# Al-Alloy 2024-T4 Cumulative Biaxial Fatigue under Complex Loading

Master's Thesis 1997 95 Pages

## Excerpt

## ABSTRACT

A study of metal cumulative fatigue damage at multilevel stress programs using 2024- T 4 AI -Alloy is carried out in this investigation. Specimens are designed and fabricated. The tests are performed under multiaxial fatigue in phase combined loading bending and torsion, with zero mean stress (R=-l) at room temperature.

Five groups of multiaxial fatigue experiments are performed, constant amplitude, linear increasing and decreasing, Low-High loading sequence, High-Low loading sequence and equivalent flight loading fatigue.

Prediction of life specimens are studied at three methods:

Palmgren-Miner, Corten - Dolan, and Marsh. The methods are conservative to some specimens and non-conservative to another. The effects of stage life (ns) have been examined and fatigue life of specimens are decreased when the stage life is decreased.

Comparisons between High-Low & Low-High loading sequences in fatigue life (multi stress level) have been investigated. The Low-High loading sequence is less dangerous than High-Low loading sequence at the same stress range (equal loading stages).

Studies the effect of stress range fatigue life have been investigated and the fatigue life is decreased with increasing the stress range fatigue (having the same lowest stress).

For the fatigue of equivalent flight loading, the life is depend basically on stress range and the cruising interval this two parameters have been studied.

LIST OF SYMBOLS

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## CHAPTER ONE AIRCRAFT FAT/GUE

### (1.1) Introduction:-

The fatigue of airframes becomes a serious problem of air safety in the 1950-60 era and it has since received a great deal of attention from Aircraft designer and Aircraft Authorities and has become the subject of extension investigation by Aeronautical Research Establishments through the world. As a result the problem of Air safety has been overcome but at considerable economic operational penalty.

In fact, with the continuing trend towards high performance aircraft, fatigue has become one of the most important design and operational consideration of both military and civil aircrafts at the present.^{1}

As a result there is a continuing effort to develop more refined methods for fatigue design and analysis and despite the immense a mount of research that has been done on this problem there is still no final solution from the engineering point of view and none of the current design and life monitoring procedures has become universally accepted. Through its service life the aircraft structure is subjected to a complex sequence of loads ranging from very frequent fluctuating loads of small amplitude up to very large loads approaching the ultimate strength. This structure may be subjected to a considerable temperature at atmospheric conditions.^{2}

The Civil Aeronautical Board requires aircraft structure as the following:

A) To be demonstrated to have satisfactory fatigue strength by comparative experience; or

B) Be analyzed and tested so that major elements are shown to have an adequate fatigue strength; or

C) The structure designed, such that, if a failure does occur, it will not become catastrophic. Similar requirements are used by airforce for military aircraft.^{3}

The basic load enrollment consists of maneuvers characteristic of the aircraft type and its mission and the guests characteristic of the atmosphere and the structure response. The applied loading can be devised from acknowledge of the flight parameter and the aircraft configuration (i.e. the details of each mission) and the member loads or stresses in the structure can then be calculated.^{4}

In most aircrafts the load applied during landing, taxing and take-off are comparable in severity to the flight loads and must be considered in estimating fatigue damage .In the design stage, representative data must be relied on but during the life the load history can be obtained from measurements within the fleet.

This may be done either by sampling to estimate the average spectrum or preferably in each individual member of the fleet.^{5}

The correlations between predicted and actual lives of component or structure parts are not good for aircraft due to some factors:- The critical sections are usually not known and can be determined by calculation only with difficulty e.g. see reference.^{6}.

The parts usually built up of sheet riveted to stiffeners etc. with the attendant danger of fretting multiple crack initiation scatter in production quality and fatigue life etc. The stress concentration factor is not known or is practically meaningless, as for lugs.

The materials are high strength Al- Ti and Fe-alloys extremely sensitive to changes in mean stress which occur twice every flight, and to small flaws. The acceleration measured at the e.g. are in some cases not related at all to the loads; the relation between load and stress at the critical section is complex in every case.^{7}

The load spectra are not well known and can be measured only with difficulty. If the present unsatisfactory situation is to be improved, one or preferably all of the above short comings must be improved.

### (1.2) Literature Survey:-

Fatigue tests under several load levels are initiated by Gassner^{8} in the forties, modes of fatigue test loading were evolving separately for research tests on small specimens and for development or substitution tests carried out on substructures or complete Airframes.

In the quest for accumulative damage theory ill fatigue under alternating stresses the amplitudes of which varied either in a stepped mode, for block loading programs, or in continuous mode beginning with the lowest or highest amplitude.

From 1959 to 1961, The Federal Aircraft Work (FEW) Emmell (Switzerland) has carried out on a flight by flight basis, the full scale fatigue test of one Pilatus P3 airframe using a repeated application of a loading cycle corresponding to a mean flight type. Some continuous flight measurements - were also used to define the load variation during the various phases of flight type.^{9}

In 1959, the Caravelle full-scale fatigue test was carried out on a flight by flight basis with a simulated flight the computed damage of which had the same value than the damage from flight loads that occur less than ten times during the expected useful life of the aircraft.^{10}

Around 1969, several another proposed carrying out fatigue test on small specimens and assemblies using random loading and this type of testing has a practical value only in the cases where the cumulative frequency of load level exceeding and the distribution of extreme values of load per flight might be similar to those of the tests.

In 1960s. Gassner et al^{11} initiated the use of standard loading program representing the flight loading of an aircraft in fatigue tests of small smooth and notched specimens, In 1969 Schutz (LBF) proposed a standard program for flight-by-flight test commercial aircraft wing components^{12}. This spectrum is similar to that for Caravelle flight loads, whereas normal or Rayleigh distribution corresponds better to flight loads of flightier aircraft.

In 1972, Barrois^{13} proposed carrying out fatigue tests of aircraft structures on a flight-by-flight basis, the global spectrum of cumulative frequency of load level exceedance being decomposed into partial spectra of the same shape for every flight their intensities being distributed among the flights as the extreme values of load per flight.

Buclt^{14} performed the effect of aircraft loading program modification on the fatigue life of 2024-T3 sheet specimens containing a central hole. The specimens subjected to and Twist and Mini - Twist spectrum tests. It was found that in particular loading program cases, rare load peaks may have not only a beneficial but also a detrimental effect on the number of simulated flights.

Lassim^{15} investigated the cumulative fatigue damage of 2024 - T4 AI-Alloy and the specimens are subjected to reversed push-pull fatigue testing, it is found that the proposed model of crack growth model method is the best method for prediction of fatigue life compared with another methods.

Abdul- Wahab^{16} investigated the aerodynamic stress applied to the aircraft wing material 2024 - T4 AI-Alloy to estimate the life of the wing in this study the safe fatigue curves for different probabilities to failure are established.

Abdul-Latif studied the fatigue of 2024 - T4 AI-Alloy, under constant and variable combined loading, type torsion & bending.^{17}

The S-N curve was established and it may be expressed mathematically by the following equation:-

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Finally, Sallman^{18} investigated the effect of mean stresses on combined loading fatigue of aircraft alloy 2024 - T4. The empirical equations which described the behavior of fatigue are derived from experimental work as

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### (1.3) Obje ctives :-

The aim from this work is to investigate the following items:

a) Study the cumulative fatigue damage of multilevel stress programs.

b) Examine the effect of stage life (ns) on fatigue life at different programs

c) Study the loading sequence effect of multilevel stress programs on fatigue properties.

d) Estimate the fatigue life of specimens by three life prediction methods.

e) Investigate specimens having loading programs similar to flight loading programs.

## CHAPTER TWO MULTIAXIAL FATIGUE

### (2.1) Introduction

Multiaxial fatigue is a subject of concern to both engineers and research scientists. In the eventuality of failure, fatigue life time is determined in the majority of cases by the applied multiaxial stress-strain state, whether generated by multiple loading or the component geometry itself.^{19}

Thus multiaxial stresses should be taken into consideration by the designer, and it is important to note the material data generated in laboratories under constrained situations (for example, uniaxial loading), cannot be used in practice without recourse to some multiaxial criterion. The majority of data available in engineers is generated under uniaxial stress cycling conditions. However, most of the structure members must be able to withstand combinations of alternating and static stresses.

Therefore, a precise knowledge of the manner in which combined stresses causes failure is of primary importance. The amount of available information on the fatigue characteristics of structural components experiencing combined loading is rather limited. Because of its importance in engineering application, it is highly desirable to make a study of the phenomenon, and, in practical, to be able to predict biaxial fatigue damage from basic uniaxial data.

Many components are exposed to varying degrees of multiaxial stress, for example components and structures found in power and chemical plants, such as pressure vessels and piping systems, aircraft structure turbine blades, and drive shafts are subjected to multiaxial stress condition during cyclic loading^{21}.In order to apply these limited fatigue data to more complex stress conditions attempts have been made to correlate multiaxial fatigue loading to an equivalent uniaxial fatigue loading condition as suggested in some design code.^{22} ^{23} ^{24}

A general state of stress in a structure component can be described by three principle stresses component and their direction as an arbitrary function of time. However in order to correlated the theoretical investigation with the experiments, it IS necessary to simplify the fluctuating stress in such a way that parameters can be identified and their influence observed. Therefore, any criteria for fracture phenomenon must be based on these three principle stresses, and the most fatigue cracks start at the surface of structural components where a biaxial state of stress exists.

The three most influencing reasons for fatigue cradles starting at the surface under biaxial loading conditions are :-

1- Many parts are stressed by bending or torsion and the highest stresses occur at the surface.

2- Surface stresses are increased by the unavoidable stress raisers, such as notches, grooves, holes and scratches.

3- Metallurgical evidence that the crystalline grams at the surface are inherently weaker under stress because there are no restricting grains adjacent to them.

The effect of combined state of stress on the fatigue metals has been investigated sporadically since 1916. The problem was too difficult and time consuming, because of the large number of tests involved, the difficulty of designing suitable testing machines, the difficulty of devising and maintaining testing techniques which do not influence the results and inherent scatter in results of fatigue tests.

Various theories have been proposed to account for failure of materials under combined loading. The commonly used and best known theories have assumed a material which is homogeneous and isotropic. The failure criteria are presented as a function of normal stresses which become the principle stresses al, a2, a3. When properly ordered o-i > ct2 > стз , various conclusions have been reached by the different investigators concerning the applicability of the well known theories of failure to the result of fatigue tests under combined stress, but none has received universal acceptance.

One of the basic reasons is that fatigue fracture initiates on a small scale at the atomic level, and many assumptions are made before it is applied in the final analysis of the actual complex structure.^{20}

### (2.2) Parameters Affecting Biaxial Fatigue :(A)- Isotropy In Biaxial Fatigue.

(B) - Mean Stress Effect.

(C) - Phase Effect.

(D) - Effect of Notches.

#### (2.2.A) Isotropy In Biaxial Fatigue:

The contribution of anisotropy to fatigue is a well-recognized fact, as the comparative fatigue strength of specimens in which the greatest principal stress is, in one case, parallel and in the other case transverse to the grain directions, is very different.

Usually fatigue strength is greater when the applied stress is parallel to the longitudinal grain direction than to the transverse direction. Findley^{25} has given an explanation for the effect of anisotropy .He suggests that anisotropy in the strength of a material may be similar to the effect of a straight groove, in that it may raise the local stress above that predicted by the formulas used for the isotropic materials for stresses normal to the direction of the groove, and not parallel to it.

Findley further argues that in specimens cut lengthwise from bar stock and tested so that the plane of bending and axis of twisting are always parallel to the length wise direction of the stock, one may expect anisotropy of the material will raise or lower the nominal stress computed from the twisting moment more than or less than the stress computed from the bending moment. The resulting bending or twisting stresses the caused by anisotropy will be liner function of the nominal bending or twisting stresses respectively and the resulting bending and twisting stresses will be the same functions of the nominal stresses regardless of the combination of bending and twisting employed.

Findley fmally concludes that under these conditions one can then reexamine all of the rational theories by multiplying the nominal twisting stress (r) by a corresponding constant for each material such that the resulting equations satisfy the conditions for pure bending and pure twisting.

Therefore, Findley suggest that one can correct for anisotropy by multiplying every value of (r) by (aeb/aet) . Where (aeb) and (aet) are the measured value of the fatigue strength in bending and in torsion respectively. The number of cycle for which (aeb) and (aet) are determined are the same as the number which is expected to produce failure under the given values of (u) and (t) .

#### (2.2.B) Mean Stress Effect :-

The majority of the fatigue investigations are carried out with completely reversed stresses or strain, i.e. the maximum and minimum values are equal in magnitude but opposite in sign, and hence the mean value is zero. In biaxial fatigue, experimental data with mean stress at static stress is indeed very scanty. In reality, engineering struck experience complex stress or strain spectrum where the mean stress mayor not be zero. If the component is subjected to mean stress (compressive or tensile), then the completely reversed results can give false information of the endurance of a structure.

The first investigation into the influence of mean stress upon biaxial fatigue results was carried out by Stulen and Cummings^{26}. The suggested, as a result of examination of available test data, that the normal stress on the critical shear plane might have a linear influence on the allowable alternating shearing stress. Findley^{27} followed the above reasoning and developed a theory which employs the linear form as where TC is the critical shear stress, GO the maximum normal stress and f and ß are constants.

The above equation assumes that the critical shear stress decreases with increase in the maximum normal stress from a value given by f when the maximum normal stress is zero. However, using the linear elastic theory, the difference between the limiting value of the alternating shearing stress on any plane and the applied alternating shearing stress on the same plane can be established.

Similarly, the differences between the normal stresses resulting from the alternating and mean stresses can be established. Findley has applied the above mentioned theory for combined torsion and axial loads and bending. He observed that the maximum stresses had a greater influence in bending them in torsion. He concluded that the fatigue strength of ductile metals in torsion was nearly independent of the mean stress, whereas the fatigue strength under axial load generally decreased with increasing tensile mean stress. He also observed a small decrease in fatigue strength for alternating torsion with superimposed static tension.

Hussain et al54 showed that increasing of mean stress will reduce the fatigue life of metals at constant alternating stress level. He investigated the effect of mean stress on fatigue life for different material.

#### (2.2.C) Phase Effect:

In certain engineering applications, structural elements encounter non-synchronous loading conditions. This is particularly for aircraft structures where loads are encountered in a random manner.

In such conditions, reduction in fatigue life may result. Most of the fatigue in uniaxial or biaxial loading conditions does not properly treat this subject because of the experimental difficulty in loading a specimen simultaneously with two independent non-synchronous stresses in two directions. The small amount of data that is available in high-cycle biaxial fatigue is either in-phase or completely out-of-phase condition.

Taira et al^{28} investigated the low cycle fatigue failure of tabular specimens subjected to combined cyclic torsion and out of phase cyclic tension-compression at a temperature of 450°C. Zamrik^{29} also studied the phase effect in low-cycle fatigue for a number of phase angles ranging from 0° to 90°.

Little^{35} analyzed Nishihara et al^{36} data on out of phase torsion and bending with respect with true shear stress amplitude and showed that the fatigue limit actually decreases as the phase difference increases.

Test results revealed the out of phase loading was more damaging than in-phase loading ,for lives between 104 and 106 cycles the damage increase as phase angle increased from 0° to 90° .^{37}

It IS desirable to have approach which will include this effect in the analysis of biaxial fatigue. The normal approach is the range calculation in interpreting the experimental data. However, when biaxial out-of phase stressing is encountered, a constant rotation of the principle axis occurs and the orientation of the element for which the principle stress occurs will vary with time. Also, if maximum and minimum values of stress occur at different times, then they can also occur in different directions. Therefore, for any range of calculations the maximum and minimum stress must be projected onto a common direction, resulting in general difficulty in determining the exact value of maximum and minimum stress.

#### (2.2.D) Effect Of Notches :-

Stress concentration has been recognized as a major factor in fatigue analysis. The ratio of maximum local stress in the region of a notch to the nominal local stress determined by elastic theory is generally high compared to the fatigue strength reduction factor which is the ratio of unnotched fatigue strength to notch fatigue strength. At the root of a notch, the stress state is triaxial due to plastic deformation at this highly stressed region.

Stu/en and Cummings^{26} have studied the effect of stress concentration under triaxial loading conditions. Their criterion for failure of notches under complex stress conditions valid for any type of notch is:

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Where al is principle maximum fluctuating stress, a3 principle minimum fluctuating stress and gN is Equivalent fluctuating stress a the notch root Where Gen = endurance limit for notched material as a nominal stress for the case of uniaxial stress.

теп = endurance limit for the same notch as a nominal stress for the case of pure torsion.

However, in full-scale complex structure, there are many additional factors that influence the local stress in relation to the applied loads, such as residual stresses from fabrication and assembly, fretting, welding, etc. In most cases the exact value of these factors cannot be established and the fatigue problem becomes more complex.

### (2.3) MULTIAXIALFATIGUE FAILURE THEORIES ;-

#### (2.3.1)Gough's Theory :-

Gough and Pollard30 were the first to do an experimental investigation of the effect of combined stresses of fatigue strength. Their investigation revealed that the ratio of endurance limits in pure torsion to the ratio of endurance limit in pure bending was not constant but varies with materials tested. They proposed an empirical relation to fit there data as is :

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The equation is known by Ellipse Quadrant relationship as the data plotted forms a part of an ellipse.

Gough^{31} has also suggested another empirical relationship to predict fatigue failure under combined stress as:

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The above relationship is known "as the Ellipse Arc Relationship.

#### (2.3.2) Maximum Shear Stress Theory :-

According to this theory, when the component is subjected to any combination of stresses, the maximum shear stress is the fatigue fracture controlling parameter. For combined bending and torsion condition the maximum shear stress is

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#### (2.3.3) Octahedral Shear Stress Theory :-

The importance of octahedral shear stress is well recognized in the elasto-plastic theory. Fatigue fracture failure criterion based on this theory seems to agree with the experimental data. The criterion is that when the octahedral shear stress reaches a critical value fracture would occur.

The general equation for octahedral shear stress is :

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The above general equation for combined bending and twisting condition reduces to :-

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#### (2.3.4) Equjyalent Stress Theories :-

It is desirable to have a unified engineering approach to fatigue fracture under multiaxial stress conditions. An approach of equivalent stress is presented in this section, so that uniaxial fatigue data can be utilized in order to predict multiaxial fatigue fracture. The equivalent stress is given as^{32}:

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Considering a component that is subjected to bending and shear stresses the equation (2-9) can be written in terms of component stress as:

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Specially for out of phase multiaxial loading most of the proposed fatigue theories are extensions of the Tresca or Von Mises criteria ..

Langer^{33} proposed of fatigue evaluation procedure seeking the highest range of shear stress in a multiaxial load history, which has been adopted in 1974 ASME Boiler and Pressure Vessel Code^{34}. For fully reversed out of phase torsion and bending, the equivalent stress based on that procedure can be obtained as:

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A modification Langer's method, an extension of Von Mises criterion, has been incorporated in ASlVIE code case^{38}. The equivalent stress based on the modification for out-of-phase torsion and bending can be expressed as following form

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In consistencies with experimental results and difficulties in implementation of most complex multiaxial fatigue criteria have prompted efforts to develop a new criterion for equivalent multiaxial loading^{39} ^{40}. The equivalent stress is used with uniaxial (S-N) data to calculate the damage of multiaxial fatigue.

### (2.4) Cumulative fatigue Damage Theories :-

#### (2.4.1) Introduction :-

In virtually every engineering application where fatigue is an important failure mode, the alternating stress amplitude may be expected to vary or change in some way during the service life. Such variations and changes in load amplitude often referred to as spectrum loading, make the direct use of standard S- N curves inapplicable because these curves are developed and presented for constant stress amplitude operation. Therefore, it becomes important to a designer to have available a theory or hypothesis,

Verified by experimental observations, that will permit good design estimates to be made for operation under conditions of spectrum loading using the standard constant-amplitude S-N curves that are more readily available.

The basic postulate adopted by all fatigue investigators working with spectrum loading is that operation at any given cyclic stress amplitude will produce fatigue damage, the seriousness of which will be related to the number of cycles of operation at that stress amplitude and also related to the total number of cycles that would be required to produce failure of an undamaged specimen at that stress amplitude.

It is further postulated that the damage incurred is permanent and operation at several different stress amplitudes in sequence will result in an accumulation of total damage equal to the sum of the damage increments accrued at each individual stress level. When the total accumulated damage reaches a critical value, fatigue failure occurs. Although the concept is simple in principle, much difficulty is encountered in practice because the proper assessment of the amount of damage incurred by operation at any given stress level Si for a specified number of cycles n, is not straight forward.

Many different cumulative damage theories have been proposed for the purposes of assessing fatigue damage caused by operation at any given stress level and the addition of damage increments to properly predict failure under conditions of spectrum loading.^{40}

#### (2.4.2) The Linear Damage Theory

The first cumulative damage theory was proposed by Palmgren in 1924 and later developed by Miner in 1945 ^{41}. This linear theory, which is still widely used, is referred to as the Palmgren- Miner hypothesis or the linear damage rule. The theory may be described using the S- N.

By definition of the S- N curve, operation at constant stress amplitude S1 will produce complete damage, or failure, in N1 cycles. Operation at stress amplitude Si for a number of cycle’s n1 smaller than Ni will produce a smaller fraction of damage, say D1. Dl is usually termed the damage fraction. Operation over a spectrum of different stress levels results in a damage fraction Di for each of the different stress levels Si in the spectrum.

When these damage fraction sums to unity, that is failure is predicted to occur if

illustration not visible in this excerpt

The Palmgren-Miner hypothesis asserts that the damage fraction at any stress level Si is linearly proportional to the ratio of number of cycles of operation to the total number of cycles that would produce failure at that stress level; that is By the Palmgren-Miner hypothesis, then, utilizing (2-14), we may write (2-15) as failure is predicted to occur if:

**[...]**

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