01 - fatigue theory (part 1)

8/9/2019 01 - Fatigue Theory (Part 1)

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Fatigue Theory

1. Introduction 2. Background

3. The History of Fatigue 4. High Cycle versus Low Cycle Fatigue 5. Summary 6. Inputs to Fatigue Life Estimation Models 7. Total Life (S‐N) Analysis 8. Stress Cycles 9. The S‐N Curve 10. Procedure for Determining the S‐N Curve 11. Limits of the S‐N Curve 12. Tensile Properties and the S‐N Curve

13. The Influence of Mean Stress 14. Factors Influencing Fatigue Life 15. Application in MSC.Fatigue 16. Crack Initiation/Strain ‐Life (ε‐ N) Analysis 17. The Microscopic Aspects of Fatigue Failure 18. The Strain ‐Life Methodology 19. Monotonic Stress ‐Strain Behaviour 20. Cyclic Stress ‐Strain Behaviour 21. The Strain ‐Life Curve 22. Strain ‐Life vs. Stress ‐Life

23.

Transition Life

24. The Effect of Mean Stress 25. Factors Influencing Fatigue Life 26. Application in MSC.Fatigue 27. The Statistical Nature of Fatigue 28. Representation of Fatigue Data on a Statistical Basis 29. The Statistical Distribution Function 30. Probability of Failure at a Finite Life 31. Probability of Failure for Infinite Life 32. Handling Statistics Under Random Loading Conditions 33. The Absolute Accuracy of Fatigue Life Estimation 34. Estimating Material Cyclic Properties From UTS & E


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1. Introduction

Fatigue theory for basic stress ‐ life (S‐N) and strain ‐ life (ε‐ N) analyses are presented in this chapter. This corresponds to the technical background for Total Life and Crack Initiation (Ch. 5). Theory and technical information for other analysis types are given in their respective chapters. However a good understanding of the other analysis types is dependent on an understanding of the basic theories presented here. For this reason, an entire chapter is dedicated to this theoretical discussion.

2. Background

Purely static loading is rarely observed in modern engineering components or structures. By far, the majority of structures involve parts subjected to fluctuating or cyclic loads. For this reason, design analysts must address themselves to the implications of repeated loads, fluctuating loads, and rapidly applied loads. Such loading induces fluctuating or cyclic

stresses that often result in failure of the structure by fatigue. Indeed, it is often said that from 80% to 95% of all structural failures occur through a fatigue mechanism. It is worth noting at the outset that the term fatigue, coined more than a hundred years ago by the French engineer Monsieur Poncelet, may not be the best choice of terminology today, since many aspects of the phenomenon are distinctly different from the biological counterpart. For example, it is next to impossible to detect any progressive changes in material behaviour during the fatigue process, and therefore failures often occur without warning. Also, periods of rest with the fatigue stress removed do not lead to any measurable healing or recovery of the material. Thus, the damage done during the fatigue process is cumulative, and generally unrecoverable. From this standpoint, the German

term Betriebsfestigkeit (operational strength) is a better descriptor of the phenomenon. However, since Betriebsfestigkeit involves 17 characters, and fatigue only 7, we shall continue to use the term fatigue! Fatigue, although a complex subject, has not been neglected by the research community. Estimates indicate that if one wished to keep up to date with all the literature published about fatigue by reading a paper each working day, one would fall behind by more than a year for each year of reading. Furthermore, attempting to catch up with the backlog would be virtually impossible. Yet the design analyst and engineer is increasingly challenged by the demands of higher performance, lower weight, and longer life, and all this at a reasonable cost and in as short a time as possible! These apparently conflicting demands can only be

overcome through a consideration of the problems associated with fatigue resistant designs. Up until recently, these problems were summarized as the following:

Life calculations are usually less accurate than strength calculations. Order of magnitude errors in life estimates are not unusual.

Fatigue properties cannot be accurately deduced from other mechanical properties; they need to be measured directly.

Full‐scale prototype testing is usually necessary to assure an acceptable life. Laboratory results of tests carried out under identical conditions may differ widely,

requiring statistical interpretation.

Materials and design geometries must often be selected to provide slow crack growth, and if possible, detection of cracks before they become dangerous.


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“Fail‐safe” design concepts must often be implemented in order to achieve acceptable reliability. That is, even if a structural element fails, the structure must remain intact and able to support the loads in the short term.

Modern advances in fatigue life estimation techniques have, to some extent, mitigated these problems. For example, these days it is usual to consider life estimates to be within a factor of two or three rather than ten. Furthermore, computerized analysis of thousands of laboratory data sets do point to acceptable empirical correlations between monotonic tensile data and fatigue parameters.

3. The History of Fatigue

For centuries, it has been known that a piece wood or metal can be made to break by repeatedly bending it back and forth with a large amplitude. However, it came as something of a surprise when it was discovered that repeated loading produced fracture even when the stress amplitude was apparently well below the elastic limit of the material. The first

fatigue investigations seemed to have been reported by a German mining engineer, W. A. S. Albert, who in 1829, performed some repeated loading tests on iron chain. Some of the earliest fatigue failures in service, occurred in the axles of stage coaches. When railway systems began to develop rapidly in the middle of the nineteenth century, fatigue failures of railway axles became a widespread problem that began to draw attention to cyclic loading effects. This was the first time that many similar components had been subjected to millions of cycles at stress levels well below the monotonic tensile yield stress. As is often the case with unexplained service failures, attempts were made to reproduce the failures in the laboratory. Between 1852 and 1870, the German railway engineer August Wöhler setup and conducted the first systematic fatigue investigation. From this point of view, he may be

regarded as the grandfather of modern fatigue thinking. He conducted tests on full‐scale railway axles and also on small scale bending, torsion, and axial cyclic loading specimens for different materials. Some of Wöhler’s data for Krupp axle steel were plotted in terms of nominal stress amplitude versus cycles to failure. This presentation of fatigue life has become very well known as the S‐N diagram. Each curve on such a diagram is still referred to as a Wöhler line. At about the same time, other engineers began to concern themselves with the problems associated with fluctuating loads in bridges, marine equipment, and power generation machines. By 1900, over 80 papers had been published on the subject of fatigue failures. During the first part of the twentieth century, more effort was placed on understanding the

mechanisms of the fatigue process rather than just observing its results. This activity finally led, in the late 1950s and early 1960s, to the development of two approaches to fatigue life estimation. One method, known as the Manson ‐Coffin local strain approach, attempts to describe and predict crack initiation whilst another is based on linear elastic fracture mechanics, LEFM, and was developed to explain crack growth. Most recently, Miller and his colleagues at Sheffield University, England, have been working on ways of finding a unified theory of metal fatigue, describing crack growth on a microscopic, macroscopic, and structural level. From this vast wealth of knowledge, one thing has become clear; modern design analysts and engineers will not create more fatigue resistant components and structures by indulging

in more experimentation, although the need for more research is ever present. From a


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practical point of view, a more profitable approach is the implementation and efficient use of the knowledge which is available today.

4. High Cycle versus Low Cycle Fatigue

Over the years, fatigue failure investigations have led to the observation that the fatigue process actually embraces two domains of cyclic stressing or straining that are distinctly different in character. In each of these domains, failure occurs by apparently different physical mechanisms: one where significant plastic straining occurs and the other where stresses and strains are largely confined to the elastic region. The first domain involves some large cycles, relatively short lives and is usually referred to as low ‐cycle fatigue . The other domain is associated with low loads and long lives and is commonly referred to as high ‐cycle fatigue . Low‐cycle fatigue is typically associated with fatigue lives between about 10 to 100,000 cycles and high ‐ cycle fatigue with lives greater than 100,000 cycles. The rules for distinguishing between these two domains are discussed later. For now, it is

enough to recognize that remedies for extending fatigue lives in each domain are different. In the high ‐cycle fatigue domain, measures such as shot peening and other surface hardening treatments or the use of higher strength materials are beneficial. For low ‐cycle fatigue, where ductility and resistance to plastic flow are important, these measures are inappropriate.

5. Summary

In summary, fatigue analysis may be thought of as a process of initiating and then growing a crack which finally causes the structure to break into two or more pieces. This process can

be represented by the following equation:

Total Life = Crack Initiation + Crack Growth

or

(15 ‐1) Nf Total fatigue cycles to failure Ni Number of cycles to initiate an engineering crack

Np Number of cycles to propagate a crack to final fracture

The mathematical models used to simulate the initiation and propagation processes are quite different. The initiation phase is usually modelled using strain ‐ life and cyclic stress ‐strain curves while the propagation phase uses crack growth rate versus stress intensity curves. The original work on fatigue life estimation did not make a distinction between crack initiation and growth. Stress was used as the control parameter and stress ‐ life (or S‐N) curves characterized the material or component response to cyclic loading. This approach was used widely until the 1970s when the new methods started to be used by industry.

Unfortunately, there are quite a few inherent problems with the stress ‐ life approach which cause a large amount of scatter in the experimental fatigue life results. In some companies,


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due to the extensive use of the stress ‐ life method in past years, a large quantity of S‐N curves and experience have been accumulated. In this case, the stress ‐ life method offers compatibility with previous work and may therefore be more appropriate. The stress ‐ life approach is still used today for situations which are not easily modelled using the local strain methods, such as welded structures and cast irons. Due to the inherent defects in these materials which act more like cracks, fatigue damage should really be modelled with LEFM crack propagation tools. Another group of materials which appear to be best modelled using the stress ‐ life approach are anisotropic and inhomogeneous materials such as composites. However, there are complications even with composites, since the mean stress correction models used in connection with metals do not apply to all composites.

6. Inputs to Fatigue Life Estimation Models

The fatigue life estimation process requires three main inputs and the process is usually

illustrated with a diagram known as the “Five Box Trick” as shown in Figure 15 ‐1.

Figure 15 ‐1 Schematic Illustration of Fatigue Life Estimation

From this diagram, it is easy to see the necessary inputs:

Materials properties Loading in the form of load time histories Local stress ‐strain information from a linear elastic analysis

These inputs are then processed using various fatigue life estimation tools which are described in more detail below. It is important to understand the nature of these inputs so that the fatigue life estimation is meaningful.

6.1. Material Properties

When considering fatigue, it is not sufficient to characterize a material purely in terms of the Young’s modulus and Poisson’s ratio. The chemical composition, heat treatment, and microstructure will all change the way in which the material responds to cyclic loading. A summary of the materials data types for fatigue life estimation is given below.


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6.2. Stress ‐Life Data

For this model, the fatigue response for a material used in a particular construction is the stress amplitude versus log cycles to failure curve. In addition, the Young’s modulus may be required to convert stress to strain or vice versa. The S‐N curve parameters are listed in Table 15 ‐1 and typical S‐N curve is shown in Figure 15 ‐2.

Parameter Name S.I. Units Imperial Units Parameter

Stress range intercept First fatigue strength exponent

Fatigue transition point Second fatigue strength exponent

Stress range fatigue limit Standard error of Log(N)

MPa or MN/m 2 none cycles none

MPa or MN/m 2 none

PSI none none none PSI

none

SRI1 b1

NC1 b2 FL SE

Table 15 ‐1Stress ‐Life Data Parameter Description

Figure 15 ‐2 Example of a Stress ‐Life Plot


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6.3. Crack Initiation Data

There are two basic sets of materials properties used for this fatigue life estimation model. They are the cyclic stress ‐strain curve and the strain ‐ life curve. These are characterized by the parameters in Table 15 ‐2 with a typical strain ‐ life plot shown in Figure 15 ‐3.

Parameter Name S.I. Units Imperial Units Parameter

Fatigue strength coefficient Fatigue strength exponent Fatigue ductility coefficient Fatigue ductility exponent

Young’s modulus Cyclic strain hardening exponent

Cyclic strength coefficient

Cut off in reversals

MPa or MN/m 2 none none none

MPa or MN/m 2 none

MPa or MN/m 2

none

PSI none none none PSI

none PSI

none

σ f' b

ε f' c E n' K'

Rc

Table 15 ‐2

Figure 15 ‐3

Example of a Strain ‐Life Plot


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6.4. Crack Growth Data

The primary piece of information used in this model is the Paris curve which defines crack growth rate as a function of the stress intensity range. In addition, the fracture toughness and the threshold stress intensity range is required. These parameters are listed in Table 15 ‐3 with a typical da/dN ‐Delta K plot in Figure 15 ‐4.

Parameter Name S.J. Units Imperial Units Parameter

Paris law coefficient Paris law exponent

Delta K threshold at R=0 Delta K threshold at R‐1

Fracture toughness Stress ratio at threshold knee

Stress corrosion threshold Unnotched fatigue strength

m/cycle none

MPa m 1/2 MPa m 1/2 MPa m 1/2

none

MPa m1/2

MPa

in/cycle none

KSI in1/2 KSI in1/2 KSI in1/2

none

KSI in1/2

PSI

C m D0 D1 K1C Rc

K1SCC FL

Table 15 ‐3

Figure 15 ‐4 Example of a Delta K Apparent Plot

6.5. Loading

Loading time histories are measurements of loading for a period of time which must be long enough to ensure that the measurement reflects a typical duty cycle. The loading measurements will be made in the same attitude as the loading applied in the FEA. For example, an axle may experience bending loads in at least two planes together with a torsion load, or a pressure vessel may experience both pressure and temperature variations

in time.

Simultaneous

measurement

of

these

loads

for

a

typical

or

worst

case

event

would

be made.


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Measurement of any physical phenomenon requires a transducer, the electronics to drive the transducer and a recording device. A typical combination would be a load cell, amplifier and FM tape recorder. Most engineering companies will have some capabilities and experience in doing this kind of work even if the group carrying out the finite element analysis does not. It is important that these two groups work together in order to achieve the best advantage of the MSC.Fatigue package. If it is not feasible to measure load or a related parameter, it may be necessary to synthesize a load time history. There are tools in the MSC.Fatigue package to allow you to create, manage, and graphically edit time histories (PTIME). In some cases) the load variation with time may be trivial and easily created artificially, (e.g., the variation of pressure in a vessel). However, it is important to remember that many fatigue problems occur because an unexpected load combination occurred. For this reason, it is a truism that “a measurement is worth a thousand guesstimates.”

6.6. Rainflow Cycle Counting

In reality, components are rarely subjected to purely constant amplitude loading, and so for many years various methods for extracting “equivalent” constant amplitude cycles from a random loading sequence were devised. Methods such as level crossing, range ‐pair, range ‐mean, and rainflow all attempt to reduce a random sequence of peaks and valleys to a set of equivalent constant amplitude cycles. The concept of an equivalent constant amplitude cycle is perhaps, at first, difficult to understand. On the one hand, the most obvious cycle to be found in any time series will be the one that includes the maximum and minimum data values. On the other hand, the problem is how to define all the other cycles which are present.

Figure 15 ‐5 The Extraction of a Fatigue Cycle


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A reversal can be easily understood as a change in the sign of the loading direction. Thus, a constant amplitude signal would consist of two equal and opposite reversals repeated continuously. In the simplest terms, therefore, a random time series may be considered to be a pair of reversals: the reversal from the maximum to the minimum in the signal and the reversal from minimum to the maximum in the signal, with all the other reversals effectively “interrupting” these two. From the point of view of a material, this phenomenon is often referred to as “material memory”, because a material subjected to a sequence of reversing loads, apparently interprets each closed cycle (matching pair of reversals), as a temporary interruption of a larger strain range, and “remembers” which hysteresis limb applies for this larger event. Figure 15 ‐5 shows a strain sequence of four turning points, and the stress ‐strain response of a material to this sequence. The closed hysteresis loop is a cycle, which may be characterized in terms of its strain range and mean strain. If the stress axis of this diagram is ignored and only successive strain ranges are considered then an algorithm can be developed which will extract cycles from a signal whatever its units. The rainflow algorithm

is able to extract cycles in the way described above, classify them in terms of their range and mean value and store them in a range mean matrix. The term “rainflow” is derived from an algorithm in which cycles are extracted through a consideration of rain drops flowing down a pagoda roof (the originators of this algorithm were Japanese). Modern algorithms no longer use this concept although the generic name “rainflow” still persists.

6.7. The Rainflow Procedure

The rainflow algorithm used by MSC.Fatigue is based on the standard practice for cycle

counting in fatigue analysis as defined by the ASTM designation E 1049 ‐85, (See ASTM standards Vol. 03.01). Note that the form of the algorithm used relies on the fact that the analysis starts at the absolute maximum value in the data set. Under these circumstances, the rainflow cycle count will be identical to a range pair cycle count which itself starts at the largest value. Differences in the results produced by the two procedures only arise if processing starts at some value other than the absolute maximum. It is possible to illustrate the rainflow cycle counting procedure used by considering the simple strain time series shown in Figure 15 ‐6.


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Figure 15 ‐6

Illustration of the Rainflow Procedure

The local stress ‐strain response to the nominal strain time record, shown to the left, reveals that four equivalent constant amplitude cycles are present (i.e., B‐C‐B, E‐F‐E, G‐H‐G and the outside loop A‐D‐A). The small event B‐C‐B is treated as a interruption to the large overall event A‐D‐A. It is important to note that although, the events B‐C‐B and G‐H‐G appear very similar in the nominal strain record and would be counted as equally damaging by say the range ‐mean method, the local mean stresses and plastic strains are quite different and so the damage contribution from each would also be different. It should also be noted that local mean stress cannot be calculated directly from nominal mean strains. The main thrust

behind

the

rainflow

cycle

counting

method,

therefore,

is

to

treat

small

cycles

as

interruptions of larger ones.

6.8. Local Stress Information

The finite element analysis (FEA) provides a link between applied loads and the stress response at regular locations across the structure. This obviates the need for an engineer to obtain an approximate stress concentration factor as is normally the case in fatigue analysis. Moreover, the FEA allows the engineer to investigate the fatigue performance for a range of load combinations.


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6.9. Linear Static Stress Analyses

One of the principles of the MSC.Fatigue package is that the load cases being considered must be analysed separately unless certain conditions apply (i.e., a stress analysis will be required for each discrete load applied to the structure). The results from the stress analysis provide a calibration between the applied load and the local stresses across the structure. Linear elastic stress analysis is used because the fatigue analysers in MSC.Fatigue carry out various manipulations which include superimposing the FEA stress analysis results and carrying out the elastic ‐plastic transformations where necessary. From this, local strain ‐ or stress ‐ time histories at each location representing the strain response due to the simultaneous combination of all the load time histories are obtained.

6.10. Types of Load Cases in FEA

It is important that the units of the loads applied in the FEA correspond to those of the load

time histories. To understand the reasoning below, it is necessary to define the following terms:

LOAD a value of an applied force in units of force (kN,N, etc). LOADING a generic term for any external or internal effect which causes straining

of a structure.

FEA is not acutely dependent on the actual units of force and length, and hence of stress. The calculation of life in MSC.Fatigue is carried out in S.I. units. However, MSC.Fatigue allows you to work in a range of unit systems.

Nevertheless, it is still the responsibility of the engineer to ensure the inputs are provided in a consistent manner. The time history used in a fatigue calculation must be a representation of the time variation in the loading applied in the FEA. For simple cases, this implies a force time history corresponding to a time variation in the point loading used in the FEA. There are a number of different kinds of loading possible, each one requiring a different type of time history as shown in Table 15 ‐4.


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Loading Time History Used

Point Loads Pressure Temperature

Accelerations Distributed LoadMoment Scalar

Load time history at point load application position Pressure time history Temperature time history

Acceleration time history A related parameter time history 1 Moment time history Any time history to define a scaled variation of the actual FEA stress ‐strain

Table 15 ‐4

A load time history will not be required if you wish to define a FE load case as a simple static offset of the stress ‐strain responses since this does not vary with time though the load case must be isolated and dealt with in a separate FEA. Pressures may be applied and pressure time histories used in the fatigue analysis. Pressures and loads may be mixed in a fatigue analysis as long as the units are the same. Any loads which vary in proportion to and in‐phase with each other may be combined together in one FEA. However, the load time history must be a scalar which represents the proportion of these loads acting on the structure at each increment of time. Note that, if you plan on using linear elastic results from a transient dynamic or forced vibration FE analysis, none of this applies since you have already defined your time variation in your FE analysis.

7. Total Life (S‐N) Analysis

It has been recognized, since 1830, that a metal subjected to a repeated or fluctuating load will fail at a stress level lower than that required to cause fracture on a single application of the load. The nominal stress (S‐N) method was the first approach developed to try to understand this failure process and is still widely used in applications where the applied stress is nominally within the elastic range of the material and the number of cycles to failure is large. From this point of view, the nominal stress approach is best suited to that area of the fatigue process known as high ‐cycle fatigue . The nominal stress method does not work well in the low ‐cycle region where the applied strains have a significant plastic component. In this region, a strain ‐based methodology must be used.

8. Stress Cycles

Before looking in more detail at the nominal stress procedure it is worth considering the general types of cyclic stresses which contribute to the fatigue process. Figure 15 ‐7 details some typical fatigue stress cycles.

1The distributed load is a special case which must be treated by considering a time history of a related quantity such as deflection. In this case you must specify the magnitude of the deflection at a given location as calculated by the FEA and the corresponding deflection time history.


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Figure 15 ‐7 Typical Fatigue Stress Cycles, (a) Fully Reversed (b) Offset, (c) Random

In Figure 15 ‐7(a) illustrates a fully‐reversed stress cycle with a sinusoidal form. This is an idealized loading condition typical of that found in rotating shafts operating at constant speed without overloads. For this kind of stress cycle, the maximum and minimum stresses are of equal magnitude but opposite sign. Usually tensile stress is considered to be positive and compressive stress negative. Figure 15 ‐7 (b) illustrates the more general situation where the maximum and minimum stresses are not equal. In this case they are both tensile and so define an offset for the cyclic loading. Figure 15 ‐7 (c) illustrates a more complex, random loading pattern which is more representative of the cyclic stresses found in real structures. From the above, it is clear that a fluctuating stress cycle can be considered to be made up of two components, a static or steady state stress , and an alternating or variable stress amplitude , . It is also often necessary to consider the stress range , , which is the algebraic difference between the maximum and minimum stress in a cycle.

(15 ‐2) The stress amplitude, , then is one half the stress range.

(15 ‐3) The mean stress, is the algebraic mean of the maximum and minimum stress in the cycle.

(15 ‐4) Two ratios are often defined for the representation of mean stress, the stress or R ratio, and

the amplitude ratio A.


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(15 ‐5) (15 ‐6) Table 15 ‐5 illustrates some R values for common loading conditions.

R ratio Loading Condition

R > 1 R = 1 0 < R < 1 R = 0 R = ‐1 R < 0

R infinite

Both σ max and σ min are negative. Negative mean stress. Static loading. Both σ max and σ min are positive; Positive mean stress, | σ max | > | σ min |. Zero to tension loading, σ min = 0 Fully‐ reversed loading, | σ max | = | σ min |; zero mean stress. | σ max | < | σ min |, σ max approaching zero.

σ max equal to zero.

Table 15 ‐5

9. The S‐N Curve

Between 1852 and 1870, the German railway engineer August Wöhler set up and conducted the first systematic fatigue investigation.

Figure 15 ‐8

S‐N

Data

Reported

by

Wöhler

(Note, 1 centner = 50 Kg, 1 zoll = 1 inch, 1 centner / 0.75 .)


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Wöhler conducted cyclic tests on full‐scale railway axles and also on small ‐scale bending, torsion, and push ‐pull specimens of several different materials. Some of Wöhler’s data, shown in Figure 15 ‐8, are for Krupp axle steel and are plotted, in terms of nominal stress vs. cycles to failure, on what has become known as the S‐N diagram. Typically, the S‐N relationship is determined for a specific value of Sm, R or A. Note that in dealing with the nominal stress approach, the convention is that nominal stress is usually referred to as S and localized stress by the Greek counterpart, .

10. Procedure for Determining the S‐N Curve

Most determinations of fatigue properties have been made in completely reversed bending (i.e., R = –1), by means of the so ‐called rotating bend test. One example is the R. R. Moore test, which uses four ‐point loading to apply a constant moment to a rotating (1750 rpm) cylindrical hourglass ‐shaped specimen. See Figure 15 ‐9. Specimens, which are typically between 6 to 8 mm in diameter in the test section, are usually polished to a mirror finish

prior to testing.

Figure 15 ‐9 The R.R. Moore Fatigue Testing Machine

The stress level at the surface of the specimen is calculated using the elastic beam equation, even if the resulting value exceeds the yield strength of the material.

(15 ‐7)

where:

S the nominal stress acting normal to the cross ‐section M the bending moment c the distance of the surface from the neutral axis I the moment of inertia

For the circular section of the R. R. Moore specimen, the beam equation reduces to:

(15 ‐8)


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where:

d the diameter of the specimen.

The usual laboratory procedure for determining an S‐N curve is to test the first specimen at a high stress, about two thirds of the static tensile strength of the material, where failure is expected in a fairly small number of cycles. The test stress is decreased for each succeeding specimen until one or two specimens do not fail before at least 10 7 cycles. For materials which exhibit it, the highest stress at which no failure occurs, a runout is taken to be the fatigue limit. For situations where an infinite life design requires a probability of survival to be associated with it, more complex testing and analysis procedures, such as the Prot and staircase methods, have been developed to determine the mean and variance of the fatigue limit. For materials which do not exhibit a fatigue limit, tests are usually terminated between 10 7 and 10 8 cycles, and the concept of an endurance limit at either 10 7 or 10 8 cycles defined.

The S‐N curve is usually determined through the use of about 15 specimens. However, it is generally found that results are accompanied by a large amount of scatter and some form of statistical analysis should be applied; see The Statistical Nature of Fatigue, 1390 for more details. S‐N data are nearly always presented in the form of a log ‐ log plot of alternating stress, amplitude Sa or range Sr, versus cycles to failure, with the actual Wöhler line representing the mean of the data. Certain materials, steels for example, display a fatigue limit, Se, which represents an alternating stress level below which the material has an infinite life. For most engineering purposes, infinite is taken to be 1 million cycles. Great care must be exercised when designing on the basis of a fatigue limit, since it has a nasty habit of disappearing due

to periodic overloads, corrosion, and elevated temperature.

Figure 15 ‐10 Idealized Form of the S‐N Curve

When plotted on log ‐ log scales, the relationship between alternating stress, S, and number of cycles to failure, N can be described by a straight line, Figure 15 ‐10. The slope of the line, b, (after Basquin, a prominent worker who first proposed the law) can be derived from the following:


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(15 ‐9)

(15 ‐10) log · log log

(15 ‐11) log log log (15 ‐12) Sometimes, for convenience, the term 1/b is replaced by the letter k,

(15 ‐13)

The above

equation

says

that

if

we

know

the

Basquin

slope,

b,

and

any

other

coordinate

pair

(No, So), then for a given stress amplitude S, the number of cycles can be calculated directly. Typically, No is taken to be 10 6 cycles and the corresponding stress amplitude is taken to be an endurance limit, usually denoted as Se or S6, so that the above equation may be rewritten as:

(15 ‐14) ·10

Example 1: A Simple Life Estimation For a material with an endurance limit of 250 MPa, and a Basquin slope, b, of ‐0.1, calculate number of cycles to failure at a stress amplitude of 300 MPa. Under these conditions, (15 ‐14) gives:

300250 ·10 161000


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11. Limits of the S‐N Curve

As previously mentioned, the S‐N approach is applicable to situations where cyclic loading is essentially elastic.

Figure 15 ‐11 Typical S‐N Curves for Ferrous and Nonferrous Metals

This means that the S‐N curve should be confined on the life axis to numbers greater than about 10,000 cycles in order to ensure no significant plasticity is occurring. Indeed, great care must be taken in using the above S‐N equations in situations where lives less than 10,000 cycles are being estimated. Figure 15 ‐11 shows typical S‐N curves for both ferrous and nonferrous metals. The points to note in Figure 15 ‐11 are the limits of the logN axis, the presence of a fatigue limit for the mild steel and the absence of a fatigue limit for the aluminium alloy. Because both materials represented in Figure 15 ‐ 11 have relatively low yield stresses, the life axis is confined to begin at 10 5 cycles at which point the alternating stress is about 350 and 300 MPa respectively for the two alloys.

12. Tensile Properties and the S‐N Curve

Through many years of experience, particularly with steels, empirical relationships between fatigue and tensile properties have been developed. These relationships are not soundly based in science. However, they remain useful tools for engineers for assessing fatigue performance. When the S‐N curves for a number of different steels of varying strengths are plotted as the ratio of endurance limit (i.e., the stress amplitude at 10 6 cycles), S6, to ultimate tensile strength, Su, all the curves tend to all fall onto a single curve which implies that:

(15 ‐15) 0.5 · 1400 and

(15 ‐16) 700 1400


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Figure 15 ‐12 Generalized S‐N Curve for Wrought Steels

In addition to this, the stress at 10 3 cycles, S3, can be approximated by 0.9 Su and so, utilizing these approximations, a generalized S‐N curve can be generated for wrought steels, see Figure 15 ‐12. Methods of representing the S‐N curve in the range 1 to 10 3 cycles have been developed but they must be treated with extreme caution. They usually use some percentage of the ultimate strength, Su, or true fracture stress, , as a measure of the stress amplitude at either 1 or 1/4 cycles. The main difficulty with employing this approach is that the deduced S‐N curves are extremely flat in the low ‐cycle region, and this makes estimates of life particularly inaccurate. The reason for this apparent flatness is the large plastic strain which

results from the high load levels. Low‐cycle fatigue analysis is best treated by strain ‐based procedures which account for rather than ignore the effects of plasticity.

13. The Influence of Mean Stress

As previously mentioned, most basic fatigue data are collected in the laboratory by means of testing procedures which employ fully reversed loading (i.e., R = –1). However, most realistic service situations involve nonzero mean stresses. It is, therefore, very important to know the influence that mean stress has on the fatigue process so that the fully reversed laboratory data can be usefully employed in the assessment of real situations.


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Figure 15 ‐13 High‐Cycle Fatigue Data Showing the Influence of Mean Stress

Fatigue data collected from a series of tests designed to investigate different combinations of stress amplitude and mean stress are characterized in Figure 15 ‐13 for a given number of cycles to failure. The diagram plots the mean stress, both tensile and compressive, along the x‐axis and the alternating constant stress amplitude along the y‐axis. This kind of representation was first proposed by Haigh and is therefore commonly referred to as the Haigh diagram. The stress amplitude at zero mean stress, Sn, corresponds to the stress amplitude at N cycles to failure as measured by the fully‐ reversed fatigue test. The failure data points tend to follow a curve which if extrapolated would pass through the ultimate tensile strength, Su,

on the mean stress axis. Notice that the influence of mean stress is different for compressive and tensile mean stress values. Failure appears to be more sensitive to tensile mean stress, than compressive mean stress. When available, data of the type illustrated above are collated into what are commonly referred to as master diagrams for a particular material. Figure 15 ‐14 illustrates the master diagram for SAE 4340 from the U.S. Department of Defense MIL Handbook ‐5.


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Figure 15 ‐14 Master Diagram for SAE 4340

Since the tests required to generate a Haigh or master diagram are quite expensive, several empirical relationships which relate alternating stress amplitude to mean stress have been developed. These relationships characterize a material through its ultimate tensile strength, Su, and are very convenient. For infinite life design strategies, the methods use various

curves to connect the endurance limit, Se, on the alternating stress axis to either the yield stress, Sy, ultimate strength, Su, or true fracture stress, , on the mean stress axis. Of all the proposed relationships, two have been most widely accepted (i.e., those of Goodman and Gerber).

(15 ‐17) Goodman: 1

(15 ‐18) Gerber: 1 Experience has shown that actual test data tends to fall between the Goodman and Gerber curves (Goodman joining Se to Su by means of straight line and Gerber by means of a parabola). For most design situations where R « 1 (i.e., small mean stress in relation to the alternating stress), there is little difference between the two relationships. However, when R approaches 1 (i.e., nearly equal mean and alternating stresses, the two relationships show considerable differences). Unfortunately, little or no experimental data exist to support one approach over the other and so typically, the recommendation would be to select the approach which provides the most conservative lives in a given situation.


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Example 2: Correcting for Mean Stress Effects A component is subjected to a maximum cyclic stress of 750 MPa and a minimum of 70 MPa. The steel from which it is manufactured has an ultimate tensile strength, Su, of 1050 MPa and a measured endurance limit, S6, of 400 MPa. The fully reversed stress at 1000 cycles is 750 MPa. Using both the Goodman and Gerber mean stress correction procedures calculate the component life.

The first step is to calculate the stress amplitude, Sa, and the mean stress, Sm, using equations (15 ‐3) and (15 ‐4).

(15 ‐19) 340 (15 ‐20) 410

A Haigh diagram for the Goodman correction procedure can now be constructed for constant lives of 10 6 and 10 3 cycles. This is done by connecting the endurance limit, S6, and the stress at 1000 cycles, S3, respectively, on the alternating stress axis with the ultimate tensile strength, Su, on the mean stress axis (see Figure 15 ‐15). The stress conditions on the component calculated above, Sa = 340 and Sm = 410, can be plotted on the diagram and a line drawn from Su to the alternating stress axis. This line represents the constant life line for the component at all combinations of stress amplitude and mean stress. The line intersects the fully reversed axis at a stress Sn = 558 MPa.

Figure 15 ‐15 The Haigh Diagram

It should be noted that this stress can also be calculated directly from the Goodman equation (15 ‐17):

(15 ‐21) 1

1


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557.8 It will be recalled that the S‐N curve is given by equation (15 ‐ 13),

and so for the conditions defined at 10 3 and 10 6 cycles,

log and so

log 0.091 and for Sn = 557.8, the life can be calculated from:

· 10 26000

The Gerber correction can be used in a similar way, i.e.,

1

1

401.2 and

· 10 973000


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It is very important to remember that all the modification factors are empirical, conservative and mostly only applicable to steels. They provide little or no fundamental insight into the fatigue process itself other than providing approximate trends. In particular, they should not be used in areas outside their measured applicability.

14.1. The Influence of Component Size

Fatigue in metals results from the nucleation and subsequent growth of crack ‐ like flaws under the influence of an alternating stress field. This view leads to the concept of failure commencing from the weakest link , the most favourably orientated metal crystal for example, and then growth through less favourably orientated grains until final failure. Intuitively, it would seem reasonable to suppose that the larger the volume of material subjected to the alternating stress, the higher the probability of finding the weakest link sooner. Actual test data do confirm the presence of a size effect particularly in the case of bending and torsion.

The stress gradient built up through the section, in bending and to a lesser extent in torsion, concentrates more than 95% of the maximum surface stress to a thin layer of surface material. In large sections, this stress gradient will be less steep than in smaller ones, and so the volume of material available which could contain a critical flaw will be greater leading to reduced fatigue strength. The effect is quite small for axial tension where the stress gradient is absent. The value for Csize can be estimated from one of the following, if the diameter of the shaft is < 8mm:

(15 ‐24) 1 if the diameter is between 8 mm and 250 mm:

(15 ‐25) 1.189. The effect of size is particularly important for the analysis of rotating shafts such as might be found in vehicle powertrains. For situations where components do not have a round cross section, an equivalent diameter, deq, can be calculated for a rectangular section width, w and thickness, t, undergoing bending from:

(15 ‐26) 0.65 · · 14.2. The Influence of Loading Type

Fatigue data measured according to one regime, axial tension for example, may be “corrected” to represent the data that would have been obtained had the test been carried out in some other loading methodology such as torsion or bending. Recall that the R.R Moore test calls for fatigue tests to be carried out under conditions of fully‐ reversed bending. The values of Cload to be used in conjunction with the endurance limit, Se, in moving from

one loading condition to another are detailed in Table 15 ‐6.


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Figure 15 ‐17 Surface Finish Correction Factor for Steel Components

Figure 15 ‐18 The Effect of Surface Roughness on Surface Finish Factor

It is worth noting that some of the curves presented in Figure 15 ‐17 include effects other than just surface finish. For example, the forged and hot ‐rolled curves include the effect of decarburization. Other diagrams present the surface factor in a more quantitative way by using a quantitative measure of surface roughness such as RA, the root mean square, or AA the arithmetic average, see Figure 15 ‐18. Values of surface roughness associated with each of the manufacturing processes are readily available in handbooks, as an example consider Table 15 ‐8.


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Type of finish Surface Roughness (microns)

Lathe ‐ formed Partly hand polished

Hand Polished

Ground Superfinished

Ground and polished

2.67 0.15 0.13

0.18 0.18 0.05

Table 15 ‐8

Figure 15 ‐19 Residual Stress in a Beam

14.4. The Influence of Surface Treatment

As in the case of surface finish, surface treatment can have a profound influence on fatigue strength, particularly the endurance limit. Surface treatments can be divided broadly into mechanical, thermal, and plating processes. The important point to note with all three is that the net effect of the treatment is to alter the state of residual stress at the free surface. In the first two processes, by providing a compressive layer, and in the case of plating, by providing a tensile residual stress. Residual stresses arise when plastic deformation is not uniformly distributed throughout the entire cross ‐section of the component being deformed. Figure 15 ‐19represents a metal bar whose surface has been deformed in tension by bending so that part of it has undergone plastic deformation. When the external force is removed, the regions which have been deformed plastically prevent the adjacent elastic regions from complete elastic recovery to the unstrained condition. In this way, the elastically deformed regions are left in residual tension, and the plastically deformed regions must be in a state of residual compression. For many purposes, residual stress can be considered identical to the stresses produced by an external force. Thus, the presence of a compressive residual stress at the surface of a component will have the effect of decreasing the likelihood of fatigue failure.


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Figure 15 ‐20 Superposition of Applied and Residual Stresses

Figure 15 ‐20 illustrates the effect schematically. Figure 15 ‐20(a) shows an elastic stress distribution in a beam with no residual stress. The typical residual stress distribution associated with shot peening is detailed in Figure 15 ‐20(b). Note that the compressive stress at the surface must be compensated by an equivalent tensile stress over the interior of the cross section. In Figure 15 ‐20(c), the distribution due the algebraic summation of the residual and applied stresses is shown. Observe that the maximum tensile stress at the surface has been reduced by the amount of the residual stress. Furthermore, note that the peak tensile stress has now been moved to the interior of the beam. The magnitude of this stress will depend on the gradient of the applied stress and the residual stress distribution. Also note that under these conditions, subsurface crack initiation becomes a possibility.

14.5. Mechanical Treatments

The main commercial methods for introducing residual compressive stresses are cold rolling and shot peening. Although some alteration in the strength of the material occurs as a result of work hardening, the improvement in fatigue strength is due mainly to the compressive surface stress. Surface rolling is particularly suited to large parts and is frequently used in critical components such as crankshafts and the bearing surface of railway axles. Bolts with rolled threads typically possess twice the fatigue strength of conventionally machined threads. Shot peening, which consists of firing fine steel or cast iron shot against the surface of a

component, is particularly well suited to processing small mass produced parts.


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It is important to remember that cold rolling and shot peening have their greatest effect at long lives. At short lives they have little or no effect. As with other modifying factors, the effect of these mechanically induced compressive stresses can be accounted for by the use of correction factors which can be used to adjust the endurance limit Se. Typically the factor associated with peening is about 1.5 ‐2.0.

14.6. Plating

Chrome and nickel plating of steel components can more than halve the endurance limit due to the creation of tensile residual stresses at the surface. These tensile stresses are a direct result of the plating process itself. As in the case of mechanically induced surface stresses, the effect of plating is most pronounced at the long life end of the spectrum and also with higher strength materials. The deleterious effect of plating can be reduced by introducing a compressive residual stress prior to the plating process by either shot peening or nitriding. An alternative

approach might be to anneal components after plating and thereby relieve the tensions. Electroplating may also cause a shortening of the fatigue life through the mechanism of hydrogen embrittlement. Hydrogen ions are produced at the surface of the metal during plating. For high ‐strength steels, these ions enter the large steel microstructure and then combine into the molecular hydrogen, H2, causing high stresses under cyclic loading and accelerating fatigue failure. Such effects are difficult to model analytically. In practice, the preventative action of baking the component after plating to release the hydrogen is often used.

14.7. Thermal Treatments

Thermal treatments are processes which rely on the diffusion of either carbon, carburizing , or nitrogen, nitriding , onto and into the surface of a steel component. Both species of atoms are interstitial (i.e., they occupy the spaces between adjacent iron atoms), and thereby both increase the strength of the steel and, through volumetric changes, cause a compressive residual stress to be left on the surface. Carburizing is commonly carried out by packing the steel components within boxes which contain carbonaceous solids, sealing to exclude the atmosphere and heating to about 900° C for a period of time which depends on the depth of the case required. Alternatively, components may be heated in a furnace in the presence of a hot carburizing gas such as methane. This process has the advantage that it is quicker and

more accurate. In addition, the carburizing cycle may be followed up by a diffusion cycle, with no carburizing agent present, which allows some of the carbon atoms to diffuse further into the component and so reduce gradients. The nitriding process is very similar in nature to gas carburizing, except that, in this case, ammonia gas is used and the components are soaked at lower temperatures. Typically 48 hours at about 550 degrees centigrade will provide a nitrided case depth of about 0.5 mm. Nitriding is particularly suited to the treatment of finished notched components such as gears and slotted shafts, the effectiveness of the process is illustrated in Table 15 ‐9.


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Geometry Endurance Limit (MPa)

not nitrided nitrided

Unnotched Semi ‐circular notch

V notch

310 175 175

620 600 550

Table 15 ‐9

Example 3: To Shot Peen or Not? From a particular vehicle, as heat treated, low alloy leaf springs (manufactured to a tensile strength of 1500 MPa), are failing in service, albeit after some considerable time, but nevertheless within the lifetime of the vehicle. Service measurement has shown that the loading is of the type from zero to a maximum stress of about 770 MPa (i.e., R = 0). The question to be answered is, will shot peening the springs prior to installation make the

problem go away? The springs have a section measuring 40 mm by 5 mm. From tables of surface roughness, Csur has been determined to be about 0.75 for the “as heat treated” condition and about 0.58 for the “shot peened” condition. Measurements have shown that shot peening introduces a compressive residual stress of about 550 MPa. From the ultimate strength, Su = 1500 MPa, S'e can be estimated to be 750 MPa. For the “as heat treated” condition the modification factor required to correct for the section size of the springs can be estimated by first calculating the equivalent diameter, deq, associated with the rectangular cross section from (15 ‐26):

deq2 = 0.65 w t = 0.65 x 40 x 5 = 130 deq = 11.4 mm

and since deq > 8 mm, Csize can now be computed from (15 ‐25):

Csize = 1.189 x 11.4 ‐0.097 = 0.94

Since, in operation, the leaf springs are loaded only in bending, CL is equal to 1.0. We are now in a position to calculate the modified magnitude of endurance limit, Se.

Se =

S'e

x the

product

of

all

the

modifying

factors Se = 750 x 0.94 x 1.0 x 0.75

Se = 529 MPa

For the loading condition defined by R = 0, the amplitude of the alternating stress, Sa, and the mean stress, Sm, must be equal. By setting Sa = Sm, the allowable stress level, S, for the as heat ‐ treated material can now be calculated from the Goodman equation (15 ‐17).

(Sa / Se) + (Sm / Su) = 1 (S / 529) + (S / 1500) = 1

S =

391

MPa


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and the maximum allowable stress for infinite life would be

Smax = Sa + Sm = 2 S = 782 MPa

This figure is uncomfortably close to the operational maximum stress of 770 MPa and so clearly we do not have an infinite life design; we must expect to have some failures in service. The procedure so far has been useful since it has helped us verify the calculation procedure being used and provided some confidence in the methodology. We are now in a position to estimate the effect of shot peening. In the shot peened condition, the modified magnitude of endurance limit, Se is given by:

Se = S'e x the product of all the modifying factors Se = 750 x 0.94 x 1.0 x 0.58 Se = 409 MPa

As in the as heat ‐ treated case, for R = 0, the amplitude of the alternating stress, Sa, and the mean stress, Sm, are equal and the Goodman equation can be used to calculate the allowable stress. By subtracting the compressive residual stress from the mean stress, the peening operation is taken into account,

1

1 439

and the maximum allowable stress for infinite life would be

Smax = Sa + Sm = 2 S = 878 MPa

The calculated maximum working stress for shot peened springs is significantly higher than both the equivalent as heat treated stress and the operational stress. From this, we could justifiably conclude that shot peening to the extent which produces residual compression of

550 MPa would be sufficient to provide the leaf spring with an infinite service life.


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15. Application in MSC.Fatigue

MSC.Fatigue supports two types of stress ‐ life curve. The first is the nominal or component S‐N curve where the stress plotted on the Y‐axis is not the actual stress where failure occurs (i.e., there is a built ‐ in geometric stress concentration factor, Kt ). The use of this type of S‐ N curve is only valid when the stress in a nominal or reference area is used to compute lives. This approach takes no account of localized plasticity effects such as might be found at a stress ‐ raiser. The effects of such discontinuities must be taken into account through the S‐ N damage curve itself. Each S‐N curve, therefore, must reflect the overall “structural” characteristics of the detail under investigation. The second type of S‐N curve is the local stress or material S‐N curve. This curve has associated with it a Kt value of 1.0 (i.e., the stress on the Y‐axis is a local stress which causes fatigue). No reference area is needed as the results at each node or element may be used directly. The S‐N approach must be used when basic material data are unavailable, or one or more of

the basic assumptions embodied in the more fundamental local stress ‐strain approach are considered invalid. Typical examples would be the treatment of welded structures or complex joints, rivets or spot welds and composites. In addition) some companies, due to the extensive use of the stress ‐ life method in past years, a large quantity of S‐N curves and experience has been accumulated. In this case, the stress ‐ life method offers compatibility with previous work and so may be more appropriate.

15.1. The Measured Loading Environment

MSC.Fatigue requires a representation of a load time history in order to define the loading

environment to which a component or structure is subject, such as shown in Figure 15 ‐21.

Figure 15 ‐21 A Load Time Series


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The equation of the second life line will of course be defined by:

(15 ‐29) ∆ 2

If b2

is

non

‐zero,

a

stress

‐based

fatigue

limit,

FL,

may

be

defined

to

cutoff

any

further

fatigue damage; all three features can be distinguished in Figure 15 ‐22. Because of the complexity and often the variability of the test pieces used, together with the difficulties of controlling local stresses and strains, the degree of scatter in the measured lives can be quite high. It is usual, therefore, to apply some statistical methods to account for this variability. MSC.Fatigue takes this scatter into account through a consideration of the standard error of log (Nf). The standard error of log (Nf) is calculated in the following way:

(15 ‐30) 1 SE Standard error of log(Nf) SDx Standard deviation of log(Nf) from the regression analysis r Correlation coefficient from the regression analysis

By the use of this statistic, a whole family of life curves, centered around the “mean life line”, may be constructed. It is usual to consider three standard deviations on either side of the mean. The lower limit represents 99.9% confidence that, for a given stress range, failure will not occur after less cycles than those which correspond to that stress. The mean life line provides 50% confidence and in the case of the upper line only 0.15% confidence that the component will survive.

Figure 15 ‐23 The 3‐SD Confidence


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15.3. Analytical Procedure

Since the stress ‐ life method in MSC.Fatigue makes no attempt to model localized plasticity, the analytical procedure used is relatively straightforward and may be summarized as follows:

1. By means of linear static FEA, derive the local stress time history from the load time histories. This includes superpositioning of multiple FEA/load time history load cases (or use stress time history directly from linear transient or forced vibration FE analysis). However, it is important to ensure that the S‐N data applies to the situation being modelled; most S‐N curves are for nominal stress, not local stress.

2. Extract the fatigue cycles in the local stress time history by means of the rainflow algorithm.

3. Assess the damage contribution of each cycle by referring to the selected damage curve.

4. Linearly sum the damage associated with each cycle by using Miner’s rule.

15.4. Account for Nominal Mean Stress

Nominal S‐N curves can be either structure, component or specimen based and are used to provide damage data for complex situations which otherwise could not be characterized. This gives rise to different procedures for accessing the S‐N curve, depending on the situation being considered. Indeed, the British Standards Institution has tried to formalize some of these procedures in the document BS5400 Pt. 10. The main differences between the various procedures lie in the treatment of mean stresses and the damage contribution of cycles with “small” ranges. The total life analyser in MSC.Fatigue offers five different

procedures: 1. Ignore the Effect of Mean Stress

In this mode of operation, the stress ‐ life analysers in MSC.Fatigue estimate the damage for each cycle and takes no account of its mean stress. If the stress range of the cycle is below the fatigue limit then no damage will be accumulated irrespective of the relative magnitude of any other cycles which may be present.

2. Account for Mean Stress According to BS5400 Pt.10, Welded In this mode of operation, the stress ‐ life analysers in MSC.Fatigue estimate the damage for each cycle and do not take into account the effects of mean stress. Specific S‐N curves are used for the BS5400 welded analysis and these are referred to

as weld classes. These damage curves are known to MSC.Fatigue by the following names: CLASSB, CLASSC, CLASSD, CLASSE, CLASSF, CLASSF2, CLASSG, CLASSW, CLASSS and CLASST

3. Account for Mean Stress According to BS5400 Pt.10, Non ‐Welded In this mode of operation, the stress ‐ life analysers in MSC.Fatigue estimate the damage for each cycle and takes account of the effects of mean stress by reducing compressive stress amplitudes to 60% of their measured values. Specific S‐N curves are used for the BS5400 non ‐welded analysis. These are referred to as classes and are known to MSC.Fatigue by the following names:

CLASSB, CLASSC,

CLASSD,

CLASSE,

CLASSF,

CLASSF2, CLASSG, CLASSW, CLASSS and CLASST


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These S‐N curves are in fact the same as those used in the welded analysis. The difference in the non ‐welded analysis lies in the treatment of compressive stresses.

4. Account for Mean Stress According to Goodman In this mode of operation, the stress ‐ life analysers in MSC.Fatigue estimate the damage for each cycle and account for the effects of mean stress by reducing the allowable applied stress amplitudes in a linear way for both positive and negative mean stresses. This procedure was first proposed by Goodman and can be summarized as follows:

(15 ‐31) 1 Sa The allowable stress amplitude. So The allowable stress amplitude at zero mean stress. Sm The mean stress.

Su

The ultimate

tensile

stress.

If the stress range of the cycle is below the fatigue limit, as defined by the FL parameter, then no damage will be accumulated irrespective of the relative magnitude of any other cycles which may be present.

Account for Mean Stress According to Gerber In this mode of operation, the stress ‐ life analysers in MSC.Fatigue estimate the damage for each cycle and account for the effects of mean stress by reducing the allowable applied stress amplitudes in a quadratic way for both positive and negative

mean stresses.

This

procedure

was

first

proposed

by

Gerber

and

can

be

summarized

as follows:

(15 ‐32) 1 Sa The allowable stress amplitude. So The allowable stress amplitude at zero mean stress. Sm The mean stress. Su The ultimate tensile stress.

If the stress range of the cycle is below the fatigue limit, as defined by the FL parameter, then no damage will be accumulated irrespective of the relative magnitude of any other cycles which may be present.

15.5. Linear Damage Summation

A constant amplitude, S‐N curve represents a set of tests at constant stress range (∆) together with associated lives. Operation at a stress range (∆ ) will result in failure in say N1 cycles. Operation at the same stress range for a number of cycles less than N1, N j say, will result in a smaller fraction of damage, D j, which is often referred to as a partial damage.

Operation over a spectrum of different stress ranges, results in a partial damage


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contribution Di from each stress cycle, ∆ . Failure is then predicted when the sum of these partial damage fractions reaches unity so that:

(15 ‐33) ⋯ 1 The Palmgren ‐Miner rule asserts that the partial damage at any stress range ∆ , is linearly proportional to the ratio of the number of cycles of operation to the total number of cycles that would produce failure at that stress level, ∑ , . For example: (15 ‐34) Failure is predicted then if:

(15 ‐35) ⋯ 1 or

(15 ‐36) ∑ 1 The above equations represent a statement of the linear damage rules used by both the stress ‐ life and local strain fatigue life processors in MSC.Fatigue. Experience shows that linear damage summation is somewhat of an over simplification of reality. The most important shortcoming is that no account is made of the sequence in which strain levels are experienced, and damage is assumed to accumulate at the same rate for a given strain level regardless of pre ‐history. In particular, it appears that large strain amplitudes which precede smaller ones, cause the smaller cycles to become more damaging than expected. The net result is that in the first instance, the Miner’s sum ∑ , is measured to be less than 1 and in the latter case it becomes greater than 1. Since most service environments involve quasi ‐random loading sequences, the use of the Palmgren ‐Miner linear damage rule summing to a constant of 1 is mostly satisfactory.

Important: BS 5400 Pt 10 attempts to cope with these effects by asserting that if there are no cycles with a stress range greater than the stress

corresponding to 1 x 107

cycles, then no damage will be accumulated. If, on the other hand, even one cycle exceeds this stress, then damage will be accumulated for ALL other cycles. This correction takes into accountthe observed behaviour when a crack growing cycle occurs; subsequent previously non ‐damaging cycles start to increment the crack size (i.e.,cause damage). This is particularly true for welded structures which are always already cracked. The choice of the 1E7 cycles point reflects the fact that the stress at this point is close to the threshold for crackadvance. If, on the other hand, even one cycle exceeds this stress, then damage will be accumulated for ALL other cycles.


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Figure 15 ‐24 Factor of Safety Illustrated on the Goodman Diagram

15.8. Gerber Based Factor

This is the same as the Goodman except the Gerber mean stress correction is applied.

(15 ‐39) Factor of Safety =

By also calculating the factor of safety for each cycle in a random loading history, the stress ‐based factor of safety may be used with realistic loading in a meaningful way. MSC.Fatigue calculates factors of safety for all cycles in a time history at each location (node or element) and reports the lowest value computed. The correction factors for surface treatment and finish are obtained from the Lipson (9) empirical data and are related to the ultimate strength of the material.

15.9. Life‐Based Calculation

The life based calculation is in fact a calculation to find the magnitude of stress scaling factor which will cause failure in a specific design life. Since the scaling factor is applied to all cycles it is an overall factor of safety. This approach is far better suited to calculating meaningful factors of safety for random loading. This approach to calculating a scaling factor is used in MSC.Fatigue in the Factor of Safety module for multi ‐ location computation and also in Design Optimization where the scaling parameter optimization calculates the value of a factor at one location (node/element).

15.10. Other Factors Influencing Safety

The calculations

described

above

compute

a

stress

scaling

factor

either

using

a

stress

based

or life based methodology. It should be remembered that there are other factors


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contributing to the life of the product. For example, a change in the material may reduce the factor of safety. This may be explored more fully using the Design Optimization tools in MSC.Fatigue or by submitting a new job with edited input data.

01 - fatigue theory (part 1)

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