Fatigue life is the number of loading (stress or strain) cycles of a specified character that a specimen sustains before failure of a specified nature occurs. For some metals, notably steel and titanium, there is a theoretical value for stress amplitude below which the material will not fail for any number of cycles, called a fatigue limit, endurance limit, or fatigue strength.

The majority of engineering failures are caused by fatigue. There is very little warning or no warming at all before fatigue failure, therefore the consequences are often catastrophic.

Engineers have used any of three methods to determine the fatigue life of a material: the stress-life method, the strain-life method, and the linear-elastic fracture mechanics method. One method to provide material properties for fatigue is the Uniform Material Law (UML). UML was developed for fatigue life prediction of aluminium and titanium alloys by the end of 20th century and extended to high-strength steels, and cast iron. As per the UML, steels are classified into unalloyed, low alloyed and high alloyed. The UML was derived from the unalloyed and low alloyed data. The estimation for high alloyed steel is not provided by UML. Unalloyed steels consist of steels such as the SAE 10xx series, and HSLA. Low alloy consist of steels such as 4130, 8620.

In materials science, fatigue is the weakening of a material caused by repeatedly applied loads called cyclic loads. It is the progressive and localized structural damage that occurs when a material is subjected to cyclic loading. The nominal maximum stress values that cause such damage may be much less than the strength of the material typically quoted as the ultimate tensile stress limit, or the yield stress limit.

Fatigue occurs when a material is subjected to repeated loading and unloading. If the loads are above a certain threshold, microscopic cracks will begin to form at the stress concentration regions such as the surface, persistent slip bands (PSBs), and grain interfaces. This is the crack initiation phase N_{i}. Eventually a crack will reach a critical size, the crack will propagate suddenly, and the structure will fracture. This is the crack propagation phase N_{p}. The total fatigue life N_{f} is the summation of the two phases:

N_{f} = N_{i} + N_{p}

The shape of the structure and surface finish of the material will significantly affect the fatigue life, especially at the crack initiatin phase. Square holes or sharp corners will lead to elevated local stresses where fatigue cracks can initiate. Round holes and smooth transitions or fillets will increase the fatigue life of the structure. Smooth surfaces will also improve fatigue life. In the crack propagation phase however, the surface condition is insignificant. For machine components the crack initiation phase accounts for most of the life; for welded connections, small crack-like imperfections are generally present during welding process, the entire life is considered to be spent mostly (90% to 100%) in the propagation phase.

In fatigue life analysis, it is important to distinguish stress range Δσ, stress amplitude σ_{a}, and maximum stress σ_{max}. It is the stress range (i.e. the fluctuation of the stresses), rather than maximum stress, that drives fatigue damage. Going hand-in-hand, the stress amplitude σ_{a} and mean stress σ_{m} are the important parameters in fatigue life analysis.

The stress range Δσ is the difference between the maximum stress σ_{max} and minimum stress σ_{min} in a loading cycle:

Δσ = σ_{max} – σ_{min} = 2σ_{a}

The effect of mean stress is expressed by a stress ratio R:

R = σ_{min} / σ_{max} = (σ_{m} – σ_{a}) / (σ_{m} + σ_{a})

Referring to the figure below, the most commonly applied stress ratios in fatigue tests are R = -1 (fully reversed loading) and R = 0 (pulsation tension).

It should be noted that some fatigue curves (S-N curves or ε-N curves) are expressed by stress (strain) ranges; on the other hand, some use stress (strain) amplitude in the curves. Because the difference between stress (strain) range and stress (strain) amplitude is 2 times, close attention should be paid to which fatigue curve is being used.

The following figure shows definitions of the bilinear S-N curve for structural steel. The stress is expressed in terms of stress amplitude σ_{a.} However, the stress may also be given in terms of stress range Δσ. Both stress and fatigue life are in log scale.

In this case, the primary (pre-knee) and secondary (post-knee) relationships are

The parameters, such as m_{1,} m_{2,} C_{1} and C_{2} etc., are experimentally obtained from tests. In practice, these curves are codified to account for worst case detrimental effects that may arise due to variance in material quality, fabrication, workmanship, accuracy of analyses etc.

Note that horizontal dashed line in the S-N curve above represents the fatigue limit or Constant Amplitude Fatigue Limit (CAFL) for constan amplitude loading. The difference between Constant Amplitude (CA) loading and Variable Amplitude (VA) loading is as shown below:

Constant Amplitude (CA) loading

Variable Amplitude (VA) loading

It should be noted that not all constant amplitude loading result in CAFL. For instance, some materials do not exhibt a fatigue limit in a corrosive environment.

### Mean Stress Effects

Fatigue life generally depends on the mean stress values. Various criteria have been proposed to deal with the mean stress effect on fatigue life, including the following:

**Soderberg criterion**

Soderberg criterion is very conservative. It is not bounded when using nagative (compressive) mean stresses.

**Goodman criterion**

Goodman criterion is usually a good choice for brittle materials. It is not bounded when using nagative (compressive) mean stresses. For ductile materials however, because the Goodman method employs the ultimate tensile strength, the results are highly inaccurate and should not be used.

**Gerber criterion**

Gerber criterion is usually a good choice for ductile materials. It is bounded when using nagative (compressive) mean stresses.

**Morrow criterion**

where σ_{f} is true fracture strength, which is often not available, but can be taken as σ’_{f} = fatigue strength coefficient, which is the stress intercept at N_{f} = 0.5 cycles. The σ_{f} = σ’_{f} estimate is quite good for steels, but often gives grossly non-conservative life estimates for aluminium alloys.

The Morrow mean stress correction in the above form was included by Morrow into the elastic term of the strain-life equation given as

The morrow model predicts that the mean stress has a significant effect on longer lives, where the elastic strain amplitude dominate. The model also predicts that the mean stress has little effect on shorter lives, where the plastic strain is large. The prediction trend of the Morrow mean stress correctin model is consistent with observations that the mean stress has greater impact at longer lives.

**Smith-Watson-Topper (SWT) criterion**

An extenstion of the SWT parameter can be aplied to the strain-life equation by replacing σ_{a} in SWT equation given above with the strain amplitude ε_{a}. The product of the maximum tensile stress σ_{max} and the strain amplitude ε_{a} in the strain life model controls the influence of both the mean stress and the strain amplitude:

This model gives good estimation of mean stress effect in the long life regime, but it is conservative in the low cycle fatigue region. SWT method provides good results in most cases, and for aluminium alloys it is more accurate than the Morrow equation.

**Walker equation**

Walker equation is similar to SWT but involves an additional material property γ. The equation is expressed as

Values of γ varies from 0 to 1; for metals the γ vaules are in the range 0.4 to 0.8. If γ = 0.5, the Walker equation is reduced to SWT criterion. With the adjustable fitting parameter γ, Walker equation gives superior results. Relatively high strength aluminium allows have γ = 0.5; higher values of γ apply for relatively low-strength aluminium alloys. For both steel and aluminium alloys, there is a trend of decreasing γ with increasing strength, indicating an increasing sensitivity to mean stress.

For elastic-plastic fatigue model, the walker equation can be modified as

The modified model combines the advantages of the Walker material parameter γ and the SWT parameter.

**Manson-Halford criterion**

Manson and Halford suggested that both elastic and plastic terms of the strain-life equation should be modified to account for the mean stress effects and maintain the independence of the elastic to plastic ratio from the mean stress.

The Manson-Halford criterion tends to overestimate the mean stress effect on short lives, where the plastic strain dominates.

In the equations above,

σ_{a} = stress amplitude

ε_{a} = strain amplitude

σ_{m} = mean stress

σ_{ar} = equivalent fully reversed (zero mean stress) stress amplitude resulting in the same fatigue life as the σ_{a} – σ_{m} combination

σ_{u} = material ultimate strength

σ_{y} = material yield strength

σ_{f} = true fracture strength

σ_{max} = maximum stress

R = stress ratio

E = modulus of elasticity

σ’_{f} = fatigue strength coefficient

b = fatigue strength exponent

ε’_{f} = fatigue ductility coefficient

c = fatigue ductility exponent

N_{f} = number of cycles to failure

Fatigue life assessements are regulated by design codes in different sectors, the most widely used codes being as listed below:

**ANSI/AISC 360, Specification for Structural Steel Buildings**

This design code covers industrial building sector with highly cyclic live loading, such as bridge cranes, crane runways, monorails, lifting beams, lifting lugs (lifting eyes), manufacturing equipment, etc. AISC 360 also covers highway bridges, railway bridges and pedestrian bridges in transportation sector. AISC 360 deals with steel members, welded components, bolts and threaded parts.

**ASME BTH-1, Design of Below-the-Hook Lifting Devices**

ASME BTH-1 deals with steel members, welded components, bolts and threaded parts of below-the-hook lifting devices, such as lifting beams, spreader beams, lifting lugs (lifting eyes), hooks, slings and rigging hardware, etc.

**EN 1993-1-9, Design of steel structures — Part 1-9: Fatigue**

EN 1993-1-9 gives methods for the assessment of fatigue resistance of members, connections and joints subjected to fatigue loading. These methods are derived from fatigue tests with large scale specimens, that include effects of geometrical and structural imperfections from material production and execution. Fatigue strengths are determined by considering the structural detail together with its metallurgical and geometric notch effects. In the fatigue details presented in this design code the probable site of crack initiation is also indicated. The assessment methods presented in this code use fatigue resistance in terms of fatigue strength curves for (1) standard details applicable to nominal stresses; (2) reference weld configurations applicable to geometric stresses.

**DNVGL-RP-C203, Fatigue design of offshore steel structures**

This recommended practice presents recommendations in relation to fatigue analysis based on fatigue tests (S-N data) and fracture mechanics. DNVGL-RP-C203 is valid for carbon manganese steel (C-Mn) in air with yield strength less than 960 MPa. For carbon and low alloy machined forgings for subsea applications, the S-N curves are valid for steels with tensile strength up to 862 MPa in air environment. For steel (C-Mn) materials in seawater with cathodic protection or steel with free corrosion, the recommended practice is valid up to 690 MPa. This limit applies also to the carbon and low alloy machined forgings for subsea applications. This recommended practice is also valid for bolts in air environment or with protection corresponding to that condition of grades up to 10.9, ASTM A490 or equivalent. This recommended practice may be used for stainless steel.

**API-2A-WSD, API Recommended Practice 2A-WSD: **

**Planning, Designing, and Constructing Fixed Offshore Platforms—Working Stress Design**

This recommended practice is based on global industry best practices and serves as a guide for those who are concerned with the design and construction of new fixed offshore platforms and for the relocation of existing platforms used for the drilling, development, production, and storage of hydrocarbons in offshore areas.

Nontubular members and connections in deck structures, appurtenances and equipment; and tubular members and attachments to them, including ring stiffeners, may be subject to variations of stress due to environmental loads or operational loads. Where variations of stress are applied to conventional weld details, the associated S-N curves provided in AWS D1.1 are used. For service conditions where details may be exposed to random variable loads, seawater corrosion, or submerged service with effective cathodic protection, the fatigue life reduction factors are applied accordingly.

For tubular connections, this recommended practice provides S-N curves for welded joints (WJ) and cast joints (CJ) in air environment, seawater with cathodic protection, and seawater free corrosion conditions.

**ISO 19902, Petroleum and natural gas industries — Fixed steel offshore structures**

This International Standard specifies requirements and provides recommendations applicable to the following types of fixed steel offshore structures for the petroleum and natural gas industries:

- caissons, free-standing and braced;
- jackets;
- monotowers;
- towers.

In addition, it is applicable to compliant bottom founded structures, steel gravity structures, jack-ups, other bottom founded structures and other structures related to offshore structures (such as underwater oil storage tanks, bridges and connecting structures), to the extent to which its requirements are relevant.

ISO 19902 provides S-N curves for tubular joints (TJ), cast joints (CJ) and other joints (OJ) in air environment, seawater with cathodic protection, and seawater free corrosion conditions.

**EN 12952-3, Water-tube boilers and auxiliary installations — Part 3: **

**Design and calculation for pressure parts of the boiler**

Water-tube boiler pressure parts shall be designed in accordance with the requirements of this European Standard. Boiler components are deemed to be exposed to cyclic loading if the boiler is designed for more than 500 cold start-ups. The design rules presented in Annex B, Fatigue cracking – Design to allow for fluctuating stress, apply to the design of pressurized components of boilers made from ferritic and austenitic rolled or forged steels. These rules allow for the fluctuating stresses occurring at the most highly stressed points as a result of internal pressure and differences in temperature and/or the addition of external forces and moments. The maximum shear stress theory shall be used in the determination of the decisive cyclic stress amplitude and the mean cyclic stress. Stress intensity, which is two times of shear stress, is also used in the analysis. The controlling stress fange is divided into elastic range, partly elastic range, and fully plastic range.

Cyclic stress range and mean cyclic stress shall be increased to account for the notch effect (micro notch effect) associated with surface and weld influences. Here, the governing factor in each case is the final state following manufacture.

In the case of a load-cycle temperature t* ≥ 100 °C, the reduction in the fatigue strength caused by the temperature shall be taken into account by means of a correction factor *C*_{t*} for ferritic and austennitic alloys.

**EN 13445-3, Unfired pressure vessels Part 3: Design**

Clause 18, detailed assessment of fatigue life, specifies requirements for the detailed fatigue assessment of pressure vessels and their components that are subjected to repeated fluctuations of stress. The Tresca criterion is applied in this clause but use of the ‘von Mises’ criterion is also permitted.

At structural discontinuities, the following stresses are defined:

**Nominal stress** – stress which would exist in the absence of a discontinuity. Nominal stress is a reference stress (membrane + bending) which is calculated using elementary theory of structures. It excludes the effect of structural discontinuities (e.g. welds, openings and thickness changes). The use of nominal stress is permitted for some specific weld details for which determination of the structural stress would be unnecessarily complex. It is also applied to bolts. The nominal stress is the stress commonly used to express the results of fatigue tests performed on laboratory specimens under simple unidirectional axial or bending loading. Hence, fatigue curves derived from such data include the effect of any notches or other structural discontinuities (e.g. welds) in the test specimen.

**Structural stress** – linearly distributed stress across the section thickness which arises from applied loads (forces, moments, pressure, etc.) and the corresponding reaction of the particular structural part. Structural stress includes the effects of gross structural discontinuities (e.g. branch connections, cone/cylinder intersections, vessel/end junctions, thickness change, deviations from design shape, presence of an attachment). However, it excludes the notch effects of local structural discontinuities (e.g. weld toe) which give rise to non-linear stress distributions across the section thickness. For the purpose of a fatigue assessment, the structural stress shall be evaluated at the potential crack initiation site. Structural stresses may be determined by one of the following methods: numerical analysis (e.g. finite element analysis), strain measurement or the application of stress concentration factors to nominal stresses obtained analytically. Under high thermal stresses, the total stress rather than the linearly distributed stress should be considered.

**Notch stress** – total stress located at the root of a notch, including the non-linear part of the stress distribution. Notch stresses are usually calculated using numerical analysis. Alternatively, the nominal or structural stress is used in conjunction with the effective stress concentration factor K_{f}.

Distribution of nominal, structural and notch stress at a structural discontinuity

A fatigue assessment shall be made at all locations where there is a risk of fatigue crack initiation. It is recommended that the fatigue assessment is performed using operating rather than design loads. In fatigue, welds behave differently from plain (unwelded) material. Therefore the assessment procedures for welded and unwelded material are different. Plain material might contain flush ground weld repairs. The presence of such repairs can lead to a reduction in the fatigue life of the material. Hence, only material which is certain to be free from welding shall be assessed as unwelded.

**ASME Section VIII, Rules for Construction of Pressure Vessels Division 2 —**

**Alternative Rules**

ASME BPVC.VIII.2 contains mandatory requirements, specific prohibitions, and nonmandatory guidance for the design, materials, fabrication, examination, inspection, testing, and certification of pressure vessels and their associated pressure relief devices. Pressure vessels are containers for the containment of pressure, either internal or external. This pressure may be obtained from an external source or by the application of heat from a direct or indirect source as a result of a process, or any combination thereof. These vessels shall be designated as either a Class 1 or Class 2 vessel in conformance with the User’s Design Specification.

Class 1 Vessel – a vessel that is designed using the allowable stresses from Section II, Part D, Subpart 1, Table 2A or Table 2B.

Class 2 Vessel – a vessel that is designed using the allowable stresses from Section II, Part D, Subpart 1, Table 5A or Table 5B.

Fatigue provisions are in Part 5 – Design by Analysis Requirements, provides requirements for design of vessels and components using analytical methods. A fatigue evaluation shall be performed if the component is subject to cyclic operation. The evaluation for fatigue is made on the basis of the number of applied cycles of a stress or strain range at a point in the component. The allowable number of cycles should be adequate for the specified number of cycles as given in the User’s Design Specification.

**ASME Section VIII, Rules for Construction of Pressure Vessels Division 3 —**

**Alternative Rules for Construction of High Pressure Vessels**

The rules of ASME BPVC.VIII.3 constitute requirements for the design, construction, inspection, and overpressure protection of metallic pressure vessels with design pressures generally above 70 MPa (10 ksi). Pressure vessels within the scope of this Division are pressure containers for the retainment of fluids, gaseous or liquid, under pressure, either internal or external. This pressure may be generated by: an external source; the application of heat from direct source or indirect source; a process reaction; or any combination thereof.

Article KD-3 Fatigue Evaluation presents a traditional fatigue analysis design approach. In accordance with KD-140, if it can be shown that the vessel will fail in a leak‐before‐burst mode, then the number of design cycles shall be calculated in accordance with this Article.

Cyclic operation may cause fatigue failure of pressure vessels and components. While cracks often initiate at the bore, cracks may initiate at outside surfaces or at layer interfaces for autofrettaged and layered vessels. In all cases, areas of stress concentrations are a particular concern. Fatigue‐sensitive points shall be identified and a fatigue analysis made for each point. The result of the fatigue analysis will be a calculated number of design cycles Nf for each type of operating cycle, and a calculated cumulative effect number of design cycles when more than one type of operating cycle exists. The resistance to fatigue of a nonwelded component shall be based on the design fatigue curves for the materials used. Fatigue resistance of weld details shall be determined using the Structural Stress method, which is based on fatigue data of actual welds.

The theory used in this Article postulates that fatigue at any point is controlled by the alternating stress intensity S_{alt }and the associated mean stress σ_{nm }normal to the plane of S_{alt}. They are combined to define the equivalent alternating stress intensity S_{eq}, which is used with the design fatigue curves to establish the number of design cycles N_{f}.

**ASME Section III — Rules for Construction of Nuclear Facility Components — Appendices**

ASME BPVC.III.A XIII-3500 Analysis for Fatigue Due to Cyclic Operation covers fatigue life analysis. The design fatigue curves used in conjunction with XIII-3500 are those in Mandatory Appendix I. When more than one curve is presented for a given material, the applicability of each is identified. Where curves for various strength levels of a material are given, linear interpolation may be used for intermediate strength levels of these materials. The strength level is the specified minimum room temperature value. The maximum possible effect of mean stress is included in the fatigue design curves.