Chapter 6

FATIGUE

6.1 Introduction to fatigue:

Fatigue is a phenomenon caused by repetitive loads on a structure. It depends on the

magnitude and frequency of these loads in combination with the applied materials and

structural shape. Structural members are frequently subjected to repetitive loading over a long

period of time. For example, the members of a bridge structure suffer variations in loading

possibly thousands of times a day as traffic moves over the bridge. In these circumstances a

structural member may fracture at a level of stress substantially below the ultimate stress for

non-repetitive static loads; this phenomenon is known as fatigue.

Fatigue cracks are most frequently initiated at sections in a structural member where

changes in geometry, e.g. holes, notches or sudden changes in section, cause stress

concentrations.

Although adequate precautions are taken to ensure that an aircraft‟s structure

possesses sufficient strength to withstand the most severe expected gust or maneuver load,

there still remains the problem of fatigue. Practically all components of the aircraft‟s structure

are subjected to fluctuating loads which occur a great many times during the life of the

aircraft. It has been known for many years that materials fail under fluctuating loads at much

lower values of stress than their normal static failure stress.

Prior to the mid-1940s little attention had been paid to fatigue considerations in the

design of aircraft structures. It was felt that sufficient static strength would eliminate the

possibility of fatigue failure. However, evidence began to accumulate that several aircraft

crashes had been caused by fatigue failure. The seriousness of the situation was highlighted

in the early 1950s by catastrophic fatigue failures of two Comet airliners. These were caused

by the once-per-flight cabin pressurization cycle which produced circumferential and

longitudinal stresses in the fuselage skin. Although these stresses were well below the

allowable stresses for single cycle loading, stress concentrations occurred at the corners of the

windows and around rivets which raised local stresses considerably above the general stress

level. Repeated cycles of pressurization produced fatigue cracks which propagated

disastrously, causing an explosion of the fuselage at high altitude.

Despite the fact that the causes of fatigue were reasonably clear at that time its

elimination as a threat to aircraft safety was a different matter. The fatigue problem has two

major facets: the prediction of the fatigue strength of a structure and knowledge of the loads

causing fatigue. Information was lacking on both counts.

Fatigue Loading

Fatigue loading is primarily the type of loading which causes cyclic variations in the

applied stress or strain on a component. Thus any variable loading is basically a fatigue

loading.

Variable Loading

Variable loading results when the applied load or the induced stress on a component is

not constant but changes with time i.e. load or stress varies with time in some pattern. Most

mechanical systems and devices consists moving or rotating components. When they are

subjected to external loadings, the induced stresses are not constant even if the magnitude of

the applied load remains invariant.

6.1.2: Fatigue failure:

Often machine members subjected to such repeated or cyclic stressing are found to

have failed even when the actual maximum stresses were below the ultimate strength of the

material, and quite frequently at stress values even below the yield strength. The most

distinguishing characteristics are that the failure had occurred only after the stresses have

been repeated a very large number of times. Hence the failure is called fatigue failure.

Fatigue Failure- Mechanism:

A fatigue failure begins with a small crack; the initial crack may be so minute and

cannot be detected. The crack usually develops at a point of localized stress concentration

like discontinuity in the material, such as a change in cross section, a keyway or a hole. Once

a crack is initiated, the stress concentration effect become greater and the crack propagates.

Consequently the stressed area decreases in size, the stress increase in magnitude and the

crack propagates more rapidly. Until finally, the remaining area is unable to sustain the load

and the component fails suddenly. Thus fatigue loading results in sudden, unwarned failure.

Fatigue failure stages:

Thus three stages are involved in fatigue failure namely

Crack initiation

Crack propagation

Fracture

Crack Initiation

• Areas of localized stress concentrations such as fillets, notches, key ways, bolt holes and

even scratches or tool marks are potential zones for crack initiation.

• Crack also generally originate from a geometrical discontinuity or metallurgical stress raiser

like sites of inclusions

• As a result of the local stress concentrations at these locations, the induced stress goes

above the yield strength (in normal ductile materials) and cyclic plastic straining results due

to cyclic variations in the stresses. On a macro scale the average value of the induced stress

might still be below the yield strength of the material.

Crack Propagation

• This further increases the stress levels and the process continues, propagating the cracks

across the grains or along the grain boundaries, slowly increasing the crack size.

• As the size of the crack increases the cross sectional area resisting the applied stress

decreases and reaches a threshold level at which it is insufficient to resist the applied stress.

Final Fracture

• As the area becomes too insufficient to resist the induced stresses any further a sudden

fracture results in the component.

6.1.3 Basic features of failure appearance:

A fatigue failure, therefore, is characterized by two distinct regions. The first of these

is due to progressive development of the crack, while the second is due to the sudden

fracture. The zone of sudden fracture is very similar in appearance to the fracture of a brittle

material, such as cast iron, that has failed in tension. The crack propagation zone could be

distinguished from a polished appearance. A careful examination (by an experienced person)

of the failed cross section could also reveal the site of crack origin.

6.1.4 Fatigue life:

Fatigue life, Nf is defined as the number of stress cycles of a specified character that a

specimen sustains before failure of a specified nature occurs.

6.1.5 S-N Curve:

A graph of failure stress against number of repetitions of the stress has the typical form

shown in figure 6.1. For some materials, such as mild steel, the curve (usually known as an S-

N curve or diagram) is asymptotic to a certain minimum value, which means that the material

has an actual infinite life stress. Curves for other materials, for example aluminum and its

alloys, do not always appear to have asymptotic values so that these materials may not

possess an infinite life stress.

Fig.6.1: Typical form of S-N diagram

6.1.6 Miner's Rule:

In 1945, M. A. Miner popularized a rule that had first been proposed by A. Palmgren

in 1924. The rule, variously called Miner's rule or the Palmgren-Miner linear damage

hypothesis, states that where there are k different stress magnitudes in a spectrum, Si (1 ≤ i ≤

k), each contributing ni (Si) cycles, then if Ni(Si) is the number of cycles to failure of a

constant stress reversal Si, failure occurs when:

C is experimentally found to be between 0.7 and 2.2. Usually for design purposes, C is

assumed to be 1.

This can be thought of as assessing what proportion of life is consumed by stress

reversal at each magnitude then forming a linear combination of their aggregate.

Though Miner's rule is a useful approximation in many circumstances, it has two major

limitations:

It fails to recognize the probabilistic nature of fatigue and there is no simple way to

relate life predicted by the rule with the characteristics of a probability distribution.

There is sometimes an effect in the order in which the reversals occur. In some

circumstances, cycles of low stress followed by high stress cause more damage than would be

predicted by the rule. It does not consider the effect of overload or high stress which may

result in a compressive residual stress. High stress followed by low stress may have less

damage due to the presence of compressive residual stress.

6.1.7 Factors that affect fatigue-life:

Cyclic stress state: Depending on the complexity of the geometry and the loading, one

or more properties of the stress state need to be considered, such as stress amplitude, mean

stress, biaxiality, in-phase or out-of-phase shear stress, and load sequence,

Geometry: Notches and variation in cross section throughout a part lead to stress

concentrations where fatigue cracks initiate.

Material Type: Fatigue life, as well as the behavior during cyclic loading, varies

widely for different materials, e.g. composites and polymers differ markedly from metals.

Residual stresses: Welding, cutting, casting, and other manufacturing processes

involving heat or deformation can produce high levels of tensile residual stress, which

decreases the fatigue strength.

Size and distribution of internal defects: Casting defects such as gas porosity, non-

metallic inclusions and shrinkage voids can significantly reduce fatigue strength.

Direction of loading: For non-isotropic materials, fatigue strength depends on the

direction of the principal stress.

Environment: Environmental conditions can cause erosion, corrosion, or gas-phase

embrittlement, which all affect fatigue life. Corrosion fatigue is a problem encountered in

many aggressive environments.

Temperature: Higher temperatures generally decrease fatigue strength.

6.1.8 Precautions against fatigue failure:

Fatigue cracks that have begun to propagate can sometimes be stopped by drilling

holes, called drill stops, in the path of the fatigue crack. This is not recommended as a general

practice because the hole represents a stress concentration factor which depends on the size of

the hole and geometry. There is thus the possibility of a new crack starting in the side of the

hole. It is always far better to replace the cracked part entirely.

6.2 Fatigue life calculations:

From the stress analysis of the lug joint the maximum tensile stress location is

identified. A fatigue crack will always initiate from the location of maximum tensile stress.

From the stress analysis it is found that such a location is at the pin hole.

Normally aircraft wing experiences variable spectrum loading during the flight. A

typical fighter aircraft flight load spectrum is considered for the fatigue analysis of the lug

joint. Calculation of fatigue life is carried out by using Miner`s Rule.

For the fatigue calculation the variable spectrum loading is simplified as block

loading. Each block consists of load cycles corresponding to 100 flights. Each block consists

of 1,03,050 cycles. The aircraft considered for the current work is designed for 5,000 flights.

Damage calculation is carried out for the complete service life of the aircraft.

The load magnitudes range considered for the fatigue analysis is given in Table 6.1.

Table 6.1: Load Magnitude Range

In the above mentioned load cycles the term „g‟ corresponds to the acceleration due to

gravity. The load corresponding to 1g is equivalent to the weight of the aircraft.

6.2.1 Maximum stress obtained by analysis of wing fuselage lug attachment

bracket:

In above analysis of wing fuselage lug joint, we considered the maximum loading

condition and factor of safety. Therefore by considering the maximum conditions the stress in

the structure is 69.5kg/mm2. Stress values at various „g‟ conditions are as shown in table 6.2.

Table 6.2: Stress values at various “g” conditions

“g” conditions Stress in kg/mm2 Stress in ksi Stress in N/mm

2

0.5 g 5.79 8.23 56.82

1.0 g 11.58 16.49 113.63

1.5 g 17.38 24.75 170.45

2.0 g 23.17 32.99 227.26

2.5 g 28.96 41.24 284.08

3.0 g 34.75 49.49 340.89

0.5 g to 1.0 g 15000

1.0 g to 1.5 g 25000

1.5 g to 2.0 g 10000

2.0 g to 2.5 g 12000

2.5 g to 3.0 g 15000

3.0 g to 3.5 g 20000

3.5 g to 4.0 g 5000

4.0 g to 4.5 g 450

4.5 g to 5.0 g 350

5.0 g to 5.5 g 250

5.5 g to 6.0 g 150

3.5 g 40.54 57.74 397.71

4.0 g 46.33 65.99 454.53

4.5 g 52.13 74.24 521.16

5.0 g 57.92 82.49 568.16

5.5 g 63.71 90.73 624.98

6.0 g 69.5 98.98 681.79

The maximum stress value obtained from the analysis is corresponding to 6 g

condition. Therefore the stress value corresponding to 0.5 g condition is obtained as

(69.5/6)*0.5

The Stress at 0.5 g in kg/mm2 =5.79kg/mm

2.

Stress in KSI =5.79× (25.4)2/.453

=8248.4347 PSI

=8.23 KSI.

Stress in N/mm2 =5.79×9.81

=56.82 N/mm2.

Similar calculations are carried out to obtain the stress values corresponding to

different g conditions and the values are tabulated in the above table 6.2.

6.2.2 Stress cycle:

The Stress is the time function which is repeated periodically and identically as shown in

figure 6.2

Fig.6.2: Stress cycle

Fmax (maximum stress) – the highest algebraic value of stress cycle, for either

tensile stress (+) or compressive stress (-).

Fmin (minimum stress) – the lowest algebraic value of stress cycle, for either

tensile stress (+) or compressive stress (-).

Fa (alternating stress or variable stress).

Fa=

Fm (mean stress):

Fm=

R (stress ratio):

R=

6.2.3 Calculation of the stress ratio and amplitude stress (sa)

Table 6.3: Amplitude stress and Stress ratio values at various ranges of “g” conditions

“g” conditions R = min stress/max

stress

Sa= σmax-σmin/2

In ksi

Sa= σmax-σmin/2

in N/mm2

0.5 g to 1.0 g .5 4.13 28.41

1.0 g to 1.5 g .66 4.13 28.41

1.5 g to 2.0 g .75 4.12 28.41

2.0 g to 2.5 g .8 4.13 28.41

2.5 g to 3.0 g .83 4.13 28.41

3.0 g to 3.5 g .86 4.13 28.41

3.5 g to 4.0 g .88 4.13 28.41

4.0 g to 4.5 g .88 4.13 33.32

4.5 g to 5.0 g .9 4.13 23.5

5.0 g to 5.5 g .91 4.12 28.41

5.5 g to 6.0 g .92 4.13 28.41

1) A typical calculation of stress ratio and stress amplitude for the range 0.5 g to 0.75 g

Stress ratio=min stress/max stress

=5.79/11.58

=.5

Stress amplitude in ksi=σmax-σmin/2.

=11.58-5.79/2

=4.13 ksi.

Stress amplitude in N/mm2

= σmax-σmin/2

=113.63-56.82/2

= 28.41 N/mm2

The stress amplitude of value of ranges „g‟ is obtained by considering maximum

design load. The obtained value of stress ratio and stress amplitude by using the life diagram

fatigue behavior of low carbon and alloy steel– AISI-4130, heat treated alloy. The fatigue life

at crack initiation is predicted.

6.2.4 Calculation of the fatigue cycles by using S-N Diagram for all ranges

from 0.5 “g” to 6 “g”.

Fig.6.3: Typical S-N Diagram for un-notched fatigue behavior of Low Carbon and alloy

Steel– AISI-4130, Heat treated alloy.

Using the maximum stress and value of R from Table6.3 in the S-N curve given in

Figure 6.3 The fatigue cycle for various stress levels are found out. Table6.4 gives the range

of g, the actual number of cycles in that range and the fatigue cycle.

Table 6.4: the range of “g”, the actual number of cycles in that range and their fatigue cycle.

Range of g Actual no of cycles Fatigue cycle by using graph

0.5 g to 1.0 g 15000 >108

1.0 g to 1.5 g 25000 >108

1.5 g to 2.0 g 10000 >108

2.0 g to 2.5 g 12000 >108

2.5 g to 3.0 g 15000 56×106

3.0 g to 3.5 g 20000 5.7×106

3.5 g to 4.0 g 5000 58×103

4.0 g to 4.5 g 450 104

4.5 g to 5.0 g 350 5.7×103

5.0 g to 5.5 g 250 4800

5.5 g to 6.0 g 150 3700

The simplest and most practical technique for predicting fatigue performance is the

palmgren-Miner hypothesis. The hypothesis contends that fatigue damage incurred at a given

stress level is proportional to the number of cycles applied at that stress level divided by the

total number of cycles required to cause failure at the same level. If the repeated loads are

continued at the same level unit failure occurs, the cycles ratio will be equal to one

From Miner‟s equation,

∑ ni/Nf= C

Where ni= Applied number of cycles

Nf= number of cycles to failure

6.2.5 Calculation of the damage accumulated from miner’s formula for all

ranges from 0.5 “g” to 6 “g”.

LOAD CASE 1: Damage accumulated for 0.5 to 1

D1= n1/N1

= 15,000/108

=1.5×10-4

LOAD CASE 2: Damage accumulated for 1 to 1.5

D2= n2/N2

=25,000/108

=2.5×10-4

LOAD CASE 3: Damage accumulated for 1.5 to 2

D3= n3/N3

=10,000/108

= 1×10-4

LOAD CASE 4: Damage accumulated for 2 to 2.5

D4= n4/N4

=12,000/108

=1.2×10-4

LOAD CASE 5: Damage accumulated for 2.5 to 3

D5= n5/N5

=15000/56×106

=2.68×10-4

LOAD CASE 6: Damage accumulated for 3 to 3.5

D6= n6 /N6

=20000/5.7×106

=3.5×10-3

LOAD CASE 7: Damage accumulated for 3.5 to 4

D7= n7/N7

=5000/58×103

=86.2×10-3

LOAD CASE 8: Damage accumulated for 4 to 4.5

D8= n8/N8

=3000/104

=0.3

LOAD CASE 9: Damage accumulated for 4.5 to 5

D9= n9/N9

=1000/5.7×103

=0.175

LOAD CASE 10: Damage accumulated for 5 to 5.5

D10= n10/N10

=250/6000

=41.67×10-3

LOAD CASE 11: Damage accumulated for 5.5 to 6

D11= n11/N11

=150/380

=39.47×10-3

The damage calculated at different g range with reference from S-N data is tabulated

in the table below.

Table 6.5 the range of “g”, the damage accumulated from miner‟s formula

Total damage accumulated for all load case is given by

Da=D1+D2+D3+D4+D5+D6+D7+D8+D9+D10+D11

=1.5×10-4

+2.5×10-4

+1×10-4

+1.2×10-4

+2.68×10-4

+3.5×10-3

+86.2×10-3

+.045+.0614+.0521

+.0405

Da=.2896

Considering the correction factors for finding out the total damage accumulated.

Correction factors for fatigue life calculations of wing fuselage Lug attachment bracket.

1) Surface roughness correction factor=0.8

2) Type of loading=1

3) Correction factor for reliability in design=0.897

Therefore the total damage accumulated by considering the correction factors is D=.2896 /

(.8×1×.897)

D=.4036

Range of

“g”

Applied no of

cycles

„Ni‟

No of cycles to

failure from graph

„Nf‟

Damage

accumulated from

miner‟s formula

„Di‟

0.5 g to 1.0 g 15000 >108 1.5×10

-4

1.0 g to 1.5 g 25000 >108 2.5×10

-4

1.5 g to 2.0 g 10000 >108 1×10

-4

2.0 g to 2.5 g 12000 >108 1.2×10

-4

2.5 g to 3.0 g 15000 56×106 2.68×10

-4

3.0 g to 3.5 g 20000 5.7×106 3.5×10

-3

3.5 g to 4.0 g 5000 58×103 86.2×10

-3

4.0 g to 4.5 g 450 104 0.045

4.5 g to 5.0 g 350 5.7×103 .0614

5.0 g to 5.5 g 250 4800 .0521

5.5 g to 6.0 g 150 3700 .0405

Total damage accumulated is .4036, which is less than 1. Therefore a crack will not

get initiated from the location of maximum stress in the lug for given load spectrum.

CONCLUSIONS

Stress analysis of the wing fuselage lug attachment bracket is carried out and

maximum tensile stress is identified at one of the lug-holes.

FEM approach is followed for the stress analysis of the wing fuselage lug attachment

bracket.

A validation for FEM approach is carried out by considering a plate with a circular

hole.

Maximum tensile stress of 69.5 kg/mm2

is observed in the lug.

Several iterations are carried out to obtain a mesh independent value for the maximum

stress.

A fatigue crack normally initiates from the location maximum tensile stress in the

structure.

The fatigue calculation is carried out for an estimation of life to crack initiation.

From the calculations maximum damage fraction of 0.4035. The value of damage

fraction is much less than 1. hence the crack will not get initiated for the given load

spectrum

Since the damage accumulated is less than the critical damage the location in the wing

fuselage lug attachment bracket is safe from fatigue considerations.

SCOPE FOR FUTURE WORK

Fatigue crack growth analysis can be carried out on the wing fuselage lug attachment

bracket in the bottom skin of the wing

Damage tolerance evaluation for the lug can be carried out for given load spectrum

A structural testing of the wing fuselage lug attachment bracket can be carried out for

the complete validation of all theoretical calculations.

REFERENCES

[1] Application of finite element analysis techniques for predicting crack propagation in

lugs. O. Gencoz, U.G. Goranson and R.R. Merrill, Boeing Commercial Airplane

Company, Seattle, Washington, 98124, USA.

[2] Experimental characterization of cracks at straight attachment lugs, Gianni Nicoletto,

Bologna, Italy.

[3] Damage tolerance assessment of aircraft attachment lugs. T.R. Brussat, K. Kathiresan

and J.L. Rudd, Lockheed-California Company, Burbank, CA 91520, U.S.A., AT&T

Bell Laboratories, Marietta GA 30071, U.S.A., AFWAL/FIBEC, Wright-Patterson Air

Force Base, OH 45433, U.S.A.

[4] Stress intensity factors for cracks at attachment lugs. R. Rigby and M. H. Aliabadi,

British Aerospace, Filton, Bristol BS99 7AR, U.K., Wessex Institute of Technology,

Ashurst Lodge, Ashurst, Southampton S040 7AA, U.K.

[5] Failure in lug joints and plates with holes. J. Vogwell and J. M. Minguez School of

Mechanical Engineering, University of Bath, Bath BA2 7AY, U.K., Facultad de

Ciencias, Universidad Del Pais Vasco, Bilbao, Spain.

[6] Aircraft landing gear failure: fracture of the outer cylinder lug. C. R. F. Azevedo, E.

Hippert, Jr. , G. Spera and P. Gerardi, Laboratory for Failure Analysis, Instituto de

Pesquisas Tecnológicas, PO Box 0141, São Paulo, Brazil.

[7] Finite element mesh refinement criteria for stress analysis. Madan G. Kittur and

Ronald L. Huston, Aero Structures, Inc., 1725 Jefferson Davis Highway, Suite 704,

Arlington, VA 22202, U.S.A., Department of Mechanical and Industrial Engineering,

University of Cincinnati, Cincinnati, OH 45221-0072, U.S.A.

[8] Fatigue crack growth behavior of Al7050-T7451 attachment lugs under flight

spectrum variation. Jong-Ho Kim, Soon-Bok Lee and Seong-Gu Hong.

[9] Fatigue crack growth in lugs. J. SCHIJVE and A. H. W.HOEYMAKERS,

Department of Aerospace Engineering, Delft University of Technology, The

Netherlands.

[10] Fatigue crack growth of a corner crack in an attachment lugs. K. Kathereean, H.S

Pearson, and G.J. Gilbert, Department 72-77, zone 415, Lock Head, Georgia

Company, Marietta, Georgia, 30063, USA.

[11] Fatigue crack growth of corner cracks in lug specimens. S.Freidrich, J. Schijve. Delft

university of technology, Department of aerospace engineering, Netherlands.

Jan 1983.

[12] Stress analysis for a lug under various conditions. G. S. Wang, The Aeronautical

Research Institute of Sweden, Bromma, Sweden.

[13] Fatigue crack growth from flaws in combat aircraft, L. Molent,

Air Vehicles Division, Defence Science and Technology Organization, 506 Lorimer

Street, Fishermans Bend 3207, Australia.

[14] Critical parameters for fatigue damage. A.K. Vasudevan, K. Sadananda, G. Glinka

a. Materials Division, Code-332, Office of Naval Research, Arlington, VA 22217-

5660, USA.

b. Materials Science and Technology, Code-6323, Naval Research, Laboratory,

Washington, DC 20375, USA.

c. Department of Mechanical Engineering, University of Waterloo, Waterloo, Ontario

N2L 3G1, Canada.