For an axially symmetric cross section, the weight of a strut, w, is given by:

(Eq 16)

Equation 16 shows that the weight of an axisymmetric strut can be reduced by reducing or by increasing E of the material, or both However, reducing is more effective, as E is present as the square root In the case of a panel subjected to buckling, it can be shown that the weight is proportional to ( /E1/3)

Reference cited in this section

1 M.M Farag, Selection of Materials and Manufacturing Processes for Engineering Design, Prentice Hall,

London, 1989

Properties Needed for the Design of Static Structures

Mahmoud M Farag, The American University in Cairo

Types of Mechanical Failure under Static Loading at Normal Temperatures

Generally, a component can be considered to have failed when it does not perform its intended function with the required efficiency The general types of mechanical failure encountered in practice are:

• Yielding of the component material under static loading Yielding causes permanent deformation that

could result in misalignment or hindrance to mechanical movement

• Buckling This type of failure takes place in slender columns when they are subjected to compressive

loading, or in thin-walled tubes when subjected to torsional loading

• Failure by fracture due to static overload This type of failure can be considered an advanced stage of

failure by yielding Fracture can be either ductile or brittle

• Failure due to the combined effect of stresses and corrosion This usually takes place by fracture due to

cracks at stress concentration points, for example, caustic cracking around rivet holes in boilers

Of the above types of mechanical failure, the first two do not usually involve actual fracture, and the component is considered to have failed when its performance is below acceptable levels On the other hand, the latter two types involve actual fracture of the component, and this could lead to unplanned load transfer to other components and perhaps other failures

Properties Needed for the Design of Static Structures

Mahmoud M Farag, The American University in Cairo

Causes of Failure of Engineering Components

As discussed earlier, the behavior of a material in service is governed not only by its inherent properties, but also by the stress system acting on it and the environment in which it is operating Causes of failure of engineering components can

be classified into the following main categories:

• Design deficiencies Failure to evaluate working conditions correctly due to the lack of reliable

information on loads and service conditions is a major cause of inadequate design Incorrect stress analysis, especially near notches, and complex changes in shape could also be a contributing factor

• Poor selection of materials Failure to identify clearly the functional requirements of a component could

lead to the selection of a material that only partially satisfies these requirements As an example, a material can have adequate strength to support the mechanical loads, but its corrosion resistance is insufficient for the application

• Manufacturing defects Incorrect manufacturing could lead to the degradation of an otherwise

satisfactory material Examples are decarburization and internal stresses in a heat-treated component Poor surface finish, burrs, identification marks, and deep scratches due to mishandling could lead to

failure under fatigue loading

• Exceeding design limits and overloading If the load, temperature, speed, and so forth, are increased

beyond the limits allowed by the factor of safety in design, the component is likely to fail Subjecting the equipment to environmental conditions for which it was not designed also falls under this category

An example here is using a freshwater pump for pumping seawater

• Inadequate maintenance and repair When maintenance schedules are ignored and repairs are poorly

carried out, service life is expected to be shorter than anticipated in the design

As this article has described, various material properties influence the design of components The type of property and the sensitivity of the design to variations in this property depend on the component geometry and type of load Underestimation of the load and/or overestimation of the material property and service conditions could lead to failure in service

Properties Needed for the Design of Static Structures

Mahmoud M Farag, The American University in Cairo

References

1 M.M Farag, Selection of Materials and Manufacturing Processes for Engineering Design, Prentice Hall,

London, 1989

2 R.E Peterson, Stress-Concentration Design Factors, John Wiley and Sons, 1974

3 J.E Shigley and L.D Mitchell, Mechanical Engineering Design, 4th ed., McGraw-Hill, 1983

4 W.C Young, Roark's Formulas for Stress and Strain, 6th ed., McGraw-Hill, 1989

Properties Needed for the Design of Static Structures

Mahmoud M Farag, The American University in Cairo

Selected References

• V.J Colangelo and F.A Heiser, Analysis of Metallurgical Failures, John Wiley and Sons, 1987

• N.H Cook, Mechanics and Materials for Design, McGraw-Hill, 1985

• F.A.A Crane and J.A Charles, Selection and Use of Engineering Materials, Butterworths, 1984

• M.M Farag, Materials Selection for Engineering Design, Prentice Hall, London, 1997

Design for Fatigue Resistance

Erhard Krempl, Rensselaer Polytechnic Institute

Introduction

FATIGUE is a gradual process caused by repeated application of loads, such that each application of stress causes some degradation or damage of the material or component An appropriate measure of deterioration or damage has not been found, and this fact makes fatigue design difficult

The design of components against fatigue failure may involve several considerations of irregular loading, variable temperature, and environment In this article, the effect of environment is excluded The effects of environment on material performance are considered elsewhere in this Volume

The main objective here is the discussion of design considerations against fatigue related to material performance under mechanical loading at constant temperature (isothermal fatigue, or simply fatigue) In this article, periodic loading of specimens is considered, and the material properties related to fatigue derived from these tests are discussed

Design methods considering the irregular nature of actual load applications, which requires the statistical treatment of the load histories and their translation into load spectra, cycle counting methods, and damage accumulation will not be

discussed These topics are addressed in more detail in Fatigue and Fracture, Volume 19 of ASM Handbook (Ref 1)

This article reviews "traditional" methods of fatigue design In recent years, the fracture mechanics approach to crack propagation has gained acceptance in the prediction of fatigue life In this approach, crack initiation is neglected and crack propagation to final failure is considered This method can give a conservative estimate of fatigue life There has been considerable success with this method However, the treatment of short cracks, the propagation of cracks for

negative R-ratios, and for irregular loading, crack closure effects and multiaxial loadings provide significant challenges

Reference 2 gives an introduction to the current research in this area

In addition, the design methods reviewed in this article focus principally on smooth and notched components Mechanically fastened joints and welded joints require special attention In particular, such joints tend to negate alloy and composition effects More detailed information on fatigue of welds and mechanical joints is contained in Ref 1, 3, and 4

References

1 Fatigue and Fracture, Vol 19, ASM Handbook, ASM International, 1996

2 M.R Mitchell and R.W Landgraf, Ed., Advances in Fatigue Lifetime Predictive Techniques, STP 1122,

ASTM, 1992

3 H.O Fuchs and R.I Stephens, Metal Fatigue in Engineering, John Wiley & Sons, 1980

4 M.L Sharp and G.E Nordmark, Fatigue Design of Aluminum Components and Structures, McGraw-Hill,

1996

Design for Fatigue Resistance

Erhard Krempl, Rensselaer Polytechnic Institute

The Fatigue Process

The fatigue process consists of a crack initiation and a crack propagation phase The demarcation between these two phases is, however, not clearly defined There is no general agreement when (or at what crack size) the crack initiation process ends, and the crack growth process begins Nonetheless, the separation of the fatigue process in initiation and propagation phases has been developing during the last forty years The tendency is now to investigate crack growth of preexisting flaws and to neglect the crack initiation process This method has been driven by the development of the fracture mechanics approach Previously, the cycles-to-failure needed to completely separate the test specimens were reported, and no separation in crack initiation and crack growth life had been made

For metals and alloys, two regimes of the fatigue phenomenon are generally considered high-cycle fatigue and low-cycle fatigue

High-cycle fatigue involves nominally linear elastic behavior and causes failure after more than approximately 100,000 cycles As the loading amplitude is decreased, the cycles-to-failure increase For many alloys, a fatigue (endurance) limit exists beyond 106 cycles The endurance or fatigue limit represents a stress level below which fatigue life appears to be

infinite upon extrapolation However, fatigue limits from controlled laboratory tests (often limited from 106 to 108 cycles)

are not necessarily valid in application environments For example, a fatigue limit can be eradicated by one large overload

or the onset of additional crack-initiation mechanisms such as corrosion at the surface Moreover, even though testing ceases at 106 or 108 cycles for practical reasons, it is known that fatigue limits cannot be extrapolated to infinity In some cases, there may be a change in failure mechanisms at lives above 106 or 108 cycles

Due to the nominally elastic behavior, dissipation is small and high-frequency testing at small amplitudes can be performed without causing self-heating of the specimen In mechanical testing machines, frequencies up to 300 Hz are possible Usual testing speeds, however, are below 100 Hz At this frequency, approximately 8.6 × 106 cycles are accumulated per day To impose 8.6 × 108 cycles, more than a hundred days of testing time at 100 Hz are required It is therefore no surprise that fatigue data extending to more than 108 cycles are hard to find

Stresses involving inelastic deformation so that a significant stress-strain hysteresis loop develops during cyclic loading lead to low-cycle fatigue failure, usually in less than 100,000 cycles Cyclic inelastic deformation causes dissipation of energy, which can lead to significant self-heating of a specimen if a high frequency is used for testing This is one reason why low-cycle fatigue testing is usually performed at frequencies below 1 Hz Low-cycle fatigue investigations started in the 1950s in response to failures that were found in power-generation equipment The problems were caused by frequent start/stop operations and thermal stresses induced by temperature changes

The simultaneous advent of servocontrolled testing machines, using feedback control, clip-on extensometers, and computer control made testing relevant to the engineering problem at hand and established low-cycle fatigue testing as a separate discipline Complex waveforms can be imposed in both displacement (strain) or load (stress) control; no backlash exists upon zero-load crossing with modern machines Reliable stress-strain data are generated Computer control and computer analysis of data permit a detailed correlation between deformation behavior and fatigue life

Because of the nominally elastic behavior during high-cycle fatigue loading, cracks will initiate from defects such as inclusions, second-phase particles, and other stress concentrators, and inelastic deformation is restricted to these sites Persistent slip bands, which cover a large area of the specimen gage section volume, are not to be expected because the nominal behavior is elastic in high-cycle fatigue loading Therefore, initiation from persistent slip bands, which is the predominant mechanism for low-cycle fatigue crack initiation, can only occur in very few, highly stressed grains These sites will become less frequent as the load amplitude is decreased Defects are normally randomly distributed in a material and vary in their size and distribution from specimen to specimen Therefore, crack initiation and propagation are not going to be identical in different specimens A comparatively large scatter of high-cycle fatigue data is to be expected The nominally inelastic deformation in low-cycle fatigue loading causes many persistent slip bands to develop Crack initiation and propagation can be more uniformly distributed than in high-cycle fatigue loading Therefore, the low-cycle-fatigue data scatter is expected to be less than that of the high-cycle-fatigue data

Nomenclature. This article uses, unless stated otherwise, engineering stress and strain and cycles to failure N,

where failure denotes separation of the test specimens Results from constant amplitude, periodic loadings are the basis of most of the discussions

A periodic loading of stress (sinusoidal, triangular, or other) is imposed, and test results are reported in terms of stress In reference to Fig 1, the maximum and minimum stress is designated by max and min, respectively The mean stress

mean and the stress range are given by (a tensile stress is introduced as a positive quantity and a compressive stress is defined as a negative quantity):

mean = ( max + min)/2 and = ( max - min) (Eq 1)

respectively The stress amplitude or alternating stress is a = /2 The stress range and the stress amplitude are always positive; the mean stress can change sign A completely reversed loading has zero mean stress

Fig 1 Schematic showing the imposed periodic stress and the definition of terms

In addition to these quantities the R-ratio:

and the A-ratio:

are used A simple calculation shows that:

and that:

Either one of the three expressions mean, A, or R can be used to describe the loading For completely reversed loading

mean = 0, R = -1, and A = ; for tension/tension loading mean > 0, 0 < R < 1, and A > 0 Similar values hold for other

types of loading that are less frequently employed

Design for Fatigue Resistance

Erhard Krempl, Rensselaer Polytechnic Institute

High-Cycle Fatigue

To establish a fatigue curve several identical specimens are needed The first specimen is subjected to a given loading intended to result in a number of cycles to failure For the next specimen, the loading is either increased or decreased, and

the number of cycles is observed again In such a way the fatigue or endurance curves, also referred to as S-N curves,

shown in Fig 2, are established The data are for steels and other metallic alloys subjected to completely reversed loading The decrease of the maximum strength with cycles is evident for every alloy For steels, the endurance limit, or fatigue limit the stress below which no fatigue failure is expected no matter how many cycles are applied is well pronounced

Fig 2 S-N diagram for various alloys subjected to completely reversed loading at ambient temperature The

decrease in fatigue strength with cycles and the endurance limit of some steels is shown Source: Ref 6

Under certain conditions, an endurance limit may be observed in steels at ambient temperature, but it may not be present

at elevated temperatures, or may be eradicated with an overload or the onset of corrosion In other alloys, such as hardening aluminum alloys for example, endurance limits at 108 cycles, are not observed Thus, the endurance limit is not

age-an inherent property of metallic alloys

Fatigue Strength and Tensile Strength. Figure 2 clearly demonstrates that the fatigue performance increases with

an increase in tensile strength The increase of the fatigue strength with tensile strength (Fig 3) is true for specimens with good surface finish and without stress concentrators and only up to a certain hardness where flaws do not govern behavior In the presence of notches or of corrosive environment, the fatigue strength does not improve substantially with

an increase of tensile strength Notches and stress raisers are likely going to be present in actual components, and it may

not be possible to achieve the desired fatigue strength by selecting an alloy with increased tensile strength without changing the geometry

Fig 3 The relation between fatigue strength and tensile strength of polished, notched specimens and of

specimens subjected to a corrosive environment Source: Ref 7

Figure 4(a) shows the relation between tensile strength and the fatigue strength for wrought steels, and it is seen that the endurance ratio (fatigue strength at the endurance limit or at a given number of cycles/tensile strength) is between 0.6 and 0.35 A comparable relation holds for aluminum alloys (see Fig 4b) with the endurance ratio between 0.35 and 0.5

Fig 4 Alternating fatigue strength in rotating bending (stress amplitude) and tensile strength Wrought steels,

fatigue strength between 10 7 to 10 8 cycles Source: Ref 8

For steels, the fatigue strength at high number of cycles is approximately 50% of the tensile strength

Data Scatter. For most of the fatigue curves in Fig 2, only one test per stress level was performed and the fatigue

(S-N) curve was drawn through these data points, giving the impression that there is a unique relationship between cycles to failure and fatigue strength If more than one specimen is tested at the same stress level, so that the loading and testing conditions are duplicated within experimental accuracy, different cycles to failure will in general be found It is then possible to plot a histogram of number of failed specimens in a certain cycle interval, as shown in Fig 5 Normalization

of the histogram is accomplished by dividing the ordinate by the total number of specimens, 57 in this case Then the ordinate gives the percentage of specimens that fail at a given number of cycles to failure This method of testing and plotting reveals that fatigue is associated with scatter The scatter has been found in carefully performed tests and has been established as a basic property of the fatigue strength (Note also the difference in the histogram when plotted on a linear and a log scale.)

Fig 5 Histograms showing fatigue-life distribution for 57 specimens of a 75S-T6 aluminum alloy tested at 30

ksi Note the influence of a linear (a) or logarithmic (b) plot of cycles to failure N on the shape of the histogram

Source: Ref 10

To appreciate the effects of scatter it is advantageous to keep the relation shown in the schematic of Fig 6 in mind In this figure, a normalized histogram as shown in Fig 5 is superposed on the fatigue curve at three different stress levels The ordinates of the bell-shaped curves give the percentage of specimens expected to fail at a given number of cycles The

peaks of the distributions can be connected by a fatigue curve This curve would indicate the number of cycles to failure

at which the largest percentage of specimens are expected to fail If only a few tests are run, the results are expected to be close to this curve The width of the band in Fig 6 indicates that all specimens are expected to fail in this interval The distributions become broader as the stress levels decrease, indicating an increase in scatter The reason for this observation has been discussed in the section "The Fatigue Process" in this article

Fig 6 Schematic showing the fatigue curve with the distribution of lives at three different stress levels Note

that the distribution widens as the stress level decreases Source: Ref 11

It is possible to treat fatigue as a probabilistic process and to use methods of probability theory for design However, a probabilistic design of components is prevalent if a large number of components are involved and is then part of a reliability analysis In the majority of cases, a probabilistic fatigue design is not performed because of the expense and time involved in getting the data However, it is always important to keep the nondeterministic nature of fatigue strength

in mind A discussion of probabilistic design methods for fatigue can be found in Ref 12, 13, and 14 Statistical analysis

of fatigue data is described in Ref 15 and 16

Mean Stress Effects. Fatigue life is affected by the presence of a mean stress as shown schematically in Fig 7 A tensile mean stress reduces the life at a given amplitude and a compressive mean stress increases it

Fig 7 Schematic showing the influence of mean stress on fatigue life Sm , mean stress Source: Ref 9

Test data, which most likely pertain to endurance or to a life around the endurance limit, for example, 108 cycles, are shown for steels and aluminum alloys in Fig 8 The beneficial effects of a compressive mean stress and the deleterious

effect of a tensile mean stress are evident It is interesting to observe that in this normalized plot very little difference exists between steels and aluminum alloys

Fig 8 The influence of compressive and tensile mean stresses on the fatigue strength amplitude of steels and

aluminum and alloys Source: Ref 17

Figure 9 shows constant fatigue-life diagrams from 104 to 107 cycles for smooth specimens (solid lines) and for notched specimens (dashed lines) for an age-hardened aluminum alloy Both maximum/minimum stress coordinates and

alternating/mean stress coordinates are used for plotting In addition, lines of constant A and R ratios are also entered The

lines for the smooth specimens end at mean stress of 82 ksi, which corresponds to the ultimate strength of the aluminum alloy In plotting the data, the authors have used the relationships given in Eq 1, 2, 3, 4, and 5 Although the diagram is very busy, it shows, in addition to the fatigue information, how the same data can be plotted differently

Fig 9 Constant-life fatigue diagrams for unnotched (solid lines) and notched (dashed lines) 7075-T6 specimens

with an ultimate tensile strength of 82 ksi The stress concentration factor for the notched specimens is Kt = 3.4 Data were obtained at 2000 cycles/min 1 ksi = 6.895 MPa Source: Ref 17

Figure 9 demonstrates again that for a given fatigue life the allowable amplitude decreases as the mean stress increases

The influence of mean stress in high-cycle fatigue at the endurance limit is often predicted using only the engineering

ultimate tensile or yield strength as shown schematically in Fig 10 The fatigue strength amplitude for R = -1 is plotted on

the ordinate and the ultimate stress and the yield strength are marked on the abscissa Straight lines are then drawn to the endurance at zero mean stress These lines are known as Soderberg and modified Goodman lines as seen in Fig 10 The modified Goodman line is sometimes too conservative, and therefore the Gerber parabola is drawn as shown Forrest (Ref 18) shows that the data fall largely between the Gerber parabola and the modified Goodman line This empirical construction gives a good prediction of the mean stress effect for positive mean stresses

Fig 10 Schematic showing the construction of Soderberg and modified Goodman lines and of the Gerber

parabola Note that this construction is a prediction of the mean stress influence on fatigue Source: Ref 18

Stress Concentration. The equilibrium conditions of mechanics result in a constant stress for a bar of uniform cross section and loaded in the direction of the axis of the bar In bending or torsion, a linear distribution of stress across the section is computed under the assumption of linear elastic behavior For a real specimen, the actual distribution of stresses varies from grain to grain and at the grain boundaries and cannot be determined accurately, if at all, without knowing the exact location, the individual properties, and the orientations of the grains

The average of the actual distribution must be equal to the stress derived from continuum analysis, and this average stress

is used in calculations

If changes in cross section occur in a component, the stress distribution calculated by elastic stress analysis is no longer uniform and no longer one-dimensional An example of an actually calculated linear elastic stress distribution for a deeply

notched bar under axial load P is given in Fig 11 The graph is normalized and is valid for any material that exhibits

linear elastic behavior

Fig 11 Stress distribution in a deeply notched bar Sl, Sc, and Sr are the longitudinal, the circumferential, and

the radial stress, respectively The nominal stress is given by Sn = (4P)/( d2 ) Source: Ref 19

The principal normal stresses plotted are the longitudinal stress Sl, the circumferential stress Sc, and the radial stress Sr

The variation of these stresses are shown as a function of the radius The distribution is symmetric to the center line at r =

0, and no shear stresses are present due to symmetry The radial stress Sr is 0 at the notch root, and the state of stress is two-dimensional there but changes to a three-dimensional one in the interior The longitudinal stress reaches its highest

value in the notch root, which is 2.44 times the nominal stress Sn based on the minimum cross section This geometry is said to have a stress concentration factor of 2.44 (The solution used in Fig 11 is for a deep notch; the outer diameter is very large and need not be specified.)

The stress concentration factor Kt is defined as the ratio of the highest longitudinal stress over the net section nominal stress (sometimes the nominal stress is based on the unnotched section) It simply measures how many times the notch-root longitudinal stress is greater than the nominal stress Effects of multiaxiality are not included

For many geometries and simple loading cases the stress concentration factor has been determined, for example, in Ref 20 and 21 If in addition to the stress concentration factor the stress distribution is desired, it can be determined by experiments or by analysis, or, of course, by finite-element calculations

The stress concentration factor does not give any information on the fatigue strength of a notched member To find out how weakening the notch is going to be in cyclic loading, a fatigue test must be performed As in the case of smooth specimens a fatigue curve is obtained Using the smooth and the notched specimen data obtained under the same loading

conditions a fatigue strength reduction factor Kf can be calculated by:

(Eq 6)

where the notched specimen strength is based on the nominal stress usually referred to the minimum cross section Most

of the tests are done for completely reversed conditions, but results with mean stress can also be analyzed as long as the

conditions for the smooth and the notched specimens are the same In contrast to the stress concentration factor Kt, the

fatigue reduction factor Kf is material specific

The results shown in Fig 9 permit the calculation of the fatigue strength reduction factor Kf based on stress amplitude (alternating stress) for several values of the mean stress at different cycles to failure It can be seen that:

always This has been found true in many other studies and a "fatigue notch sensitivity index" q is in use:

q = (Kf - 1)/(Kt - 1) (Eq 8)

The index varies from 0 for a material with no notch sensitivity to full sensitivity if q = 1

A conservative estimate for the fatigue strength reduction factor Kf is the assumption that it is equal to the stress

concentration factor Kt or that q = 1

This estimate is too conservative in some cases and does not consider the multiaxial state of stress in the notch root Other, more complicated methods of predicting fatigue strength of notched members have been proposed: see Ref 3, 22,

23, and 24

Multiaxial Fatigue. Components can be subjected to complex loadings in the presence of multiaxial states of stress and design methods must be developed for these conditions Multiaxial fatigue tests are performed to elucidate the basic fatigue properties from which life-prediction methods can be derived A thin-walled tube subjected to axial and torsional loading and bars subjected to bending and torsion are examples of possible specimens and loading conditions As a first step, it is desirable to have a measure of stress that can correlate the data obtained under different conditions

A thin-walled tube subjected to axial and torsional loading is considered to elucidate the approach For the tube, the torque generates shear stresses , and the axial forces induces a normal stress in planes perpendicular to the axis For a sinusoidal loading:

= 0 sin t

= 0 sin ( t + )

(Eq 9)

where the subscript 0 indicates amplitude and where and denote the frequency and the phase shift, respectively

If the phase shift is 0, proportional loading is imposed Fatigue curves can be obtained for axial and torsional loading and combined loading Figure 12(a) shows a schematic plotting multiaxial fatigue results for constant amplitude,

completely reversed loading The usual S-N curves are obtained for axial and torsion loading Iso cycles-to-failure curves are obtained for N = 104 and N = 108 cycles They represent the combination of axial and shear stress that lead to failure in the stated number of cycles These curves are replotted in Fig 12(b)

Fig 12(a) Schematic showing S-N curves for multiaxial loading and surface and equal cycles to failure

Fig 12(b) The results of Fig 12(a) replotted as iso-cycles-to-failure curves

A multiaxial stress criterion should be able to reproduce the iso cycles-to-failure curves shown in Fig 12(b)

The von Mises or effective stress criterion is in indicial notation:

(Eq 10)

where ij is the stress matrix, ij is the Kronecker delta, and is the effective stress, which is a function of cycles to

failure N

If Eq 10 is specialized for the axial-torsion case, the equation of iso cycles to failure curves is given by:

2

It is seen that the von Mises criterion requires that the iso cycles to failure curves are ellipses that vary as a function of N

Other criteria, the Tresca criterion for example, result in different iso-cycle curves

The von Mises criterion has been used as a model of a yield surface in plasticity Plastic deformation is known to be insensitive to superposed pressure up to large pressures As a consequence, Eq 10 would predict a 0 right-hand side for pure hydrostatic loading Fatigue on the other hand has been shown to be sensitive to superposed hydrostatic pressure; see Ref 25 for a summary Therefore, the von Mises criterion and other yield criteria are not suitable for correlating fatigue data on principal grounds It is therefore not surprising that the use of yield criteria for correlating fatigue data has not always shown good results (see Ref 25) Despite these fundamental problems, the yield criteria continue to be used in design codes

Loading with constant normal and shear stress amplitudes is assumed Proportional loading is obtained for zero phase shift, = 0 It can be shown easily that in this case the planes of maximum shear stress amplitude are stationary If, on the other hand, a phase shift between axial and shear loading exists, 0, nonproportional loading takes place and the planes of maximum shear stress amplitude change during a cycle

Because of the rotation of the planes of maximum shear stress in out-of-phase loading, 0, there are more potential initiation sites in nonproportional than in proportional loading Different fatigue lives are to be expected, even if axial and shear stress amplitudes are the same This is indeed the case

These and other observations and considerations have led to "critical plane approaches" (see Ref 26)

Additional overviews of multiaxial fatigue are given in Ref 27, 28, and 29 A review paper on multiaxial fatigue and notches is given in Ref 30

References cited in this section

3 H.O Fuchs and R.I Stephens, Metal Fatigue in Engineering, John Wiley & Sons, 1980

6 H.E Boyer, Ed., Atlas of Fatigue Curves, American Society for Metals, 1986, p 30

7 P.G Forrest, Fatigue of Metals, Pergamon Press, 1962, p 59

8 P.G Forrest, Fatigue of Metals, Pergamon Press, 1962, p 58

9 P.G Forrest, Fatigue of Metals, Pergamon Press, 1962, p 73

10 P.G Forrest, Fatigue of Metals, Pergamon Press, 1962, p 47

11 P.E.K Frost, J Marsh, and L.P Cook, Metal Fatigue, Clarendon Press, 1974, p 241

12 P.E.K Frost, J Marsh, and L.P Cook, Metal Fatigue, Clarendon Press, 1974, p 417-427

13 H.O Fuchs and R.I Stephens, Metal Fatigue in Engineering, John Wiley & Sons, 1980, p 94-98

14 H.O Fuchs and R.I Stephens, Metal Fatigue in Engineering, John Wiley & Sons, 1980, p 304-308

15 P.S Veers, Statistical Considerations in Fatigue, Fatigue and Fracture, Vol 19, ASM Handbook, ASM

International, 1996, p 295-302

16 R.C Rice, Fatigue Data Analysis, Mechanical Testing, Vol 8, 9th ed., Metals Handbook, 1985, p 695-719

17 H.O Fuchs and R.I Stephens, Metal Fatigue in Engineering, John Wiley & Sons, 1980, p 75

18 P.G Forrest, Fatigue of Metals, Pergamon Press, 1962, p 94-102

19 H.J Grover, "Fatigue of Aircraft Structures," NAVAIR 01 1A 13, Naval Air Systems Command, U.S Government Printing Office, 1966, p 59

20 W.D Pilkey, Peterson's Stress Concentration Factors, 2nd ed., John Wiley & Sons, 1997

21 E.A Avallone and T Baumeister III, Ed., Mark's Standard Handbook for Mechanical Engineers, 10th ed.,

McGraw-Hill, 1996

22 P.G Forrest, Fatigue of Metals, Pergamon Press, 1962

23 H.J Grover, "Fatigue of Aircraft Structures," NAVAIR 01 1A 13, Naval Air Systems Command, U.S Government Printing Office, 1966

24 N.E Dowling, Mechanical Behavior of Materials: Engineering Methods for Deformation, Fracture, and

Fatigue, Prentice-Hall, 1993

25 E Krempl, The Influence of State of Stress on Low-Cycle Fatigue of Structural Materials A Literature

Survey and Interpretive Report, STP 549, ASTM, 1974

26 M.W Brown and K.J Miller, A Theory of Fatigue under Multiaxial Stress-Strain Conditions, Proc Inst

Mech Eng., Vol 187, 1978, p 745-755

27 Y.S Garud, Multiaxial Fatigue: A Survey of the State of the Art, J Test Eval., Vol 9, 1981, p 165-178

28 F Ellyin, Fatigue Damage, Crack Growth and Life Prediction, Chapman & Hall, 1997

29 D.L McDowell, Multiaxial Fatigue Strength, Fatigue and Fracture, Vol 19, ASM Handbook, ASM

International, 1996, p 263-273

30 S.M Tipton and D.V Nelson, Advances in Multiaxial Fatigue for Components with Stress Concentrations,

Int J Fatigue, in press

Design for Fatigue Resistance

Erhard Krempl, Rensselaer Polytechnic Institute

Low-Cycle Fatigue

Low-cycle fatigue loading usually involves bulk inelastic deformation behavior, and a mechanical hysteresis loop develops between stress and strain Low-cycle fatigue testing is performed at low frequencies, usually below 1 Hz The demands of the application and the development of clip-on axial and diametral extensometers and of servocontrolled testing machines have established strain control as the standard practice in low-cycle fatigue

Low-cycle fatigue investigation was in part prompted by component failures in the power-generation industry These failures at stress risers (notches, fillets) were caused by few, repeated thermal stresses due to startup and shutdown operations The elastically stressed neighborhood limited the strains of the inelastic deformation region associated with stress riser The plastic zone is considered to be under strain control by the surrounding elastic material

In testing, strain control eliminates potentially large strains caused by small errors in controlling the load and by the small tangent modulus in the plastic range Strain-controlled testing limits the strains automatically

Figure 13 explains the terminology used in low-cycle fatigue testing The terms are essentially the same as shown in Fig

1, except that strain is replaced by stress As in the case of high-cycle fatigue, completely reversed testing is done in the majority of cases In contrast to high-cycle fatigue where the deformation behavior is nominally linearly elastic, nonlinear, inelastic behavior is encountered in low-cycle fatigue testing A hysteresis loop develops, and terms related to the stable hysteresis loop are explained in Fig 14

Fig 13 Definition of terms in low-cycle fatigue testing Source: Ref 31

Fig 14 Definition of terms related to the stable hysteresis loop Source: Ref 32

Deformation Behavior

Figure 15 shows hysteresis loops for copper at three different levels of cold work The curves on the left pertain to a fully annealed material (no prior cold work), and the stress range increases considerably The increase of stress range diminishes from cycle to cycle, and finally the hysteresis loop is traced over and over again (not shown) A stable state, shakedown, a cyclic steady state, or cyclic neutral behavior has been reached

Fig 15 Hysteresis loops for copper with varying degree of prior cold work Source: Ref 33

The hysteresis loops of the partially annealed material reach a higher stress amplitude than the ones for the fully annealed material Hardening continues and is followed by softening, a decrease in stress range (see the cycle numbers written on top right; the stress level at first reversal is higher than that at the 4054th) There may have been a quasi-steady-state between the 2000th and 4000th cycle, where the maximum stress changed very little when measured on a per cycle basis

The fully cold-worked specimen, hysteresis loops on the right, cyclically softens, and the stress range decreases The tests depicted in Fig 15 were done in strain control The strain amplitude was kept constant All other quantities, the stresses, especially the stress range, and the plastic strain range (the width of the hysteresis loop at zero stress) can change with cycles Figure 15 shows that these changes can be significant

All dependent variables (all those that are not enforced, or controlled) can change in response to the internal microstructural rearrangements and are macroscopic indicators of such changes If the hysteresis loop changes from cycle

to cycle, a new material forms at each cycle, in principle While the changes and therefore the differences in the material from cycle to cycle are small, they can be significant between the initial condition and steady-state condition

Figure 15 illustrates a general trend found in cyclic inelastic deformation Annealed materials tend to harden cyclically In contrast, cyclic softening is observed in cold-worked materials The trend is not always uniform; sometimes cyclic hardening (softening) is followed by softening (hardening) An example is given in Fig 15 for the partially annealed specimen and in Fig 16 for type 304 stainless steel At the high strain ranges, cyclic hardening is observed followed by significant softening At the low strain ranges the initial behavior is cyclic neutral and is followed by softening It should

be noted that the initial behavior is for a small number of cycles only and that the drops in stress range appear exaggerated due to the vertical scale starting at 400 MPa

Fig 16 Stress range versus cycles for type 304 stainless steel showing initial hardening or neutral behavior

followed by softening Source: Ref 34

Designers allow for the difference of a cyclically loaded material and the material when cycling started by constructing the so-called cyclic stress-strain diagrams Here the stress amplitude at cyclic steady state, or, if such a steady state does not exist, the stress amplitude at half the number of cycles to failure is determined The corresponding strain amplitude and stress amplitude are then entered into a stress versus strain diagram Only a few points (around 5 points usually) are obtained this way, but they suffice in most cases to draw a cyclic stress-strain diagram No significant microstructural changes take place for elastic cycling, and therefore the monotonic and the cyclic stress-strain diagram have the same elastic slope Differences appear at strain amplitudes beyond the elastic range as shown in Fig 17, where the monotonic and the cyclic stress-strain diagrams for six different engineering alloys are presented When the cyclic curve is above (below) the monotonic curve, cyclic hardening (softening) is achieved For many of the alloys, the cyclic curve is equal or above the monotonic curve

Fig 17 Monotonic and cyclic stress-strain diagrams for six different engineering alloys , companion specimens; solid line, incremental step Source: Ref 35

For an analytical representation, the Ramberg-Osgood formula is used for both the cyclic and the monotonic stress strain diagram For the monotonic case:

(Eq 12)

and for the cyclic condition:

(Eq 13)

where E and K are the elastic modulus and strength constant with dimensions of stress, respectively, and n is the

dimensionless strain-hardening exponent Primed quantities refer to the cyclic stress-strain diagram, and and denote the strain and stress range, respectively Table 1, from Ref 38, gives typical data for some steels and some aluminum alloys

Table 1 Monotonic and cyclic fatigue properties of selected engineering alloys

-274 0.33

4340 Q&T 1241 1172/758 1579/ 0.066/0.14 0.84/0.73 1655/1655

-0.076

0.62

-492 0.40

4340 Q&T 1469 1372/827 /0.15 0.48/0.48 1560/2000

-0.091

0.60

-467 0.32

9262 Annealed 924 455/524 1744/1379 0.22/0.15 0.16/0.16 1046/1046 - - 348 0.38

0.071 0.47

9262 Q&T 1000 786/648 ./1358 0.14/0.12 0.41/0.41 1220/1220

-0.073

0.60

-37 0.33

2024-T3 469 379/427 455/655 0.032/0.065 0.28/0.22 558/1100

-0.124

0.59

-176 0.30

These values do not represent final fatigue design properties HR, hot rolled; CD, cold drawn; Q&T, quenched and tempered

Ultimate engineering strength, Su; engineering yield strength, Sy; fatigue strength, S'f All other quantities are defined in Eq 15 and 16 Source: Ref 36, 37

Equations 12 and 13 are the simplest representation of stress-strain diagrams They are only valid for monotonic loading

If the hysteresis loop and its evolution have to be described and modeled, incremental plasticity or unified state variable theories are needed (see Ref 39)